Strangeness and Spin in Fundamental Physics Stranezza E Spin in fisica Fondamentale: Proceedings of the International School of Physics

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Strangeness and Spin in Fundamental Physics, Course CLXVII

M. ANSELMINO et al., Editors

IOS Press

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` ITALIANA DI FISICA SOCIETA

RENDICONTI DELLA

SCUOLA INTERNAZIONALE DI FISICA “ENRICO FERMI”

CLXVII Corso a cura di M. Anselmino e T. Bressani Direttori del Corso e di A. Feliciello e Ph. G. Ratcliffe

VARENNA SUL LAGO DI COMO VILLA MONASTERO

19 – 29 Giugno 2007

Stranezza e spin in ﬁsica fondamentale 2008

` ITALIANA DI FISICA SOCIETA BOLOGNA-ITALY

ITALIAN PHYSICAL SOCIETY

PROCEEDINGS OF THE

INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

Course CLXVII edited by M. Anselmino and T. Bressani Directors of the Course and A. Feliciello and Ph. G. Ratcliffe

VARENNA ON LAKE COMO VILLA MONASTERO

19 – 29 June 2007

Strangeness and Spin in Fundamental Physics 2008

AMSTERDAM, OXFORD, TOKIO, WASHINGTON DC

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INDICE

M. Anselmino and T. Bressani – Preface . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. XIII

Gruppo fotograﬁco dei partecipanti al Corso . . . . . . . . . . . . . . . . . . . . . . . . . .

XVI

STRANGENESS PHYSICS

T. Bressani – Strange nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Production reactions of hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The experimental eﬀort in hypernuclear physics: from craftsmen to factories Hypernuclear spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak decay of Λ-hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron-rich Λ-hypernuclei and ΛΛ-hypernuclei . . . . . . . . . . . . . . . . . . . . . . Exotic strange nuclei and more . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Σ-hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Proton rich S = −1 nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. S = 1 hyponuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Supernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 10 23 30 41 45 45 46 46 46 47

E. Friedman and A. Gal – Strange atoms, strange nuclei and kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Exotic atom methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Wave equations and optical potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Nuclear densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Radial sensitivity in exotic atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4. Nonperturbative aspects of exotic atoms . . . . . . . . . . . . . . . . . . . . . . . 3. K − atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Fits to K − -atom data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Deeply bound K − atomic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54 55 55 56 57 58 58 58 62 VII

VIII

¯ nuclear interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. K . 4 1. The K − p interaction near threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ 4 2. K-nucleus potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Deeply bound K − nuclear states in light nuclei . . . . . . . . . . . . . . . . . . . ¯ quasi-bound nuclear states . . . . . . 4 4. RMF dynamical calculations of K . 4 5. Kaon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Σ hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Fits to Σ− atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Evidence from (π − , K + ) spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

indice pag.

64 64 66 68 69 71 73 73 74 77

E. Nappi – Strangeness in hot and dense nuclear matter . . . . . . . . . . . . . . . .

83

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The quark-gluon plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. The QCD-vacuum and the chiral symmetry . . . . . . . . . . . . . . . . . . . . . 3. Ultra-relativistic nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Space-time evolution of nuclear collisions . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Experimental probes and QGP’s signatures . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Global observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Electro-magnetic probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. Quarkonium state suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4. Low-mass vector mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5. Particle production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Strangeness enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Strangeness studies at SPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Strangeness studies at RHIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Features of QGP at RHIC energies and prospect at LHC . . . . . . . . . . . . . . . . 5 1. LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. ALICE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1. The ALICE layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. Performance of the main sub-detector systems for the strangeness studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 86 87 89 90 92 93 93 94 97 97 99 100 104 105 107 110 112

J. Schaffner-Bielich, S. Schramm and H. St¨ ocker – Strangeness in relativistic astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperons in neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strange bosons in hadron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strange quark matter in compact stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. General properties of quark stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Hybrid stars: compact stars with quark and hadron matter . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114 116

119 120 120 130 136 136 139 142

indice T. Nagae – The J-PARC strangeness physics program . . . . . . . . . . . . . . . . . 1. 2. 3. 4.

Status of J-PARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief history of strangeness nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . Hadron Experimental Hall and K1.8 beam line . . . . . . . . . . . . . . . . . . . . . . . Strangeness nuclear physics program at J-PARC . . . . . . . . . . . . . . . . . . . . . . . 4 0.1. E05: spectroscopic study of Ξ-hypernuclei . . . . . . . . . . . . . . . . . 4 0.2. E13: Gamma-ray spectroscopy of light hypernuclei . . . . . . . . . . 4 0.3. E15: A search for deeply-bound kaonic nuclear states . . . . . . . 4 1. Other experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G. Bendiscioli – Strangeness production in antiproton-4 He annihilation at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. An overview of experiments and theoretical predictions . . . . . . . . . . . . . . . . . 2 1. p ¯ -nucleon annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. p ¯ -nucleus annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. HG and QGP formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3. Pentaquark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ 2 2.4. K-few nucleon bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.5. Antiproton-4 He annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Investigation of the p ¯ -4 He annihilation at rest with the Obelix spectrometer . 3 1. Spectrometer and data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Charged-kaon identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Identiﬁcation of single- and multi-nucleon annihilations . . . . . 3 1.3. Neutral kaon and Λ identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4. Apparatus resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Excess in strangeness production . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Pentaquark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. K− pn, K− pnn and K− d bound states (DBKS) . . . . . . . . . . . . . 3 2.4. S = −2 strangeness production and 2K− nn and 2K− nnp . . .

A. Feliciello – Strangeness nuclear physics at FINUDA . . . . . . . . . . . . . . . 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The FINUDA experiment at DAΦNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data taking and apparatus performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physics results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Hypernuclear spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Hypernucleus non-mesonic decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. neutron-rich Λ-hypernuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Deeply bound K-nucleus states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Low-energy K + -nucleus interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

pag. 145 145 147 148 150 150 153 155 157 157

159 159 160 160 161 164 164 167 169 171 173 173 175 178 184 187 187 187 191 197 205

219 219 221 223 226 230 231 238 242 246 250 256

indice

X

SPIN PHYSICS

E. Leader – The longitudinal spin structure of the nucleon . . . . . . . . . . . . .

pag. 263

1. Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. General formalism in one photon exchange approximation . . . . . . . . . . 1 2. Polarized DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The simple parton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Longitudinal polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. What about g2 (x)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The spin crisis in the parton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Simple parton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Resolution of the spin crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Eﬀect of anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. A surprising aspect of this result! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5. A note on angular momentum sum rules . . . . . . . . . . . . . . . . . . . . . . . . 3 6. Is the spin crisis really resolved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Field-theoretic generalization of the parton model . . . . . . . . . . . . . . . . . . . . . . 4 1. QCD corrections and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Structure of G1 (x, Q2 ) at and beyond leading order . . . . . . . . . . . . . . 5. The polarized strange quark density: attempts to measure Δs(x) . . . . . . . . . 5 1. Results from polarized DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Results from SIDIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. A last word on the “spin crisis” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 264 266 267 268 269 269 272 273 274 275 275 275 277 279 281 282 283 284 285

M. Anselmino – The transverse spin structure of the nucleon . . . . . . . . . . .

287

1. Introduction and lecture plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The collinear proton spin conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. The helicity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. The transversity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The (problem of) transverse single-spin asymmetries . . . . . . . . . . . . . . . . . . . 3 1. Experimental results on SSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. SSA in SIDIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Parton model with intrinsic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Spin eﬀects in TMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Sivers and Collins eﬀect in SIDIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Comments on Sivers distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5. Quark-quark correlator with intrinsic motion and SIDISLAND . . . . . 5. SSA in hadronic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. SSA and transversity in Drell-Yan processes . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 288 289 289 292 295 297 298 301 302 305 306 309 311 313

D. Hasch – Transverse spin phenomena in DIS—Experiments . . . . . . . . . . .

317

1. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A brief introduction to transversity and its friends . . . . . . . . . . . . . . . . . . . . . 2 1. Parton distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 318 318

indice

XI

. 2 2. Fragmentation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How to measure tranversity and its friends in DIS? . . . . . . . . . . . . . . . . . . . . 3 1. The generalised semi-inclusive DIS cross-section . . . . . . . . . . . . . . . . . Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Instrumental eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Asymmetry amplitudes extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Charge conjugation and isospin symmetry . . . . . . . . . . . . . . . . . . . . . . . 4 4. Lepton beam and virtual photon asymmetries . . . . . . . . . . . . . . . . . . . . 4 5. Contribution from exclusive channels . . . . . . . . . . . . . . . . . . . . . . . . . . The Collins asymmetries and how to interpret them . . . . . . . . . . . . . . . . . . . . 5 1. Collins asymmetries from a deuterium target . . . . . . . . . . . . . . . . . . . . . 5 2. What about the Collins fragmentation function? . . . . . . . . . . . . . . . . . . 5 3. A brief look at model predictions for transversity . . . . . . . . . . . . . . . . . 5 4. First glimpse of transversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transversity from two-hadron production . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sivers asymmetries and how to interpret them . . . . . . . . . . . . . . . . . . . . . 7 1. Sivers asymmetries from a deuterium target . . . . . . . . . . . . . . . . . . . . . 7 2. A brief look at models and extractions of the Sivers function . . . . . . What should come next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 320 322 323 323 325 325 327 327 328 328 329 332 333 335 336 337 339 342 342 344

W. Vogelsang – QCD spin physics in hadronic interactions . . . . . . . . . . . .

349

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. QCD perturbation theory and its applications . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Perturbative QCD in e+ e− annihilation . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Factorized deep-inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Factorized pp scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Longitudinally polarized pp collisions at RHIC . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Results from unpolarized pp scattering at RHIC . . . . . . . . . . . . . . . . . . 3 2. Probing the spin structure of the nucleon in polarized pp collisions . . 3 3. Access to Δg in polarized proton-proton scattering at RHIC . . . . . . . . 3 4. Weak boson production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5. Global analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Collisions of transversely polarized protons . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Transversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Single-transverse spin asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Single transverse-spin asymmetry in high-⊥ pion production in pp collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Single transverse-spin asymmetries in two-scale situations . . . . . . . . . . 4 5. Relation between mechanisms for single-spin asymmetries . . . . . . . . .

349 351 351 353 353 355 355 355 360 361 361 363 363 366

3. 4.

5.

6. 7.

8.

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N. Saito – Spin structure of the nucleon: RHIC results and prospects for J-PARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Polarized parton distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experimental eﬀorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Electromagnetic interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

379 381 382 384

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. 3 2. Strong interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Weak interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gluon polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transverse-spin phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J-PARC: high-intensity proton beam facility . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Day-1 experiments with 50 GeV proton synchrotron . . . . . . . . . . . . . . . 6 2. Possible spin physics at J-PARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 384 385 387 388 389 390 390 393 394 395 395

R. Bertini – The nature of spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. SPIN and polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic moments, Stern-Gerlach and polarised source . . . . . . . . . . . . . . . .

399 401 403

4.

5. 6.

7.

H. Avakian – Studies of semi-inclusive and hard exclusive processes at JLab 1. SIDIS and the transverse structure of the nucleon . . . . . . . . . . . . . . . . . . . . . . 1 1. Present experimental results on spin-azimuthal asymmetries . . . . . . . . 1 1.1. Transversely polarized target . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Longitudinally polarized target . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3. Unpolarized target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. TMD Measurements with JLab at 12 GeV . . . . . . . . . . . . . . . . . . . . . . 2. GPDs and quark distributions in transverse space . . . . . . . . . . . . . . . . . . . . . 2 1. Present experimental results on hard exclusive processes . . . . . . . . . . . 2 2. GPD measurements with Jeﬀerson Laboratory at 12 GeV . . . . . . . . . . 2 3. GPD studies with a transversely polarized target . . . . . . . . . . . . . . . . 3. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M. Boglione – Global ﬁt for the simultaneous determination of the transversity distribution and the Collins fragmentation functions . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6.

407 407 410 410 411 412 413 417 421 422 425 427

431

Distribution and fragmentation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transversity and Collins functions from SIDIS processes . . . . . . . . . . . . . . . Collins functions from e+ e− processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transversity and Collins functions from a global ﬁt . . . . . . . . . . . . . . . . . . . Predictions for ongoing and future experiments . . . . . . . . . . . . . . . . . . . . . . . Comments and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 440 444 451 457 460

Elenco dei partecipanti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface

The CLXVII Course “Strangeness and Spin in Fundamental Physics” of the “Enrico Fermi” School, held in Varenna from June 19 to 29, 2007, was dedicated to the discussion of the role played by two subtle and somehow puzzling quantum numbers, the Strangeness and the Spin, in Fundamental Physics. They both relate to basic properties of the fundamental quantum ﬁeld theories describing strong and electro-weak interactions and to their phenomenological applications. In some instances, like the partonic spin structure of the proton, they are deeply correlated. The concept of Strangeness was introduced in 1953 in order to explain experimental observations in Particle Physics, which could not be interpreted without the introduction of such a new quantum number. Strangeness was the paradigm for the introduction, in the following decades, of other new quantum numbers (Charm, Beauty, . . . ) and may be considered as the cornerstone for the building up of the presently accepted theory of strong interactions, QCD, and of the quark content of matter. However, it was soon realized that Strangeness plays a crucial role not only in elementary systems, but also in much more complicated many-body ensembles (nuclei, atoms, neutron stars, . . . ). Due to the circumstance that the mass of the s-quark is somewhat between the masses of the light u- and d-quarks (from which our world is built) and those of the heavy c-, b- and t-quarks, exotic many-body systems containing s-quarks may be assembled and carefully examined experimentally. The study of their properties is of fundamental importance for understanding their underlying structure. The physics of Strangeness then encompasses the ﬁelds of Elementary Particles, Nuclear Physics, Atomic Physics and Astrophysics, with close and important links between them. The concrete demonstration of the great importance gained by the Strangeness Physics is given by the great number of large experiments partially or fully devoted to it at large Laboratories like RHIC-BNL, TJNAF, DAPHNE, GSI, Nuclotron-Dubna. Even more, the powerful complex of accelerators J-PARC, under completion at Tokai (Japan) which will supply in two years beams of mesons, in particular Kaons, of unprecedented intensity and purity, will dedicate a large amount of time to Strangeness Physics studies. Half of the lectures and seminars of the Course covered many aspects of the Strangeness Physics, and were given by world-recognized experts in the ﬁeld: Professors Bendiscioli, Bertini, Bressani, Feliciello, Gal, Guaraldo, Nagae, Nappi and Schramm. XIII

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Preface

The interest in Spin Physics has enormously grown in the last 15-20 years, both from the experimental and theoretical points of view. In all high-energy experimental facilities there are ongoing or planned spin-dedicated experiments with collisions of polarized particles: HERMES at DESY, STAR and PHENIX at RICH-BNL, several experiments at TJNAF. Even BELLE, at KEK, studies spin eﬀects in the fragmentation of quarks produced in e+ e− interactions. Collisions between polarized protons and polarized antiprotons have been proposed for future experiments at GSI: these would explore entirely new aspects of the strong interactions and QCD. Experiments with polarized protons are also planned at J-PARC. The many puzzling results recently obtained by measuring several spin asymmetries have stimulated enormous progress in the study of the spin structure of protons and neutrons. Intense theoretical activity has discovered new features of non-perturbative QCD, like strong correlations between the spin and the intrinsic motions of quarks inside the nucleons. The high-energy spin physics community is aiming at reaching a full understanding of the proton structure, which not only takes into account the longitudinal degrees of freedom, but also transverse motions, spin and orbital angular momenta of quarks and gluons. The other half of the lectures and seminars of the Course was devoted to many facets of high-energy spin physics, and were given by excellent speakers and active researchers in the ﬁeld: Professors Anselmino, Avagyan, Boglione, Hash, Jaﬀe, Leader, Saito and Vogelsang. The purpose of the Course was that of providing a complete, updated and critical account of the most recent and relevant discoveries in the above ﬁelds, both from the experimental and theoretical sides, and was fully achieved. The Course was very timely and important for the Enrico Fermi School, focused on education and information and providing a careful analysis of the available data, predictions for ongoing experiments and suggestions for future plans. In conclusion the organizers of the Course warmly thank the European Project Hadron Physics of the FP6 framework and the Project HYPERGAMMA of PRIN (Italian Ministry of Research) for the generous ﬁnancial support to the School, which made its organization possible. Special thanks are due to Miss B. Alzani for the continuous and invaluable help during the Course, as well as to Miss G. Bianchi Bazzi, Miss R. Brigatti, Mr. D. Caffarri of the Villa Monastero organization. Mrs. M. Missiroli and Mrs. C. Vasini are also acknowledged for their precious activity before and after the completion of the Course. Finally special acknowledgements are due to Dr. A. Feliciello and Prof. Ph. G. Ratcliffe, Scientiﬁc Secretaries of the Course, for their continuous, patient and precious work at all stages of the scientiﬁc and practical organization.

M. Anselmino and T. Bressani

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Società Italiana di Fisica SCUOLA INTERNAZIONALE DI FISICA «E. FERMI» CLXVII CORSO - VARENNA SUL LAGO DI COMO VILLA MONASTERO 19 - 29 Giugno 2007

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1) D. Calvo 2) S. Bufalino 3) S. Sarkar 4) O. B. Samoylov 5) X. Lu 6) L. L. Pappalardo 7) F. Giordano

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4

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8) T. Takahashi 9) M. Pincetti 10) A. López Ruiz 11) C. Höppner 12) G. Serbanut 13) C. Türk 14) S. Melis

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15) T. Katori 16) A. Mastroserio 17) V. Lucherini 18) H. Fujioka 19) K. Moriya 20) G. Simonetti 21) B. Dalena

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22) D. Faso 23) A. Bolshakova 24) E. Zemlyanichkina 25) K. Szymanska 26) S. Petrochenkov 27) I. Gnesi 28) A. Prokudin

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29) S. Pereira 30) D. Caffarri 31) G. Bianchi Bazzi 32) B. Alzani 33) Ph. G. Ratcliffe 34) A. Feliciello

35) D. Hasch 36) M. Anselmino 37) T. Bressani 38) A. Gal 39) G. Bendiscioli 40) F. Viscardi

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STRANGENESS PHYSICS

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Strange nuclei T. Bressani Dipartimento di Fisica Sperimentale, Universit` a di Torino - Torino, Italy INFN, Sezione di Torino - Torino, Italy

Summary. — The more important steps in hypernuclear physics are brieﬂy summarized, mostly from the experimental point of view. Last results on diﬀerent items, like spectroscopy, weak-decay modes and other more exotic systems are discussed.

1. – Introduction The ﬁrst strange particle was observed in 1947 [1], the hyphotesis of an associated production of the newly discovered particles was put forward in 1951 [2, 3], and in 1953 two Polish physicists, M. Danysz and J. Pniewski [4], in the analysis of events recorded by a stack of photographic emulsions exposed to the cosmic radiation at about 26 km from the Earth’s surface with a baloon, interpreted an event with unusual features as signature of a nuclear fragment containing one (we call it now a Λ-hyperon) of these still mysterious particles. It was the ﬁrst example of what we call now a hypernucleus. In the same year Gell-Mann introduced the concept and name of strangeness S [5]. There is an interesting hystorical review of the very exciting research activity in these years with many unknown apects and anedocts done by Wrobleski, a student of Danysz and Pniewski at that time, written for celebrating the 50th year of the discovery of hypernuclei [6]. I recalled these quite well-known dates for the sake of stressing the intuition and intellectual courage of Danysz and Pniewski in accepting very novel ﬁndings and concepts of particle physics and c Societ` a Italiana di Fisica

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apply them to the study of other more complex systems, the nuclei. Once the concept of S was well understood and fully accepted, the approach of exploiting this new quantum number for the study of other complex systems, from atoms to stars, was followed with very interesting results, that will be outlined in several lectures of this dedicated Course of the Varenna School. In my lectures I will review the present status and the future perspectives of hypernuclear physics. A half of the Course CLVII of this School, held from June 22 to July 2, 2004, was dedicated to hypernuclear physics, and the Proceedings [7] provide a quite complete and updated review of the ﬁeld. My lectures summarize the main items, with recent updates and are related to other lectures or review talks that I gave to this School [8, 9] and elsewhere [10-12]. Hypernuclei are nuclear systems in which one or more nucleons are replaced by one or more hyperons. As previously said, the more known and studied for a long time (54 years) are Λ-hypernuclei, in which one Λ-hyperon replaces a nucleon of the nucleus; they are indicated by the suﬃx Λ preceding the usual notation of nuclei: A Λ Z. In the following, for simplicity, Λ-hypernuclei will be simply called hypernuclei. The complete notation will be used for other more exotic and less known systems. Hypernuclei are stable at the nuclear time scale (10−23 s) since the Λ particle, the lightest of the hyperons, maintains its identity even if embedded in a system of other nucleons; the only strong interaction which conserves strangeness and is allowed is, in fact, elastic scattering. Similarly, two Λ’s may stick to a nuclear core, forming the so-called double Λ-hypernuclei, indicated by the symbol A ΛΛ Z. Two candidates were found with the emulsion technique, in the early 60’s, and only in the past few years this old observation was conﬁrmed by modern counter experiments. Σ-hypernuclei do not exist, at least as nuclear systems surviving for times longer than 10−23 s. The reason is that the strong ΣN → ΛN conversion occurring in nuclei prevents the observation of narrow nuclear states like for hypernuclei, with the exception of 4Σ He, that will be discussed in sect. 7. A similar situation occurs for the Ξ-hypernuclei, whose importance lies in the fact that the conversion reaction Ξ− p → ΛΛ in nuclei may be used as a breeder for the formation of ΛΛ-hypernuclei. We may also think of Ω-hypernuclei, possibly breeders of 3Λ-hypernuclei through a sequence of the two conversion reactions Ω− n → ΛΞ− , Ξ− p → ΛΛ occurring in the same nucleus. Finally I remind that theoretical speculations, starting from the analogies with the physics of strange nuclei, deal with the possible existence of nuclear systems with a C or B quantum number. No experiment was done or is planned on this subject. It is clear that from its birth hypernuclear physics was interfacing particle and nuclear physics. This circumstance was not so useful for the experimental eﬀort, at least up to the last decade of the past Millennium. Often the beam time allocations at the various Laboratories and the ﬁnancial and human resources granted by the Scientiﬁc Agencies, separated for ﬁelds of competence, were scarce since hypernuclear physics, even though scientiﬁcally interesting, was not considered to belong deﬁnitely to one ﬁeld. Then a destructive interference eﬀect often occurred. Hopefully in the last two decades, thanks to the excellent scientiﬁc results provided with great skillfullness by the researchers, the interference turned out from destructive to partially constructive. I will return on this point at the end of this review. The above considerations led also to suggest several

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Strange nuclei

slogans or logos to emphasize the speciﬁc features of hypernuclear physics. The ﬁrst logo that I remember in the 60’s, was an animal resulting from a genetic manipulation between a ﬁsh and a chicken (nuclear and particle physics). A second, in the 80’s, was nuclei with a tracer; the Λ was resembled to a radioactive atom embedded in a complex molecole (the nucleus). A third one, quite recent, is: nuclei with a third dimension; the deﬁnition comes from the representation (logo) in which, besides the usual (Z, N = A − Z)-plane in which the ordinary nuclei lie, a third axis, lSl (or better the number of speciﬁc hyperons) is added. A last analogy with a real animal, the bat, was used by me for the logo of FINUDA, shown in sect. 3. The bat is the only mammal (nuclear physics) able to ﬂy (particle physics). The paper is organized in the following way. Section 2 describes the production reactions of hypernuclei and sect. 3 provides a personal overview of the evolution of hypernuclear physics in the years from the experimental side. Section 4 summarizes the main results so far achieved on hypernuclear spectroscopy, and sect. 5 those on the weak decays. Section 6 points out brieﬂy the last results achieved in the quite elusive ﬁeld of neutron-rich hypernuclei and ΛΛ-hypernuclei. Section 7 ﬁnally provides a very short overview on the possible existence of nuclear systems with other strange and more exotic particles. 2. – Production reactions of hypernuclei Recent overviews on the diﬀerent aspects of strange nuclei production can be found in [13] and [14]. In the ﬁrst stage of hypernuclear physics (1953-1972), experiments were carried out with visualizing techniques and no special care was devoted to the quality or the energy of the incident beam. The main requirement was that the incident particle would have enough energy to produce a Λ, which could eventually stick to one of the fragments of the nucleus disintegrated by the projectile. This fact was also the reason for which hypernuclei were often called “hyperfragments”. I may also remark that the major part of the experimentalists carrying out this research in these years had a formation and an approach typical of particle physics. From 1972 two-body reactions producing Λ’s on a nuclear target were studied. The two-body reactions that led to practically all the present bulk of experimental information are 1. The “strangeness exchange” reaction: (1)

K− + N → Λ + π

exploited mainly in the K − +n → Λ+π − charge state, for evident reasons of easiness for spectroscopizing the ﬁnal state π. The reaction can be seen as a transfer of the s-quark from the incident meson to the struck baryon. 2. The “associated production” reaction (2)

π+ + n → Λ + K + .

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T. Bressani

This reaction can be seen as a creation of a (s¯ s) pair by the incident meson. Since 1972 the major part of experimentalists involved in strange nuclei physics had a formation and an approach typical of nuclear physics. Quite recently a third kind of reaction, the electroproduction of strangeness on protons in the very forward direction was exploited: (3)

e + p → e + Λ + K + .

In these kinematical conditions the virtual photons associated to the reaction (3) can be regarded as quasi-real and (3) is often rewritten as a two-body photoproduction reaction: (4)

γ + p → Λ + K +.

It must be noticed that, whereas with reactions (1) and (2) it is possibile to replace a neutron in the target nucleus by a Λ, with reaction (3) a proton is replaced by a Λ, and the neutron-rich mirror hypernucleus is obtained out from the same nuclear target. Just for example, with reaction (1) and (2) on a 12 C target 12 Λ C is produced, with reaction (3) 12 B. Λ Experiments with projectiles diﬀerent than those for reactions (1), (2) and (3), like protons [15], antiprotons [16] or relativistic ions [17] were also performed, but they did not supply systematic series of results. Each of the aforesaid reactions has its own characteristics in the elementary crosssection, internal quantum numbers transfer, momentum transfer, absorption of incoming and outcoming particles in nuclear matter. However, the most important parameter in determining the selectivity of the diﬀerent reactions is the momentum transfer. Figure 1 shows the momentum transferred to the Λ-hyperon, qΛ , as a function of the momentum of the projectile in the laboratory frame, pLab , for reactions (1) and (2), as well as for photoproduction (4) at θLab = 0◦ . A striking kinematical diﬀerence appears: for (1), which is exoenergetic, there is a value of pLab (505 MeV/c, called also “the magic momentum” [18])(1 ) for which qΛ vanishes and recoilless production takes place; for (2), which is endoenergetic, qΛ decreases monotonically with pLab , staying always at values exceeding 200 MeV/c. A further degree of freedom in selecting qΛ is given by the detection angle of the produced meson. It is also quite interesting to notice that qΛ for (1) with K − at rest is of the same order of magnitude as for (2). In order to get insight on how the features of the elementary reaction aﬀect the production of hypernuclei in well deﬁned states the impulse approximation may be used. The two-body reaction t-matrix in the nuclear medium is replaced by the free-space (1 ) I recall here that the deﬁnition of “magic momentum” used in the following years to describe the kinematical conditions for recoilless production in hadron-induced reactions on nuclei was inspired by the popular song in the ’60s “Magic moment” by Perry Como.

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Fig. 1. – Kinematics of the reactions (1), (2) and photoproduction (4) for forward Λ emission (θL = 0◦ ). The momentum transferred to the hyperon is plotted as a function of the incoming particle (K − , π + or γ) in the laboratory system.

t-matrix at the same incident momentum and the diﬀerential cross-section for the reaction − K− π A A (5) + Z →Λ Z + π+ K+ may be written as (6)

dσ(θ)/dΩL = ξ [dσ(θ)/dΩL ]free Neﬀ (i → f, θ).

In (6) dσ(θ)/dΩL is the Lab. cross-section for the production of the hypernucleus A ΛZ in a given ﬁnal state f , ξ a kinematic factor arising from two-body to many-body frame transformation, [dσ(θ)/dΩL ]free the cross-section for the elementary (or free) reactions (1) and (2) and Neﬀ (i → f, θ) the so-called “eﬀective nucleon number”. Equation (6), quite simple and understandable at ﬁrst sight, was used by Bonazzola et al. [19] for the ﬁrst time to describe the production of hypernuclei by the (K − , π − ) reaction in ﬂight. All the complications and diﬃculties related to the many-body stronginteraction system are obviously contained in the term Neﬀ (i → f, θ) (the nuclear physics “black box”). In the simplest Plane Wave Approximation (PWA) Neﬀ (i → f, θ) can be furthermore written as (7)

Neﬀ (i → f, θ) = (Clebsh-Gordan coeﬃcients) × F (q),

in which F (q) is a form factor. It is possible to see in the frame of the independent particle model that F (q) is related to q by simple relationships.

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Equations (6), (7) provide the essential features of reactions like (5) and corroborate the statement that the momentum transfer is the most important parameter determining the hypernuclei production. PWA is only a rough approximation for describing hypernuclei production reactions and is unable to provide reliable values of the cross-sections. A better insight is given by the Distorted Wave Impulse Approximation (DWIA). The K and π distorted waves are calculated separately by solving the Klein-Gordon equation with the use of appropriate optical potentials. The meson potentials are conventionally taken to be proportional to the nucleon density. DWIA calculations are quite successful in describing the shape of the angular distribution as well as the absolute value of the cross-section (within ∼ 25%) and are generally used to determine the spin of the produced hypernuclear ﬁnal states. Similar theoretical considerations may be applied also for describing the electroproduction of hypernuclei by means of the elementary reaction (3), even though the relevant expressions are somehow complicated by the three-body ﬁnal-state kinematics and by the dynamics described by the electromagnetic interaction. The relevant relationships can be found, e.g., in [13] and [14]. From the above considerations and from ﬁg. 1 it appears that with the (K − , π − ) reactions in-ﬂight substitutional states of the hypernucleus in which a neutron of the target nucleus is converted into a Λ-hyperon in the same orbit with no orbital angular-momentum transfer are preferentially populated. On the other hand, the (π + , K + ) and (e, e K + ) reactions, and also the (K − , π − ) reaction at rest transfer a signiﬁcant (200–300 MeV/c) recoil momentum to the hypernucleus, and then several high spin hypernuclear states are simultaneously produced. This feature is of paramount importance for hypernuclear spectroscopy studies (see sect. 4) and also for investigations on the weak decay, which occurs mainly for hypernuclei in the ground state (see sect. 5). Furthermore, since the spin-ﬂip amplitudes in reactions (1) and (2) are generally small, they populate mainly non spin-ﬂip states of hypernuclei. On the contrary, since the amplitude for reaction (3) has a sizable spin-ﬂip component even at zero degrees due to the spin 1 of the photon, both spin-ﬂip and non spin-ﬂip ﬁnal hypernuclear states may be populated in electroproduction. From the above considerations it appears that in principle we have at disposal all the tools needed for a complete study of all the hypernuclei we wish to study. In practice the experimenters have to face many other constraints in designing their detectors. They are: a) the values of the cross-sections for producing hypernuclei with reactions (1), (2) and (3), b) the presence of physical backgrounds in which the signals due to hypernuclei’s formation could be blurred, c) the intensity and energy resolution of the beams of projectiles (K − , π + , e), d) the availability of the above beams at the diﬀerent laboratories, in competition with other experiments in nuclear and particle physics. A simple answer to question (a) is given by ﬁg. 2.

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Fig. 2. – Hypernuclear production cross-section for typical reactions vs. momentum transfer. Slightly modiﬁed from ref. [14].

It must be noticed that the production of hypernuclei by means of the (K − , π) reaction with K − at rest is reported too on a cross-section scale, which is formally wrong since the production by particles at rest is deﬁned by a capture rate and not by a crosssection. I inserted this information too on the plot, with a suitable normalization on similar reactions. Unfortunately the intensities and the qualities (point (c)) of the beams scale inversely to the values of the production cross-sections. The electron beams were always excellent concerning the intensity and the energy resolution. Unfortunately in the past years the duty cycle of the machines was very low, and the smallness of the cross-section joined to the need of measuring the e and K + from (3) in coincidence did not allow to study the production of hypernuclei. Only the advent of CEBAF a machine with an excellent duty cycle, allowed the start up of experiments on hypernuclei. The best projectiles for producing hypernuclei, and in general strange nuclei, are K − . Unfortunately the intensity of such secondary beams is quite low, and even worse the quality (contamination by π − , energy spread). For these reasons the magnetic spectrometers that were designed were quite complicated and the running times needed to obtain adequate statistics long. The poweful complex of machines J-PARC, that will be operational in two years, and for which strange nuclei and atoms physics will be one of the top priorities, will allow a real breakthrough in this ﬁeld. A detailed account on the strangeness physics experiments at J-PARC is given by T. Nagae at this School [20]. Production of hypernuclei by pion beams is midway concerning both the cross-sections and the quality

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of the beams. For these reasons experiments using reaction (2) performed mostly at KEK with the SKS spectrometer produced the major part of experimental information in the last ten years by means of advanced state-of-art technologies at an old machine, the KEK 12 GeV ProtonSynchrotron. Overviews of the measurements done with the SKS can be found in refs. [14] and [21]. A special discussion must be given in relation to the use of the (K − , π − ) reaction at rest. Looking at ﬁgs. 1 and 2 this method seems the easier to exploit, since no experimental information on the momentum of the incoming particle is needed. The main reason for having not been so widely used stems in point (b). Taking 12 again as example the production of 12 C target with reaction (1) with K − at Λ C from a rest: (8)

− − Kstop + 12 C → 12 Λ C+π

at ﬁrst sight one could think that the simple spectroscopy of the π − from (8) should − be enough to get information on the excitation spectrum of 12 Λ C. The fastest π should 12 be the one related to the formation of Λ C in the ground state. Unfortunately other processes following the capture of K − at rest produce π − with momenta even larger than those from reactions like (8). The more dangerous for hypernuclear spectroscopy is the capture of a K − by a correlated (np) pair in the nucleus: (9)

K − + (np) → Σ− + p

followed by the decay in ﬂight of the Σ− into π − + n. This process leads to the production of π − with a ﬂat momentum spectrum extended beyond the kinematical limit due to the two-body production of hypernuclei. This drawback was clearly understood in the ﬁrst series of experiments on hypernuclei with K − at rest, done at KEK [22], and the background reasonably modeled by accurate simulations [23]. This circumstance generated shadows and lights on the subsequent experiments with K − at rest, described in the following section 3. 3. – The experimental eﬀort in hypernuclear physics: from craftsmen to factories Like many other ﬁelds of physics, experimental research on hypernuclei started as a modest eﬀort of a few small experimental teams, focusing their interest on well-deﬁned aspects. Up to the early 70’s, visualizing techniques (photographic emulsions or bubble chambers ﬁlled with 4 He or heavy liquids) were the experimental tool used to study the properties of hypernuclei. Cosmic rays and afterwards K − beams were employed for producing hypernuclei. Notwithstanding the inherent weakness of this technique (low statistics), about 20 hypernuclei were unambiguosly identiﬁed by means of the kinematic analysis of the disintegration star. The Λ binding energy, BΛ , of the A Λ Z hypernuclear

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Fig. 3. – Variation of the BΛ (g.s.) values with the hypernuclear mass number A, taken from ref. [24].

ground state, deﬁned as (10)

BΛ = Mcore + MΛ − Mhyp ,

where Mcore is the mass (in MeV/c2 ) of the A−1 Z nucleus well known from nuclear physics, MΛ is the Λ-hyperon mass and Mhyp is the measured A Λ Z hypernuclear mass, was determined for twenty hypernuclei. It was found that BΛ varies linearly with A, with a slope of around 1 MeV/(unit of A), saturating at about 23 MeV for heavy hypernuclei. This result is perhaps the more interesting and even today the better values for BΛ , at least for light hypernuclei, are those deduced from emulsion experiments, due to the extraordinary precision of this technique. Figure 3 shows the BΛ values deduced from emulsion experiments [24].

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Fig. 4. – Pictorial view of the experimental set-up, which was used for the ﬁrst study of the production of hypernuclei by K − in ﬂight. From ref. [27].

In the early 70’s the study of hypernuclei was started by means of electronic techniques (magnetic spectrometers), following the progress in the design of new intense K − beams and pioneering the use of the newly developed multiwire proportional chambers [25]. However, the magnetic spectrometers that were used by the diﬀerent groups were not designed speciﬁcally and optimized for hypernuclear physics. The ﬁrst measurement of the π − spectra emitted from K − capture at rest by 12 C was performed by means of an apparatus dedicated to the measurement of the Ke2 decay, with some modiﬁcations of the target assembly and the trigger [26]. The energy resolution on the hypernuclear energy levels was 6 MeV FWHM. A similar resolution was attained in the ﬁrst experiment on the production of hypernuclei with the reaction (1) in ﬂight, at 390 MeV/c. The apparatus was realized [27] speciﬁcally for the study of the production of hypernuclei, but using pre-existing general purpose magnets, as shown by ﬁg. 4. It is interesting to notice that some experimental techniques pioneered with this detector were recently applied in the design of the FINUDA spectrometer [28]. The energy resolution on the hypernuclear energy levels was about 6 MeV FWHM. Figure 5 shows the excitation spectrum measured on a 16 O target [29]. Gamma-rays spectra emitted from low-lying excited hypernuclear states were too measured from A = 4 systems in a pilot experiment performed with NaI detectors [30] at CERN. A double spectrometer for the study of the (K − , π − ) reaction on some nuclear targets at 900 MeV/c was assembled, soon afterwards always at CERN, again using pre-existing beam elements and magnets, providing a better resolution (5 MeV) and cleaner spectra with higher statistics [31]. The ﬁrst magnetic spectrometer which was speciﬁcally designed for hypernuclear physics, SPES 2, had a quite funny story. It had a quite sophisticated and nice mag-

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Fig. 5. – Spectrum of

16

O after subtraction of the background. From ref. [29].

netic optics, and was assembled in 1976 and afterwards used at the Proton Synchrotron SATURNE in Saclay in an attempt to produce hypernuclei by the (p, p K + ) three-body reaction in the forward direction on nuclear targets. Unfortunately the experiment failed, due to the overwhelming background, but the spectrometer was so properly designed that, transported at CERN and installed at a K − beam in 1978, produced an excellent amount of data on hypernuclear physics by the (K − , π − ) reaction at 900 MeV/c [32]. Furthermore it was moved at the LEAR Complex at CERN and used as a large acceptance spectrometer for the measurement of elastic and inelastic scattering of p¯ by nuclei. At the end it was transported to Japan, where it is waiting again to serve as a still now excellent spectrometer for hypernuclear physics [20]. Old soldiers never die! A great step of quality in hypernuclear physics was made possible by these experiments. Production of excited states was observed for several hypernuclei, mostly in the p-shell, and their quantum numbers determined in some cases. These data constituted the input for several theoretical investigations on the description of the hypernuclear excited states in terms of microscopic models starting from the ΛN potential. The more important result from this series of measurements is the observation that the spin-orbit term in the hypernuclear optical potential is consistent with zero. This conclusion was recently questioned by recent experiments on heavier hypernuclei, but the original hypothesis of a very small value seems still the more valid. The latest experiments performed by the Heidelberg-Saclay-Strasbourg Collaboration with the SPES II spectrometer claimed also the observation of unexpected exotic nuclei, the Σ-hypernuclei [33]. This discovery seemed conﬁrmed by a subsequent low statistics experiment with stopped K − performed at KEK [34]. Looking at the very promising results obtained at CERN, the BNL management decided to start a strong eﬀort on hypernuclear physics. The ﬁrst step was to setup a focusing spectrometer system placed on a rotatable platform, called Moby-Dick (presumably because of its size). A suitable beam-line was designed to be operated in

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Fig. 6. – The excitation spectrum for the 89 Y(π + , K + )89 Λ Y reaction at pπ = 1.05 GeV/c and θK = 10◦ , from the AGS experiment. The curves represent theoretical calculations. (From ref. [35].)

connection to the spectrometer. From the late 70’s BNL produced an impressive set of beautiful data, practically on all the experimental aspects of hypernuclear physics. The greater amount of data was obtained by means of the (K − , π − ) reaction, thanks to the fact that BNL had the better K beams in the world. However, BNL pioneered also to exploit the associated production reaction (2) on nuclei in a very eﬃcient and controlled way. As a matter of fact, the spectra obtained on heavy targets, like 89 Y showed for the ﬁrst time, in a very impressive way, the validity of the single-particle model even for the inner shells [35]. Figure 6 shows the spectrum, in which peaks corresponding to the Λ sitting in s-, p-, d- and f -orbits are clearly seen. This spectrum is quite popular even in the textbooks since it is the best evidence for the validity of the nuclear shell model. It is diﬃcult to enumerate here all the contributions to hypernuclear physics provided by the BNL experiments, without omissions. They can be found, e.g., in the review by Chrien [36]. First of all, I would recall the series of experiments, with good statistics, that proved the non-existence of Σ-hypernuclei, in contrast to the previous claims. As second item, I was impressed by the ﬁrst measurement of Non-Mesonic (NM) weak decays in coincidence [37]. It pioneered the technique which was afterwards exploited by the SKS spectrometer at KEK and now by FINUDA at LNF (sect. 5). Furthermore, several BNL experiments attempted to look for ΛΛ-hypernuclei, for which no other candidates were found after the initial discovery in 1963 with emulsion techniques. In conclusion my impression is that BNL pionereed practically all the experimental techniques that are the

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Fig. 7. – Schematic drawing of the SKS spectrometer system with the beam line spectrometer. From ref. [14].

basis for the present and future installations, but did not perform the systematic study that is the mission of a “Hypernuclei’s factory”. At the 12 GeV Proton Synchrotron of KEK the experimental eﬀort started with a series of experiments in which the π spectra emitted following the capture of K − at rest by several nuclei were measured with a magnetic spectrometer. The apparatus was at the beginning that one used for the study of rare and exotic decays of K + . In some sense the situation was similar to that one occurred in 1973 for the ﬁrst CERN experiment [23], but the results, immediate and also far reaching, completely diﬀerent. Exploiting the better beam quality, the use of dedicated detectors for coincidence measurements and also the physics information gained by more than ten years of experimental results with the in-ﬂight reactions, the KEK results showed that reaction (1) with K − at rest could provide results on hypernuclear physics of very high quality, in contrast to the initial perception. The main results from the KEK experiment are summarized in ref. [23]. They were the trigger for the construction of two powerful “Hypernuclear Factories”. The ﬁrst one is not a direct continuation of the ﬁrst KEK experiment with stopped K − , but perhaps the result of the great enthusiasm and interest for hypernuclear physics gained by many clever and young Japanese physicists. At the end of the 90’s, a powerful detector dedicated to hypernuclear physics was approved for installation at KEK. It was based on a large superconducting dipole magnet with a pole gap of 50 cm and a maximum ﬁeld of 3 T, optimized to provide a large

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Fig. 8. – Missing mass spectrum of the the SKS spectrometer. From ref. [38].

T. Bressani

12

C(π + , K + )12 Λ C reaction at the KEK 12 GeV PS using

solid angle (100 msr) for the reaction (π + , K + ) on nuclei. A dedicated new beam line, providing ∼ 2×106 π + /spill at about 1 GeV/c was put into operation. The spectrometer, named SKS (acronym for Superconducting Kaon Spectrometer) was equipped with stateof-art drift chambers and detectors for particle identiﬁcation [21]. Figure 7 shows a sketch of the detector. The ﬁnal resolution of the device was about 1.5 MeV on the hypernuclei’s energy spectrum as shown in ﬁg. 8 [38]. 12 As a ﬁrst step, excitation energy spectra were measured for 7Λ Li, 9Λ Be, 10 Λ B, Λ C, 13 16 28 51 89 139 208 Λ C, Λ O, Λ Si, Λ V, Λ Y, Λ La, Λ Pb. It is the more systematic study ever done, and conﬁrmed the previous result on the physical reality of single particle states deeply located in nuclei as heavy as 208 Pb. This bank of data allowed a reliable determination of the parameters of the potential well seen by the Λ in a hypernucleus. Furthermore, thanks to the improved energy resolution, a splitting of the peaks corresponding to the Λ in one shell was observed. The splitting seemingly increased with the angular momentum of the observed shell-model orbit, asking then for a non-zero value of the spin-orbit term. This question is still now open. After this series of systematic measurement, the SKS was used to identify hypernuclei in well-deﬁned states (ground or excited) and study their decay by other arrays of complex detectors. This was possible thanks to the space left free around the targets used in the (π + , K + ) reaction. The second hypernuclear factory which was built taking into account the very interesting results obtained by using K − at rest is FINUDA. FINUDA (acronym for FIsica NUcleare a DAΦNE) may be considered the more complete hypernuclear factory built up to now and is completely diﬀerent from all the other detectors and spectrometers de-

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interaction/target region end cap clepsydra straw tubes LMDC beam pipe external TOF superconducting coil compensating magnets magnet yoke

Fig. 9. – Global view of the FINUDA detector with the superconducting solenoid and the iron yoke.

scribed up to now. It is installed at an (e+ , e− ) collider, DAΦNE at Laboratori Nazionali di Frascati. Thanks to the large angular coverage for detection of charged and neutral particles from the formation and decay of hypernuclei, many observables (excitation energy spectra, partial decay widths for mesonic, NM decay and multi-nucleon absorption), are measured simultaneously, and will provide hopefully a bulk of data of unprecedented completeness, precision and cleanliness. Furthermore, spectra corresponding to diﬀerent targets can be measured simultaneously, avoiding possible systematic errors in comparing the properties of diﬀerent hypernuclei. Under this point of view, FINUDA is the modern electronic experiment approaching more closely the original visualizing technique by Danysz and Pniewski. The idea of performing an experiment on the production and decay of hypernuclei, which is inherently a ﬁxed-target physics, at a collider seems not good at ﬁrst sight. The main decay channel of the Φ-meson, produced at a rate of ∼ 4.4 × 102 s−1 at the luminosity L = 1032 cm−2 s−1 , is (K + , K − ), ∼ 49%. Since the Φ-meson is produced at rest, DAΦNE at LNF is a source of ∼ 2.2 × 102 (K + , K − ) pairs/s, which are collinear, background free, and, very important, of very low energy (∼ 16 MeV). The low energy of the produced charged kaons is the key advantage for an experiment on hypernuclei production and decay by means of the strangeness-exchange reaction with K − at rest: (11)

− − Kstop +A Z →A Λ Z +π

− in which A Z indicates the nuclear target and A Λ Z the produced hypernucleus. The K 2 can be stopped in very thin targets (a few hundreds of mg/cm ), in contrast to what happens in experiments with stopped K − at hadron machines, where from 80% up to 90% of the incident K − beam is lost in the degraders facing the stopping target. Furthermore,

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Fig. 10. – Pictorial view of the manifold physical information that can be obtained from the − + 6 Li reaction. Kstop

the cylindrical symmetry of the interaction region at a collider allows the construction of high-resolution spectrometers with very large (> 2πsr) solid angles for detecting the π − from (11). Last but not least, the use of thin targets introduces lower cuts on the measurement of the energy spectra of charged particles (π − , p, d) from the weak decay of the hypernuclei produced by (11) and they are detected by the same arrays of detectors with large solid angles (> 2πsr). I put forward these considerations in 1991 [39] and a proposal was afterwards elaborated [40] and soon accepted by the Scientiﬁc Committee of LNF. Figure 9 shows a sketch of the detector, immersed in a superconducting solenoid which provides a homogeneous magnetic ﬁeld of 1.1 T inside a cylindrical volume of

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Fig. 11. – Logo of the FINUDA experiment.

146 cm radius and 211 cm length. (e+ , e− ) collide at the center of the magnet with an energy of 510 MeV each. A. Feliciello will give a detailed description of the FINUDA’s experimental and technical features as well as an account of the physics results so far reached [28]. The amount of physical information that can be gained on a single target in a single run is represented pictorially by ﬁg. 10 for the 6 Li nucleus [41]. I emphasize that diﬀerent experimental features relevant to diﬀerent measurements (e.g., momentum resolution for the diﬀerent particles vs. the statistical signiﬁcance of the spectra) may be optimised out from the same data bank by applying with software diﬀerent selection cryteria. The approach is then quite diﬀerent from that followed by the SKS hypernuclear factory, in which diﬀerent hardware layouts were used in subsequent runs in order to obtain the excellent results, some of which I will present in the following sections. 6 Li is a particulary interesting target, since its simple structure allows to reach and study also the 6Λ He hypernucleus, without performing the spectroscopy of the π 0 emitted in the formation reaction [42]. I had in mind these unique features when I drawn the logo of FINUDA (ﬁg. 11). I represented FINUDA by a bat for the following analogies: a) it is the only mammal capable of ﬂying, b) it is an animal exploiting very sophisticated technologies, like orientation in the dark by detecting and measuring ultrasonic waves reﬂection, c) even being a formidable and ecologically correct destroyer of harmful insects, it was heavily persecuted, and only recently its usefulness was recognized. The four constellations appearing in the sky represent 6 Li at the top, and the three mass 6 hypernuclei of ﬁg. 10 at the bottom. Let me now emphasize a peculiar feature of FINUDA, not adopted in the detectors so far mentioned. It is the presence of a powerful vertex detector, composed by two barrel-shaped arrays of Si Microstrip detectors that allow the separation of the primary

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Fig. 12. – a) Negative pion momentum spectra measured at KEK and c) with FINUDA. b) The total energy of the neutral pion measured by ref. [44]. The inset in a) shows an enlarged view of the spectrum in the region corresponding to the bound hypernuclear state production. The dotted line in c) represents a Monte Carlo simulation of the background. From ref. [43].

interaction vertex of reaction (1) from the secondary vertex due to the decay of the produced hyperons Λ and Σ with a spatial resolution of 3 mm. For studies of hypernuclear spectroscopy it is of paramount importance to measure the hypernuclear energy spectra minimizing the background, that should forbid the observation of states produced with a low strength. To show this eﬀect let me discuss [43] similar spectra of pions emitted following the interaction of K − stopped in 12 C targets in experiments performed at KEK [22], BNL [44] and with FINUDA [45]. At BNL π 0 were spectroscopized instead than π − with the aim of studying the structure of mirror hypernuclei (12 Λ B in this case). Upon proposal, the BNL experiment looked very promising and challenging concerning resolution and rates, comparable with those of competing experiments (see, e.g., [46]). Unfortunately the experimental results were quite decepting. The energy resolution was about 2.2 MeV, which may be considered reasonable if one takes into account that the ﬁne spectroscopy of π 0 is very much more diﬃcult than that of π − . However, the spectrum of π 0 carrying the information on the discrete levels of the ﬁnal hypernucleus 12 Λ B was superimposed to a large background and this circumstance led the authors to conclude their paper with the following statement: “Finally, given the observed level of background compared to the signal, it appears that the usefulness of this reaction for hypernuclear spectroscopy is limited”. This statement is uncorrect, since the beam and detector’s design and performances were the limiting factor. As discussed at the end of sect. 2, the more signiﬁcant contribution in the bound hypernuclear states region comes from the absorption of the K − on a (N N ) pair, producing a Σ± and a nucleon. The Σ decays in ﬂight into N + π, and the π’s from this process have a broad ﬂat spectrum extending beyond the kinematical limit corresponding to the formation of the hypernucleus in the ground state (12 Λ C in the example). The amount of this background is diﬀerent in the three spectra shown in ﬁg. 12 and is closely related to the experimental conditions, namely the purity of the K − beam, the precision on the determination of the stopping point inside the target and consequently the capability of

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Fig. 13. – Λ-binding energy spectrum of 12 Λ C measured by FINUDA. The solid line represents the best ﬁt obtained with seven Gaussian function. From ref. [45].

distinguishing secondary vertices (as in the case of π − from Σ− decay). FINUDA, which is a third-generation experiment equipped with a vertex detector, provides the better conditions. Thanks to these features FINUDA provided nice results on hypernuclear spectroscopy featuring the best energy resolution on the hypernuclear energy levels (1.29 MeV FWHM) so far reached with magnetic spectrometers for mesoninduced reactions like (1) and (2). Figure 13 shows the excitation energy spectrum measured for 12 Λ C [45]. A short discussion on the interpretation of such a spectrum will be given in the next section 4. There is still one option missing in the detection capabilities of FINUDA: that of the γ-rays spectroscopy with excellent energy resolution with High-Purity Ge detectors (HPGe), pionereed at KEK [47]. Feasibility studies for implementing the existing complex of detectors of FINUDA with HPGe have been performed [48, 49], as well as an R&D study of the performances of HPGe detectors in strong magnetic ﬁelds [50]. As mentioned in sect. 2, only very recenly reaction (3) has been exploited at TJNAF for the production of hypernuclei. Two performing complexes of spectrometers were installed in Hall A and Hall C, respectively. Since reaction (3) leads to three bodies in the ﬁnal state, two magnetic spectrometers with large acceptance had to be installed for the event-by-event reconstruction. A further diﬃculty was represented by the necessity of using a splitter magnet after the target, keeping at minimum the disturbance introduced by such an element in the magnetic optics of the spectrometers. Figure 14 shows a sketch of the spectrometer system installed in Hall C.

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Fig. 14. – Hypernuclear spectrometer system in JLab Hall C for E89-009. From ref. [14].

Fig. 15. – Missing mass spectrum for the 12 C/CHx (e, e K + ) reaction. The shaded histogram is the accidental background. From ref. [51].

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Fig. 16. – The 12 Λ B excitation-energy spectrum. The best ﬁt (solid curve) and a theoretical prediction (dashed curve) are superimposed on the data. From ref. [52].

A pilot experiment performed on a 12 C target produced very interesting results, showing an energy resolution on the hypernuclear energy levels of 0.9 MeV FWHM (see ﬁg. 15). A better resolution of at least 0.67 MeV FWHM was very recently attained by using the spectrometer system installed in Hall A, in a measurement on a 12 C target, with good statistics (ﬁg. 16). 4. – Hypernuclear spectroscopy In the following the term “spectroscopy” will be used in a broad sense, to indicate the energy levels of a many-body system, composed by Z protons, A − Z − 1 neutrons and a Λ-hyperon. As mentioned in sect. 3, the ground state of such a system is given by a conﬁguration in which the (A − 1) nucleons occupy the ground state of the residual nucleus (A−1) Z and the Λ-hyperon is in its lowest energy level (1s). Λ-hypernuclei known up to now are 35 and are shown in ﬁg. 17, called the Segr`e hypernuclei chart. BΛ increases linearly with A, with a slope of about 1 MeV per A unit, and saturates at about 23 MeV for heavy hypernuclei. This trend suggests a simple model in which the Λ-hyperon is conﬁned in a potential well, with radius equal to the radius of the (A−1) Z nucleus and with a width of about 28 MeV, i.e. about one half of the width of the potential well of a normal nucleus (55 MeV). This hypothesis is consistent with the fact that the strength of the ΛN interaction is about one half of the N N one. It is also possible to represent the excited states of a hypernuclear system following the elementary models developed for the normal nuclear excited states. The simplest one is certainly the single-particle shell model and, as an example, the possible excited states of 12 Λ C are shown ﬁg. 18.

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Fig. 17. – Λ hypernuclear chart. The experimentally identiﬁed Λ-hypernuclei and the experimental methods used to study them (reaction spectroscopies of (K − , π−), (π + , K + ), (e, e K + ) etc., γ-spectroscopy, and the emulsion method) are shown. From ref. [14].

In a) the baryon conﬁguration corresponding to the ground state is reported; in b) the (A − 1) nucleons are in an excited state and the Λ-hyperon is in the ground state; in c) the (A − 1) nucleons are in the ground state of the residual nucleus and the Λ is in one of the single-particle excited states of the shell model; in d) both the (A − 1) nucleons and the Λ are in an excited state. To simplify the representation, diﬀerent conﬁgurations of neutrons only have been shown to represent the global nucleon system. At ﬁrst sight, it should seem unreasonable to study nuclear systems composed by three diﬀerent kinds of baryons, being already very diﬃcult to get a deep understanding of those composed by two baryons only. But we must remind that in hypernuclei only one, distinguishable, baryon is added, allowing to examine the true single-particle states of the Λ embedded in the average potential of the other (A − 1) nucleons, without the distortion due to the Pauli principle and to other side eﬀects that cannot be avoided when considering ordinary nuclei. A possibile answer to this question was given by H. Feshbach, in 1973 [53] just on the appearance of the ﬁrst results on hypernuclei’s excitation spectra, with poor resolution and marginal statistics (see, e.g., ﬁg. 5). He writes: “Since we have nearly zero information, we shall esamine what we might learn if all possibile experimental information regarding hypernuclei were available. Suppose we have the energy spectrum, the electromagnetic properties of each bound level, i.e. their static electric and magnetic multipole moments, their Coulomb energy. Suppose we know the transition multiple moments as well: Finally these can decay —and suppose we know the lifetimes and disintegration products. What would we learn about nuclei from these data?” We have today these data for some hypernuclei, and I will present the rich physics information we obtained. It is now easy to understand why with reaction (1), close to the magic momentum, it is hard to produce hypernuclei in the ground state. Recalling that this reaction simply

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25

Fig. 18. – Simpliﬁed representation of the single-particle states of the nucleons and of the Λhyperon in 12 Λ C. a) Ground state; b) ﬁve neutrons in an excited state and the Λ-particle in the ground state; c) ﬁve neutrons in the ground state and the Λ-particle in one of the states allowed by the single-particle shell model; d) both ﬁve neutrons and the Λ-particle in an excited state.

changes a neutron to a Λ, it will convert a neutron in the 1s1/2 or 1p3/2 state in a Λ with the same quantum numbers. Considering again 12 C and 12 Λ C, from ﬁg. 18 it is straightforward to check that such conﬁgurations do not correspond to 12 Λ C in the ground state. On the other hand, with reaction (2) or reaction (1) with K − at rest, it is possible to convert a neutron in the 1p3/2 state to a Λ in the 1s1/2 state, due to the transferred momentum, producing a 12 Λ C hypernucleus in the ground state. The plainest demonstration of the validity of this simple representation is given by the experimental data. In ﬁg. 19 a recent excitation spectrum of 89 Λ Y is shown; it has been obtained with reaction (2) at KEK [54]. It is known, from the independent particle model, that it is possible to transform a neutron in the g9/2 state into a Λ-hyperon in the s, p, d, f states. In the ﬁgure, in fact, it is possible to observe four well-separated peaks corresponding to these conﬁgurations. With a worse energy resolution, a structure with six peaks has been observed in the 208 Pb excitation spectrum, corresponding to the s, p, d, f , g, h states [54]. These measurements are the most spectacular conﬁrmation of the physical validity of the independent particle model for nuclear physics. Figure 20 shows the hypernuclear mass number dependence of the binding energy for each Λorbital, including the s-wave. The dashed lines represent the calculation with a WoodsSaxon potential for a Λ-hyperon with the parameters given in ref. [55]. In ordinary nuclei, even with the best state-of-art experimental techniques, it has never been possible to observe contemporaneously the existence of nucleons in more than two well-deﬁned states, corresponding, moreover, only to the more external orbits. This is a consequence of the fragmentation of the single-particle states intensity due to various eﬀects, and the experimental observation is a continuous spectrum, without any structure. A recent unexpected result, clearly visible in the spectra of ﬁgs. 8, 13 and 16, taken from the

26

Fig. 19. – Missing mass spectrum of

T. Bressani

89 Λ Y

(KEK E369). From ref. [54].

− same target with diﬀerent detectors and reactions (π + , K + , SKS), (Kstop , π − , FINUDA), 12 (e, e p, Hall A TJNAF) on the same C target is the occurrence of a considerable strength of excitation in between the two large peaks corresponding to the Λ in s- and p-orbitals, not predicted by theoretical calculations. This observation opens the door for the study of mixed Λ-excited nuclear core conﬁgurations.

Fig. 20. – Hypernuclear mass dependence of a Λ-hyperon binding energy in various orbits. From ref. [54].

27

Strange nuclei

The problem of the experimental resolution is of paramount importance for precise spectroscopic studies. Magnetic spectrometers, at present, provide a ΔE of 1.3 MeV. A reduction of about a factor 2 (ΔE of 0.7 MeV) is expected for the forthcoming experiments at DAΦNE and TJNAF, and a ΔE = some tenths of MeV is discussed for J-PARK [20], a remarkable result but still insuﬃcient to obtain the precision asked for the determination of important physical parameters. Only the γ-spectroscopy is able to reach resolutions of the order of the keV, but this approach is experimentally hard, since the counting rates are strongly reduced (coincidence experiments) and, furthermore, only the low-lying excited states, not decaying by nucleon emission, can be studied. The gross features of the data are reasonably well reproduced by an eﬀective one-body hyperon-nucleus interaction constructed on potential models of the hyperon-nucleon Y N interaction. The eﬀective two-body ΛN interaction in hypernuclei can be expressed as (12)

VΛN (r) = V0 (r) + Vσ (r) sN · sΛ + VΛ (r) lN Λ · sΛ + VN (r) lN Λ · sN +VT (r)[3(σN · r )(σΛ · r − σN · σΛ )]

with a straightforward meaning of the various terms. The starting point of these models is a good N N interaction generated by the exchange of nonets of mesons, with SU (3)f constraints for the coupling constants. Their predictions are ﬁtted simultaneously to the abundant N N data and the very scarce Y N free-scattering data. However, until now, it has not been possible to obtain a reliable and unambiguous Y N interaction model. The improvement of the Y N interaction models would need precise data on the free Y N interaction, which are very diﬃcult to obtain, due to low hyperon beam intensities and short lifetimes of the hyperons. In particular, production and scattering in the same target are almost automatically required. At present, the experimental data on ΛN and ΣN scattering consist of not more than 850 scattering events, in the momentum region from 200 to 1500 MeV/c. The low-energy data, in particular, fail to adequately deﬁne even the relative sizes of the dominant s-wave spin-triplet and spin-singlet scattering lengths and eﬀective ranges. This is the reason why a systematic and detailed spectroscopic study of hypernuclei with high-resolution experiments, with single and even multiple strangeness contents, would oﬀer the possibility of increasing the experimental information on Y N interaction, and for the ﬁrst time, allow both the study of dynamics of systems carrying SU (3)f ﬂavor symmetry, in the non-perturbative QCD sector, and the generalization of the baryonbaryon interaction to the full SU (3)f family, including hyperons [56]. The V (0) term was determined by the measurement of the binding energy for each Λ-orbital, discussed previously [54]. The spin dependent parts of the Y N interaction are of major interest, since they are intimately related to the modelling of the short-range part of the interaction. In particular, for the ΛN interaction, the spin-orbit interaction was found to be smaller than that for the nucleon, which amounts to 3–5 MeV. In fact, as already mentioned in sect. 3,

28

T. Bressani

Fig. 21. – γ-ray energy spectrum from the de-excitation of 9Λ Be obtained at BNL. From ref. [58].

old experiments based on the hypernucleus formation kinematics from reaction (1), with energy resolution of the order of 5 MeV were not able to measure any spin-orbit splitting. A closer inspection of the spectrum of ﬁg. 19 (and analogous one for 51 Λ V, . . .) led to the conclusion that the peaks corresponding to the states p, d, f have a width larger than the experimental resolution and indicate the existence of doublets of states, not fully resolved. The splitting of doublets seems to increase with the angular momentum of the singleparticle Λ-state. Such an observation lead the authors to the immediate conclusion that the spin-orbit term in the Λ-nucleus optical potential, that was developed in complete analogy to the nucleon-nucleus optical potential, must have a strength of a few MeV, in contrast to the previous determination that assesses that it was consistent with zero. However, a further theoretical analysis of the spectra indicated that a splitting may also be expected from other nuclear structure eﬀects, still maintaining the spin-orbit term in the Λ-nucleus potential equal to zero. Calculations of the spin-orbit component of the ΛN interaction using meson exchange theories, lead to predictions of the spin-orbit splitting for the 5/2+ –3/2+ doublet in 9Λ Be of 80–160 keV, depending on the interaction used [57]. On the other hand, when a quarkmodel–based spin-orbit force is used, which naturally accounts for the short-range part of the interaction, much smaller values of 30–40 keV are obtained. The two types of models, in fact, predict diﬀerent features for the antisymmetric part of the spin-orbit force. However, it needs to be said that quark models have yet to provide an extensive and satisfactory description of the Y N interaction. A clever combination of the best state-of-the-art technologies in magnetic spectrometry and high-rate, high-resolution γ-ray spectroscopy with HPGe detectors in coincidence allowed a real breakthrough in hypernuclear spectroscopy. This new technique allowed the energy resolution on low-lying hypernuclear levels to be improved from a few MeV to a few keV, even if the resulting count rate were still quite low. Figure 21 reports a measurement of the splitting of the 5/2+ –3/2+ doublet in 9Λ Be by the BNL-AGS E930 experiment [58]; the measured spacing is 31 ± 2 keV, incompatible with the prediction of the meson exchange models.

Strange nuclei

29

Fig. 22. – a) Level scheme and γ transition of 7Λ Li; b) measured γ-ray spectrum of 7Λ Li. The two insets show views of the two portions of the spectrum. From ref. [47].

At KEK the spin orbit splitting of the 3/2− –1/2− doublet in 13 Λ C was measured, though with a coarse resolution on γ-rays [59], and a doublet spacing of 152 ± 54(stat) ± 36(sys) keV was determined. The splitting predicted for this transition by meson exchange models is 390–780 keV, depending on the interaction used, and 150–200 keV by quark-based models [60]. Let me now turn to another example of the importance of precise measurements of lowlying hypernuclear excited states by means of coincidence high-resolution γ-spectroscopy. We will see how from a single experiment we will get important insight on the ΛN elementary interaction and further. It is possible to deduce the strength of the spin-spin term by the spacing of appropriate levels in 7Λ Li. When a Λ in the state 1s of a hypernucleus is coupled to a nuclear core with J = 0, the spin interaction leads to a doublet (J − 1/2, J + 1/2) of states whose splitting may be directly linked to the spin-spin part of the elementary ΛN interaction. Figure 22a) [47] shows how we get the low-lying part of the 7Λ Li excitation spectrum. By coupling the Λ in the 1s ground state to the 1+ ground state of the core 6 Li, we expect a doublet 1/2+ (ground state of 7Λ Li), 3/2+ at some hundreds of keV. By coupling the Λ, always in the 1s ground state, to the ﬁrst 3+ excited state of 6 Li, we expect a doublet of states 5/2+ , 7/2+ of 7Λ Li. All the excited states of 7Λ Li decay to the ground state by γ-transitions of magnetic dipole (M 1) or electric quadrupole (E2), but it is not possible to separate the diﬀerent components of the doublets by the analysis of the formation reaction since the energy resolution due to the magnetic spectrometer is lower by more than one order of magnitude than the expected level’s spacing. The measured γ-ray spectrum is reported by ﬁg. 22b) [47]. We clearly observe the presence of γ-rays at 692 keV (spin-ﬂip M 1, 3/2+ → 5/2+ ) and 2050 keV (E2, 5/2+ → 1/2+ ). The spacing of the doublet (3/2+ , 1/2+ ) is determined almost completely from the strength of the spin-spin ΛN interaction. A value of 0.5 MeV was inferred from this experiment, which is of great importance in the study of the potential models for the ΛN interaction.

30

T. Bressani

Perhaps more important is the physical information deduced from the analysis of the transition 5/2+ → 1/2+ at 2050 keV. It can be noticed preliminarly that the experimental spectra present for both transitions the so-called Doppler shift, due to the circumstance that the hypernuclei 7Λ Li directly produced in their excited states in the two-body reaction π + + 7 Li → 7Λ Li+K + decay by γ emission in ﬂight. Correcting the spectra event by event for this eﬀect, the γ-line at 692 keV becomes a narrow peak, with a width comparable with the experimental resolution, and that means that the lifetime of the state is less than the time spent by the recoiling 7Λ Li for stopping in the target (a few ps). On the contrary, the γ-line at 2050 keV has the more complex structure of a narrow spike superimposed to a broad bump, the ﬁrst corresponding to decays of 7Λ Li at rest, the second in ﬂight. That means that the lifetime is comparable to the stopping time and this circumstance allowed the authors to determine, by the method known in nuclear physics as Doppler attenuation method, the lifetime: 5.8 ps with a ∼ 20% total error. The measured energy and lifetime of the 7Λ Li may be used to determine the reduced transition probability, +0.5 2 B(E2). They found B(E2) = 3.6 ± 0.5−0.4 e fm4 that, compared with the well-known 2 4 B(E2) = 10.9 ± 0.9 e fm of the core nucleus 6 Li (3+ → 1+ ), indicates a signiﬁcant shrinkage (∼ 20%) of 7Λ Li size from the 6 Li size. It is an evidence of the “glue-like” role of the Λ, predicted by several authors. The above examples showed the great potentiality of γ-spectroscopy of hypernuclei. Stimulated by the beautiful results the authors started a strong eﬀort of measuring γ-transitions in several hypernuclei of the p-shell, selected as to be sensible only to one of the terms of the spin-dependent parameters in the eﬀective potential. A ﬁrst reasonable determination was reached, even though with some inconsistency. A full account of these results is given in [14]. A very ambitious research program is under development and will be performed at J-PARC [20]. In addition to continuing the precise measurement of ΛN eﬀective potential and nuclear parameters by means of the “impurity” Λ, other very interesting eﬀects were anticipated. For instance, it is proposed to measure the magnetic momentum of the Λ in a nuclear medium, that is expected to be diﬀerent from that of the free Λ, as the mass and the size. B(M 1) is proportional to (gN −gΛ )2 , where gN and gΛ denote eﬀective g-factors of the nucleus and the Λ, and can be measured for some selected M 1 transitions by the Doppler shift attenuation method or, in case of larger lifetimes, by the time spectra of the γ-weak decay coincidence. 5. – Weak decay of Λ-hypernuclei In addition to information on nuclear structure and the Y N interaction, hypernuclei may give access to experimental information not otherwise accessible by their decays, in particular the NM decay. The importance of such process was pointed out just at the ﬁrst beginning of hypernuclear physics. As a matter of fact, Cheston and Primakoﬀ gave a quantitative discussion of the possibility that a Λ-hyperon bound to nucleons might undergo NM decays in 1953 [61]. Let me recall that a free Λ-hyperon decays almost at 100% into a pion and a nucleon

Strange nuclei

31

(Λ → pπ − (∼ 64%), Λ → nπ 0 (∼ 36%)), with a release of kinetic energy of about 5 MeV to the nucleon, corresponding to a ﬁnal momentum of about 100 MeV/c. The decay may occur, in principle, with isospin change ΔI = 1/2 or ΔI = 3/2. However, the experimental decay branching ratios of the Λ and other hyperons imply a dominance by a factor 20 of the ΔI = 1/2 component over the ΔI = 3/2 one. The ΔI = 1/2 rule is based on experimental observations but its dynamical origin is not yet understood on theoretical grounds. It is also valid for the decay of the Σ-hyperon and for pionic kaon decays (namely in non-leptonic strangeness-changing processes). Actually, this rule is slightly violated in the Λ free decay, and it is not clear whether it is a universal characteristic of all non-leptonic processes with ΔS = 0. As a consequence, in nuclei the mesonic decay is disfavoured by the Pauli principle, particularly in heavy systems. It is strictly forbidden in normal inﬁnite nuclear matter (where the Fermi momentum is kF0 270 MeV/c), while in ﬁnite nuclei it can occur because of three important eﬀects: 1) In nuclei the hyperon has a momentum distribution (being conﬁned in a limited spatial region) that allows larger momenta to be available to the ﬁnal nucleon. 2) The ﬁnal pion feels an attraction by the medium such that for ﬁxed momentum q it has an energy smaller than the free one and consequently, due to energy conservation, the ﬁnal nucleon again has more chance to come out above the Fermi surface. Indeed it has been shown that the pion distortion increases the mesonic width by one or two orders of magnitude for very heavy hypernuclei (A 200) with respect to the value obtained without the medium distortion. 3) At the nuclear surface the local Fermi momentum is considerably smaller than kF0 , and the Pauli blocking is less eﬀective in forbidding the decay. In any case the mesonic width rapidly decreases as the nuclear mass number A of the hypernucleus increases. Even though the Λ-particle has a lifetime of the order of 2.76 × 10−10 s due to its stability against strong interaction, it is still so diﬃcult to perform ΛN scattering experiments that the amount and accuracy of the data are limited. Therefore, to produce hypernuclei and to investigate their structures has been the most practical approach to study the ΛN interaction. In the calculations starting from the realistic ΛN two-body interaction, the existence of a central repulsion in the hyperon-nucleus potential has been discussed by many authors. The strength of long-range attraction of the Y N interaction is much weaker than that of the N N interaction and it is almost counterbalanced by the short-range repulsion. Reﬂecting this situation, there remains inner repulsion also in the hyperon-nucleus potential which is constructed from the Y N interaction using the folding procedure. This eﬀect can be experimentally detected most distinctly in the case of very light (A = 4–5) hypernuclear systems. A Λ-hyperon is pushed outward from a core nucleus due to this repulsion. Consequently, the overlap of the wave function of the hyperon

32

T. Bressani

with the nucleus is considered to be much reduced. It has been realized that observables concerning hypernuclear decay provide signiﬁcant information on the Λ single-particle (s.p.) potential. The π-mesonic decay of hypernuclei, as said, undergoes Pauli suppression and the mesonic decay rate is thus sensitive to the extent of the overlap between the wave functions of Λ and core nucleus, which reﬂects the potential shape felt by Λ in nuclei. If the mesonic decay rate can be precisely measured, it is expected to be an essential complement of knowledge of the Λ single particle potential, the shape of which cannot be investigated from the Λ binding energy. Accurate measurements of the π-mesonic decay width of light s-shell Λ-hypernuclei were performed at KEK with the SKS spectrometer in E462 experiment [62] and the existence of an inner repulsive core was established. In hypernuclei the weak decay can occur through processes which involve a weak interaction of the Λ with one or more nucleons. Sticking to the weak hadronic vertex Λ → πN , when the emitted pion is virtual, then it will be absorbed by the nuclear medium, resulting in a non-mesonic process of the following type: (13)

Λn → nn

(Γn ),

(14)

Λp → np

(Γp ),

(15)

ΛN N → nN N

(Γ2 ).

The total weak decay rate of a Λ-hypernucleus is then (16)

ΓT = ΓM + ΓNM ,

where ΓM and ΓNM indicate, respectively, the mesonic and non-mesonic decay widths and can be written as (17)

ΓM = Γπ− + Γπ0 ,

(18)

ΓNM = Γ1 + Γ2 ,

(19)

Γ1 = Γn + Γp

and the lifetime is τ = h ¯ /ΓT . The channel (15) can be interpreted by assuming that the pion is absorbed by a pair of nucleons, correlated by the strong interaction. This term, often referred as “two-nucleon-induced decay” has been proposed in addition to the two-body reaction Λ + N → N + N by theoretical considerations [63], but a convincing experimental evidence has not yet been put forward. Obviously, the NM processes can also be mediated by the exchange of more massive mesons than the pion as shown in ﬁg. 23. The non-mesonic mode is only possible in nuclei and, nowadays, the systematic study of the hypernuclear decay is the only practical way to get information on the weak process ΛN → N N (which provides the ﬁrst extension of the weak ΔS = 0 N N → N N interaction to strange baryons), especially on its parity-conserving part.

Strange nuclei

33

Fig. 23. – One-nucleon (on the left) and two-nucleons (on the right) induced Λ decay in nuclei.

The ﬁnal nucleons in the NM processes are emitted with large momenta: disregarding the Λ and nucleon binding energies and assuming the available energy Q = mΛ − mN 176 MeV to be equally splitted among the ﬁnal nucleons, it turns out that pN 420 MeV for the one-nucleon–induced channels (eqs. (13) and (14)) and pN 340 MeV in the case of the two-nucleon–induced mechanism (eq. (15)). Therefore, the NM decay mode is not forbidden by the Pauli principle: on the contrary, the ﬁnal nucleons have great probability to escape from the nucleus. The NM mechanism dominates over the mesonic mode for all but the s-shell hypernuclei. Only for very light systems the two decay modes are competitive. Several theoretical models have been developed in order to explain the weak-decay mechanism of hypernuclei; the main diﬀerences between these models must be referred to the nature of the exchange potential. The ﬁrst calculations of the mesonic rate for light hypernuclei date at the end of the 50’s [64]. The Pauli blocking eﬀect for nuclear decay was estimated and used in order to assign the spin to the ground state of s-shell hypernuclei. As mentioned before the possibility of NM hypernuclear decay was suggested for the ﬁrst time in 1953 [61] and interpreted in terms of the free space Λ → N π decay, where the pion was considered as virtual and then absorbed by a bound nucleon. In the 60’s Block and Dalitz [65] developed a phenomenological model, which has been more recently updated [66]. Within this approach, some important characteristics of the NM decays (for instance the validity of the ΔI = 1/2 rule) of s-shell hypernuclei can be reproduced in terms of elementary spin-dependent branching ratios for the Λn → nn and Λp → np processes, by ﬁtting the available experimental data. After the ﬁrst analysis by Block and Dalitz, microscopic models of the ΛN → nN interaction began to be developed. The ﬁrst of these model was the One-Pion-Exchange (OPE) model. The OPE model is based on the ΛN π weak vertex followed by the absorption of the virtual pion by a second nucleon of the nuclear medium. The results of the decay width calculation for the one-nucleon–induced NMWD were not realistic also because the employed ΛN π coupling was too small to reproduce the Λ free lifetime. Afterwards, in order to improve the OPE model, mesons heavier than the pion were

34

T. Bressani

introduced as mediators of the ΛN → N N interaction. McKellar and Gibson [67] evaluated the width for a Λ in nuclear matter, adding the exchange of the ρ-meson and taking into account ΛN relative s-states only. They calculated the ΛN ρ weak vertex assuming the ΔI = 1/2 rule and made the calculation by using the two possible relative signs (being at the time unknown and not ﬁxed by their model) between the pion and ρ potentials. It is important to note that for mesons heavier than the pion, no experimental indication supports the validity of the ΔI = 1/2 rule for their couplings with baryons. Some years later, Nardulli [68] determined the relative sign (−) between ρ and π exchange implementing the available information from weak non-leptonic and radiative decays. References [67, 68] obtained a non-mesonic width in the (ρ + π) exchange model smaller than the OPE one. In 1986 Dubach et al. [69] extended the OPE model to the more complete OME (One Meson Exchange), in which diﬀerent exchanged mesons (π, ρ, K, K ∗ , ω and η) were considered. For most of the OME calculation the ΔI = 1/2 rule was assumed for the meson coupling with baryons. This assumption is justiﬁed for the ΛN π weak vertex, but there is no experimental evidence to assume it for other vertices like ΛN ρ which is not accessible experimentally. Another theoretical approach to the short-range nature of the ΛN weak interaction is the quark current interaction model. Cheung, Heddele and Kisslinger [70, 71] considered the hybrid quark hadron model for the NM decay. In this model the decay is explained by two separate mechanisms which have diﬀerent interaction ranges: the long-range term (r ≥ 0.8 fm) was described by OPE with ΔI = 1/2 rule, whereas the short-range interaction was described by a six-quark cluster model including both ΔI = 1/2 and ΔI = 3/2 contributions. More recently Inoue et al. calculated the NM decay width with a direct quark (DQ) model [72] where a four-quark interaction between the costituent quarks of baryons caused the transition without exchanging mesons. The four-quark vertex was obtained from the low-energy eﬀective Hamiltonian in the perturbative QCD theory which contains both ΔI = 1/2 and ΔI = 3/2 transitions. These authors stressed that the DQ mechanism gives a signiﬁcant ΔI = 3/2 contribution in the l = 0 channel. This model was further extended to combine with the OME calculation [73] in order to take into account the long-range interaction which is not included in the DQ mechanism. All these theoretical eﬀorts done in the last years were devoted to the solution of an important question concerning the weak-decay rates. In fact the study of the NMWD was characterized by a long-standing disagreement between theoretical estimates and experimental determinations of the ratio Γn /Γp between the neutron- and the protoninduced decay widths. It is worth recalling here that, a few years ago, all theoretical calculations appeared to strongly underestimate the available central data measured in several hypernuclei: (20)

Γn Γp

Theory

Γn Γp

Exp ,

0.5 ≤

Γn Γp

Exp ≤ 2.

Nowadays it seems that hypernuclear physics is going toward a solution of the Γn /Γp

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35

Fig. 24. – Lifetime of the Λ-hyperon as measured for several Λ-hypernuclei.

puzzle in the NM weak decay thanks also to the inclusion in the ΛN → nN transition potential of mesons heavier than the pion, the inclusion of interaction terms that explicitly violate the ΔI = 1/2 rule and the description of the short-range baryon-baryon interaction in terms of quark degrees of freedom, which automatically introduces ΔI = 3/2 contributions. Since the NM channel is characterized by a large momentum transfer, the details of the hypernuclear structure do not have a substantial inﬂuence (then providing useful information directly on the hadronic weak interaction). On the other hand, the N N and ΛN short-range correlations turn out to be very important. In fact the large momentum transfer in NM decay processes implies that they probe short distances and might, therefore, elucidate the role of explicit quark/gluon substructure of the baryons. Furthermore, the fundamental question as to whether the ΔI = 1/2 rule, which governs pionic decay, applies to the NM weak decays may also be addressed. Recent review papers on this topic are due to Alberico and Garbarino [74] and to Outa [75]. The ﬁrst observable that can be measured is the lifetime τΛ of a given hypernucleus for weak decay, otherwise referred as the lifetime of Λ in nuclei. Recently measurements of τΛ of reasonable quality (errors of ∼ 10%) were performed [15, 76, 77] and the results are summarized by ﬁg. 24. It should be noticed that all the experimental data points show a smoothly decreasing behaviour as a function of the mass number A, with the exception of the value for 16 O and that for the heavier targets (Bi, U). Whereas for all the experiments in agreement the hypernuclei were produced by means of reactions (1) and (2), the experiment on 16 Λ O 16 was performed by Nield et al. [17] using a beam of relativistic O ions and that on the heavier targets by means of delayed ﬁssion induced by protons. The datum on 16 Λ O is very likely aﬀected by some experimental drawback, the datum for the heavy targets is quite recent [15] and the experiment was carefully done. The

36

T. Bressani

Fig. 25. – Total NM decay widths of Λ-hypernuclei. For details see the text. From ref. [78].

conclusion is that for the medium A hypernuclei τΛ is about 80% that of the free Λ, falling at about 50% for the heaviest targets. Based on the results of lifetime measurements and mesonic decay branching ratio in the mass number range of 12 ≤ A ≤ 56, the mass number dependence of the NM decay widths is plotted in ﬁg. 25 [78]. Closed circles are the data from [78] and the open circles are previous experimental data in which the hypernuclear production was explicitly identiﬁed. The open diamonds are experimental data of lifetimes measurements for very heavy hypernuclei in the mass region of A ∼ 200 with recoil shadow method on p + Bi and p + U reactions at COSY [15] in which the production of strangeness was not explicitly identiﬁed. The plotted data around A ∼ 200 in ﬁg. 25 were the ones converted from the results of lifetimes measurements, assuming that only NM decay occurs in heavy Λ-hypernuclei. The lines represent theoretical calculations by diﬀerent authors. As said, up to very recent times, the main challenge of hypernuclear weak-decay studies has been to provide a theoretical explanation of the large experimental values for the ratio Γn /Γp between the neutron- and the proton-induced decay widths. Until recently, the large uncertainties involved in the extraction of the ratio from data did not allow to reach any deﬁnitive conclusion. The data were quite limited and not precise due to the diﬃculty of detecting the products of the NM decays, especially the neutrons. In ﬁg. 26 the Γn /Γp ratio of Λ-hypernuclei as a function of the mass number A is reported. The closed and the open circles are the results of the E307 experiment [78], and they were obtained assuming, respectively, the “1N only” and “1N and 2N ” processes. The open squares are previous experimental data by Szymanski et al. [37] and Noumi et al. [79]. Theoretical calculations by diﬀerent authors are plotted in the same ﬁgure.

Strange nuclei

37

Fig. 26. – The Γn /Γp ratio of Λ-hypernuclei. From ref. [78].

The DQ current exchange model was applied to explain the very short-range part of the NM decay in light hypernuclei and nuclear matter, incorporating it with the contribution of π- and K-meson exchanges. Systematic studies were performed about various combinations of meson exchange potentials. It was also pointed out that twonucleon–induced NM decay could play an important role in the NM weak decay even if no experiment has so far explicitly measured the contribution of the two-nucleon process. The results of the NM decay widths and the Γn /Γp ratios provide good criteria to test the short-range nature of the NM decay, which is essentially important to understand its mechanism. However, as shown in ﬁgs. 25 and 26, there is no theoretical calculation that explains both the NM decay width and the Γn /Γp ratio consistently. For instance, the quark-current exchange model gives the Γn /Γp ratio in nuclear matter comparable to the present experimental data, but overestimates the NM decay width. The conventional meson exchange models, in which the contribution of heavy mesons is included, provide Γn /Γp ratios larger than those by the OPE model, though the results of OME models are still smaller than the experimental data. As said before among the mechanisms explored to remedy the puzzle, DQ model approaches have been adopted [73] and the results of these calculations, which yield a large violation of the ΔI = 1/2 rule, provide, on the contrary, signiﬁcant larger values for the ratio Γn /Γp ∼ 0.4. A similar experimental value of 0.4, though still aﬀected by considerably high errors (∼ 20%), was reported by recent measurements at KEK, in which nucleon spectra in coincidence with the K + corresponding to the formation of the hypernucleus in the ground state produced by the (π + , K + ) reaction were reported. Impressive, for the quality of the data, not for the statistics, are the data reported by refs. [75,80] and [81] for 5Λ He and 12 Λ C, in which the spectra of both nucleons in coincidence were measured, applying also an angular correlation (quasi back-to-back) cut in order to

38

T. Bressani

Fig. 27. – (a),(b): the yields of the np and nn coincidence events as a function of the energy sum of the pair nucleons. (c),(d): the upper panel shows the yields of the N N coincidence events plotted as a function of the opening angle between the two nucleons. The lower panel depicts the normalized yields of the np and nn coincidence per NMWD. From ref. [75].

eliminate events in which one of the nucleons suﬀered a ﬁnal-state interaction. Figure 27 shows the ﬁnal experimental results. In conclusion, I may remark that the “Γn /Γp puzzle” is partially solved. Errors are still very large and the threshold on detection for proton is still quite high, avoiding a very precise comparison with theory. These circumstances did not allow to determine the decay width of the two-body process Γ2 : Λ + (N N ) = n + N + N , which is predicted to amount up to ∼ 15% of the total decay width [74]. More puzzling is still the situation concerning the nucleon spectra emitted in the nonmesonic weak decays, in particolar those relative to the protons. An interesting series of measurements was recently performed by the SKS Collaboration [75, 81, 82], and the ﬁnal results are summarised by ﬁg. 28. I remark that the proton spectra have a low-energy cut at 35 MeV, due to the use of quite thick targets in order to increase the counting rates. Obviously the neutron spectra did not suﬀer from such a drawback. All the measured nucleon spectra exhibit a smooth behaviour, without memory of an expected enhancement around 80 MeV, the energy of the nucleons from NMWD. This eﬀect is visibile even for the light hypernucleus 5Λ He. The experimental spectrum does not agree with the spectrum calculated by Garbarino et al. [83], who take into account both ﬁnal-state interactions and two-nucleon–induced NMWDs, but in which there is still the memory of the peak at 80 MeV. I notice that the same authors calculate nucleon spectra emitted from heavier hypernuclei, like 12 Λ C,

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39

Fig. 28. – Neutron (ﬁlled circle) and proton (open circle) energy spectra per NMWD of (a) 5 12 89 Λ He and (b) Λ C with the neutron spectrum of (c) Λ Y. The errors are statistical. The dashed histogram in the top ﬁgure shows the neutron spectrum per NMWD calculated by Garbarino et al. [83] in which FSI eﬀect and the 2N-induced process were taken into account. From ref. [82].

in which the memory of the peak at 80 MeV is lost, due to the overwhelming eﬀect of FSI. Very recently FINUDA has presented the measurement of the proton spectra from NMWD of 5Λ He, 7Λ Li and 12 Λ C [84] shown by ﬁg. 29. The data are of high quality due to the speciﬁc features of the FINUDA spectrometer, i.e., the transparency, the use of very thin nuclear targets and the measurement of the proton energies by magnetic analysis. For these reasons the threshold on the measurement of the proton energy is 15 MeV. For 7 Λ Li there are no measurements nor theoretical calculations in spite of the fact it is the best known hypernucleus from the spectroscopic point of view [23]. The spectra show a similar shape, i.e., a peak around 80 MeV, corresponding to about a half of the Q-value for the Λp → np weak reaction, with a low energy rise, due to the

40

T. Bressani

Fig. 29. – Proton energy spectrum after the background subtraction from NMWD of a) b) 7Λ Li and c) 12 Λ C. From ref. [84].

5 Λ He,

FSI and/or to two-nucleon induced weak decays. The shape of the spectrum for 5Λ He is diﬀerent from the experimental one by Okada et al. [82] as well as from the theoretical one [83] shown in ﬁg. 28a), even though the diﬀerence is not so severe. The situation for 12 Λ C is completely diﬀerent. The new data are in complete disagreement with the previous experimental data [82] (ﬁg. 28b)) and from the theoretical calculation [83]. Concerning the discrepancy between the two sets of experimental data, I may remark that in [82] the proton energy was measured by a combination of time-of-ﬂight and total energy deposit measurements. The energy loss inside the thick targets was corrected event-by-event. The energy resolution became poorer in the high energy region, especially

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41

above 100 MeV, with the consequence that the spectra could be strongly distorted. On the contrary with FINUDA the proton momenta are measured by means of a magnetic analysis, with an excellent resolution (2% FWHM) and no distortion at all on the spectra is expected, in particular in the high-energy region. I believe that the spectrum of FINUDA is the genuinely undistorted one, even though with a limited statistics. The spectra for 5Λ He, 7Λ Li and 12 Λ C look quite similar, in spite of the large mass number diﬀerence of these nuclei. If the low energy rises were predominantly due to FSI eﬀects, one should naturally expect that the broad peak structure at 80 MeV (coming from clean Λp → np weak processes broadened by the Fermi motion of nucleons) would be smeared out for the heavier nuclei. Another eﬀect that should be the origin of the low energy rise is a large contribution of the two-nucleon–induced weak process Λnp → nnp. If the weak decay Q-value (160 MeV) is shared by three nucleons, a low energy rise may exist even for the very light s-shell hypernuclei. The FINUDA data seem to agree with the hypothesis of a substantial contribution of the two nucleon induced NMWD. Indeed recently Bhang et al. [85] suggested a contribution of the two nucleon induced weak decay process as large as 40% of the total NMWD width for 12 Λ C. As a ﬁnal remark concerning the amount of FSI introduced in the theoretical calculations [83], I suspect that it is too large, taking into account the considerations given by ¯ the same authors [86] in explaining the interpretation of a possibile (Kpp) deeply bound state [87]. A. Feliciello will give a more detailed report on this item [28]. 6. – Neutron-rich Λ-hypernuclei and ΛΛ-hypernuclei Nuclei at the most extreme N/Z ratio are found at small proton and neutron numbers. Among the bound nuclei 8 He has the largest N/Z value reached so far, N/Z = 3. Still larger values, however, may be obtained in forming resonant nuclear systems beyond the drip lines; 5 He [88] and 10 He [89] were formed as quasi-bound nuclear states. One of the most exciting recent discoveries in nuclear physics has been the observation of the halo phenomenon, i.e. that some of the nucleons extend far outside the region of their nuclear core. In the archetype halo nucleus 11 Li, the two outermost neutrons occupy a volume almost comparable in size to that of the much heavier nucleus 208 Pb. The halo phenomenon is now interpreted as a typical quantum-mechanical eﬀect, i.e. very weakly bound valence nucleons penetrate into the classically forbidden region beyond the potential barrier. Low-angular-momentum orbits contribute preferentially to the formation of extended halos due to their low centrifugal barrier. It was ﬁrst stressed by Majiling [90] that hypernuclei may be even better candidates for exhibiting larger values of N/Z and halo phenomena. The glue role of the Λ (compression of the nuclear core and addition of the extra binding energy BΛ ) as well as the reliable picture in terms of single-particle states are the main arguments supporting such a prediction. Existence of hypernuclei like 7Λ H, with a value of N/Z = 5 (or (Λ+N )/Z = 6) and halo hypernuclei like 7 9 Λ He and Λ He was predicted on the basis of calculations following simpliﬁed assumptions.

42

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Fig. 30. – Missing mass spectrum of the (π − , K + ) reaction on a 10 B target at 1.2 GeV. An expanded view near the Λ bound region is represented in the inset. From ref. [93].

Unfortunately the production of neutron-rich hypernuclei must proceed through twostep reactions occurring on two nucleons of the same nucleus (e.g., K − p → Λπ 0 , π 0 p → nπ + ) with inevitable low cross-sections or capture rates. The ﬁrst experimental attempt to produce neutron-rich hypernuclei via the reaction (21)

− A + Kstop +A Λ Z → Λ (Z − 2) + π

16 was carried out at KEK [91]. Upper limits for the production of 9Λ He, 12 Λ Be and Λ C from 9 12 16 Be, C and O targets were determined. The measured capture rates are in the range (0.6–2.0) × 10−4 ; it must be noticed that the theoretical values calculated by Tetryakova and Lanskoy [92] are in the range (10−6 –10−7 ), i.e. at least one order of magnitude less than the experimental upper limits and three orders of magnitude less than the usual one-step reactions like (11) on the same targets. An experiment performed at KEK with the SKS Spectrometer reported the pro− + 10 duction of 10 B target [93]. The two-step reactions Λ Li in the (π , K ) reaction on a occurring on two nucleons of the same nucleus are in this case (π − p → π 0 n, followed by π 0 p → K + Λ, or π − p → K 0 Λ, followed by K 0 p → K + n). No deﬁned peak was observed (see ﬁg. 30), due perhaps to the limited resolution imposed by the use of a thick (about 3.5 g/cm2 ) target. Then the measured cross-section in the region of the bound states of 10 Λ Li was integrated over all possibile states and no comparison with theoretical predictions could be done. A considerable interest was focused from the theoretical side on the possibile existence of the strange heavy isotopes of hydrogen, 6Λ H and 7Λ H, whose production should be observed in reactions like (21) on 6 Li and 7 Li targets. The case of 6Λ H is very interesting; in fact theoretical calculations predict the existence of a bound single-particle state with a binding energy of 5.8 MeV from the (5 H+Λ) threshold when the reaction is supposed to

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43

occur not as a two-step process but as a one-step reaction (K − +p → Σ− +π + ; Σ− p ↔ Λn) with a Λ-Σ coupling term [94]. A recent review on the interest of hunting 6Λ H is due to Majiling [95]. A recent study of the reaction (21) on 6 Li and 7 Li targets was performed by FINUDA [96]. Upper limits of 2.5 × 10−5 for the production of 6Λ H and 4.5 × 10−5 for 7Λ H were reported. A. Feliciello [28] gives more details on this measurement. The case of double Λ-hypernuclei is quite unique. We remind that they were discovered ten years (1963) after the discovery of Λ-hypernuclei (1953), but the two events found in emulsions were for long time the only proof of their existence. Several counter experiments designed to detect them were unsuccessful, and only in the last few years new experiments at KEK (E176 and E373) and at BNL (E906) conﬁrmed their existence, by the detection of four more events. The reason for such an obstinacy in searching for ΛΛ-hypernuclei is also linked to the circumstance that they could be the breeder for the production of the H-particle, the advocated paradigm of the dybarions, predicted by many-quark bag models. Recent contributions in the ﬁeld may be found in [97, 98]. The study of the production of Λ-hypernuclei and other S = −2 nuclear systems is one of the leading items for the experimentation at the new J-PARC facility, that will start to provide beams in a couple of years from now. A detailed overview of the physics program at J-PARC is given by T. Nagae at this School [20] and I refer to his lecture for more details. I would only mention that an interesting program on the physics of ΛΛ-hypernuclei was proposed and approved at the future FAIR facility in Darmstadt (Germany) [99]. An important component of FAIR will be the High Energy Storage Ring (HESR) for high-intensity, phase-space cooled p¯ between 1.5 and 15 GeV/c. The expected luminosity for the stored p¯ impinging on a nuclear cluster or wire target in the ring is about 1032 cm−2 s−1 . A ¯ pairs will be produced preliminary account on this subject can be found in [100]. Ξ-Ξ

Fig. 31. – Various steps of the production of ΛΛ-hypernuclei by p¯ annihilation. From ref. [100].

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Fig. 32. – Simple representation of ΛΛ-hypernucleus.

abundantly by p-nucleus ¯ collisions close to threshold. The trigger will be based on the detection of high-momentum anti-hyperons at small angles or of positive kaons produced by the anti-hyperons in the nuclei composing the primary target. A trigger selecting two K + should provide high counting rates. A realistic calculation of the expected rates was recently done by Ferro et al. [101]. The associated Ξ− will be decelerated and subsequently absorbed by nuclei of a secondary target. Ξ− p → ΛΛ conversion (Q = 28 MeV) on protons of the absorbing nuclei will produce ΛΛ-hypernuclei: Figure 31 shows schematically the described steps. In particular, the interesting results obtained by the analysis of γ-ray spectra from Λ-hypernuclei in order to get unique information on the N N interaction led to the proposal of using this method for getting information on the ΛΛ elementary interaction by the analysis of γ-ray spectra from selected ΛΛ-hypernuclei [99]. It is a quite ambitious proposal since it combines the diﬃculties of producing abundantly ΛΛ-hypernuclei with those of the coincidence γ-spectroscopy.

Fig. 33. – Schematic view of the PANDA detector (right) with an enlarged view of the layout for ΛΛ-hypernuclei production and spectroscopy. From ref. [100].

45

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Considering the eﬀect of the nuclear core compression discussed previously, it is straightforward to conclude that the simple guess (22)

ΔBΛΛ ≡ BΛΛ − 2BΛ ≡ − VΛΛ

in which BΛΛ is the binding energy of the ΛΛ-hypernucleus A ΛΛ Z and BΛ that of the hypernucleus A−1 Z (see ﬁg. 32), is not valid, but one needs a systematic study as a Λ function of A. An advanced segmented array of HPGe detectors, similar to that envisaged for use at J-PARC will be used. No harm is expected for the operation of HPGe detectors in the fringing ﬁeld of the PANDA detector, which will provide the trigger [50]. Figure 33 shows a schematic layout for the proposed experiment. 7. – Exotic strange nuclei and more In this section I shall very brieﬂy mention attempts to observe or speculate about nuclear systems containing S = −1 particles diﬀerent than Λ-hyperons as well as other baryons with quantum numbers diﬀerent than S = −1. These attempts were done taking as a paradygm Λ-hypernuclei, with the related experimental and mainly theoretical methods. . 7 1. Σ-hypernuclei. – As I mentioned in sect. 3, a great surprise was spurred in 1980 by the claim of the Heidelberg-Saclay-Strasbourg Collaboration [33] of the observation of relatively narrow (8 MeV) peaks in the unbound region of the 9 Be (K − , π − ) spectrum measured at CERN. The kinematics was similar to that of reaction (1), shifted at lower momenta due to the higher mass of the Σ-hyperon. Some other experiments at CERN, BNL and KEK, always with low statistics, seemed to conﬁrm the existence of these narrow strucures, however not in agreement with each other. The theoretical and experimental situation in these years was well described by Dover, Millener and Gal [102]. A high-statistics experiment carried out at BNL ten years later [103] did not ﬁnd any narrow structure, with the exception of 4Σ He, and put the tomb stone on Σ-hypernuclei. An intersting critical review, with the provocative title: “The story of Σ-hypernuclei-A modern fable” is due to Chrien [104]. Evidence of a bound state of 4Σ He was reported by − Hayano et al. [105] in the (Kstop , π − ) reaction. This result was conﬁrmed by a subsequent experiment at BNL, carried out with K − of 600 MeV/c [106]. As shown by ﬁg. 34 the existence of the bound state is signalled by a clear peak below the Σ-binding threshold in the 4 He(K − , π − ) reaction. No peak was found in the 4 He(K − , π + ) reaction. Since the 4 He(K − , π − ) reaction can populate both the T = 3/2 and T = 1/2 states, while the 4 He(K − , π + ) reaction can only produce the T = 3/2 ﬁnal state, the experimental data suggest that this 4Σ He bound state has T = 1/2, as predicted by Harada et al. [107]. The unique nature of this bound state is due to the large isospin dependence in their potential. The T = 1/2 state is attractive, while the T = 3/2 state is strongly repulsive.

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Fig. 34. – Spectra of the 4 He(K − , π ± ) reaction at KEK at pK = 600 MeV/c. A clear peak is obtained only in the (K − , π − ) spectrum. From ref. [106].

. 7 2. Proton rich S = −1 nuclei. – One of the more controversial recent issues in hadronic physics is the possibile existence of the so-called kaonic nuclear clusters, often referred to also as deeply bound kaon states. They are strange (S = −1, −2) systems ¯ This topic is connected with the composed by nucleons strongly bound to one or two K. possibile existence of unusual nucleon bound states like pp or ppp, with the possibility ¯ More details on this that a high density nuclear medium will be created around the K. subject are given in the lectures of A. Gal [108], G. Bendiscioli [109] and A. Feliciello [28] at this School and in [110]. . 7 3. S = 1 hyponuclei. – Following the big excitement spurred by the claimed discovery of the pentaquark Z + (1530), Miller [111] put forward the hypothesis that Z + could be bound in nuclei, due to a strongly attractive potential. Today the prevailing attitude is that the Z + does not exist, and then there is no much hope for the existence of these nuclear systems, sometimes referred to as “Hyponuclei = Hypothetical Nuclei” [112]. . 7 4. Supernuclei. – The possibile existence of bound systems in which a Λ+ c (and also a Λb ) replaces a nucleon of a nucleus, in analogy to what happens for ordinary Λ-hypernuclei was put forward by Tyapkin [113] in 1975. Following this idea, several

47

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Fig. 35. – Kinematics in the forward direction for the reaction (1) (I, dashed lines) (23) (II continuous line), (24) (III, dot-dashed lines). From ref. [116].

authors and in particolar Starkov and Tsarev [114] developed theoretical models for describing the possibile properties of these objects, sometimes referred to also as “Supernuclei”. There is up to now no experimental evidence of the existence of these bound systems, apart from three ambiguous candidates of Λ+ c -hypernuclei reported by an emulsion experiment [115] with 250 GeV protons. An alternative idea to detect these systems was put forward by Bressani and Iazzi [116], in connection with the proposal for a singlepass collider for beauty and charm physics. It was based on the observation that the two reactions: (23)

+ D + + p → Λ+ c +π

and (24)

B − + n → Λb + π −

have kinematical features similar to reaction (1). In particolar for reaction (23) there is a “magic momentum” at about 600 MeV/c, for reaction (24) at about 860 MeV/c (see ﬁg. 35). A guess of the expected counting rate was about 4 events/day, but with a luminosity of 1033 cm−2 s−1 . If we consider that the luminosity at present or upgraded beauty/charm factories is from 3 to 4 orders of magnitude larger, a confortable counting rate should be expected. Incidentally, this idea of performing exotic nuclear physics at an (e+ e− ) collider was the precursor of the FINUDA experiment. 8. – Conclusions It is somehow surprising to remark that, in a period in which nuclear physics research is declining in ﬁnancial support, the subﬁeld of hypernuclear physics shows a growth:

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there are at least two reasons for such a tendency. The ﬁrst one is certainly of scientiﬁc origin; as I have brieﬂy discussed, many interesting open problems are still left in hypernuclear physics and now the present technology of machines and detectors seems able to give an answer. The second one is perhaps more psychological. The community of nuclear physicists is now dispersed in a range of activities very much broader than in the past. There is still a core of physicists (the integralists) for which nuclear physics is “by deﬁnition” the science studying the properties of the many-body systems composed by the nucleons of the nucleus. Hypernuclear spectroscopy is well ﬁtting this deﬁnition. There is also a considerable group of physicists (the fundamentalists) for which nuclear physics has the objective of understanding the fundamental interactions of hadrons. For this reason topics like hadron spectroscopy or nucleon structure are now considered to belong to nuclear physics. Non-mesonic decay of hypernuclei is an item well suited to fundamentalists. Finally physicists attracted by the search of new objects (the exoterics) may ﬁnd a very large ﬁeld of activity in hunting the many exotic system that may be sought, in the hope of catching at least one of them. I hope that this School will contribute to enlarge even more the community of hypernuclear physicists. ∗ ∗ ∗ I am very indebted to Dr. S. Bufalino for her unvaluable help in preparing this manuscript. This work has been partially granted by Progetti di Ricerca di Interesse Nazionale 2005 of the Italian Government (Hypergamma Program). REFERENCES [1] [2] [3] [4] [5] [6] [7]

[8]

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[52] Iodice M. et al., Phys. Rev. Lett., 99 (2007) 052501. [53] Feshbach H., in Proceedings of the Summer Study Meeting on Nuclear and Hypernuclear Physics with Kaon Beams, BNL Internal Report, BNL 18335 (1973) pp. 185-199. [54] Ajimura S. et al., Nucl. Phys. A, 585 (1995) 173c. [55] Motoba T. et al., Phys. Rev. C, 38 (1988) 1322. [56] Hungherford III E. V., Phys. Rep., 257 (1995) 349. [57] Hiyama E. et al., Phys. Rev. Lett., 85 (2001) 270. [58] Akikawa K. et al., Nucl. Phys. A, 691 (2001) 4255. [59] Ajimura S. et al., Phys. Rev. Lett., 86 (2001) 4255. [60] Tamura K. et al., Nucl. Phys. A, 691 (2001) 86c. [61] Cheston W. B. and Primakoff H., Phys. Rev., 92 (1953) 1537. [62] Kameoka S. et al., Nucl. Phys. A, 754 (2005) 173. [63] Ericson M., Nucl. Phys. A, 547 (1992) 127c. [64] Dalitz R. H., Phys. Rev., 112 (1958) 605; Dalitz R. H. and Liu L., Phys. Rev., 116 (1959) 1312. [65] Block M. M. and Dalitz R. H., Phys. Rev. Lett., 11 (1963) 96. [66] Alberico W. M. and Garbarino G., Phys. Rev. Lett. B, 486 (2000) 362. [67] McKellar B. H. J. and Gibson B. F., Phys. Rev. C, 30 (1984) 322. [68] Nardulli G., Phys. Rev. C, 38 (1988) 832. [69] Dubach J. F. et al., Nucl. Phys. A, 450 (1986) 71c. [70] Cheung C. Y. et al., Phys. Rev. C, 27 (1983) 335. [71] Heddle D. P. and Kisslinger L. S., Phys. Rev. C, 3 (1985) 608. [72] Inoue T. et al., Nucl. Phys. A, 633 (1998) 312. [73] Sasaki K. et al., Nucl. Phys. A, 669 (2000) 331; 678 (2000) 455. [74] Alberico W. M. and Garbarino G., Phys. Rep., 369 (2002) 1. [75] Outa H., in Varenna 2004, pp. 219-258. [76] Park M. et al., Phys. Rev. C, 61 (2000) 054004. [77] Bhang H. et al., Phys. Rev. Lett., 81 (1998) 4321. [78] Sato Y. et al., Phys. Rev. C, 71 (2005) 025203. [79] Noumi H. et al., Proceedings of the IV International Symposium on Weak and Electromagnetic Interactions in Nuclei, edited by Ejiri H., Kishimoto T. and Sato T. (World Scientiﬁc, Singapore) 1995, p. 550. [80] Kang B. H. et al., Phys. Rev. Lett., 96 (2006) 062301. [81] Kim M. J. et al., Phys. Lett. B, 641 (2006) 28. [82] Okada S. et al., Phys. Lett. B, 597 (2004) 249. [83] Garbarino G. et al., Phys. Rev. C, 69 (2004) 054603. [84] Agnello M. et al., Nucl. Phys. A, 804 (2008) 151. [85] Bhang H. et al., Eur. Phys. J. A, 33 (2007) 259. [86] Magas V. K. et al., Phys. Rev. C, 74 (2006) 025206. [87] Agnello M. et al., Phys. Rev. Lett., 94 (2005) 212303. [88] Korsheninnikov A. A. et al., Phys. Rev. Lett., 87 (2001) 09251. [89] Von Oertzen W. et al., Nucl. Phys. A, 588 (1995) 129c. [90] Majiling L., Nucl. Phys. A, 585 (1995) 211c. [91] Kubota K. et al., Nucl. Phys. A, 602 (1996) 327. [92] Tretyakova T. YU. and Lanskoy D. E., Phys. At. Nuclei, 66 (2003) 1651. [93] Saha P. K. et al., Phys. Rev. Lett., 94 (2005) 052502. [94] Saha P. K. et al., Nucl. Phys. A, 691 (2001) 51c. [95] Majiling L., in HYP 2006, p. 149. [96] Agnello M. et al., Phys. Lett. B, 640 (2006) 145. [97] Yoon C. J. et al., in HYP 2006, p. 121.

Strange nuclei

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[98] Hiyama E., in HYP 2006, p. 141. [99] An International Accelerator Facility for Beams of Ions and Antiprotons, Conceptual Design Report GSI, (2001) 222. [100] Pochodzalla J., in Varenna 2004, pp. 259-282. [101] Ferro F. et al., Nucl. Phys. A, 789 (2007) 209. [102] Dover C., Millener D. J. and Gal A., Phys. Rep., 184 (1989) 1. [103] Sawafta R., Nucl. Phys. A, 585 (1989) 103c. [104] Chrien B., Strangeness Nuclear Physics, Proceedings of the APTCP Workshop SNP’99, Seoul, 19-22 Feb. 1999, edited by Il-Tong-Cheon, Woo Hong S. and Motoba T. (World Scientiﬁc, Singapore) 1999, pp. 217-136. [105] Hayano R. S. et al., Nuovo Cimento A., 102 (1989) 437; Phys. Lett. B, 231 (1989) 355. [106] Nagae T. et al., Phys. Rev. Lett., 80 (1998) 1605. [107] Harada T. et al., Nucl. Phys. A, 507 (1990) 715. [108] Gal A., this volume, p. 53. [109] Bendiscioli G., this volume, p. 159. [110] Bressani T., Nucl. Phys. News Int., 17 (2007) 11-17. [111] Miller G., Phys. Rev. C, 70 (2004) 022202(R). [112] Gal A., private communication. [113] Tyapkin A. A., Sov. J. Nucl. Phys., 22 (1975) 89. [114] Starkov N. I. and Tsarev V. A., Proceedings of the 1986 INS International Symposium on Hypernuclear Physics, Tokyo, August 20-23, 1986, edited by Bando H., Hashimoto O. and Ogawa K. (INS, University of Tokyo) 1986, p. 583. [115] Batusov Yu. A. et al., JETP Lett., 33 (1981) 56. [116] Bressani T. and Iazzi F., Nuovo Cimento A, 102 (1989) 697.

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Strange atoms, strange nuclei and kaon condensation E. Friedman and A. Gal Racah Institute of Physics, The Hebrew University - Jerusalem 91904, Israel

Summary. — Analyses of strong-interaction data consisting of level shifts, widths and yields in strange atoms of K − mesons and Σ− hyperons are reviewed. Recent results obtained by ﬁtting to comprehensive sets of data across the periodic table in terms of density-dependent optical potentials are discussed. The introduction of density dependence generally improves signiﬁcantly the ﬁt to the data, leading to novel results on the in-medium hadron-nucleon t matrix t(ρ) over a wide range of densities up to central nuclear densities. A strongly attractive K − -nuclear potential of order 150–200 MeV in nuclear matter is suggested by ﬁts to K − -atom data, ¯ condensation and on the evolution of with interesting possible repercussions on K strangeness in high-density stars. The case for relatively narrow deeply bound K − atomic states is made, essentially independent of the K − potential depth. In view ¯ deeply bound states, of the recently reported inconclusive experimental signals of K ¯ dynamical models for calculating binding energies and widths of K-nuclear states are discussed. Lower bounds on the width, ΓK¯ 50 MeV, are established. For Σ− atoms, the ﬁtted potential becomes repulsive inside the nucleus, in agreement with recently reported (π − , K + ) spectra from KEK, implying that Σ hyperons generally do not bind in nuclei. This repulsion signiﬁcantly aﬀects calculated compositions and masses of neutron stars.

c Societ` a Italiana di Fisica

53

54

E. Friedman and A. Gal

1. – Introduction An exotic atom is formed when a negatively charged particle stops in a target and is captured by a target atom into an outer atomic orbit. It will then emit Auger electrons and characteristic X-rays whilst cascading down its own sequence of atomic levels until, at some state of low principal quantum number n, the particle is absorbed due to its interaction with the nucleus. The lifetimes of the particles considered here, namely K − and Σ− , are much longer than typical slowing down times and atomic time scales. Therefore, following the stopping of the hadron in matter, well-deﬁned states of an exotic atom are established and the eﬀects of the hadron-nucleus strong interaction can be studied. The overlap of the atomic orbitals with the nucleus covers a wide range of nuclear densities thus creating a unique source of information on the density dependence of the hadronic interaction. In the study of strong-interaction eﬀects in exotic atoms, the observables of interest are the shifts ( ) and widths (Γ) of the atomic levels caused by the strong interaction with the nucleus. These levels are shifted and broadened relative to the electromagnetic case but the shifts and widths can usually only be measured directly for one, or possibly two levels in any particular hadronic atom. The broadening due to the nuclear absorption usually terminates the atomic cascade at low n thus limiting the experimentally observed X-ray spectrum. In some cases the width of the next higher n + 1 “upper” level can be obtained indirectly from measurements of the relative yields of X-rays when they depart from their purely electromagnetic values. As the atomic number and size of the nucleus increase, so the absorption occurs from higher n-values as shown for K − atoms and for Σ− atoms in the corresponding sections. Shifts and widths caused by the interaction with the nucleus may be calculated by adding an optical potential to the Coulomb interaction. The study of the strong interaction in exotic atoms thus becomes the study of this additional potential, as reviewed in great detail by Batty, Friedman and Gal [1] and very recently by Friedman and Gal [2]. On the experimental side, studies of strong-interaction eﬀects in exotic atoms have been transformed over the years with the introduction of increasingly more advanced X-ray detectors and with increasing the eﬃciency of stopping the hadrons, such as with a cyclotron trap [3]. The present lectures focus particularly on the physics of the strong interaction which can be deduced by studying strange atoms, a term which is used for exotic atoms formed initially by stopping K − mesons in matter. The importance of the strange-atoms subject stems from the progress made in recent years in quantifying medium modiﬁcation eﬀects on the hadron-nucleus interaction which has enabled one to achieve improved ﬁts to existing data within the framework of commonly accepted models [2]. These modiﬁed interactions obey a low-density limit which has not always been enforced in earlier analyses, since it is not always relevant to the higher-density regime explored, where new features of the hadron nucleus interaction may become signiﬁcant to other ﬁelds such as astrophysics. In the next section we will outline the methodology of exotic-atom studies, including common tools such as wave equations and optical potentials. Of prime importance is the

55

Strange atoms, strange nuclei and kaon condensation

dependence of the optical potential on the model of nuclear density used, with emphasis placed on the overlap region between the atomic state studied and the nucleus. Radial sensitivity will be deﬁned, to serve as a guide. Finally the extreme non-perturbative nature of exotic atoms will be discussed. Readers who are not concerned about these tools of analyzing exotic atoms may skip the next section and go immediately to the sections dealing with K − atoms and nuclei and with Σ hyperons. 2. – Exotic atom methodology . 2 1. Wave equations and optical potentials. – The interaction of hadrons at threshold with the nucleus is customarily described by a Klein-Gordon (KG) equation which for exotic-atom applications is of the following form: (1)

2 ∇ − 2μ(B + Vopt + Vc ) + (Vc + B)2 ψ = 0

( = c = 1),

where μ is the hadron-nucleus reduced mass, B is the complex binding energy and Vc is the ﬁnite-size Coulomb interaction of the hadron with the nucleus, including vacuumpolarization terms, added according to the minimal substitution principle E → E − Vc . A term 2Vc Vopt and a term 2BVopt were neglected in eq. (1) with respect to 2μVopt ; the term 2BVopt has to be reinstated in studies of deeply bound states. The optical potential Vopt is of the tρ(r) generic class which for underlying s-wave hadron-nucleon interactions assumes the form (2)

A−1 μ 2μVopt (r) = −4π 1 + {b0 [ρn (r) + ρp (r)] + τz b1 [ρn (r) − ρp (r)]} . A M

Here, ρn and ρp are the neutron and proton density distributions normalized to the number of neutrons N and number of protons Z, respectively, M is the mass of the nucleon and τz = +1 for the negatively charged hadrons considered in the present lectures. In the impulse approximation, b0 and b1 are minus the hadron-nucleon isoscalar and isovector scattering lengths, respectively, which are complex for the absorptive strong interactions of K − mesons and Σ− hyperons. Generally these “one-nucleon” parameters are functions of the density ρ, but often the density dependence may be approximated by ﬁtting eﬀective values for b0 and b1 to low-energy data. The extension of the threshold KG equation (1) and the optical potential (2) for scattering problems is straightforward [1,2]. The use of the KG equation rather than the Dirac equation for fermions, such as Σ hyperons, is numerically justiﬁed when ﬁne-structure eﬀects are negligible or are treated in an average way, as for the X-ray transitions considered here. The leading j dependence (j = l ± 21 ) of the energy for solutions of the Dirac equation for a point-charge 1/r potential goes as (j + 12 )−1 , and on averaging it over the projections of j gives rise to (l + 12 )−1 which is precisely the leading l dependence of the energy for solutions of the KG equation. The higher-order contributions to the spin-orbit splitting are suppressed by O(Zα/n)2 which is of order 1% for the high-n X-ray transitions encountered for

56

E. Friedman and A. Gal

Σ hyperons. This is considerably smaller than the experimental errors placed on the measured X-ray transition energies and widths. . 2 2. Nuclear densities. – The nuclear densities are an essential ingredient of the optical potential. The density distribution of the protons is usually considered known as it is obtained from the nuclear charge distribution by unfolding the ﬁnite size of the charge of the proton. The neutron distributions are, however, generally not known to suﬃcient accuracy. For many nuclei there is no direct experimental information whatsoever on neutron densities and one must then rely on models which sometimes give conﬂicting results for the root-mean-square (r.m.s.) radii. Given this unsettled state of aﬀairs, a semi-phenomenological approach was adopted that covers a broad range of possible neutron density distributions. Experience with pionic atoms showed [2] that the feature of neutron density distributions which is most relevant in determining strong-interaction eﬀects in pionic atoms is the radial extent, as represented, e.g., by rn , the neutron density r.m.s. radius. A linear dependence of rn − rp on (N − Z)/A has been successfully employed in p¯ studies [4-6], namely rn − rp = γ

(3)

N −Z + δ, A

with γ close to 1.0 fm and δ close to zero. Expression (3) has been adopted in analyzing strange atoms and, for lack of better global information about neutron densities, the value of γ was varied over a reasonable range in ﬁtting to the data. This procedure is based on the expectation that for a large data set over the whole of the periodic table some local variations will cancel out and that an average behavior may be established. Phenomenological studies of in-medium nuclear interactions are based on such averages. In order to allow for possible diﬀerences in the shape of the neutron distribution, a two-parameter Fermi (2pF) distribution was used both for the known proton (unfolded from the charge distribution) and for the unknown neutron density distributions (4)

ρn,p (r) =

ρ0n,0p , 1 + exp((r − Rn,p )/an,p )

in the “skin” form of ref. [4]. In this form, the same diﬀuseness parameter for protons and neutrons, an = ap , is assumed and the Rn parameter is determined from the r.m.s. radius rn deduced from eq. (3) where rp is considered to be known. It was checked that for K − atoms and for Σ− atoms the assumption of “skin” form was as good, or better, than assuming other (notably “halo”) forms. Another sensitivity that may be checked in global ﬁts is to the radial extension of the hadron-nucleon interaction when folded together with the nuclear density. The resultant “ﬁnite range” density is deﬁned as (5)

F

ρ (r) =

dr ρ(r )

2 2 1 e−(r−r ) /β , π 3/2 β 3

Strange atoms, strange nuclei and kaon condensation

57

assuming a Gaussian interaction. It was found that K − and Σ− atoms do not display sensitivity to ﬁnite-range eﬀects. . 2 3. Radial sensitivity in exotic atoms. – The radial sensitivity of exotic atom data was addressed before [1] with the help of a “notch test”, introducing a local perturbation into the potential and studying the changes in the ﬁt to the data as a function of position of the perturbation. The results gave at least a semi-quantitative information on what are the radial regions which are being probed by the various types of exotic atoms. However, the radial extent of the perturbation was somewhat arbitrary and only very recently that approach was extended [7] into a mathematically well-deﬁned limit. In order to study the radial sensitivity of global ﬁts to exotic atom data, it is necessary to deﬁne the radial position parameter globally using as reference, e.g., the known charge distribution for each nuclear species in the data base. The radial position r is then deﬁned as r = Rc + ηac , where Rc and ac are the radius and diﬀuseness parameters, respectively, of a 2pF charge distribution [8]. In that way η becomes the relevant radial parameter when handling together data for several nuclear species along the periodic table. The value of χ2 is regarded now as a functional of a global optical potential V (η), i.e. χ2 = χ2 [V (η)], where the parameter η is a continuous variable. It leads to [7] (6)

2

dχ =

dη

δχ2 δV (η), δV (η)

where (7)

χ2 [V (η) + V δσ (η − η )] − χ2 [V (η)] δχ2 [V (η)] = lim lim σ→0 V →0 δV (η ) V

is the functional derivative (FD) of χ2 [V ]. The notation δσ (η − η ) stands for an approximated δ-function and V is a change in the potential. From eq. (6) it is seen that the FD determines the eﬀect of a local change in the optical potential on χ2 . Conversely it can be said that the optical potential sensitivity to the experimental data is determined by the magnitude of the FD. Calculation of the FD may be carried out by multiplying the best-ﬁt potential by a factor (8)

f = 1 + δσ (η − η )

using a normalized Gaussian with a range parameter σ for the smeared δ-function, (9)

δσ (η − η ) = √

2 2 1 e−(η−η ) /2σ . 2πσ

For ﬁnite values of and σ the FD can then be approximated by (10)

1 χ2 [V (η)(1 + δσ (η − η ))] − χ2 [V (η)] δχ2 [V (η)] ≈ . δV (η ) V (η )

58

E. Friedman and A. Gal

The parameter is used for a fractional change in the potential and the limit → 0 is obtained numerically for several values of σ and then extrapolated to σ = 0. . 2 4. Nonperturbative aspects of exotic atoms. – The optical potentials used to calculate the shifts and widths of atomic energy levels are conﬁned to a small region of the atom and thus lead to very small energy shifts compared to the corresponding binding energies. However, they greatly modify the wave function locally and this modiﬁcation makes the strong interaction eﬀects non-perturbative. For example, in kaonic atoms all the measured level shifts are repulsive, yet “teﬀ ρ” ﬁts to the data invariably lead to attractive potentials. The net repulsive shift is due to the imaginary part of the potential being comparable in magnitude to the real part. Repulsive shifts may also arise from dominantly real attractive optical potentials that are suﬃciently strong to bind nuclear states. A nuclear state generated by the optical potential gives rise to a node inside the nucleus for the atomic wave function by orthogonality. The strong modiﬁcation of atomic wave functions due to binding nuclear states by Vopt may also give rise to irregularities in the parameters of Vopt obtained from ﬁts to data. This eﬀect was ﬁrst observed by Krell [9] for kaonic atoms. These irregularities can be explained by large variations in the atomic wave functions such that additional nodes may be accommodated within the nucleus. A more comprehensive discussion of these nonperturbative aspects is found in ref. [10]. 3. – K − atoms . 3 1. Fits to K − -atom data. – The K − -atom data used in global ﬁts [1] are shown in ﬁg. 1, spanning a range of atomic states from 2p in Li to 8j in U, with 65 level-shifts, widths and transition yields data points. We note that the shifts are “repulsive”, largely due to the substantial absorptivity of the K − -nuclear interaction. It was shown already in the mid 1990s [1] that although a reasonably good ﬁt to the data is obtained for a tρ potential, eq. (2), with an eﬀective complex parameter b0 corresponding to attraction, greatly improved ﬁts are obtained with a density-dependent potential, where the ﬁxed b0 is replaced by (11)

b0 + B0 [ρ(r)/ρ0 ]α ,

with b0 , B0 and α ≥ 0 determined by ﬁts to the data. Fitted potentials of this kind are marked DD. This parametrization oﬀers the advantage of ﬁxing b0 at its (repulsive) free-space value in order to respect the low-density limit, while relegating the expected in-medium attraction to the B0 term which goes with a higher power of the density. The departure of the optical potential from the ﬁxed-t tρ approach was recently given a more intrinsically geometrical meaning within a model [11] where, loosely speaking, Vopt follows the shape of a function F (r) inside, and the shape of [1 − F (r)] outside the nucleus: (12)

b0 → B0 F (r) + b0 [1 − F (r)],

F (r) =

ex

1 , +1

59

Strange atoms, strange nuclei and kaon condensation Kaonic atoms 10 4

(a) n=2

−Shift (eV)

10 3

n=3 n=5

n=4

n=6

10 2

n=7

10

1 0

10

20

30

40

50

60

70

80

90

100

90

100

Z

(b)

10 4

n=2

n=3

n=5

n=4

n=6

10 3

Width (eV)

n=7 10 2

10

n=8 1

10

-1

0

10

20

30

40

50

60

70

80

Z

Fig. 1. – Shift and width values for kaonic atoms. The continuous lines join points calculated with a best-ﬁt DD optical potential.

with x = (r − Rx )/ax . Then clearly F (r) → 1 for (Rx − r) ax , which deﬁnes the internal region. Likewise [1 − F (r)] → 1 for (r − Rx ) ax , which deﬁnes the external region. Thus Rx forms an approximate border between the internal and the external regions, and if Rx is close to the radius of the nucleus and ax is of the order of 0.5 fm, then the two regions will correspond to the high density and low density regions of nuclear matter, respectively. This is indeed the case, as found in global ﬁts to kaonic atom data [11]. The parameter b0 represents the low-density interaction and the parameter B0 represents the interaction inside the nucleus. We note that, unlike with pionic and antiprotonic atoms, the dependence of kaonic atom ﬁts on the r.m.s. radius of the neutron distribution is weak, and the explicit inclusion of isovector terms, such as b1 of eq. (2), has only marginal eﬀect.

60

E. Friedman and A. Gal 0 Ni K

−

−50 F (84)

VR (MeV)

tρ (130) −100

FB (84) −150

DD (103) −200 0

2

4

6

r (fm) 15 chiral (Baca et al.)

kaonic atoms

2

Χε

10

5 F

0

10

100 Z

¯ 58 Ni potential obtained in a global ﬁt to K − -atom data Fig. 2. – Top: real part of the Kusing the model-independent FB technique [7], in comparison with other best-ﬁt potentials and χ2 values in parentheses. Bottom: contributions to the χ2 of K − atomic shifts for the deep density-dependent potential F from ref. [11] and for the shallow chirally based potential from ref. [12].

Figure 2 (top) shows, as an example, the real part of the best-ﬁt potential for 58 Ni obtained with the various models discussed above, i.e. the simple tρ model and its DD extension, and the geometrical model F, with the corresponding values of χ2 for 65 data points in parentheses. Also shown, with an error band, is a Fourier-Bessel (FB) ﬁt [7] that is discussed below. We note that, although the two density-dependent potentials marked

Strange atoms, strange nuclei and kaon condensation

61

DD and F have very diﬀerent parametrizations, the resulting potentials are quite similar. In particular, the shape of potential F departs appreciably from ρ(r) for ρ(r)/ρ0 ≤ 0.2, where the physics of the Λ(1405) is expected to play a role. The density dependence of the potential F provides by far the best ﬁt ever reported for any global K − -atom data ﬁt, and the lowest χ2 value as reached by the model-independent FB method. On the right-hand side of the ﬁgure are shown the individual contributions to χ2 of the shifts for the deep F potential and the shallow chirally based potential (of depth about 50 MeV) due to Baca et al. [12]. It is self-evident that the agreement between calculation and experiment is substantially better for the deep F potential than for the shallow chiral potential. The question of how well the real part of the K − -nucleus potential is determined was discussed in ref. [7]. Estimating the uncertainties of hadron-nucleus potentials as a function of position is not a simple task. For example, in the “tρ” approach the shape of the potential is determined by the nuclear density distribution and the uncertainty in the strength parameter, as obtained from χ2 ﬁts to the data, implies a ﬁxed relative uncertainty at all radii, which is, of course, unfounded. Details vary when more elaborate forms such as DD or F are used, but one is left essentially with analytical continuation into the nuclear interior of potentials that might be well determined only close to the nuclear surface. “Model-independent” methods have been used in analyses of elastic scattering data for various projectiles [13] to alleviate this problem. However, applying, e.g., the Fourier-Bessel (FB) method in global analyses of kaonic atom data, one ends up with too few terms in the series, thus making the uncertainties unrealistic in their dependence on position. This is illustrated in ﬁg. 2 by the FB curve, obtained by adding a Fourier-Bessel series to a tρ potential. Only three terms in the series are needed to achieve a χ2 of 84 and the potential becomes deep, in agreement with the other two “deep” solutions. The error band obtained from the FB method [13] is, nevertheless, unrealistic because only three FB terms are used. However, an increase in the number of terms is found to be unjustiﬁed numerically. The Functional Derivative (FD) method for identifying the radial regions to which . exotic atom data are sensitive was described in detail in subsect. 2 3. This method was applied in ref. [7] to the F and tρ kaonic atom potentials and results are shown in ﬁg. 3 where η is a global parameter deﬁned by r = Rc + ηac , with Rc and ac the radius and diﬀuseness parameters, respectively, of a 2pF charge distribution. From the ﬁgure it can be inferred that the sensitive region for the real tρ potential is between η = −1.5 and η = 6, whereas for the F potential it is between η = −3.5 and η = 4. Recall that η = −2.2 corresponds to 90% of the central charge density and η = 2.2 corresponds to 10% of that density. It therefore becomes clear that within the tρ potential there is no sensitivity to the interior of the nucleus whereas with the density-dependent F potential, which yields greatly improved ﬁt to the data, there is sensitivity to regions within the full nuclear density. The diﬀerent sensitivities result from the potentials themselves: for the tρ potential the interior of the nucleus is masked essentially by the strength of the imaginary potential. In contrast, for the F potential not only is its imaginary part signiﬁcantly smaller than the imaginary part of the tρ potential [11] but

62

E. Friedman and A. Gal 40 F potl.

Comp 20 Re

tρ potl.

0

2

δΧ /(δV/V)

Re

Comp

−20

−40

−6

−4

−2

0

2

4

6

8

η

Fig. 3. – Functional derivatives of kaonic atoms χ2 with respect to the fully complex (Comp, dashed lines) and real (Re, solid lines) potential as a function of η, where r = Rc + ηac , with Rc and ac the radius and diﬀuseness parameters, respectively, of a 2pF charge distribution. Results are shown for the tρ and for the F potentials of ref. [11] obtained from global ﬁts to kaonic atom data.

also the additional attraction provided by the deeper potential enhances the atomic wave functions within the nucleus [1] thus creating the sensitivity at smaller radii. As seen in the ﬁgure, the functional derivative for the complex F potential is well approximated by that for its real part. The optical potentials derived from the observed strong-interaction eﬀects in kaonic atoms are suﬃciently deep to support deeply bound antikaon nuclear states, but it does not necessarily imply that such states are suﬃciently narrow to be resolved unambiguously from experimental spectra. Moreover, choosing between the very shallow chirally motivated potentials [14, 15], the intermediate chiral potentials of depth around 100 MeV [16] or the deep phenomenological potentials of type F adds appreciable ambiguity to predictions made for such states. It should also be kept in mind that these depths ¯ potentials at threshold, whereas the information required for K-nuclear ¯ relate to K quasibound states is at energies of order 100 MeV below threshold. Predictions become model independent only when it comes to “deeply bound” K − atomic states, as discussed below. . 3 2. Deeply bound K − atomic states. – Somewhat paradoxically, due to the strong absorptive imaginary part of the K − -nucleus potential, relatively narrow deeply bound atomic states are expected to exist which are quite independent of the real potential. Such states are indeed found in numerical calculations as demonstrated in ﬁg. 4 where calculated binding energies and widths of atomic states of K − in 208 Pb are shown for several l-values, down to states which are inaccessible via the X-ray cascade. For 208 Pb,

63

Strange atoms, strange nuclei and kaon condensation 0 Pb K

−

energy (MeV)

−2

−4

−6

−8

−10

2

0

4

6

l value

Fig. 4. – Calculated energies of K − atomic states in 208 Pb. The lowest energy for each l value corresponds to n = l + 1. The bars represent the widths of the states.

the last observed atomic circular state is the 7i, corresponding to l = 6. The general physics behind this phenomenon is similar to that responsible for the deeply bound pionic atom states, although there are diﬀerences in the underlying mechanisms. The mechanism behind the pionic atom deeply bound states is simply the repulsive real part of the s-wave potential which expels the atomic wave function ψatom from the nucleus, thus reducing the overlap between ψatom and the imaginary potential. This reduction, according to (13)

Γ = −2

2

|ψatom | ImVopt dr , 2 |ψatom | dr

results in a reduced width for atomic states. Equation (13) holds exactly for a Schr¨ odinger equation, with only small changes for a KG equation, see refs. [17, 18]. In contrast, phenomenological kaonic atom potentials are attractive, but the strengths of the imaginary potential are such that the decay of ψatom as it enters the nucleus is equivalent to repulsion, resulting in narrow atomic states due to the reduced overlap as discussed above. It is seen from ﬁg. 4 that there is a saturation phenomenon where widths hardly increase for l ≤ 2, contrary to intuitive expectation. The repulsive eﬀect of suﬃciently strong absorption is responsible for the general property of saturation of widths of atomic states and also for saturation of reaction cross-sections above threshold, observed experimentally for antiprotons [19]. The left-hand side of ﬁg. 5 shows the saturation of widths as a function of the absorptive strength parameter Im b0 of Vopt , eq. (2), for the 2p state of kaonic atoms of 208 Pb. For small values of Im b0 the calculated width increases linearly, but already

64

E. Friedman and A. Gal 10.0

−ε

−

PbK 2p 0.2

Coul.

−

Γ

Comp. −1

|R| (fm )

1.0

Im

2

− ε, Γ (MeV)

Pb K 2p

0.1 0.01

0.10 Im b 0 (fm)

1.00

Re

0.1

0

0

10

20

r (fm)

Fig. 5. – Left: saturation of width Γ for the 2p “deeply bound” K − atomic state in 208 Pb as a function of absorptivity Im b0 , for Re b0 = 0.62 fm. Right: wave functions for this state, see text.

at 20% of the best-ﬁt value of 0.9 fm saturation sets in and eventually the width goes down with further increase of the absorption. Note that the real part of the binding energy, represented here by the strong-interaction level shift , hardly changes with Im b0 . The right-hand side of ﬁg. 5 shows radial wave functions for the 2p atomic K − state in 208 Pb for several combinations of potentials. The dashed curve marked “Coul” is for the Coulomb potential only, and with a half-density radius for 208 Pb of 6.7 fm it clearly overlaps strongly with the nucleus. Adding the full complex optical potential the solid curve marked “Comp” shows that the atomic wave function is expelled from the nucleus, and the dotted curve marked “Im” shows that this repulsion is eﬀected by the imaginary part of the potential. Clearly the overlap of the atomic wave function with the nucleus is dramatically reduced compared to the Coulomb-only situation. An interesting phenomenon is displayed by the dot-dashed curve marked “Re”. It shows the atomic wave function when the real potential is added to the Coulomb potential, demonstrating signiﬁcant repulsion of the atomic wave function by the added attractive potential. The explanation for this bizarre result is provided by the three small peaks inside the nucleus which are due to the orthogonality of the atomic wave function and strongly bound K − nuclear wave functions having the same l-values. This extra structure of the atomic wave function in the interior eﬀectively disappears when the imaginary potential is included. ¯ nuclear interactions 4. – K . 4 1. The K − p interaction near threshold . – The K − p data at low energies provide a ¯ system good experimental base upon which models for the strong interactions of the KN

65

Strange atoms, strange nuclei and kaon condensation 300

200 −

+

−

0

0

−

+

πΣ

Kp 150 σT (mb)

σT (mb)

200 100

100 50

0

0

_0 Kn

πΣ

80

σT (mb)

σT (mb)

40 60 40

20 20 0

0 0

πΛ

πΣ

80

σT (mb)

σT (mb)

40 60 40

20 20 0

50

100 150 pL (MeV/c)

200

0

50

100 150 pL (MeV/c)

200

Fig. 6. – Calculations from ref. [15] of cross-sections for K − p scattering and reactions. The dashed lines show free-space chiral-model coupled-channel calculations. The solid lines show chiral-model coupled-channel calculations using slightly varied parameters in order to ﬁt also the K − -atom data for a (shallow) optical potential calculated self consistently.

have been developed. Near threshold the coupling to the open πΣ and πΛ channels is extremely important, as may be judged from the size of the K − p reaction cross-sections, particularly K − p → π + Σ− , with respect to the K − p elastic cross-sections shown in ¯ ﬁg. 6. By developing potential models, KN amplitudes are obtained that allow for analytic continuation into the nonphysical region below K − p threshold. Using a K-matrix analysis, this was the way Dalitz and Tuan predicted the existence of the Λ(1405) πΣ, I = 0 resonance in 1959 [20]. A recent example from coupled-channel potential model calculations [21-24], based on low-energy chiral expansion of meson-baryon potentials in the S = −1 sector, is shown in ﬁg. 7 where the real and imaginary parts of the resulting K − p elastic-scattering

66

E. Friedman and A. Gal

Fig. 7. – Real and imaginary parts of the K − p forward elastic-scattering amplitude, ﬁtted within a NLO chiral SU (3) coupled-channel approach to K − p scattering and reaction data. The line denoted WT is the (real) LO Tomozawa-Weinberg K − p driving-term amplitude. The DEAR measurement [25] value for aK − p is shown with error bars. Figure taken from ref. [26], based on the work of ref. [22].

amplitude, continued analytically below the K − p threshold, are plotted. The line marked WT stands for the leading Weinberg-Tomozawa (WT) nonresonant K − p amplitude below threshold when channel-coupling eﬀects are switched oﬀ. The ﬁgure demonstrates that the Λ(1405) resonance is generated dynamically within the coupled-channel calculation. A discrepancy with Im aK − p deduced from the DEAR measurement [25] is highlighted in this ﬁgure. In contrast, the purely I = 1 K − n amplitude does not show such resonance eﬀects below threshold, and its chiral model dependence is considerably weaker than the model dependence of amplitudes aﬀected by the Λ(1405) resonance, e.g., the K − p elastic-scattering amplitude shown in ﬁg. 7. . ¯ ¯ physics, as demon4 2. K-nucleus potentials. – The gross features of low-energy KN strated in the previous section by chiral coupled-channel ﬁts to the low-energy K − p scattering and reaction data, are encapsulated in the Lowest-Order (LO) WT vector term of ¯ the chiral eﬀective Lagrangian [27]. The Born approximation for the K-nuclear optical potential VK¯ due to the driving-term WT interaction yields then a sizable attraction: (14)

VK¯ = −

3 ρ ρ ∼ −55 MeV 8fπ2 ρ0

for ρ0 = 0.16 fm−3 , where fπ ∼ 93 MeV is the pseudoscalar meson decay constant. Iterating the TW term plus Next-to-Leading-Order (NLO) terms, within an in-medium ¯ −πΣ−πΛ data near the KN ¯ threshold, coupled-channel approach constrained by the KN ¯ attraction as may be seen by inspecting ﬁg. 7. It is found roughly doubles this K-nucleus

Strange atoms, strange nuclei and kaon condensation

67

(e.g., ref. [16]) that the Λ(1405) quickly dissolves in the nuclear medium at low density, so that the repulsive free-space scattering length aK − p , as a function of ρ, becomes attractive well below ρ0 . Since the purely I = 1 attractive scattering length aK − n is ¯ isoscalar scattering length only weakly density dependent, the resulting in-medium KN 1 − − b0 (ρ) = 2 (aK p (ρ) + aK n (ρ)) translates into a strongly attractive VK¯ : VK¯ (r) = −

(15)

2π b0 (ρ) ρ(r), μKN

ReVK¯ (ρ0 ) ∼ −110 MeV .

However, when VK¯ is calculated self-consistently, including VK¯ in the propagator G0 used in the Lippmann-Schwinger equation determining b0 (ρ), one obtains ReVK¯ (ρ0 ) ∼ −(40–60) MeV [14, 15, 28, 29]. The main reason for this weakening of VK¯ , approximately going back to that calculated using eq. (14), is the strong absorptive eﬀect which VK¯ ¯ TW potential. exerts within G0 to suppress the higher Born terms of the KN Additional considerations for estimating VK¯ are listed below. – QCD sum-rule estimates [30] for vector (v) and scalar (s) self-energies: (16) (17)

¯ ∼ − 1 Σv (N ) ∼ − 1 (200) MeV = −100 MeV , Σv (K) 2 2 m 1 s ¯ ∼ Σs (K) (−300) MeV = −30 MeV , Σs (N ) ∼ MN 10

where ms is the strange-quark (current) mass. The factor 1/2 in eq. (16) is due ¯ meson out of two possible, and the to the one nonstrange antiquark q¯ in the K minus sign is due to G-parity going from q to q¯. This rough estimate gives then VK¯ (ρ0 ) ∼ −130 MeV. – The QCD sum-rule approach essentially reﬁnes the mean-ﬁeld argument [31, 32] (18)

VK¯ (ρ0 ) ∼

1 (Σs (N ) − Σv (N )) ∼ −170 MeV , 3

¯ meson, where the factor 1/3 is again due to the one nonstrange antiquark in the K but here with respect to the three nonstrange quarks of the nucleon. – The ratio of K − /K + production cross-sections in nucleus-nucleus and protonnucleus collisions near threshold, measured by the Kaon Spectrometer (KaoS) Collaboration [33] at SIS, GSI, yields an estimate VK¯ (ρ0 ) ∼ −80 MeV by relying on BUU transport calculations normalized to the value VK (ρ0 ) ∼ +25 MeV. Since ¯ N → Y N absorption processes apparently were disregarded in these calculaKN tions, a deeper VK¯ may follow once nonmesonic absorption processes are included.

68

E. Friedman and A. Gal

Table I. – Binding energies (B) and widths (Γ) calculated for K − pp (in MeV). Channels ref. B Γ

Single channel

Coupled channels

ATMS [35]

AMD [36]

Faddeev [37, 38]

Faddeev [39, 40]

48 61

20–50 –

50–70 90–110

60–95 45–80

. 4 3. Deeply bound K − nuclear states in light nuclei. – The ﬁrst prediction of a ¯ K-nuclear quasi-bound state was made by Nogami [34] as early as 1963, arguing that the I = 1/2, L = S = 0 state of the K − pp system could be bound by about 10 MeV. Recent calculations conﬁrm this prediction, with higher values of binding energies but also with substantial values for the (mesonic) width of this state, as summarized in table I. We note that the Faddeev calculations listed in the table account rigorously for the strong ¯ → ΣN coupling, but all the calculations overlook the KN ¯ N → Y N couI = 0 KN pling to nonmesonic channels which are estimated to add conservatively 20 MeV to the ¯ overall width. If K − pp, the lightest possible K-nuclear system, is indeed bound, then it ¯ is plausible that heavier K-nuclear systems will also possess quasi-bound states and the remaining question is whether these states are suﬃciently narrow to allow observation . and identiﬁcation. Unlike the saturation of width in K − atoms, discussed in subsect. 3 2, ¯ no saturation mechanism holds for the width of K-nuclear states which retain very good overlap with the potential. Ongoing experiments by the FINUDA spectrometer collaboration at DAΦNE, Frascati, already claimed evidence for a relatively broad K − pp deeply bound state (B ∼ − 115 MeV) by observing back-to-back Λp pairs from the decay K − pp → Λp in Kstop reactions on Li and 12 C [41], but these pairs could naturally arise from conventional absorption processes at rest when ﬁnal-state interaction is taken into account [42]. In− deed, the Kstop pn → Σ− p reaction observed recently in 6 Li [43] does not require any − K d quasi-bound state. It is worth noting, however, that in order to search for a K − pn bound state which is charge symmetric to the K − pp quasi-bound state discussed above, one should use a 7 Li target to look for back-to-back Σ− p pairs. Very recently, a Λp narrow peak has been reported in p¯ annihilation on 4 He from the OBELIX spectrometer data at LEAR, CERN [44], corresponding to a yet deeper K − pp quasi-bound state (B ∼ 160 MeV) if this interpretation is valid, given the reservations mentioned above. A ¯ N ]I=1 }I=1/2 ) deﬁnitive study of the K − pp quasi-bound state (or more generally {K[N could be reached through fully exclusive formation reactions, such as (19)

¯ N ]I=1 }I=1/2,I =+1/2 , K − + 3 He → n + {K[N z

¯ N ]I=1 }I=1/2,I =−1/2 , p + {K[N z

the ﬁrst of which is scheduled for day-one experiment in J-PARC [45]. We note that the large widths calculated for the K − pp quasi-bound state could make it diﬃcult to identify the state experimentally [46].

Strange atoms, strange nuclei and kaon condensation

69

¯ The current experimental and theoretical interest in K-nuclear bound states was triggered back in 1999 by the suggestion of Kishimoto [47] to look for such states in (K − , p) reactions in ﬂight, and by Akaishi and Yamazaki [48, 49] who suggested to look ¯ N N I = 0 state bound by over 100 MeV for which the main KN ¯ → πΣ decay for a KN channel would be kinematically closed. In fact, Wycech had conjectured that the width of such states could be as small as 20 MeV [50]. Evidence claimed initially for relatively − − narrow states in the inclusive (Kstop , p) and (Kstop , n) spectra on 4 He has recently been withdrawn [51,52], just to be replaced by a complementary low-statistics Λd narrow peak ¯ NN I = 0 reported in p¯ annihilation on 4 He [44], corresponding to a quasi-bound KN state with B ∼ 120 MeV. Such correlated Λd pairs could arise from secondary threenucleon absorption processes, as recently discussed by the FINUDA [53] and the KEK [54] − Collaborations in Kstop reactions on 6 Li and 4 He, respectively. On heavier targets, enhancements have been observed in the (K − , n) in-ﬂight spectrum on 16 O [55], but subsequent (K − , n) and (K − , p) reactions on 12 C at plab = 1 GeV/c have not disclosed ¯ any peaks beyond the appreciable strength observed below the K-nucleus threshold [56]. ¯ It is clear that the issue of K nuclear states is far yet from being experimentally resolved and more dedicated, systematic searches are necessary. . ¯ quasi-bound nuclear states. – In this model, 4 4. RMF dynamical calculations of K spelled out in refs. [11, 57, 58], the (anti)kaon interaction with the nuclear medium is incorporated by adding to LN the Lagrangian density LK : (20)

¯ μ K − m2K KK ¯ − gσK mK σ KK ¯ . LK = Dμ∗ KD

The covariant derivative Dμ = ∂μ + igωK ωμ describes the coupling of the (anti)kaon to the vector meson ω. The (anti)kaon coupling to the isovector ρ meson was found to ¯ meson induces additional source terms in the equations have negligible eﬀects. The K of motion for the meson ﬁelds σ and ω0 . It thus aﬀects the scalar S = gσN σ and the vector V = gωN ω0 potentials which enter the Dirac equation for nucleons, and this leads to rearrangement or polarization of the nuclear core, as shown on the left-hand side of ﬁg. 8 for the calculated average nuclear density ρ¯ = A1 ρ2 dr as a function of BK − for K − nuclear 1s states across the periodic table, and on the right-hand side of the ﬁgure for the density of K40− Ca for several 1s K − nuclear states with speciﬁed BK − values [11]. It is seen that in the light K − nuclei, ρ¯ increases substantially with BK − to values about ¯ The increase of the central nuclear densities is bigger, 50% higher than without the K. up to 50–100%, and is non-negligible even in the heavier K − nuclei where it is conﬁned to a small region of order 1.5 fm. Furthermore, in the Klein-Gordon equation satisﬁed ¯ the scalar S = gσK σ and the vector V = −gωK ω0 potentials become state by the K, dependent through the dynamical density dependence of the mean-ﬁeld potentials S and V , as expected in a RMF calculation. An imaginary ImVK¯ ∼ tρ was added, ﬁtted to the K − atomic data [59]. It was then suppressed by an energy-dependent factor f (BK¯ ), considering the reduced phase-space for the initial decaying state and assuming two-body ¯ → πY mesonic modes (80%) ﬁnal-state kinematics for the decay products in the KN ¯ N → Y N nonmesonic modes (20%). and in the KN

70

E. Friedman and A. Gal

0.4

0.20

40

C

Ca

0.3

0.16

ρ (fm )

_

O

BK- = 40 MeV BK- = 80 MeV BK- =120 MeV BK- =160 MeV BK- =200 MeV

-3

-3

ρ (fm )

0.18

0.14

0.2

__

no K

Ca

0.1

0.12

Pb 0.10 0

50

100 150 BK- (MeV)

200

250

0 0

1

2

3 r (fm)

4

5

6

Fig. 8. – Left: dynamically calculated average nuclear density ρ¯ of 1s K − -nuclear states in the nuclei denoted, as a function of the 1s K − binding energy. Right: dynamically calculated nuclear density ρ of K40− Ca for several 1s K − nuclear states with speciﬁed BK − values [11].

The RMF coupled equations were solved self-consistently. For a rough idea, whereas 1s − the static calculation gave BK 1s state in 12 C, using the values − = 132 MeV for the K atom atom − 1s gωK , gσK from the K -atom ﬁt, the dynamical calculation gave BK − = 172 MeV. In 1s order to scan a range of values for BK − , the coupling constants gσK and gωK were varied in given intervals of physical interest. An example is shown in ﬁg. 9. Beginning approximately with 12 C, the following conclusions may be drawn: ¯ binding energy BK¯ saturates as a function of – For given values of gσK , gωK , the K A, except for a small increase due to the Coulomb energy (for K − ). – The diﬀerence between the binding energies calculated dynamically and statically, dyn stat BK − BK ¯ for a given value ¯ , is substantial in light nuclei, increasing with BK ¯ of A, as shown in the upper panels of ﬁg. 9, and decreasing monotonically with A for a given value of BK¯ . It may be neglected only for very heavy nuclei. The s.p. same holds for the nuclear rearrangement energy BK ¯ which is a fraction of ¯ − BK dyn stat − B . BK ¯ ¯ K – The functional dependence of the width ΓK − (BK − ), shown for K12− C in the lower panels of ﬁg. 9 follows the shape of the suppression factor f (BK − ) which falls oﬀ ¯ → πΣ gets switched oﬀ, rapidly until BK − ∼ 100 MeV, where the dominant KN − and then stays rather ﬂat in the range BK ∼ 100–200 MeV where the width is ¯ N → Y N absorption modes. The widths calculated dynamdominated by the KN ically in this range are considerably larger than if calculated statically. Adding ¯ → πΛ secondary the residual width neglected in this calculation, due to the KN mesonic decay channel, and assigning these two-nucleon absorption modes a ρ2

71

Strange atoms, strange nuclei and kaon condensation

200

BK- (MeV)

12

C

100

0

ΓK- (MeV)

150 100 50

0.4

0.6

αω

0.8

0

0.2

0.4

ασ

0.6

0.8

1

Fig. 9. – 1s K − binding energy and width in K12− C calculated statically (open circles) and dynamically (solid circles) for the nonlinear RMF model NL-SH [60] as a function of the ωK and σK coupling strengths: αω is varied in the left panels as indicated, with ασ = 0, and ασ is varied in the right panels as indicated, with αω = 1. The dotted line shows the calculated binding energy when the absorptive K − potential is switched oﬀ in the dynamical calculation.

density dependence, a lower limit of ΓK¯ 50 MeV is obtained for deeply bound states in the range BK − ∼ 100–200 MeV [58]. . 4 5. Kaon condensation. – The possibility of kaon condensation in dense matter was proposed by Kaplan and Nelson [61,62], with subsequent works oﬀering related scenarios in nuclear matter [63, 64]. Neutron stars, with a density range extending to several times nuclear-matter density, have been considered extensively as the most natural dense systems where kaon condensation is likely to be realized. It is commonly accepted that under some optimal conditions, kaon condensation could occur at densities above 3ρ0 depending on the way hyperons enter the constituency of neutron stars. However, our concern here is not with neutron stars where time scales of the weak interactions are operative, enabling the transformation n → p + K − or a rare weak decay such as e− → K − + νe to transform “high-energy” electrons to antikaons once the eﬀective mass of K − mesons dropped below 200 MeV approximately. Our concern here is limited to laboratory strong-interaction processes where hadronization and equilibration time scales in collisions leading to dense matter are much shorter, of order fm/c. If antikaons bind ¯ meson in strongly to nuclei, then one might ask whether or not the binding energy per K ¯ nuclear states increases signiﬁcantly upon adding a large number of K ¯ mesons, so multi-K

72

E. Friedman and A. Gal 200

0.30

208

0.20

-

Pb + κK

208

Pb + κK

-3

ρN (fm )

0.25

0.15

150

0.10

BK (MeV)

0.05 0.00

-3

ρK- (fm )

0.16

-

no K 2K 4K 6K 8K 10K 12K 14K 16K

0.12 0.08 0.04 0.00

0

2

4

6

r (fm)

8

100

50

-

K 0 K

10

0

1

2

3

κ

4

5

6

¯ density ρK¯ (bottom panel) in 208 Pb + Fig. 10. – Left: nuclear density ρN (top panel) and 1s K − 208 − κK , starting with BK − = 100 MeV in Pb + 1K . The dotted curve stands for the 208 Pb ¯ where ¯ ¯ binding energy BK¯ in 208 Pb + κK, density in the absence of K mesons. Right: 1s K ¯ = K − (circles) and K ¯ 0 (squares). Figure taken from ref. [58]. K

¯ mesons provide the physical degrees of freedom for self-bound strange hadronic that K systems. Precursor phenomena to kaon condensation in nuclear matter would occur ¯ meson beyond some threshold value of strangeness, if the binding energy BK¯ per K 2 2 exceeds the combination mK c + μN − mΣ c 240 MeV, where μN is the nucleon chemical potential. Furthermore, once BK¯ mK c2 + μN − mΛ c2 320 MeV, Λ, Σ and Ξ hyperons would no longer combine macroscopically with nucleons to compose the more conventional kaon-free form of strange hadronic matter [65]. ¯ nuclear conﬁgurations, ﬁnding that the Gazda et al. [58] recently calculated multi-K ¯ densities behave regularly upon increasing the number of antikaons emnuclear and K bedded in the nuclear medium, without any indication for abrupt or substantial increase of the densities. The central nuclear densities appear to saturate at approximately 50% higher values than the central nuclear density with one antikaon, as shown on the left¯ binding energy hand side of ﬁg. 10 for multi-K − 208 Pb nuclei. Furthermore, the K ¯ mesons embedded in the nuclear medium. saturates upon increasing the number of K The heavier the nucleus is, the more antikaons it takes to saturate the binding energies, but even for 208 Pb the number required does not exceed approximately 10, as shown on the right-hand side of ﬁg. 10. We note that the interaction between antikaons in this extended RMF calculation is mediated by isoscalar boson ﬁelds: vector ω and φ, and scalar σ. The binding-energy saturation owes its robustness to the dominance of the re-

Strange atoms, strange nuclei and kaon condensation

73

pulsive vector interactions over the attractive scalar interactions for antikaon pairs. The ¯ binding energies do not exceed the range of values 100–200 MeV saturated values of K considered normally as providing deep binding for one antikaon. This range of bind¯ nuclei comfortably above the range of energies ing energies leaves antikaons in multi-K ¯ nuclei may oﬀer where hyperons might be relevant. It is therefore unlikely that multi-K precursor phenomena in nuclear matter towards kaon condensation. 5. – Σ hyperons . 5 1. Overview . – One Boson Exchange (OBE) models ﬁtted to the scarce low-energy Y N scattering data produce within a G-matrix approach, with one exception (Nijmegen Model F), as much attraction for the Σ nuclear potential as they do for the Λ nuclear potential, see ref. [66] for a review of “old” models and ref. [67] for the latest state of the art for Nijmegen models. Indeed, the best-ﬁt teﬀ ρ potential for Σ− atoms was found by Batty et al. [68, 69] to be attractive and absorptive, with central depths for the real and imaginary parts of 25–30 MeV and 10–15 MeV, respectively. It took almost a full decade, searching for Σ hypernuclear bound states at CERN, KEK and BNL, before it was realized that except for a special case for 4Σ He, the observed continuum Σ hypernuclear spectra indicate a very shallow, or even repulsive Σ nuclear potential, as reviewed by Dover et al. [70]. These indications have received ﬁrm support with the measurement of several (K − , π ± ) spectra at BNL [71] followed by calculations for 9 Be [72]. Recently, with measurements of the Σ− spectrum in the (π − , K + ) reaction taken at KEK across the periodic table [73, 74], it has become established that the Σ nuclear interaction is strongly repulsive. In parallel, analyses of Σ− -atom in the early 1990s, allowing for density dependence or departure from the tρ prescription, motivated mostly by the precise data for W and Pb [75], led to the conclusion that the nuclear interaction of Σs is dominated by repulsion [76-78], as reviewed in ref. [1]. This might have interesting repercussions for the balance of strangeness in the inner crust of neutron stars [79], primarily by delaying the appearance of Σ− hyperons to higher densities, if at all, as discussed below. The inability of the Nijmegen OBE models, augmented by G-matrix calculations [67], to produce Σ nuclear repulsion is a serious drawback for these models at present. This problem apparently persists also in the Juelich model approach [80]. The only theoretical works that provide exception are SU (6) quark-model RGM calculations by the Kyoto-Niigata group [81], in which a strong Pauli repulsion appears in the I = 3/2, 3 S1 − 3 D1 ΣN channel, and Kaiser’s SU (3) chiral perturbation calculation [82] which yields repulsion of order 60 MeV in nuclear matter. Since Σ− is the ﬁrst hyperon (as a function of density) to appear in neutron stars when the hyperon interactions are disregarded, it is natural to expect that the composition of neutron-star matter depends sensitively on the Σ− -hypernuclear potential. For attractive Σ− -hypernuclear potentials of the order of 30 MeV depth, as for Λ hyperons in Λ hypernuclei, the Σ− is indeed the ﬁrst hyperon to appear, at density lower than twice nuclear matter density. However, for a repulsive potential, the situation reverses dramatically as shown in ﬁg. 11. Incidentally, a K − condensed phase might then appear

74

E. Friedman and A. Gal

10

0

n

GM1

p 10

−1

Baryon fraction

e

10

−2

UΣ=+30 MeV

μ

UΞ=−18 MeV 0

10

10

Ξ

−3

Λ

Ξ

−

−4

0.0

0.3

0.6

0.9

1.2

1.5

−3

Density (fm ) Fig. 11. – Fractions of baryons and leptons in neutron-star matter for a RMF calculation using set GM1 with weak Y Y interactions [83]. Figure taken from ref. [84].

at density between 3 to 4 times nuclear matter density replacing Ξ− hyperons and second in strangeness only to Λ hyperons [85]. Below we brieﬂy review and update the Σ− atom ﬁts and the recent (π − , K + ) KEK results and their analysis. . 5 2. Fits to Σ− atoms. – The data used in the Σ− -atom ﬁts are shown in ﬁg. 12 representing all published measurements from C to Pb inclusive. The data are relatively inaccurate, reﬂecting the diﬃculty in making measurements of strong-interaction eﬀects in Σ− atoms where most of the X-ray lines are relatively weak and must be resolved from the much stronger K − atomic X-ray transitions. Batty et al. [76, 77] analyzed the full data set of Σ− atoms, consisting of strong-interaction level shifts, widths and yields, introducing a phenomenological Density Dependent (DD) potential of the isoscalar form (21)

α

VΣ (r) ∼ [b0 + B0 (ρ(r)/ρ(0)) ] ρ(r),

α > 0,

and ﬁtting the parameters b0 , B0 and α to the data, greatly improved ﬁts to the data are obtained. Isovector components are readily included in eq. (21) but are found to have a marginal eﬀect. Note, however, that the absorption was assumed to take place only on protons. The complex parameter b0 may be identiﬁed with the spin-averaged Σ− N scattering length. For the best-ﬁt isoscalar potentials, ReVΣ is attractive at low densities outside the nucleus, changing into repulsion in the nuclear surface region. The precise magnitude and shape of the repulsive component within the nucleus is not determined by the atomic data. The resulting potentials are shown in ﬁg. 13 (DD, solid lines), where

75

Strange atoms, strange nuclei and kaon condensation Sigma atoms 10 3

n=3

n=9

n=4

(a)

Shift (eV)

10 2

10

n=10 n=5

1

10

10

n=6

-1

-2

0

10

20

30

40

50

60

70

80

90

Z

10 3

n=3

n=4

(b)

Width (eV)

10 2

n=9

10

n=5 1

10

10

n=10 n=6

-1

-2

0

10

20

30

40

50

60

70

80

90

Z

Fig. 12. – Shift and width values for sigma atoms. The continuous lines join points calculated with a best-ﬁt DD optical potential, see next ﬁgure.

it is worth noting that the transition from attraction to repulsion occurs well outside of the nuclear radius, hence the occurrence of this transition should be largely model independent. To check this last point we have repeated the ﬁts to the atomic data with . the “geometrical model” F of subsect. 3 1, using separate tρ expressions in an internal and an external region, see eq. (12). The neutron densities used in the ﬁts were of the skin type, with the rn − rp parameter eq. (3) γ = 1.0 fm. The ﬁts deteriorate signiﬁcantly if the halo type is used for the neutron density. The ﬁt to the data is equally good with this model as with the DD model (χ2 per degree of freedom of 0.9 here compared to 1.0 for the DD model) and the potentials are shown as the dashed lines in ﬁg. 13. The half-density radius of the charge distribution is indicated in the ﬁgure. It is clear that

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VR (MeV)

Ca Σ

−

DD

60 F 20

−20 100

Rc

3

4

5

6

7

−

VR (MeV)

Pb Σ 60

DD F

20

−20

Rc

5

6

7 r (fm)

8

9

Fig. 13. – ReVopt for DD (solid) and for the geometrical model F (dashed) Σ− nuclear potentials ﬁtted to Σ− atomic data. Vertical bars indicate the half-density radius of the nuclear charge distribution.

both models show weak attraction at large radii, turning into repulsion approximately one fm outside of that radius. Further insight into the geometry of the Σ-nucleus interaction is gained by inspecting the Functional Derivatives (FD) of χ2 with respect to the optical potentials, see sub. sect. 2 3. Figure 14 shows the FDs based on the best ﬁt of the geometrical model F as discussed above. From the diﬀerences between the FD with respect to the full complex potential and the FD with respect to the real potential it is concluded that both real and imaginary parts play similar roles in the Σ-nucleus interaction. The bulk of |FD| is in the range of 0.5 ≤ η ≤ 6, covering the radial region where the weak attraction turns into repulsion. Obviously no information is obtained from Σ− atoms on the interaction inside the nucleus. It is also interesting to note quite generally that such potentials do not produce bound states, and this conclusion is in agreement with the experimental results from BNL [71] for the absence of Σ hypernuclear peaks beyond He. Some semi-theoretical support for this ﬁnding of inner repulsion is given by RMF calculations by Mareˇs et al. [78] who generated the Σ-nucleus interaction potential in terms of scalar (σ) and vector (ω, ρ) meson mean-ﬁeld contributions, ﬁtting its coupling constants to the relatively accurate Σ− atom shift and width data in Si and in Pb. The obtained potential ﬁts very well the whole body of data on Σ− atoms. This potential, which is generally attractive far outside the nucleus, becomes repulsive at the nuclear

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Strange atoms, strange nuclei and kaon condensation 5 Comp

Real

2

δΧ /(δV/V)

0

−5

−10

−4

−2

0

2 η

4

6

8

Fig. 14. – Functional derivatives of χ2 with respect to the real (solid) and with respect to the full complex (dashed) optical potentials for the best-ﬁt F potential.

surface and remains so inward in most of the acceptable ﬁts, of order 10–20 MeV. The Pb data [75] are particularly important in pinning down the isovector component of the potential which in this model is sizable and which, for Σ− , acts against nuclear binding in core nuclei with N − Z > 0, countering the attractive Coulomb interaction. On the other hand, for very light nuclear cores and perhaps only for A = 4 hypernuclei, this isovector component (Lane term) generates binding of Σ+ conﬁgurations. In summary, the more modern ﬁts to Σ− atom data [76-78] and the present ﬁts with the geometrical model support the presence of a substantial repulsive component in the Σ-nucleus potential which excludes normal Σ-nuclear binding, except perhaps in very special cases such as 4 Σ He [86-89]. . 5 3. Evidence from (π − , K + ) spectra. – A more straightforward information on the nature of the Σ-nuclear interaction has been provided by recent measurements of inclusive (π − , K + ) spectra on medium to heavy nuclear targets at KEK [73, 74]. The inclusive (π − , K + ) spectra on Ni, In and Bi are shown in ﬁg. 15 together with a ﬁt using WoodsSaxon potentials with depths V0 = 90 MeV for the (repulsive) real part and W0 = −40 MeV for the imaginary part. These and other spectra measured on lighter targets suggest that a strongly repulsive Σ-nucleus potential is required to reproduce the shape of the inclusive spectrum, while the sensitivity to the imaginary (absorptive) component is secondary. The favored strength of the repulsive potential in this analysis is about 100 MeV, of the same order of magnitude reached by the DD Σ− atomic ﬁt potential shown in ﬁg. 13 as it “enters” the nucleus inward. The general level of agreement in the ﬁt shown in ﬁg. 15 is satisfactory, but there seems to be a systematic eﬀect calling for

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Fig. 15. – Inclusive (π − , K + ) spectra on Ni, In and Bi, ﬁtted by a Σ-nucleus WS potential with depths V0 = 90 MeV, W0 = −40 MeV [74].

more repulsion, the heavier is the target. We conclude that a strong evidence has been ﬁnally established for the repulsive nature of the Σ-nucleus potential. More sophisticated theoretical analyses of these KEK (π − , K + ) spectra [90-93] have also concluded that the Σ-nuclear potential is repulsive within the nuclear volume, although they yield a weaker repulsion in the range of 10–40 MeV. An example of a recent analysis of the Si spectrum is shown in ﬁg. 16 from ref. [90] where six diﬀerent Σ-nucleus potentials are tested for their ability within the Distorted Wave Impulse Approximation (DWIA) to reproduce the measured 28 Si(π − , K + ) spectrum [74]. This particular DWIA version was tested on the well-understood 28 Si(π + , K + ) quasi-free Λ hypernuclear spectrum also taken at KEK with incoming pions of the same momentum plab = 1.2 GeV/c.

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Fig. 16. – Comparison between DWIA calculations [90] and the measured 28 Si(π − , K + ) spectrum [74] using six Σ-nucleus potentials, (a)-(c) with inner repulsion, (d)-(f) fully attractive. The solid and dashed curves denote the inclusive and Λ conversion cross-sections, respectively. Each calculated spectrum was normalized by a fraction fs . The arrows mark the Σ− − 27 Alg.s. threshold at ω = 270.75 MeV.

Potential (a) is the DD, type A’ potential of ref. [77], (b) is one of the RMF potentials of ref. [78], that with αω = 1, and (c) is a local-density approximation version of a G matrix constructed from the Nijmegen model F. These three potentials are repulsive within the nucleus but diﬀer considerably there from each other. Potentials (d)-(f) are all attractive within the nucleus, with (f) being of a teﬀ ρ form. All of the six potentials are attractive outside the nucleus, as required by ﬁts to the “attractive” Σ− atomic level shifts. The ﬁgure shows clearly, and judging by the associated χ2 /N values, that fully

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attractive potentials are ruled out by the data and that only the “repulsive” Σ-nucleus potentials reproduce the spectrum very well, but without giving preference to any of these potentials (a)-(c) over the other ones in this group. It was shown by Harada and Hirabayashi [93], furthermore, that the (π − , K + ) data on targets with neutron excess, such as 209 Bi, also lack the sensitivity to conﬁrm the presence of a sizable (repulsive for Σ− ) isovector component of the Σ nucleus interaction as found in the Σ− -atom ﬁts [76-78]. ∗ ∗ ∗ Special thanks are due to the Directors of Course CLXVII “Strangeness and spin in fundamental physics” M. Anselmino and T. Bressani, to the Scientiﬁc Secretary A. Feliciello and to the School Secretary Ms. B. Alzani and her crew. This Review was supported in part by the Israel Science Foundation grant 757/05. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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[30] Drukarev E. G., private communication (May 2006). ¨ cker H. and Greiner W., Phys. Lett. B, [31] Schaffner J., Gal A., Mishustin I. N., Sto 334 (1994) 268. [32] Brown G. E. and Rho M., Nucl. Phys. A, 596 (1996) 503. [33] Scheinast W. et al., Phys. Rev. Lett., 96 (2006) 072301 and references therein. [34] Nogami Y., Phys. Lett., 7 (1963) 288. [35] Yamazaki T. and Akaishi Y., Phys. Lett. B, 535 (2002) 70. ´ A. and Weise W., arXiv:nucl-th/0701050, presented at HYP06, Mainz (October [36] Dote 2006). [37] Shevchenko N. V., Gal A. and Mareˇ s J., Phys. Rev. Lett., 98 (2007) 082301. ´vai J., Phys. Rev. C, 76 (2007) 044004. [38] Shevchenko N. V., Gal A., Mareˇ s J. and Re [39] Ikeda Y. and Sato T., arXiv:nucl-th/0701001, presented at HYP06, Mainz (October 2006). [40] Ikeda Y. and Sato T., Phys. Rev. C, 76 (2007) 035203. [41] FINUDA Collaboration (Agnello M. et al.), Phys. Rev. Lett., 94 (2005) 212303. [42] Magas V. K., Oset E., Ramos A. and Toki H., Phys. Rev. C, 74 (2006) 025206. [43] FINUDA Collaboration (Agnello M. et al.), Nucl. Phys. A, 775 (2006) 35. [44] Bendiscioli G., Bressani T., Fontana A., Lavezzi L., Panzarasa A. and Rotondi A., Nucl. Phys. A, 789 (2007) 222. ´ P., Gal A. [45] Nagae T., in Topics in Strangeness Nuclear Physics, edited by Bydˇ zovsky and Mareˇ s J., Lect. Notes Phys., Vol. 724 (Springer, Berlin Heidelberg) 2007, pp. 81-111. [46] Koike T. and Harada T., Phys. Lett. B, 652 (2007) 262. [47] Kishimoto T., Phys. Rev. Lett., 83 (1999) 4701. [48] Akaishi Y. and Yamazaki T., in Proceedings of III International DAΦNE Workshop, Frascati Physics Series, Vol. XVI, edited by Bianconi S. et al., LNF, Frascati, (1999), pp. 59-74. [49] Akaishi Y. and Yamazaki T., Phys. Rev. C, 65 (2002) 044005. [50] Wycech S., Nucl. Phys. A, 450 (1986) 399c. [51] KEK E549/570 Collaboration (Iwasaki M. et al.), arXiv:0706.0297 [nucl-ex], plenary talk given by Iwasaki M. at HYP06, Mainz (October 2006). [52] KEK E549 Collaboration (Sato M. et al.), Phys. Lett. B, 659 (2008) 107. [53] FINUDA Collaboration (Agnello M. et al.), Phys. Lett. B, 654 (2007) 80. [54] KEK E549 Collaboration (Suzuki T. et al.), Phys. Rev. C, 76 (2007) 068202. [55] AGS E930 Collaboration (Kishimoto T. et al.), Nucl. Phys. A, 754 (2005) 383c. [56] KEK E548 Collaboration (Kishimoto T. et al.), Prog. Theor. Phys., 118 (2007) 181. [57] Mareˇ s J., Friedman E. and Gal A., Phys. Lett. B, 606 (2005) 295. [58] Gazda D., Friedman E., Gal A. and Mareˇ s J., Phys. Rev. C, 76 (2007) 055204. ´ A., Phys. Rev. C, 60 (1999) 024314. [59] Friedman E., Gal A., Mareˇ s J. and Cieply [60] Sharma M. M., Nagarajan M. A. and Ring P., Phys. Lett. B, 312 (1993) 377. [61] Kaplan B. D. and Nelson A. E., Phys. Lett. B, 175 (1986) 57. [62] Nelson A. E. and Kaplan B. D., Phys. Lett. B, 192 (1987) 193. [63] Brown G. E., Lee C.-H., Rho M. and Thorsson V., Nucl. Phys. A, 567 (1994) 937. [64] Lee C.-H., Brown G. E., Min D.-P. and Rho M., Nucl. Phys. A, 585 (1995) 401. [65] Schaffner-Bielich J. and Gal A., Phys. Rev. C, 62 (2000) 034311. [66] Dover C. B. and Gal A., Prog. Part. Nucl. Phys., 12 (1984) 171. [67] Rijken Th. A. and Yamamoto Y., Phys. Rev. C, 73 (2006) 044008. [68] Batty C. J., Phys. Lett., 87B (1979) 324. [69] Batty C. J., Gal A. and Toker G., Nucl. Phys. A, 402 (1983) 349. [70] Dover C. B., Millener D. J. and Gal A., Phys. Rep., 184 (1989) 1. [71] Bart S. et al., Phys. Rev. Lett., 83 (1999) 5238.

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[72] Dabrowski J., Phys. Rev. C, 60 (1999) 025205. [73] KEK E438 Collaboration (Noumi H. et al.), Phys. Rev. Lett., 89 (2002) 072301; ibid., 90 (2003) 049902(E). [74] KEK E438 Collaboration (Saha P. K. et al.), Phys. Rev. C, 70 (2004) 044613. [75] AGS E723 Collaboration (Powers R. J. et al.), Phys. Rev. C, 47 (1993) 1263. [76] Batty C. J., Friedman E. and Gal A., Phys. Lett. B, 335 (1994) 273. [77] Batty C. J., Friedman E. and Gal A., Prog. Theor. Phys. Suppl., 117 (1994) 227. [78] Mareˇ s J., Friedman E., Gal A. and Jennings B. K., Nucl. Phys. A, 594 (1995) 311. [79] Balberg S. and Gal A., Nucl. Phys. A, 625 (1997) 435. [80] Haidenbauer J. and Meissner Ulf-G., Phys. Rev. C, 72 (2005) 044005. [81] Kohno M., Fujiwara Y., Fujita T., Nakamoto C. and Suzuki Y., Nucl. Phys. A, 674 (2000) 229. [82] Kaiser N., Phys. Rev. C, 71 (2005) 068201. [83] Schaffner J. and Mishustin I. N., Phys. Rev. C, 53 (1996) 1416. [84] Schaffner-Bielich J., arXiv:astro-ph/0703113, plenary talk at HYP06, Mainz (October 2006). [85] Ramos A., Schaffner-Bielich J. and Wambach J., Lect. Notes Phys., Vol. 578 (2001), pp. 175-202. [86] Hayano R. S., Ishikawa T., Iwasaki M., Outa H., Takada E., Tamura H., Sakaguchi A., Aoki M. and Yamazaki T., Phys. Lett. B, 231 (1989) 355. [87] Harada T., Shinmura S., Akaishi Y. and Tanaka H., Nucl. Phys. A, 507 (1990) 715. [88] AGS E905 Collaboration (Nagae T. et al.), Phys. Rev. Lett., 80 (1998) 1605. [89] Harada T., Phys. Rev. Lett., 81 (1998) 5287. [90] Harada T. and Hirabayashi Y., Nucl. Phys. A, 759 (2005) 143. [91] Kohno M., Fujiwara Y., Watanabe Y., Ogata K. and Kawai M., Prog. Theor. Phys., 112 (2004) 895. [92] Kohno M., Fujiwara Y., Watanabe Y., Ogata K. and Kawai M., Phys. Rev. C, 74 (2006) 064613. [93] Harada T. and Hirabayashi Y., Nucl. Phys. A, 767 (2006) 206.

Strangeness in hot and dense nuclear matter E. Nappi INFN, Sezione di Bari - 70124 Bari, Italy

Summary. — Ultra-relativistic heavy-ion collisions are believed to provide the extreme conditions of energy densities able to lead to a transition to a short-lived state, called Quark-Gluon Plasma (QGP), where the quarks are no longer bound inside hadrons. The studies performed so far, formerly at SPS (CERN) and later at RHIC (BNL), allowed to achieve a multitude of crucial results consistent with the hypothesis that a new phase of the QCD matter has been indeed created. However, the emerging picture is that of the formation of a strongly interacting medium with negligibly small viscosity, a perfect liquid, rather than the ideal perturbative QCD parton-gas predicted by most theorists. The head-on collisions between lead nuclei at the unprecedented energies of the forthcoming Large Hadron Collider (LHC) at CERN, due to start in 2008, will allow to measure the properties of compressed and excited nuclear matter at even higher initial densities and temperatures, far above the predicted QCD phase transition point. The longer duration of the quark-gluon plasma phase and the much more abundant production of hard probes, which depend much less on details of the later hadronic phase, will likely provide a consistent and uncontroversial experimental evidence of the QGP formation. Among the signals that witness the change in the nature of the state of nuclear matter, the chemical equilibrium value of the strangeness plays a key role since it is directly sensitive to the matter properties and provides information on the link between the partonic and the hadronic phases. The aim of this course is to overview the underlying goals, the current status and the prospect of the physics of the nucleus-nucleus collisions at ultrarelativistic energies. Among the experimental methods adopted to investigate the challenging signatures of the QGP formation, emphasis on those related to the strangeness ﬂavour will be given.

c Societ` a Italiana di Fisica

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1. – Introduction According to the Quantum Chromo-Dynamics (QCD), the current theory of the strong interactions, spin-1/2 quarks are the elementary constituents of hadronic matter and they interact by exchanging massless vector ﬁelds, called gluons, coupled to the matter ﬁelds through a set of three “colour” charges (the concept of quark carrying a “colour” charge led to the name QCD). Although QCD was formulated by analogy with Quantum Electro-Dynamics (QED) where the role of gluons is played by photons, carriers of the electromagnetic interaction, a profound diﬀerence exists between the two theories: gluons are directly coupled among themselves, whereas the photons are electrically neutral and therefore can couple only through electron loops. One of the consequences of the direct coupling among gluons is that the ﬁeld between two colour sources is limited to a thin string about 1 fm (10−15 m) across, with a resulting eﬀective potential increasing linearly with r (the string tension). This peculiar behaviour, called “colour conﬁnement”, entails that the strong interaction remains constant as the sources move apart whereas in the case of electrical charges the ﬁeld falls oﬀ at large distances. The colour conﬁnement is the most evident manifestation of the complicated structure of the QCD vacuum that, acting as a “colour” dielectric, prevents the strong ﬁeld to . extend away from any set of colour sources (see subsect. 2 1). Consequently hadrons are a globally neutral system with respect to colours (colour singlets). Combination of three quarks(1 ) carrying each a diﬀerent colour makes up colourless non-integer spin hadrons, called baryons. Integer spin hadrons (mesons) are instead made up of combinations of a quark of a certain colour and an antiquark with the corresponding anti-colour. Both quarks and gluons cannot be observed outside the hadrons; in fact by attempting to pull apart from each other two quarks, the interaction energy would increase up to a point where new quark pairs are created from the vacuum thus neutralizing the colour of the previous quarks and forming new colour neutral objects (hadronization mechanism). Whilst any QED phenomenon can be calculated in a perturbative way, such an approach is applicable in QCD only to the strong-interaction processes at large momentum transfer (or equivalently at short distance) where quarks and gluons appear to be weakly coupled due to the logarithmic decrease of the interaction strength between two coloured objects as they get closer (asymptotic freedom). For distances approaching the typical hadron size (∼ fm), the eﬀective coupling constant is larger than one, because of the QCD vacuum structure, thus requiring a nonperturbative approach. In fact, perturbation theory to any ﬁnite order in QCD depicts only a particle spectrum made of quarks and gluons in contrast to the conﬁnement property. (1 ) Six varieties of quarks, called “ﬂavors”, can account for the properties of the entire multitude of hadrons. Flavors are labelled u (up), c (charm), t (top), d (down), s (strange) and b (bottom).

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Fig. 1. – Lattice-QCD calculations of the energy density plotted in units of T 4 ; εSB is the energy density of an ideal Stefan-Boltzmann gas system of non-interacting quarks and gluons.

A very promising non-perturbative technique is the lattice-QCD: a Monte Carlo simulation that approximates the continuous space-time of real world into a discrete lattice where the average value of each observable is calculated as a function of the temperature in grand-canonical ensembles. Good statistics for accurate predictions on large lattices, with a grid spacing rather small to capture any details in the structure of the quantum ﬁeld, require the employment of a huge computer power and a long running time. Despite this diﬃculty, computations with three dynamical light quark ﬂavors on the lattice [1] revealed interesting insights into the behaviour of hadronic matter under extreme conditions of density and indicated that a phase transition from a hadronic gas to a plasma of quarks and gluons (QGP), within which colour freely propagates, is expected at low quark chemical potential and at a temperature(2 ) of Tc ∼ 170 MeV which corresponds to an energy density εc = 0.6 GeV/fm3 (ﬁg. 1). This paper consists of four parts: after an introduction on the underlying physics concepts, the main observables and signatures investigated in the study of ultrarelativistic heavy-ion collisions are described in the framework of the results already achieved at CERN-SPS and BNL-RHIC. The successive two parts are devoted to the role played by the strangeness in the research programme carried out so far. Finally, the related experimental issues and the prospect at the forthcoming facility LHC, and an in-depth description of the ALICE experiment at LHC are provided. (2 ) Temperature is deﬁned by the relationship: kT = average kinetic energy per nucleon (k is the Boltzmann constant of proportionality: 1 MeV ∼ 1.2 × 1010 K). Tc (QGP) ∼ 170 MeV ∼ 2040 109 K (much more than the temperature of supernova, the hottest process active nowadays in the Universe).

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μB Color Superconductor

Ordinary matter

Triple point ?

Hadronic Fluid

Quark-Gluon Plasma

Critical End point Big Bang

Vacuum

~ 173 MeV

Temperature

Fig. 2. – Phase diagram of nuclear matter showing regions of hadronic, deconﬁned matter and colour superconductivity. Ordinary nuclear matter density is ρ0 = 0.14 nucleons · fm−3 , μB is the baryonic chemical potential.

The present paper is not meant to cover all the aspects of the investigation of the phase transition of hadronic matter to the QGP but rather to provide some context of the ﬁeld. Interested readers are referred to the excellent books of Hwa [2], Letessier and Rafelsky [3] and Shuryak [4]. 2. – The quark-gluon plasma Following the lattice-QCD predictions, the diagram for the envisaged phases of nuclear matter is shown in ﬁg. 2. Whilst in the low temperature and baryon density region, the basic degrees of freedom are hadronic (nucleons, mesons. . . ), at high temperature and baryon density the basic degrees of freedom become those of quarks and gluons. Colour superconductivity [5] is expected at large baryonic chemical potentials and small temperatures. QGP is not a mere “lab-creation” but it is supposed to be the primordial soup that originated the hadronic matter a few microseconds after the Big Bang, in the process of hadro-synthesis, as shown in ﬁg. 3. As the early Universe expanded and cooled down, the matter, characterized by a relatively small net baryon density, underwent through the hadronization process, following a downward trajectory practically along the vertical axis of the phase diagram. A dense state of matter is thought to exist also in the interior of neutron stars [6]. Therefore, the study of the properties of dense matter at low temperature will allow not only to achieve a more accurate knowledge of the dynamics of the evolution of neutron stars in supernovae, as a consequence of the huge compression of matter by gravitational collapse, but also to evaluate with better accuracy the correlation between the stability of neutron stars and their maximum mass.

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Fig. 3. – The transition from the primordial plasma of quarks and gluons to ordinary matter occurred some 10−5 s after Big Bang when the temperature lowered to about 1012 K. The inverse process is expected to occur nowadays at RHIC (and in near future at LHC).

Finally, the study of strongly interacting matter under extreme conditions of high energy density, beyond its paramount importance of addressing a new phase of matter and the above-mentioned exciting issues in astrophysics and cosmology, is expected to reveal key insights into the fundamental questions of conﬁnement and broken chiralsymmetry. In fact, in the framework of statistical QCD, the phase transition from the ordinary nuclear matter to the QGP is accompanied by chiral-symmetry restoration resulting in modiﬁed nuclear states and excitation of vacuum. . 2 1. The QCD-vacuum and the chiral symmetry. – According to the ﬁeld theories, the phenomenon of quantum ﬂuctuations led to a revision of the concept of vacuum being it no more an empty space as in the classical idea but a complex entity ﬁlled by virtual particle-antiparticle pairs. However, the resulting eﬀect of vacuum polarization has a completely diﬀerent behaviour in QED with respect to QCD, because in the latter theory the polarization from quark-antiquark pairs is not suﬃcient to oﬀset the eﬀect of interaction with the virtual gluons ﬁlling the vacuum. Hence, the property of gluons to carry colour charge entails the remarkable eﬀect of “anti-screening” of pair creation oppositely to the screening eﬀect of pair creation in the QED vacuum. Hadrons can therefore be imaged as bubbles within the “anti-screening” vacuum behaving as a liquid, i.e. as small regions of space where the colour ﬂux tubes between the constituent quarks are severely conﬁned, the whole system having neutral colour. Such a behaviour inspired the “Bag model” in which quarks and gluons modify the vacuum in their vicinity, carving out regions of perturbative vacuum (bags) immersed in the normal non-perturbative vacuum: almost massless quarks and gluons propagate freely in the bags, but are strongly repelled by the non-perturbative vacuum. The bag constant B

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measures the excess energy density of the perturbative vacuum, and hence is a measure of the pressure of the normal vacuum on the bubble of perturbative vacuum. In this model, the deconﬁnement occurs at a critical temperature such that the corresponding internal pressure compensates the external pressure. A value of 170 MeV is obtained, very similar to that calculated in the lattice theories. The states of virtual excitations that dress quarks with dense gluon and quarkantiquark clouds modify the value of the quark mass depending on the distance at which the mass is measured: the shorter the distance, the smaller the mass. The two following extreme cases are particularly relevant: – at very short distances (of the order of 10−14 cm), gluon inﬂuence is negligible owing to the asymptotic freedom: the mass of light quarks u and d is close to zero (mu ∼ 5 MeV/c2 and md ∼ 7 MeV/c2 ). Quarks are called “current quarks”. – At large distances, quarks acquire a large eﬀective mass via interactions between themselves and the surrounding gluon clouds. The eﬀective mass value for u and d quarks is about 300 MeV/c2 , i.e. 1/3 of the nucleon mass by assuming that the mass of a nucleon is the sum of the masses of the three non-relativistic constituent quarks. The QCD Lagrangian with massless u and d quarks reﬂects the existence of an underlying chiral symmetry since it consists of two components related to the splitting into two “handedness” terms. This is a consequence of the fact that massless particles possess a speciﬁc conserved helicity (the projection of the spin of a particle on its momentum) deﬁned as left-handed (L) if the directions of spin and momentum are opposite and as right-handed (R) if they are identically oriented. Helicity cannot be deﬁned in a Lorentz-invariant manner for particles with nonzero mass, for instance a right-handed electron can be converted into a left-handed one by changing the reference frame. On the contrary, massless particles move at the speed of light and since there is no observer travelling faster to reverse the direction of motion, the helicity of massless particles is Lorentz invariant. This is similar to the case of real photons that can have only two transverse polarizations. Emission and absorption of vector gluons by colour charges of quarks do not change quark helicities: the QCD Lagrangian of massless quarks naturally factorizes into two symmetric terms, one of which contains left-handed quarks uL , dL and the other, righthanded quarks uR , dR ; left-handed quarks interact only with left-handed antiquarks and similarly for the right-handed ones. Hence, although the Lagrangian has a deﬁnite symmetry, the physical states (nonzero mass nucleons) do not have it, with the net result that chiral symmetry is spontaneously broken through the creation of a vacuum scalar condensate (left-handed quarks and right-handed antiquarks and viceversa) that couples to the hadrons thus providing most of their mass. As the QGP phase occurs, the chiral symmetry is expected to be restored (or partially restored) since the new degrees of freedom are (almost) massless quarks and gluons acting as free particles. The progressive disappearance of the quark condensate (partial

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restoration of the chiral symmetry) in a dense and hot medium entails a decrease in the mass value of vector mesons (ρ, ω, φ. . . ), maybe accompanied by an increase in the width . of the resonance peak (see subsubsect. 3 2.4). 3. – Ultra-relativistic nuclear collisions As shown in the phase diagram of matter (ﬁg. 2), ordinary nuclei are located in a region deﬁned by a temperature much smaller than the proton and pion rest masses and a baryon density(3 ) of about 0.14 nucleons/fm3 corresponding to an energy density of 130 MeV/fm3 (ρ = 2.3 · 1014 g/cm3 ). This means that distances between nucleons are larger than their radius (≈ 0.8 fm), whereas the transformation of ordinary matter into QGP requires that the nucleon wave functions signiﬁcantly overlap each other, condition likely reachable by smashing together, at relativistic energies, heavy nuclei that thus become very hot and very dense as a consequence of the substantial squeezing occurring in the collision. An energy density many times higher than that of ordinary nuclear matter will be achieved in this way and hence nucleons will lose their identity by melting into a soup of quarks and gluons through a process that reverses the early Universe history. . As will be described in subsect. 3 2, from an experimental point of view, the most convenient way to study QGP complicated behaviour is to produce a high-energy density system in thermal equilibrium with a rather long lifetime. This entails that the deconﬁnement of quarks and gluons must preferably occur over a suﬃciently large volume, compared to their scattering length, and hence in interactions between extended hadronic objects as heavy nuclei. The several rescattering processes experienced by the produced particles will redistribute the available centre-of-mass energy in higher degrees of freedom thus producing a state of thermal equilibrium. On the contrary, although the energy densities achieved in collision e+ e− or pp could be as high as in colliding heavy nuclei, tiny projectiles as leptons and nucleons are unable to create QGP since the overall size of the interaction region would be too small to study the eﬀects of deconﬁnement. Moreover, because of the rather short range of the strong interactions and the subsequent evolution of the QGP, the relevant experimental observables come mainly from the interior of the dense energy region whilst background is essentially originated on the surface. Consequently the signal over background ratio is proportional to the colliding object’s volume over surface ratio, thus favouring, also in this case, the employment of . heavy nuclei (subsect. 3 2). (3 ) For a nucleus of atomic mass A and radius R ∼ 1.2A1/3 fm, the density of nucleons ρ0 is almost constant (no dependence on A): ρ0 = .

A nucleons ∼ nucleons = 0.14 (4/3)πR3 fm3

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Fig. 4. – Schematics of a collision, with an impact parameter b, between two nuclei in the centreof-mass system. The hot volume created by the participants consists of two regions: the central region (centred at y = 0) almost baryon-free at the energy regimes of RHIC and LHC and the fragmentation region (at high rapidity) rich in nucleons.

A big step forward in the QGP research was established when a collision energy well above 100 GeV per nucleon was reached in the centre-of-mass frame. Such condition was obtained by using a heavy-ion collider, this being the most eﬀective way to provide large energy to the nuclear interactions, whereas in ﬁxed-target collisions part of the incident energy is lost for the movement of the system as a whole. The deployment of RHIC at BNL and the perspective to operate LHC at CERN with heavy nuclei, make available a centre-of-mass energy in the nucleon-nucleon system increased with respect to SPS by over 10 and 300 times, respectively, hence allowing to reach initial values of energy and temperature substantially higher than the critical ones. . 3 1. Space-time evolution of nuclear collisions. – Scattering reactions between heavy nuclei are characterized by the number of participant and spectator nucleons. A simple geometrical description, based on the impact parameter b, is indeed possible owing to the relevant size of heavy nuclei whose radius is larger than the interaction length: only some nucleons of both nuclei do participate in the interaction and therefore are called “participants”, while the others, not in geometrical overlap with each other, remain as “spectators” (ﬁg. 4). Central collision events (b ∼ 0) are the best candidate for searching QGP because grazing or peripheral collisions at large b do not provide the geometrical overlap of enough nucleons mandatory to achieve a high energy density in a large volume. The theoretical interpretation of the space-time evolution of a collision between two nuclei at very high energies is based on the assumption that, as soon as the nucleons penetrate through each other, quarks and gluons experience hard scatterings. The kinetic energy of the incoming nuclei is transferred from the longitudinal into the accessible transverse degrees of freedom. After the “formation time” (about 1 fm/c), quarks and gluons materialize out of the highly excited colour ﬁeld and thermal equilibrium is approached via reaction between

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Fig. 5. – Space-time diagram of the evolution of an ultra-relativistic nuclear collision as viewed in the nucleon-nucleon centre-of-mass system.

individual pairs leading to the creation of the so-called “ﬁreball”. At this point the system expands rapidly, mainly along the longitudinal direction, and cools down thus reaching the transition temperature Tc for the creation of QGP. In the subsequent mixed phase, hadronization starts in the “ﬁreball” that still expands, likely in an ordered motion (large outward ﬂow) through a hadron gas phase (ﬁg. 5) until the “freeze out” is achieved when interactions cease and particles freely leave the reaction region and eventually can be detected by the experimental instrumentation. Each phase has associated speciﬁc signals that can be observed by experiments . as it will be illustrated in the next section (subsect. 3 2). Ultrarelativistic nuclear collisions are conveniently described in the rapidity(4 ) variable: the baryons appear in the centre-of-mass frame, predominantly at the rapidities of the initial beams, while in the central rapidity region one expects the bulk of created particles. As already mentioned, according to theoretical predictions, QGP may occur at about 0.6 GeV/fm3 (which is about ﬁve times the nucleus density). (4 ) Rapidity is a Lorentz-invariant longitudinal “velocity” deﬁned by y=

“p ” 1 E + pL L , = tanh−1 ln 2 E − pL E

where E is the particle’s energy and pL is its longitudinal momentum parallel to the beam. In the case of transverse momentum pT = 0 and pL /E = β 1, y ≈ β (particle velocity). It is an additive quantity under Lorentz transformation along the beam axis (the shape of distributions in this variable is invariant under a Lorentz transformation): passing from the center-of-mass system (c.m.s.) to the laboratory system (l.s.) the rapidity distribution is the same, with the y-scale displaced by an amount equal to ycm .

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The attained energy density ε, deﬁned as the amount of the energy made available in the collision in the volume of the interaction region, is experimentally estimated from the following equation based on Bjorken’s model [7]: ε=

particle’s average energy × number of particles 1 dET = . 2/3 dy 2 interaction volume cτ0 πr0 Ap 1/3

Here τ0 is the formation time of the QGP state (typically ∼ 1 fm/c) and r0 Ap is the projectile’s nuclear radius. The transverse energy, ET , is the energy lost by the incident baryons that is redistributed among many particles emitted at polar angles θi . It is deﬁned as ET =

Ei sin ϑi ,

i

where Ei is the kinetic energy for baryons and the total energy for all other particles. The highest transverse energies correspond to the most violent central collisions where the conditions to create the QGP are more likely to develop. . 3 2. Experimental probes and QGP’s signatures. – The main diﬃculty in hunting the QGP arises from the fact that once formed, it blows away almost immediately as the high-density nuclear matter expands, cools down and hadronizes. The current experimental approach aims at identifying observables that either decouple at diﬀerent times from the expansion or are more sensitive to the early and hot stages of matter. Moreover, since theorists are not yet able to deﬁne any single deﬁnitive signature for the QGP, many experimental observables must be measured to test unambiguously the plasma formation. The application of this strategy in the experiments performed for several years at CERN and BNL provided the evidence that the onset of new collective phenomena has been achieved. Ideally QGP is similar to a black body and in principle it is expected to obey the equation of Stefan-Boltzmann law for a system of non-interacting particles (ε ∝ T 4 ) and hence, the entire hot volume of the interaction region will likely radiate a large number of photons and lepton pairs at low invariant mass. Oppositely, pions will be emitted mainly from the surface at lower temperature than photons and they will cease to interact when the density has fallen enough (“freeze-out” phase). In a few 10−23 s, most of the particles undergo signiﬁcant reinteractions between the time of their production and their ﬁnal detection. Consequently QGP’s transient existence must be inferred from what remains once everything interesting has already happened, thus making the “direct” identiﬁcation of the QGP quite a hard task. It is worthwhile noticing that the large number of particles per event produced at the colliders will allow the study of non-statistical ﬂuctuations in a number of observables on an event-by-event basis. Anomalous ﬂuctuations are in fact associated with critical phenomena in the vicinity of a phase transition [8].

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In the following subsections, the most signiﬁcative QGP’s signatures will be reviewed in the light of the great outcomes from SPS and RHIC runs. . 3 2.1. Global observables. The thermodynamic properties, the ratio S/A (entropy per baryon), the energy density, the temperature and the chemical potential are inferred by measuring the rapidity distributions and the transverse mass of hadrons and their relative production ratios. According to the Bjorken model, events in which QGP has occurred are likely characterized by the presence of a large amount of transverse energy and secondary particles and hence by a large amount of energy deposited during the collision. Experimental estimates from calorimetric study of bulk transverse energy production and from hadron yield measurement in full phase space indicate that the theoretical density has been reached at SPS [9, 10], in sulphur-induced reactions, and far surpassed at RHIC [11]. In fact, from the value measured by the experiment NA49, one calculates an energy density of the order of 2–3 GeV/fm3 in central collisions [9], well above the critical energy density evaluated via lattice-QCD simulations. Experimental data from RHIC have shown an initial energy density of about 15 GeV/fm3 corresponding to twenty times the expected transition energy density. A valuable tool for understanding hadronic reaction is the measurement of inclusive transverse-momentum distribution. This is certainly true even in the study of heavy-ion collisions where a dramatic increase of pT could be the signal of the phase transition from ordinary matter to QGP. A comparison of particle spectra at high pT allows to disentangle possible phase transition eﬀects from nuclear or hadronic medium eﬀects; in fact, large diﬀerences in the high-pT tails of spectra are predicted depending upon various medium eﬀects. Data are properly plotted against the transverse mass, mT = (p2T +m2 )1/2 , rather than pT , the slope of such distributions being proportional to the inverse of the temperature T in the truly thermal case(5 ). SPS [12] and RHIC data [13] showed temperatures in agreement with Monte Carlo lattice predictions as can be seen in ﬁg. 6. . 3 2.2. Electro-magnetic probes. Probes of the hot initial stage of the collision, where the most interesting phase of the nuclear matter has occurred, are based on particles with long mean free paths like direct photons that succeed to escape the plasma medium, while it is still hot, with a small probability of interacting in the outer freeze-out region since they do not undergo any strong interaction. The related momentum spectrum should reﬂect the initial temperature since direct photons are created by quarks thermalizing through collisions in the plasma. These electromagnetic probes are therefore acting as a “thermometer” that would keep the memory of the temperature in which the thermal emission was created and (5 )

1 dn mT dmT

∝ e−

mT T

.

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Fig. 6. – Transverse mass spectra for positive (a) and negative (b) hadrons in central Pb-Pb collisions as from NA44 data. Fits by collective radial ﬂow model are also shown: the inverse slope increases linearly with the particle mass as a result of the fast ﬁreball expansion.

hence, can serve as a diagnostic measure of the state of the system before it cooled and hadronized. Unfortunately the shape of photon spectra is given by the laws of black-body radiation and therefore does not provide any contents of the dynamics of the emitting source. Furthermore, there are also many other processes like the decay of π 0 and η mesons, which can produce photons, resulting in a huge amount of background. Because of the relevant production rates of photons from hadronic reactions in the hadron gas, experiments at SPS and RHIC were not able to provide unambiguous evidence of substantial thermal emission from the hot initial reaction volume, however this observable will become clearer at LHC as both the QGP’s temperature and lifetime will increase. . 3 2.3. Quarkonium state suppression. Long-lived bound states of heavy quarks (J/ψ, χ, ψ , Υ) mainly originate from gluon-gluon fusion, generating either a c-¯ c or a b-¯b pair that becomes a bound state at very early times in the collision, when the temperature is still above the charm or bottom production threshold. In this production mechanism, it is noteworthy that the quark pair stays separate until it attains the binding distance of the relative quarkonium state. Therefore, when the reaction takes place in a medium with a high density of colour sources (quarks and

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Fig. 7. – J/ψ yield, normalized to Drell-Yan pair production, as a function of initial energy density.

gluons), a suppression of such resonances is expected since the two charmed or beauty quarks are screened from each other by the dense colour medium and cannot bind, the size of the bound state being larger than the screening radius. By the time the density has decreased enough for them to bind, they meet other quarks in the dense medium, thus eventually forming two charmed mesons D. The analogy with the QED case is the Debye screening. In its ground state, hydrogen atom is an insulator: the electron is bound to the proton and both experience little screening from other nearby electric charges because the atomic spacing is large. By increasing the gas pressure and therefore the density, the attractive Coulomb ﬁeld between the proton and the electron is screened by the presence of other protons and electrons in the dense system. Consequently the Debye radius decreases and the electron becomes quasi-free. As soon as the Debye radius becomes comparable with the electron orbital radius, the electron will be no more able to recognize its own proton and the hydrogen atom will eventually transform into a metallic conducting object (Mott transition). Based on similar arguments, in 1986, Helmut Satz of Bielefeld University, together with Tetsuo Matsui [14], predicted that deconﬁnement would be signalled by the melting of heavy quarkonium states if the temperature of formation of the QGP is greater than a critical deconﬁnement temperature Tc , there should be a distance λD such that, for

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√ Fig. 8. – Suppression of J/ψ in lead-lead (Pb-Pb) collisions at the SPS ( SNN = 17 GeV) and √ in gold-gold (Au-Au) at RHIC ( SNN = 200 GeV) as a function of the reaction centrality (the number of participant nucleons).

distances greater than λD the strong (colour) forces that bind the c-¯ c or the b-¯b pair together become screened and therefore no bound state will be formed. Quite in line with this observation the predicted pattern of charmonium suppression, due to a transient medium made up of constituents, was measured at SPS by the experiment NA50 (ﬁg. 7) using a muon spectrometer [15]. At RHIC, this experimental observables, surprisingly, showed an amount of suppression almost identical to that found at the SPS (ﬁg. 8) although the medium created at RHIC is much denser and hotter than that at the SPS and most models predicted a stronger depletion. It seems that the increase in the temperature from the SPS to RHIC is not enough high to dissolve all the quarkonium states, therefore the J/ψ is then only indirectly suppressed due to the lack of feed-down contributions from the dissolved χc and ψ states. It has also been argued alternatively that direct J/ψ suppression is partially counterbalanced by the recombination of charm and anticharm quarks in the thermal bath (up to ten charm-anticharm pairs are produced in central gold-gold collisions at RHIC). √ A natural explanation of the similarity of the suppression at the two energies ( SNN = 17 and 200 GeV) is put forward by recent lattice gauge calculations on heavy quarkonium at ﬁnite temperature. They show that, in contrast to early calculations, the ground states (J/ψ, Υ) survive at least up to twice the critical QCD temperature, whereas excited states such as the ψ and χc melt around Tc . However, it must be noticed that in lattice QCD the J/ψ mesons are produced at rest, whereas in reality they will move at high velocities; from the viewpoint of the mesons, they will be in a “wind” of hot QGP.

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Fig. 9. – Debye length as a function of the ratio between the melting temperature and the transition temperature. Small-size quarkonia break up at lower temperatures.

In ﬁg. 9 it is shown how bound states with larger size (or equivalently less tight) ﬁrst disappear, ones with smaller size disappear at higher T . The Υ ground state melts at a temperature around two times the transition temperature. Such a high temperature will be likely reached at LHC. . 3 2.4. Low-mass vector mesons. The study of vector mesons represents an eﬀective probe of QGP formation since their properties such as mass, width, and branching ratios are expected to be sensitive to strong in-medium eﬀects and to changes in the quark masses, if chiral symmetry were partially restored. The identiﬁcation of vector mesons through their leptonic decay modes is diﬃcult because of the presence of lepton pairs, produced via electromagnetic interaction of quarks and antiquarks (Drell-Yan pairs), totally independent of whether the system attains a thermalized QGP that goes through a phase transition to hadron gas. However, in the 1–1.5 GeV/c2 invariant-mass region, Drell-Yan background is negligible, hence light mesons (ρ, ω and φ), decaying in lepton pairs with lifetimes of the order of the expansion time scale, are more easily identiﬁed. Drastic changes have been observed in electron-positron pair mass spectra thus reﬂecting the in-medium meson properties [16] (ﬁg. 10). These observations are being further pursued at RHIC. . 3 2.5. Particle production. The relative abundances of the various particle species created in the course of the AA collisions provide information on the nature of the medium from which they originate. According to the current model of the QGP formation in heavy-nucleus collisions, the phase transition is followed by a hadronization process fea-

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Fig. 10. – Background subtracted e+ e− -pair invariant-mass spectra, normalized to the charged particle density, as obtained by the experiment NA45 in central Pb-Au collisions. Estimated contribution from meson decay is shown by the solid line.

turing a chemical equilibrium of the hadronic constituents. In such a chemical equilibrium, the probability that a quark q hadronizes into a given particle specie h is governed by the Boltzmann law and, therefore, by the temperature T , the respective particle’s mass mh and the chemical potential μq according to the following relationship: ρ(h) mh 3/2 −(mh −μq )/T = e . ρ(q) T The chemical potential is deﬁned as the energy that must be fed to the system for increasing of one unit the number of quarks. It corresponds to the Fermi energy at T = 0 and is given by the diﬀerence between the number of quarks and that of anti-quarks of the system and it increases as the density ρ goes up: ρ = T /V × ∂ ln Z/∂μ (V and Z are the volume and the partition function of the system, respectively), ρ increases as the nucleons are compressed.

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In this respect, great interest raised the papers published by P. Koch, B. Muller and J. Rafelski [17-20] in the 80s. They predicted an enhancement of strange particles (and in particular of anti-hyperons) in comparison to the production observed in hadronic collisions due to the lower production energy threshold and the strong dependence of the thermal production on a high power of the strangeness density, whereas the nonequilibrium production in the hadron-gas phase is suppressed owing to the large mass. 4. – Strangeness enhancement Among particle species, strange particles freeze-out (chemically) at the early stage of system evolution and provide information on the high density stage of the collision, being the strangeness aﬀected only by the annihilation process s (¯ s) that goes into q (¯ q) which happens in reactions of this type: Λ + K + → n + π + . The concentration of s and s¯ quarks, before the hadronization phase of QGP, reaches a maximum for the following two main reasons: – once chiral restoration takes place, the strange-quark mass value from 500 MeV/c2 decreases to about 150 MeV/c2 , thus favouring the strangeness production since the temperature of the plasma will be signiﬁcantly higher than the strange-quark current mass; – the chemical potential for u and d quarks, which corresponds to the Fermi level, EF , at zero temperature, may have a value comparable to the lighter strange quark mass. Hence, again strange quarks pairs, produced in the hot stage of the reaction, are favoured in the baryon-rich environment of an ion collision since the Pauli exclusion principle pushes the creation of u-¯ u and d-d¯ pairs, via gluon-gluon fusion, to higher energy. EF > 2ms , making easier to produce s-¯ s pairs than other q-¯ q pairs. As the temperature drops during the plasma expansion, most of the abundantly produced strange quarks and anti-quarks will not be able to encounter and annihilate into lighter quarks because they are diluted in a non-strange medium. However, given the weak concentration of strange particles in comparison to the high number of not-strange particles, the rate of this annihilation is low. Therefore, each member of the pair, remaining frozen out of equilibrium, interacts with quarks of diﬀerent ﬂavour and eventually appears as a strange hadron in the ﬁnal state. The chemical equilibrium is expected to proceed much faster in the QGP rather than by rescattering in a hadron gas as a result of the equilibration of strange and light quark ﬂavours. In fact, light hyperons, like the Λ (uds), with a mass of 1115 MeV, are produced copiously in a hadronic medium as soon as the formation threshold is reached. On the contrary, heavier hyperons: Ξ (qss) with a mass of 1315 MeV and Ω (sss) with a mass of 1672 MeV and their respective anti-particles are not favoured. The direct production of Ω requires a very high energy (threshold of the order of 3 GeV), whereas for its indirect production in a hadron gas, a sequence of three reactions (π+N → K +Λ; π+Λ → Ξ+K;

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π + Ξ → Ω + K) is necessary for a total average time of about 100 fm/c, which is much longer than the freeze-out times. Consequently the hyperon concentration has no time to reach the equilibrium saturation, while, in dense nuclear matter, the chemical equilibration times of strange quarks, through gluon fusion (much more probable given the gluon degeneration in eight colours and because they are massless) and light quark annihilation mechanisms, will be very fast, of the order of 5–6 fm/c, which corresponds to the lifetime of QGP. Still more unlikely is the production of anti-hyperons which, being made by quarks not present in the incoming beam particles, requires interactions involving anti-nucleons. This trend is larger as anti-hyperon mass increases because of the very high production threshold. This diﬀerent behaviour of multi-strange anti-baryons in QGP vs. hadron gas is therefore a strong probe. . 4 1. Strangeness studies at SPS . – The following experiments at the SPS included studies on the strange-particle production in their programmes: NA35, NA36, WA85, WA94, WA97 and NA49 with S and Pb beams at 200 AGeV and 160 AGeV, respectively. However, the most relevant results were achieved by NA57 as it will be illustrated in the following. NA57 set-up was designed to reconstruct the charged tracks emerging from strange particle decays in a telescope entirely made of hybrid silicon pixel detector planes of 5 × 5 cm2 cross-section, for a total of about 106 channels. The whole apparatus was placed inside the 1.4 T magnetic ﬁeld perpendicular to the beam line of the GOLIATH magnet in the CERN North experimental Area. The bulk of the detectors was closely packed in an approximately 30 cm long compact part used for pattern recognition. The telescope was placed above the beam line (ﬁg. 11) with an inclination angle and distance of the ﬁrst plane from the target such to accept particles produced in about a unit of rapidity around mid-rapidity, with transverse momentum above a few hundred MeV/c. In such conditions the track densities reached about 10 cm−2 for central Pb-Pb collisions at 158 GeV/c. To improve the momentum resolution of high-momentum tracks a lever arm detector (an array of four double-sided silicon micro-strip detectors) was placed downstream of the tracking telescope. An array of six scintillation counters, named Petals, located 10 cm downstream of the target, provided a fast signal to trigger on central collisions. The Petals covered the pseudorapidity region 1 < η < 2 and their thresholds were set so as to accept events with track multiplicities above an adjustable limit. This was tuned so that the triggered event sample corresponds to approximately the most central 60% of the Pb-Pb inelastic crosssection. The thickness of the Pb target was of 1% interaction length, which made the trigger rate and the data acquisition capability properly matched. The centrality of the collisions was determined on-line by analysing the charged particle multiplicity measured by two stations of Microstrip Silicon Detectors (MSD) which sampled the pseudo-rapidity intervals 1.9 < η < 3 and 2.4 < η < 3.6. Strange baryons decay into stable particles (p, π and K) via the weak interaction (s quark goes into a lighter quark), therefore their lifetime is enough long (about 10−10 s) to

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Fig. 11. – Experimental layout of the NA57 experiment.

cover a ﬂight distance of some cm before decaying, thus enabling researchers to distinguish the secondary vertexes from the primary interaction vertex searching for trajectories not pointing to the main vertex of the collision. In NA57, the hyperons were searched for by combining any pair of oppositely charged tracks in an event to ﬁnd what is usually called a V0 topology: the two tracks were required to cross in space within a predetermined tolerance (distance of closest approach smaller than 0.3 mm in the Pb-Pb sample, 1 mm in pBe and pPb), in a ﬁducial region deﬁned to be between the target and the telescope. The cascade candidates were selected by combining a V0 (required not to point to the vertex) with a charged track of the proper sign and requiring a set of conditions similar to those employed for the V0s (e.g., on the distance of closest approach between the V0 and the charged track, on the ﬁducial region, on the impact parameter at the vertex). In addition, the V0 decay vertex was required to be located downstream of the cascade decay vertex and the candidates were required not to be kinematically ambiguous with the decay of a Λ. The trend in the strangeness production in pp, pA and AA collisions was studied as a function of energy and system size. Data were interpreted by making use of statistical models that describe particle production in terms of a few parameters such as temperature and entropy. Figure 12 shows as an example a sketch of the reconstruction of the cascade decay for an Ω and ﬁg. 13 shows the mass plots of hyperons detected by NA57. The NA57 experimental results concerning the strangeness and strange antibaryon enhancement agreed with the theoretical expectations that the QGP state was formed.

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Fig. 12. – Reconstruction of the V0 decay vertex in the NA57 experiment.

Fig. 13. – Mass plots of strange particles as reconstructed by the NA57 experiment.

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Fig. 14. – Strange particle production in Pb-Pb, p-Pb and p-Be as measured at the NA57 experiment at SPS. Hyperon and anti-hyperon yields as a function of participants nucleons, normalized to the corresponding yield in p-Be collisions.

In February 2000, this intriguing experimental result obtained at the SPS over several years was folded into a public announcement stating that the formation of a new phase of matter was their best explanation. SPS data showed that the total strangeness content of the ﬁnal hadronic phase is consistent with the input into hadronization expected from a ﬂavour coalescence mechanism typical of a non-QCD perturbative quark-hadron transition (ﬁg. 14) [21]. A further ﬁngerprint [22] of the hadronization process has been found by plotting the overall set of particle over its relative antiparticle ratios for various species from pions to Ω hyperons (ﬁg. 15), revealing a remarkable order in the hadronic population and temperature without the rather prominent ﬂuctuations occurring in hadronic collisions. Such a regularity among all the produced hadrons showing an apparent thermal equilibrium of all species can be understood as a consequence of the transition from QGP to the hadron gas phase. The bulk of the observed hadrons with low transverse momenta (pT < 2 GeV/c) are produced from matter that seems to be well equilibrated by the time it dresses up into hadrons. In other words, statistical hadronization models reproduce hadron yields and ratios well, and in terms of only a few ﬁtted parameters, such as temperature and chemical potentials. The phase boundary appears to be located at about 170 MeV, corresponding to an energy density of about 0.6 GeV/fm3 in agreement with the lattice QCD prediction.

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Fig. 15. – Compilation of particle ratios as measured at SPS (Pb-Pb central collisions). A comparison has been made with a model of a hadron resonance gas in chemical equilibrium at T = 168 MeV and baryochemical potential of 266 MeV.

. 4 2. Strangeness studies at RHIC . – RHIC data on copper-copper collisions at 200 GeV ¯ and Ξ, ¯ with respect to proton-proton collisions showed enhancements of the Λ, Ξ, Λ (ﬁg. 16), similar to those found (at a given number of participant nucleons) in gold-gold collisions at the same energy and in lead-lead collisions at the energies of CERN’s SPS. Moreover, it was conﬁrmed with higher accuracy and better systematics that hadron

Fig. 16. – Comparison between results at NA57 and RHIC. Within error, the strangeness enhancement at RHIC and SPS appears the same.

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abundances ratios are characterized by a chemical equilibrium distribution with “chemical freeze-out” temperature of 160–170 MeV [23]. Therefore, this value seems to be independent of the collision system and centrality and agrees with the value for the quark-hadron phase transition temperature predicted by lattice-QCD. One of the most striking early results at RHIC was the discovery that hadrons carrying a high transverse momentum are suppressed in hot QCD matter, whereas high-pT photons remain essentially unaﬀected, leading to a picture of a dense medium opaque to partonic, coloured projectiles but relatively transparent to photons. These studies provide a direct measurement of the parton number density and transport properties of the system that is produced. Vigorous theoretical and experimental eﬀorts are under way to understand the observed parton energy loss in terms of perturbative QCD (pQCD). Various groups have described the suppression of light hadrons in terms of radiative energy loss by gluon bremsstrahlung. Comparisons between nuclear collisions and scaled nucleon-nucleon collisions are expressed by the nuclear modiﬁcation factor RAA (pT ): RAA (pT ) =

1 Ncoll C

×

C /dpT dy d2 NAA . 2 d Npp /dpT dy

Similar eﬀects are observed when peripheral nuclear collisions are compared to central collisions, expressed by the central-to-peripheral nuclear modiﬁcation factor RCP (pT ): RCP (pT ) =

Ncoll P d2 N C /dpT dy . × 2 AA P /dp dy Ncoll C d NAA T

At SPS energy, NA57 observed a diﬀerence between mesons (K 0 ) and baryons (Λ) similar to that observed at RHIC energies. For low-pT values, the values are close to participant ¯ corresponds to the lower value of scaling and the marked diﬀerence between Λ and Λ, ¯ the Λ/Λ yield ratio reported earlier. Comparisons with STAR results from RHIC and WA98/NA57 results from SPS are shown in ﬁg. 17 [24]. The main conclusion that may be drawn is that the bulk of particles formed in heavyion collisions shows a hydro-like behaviour with a high level of thermalization and collectivity. A deeper understanding of the current QGP picture will be reached at LHC as it will be described in the next section. 5. – Features of QGP at RHIC energies and prospect at LHC Hard probes (heavy ﬂavours, direct photons and photon tagged jets, jets and di-jets) have played a central role at RHIC by providing valuable information and insights on the early and thermalized stages of the collision phase since their production occurs on so small temporal and spatial scales to entail that any modiﬁcation of their known properties is caused by the medium they have subsequently traversed. The medium-induced modiﬁcations to the propagation of high-pT partons generated in hard scatterings at the initial stages of the nucleus-nucleus collisions is one of the most

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Fig. 17. – Nuclear modiﬁcation factor showing a diﬀerent suppression behaviour between strange and non-strange particles.

dramatic pieces of evidence for the strong-coupling nature of the quark-gluon matter produced at RHIC. Partons lose part of their energy so strongly that they are stopped within much less than a nuclear diameter, “quenching” the jet of hadrons that would normally materialize from the liberated quark or gluon. This results in a softening of transverse-momentum particle spectra and, as a by-product, the coplanarity of the two jets from the quarks or gluon scattering is also destroyed entailing an azimuthal asymmetry. Analysing the results from two- and three-particle correlation studies one ﬁnds that this energy reappears as softer particles far from the initial parton direction both in azimuth and rapidity space [25]. A peak in particle production appears on a cone at an angle of about one radian from the direction of the propagating parton (ﬁg. 18). A possible explanation would be the generation of a shock wave in the medium akin to Machian waves or Cherenkov radiation patterns, indicative of very fast particles moving through a medium faster than sound or light. The second important indication of the formation of thermalized collective QCD matter at RHIC is the observation of robust collective hydrodynamic ﬂow ﬁelds in the form of large anisotropies in the ﬁnal particle yields with respect to the reaction plane. In fact, if the medium expands collectively, the pressure gradients present in non-central collisions—with an initial lens-shaped overlap area—result in momentum anisotropies in the ﬁnal state. Fluid-dynamics calculations indicate that such gradients must develop very early in the collision, during the high-density partonic phase, in order to reproduce the strong “elliptic ﬂow” seen in the data (ﬁg. 19) [26]. The resulting enormous collective motion of the medium is in agreement with near-zero viscosity hydrodynamic behaviour (often characterized as “perfect ﬂuid” behaviour), pointing to rapid thermalization and strong coupling of the matter.

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Fig. 18. – Azimuthal angle correlations with respect to a semi-hard trigger hadron at Δφ = 0 in central gold-gold (Au-Au) collisions. The strong suppression of the yields of leading-hadrons indicates that fast quarks and gluons lose a sizeable amount of energy when traversing dense matter.

. 5 1. LHC . – As previously shown, the wealth of data from experiments at RHIC, together with that from the ﬁxed-target programme at CERN indicate that a very rapid thermalization occurs in the collisions, after which a strongly interacting medium characterized by a large transport coeﬃcients, an energy densities as high as 30 GeV/fm3 and negligibly small viscosity, a perfect liquid, seems to be produced. However, the detailed study of the transport properties of this medium and the potential observation of the anticipated weakly interacting quark-gluon plasma will require key measurements at the Large Hadron Collider (LHC), operated with Pb on Pb at a center-of-mass energy of 2.75 TeV/nucleon with a luminosity of 1027 cm−2 s−1 . At LHC, it will also be possible to accelerate lighter nuclei (at higher luminosity) and asymmetric systems like pA interactions [27]. This forthcoming facility, scheduled to start operation in 2008, will be the ultimate facility for searching QGP enabling to explore regions of energy and particle density which are signiﬁcantly beyond those achieved at SPS and RHIC (ﬁg. 20). It will also allow to study new signatures arising from very high momentum transfer processes whose cross-sections, at the LHC energies, are large enough to comfortably enable detailed experimental studies. The anticipated very high initial temperatures will result in a stretched hadronization time that will enable scientists to make direct observations of the plasma and to

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Fig. 19. – Elliptic ﬂow coeﬃcient for particles compared to hydrodynamical calculations.

Fig. 20. – Initial energy density and temperature as expected at LHC. Estimates made at BNLAGS, CERN-SPS and BNL-RHIC are also shown.

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carry out a more quantitative characterization of the phase transition evolution. Therefore the main advantage oﬀered by the LHC will be the opportunity to investigate the non-perturbative QGP physics by means of perturbative probes whose predictions are supported by solid theoretical calculations, being a ﬁrm ground for testing experimentally well-established theoretical models, since quark and gluon scattering cross-sections at large transferred momentum can be calculated in perturbative QCD. The early evolution of Pb-Pb collisions at the LHC energy of 5.5 TeV in the nucleonpair center-of-mass, about thirty times higher than at RHIC, will mostly be determined by a strong nuclear shadowing. As anticipated by classical QCD, in the regime of low √ Bjorken-x (∼ xT = 2pT / s = 10−4 –10−3 ), made accessible because of the large energy attainable at the LHC, one of the two colliding lead nuclei will resolve the other incoming nucleus as a superposition of a large number of soft gluons which, being so densely packed, will eventually overlap and merge. As a consequence of the large number of energy and momentum exchanges, the thermal equilibration of the medium will be quickly achieved, on a time scale of 0.1 fm/c, leading to a very high initial energy density and temperature. The energy density, of the order of 10 GeV/fm3 , will be about 50% higher than that reached at RHIC while the initial temperature will be greater by more than a factor of two. This high initial temperature will extend the lifetime and the volume of the deconﬁned medium which will subsequently expand while cooling down to the freeze-out temperature. Compared to RHIC, the ratio of the lifetime of the quark-gluon plasma state to the time for thermalization is expected to be larger by an order of magnitude. As a result, the “ﬁreballs” created in heavy-ion collisions at the LHC will spend nearly their entire lifetime in a purely partonic state, widening the time window available to experimentally probe the quark-gluon plasma state. In conclusions, the LHC will create a hotter, larger and longer-living QGP state than the present heavy-ion facilities, approaching the most favourable conditions for providing a consistent and uncontroversial experimental evidence of the phase transition. Moreover, since the baryon content of the system after the collision is expected to be concentrated rather near the rapidity of the two colliding nuclei, mid-rapidity nuclear matter at the LHC will be much more baryon-free than at RHIC and closer to the conditions simulated in lattice QCD for a vanishing baryochemical potential and such to reproduce conditions similar to those present in the early universe, allowing a more direct comparison with theory predictions. The experiments will not probably see a real phase transition between the hadronic and quark-gluon descriptions; it is more likely to be a crossover that may not have a distinctive experimental signature at high energies. However, it may well be possible to see quark-gluon matter in its weakly interacting high-temperature phase. The other parameters relevant to the formation of QGP will also be favourable (table I): the size and lifetime of the system will improve by a large factor compared to Au-Au collisions at RHIC thus allowing to probe QGP in its asymptotically StefanBoltzmann free “ideal gas” form. In particular, the lifetime is expected to be much longer than the relaxation times (which decrease in a denser system) thus allowing the system to reach and maintain thermal equilibrium throughout the expansion.

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Table I. – Global features of central ultra-relativistic nuclear collisions as anticipated at RHIC 1/2 and LHC. SPS values are also shown for comparison. (sNN = centre-of-mass energy in the nucleon-nucleon system; dNch /dy = charged multiplicity per unit of rapidity; Vf = ﬁreball volume; ε = energy density; τQGP = QGP lifetime).

LHC will make it possible to carry out a large number of measurements, including much-improved determinations of leptonic and hadronic signals, and precise studies of even heavier quarkonium bound states like the Υ, Υ , Υ especially at LHC where the initial temperature will likely be larger than the melting point of this resonance family . (subsubsect. 3 2.3). At the LHC energies, high-pT particle production is dominated by scattering and fragmentation of gluons that slow down faster (about 250 MeV/fm), in the hot interaction region, owing to a colour charge larger than that of quarks. The energy loss mechanism will eventually depend sensitively on the Debye screening scale of the medium and will also reduce the background from high-pT π 0 ’s, making the observation of thermal radiation from QGP easier. . 5 2. ALICE . – Currently, more than 1000 physicists from 90 institutes in 30 countries are involved in building ALICE, the only experiment at LHC speciﬁcally devoted to investigate heavy ion collisions [28]. The uneasy role of being alone for this kind of physics (apart for the CMS and ATLAS experiments which have some very limited capabilities for nuclear collision studies) conditioned its design. In fact, ALICE is meant as a general-purpose experiment with a single set-up whose design has been optimized to study the full spectrum of observables (hadrons, photons and leptons) together with a global survey of the events over a large phase space.

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Designed to cover a wide range of momenta (from 100 MeV/c up to 100 GeV/c) and identiﬁed particles (from the electron to the Υ resonances), ALICE will be able to cope with the highest particle density anticipated for Pb-Pb collisions at the LHC. Its apparatus has also been optimized to study interactions between nucleon-nucleon, at lower luminosity than the other LHC experiments, and proton-nucleus, to be used as reference data for the Pb-Pb collisions. Collisions between lighter nuclei than lead are also envisaged to investigate the inﬂuence of the energy density and volume on the phase transition properties. In contrast to p-p physics, which requires high luminosity to search for rare events produced with cross-sections of the order of the pb, the measurement of observables related to the deconﬁned plasma of quarks and gluons and its time evolution can be performed in nuclear collisions at relatively low luminosity. The LHC design luminosity for Pb beams is L = 1027 cm−2 s−1 corresponding to an interaction rate of about 104 Hz of which only a small fraction, approximately 5%, accounts for the most interesting central collisions with maximum particle production. Since ﬂuences and doses roughly scale with luminosity, it follows that requirements on detector electronics and radiation damage issues are much less relevant than those in pp experiments thus allowing to exploit, in the ALICE layout, the outstanding tracking capability of slow devices like the Time Projection Chamber (TPC) and the silicon drift detector. On the other hand, nuclear cross-sections ranging between 10 mb and the barn entail the production of a large amount of particles per Pb-Pb interaction. The high multiplicities anticipated at LHC combined with the large rapidity range to be covered have represented the main technical challenges in designing the ALICE apparatus. The average chargedparticle multiplicity per unit rapidity plays an important role not only in assessing the detector performance but also physics-wise, because it largely determines the accuracy with which many observables can be measured, especially in the event-by-event studies of ﬂuctuations. Initial theoretical estimates for the rapidity density of produced charged hadrons for central Pb + Pb collisions ranged from ∼ 2000 to 8000 per unit of rapidity at mid-rapidity. The high multiplicities in central nucleus-nucleus collisions typically arise from the large number of independent and successive nucleon-nucleon collisions, occurring when many nucleons interact several times on their path through the oncoming nucleus. A scaling function, based on the parton saturation and classical QCD, has recently been derived to predict the charged multiplicity as a function of beam energy. It has been successfully used for evaluating the multiplicity at full RHIC energy (200 GeV) based on the 130 GeV data [29]. Extrapolation of this model to LHC energies yields a multiplicity density at the lower end of the predicted range (2000 or less). However, since the diﬀerence in energy between SPS to RHIC is smaller than that between RHIC to √ LHC, extrapolation to the center-of-mass energy per nucleon pair of sNN = 5.5 TeV is quite inaccurate and evenly ranges between 1500 and 3000 charged particles per unit of rapidity. In view of this uncertainty, the ALICE detector granularity has been optimized to provide the best performance at a charged particle rapidity density of 4000, assuring a large safety margin, and to maintain the occupancy at the highest predicted multiplicity at such a level to still allow the event analysis.

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Fig. 21. – Axonometric view of the ALICE layout.

An irrenounceable feature of heavy-ion experiments is the capability to identify the charged hadrons in the full range of momentum because it allows to investigate many new event-by-event observables and plays an important role in the study of the properties of the quark-gluon plasma during its dynamical evolution toward the freeze-out phase when the re-hadronization occurs. At LHC energies, more than 97% of the particles will be produced with a pT < 2 GeV/c, the bulk of the remaining 3% will be in the range 2–5 GeV/c although a not negligible fraction (∼ 0.02%) will have a pT > 5 GeV/c. Therefore ALICE will employ more than one method to perform an eﬃcient and unambiguous particle identiﬁcation (PID) and envisages two dedicated and complementary systems for di-electrons (a Transition Radiation Detector, TDR) and di-muons (a forward muon arm spectrometer). Other design considerations are dictated by the need for special physics triggers to select the events of interest for storage. The trigger capabilities become indispensable for the most interesting rare probes in view of the large data volume and slow readout of the central TPC detector (about 25 Hz for central Pb+Pb collisions). . 5 2.1. The ALICE layout. The ALICE experiment is ern, 45 m below ground level, previously used by the Intersection Point (IP) 2 of the LHC machine. From the artist view of the layout, shown in ﬁg. 21, ALICE consists of two main components: a 2π barrel

housed in the underground cavLEP L3 experiment, located at one immediately recognizes that detector system at mid-rapidity

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(polar angles from 45◦ to 135◦ ) embedded within the 0.2–0.5 T uniform solenoidal ﬁeld provided by the large magnet formerly used in LEP’s L3 experiment and a forward muon spectrometer (from 2◦ to 9◦ polar angles). The following detectors, starting from the beam axis, are sheathed in the L3 magnet: – An Inner Tracking System (ITS) [32] with six cylindrical layers of highly accurate position-sensitive silicon detectors designed to track charged particles emerging from the main interaction vertex and to identify decay products of short-lived secondary particles having strange and charm quark content. – A cylindrical, large volume, Time Projection Chamber (TPC) [33] envisaged to determine the charged-particle trajectories curving in the magnetic ﬁeld, allowing particle momentum and charge to be measured. – Two highly segmented particle identiﬁcation barrel arrays for hadrons (by means of Time Of Flight, TOF) [34] and electrons (by means of a Transition Radiation Detector, TRD) [35]. – A single arm array of seven Ring Imaging CHerenkov detectors (RICH) optimized to perform an inclusive identiﬁcation of high momentum charged hadrons (High Momentum Particle Identiﬁcation Detector, HMPID) [30]. – A single-arm high-resolution electromagnetic calorimeter (PHOton Spectrometer, PHOS) [31], consisting of 17 k lead tungstate crystals (PbWO4 ) readout by silicon photodiodes designed to search for direct photons and to measure π 0 and η spectra at high momenta. It is located in the bottom part of the central barrel region (−0.12 < η < 0.12 and 40 < φ < 140), at 4.6 m from the vertex and covers 8 m2 . A charged-particle veto detector will be placed in front of the PHOS with the aim to identify charged particles reaching the calorimeter. Several US groups have expressed interest to join ALICE by submitting a ﬁnancial request to DOE, still pending, for building a large (13248 channels) acceptance, moderate resolution electromagnetic calorimeter (EMCal). EMCal will extend the measured momentum range for photons and electrons to provide a measurement of jet energy and jet fragmentation functions. The central detectors are supported inside the L3 solenoid by a cylindrical, metallic structure, 7 m long and with a diameter of 8.5 m, called space-frame. The forward muon spectrometer [37], featuring measurements of quarkonia states with a Υ mass resolution of about 100 MeV/c2 , consists of a complex arrangement of absorbers (reaching totally about 18λint ), a large dipole magnet (3 Tm integral ﬁeld), ten stations of thin Multi-Wire Proportional Chambers (MWPC) equipped with highly segmented cathode planes for tracking and four Resistive Plate Chambers (RPCs) for muon identiﬁcation and triggering. Four small and very dense zero-degree calorimeters [38], made of tungsten and lead with embedded quartz ﬁbres read out by photomultipliers, are located about 100 m downstream the machine tunnels on both sides of the interaction region to measure and trigger

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Fig. 22. – Momentum resolution as a function of the transverse momentum.

upon the impact parameter. A preshower detector called PMD (Photon Multiplicity Detector) [39] has been designed to measure photon multiplicity in the forward region. It consists of a lead converter sandwiched between two planes of high-granularity gas proportional counters. Additional detectors [40], designed to provide fast trigger signals and event multiplicity at large rapidity and an array of scintillators (ACCORDE) on top of the L3 magnet, envisaged to trigger on cosmic rays, complete the ALICE set-up. The experiment was approved in February 1997. The ﬁnal designs of the diﬀerent detector systems have been comprehensively described in the Technical Design Reports (TDR) [30-41], since mid 1998. Installation is well advanced for all of the detector systems and the service structures, and the pre-commissioning work is in progress. . 5 2.2. Performance of the main sub-detector systems for the strangeness studies. The ALICE layout features a high eﬃciency and precision in the identiﬁcation of strange particles. By combining the information from the tracking systems and the PID detectors, the secondary vertices and the identiﬁed decay products are used to detect charged and neutral kaons and hyperons with momenta beyond 10 GeV/c. The tracking capability of the TRD, in junction with ITS and TPC, will valuably improve the predicted momentum resolution for charged particles as shown in ﬁg. 22. The results indicate that the ALICE tracking system will reach a good momentum resolution up to ∼ 100 GeV/c, even at the highest multiplicities, to enable detailed jet fragmentation studies [42]. The momentum intervals covered by the various particle identiﬁcation devices of ALICE are shown in ﬁg. 23.

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Fig. 23. – Momentum intervals in which the various ALICE PID detectors allow to achieve a 3σ separation level.

In the momentum regions where more PID detectors are simultaneously able to provide a response, a Bayesian approach to combine the various signals is currently under study [43]. Results obtained so far are quite promising as shown in ﬁg. 24. In the pT interval from 1 to 6 GeV/c, ALICE envisages to reconstruct 13 Λ/event, 0.1 Ξ/event and 0.01 Ω/event. Mass plot and pT distribution of simulated Λ, as reconstructed by ALICE, are shown in ﬁg. 25.

Fig. 24. – Eﬃciency and contamination for kaon identiﬁcation as obtained by combining together three PID detectors.

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Fig. 25. – Λ invariant-mass spectrum and pT distribution corresponding to the reconstruction of 300 simulated events Pb-Pb at LHC.

6. – Conclusions Collisions between heavy nuclei represent a powerful tool to investigate the behaviour of nuclear matter at densities higher than the ground state density, where, according to the lattice-QCD calculations, a primordial non-hadronic bulk phase of strongly interacting matter, named QGP, will likely occur. This phase of nuclear matter, being an ideal testing ground for fundamental concepts of QCD, requires an experimental insight. For this reason, a large community of physicists is engaged on such a very promising research located at the interface between particle and nuclear physics. The Relativistic Heavy Ion Collider (RHIC) at BNL has conﬁrmed the rich harvest of data from the ﬁrst running periods at CERN-SPS and new ﬁndings on high transverse momentum (pT ) physics have opened up a new avenue of enquiry. From the ideal hydrodynamics description of collective elliptic ﬂow and the large energy loss suﬀered by energetic quarks and gluons, the emerging picture in ultra-relativistic heavy-ion collisions at RHIC energies is leading to the formation of a short-lived state of matter whose characteristics are incompatible with the presence of hadrons. The observed anisotropies (in ﬂow) for diﬀerent particle species indicate that the produced medium instead of being regarded as a weakly interacting gas of quasi-particles is now viewed as a near-perfect liquid. The above interpretative diﬃculties met in drawing conclusions on the character of the strong interaction from the analysis of RHIC data will likely be overcome at the LHC in view of the unprecedented energies at which nuclei will be accelerated. ∗ ∗ ∗ I am very grateful to Prof. T. Bressani, for having invited me to give these lectures on nuclear matter at extreme conditions of density and temperature. I hope that I succeeded to draw some of the students to study in depth such a challenging but fascinating subject.

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REFERENCES [1] Blum T. et al., Phys. Rev. D, 51 (1995) 5153. [2] Hwa R. C., Quark-Gluon Plasma, Vol. 1&2 (World Scientiﬁc, Singapore) 1990, 1995. [3] Letessier J. and Rafelski J., Hadrons and Quark-Gluon Plasma, Cambridge Monographs, Vol. 18 (Cambridge University Press) 2005. [4] Shuryak E. V., The QCD Vacuum, Hadrons and Superdense Matter, World Scientiﬁc Notes in Physics, Vol. 71 (World Scientiﬁc) 2004. [5] Alford M., Rajagopal K. and Wilczeck F., Nucl. Phys. B, 537 (1998) 443. [6] Baym G., Nucl. Phys. A, 590 (1995) 233c. [7] Bjorken J. D., Phys. Rev. D, 27 (1983) 140. [8] Landau L. D. and Lifshitz E. M., Statistical Physics (Addison-Wesley) 1968. [9] NA49 Collaboration (Alber T. et al.), Phys. Rev. Lett., 75 (1995) 3814. [10] WA98 Collaboration (Aggarwal M. et al.), Nucl. Phys. A, 610 (1996) 200c. [11] Adcox K. et al., Phys. Rev. Lett., 87 (2001) 052301. [12] NA44 Collaboration (Murray M. et al.), Phys. Rev. Lett., 78 (1997) 2080. [13] Adler S. S. et al., Phys. Rev. C, 69 (2004) 034909. [14] Matsui T. and Satz H., Phys. Lett. B, 178 (1986) 416. [15] NA50 Collaboration (Abreu M. C. et al.), Phys. Lett. B, 477 (2000) 28. [16] CERES Collaboration (Lenkeit B. et al.), Nucl. Phys. A, 661 (1999) 23c. [17] Rafelski J. and Muller B., Phys. Rev. Lett., 48 (1982) 1066. [18] Rafelski J. and Muller B., Phys. Rev. Lett., 56 (1986) 2334. [19] Koch P., Muller B. and Rafelski J., Phys. Rep., 142 (1986) 167. [20] Rafelski J., Phys. Lett. B, 262 (1991) 333. [21] WA97 Collaboration (Lietava R. et al.), J. Phys. G, 25 (1999) 181. [22] Braun-Munzinger P., Heppe I. and Stachel J., Phys. Lett. B, 465 (1999) 15. [23] Ullrich T. S., Nucl. Phys. A, 715 (2003) 399c. [24] STAR Collaboration (Adler C. et al.), Phys. Rev. Lett., 89 (2002) 202301. [25] Salgado C. A. and Wiedemann U. A., Phys. Rev. Lett., 93 (2004) 042301. [26] PHENIX Collaboration (Adcox K. et al.), Phys. Rev. Lett., 89 (2002) 212301. [27] http://lhc.web.cern.ch/lhc/. [28] ALICE Technical Proposal, CERN/LHCC 95-71. [29] Karsch F., Laermann E. and Peikert A., Nucl. Phys. B, 605 (2001) 579. [30] ALICE Collaboration, CERN/LHCC 98-19, ALICE TDR 1 (1998). [31] ALICE Collaboration, CERN/LHCC 99-4, ALICE TDR 2 (1999). [32] ALICE Collaboration, CERN/LHCC 99-5, ALICE TDR 3 (1999). [33] ALICE Collaboration, CERN/LHCC 99-12, ALICE TDR 4 (1999). [34] ALICE Collaboration, CERN/LHCC 99-22, ALICE TDR 5 (1999). [35] ALICE Collaboration, CERN/LHCC 99-32, ALICE TDR 6 (1999). [36] ALICE Collaboration, CERN/LHCC 2000-001, ALICE TDR 7 (2000). [37] ALICE Collaboration, CERN/LHCC 2000-012, ALICE TDR 8 and CERN/LHCC 2002-016, Addendum to ALICE TDR 8 (2000). [38] ALICE Collaboration, CERN/LHCC 2001-21, ALICE TDR 9 (2001). [39] ALICE Collaboration, CERN/LHCC 2003-062, ALICE TDR 10 (2004). [40] ALICE Collaboration, CERN/LHCC 2004-025, ALICE TDR 11 (2004). [41] ALICE Collaboration, CERN/LHCC 2005-018, ALICE TDR 12 (2005). [42] ALICE Collaboration, J. Phys. G: Nucl. Part. Phys., 30 (2004) 1517. [43] ALICE Collaboration, J. Phys. G: Nucl. Part. Phys., 32 (2006) 1295.

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Strangeness in relativistic astrophysics J. Schaffner-Bielich Institut f¨ ur Theoretische Physik/Astrophysik, J. W. Goethe Universit¨ at Max-von-Laue Str. 1, D-60438 Frankfurt/Main, Germany

S. Schramm Center for Scientiﬁc Computing and Institut f¨ ur Theoretische Physik/Astrophysik J. W. Goethe Universit¨ at - Max-von-Laue Str. 1, D-60438 Frankfurt/Main, Germany

¨ cker H. Sto Gesellschaft f¨ ur Schwerionenforschung - D-64291 Darmstadt, Germany Frankfurt Institute for Advanced Studies (FIAS), J. W. Goethe Universit¨ at Ruth-Moufang Str. 1, D-60438 Frankfurt/Main, Germany Institut f¨ ur Theoretische Physik/Astrophysik, J. W. Goethe Universit¨ at Max-von-Laue Str. 1, D-60438 Frankfurt/Main, Germany

Summary. — In these lecture notes, the role of strangeness in relativistic astrophysics of compact stars is addressed. The appearance of strange particles, as hyperons, kaons, and strange quarks, in the core of compact stars is examined and common features as well as diﬀerences are presented. Impacts on the global properties of compact stars and signals of the presence of exotic matter are outlined for the various strange phases which can appear in the interior at high densities.

c Societ` a Italiana di Fisica

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¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

1. – Introduction Neutron stars are the ﬁnal endpoint in the evolution of stars more massive than eight solar masses. They are born in a spectacular explosive event, a core-collapse supernova, which can outshine an entire galaxy in its brightness. A surprisingly good estimate about the characteristic masses and radii can be made by adopting the argument of Landau [1]. The delicate balance between gravitational energy and Fermi energy can only be achieved up to a certain critical number of nucleons (fermions) which is N ≈ (MP /mN )3 ≈ 1057 , where MP stands for the Planck mass and mN for the nucleon mass. The corresponding maximum mass for neutron stars amounts to Mmax ≈ Mp3 /m2N ≈ 1.8M . This mass estimate is exactly in the region of maximum masses presently discussed in the literature with much more reﬁned theoretical models of neutron star matter. There are excellent modern textbooks on compact stars available, which discuss the physics of neutron stars and its various phases in great detail. We refer the interested reader to the books by Norman Glendenning [2] and by Fridolin Weber [3] for a thorough treatment of this exciting ﬁeld in the interplay of physics and astrophysics. Most recently, a new textbook on compact stars appeared which updates and complements the existing ones [4]. A recent review article on the relation between the nuclear equation of state and neutron stars can be found in [5], a nice popular article on neutron stars in [6]. In these lecture notes, we discuss the global properties of neutron stars, masses and radii, and how those change for diﬀerent compositions in their interior. In particular, we address the appearance of new and exotic phases in dense neutron star matter and how they can modify the mass-radius diagram for compact stars. In sect. 2, the baryons of the SU (3) octet are treated together in a chiral SU (3) × SU (3) eﬀective Lagrangian which is motivated from the approximate symmetries of the underlying theory of strong interactions, quantum chromodynamics (QCD). When the properties of hyperons are tied to the known experimental data on hypernuclei, hyperons are present in neutron star matter above twice normal nuclear matter density aﬀecting the properties of massive neutron star conﬁgurations. In sect. 3, strange mesons, the K − , and their presence in neutron star matter by forming a Bose-Einstein condensate are addressed. A relativistic ﬁeld-theoretic approach is utilised to outline the basic features of kaon condensation for compact stars. Ultimately, hadrons have to be described in terms of quark degrees of freedom at high densities. The phase transition from the chirally broken hadronic phase to the approximately chirally restored quark matter phase is examined within perturbative QCD and the MIT bag model to illustrate the main features of a strong ﬁrst order phase transition in compact star matter in sect. 4. Characteristics of the mass-radius relation are highlighted which signal the presence of exotic matter in the core of compact stars. 2. – Hyperons in neutron stars QCD with massless quarks is chirally symmetric, which means that all left-handed and right-handed quarks decouple. This statement is actually true for all vector interactions.

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Left- and right-handed quarks are deﬁned by the relations 1 (1 − γ5 )q ∼ (3, 0), 2 1 qR = (1 + γ5 )q ∼ (0, 3). 2 qL =

(1)

Splitting a spinor Ψ to left and right-handed components one gets (2)

¯ μ Aμ Ψ = (L ¯ + R)γ ¯ μ Aμ (L + R) = Lγ ¯ μ Aμ L + Rγ ¯ μ Aμ R. Ψγ

The mass term for quarks violates chiral symmetry as ¯ = m(L ¯ + R)(L ¯ ¯ + RL) ¯ mΨΨ + R) = m(LR

(3)

and quarks with diﬀerent chirality mix with each other. The explicit breaking of chiral symmetry is small, however, as mu,d ≈ 10 MeV and also ms ≈ 100 MeV is much smaller than the nucleon mass mN ≈ 1 GeV so that chiral symmetry is a useful tool. QCD has a complex and nontrivial structure, quark and gluon condensates are present in the vacuum, in particular the light quark condensate σ, the strange quark condensate ζ and the gluon condensate χ. The nonvanishing vacuum expectation values of these condensates actually generate most of the masses of the hadrons (except for the pseudo-Goldstone bosons). In constructing an eﬀective chiral Lagrangian for the description of neutron star matter, one has to consider composite quark ﬁelds for the meson and baryon ﬁelds. First, consider the spin-zero mesons. Assuming that they are s-wave bound states, then the only spinless objects we can form are (4)

q R qL

q L qR .

The combinations q L qL and q R qR vanish, since the left and right chiral subspaces are orthogonal to each other. The resulting representation in chiral SU (3)×SU (3) symmetry is then (3, 3∗ ) and (3∗ , 3), respectively. The antiparticles belong to the conjugate representation. Hence, nonets of pseudoscalar and scalar particles have to be considered. For the vector mesons, one has to construct vector-like quantities out of the quark ﬁelds qL and qR . Again assuming s-wave bound states, the only vectors which can be formed are (5)

q L γμ qL

q R γμ qR .

This suggests assigning the vector and axial vector mesons to the representation (3 × 3∗ , 0) ⊕ (0, 3 × 3∗ ) = (8, 1) ⊕ (1, 8), an octet and a singlet state. The meson ﬁelds can be grouped conveniently into matrices under ﬂavour SU (3), here for the scalar ﬁelds and pseudoscalar ﬁelds (6)

8

a=0

(q L λa qR + q L λa γ5 qR ) ≡

8

a=0

(ξa λa + iπa λa ) = Σ + iΠ = M

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¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

and correspondingly for the vector and axial vector ﬁelds as well as for the baryon ﬁelds. Out of all these ﬁelds one constructs a Lagrangian which obeys the chiral SU (3) symmetry (7)

L = Lkin + LBM + LBV + Lvec + L0 + LSB + Llep ,

with the usual kinetic terms for baryons (Lkin ), spin-0 ﬁelds (L0 ), vector mesons (Lvec ) and leptons, electrons and muons (Llep ). The baryons couple with Yukawa-type interactions to the spin-0 mesons (LBM ) and to spin-1 mesons (LBV ). An explicit chiral symmetry-breaking term is introduced also (LSB ). For homogeneous matter and in the mean-ﬁeld approximation, derivatives of the boson ﬁelds vanish and the ﬁelds with unnatural parity (pseudoscalars and axial vector ﬁelds) vanish. One arrives at the following expressions for the various terms in the Lagrangian:

(8) LBM + LBV = − ψ i m∗i + giω γ0 ω 0 + giφ γ0 φ0 + gN ρ γ0 τ3 ρ0 ψi , i

χ2 χ2 1 1 1 χ2 2 2 m ρ + g44 (ω 4 + 2φ4 + 6ω 2 ρ2 + ρ4 ), Lvec = m2ω 2 ω 2 + m2φ 2 φ2 + 2 χ0 2 χ0 2 χ20 ρ 4 σ 1 2 2 2 2 2 2 4 L0 = − k0 χ (σ + ζ ) + k1 (σ + ζ ) + k2 + ζ + k3 χσ 2 ζ 2 2 1 σ2 ζ δ χ4 −k4 χ4 − χ4 ln 4 + ln 2 , 4 χ0 3 σ0 ζ0 2 √ 2 χ 1 LSB = − 2mK fK − √ m2π fπ ζ , m2π fπ σ + χ0 2

μ ψ l [iγμ ∂ − ml ]ψl . Llep = l=e,μ

Nonvanishing vacuum expectation values are generated by spontaneous breaking of chiral symmetry, so there appears a light quark condensate and a strange quark condensate. In addition, eﬀects from the nonvanishing gluon condensate are taken into account by introducing a scalar χ ﬁeld, which is a chiral singlet. These condensates generate the hadron masses. So hadron masses are not additional input parameters in this eﬀective ﬁeld theory and come out automatically by considering chiral SU (3) symmetry, contrary to, e.g., the standard relativistic mean-ﬁeld theory (there is for example no explicit mass term for baryons in the Lagrangian!). The expressions for the baryon masses read, for example, (9)

√ 1 S m∗N = m0 − gO8 (4αOS − 1) 2ζ − σ , 3 √ 2 S ∗ mΛ = m0 − gO8 (αOS − 1) 2ζ − σ , 3 √ 2 S ∗ mΣ = m0 + gO8 (αOS − 1) 2ζ − σ , 3 √ 1 S ∗ mΞ = m0 + gO8 (2αOS + 1) 2ζ − σ . 3

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Strangeness in relativistic astrophysics

Fig. 1. – The hadron masses as generated by the vacuum expectation values of the chiral SU (3) model in comparison to the experimental data.

√ √ S with m0 = gO1 ( 2σ + ζ)/ 3. The generated hadron mass spectrum is compared to the experimental data in ﬁg. 1. For determining the necessary input to the Tolman-Oppenheimer-Volkoﬀ (TOV) equation, the relation between pressure and energy density has to be calculated. The thermodynamic grandcanonical potential Ω/V = −Lvec − L0 − LSB − Vvac −

i

γi (2π)3

3

d

k [Ei∗ (k)

−

μ∗i ] −

1 1 3 π2 l

dk k4 k 2 + m2l

can be derived by standard methods. The particle energy of the baryons depends now on the expectation values of the meson ﬁelds in the medium (10)

Ei∗ (ki ) =

ki2 + m∗2 i ,

so that the baryons acquire an eﬀective mass m∗ and an eﬀective chemical potential (11)

μi = Ei∗ (kF,i ) + giω ω0 + giφ φ0 + giρ I3i ρ0 ,

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¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

which is shifted by the vector potentials. All thermodynamic quantities, as the number density n and the energy density , can be extracted from the thermodynamic potential via (12)

p=−

Ω , V

ni =

∂p , ∂μi

= −p +

μi ni .

i

The coupling constants of the scalar mesons to the baryons are ﬁxed by determining the baryon masses in vacuum. The vector coupling constants are automatically given by the SU (3) symmetry relation: (13)

gΛω = gΣω = 2gΞω

2 V = gN ω = 2gO8 ; 3

gΛφ = gΣφ

√ 2 gΞφ = gN ω . = 2 3

which are actually the SU (6) symmetry relations known from the quark model (ideal mixing is assumed which is a very good approximation for the vector meson nonet). The chiral eﬀective model gives a good description of nuclear matter as well as of the properties of nuclei. The nuclear matter properties are a binding energy of EB /A = −16 MeV at n0 = 0.15 fm−3 with an eﬀective mass of m∗N /mN = 0.61, a compression modulus of K = 276 MeV, and an asymmetry term of asym = 40.4 MeV. Hyperons are automatically included by adopting consistently SU (3) symmetry for the chiral eﬀective Lagrangian from the beginning. The computed single-particle energy levels of hypernuclei are depicted in ﬁg. 2 and compared with the experimental hypernuclear data. One sees an overall agreement with the data, just reﬂecting the fact that Λ hypernuclei are well described by a potential depth of the Λ at saturation density of about −30 MeV. The situation for the other hyperons, Σ and Ξ, is far less clear. The Σ hypernuclear potential is likely to be repulsive, while there is experimental evidence that the Ξ hypernuclear potential is attractive, although signiﬁcantly less in comparison to the one for Λ hyperons. In the chiral eﬀective model, the hyperon potential are ﬁxed already by the parameters and the SU (3) symmetry relations. The hyperon potential for the Σ comes out to be only barely repulsive in the chiral model, which will be important for the composition of neutron star matter. Hyperons can appear in neutron star matter by virtue of the conditions of βequilibrium [7, 8]. The timescale for weak interaction rates is of the order of 10−10 s (decay timescale for hyperons) to 10−8 s (decay timescale for kaons) in vacuum. In matter, similar rates are expected, as for example hyperons in hypernuclei have similar to just slightly smaller lifetimes compared to the hyperon lifetime in vacuum. Neutron stars are known to be quite old, some of them have characteristic ages close to the age of the universe of 1010 years, plenty of time to reach equilibrium for weak interactions, β-equilibrium. Hence, Λ-hyperons can appear in dense neutron star matter if their effective energy in the medium reaches the baryochemical potential of neutrons at some baryon density n: EΛ∗ (n) = μΛ = μn . The relations between the chemical potentials of all particles are determined by the conserved charges of the particles, which for neutron star

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Strangeness in relativistic astrophysics

5

Binding Energy (MeV)

h

g

0

f

-5

d

-10

p

-15

s

-20

28 40 51

-25 89 209

-30 0.0

16

12

C

C

Si

Ca

V

Y

nuc

C1 Exp.

Pb

0.05

13

O

0.1

A

0.15

0.2

-2/3

Fig. 2. – The single-particle energy levels of various hypernuclei for the SU (3) chiral model compared to the experimental data (taken from [9]).

matter are just baryon number and charge (not strangeness, contrary to, e.g., heavy-ion reactions). Hence, the chemical potential of all particles can be ﬁxed by (14)

μi = Bi · μB + Qi · μQ ,

where Bi and Qi are the baryon number and the charge of the particle i, respectively. The eﬀective energy for hyperons increases less rapidly than the one for nucleons, due to the SU (3) symmetry of the vector coupling constant of the vector potential, which dominates the high-density behaviour. Therefore, modern calculations of the composition of neutron star matter predict that hyperons appear around 2n0 , where n0 stands for the normal nuclear matter density. Figure 3 shows the composition as calculated from the chiral eﬀective model presented above. Neutrons are the main component at low densities. Protons and electrons appear then with equal amounts, so as to conserve charge neutrality. Then at about 2.5n0 ﬁrst the Σ− - then the Λ-hyperons are present in the matter, reaching fractions of around 10% at larger densities. The Σ− appears before the Λ despite its heavier mass as negatively charged particles are favoured in neutron star

126

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

10

0

n

5

p

Particle fraction

2

10

Chiral

-1

-

e

5

-

2

10

-2

0

0

+

5

2

10

-3

10

0

n

5

p

Particle fraction

2

10

TM1

-

-1

5

0 0 +

2

10

-2 5

e

2

10

-3

0

2

4

6

8

10

12

B/ 0 Fig. 3. – The composition of neutron star matter as a function of density for the chiral SU (3) model (upper plot) and the relativistic mean-ﬁeld model using the parameter set TM1 (lower plot). Hyperons appear around twice normal nuclear matter density (taken from [10]).

Strangeness in relativistic astrophysics

127

Fig. 4. – The equation of state of neutron star matter in β-equilibrium with and without hyperons (taken from [10]). Thick lines show the results for the chiral model, thin lines the ones for the RMF model using set TM1. Hyperons substantially reduce the pressure for a given energy density, i.e. they soften the equation of state.

matter, so as to balance the positive charge of the protons. The asymmetry energy of nucleons drives the system to isospin-neutral matter, i.e. equal amounts of neutrons and protons. Therefore, also the Ξ− -hyperons appears at lower densities than the Ξ0 , here slightly above 3n0 already. For comparison, the composition for the standard Relativistic Mean-Field (RMF) model is plotted also in ﬁg. 3. Here, equal hyperon potentials have been adopted for all hyperons. The pattern of the onset of hyperon populations is quite similar in the two models up to 4n0 . For larger densities, the other hyperons are present at lower densities in the RMF model compared to the chiral model. However, the maximum density reached in neutron star conﬁgurations is about 6n0 , so that the diﬀerence between those two models will be only important for the most massive neutron star conﬁgurations close to the maximum mass. Hyperons have a dramatic eﬀect on the nuclear Equation of State (EoS), the relation between the pressure and the energy density of matter, which serves as the crucial input to the structure equations for compact stars, the Tolman-Oppenheimer-Volkoﬀ equations. Figure 4 shows the equation of state for the chiral model (thick lines) and the RMF model (thin lines). The upper curves are calculated with nucleons and leptons only,

128

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

Fig. 5. – The mass-radius relation for the relativistic mean-ﬁeld model using the parameter set TM1 (thin lines) and for the chiral SU (3) model (thick lines) with and without hyperons (taken from [10]). Hyperons substantially reduce the maximum possible mass.

the lower ones include eﬀects from the presence of hyperons in dense matter. One sees that the hyperons substantially lower the pressure for a given energy density. This eﬀect is stronger than the diﬀerence between the two models used here. In particular, the high-density EoS with hyperons is nearly the same in both models. The lowering of the pressure has a destablizing eﬀect for neutron stars, as pressure is needed to counteract the attractive pull of gravity. Hence, one expects that the maximum possible mass for compact stars with hyperons is signiﬁcantly lower than the one for neutron stars with just nucleons and leptons. Figure 5 demonstrates this eﬀect of hyperons on the mass-radius diagram of compact stars. Matter with nucleons and leptons only arrive at maximum masses of 1.84M for the chiral and 2.16M for the RMF model, respectively. The maximum masses with hyperons included are reduced to 1.52M for the chiral and 1.55M for the RMF model. Note that the radii are diﬀerent already at lower masses which is due to the diﬀerence in the EoS slightly above normal nuclear saturation density. The drastic reduction of the maximum mass due to hyperons is known to be a quite generic feature, see, e.g., [11].

Strangeness in relativistic astrophysics

129

Fig. 6. – The mass-radius relation for diﬀerent strength of the attractive hyperon-hyperon interactions as given by the coupling constant gσ∗ (taken from [12]). A new stable solution appears in the mass-radius relation with similar masses but smaller radii. Selfbound hyperon stars are shown by long-dashed lines.

Finally, the transition to hyperon matter is studied by tuning up the hyperon-hyperon interaction, which is scarcely known from the few double Λ hypernuclear events available. With increasing hyperon-hyperon attraction, the transition to hyperons becomes a ﬁrst-order phase transition. The mixed phase, with slowly rising pressure as a function of energy density, will make the compact star less stable. The onset of the pure hyperonic phase with a steeply rising pressure can lead to another stable sequence of compact stars. For a suﬃciently huge attraction between hyperons, hyperonic matter becomes more stable than pure nucleonic neutron star matter and the mass-radius relation changes to the one for self-bound stars, i.e. the mass increases from the origin with the radius as R3 , so that the average density in the star is nearly constant. Figure 6 depicts the massradius diagram for an increased hyperon-hyperon attraction. For moderate attraction, the mass-radius relation has two distinct branches with correspondingly two maximum masses. The new branch at smaller radii is another stable solution to the TOV equations, constituting the so-called third family of compact stars [13, 14]. For even larger hyperon

130

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

attraction, the mass-radius relation changes to the one for self-bound stars: the massradius relation starts at the origin. Self-bound stars can still possess an outer nuclear crust, which is determined by the low-density nuclear EoS below neutron-drip density. In that case, the mass-radius relation for the low-mass conﬁgurations interpolates between the self-bound star conﬁguration and the normal low-mass neutron star sequence. 3. – Strange bosons in hadron stars Besides hyperons, other hadronic particles can be present at high density in the core of neutron stars. In the following, we discuss how bosons and the phenomenon of BoseEinstein condensation can be described in a simple relativistic ﬁeld-theoretical approach, here for the case of kaon condensation in neutron star matter [15, 16]. We follow the ﬁeld-theoretical model of refs. [17, 18]. In dense matter, kaons, and in particular the negatively charged K − , can appear in neutron star matter. For that to happen, it must be energetically favoured to replace electrons by K − -mesons which translates to the condition that the eﬀective energy of the K − in the dense medium must be equal to the chemical potential of electrons, ∗ (n) = μe . The in-medium shift of the mass and the energy of the kaons can be EK modelled by Yukawa coupling terms to scalar and vector ﬁelds generated by the nuclear matter. Here, for simplicity, we adopt the standard relativistic mean-ﬁeld model for the nuclear part and just add those coupling terms for the kaon ﬁeld to the Lagrangian: (15) L =

B

¯ B iγμ ∂ μ − mB + gσB σ − gωB γμ Vμ − gρB τB R μ ΨB + 1 ∂μ σ∂ μ σ Ψ 2

1 1 1 μν 1 2 μ 1 + mρ R μ R , − m2σ σ 2 − U (σ) − Vμν V μν + m2ω Vμ V μ + U (V ) − R μν R 2 4 2 4 2 where the self-interaction terms between the meson ﬁelds read (16)

U (σ) =

1 1 bm(gσ σ)3 + c(gσ σ)4 , 3 4

U (V ) =

d (Vμ V μ )2 . 4

If the critical condition for the onset of kaon condensation is fulﬁlled, (17)

ωK = μK − = μe ,

then processes like (18)

e− → K − + νe ,

n → p + K−

produce kaons in the dense medium. The Lagrangian for the kaon ﬁeld can be cast in the form (19)

LK = Dμ∗ K ∗ Dμ K − m∗K 2 K ∗ K,

131

Strangeness in relativistic astrophysics

where the vector ﬁelds are coupled minimally (20)

μ Dμ = ∂μ + igωK Vμ + igρK τK R

and the eﬀective mass of the kaon is deﬁned as a linear shift of the mass term by the scalar ﬁeld m∗K = mK − gσK σ.

(21)

The in-medium eﬀective energy of the kaon can be read oﬀ from the energy-momentum relation by using a plane wave ansatz for the kaon ﬁeld: (22)

ωK = mK − gσK σ − gωK V0 − gρK R0,0 .

The total energy density and pressure can be calculated with standard techniques from the energy-momentum tensor (assuming an ideal ﬂuid), so that (23)

= N + K + e,μ ,

(24)

p = pN + pe,μ .

The explicit expression for the energy density originating from kaon condensation reads simply K = m∗K nK ,

(25)

where nK stands for the number density of kaons. Note that there is no direct contribution from the kaon condensate to the pressure, as it vanishes for a Bose-Einstein condensate. Finally, one has to ﬁx the coupling constants for the kaons. We adopt again symmetry arguments to relate the vector coupling constants for the kaons to the ones of the nucleon (26)

gωK =

1 gωN 3

and gρK = gρN .

Here, the coupling to the vector mesons are assumed to be universal, which is the case when gauging the Lagrangian, so that the coupling constants are just related by isospin symmetry. An extension to SU (3) is straightforward, but not necessary here, as we discuss ﬁrst neutron star matter with nucleons, leptons and kaons only and ignore the eﬀects from hyperons. The scalar coupling constant can be related to the relativistic kaon in-medium potential via (27)

UK (ρ0 ) = −gσK σ(ρ0 ) − gωK V0 (ρ0 )

which is a more appropriate (and pragmatic) control parameter. The kaon potential is not well known, but likely to be considerably attractive at high densities. Coupled

132

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

700.0

TM1 600.0

+

Kaon energy ωΚ [MeV]

K 500.0 400.0

-

K 300.0 200.0 100.0 0.0 0.0

RMF ChPT (Σ=270 MeV) ChPT (Σ=450 MeV) ChPT (D=0) coupled channel 1.0 2.0 Density ρ/ρ0

3.0

Fig. 7. – The kaon and antikaon energy in dense nuclear matter (taken from [24]). The kaon energy is shifted up, while the antikaon energy is greatly reduced as a function of density due to an overall strongly attractive antikaon-nucleon potential. RMF: relativistic mean-ﬁeld model, CHPT: chiral perturbation theory with diﬀerent values of the Σ term and for a vanishing range term, D = 0 (see [24] for details).

channel calculations for kaons in dense matter arrive at values of up to −120 MeV at normal nuclear matter density [19-21] although much shallower potentials are found in self-consistent treatments, see, e.g., [22, 23]. Figure 7 shows the in-medium energy of kaons and antikaons as a function of baryon density for diﬀerent approaches. The model outlined here is denoted by the label RMF. The prediction from the RMF approach are in good agreement with the one from Chiral Perturbation Theory (ChPT), where several cases are shown but not discussed here in more details. The kaon energy is shifted up in the nuclear medium due to the repulsive vector potential. The antikaon, on the other hand, experiences an attractive vector potential, so that the eﬀective energy decreases dramatically as a function of density.

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Strangeness in relativistic astrophysics

-3

pressure [MeV fm ]

300

200

nucleon phase kaon phase mixed phase UK=-80 MeV UK=-100 MeV

100

UK=-120 MeV

UK=-140 MeV 0 0

500

1000

1500 -3

energy density [MeV fm ] Fig. 8. – (Colour on-line) The equation of state for nucleon star matter with kaon condensation (from [18]). For large attractive kaon potentials, the transition to kaon condensation is of ﬁrst order. The pressure decreases as function of energy density at the onset of kaon condensation (red dotted lines) indicating an instability. A Gibbs construction has to be used for the description of the mixed phase (blue dashed lines).

For comparison, a coupled channel calculation for the K − is also plotted, with rather similar results to the ones of the RMF model. The appearance of kaon condensation in neutron star matter is accompanied by a strong ﬁrst-order phase transition. The Gibbs criteria for handling a phase transition with two conserved charges, baryon number and charge, states that the pressure in the two phases has to be equal for the same chemical potentials (28)

pI = pII ,

μIB = μII B,

μIe = μII e.

The equation of state for neutron star matter is depicted in ﬁg. 8. The solid line stands for the purely nucleonic EoS, the dotted lines for the pure kaon condensed phase with nucleons and kaons for diﬀerent values of the kaon potential. The pure kaon condensed phase is seen to be unstable, there is a region where the pressure decreases with increasing energy density. The Gibbs construction for the mixed phase of pure nucleon matter and pure kaon condensed matter (dashed lines) interpolates between the two phases in a thermodynamical consistent way, so that the pressure is continuously rising as a function of energy density.

134

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

10

0

Ŧ3

Population (fm )

n

10

p

Ŧ1

Ŧ

K

UK=Ŧ120 MeV

10

Ŧ2

P

10

e

Ŧ

Ŧ3

0.0

0.3

0.6

0.9

1.2

Ŧ3

Density U (fm ) Fig. 9. – The population of nuclear star matter with kaon condensation for a kaon optical of U = −120 MeV at n = n0 (taken from [18]). Kaons appear around 3n0 and replace the electrons. The proton fraction balances the negative charge of the kaons and can be even larger than the neutron fraction for high densities.

The composition of neutron star matter with kaon condensation is shown in ﬁg. 9 for a kaon potential of UK = −120 MeV at n0 . The K − set in at 3n0 and reach a very high fraction already for slightly larger densities. As soon as a the K − are present, the electron and muon fraction decreases accordingly. At 5n0 there are equal fractions of neutrons, protons and K − , so neutron star matter would be more aptly called nucleon star matter. At even larger densities, the K − dominate the population together with the protons, which ensures overall charge neutrality. The K − in combination with protons are favoured in comparison to the neutrons as they are deeply bound in dense matter. Figure 10 depicts the corresponding mass-radius relation by using the equations of state as seen in ﬁg. 8. The lower the kaon optical potential, the lower is the critical density for the onset of kaon condensation. Hence, the mass-radius relation will deviate from the one of the canonical neutron star at lower masses for larger values of the kaon potential. As kaons are in a Bose-Einstein condensate and do not give a direct contribution to the pressure, less mass can be supported against the gravitational pull. For low values of the kaon potential, the mass-radius curve simply gets unstable as soon as kaons are part of the composition in the core of the compact star. For the case UK = −130 MeV, the mass slowly changes as a function of radius and very small radii well below 10 km are reached. High compression is needed to stabilise a kaon condensate compact star. For even more

Strangeness in relativistic astrophysics

135

Fig. 10. – The mass-radius relation for kaon-condensed nucleon stars (taken from [18]). The maximum mass is reduced and extremely small radii of about 8 km are possible when kaon condensation is present.

attraction, the case UK = −140 MeV, there appears a signiﬁcant diﬀerence in the massradius curves between the (thermodynamically inconsistent) Maxwell construction and the Gibbs construction. A strong ﬁrst-order phase transition is present in the core of the compact star from neutron star matter to kaon condensed star matter. Interestingly, the deviations in the mass-radius curve due to the diﬀerent descriptions of the mixed phase remains to be just in some intermediate region of the mass-radius curve. For the last sequence of stable compact star conﬁguration up to the maximum mass, the two descriptions give about similar results for the mass-radius curve. The unstable region in the case of the Maxwell construction, indicated by a change of the slope of the massradius diagram, is absent in the case of the Gibbs construction. Note that for the Gibbs construction, the kaon condensed phase appears at much lower values of the mass of the compact star compared to the Maxwell construction. We will discuss phase transitions in more detail with regard to the onset of the quark matter phase at high densities, to which we turn now.

136

¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

4. – Strange quark matter in compact stars In this section, we discuss the properties of compact stars with quark matter. First, pure quark stars are addressed and the mass-radius diagram for so-called self-bound stars. Then, the quark matter equation of state is matched to the one of the low-density hadronic equation of state. Compact stars with both types of matter, hadronic and quark matter, are dubbed hybrid stars. . 4 1. General properties of quark stars. – QCD predicts that quarks at large energy scales are asymptotically free. For high energy-density matter one considers a gas of free quarks with corrections from one-gluon exchange. The pressure at zero temperature and ﬁnite quark-chemical potential μ reads for massless quarks

(29)

p(μ) =

Nf μ4 4π 2

¯ α Λ 2 αs 2 αs s − G + Nf ln + 11 − Nf ln 1−2 , π π 3 μ π

where Nf stands for the number of ﬂavours (here Nf = 3). The constant G is scheme dependent and in the MS scheme given by G = G0 − 0.536Nf + Nf ln Nf , G0 = 10.374 ± 0.13 [25]. The renormalisation scale Λ will be ﬁxed to be proportional to the quarkchemical potential. The expression for the pressure of massive quarks can be found in [26]. In compact star matter, weak reactions are in equilibrium so that d −→ u + e− + ν¯e− ,

(30)

s −→ u + e− + ν¯e− , s + u ←→ d + u, and the chemical potentials for down and strange quarks must be equal, while the one for up quarks is just shifted by the electrochemical potential: (31)

μs = μd = μu + μe .

Note that this amounts to baryon and charge number conservation, the chemical potentials of the quarks can be also expressed in terms of the baryochemical potential and the electrochemical, as done in the previous sections. This fact will be beneﬁcial for our discussion of the matching of the two phases in the next subsection, for example the connection μn ≡ μu + 2μd can then be easily made. From the pressure one can compute the number density n and energy density via the standard thermodynamic relations for each quark separately

(32)

nf (μf ) =

dpf (μf ) dμf

and f (μf ) = −pf (μf ) + μf ρf (μf ).

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Strangeness in relativistic astrophysics

In addition, pure quark star matter should be charge neutral so that for the total charge density (33)

nc =

2 1 1 nu − nd − ns − ne = 0, 3 3 3

where ni stands for the number density of the species i. Finally, the total energy density and pressure of the quark matter is (34)

Q = u + d + s + e ,

(35)

pQ = p u + p d + p s + p e .

For comparison, one arrives at the MIT bag model by setting the coupling constant to zero and shifting the energy density and pressure of the quark matter by the bag constant B which describes the non-perturbative aspects of the QCD vacuum: (36)

bag = free + B

pbag = pfree − B

which can be combined to (37)

pbag =

1 4 bag − B 3 3

as pfree = free /3. In the following we consider quark matter of massless up, down, and strange quarks. The current strange quark mass of about 100 MeV is smaller than the scale of the quarkchemical potential of 300 MeV and more. Corrections are of the order of (ms /μ)2 and can be safely ignored for our purposes. For the case of three-ﬂavour massless quarks quark matter consists of equal number densities of up, down and strange quarks and is charge neutral by itself. Hence, there are no electrons in such idealised quark matter (in fact the corrections from the ﬁnite strange quark mass are small). The chemical potentials of all three quark species are the same and just given by the quark-chemical potential. The quark number density and the energy density is then determined by (38)

n=

dp dμ

= −p + μ · n

which ﬁxes the quark matter equation of state in parametric form. The pressure actually vanishes for some critical chemical potential. Hence, there is a energy density jump from the vacuum to the energy density of quark matter. Pure quark matter is then stable at this characteristic energy density. This feature of the quark matter equation of state can be easily seen for the MIT bag model but is also present in the interacting model. The characteristic energy density is then ﬁxed by the bag constant, as bag = 4B for vanishing pressure.

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1/4

B =145 MeV

2

/=3P

M/Msun

1.5 1/4

B =200 MeV

1

/=2P

0.5

0 0

5

10

15

Radius (km) Fig. 11. – (Colour on-line) The mass-radius relation of pure quark stars in two diﬀerent approaches: for the MIT bag model (green dashed curves) and within perturbative QCD calculation (blue solid lines). The general form of the mass-radius relation is remarkably similar.

Figure 11 depicts the mass-radius relation for compact stars consisting of pure quark matter. The solid lines show the interacting case for the perturbative QCD equation of state, the dashed line the one for the MIT bag model for so-called strange stars [27, 28]. As the pressure vanishes at some ﬁnite value of the energy density, the quark stars are stabilised and have about similar densities. Hence, the mass increases with the radius of the quark star as R3 starting at the origin. There is a maximum mass, as at some critical number of quarks the gravitational pull cannot be counteracted by the Fermi pressure. The curves can be scaled into each other. For the MIT bag model, maximum mass and the corresponding radius scale as B −1/2 , smaller values of the MIT bag constant give larger maximum masses and larger radii. The results for the perturbative QCD model are strikingly close to the simple MIT bag model. Here larger choices for the renormalisation scale result in larger maximum masses and radii. Note that pure quark stars can only exist if strange quark matter is more stable than ordinary nuclear matter, so that any hadronic mantle is transformed to strange quark matter, which is the so-called BodmerWitten hypothesis [29, 30].

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. 4 2. Hybrid stars: compact stars with quark and hadron matter . – It is more likely that strange quark matter is not absolutely stable, so that the quark core is surrounded by hadronic matter. The compact star is then dubbed a hybrid star. The transition from one phase to the other in compact star matter has to be modelled via the Gibbs condition for phase transitions [31] which states that in phase equilibrium the pressure of the two phases is the same for the same chemical potentials of the two phases (39)

pH (μB , μe ) = pQ (μB , μe ).

Here, the index Q denotes the quantity in the quark phase, H the one in the hadronic phase. There are two conserved charges for compact star matter, the baryon number and charge, so that there are two independent chemical potentials. A Maxwell construction could only ensure that one chemical potential is continuous throughout the phase transition while the other would jump discontinuously. The volume fraction of the quark phase (40)

χ=

VQ VQ + VH

can be used to calculate the total energy density in the mixed phase (41)

mix = χ Q + (1 − χ) H .

The charge neutrality is now guaranteed globally, i.e. the Gibbs construction allows for highly charged phases whose charges balance each other. The Gibbs construction can be visualised by the intersection of the two planes in a diagram where the pressure is plotted as a function of the two chemical potentials, see ﬁg. 12. In the plane of the pressures there is one line which indicates locally charge neutral matter of a single component, quark or hadron matter. The line of the intersection of the two pressure planes deﬁnes the mixed phase, the pressure is equal in the two phases. One realizes that the line of intersection is oﬀ the line of charge neutral matter for the quark or the hadron phase alone. The volume fraction is now chosen in such a manner that the total mixed phase has a vanishing charge density ρ and is charge neutral globally: (42)

ρmix = χ ρQ + (1 − χ) ρH = 0.

At the beginning of the mixed phase, highly negative charged quark matter bubbles appear while hadron matter is slightly positively charged. At the end of the mixed phase, quark matter is slightly negative charged while there are small bubbles of highly positive charged hadron matter. This mismatch of charges is energetically favoured, hadron matter is more stable for about equal numbers of protons and neutrons due to the asymmetry energy, so that a positive charge is advantageous. The corresponding composition for hybrid stars is depicted in ﬁg. 13 as a function of the radius for diﬀerent choices of the MIT bag constant. For low values of the bag

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¨ cker J. Schaffner-Bielich, S. Schramm and H. Sto

1400 1200 1000

300

200 0

100 00

0 200 150 100 50 0 Fig. 12. – The Gibbs phase construction for two chemical potentials by looking at the pressure in 3D. The surface of the hadronic pressure cuts the one of the quark matter pressure along the phase coexistence line where both pressures are equal. The overall charge neutrality condition ﬁxes a line in the hadronic pressure area and also in the quark one. Charge neutrality is ensured along the mixed phase line by adjusting the volume fractions of hadron and quark matter (taken from [14]).

constant, the core consists of pure quark matter which can be the dominant part in the overall composition of the hybrid star. A sizable mixed phase exists, in particular for large values of the bag constant, where the pure quark core is absent in the hybrid star. For suﬃciently large values of the bag constant, the mixed phase as well as the pure quark phase is not present in the compact star, which is then an ordinary neutron star without any quark matter component in its core. The right plot shows the case, when the quarks have a quasi-particle eﬀective mass due to interactions via gluon exchange which is parameterised by the coupling constant g. The onset of the mixed phase and the pure quark phase is shifted to lower values of the bag constant. Note that the composition of a compact star with quark matter is highly sensitive to the choice of the bag constant. Anything between a pure quark star and an ordinary compact star of hadronic matter only is possible due to our present limited knowledge about the properties of quark matter at extreme densities. Finally, the mass-radius diagram for hybrid stars is shown in ﬁg. 14 for diﬀerent values of the bag constant. For large radii, a mixed phase is formed in the core of the

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g=0.

g=2.

190

190

185

185

180

180

175

175

170

170

165

165 0

2.5

5

7.5 10 12.5 15

0

2.5

5

7.5 10 12.5 15

Fig. 13. – The composition of hybrid stars for diﬀerent values of the MIT bag constant B (taken from [14]). For large values of B, the compact star is purely hadronic. A pure quark core is present for small values of B. Note, that for even lower values of B, the pure quark phase will extend up to the surface of the compact star and a pure quark star (a so-called strange star) is formed.

compact star. The mass is decreasing for lower values of the bag constant (note the small diﬀerences in the values for the bag constant). For some values of the bag constant, the mass decreases as a function of the central energy density (for smaller radii) after reaching a maximum mass. This region is unstable with respect to radial oscillations. However, at even larger central energy densities (smaller radii), the mass starts to increase again and a new sequence of stable solutions appear. The compact stars corresponding to these new stable sequence constitute a third family of compact stars, besides white dwarfs and ordinary neutron stars. They are stabilised by the presence of a pure quark phase which has a suﬃciently stiﬀ equation of state so as to support very dense compact star conﬁgurations. It is possible to have two compact stars with identical masses but with diﬀerent radii, so-called compact star twins [13, 14]. In recent years, the properties of QCD at high densities have been explored in connection with the phenomenon of colour-superconductivity [32, 33], for reviews see, e.g., [34, 35]. A rich phase structure exists in the QCD phase diagram at high densities and low temperatures right in the regime of compact star physics. In particular, several

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1.45 1.4 o

1.35

o

1.3 1.25 1.2 1.15

10

11

12

13

14

15

16

Fig. 14. – The mass-radius relation of hybrid stars, compact stars with hadronic and quark matter (taken from [14]). Conﬁgurations exists where the core consists just of the mixed phase (mixed core: MC) or where the core contains pure quark matter (quark core: QC). In the latter case, the compact star with a quark core can constitute a third family of compact stars with similar masses but smaller radii than the mixed core counterparts.

phase transitions might be present in compact star matter, so that the quark matter core of hybrid stars might contain more than one phase transition. This research ﬁeld is rapidly evolving and is poised for new discoveries for the properties of compact stars with quark matter. We refer the interested reader to the above mentioned review articles for further reading. 5. – Conclusions Compact stars can have a rich structure with new and exotic phases being present in their cores. Hyperons, heavy baryons with strangeness, can appear in neutron star matter. As a new fermionic degree of freedom, hyperons lower the overall pressure so that the maximum mass of a compact star with hyperons is substantially smaller than the one of a neutron star consisting of nucleons and leptons only. Also, negatively charged Bose-Einstein condensate could be formed via kaon condensation at high densities. As the Bose-Einstein condensate does not give a direct contribution to the pressure of the system, the stable sequence of compact stars in the mass-radius diagram simply stops as soon as the kaon condensed phase appears or extremely dense compact stars are

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generated with unusually small radii. Finally, at high densities quark matter can be present. Depending on the onset of the phase transition and the strength of the ﬁrstorder phase transition from the hadronic to the quark matter phase, a third family of compact stars is found in the mass-radius diagram. The compact stars of this third family have a core of pure quark matter and smaller radii than ordinary hybrid stars. The physics of dense quark matter in compact star opens exciting new perspectives in exploring the phase diagram of QCD at high densities. The new research facility FAIR at GSI Darmstadt will explore this fascinating regime of strong interaction physics in the near future with relativistic heavy-ion collisions with bombarding energies tuned so as to create compressed baryonic matter at highest baryon densities.

REFERENCES [1] Landau L. D., Physik. Zeits. Sowjetunion, 1 (1932) 285. [2] Glendenning N. K., Compact Stars—Nuclear Physics, Particle Physics, and General Relativity, 2nd ed. (Springer, New York) 2000. [3] Weber F., Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics (Institute of Physics, Bristol) 1999. [4] Hansel P., Potekhin A. Y. and Yakovlev D. G., Neutron Stars 1: Equation of State and Structure, Vol. 326 of Astrophysics and Space Science Library (Springer, New York) 2007. [5] Lattimer J. M. and Prakash M., Phys. Rep., 442 (2007) 109, astro-ph/0612440. [6] Lattimer J. M. and Prakash M., Science, 304 (2004) 536, astro-ph/0405262. [7] Ambartsumyan V. A. and Saakyan G. S., Sov. Astron., 4 (1960) 187. [8] Glendenning N. K., Astrophys. J., 293 (1985) 470. [9] Beckmann C. et al., Phys. Rev. C, 65 (2002) 024301, nucl-th/0106014. ¨ cker H. and Greiner W., [10] Hanauske M., Zschiesche D., Pal S., Schramm S., St o Astrophys. J., 537 (2000) 958, arXiv:astro-ph/9909052. [11] Glendenning N. K. and Moszkowski S. A., Phys. Rev. Lett., 67 (1991) 2414. ¨ cker H. and Greiner W., Phys. Rev. Lett., [12] Schaffner-Bielich J., Hanauske M., Sto 89 (2002) 171101, astro-ph/0005490. [13] Glendenning N. K. and Kettner C., Astron. Astrophys., 353 (2000) L9, astroph/9807155. [14] Schertler K., Greiner C., Schaffner-Bielich J. and Thoma M. H., Nucl. Phys. A, 677 (2000) 463, astro-ph/0001467. [15] Kaplan D. B. and Nelson A. E., Phys. Lett. B, 175 (1986) 57. [16] Brown G. E., Kubodera K., Rho M. and Thorsson V., Phys. Lett. B, 291 (1992) 355. [17] Glendenning N. K. and Schaffner-Bielich J., Phys. Rev. Lett., 81 (1998) 4564. [18] Glendenning N. K. and Schaffner-Bielich J., Phys. Rev. C, 60 (1999) 025803. [19] Koch V., Phys. Lett. B, 337 (1994) 7. [20] Waas T., Kaiser N. and Weise W., Phys. Lett. B, 379 (1996) 34. [21] Tolos L., Ramos A., Polls A. and Kuo T. T. S., Nucl. Phys. A, 690 (2001) 547. [22] Lutz M., Phys. Lett. B, 426 (1998) 12. [23] Tolos L., Ramos A. and Oset E., Phys. Rev. C, 74 (2006) 015203, nucl-th/0603033. [24] Schaffner-Bielich J., Mishustin I. N. and Bondorf J., Nucl. Phys. A, 625 (1997) 325, nucl-th/9607058.

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[25] Fraga E. S., Pisarski R. D. and Schaffner-Bielich J., Phys. Rev. D, 63 (2001) 121702(R), hep-ph/0101143. [26] Fraga E. S. and Romatschke P., Phys. Rev. D, 71 (2005) 105014, hep-ph/0412298. [27] Haensel P., Zdunik J. L. and Schaeffer R., Astron. Astrophys., 160 (1986) 121. [28] Alcock C., Farhi E. and Olinto A., Astrophys. J., 310 (1986) 261. [29] Bodmer A. R., Phys. Rev. D, 4 (1971) 1601. [30] Witten E., Phys. Rev. D, 30 (1984) 272. [31] Glendenning N. K., Phys. Rev. D, 46 (1992) 1274. [32] Alford M., Rajagopal K. and Wilczek F., Phys. Lett. B, 422 (1998) 247, hepph/9711395. ¨fer T., Shuryak E. V. and Velkovsky M., Phys. Rev. Lett., 81 (1998) [33] Rapp R., Scha 53, hep-ph/9711396. [34] Rischke D. H., Prog. Part. Nucl. Phys., 52 (2004) 197. ¨ fer T., arXiv:hep-ph/0709.4635 [35] Alford M. G., Schmitt A., Rajagopal K. and Scha (2007).

The J-PARC strangeness physics program T. Nagae(∗ ) High Energy Accelerator Research Organization (KEK) J-PARC Project Oﬃce, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

Summary. — A high-intensity proton accelerator complex, J-PARC, is now under construction in Japan. We expect that the ﬁrst beam would be delivered to the Hadron Experimental Hall around the end of 2008. The world highest intensity K − beams will be available, which will open new opportunities to conduct various types of strangeness nuclear physics experiments. In this lecture several examples from the initial experimental program are introduced.

1. – Status of J-PARC The accelerators of J-PARC [1] were designed to provide megawatt-class proton beams at three diﬀerent energies with the following components: i) a proton linac (with normal conducting cavities) to inject beams to a 3 GeV proton synchrotron (PS), ii) a 3 GeV PS operating at 25 Hz with 1 MW of beam power to be used primarily for materials and life sciences with neutrons and muons, and iii) a 50 GeV PS with slow extraction for secondary beams and fast extraction for neutrino beams to Super-Kamiokande. (∗ ) Present address: Kyoto University, Department of Physics, Kitashirakawa, Kyoto 606-8502, Japan. c Societ` a Italiana di Fisica

145

146

T. Nagae Hadron Experimental Area 3 GeV PS Material and Life Science (25Hz) Experimental Area

Phase 1 Phase 2 R&D for Nuclear Transmutation Linac (Superconducting)

50 GeV PS

Linac (Normal Conducting) Neutrinos to SuperKamiokande

Fig. 1. – Schematic layout of the J-PARC facility, which also shows the two parts of the project in Phase 1 and Phase 2.

The J-PARC project is split into two phases according to the construction budget (ﬁg. 1). The total budget proposed was about 189 billion yen. The Phase 1 budget, which amounts to 151 billion yen, has been approved. It covers most of the accelerator components and part of the experimental facilities. The proton linac energy will be limited to 181 MeV during the initial stage, but will be increased rather easily to 400 MeV in the near future by adding some linac components. A superconducting proton linac that will further increase the beam energy from 400 MeV to 600 MeV is planned for Phase 2 together with basic R&D facilities for nuclear transmutation. The 50 GeV PS will be operated at 30 GeV in the beginning. Because of the limited linac energy, the beam power from the 3 GeV PS will be reduced to ∼ 0.6 MW. The beam intensity in the 50 GeV PS will not be aﬀected if we could inject more beam pulses from the 3 GeV PS than the original design. There is an idea to double the harmonic number of the 50 GeV PS for this purpose. The experimental facilities to be constructed in Phase 1 are the Material and Life Science Facility for pulsed neutron and muon beams, the Hadron Experimental Hall for kaon and pion beams, and the Neutrino Experimental Facility for neutrino oscillation experiments. In Phase 2, the Hadron Experimental Hall will be extended to provide more secondary beam lines, and some new neutron and muon beam lines will be installed in the Material and Life Science Facility. The equipment required for the 50 GeV operation of the 50 GeV PS will also be installed during Phase 2. The beam commissioning of the proton linac was started in the fall of 2006, and the beam was successfully accelerated up to the design energy of 181 MeV in January, 2007. We expect that the beam from the linac will be delivered to the 3 GeV PS in September, 2007. Crews have been very busy with the installation and ﬁnal local testing of the accelerator apparatus in the 3 GeV PS tunnel. Most of the major magnets are already in the 50 GeV PS tunnel, whereas some of the injection and extraction devices are still under construction. The Hadron Experimental Hall (ﬁg. 2) is almost completed as of June, 2007. The civil engineering for the neutrino beam facility is now in rapid progress.

The J-PARC strangeness physics program

147

Fig. 2. – The Hadron Experimental Area: a view from the outside (left) and a view inside (right) in June, 2007.

The current goal of the project is to start the beam delivery to the Material and Life Science Facility in the middle of 2008 and to the Hadron Experimental Hall at the end of 2008. The beam delivery to the neutrino beam line will begin in the spring of 2009. When beam delivery begins, the expected beam power will be about 1% of the full beam power. We plan to open the facilities to general neutron users when the beam power reaches the 10% level, which may take several months to achieve. The beam commissioning of the secondary beam lines at the Hadron Experimental Hall is expected to start at a lower beam power level. 2. – Brief history of strangeness nuclear physics Strangeness nuclear physics has evolved from traditional hypernuclear physics since 1990s. Hypernuclear physics research has rather long history since the ﬁrst discovery of hyperfragment in 1953 [2]. It was discovered in a nuclear emulsion produced by a highenergy cosmic ray. It should be noted this was just after a few years from the discovery of strange V particles, now known as Λ hyperon and kaons. In 1960s, the stopped K − reaction was used to produce hyperfragments in nuclear emulsions and nuclear bubble chambers. While the K − beam intensity at that time was very much limited, the stopped K − reaction eﬃciently produced various light hyperfragments. The hyperfragments were identiﬁed from weak decay patterns in the detector. The very good position resolution of such detectors (1–10 μm) enabled us to observe the hyperfragment running in the detector. Various light hyperfragments were observed, and the binding energies of the ground states were measured. In some cases, the spin of the state was assigned with a help of weak decay characteristics. In 1970s, the in-ﬂight (K − , π − ) reaction was applied for the production of hypernuclei at CERN [3]. The reaction has a very small (≤ 100 MeV/c) recoil momentum for the Λ hyperon produced. Therefore, high sticking probability of the Λ hyperon to the recoil nucleus was expected for so-called substitutional states; for example, a neutron in p1/2 orbit is converted to a Λ hyperon in p1/2 orbit in the case of 12 Λ C. Since the produced hypernucleus is identiﬁed as a missing-mass in the (K − , π − ) reaction, not only the ground

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state but also the excited states can be identiﬁed in this method. It was found that the spin-orbit splitting of 16 Λ O was very small compared with normal nuclei [4]. In the later stage, the method was applied for heavy Λ hypernuclei. However, it was not successful to observe deeply bound states of heavy Λ hypernuclei, although the states near the binding threshold were highly excited. At the end of the (K − , π − ) reaction experiments at CERN-PS, they took data in the Σ production region, and found some narrow resonance states indicating the excited levels of Σ hypernuclei. However, it was naively expected that the width of the Σ hypernuclei should be large (≥ 20 MeV) because of the strong conversion process of ΣN → ΛN in Σ hypernuclei. This puzzle of the Σ hypernucleus width was one of main topics in hypernuclear physics in 1980s [5]. After CERN-PS, the Alternating Gradient Synchrotron (AGS) of Brookhaven National Laboratory (BNL) and the 12 GeV PS at KEK were the two major accelerator facilities used for hypernuclear physics. BNL-AGS operated at 24 GeV could provide high-intensity K − beams. In the early stage, they also observed several narrow Σ states. At KEK-PS, the stopped K − beam method was revived because of the limited intensity at KEK-PS. They also reported some narrow structures for a Σ hypernucleus. Unfortunately, those observations were denied in both places in the later high-statistics data. Nevertheless, existence of one bound state of Σ hypernucleus, 4Σ He, was established; once claimed at KEK-PS [6], and conﬁrmed at BNL-AGS [7]. In 1990s, several new attempts to extend the investigations in hypernuclear physics were carried out. At BNL-AGS, a new 2 GeV/c kaon beam line, D6 [8], was constructed. It was equipped with double-stage electrostatic separator system to realize the K − /π − ratio of ≈ 1. This beam line provided a unique tool to investigate the strangeness S = −2 system [9]. The main focus was the search for H dibaryon. Spectroscopic studies on the S = −2 systems were not carried out there because of the limited energy resolution of the spectrometer system. At KEK-PS, the SKS spectrometer [10] played an essential role to conduct the hypernuclear programs in Japan. Further, the Hyperball detector has opened a new regime of hypernuclear spectroscopy in precision [11]. In 2000, Thomas Jeﬀerson Laboratory (JLab) started hypernuclear spectroscopy with a diﬀerent reaction of (e, e K + ) [12,13]. The ﬁrst measurement demonstrated a good energy resolution of 0.7 MeV. In 2004, the FINUDA experiment at Frascati also completed − its ﬁrst run for (Kstop , π − ) spectroscopy with 1.3 MeV resolution [14]. Thus, the hypernuclear spectroscopy has been in the new stage in energy resolution, and new energy levels have been resolved. 3. – Hadron Experimental Hall and K1.8 beam line The Hadron Experimental Hall at J-PARC has a size of 58 m[W] × 56 m[L]. The ﬂoor level is 6.4 m underground. The primary beam from the 50 GeV PS will be delivered to the Hadron Experimental Hall through a 180 m long switchyard tunnel. The beam will hit a production target, T1, to provide secondary beams. A charged-kaon beam line, K1.8, with the nominal beam momentum of 1.8 GeV/c will be constructed in the hall. Design work of the K1.8 beam line was already completed by the J-PARC hadron

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SKS+

Beam Q12,13 Analyzer ES2

Q10,11

Q8,9

D4

D3 Q7

MS2

Q5,6

T1 target

ES1

MS1

F.F.

Q3,4 Q1,2 D1 D2

Fig. 3. – A layout of K1.8 beam line and K1.8 experimental area.

beam line construction group. Figure 3 shows a beam line layout together with the K + spectrometer in the K1.8 experimental area. Figure 4 shows a calculated beam envelope of the K1.8 beam line. The beam line has two stages of electrostatic separators (ES1 and ES2) with two mass slits (MS1 and MS2)

(cm) 25

K1.8 Beam Line Optics D 1

Q 1

Q 2

D 2

Q 3

IF I F

O 1

S Q 2 5

C M 1

Vertical

W C N O D L W

Q S 4 1

Q 6

D 3

F:5.656m M:0.6@MS1 C M 2

(Beam envelope to the 1st Order)

Q S 7 3

M S 1

S Q 4 8

Q 9

Q 1 0

Q 1 1

F:5.038m M:-0.8@MS2

O C 2 M 3

C M 4

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D D 4 4 C D

-25

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FWCIOCCMOCCOMV FNOF1MMS2MM3SI DL 121 34 2 W

QQDQQ ES1

ES2

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0

25 (cm)

1 3 . 0 9 9

7 . 6 0 0

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9 . 7 8 9

7 . 9 9 5

1 5 . 0 1 0

1 5 . 0 1 0

1 5 . 0 1 0

1 5 . 0 1 0

6 . 7 0 1

Total Length: 45.853 m

8 . 5 0 0

Fig. 4. – Beam envelope of the K1.8 beam line calculated by TRANSPORT.

25

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in order to separate kaons from pions and other particles at the level of K − /π − ratio greater than 5. 4. – Strangeness nuclear physics program at J-PARC Proposals for nuclear and particle physics experiments at J-PARC were called for in the fall of 2005. By the end of April, 2006, we received twenty proposals, including four letters of intent. Around ten were related to Strangeness Nuclear Physics. Fourteen proposals were considered at the ﬁrst PAC meeting held at the end of June, 2006. The PAC approved three experiments at that time: one neutrino oscillation experiment and two strangeness nuclear physics experiments. The approved nuclear physics experiments are: E05:

E13:

Spectroscopic study of Ξ-hypernucleus, (T. Nagae),

12 Ξ Be,

via the

12

C(K − , K + ) reaction

Gamma-ray spectroscopy of light hypernuclei (H. Tamura).

In addition to these two approved experiments, the Committee also selected the following ﬁve experiments for stage-1 (scientiﬁc merit) approval in strangeness nuclear physics; E03:

Measurement of X-rays from Ξ− Atom (K. Tanida),

E07:

Systematic study of double strangeness system with an emulsion-counter hybrid method (K. Imai, K. Nakazawa, H. Tamura),

E15:

A search for deeply bound kaonic nuclear states by in-ﬂight 3 He(K − , n) reaction (M. Iwasaki, T. Nagae),

E17:

Precision spectroscopy of kaonic 3 He 3d → 2p X-rays (R. S. Hayano, H. Outa),

E19:

High-resolution search for Θ+ pentaquark in π − p → K − X reaction (M. Naruki).

Five experiments, E05, E13, E15, E17, and E19, were also categorized as “Day-1 experiments” in the Hadron Experimental Hall. Among them, E05 has the ﬁrst priority and E13 the second priority. The 2nd PAC meeting was held in January, 2007, and several other proposals were approved. Here I mainly introduce the E05, E13, and E17 experiments. . 4 0.1. E05: spectroscopic study of Ξ-hypernuclei. The (K − , K + ) reaction is one of the best tools to create Strangeness S = −2 systems through the elementary process, K − p → K + Ξ− , the cross-section of which has a broad maximum in the forward direction for a K − momentum around 1.8 GeV/c [15] as shown in ﬁg. 5. This reaction has been used for studies of S = −2 systems so far. The Ξ-hypernuclei will play an important role in the investigations of S = −2 systems as the entrance channel to the S = −2 world. In ﬁg. 6, typical energy spectrum

The J-PARC strangeness physics program

151

Fig. 5. – The incident momentum dependence of the forward angle laboratory cross-section for K − p → K + Ξ− taken from ref. [15].

and decay threshold for Ξ- and double-Λ hypernuclear conﬁgurations are shown. Produced Ξ-hypernuclear states eventually decay into several forms of double-Λ systems through a strong conversion process, Ξ− p → ΛΛ. Moreover, Ξ-hypernuclei give valuable information on the S = −2 baryon-baryon interactions such as ΞN , and ΞN → ΛΛ. There are some hints for the existence of Ξ-hypernuclei in existing emulsion data, however it is still not conclusive. Some information on the Ξ-nucleus potential has been obtained from the production rate and spectrum shape in the bound region of the Ξhypernucleus via the 12 C(K − , K + ) reaction [16,17]. In these experiments, Ξ hypernuclear states were not clearly observed because of the limited statistics and detector resolution. From the data analysis, however, the potential depth, VΞ , was favored to be ≈ 14 MeV for A = 12 when a Woods-Saxon–type potential shape is assumed. In this experiment, the bound states of Ξ hypernuclei will be observed as clear peaks with good energy resolution. The peak position will give us direct information on the depth of the Ξ-nucleus potential. The width of the bound-state peak also provides us with information on the imaginary part of the Ξ-nucleus potential, or Ξ− p → ΛΛ conversion. For this purpose, we need high-resolution spectrometers for both the K − beam and scattered K + ’s.

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Fig. 6. – Typical energy spectrum and decay threshold for S = −2 system; Ξ- and double-Λ hypernuclear conﬁgurations.

A new kaon beam line K1.8 with a maximum beam momentum of 1.8 GeV/c has been designed for the experiment. The beam line provides a high-intensity (1.4×106 K − /spill) and high-purity K − beam. The beam line has two stages of electrostatic separators with two mass slits in order to separate kaons from pions and other particles and achieve a K − /π − ratio greater than 5. The beam analyzer located after the last mass slit is comprised of QQDQQ magnets and four sets of tracking detectors. The expected momentum resolution, Δp/p, is 1.4×10−4 root-mean-square when a position resolution of 200 μm is realized in the tracking detectors placed before and after the QQDQQ system. For the K + spectrometer, the existing SKS spectrometer will be used with some modiﬁcations. A dipole magnet with ≈ 1.5 T will be added at the entrance of the SKS magnet as shown in ﬁg. 7. A simulation shows that the spectrometer, called SKS+, has a solid angle of ≈ 30 msr with an angular range from 0◦ to 10◦ , and a momentum resolution of Δp/pFWHM = 0.17%. The overall energy resolution is expected to be better than 3 MeV FWHM including the energy-loss straggling in the target. The production cross-sections of the Ξ-hypernuclei in the (K − , K + ) reaction have been calculated by several theorists within the framework of the distorted-wave impulse approximation (DWIA) [18-20]. Also, the previous experimental studies reported the

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Fig. 7. – SKS+ Spectrometer in consideration. SDC1-6 are tracking drift chambers. TOF is a time-of-ﬂight counter wall, and AC is an aerogel Cherenkov detector system.

cross-sections [16, 17]. Based on these values, the yield of estimated to be ≈ 190 events/month.

12 Ξ Be

with a

12

C target is

. 4 0.2. E13: Gamma-ray spectroscopy of light hypernuclei. Many γ-ray transitions for various Λ hypernuclei are expected to be observed at J-PARC. Abundant production of Λ hypernuclei will be possible in the (K − , π − ) reactions using the high-intensity K − beams. As for the Day 1 experiment, the (K − , π − ) reaction at 1.5 GeV/c will be used to produce Λ hypernuclei at the K1.8 beam line together with the SKS spectrometer. The maximum K − beam intensity at Day 1 will be 0.5 × 106 per spill at 1.5 GeV/c. The incident K − momentum is selected to be 1.5 GeV/c where both the (K − , π − ) crosssection and the spin-ﬂip amplitude are reasonably large (ﬁg. 8) [21]. The SKS magnet will be set to 2.7 T for the scattered pion momentum of ∼ 1.4 GeV/c. The tracking detectors at the exit of the SKS magnet should be replaced with larger ones to keep the solid angle acceptance ≥ 100 msr. A new germanium detector, Hyperball-J (ﬁg. 9), is going to be constructed for the experiment. It consists of about thirty sets of Ge detectors that have a photo-peak eﬃciency of about 75% relative to a 3 × 3 NaI detector. Each Ge detector is surrounded with fast PWO counters for background suppression. The photo-peak eﬃciency is expected to be better than 5% at 1 MeV at a distance of ∼ 15 cm from the target. Various interesting subjects are proposed for the Hyperball-J detector. One important subject is to measure the transition probabilities (B(M 1)) of the Λ spin-ﬂip M 1 transitions and probe the g-factor of a Λ inside a nucleus. A measurement of the M 1(3/2+ → 1/2+ ) transition of 7Λ Li is proposed. Considering the estimated lifetime of the 3/2+ state, ∼ 0.5 ps, a Li2 O target with a density of 2.01 g/cm3 in granular powder form is selected so that the Doppler-Shift Attenuation method [22] can be applied. Since

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Fig. 8. – The incident K − momentum dependence of the laboratory cross-sections and polarizations for the elementary K − n → Λπ − process [21].

Fig. 9. – Schematic view of the proposed Hyperball-J detector. The bottom half is shown.

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the stopping time of the recoiling 7Λ Li∗ would be 2-3 times longer than the lifetime, we will observe a mixture of line shapes composed of a sharp peak and a Doppler-broadened peak. From the mixing ratio, we can estimate the lifetime knowing the stopping time. − + + It is proposed to take data with a 19 F target to detect both 19 Λ F(1/2 → 3/2 , 1/2 ) transitions. This would determine the ground-state doublet spacing in the sd-shell region for the ﬁrst time. It is also planned to measure the M 1 γ-ray in 4Λ He(1+ → 0+ ) with high precision. An extremely large charge-symmetry breaking is reported between 4Λ He and 4 Λ H [23, 24]. An improved measurement would clarify the eﬀect. Also, this measurement is useful to conﬁrm the spin-ﬂip and non-spin-ﬂip amplitudes in the (K − , π − ) reaction in this momentum range. . 4 0.3. E15: A search for deeply bound kaonic nuclear states. Conﬁrming the existence of deeply bound kaonic nuclear states, which have been suggested as narrow states for speciﬁc ﬁnite nuclear systems [25], is now an important experimental subject. There are some experimental data suggesting the existence of such bound states [26, 27]. However, on the theoretical side, there have been a lot of discussions regarding the depth of the K − nucleus potential, whether it is deeply attractive (−Re Vopt (ρ0 ) ≈ 150–200 MeV) [28, 29] or much shallower (−Re Vopt (ρ0 ) ≈ 50–75 MeV) [30-34]. Both types of potentials reasonably reproduce the shifts and widths of the kaonic X-ray data [35]. There are also concerns that the widths of the bound states would be too broad to be separately observed as a clean bound-state peak. Moreover, there is an issue concerning whether such a deeply bound state, if it exists, would give rise to the formation of a dense nuclear system by strongly attracting the nucleons. If this is true, the formation of deeply bound kaonic states would give us a unique opportunity to investigate hadrons in dense nuclei. Therefore, it is an urgent task to conﬁrm experimentally if such a bound state exists or not. The K − pp system is important because it would be the simplest and lightest kaon bound state, if it exists. In this experiment, the mass of the K − pp system will be measured in both a missing-mass measurement and an invariant-mass measurement. Therefore, a clean and unambiguous identiﬁcation of the bound state will be possible. Here, we use the in-ﬂight (K − , n) reaction on 3 He at 1 GeV/c, where the crosssection of K − + n → n + K − has a broad maximum of ∼ 5 mb/sr (ﬁg. 10). The momentum of the neutron emitted in the forward direction is measured with a timeof-ﬂight counter wall. This neutron counter array consists of plastic scintillators of 20 cm[W] × 5 cm[T] × 150 cm[L] with a total thickness of 40 cm. The K − pp mass is thus measured as a missing mass. At the same time, the target region is covered by a cylindrical detector system with a large acceptance which is placed into a solenoidal magnetic ﬁeld. Thus, most of the charged particles produced in the decay of the K − pp system are detected. Here, a decay mode of K − pp → Λ + p followed by the Λ → p + π − decay is detected, and the K − pp mass is reconstructed as an invariant mass of a Λ and a proton. A schematic view of the experimental set-up is shown in ﬁg. 11. The designed missingmass resolution is ≈ 28 MeV FWHM with a ﬂight path of ∼ 12 m, and the invariant-mass resolution is ≈ 40 MeV FWHM.

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8

-,p)Kp(K,p)K p(K -,n)Kn(K,n)K n(K _ p(K,n)K p(K-,n)K0

7 6

mb/sr

5 4 3 2 1 0 200

400

600

800

1000

1200

1400

1600

PK (MeV/c) Fig. 10. – The incident K − momentum dependence of the K − N backward scattering crosssections.

Fig. 11. – Schematic view of the experimental set-up of E15. A cylindrical detector system CDS is composed of a cylindrical drift chamber CDC and a hodoscope CDH.

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. 4 1. Other experiments. – Here, I brieﬂy mention the other experiments approved, at least at the level of Stage 1, at J-PARC. An emulsion-counter hybrid measurement (E07) to look for double strangeness systems is proposed, aiming for ≈ 100 candidate events of double-Λ hypernuclei and to identify more than ten double-Λ hypernuclei species for the ﬁrst time. The experiment does not need the full beam intensity at the K1.8 beam line, but need a high ratio of K − /π − ≥ 5. It also requires a large amount of emulsions. A fast scanning system for emulsions has been developed for eﬃcient and quick analysis. X-ray measurements of Ξ− atoms will provide us with valuable information of Ξ nucleus potentials from their energy shifts and widths. However, they require the full beam intensity of K − at K1.8, or even more. The ﬁrst measurement on an Fe target (E03) is proposed with sensitivities on the energy shift of ≈ 0.05 keV and on the width of ≈ 1 keV. A precision measurement of X-rays of Kaonic 3 He atom (E17) is also proposed to observe the energy shift with ±2 eV accuracy. It would be sensitive to the depth of the K − -3 He potential, which is related to the existence of kaonic nuclear bound state, too. The data will be obtained in a short period. A pentaquark search with the p(π − , K − ) reaction (E19) is proposed by using the SKS spectrometer with an energy resolution of 2.5 MeV FWHM. A narrow pentaquark width, if it exists, could be determined with an experimental resolution down to 2 MeV. An experimental search (E10) for neutron-rich Λ hypernuclei, such as 6Λ H and 9 − + Λ He, is proposed with the double-charge-exchange reaction of (π , K ). It requires − a high-intensity π beam because of the small production cross-sections through the two-step reaction. 5. – Conclusions The construction of the high-intensity proton accelerator complex, J-PARC, is at the ﬁnal stage. The beam commissioning has been in progress since the fall of 2006. Three experimental facilities are to be constructed during Phase 1 of the project: the Hadron Experimental Hall, the Neutrino Experimental Facility, and the Material and Life Science Facility. By improving the intensity step by step, kaon beams, neutrino beams, and slow neutron and muon beams will soon be available with the world’s highest intensities. Various interesting experimental programs in strangeness nuclear physics are waiting for beam. New types of spectroscopic studies are proposed for Ξ hypernuclei, doubleΛ hypernuclei, and kaonic nuclei. A systematic extension of hypernuclear γ-ray spectroscopy to a wide range of mass number will also be conducted at J-PARC. ∗ ∗ ∗ This work is partially supported by the Grant-in-Aid for Scientiﬁc Research on Priority Areas, Area Number 449, by Ministry of Education, Culture, Sports, Science and Technology, Japanese Government.

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Nagamiya S., Nucl. Phys. A, 755 (2005) 219; 774 (2006) 895. Danysz M. and Pniewski J., Philos. Mag., 44 (1953) 348. Povh B., Prog. Part. Nucl. Phys., 18 (1987) 183. ¨ ckner W. et al., Phys. Lett. B, 79 (1978) 157. Bru Dover C. B., Millener D. J. and Gal A., Phys. Rep., 184 (1989) 1. Hayano R. S. et al., Nuovo Cimento A, 102 (1989) 437; Phys. Lett. B, 231 (1989) 355. Nagae T. et al., Phys. Rev. Lett., 80 (1998) 1605. Pile P. H. et al., Nucl. Instrum. Methods A, 321 (1992) 48. Chrien R. E., Nucl. Phys. A, 691 (2001) 501c. Fukuda T. et al., Nucl. Instrum. Methods A, 361 (1995) 485. Tamura H. et al., Phys. Rev. Lett., 84 (2000) 5963. Hungerford E. V., Nucl. Phys. A, 691 (2001) 21c. Hashimoto O. and Tamura H., Prog. Part. Nucl. Phys., 57 (2006) 564. Agnello M. et al., Phys. Lett. B, 622 (2005) 35. Dover C. B. and Gal A., Ann. Phys. (N.Y.), 146 (1983) 309. Fukuda T. et al., Phys. Rev. C, 58 (1998) 1306. Khaustov P. et al., Phys. Rev. C, 61 (2000) 054603. Dover C. B., Gal A. and Millener D. J., Nucl. Phys. A, 572 (1994) 85. Ikeda K., Fukuda T., Motoba T., Takahashi M. and Yamamoto Y., Prog. Theor. Phys., 91 (1994) 747; Yamamoto Y., Motoba T., Fukuda T., Takahashi M. and Ikeda K., Prog. Theor. Phys. Suppl., 117 (1994) 281. Tadokoro S., Kobayashi H. and Akaishi Y., Phys. Rev. C, 51 (1995) 2656. ¯ H., Phys. Lett. B, 232 (1989) 24. Kishimoto T., Ejiri H. and Bando Tanida K. et al., Phys. Rev. Lett., 86 (2001) 1982. Bedjidian M. et al., Phys. Lett. B, 62 (1976) 467. Bedjidian M. et al., Phys. Lett. B, 83 (1978) 252. Akaishi Y. and Yamazaki T., Phys. Rev. C, 65 (2002) 044005. Agnello M. et al., Phys. Rev. Lett., 94 (2005) 212303. Kishimoto T. et al., Nucl. Phys. A, 754 (2005) 383c. ´ A., Phys. Rev. C, 60 (1999) 024314. Friedman E., Gal A., Mareˇ s J. and Cieply Friedman E., Gal A. and Batty C. J., Nucl. Phys. A, 579 (1994) 518. ´ A., Friedman E., Gal A. and Mareˇ Cieply s J., Nucl. Phys. A, 696 (2001) 173. Kaiser N., Siegel P. B. and Weise W., Nucl. Phys. A, 594 (1995) 325. Waas T., Kaiser N. and Weise W., Phys. Lett. B, 365 (1996) 12; 379 (1996) 34. Ramos A. and Oset E., Nucl. Phys. A, 671 (2000) 481. Baca A., Garc´ıa-Recio C. and Nieves J., Nucl. Phys. A, 673 (2000) 335. Batty C. J., Friedman E. and Gal A., Phys. Rep., 287 (1997) 385.

Strangeness production in antiproton-4 He annihilation at rest G. Bendiscioli Dipartimento di Fisica Nucleare e Teorica dell’Universit` a di Pavia and INFN Sezione di Pavia, Pavia, Italy

Summary. — This paper reports an overview of experimental results and theoretical predictions about the p ¯ annihilation at rest on nuclei. It stresses that the annihilation on 4 He creates an environment very favourable to the formation of strange quarks and describes results of recent analyses on this process made on data collected with the spectrometer Obelix at the LEAR accelerator of CERN. The results concern the high strangeness production in speciﬁc reaction channels, double-strangeness production and signals for the production of the pentaquark Θ+ (1539) and antikaon-few-nucleon bound states like K− pn, K− pnn and K− d.

1. – Introduction Antiproton annihilation on nuclei has been studied widely both experimentally and theoretically. Many data can be explained in terms of a two-step mechanism consisting of p ¯ annihilation on a single nucleon followed by the interaction of the annihilation products, mostly pions, with the residual nucleus according to standard meson-nucleon physics. However, there is a variety of p ¯-nucleus annihilations which have no relation with those on free nucleons as they require necessarily the involvement of more than one nucleon. This gives rise to “unconventional” hypotheses on the annihilation mechanism. The most appealing one predicts that the presence of several nucleons in the annihilation region c Societ` a Italiana di Fisica

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gives rise to the formation of a quark-gluon plasma (QGP), a state of the matter which is supposed to exist in some early stage of our Universe. A speciﬁc signature of QGP formation is expected to be a “high” production of strangeness. On the other side, an environment rich of strange quarks appears to be favourable to the formation of strange baryons and of antikaon-nucleon bound systems, predicted by several theoretical investigations. The main part of this paper is devoted to recent results on strangeness production in the p ¯-4 He annihilation at rest reported in refs. [1-7]. The rough data were collected with the Obelix spectrometer [8] working at CERN in the years 1990-1996. The investigations contribute to clarify still open problems concerning the formation of quark-gluon plasma, pentaquarks and bound kaon-nucleon systems. Many questions about possible highly excited or unusual states of the hadronic matter are the same as in the heavy-ion collision ﬁeld, see [9]. 2. – An overview of experiments and theoretical predictions In this review the attention is driven to p ¯ annihilation at rest on free protons and on nuclei. The restriction “at rest” minimizes the delivered energy to two nucleon masses ¯ and other hyperonand reduces the number of ﬁnal channels, excluding, for example, ΛΛ antihyperon pairs. . 2 1. p ¯-nucleon annihilation. – When an p ¯ and a p interact, they annihilate producing an energy blob which decays directly or indirectly into pions and kaon pairs. This process is called single-nucleon annihilation (SNA); according with the number of nucleons involved and of the baryonic number of the annihilating system it is also called an A = 1 or B = 0 process. ¯ pair in about 5% of On the average, ﬁve mesons per event are produced, with a KK the events ([10] and references quoted therein). In p ¯p annihilation the average number of π + is 1.5, as well as that of π − , and that of π 0 is 2; the number of π − varies from 0 to 4 with the frequency shown in ﬁg. 1. In p ¯n annihilation the number of π − is 2 and that of

Fig. 1. – π − multiplicity distribution in hydrogen for annihilations at rest and in some nuclei for annihilations in ﬂight (∼ 600 MeV/c) [10]. The latter ones are very similar to those for annihilations at rest.

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Fig. 2. – Fusion of a proton-antiproton pair into a quark blob and decay into pions.

π + 1, due to the electric charge conservation. Pions and kaons are produced directly or indirectly as decay products of heavier mesons (ρ, ω, η, . . . φ, . . . K∗ , . . .). Following [11,12] and [13], we may say that in the annihilation process the two interacting baryons loose their identity and fuse into a highly excited blob, sometimes called a “ﬁreball”, with an energy equal to twice the nucleon mass and a baryonic number B = 0 (ﬁg. 2). In principle, it may consist either of a gas of individual hadrons (hadronic gas, HG) or of a plasma of deconﬁned or free quarks (q, q ¯) and gluons (g) (quark-gluon plasma, QGP). ¯ In the hadronic gas, strange particle pairs can be produced via reactions like ππ → KK. The hadronic gas can be transformed into QGP. In a QGP q, q ¯ and g may interact with each other annihilating and/or producing additional q¯ q pairs and g. Strange quarks can be produced and absorbed via the reactions ¯ ↔ s¯s, u¯ dd u ↔ s¯s and gg ↔ ¯ss. Due to their higher mass, ¯ss pairs are produced with ¯ The study of the features of these energy blobs shows lower probability than u¯ u and dd. that in QGP ¯ss pairs are produced with higher probability (90%) by gg annihilations than by q ¯q annihilations [13]. Moreover, in QGP the strangeness density is expected up ¯ pair in a to 10 times larger than in HG [12]. A reason for this is that the creation of KK conventional hadronic collision requires at least 700 MeV, while the creation of s¯s pair in QGP requires only 2ms ≈ 300 MeV. Non-strange mesons and kaon pairs are emitted after the energy blob deexcites. The variety and energy of the ﬁnal products are determined by the conservation laws of energy, electric charge, baryonic number and strangeness. . 2 2. p ¯-nucleus annihilation. – In the case of an p ¯ impinging on a nucleus, the panorama of observed ﬁnal states is wider than in the case of annihilation on free nucleons. Many ﬁnal states are similar to those on free nucleons, but some are substantially diﬀerent. The latter can be, for instance, those with only one meson, like the Pontecorvo reactions p ¯ d → π − p, p ¯3 He → π − d and pd → KΛ [14]. Moreover, in the third reaction a hyperon is present, which is not allowed in p ¯p annihilation. Some prominent features of p ¯-nucleus annihilation are the following [10]: i) The number of π − per event (or π − multiplicity) varies from 0 to 4 with the frequency shown in ﬁg. 1. The π + , π − and π 0 mean multiplicity decrease with A; in particular, the π ± mean multiplicity decreases from 3.1 to 2.5 and saturates above A = 60 (see ﬁg. 3). The decrease is due to the pion absorption by nucleons.

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Fig. 3. – (a) π ± multiplicity distribution vs. A. The full line is the result of a best-ﬁt calculation. The dashed line is the behaviour expected neglecting FSI [10]. (b) Detail of (a) for A < 18.

ii) The ratio π + /π − decreases quickly from 1 to 0.7 below A = 40 and saturates above (ﬁg. 4). The main reason for this is the increase of neutron excess with A. Correspondingly, the diﬀerence nπ− − nπ+ increases from zero to 0.5. iii) A fraction of the annihilation energy is transferred to the residual nucleons. The meson energy spectra are lowered in the high-energy region and those of the emitted nucleons are enriched above the distribution expected according to the Fermi motion in the nuclei. iv) K0S production decreases from 2% to 1% up to A ≈ 10 and, correspondingly, Λ production increases regularly from 0 to ≈ 2% (ﬁg. 5). Many data can be explained within a conventional framework where annihilation on nuclei develops in two steps: annihilation on a single nucleon (SNA, as described in . subsect. 2 1), followed by the interaction of the mesons produced with the residual nucleus (ﬁnal-state interaction, FSI) (see ﬁg. 6). Less conventional annihilation mechanisms are

Fig. 4. – nπ+ /nπ− and nπ− − nπ+ vs. A for annihilations at rest. The dotted lines give the behaviour estimated neglecting FSI [10].

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Fig. 5. – Inclusive yields (%) for Λ and K0S at rest (•) and at 600 MeV/c (◦) on diﬀerent nuclei [10].

based on the formation of HG or QGP with baryonic number B ≥ 1. The diﬀerent mechanisms may lead to the same ﬁnal states, though with diﬀerent relative branching ratios; this makes it very diﬃcult to disentangle the diﬀerent contributions in the data.

Fig. 6. – p ¯ annihilation of four nucleons. Pictorial representation of (a) single-nucleon annihilation on a proton with the other three nucleons as spectators, (b) single-nucleon annihilation on a proton followed by ﬁnal-state interaction involving the other proton with the two neutrons as spectators and (c) multinucleon annihilation involving directly two protons. The nucleon systems on the right side represent the spectators.

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. 2 2.1. FSI. The mesons emitted may interact with the residual nucleons according to the known physics of the meson-nucleon interactions. As their average momentum is about 300 MeV/c, they may interact in a resonant way and be strongly absorbed by nucleons. A summary of FSI eﬀects is given in [2]. The main eﬀects on strangeness production are the following: i) Antikaons and nucleons can fuse into hyperons: K− p → Λ 0 π 0 , K0 n → Λ0 π 0 , K− n → Σ− π 0 , etc. ¯ 0 , K− ) but not the kaons (K0 , K+ ); hence These reactions involve only the antikaons (K the ﬁnal number of antikaons is reduced with respect to the ﬁnal number of kaons. ii) Kaon-nucleon charge exchange can occur: K+ n → K0 p, K0 p → K+ n. The two reactions deplete and feed, respectively, the K+ population, with negligible global eﬀect. iii) Energetic mesonic resonances may fuse with nucleons producing kaon-antikaon and hyperon-kaon pairs: ωp → Λ0 K+ , ωp → Σ+ K0 , ¯ ωN → NKK, etc. These reactions increase the number of strange-antistrange particle pairs with respect to annihilation on free nucleons and, as far as the charged kaons are concerned, they ¯ pairs or single K+ , but no single K. ¯ produce K+ K SNA followed by FSI, including reactions like the above ones, involves several nucleons. It is therefore a particular B > 0 process. This annihilation scheme has been followed . in a number of theoretical studies (see [15-19], papers quoted in subsubsect. 2 2.2 and references quoted therein). . 2 2.2. HG and QGP formation. When a p ¯ impinges on a system of A nucleons, it may fuse with them forming a blob with energy (A + 1)mp and baryonic number B = A − 1 ≥ 1. This process will be called multinucleon annihilation (MNA). The energy blob may consist of HG or of QGP. For the sake of simplicity, we consider in the following the case B = 1.

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Fig. 7. – On the left side: (a) kaon production as a function of the number of annihilating nucleons. (b) Kaon production per annihilating nucleon. (c) Antikaon production. The diﬀerence between (a) and (c) gives the production of kaon-antihyperon pairs [16]. On the right side: ratio between the number of strange antiquarks and the total number of antiquarks per event (R = ¯s/¯q) as a function of the baryonic number B of the energy blob [16].

Assuming that this energy blob evolves similarly to the B = 0 one described in . subsect. 2 1, it is reasonable to expect that the higher available energy allows the creation ¯ or s¯s pairs higher than in the B = 0 energy blob. As a consequence, of a number of KK we expect also that the deexcitation delivers a higher number of strange particle pairs per event. Finally, the baryonic number conservation law requires that a baryon (nucleon or hy¯ pairs, also YK pairs peron) be present among the ﬁnal products; therefore, beside KK ¯ can be produced, although YY emission is forbidden. The de-excitation products of the B = 1 energy blob may interact with the residual nucleus as described in subsub. sect. 2 2.1. However, we stress that MNA without FSI can produce ﬁnal states similar ¯ pairs). to those given by SNA plus FSI (in particular more strangeness and YK Strangeness production from B ≥ 1 energy blobs formed by p ¯ annihilation has been studied by [13] and [16-18] assuming statistical mechanisms of the blob de-excitation. The main theoretical predictions concerning strangeness emission from HG are the following: i) The relative probability of forming B = 1 energy blobs may be far from negligible (say 10%) [16, 18]. According to [16] (see ﬁg. 7), kaon production (i.e. total strangeness production) as function of the number A of nucleons involved in the annihilation increases more than three times from A = 1 to A = 2, and at lower pace when A increases. At the same time, kaon production per nucleon increases from A = 1 to A = 2 and then decreases. Antikaon production is almost constant and much lower than kaon production for A ≥ 2. The ratio of antistrange quarks to the total antiquark content of the produced particles R = ¯s/¯ q increases with B by a factor of seven between B = 0 and B = 4 [16] (see ﬁg. 7). Correspondingly, the fraction of non-strange channels decreases with B. In

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Fig. 8. – Total strangeness per baryon number as a function of the nucleon number density with A/Vρ0 as a parameter. In all cases T = 160 MeV and ρ0 = 0.17 fm−3 [13].

this study, the strangeness increase with A is essentially due to the channels containing ¯ ¯ channels are depleted. This is essentially due to the fact that the YK, while the KK transformation of a nucleon into a hyperon requires only 35 to 50% of the energy necessary to create a kaon [16]. ii) According to [13], strangeness production per nucleon depends sensitively on the nucleon density. In annihilation on light nuclei the strangeness abundance may increase remarkably between A = 1 and A = 6, and is quite constant for A ≥ 7 (ﬁg. 8). In particular, the strangeness production increases by a factor of about 10 going from A = 1 to A = 4. This result can be taken as an upper limit on strangeness production in the hadronic picture. Any substantial excess would be a signal of QGP occurrence. Note that the behaviour of strangeness per nucleon is quite diﬀerent from that predicted in ﬁg. 7 (line b). iii) Nuclei are rather transparent to K+ so that K+ features should reﬂect unambiguously the mechanism of their production [18,19]. For this reason the measurement of K+ is more convenient than that of K0S , which are the only ones measured up to now. In fact ¯ 0 , and K ¯ 0 are strongly absorbed. the latter ones are a mixture of K0 and K iv) If the energy and baryon density is high enough, the energy blob may become a QGP, where the individual hadrons dissolve in a new phase consisting of almost-free quarks and gluons. Strange quarks (s¯ s) are produced mainly by gluon-gluon annihi¯ and ΛK pairs. The occurrence of QGP lations [20]. They appear ultimately as KK (or of other unconventional phenomena) would be revealed by a strong enhancement of strangeness production over the statistical estimations. It was suggested that the best way to look for HG or QGP is to measure KS0 , Λ and + K production with a systematic study of annihilation on light nuclei with A = 2, 3, 4, etc. [12, 19]. Indeed, some experiments concerning p ¯ annihilation on nuclei with A = 3–208 at rest and in ﬂight (pp¯ = 600–4000 MeV/c) were carried out (see [10]). However, K0 and Λ yield can be explained in terms of annihilation on a single nucleon followed by

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Fig. 9. – Decuplet of baryons made of u, d and s quarks.

rescattering and does not require any exotic process [17, 19, 21], although [22] recognizes in the p ¯-Ta annihilation at 4 GeV/c the signatures of QGP formation. Finally, data on p ¯d annihilation were interpreted theoretically partly within a conventional frame, partly in an exotic frame and partly in both frames [13, 23-28]. All considered, the formation of high-energy blobs is still uncertain. . 2 2.3. Pentaquark. The well-established hadrons are either combination of three valence quarks (baryons) or of a quark and an antiquark (mesons), but the theory of the strong interactions allows for other types of hadrons, for instance baryons made of four quarks and one antiquark (pentaquarks, see ﬁg. 9). The existence of pentaquarks has been investigated for more than 30 years without convincing experimental evidence, but recently a theoretical paper [29] has reawakened the interest on it and has led to a new round of experimental investigations. Reference [29] predicts an exotic baryon with quark content uudd¯s, strangeness S = +1, mass about 1530 MeV, width less than 15 MeV and decay modes into nK+ and pK0 . These predictions are based on the chiral soliton model, but alternative explanations have also been suggested in [30-32] invoking “molecules” of various tightly-coupled quark conﬁgurations, and in [33-35]. Following these speculations, many experimental groups have looked for signals of the predicted baryon employing various beams, targets and detector conﬁgurations and a considerable number of them have observed a narrow resonance (called Θ+ ) either in the nK+ or in the pK0S invariant-mass distribution [36-49]. The measured values of the Θ+ mass and width are spread as shown in table I, where also the relevant statistical signiﬁcance is quoted. Most of the measured widths are consis-

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Table I. – Summary of experiments which reported Θ+ observation. X = anything; (∗ ) A = ∗∗ ) A = p, d, Ne. The statistical signiﬁcance is evaluated according to the relation [50] C, Si, Pb; (√ S = NΘ+ / Ntot + Nb , where NΘ+ is the number of signal events, Nb is that of the background events under the signal and Ntot is the sum of the previous numbers (ref. [50]). Experiment

Reaction

Mass (MeV)

Width Γ (MeV)

S

Ref.

a

ZEUS

ep → (pK0S )X

1522 ± 3

8±4

4.7

[45]

b

HERMES

γd → (pK0S )nsp

1526 ± 3 1528 ± 2.6 ± 2.1

13 ± 9 8±2

2.7

[43] [44]

c

SVD-2

pA → (pK0S )X(∗ )

1526 ± 3 ± 3 1523 ± 2 ± 3

< 24 < 14

2.4 5.9

[46a)] [46b)]

d

NOMAD

νA → (pK0S )X

1528.7 ± 2.5

Few MeV

2.7

[48]

e

JINR

p(C3 H8 ) → (pK0S )X

1540 ± 8

9.2 ± 1.8

4.1

[47]

f

COSY-TOF

pp → (pK0S )Σ+

1530 ± 5

18 ± 4

3.7

[45b)]

g

ITEP

νμ (¯ νμ )A → (pK0S )K− X(∗∗ )

1532.2 ± 1.3

< 12

3.5

[41]

1539 ± 2

<9

2.7

[38]

1540 ± 6

< 25

4.3

[40]

h

DIANA

K Xe → +

(pK0S )X

γp → (nK

+

)K0S

i

SAPHIR

l

LEPS

γ 12 C → (nK+ )K− X γd → (nK+ )K− X

1540 ± 10 ≈ 1530

< 25

2.6

[36] [37]

m

CLAS-d

γd → (nK+ )K− p

1542 ± 5

< 21

3.5

[39](a)

n

CLAS-p

γp → (nK+ )K− π +

1555 ± 10

< 26

3.9

[42]

o

JINR

np → (nK+ )pK−

1541 ± 5

< 11

1560.0 ± 3.7

Very narrow

OBELIX

(a)

p ¯ He → → → 4

(pK0S )K0S X (pK0S )K− X (pK0S )ΛX

[49] 2.7

[5]

These results have not been conﬁrmed by a recent analysis of high-statistics data by the same group [71].

tent with the experimental resolution, therefore they are upper limits for the true width, which might be much narrower (≤ 1–4 MeV) [51]. In spite of the above results, the existence of the Θ+ pentaquark is still under discussion for theoretical and experimental reasons [50-57]. From the experimental point of view, the main reason of perplexity is that there is also a considerable number of experiments that have not observed it [58-70]. Moreover, two groups have conﬁrmed their previous observations of Θ+ [37, 46b)], while another one has found no signal analyzing new data samples with high statistics [71].

Strangeness production in antiproton-4 He annihilation at rest

169

Fig. 10. – Calculated K− N and K− -nucleus potential and bound levels. The shaded zones indicate the widths; from ref. [73].

. ¯ 2 2.4. K-few nucleon bound states. Recently a considerable interest has arisen about the possible existence of strange (S = −1, −2) systems composed by nucleons strongly ¯ called deeply-bound K ¯ nuclear states (DBKS). The production of bound to one or two K, DBKS is connected with the possible existence of unusual nucleon bound states like pp and ppp, with the possibility that a high-density nuclear medium will be created around ¯ that could be a seed for the understanding of the dense nuclear matter in the the K, neutron stars ([72] and reference quoted therein). ¯ Akaishi and Yamazaki [72-74] have derived a K-nucleus potential from a phenomeno¯ ¯ ¯ scattering lengths, X-ray logical K-nucleon (KN) potential so as to account for free KN shifts of the kaonic hydrogen atom and the energy and width of the Λ(1405). Λ(1405) is assumed to be a bound K− -p system rather than an excited three-quark state with = 1. The I = 0 potential, which turns out to be much deeper than the usual nucleon-nucleon potential, produces the unstable bound state Λ(1405) with a binding energy of 27 MeV and width of 40 MeV (ﬁg. 10), while the I = 1 interaction does not produce any bound state. The predicted binding energies are rather large (of the order of −100 MeV) and their widths quite narrow (20–30 MeV) (see ﬁg. 10 and table II). The widths turn out to be narrow as the binding energies are so large that the main decay channel K− p(I = 0) → Σπ is forbidden energetically and the decay to Λπ is suppressed by the isospin selection rules. The predicted central nucleon density is 4–9 times as much as the normal nuclear ¯ clusters should density (ρ0 = 0.17 fm−3 ); at these values one may also argue that the K ¯ be in a deconﬁned quark-gluon plasma. The binding of two K turns out to be stronger ¯ than that of one K.

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G. Bendiscioli

¯ clusters. M = total mass (MeV), EK = total binding Table II. – Summary of predicted K energy (MeV), ΓK = decay width (MeV), ρ(0) = nucleon density at the centre of the system (fm−3 ), Rrms = root-mean-square radius of the nucleon system (fm). ¯ cluster K

M c2 (MeV)

EK (MeV)

ΓK (MeV)

ρ(0) (fm−3 )

Rrms (fm)

pK− ppK− pppK− ppnK− ppppK− pppnK− ppnnK−

1407 2322 3211 3192 4171 4135 4135

27 48 97 118 75 113 114

40 61 13 21 162 26 34

0.59 0.52 1.56 1.50 1.68 1.29

0.45 0.99 0.81 0.72 0.95 0.97 1.12

ppK− K− ppnK− K− pppnK− K−

2747 3582 4511

117 221 230

35 37 61

2.97 2.33

0.69 0.73

¯ clusters have been proposed. For Diﬀerent ways to put in evidence the existence of K example, we mention the following ones. i) The study of K− -nucleus reactions like K− + A →K− A + π − [73, 75]. ii) The study of the discrete component of the energy distribution of the neutrons emitted in reactions where the antikaons are captured at rest, like K− (at rest)+ 4 He →K− 3 He + n [72, 75]. iii) The study of proton-nucleus reactions like pA → ΛpX [76]. iv) The study of K− -nucleus interactions where a nucleon N is knocked out and the − K is captured by the residual nucleus: K− + A → (K− + B) + N →K− B + N [75, 77]. v) Bound states are expected to be created in heavy ion collisions through diﬀerent mechanisms: for instance, in a collisional capture process and directly from a quarkgluon plasma (QGP). In the former case, individual antikaons can be captured by ¯ clusters as in i) (cascade collisional nucleons producing hyperons and then forming K capture processes). Alternatively, if a quark-gluon plasma (QGP) is produced in the ¯ clusters (direct formation of K ¯ ion collisions, the s-quarks can act as seeds for K clusters from QGP) [75, 78]. Preliminary results were already obtained by the FOPI experiment [79, 80] in the study of Ni + Ni and Al + Al collisions with the production of ΛX (X = p, d, t, etc.). In particular, refs. [81] and [82] have stressed that bound states could be revealed indirectly by the production of pairs of K+ . K+ have a probability ¯ in particular they cannot be of interacting with the nuclear medium much less than K, absorbed by nucleons, while the antikaons can be absorbed to form hyperons. Therefore K+ pairs (S = 2) could carry information on the story of the associated S = −2 systems. However, the appearance of K+ pairs is a necessary but not suﬃcient condition for the existence of bound states and the associated S = −2 systems could be produced directly from QGP [72] or via other mechanisms. The production of K+ pairs is expected also

Strangeness production in antiproton-4 He annihilation at rest

171

in antiproton annihilations at rest on nuclei [81, 82] and was indeed observed in a Xe bubble chamber experiment [83]. The existence of DBKS has been investigated also by theoretical approaches diﬀerent from [73] and [74], with the result that the expected binding potentials are shallower and the widths of the bound states so large as to prevent their experimental observation [84-88]. Signals with features compatible with the theoretical expectations for the light nuclei have been reported for the K− ppn (I = 0) system in an experiment [89-91] with K− stopped in a liquid 4 He target and exploiting the missing-mass method. A signal corresponding to a (K− pp) aggregate has been reported by Agnello et al. [92] in another experiment with K− stopped in thin solid targets of light nuclei (6 Li, 7 Li, 12 C) and exploiting the invariant-mass (IM) method. The reported binding energy and width +3 +14 +2 are B = −115+6 −5(st) −4(syst) MeV and Γ = 67−11(st) −3(syst) MeV, respectively. Finally, − a hint for the existence of 15 binding energy of −90 (or −130) MeV has K− O with a K been reported by Kishimoto et al. [93] by studying the (K− , n) reaction at 930 MeV/c, again based on a missing-mass analysis. However, these experimental results have been interpreted also without recalling the existence of DBKS. In particular, Magas et al. [94] have put forward the hypothesis that the results from [92] could be explained in terms of two-nucleon K− absorption and rescattering of the produced particles. Reference [76] has observed structures in the Λp and Λpp invariant mass distributions in p-12 C interactions in a propane bubble chamber exposed to a 10 GeV/c proton beam; they have been interpreted as the decay product of exotic (S = −2) dibaryons H0,+ [95]. Clearly, the quest for the existence of such states is still open and experimental conﬁrmations are eagerly awaited. . 2 2.5. Antiproton-4 He annihilation a) According to the theoretical studies mentioned previously, a quark-gluon plasma may occur with higher probability if p ¯ annihilate in a high-density region of the nuclear matter, i.e. close to the centre of the nucleus. Antiprotons may annihilate in ﬂight and at rest. Since the annihilation cross-section is higher at lower p ¯ momenta (ﬁg. 11), p ¯ can reach the high density region only if they have high enough an initial momentum, which decreases by scattering as the p ¯ moves into the nuclear matter. Low-momentum p ¯ have a high probability to annihilate on the nuclear surface, where the density is low. Annihilations on the surface occur with high probability also in annihilations at rest, which develop as follows. Antiprotons, slowed down traversing the target material, are captured in atomic levels forming antiprotonic atoms and ﬁnally reach by a de-excitation cascade the nuclear surface, where annihilate. In this context, favourable conditions for the formation of B ≥ 1 ﬁreballs are oﬀered by the annihilation at rest on 4 He nuclei, where the nucleons are strongly bound in a small volume (radius ≈ 1 fm) in the same quantum state. They are so close to each other that very likely they are all involved if one of them is “touched” by the p ¯. Note that the average formation length for annihilation pions, that is the length travelled freely by two quarks before condensing into a pion, is of the order of the radius of the 4 He nucleus ([4]

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G. Bendiscioli

Fig. 11. – The p ¯ -nucleus annihilation cross-section vs. p ¯ antiproton momentum for diﬀerent nuclei [10].

and references quoted therein). After all, the description of annihilation in terms of two successive distinct processes like SNA and FSI seems to ﬁt poorly. Finally, the small number of nucleons facilitates the identiﬁcation of the ﬁnal products of annihilation (p, d, t and 3 He, [96]) and hence of B = 0 and B ≥ 1 processes. b) Annihilation on 4 He has been investigated by the Obelix Collaboration [1-4] with a particular attention to the B = 0 and B ≥ 1 annihilations, and in a streamer chamber experiment, most results of which are quoted in [10]. Clear evidence of B > 0 annihilations without strangeness production was found both in four-prong events and in ﬁve-prong events: in particular, baryonic resonances (Δ++ → π + p, Δ0 → π − p), Pontecorvo-like reactions (2π − 3π + , 2π − 3p, etc.) [2] and meson absorption on nucleon pairs [1]. Kaon production was investigated in [3, 4]. It has been observed that in some reaction channels the strangeness production is deﬁnitely higher than that predicted by intranuclear cascade models and than that measured in heavy-ion collisions. This indicates that the energy blob produced by the annihilation involving more than one nucleon is an environment rich of strange quarks, as demonstrated by the high production of π + π − π − K+ and π + π + π − K− (higher by a factor of 22 and 13, respectively, than the production in annihilation on hydrogen). Although these results cannot be considered a deﬁnitive evidence of the formation of QGP, they are surely in favour of it. These features suggest that the environment produced by the p ¯4 He annihilation could be suitable for the creation of bound nucleon systems with strange content, similarly to the case of the heavy-ion interactions.

Strangeness production in antiproton-4 He annihilation at rest

173

Fig. 12. – Obelix spectrometer: view with a lateral supermodule of the EM calorimeter shifted in maintenance position. (1) Open-ﬁeld magnet; (2) target (diameter 6 cm, length 60 cm); (3) spiral projection chamber (SPC, external diameter 30 cm, length 60 cm); (4) jet drift chambers (JDC, internal diameter 40 cm, external diameter 160 cm, length 140 cm); (5) high-angular-resolution gamma-ray detector (HARGD, four supermodules 300 × 400 × 80 cm3 each); time-of-ﬂight system: (6) internal scintillator barrel (internal diameter: 36 cm, thickness: 1 cm, length: 80 cm, 30 elements), (7) external scintillator barrel (internal diameter: 270 cm, thickness: 4 cm, 84 elements) [8].

3. – Investigation of the p ¯-4 He annihilation at rest with the Obelix spectrometer . 3 1. Spectrometer and data collection. – The raw data were collected with the magnetic spectrometer Obelix [8] using the 105 MeV/c antiproton beam extracted from LEAR at CERN. The spectrometer (see ﬁgs. 12 and 13), with cylindrical symmetry around the beam line coincident with the magnetic ﬁeld axis, consisted of a gas target, a spiral projection chamber (to measure event vertex and prong multiplicity and to detect slow particles without measuring the momentum) and two scintillator barrels (for time-of-ﬂight measurements) separated by six jet-drift chambers (for momentum, trajectory length and speciﬁc ionization measurements). The angular acceptance of the apparatus was Ω/4π = 70%. The error on the momentum measurement depends on the momentum value and orientation and is of the order of 2% [8]. The beam was slowed down to stop in the target centre. The 4 He target was at NTP, the H target at three atmospheres. Since the detector was optimized for meson spectroscopy studies, i.e. the detection of several particles originating from a single vertex, both the hardware and the software were not suited to the identiﬁcation of

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G. Bendiscioli

Fig. 13. – Section of the magnet with the behaviour of the magnetic ﬁeld lines.

secondary vertices. Then the detection of long-living particles like Λ and K0 was limited only to those decaying within a ﬁducial volume around the primary vertex, a sphere with a radius of 5 mm. Events with four prongs (two negative and two positive tracks) and with ﬁve prongs (two negative and three positive tracks) connected to the annihilation vertex were considered. The total numbers of four and ﬁve prong events in helium are 238 746 and 47 299, respectively. The four prong events in hydrogen are 607 381 and serve as reference, as they concern annihilations on a single nucleon. The apparatus had a limited acceptance in the low momentum region: the minimum momentum measured by the four jet-drift chambers is ∼ 80 MeV/c for pions, ∼ 100 MeV/c for kaons and ∼ 150 MeV/c for protons. For the kinematical calculations the momenta were corrected for the energy loss in the materials between the event vertex and the jet-drift chambers. The minimum momenta at the vertex are ∼ 100 MeV/c for pions, ∼ 150 MeV/c for kaons and ∼ 300 MeV/c for protons. The analysed reaction channels are the following: Measured four prongs (1)

p ¯ p → 2π + 2π − X, π + π − π ∓ K± X, π + π − K+ K− X,

(2)

p ¯4 He → 2π + 2π − ps 2nX, π + π − π ∓ K± ps 2nX, π + π − K+ K− ps 2nX,

Measured ﬁve prongs (3)

p ¯4 He → 2π + 2π − p2nX, π + π − π ∓ K± p2nX, π + π − K+ K− p2nX,

where X stands for any undetected neutral mesons like π 0 and K0L , 2n are undetected neutrons, p is a fast proton (with momentum pp > 300 MeV/c) and ps is a slow proton (i.e.

Strangeness production in antiproton-4 He annihilation at rest

175

with momentum < 300 MeV/c, which can be undetected or detected with unmeasured momentum). In the data analyses three basic problems had to be faced: i) the identiﬁcation of the charged kaons, ii) the identiﬁcation of neutral kaons and Λ and iii) the identiﬁcation of annihilations on single- and multi-nucleon annihilations. K± are a small number of the particles (∼ 1%) embedded in a great number of π ± , protons and deuterons; therefore the K± identiﬁcation must be very careful to avoid a heavy contamination by pions and protons. K0S are identiﬁed through decays into π + π − occurring mostly inside the annihilation blob, as secondary vertices are not detected with our apparatus: as a consequence pions may loos memory of their origin due to interaction with the residual nucleons. The same problem exists for Λ detected via the decay into pπ − . Finally, the neutrons and the neutral mesons are not detected, so suitable procedures must be devised to disentangle annihilations with and without neutral mesons and with the involvement of diﬀerent numbers of nucleons. . 3 1.1. Charged-kaon identiﬁcation. The charged particles were identiﬁed through the independent measurements of velocity (β), momentum (p) and speciﬁc energy loss (dE/dx) following the criteria outlined in details in [4]. These quantities are given by independent measurements but, if correctly measured, must show the following correlations: (4) (5)

(6)

dE 4πN z 2 e4 Zc2 2me c2 β 2 2 = ln −β , dx me β 2 A I(1 − β 2 ) p . β= 2 p + M2 ⎡ ⎛ ⎤ ⎞ 2 2 2 p +M 4πN z 2 e4 Zc2 p2 + M 2 ⎣ ⎝ 2me c p2 ⎠ p2 + M 2 ⎦ dE = ln , − 2 2 dx me A p2 p2 I(1 − p +M ) p2

The ﬁrst relation is independent of the mass; hence, if the measurement errors are small enough, the data of all tracks are distributed close to the corresponding functional line. The second relation (as well as the third one) depends on the mass, and describes a family of lines, one for each mass. If the errors are small enough, the data should accumulate around these lines. Unfortunately, the errors (on p, β, dE/dx or on a combination of them) are such that the data are scattered also far from the ideal lines. Moreover, for a lot of tracks β is unphysical, that is β > 1. The scatter plots dE/dx vs. β, β vs. p and dE/dx vs. p for the positively charged particles are given in ﬁg. 14. K+ are visible up to about 450 MeV/c in the β vs. p plot, but not at all in that for dE/dx vs. p. For a number of tracks both dE/dx and β were measured, for other only dE/dx due to the orientation, the length or the momentum. The picking up of the kaons from these scatter plots was made in steps starting with the prongs with measured dE/dx and β. i) A narrow band was selected empirically around the calculated dE/dx vs. β line as shown in ﬁg. 14a; tracks out of the band were neglected in the analysis.

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G. Bendiscioli

Fig. 14. – Tracks with both dE/dx and β measured for positively-charged particles. (a) dE/dx vs. β. The light gray lines represent eq. (1). The dark gray lines delimit the region of the well-measured tracks (see text). (b) dE/dx vs. p and (c) β vs. p. dE/dx is in arbitrary units. The lines in (b) and (c) represent the families corresponding to eqs. (2) and (3) for π + , K+ , p and d, respectively [4].

ii) Since the eﬃciency of dE/dx vs. β decreases as β increases, we used this tool below a maximum β value (βmax ) determined empirically with the help of bubble chamber data on neutral kaons, which are assumed to hold for charged kaons too. Tracks with β > βmax were excluded to be kaons. To ﬁx βmax we had recourse to the fact that pions and kaons produced in p ¯ annihilations at rest in deuterium have practically the same momentum distribution with a maximum around 300 MeV/c (ﬁg. 15a, from bubble chambers). Therefore, they have quite diﬀerent velocity distributions with maxima at β = 0.9 and β = 0.45, respectively (ﬁg. 15b). The large majority of kaons are distributed below β = 0.7 and the large majority of pions above this value; therefore, below β = 0.7

Fig. 15. – K0 and π ± momentum (in GeV/c) distribution (a) and β distribution (b) deduced from p ¯ d annihilation data [4, 97].

Strangeness production in antiproton-4 He annihilation at rest

177

Fig. 16. – Track distribution for positively charged particles in four-prong events after the cut in the dE/dx vs. β plot (see ﬁg. 14a) and with β < 0.7. The diﬀerent particles are accumulated around the corresponding theoretical lines. Kaons are denoted as points between the dotted lines [4].

a possible contamination of the kaon distribution by pions is strongly reduced. Note that also the number of pions below βmax is much higher than the total number of kaons, as these are present in only 5% of all annihilations. However, most incorrect mass identiﬁcations (with particular attention to false kaons in the annihilations without π 0 production) . can be removed by the kinematical event analysis described in subsubsect. 3 1.2. Tracks with β > βmax were excluded to be kaons. iii) Considering only the tracks within the band in the dE/dx vs. β plot chosen at point i) (ﬁg. 14a) and with β < βmax = 0.7, the dE/dx vs. p and β vs. p plots appear as in ﬁg. 16, where the kaon distributions are clearly visible. Moreover these tracks appear to be kaon tracks in both plots. In this way we have identiﬁed a set of “high quality” tracks which allow us to deﬁne two kaon bands in the two scatter plots dE/dx vs. p and β vs. p. The band size in ﬁg. 14a (as well as those in ﬁg. 16) was chosen as a compromise between a high reduction of the contamination by pions and protons and the preservation of the statistics. Thus in the β vs. p plane and in the dE/dx vs. p plane we can recognize regions where the kaon presence is more probable and the contaminations by other particles lower. If dE/dx and β give results in disagreement, we assume that β prevails on dE/dx. Clearly, the mass identiﬁcation for the latter tracks is less reliable. iv) As far as the tracks without the β measurement are concerned, in the scatter plots dE/dx vs. p the tracks in the kaon band deﬁned in ﬁg. 14a were considered and, assuming for the mass the kaon mass, the dE/dx vs. βcalc scatter plot was built up, where βcalc is the velocity deduced from the measured p value. βcalc plays a role similar (not exactly the same) to that of the measured β. The tracks are assumed to be kaons if they fall in the bands of the dE/dx vs. p and dE/dx vs. βcalc plots deﬁned previously.

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G. Bendiscioli

. 3 1.2. Identiﬁcation of single- and multi-nucleon annihilations a) Kinematics In order to identify annihilations involving one or several nucleons and to put in evidence the diﬀerent roles of the unseen particles (neutrons and neutral mesons), a kinematical analysis of each event according to the criteria outlined in [2] and [4] is powerful. We denote with A the number of nucleons of a nucleus involved in the annihilation process; the reminder nucleons are assumed to behave like spectators and are neglected in the analysis. As a ﬁrst approximation, their momentum is assumed to be zero. If, for any event, ptot is the total measured momentum (i.e. that of the charged particles including mesons and protons with momentum > 300 MeV/c), Etot is the corresponding total energy (evaluated according the previous mass identiﬁcation), E0 is the total energy of the particles partecipating to the annihilation process and (7)

MU =

2 − p2 EU U

is the invariant mass of the unseen ﬁnal particles (π 0 , K0 , n), the momentum and energy conservation laws (8)

E0 = Etot + EU ,

(9)

0 = ptot + pU

lead to the relation (10)

ptot =

(E0 − Etot )2 − MU2 .

Neglecting the binding energy of the nuleons in a nucleus and denoting with mp the average nucleon mass, in the case of annihilation on 4 He, E0 may vary from ∼ 2mp to ∼ 5mp . The baryonic number of a process (B = A−1) denotes the number of the residual nucleons. For a ﬁxed value of MU , ptot is a decreasing function of Etot and reduces to 0 for Etot = E0 − MU . The slow protons (pp < 300 MeV/c) are included among the spectators. A relation with signiﬁcance and features similar to those of eq. (10) is that 2 − p2 : between ptot and the invariant mass Mtot = Etot tot (11)

ptot =

2 − M 2 ]2 [E02 + Mtot U 2 . − Mtot 4E02

For constant MU , ptot is a decreasing function of Mtot and approaches zero for Mtot = E0 − MU . The physical events are represented in the (ptot , Etot ) plane and in the (ptot , Mtot ) one by points distributed in the left-side arias of the B = 3 lines in ﬁgs. 17-18 and 20; moreover, in each area they are distributed in diﬀerent regions depending on the number and type of neutral mesons and of charge particles and on the number of nucleons involved

Strangeness production in antiproton-4 He annihilation at rest

179

Fig. 17. – ptot vs. Etot for annihilations without (a) and with (b) a fast proton. (a) 4π momentum transfer ptot vs. 4π energy Etot . B = 0 indicates the accumulation point of single-nucleon annihilations without π 0 ; B = 1, 2, 3 indicate the accumulation lines of the multinucleon annihilations. π 0 indicates the accumulation line of the single-nucleon annihilations with production ¯ pp or p ¯pn, B = 2 means p ¯ ppn or p ¯pnn, B = 3 means p ¯ ppnn. (b) 4πpf of one π 0 . B = 1 means p momentum p vs. 4πpf energy E [4].

Fig. 18. – (a) 4π momentum vs. 4π invariant mass M . (b) The B = 1 line is the accumulation site for all the single-nucleon annihilations without π 0 and multinucleon annihilations (for instance, B > 1) with diﬀerent maximum 4π invariant masses (the B = 2 lines represent three decreasing values of M4π max < 2mp ) [2].

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Table III. – Parameters to be introduced in eqs. (10) and (11) in the case of annihilations with detection of four charged mesons without production of neutral mesons and with minimum value for MU (see text) (lines from 2 two 5). Line 6: B = 0 annihilations with production of one π 0 .

in the annihilations, as explained in the following. The right-side areas of the B = 3 lines are unphysical. i) To clarify in a simple way the implications of eqs. (10) and (11), the case is discussed where no fast proton is observed (i.e. one of the 2 protons is a spectator) and the measured charged particles are four pions (2 positive and 2 negative). In this case, annihilations on 4 He may involve 1 to 3 nucleons and the baryonic number B may be 0 to 2. Let us suppose that no π 0 is produced and the residual nucleons are emitted jointly, so that their IM is minimum being simply the sum of their masses (MU = E0 − MUmin ≈ max max Bmp ) and Etot is maximum (Etot = E0 − MUmin ). The value of E0 , MUmin and Etot for the diﬀerent numbers of involved nucleons (or B values) are reported in table III. max One can see that in all cases Etot = 2mp . Therefore, all the lines with diﬀerent MU max described by eq. (10) depart from the point ptot = 0, Etot = 2mp , independently of the value of MU . This is shown in ﬁg. 17a. For A = 1 (or B = 0) the line described by eq. (10) reduces simply to the point ptot = 0, Etot = 2mp . This is the accumulation point for annihilations on free nucleons without production of neutrals. Now let us consider the case where the annihilation is on a single nucleon (B = 0) with emission of one π 0 . With reference to table III, line 6, the line according to eq. (10) has its origin in the point ptot = 0, Etot = 2mp − mπ , as shown in ﬁg. 17a. A similar discussion can be made for eq. (11) too (see ﬁg. 18a). ii) Similar considerations can be made for the annihilations where the measured charged particles are 4 pions and one fast proton (see ﬁg. 17b), three charged pions and two protons and two charged pions and three protons. iii) Annihilations with production of K0 or several π 0 fall on the left-side area of the 0 π line.

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Fig. 19. – ptot (GeV/c) vs. Etot (GeV) for annihilations in 1 H into four charged pions. The B = 1 and π 0 lines are deﬁned in ﬁg. 17a.

iv) The missing of some charged particles has the same eﬀect on the event point position as that of the neutral mesons. In ﬁgs. 17 and 18 the B = 1, 2, 3 lines are plotted assuming that the residual nucleons move jointly, that is without kinetic energy in their centre of mass system, so that their max IM is minimum and Etot maximum; in the contrary case Etot < Etot and the lines are shifted toward the left side (ﬁg. 18b). If MU for a given B is spread within some interval, the B lines are transformed into bands and overlap each other (see ﬁg. 18b). b) Comparison between kinematics predictions and data The ﬁtting between the schemes of ﬁgs. 17 and 18 and the experimental distributions is put in evidence by the examples in ﬁgs. 19, 20 and 21. Figure 19 concerns annihilations in hydrogen, which are, by deﬁnition, single-nucleon annihilations (B = 0). One can see an accumulation spot due to annihilation without neutral meson production around ptot ∼ 0 MeV/c and Etot ∼ 2mN ∼ 1876 MeV. An accumulation due to production of one π 0 along the π 0 line and one accumulation due to several π 0 and/or to K0 production on the left-side region of this line. As expected, no accumulation appears along the B = 1 line, as no neutron is involved. Moreover, events are distributed randomly in the unphysical region on the right side region of the π 0 line. Figure 20 concerns annihilations on helium. There are three sets of data: i) annihilations without fast proton but with one visible slow nuclear particle (they are indicated with 4π1ps ) (ﬁg. 20b); ii) annihilations without fast proton and without visible slow nuclear particle (they are indicated with 4π0ps ) (ﬁg. 20a);

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Fig. 20. – 4π momentum p vs. 4π invariant mass M scatter plots for 4π0ps (a), 4π1ps (b) and 4πpf events (c); 4πpf momentum p vs. 4πpf energy E for 4πpf events (d). The lines are deﬁned in ﬁg. 17 and identify events with one π 0 and without π 0 and with B = 1, 2, 3, respectively. The spot around p(4π) ∼ 0, M (4π) ∼ 1876 MeV in (a) and (b) (missing in (c), see text) identiﬁes events with B = 0. In (d), where the measured fast proton is included, the distribution along the B = 1 line of (c) reduces to the B = 1 spot at P (4πpf ) ∼ 0, E(4πpf ) ∼ 3mN ∼ 2814 MeV. In each plot, the B = 3 line represents the kinematical limit of the events; the events on the right are due to experimental errors or to wrong mass identiﬁcation [2].

iii) annihilations with a measured fast proton (pf > 300 MeV/c; they are indicated with 4πpf ), (ﬁg. 20c and d). The events in the ﬁrst set are like those in the second one; simply, the slow nuclear particles are not visible due to the geometrical detector acceptance or to the lower energy, which forbids them to pass the material between the vertex and the drift chambers. Likely the invisible nuclear particles are mostly tritons, since their energy does not exceed 7 MeV (about 200 MeV/c) [96].

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Fig. 21. – Evidence of the Fermi motion in the scatter plots p4π vs. E4π for annihilations in hydrogen (a) and in 4 He (a) and (b). In (b) the spectator proton is unseen and in (c) is seen. The ﬁgures are details of ﬁgs. 19 and 20b,c.

The experimental points are spread randomly around the ideal accumulation sites deﬁned above due to i) the momentum measurement errors, ii) the interaction among the ﬁnal particles, iii) an incorrect particle identiﬁcation and iv) the Fermi motion of the annihilating nucleons inside the nucleus. In particular, points i) and iii) can populate the unphysical region deﬁned above, where the energy is evaluated in excess: indeed, the energy is overestimated if the measured momenta are higher than the real ones and a heavier mass is attributed to a lighter particle, a kaon mass to a pion or a proton mass to a kaon. Concerning point iv), previously (see eq. (9)) it has been assumed that the total momentum p is zero. In reality this is not the case due to the motion of the antiproton in the antiprotonic atom and, mainly, of the nucleons inside the nucleus. For the sake of simplicity, let us consider the annihilation on a single nucleon without production of π 0 . The initial momentum pp¯N of the annihilating pair is determined by the momentum pF of the nucleon bound in the nucleus and by the p ¯ momentum in the atomic orbit, which is negligible with respect to of the previous one; pp¯N ≈ pF . On the other side pp¯N must be equal to the ﬁnal momentum, which is expressed by the momentum of the charged pions p4π , that is, p4π = pF . Thus the measured momentum distribution reﬂects the momentum distribution of the annihilating nucleon bound in the 4 He nucleus. This momentum is also equal to that of the spectator nucleons. The evidence of the Fermi motion is stressed in ﬁg. 21, which shows details of ﬁgs. 19 and 20a,b. This ﬁgure compares the 4π momentum distributions for annihilations on a single nucleon in hydrogen (a) (no Fermi motion), in helium for events with unseen spectator proton (b) and with seen spectator proton (c). The accumulation spot for the B = 0 events is centred close to zero momentum

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Fig. 22. – Empirical deﬁnition of diﬀerent types of events (see text). The area between the dotted lines include the lines B = 1, 2 and in ﬁg. 14a. The segment separating the B = 0 and B > 0 regions corresponds to ptot = 250 MeV/c.

(< 0.05 GeV/c) in absence of Fermi motion and is centred at increasing momenta (∼ 0.15 GeV/c and ∼ 0.25 GeV/c) while the spectator momentum increases. This considered, the diagrams in ﬁg. 17 are not expected to match fully the data in any case; nevertheless they are very useful as a guide to interpret them as is shown in the next section. c) Event classiﬁcation In practice, the schemes of ﬁgs. 17 have been used as follows. In the ptot , Etot plane four regions are identiﬁed, as exempliﬁed in ﬁg. 22. In the B = 0 region they are accumulated mostly single-nucleon annihilations without neutral mesons production, in the B > 0 one multi-nucleon annihilations without neutral mesons are the majority and in the π 0 K0 region annihilation with neutral mesons prevail. The width of the band containing events without neutral mesons has been suggested by the size of the B = 0 spot in ﬁg. 21. The separation between the B = 0 and B > 0 regions has been set at ptot = 0.25 GeV/c. Clearly, in the reality the diﬀerent event distributions are not separated sharply, but with a smooth transition from one to the other; therefore, the deﬁnition of the regions is somewhat arbitrary. For this reason, we have changed their size in order to estimate systematic eﬀects on the physical results. . 3 1.3. Neutral kaon and Λ identiﬁcation. The production of K0S and Λ in p ¯-4 He annihilations has been measured in a previous experiment carried out with a visualizing technique (streamer chamber) [98]. The K0S and Λ production frequencies (mean number of K0S and Λ per event) have been deduced observing the decays K0S → π + π − and Λ → pπ − far from the annihilation vertex (V events); they are 1.07 ± 0.11% and 1.12 ± 0.12%, respectively. On the contrary, the present data concern decays occurred inside a volume around the annihilation vertex,

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Fig. 23. – π + π − π − K+ plus π + π + π − K− events: dependence of the K0S peak and of the π + π − IM distribution on the number of nucleons involved in the annihilation process identiﬁed by the baryonic number B (left side) and on the presence of neutral mesons (right side) [3].

a ﬁducial sphere of 5 mm of radius. According to [98], less than ≈ 40% of the K0S and less than ≈ 57% of Λ decays occur inside the ﬁducial volume. In the V events, the decay pions and proton do not undergo interactions, while in the latter case they can interact with the other ﬁnal particles in the annihilation energy blob (FSI) losing memory of their origin. Moreover, in the latter case, the pions from K0S are mixed with phase-space pions and pions from other resonances, so that K0S are put in evidence by a peak over a background in the π + π − invariant-mass distribution. This distribution is changed from the annihilation on hydrogen to that on helium due to FSI, as put in evidence in ﬁg. 23: this concerns annihilations on hydrogen with and without production of charged mesons and similar annihilations on helium with B = 0 and B ≥ 1 selected according to subsect. 3.2.1. One can see that the prominence of the K0S peak decreases from hydrogen (pure B = 0 annihilations) to helium with B ≥ 1 and, correspondingly, the distribution

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Fig. 24. – (a) π − π + invariant-mass distribution for all the events (full histogram), the relevant π + π + plus π − π − distribution (dotted histogram) and their diﬀerence. The dotted histogram is normalized in order to erase the diﬀerence on the left-side region. The full vertical lines indicate the K0S and the ρ masses and the dashed ones the ±20 MeV interval around the K0S mass. The total number of entries concerns the full histogram. (b) Signature of K0S in the π + π − invariantmass distribution for the ﬁnal state pπ − π − π + π + with an unseen fast neutron and no π 0 . π − π + invariant-mass distribution (full); π + π + plus π − π − invariant-mass distribution (dotted) and their diﬀerence. The total number of entries concerns the full histogram [5].

shifts toward lower energies increasing B and the presence of neutral mesons. The latter eﬀect reﬂects the pion energy loss due to FSI. Beside FSI and the involvement of several nucleons in the annihilation, the detection of K0S is obstacled by other factors. The π + π − decay mode occurs only in 68.6% of the cases and a large fraction of K0S [98] decays far from the annihilation ﬁducial volume (a sphere of ∼ 0.5 cm). Finally, combinatorial eﬀects increase the background when more than one π + π − pair per event can contribute to the IM distribution. It is evident that the K0S signals may be completely cancelled, but this depends on the type of reaction channel considered. In spite of this, some information on the K0S can be carried out comparing the π + π − invariant-mass distribution with the space-fase like (or background) π ± π ± IM distribution. Consider, for instance, the full set of p π + π − π + π − events used to look for θ+ signals . in subsubsect. 3 2.2. The π + π − IM distributions, shown in ﬁg. 24, do not display any K0S peak. However, a large bump appears in the diﬀerence between the π + π − distribution and the π + π + plus π − π − one. Beside a bump around the K0S mass position there is a bump around the ρ(770) mass, as expected. Note that in this example the evidence of the bumps is reduced by combinatorial eﬀects, which increase the background with four entries for each event, while the number of K0S per event is one or two and that of ρ is one. Diﬃculties greater than or similar to those mentioned for the K0S are encountered in the Λ identiﬁcation through the pπ − decay mode.

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Fig. 25. – K0S peak in the annihilation reaction p ¯ p → 3π ± K± [5].

. 3 1.4. Apparatus resolution. In order to select intervals in the π − π + IM distribution richer of K0S and intervals in the pπ − IM distribution richer of Λ, the resolution of the apparatus has been measured by ﬁtting the K0S peak in the distribution shown in ﬁg. 25. This calculation concerns annihilations in hydrogen for events with production of K± . These are favourable conditions as the momenta of the π ± pair are not aﬀected by rescattering on the ﬁnal particles and the π ± pair background is reduced. The resolution becomes worse in 4 He and in the absence of charged kaons, as pointed out in [3] and stressed in ﬁg. 23. The calculation with a polynomial to describe the background plus a Gaussian to describe the K0S peak led to m(K0S ) = 494.08 ± 0.14 MeV and σGaussian = 9.3 ± 0.7 MeV [5]. As K0S is very narrow, σGaussian = 9.3 MeV (or Γ(FWHM) = 22 MeV) is the intrinsic resolution of our apparatus. This result is in agreement with that obtained in the study of the annihilation p¯p → KS KL [99]. Concerning Λ, it is not produced in p ¯ annihilation at rest on protons and no information on the resolution on the Λ identiﬁcation is available from previous measurements of the annihilation on 4 He. We can argue some information from the resolution on the measurement of the K0S (see above), φ, ω and η meson masses obtained with our apparatus. In [3] evidence of the narrow φ(1020) meson was found in the K+ K− IM in p ¯ -4 He annihilations and the width (FWHM) turns out to be about 21 MeV. In a search on the n ¯-hydrogen annihilation it was found Γφ = 12 ± 9 MeV, Γη = 21 ± 4 MeV and Γω = 45 ± 3 MeV [100]. . 3 2. Experimental results . 3 2.1. Excess in strangeness production [3, 4] a) Introduction In order to study the dependence of the strangeness production on the number of nucleons involved in the annihilation process, the production of K± , K0s , K0∗ and φ in annihilations on H2 and 4 He has been investigated. Here only the most interesting results

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are reported, which concern the production of K± . The following reactions have been . selected from those listed in subsect. 3 1: Four prongs (12)

p ¯ p → π + π − π ∓ K± X, π + π − K+ K− X,

(13)

p ¯4 He → π + π − π ∓ K± ps 2nX, π + π − K+ K− ps 2nX.

Five prongs (14)

p ¯4 He →, π + π − π ∓ K± ps 2nX, π + π − K+ K− p2nX,

where X indicates possible neutral mesons. π ∓ , K± and p are detected through speciﬁc . energy loss, momentum and velocity measurements as described in subsubsect. 3 1.1. The 2π + 2π − , π + π − π ∓ K± , π + π − K+ K− systems will be indicated also with the simpler notations 4π, 3πK and 2π2K, respectively. Moreover, the set of reaction with production of positive kaons (i.e. 3πK+ ) will be indicated with K+ , that of negative kaons with K− . b) Charged-kaon production The goal of the analysis was a comparison between the strangeness production in 4 He events with that in hydrogen. Annihilations in 4 He may involve diﬀerent numbers of nucleons, as can be put in evidence by the event positions in the ptot , Etot plane (see . subsubsect. 3 1.2), while annihilations in hydrogen involve only one nucleon. To this aim, 4 the events in He were divided into diﬀerent sets according to (a) the presence or absence . of neutrals, (b) four or ﬁve prongs, (c) B = 0 or B ≥ 1 as described in subsubsect. 3 1.2. For each set the “normalized” ratio (or enhancement factor) RN = (frequency of events in 4 He with B = 0, or B = 1 or B > 1/(frequency of events in 1 H with B = 0 by deﬁnition) has been evaluated: RN (Y, He, B) = R(Y, He, B)/R(Y, H, B = 0), where Y indicates 3πK+ , 3πK− , 2π2K, 3πK+ X, 3πK− X, 2π2KX. The RN values are reported in ﬁg. 26. The errors on the normalized ratios were estimated in ref. [4] and include statistical and systematic eﬀects. The latter ones depend on the criteria adopted for the mass iden. tiﬁcation and on the deﬁnition of the regions in the p vs. E plane (see subsubsect. 3 1.2). Diﬀerent assumptions produce diﬀerent data samples of a same reaction, which diﬀer for the statistics and for the contamination level by wrong mass identiﬁcations. Nevertheless, the normalized ratios deﬁned above are little sensitive to the change of the event selection and in all cases the total error is less than ±15% [4]. Figure 26 shows that: i) In the events without neutrals, RN increases strongly both for 3πK+ and 3πK from pure B = 0 annihilations (H target) to a mixture of B = 0 and B ≥ 1 annihilations (four-prong events in He) and to pure B ≥ 1 annihilations (ﬁve-prong events in He) and decreases for 2π2K; 3πK+ increases about twice 3πK; RN for K+ and K− is near

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Fig. 26. – Charged kaon production. RN = ratio between He and H yields (see text). (a) 4π inclusive events with neutrals; (b) 4π exclusive events without neutrals. The lines join RN values concerning the production of 3πK+ , 3πK− , 2π2K in reactions with diﬀerent numbers of prongs (four or ﬁve) and B values: (a) and (b) B4 = 0 (H), B4 ≥ 0 (He), B5 ≥ 1 (He); (b) B4 = 0 (H), B4 ∼ = 0 (He), B4 ≥ 1 (He), B5 ≥ 1 (He) [3].

constant with B. This was pointed out and discussed in [4]. Note that the lower increase of the K− production must be attributed to the K− absorption by nucleons. ii) In the events with neutrals, RN increases weakly for 3πK+ and is constant for 3πK− ; RN for 2π2K behaves like for the events without neutrals. This aﬀects RN for all the events, where the RN values for 3πK+ , 3πK− , K+ and K− are smaller than in the case of the events without neutrals. c) Discussion i) The most interesting result of the analysis is the observation of the strong increase of the strangeness production in the reaction channels 3πK+ , 3πK− and 2πφ (not shown in ﬁg. 26). ii) As observed in refs. [101] and [19], signals of phenomena leading to a high strangeness production may not be displayed by any annihilation reaction, but only by speciﬁc channels. This is conﬁrmed by our data, where, for example, RN for K+ increases strongly in the 3πK+ channel (by a factor of 22) and weakly in the 3πK− X channel (by a factor of 2) and decreases in the 2πK+ K− channel. As a consequence, the total K+ production increases by a factor of 4.5, much less than 22. Therefore, the measured strangeness increase depends strongly on the reaction channel detected. iii) Extrapolating the predictions of [13] and [21] in the HG framework, it would require more than four nucleons participating to the annihilation. In this framework, it is natural to conclude that the value of 22 may be a signal of the occurrence of a QGP phase. iv) In any annihilation energy equal to about 1876 MeV is delivered, independently of the way by which the process develops. In B ≥ 1 annihilations some energy amount is delivered to the nucleons and subtracted to other particles. This reduces the phase space of the mesons and, the higher the meson mass is, the smaller is the probability that it is emitted. The minimum energy required for the production of K0s K0s is about

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¯ 0∗ K0∗ about 990 MeV, for πK0s K± about 1100 MeV, for ππφ about 1300 MeV and for K 0∗ 0∗ ¯ 1782 MeV. The large fraction of the total energy required by K K explains the strong decreasing of 2πK+ K− production from B = 0 to B ≥ 1. This argument explains also why the strangeness production with neutrals (π + π − π ∓ K± X, see ﬁg. 26) increases less than that without neutrals (π + π − π ∓ K± ). i) An additional comparison term for “high” strangeness production is given by the experiments on high-energy heavy-ion collisions. The experiments show that the ratio (strange particles)/(non-strange particles) is higher than in the nucleon-nucleon collision and the increase depends on energy, size of the interacting nuclei and collision geometry (i.e. peripheral or central collisions). However, in spite of a great lot of data and theoretical eﬀorts, the origin of the increase is not yet fully understood [9, 102]. Compared with the nucleon-nucleon collision, the ratios K+ /π + , K− /π + and φ/π + obtained in C-C, Si-Si, S-S and Pb-Pb collisions at 158 A GeV are higher by an enhancement factor between ≈ 1.2 and ≈ 2.5; for the same nucleus-nucleus interaction, the ratio is higher for central collisions, which involve a higher number of nucleons, than for peripheral collisions. The dependence of the strangeness production in central Pb-Pb collisions on the interaction energy has been investigated through measurements at 40, 80 and 158 A GeV. The ratio (K+ /π + )Pb-Pb /(K+ /π + )pp turns out to be higher at 40 A GeV (about 2.5). Taking into account also data on Ag-Ag collisions at energies below 40 A GeV [103], the K+ /π + ratio displays a steep increase in the low-energy region up to a maximum around 40 A GeV followed by a decrease to a near-constant behaviour. This characteristic energy dependence is quite diﬀerent from that observed in the nucleon-nucleon collision, which increases monotonically [104]. The steepening has been interpreted as a signature of the activation of a transient state of deconﬁned matter [105], but other explanations have been given [9]. For any nucleus-nucleus collision, the ratio (strange particles)/(nonstrange particles) is deduced for event subsets with diﬀerent numbers of participants and increases with this number. However, within each subset no exclusive reaction channel is selected and this may hide possible signals of high strangeness production, as shown in the helium case. The above enhancement factors have a meaning similar to that of the normalized ratios RN for the reactions in He with production of K+ , where the number of π + per event is exactly one. In conclusion, the enhancement factors for K+ from He (B ≥ 1) reactions are deﬁnitely higher than those obtained in the heavy-ion experiments. A similar observation holds also for the K− production from He (B ≥ 1) reactions without neutrals. The diﬀerent strangeness production in annihilation at rest on hydrogen and helium could arise also from the diﬀerent angular momentum of the initial states. Eﬀectively, while in hydrogen at three atmospheres annihilations proceed with nearly the same probability from P and S-levels, there are indications that in helium at NTP the S-level prevails [2]. On the other side, the selection rules based on angular momentum, parity, G-parity and isotopic spin are much more selective on two- and three-body systems than in more numerous systems, like our ﬁnal states with at least four particles. Therefore, we think that the

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selection rules do not aﬀect substantially the diﬀerence of the strangeness productions. d) Conclusions Strangeness production increases remarkably with B in the π + π − π ∓ K± channels (without neutrals); in the π + π − π − K+ X channel increases moderately; in the π + π − π + K− X one is constant and in all other channels decreases. The maximum increase (by a factor of (more than) 22) is observed for the exclusive 3πK+ (B ≥ 1) channel. K+ give the cleanest information on the strangeness content in the energy blob from which they have origin as they undergo weakly to ﬁnal-state interaction and have no competition with hyperon production. Our analysis shows that inclusive annihilation reactions may be not suitable to put in evidence high strangeness production associated to exclusive reaction channels of low probability. The 3πK+ enhancement factor of 22 is deﬁnitely higher than values predicted for the strangeness production from a hadron gas with baryonic number less than 6. Therefore, it could be an evidence of the formation of quark-gluon plasma. Also the increase of the φ production is on this line, but the meaning of our result is limited by the poor statistics. . 3 2.2. Pentaquark [5]. Signals of the pentaquark have been looked for through the decay Θ+ → pK0S → pπ + π − in the following reaction channels with four and ﬁve prongs . (selected among those listed in subsect. 3 1): p ¯4 He → (p2π + 2π − )2nX (15)

(pπ + π − K− )ps nX, (p2π + π − K− )2nX, (ppπ + 2π − )nX

The ﬁnal states can be reached directly or via intermediate states with strange and non-strange content as shown in table IV. π ∓ , K± and p were identiﬁed through speciﬁc . energy loss, momentum and velocity measurements, as described in subsubsect. 3 1.1; K0 ¯ 0 ), Λ and Σ+ were detected via their π + π − , pπ − and pK0S decay modes, respectively, (K . with the diﬃculties outlined in subsubsect. 3 1.3; the N and Δ baryons were recognized via the p2π and pπ ones. In the following some details are given about the investigation on the event set with higher statistics, which allows investigating subsamples with diﬀerent features. a) pπ − π − π + π + events As shown in table IV, the production of the Θ+ requires necessarily the presence of two K0S , while is excluded from reactions with a K0S and Λ pair. Therefore Θ+ have been looked for in the pπ + π − invariant-mass (IM) distribution within the event subset with two π + π − pairs with invariant mass close to the K0S mass. This choice maximizes the visibility of the Θ+ , if it exists. The results of the research are summarized in ﬁg. 27. The data analysis has been supported by comparisons between the pπ + π − IM distribution and the background-like pπ + π + one, as well as by Monte Carlo simulations.

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Table IV. – Analyzed ﬁnal states and possible intermediate states of the p ¯4 He annihilation involving Θ+ , kaons, N, Δ, Λ and Σ baryons. Additional neutrals (neutrons and, in some cases, mesons) are present. Intermediate states with undetected K0L are not included for the sake of simplicity (for instance, Θ+ π + π − (K0L ) → pK0S π + π − (K0L )). The considered N and Δ baryons are the following: N(1440), N(1520), N(1535), Δ(1600) and Δ(1620); they decay mainly into p(n)2π and p(n)π. X = 0, 1, 2π 0 .

A

n(K0S )

Final states

Possible intermediate states

pπ − π − π + π +

N(Δ) → pπ + π − π + π − X

0

pK0S K0S

2

Σ+ (1660)K0S → (pK0S )K0S

2

Θ+ K0S → (pK0S )K0S Λ(1520)K0S π +

B

C

pπ − π + K−

pπ − π + π + K−

ppπ + π − π −

→ [Σ π

−

]K0S π +

−

−

→ [(pπ )π ](π π )π 0

+

+

1

Λ(1520)K0S π + → [Σ0 π 0 ]K0S π + → [(Λγ)π 0 ]K0S π +

1

Λ(1520)K0S π + → [Λπ 0 π 0 ]K0S π +

1

Λ(1115)K0S π +

1

Σ0 K0S π + → (Λγ)K0s π +

1

Σ+ K0S π − π + → (pπ 0 )(π − π + )π − π +

1

Θ+ K− → (pK0S )K−

1

pK0S K−

1

Σ+ (1660)K− → (pK0S )K−

1

Θ+ K− π + → (pK0S )K− π +

1

pK0S π + K−

1

¯ 0∗ pK0S K

D

2 +

→

pK0S K− π +

1

Σ+ (1660)K− π + → (pK0S )K− π +

1

Diﬀerent combinations of p, N, Δ and charged pions

0

Θ+ Λ(1115) → (pK0S )Λ

1

Θ+ Σ0 → (pK0S )Λγ

1

pΛ(1115)K0S

1

Σ

+

K0S pπ −

→ pπ

0

K0S pπ −

1

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Fig. 27. – pπ + π − IM distributions: (a) for events with two π + π − pairs with mass close to the K0S mass (in the interval 495 ± 20 MeV); (b) for all the events; it is very close to the relevant pπ + π + plus pπ − π − invariant-mass distribution; (c) for events with at least one π + π − pair with IM mass close to the K0S mass; (d) for events with one π + π − pair with IM close to the K0S mass and one pπ − pair close to the Λ mass (in the interval 1115 ± 20 MeV); (e) for events with two π + π − pairs with IM outside the interval 495 ± 20 MeV [5].

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i) Figure 27a shows the pπ + π − invariant-mass (IM) distribution for the events with two π + π − pairs with IM in the interval 495 ± 20 MeV. One can see two peaks, one close to 1500 MeV and the other close to 1550 MeV. The latter one is assumed to be associated to the Θ+ pentaquark. The interval ±20 MeV around the K0S mass is about twice the apparatus resolution. While enlarging the interval, the peak at 1550 MeV disappears progressively. ii) As pointed out in table IV, contributions to the pπ + π − IM distribution come ¯ 0 , Σ+ (1660)K0 and N and Δ resonances. from the pπ + π − π + π − phase-space, from pK0S K S S The N and Δ have full widths ranging between 100 and 450 MeV and two of them have their mass close to that expected for the Θ+ : N+ (1520), which has a probability of 45% to decay into p(n)2π, a mass in the interval 1515–1530 MeV and a full width of 100–135 MeV, and N(1535), which has a much lower probability to decay into p(n)2π (1–10%), a mass in the interval 1520–1555 MeV and a full width of 100–250 MeV [98]. Considering the apparatus resolution and the statistics, only N(1520) is observable and is just the peak close to 1500 MeV in ﬁg. 27a. iii) No peak is visible in the inclusive pπ + π − distribution concerning the whole pπ + π − π + π − sample (ﬁg. 27b). It is very close to the relevant pπ + π + plus pπ − π − invariant-mass distribution and to phase space distribution obtained by Monte Carlo calculations for the reaction p ¯2p → pπ + π − π + π − 2π 0 , where 2π 0 accounts approximately for the energy and momentum carried by the unseen particles in the reactions on helium. iv) According to table IV, the Θ+ peak is not visible in the pπ + π − distribution concerning all events where at least one IM falls in the 495 ± 20 MeV interval (ﬁg. 27c). v) According to table IV, Θ+ is not present in the pπ + π − distribution concerning events with a K0S and Λ(1115) pair (ﬁg. 27d). vi) Owing to the four π + π − combinations, each pπ + π − π + π − event may contribute to the pπ + π − IM distribution with two or with four entries with both π + π − pairs with IM close to the K0S mass. However, as a matter of fact, only few events over 375 contribute with four entries; therefore, the entries are practically twice the events. Of course, entries relevant to uncorrelated particles are distributed at random according to the phase-space in the histograms, while those arising from the resonance decay are concentrated around the proper mass. vii) In conclusion, the pπ + π − IM distribution for the subset of the pπ − π − π + π + events, where the Θ+ may be expected, shows a peak close to 1550 MeV with a width of the order of 30 MeV. b) ppπ + π − π − , pπ − π + K− and pπ − π + π + K− events These event samples separately contribute little to the pπ + π − statistics; all together contribute with a statistics about 1/2 that of pπ + π + π − π − . In ppπ + π − π − , Θ+ is expected only in the presence of a K0S and Λ pair and in pπ − π + K− and pπ − π + π + K− , in the presence of only one K0S . The pπ + π − IM distribution for the subsets of events selected according to these restrictions resembles that in ﬁg. 1a and has been summed up to it (ﬁg. 28).

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Fig. 28. – Best ﬁt to the pπ + π − IM distribution (a) and to the corresponding phase-space like pπ ± π ± one (b) [5].

c) Summary Figure 28 reports the pπ + π − IM distribution for the selected events with the similar distribution relevant to pπ + π + and pπ − π − . The histogram of ﬁg. 28a has been ﬁtted in diﬀerent ways with the ﬁnal typical result shown in ﬁg. 28a (χ2r = 0.68, P = 0.83, NΘ+ = 46). The ﬁt on the Θ+ peak leads to m(Θ+ ) = (1560.0 ± 3.7) MeV, σGaussian = (10.76 ± 4.51) MeV. The σ value is equal within the errors to our resolution (∼ 9.3 MeV, see subsub. sect. 3 1.3). d) Statistical analysis Once veriﬁed that the observed peak is not the eﬀect of the apparatus acceptance or an indirect signature of known physical signals, it has been veriﬁed that it is not the trivial eﬀect of a statistical ﬂuctuation in diﬀerent ways with consistent results. According to the most conventional one, the statistical signiﬁcance of the Θ+ peak can be evaluated by the ratio (16)

S=

NΘ + , σ Θ+

where σΘ+ includes all uncertainties on NΘ+ (due to statistics, entry selection, background estimation and best ﬁt procedures). Equation (16) measures how much the peak diﬀers from zero in units of its own standard deviation. When NΘ+ comes from the diﬀerence NΘ+ = N − Nb between the observed events in the peak region and the background events under the peak and the Poisson statistics can be applied, the uncertainty is given by the relation (17)

σ Θ+ =

2 = σb2 + σN

Nb + N

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G. Bendiscioli

and the signiﬁcance is (18)

S1 = √

NΘ + . Nb + N

As NΘ+ = 46, the total number of the entries in the four bins containing the peak is N = 206 and the number of background entries under the peak is Nb = 160, it turns out S = 2.4σ. This value is likely an underestimation as the calculation ignores that the number of the entries is about twice that of the events and the entries coming from the background events are distributed diﬀerently from those coming from the peak events. This considered, the signiﬁcance turns out to be a bit higher, namely of 2.7σ, which means that the peak is a statistical ﬂuctuation with the probability of 0.35%. Compared . with the signiﬁcance of the other experiments (see table I in subsubsect. 2 2.4), the present S value is close to the lower values. e) Discussion . The experimental concerns about Θ+ (see table II in subsubsect. 2 2.4) include the statistical signiﬁcance of the observed peaks, the discrepancy between the mass measured in the nK+ and pK0S → pπ + π − decay modes, the physical meaning of the cuts used to enrich and even to see the signal, the background not completely understood (kinematical reﬂections included), nuclear target eﬀects and, overall, the fact that several experiments did not see it. Detailed discussions of all this can be found in [50, 53, 54] and [56, 57]. The above analysis allows clarifying some points. To this aim, let us consider the identiﬁcation of Θ+ through the pK0S decay mode and, for the sake of simplicity, assume that two strange particles at most are produced. The probability to detect the Θ+ , if it exists, increases if it is possible to reject the events where the presence of Θ+ can be excluded a priori. According with table IV, pπ + π − may appear in the ﬁnal state of a lot of intermediate states, strange and not strange, with S = +1 and S = −1. The ﬁnal states compatible with the presence of Θ+ are only those including two π + π − pairs both ¯ 0 → (pK0 )K ¯ 0 → pK0 K0 ) and one π + π − with IM ≈ 495 MeV (through the decays Θ+ K S S − pair with IM ≈ 495 MeV in company with one K . ¯ 0 (hence On the contrary, a π + π − pair in a ﬁnal state with a K+ is the signature of K + − + + − pπ π cannot be the memory of Θ ) and, in company with K K , cannot originate from a strange particle. Therefore, the detection of only one π + π − pair accumulates a lot of background, which may sink the Θ+ peak. If the ﬁnal state contains more than one proton or more than two charged pions, the background increases more and more due to combinatorial eﬀects. In conclusion it is likely that the Θ+ peak, if it exists, may not be observed in inclusive reactions, as it is demonstrated by ﬁg. 27. This is just the case of most of the experiments which did not observe the Θ+ . Most of them detected only pπ + π − in high multiplicity ﬁnal states and, in some cases, unclean incoming beams were used (unknown mixtures of proton, pions and kaons) [53, 54]. As far as the Θ+ detection via the nK+ decay mode is concerned (see reactions i–o in table I), the Θ+ strangeness is +1 without ambiguity. Θ+ was observed via the nK+ or the pK0S → pπ + π − decay modes in the experiments

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summarized in table I. In the reactions f–h the strangeness carried by K0S is deﬁned through the incoming K+ and the outgoing K− and Σ+ , respectively. Such a check is lacking in reactions a–e. f) Conclusion A signal of the existence of the Θ+ via its pK0S → pπ + π − decay mode is found analysing the pπ + π − invariant-mass distribution in the annihilation reactions at rest. The research has been motivated by the fact that these reactions, where two to four nucleons are involved in the annihilation, assure favourable conditions for the creation of strange particles. The favourable features of our analysis can be summarized as follows. i) The reactions are (partially) exclusive; ii) the Θ+ signal is found in subsets of reactions where it is expected and is absent where it is not expected; iii) reﬂections of other resonances are likely absent (not discussed in this review); iv) the background is fairly understood. A peak has been found around the invariant mass value 1560.0 ± 3.7 MeV with a very narrow width. The mass value agrees with the values obtained in experiments which studied the Θ+ → nK+ decay mode. The weak point of this analysis is the poor statistics. For this reason the features of the invariant-mass distributions have studied carefully in order to evaluate correctly the statistical signiﬁcance, which turns out to be 2.7. It is at the threshold limit to claim evidence for Θ+ and implies that the Θ+ signal is not a trivial statistical ﬂuctuation with a probability of 99.65%. . 3 2.3. K− pn, K− pnn and K− d bound states (DBKS) [6]. Signals of the formation ¯ of the bound Kpp system (called 2K ¯ H in the following) has been searched for in the 4 antiproton- He annihilation at rest through the decay mode 2K ¯ H → Λp, already studied ¯ in [92], and of the bound Kppn system (called 3K ¯ H in the following) through the decay mode 3K ¯ H → Λd, where d is a deuteron. To this aim, the following ﬁve-prong reactions . (selected from those listed in subsubsect. 3 1) have been analyzed assuming that the ﬁnal states are reached through intermediate states with the presence of a K0 -Λ pair, where Λ originates from the decay of a bound system and K0 is required by the strangeness conservation: (19)

0 0 − + − p ¯4 He → 2K ¯ HnK X → ΛpnKS X → pπ pπ π nX,

(20)

0 0 + − 0 + − − + − p ¯4 He → 2K ¯ Hnπ π K X → Λpπ π nKS,L X → pπ pπ π nKS,L X,

(21)

0 0 − + − p ¯4 He → 3K ¯ HK X → ΛdKS X → pπ dπ π X,

(22)

+ − 0 + − 0 − + − 0 p ¯4 He → 3K ¯ Hπ π K X → Λdπ π KS,L X → pπ dπ π KS,L X,

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Fig. 29. – (a) ptot (GeV/c) vs. Etot (GeV) from ref. [6]. Events around the point ptot ≈ 0.2 GeV/c, ¯ 4 He → 2pπ + 2π − ns , where ns is a Etot ≈ mp¯ + 2mp + mn ≈ 3.727 GeV are annihilations p spectator neutron, that is the neutron does not participate in the annihilation process; the events along the full line are annihilations p ¯ 4 He → 2pπ + 2π − nf , where nf is a fast neutron; events in the left side region of the full line are annihilations ¯ p4 He → 2pπ + 2π − nX, where X includes neutral mesons (π 0 , K0 , etc.) and n may be a fast or a spectator neutron [6]. (b) White histogram: pπ − IM distribution for all events (4 entries per event). The full vertical line puts in evidence the Λ mass, the dotted lines deﬁne the interval 1115 ± 30 MeV. Grey histogram: pπ + IM distribution (2 entries per event); the total entries are 15770; this histogram is normalized in order to be superimposed to the white histogram at the lower energies [6].

X is any number of undetected π 0 , n an undetected neutron and d a deuteron. In reactions (19) and (21) K0S is detected through the π + π − decay mode (with the diﬃculties . outlined in subsubsect. 3 1.3), in reaction (20) and (22) K0S is undetected, as well as K0L . The number of nucleons involved in the reactions (19) and (20) may be 3 or 4, in the reactions (21) and (22) is 4. Λ is detected through the pπ − decay mode with diﬃculties similar to those concerning K0S . a) Search for 2K ¯H According with eqs. (19)–(22), the existence of the bound system 2K ¯ H is expected to be revealed by a peak in the invariant-mass (IM) distribution of the 2pπ − system in a subset of events with the pπ − IM within an interval around the Λ mass (1115 ± 30 MeV, see ﬁg. 29b). Note that the invariant mass of the 2K ¯ H must be less than the sum of the masses of the constituent free particles, namely less than 2373 MeV. Considering events placed in diﬀerent regions of the ptot vs. Etot plane in ﬁg. 29a, we have veriﬁed that the events in a band around the line in this ﬁgure, that is the events without production of neutral mesons, contribute very little the pπ − IM distributions below 2371.3 MeV; the selected band width in the ptot vs. Etot plane is 3727 ± 150 MeV. For this reason these events were excluded from the following analysis.

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199

− For the identiﬁcation of 2K ¯ H, 2pπ IM distributions have been considered under diﬀerent event selection criteria and compared with the experimental phase-space like distributions relevant to the uncorrelated particles 2pπ + . Furthermore, in order to get guidance on the interpretation of the data, some experimental IM and angular distributions have been compared with those deduced by Monte Carlo calculations.

The results are summarized in ﬁgs. 29 and 30. The unconditioned 2pπ − IM distribution does not show any signiﬁcant signal, looking very similar to the distribution of the 2pπ + IM (ﬁg. 30a). Instead, a narrow peak close to 2200 MeV is clearly visible in subsets of events selected with the pπ − IM (ﬁg. 30b) close to the Λ mass, that is in intervals varying from about 1115 ± 10 MeV to 1115 ± 70 MeV. For narrower intervals the peak disappears because of the low statistics and our resolution, for larger intervals the peak is sunk by the background. For the ﬁnal analysis we have chosen as emblematic the interval 1115 ± 30 MeV (ﬁg. 29b), which may appear to be quite large. The reasons of this choice are two: the peak has its maximum evidence and a large interval avoids possible momentum cuts in the analysed events, since the resolution on the invariant mass of a particle, as measured by OBELIX, depends also on its momentum. We stress that the peak is absent from the experimental phase-space like 2pπ + IM distribution (ﬁg. 30c), where the pπ + IM has been selected close to the Λ mass. As the entries in the histogram of ﬁg. 30b are twice the events, the prominence of the peak is depressed, as the signals in the peak are one per event. Of course, this is true if the peak is not a trivial statistical ﬂuctuation. The solid line in ﬁg. 30b is the result of a best ﬁt calculation in the interval 2100–2400 MeV with a phase-space like function plus a gaussian to account for the peak. The phase-space like function ﬁts also the 2pπ + IM distribution in the same energy interval (ﬁg. 30c). At higher energies the 2pπ + IM distribution deviates from the 2pπ − IM one as a consequence of the discrepancies between the relevant distributions in ﬁg. 29b (eﬀects of the N∗ and Δ production in 2pπ − ). The mass of the presumed peak turns out to be 2212.1±4.9 MeV, the width (FWHM) 24.4 ± 8.0 MeV; likely, owing to the apparatus resolution, 24.4 MeV is an upper limit. The reactions (19) and (21) require the production and the detection of a K0S , the . evidence of which is questionable for the reasons explained in subsubsect. 3 1.3. Any+ − way, some evidence can be found. Figure 31a shows the π π IM distribution and the correspondent phase-space like π + π + one for all the events and ﬁg. 31b their diﬀerence; ﬁg. 31c shows π + π − IM distribution for the events with the 2pπ − IM around the peak in ﬁg. 30b (in the interval 2200–2240 MeV). In spite of the similarity of the two distributions in ﬁg. 30a, their diﬀerence displays a peak just in correspondence of the K0S mass (497 MeV). Figure 30c has a (large) maximum around the same energy. b) Angular correlations and background reduction To have a conﬁrmation that the observed peak is compatible with the processes summarized in reactions (19) and (20), some angular correlations have been examined, in particular those expressed by the distributions of the laboratory angle Φ between a proton and a pπ − pair (or Λ) and of the angle Θ between 2pπ − (or Λp) and π + π − (or K0S ). Concerning the angle Θ between 2pπ − and π + π − , for the ideal case of annihilations at

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G. Bendiscioli

Fig. 30. – (a) All the events. Full histogram: 2pπ − IM distribution (2 entries per event). Dotted histogram: phase-space like 2pπ + IM distribution (1 entry per event); the total entries are 7885. These histograms are normalized to the same area. (b) 2pπ − IM distribution for the events with the pπ − IM close to the Λ mass (1115 ± 30 MeV); the solid curve is a ﬁt with a phase-space line deduced from (c) plus a Gaussian to account for the peak. (c) 2pπ + IM distribution with the pπ + IM close to the Λ mass (1115 ± 30 MeV). The solid curve represents a best-ﬁt function. ¯ 2pn → 2pπ + 2π − K0S π 0 (d) 2pπ − IM distributions from Monte Carlo calculations: a) reaction p with nucleon initial momentum equal to 750 MeV/c; b) reaction p ¯2pn → 2pπ + 2π − π 0 with nucleon initial momentum equal to zero; c) reaction p ¯ 2pn → 2pπ + 2π − K0S π 0 with nucleon initial momentum equal to zero [6].

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201

rest on three free nucleons without production of unseen particles, 0 0 − + − p ¯ pnn → 2K ¯ HK → ΛpKS → pπ pπ π ,

the cos Θ distribution should be peaked at cos Θ = −1. The experimental distribution conserves some memory of this showing a broad peak around cos Θ ≈ −0.75 (white histogram in ﬁg. 32a). The gross features of this distribution obtained with a Monte Carlo calculations are shown in ﬁg. 32c. A behaviour similar to the previous one is displayed by the relevant background distribution, but the maximum is shifted at smaller angles, around cos Θ = −0.65 (grey histogram in ﬁg. 32a). This suggests that the 2K ¯ H signal should be more prominent at larger angles. The same suggestion is given also by the comparison between the cos Θ distributions with the additional constraint that the ppπ − and ppπ + invariant masses are in the interval 2200–2240 MeV (ﬁg. 32b). Selecting, for example, events with cos Θ < −0.4 one obtains the white histogram of ﬁg. 32d, where the background is decreased (compared to ﬁg. 30b) by a factor of 0.56 and the number of events in the peak only by a factor of about 0.75 (about 50 entries against 67). After the same selection, the background distribution in ﬁg. 30b transformed into the grey one in ﬁg. 32d. c) Search for 3K ¯H The data concerning the reactions (21) and (22) have been analyzed with the same criteria adopted for the reactions (19) and (20). Unfortunately, the statistics is very poor. The main features of the data are shown in ﬁg. 33. No peak is present in the full pdπ − IM distribution in ﬁg. 33b instead, a narrow peak appears in the pdπ − IM distribution for events with the π − p IM around the Λ mass (1115 ± 30 MeV) (ﬁg. 33c): the number

Fig. 31. – (a) π + π − IM distribution and the corresponding phase-space like π + π + one for all the events. (b) Diﬀerence between the distributions in (a). (c) π + π − IM distribution for the events with the 2pπ − IM in the interval 2200–2240 MeV in ﬁg. 30b.

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Fig. 32. – cos Θ distributions. (a) White histogram: all events in ﬁg. 30b; grey histogram: background events from ﬁg. 30c. (b) White histogram: events in ﬁg. 30b with the additional constraint that the ppπ − IM is in the interval 2200–2240 MeV; grey histogram: events in ﬁg. 30c with the additional constraint that the ppπ + IM is in the interval 2200–2240 MeV; total number of entries 216. (c) Monte Carlo calculations: a) reaction p ¯ 2pn → 2pπ + 2π − K0S π 0 with nucleon initial momentum equal to 750 MeV/c; b) reaction p ¯2pn → 2pπ + 2π − K0S π 0 with nucleon initial momentum equal to zero; c) reaction p ¯ 2pn → 2pπ + 2π − π 0 with nucleon initial momentum equal − to zero. (d) 2pπ (white histogram) and 2pπ + (grey histogram): IM distributions for the events of ﬁgs. 30b and c with the requirement of cos Θ < −0.4; total number of entries 1135 [6].

Strangeness production in antiproton-4 He annihilation at rest

203

Fig. 33. – Reactions p ¯ 4 He → pπ − dπ + π − X and p ¯4 He → pπ − dπ + π − K0S,L X. (a) ptot vs. Etot scatter plot. The points around ptot ≈ 0.1 GeV/c, Etot ≈ mp¯ + 2mp + 2mn − B4 He ≈ 4.666 GeV should identify annihilations p ¯ 4 He → pdπ + 2π − , the other points should identify annihilations 4 + − p ¯ He → pdπ 2π Z, where Z means neutral mesons. (b) pdπ − IM distribution (two entries per event). (c) pdπ − IM distribution for the events with pπ − IM close to Λ mass (1115 ± 30 MeV) for bin of 60 MeV; (d) as in (c) but with bin of 30 MeV [6].

√ of the peak entries is ∼ 17 ± 41 over a total number of 29 entries. Considering also the IM distribution with bins of 30 MeV (ﬁg. 33d), instead of 60 MeV as in ﬁg. 33c, one deduces that the peak is centred at 3190 ± 15 MeV, below the mass of the free particles mp +md +mK ¯ = 3311 MeV, and the full width (FWHM) is less than ≈ 60 MeV. Assuming that the peak is a signal corresponding to the decay of a DBKS, the binding energy should be −121 ± 15 MeV. The statistical signiﬁcance is of ∼ 2.6σ and is near the same selecting diﬀerent intervals around the Λ mass (±15 MeV, ±20 MeV, ±30 MeV, ±40 MeV). d) Statistical signiﬁcance The statistical signiﬁcance of the peaks in ﬁgs. 32b and d has been evaluated in diﬀerent ways, which agree for a value ≥ 3σ. The simplest estimation for the peak in ﬁg. 30b is as follows. The number of signal events in the three bins covered by the peak in ﬁg. 30b is 67±22 and the number of signal events in the bin where the peak is maximal is 45±15. The errors are the total errors evaluated by MINUIT. The statistical signiﬁcance S = N/σN is 3.0 for the full peak and for its maximum. Note that the frequency of these

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events is of the order of 0.67%, compatible with the total frequency for the production of Λ (1.1%). A similar evaluation for the peak with reduced background in ﬁg. 32d leads to S = 3.7. The statistical signiﬁcance for the 3K ¯ H peak (ﬁg. 32c) is S = 2.6σ. e) Production rates Considering that the p ¯4 He annihilation into 2p3πX has an absolute branching ratio of (0.856 ± 0.001)% [2], the frequency of the events in the 2K ¯ H peak is (67 ± 22)/10132 = (0.66 ± 0.22)%, that the probability of the Λ decay into pπ − is of 64% and that the Λ detection eﬃciency is less than 57% [98], a rough estimation of the lower limit for the −4 production of the peak interpreted as due to the production of 2K . ¯ H is 1.5 10 The absolute branching ratio for the pd3πX production is unknown and can be estimated multiplying the ratio (number of pd3πX events)/(number of 2p3πX events) by the absolute branching ratio for the 2p3πX events: 623/10232 × 0.856% = 0.053%. Hence, √ considering the frequency of the events in the Λd peak (17 ± 41/623), the frequency of Λ → pπ − decay mode (64%) and the Λ detection eﬃciency (< 57%), the branching −4 . ratio for the 3K ¯ H production is at least 0.39 10 f) Conclusions Summarizing, with a statistical signiﬁcance of 3.7σ, evidence is found for a peak in the 2pπ − IM distribution at 2212.1 ± 4.9 MeV with a width < 24.4 ± 8.0 MeV, which can be attributed to the presence of Λp systems produced by the decay of the system 2 ¯ H, having a bound energy of −160.9 ± 4.9 MeV. Table V compares these results with K the few other experimental results and with some theoretical predictions. The B value is higher than that predicted in [73] for the K− pp system; the Γ value is lower than the prediction. The B value is higher also than that measured in [92] (more than 5σ) and the Γ value is lower. The diﬀerences could be due to the fact that in [92] the target nuclei are other than 4 He, namely 6 Li, 7 Li, 12 C, and the IM has been evaluated by adding up the contributions from the three targets. Indeed, several eﬀects, like the variation of the mass and the width of a resonance in diﬀerent nuclear media, diﬀerent initial states of the nuclear targets, etc., should be advocated (see ref. [106] and references quoted therein). The understanding of the diﬀerence between the present results and that in [92] is not straightforward also for the diﬀerent mechanisms leading to the bound Λ-nucleon system, antiproton annihilation in one case, K− capture in the other. In spite of the low statistics, a peak is observed also in the pπ − d IM distribution centred around 3190 ± 15 MeV with a statistical signiﬁcance of 2.6σ. If it is really a signature of the 3K ¯ H bound state, its binding energy and width (−121 ± 15 MeV and less than ≈ 60 MeV) agree fairly with the predictions by [73] (−108 MeV and 20 MeV, respectively). The value of B is less than that found in [91]. Another evidence for a peak in the Λd system has been claimed very recently [107]: the peak should be produced following the absorption of K− at rest by 6 Li nuclei. On the other side, as mentioned in the introduction, ref. [94] puts forward the hypothesis that the peak observed in [92] is simply due to FSI. Likely, such an explanation cannot be raised for this present experiment for at least two reasons. The former one is that the number of ﬁnal residual nucleons is lower (one in the 4 He case and from 4 to 10, depending on the target, in the experiment [92]). The latter one, perhaps the more clear reason,

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Strangeness production in antiproton-4 He annihilation at rest

Table V. – Comparison between experimental data and predictions.

p ¯4 He −

6

Stopped K ( Li, 7 Li, 12 C)

p ¯4 He −

Stopped K

4

( He)

B (MeV)

Γ (MeV)

Statistical signiﬁcance

Ref.

K− pp

−160.9 ± 4.9

< 24.4 ± 8.0

3.7σ

[6]

−

K pp

+3 −115+6 −5(st) −4(syst)

+2 67+14 −11(st) −3(syst)

[92]

K− pp

−48

61

[72]

K− ppn

−121 ± 15

< 60

−

K ppn

O(K− , n) 15 K− O

2.6σ

[6]

< 21.6

[89, 91]

K− ppn

−108

20

[73]

K− p

−27

40

[72]

−

−86

34

[72]

K ppnn

16

−169.3 ±

2.3(st) +3.0 −0.8(syst)

−90(130)

[93]

is that, without invoking angular correlations, FSI produces in the IM spectrum of the Λp system a large plateau extending from ∼ 2100 to ∼ 2300 MeV. Imposing an angular correlation corresponding to that of the experiment [92], the large plateau is transformed into an asymmetric peak, centred at about the same value found in the measurement. The IM spectrum of ﬁg. 30b was obtained without imposing any angular correlation and then it is hard to think that FSI could be the responsible of such a narrow peak. An interesting alternative explanation that could reconcile qualitatively the present experimental observations with the theoretical predictions (see table V), is that put forward by [108]. According with this idea, the peak observed in the Λd IM could be due to the decay of a true T = 0 K− ppn DBKS, whereas the peak observed in the Λp IM could be a fake peak due to the correlation eﬀect between the Λp system and the neutron produced in the three-body decay of the parent K− ppn state. Unfortunately such a neutron could not be detected and such a hypothesis cannot be veriﬁed. . 3 2.4. S = −2 strangeness production and 2K− nn and 2K− nnp [7]. The strangeness conservation requires that the creation of two K+ , that is of two ¯s quarks, is accompanied by the simultaneous creation of two s quarks; ﬁnally, s may materialize into a pair of antistrange particles, antikaons or hyperons. In the annihilation at rest, the energy delivered to the ﬁnal particles is equal to the p ¯p or p ¯n system mass (∼ 1876 MeV) and this excludes that the ﬁnal products include, beside two kaons, two antikaons, as an energy of

206

G. Bendiscioli

Table VI. – p ¯4 He annihilation reactions into four and ﬁve prongs with production of two K+ admitted by the energy conservation, i.e. with threshold energy Es less than the p ¯4 He energy (4664 MeV). The ﬁnal states of reactions (1), (3)–(5) include two neutral mesons and their . representative points in the ptot vs. Etot plane of ﬁg. 17 of subsubsect. 3 1.2 are distributed in 0 the left-side areas of the π lines. The representative points of the reactions (2) and (6), (7) are distributed along the B lines. n means neutron and ps means slow proton, i.e. with momentum less than 300 MeV/c, the minimum value for the protons to reach the spectrometer. (4 prongs) p ¯4 He → 2K+ 2π − X Intermediate states according to strangeness and energy cons.

Es (MeV)

1

2K+ ΛΛ2π − ps → 2K+ (nπ 0 )(nπ 0 )2π − (ps) → 2K+ 2π − (2nps 2π 0 )

4436.4

2

2K+ Σ− Σ− ps → 2K+ (nπ − )(nπ − )ps → 2K+ 2π − (2nps )

4320.3

p ¯4 He → 4665.9 MeV

(5 prongs) p ¯4 He → 2K+ π − π − (K− )pX Intermediate states according to strangeness and energy cons.

p ¯4 He → 4665.9 MeV 3 4

2K+ K0 Λ2nπ − → 2K+ (pπ − )2nπ − K0 → 2K+ p2π − (2nK0 ) +

2K ΛΛnπ

−

−

→ 2K (nπ )(pπ )nπ

−

+

+

+

0

−

0

0

−

0

−

+

2K ΛΛn2π π

6

2K+ Σ− Σ+ nπ − → 2K+ (nπ − )(nπ + )nπ − → 2K+ 2π − π + (3n)

7

2K Σ Λn → 2K (nπ )(pπ )n → 2K p2π (2n)

8

2K K Λ2n → 2K K (pπ )2n → 2K K pπ (2n) +

−

−

+

+

−

−

−

−

−

+

+

4297.8

→ 2K π 2π (3n2π ) +

5

+

4619.4

→ 2K p2π (2nπ )

→ 2K (nπ )(nπ )nπ +

−

+

−

−

0

4437.4 4461.3 4240.0 4475.8

at least 1974 MeV should be required. Instead, the same amount of annihilation energy plus the involvement of some nucleons allows the presence among the ﬁnal products of hyperons, which can be detected through their decay modes. The p ¯4 He annihilation + reactions into four and ﬁve prongs with production of two K admitted by the energy conservation, i.e. with threshold energy Es less than the p ¯4 He energy (4666 MeV), are + + ¯ listed in table VI. K K can be associated to ΛΛ, K Λ, ΛΣ and ΣΣ. Although events with production of one K− are observed, events with production of two K− are forbidden because two K− must necessarily be accompanied by two K+ , which is energetically forbidden. The observation of events with two K− is an evidence of wrong mass identiﬁcations. These fake 2K− events allow estimating the amount of wrong 2K+ events. S = −2 strangeness production has been looked for in the following reactions selected . among those listed in subsect. 3 1: (4 prongs) p ¯4 He → 2K+ 2π − X, (5 prongs) p ¯4 He → 2K+ π − pX. Particular care has been devoted to the K± identiﬁcation following the criteria described

Strangeness production in antiproton-4 He annihilation at rest

207

Fig. 34. – (a) dE/dx vs. p with dE/dx in arbitrary units: the crosses identify kaons and are all inside the kaon band; the dots identify pions and are spread also out of the proper regions (below the kaon band) due to less restrictive selection criteria (see text). (b) β vs. p: the lines are the calculated β vs. p relations for the diﬀerent particles.

. in subsubsect. 3 1.1. The results of the particle identiﬁcation are exempliﬁed in ﬁg. 34 for helium and in ﬁg. 35 for hydrogen. Note that the kaons are correctly distributed within the kaon band in ﬁgs. 34a, and along the calculated kaon line in ﬁgs. 34b. Some pion points are spread also out of the pion region due to the less restrictive criteria adopted for the pion identiﬁcation: in eﬀect, they are assumed to be π + (π − ) all tracks not identiﬁed as K+ (K− ) or protons or deuterons. The above mass identiﬁcation procedure allows to ﬁnd the number of the possible + 2K events and that of the forbidden 2K− ones. The number of true 2K+ events is

Fig. 35. – Fake 2K+ 2π − plus 2K− 2π + events in hydrogen: (a) dE/dx vs. p with dE/dx in arbitrary unit and (b) βmeasured vs. p; triangles represent π ± and circles K± .

208

G. Bendiscioli

Table VII. – 2K+ 2π − events. Numbers (N ) and fractions (F ) of 2K+ events and of fake 2K− events in 4 He; for hydrogen the ratio 2K− /2K+ is given. The fractions are referred to the total number of four-prong events in helium (238 746). Helium

Hydrogen

2K+

2K−

Diﬀerence 2K+ − 2K−

Diﬀerence 2K+ − 2K−

2K− /2K+

N

N

N

F (10−5 )

F (10−2 )

47

11 (13 ± 4)(a)

34 ± 8(a)

14 ± 3(a)

87 ± 10

(a)

Before subtracting, the numbers of 2K − have been corrected for the K− defect put in evidence by the −

2K /2K+ ratio in hydrogen (see text).

evaluated as the diﬀerence between 2K+ and 2K− . In the hydrogen case, both K+ K+ and K− K− events are fake; moreover, the number of K− K− is expected to be somewhat lower as the K− can be absorbed in the materials between the event vertex and the jet-drift chambers, as put in evidence in [4]. a) 2K+ 2π − events The numbers of the 2K+ 2π − events in helium are reported in table VII, together with the numbers of the fake 2K− 2π + events; for the fake 2K+ and 2K− in hydrogen the ratio is given. The hydrogen data show that the fake 2K− events are underestimated by a factor of 87% with respect to the fake 2K+ , a value consistent with the ratio K− /K+ ∼ 0.93 for the single kaon production [4]; this allows to correct the numbers of helium fake events which become 13 ± 4. By subtracting from the observed numbers of 2K+ events the corrected 2K− background events, we found the numbers of the true 2K+ events, that is 34 ± 8. A check on these events is made in ﬁg. 36: it shows the event point distribution in . the (ptot , Etot ) plane described in ﬁg. 17 of subsubsect. 3 1.2 both for the true and the fake events. The plane in ﬁg. 36 is divided in three regions by a band containing the B = 1, 2, 3 lines. One can see that the events in helium are distributed about one half inside the band and one half in the right-side area (the “unphysical” area), while the leftside area is empty. According to [4], the band contains mostly physical events without production of neutral mesons. Events are distributed in the “unphysical” region of ﬁg. 36 if the total energy is evaluated in excess; this can be due to i) an error in the momentum measurement, ii) an incorrect particle identiﬁcation (attribution of a heavier mass to a lighter particle: kaon mass to a pion or proton mass to a kaon) and iii) some physical facts neglected in building up ﬁg. 36. As far as the errors due to points i) and ii) are concerned, we observe that also the fake events in ﬁg. 36b are distributed in the band and in the right-side region, in spite that the bulk of the annihilation events is distributed in the left-side region; therefore these errors should be corrected by the previous background subtraction. About point iii), note that

Strangeness production in antiproton-4 He annihilation at rest

209

Fig. 36. – 2K+ 2π − events. ptot vs. Etot for the 2K+ (a) and the 2K− (b) events. The full . lines labelled π 0 and B = 1, 2, 3 are deﬁned in ﬁg. 17 of subsubsect. 3 1.2; the bands within the dotted lines enclose mostly events without production of neutral mesons (see text and ﬁg. 34). The stars indicate the events with pion and kaon momenta less than 400 MeV/c.

in the hydrogen case the momenta do not exceed 500 MeV/c (see ﬁg. 35), while in the helium case they reach 750 MeV/c (ﬁg. 34). At a ﬁrst glance this is unexpected, as in an annihilation on a single nucleon all the annihilation energy is delivered to the pions, while in an annihilation involving several nucleons an amount of energy is delivered to the residual nucleons too. However, this momentum reduction can be overcompensated by momenta due to the Fermi motion of the annihilating nucleons, which are neglected in the diagrams in the reference scheme in ﬁg. 36: this explains why several events look . like unphysical, as anticipated in subsubsect. 3 1.2. A pion with a typical momentum of 300 MeV/c has energy of 330 MeV; if its momentum is increased to 700 MeV/c, its energy becomes 713 MeV, with an increase of 383 MeV. To point out the eﬀects of the momentum overestimation, ptot vs. Etot plots have been built up selecting events with pion and kaon momenta less than a maximum value pmax . A typical result is shown by stars in ﬁg. 36a for pmax = 400 MeV/c; one can see that the events in the “unphysical” region are drastically reduced. Recall that a high Fermi momentum means a small distance among the interacting particles, which agrees with the idea that several nucleons are involved in the annihilation if they are very close each other. This considered, the events after the background subtraction may be considered as true 2K+ events in spite of the position in the (ptot , Etot ) plane of ﬁg. 36a. An interesting point is that the region where the events with production of neutral mesons are expected is empty: this means that only reaction (2) (2K+ Σ− Σ− ps → 2K+ 2π − (2nps )) in table VI and not reaction (1) contributes to the 2K+ production. The frequency with respect to the four prong events (see table VII) is (14±3) 10−5 . As the frequency of the four-prong events in the annihilation on 4 He at rest is (12.26 ± 0.63)% [96], the absolute yield for the ﬁnal state 2K+ 2π − is Y (2K+ Σ− Σ− ps → 2K+ 2π − (2nps )) = (0.172 ± 0.038) × 10−4 . This value has to be taken as a lower limit since the momenta are measured only above

210

G. Bendiscioli

Table VIII. – Five-prong events. Numbers (N ) and fractions (F ) of 2K+ events and of fake 2K− events in 4 He. The fractions are referred to the total number of ﬁve-prong events (47 299). 2 K+ π + 2π −

2 K+ p2π −

2 K+ K− pπ −

2K+

2K−

2K+ 2K−

2K+

N

N

N

1

17

37 36 ± 6

(a)

(a)

16 ± 4

N

N

1

4

(a)

2K− N

2 K+ π + 2π −

2 K+ p2π −

2 K+ K− pπ −

2K+

2K+

2K+

F (10−3 )

F (10−3 )

F (10−3 )

0.76 ± 0.13

0.34 ± 0.08

0.08 ± 0.04

0

4±2

(a)

Before subtracting, the numbers of 2K − have been corrected for the K− defect put in evidence by the −

2K /2K+ ratio in hydrogen (see table II and text).

. a minimum value (see subsect. 3 1) and since the event selection does not assure that all events are picked up. As expected, this yield is much smaller than that for the fourprong annihilation with production of one K+ , no π 0 and with the involvement of 2 or 3 nucleons p ¯ + 4 He → K+ 2π − π + (ps 2n), the frequency of which within the four-prong events is (4.07 ± 0.14)% [4]; that is, Y = (5.0 ± 0.3)×10−3 . b) 2K+ p2π − , 2K+ π + 2π − and 2K+ K− pπ − events The ﬁve-prong events have been processed in a way similar to that utilized for the four-prong ones. The results are summarized in table VIII. The situation is similar to that of the four-prong case: the regions, where events with production of neutral mesons are expected, are empty (see ﬁg. 37); hence only the

Fig. 37. – Five-prong events. ptot vs. Etot for (a) 2K+ π + 2π − events, (b) 2K+ p2π − events and (c) 2K+ K− pπ − events. The regions between the two lines include mostly events without production of neutral mesons (see text and caption in ﬁg. 36). The stars indicate the events with pion and kaon momenta less than 400 MeV/c.

Strangeness production in antiproton-4 He annihilation at rest

211

Fig. 38. – pπ − IM distribution for the 2K+ p2π − events.

reactions 2K+ Σ− Σ+ nπ − → 2K+ 2π − π + (3n), 2K+ Σ− Λn → 2K+ p2π − (2n) 2K+ K− Λ2n → 2K+ K− pπ − (2n) (reactions (6),(7) and (8) in table VI) are observed. Again, the event points are distributed also (or mainly) in the unphysical regions. However, the fake events are few, as far as it concerns the wrong particle identiﬁcation (see table VIII), and the kaons are distributed correctly. Also in these cases, the events with pion and kaon momenta less than 400 MeV/c are inside or closer to the physical region (see stars in ﬁg. 37). All considered, we may assume that all the observed events are true 2K+ . Considering that the frequency of the ﬁve-prong annihilation events in 4 He is (35.68± 0.93)% [96], the absolute yields for the three types of ﬁve-prong events are Y(2K+ π + 2π − ) = (2.71 ± 0.47) × 10−4 , Y(2K+ p2π − ) = (1.21 ± 0.29) × 10−4 , Y(2K+ K− pπ − ) = (0.28 ± 0.14) × 10−4 . These yields are much smaller than that for the ﬁve-prong annihilation with production ¯ + 4 He → K+ 2π − π + p(2n), of one K+ , no π 0 and with the involvement of 2 to 4 nucleons p the frequency of which within the ﬁve-prong events is (6.07 ± 0.29)% [4]; that is, Y = (2.17 ± 0.12) × 10−2 . Note that reaction (7) in table VI requires the production of Λ: indeed hints of this are displayed by the pπ − IM distributions in ﬁg. 38, which reveal an accumulation of events around 1115 MeV.

212

G. Bendiscioli

The number of entries is twice that of the events, which increases the background entries and masks the Λ accumulation. Finally, recall that the pπ − pairs can lose memory of their origin from Λ due to the interactions among the ﬁnal particles [6]. The yields found in this work are somewhat larger than those concerning other reactions involving more than one nucleon studied with the same apparatus: for instance, the two body reactions without production of strange particles p ¯ + d → p + π− −5 (Y = (1.2 ± 0.14) × 10 [109]) and that with production of hidden strangeness p ¯ + d → φ + n (Y = (3.56 ± 0.20) × 10−6 [110]). c) Kinematical analysis of the four-prong events The production of 2K+ requires necessarily the involvement in the annihilation of more than one nucleon, but the mechanisms through which the annihilation develops may be diﬀerent; for instance, rescattering cascades, formation and decay of a quark-gluon plasma, formation and decay of bound antikaon-nucleon systems. In this Section some eﬀorts are reported to single out in the data features compatible with the formation of bound systems by means of an analysis based on the three-body kinematics. A clean way to ascertain the existence of double kaonic nuclei has been outlined in [111]. It is based on the correlation of the energies of the 2K+ , that exhibit peculiar features. However, to follow this approach the statistics should be very much larger than the present one and complete information on all the particles produced in the analyzed reactions should be available. Then a simpliﬁed approach is followed, which does not allow to quote errors and to evaluate the statistical signiﬁcance of the diﬀerent distributions. Nevertheless, also qualitative or semi qualitative results may give interesting physical information. c.1) Following this idea and following [111], it assumed that the annihilation energy blob (23)

E0 ≈ mp¯ + (mp + 2mn ) − 28 MeV = 2mp + 2mn − 28 MeV

materializes into a K+ pair and a bound system made of two K− and two neutrons, 4 ¯ which will be indicated with (2K2n) B ; 28 MeV is the He binding energy. The mass of the bound system is set (24)

= 2mK m2K2n ¯ ¯ + 2mn − B,

where B is its binding energy. The invariant mass (IM) of the K+ pair varies from the minimum value MImin = 2mK , when the two K+ move jointly against the bound system, to the maximum when the bound system is at rest and the two K+ move back-to-back max (pK1 + pK2 = 0): IMmax = Ekk = E0 − m2K2n ¯ ; that is (25)

987.3 MeV < IM < 861.2 MeV + B.

This relation indicates that B must be > 126.1 MeV. These considerations show that ¯ information on the binding energy of the (2K2n) B system, hence on its existence, can be + obtained from the features of the 2K IM distribution.

Strangeness production in antiproton-4 He annihilation at rest

213

Fig. 39. – Four-prong events. 2K+ invariant-mass distributions. White histograms: (a) events with pion and kaon momenta less than 400 MeV/c, (b) all events. The grey histograms are the diﬀerence between the 2K+ distributions and the fake 2K− ones. In the diﬀerence histograms few negative values are omitted.

By the way, note that, if the interpretation of the K− K− nn system is correct, a bound system of two neutrons without the participation of protons and with negative charge −2 is observed. Figure 39 shows the IM distributions for the diﬀerence 2K+ 2π − minus 2K− 2π + for diﬀerent event sets. One can see that the IM distributions are extended between 987.3 MeV, the minimum expected value, and ∼ 1300 MeV, so that B could reach a value of ∼ 1300 − 861.2 = 438.8 MeV. c.2) The IM distributions in ﬁg. 39 may be considered as due to an over imposition of two distributions, a phase space like one due to uncorrelated four particles (2K+ Σ− Σ− ) and one due to correlated three particles (2K+ X, X = (2K− 2n)B ). Figure 40a shows the 2K+ IM distribution evaluated for diﬀerent values of the binding energy B of X; ﬁg. 40b shows the phase-space like IM mass distribution. As our apparatus did not allow to measure kaons with momenta less than ∼ 150 MeV/c, we have calculated the distributions for pK > 0, 100 MeV/c and 150 MeV/c. The calculations show that i) the phase space distributions are quite diﬀerent from those for the correlated particles, in particular are much wider, ii) the width of the distribution for the correlated particles increases with the B value, iii) the distributions for the correlated particles are conﬁned at the lowest energies, particularly for the smaller B values, iv) the cuts on the lower momenta do not change the widths, but wear away the distributions, particularly in the lower energy side for the correlated particles; the correlated particle distribution disappears for pK > 150 MeV/c and B = 126.2 MeV. Figure 41 shows three examples of sum of the phase space distribution and of the

214

G. Bendiscioli

Fig. 40. – (a) Evaluation of the K+ K+ IM distribution for binding energy B = 126.2, 150, 175 and 200 MeV. (b) Four-prong events: phase space distributions for diﬀerent lower limits of pK . (c) Five-prong events: phase space distributions for diﬀerent lower limits of pK . White histograms: pK ≥ 0; grey: pK ≥ 100 MeV/c; black: pK ≥ 150 MeV/c.

correlated particle distribution, calculated for diﬀerent B and pK values, which reproduce fairly the prominent features of ﬁg. 39b; the distributions are given with the same bin as for the experimental one. We call the attention to ﬁg. 41c, which has been calculated for pK > 150 MeV/c: it indicates that the system (2K− 2n)B has a binding energy of the order of 150 MeV (166 MeV in the example) and contributes little to the full distribution (about 10% in the example). The value of 150 MeV is comparable with values predicted ¯K ¯ and ppnK ¯K ¯ (117 and 221 MeV, respectively). in table VI for the bound systems ppK The above consideration can be extended to the ﬁve-prong events, which may be considered as due to an over imposition of a phase space like distribution due to ﬁve uncorrelated particles (2K+ Σ− Λn) and one due to three correlated particles (2K+ X,

Strangeness production in antiproton-4 He annihilation at rest

215

Fig. 41. – Calculated K+ K+ IM distributions for four-prong events. Phase space plus correlated particle distribution: (a) B = 150 MeV without a cut on pK ; (b) B = 126.2 MeV and pK > 100 MeV/c; (c) B = 166 MeV and pK > 150 MeV/c. Grey histograms: correlated particle distributions; dotted histograms: phase space.

X = (2K− 2np)B ). The (2K− 2np)B system could have a binding energy of 150 MeV, as in the four-prong case, and contribute largely to the full distribution (about 50%). e) Conclusions Events of p ¯-4 He annihilation at rest with production of two K+ have been analysed. According to the energy and strangeness conservation laws, these events develop via the following intermediate states with the quoted yields (lower limits): 2K+ Σ− Σ− ps → 2K+ 2π − (2nps ) (0.17 ± 0.04) 10−4 , 2K+ Σ− Σ+ nπ − → 2K+ π + 2π − (3n) (2.71 ± 0.47) 10−4 , 2K+ Σ− Λn → 2K+ p2π − (2n) (1.21 ± 0.29) 10−4 and 2K+ K− Λ2n → 2K+ K− pπ − (2n) (0.28 ± 0.14) 10−4 . The production of 2K+ requires necessarily the involvement in the annihilation of more than one nucleon. The mechanisms through which the annihilation develop may be diﬀerent; for instance, rescattering cascades, formation and decay of quark-gluon plasma, formation and decay of bound antikaon-nucleon systems. In spite of the poor statistics, it has been put in evidence that the 2K+ IM distributions have features compatible with the idea of the formation of the bound systems nnK− K− and nnpK− K− , which decay through the intermediate states quoted above. A rough estimation of the binding energy and of the frequency indicates that B is of the order of 150 MeV. This ¯K ¯ value is comparable with theoretical predictions relevant to the bound systems ppK ¯K ¯ (221 MeV). (117 MeV) and ppnK

216

G. Bendiscioli

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Strangeness nuclear physics at FINUDA A. Feliciello INFN, Sezione di Torino - Via P. Giuria 1, 10125 Torino, Italy

Summary. — The FINUDA Experiment, installed at DAΦNE, completed its ﬁrst two data taking campaigns. A selection of the most interesting results is presented.

1. – Introduction FINUDA is a nuclear-physics experiment devoted to the study of single Λ-hypernucleus production and decay. As recalled by Prof. T. Bressani [1], a hypernucleus is a nucleus where one or more nucleons are replaced by one or more hyperons, usually Λ particles. The result is a manybody nuclear system with an explicit strangeness content: for this reason hypernuclear physics is often referred to as strangeness nuclear physics. The interest in investigating hypernuclei was triggered by the ﬁrst observation of such an exotic object in a stack of photographic emulsions, exposed to cosmic rays, made in 1953 by M. Danysz and J. Pniewski [2]. The Λ particle bound to a nucleus represents an excellent probe of nuclear properties. Its mass (MΛ = 1115 MeV) exceeds the mass of a nucleon by less than 20%. Even though its coupling with other nucleons is weaker than the nucleon-nucleon (N N ) interaction, one can expect that its behavior inside a nucleus is very similar to the one of a neutron. However this neutron has the strangeness quantum number S = −1, that makes the hyperon distinguishable from the other nucleons. Since the strangeness quantum number c Societ` a Italiana di Fisica

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is conserved by both strong and electromagnetic interactions, the hyperon embedded in nuclear medium lives long enough to give origin to sharp nuclear energy levels. Actually hypernuclear spectroscopy is the main subject of this research ﬁeld. However, hypernuclear physics oﬀers the opportunity to address a wide spectrum of fundamental questions, ranging from genuine nuclear physics to particle physics. On one side, spectroscopic investigation of hypernuclei represents the only practical way to extract information about hyperon-nucleon (Y N ) low-energy interaction. As a matter of fact, the hyperon lifetime (∼10−10 s) is too short to allow for direct Y N scattering experiments, which are, in principle, the most natural way to provide basic information about that interaction. This experimental solution becomes deﬁnitively unpractical in the case of hyperon-hyperon (Y Y ) interaction. The ultimate goal of these measurements is the precise determination of the strength of the spin dependent terms of the Y N eﬀective interaction. Following a phenomenological approach [3, 4], in the case of p-shell Λ-hypernuclei, such a potential can be written in the form (1)

VΛN (r) = V0 (r) +Vσ (r)sN · sΛ +VΛ (r)lΛN · sΛ +VN (r)lΛN · sN +VT (r)[3(σN · r)(σΛ · r − σN · σΛ )].

For each of the ﬁve terms in eq. (1), low-lying level energies can be described in terms of radial integrals over the sΛ pN wave function. These integrals, commonly denoted as V , Δ, SΛ , SN and T , can be then evaluated on the basis of measured observables [5]. The comparison of experimental results with theoretical predictions allows then to discriminate among the diﬀerent models depicting the Y N interaction and to improve the understanding of the baryon-baryon force in the SU (3) framework. On the other side, another important and, to some extent, unique source of information is represented by the observation of hypernucleus decay modes [6]. In particular, it is possible to have access to the four fermions, strangeness-changing, baryon-baryon weak interaction (2)

Λ + N → N + N,

which can occur only inside hypernuclei and then to investigate the question about the validity of the ΔI = 1/2 rule in weak interaction [7]. Indeed, the opening of the decay channel (2) is the most evident manifestation of the medium eﬀect, that is the modiﬁcation of the properties of a particle when embedded in the nuclear medium. As

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it is well known, a Λ-hyperon decays in free space through the mesonic channels (3)

Λ → p + π−

(BR = 63.9%),

(4)

Λ → n+π

(BR = 35.8%).

0

On the contrary, due to the Pauli exclusion principle, in all but the lightest hypernuclei these decay modes are more and more suppressed as A increases and leave room to the proton- or neutron-stimulated non-mesonic channels (5)

Λ + p → p + n,

(6)

Λ + n → n + n.

The Pauli principle plays as well a fundamental role in the so-called impurity physics, that is the study of the nuclear matter modiﬁcations induced by the presence of a strange particle. Since Λ-hyperon does not suﬀer from Pauli blocking, it can share space and momentum coordinates with the ordinary nucleons. Then it can penetrate into nucleus interior providing on the one hand a sensitive probe of the nuclear medium and forming, on the other hand, deeply bound hypernuclear states. The most spectacular eﬀect so far observed is the so-called glue-like role of the Λ. By precisely measuring the reduced probability of electric quadrupole γ-ray transitions (B(E2)), it has in fact inferred that the size of 7Λ Li hypernucleus is ∼ 20% smaller with respect to 6 Li ordinary nucleus [8]. Such a behavior can be considered a precursor of matter condensation induced by strange particles. This extraordinary binding capability of the Λ oﬀers as well the opportunity to look for neutron-rich Λ-hypernuclei, that is bound nuclear systems with extreme neutron to proton (N/Z) ratios [9]. A comprehensive review about hypernuclear physics can be found in ref. [1]. 2. – Experimental challenges From the experimental point of view the ﬁrst challenge to be faced with is the hypernucleus production. Since the ﬁnal goal is to put an explicit strangeness content inside a nucleus, the most natural and intuitive solution is to irradiate a nuclear target with a beam of strange particles. As far as single Λ-hypernuclei are concerned, the strangeness exchange reaction (7)

− K − + AZ → A ΛZ + π

both in-ﬂight and at rest, is historically the ﬁrst used in the pioneering measurements performed at CERN in the 70s [10-12]. Afterwards, several experimentalists began to use the more eﬃcient strangeness associated production (8)

+ π+ + A Z → A ΛZ + K

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both at BNL [13] and KEK [14] laboratories. The recent improvement of the e− beams’ quality and, moreover, of their duty cycle, made the so-called electro-production of hypernuclei possible at a reasonable rate [15], despite of the low cross-section value of process

(9)

e− + A Z → e− + K + + A Λ (Z − 1).

Finally hypernuclei can be produced also in p annihilations [16] or in heavy-ion collisions [17-19]. Regardless of the physical process exploited to produce a hypernucleus, it can be then considered the outcome of a genetic engineering manipulation, applied to the nuclear physics domain. Each of the above mentioned processes has its advantages and its drawbacks from both the physical and the technical point of view. Recent comparative and critical reviews of the diﬀerent exploitable methods can be found in refs. [1, 20, 21]. The second, fundamental ingredient required to carry on a signiﬁcant program of hypernuclear physics is, obviously, the realization of a spectrometer featuring the best momentum resolution and the maximum solid angle coverage. Traditionally these studies have been performed with magnetic spectrometer. Several good apparatuses were actually designed and put in operation. Typical values for energy resolution and angular acceptance were 2 MeV and few hundreds msr, respectively. However, it is worthwhile to remind that sometimes the ﬁnal performance of experiments was hampered essentially due to the unsatisfactory beam characteristics. At hadronic machines, to achieve a reasonable event rate, the thickness of the hypernucleus production target has to be usually of few tens g/cm2 , in order to partially oﬀset the low ﬂux of the incident particles in reactions (7), (8) and (9). This forced solution has a direct impact on the achievable resolution on the momentum of the particles tagging the hypernucleus formation. The multiple Coulomb scattering suﬀered by such particles, emerging from bulky targets, makes useless the spectrometer capabilities, so preventing a precise determination of the energy level scheme of the produced hypernuclei. The lack of suitable intensity and quality beams conditioned the detector technological development as well. γ-ray spectroscopy is a powerful and consolidated methodology used in nuclear physics which, thanks to the excellent energy resolution typical of High-Purity Germanium (HPGe) crystals, allowed to get a deep insight into the nuclear structure. However, low incident particle ﬂux and high level of contaminating hadronic particles prevented for long time the porting of this technique to the hypernuclear ﬁeld. Only very recently the coupling of large γ-ray arrays [22] and magnetic spectrometer has been made possible, bringing great progress in hypernuclear physics by revealing precise structure of several light Λ-hypernuclei with an energy resolution in the keV range [23].

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223

Fig. 1. – Global view of the FINUDA apparatus.

3. – The FINUDA experiment at DAΦNE The DAΦNE e+ e− collider [24, 25], that is the double annular Φ-factory for Nice Experiments currently operating at the INFN National Laboratories of Frascati (LNF), is the unconventional playground where the FINUDA Collaboration is successfully carrying on its hypernuclear studies program. Single Λ-hypernuclei are produced by stopping the K − following the main φ resonance decay, through the reaction (10)

− − Kstop + AZ → A ΛZ + π .

Also in this case the beam characteristics and, moreover, the accelerator type strongly inﬂuenced the experiment concept and the apparatus design. Usually hypernuclear experiments are carried out at hadronic machines, delivering kaon and/or pion beams. Then, the apparatus layout is the one of a ﬁxed-target experiment and the spectrometer has the typical one-arm conﬁguration. On the contrary the FINUDA apparatus has the typical architecture of a collider detector (see ﬁg. 1). The ﬁrst, striking advantage of such a conﬁguration is the large solid angle coverage (ΔΩ > 2π sr) with respect to the small acceptance (ΔΩ ≈ 100 msr) that characterizes past and even next-generation experiments [26]. This means that the FINUDA spectrometer allows to measure in coincidence the particles tagging the hypernucleus formation and all the products, both charged and neutral, following its decay, namely pions from the mesonic channel (see eq. (3)), (11)

A ΛZ

→ A (Z + 1) + π − ,

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and nucleons from the non-mesonic ones (see eqs. (5), (6)), (12) (13)

A ΛZ A ΛZ

→

(A−2)

(Z − 1) + p + n,

→

(A−2)

Z + n + n.

An important consequence is that many observables relative to the produced hypernuclei, like excitation spectrum, lifetime and partial decay widths for the diﬀerent decay channels, can be measured in the same run with high statistics and high energy resolution. Moreover the FINUDA apparatus has been designed in such a way that up to 8 solid targets of diﬀerent material can be installed and simultaneously exposed to the K − ﬂux. This smart solution is a further key feature that makes it possible to signiﬁcantly reduce the possibility of systematic errors in comparing properties of diﬀerent Λ-hypernuclei. The FINUDA apparatus ﬁnal conﬁguration was the outcome of a long and systematic work of modeling and optimization. During the design phase, nothing was left to chance. Thanks to an intensive and extensive use of a Monte Carlo program simulation based on the 3.21 release of the GEANT package [27], every detail was carefully studied in order to achieve the maximum performance with the available state-of-art detector technology. A sketch of the FINUDA spectrometer is shown by ﬁg. 1. The various sub-detectors are housed inside the coil of a superconducting solenoid, providing a magnetic ﬁeld of 1.0 T in a cylindrical volume with 1.46 m radius and 2.11 m length, that is of ∼ 14 m3 . Particular care was devoted to its design in order to achieve a ﬁeld homogeneity within 2% over the whole tracking region. This feature was also required by the fact that the FINUDA magnet must be considered as an element of the DAΦNE machine magnetic lattice. Then it has been installed and precisely aligned in such a way as its geometrical center coincides with the (e+ , e− ) colliding region, where φ-mesons are formed and decay. On the basis of their speciﬁc function, the diﬀerent components of the FINUDA spectrometer can be logically divided in three main groups. 1) The interaction/target region. A sketch of this innermost and essential part of the apparatus is shown by ﬁg. 2. Since K − ’s necessary to induce the Λ-hypernucleus formation reaction (10) are not directly delivered as primary beam by the accelerator, the selection of kaons against other φ decay products and background particles is a fundamental prerequisite. A barrel of 12 thin plastic scintillator staves (toﬁno), installed immediately outside the DAΦNE beam pipe, is devoted to this crucial task. Thanks to their high ionizing power, (K + , K − ) pairs are identiﬁed by toﬁno which provides as well the timing signal for the trigger logic and the start for time-of-ﬂight (ToF) measurements with a resolution of σ ≈ 300 ps. The (K + , K − ) trajectories are then measured with an accuracy of σ ≈ 30 μm by a ﬁrst, octagonal array of silicon microstrip detectors (ISIM), surrounding the scintillator barrel. ISIM allows also dE/dx measurements with an energy resolution of 20% FWHM; this information is used to endorse the particle identiﬁcation based on the toﬁno response. Finally a very thin target slice (0.2–0.3 g/cm2 ) is placed at a distance of

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Fig. 2. – Magniﬁed view of the FINUDA interaction/target region.

a few millimeters with respect to the external surface of each ISIM module. The spatial information delivered by ISIM represents the starting point to extrapolate the path of kaons inside the target module and then to estimate the coordinates of their interaction vertex. The precision on the localization of the (K + , K − ) stopping point turns out to be of few hundreds microns, essentially because of the Coulomb multiple scattering suﬀered by very slow particles (pK ± ≈ 127 MeV/c) in the last part of their trajectory. 2) The tracking device. Four diﬀerent layers of position sensitive detectors are in charge of accurately measuring charged particle trajectories. They are mounted around the apparatus geometrical axis, with cylindrical symmetry and hold in place by a sturdy spaceframe (clepsydra), anchored to the solenoid cryostat (see ﬁg. 1). They are fully contained in a helium chamber of ∼ 8 m3 volume, in order to minimize the eﬀect of the Coulomb multiple scattering. The trajectories of charged particles following the hypernucleus formation and its decay are measured as soon as they leave the target by a second array of 10 Silicon microstrip detectors (OSIM), surrounding the target array (see ﬁg. 2); the second and the third point coordinates are provided by two sets of 8 planar, Low-Mass Drift Chambers (LMDC), ﬁlled with a (70% He–30% C4 H10 ) mixture and featuring a spatial resolution of σρϕ ≈ 150 μm and σz ≈ 1 cm (corresponding to the canonical 1% of the wire length); ﬁnally a complex structure made of 2444 Straw Tube Detectors (STB), arranged in six layers of both longitudinal and stereo elements, allows to measure the coordinates of

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the fourth point at the most external radius of the spectrometer with a spatial resolution of σρϕ ≈ 150 μm and σz ≈ 500 μm. With the magnetic ﬁeld intensity set at B = 1.0 T, the design momentum resolution of the spectrometer for pions emitted after hypernucleus formation and having a momentum value typically lying in a range centered around 270 MeV/c is Δp/p ≈ 0.4% FWHM. This ﬁgure translates into an energy resolution on hypernuclear energy levels better than 1 MeV FWHM. As far as the hypernucleus decay product detection is concerned, protons emitted in the Λ-hypernucleus non-mesonic decay (12) are tracked with an acceptance of ∼ 30% and an energy resolution ΔE ≈ 1.3 MeV FWHM for a typical proton energy value of 80 MeV. 3) The neutron detector. A second, larger plastic scintillator barrel (tofone) consisting of 72 staves 10 cm thick, encloses the whole spectrometer (see ﬁg. 1). It accomplishes a twofold task. On the one hand, it integrates the information coming from toﬁno, so providing signals for the ﬁrst-level trigger and allowing for measurements of the time of ﬂight of charged particles following formation and decay of hypernuclei. On the other hand, tofone has been designed to detect neutrons emitted in Λ-hypernucleus non-mesonic decays (12) and (13) with an eﬃciency of ∼ 12%, an acceptance of ∼ 70% and an energy resolution ΔE ≈ 8 MeV FWHM, again for a typical neutron energy of 80 MeV. Such a complex apparatus would have been jut a collection of excellent detectors without an appropriate trigger capability. Actually, the implemented trigger logic is a further FINUDA distinctive feature that makes the described spectrometer a real powerful tool for strangeness nuclear physics. The candidate events are selected by requiring two ﬁred, opposite (back-to-back) toﬁno slabs, giving signal amplitude above an energy threshold accounting for the high ionization of very slow kaons, in coincidence with a signal from the tofone barrel within a narrow temporal window. Figure 3 shows one of the selected events. Further details about the FINUDA Experiment and its physics program can be found in refs. [28, 29]. 4. – Data taking and apparatus performance As mentioned in sect. 3, the FINUDA apparatus is deeply integrated with the DAΦNE collider. Then it is not surprising that the experiment data taking is characterized by a continuous and intense exchange of information with the machine team. Many experimental tests are performed in order to monitor and to keep under control the accelerator behavior [30]. Their feedbacks are sent to operators on shift to improve the beams’ tuning and to achieve the maximum common performance. The main measurements routinely performed are the following: – the luminosity of the DAΦNE collider was continuously evaluated by looking at Bhabha scattering events; on average it turned to be in good agreement (within 10%) with the value logged by machine operators;

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Fig. 3. – Event display of a candidate hypernuclear event on a 6 Li target.

– the contour of e+ e− interaction volume was continuously monitored as well, paying attention to any modiﬁcation related to a possible machine malfunctioning; – the energy of the circulating beams was measured on-line by looking again at Bhabha scattering events and through the reconstruction of the KS0 → π + + π − invariant mass, where KS0 ’s are due to the φ → KS0 + KL0 decay. The reconstruction process of the φ formation point and of the (K + , K − ) pair initial direction and momentum uses the coordinates of the points where kaons cross the ISIM modules, identiﬁed thanks to their high stopping power. The procedure is based on a twin-helix algorithm which accounts for the kinematics of φ decay, the average value of the φ mass, the colliding angle between e+ and e− beams (12.5 mrad, measured by using Bhabha scattering events) and the geometrical conﬁguration of the vertex region. The coordinates of the stopping points of the kaons inside the targets are computed by means of a tracking procedure based on the GEANE package [31], which performs a numerical integration of the trajectory starting from the assumed φ formation point, taking into account the kaon initial direction and momentum and accounting for geometrical structure and material budget of the FINUDA interaction/target region. The beam crossing angle determines a small total momentum of the produced φ (boost = 12.3 MeV/c), directed towards the positive x side. This boost adds to the 127 MeV/c average momentum of the kaons following φ decay, introducing a left-right

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Fig. 4. – Scatter-plot of the reconstructed y vs. x coordinates of the K − stopping points. The most external octagonal silhouette corresponds to the position of the 8 target modules, where most of the negative kaons go to rest. Stops in the microstrip ISIM modules are distributed mostly on the left side of the apparatus: as explained in the text, this asymmetry is due to the φ boost, which is directed towards the positive x side. Finally, the vanishing circular proﬁle put into evidence the toﬁno presence.

asymmetry clearly visible in ﬁg. 4, which shows the scatter-plot of the reconstructed y vs. x coordinates of the K − stopping points. The distribution of points on the outer octagonal silhouette reproduces the positions of the 8 targets, where most of the K − stop (∼ 75% of all K − ’s interacting in the apparatus). A smaller accumulation of stopping points (∼ 10%) also occurs on the left-side ISIM modules, providing an additional, even though statistically limited, data sample in supplementary 28 Si targets. Finally the remaining thickened points partially depict toﬁno. The selected events corresponding to a Λ-hypernucleus production are characterized by the simultaneous presence of K + and K − particles. The K + fully reconstruction enables the K − tagging and moreover oﬀers the possibility to perform an accurate and continuous in-beam calibration of the FINUDA spectrometer. The positive kaons which stop in one of the modules of the target array decay at rest with a mean life of 12.4 ns. The two most frequent two-body decays (14)

K + → μ+ + νμ

(BR = 63.52%)

K + → π+ + π0

(BR = 21.16%)

and (15)

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Fig. 5. – Momentum distribution of the positive tracks coming from stopped K + (from ref. [34]). The peak at ∼ 236 MeV/c corresponds to the two-body decay K + → μ+ + νμ , while the one at ∼ 205 MeV/c corresponds to the two-body decay K + → π + + π 0 .

are a source of monochromatic particles, capable in principle of fully crossing the apparatus, with momenta 235.5 MeV/c for the μ+ and 205.1 MeV/c for the π + . By reconstructing such events, the absolute scale for the momentum measurement was determined with a precision better than 200 keV/c, even in the simpliﬁed hypothesis of a perfect constant magnetic ﬁeld of 1.0 T, directed along the z-axis, over the whole tracking volume. The precision achieved can be then assumed as the systematic error on the measurement of the particles’ momenta in the range between 200 MeV/c and 300 MeV/c. This experimental result, reported in ﬁg. 5, has been obtained by selecting only high-quality tracks. The criteria to classify a track as high-quality one are the following: they must be emitted in the forward hemisphere, with respect to the direction of the incident K + , and they have to cross the whole spectrometer without impacting on the structural elements of the apparatus, so giving a signal on all the four tracking detectors described in sect. 3 (see ﬁg. 3). The tails on the left side of the two peaks are due to several contributions. The main part is played by instrumental eﬀects due to the energy loss suﬀered by particles crossing the edges of the drift chambers and their support. Moreover in this energy range two + additional K + decay channel open: the Ke3 mode (BR = 4.8%), originating a continuum spectrum of positrons (which cannot be distinguished from μ+ ’s) ending at 228 MeV/c, + and the Kμ3 one (BR = 3.2%), which gives again a continuum spectrum with end point at 215 MeV/c. By analyzing all these diﬀerent contributions to the peak shape, one can conclude that the asymmetry aﬀects, overall, the Gaussian line shape at the level of ∼ 4%. From the width of the μ+ signal the preliminary momentum resolution of the FINUDA spectrometer can be estimated to be Δp/p ≈ 0.6% FWHM, which translates into

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Fig. 6. – Schematic frontal cross-section of the FINUDA interaction/target region, with the target set used in the ﬁrst (left) and the second (right) data taking period.

an energy resolution on hypernuclear energy level of ∼ 1.3 MeV FWHM. It is expected that the momentum resolution should improve to the design value of ∼ 0.4% FWHM once the ﬁnal detector calibration and alignment will be performed and the mapped magnetic ﬁeld will be made available to the reconstruction and ﬁtting procedures. 5. – Physics results As mentioned in sect. 1, the central item of the FINUDA physics program is the production of several hypernuclear species and a systematic study of their properties. The ﬁrst data sample was taken from December 2003 to March 2004. An integrated luminosity of 190 pb−1 was collected, corresponding to ∼ 37 × 106 logged events. Five diﬀerent types of target were used: 2× 6 Li (isotopically enriched to 90%), 1× 7 Li (natural isotopic abundance), 3 × 12 C, 1 × 27 Al and 1 × 51 V (see ﬁg. 6). The ﬁrst, important result of this exploratory run is represented by the demonstration of the validity of the intuition of using low-energy kaons from φ resonance decay, in order to produce single Λ-hypernuclei [32]. Initially, the attention was focussed on spectroscopic study of produced Λ-hypernuclei and on their decay modes. However, thanks to the excellent FINUDA tracking performance and the capability of identifying all particles involved in each event, it was possible as well to search for neutron-rich Λ-hypernuclei and to address one of the hottest topics in strangeness nuclear physics, that is the formation of (deeply) bound K-nucleus states. Finally an interesting investigation on low-energy K + -nucleus interaction has been performed.

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Fig. 7. – Spectrum of the momentum of the π − emitted from the interaction vertex of a K − onto a 12 C target (from ref. [34]). The dashed line represents the contribution due to K − absorption by two nucleons (process (21)).

. 5 1. Hypernuclear spectroscopy. – 12Λ C is one of the ﬁrst hypernuclear systems extensively studied. Therefore, it traditionally represents the natural reference point for hypernuclear physics. This was an additional reason behind the choice of having three 12 C targets installed during the ﬁrst FINUDA run [33]. In order to evaluate the capabilities of FINUDA to yield relevant spectroscopic parameters, the analysis started just from the data collected on 12 C targets. The obtained results were reported in ref. [34]. It was possible to merge the spectra of only two, out of the three available, 12 C targets since the third one showed a slight systematic energy displacement of 0.5 MeV. This is an example of the possible drawbacks of the FINUDA architecture, that is of having a composite apparatus requiring a careful alignment procedure in order to get coherent information from the several discrete components. The requirement of high-quality tracks (long tracks crossing the whole spectrometer with a hit on each tracking detector, i.e. OSIM, LMDC’s and STB) reduced the analyzed data to ∼ 40% of the whole available sample of events with vertex coming from a 12 C target. Figure 7 shows the momentum distribution coming from the two selected 12 C targets. Besides the hypernuclear production reaction (10), several process produce π − after K − absorption and reproduce well the experimental spectrum [35]: (a) quasi-free Σ+ , Σ0 and Λ production: (16)

K − + p → Σ+ + π − ,

(17)

K − + n → Σ0 + π − ,

(18)

K − + n → Λ + π− ;

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(b) quasi-free Λ decay: Λ → p + π− ;

(19) (c) quasi-free Σ− production: (20)

K − + p → Σ− + π + Σ− → n + π − ;

(d) two-nucleon K − absorption: (21)

K − + (N N ) → Σ− + N , Σ− → n + π − .

All the above-mentioned background reactions were simulated in detail by means of the FINUDA Monte Carlo program. The simulated events were reconstructed by the same program used for the real events, adopting the same selection criteria, in order to accurately take into account the geometrical acceptance and the reconstruction efﬁciency of the apparatus. In particular, the size of the spectrometer and the value of the magnetic ﬁeld determine an acceptance momentum cut at ∼ 180 MeV/c for four-hits tracks (see ﬁg. 7), which naturally excluded most of the low-energy π − ’s produced in background processes (a)-(d). However, in the momentum region above ∼ 260 MeV/c, where signals corresponding to 12Λ C bound states formation are expected, there is the insidious contribution from process (d). It is worthwhile to note that both processes (c) and (d) are due to Σ− in ﬂight decay but the π − momentum distribution from the process (c) is peaked at 190 MeV/c and goes to zero beyond 260 MeV/c. The dashed line in ﬁg. 7 represents the contribution due to the process (d), normalized to the number of entries in the (275–320) MeV/c momentum range, that is beyond the physical region for the production of Λ-hypernuclei via the reaction (10). In order to obtain the Λ binding energy distribution, the estimated (d) process contribution was subtracted from the experimental π − momentum distribution; then momenta were converted into binding energy (−BΛ ). The two prominent peaks, as can be seen in both ﬁgs. 8 a) and 8 b) at BΛ ≈ 11 MeV (ground state) and BΛ ≈ 0 MeV, were already observed in previous experiments [11, 36, 37] and interpreted as (νp−1 1/2 , Λs) and −1 (νp3/2 , Λp), where ν = nucleon. The experimental energy resolution was determined by ﬁtting the BΛ ≈ 11 MeV peak with a Gaussian curve (χ2 /d.o.f. = 1.71): it turned out to be 1.29 MeV FWHM. In this respect, it is worthwhile to remind that for 12Λ C an excitation spectrum with a 1.45 MeV FWHM resolution was recently obtained at KEK using the (π + , K + ) reaction at 1.05 GeV/c by the E369 Collaboration [38]. The 12Λ C ground state is assumed to be a single state. Indeed it is known that it consists of a (1− , 2+ ) doublet, but theoretical calculations predict splitting of 70 keV [39], 80 keV [40] and 140 keV [41] between them, that is one order of magnitude smaller than the presently

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Fig. 8. – Λ binding energy spectrum of 12Λ C measured by the FINUDA Experiment (from ref. [34]). a) The solid line represents the result of a ﬁt with six Gaussian functions (#1– #6), as explained in the text; b) the solid line represents the results of a second ﬁt with seven Gaussian functions as explained in the text. The dot-dashed line starting at BΛ = −1 MeV represents the contribution from the quasi-free Λ. The dotted lines represent the result of a Gaussian ﬁt on every single peak.

achieved instrumental resolution. The peak at BΛ ≈ 0 MeV has a more complex structure and an attempt to disentangle diﬀerent contributions was done. The experimental spectrum showed in ﬁg. 8 closely resembles the one from E369 Experiment [38]. This is expected, as the production of hypernuclear states is, in ﬁrst approximation, determined by the momentum transferred to Λ’s, which is grossly comparable for both experiments (∼ 250 MeV/c in the FINUDA case, ∼ 350 MeV/c for E369). The ∼ 100 MeV/c diﬀerence may account for the diﬀerent yield of the two main peaks. The absolute values of the capture rates for the diﬀerent peaks could be obtained in a simple way by the method of the K − tagging. Indeed, in the events where the K + is seen to decay in the Kμ2 or Kπ2 decay mode, with the produced μ+ or π + crossing the spectrometer and hitting the tofone barrel, one can assume that the trigger condition on the prompt tofone coincidence has been satisﬁed by charged products of K + decay. Hence, in these events triggered by K + decay products, the interactions of the corresponding K − in the targets are observed without any trigger bias. Using this subsample of events, the number of K − stopping in the target can be directly counted and the number of π − produced by the K − interactions can be accurately determined by only applying the correction for the apparatus acceptance for π − of selected momentum and for detector eﬃciency. The acceptance was calculated by using the FINUDA Monte Carlo program, while the detector eﬃciency was determined by calibration data. The value obtained for the 12Λ C ground-state formation is (1.01 ± 0.11stat ± 0.10syst ) × −3 10 /(stopped K − ). It agrees very well with the value (0.98 ± 0.12) × 10−3 /(stopped

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Table I. – Results from −BΛ spectrum ﬁts (from ref. [34]): the upper part of the table refers to a ﬁt performed with the same energy level scheme determined by E369 Experiment [38], with six Gaussian functions; the lower part reports the results of a ﬁt with seven hypernuclear levels. The rightmost column lists the capture rates corresponding to each peak. The errors quoted for peaks (#2–#6) in the upper panel and (#2–#7) in the lower one do not include the error on the 12 Λ C ground-state capture rate. The errors on the rates of peaks #6 and #7 take into account the error on the subtracted background. Peak number

−BΛ (MeV) (ﬁxed at E369 values)

1 2 3 4 5 6

−10.76 −8.25 −4.46 −2.70 −0.10 +1.61

Peak number

−BΛ (MeV)

1 2 3 4 5 6 7

−10.94 ± 0.06 −8.4 ± 0.2 −5.9 ± 0.1 −3.8 ± 0.1 −1.6 ± 0.2 +0.27 ± 0.06 +2.1 ± 0.2

Capture rate/(stopped K − ) (×10−3 ) 1.01 ± 0.11stat ± 0.10syst 0.23 ± 0.05 0.62 ± 0.08 0.45 ± 0.07 2.01 ± 0.14 0.57 ± 0.11 Capture rate/(stopped K − ) (×10−3 ) 1.01 ± 0.11stat ± 0.10syst 0.21 ± 0.05 0.44 ± 0.07 0.56 ± 0.08 0.50 ± 0.08 2.01 ± 0.17 0.58 ± 0.18

K − ) measured at KEK [35]; it is worthwhile to remind that the ﬁrst-generation CERN Experiment reported the value (2 ± 1) × 10−4 /(stopped K − ) [11]. In between the two main signals there are also indications of other states produced with weaker strength. In order to reproduce, at least qualitatively, this spectrum six Gaussian functions were used, centered at the BΛ values reported in ref. [38]; their widths were ﬁxed to σ = 0.55 MeV, corresponding to the experimental resolution. The abscissa scale is aﬀected only by a scale error of ±80 keV. The result of this ﬁt is shown in ﬁg. 8 a). The spectrum is not well reproduced, the resulting reduced χ2 /d.o.f. is 3.8 (for 64 d.o.f.) and, in particular, the region −10 MeV < −BΛ < −5 MeV is poorly ﬁtted. The capture rates for these diﬀerent contributions, normalized to the ground-state one, are reported in the second column of the upper part of table I. A better χ2 /d.o.f. = 2.3 is obtained by adding a further contribution and leaving the positions of the seven levels free (57 d.o.f.). Their values are listed in the second column of the lower part of table I. The capture rates for these diﬀerent contributions are again normalized to the capture rate for the 12Λ C ground-state formation. The results of this second ﬁt are shown in ﬁg. 8 b). A contribution from the quasi-free Λ production,

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starting from BΛ = 0 and properly smeared by taking into account the instrumental resolution, was included in both ﬁts. The peaks #2 and #3 can be attributed to the 11 C core exited states at 2.00 MeV and 4.80 MeV. The excitation of these states was predicted in several theoretical calculations [42, 43]; their energies may be sensitive to the ΛN interaction matrix elements. However, the peak #4 and the newly observed #5 are not explained in such a simple way. Excluding from the ﬁt procedure the peak #5, the value for χ2 /d.o.f. worsened to 3.3. There exist several positive-parity excited states of the 11 C core in this energy region which could contribute to these structures [43]. Finally, it can be noticed that the integrated strength for the excitation of all these weakly excited states compared to that of the two main peaks is more than twice larger than the one reported by E369 [38]. The sum of the capture rates for the BΛ = 0.27 MeV and BΛ = 2.10 MeV states is (2.59 ± 0.19stat ) × 10−3 /(stopped K − ) and agrees with the KEK result (2.3 ± 0.3) × 10−3 /(stopped K − ) [35], in which the contributions for the two levels were not resolved. The CERN Experiment [11] measured (3 ± 1) × 10−4 /(stopped K − ). In the FINUDA analysis these states are indeed resolved though, inevitably, strongly correlated in the ﬁt. Nevertheless it is remarkable that the relative intensities for the contribution at BΛ = 0.27 MeV and BΛ = 2.10 MeV are close to the values found by Dalitz et al. in an old emulsion experiment [44]. Theoretical calculations for the ground-state formation quote the values 0.33 × 10−3 /(stopped K − ) [45], 0.23 × 10−3 /(stopped K − ) [46] and 0.12 × 10−3 /(stopped K − ) [47]. Analogous predictions for the capture rate leading to states in which the Λ in a p state quote, respectively, 0.96 × 10−3 /(stopped K − ) [48], according to the theoretical prediction of ref. [45], and 0.59 × 10−3 /(stopped K − ) following ref. [47]. As a general remark it may be noticed that the values measured by the FINUDA Experiment are larger by factor (3–6) as compared with theoretical predictions. Finally, the pattern of the relative strength for the excited-core states is also signiﬁcantly larger than the results of the theoretical calculation reported in ref. [49]. Interesting preliminary results were obtained as well for 7Λ Li hypernucleus [50]. Figure 9 shows the π − momentum distribution related to high-quality, negatively charged, tracks emerging from the 7 Li target. A clear peak at ∼ 270 MeV/c is visible. It can be interpreted as the signature of 7Λ Li formation, given the Λ binding energy of the 7Λ Li ground state (5.58 ± 0.03 MeV [51]) and the excitation energies measured via γ-spectroscopy [52-54]. The binding energy −BΛ corresponding to the signal region is plotted in ﬁg. 10. Two peaks emerges from the background, revealing the presence of at least two 7Λ Li hypernucleus states or complexes. Like in the case of 12Λ C, the FINUDA Monte Carlo program has been used in order to get an estimation of the background underneath the bound region. After a detailed analysis, processes (18) and (21) were found to eﬀectively contribute. K − in-ﬂight decay was then recognized as an additional background source, when it occurs after that K − generates the trigger, by interacting in the toﬁno, and before it enters one of the targets. μ’s produced in such a way in the vicinity of the target can be reconstructed as π’s and can originate a fake entry in the signal region. The BΛ distribution corresponding to each

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Fig. 9. – Spectrum of the momentum of the π − emitted from the interaction vertex of a K − onto a 7 Li target (from ref. [50]).

of these three contributions was ﬁnally ﬁtted in the range −20 MeV < BΛ < +2 MeV by choosing the most proper curve (exponential or polynomial) to reproduce its shape. The experimental data was thus ﬁtted to the sum of the curves, representing the three relevant background sources, and of two Gaussian functions, for the signals. The weight of the ﬁrst two background reactions (18) and (21) was left free to scale, while the one of the K − in-ﬂight decay was ﬁxed to the experimental value of the K + in-ﬂight decay inferred from the FINUDA data (same sample used for signal extraction) corrected for the diﬀerent reconstruction eﬃciency. The width of both Gaussian functions was ﬁxed to the FINUDA resolution of 0.46 MeV, corresponding to a 1.1 MeV FWHM. This better performance can be to a large extent explained by the fact that data were collected on a single 7 Li target, making the analysis much simpler than in the case of the abovedescribed three 12 C targets. Moreover, it is worthwhile to remind that the best previous 7 Λ Li hypernuclear spectrum measured with a magnetic spectrometer was obtained by the E336 Experiment at KEK [55, 56], exploiting the (π + , K + ) production process, with an energy resolution of 2.2 MeV FWHM. The result of the ﬁt, with a reduced χ2 of 1.2, for the Λ binding energy distribution is showed in ﬁg. 10. Two clear peaks are clearly identiﬁed at −5.33 ± 0.13stat MeV and −3.68 ± 0.15stat MeV. Their features are summarized in table II. The ΔBΛ between the two signals amounts to 1.69 MeV. As recalled in sect. 4, the absolute scale of the momentum was determined with a precision better than 200 keV/c, corresponding to

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Fig. 10. – Λ binding energy spectrum of 7Λ Li measured by the FINUDA Experiment (from ref. [50]). The superimposed ﬁt, with a reduced χ2 of 1.2, is the result of the sum of two Gaussian functions and a model for the background (see text for details).

∼ 0.18 MeV for the energy absolute scale. This value accounts for the systematic error. The energy level scheme of 7Λ Li is well known thanks to γ-ray spectroscopy campaigns carried out by the Hyperball Collaboration [52] (see ﬁg. 11). Such measurements allowed to infer the excitation energies from the ground state, established at BΛ = 5.58 ± 0.03 MeV [51]. However it must be noticed that this value was calculated on the basis of the measured π + beam momentum and by using the masses of the K − , Λ and nuclei known in 1970 [57-59]. By taking into account all the correction due to improvements in the measurement of these parameters, it is reasonable to estimate that the up-to-date value for BΛ of the 7Λ Li ground state is 5.79 MeV.

Table II. – Features of the 7Λ Li levels observed by the FINUDA Experiment (from ref. [50]). Peak number

−BΛ (MeV)

Yield (events)

Capture rate/(stopped K − ) ()

1 2

−5.33 ± 0.13stat ± 0.18syst −3.68 ± 0.15stat ± 0.18syst

52 ± 11 44 ± 10

0.47 ± 0.12stat ± 0.11syst 0.39 ± 0.11stat ± 0.11syst

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Fig. 11. – 7Λ Li energy level scheme measured by the Hyperball Collaboration (from ref. [52]).

Given the total error on the FINUDA measurement, it would be diﬃcult to assert if the peak #1 is due to the production of the ground state, of the ﬁrst excited state (3/2+ ) or of a mixture of the two. In the same way it is not possible to state if the peak #2 is due to the production of the 5/2+ state, of the 7/2+ one or of a mixture of the − two. However in the (Kstop , π − ) reaction the spin-ﬂip amplitude is expected to be quite small and then the spin-ﬂip states (3/2+ and 7/2+ ) cannot be populated [3]. Then, most likely, the two observed signals can be attributed to the 1/2+ and 5/2+ states. This is the ﬁrst time that such hypernuclear states are so clearly visible in the momentum spectrum of π − coming from the formation reaction (10), allowing the absolute binding energy determination. The limited statistics prevented the search for further states and did not allow a better insight of the observed ones. However, the estimated production rates, reported in table II, are in good agreement with earlier calculations by Dalitz and Gal [3]. . 5 2. Hypernucleus non-mesonic decay. – As brieﬂy mentioned in sect. 1, the study of Λ-hypernucleus decay modes represents an intriguing investigation ﬁeld. In particular the determination of the relative weights of the diﬀerent decay channels represented a long-standing puzzle [60]. For this reason, a preliminary study of the energy spectrum of proton emitted in 12Λ C ground-state weak decay was carried out [61]. To this purpose, the same data sample used for the spectroscopic studies was analyzed. Figure 12 shows the momentum distribution of π − coming from the three 12 C targets. By comparing ﬁgs. 7 and 12 one immediately notices the striking diﬀerence in the quality of the two spectra and in the number of the entries. Since for the Λ-hypernucleus decay mode observation the FINUDA maximum performance is unnecessary, the latter spectrum has indeed obtained by loosening the selection criteria adopted for hypernuclear spectroscopy. This time, backward tracks, that is π − ﬂying in opposite direction with respect to the incident K − and then backcrossing the interaction/target region, are accepted. In addition the cut applied on the track ﬁt χ2 was partially weakened. The result was an increase of statistics by a factor

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Fig. 12. – Momentum spectrum of π − from 12 C targets obtained by applying looser selection criteria with respect the quality cuts leading to the analogous spectrum of ﬁg. 7 (from ref. [61]).

∼ 6 as well as a signal/background ratio worsening. This possibility of getting from the same data sample excitation energy spectrum of quite diﬀerent quality by just changing the data selection criteria put in evidence another FINUDA distinctive feature. It is worthwhile to remind that in other experimental situations, like for instance that of the SKS spectrometer [62], data at high resolution or at high statistics requires diﬀerent set-ups, that is targets of diﬀerent thickness, and dedicated runs. Figure 13 shows how the spectrum of ﬁg. 12 looks like when the additional requirement of a proton detected in coincidence is applied. The shaded area indicates the simulated background contribution from process (21), ﬁltered through the same selection criteria of the real data and normalized in area beyond 277 MeV/c. The agreement in the overlap region is quite good, with the exception of the range 277–285 Mev/c, where the experimental spectrum exhibits a small excess of events. Such a discrepancy was discovered to be due to in-ﬂight K − induced reactions. It was veriﬁed that the contribution of these events can be reduced by applying dedicated selection criteria which, however, would have dramatically reduced the available statistics. The left panel of ﬁg. 14 shows the proton energy spectrum emitted in coincidence with a π − coming from the 12Λ C ground-state region. It is compared to the energy spectrum of protons generated by the two-nucleon K − absorption mechanism, evaluated by means of the FINUDA simulation program in correspondence of the π − momentum range 270–277 MeV/c (black area in ﬁg. 13). Both histograms are acceptance corrected. It is worthwhile to remark that the main feature of the measured spectrum, that is a broad bump centered at ∼ 120 MeV, was already observed in another study of the proton spectra emitted following the capture of K − [63]. The right panel of ﬁg. 14 shows the diﬀerence between the two distributions reported in the left part of the same ﬁgure. The errors are statistical only. An encouraging conﬁrmation of this result comes from a simple, experimental approach to the problem of the correct estimation of the background

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Fig. 13. – Momentum spectrum of π − of ﬁg. 12 with the additional requirement of a proton detected in coincidence (from ref. [61]). The grey area represents the π − following two-nucleon K − absorption (process (21)). The peak at ∼ 273 MeV/c (black bins) corresponds to the 12Λ C ground-state region.

contribution to the spectrum of ﬁg. 13. By subtracting the proton spectrum measured in correspondence of the side-bins (277–282 MeV/c) of the π − spectrum, one obtains a spectrum fully compatible with the one reported in the right panel of ﬁg. 14, even though characterized by larger errors.

Fig. 14. – Left: energy spectrum of the protons emitted in coincidence with π − from the 12Λ C ground-state region (solid circles); energy spectrum of the protons following two-nucleon K − absorption, emitted in correspondence of the black area of ﬁg. 13 (solid squares). Right: diﬀerence between the two distributions shown in the left panel. (From ref. [61].)

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Fig. 15. – Comparison between proton energy spectrum measured by the FINUDA Experiment (black circles) and the one obtained at KEK with the SKS spectrometer (gray squares) [69] (from ref. [61]). The two spectra are normalized in area beyond 35 MeV.

The absolute branching ratios Γp for the proton-stimulated weak decay turned out to be (0.41 ± 0.15) × ΓΛ , for protons of energy larger than 20 MeV. As a general comment, one can say that the proton spectrum of the right panel of ﬁg. 14 has the shape that one can naively expect from a simple Λ + n → n + p weak decay occurring in nuclear matter. The bell-shaped spectrum, centered around 80 MeV (Q-value for the weak-decay reaction), is well explained by the Fermi motion of the interacting baryons in the nucleus, giving a width of ∼ 60 MeV. The low energy rise can be explained both in terms of Final State Interaction (FSI) of the emitted protons and by the two-nucleon induced decay Λ + (np) → n + n + p. Early theoretical calculation [64] predicted such naive spectrum, even though inconsistent with pioneer emulsion experiments [65] and counter experiments, characterized by a quite large energy threshold for proton detection [66]. In the recent past years a remarkable eﬀort was done, both on the experimental side at KEK with the SKS spectrometer and on the theoretical side, that partially shed light on several aspects of the problem. Quite recent reviews can be found in refs. [67] and [6,68], respectively. A high-statistics proton spectrum, measured from the 12Λ C ground-state decay was recently published [69]: it looks quite ﬂat and with an energy threshold of ∼ 30 MeV. Figure 15 shows a comparison between the spectrum measured by FINUDA and the one obtained with SKS, normalized in area beyond 35 MeV: at glance they look quite inconsistent. A reduced χ2 test applied to test the compatibility of the two distributions gave a value of 5.0, corresponding to a conﬁdence level lower than 10−7 . The same check done comparing the initial distribution reproduced the left panel of ﬁg. 14 and the SKS spectrum delivered a reduced χ2 value of 23.1, putting in evidence an even bigger inconsistency between the two sets of data.

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In order to understand the reasons behind this discrepancy several considerations were put forward. The FINUDA data are certainly superior as far as the quality of the experimental apparatus is concerned. The proton energy values measured by the FINUDA magnetic spectrometer, capable of providing a resolution better than 1% FWHM, are more precise than those obtained with a range telescope like in the SKS case. Another advantage deriving from the use of very thin stopping targets in FINUDA is the energy threshold for proton detection as low as 20 MeV, value that could be further lowered to ∼ 5 MeV by adopting an alternative, more ﬂexible pattern recognition strategy. On the contrary, the thickness of the nuclear targets used for SKS was quite high in order to increase the coincidence counting rate and then the raw data had to be signiﬁcantly corrected to account for the degradation of the momentum of charged particle crossing such not negligible amount of material. Finally it must be noticed that a recent theoretical work [70] put in evidence a strong disagreement between the proton spectrum reported by the SKS Collaboration and the performed calculations. A possible explanation suggested by the authors is the possibility of an underestimation of the number of protons in the experimental data. . 5 3. neutron-rich Λ-hypernuclei. – As pointed out in sect. 1, Λ-hypernuclei may be even better candidates than ordinary nuclei to exhibit extreme values of (N/Z) and halo phenomena [9]. As a matter of fact, a Λ-hypernucleus is more stable than an ordinary nucleus due to the shrinking on the nuclear core [8] and to the addition of extra binding energy from the Λ hyperon [71]. From the hypernuclear physics point of view, the attempt to extend investigation towards the limit of nuclear stability can provide more information both on baryon-baryon interaction and on the behavior of hyperons in a medium with much lower density than ordinary Λ-hypernuclei. Furthermore, the role of the three-body ΛN N force related to the coherent “Λ-Σ coupling” has connections with nuclear astrophysics [72], as previously suggested on the basis of theoretical calculations about high-density nuclear matter (neutron stars) [73,74]. In particular, there is a strong interest about the possible existence of 6Λ H; in fact, theoretical calculations predict the existence of a stable single-particle state with a binding energy of 5.8 MeV from the (5 H + Λ) threshold (+1.7 MeV [75]), when the Λ-Σ coupling term is considered. Without this coupling force the state would be very close to the (4Λ H + n + n) threshold [76, 77]. Experimentally, the production of neutron-rich Λ-hypernuclei is more diﬃcult than standard Λ-hypernuclei, whose one-step direct production reactions (7), (8), (9), access only a limited region in the hypernuclear chart, rather close to the stability line. On the contrary, neutron-rich Λ-hypernuclei can be produced via diﬀerent reactions based on Double Charge Exchange (DCX) mechanism, such as (π − , K + ) and (K − , π + ). The latter reaction proceeds through two elementary interactions: (22)

K − + p → Λ + π0 π0 + p → n + π+ ,

(23)

K − + p → Σ− + π + ;

Σ− p ↔ Λn.

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Process (22) is a two-step reaction in which a ﬁrst strangeness exchange is followed by a pion charge exchange. Process (23) is a single-step reaction with a Σ− admixture, due to the Σ− p ↔ Λn coupling [78]. Owing to these features, both processes usually have lower cross-section with respect to one-step reactions. The ﬁrst experimental attempt to produce neutron-rich Λ-hypernuclei via the − (Kstop , π + ) reaction was carried out at KEK [79]. An upper limit (per stopped kaon) was obtained for the production of 12Λ Be, 9Λ He and 16Λ C hypernuclei. The results are in the range (0.6–2.0) × 10−4 , while the theoretical predictions [78] for 12Λ Be and 16Λ C lie in the interval (10−6 –10−7 ) per stopped kaon, that is at least one order of magnitude lower than the experimental results and three orders of magnitude smaller than the usual − (Kstop , π − ) one-step reaction rates on the same targets (10−3 ). Recently, another KEK experiment [80] claimed to have observed the production of 10Λ Li in the (π − , K + ) reaction on a 10 B target. The published results are not directly comparable with theoretical calculations, since no discrete structure was observed and the production cross-section was integrated over the whole bound region (0 MeV < BΛ < 20 MeV). Furthermore, the observed trend of the cross-section energy dependence strongly disagrees with theoretical predictions [81]. This puzzling situation stimulated a renewed interest about neutron-rich Λ-hypernuclei, in particular 6 H and 7 H systems. Actually for their production rates neither theoretical predictions nor experimental measurements exist. For this reason the FINUDA Collaboration carried out a study on this subject [82]. These exotic Λ-hypernuclei can in fact be produced through processes (22) and (23) with K − at rest. In both cases neutron-rich Λ-hypernuclei production would be tagged by the presence of a π + in the ﬁnal state, according to the reaction (24)

− + Kstop + AZ → A Λ(Z − 2) + π .

Once more it is worthwhile to remind that neither modiﬁed experimental set-up nor dedicated data taking were necessary in order to perform this investigation. From the same data sample used for all the described analysis, events relative to lighter targets, namely two 6 Li and one 7 Li, were selected, in order to look at the inclusive momentum spectrum of π + coming from reactions: (25)

− Kstop + 6 Li → 6Λ H + π + ,

(26)

− Kstop + 7 Li → 7Λ H + π + .

The emitted π + momenta are related to the Λ binding energies BΛ of the predicted hypernuclear ground state of 6Λ H (BΛ = 4.1 MeV [76]) and of 7Λ H (BΛ = 5.2 MeV [9]) through momentum and energy conservation and turned out to be ∼ 252 MeV/c and ∼ 246 MeV/c, respectively. The candidate events were selected by requiring a successfully reconstructed positive track associated to a K − stopped in the chosen target. The positive particle associated with such a track is identiﬁed as a π + by means of its ΔE/Δx and its ToF (“soft” cuts). The momentum spectra of the selected π + are shown in ﬁg. 16.

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Fig. 16. – Inclusive π + momentum spectra (from ref. [82]). Magniﬁed views of the region between the two arrows are shown in the insets, with the corresponding Λ binding energy values on top.

The spectra are not corrected for acceptance. This aﬀects their shape mainly in the region 180–220 MeV/c, due to the kinematic cut of the spectrometer. In the same ﬁgure a residual accumulation of events at ∼ 236 MeV/c, due to Kμ2 decay contamination, can be seen. It originates from a few K + /K − misidentiﬁed events not completely removed by ToF selection. As it can be seen in the insets, no signiﬁcant signals are observed in the region 0 MeV < BΛ < 10 MeV. The bulk of the spectrum is due to π + coming from Σ+ decay, produced in the following two quasi-free reactions: (27)

K − + p → Σ+ + π − Σ+ → n + π +

(28)

−

(∼ 130 MeV/c < pπ+ 250 MeV/c),

K + (pp) → Σ + n +

Σ+ → n + π +

(∼ 100 MeV/c < pπ+ 320 MeV/c).

More in detail, the π + counts in the momentum region of interest are mostly due to process (28), to some Kμ2 in-ﬂight decay contamination and to a small contribution from the high-momentum tail of the spectrum of the π + ’s from process (27). In order to reduce the contribution of these background events, further selection criteria were then applied, taking advantage of the tracking capability of the FINUDA

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245

Fig. 17. – Inclusive π + momentum spectra after the background reduction carried out as described in the text (from ref. [82]). The ROI’s are highlighted and shown in greater detail in the magniﬁed views of the insets.

spectrometer. To this purpose the attention was focussed on the distance between the K − absorption point and the π + origin, estimated by the reconstruction program as the point of closest approach between their two extrapolated trajectories, beyond ISIM and back from OSIM respectively, towards a plane inside the target volume. A cut on the value of this distance was found to be able to reduce the contribution from in-ﬂight Σ+ decay of reactions (27) and (28) and from Kμ2 in-ﬂight decay; in fact the π + or μ+ coming from these processes can be reconstructed some millimeters apart from the K − stopping point (whereas, the π + following the hypernuclear formation is produced at the same point in which the K − is absorbed at rest). By using two distinct simulations, one for the background and the other for the signal, a 2 mm cut (in the following referred to as “hard ” cut) in such a distance was veriﬁed to be eﬀective in selecting almost 50% of pions coming from the hypernuclear formation and 10% from background. Therefore this selection improved the signal-to-noise ratio by a factor ∼ 5. Figure 17 shows that, by applying this cut, any contribution in the high-momentum tail, due to in-ﬂight decays, is greatly reduced. The Kμ2 contamination at ∼ 236 MeV/c, produced by the vertex misidentiﬁcation, is not aﬀected by the “hard ” cut, as expected. From the inset of ﬁg. 17 for 6 Li, there is a hint for a signal at ∼ 254 MeV/c, corresponding to BΛ ≈ 5.6 MeV.

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Table III. – Upper limits (UL) at a 90% CL of neutron-rich Λ-hypernuclei production rate − , π + ) reaction on 6 Li and 7 Li target nuclei (from ref. [82]). The per stopped K − for the (Kstop two last columns show the variation in UL values obtained shifting the center of the ROI by ±1 MeV/c, respectively. Target

Λ-hypernucleus

UL (90% CL) (×10−5 )

Δ(UL) at +1 MeV/c (×10−5 )

Δ(UL) at −1 MeV/c (×10−5 )

6

Li

6 ΛH

2.5 ± 0.4stat

+0.4 −0.1 syst

−0.4

+0.0

7

Li

7 ΛH

4.5 ± 0.9stat

+0.4 −0.1 syst

−0.5

+0.1

In order to evaluate the statistical signiﬁcance of the observed signal, three diﬀerent hypotheses on the shape of background, due to the reactions (27) and (28), were taken into account. On the basis of this study an upper limit at 90% CL for the experimental production rate of the neutron-rich Λ-hypernuclei 6Λ H and 7Λ H was inferred (see table III). To achieve these results it was necessary to carefully estimate the maximum number of π + counts to be ascribed to neutron-rich Λ-hypernuclei formation. To this purpose a Region Of Interest (ROI) was deﬁned in each spectrum, centered at the π + momentum value corresponding to the predicted BΛ . The ROI width was set to ±2σp , where σp is the standard deviation of the peak momentum resolution (0.9% FWHM). It must be noted that this value was estimated by using the monochromatic μ+ peak at 236 MeV/c, without selecting high-quality tracks, in order to have as much statistics as possible to study such rare events. Finally it worthwhile to mention that the same analysis procedure was applied to the three 12 C targets. No evidence of 12Λ Be production was found. The upper limit for +0.3 −5 its production rate turned out to be (2.0 ± 0.4stat −0.1 /(stopped K − ) which syst ) × 10 −5 − improves the previous published result 6.1 × 10 /(stopped K ) [79]. . 5 4. Deeply bound K-nucleus states. – In the last few years a vivid debate was triggered by the claim about the existence of deeply bound kaonic states. Two groups of experimentalists reported about possible signatures of such objects in two diﬀerent − reactions: 16 O(K − , n) [83] and 4 He(Kstop , nX) [84, 85]. The observed signals were inter15 preted as the production of K − O and K − ppn, respectively. Afterwards, the latter group reported as well the observation of much clearer evidence of the production of a tribary− onic state S 0 (3115) in the 4 He(Kstop , pX) reaction [85, 86]. At that time, following the interpretation of the authors, one would have to accept that the K − -nucleus potential was deeper than 100 MeV, in contradiction with many theoretical predictions. However, − a new recent measurement of the neutron spectrum from the reaction 4 He(Kstop , nX) was performed at KEK and no narrow structure was found [87]. Then it is an urgent experimental subject to verify or to conﬁrm whether these signals really correspond to the production of kaon-bound states. As for the K − p interaction in free space, the existing experimental information on low-energy K − p scattering [88-90], on K − p threshold branching ratios [91-93] and on

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kaonic-hydrogen atomic X-ray measurement [94-96], leaves room for diﬀerent theoretical approaches. The information on the K − -nucleus interaction has been obtained with the kaonic atomic data [97]. However, there are two qualitatively diﬀerent predictions for the depth of the K − -nucleus potential at normal nuclear matter density: very deep attractive potential (− Vopt (ρ0 ) ≈ 150–200 MeV) [98, 99] and much shallower potential (− Vopt (ρ0 ) ≈ 50–75 MeV) [46, 100-104]. Both types of potential reasonably describe the shift and the widths of the X-ray data. Recently, an interesting idea has been put forward suggesting the possible existence of deeply-bound nuclear K states in light nuclei [105,106], for which the calculated potential is very deep. Since the potential is strongly attractive in the I = 0 KN interaction, proton-rich K nuclei, such as K − pp and K − ppn, are predicted to have large binding energies, in the range 50–100 MeV. A further interesting suggestion is that the nucleus might be shrunk because of the large binding energy and form a high-density state, which is also conﬁrmed with the method of antisymmetrized molecular dynamics for the cases of K − ppn and K − 8 Be [107]. Since there is no Pauli blocking eﬀect for a K − boson, it can attract the nucleus at its center. If such a K nucleus was observed, it would provide direct information on the K − nucleus potential at nuclear matter density. Quite naturally the FINUDA Collaboration analyzed its collected data sample in order to look for the existence of such objects [108]. It must be noticed that all of the previous measurements were based on the missing-mass method, so that there exist some ambiguities about the fact that a kaon was really bound in the system. On the contrary the FINUDA spectrometer enables to detect a Λ-hyperon through its decay to proton and π − . It this then possible to clearly identify the formation of bound states with strangeness from their decays to Λ + X. Λ-hyperons are produced with sizeable fraction in kaon absorption through various processes. The quasi-free process (29)

K− + N → Λ + π

emits a slow Λ with a momentum of ∼ 300 MeV/c. Above ∼ 400 MeV/c, the main contribution comes from the Λ-hyperons following two-nucleon absorption: (30)

K− + N N → Λ + N ,

Σ0 + N .

. As already mentioned in subsect. 5 1 the acceptance of the FINUDA spectrometers cuts the Λ-hyperons with a momentum lower than ∼ 300 MeV/c, which is restricted by the low-momentum threshold for π − from the Λ → p + π − decay. Therefore, the Λ from the quasi-free process (29) is hardly observed in the FINUDA apparatus. Λ particles can be identiﬁed by reconstructing the invariant mass of a proton and a negative pion as shown by ﬁg. 18 (a). The peak position agrees well with the known Λ mass and the width of the peak is as narrow as 6 MeV/c2 FWHM.

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Fig. 18. – (a) Invariant-mass distribution of a proton and a π − for all the events in which these two particles are observed, ﬁtted by a single Gaussian function together with a linear background in the range 1100–1130 MeV/c2 . (b) Opening angle distribution between a Λ and a proton. Solid line: 6 Li, 7 Li and 12 C targets; dashed line: 27 Al and 51 V targets. The hatched area (cos θlab < −0.8) is selected as the back-to-back event. (From ref. [108].)

When a K − interacts with two protons, one expects that a hyperon-nucleon pair (Λ+p, Σ + p or Σ+ + n) is emitted in the opposite direction, ignoring a ﬁnal-state interaction inside the residual nucleus. The angular correlation between a Λ and a proton from the same vertex in the target clearly indicates the existence of this kind of reaction (see ﬁg. 18 (b)). Even for heavy nuclei, such as 27 Al and 51 V, similar correlations were observed, which might suggest the absorption would take place at the surface of the nucleus. In the subsequent analysis only the events from light nuclear targets (6 Li, 7 Li and 12 C) and in which the Λ-p pairs are emitted in the opposite direction (cos θlab < −0.8) were retained. Since the back-to-back angular correlation between a Λ and a proton is so clear, it is naturally expected that the two particles are emitted from a “K − pp” intermediate system. The angular correlation is smeared out due to the Fermi motions of the two protons at the surface of a nucleus by which the K − is absorbed after cascading down the atomic orbits by emitting X-rays. If the reaction process was simply a two-nucleon absorption process, the mass of the system should be close to the sum of a kaon and two proton masses, namely 2.370 GeV/c2 . The initial motion of the two protons does not aﬀect the invariant-mass distribution. The invariant-mass distribution of the Λ-p pairs is shown by ﬁg. 19. A signiﬁcant mass decrease of the K − pp system with respect to its expected value is observed. It can be interpreted as a bound state composed of a kaon and two protons, hereafter abbreviated as K − pp. In the inset of ﬁg. 19, the acceptance corrected invariant-mass distribution for events with two well-deﬁned long-track protons is shown. Since the trigger and the detection 0

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249

Fig. 19. – Invariant mass of a Λ and a proton in back-to-back correlation (cos θlab < −0.8) from light targets before the acceptance correction. The inset shows the result after the acceptance correction for the events which have two protons with well-deﬁned good tracks. Only the bins in the range 2.22–2.33 GeV/c2 are used in the ﬁt. (From ref. [108].)

acceptance are monotonically increasing functions of the invariant mass in the considered mass region, the peak further shifts towards a lower mass value. The binding energy +3 +2 +14 BK − pp = 115+6 −5 stat −4 syst MeV and the width Γ = 67−11 stat −3 syst MeV are obtained from the ﬁt to a Lorentzian function (folded with a Gaussian curve with σ = 4 MeV/c2 , corresponding to the detector resolution estimated with a Monte Carlo simulation) in the range 2.22–2.33 GeV/c2 . Here, the systematic errors were estimated by changing the event selection criteria in the Λ invariant mass and in the Λ-p opening angle cut as well as by taking into account the spectrometer acceptance change due to possible systematic deviations in absolute momentum scale or reaction vertex distributions. Despite some uncertainties on absolute normalization, a rough estimation of the yield of K − pp → Λ + p is of the order of 0.1% per stopped K − . Consistency of the Monte Carlo simulation used for estimations of the acceptance and of the apparatus resolution was checked by producing the K − pp events according to the measured mass and width. The same reconstruction procedures were applied to these events: the momentum distributions for Λ’s and protons, the Λ-p opening angle distribution and the K − pp momentum distribution were all found in good agreement with the observed ones. The K − pp decay into Σ0 + p should have as well a back-to-back correlation between a Λ and a proton, but a γ from the Σ0 decay is missing when reconstructing the invariant mass (Σ0 at rest emits a 74 MeV γ). The accumulation of events in the low-mass region below ∼ 2.2 GeV/c2 may be attributed to this decay mode. From old bubble chamber and emulsion data [109], it is known that kaon two-nucleon absorption processes take place in 15–20% per stopped K − in a broad range of the periodic table. However, experimental information was statistically too limited to further

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investigate the reaction mechanism and nobody has identiﬁed such a process experimentally. Several theoretical analyses were performed in the 60s in order to understand the reaction mechanism. A more recent theoretical approach [110] roughly explained it with the two-nucleon absorption process including meson-rescattering diagrams. Here it should be noted that the main mode for the kaon two-nucleon absorption is on a pn pair, with S = 1 and I = 0, and not on a pp pair, which contributes only by ∼ 10% according to the calculation. Then, there is no a priori reason that the two-nucleon absorption processes are to be observed in the Λ-p coincidence events. On the other hand, the FINUDA detector is very sensitive to the existence of the two-nucleon mode (31)

K − + (pp) → Λ + p

since its resolution on invariant mass is ∼ 10 MeV/c2 FWHM. The eﬀect of the nuclear binding of two protons is only to shift the peak position towards lower mass values, of the order of separation energies of two protons (∼ 30 MeV), and not to broaden the peak. A sharp spike at ∼ 2.34 GeV/c2 may be attributed to this process. There could be, in addition, the two-nucleon absorption mode (32)

K − + (pp) → Σ0 + p.

In this case the Λ-p invariant mass distribution is shifted to the lower mass region by ∼ 74 MeV and broadened because a γ from the Σ0 → Λ+γ decay is missing. However the observed invariant-mass distribution is too broad to be unambiguously attributed to this process only. Furthermore, according to the old data from helium bubble chamber [111], the branching ratios of processes (30) were estimated to be 9.3% ± 2.6% and 2.3% ± 1.0% per stopped K − in 4 He, respectively. The theoretical calculation [110] also suggests that the ΛN channel has a larger branching ratio than the Σ0 N one. Therefore one can assume that the kaon two-nucleon absorption mode is not dominant in this channel, nor in the in the Σ0 + p one which has even a lower branching ratio. Thus, the observed events in the bound region were ﬁnally ascribed to the deeply bound K − pp state. . 5 5. Low-energy K + -nucleus interaction. – The charge exchange reaction induced by low energy K + was intensively studied in the 50s and early 60s by using emulsions, bubble chambers and ﬁnally counter experiments [112-116], while in subsequent years, this subject attracted only sporadic interest [117, 118]. The reason was that, contrary to K − , a low momentum K + has no access to states containing hyperons with their rich and interesting dynamics (see, for instance, refs. [100, 119, 120]) and its interaction was hence thought to simply be smoothly decreasing at low momenta. Therefore the later, more reﬁned experiments focused only on higher energies to search for possible S = +1 resonances [121-123]. When it was understood that the relative weakness of the K + strong interaction with nucleons could probe the interior of nuclei [124], an intense phase of studies devoted to this item started at intermediate energies to compare K + nucleus interactions with those on deuterons [125-133] and to look for possible evidence

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of strange quark content in nucleons. Later on, however, the interest in K + interactions on nucleons and nuclei subsided, until the very recent upsurge of activity related to the pentaquark search [134-136] (and references therein). Since DAΦNE is an excellent source of low-energy K + ’s as well, with the same K − ’s unique characteristics, the FINUDA Collaboration explored the possibility of studying the K + charge exchange reaction on medium-light nuclei from ∼ 100 MeV/c down to the threshold of the reaction, in order to provide experimental information on the scattering amplitude 21 (f1 − f0 ) of the process [137]. Here f0 and f1 denote the isospin I = 0 and isospin I = 1 amplitudes, respectively. In particular the attention was focussed on the (K + , K 0 ) charge reaction on the 7 Li target, never measured before in this K + low momentum range. The majority of the K + entering the FINUDA targets are brought to rest and then decay. For this analysis events originating from K + interactions were retained. Such events were collected using the same trigger set-up for hypernuclear data taking; hence, even in this case, no dedicated trigger was needed to study this process. Events of interest were selected by looking for a pair of positive and negative charged particle coming out from the K + vertex in the 7 Li target. Protons tracks were excluded by checking the energy loss in ISIM, with a rejection power better than 96% in the momentum region of interest [138]. The events selected by the reconstruction program were subject to topology and invariant-mass consistency checks to test the hypothesis of pions produced in the decay (33)

KS0 → π + π −

(BR = 0.69%).

Due to the very low moment of the produced KS0 (∼10–90 MeV/c), a nearly back-to-back topology for the π + π − pairs, both with momentum of ∼ 205 MeV/c, is expected. It is very important to recall that the reaction (K + , K 0 ) occurs inside a nucleus. Indeed, even the elementary process, (34)

K + + n → KS0 + p

cannot be experimentally studied on free neutrons and, actually, exiting data were obtained mainly on Deuterium targets. This means that the actual threshold of the reaction increases with respect to that of the elementary one (63.8 MeV/c), depending on the target nucleus. In the case of 7 Li, for instance, the threshold is 68.9 MeV/c. This is a crucial point for FINUDA. The K + produced by DAΦNE have a maximum momentum of ∼ 133 MeV/c but, after crossing the beam pipe and the inner detectors, they reach the targets with momenta not higher than ∼ 100 MeV/c. This reaction, therefore, turns out to be below threshold on several nuclei. Moreover, due to the low momentum of the involved K + , the Coulomb barrier plays a role in its interaction on nuclei, increasing with the Z of the target nucleus. Figure 20 shows the K + momentum threshold Q of the (K + , K 0 ) reaction for a selection of diﬀerent nuclei. In the ﬁrst FINUDA run, the

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Fig. 20. – The threshold momentum Q of the reaction (K + , K 0 ) for several nuclei as a function of their mass number A. The horizontal line indicates the maximum K + momentum reachable (in the φ boost direction) on the FINUDA targets (from ref. [137]).

experimental set-up allowed the investigation of the (K + , K 0 ) reaction only for the 7 Li target. A visual inspection of the selected data sample, allowed to discover some fake events. Among all the possible background sources, (π + , e− ) pairs originating from the K + decays at rest are by far the largest. K + decays produce, with a BR of ∼ 21%, (π + , π 0 ) pairs whose topology and momentum are similar to that of the (π + , π − ) pairs from decay (33) at rest. The π 0 decays in ∼ 10−16 mainly into γγ. One of the γ’s can be emitted in the same direction as the decaying π 0 and can create an (e+ , e− ) pair within the same target. If the e− is in turn forward emitted, it keeps the topology and the momentum of the parent π 0 , similar to the π − from the decay (33) at rest. In the range of momentum involved, the FINUDA apparatus cannot discriminate between an e− and a π − , hence this background is very insidious. An experimental signature of the presence of such a background can be seen by looking at the invariant-mass distribution of two positive charged tracks, assumed to be both π + , emitted following a K + interaction in any of the installed targets. Such events can only occur from pair creation in which the e+ , instead of the e− , is forward emitted. Figure 21 shows the invariant mass distribution of the measured (+, +) tracks (assumed to be both π + and with relative angles larger than 145◦ ) following K + ’s interactions in any of the eight targets. A wide peak, close to the KS0 nominal mass value, can be seen. This is also found in the equivalent distribution obtained using (+, −) tracks. The peak due to (+, +) tracks is depressed simply due to the lower acceptance of the spectrometer for (+, +) tracks with respect to (+, −) tracks of momenta of ∼ 200 MeV/c. At glance one can argues that the peak around the KS0 mass is too wide to be due to its decay only, since FINUDA features a resolution in this energy range better than 2 MeV/c2 [30], that is well within the bin width of the histograms presented in ﬁg. 21

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253

Fig. 21. – Invariant-mass distribution of the measured (+, −) and (+, +) (empty and shaded histograms, respectively) tracks (assumed to be π ± ) with relative angle greater than 145◦ , following K + interactions in all the installed targets (from ref. [137]). Protons have been rejected using dE/dx information provided by ISIM.

(10 MeV/c2 ). The consequence is that events occurring in the decay (33) cannot be simple selected by looking at distribution of the type shown in ﬁg. 21. The background contamination was studied thanks to the FINUDA Monte Carlo program. A number of K + → π + π 0 decays, comparable with that expected on the basis of the measured number of stopped K + , was generated. Then the distributions in relative angle and in invariant mass of the resulting (π + , e− ) events were analyzed under the hypothesis that they were (π + , π − ) pairs. A further simulation was done in order to produce a sample of decays (33), to be submitted to the same reconstruction procedure as of real data. The high resolution on the KS0 mass and on the relative angle of (+, −) or (+, +) tracks (∼ 1◦ ) should allow to distinguish the diﬀerence, if any, between the topologies of the two processes. From this study it was indeed possible to deﬁne, at a 95% CL, a background-free signal region of 494–502 MeV/c2 and 176◦ –180◦ in the (π + , π − ) invariant mass and relative angle, respectively. As already mentioned, in the FINUDA ﬁrst run only the 7 Li target was eﬀectively accessible to the reaction (K + , K 0 ) with a sizeable ﬂux of K + with momentum above the threshold. The momentum distribution of K + entering the 7 Li target exhibited the shape expected from the Landau distribution of the momentum losses, with a peak at 95 MeV/c and a width of 15.0 MeV/c FWHM. The topological analysis of the experimental data showed that for the 7 Li target ﬁve candidate (π + , π − ) pairs can be found, all of which, when displayed on an invariant mass vs. relative angle scatter-plot, lay outside the signal region expected for genuine (π + , π − ) pairs from KS0 decay almost at rest (see ﬁg. 22).

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Fig. 22. – Scatter-plot of the invariant mass vs. the relative angle for the (+, −) measured tracks, which were assumed to be π ± , following K + interactions in the 7 Li target (from ref. [137]). Protons have been rejected using dE/dx information provided by ISIM. The signal region expected (at a 95% CL) for events from (33) is indicated by the shaded area.

In other words, no good events were detected. By using again the FINUDA Monte Carlo program, that takes into account the threshold for the reaction and the K + ﬂux with the proper momentum distribution, the integrated luminosity for the 7 Li target was inferred: 16.15 × 1027 cm−2 . The FINUDA global eﬃciency (geometrical acceptance × trigger eﬃciency × detector eﬃciency) for detection of (π + , π − ) pairs from KS0 decay almost at rest was evaluated as well. It was also veriﬁed that, in the limited momentum range (∼ 10–90 MeV/c) of the involved KS0 ’s, the eﬃciency remained constant and equal to 0.099 ± 0.005. With the available statistics, on the basis the measured integrated luminosity and of the apparatus global eﬃciency, the achieved sensitivity turned out to be ∼ 0.62 mb per event. Having found no good events, and no background, the result indicates, following standard statistical analysis methods, that near the threshold the cross-section for the reaction 7 Li(K + , K 0 )7 Be is less than 2.0 mb, at a 95% CL. There are no previous measurements of the (K + , K 0 ) cross-section on nuclei close to threshold, deuteron included. Moreover, theoretical calculations are also lacking for this energy region. The charge exchange reaction of a K + on a nucleus is, of course, related to the elementary process (34). A compilation of the existing experimental data and theoretical calculations for the elementary cross-section is shown in ﬁg. 23, for K + laboratory momentum from 200 MeV/c up to 800 MeV/c. The experimental data were extracted from measurements on Deuterium or from heavier nuclei. As one can see, in the low momentum region the available data or calculations are very old and go down to a minimum momentum value as high as two times the one accessible by FINUDA. The

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Fig. 23. – Compilation of the existing data and theoretical calculations for the total crosssection of the elementary charge exchange reaction (34), for plab K + < 800 MeV/c (calculation from refs. [118] and [136] are adapted). (From ref. [137].) The threshold momentum pthr and the region explored by FINUDA area also indicated, as well as the 0.5 mb limit (horizontal dotted line). The 0.5 mb level is equal to the value obtained by dividing the FINUDA result (< 2.0 mb) by the number of neutrons in 7 Li.

data are reasonably consistent (within the experimental errors) and show a rather smooth decreasing trend in the low momentum region. In the momentum range accessible by FINUDA, indicated in the same ﬁgure, a prediction of less than 0.5 mb for the elementary charge exchange cross-section seems quite a reasonable upper limit. The 0.5 mb upper limit corresponds to the FINUDA result obtained for 7 Li divided by the number of neutrons in the nucleus. The relationship of the (K + , K 0 ) cross-section on nuclei to that for the elementary process (34) is not trivial at low momenta. The most relevant eﬀect is the Pauli exclusion principle, as the momentum of the created proton is below the Fermi value, and the crosssection would be heavily damped. At higher laboratory momenta (from 300 MeV/c up to and above the threshold for pion emission) Pauli blocking becomes progressively less eﬀective. In such a momentum region, it is known that the interaction cross-section of K + with nuclei tends to become proportional to volume, that is rather accurately (within 10%) proportional to the number of nucleons in the nucleus [139]. It is diﬃcult, without an explicit calculation, to predict the actual near-threshold value, but it is possible to try to put two limiting expectations for the possible value of

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the cross-section of (K + , K 0 ) on 7 Li close to threshold: 1) an upper bound equal to that of the corresponding cross-section for the elementary process times the number of neutrons in the target (that is no Pauli damping). 2) A lower bound equal to, or less than, the cross-section of the corresponding elementary process (that is complete Pauli damping). Taking ∼ 0.5 mb as the estimate for the near threshold elementary cross-section (see ﬁg. 23) and assuming the volume approximation, we get for the 7 Li(K + , K 0 )7 Be cross-section an upper bound value of ∼ 2 mb. In conclusion, the estimated cross-section window (≤ 0.5–2.0 mb) is compatible with the FINUDA measured upper limit, conﬁrming, for the ﬁrst time experimentally, a smooth and decreasing cross-section on 7 Li in low-momentum region. 6. – Conclusions The quality and the quantity of the obtained results allowed the FINUDA Collaboration to successfully demonstrate the validity of the idea of carrying out, for the ﬁrst time, a wide hypernuclear physics program at a Φ-factory, exploiting a completely brand new technique for Λ-hypernuclei production, featuring considerable advantages with respect to the other pursued approaches. On this basis, a second, longer data taking campaign has been carried out from October 2006 to June 2007. An integrated luminosity of ∼ 1 fb−1 was delivered by DAΦNE, corresponding to ∼ 240 × 106 recorded events. For this second run the installed target were the following: 2 × 6 Li, 2 × 7 Li, 2 × 9 Be, 1 × 13 C and 1 × D2 O (see ﬁg. 6). The motivations behind this choice are, on one side, the aim of making conclusive some of the described investigations and, on the other, of studying the properties of some new Λ-hypernuclei. Thanks to the high-statistics available, it will also possible to look at new subjects like, for instance, the search for rare hypernuclear decays [140]. The analysis of this new sizeable data sample is currently in progress and looks very promising. ∗ ∗ ∗ It is a pleasure to warmly thank Mrs. B. Alzani and her staﬀ for the excellent logistic organization of the School and for the enjoyable atmosphere she was able to create among the participants. The eﬀort made by the Italian Physical Society in order to keep alive the Enrico Fermi School tradition is deeply acknowledged.

REFERENCES [1] Bressani T., this volume, p. 3. [2] Danysiz M. and Pniewsky J., Philos. Mag., 44 (1953) 348. [3] Dalitz R. H. and Gal A., Ann. Phys. (N.Y.), 116 (1978) 167.

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SPIN PHYSICS

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The longitudinal spin structure of the nucleon E. Leader Imperial College London - Prince Consort Road, London SW7 2AZ, UK

Summary. — We review ﬁrst the parton model formalism for polarized deep inelastic lepton-hadron scattering. Topics discussed include the “spin crisis in the parton model”, the role of the axial anomaly, our knowledge of the polarized gluon number density and attempts to measure it. Secondly, going beyond the simple parton model, we discuss the evolution of parton densities, the generalization of the parton model in QCD, perturbative QCD corrections and scheme dependence. Finally we comment on our knowledge of the polarized strange quark density and attempts to learn about it from semi-inclusive deep inelastic scattering.

1. – Deep inelastic scattering Deep inelastic lepton-hadron scattering (DIS) has played a seminal role in the development of our present understanding of the sub-structure of elementary particles. The discovery of Bjorken scaling in the late nineteen-sixties provided the critical impetus for the idea that elementary particles contain almost pointlike constituents and for the subsequent invention of the parton model. DIS continued to play an essential role in the long period of consolidation that followed, in the gradual linking of partons and quarks, in the discovery of the existence of missing constituents, later identiﬁed as gluons, and in the wonderful conﬂuence of all the diﬀerent parts of the picture into a coherent dynamical theory of quarks and gluons—Quantum Chromodynamics (QCD). c Societ` a Italiana di Fisica

263

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E. Leader

k’ lepton

s’

k

s

q

X P nucleon

S

Fig. 1. – Feynman diagram for deep inelastic lepton-hadron scattering.

. 1 1. General formalism in one photon exchange approximation. – Consider the inelastic scattering of polarized leptons on polarized nucleons. We denote by m the lepton mass, k (k ) the initial (ﬁnal) lepton four-momentum and s (s ) its covariant spin fourvector, such that s · k = 0 (s · k = 0) and s · s = −1 (s · s = −1); the nucleon mass is M and the nucleon four-momentum and spin four-vector are, respectively, P and S. Assuming one photon exchange (see ﬁg. 1), the diﬀerential cross-section for detecting the ﬁnal polarized lepton in the solid angle dΩ and in the ﬁnal energy range (E , E + dE ) in the laboratory frame, P = (M, 0), k = (E, k), k = (E , k ), can be written as d2 σ α2 E Lμν W μν , = dΩdE 2M q 4 E

(1)

where q = k − k and α is the ﬁne-structure constant. The leptonic tensor Lμν is given by (2)

Lμν (k, s; k , ) =

∗ u ¯(k , s ) γμ u(k, s) u ¯(k , s ) γν u(k, s) s

and can be split into symmetric (S) and antisymmetric (A) parts under μ, ν interchange: (3)

(A) Lμν (k, s; k , ) = 2 L(S) μν (k; k ) + iLμν (k, s; k ) ,

where (4)

2 L(S) μν (k; k ) = kμ kν + kμ kν − gμν (k · k − m ), α β L(A) μν (k, s; k ) = m εμναβ s q .

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The longitudinal spin structure of the nucleon

The unknown hadronic tensor Wμν describes the interaction between the virtual photon and the nucleon and depends upon four scalar structure functions, the unpolarized functions W1,2 and the spin-dependent functions G1,2 . These must be measured and can then be studied in theoretical models, in our case in the QCD-modiﬁed parton model. These can only be functions of the scalars q 2 and q · P . Usually, people work with Q2 ≡ −q 2

(5)

and

xBj ≡ Q2 /2q · P = Q2 /2M ν ,

where ν = E − E is the energy of the virtual photon in the Lab frame; xBj is known as “x-Bjorken”, and we shall simply write it as x. One has (S) (A) (q; P ) + i Wμν (q; P, S), Wμν (q; P, S) = Wμν

(6) with

1 W (S) (q; P ) = 2M μν

(7)

(8)

qμ qν −gμν + 2 W1 (P · q, q 2 ) q

W2 (P · q, q 2 ) P ·q P ·q + Pμ − 2 qμ , Pν − 2 qν q q M2

1 W (A) (q; P, S) = εμναβ q α M S β G1 (P · q, q 2 ) 2M μν

G (P · q, q 2 ) 2 . + (P · q)S β − (S · q)P β M

Note that these expressions are electromagnetic gauge-invariant: q μ Wμν = 0.

(9) From these one has

d2 σ α2 E (S) μν(S) (A) μν(A) = W − L W . L μν μν dΩ dE 2M q 4 E

(10)

(A)

Diﬀerences of cross-sections with opposite target spins single out the Lμν W μν(A) term:

(11)

d2 σ d2 σ α2 E μν(A) 4L(A) (k, s, P, −S; k ) − (k, s, P, S; k ) = . μν W dΩ dE dΩ dE 2M q 4 E

266

E. Leader

After some algebra (for a detailed explanation of the steps involved, see [1]) one obtains (12)

d2 σ s,S d2 σ s,−S 8mα2 E − = dΩ dE dΩ dE q4 E

(q · S)(q · s) + Q2 (s · S) M G1 G 2 +Q2 (s · S)(P · q) − (q · S)(P · s) , M

which yields information on the polarized structure functions G1 (P · q, q 2 ) and G2 (P · q, q 2 ). In the Bjorken limit, or Deep Inelastic Scattering (DIS) regime, −q 2 = Q2 → ∞,

ν = E − E → ∞,

x=

Q2 Q2 = 2P · q 2M ν

ﬁxed,

the scalar functions are known to approximately scale: lim M W1 (P · q, Q2 ) = F1 (x),

(13)

Bj

lim νW2 (P · q, Q2 ) = F2 (x), Bj

(P · q)2 G1 (P · q, Q2 ) = g1 (x), Bj ν lim ν (P · q) G2 (P · q, q 2 ) = g2 (x),

(14)

lim Bj

where F1,2 and g1,2 vary very slowly with Q2 at ﬁxed x. (A) In terms of g1,2 the expression for Wμν becomes

(15)

(A) (q; P, s) Wμν

2M (S · q) P β α β 2 β 2 εμναβ q S g1 (x, Q ) + S − = g2 (x, Q ) . P ·q (P · q)

. 1 2. Polarized DIS . – The cross-section for unpolarized scattering is given by (16)

4πα2 s 2 d2 σ = F + (1 − y)F xy , 1 2 dx dy Q4

where we have used (17)

y≡

P ·q ν = E P ·k

and where s = (P + k)2 . If now we take the lepton and target nucleon polarized longitudinally, i.e. along or opposite to the direction of the lepton beam, then, under reversal

267

The longitudinal spin structure of the nucleon

of the nucleon’s spin direction the cross-section diﬀerence is given by → →

d2 σ⇒ 16πα2 d2 σ⇐ 2M 2 xy y 1 − − = − g g 1 2 . dx dy dx dy Q2 2 Q2

(18)

For nucleons polarized transversely in the scattering plane, one ﬁnds d2 σ →⇓ 16α2 d2 σ →⇑ − =− 2 dx dy dx dy Q

(19)

2M x Q

y 1 − y g1 + g2 . 2

In principle these allow measurement of both g1 and g2 , but the transverse asymmetry is much smaller and therefore much more diﬃcult to measure. Only in the past few years has it been possible to gather information on g2 which turns out to be smaller than g1 . 2. – The simple parton model In a reference frame where the proton is moving very fast, say along the OZ axis, it can be viewed as a beam of parallel-moving partons, as shown in ﬁg. 2. In the hard interaction with the photon, the quark-partons are treated as free, massless particles with momentum x P , as shown in ﬁg. 3. One ﬁnds that the antisymmetric part of the hadronic tensor is given by (20)

(A) Wμν (q

: P, S) =

e2f

f,s

1 2P · q

0

1

dx (A) δ(x − x) nf (x ; s, S) wμν (x ; q, s) , x

where wμν (x ; q, s) is the quark tensor and is just like the leptonic tensor Lμν since the quarks are treated as point-like particles, and the sum is over ﬂavours f and spin orientations s of the struck quark. The delta-function that forces x = x arises from the usual convention of treating the quarks as “free” particles on mass shell, i.e. one takes (A)

(21)

(A)

p2 = (x P )2 = 0

and

(q + p)2 = (q + x P )2 = 0,

so that (22)

q 2 + 2x q · P = 0.

Fig. 2. – Visualization of parton density q(x , s).

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E. Leader

Fig. 3. – Lepton-quark deep inelastic scattering.

With the deﬁnition of xBj (see eq. (5)) this implies (23)

−Q2 + Q2

x =0 x

or

x = x.

However, let us for the moment take p2 = m2 and p2 = m2 . One ﬁnds that (24)

(A) wμν = 2 εμναβ m sα

1−

m β m β p − q . m m

Note that because of the term in round brackets the result is not gauge invariant, i.e. q μ wμν = 0 unless m = m. But for longitudinal polarization, sα = sα L , we have (25)

α m sα L → ±p

for

m 1 p

and therefore the non-gauge invariant term vanishes because of the antisymmetry of the ε symbol. . 2 1. Longitudinal polarization. – Consider a fast moving proton, momentum along OZ, and polarized along OZ. Substituting eq. (24) into eq. (20), and comparing with eq. (15), we ﬁnd (26)

g1 (x) =

1 2 ef Δqf (x), 2 f

where (27)

Δq(x) = q(+) (x) − q(−) (x),

where q(±) (x) are the number densities of quarks whose spin orientation is parallel or antiparallel to the spin direction of the proton (see ﬁg. 4). In terms of these, the usual

269

The longitudinal spin structure of the nucleon

Fig. 4. – Visualization of the longitudinally polarized parton density Δq(x). The upper arrows show the spin direction.

(unpolarized) parton density is (28)

q(x) = q(+) (x) + q(−) (x).

. 2 2. What about g2 (x)? – For a transversely polarized quark we have seen eq. (24), that the quark tensor is only gauge-invariant if m = m. This is a bad sign! The result should not be sensitive to the precise value of the quark mass. So what happens if we take m = m? We ﬁnd g2 (x) = 0! Is this result reliable? Not in so far as it relates to the proton. The point is that the quark DIS tensor does not have a transverse asymmetry, so we cannot hope to use it to provide such an asymmetry in the proton. There are many diﬀerent, inconsistent results for g2 (x) in the literature, including this beautiful one (29)

g2 (x) =

1 2 mq ef − 1 Δq(x), 2 xM

due to Anselmino and myself [2], which, alas, should not be taken seriously. The only reliable result is the Wandzura-Wilczek relation [3]

1

g2 (x) −g1 (x) +

(30)

x

g1 (x ) dx , x

which was originally derived as an approximation in an operator product expansion approach, but which has recently been shown to be derivable directly in the simple parton model [4]. 3. – The spin crisis in the parton model The accepted expression for g1 was completely analogous to the equation for F1 , with the unpolarized quark density replaced by the (longitudinal) polarized density Δq(x). The expression for g1 is then (31)

g1 (x) =

1 2

4 1 1 Δu(x) + Δd(x) + Δs(x) + antiquarks . 9 9 9

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E. Leader

Deﬁne combinations of quark densities which have speciﬁc transformation properties under the group of ﬂavour transformations SU (3)F :

(33)

¯ u) − (Δd + Δd), Δq3 = (Δu + Δ¯ ¯ − 2(Δs + Δ¯ Δq8 = (Δu + Δ¯ u) + (Δd + Δd) s),

(34)

¯ + (Δs + Δ¯ ΔΣ = (Δu + Δ¯ u) + (Δd + Δd) s),

(32)

which transform, respectively, as the third component of an isotopic spin triplet, the eighth component of an SU (3)F octet and a ﬂavour singlet. Then (35)

g1 (x) =

1 3 1 Δq3 (x) + Δq8 (x) + ΔΣ . 9 4 4

Taking the ﬁrst moment of this yields (36)

Γ1 ≡

1 1 4 a3 + √ a8 + a0 , 12 3 3

1

g1 (x)dx = 0

where (37)

1

a3 =

dx Δq3 (x), 0

1 a8 = √ 3

1

dx Δq8 (x), 0

a0 = ΔΣ ≡

1

dx ΔΣ(x). 0

Via the operator product expansion these moments can be related to hadronic matrix elements of currents which are measurable in other processes, as will be explained below. The hadronic tensor W μν is given by the Fourier transform of the nucleon matrix elements of the commutator of electromagnetic currents Jμ (x): (38)

1 Wμν (q; P, S) = 2π

d4 x eiq·x P, S|[Jμ (x), Jν (0)]|P, S ,

where S μ is the covariant spin vector specifying the nucleon state of momentum P μ . In hard processes, x2 0 is important, so we can use the Wilson expansion. The OPE gives moments of g1,2 in terms of hadronic matrix elements of certain operators multiplied by perturbatively calculable coeﬃcient functions. The ai in eq. (36) are hadronic matrix j elements of the octet of quark SU (3)F axial-vector currents J5μ (j = 1, . . . , 8) and the 0 ﬂavour singlet axial current J5μ . The octet currents are j ¯ μ γ5 λj ψ = ψγ (39) J5μ (j = 1, 2, . . . , 8), 2

271

The longitudinal spin structure of the nucleon

where the λj are the usual Gell-Mann matrices and ψ is a column vector in ﬂavour space ⎛

⎞ ψu ψ = ⎝ ψd ⎠ ψs

(40)

and the ﬂavour singlet current is 0 ¯ μ γ5 ψ. = ψγ J5μ

(41)

j can only be proportional to S μ , and the aj are The forward matrix elements of the J5μ deﬁned by j |P, S = M aj Sμ , P, S|J5μ

(42)

0 P, S|J5μ |P, S = 2M a0 Sμ .

Analogous to eq. (39) one introduces an octet of vector currents (43)

Jμj

¯ μ = ψγ

λj 2

ψ

(j = 1, . . . , 8),

which are conserved currents to the extent that SU (3)F is a symmetry of the strong interactions. These octets of currents control the β-decays of the neutron and of the octet of hyperons which implies that the values of a3 and a8 are known from other measurements. Therefore a measurement of Γ1 can be considered as giving the value of the ﬂavour singlet a0 . Now the European Muon Collaboration, working at CERN, measured the ﬁrst moment of the spin dependent structure function g1 of the proton and in 1988 announced their startling results [5]. Knowing the values of a3 and a8 , the EMC measurement implied (44)

0. aEMC 0

But in the naive parton model (45)

a0 = ΔΣ ,

where ΔΣ is given by eq. (34). In 1974 Ellis and Jaﬀe [6] had suggested that one could ignore the contribution from the strange quark, i.e. from Δs + Δ¯ s, implying that (46)

a0 a8 0.59 .

Thus the EMC result eq. (44) is in gross contradiction with Ellis-Jaﬀe.

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It was this contradiction which at ﬁrst aroused interest in the EMC result, but it was soon realized that their result had far more serious consequences. Consider the physical signiﬁcance of ΔΣ(x). Since q± (x) count the number of quarks of momentum fraction x with spin component ±1/2 along the direction of motion of the proton (say the z-direction), the total contribution to Jz coming from a given ﬂavour quark is

1

Sz =

(47)

0

1 = 2

1 1 dx q+ (x) + − q− (x) 2 2 1

dx Δq(x). 0

It follows that a0 = 2Szquarks ,

(48)

where Szquarks is the contribution to Jz from the spin of all quarks and antiquarks. Note that a0 plays two roles: i) it measures the z component of the spin carried by the quarks, ii) it measures the expectation value of the ﬂavour singlet axial-vector current. What is the connection? Noether’s Theorem tells us that the spin density operator ρ νλ ¯ σ ψ(x). for a spin-(1/2) particle is (1/2)ψ(x)γ Having the spin in the z direction implies ρ = 0, ν = 1, λ = 2. Then we recognize the connection between the operators (49)

1¯ 1¯ ψ(x)γ 0 σ 12 ψ(x) = ψγ 3 γ5 ψ . 2 2

If we write the nucleon state as a superposition of partonic states, we ﬁnd that (50)

(axial-vector current)expectation value = 2 × (spin carried by quarks).

. 3 1. Simple parton model. – One takes p⊥ = 0 and all quarks move parallel to the parent hadron. Thus the quark, momentum p, has p = xP which implies that the orbital angular momentum carried by quarks is perpendicular to P, and hence does not contribute to Jz . Thus, in the simple parton model, one expects for a proton of helicity +1/2: Szquarks = Jz =

(51)

1 . 2

The EMC result [5] for the value of a0 , on the contrary, implied that (52)

Szquarks

! exp

= 0.03 ± 0.06 ± 0.09 .

The longitudinal spin structure of the nucleon

273

It was this highly unexpected result which was termed a “spin crisis in the parton model” [7]. . 3 2. Resolution of the spin crisis. – Take the divergence of the ﬂavour-f axial current. Using the equations of motion one ﬁnds (53)

f ∂ μ J5μ = 2imq ψ¯f (x) γ5 ψf (x),

where mq is mass of the quark of ﬂavour f . f is conserved. This would mean a In the chiral limit mq → 0 this implies that J5μ symmetry between left and right-handed quarks and ultimately a parity degeneracy of the hadron spectrum e.g. there would exist two protons, of opposite parity. Adler [8], and Bell and Jackiw [9] showed that the formal argument from the free equations of motion is not reliable, and there exists an anomalous contribution arising from the triangle diagram shown in ﬁg. 5. For the QCD case one ﬁnds (54)

f ∂ μ J5μ =

αs a " μν αs " μν , Gμν Ga = Tr Gμν G 4π 2π

" aμν is the dual ﬁeld tensor where αs is QCD analogue of ﬁne-structure constant, and G (55)

" a ≡ 1 εμνρσ Gρσ . G μν a 2

A ﬁeld vector or tensor without a colour label stands for a matrix. In this case (56)

Gμν ≡

1 λa Gaμν . 2

Fig. 5. – Feynman diagram responsible for the anomaly.

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E. Leader

The above result is actually very tricky. It is a particular limit of a non-uniform function [1]. If we take mq = 0, k 2 = 0 then the RHS of eq. (54) is multiplied by ⎛ (57)

T (m2q /k 2 ) = 1 −

2m2q /k 2 1+

4m2q /k 2

ln ⎝

1 + 4m2q /k 2 + 1 1+

4m2q /k 2

−1

⎞ ⎠.

The anomaly corresponds to T → 1 for (m2q /k 2 ) → 0. But for on-shell gluons, k 2 = 0, and mq = 0, i.e. in the limit (m2q /k 2 ) → ∞ the terms cancel, T → 0, and there is no anomaly. For gluons bound inside a nucleon one should utilize k 2 = 0 and the anomalous triangle contributes. . 3 3. Eﬀect of anomaly. – Adler’s expression for the triangle diagram, modiﬁed to QCD, gives for the forward gluonic matrix element of the ﬂavour f current (58)

f k, λ|J5μ |k, λ = −

αs g S (k, λ) T (m2q /k 2 ), 2π μ

where λ is the gluon helicity and Sμg (k, λ) ≈ λkμ

(59)

is the covariant spin vector for almost massless gluons. We can now compute the gluonic contribution to the hadronic expectation value 0 P, S|J5μ |P, S . The gluons being bound will be slightly oﬀ-shell i.e. k 2 = 0, but small. The full triangle contribution involves a sum over all quark ﬂavours. Take mu , md and ms to be k 2 whereas mc , mb and mt are k 2 . The function T (m2q /k 2 ) thus takes the values (60)

T =1

for u, d, s,

T =0

for c, b, t.

Hence the gluon contribution is [10-13] (61)

agluons (Q2 ) = −3 0 ≡ −3

αs 2π

1

dx ΔG(x, Q2 ) 0

αs (Q2 ) ΔG(Q2 ), 2π

where ΔG(x) is analogous to Δq(x) (62)

ΔG(x) = G+ (x) − G− (x).

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The longitudinal spin structure of the nucleon

So, there exists a gluonic contribution to ﬁrst moment of g1 ! (63)

Γgluons (Q2 ) = − 1p

1 αs (Q2 ) ΔG(Q2 ). 3 2π

This result is of fundamental importance. It implies that the simple parton model formulae for a0 (and hence for Γp1 ) in terms of the Δqf are incorrect. Instead, (64)

a0 = ΔΣ − 3

αs ΔG . 2π

The fundamental conclusion is that the small measured value of a0 does not necessarily imply that ΔΣ is small. . 3 4. A surprising aspect of this result! – The simple parton model is usually thought of as the limit when the QCD coupling is switched oﬀ. Moreover QCD possess the property of asymptotic freedom, i.e. the eﬀective coupling goes to zero logarithmically as Q2 → ∞. Hence we would expect that as Q2 → ∞ the term agluons (Q2 ) should vanish implying 0 a return to the simple parton model result. But . . . the anomalous gluon contribution is really anomalous! It can be shown that the ﬁrst moment ΔG(Q2 ) tends to inﬁnity logarithmically as Q2 → ∞, thus exactly cancelling the decrease in α(Q2 ) and the gluonic term survives! . 3 5. A note on angular momentum sum rules. – The “spin crisis” was signalled via the failure of (65)

Szquarks = Jz =

1 , 2

which is an intuitive statement that the angular momentum of the nucleon should be made up of the angular momentum of its constituents. This is an example of an angularmomentum sum rule, and it seems obviously true. However, such relations in a relativistic theory are very subtle. Angular-momentum sum rules require explicit expressions for the matrix elements of the angular-momentum operators and obtaining these is non-trivial. Indeed, for some decades it was believed that one could not have an angular-momentum sum rule for a transversely polarized nucleon, but this was recently shown to be incorrect [14]. . 3 6. Is the spin crisis really resolved? – The key point concerning the axial anomaly is that what is measured, a0 , is not equal to ΔΣ. Instead one has (66)

a0 = ΔΣ − 3

αs ΔG , 2π

so we could have a “big” ΔΣ and a “small” a0 , but this requires a large value of ΔG.

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E. Leader

Fig. 6. – x ΔG(x) extracted from the world DIS data by various groups.

You will see in the lectures of Saito that the measured ΔG seems to be much too small! Presently the best value for a0 is a0 0.33. If we want, say, ΔΣ 0.6 we need 2 ΔG 1.7 at Q2 = 1 (GeV/c) . But the latest value for ΔG, obtained from an analysis of the world data on polarized DIS [15] (see ﬁg. 6) is ΔG = 0.29 ± 0.32, which, even with the large error, looks much too small.

P

P’

c

cŦbar

p Fig. 7. – Feynman diagram responsible for producing back-to-back charm particles.

277

The longitudinal spin structure of the nucleon

It could be argued that the role of ΔG(x) in DIS is very indirect. Its main inﬂuence is in the evolution with Q2 , and given the limited range of Q2 available in polarized DIS experiments it is not very well determined. However there are other, more direct ways, of measuring ΔG and they support the conclusion that it is rather small, much too small to resolve the spin crisis. The “golden” method is in the reaction ← −+− → μ p → μ + two charmed particles with the charmed particles roughly back to back. Since we assume essentially zero charm content in the nucleon the only possible mechanism is the one shown in ﬁg. 7. Unfortunately, attempting to detect two charmed mesons turns out to be hopeless, from a statistics point of view, so the ΔG extraction is based mainly on detecting a single, charmed meson with large transverse momentum. The results are consistent with a vanishingly small ΔG. 4. – Field-theoretic generalization of the parton model In a ﬁeld theory whose elementary ﬁelds are quarks and gluons, i.e. in QCD, one can split the diagram for the hadronic tensor W μν into a hard part H μν , where the hard photon interacts with a quark and a soft part Φ, where the nucleon emits the quark, as shown in ﬁg. 8. Note that this is not a Feynman diagram for an amplitude. It represents something like a cross-section, and is really the imaginary part of the forward Compton scattering amplitude for γ + P → γ + P with the photon polarization tensors removed. The “blobs” H and Φ are cut diagrams and contain on-shell particles as intermediate states. For a more pedagogical explanation of all this, see Chapt. 11 of [16]. The blob H, involving hard interactions, can be treated in perturbation theory and the leading terms are just the Born terms shown in ﬁg. 9.

q

q Q

P H

D

E k

k D P, S Fig. 8. – QCD generalization of the parton model.

E P, S

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E. Leader

HP Q Fig. 9. – Born term expression for the hard part.

The blob Φ, involving soft interactions, known as the quark-quark correlator, cannot be evaluated explicitly, but its mathematical expression is (67)

Φαβ (P, S; k) =

d4 z ik.z e P, S| ψ¯β (0) ψα (z) |P, S , (2π)4

where the quark ﬁelds are interacting ﬁelds. Note that ﬂavour and the quark charge have been ignored—they are trivially reinstated at the end—and that we work with mq = 0. In terms of these W μν is given by (68)

W μν =

1 2π

d4 k H μν Φαβ (P, S; k), (2π)4 βα

where α, β are Dirac indices. Note that to regain the results of the simple parton model one approximates H by its Born terms and treats the quark ﬁelds in Φ as free ﬁelds. Note too that in this language the axial anomaly emerges from the diagrams in ﬁg. 10, in which the right-hand diagram involves the gluon-gluon correlator ΦG and the horizontal quark lines are on-shell. In the Bjorken limit the top quark line eﬀectively contracts to a point yielding the anomaly triangle.

q

2

q

= k

k G

Fig. 10. – QCD diagram leading to the anomaly contribution.

The longitudinal spin structure of the nucleon

279

(a)

(b) Fig. 11. – Example of QCD correction terms (b), to the Born approximation (a).

. 4 1. QCD corrections and evolution. – Going beyond the parton model means including various QCD corrections. For example in ﬁg. 11 we show the Born term for H and the simplest correction terms, a vertex correction and a diagram where a gluon is radiated from the active quark before it interacts with the photon. Unfortunately these correction terms are inﬁnite. There are two kinds of inﬁnity: i) the usual ultra-violet type divergences which have to be eliminated by renormalization ii) collinear divergences which occur because of the masslessness of the quarks and which are removed by a process known as factorization. In this the reaction is factorized (separated) into a hard and soft part and the inﬁnity is absorbed into the soft part which in any case cannot be calculated and has to be parametrized and studied experimentally. The point at which this separation is made is referred to as the factorization scale μ2 . Schematically, one ﬁnds terms of the form αs ln(Q2 /m2q ) which one splits as follows: (69)

αs ln

Q2 Q2 μ2 = α ln + α ln s s m2q μ2 m2q

and one then absorbs the ﬁrst term on the right-hand side into the hard part and the second into the soft part. μ2 is an arbitrary number, like the renormalization scale, and, in an exact calculation, physical results cannot depend on it. However it does mean that what we call the parton density has an extra label μ2 specifying our choice. Moreover,

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E. Leader

since we never calculate to all orders in perturbation theory, it can make a diﬀerence what value we choose. It turns out that an optimal choice is μ2 = Q2 , so the parton densities now depend on both x and Q2 , i.e. we have q(x, Q2 ) and Δq(x, Q2 ), and perfect Bjorken scaling is broken. But the variation with Q2 is gentle (logarithmic), and can be calculated via what are called the evolution equations which will be discussed later. It turns out to be crucial in handling these divergences to use the technique of dimensional regularization, which is straightforward in the unpolarized case, but which runs into a snag in the polarized case. The problem is that the generalization of γ5 in more than 4-dimensions is ambiguous. In 4-dimensions we have (70)

{γ μ , γ5 } = 0

μ = 0, 1, 2, 3.

{γ n , γ5 } = 0

n = 4, 5, . . .

If we try (71)

it leads to a contradiction when using (72)

Tr[ABC − − − X] = Tr[XABC − −−].

There is also a problem with the generalization of εμνρσ . ’t Hooft and Veltman [17] and Breitenlohner and Maison [18] suggested using (73) (74)

{γ μ , γ5 } = 0 n

[γ , γ5 ] = 0

μ = 0, 1, 2, 3, n = 4, 5, . . .

This gives rise to the MS-HVBM renormalization scheme, which, however, has a problem. The third component of the isovector axial current Jμ35 is NOT conserved, implying that a3 depends on Q2 . It turns out that this feature is linked to how the factorization between hard and soft parts is implemented and can be remedied. At present there are three schemes in use, all of them modiﬁed versions of MS-HVBM: i) The Vogelsang, Mertig, van Neerven scheme [19,20] MS-MNV. Here Jμ35 is conserved i.e. a3 is independent of Q2 . ii) The AB scheme of Ball, Forte and Ridolﬁ [21], which, in addition, has the ﬁrst moment (75)

1

dx ΔΣ(x, Q2 )

ΔΣ = 0

independent of Q2 . iii) The JET scheme of Carlitz, Collins and Mueller [12], Anselmino. Efremov and Leader [1] and Teryaev and M¨ uller [22], and which is identical to the chiral invariant scheme of Cheng [23]. In this scheme a3 and a8 are independent of Q2 as is ΔΣ, but it

281

The longitudinal spin structure of the nucleon

can be argued that the JET scheme is superior to the others in that all hard eﬀects are included in H. Of course if one could work to all orders in perturbation theory it would make no diﬀerence which scheme one used, but given that we work to leading order (LO), next to leading order (NLO), and in some cases to NNLO, the choice of scheme can be of importance. . 4 2. Structure of G1 (x, Q2 ) at and beyond leading order . – For the polarized densities the evolution equations are (76)

(77)

d αs (Q2 ) 2 Δq(x, Q ) = d ln Q2 2π

d αs (Q2 ) ΔG(x, Q2 ) = 2 d ln Q 2π

1

x

1

x

dy ΔPqq (x/y) Δq(y, Q2 ) + y +ΔPqG (x/y) ΔG(y, Q2 ) , dy ΔPGq (x/y) Δq(y, Q2 ) + y +ΔPGG (x/y) ΔG(y, Q2 ) .

The ΔP are the polarized splitting functions and are calculated perturbatively (78)

ΔP (x) = ΔP (0) (x) +

αs ΔP (1) (x), 2π

where the superscripts (0) and (1) refer to LO and NLO contributions. For details about these the reader is referred to Vogelsang [19]. Note that in LO ﬂavour combinations like qf − qf , e.g., u(x) − d(x) and valence combinations like qf − q¯f , e.g., u(x) − u ¯(x) are non-singlet and evolve in the same way, without the ΔG term in eq. (75). (There is no splitting in LO from a q to a q¯, nor from say a u to a d.) However, in NLO ﬂavour non-singlets like u(x)−d(x) and charge-conjugation non-singlets like u(x) − u ¯(x) evolve diﬀerently. The origin of this diﬀerence can be seen in ﬁgs. 12 and 13. Figure 12 shows an NLO amplitude for a quark to split into a q. ¯

Fig. 12. – NLO amplitude for q → q¯ transition.

282

E. Leader

Fig. 13. – NLO contributions to the splitting function for a q → q¯ transition.

Figure 13 shows two possible contributions to ΔPqq¯ from taking the modulus squared of this amplitude. In (a) the contribution is pure ﬂavour singlet and involves only gluon exchange, whereas in (b) the contribution is non-singlet. However, if we try to do something similar for a ﬂavour changing splitting function, e.g., ΔPdu we ﬁnd that we cannot construct the non-singlet diagram. The expression for g1 (x, Q2 ) now becomes (79)

g1 (x, Q2 ) =

1 2 eq Δq(x, Q2 ) + Δ¯ q (x, Q2 ) + 2 ﬂavours αs (Q2 ) 1 dy ΔCq (x/y) Δq(y, Q2 ) + Δ¯ + q (y, Q2 ) + 2π y x +ΔCG (x/y) ΔG(y, Q2 ) ,

where ΔCG and ΔCq are Wilson coeﬃcients evaluated from the hard part calculated beyond the Born approximation. Note that very often the evolution equations are written using the convolution notation, for example, (80)

1

ΔCq ⊗ Δq ≡ x

dy ΔCq (x/y) Δq(y). y

5. – The polarized strange quark density: attempts to measure Δs(x) Although the strangeness content of the nucleon is small, it has played a major role in provoking puzzles and controversies in our understanding of the internal structure of the nucleon, particularly as concerns the spin structure. Recall that it was a misjudgement of the signiﬁcance of strangeness in the Ellis-Jaﬀe result that was behind the original excitement generated by the famous EMC experiment in 1988. And as we shall now see there is still some mystery surrounding the polarized strange density.

The longitudinal spin structure of the nucleon

283

There are two possibilities for measuring Δs(x), via polarized DIS or via polarized semi-inclusive DIS (SIDIS). Recall that DIS only depends on Δq(x) + Δ¯ q (x). So we can obtain information on Δs(x) + Δ¯ s(x). In SIDIS we could, in principle obtain Δs(x) and Δ¯ s(x) separately, but that is for the future! . 5 1. Results from polarized DIS . – Aside from one small issue there is general agreement between several analyzes: see ﬁg. 14. What causes the disagreement at moderate to large x? Surprisingly—positivity, i.e. the requirement that (81)

|Δs(x)| ≤ s(x).

As shown in ﬁg. 15, the data seem to want a large negative Δs(x) at moderate values of x. So there is a clash with positivity and the result is that the shape of Δs(x) is sensitive to the input unpolarized density. In the ﬁgure the polarized analyses BB2 [24], AAC03 [25] and GRSV [26] utilized the unpolarized strangeness density of GRV98 [27], whereas LSS05(Set 1) [15] used the unpolarized strangeness density of MRST’02 [28]. It is seen that LSS05 is incompatible with the unpolarized GRV density.

Fig. 14. – Polarized strange quark density from various analyses of world data on DIS.

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E. Leader

Fig. 15. – Role of positivity in inﬂuencing Δq(x) at moderate values of x.

. 5 2. Results from SIDIS . – Before looking at results consider the following constraint [29] on the ﬁrst moment: δs ≡ [Δs + Δ¯ s].

(82)

We can rewrite the expression for Γp1 as (83)

Γp1 (Q2 )

1 1 5 2 a3 + a8 + 2δs (Q ) , = 6 2 6

or (84)

a8 =

1 6 6Γp1 (Q2 ) − a3 − 2δs (Q2 ) . 5 2

We know a3 very accurately. Using the measured values of Γp1 (Q2 ) we show that δs (Q2 ) ≥ 0 implies an unacceptable value for a8 . We have to decide what value to use for Γp1 (Q2 ), since the result depends on the extrapolation to x = 0. We take two extremes: i) Assume perturbative QCD holds at small x as done by SLAC experiment E155 [30], etc. This yields (85)

Γp1 (Q2 =5) = 0.118 ± 0.004 ± 0.007 .

ii) Assume Regge behaviour at small x as utilized by SLAC experiment E143 [31], etc. This gives (86)

Γp1 (Q2 =3) = 0.133 ± 0.003 ± 0.009 .

The longitudinal spin structure of the nucleon

285

Fig. 16. – x [Δs(x) + Δ¯ s(x)] from HERMES SIDIS analysis.

Results: If δs is positive we ﬁnd i) a8 ≤ 0.089 ± 0.058, ii) a8 ≤ 0.197 ± 0.068. Now to the best of our knowledge hyperon β-decay is adequately described by SU (3)F and this leads to a8 = 0.585 ± 0.025. Thus δs (Q2 ) ≥ 0 implies a dramatic breaking of SU (3)F , and we conclude that it is almost impossible to have δs (Q2 ) ≥ 0. Now HERMES has extracted Δs(x) + Δ¯ s(x) from a study of SIDIS [32]. The results are shown in ﬁg. 16. Within errors the results are consistent with zero, and HERMES quote (87)

δs (Q2 =2.5) = 0.028 ± 0.033 ± 0.009 .

The previous discussion suggests that the central value cannot be the true value unless we have totally failed to understand the connection between DIS and SIDIS. If the latter is not the case, how can we understand the HERMES results? I think it is important to remember that HERMES uses a LO method based on socalled purities. I suspect that such an approach is unreliable at the values of Q2 involved, and that the errors on the purities are somewhat underestimated in their analysis. So I strongly believe that this new “strange quark crisis” will prove to be illusory. 6. – A last word on the “spin crisis” We have seen that the hope that a large polarized gluon density could resolve the “spin crisis” is probably no longer tenable, given that recent experiments seem to be indicating quite a small value for ΔG. If that is so, how are we to resolve the crisis? The answer is, in principle, quite straightforward. In our collinear parton model we neglected transverse motion of the partons, and this transverse momentum can generate orbital angular momentum with a component in the direction of motion of the nucleon. It is a simple exercise to show that an acceptable magnitude of transverse momentum could yield enough Lz to satisfy the longitudinal angular-momentum sum rule. Against this explanation is

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the intuitive, but probably incorrect, argument that in quark models of hadrons the nucleon appears as an s-wave ground state, i.e. with zero orbital angular momentum. ∗ ∗ ∗ Almost all of the work reported here was carried out in collaboration with A. V. Sidorov (Bogoliubov Theoretical Laboratory, Dubna) and D. B. Stamenov (Institute for Nuclear Research and Nuclear Energy, Soﬁa). REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

Anselmino M., Efremov A. and Leader E., Phys. Rep., 261 (1995) 1. Anselmino M. and Leader E., Phys. Lett. B, 293 (1992) 216. Wandzura S. and Wilczek F., Phys. Lett. B, 72 (1977) 195. D’Alesio U., Leader E. and Murgia F., The importance of Lorentz invariance in the parton model, unpublished. Ashman J. et al., Phys. Lett. B, 206 (1988) 364; Nucl. Phys. B, 328 (1989) 1. Ellis J. and Jaffe R. L., Phys. Rev. D, 9 (1974) 1444; Phys. Rev. D, 10 (1974) 1669(Erratum). Leader E. and Anselmino M., Z. Phys. C, 41 (1988) 239. Adler S. I., Phys. Rev., 177 (1969) 2426. Bell J. S. and Jackiw R., Nuovo Cimento A, 51 (1969) 47. Efremov A. V. and Teryaev O. V., JINR Report E2-88-287 unpublished. Altarelli G. and Ross G. G., Phys. Lett. B, 212 (1988) 391. Carlitz R. D., Collins J. C. and Mueller A. H., Phys. Lett. B, 214 (1988) 229. Leader E. and Anselmino M., Santa Barbara Preprint NSF-88-142 unpublished. Bakker B. L. G., Leader E. and Trueman T. L., Phys. Rev. D, 70 (2004) 114001. Leader E., Sidorov A. V. and Stamenov D. B., JHEP, 06 (2005) 033. Leader E., Spin in Particle Physics (Cambridge University Press, Cambridge, UK) 2005. ’t Hooft G. and Veltman M., Nucl. Phys. B, 44 (1972) 189. Breitenlohner P. and Maison D., Commun. Math. Phys., 52 (1977) 11. Vogelsang W., Phys. Rev. D, 54 (1996) 2023. Mertig R. and van Neerven W. L., Z. Phys. C, 70 (1996) 637. Ball R. D., Forte S. and Ridolfi G., Phys. Lett. B, 378 (1996) 255. ¨ller D., Phys. Rev. D, 56 (1997) 2607. Teryaev O. V. and Mu Cheng hai-Yang., Phys. Lett. B, 427 (1998) 371. Blumlein J. and Bottcher H., Nucl. Phys. B, 636 (2002) 225. Hirai M., Kumano S. and Saito N., Phys. Rev. D, 69 (2004) 054021. ¨ck M., Reya E., Stratmann M. and Vogelsang W., Phys. Rev. D, 63 (2001) Glu 094005. ¨ck M., Reya E. and Vogt A., Eur. Phys. J. C, 5 (1998) 461. Glu Martin A. D., Roberts R. G., Stirling W. J. and Thorne R. S., Eur. Phys. J. C, 28 (2003) 455. Leader E. and Stamenov D. B., Phys. Rev. D, 67 (2003) 037503. Anthony P. L. et al., Phys. Lett. B, 493 (2000) 19. Abe K. et al., Phys. Rev D, 58 (1998) 112003. Airapetian A. et al., Phys. Rev. D, 71 (2005) 012003.

The transverse spin structure of the nucleon M. Anselmino Dipartimento di Fisica Teorica, Universit` a di Torino and INFN, Sezione di Torino Via P. Giuria 1, I-10125 Torino, Italy

Summary. — The last years have witnessed an impressive eﬀort—in theoretical work, experimental measurements and data analysis—towards a deeper and better understanding of the proton and neutron structure in terms of their elementary constituents, quarks and gluons. Much progress, beyond the usual simple QCD representation of a fast moving nucleon as a bunch of collinear partons, has been achieved. Fundamental issues, like the intrinsic and orbital motion of quarks inside a proton, its coupling to the parton spin and to the parent proton spin, have been raised and investigated. The transverse, with respect to the direction of motion, degrees of freedom of partons, both in spin and motion, are best suited to study the nucleon internal properties. A new QCD picture of protons and neutrons is slowly emerging.

1. – Introduction and lecture plan This series of three lectures focuses on our actual understanding of the proton and neutron (nucleon) structure and on the crucial role that spin plays in elementary particle physics and in all quantum ﬁeld theories, thus including the theory of strong interactions, Quantum Chromo Dynamics (QCD). The reader might ﬁnd helpful to consult, alongside with reading, for more ﬁgures and technical details, the slides of the presentation at the school, available at http://www.to.infn.it/~anselmin/ c Societ` a Italiana di Fisica

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The ﬁrst lecture brieﬂy summarizes our QCD knowledge of the nucleon structure, as developed in the past 35 years, very successful but limited to the so-called collinear conﬁguration, involving only longitudinal degrees of freedom. We then introduce the problematic of transverse Single Spin Asymmetries (SSA), a fascinating issue which played a seminal role in leading to a rediscovery of the proton structure, forcing the introduction of transverse degrees of freedom. The second lecture is devoted to the study of SSA in Semi-Inclusive Deep Inelastic Scattering (SIDIS, N → h X) and the introduction of the so-called Transverse Momentum Dependent (TMD) Parton Distribution Functions (PDF) and fragmentation functions (FF) (sometimes globally referred to simply as TMDs). Their phenomenological extraction from data and their theoretical interpretation is discussed. The third lecture deals with the (much more subtle and debated) issue of SSA in hadronic interactions, A B → h X, and Drell-Yan (DY) processes, A B → + − X, discussing the possible role that TMDs play in understanding, and possibly predicting, the experimental results. These lectures are tightly related to other series of lectures (and some seminars) presented during the school, with which they form a unique program. A comprehensive theoretical discussion of the longitudinal spin structure of the nucleon can be found in the lectures by Elliot Leader; the experimental aspects and the analysis of SSA data in SIDIS are discussed by Delia Hasch; the theoretical issues related to QCD spin physics in hadronic interactions are presented by Werner Vogelsang, while the corresponding experimental program is illustrated in the lectures by Naohito Saito. Some more speciﬁc points are further investigated in the contributions by Harut Avakian (experimental program at JLab) and Mariaelena Boglione (ﬁrst extraction of the transversity distribution). 2. – The collinear proton spin conﬁguration Information about the partonic structure of protons and neutrons has been historically mainly collected via the Deep Inelastic Scattering (DIS) of pointlike leptons oﬀ a nuclear target, N → X, and relies on the QCD parton model, which visualizes a fast-moving nucleon as a composite object made of collinear quasi-free massless quarks and gluons, each carrying a fraction x of the parent nucleon momentum (as discussed in the lectures by E. Leader). Thus, the DIS cross-section, in terms of the usual DIS variables, is simply given by (1)

dσ dˆ σ = e2q q(x, Q2 ) , 2 dx dQ2 dQ q

which allows, from a measurement of the unpolarized cross-section dσ/dx dQ2 , to obtain information on the unpolarized parton distribution functions q(x, Q2 ). dˆ σ /dQ2 is the computable cross-section for the scattering of the lepton oﬀ a parton. The Q2 dependence of the PDFs is generated and controlled by QCD eﬀects and its comparison with data is one of the most stringent QCD tests and a great success of the theory.

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However, despite an impressive agreement between data on q(x, Q2 ) and QCD predictions, the ultimate amount of information on the nucleon structure we obtain from unpolarized DIS in the collinear partonic interpretation is “only” the number density of quarks (and, indirectly, of gluons), as explored by a lepton at a certain energy, as a function of x and Q2 . The q(x, Q2 )s are by now rather well known, but no information can be extracted, for example, about the quark motion inside the nucleon. . 2 1. The helicity distribution. – Some further information about the partonic proton content can be obtained in polarized DIS; good experiments have been performed with longitudinally polarized leptons and nucleon targets. In this case, one explores the amount of partons with longitudinal spin parallel or antiparallel to the spin of the parent proton, the helicity distributions (2)

+ + Δq(x, Q2 ) = q+ (x, Q2 ) − q− (x, Q2 ),

where ± denote the helicities of the partons (subscripts) and the parent protons (superscripts). These quantities are accessed via the measurement of a double longitudinal spin asymmetry: (3)

++

dˆ σ dσ ++ dσ +− dˆ σ +− 2 2 − = e Δq(x, Q ) − . q dx dQ2 dx dQ2 dQ2 dQ2 q

Again, a good knowledge of the helicity distributions, although not as detailed as in the unpolarized case, has been obtained, and some subtle problems with the contribution of the gluons brilliantly solved by QCD [1]. However, the detailed partonic contribution to the total longitudinal spin of a proton remains largely uncertain: it appears that contributions from the orbital angular momentum of quarks and gluons are necessary to build up the total 1/2 value [2]. . 2 2. The transversity distribution. – Even if we knew exactly all the partonic contributions to the longitudinal spin of the proton, that would not exhaust our study of the internal nucleon spin structure. In fact the transversity distribution (4)

ΔT q(x, Q2 ) = q↑↑ (x, Q2 ) − q↓↑ (x, Q2 )

is a new fundamental quantity, not accessible in DIS. ΔT q, also often denoted as h1 , h1T or δq, is the analogue of eq. (2), with the helicity ± labels replaced by the ↑, ↓ labels, which denote a polarization direction transverse with respect to the proton motion. The transversity distribution is thus the diﬀerence between the number densities of quarks with spin parallel and antiparallel to the parent proton transverse spin. Notice that massless gluons cannot carry any transverse spin. An excellent review on transverse spin and transversity can be found in [3]. Notice that the helicity and transversity distributions need not be equal: they would be equal, under rotational invariance, for a proton at rest, but the parton model holds

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Fig. 1. – The handbag diagram for DIS. At leading QED order, the interaction between the lepton (not shown) and the nucleon is mediated by the exchange of a virtual photon. Thus, the DIS cross section is just the total cross section for the γ ∗ N → X process, which, by the optical theorem, is related to the forward scattering amplitude. In the parton model, at leading QCD order, the virtual photon scatters oﬀ a single quark in the nucleon, as represented in the ﬁgure. The lower blob is thus the matrix element between the nucleon initial and ﬁnal states of two quark ﬁelds, one “extracted from” and the other “replaced into” the nucleon. It is a matrix in the Dirac spinor space.

only in the so-called inﬁnite momentum frame, which spoils rotational invariance. On a more formal ground, the object which is explored in DIS is the quark-quark correlator (actually, a correlation matrix) Φ(k, P, S), graphically represented by the blob in the lower part of ﬁg. 1. Its most general expression in Dirac spinor space can be written, in the collinear case, as [3] (5)

Φ(x, S) =

1 f1 (x) n / + + SL g1L (x) γ 5 /n+ + h1T iσμν γ 5 nμ+ STν , 2

where n+ = (1, 0, 0, 1) is a null vector and the three independent functions f1 , g1L and h1T are the three leading-twist partonic distribution functions: respectively, the unpolarized one (usually denoted by q(x) for quarks and g(x) for gluons, or fq,g (x)), the helicity distributions (Δq(x) and Δg(x)) and the transversity distribution (h1T (x) = h1 (x) = Δq(x) = δq(x)), according to diﬀerent notations used in the literature. In eq. (5) we have adopted the notations of the Amsterdam group [4-6]. In terms of the quark-quark correlator the helicity and transversity distributions are shown in ﬁg. 2. The representation of the transversity distribution in the helicity basis (last line) is obtained by using the relation (6)

1 | ↑, ↓ = √ (|+ ± i |− ) 2

and it shows the chiral-odd nature of transversity, as it relates quarks with opposite helicities. It is then clear why h1 cannot be measured in DIS: the bottom blob of ﬁg. 2

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Fig. 2. – The combinations of quark-quark correlators corresponding to the helicity distribution (ﬁrst line) and the transversity distribution (second line). Notice that the unpolarized distributions are simply obtained taking the sum, rather than the diﬀerence, either in line 1 or 2. The diagram in the third line corresponds to the transversity distribution, when expressed in the helicity basis, eq. (6) of the text, showing its chiral-odd nature.

cannot be inserted in the handbag diagram of ﬁg. 1, as the QED (and QCD) interactions conserve helicity and there is no way, by photon or gluon couplings, of ﬂipping the helicity of massles quarks. A measurement of transversity requires a process in which h1 couples to another chiral-odd function. Several suggestions have been discussed in the literature. At the moment the most practicable way appears via SIDIS processes [7], in which h1 couples to a chiral-odd fragmentation function, the Collins fragmentation function, as depicted in ﬁg. 3. In principle, the cleanest and most direct way should be via the measurement

Fig. 3. – Coupling of the transversity distribution with a chiral-odd fragmentation function in polarized SIDIS. The same could be done in p p scattering, with one proton transversely polarized; in this case the virtual photon is replaced by a virtual gluon, from the unpolarized proton.

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Fig. 4. – The combination of two transversity distributions in polarized Drell-Yan processes.

of the double transverse-spin asymmetry AT T in Drell-Yan processes, which couples two transversity distributions (see ﬁg. 4), as discussed in sect. 5. So far we have only considered collinear partonic conﬁgurations, in which the relevant degrees of freedom, describing the nucleon structure, are the parton longitudinal momentum fraction x and the helicities. Yet, it is already clear that the spin transverse degree of freedom is at least as interesting, but much less known. It will be much more so when also the intrinsic transverse motions of partons, k⊥ , in addition to x, will be considered. Which requires a detour into the issue of SSA. 3. – The (problem of ) transverse single-spin asymmetries Let us consider a 2 into 2 physical process, like A B → C D, in the center-of-mass reference frame, A(p) + B(−p) → C(p ) + D(−p ), like in ﬁg. 5. We wonder whether or not the cross-section for such a process can depend on the spin polarization S of one particle only, say A; particle B is not polarized and the polarization of the ﬁnal particles is not observed.

Fig. 5. – The kinematical variables for a process A B → C D in the center-of-mass reference frame.

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Fig. 6. – A combination of Feynman diagrams contributing to a SSA in the elastic scattering of two diﬀerent spin-(1/2) Dirac particles.

The single-spin asymmetry is deﬁned as AN =

(7)

dσ ↑ − dσ ↓ , dσ ↑ + dσ ↓

where dσ stands for the diﬀerential cross-section of interest (dσ/dθ in this case) and ↑= S, ↓= −S, denote the spin directions. For a parity-invariant process there is only one way in which one can build a scalar quantity out of the available independent vectors (p, p ) and pseudovector (S), so that one expects (8)

AN ∼ S · (p × p ) = S · (p × PT ) ∼ PT sin(ΦS − Φ).

Equation (8) shows that only the polarization component perpendicular to the scattering plane (the transverse polarization) contributes to AN and that the asymmetry vanishes in the forward direction (as expected from rotational invariance). Although possible in principle, SSA asymmetries need subtle quantum mechanical eﬀects in order to be realized. This can be seen by considering, for example, the elastic scattering of two protons, p1 p2 → p1 p2 ; all related physical quantities can be expressed in terms of 5 independent helicity amplitudes [8], Hλ1 ,λ2 ;λ1 ,λ2 : (9) Φ1 ≡ H++;++

Φ2 ≡ H−−;++

Φ3 ≡ H+−;+−

Φ4 ≡ H−+;+−

Φ5 ≡ H−+;++

and it turns out that (10)

∗ AN ∝ Im Φ5 (Φ1 + Φ2 + Φ3 − Φ4 ) .

Thus, a non-vanishing AN requires both a non vanishing single helicity-ﬂip amplitude (Φ5 ) and a relative phase between Φ5 and at least one of the other amplitudes. Both features are diﬃcult to achieve in the QED or QCD interactions of elementary constituents. Let us consider the simplest example of the elastic scattering of two quarks of diﬀerent ﬂavors, q q → q q . AN is related to the imaginary part of an interference √ term, as shown in ﬁg. 6. The helicity ﬂip in the left diagram introduces a factor mq / s (helicity ﬂips in perturbative QED and QCD for a particle of mass m and energy E are suppressed by a factor m/E), while the right diagram—which indeed has an extra i

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Fig. 7. – Kinematics of the inclusive process p↑ p → π X in the center-of-mass reference frame.

factor as compared with the left one—is a higher-order one, introducing an extra factor αs , the strong-coupling constant. Altogether one expects, in perturbative QCD (pQCD) mq AN ∼ αs √ s

(11)

which is a negligible value at high energies. The same conclusion holds for any elementary 2 → 2 scattering involving leptons, quarks and gluons in pQED and pQCD [9]. For a long time it was thought that the smallness of SSAs at the partonic level should reﬂect into an analogous smallness at the hadronic level. Indeed, this was considered almost as a theorem [10]. Let us look, as a typical case, at the large PT single inclusive production in high-energy hadronic interactions, p↑ p → π X, with only one of the initial protons transversely polarized. One can deﬁne a SSA for this process as (see ﬁg. 7) (12)

AN =

dσ ↑ (PT ) − dσ ↑ (−PT ) dσ ↑ (PT ) − dσ ↓ (PT ) = ↑ ↓ dσ (PT ) + dσ (PT ) 2 dσ(PT )

where dσ(PT ) is the unpolarized cross-section and we have exploited rotational invariance, using dσ ↓ (PT ) = dσ ↑ (−PT ).

(13)

The SSA AN is then simply a left-right asymmetry, easy to measure as it implies a diﬀerent number of pions produced to the left (PT ) or to the right (−PT ), with respect to the polarized (↑) beam direction, as shown in ﬁg. 7. According to the usual collinear QCD factorization scheme pictured in ﬁg. 8 (see the lectures by W. Vogelsang) the cross section for the the large PT and high-energy p p → π X unpolarized process is given by (14)

dσ =

fa (xa ) ⊗ fb (xb ) ⊗ dˆ σ ab→cd (xa , xb ) ⊗ Dπ/c (z),

a,b,c,d=q,¯ q ,g

which shows the factorized convolution of PDFs (fa,b ) with pQCD cross-sections for the elementary processes ab → cd (dˆ σ ) and FFs (Dπ/c ). In the polarized case the diﬀerence

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Fig. 8. – Schematic representation of the hard scattering formalism for the computation of cross-sections within the QCD factorization scheme.

appearing in the numerator of AN is given, in the same scheme, by (15)

dσ ↑ − dσ ↓ =

ΔT fa ⊗ fb ⊗ [dˆ σ ↑ − dˆ σ ↓ ] ⊗ Dπ/c ,

a,b,c,d=q,¯ q ,g

which involves the convolution of transversity distributions (ΔT fa ) with SSA for the elementary process (dˆ σ ↑ − dˆ σ ↓ ). We have just seen that these are very small, eq. (11), thus leading to the conclusion that all SSAs in pQCD calculations should be negligible. . 3 1. Experimental results on SSA. – Contrary to these expectations, and despite some generalized skepticism, according to which all single spin eﬀects should somehow disappear at high energies, several SSAs have been measured and, in most cases, found to be surprisingly large. Figure 9 shows some measured value of AN for p↑ p → π X processes at diﬀerent energies; these SSAs are very large, in some cases impressively large, and √ remain sizable even at energies as high as s = 200 GeV. Their typical feature is that √ they are larger at larger values of the Feynman variable xF = 2PL / s, which implies a pion generated by the fragmentation of a large x parton; this suggests that SSA are related to the properties of valence quarks inside the polarized proton. Inclusive pion production in hadronic interactions is not the only instance where SSAs have been measured to be large; a longstanding problem is the polarization of Λ’s and other hyperons produced in the interactions of unpolarized nucleons [15]. This is a single-spin asymmetry observed when looking at the ﬁnal hadron polarization, which can be revealed only for hyperons, thanks to their parity-violating decays. SSAs have been more recently observed also in SIDIS processes, N ↑ → h X, as explained in the dedicated lectures by D. Hasch. These are rich and interesting processes, which can yield a lot of information on the nucleon structure and will be further discussed in the next section. Figure 10 shows one of the (many) ways in which a SSA can occur in SIDIS: in this case we do not look at the ﬁnal lepton and consider the process γ ∗ p → π X, which can depend on the proton spin polarization S through the usual parity-invariant mixed product of eq. (8), thus generating a SSA (16)

A ∼ S · (p × PT ) ∼ PT sin(Φπ − ΦS ).

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Fig. 9. – Selected experimental data on AN for p↑ p → π X at diﬀerent energies. Results in the √ upper left corner are from BNL-AGS at s = 6.6 GeV and 0.6 ≤ PT ≤ 1.2 Gev/c [11]; data in √ the upper right corner are from the E704 experiment at Fermilab, at s 20 GeV and 0.7 ≤ + − 0 PT ≤ 2.0 Gev/c for π (upper set), π (lower set) and π (middle set) production [12]. Finally, results in the lower part show, on the left, the values of AN as measured by the RHIC-STAR √ Collaboration, at s = 200 GeV [13]; on the right, data on the unpolarized cross section [14], in the same kinematical region, are compared with pQCD calculations, based on the factorized scheme of eq. (14), at LO (dashed lines) and NLO (solid lines).

Fig. 10. – Kinematics of the inclusive γ ∗ p → π X process in the γ ∗ p c.m. frame.

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Fig. 11. – Kinematics of SIDIS processes according to Trento conventions [18] in the γ ∗ p c.m. frame.

Such an azimuthal asymmetry has indeed been observed by the HERMES Collaboration experiment [16, 17]. 4. – SSA in SIDIS We have now to face the problem of SSA. It was sometimes argued that these singlespin eﬀects are conﬁned in the low energy domain, where pQCD cannot be at work yet, and that eventually they will all disappear at higher energies. Recent data, like the STAR RHIC results of ﬁg. 9, have clearly shown the persistence of subtle spin eﬀects at high energy, exactly in the energy range where the pQCD factorization theorem, in the collinear conﬁguration, works excellently in computing the unpolarized cross-sections. Analogously, diﬀerent sets of SIDIS HERMES data, collected in the same kinematical regions, show on one hand azimuthal asymmetries which could not be explained in collinear pQCD, while, on the other hand, are successfully used for the usual extraction of collinear unpolarized and longitudinally polarized PDFs. Let us look again at the SIDIS asymmetry, actually observed, and related to the kinematical conﬁguration of ﬁg. 10 and eq. (16). In a purely collinear conﬁguration, the virtual photon would hit a quark inside the proton, which would bounce back in the opposite direction, then fragmenting collinearly. There would be only one direction, thus forbidding, by rotational invariance, any SSA (PT = 0). In actual experiments the situation is well diﬀerent, like the one described in ﬁg. 11, in which the kinematical conﬁguration follows the so-called Trento conventions [18]. Within the parton model at leading order, that is still considering the elementary interaction as q → q and avoiding, for the moment, higher-order pQCD contributions like q → q g or g → q q¯, the ﬁnal PT of the hadron could originate from some transverse intrinsic motion of the quark inside the nucleon (k⊥ ) and/or from some transverse motion, in the fragmentation process, of the ﬁnal hadron with respect to the direction of the fragmenting quark (p⊥ ). Although the PDFs and the FFs are usually extracted from data as functions of the

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longitudinal momenta fractions only (respectively, x and z), implying an integration over the transverse motions, there might be cases—and the SSA prove to be one of them—in which the unintegrated PDFs and FFs (the transverse momentum dependent ones, the TMDs shortly) play a crucial role in explaining observed physical eﬀects. The collinear conﬁguration is a simple and useful one, well suitable and indeed successful in describing many high energy processes, but it might not be adequate for more subtle eﬀects like the spin asymmetries. There are several clear motivations, both theoretical and experimental, to introduce the intrinsic transverse motion in the parton model, yet maintaining the probabilistic interpretation of parton distribution and fragmentation functions. – A quark conﬁned in a region of the nucleon dimension (1 fm) has, under uncertainty principle, an intrinsic momentum of the order k⊥ = |k⊥ | ∼ 200 MeV/c. – A strictly collinear gluon emission from a massless quark is forbidden by the helicity conservation of the pQCD interactions, coupled to angular momentum conservation. QCD evolution builds up intrinsic motion. – As shown in ﬁg. 11, SIDIS processes in the γ ∗ N c.m. frame are not collinear at all. Similarly, DY processes, that originate from the annihilation of a q q¯ pair into a lepton pair, are not observed to happen along one direction only, as they should in a collinear conﬁguration. – Quark hadronization into jets has been clearly observed not to be collinear. Let us then try and take intrinsic motion into account. Is there a simple case, independently of SSAs, which needs an interpretation based on parton intrinsic motion, and which might give us some direct information on the amount of k⊥ carried by quarks? . 4 1. Parton model with intrinsic motion. – The answer to the previous question comes, once again, from exploring the nucleon with a pointlike lepton, and looking at the SIDIS cross section in the parton model interpretation. There is a general agreement and an accepted proof [19] that the cross-section for the SIDIS process, N → h X, can be written in a factorized form, even with TMDs, in the kinematical region of small PT and large Q2 , PT ∼ k⊥ ∼ ΛQCD Q2 (the relevant kinematical variables are deﬁned in ﬁg. 12): (17)

dσ p→hX =

fq (x, k⊥ ; Q2 ) ⊗ dˆ σ q→q (y, k⊥ ; Q2 ) ⊗ Dπ/q (z, p⊥ ; Q2 ),

q

where now all terms in the factorization, the partonic distribution and fragmentation functions and the elementary interaction, depend on the intrinsic transverse momenta. At O(k⊥ /Q) one has simple relations between the partonic and the hadronic variables (see ﬁg. 12): (18)

PT = zk⊥ + p⊥ ,

zh ≡

P · Ph = z, P ·q

xB ≡

Q2 = x, 2P · q

y≡

P · . P ·q

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Fig. 12. – Kinematics of SIDIS processes in terms of the partonic intrinsic motion.

For exact relations and a full expression of the convolution, eq. (17), with all appropriate integrations and kinematical factors see ref. [20]. It suﬃces to notice here the expression of the elementary cross-section (19)

dˆ σ q→q ∝ sˆ2 + u ˆ2

in terms of the Mandelstam variables sˆ = ( + k)2 , tˆ = ( − )2 = −Q2 and u ˆ = ( − k )2 . In the collinear conﬁguration k = xP , while in the non-collinear one the four-momentum k contains the intrinsic transverse momentum k⊥ , which has an azimuthal angle ϕ. Then

Fig. 13. – The cos Φh azimuthal dependence induced in SIDIS processes by the intrinsic motion of quarks. The ﬁt of the experimantal data, based on the assumption of a simple Gaussian k⊥ dependence, is from ref. [20].

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Fig. 14. – The PT dependence of the SIDIS cross-section computed in the parton model with TMDs below PT = 1 GeV/c and with collinear higher-order pQCD above 1 GeV/c. Details can be found in ref. [22].

one has [20] (20)

dˆ σ

q→q

k⊥ Q4 2 (2 − y) 1 − y cos ϕ , ∝ sˆ + u ˆ = 2 1 + (1 − y) − 4 y Q 2

2

which contains a dependence, of elementary kinematical origin, on the azimuthal angle ϕ of the non collinear quark. Upon integration over d2 k⊥ such a dependence produces a dependence on the azimuthal angle Φh of the observed hadron. This dependence has indeed been observed, as shown in ﬁg. 13; the explanation based on the parton intrinsic motion was ﬁrst pointed out by Cahn [21], and is now referred to as the Cahn eﬀect. Assuming a factorized Gaussian k⊥ dependence of the PDFs, with a ﬁxed, ﬂavor2 independent value of the Gaussian width k⊥

(and similarly for the FFs, with p2⊥ ) one can ﬁt the data, as shown in ﬁg. 13, obtaining [20, 22] (21)

2 k⊥

= 0.28 (GeV/c)2 ,

p2⊥ = 0.25 (GeV/c)2 .

2 There is then some direct evidence that intrinsic motion, with average values of k⊥

2 and p⊥ as expected from simple basic motivations, is visible in experimental data, and can explain in a simple way results which cannot be accommodated in the oversimpliﬁed collinear conﬁguration. One also expects that the unintegrated PDFs and FFs can be eﬀective only at small values of PT k⊥ , while at higher values genuine higher-order pQCD contributions should take over. In ref. [22] it has been shown that this is indeed the case, with a transition between TMD eﬀects and higher order pQCD at PT 1 GeV/c. In ﬁg. 14 the SIDIS cross-section as a function of PT is computed according to eq. (17)

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Fig. 15. – Feynman diagrams contributing at NLO pQCD to the large PT dependence of the SIDIS cross-section.

up to PT = 1 GeV/c, and according to the higher-order Feynman diagrams of ﬁg. 15 (with collinear PDFs and FFs) above 1 GeV/c. Notice that TMD eﬀects are negligible above that value, while genuine pQCD corrections cannot be computed below it. . 4 2. Spin eﬀects in TMDs. – Having obtained an experimental conﬁrmation of the importance of the intrinsic transverse motion, and even an estimate of its average value, we go back to our main issue of SSA. How could we introduce single-spin eﬀects in TMDs? The usual way of interpreting partonic distributions and fragmentation functions is that of inclusive cross-sections for the (although unphysical) process A → a + X, where A is a hadron and a a parton (PDF) or vice versa (FF). Let us consider, as an example, the distribution of unpolarized quarks inside a polarized proton, p(S) → q + X, ﬁg. 16. As we said several times by now, the only parity-conserving spin eﬀect can originate from the mixed product S · (p × k⊥ ). Therefore, at diﬀerent stages in the literature, four new single-spin–dependent distribution or fragmentation functions have been introduced. Let us list them here. – The most general distribution function for an unpolarized parton inside a polarized proton (or any hadron in general) can be written as (22)

1 N ˆ⊥) Δ fq/p↑ (x, k⊥ )S · (pˆ × k 2 k⊥ ⊥q ˆ ⊥ ), f (x, k⊥ )S · (pˆ × k = fq/p (x, k⊥ ) − M 1T

fq/p,S (x, k⊥ ) = fq/p (x, k⊥ ) +

⊥q (x, k⊥ ) according to the Amsterdam group notawhere ΔN fq/p↑ (x, k⊥ ) (or f1T tions) is the so-called Sivers distribution function [23], which introduces, in the only possible way, a spin S-k⊥ correlation in PDFs. M is the proton mass.

Fig. 16. – Partonic distribution function of quarks inside a polarized proton.

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– The most general distribution function for a polarized parton inside an unpolarized proton (or any hadron in general) can be written as (23)

1 fq/p (x, k⊥ ) + 2 1 = fq/p (x, k⊥ ) − 2

fq,sq /p (x, k⊥ ) =

1 N ˆ⊥) Δ fq↑ /p (x, k⊥ )sq · (pˆ × k 2 1 k⊥ ⊥q ˆ⊥) h (x, k⊥ )sq · (pˆ × k 2 M 1

where ΔN fq↑ /p (x, k⊥ ) (or h⊥q 1 (x, k⊥ ) according to the Amsterdam group notations) is the so-called Boer-Mulders distribution function [24], which introduces, in the only possible way, a parton spin sq -k⊥ correlation in PDFs. – The most general fragmentation function for an unpolarized hadron generated in the fragmentation of a polarized parton can be written as (24)

1 N Δ Dh/q↑ (z, p⊥ )sq · (pˆq × pˆ⊥ ) 2 p⊥ = Dh/q (z, p⊥ ) + H ⊥q (x, k⊥ )sq · (pˆq × pˆ⊥ ), zMh 1

Dh/q,sq (z, p⊥ ) = Dh/q (z, p⊥ ) +

where ΔN Dh/q↑ (z, p⊥ ) (or H1⊥q (z, p⊥ ) according to the Amsterdam group notations) is the so-called Collins fragmentation function [25], which introduces, in the only possible way, a parton spin sq -p⊥ correlation in FFs. – The most general fragmentation function for a polarized hadron generated in the fragmentation of an unpolarized parton can be written as (25)

1 DΛ/q (z, p⊥ ) + 2 1 = DΛ/q (z, p⊥ ) + 2

DΛ,SΛ /q (z, p⊥ ) =

1 N Δ DΛ↑ /q (z, p⊥ )SΛ · (pˆq × pˆ⊥ ) 2 p⊥ D⊥q (x, k⊥ )SΛ · (pˆq × pˆ⊥ ), zMΛ 1T

⊥q where ΔN DΛ↑ /q (z, p⊥ ) (or D1T (z, p⊥ ) according to the Amsterdam group notations) is the so-called polarizing fragmentation function [15], which introduces, in the only possible way, a hadron spin SΛ -p⊥ correlation in FFs. The choice of a label Λ for the produced hadron indeed reminds that such a fragmentation function might be responsible for the observed Λ polarization.

. 4 3. Sivers and Collins eﬀect in SIDIS . – It is easy to see that the four functions introduced above can induce azimuthal dependences—on diﬀerent combinations of the angles ΦS and Φh —in SIDIS cross-sections. Let us see how by looking ﬁrst at the Sivers function. Inserting eq. (22) in the factorized scheme of eq. (17) one has (26)

dσ ↑,↓ =

q

fq/p↑,↓ (x, k⊥ ) ⊗ dˆ σ (y, k⊥ ) ⊗ Dπ/q (z, p⊥ )

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The transverse spin structure of the nucleon

Fig. 17. – The moments of the Sivers functions as extracted in refs. [20, 27-29]. Notice that R R k2 ⊥(1)q ⊥(1/2)q ⊥q ⊥q (x, k⊥ ) and that f1T (x) = d2 k⊥ kM⊥ f1T (x, k⊥ ). Moreover, f1T (x) = d2 k⊥ 2M⊥2 f1T ⊥q N M f1T = − 2k⊥ Δ fq/p↑ . The uncertainty bands are shown, see ref. [30] for details.

and

(dσ ↑ − dσ ↓ )siv =

(27)

ˆ ⊥ ) ⊗ dˆ ΔN fq/p↑ (x, k⊥ ) S · (pˆ × k σ (y, k⊥ ) ⊗ Dπ/q (z, p⊥ )

q

≡ ΔN fq/p↑ (x, k⊥ ) ⊗ dˆ σ (y, k⊥ ) ⊗ Dπ/q (z, p⊥ ). ˆ ⊥ ) = sin(ϕ − ΦS ) and, upon integration over d2 k⊥ , this will result Notice that S · (pˆ × k in a sin(Φh − ΦS ) ﬁnal dependence. All details can be found in ref. [20]; the actual measured quantity (an azimuthal weighted cross-section diﬀerence) and the corresponding theoretical expression are explicitly given below at O(k⊥ /Q): sin(Φh −ΦS )

(28)

AU T

=

q

≡2

dΦS dΦh [dσ ↑ − dσ ↓ ] sin(Φh − ΦS ) dΦS dΦh [dσ ↑ + dσ ↓ ]

dΦS dΦh d2 k⊥ ΔN fq/p↑ (x, k⊥ ) sin(ϕ − ΦS )

q

dΦS dΦh d2 k⊥ fq (x, k⊥ )

dˆ σ q→q Dh/q (z, p⊥ ) sin(Φh − ΦS ) dQ2

dˆ σ q→q Dh/q (z, p⊥ ) dQ2

,

where p⊥ = PT − zk⊥ . The ﬁrst line of eq. (28) is the experimental quantity, while the second line contains as only unknown the Sivers distribution ΔN fq/p↑ (x, k⊥ ); by parameterizing this unknown function and best ﬁtting the data one obtains information on the Sivers function. This procedure has been performed by several groups [20, 27-29], with minor differences in the theoretical approach, and an overall agreement on the extracted Sivers functions for u and d quarks (sea quark Sivers functions have been neglected in a ﬁrst analysis). Plots and comparison between the diﬀerent extractions can be found in a common publication, ref. [30], and are shown in ﬁg. 17.

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M. Anselmino

Fig. 18. – The Collins mechanism at work in SIDIS processes. The transversely polarized quark, after interaction with the lepton via exchange of a virtual photon, retains an amount of transverse polarization and fragments into the observed ﬁnal hadron. Notice that the direction of Sq is symmetric of Sq with respect to the normal (y) to the leptonic plane and is reduced in magnitude by a depolarization factor, |Sq | = |Sq | 2(1 − y)/[1 + (1 − y)2 ] [36]. The Collins azimuthal angle between Sq and PT is ΦC = ΦS − Φh = π − ΦS − Φh .

Notice that a simple but important model of the SSA originating from the Sivers eﬀect in SIDIS, was discussed in ref. [31], looking at the proton as a simple quark-diquark model. The necessary interference between a leading and a higher-order diagram, similar to that previously discussed in ﬁg. 6, is given by adding a ﬁnal-state interaction (gluon exchange) between the struck quark and the diquark remnant of the proton. This might induce a process dependence in the Sivers functions, which would require data from SSA in Drell-Yan processes in order to be observed, and is considered as one of the most important tests of our understanding of the QCD origin of SSA [32-35]. In a way similar to the Sivers eﬀect also the Collins mechanism can originate SSA in SIDIS. A transversely polarized quark in the transversely polarized proton (the amount of which is given by h1 ) interacts with the lepton and changes (in a known way) its polarization; the ﬁnal polarized quark fragments into the observed hadron, according to eq. (24), which contains the Collins function. This process is represented in ﬁg. 18, and translates into the schematic formula

(29) (dσ ↑ − dσ ↓ )col =

h1q (x, k⊥ ) ⊗ dΔˆ σ (y, k⊥ ) ⊗ ΔN Dπ/q↑ (z, p⊥ ) sq · (pˆq × pˆ⊥ )

q

≡

h1q (x, k⊥ ) ⊗ dΔˆ σ (y, k⊥ ) ⊗ ΔN Dπ/q↑ (z, p⊥ ),

q

where dΔˆ σ ≡ dˆ σ q

↑

→q ↑

− dˆ σ q

↑

→q ↓

.

305

The transverse spin structure of the nucleon

The analogue of eq. (28) is now (details can be found in [38]) h +ΦS ) Asin(Φ ≡2 UT

(30)

=

q

dΦS dΦh [dσ ↑ − dσ ↓ ] sin(Φh + ΦS ) dΦS dΦh [dσ ↑ + dσ ↓ ]

dΦS dΦh d2 k⊥ h1q (x, k⊥ )

q

dΔˆ σ q→q N Δ Dh/q↑ (z, p⊥ ) sin(Φh + ΦS ) dQ2

dˆ σ q→q Dh/q (z, p⊥ ) dΦS dΦh d k⊥ fq (x, k⊥ ) dQ2

.

2

h +ΦS ) Also the Asin(Φ has been observed experimentally by the HERMES CollaboraUT tion [16, 17]. A ﬁt of the data in this case is more problematic, as the above equation contains two unknown functions, the transversity distribution, h1q , and the Collins function ΔN Dh/q↑ . One could assume a model value for h1 and obtain information on the Collins FF, as done in refs. [28, 29]. Recently, the HERMES and COMPASS SIDIS data have been combined with data on correlated azimuthal asymmetries measured in e+ e− annihilation into 2 pions by the Belle Collaboration at KEK, which supply direct information on the Collins functions [37]. The simultaneous ﬁt of the Belle and SIDIS data has allowed, for the ﬁrst time, the extraction of the transversity distributions for u and d quarks [7, 38].

. 4 4. Comments on Sivers distributions. – The Sivers function is, so far, the best known of the four single-spin TMDs of eqs. (22)-(25). It is thus worth making some further comments about its interpretation and physical meaning. Notice that many issues related to TMDs are all still under open debate: in particular their QCD evolution, universality and factorization properties are not yet well understood. We stick here to their intuitive parton model interpretation. – SIDIS HERMES data clearly indicate an active presence of the Sivers (and Collins) eﬀects; in that they reached a true milestone along the path of understanding the inner proton structure. The extracted Sivers functions are diﬀerent from zero in the valence quark regions, and show opposite values for the u and q quarks, somewhat reminiscent of the opposite polarization of u and d quarks in a polarized proton. – The COMPASS Collaboration found so far negligible values for azimuthal asymmetries [39]. This appears to be an unfortunate case, due to the fact that they used a deuteron polarized target. In this case, from eqs. (27) and (29), using isospin invariance, one has (31)

! 4 Du/p↑ + Dd/p↑ , ! ∝ (h1u + h1d ) 4 ΔN Du/p↑ + ΔN Dd/p↑

h −ΦS ) Asin(Φ ∝ ΔN fu/p↑ + ΔN fu/p↑ UT h +ΦS ) Asin(Φ UT

!

and the opposite contributions from u and d quarks, in addition to a kinematical region around smaller x values, induce strong cancelations.

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– The Sivers distribution function (22), even taking into account the ﬁnal state interactions which generate SSAs, has been proved to obey the sum rule [40]

(32)

dx d2 k⊥ k⊥ fa/p,S (x, k⊥ ) = 0

a

in agreement with the parton model interpretation, according to which the above quantity is just the total amount of k⊥ carried by all partons in a polarized proton, and which must sum up to zero. According to this interpretation the quantity

1 N ˆ ≡ dx d k⊥ k⊥ fa/p (x, k⊥ ) + Δ fa/p↑ (x, k⊥ ) S · (pˆ × k⊥ ) 2 π 2 ˆ − cos ΦS y) ˆ = (sin ΦS x ΔN fa/p↑ (x, k⊥ ) dx dk⊥ k⊥ 2

(33)

a k⊥

2

is the total amount of intrinsic motion carried by partons of ﬂavor a inside a ˆ+ proton moving along z and polarized along the transverse direction S = cos ΦS x ˆ Inserting into eq. (33) the Sivers functions extracted in ref. [27], one sin ΦS y. obtains [41] u ˆ − cos ΦS y), ˆ

= +0.14+0.05 k⊥ −0.06 (sin ΦS x d ˆ − cos ΦS y). ˆ

= −0.13+0.03 k⊥ −0.02 (sin ΦS x

Thus, the Burkardt sum rule (32) appears to be saturated by u and d quarks alone. – In terms of the quark-quark correlator of ﬁgs. 1 and 2, the Sivers function turns out to be the matrix element of unpolarized quark operators between proton states with opposite helicities. Such matrix elements also appear in the deﬁnition of the anomalous magnetic moments, so that one expects [42-44] a relation between the anomalous magnetic moments of u and d quarks and the corresponding Sivers functions; this would naturally explain the opposite signs they have. – The observation of a Sivers eﬀect, that is of a coupling between the proton spin and an unpolarized quark motion must be ultimately related to a coupling of the kind S · Lq : thus the Sivers eﬀects points directly towards the existence of some quark orbital angular momentum [45]. . 4 5. Quark-quark correlator with intrinsic motion and SIDISLAND. – Our knowledge about the proton structure is embedded in the PDFs. So far, we encountered 5 of them, the unpolarized, helicity, transversity, Sivers and Boer-Mulders distributions, each of which has a partonic interpretation. However, they are not the whole story. This can

The transverse spin structure of the nucleon

307

be seen by considering the quark-quark correlator, eq. (5), with the addition of intrinsic motion. This allows a richer structure in the Dirac spinor space, which is given by (34)

μ ν ρ σ 1 k⊥ · ST ⊥ ⊥ μνρσ γ n+ k⊥ ST + SL g1L + g1T γ 5 /n+ Φ(x, k⊥ ) = n+ + f1T f1 / 2 M M μ ν k⊥ · ST ⊥ iσμν γ 5 n+ k⊥ 5 μ ν ⊥ h1T + h1T iσμν γ n+ ST + SL h1L + M M μ ν σμν k⊥ n+ + h⊥ . 1 M

The above correlator contains, at leading twist, 8 independent structure functions; let us list them, stressing their partonic meaning and pointing out the alternative notations existing in the literature. – f1 (x, k⊥ ) is the unpolarized unintegrated PDF. Upon integration over d2 k⊥ gives the usual x-dependent PDF. Common alternative notations are q(x) for quarks, g(x) for gluons, fa (x) for a generic parton a. – g1L (x, k⊥ ) is the unintegrated helicity distribution. Upon integration over d2 k⊥ gives the usual x-dependent helicity distribution, eq. (2), also denoted by Δq(x) for quarks, Δg(x) for gluons, Δfa (x) for a generic parton a, sometimes also g1 (x) [46]. – h1T (x, k⊥ ) and h⊥ 1T (x, k⊥ ) are unintegrated transversity distributions. When suitably combined, upon integration over d2 k⊥ give the usual x dependent transversity distribution, eq. (4), also denoted by h1 (x), ΔT q(x) or δ(x). Such a distribution does not exist for (massless) gluons. ⊥ – f1T (x, k⊥ ) is usually denoted as the Sivers function, eq. (22). It vanishes when k⊥ = 0. It is related by some constant factors to the distribution ΔN fq/p↑ (x, k⊥ ) which gives the diﬀerence between number densities of unpolarized partons inside a transversely polarized proton, fq/SN − fq/−SN , eq. (22), where SN is a unit (pseudo)vector orthogonal to the (p k⊥ ) plane. Notice that in the original Sivers paper [23], this distribution diﬀerence was denoted by ΔN Gq/p↑ .

– h⊥ It van1 (x, k⊥ ) is usually denoted as the Boer-Mulders function, eq. (23). ishes when k⊥ = 0. It is related by some constant factors to the distribution ΔN fq↑ /p (x, k⊥ ) which gives the diﬀerence between number densities, inside an unpolarized proton, of transversely polarized quarks, fq,sqN − fq,−sqN , eq. (23), where sqN is a unit (pseudo)vector orthogonal to the (p, k⊥ )-plane. ⊥ – g1T (x, k⊥ ) is related to the distribution of longitudinally polarized partons inside a transversely polarized proton. It vanishes in the collinear limit.

– h⊥ 1L (x, k⊥ ) is related to the distribution of transversely polarized quarks inside a longitudinally polarized proton. It vanishes in the collinear limit.

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Notice that f1 , g1L and h1T are the only distributions which survive in the absence of intrinsic k⊥ . In terms of a p → q + X process, f1 is an unpolarized cross-section, g1 and h1 are related to double-spin eﬀects (longitudinal-longitudinal and transverse⊥ ⊥ transverse), f1T and h⊥ 1 are related to single transverse spin asymmetries, and h1T , ⊥ g1T , h⊥ 1L are, again, double-spin asymmetries with diﬀerent initial-ﬁnal directions. For a comprehensive deﬁnition of all structure functions in terms of amplitudes for the polarized p → q + X process and polarized partonic distributions see ref. [47], where all polarized gluon distributions (including linear polarization) are also discussed. Concerning the fragmentation functions the situation is much simpler, if we consider only the fragmentation of a quark into a spinless particle, like a pion or a kaon. In this case eq. (24) is the full, leading-twist, expression for the FF, which contains two distributions: the unpolarized FF Dπ/q (z, p⊥ ) and the Collins function H1⊥ (z, k⊥ ), related by some factors to the Collins distribution ΔN Dπ/q (z, p⊥ ). The latter gives the diﬀerence between two number densities of pions, resulting from the fragmentation of polarized quarks, fq,sqN − fq,−sqN , where sqN is a unit (pseudo)vector orthogonal to the (pq , p⊥ )-plane. Equation (34) gives the most general expression of the spin and k⊥ -dependent quarkquark correlator at leading-twist (no hard gluon exchanged between the struck quark and the proton remnants, see ﬁg. 1). The 8 independent functions which appear in it, and the 2 independent FF of eq. (24) (corresponding to the upper blob in ﬁg. 3), will also appear in the most general expression of the SIDIS cross-section; we have already seen how this happens with the Sivers, eq. (27), and the Collins, eq. (29), functions. The most general expression of the polarized SIDIS cross-section, with leading-twist PDFs and FFs, and up to O(k⊥ /Q) kinematical corrections, like in eq. (20), can be written as [48, 4, 5, 49]: (35)

1 1 3 cos Φh dσU2 U + λ sin Φh dσLU dσ = dσU0 U + cos 2Φh dσU1 U + Q Q

1 1 4 5 6 7 sin Φh dσU L + λ dσLL + cos Φh dσLL +SL sin 2Φh dσU L + Q Q +ST sin(Φh − ΦS ) dσU8 T + sin(Φh + ΦS ) dσU9 T + sin(3Φh − ΦS ) dσU10T

1 11 12 + sin(2Φh − ΦS ) dσU T + sin ΦS dσU T Q

! 1 13 14 15 +λ cos(Φh − ΦS ) dσLT + cos ΦS dσLT + cos(2Φh − ΦS ) dσLT Q

I (I = 0–15) are particular where λ is the lepton helicity and the various terms dσXY convolutions of the PDFs, FFs and the elementary interaction, with X, Y = U, L, T indicating, respectively, the polarization of the lepton and the proton (U stands for unpolarized, L for longitudinally and T for tranversely polarized). For example, sin(Φh − ΦS ) dσU8 T = (dσ ↑ − dσ ↓ )siv , eq. (27) and sin(Φh + ΦS ) dσU9 T = (dσ ↑ − dσ ↓ )col , eq. (29). Equation (35) contains a wealth of information on the nucleon structure; the single pieces, which give access to particular distribution functions, can be isolated by appropri-

The transverse spin structure of the nucleon

309

ate azimuthal weighting of the cross section, like in eqs. (28) and (30). Dedicated eﬀorts in HERMES, COMPASS and JLab experiments are concentrated on the exploration of the proton and neutron structure, via eq. (35). 5. – SSA in hadronic interactions We would like now to use these new PDFs and FFs in order to understand and explain, in terms of elementary pQCD dynamics coupled to non-perturbative information on the nucleon structure, the SSAs observed in hadronic interactions. The approach we follow here is based on a straightforward generalization of the collinear factorization scheme of eq. (14) and ﬁg. 8, similarly to what has been done in SIDIS, eq. (17). However, while the QCD factorization with TMDs has been formally proven, at least in particular kinematical regions, for SIDIS [19] and, before that, for Drell-Yan processes [50], a formal proof (or deﬁnite disproof) is still lacking for single particle inclusive production, A B → C X. We simply adopt such a factorization as a reasonable, physically very intuitive, working model, assuming for p p → π X: (36) dσ =

# $ fa/p (xa , k⊥a ) ⊗ fb/p (xb , k⊥b ) ⊗ dˆ σ ab→cd (xa , xb , k⊥a , k⊥b ) ⊗ Dπ/c (z, p⊥ ) . a,b,c,d=q,¯ q ,g

Notice that the issue of spin and SSAs in QCD hadronic interactions will be much further discussed in the lectures by W. Vogelsang, with alternative—although somewhat related—approaches, and we refer the readers to his lectures [51]. We also recommend consulting a very useful and comprehensive review paper on azimuthal and SSAs in hard scattering processes which has just appeared [52]. In the scheme of eq. (36) spin eﬀects can easily enter via the PDFs and FFs of eqs. (22)-(25). Thus, one could have SSA for the p↑ p → π X process originating from Sivers, Collins and Boer-Mulders eﬀects, leading to, schematically (37)

dσ − dσ = σ (k⊥ ) ⊗ Dπ/c ΔN fa/p↑ (k⊥ ) ⊗ fb/p ⊗ dˆ ↑

↓

a,b,c

+h1a ⊗ fb/p ⊗ dΔˆ σ (k⊥ ) ⊗ ΔN Dπ/c↑ (p⊥ )

+h1a ⊗ ΔN fb↑ /p (k⊥ ) ⊗ dΔ σ ˆ (k⊥ ) ⊗ Dπ/c .

The ﬁrst line shows the Sivers eﬀect, the second the (chiral-odd) Collins function coupled to the (chiral-odd) transversity distribution and the third line couples the (chiral-odd) Boer-Mulders function to the transversity again. dΔˆ σ and dΔ σ ˆ are double spin asymmetries of the elementary interactions. The SSA generated by the ﬁrst two lines have been studied in the past [53].

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M. Anselmino

In eq. (37) the intrinsic motion has been taken into account in the elmentary interactions and in one of the other terms at a time (Sivers, or Collins or Boer-Mulders function). More recently, the scheme has been fully developed by taking into account the intrinsic motion in all PDFs and FFs, with exact kinematics [47]. The scheme of eq. (36) has been expanded with the inclusion of spin, so that, for the general A(SA )+B(SB ) → π+X process, one has

a/A,S b/B,S (38) dσ (A,SA )+(B,SB )→C+X = ρλ ,λ A fa/A,SA (xa , k⊥a ) ⊗ ρλ ,λ B fb/B,SB (xb , k⊥b ) a

a

b

b

C C ˆ ∗ab→cd ˆ λab→cd M ⊗M λc ,λ ;λa ,λ ⊗ Dλ ,λ (z, p⊥ ), c ,λd ;λa ,λb

λ ,λ

d

b

c

c

ˆ, which contains, through the helicity density matrices ρ and the helicity amplitudes M all the information about the parton polarization states and the elementary polarized interactions. The terms ρ f , and similarly the polarized fragmentation process described by the D with helicity indices, are related to the 8 PDFs and 2 FFs of eqs. (34) and (24). All details, with all appropriate exact kinematical factors, can be found in ref. [47]. We stress here one important consequence of this approach. The elementary amplitudes are computed with exact kinematics, taking into account all k⊥ s and p⊥ . Thus, the elementary scatterings, in this non-collinear conﬁguration, are not, in general, planar processes. As a result, all phases attached to the spinors entering the Feynman diagram computations, which would cancel in the planar case being the same, do not cancel out. Integration over d2 k⊥ , and then over phases, strongly suppresses many of the contributions [54]. It turns out that the main contribution to SSAs in p↑ p → π X processes is generated by the Sivers eﬀect, according to the expression (39)

dσ ↑ − dσ ↓ =

# N $ Δ fa/p↑ (xa , k⊥a ) ⊗ fb/p (xb , k⊥b ) ⊗ dˆ σ ab→cd (k⊥a , k⊥b ) ⊗ Dπ/c (z, p⊥ ) .

a,b,c,d=q,¯ q ,g

The above expression was used, parameterizing the unknown Sivers functions, to ﬁt the E704 data and, with the extracted Sivers functions, to compute SSAs for the STAR experiment at RHIC; the result [55], in remarkable agreement with data, is shown in ﬁg. 19. Thus, there seems to be an indication that the Sivers eﬀect, considered as an intrinsic property of a polarized proton, might be at work in building observable SSAs, both in SIDIS and pp interactions. However, there is no unanimous consensus on this. While the mechanism itself is accepted as a source of single-spin eﬀects, some fundamental issues still remain unclear: are the Sivers functions in SIDIS and hadronic interaction the same (universality)? how do they evolve with Q2 ? is factorization indeed a good model supported by QCD? A lot of theoretical and experimental activity on such problems is ongoing and some answers, together with new problems, might arrive soon. For example, in a very recent paper it was shown, with a purely phenomenological analysis, that the same Sivers functions extracted from SIDIS experiments can, at least qualitatively, explain the RHIC pp data on AN [57].

311

The transverse spin structure of the nucleon

Fig. 19. – The left plot is a ﬁt of the E704 data [12] with Sivers eﬀect. The right plot is the computed value of AN using the Sivers functions extracted from the ﬁt [55], compared with STAR data [56].

. 5 1. SSA and transversity in Drell-Yan processes. – Let us now consider Drell-Yan processes, A B → + − X, ﬁg. 20. Similarly to SIDIS, there is a general consensus that a TMD factorization holds for the corresponding cross-section: (40)

dσ DY =

fq/p (x1 , k⊥1 ; Q2 ) ⊗ fq¯/p (x2 , k⊥2 ; Q2 ) ⊗ dˆ σ qq¯→

+ −

.

q

As we have seen, the most general expression of the spin and k⊥ -dependent PDFs contains, at leading twist, 8 independent TMDs; their convolutions, can, in principle, be accessed in DY processes, which would be a precious source of information on the nucleon structure. A most general expression like eq. (35) could be written and exploited for DY. However, these processes are relatively rare and diﬃcult to study; a dedicated machine with, possibly, polarized antiprotons would be an ideal support [58]. As an example of particular azimuthal dependences that single out speciﬁc TMDs we can start from the unpolarized DY cross-section; its most general form can be written

Fig. 20. – Kinematics of Drell-Yan processes.

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M. Anselmino

Fig. 21. – The Collins-Soper reference frame for Drell-Yan processes.

as: (41)

1 dσ 3 1 ν = 1 + λ cos2 θ + μ sin2 θ cos ϕ + sin2 θ cos 2ϕ , σ dΩ 4π λ + 3 2

where θ and ϕ are angles deﬁned in the so-called Collins-Soper frame, deﬁned in ﬁg. 21. Notice that in the collinear parton model with massless quarks one has λ = 1, μ = ν = 0. However, large values of ν have been experimentally observed (see, for example ref. [52] and references therein). According to the partonic expression with TMDs of eq. (40), a cos 2ϕ dependence could naturally arise from the convolution of two BoerMulders functions [24, 59]: (42)

dσ cos 2ϕ ∝

⊥ h⊥ σ ↑↑ − dˆ σ ↑↓ ], 1q (x1 , k⊥1 ) ⊗ h1¯ q (x2 , k⊥2 ) ⊗ [dˆ

q

σ ↑↓ ], where the arrows refer to the as the double elementary spin asymmetry [dˆ σ ↑↑ − dˆ transverse q and q¯ polarizations, has indeed a cos 2ϕ azimuthal dependence. Similarly, by carefully selecting the azimuthal dependences, one can single out other TMDs. For example a SSA in the diﬀerential cross-section d4 σ/d4 q, where q μ is the virtual photon (or lepton pair) four-momentum, can isolate the eﬀect of the Sivers mechanism: (43)

(dσ ↑ − dσ ↑ )DY,siv ∝

ΔN σ, q/p↑ (x1 , k⊥1 ) ⊗ fq¯/p (x2 , k⊥2 ) ⊗ dˆ

q

which, inserting the SIDIS Sivers functions, turns out to be sizeable at RHIC or PAX [60]. Such a measurement is of crucial importance, as it would allow to test the “QCD predicted” change in sign between the SIDIS and DY Sivers functions [33]. Finally, we point out that DY processes with transversely polarized proton and antiproton would oﬀer the ideal channel to access the transversity distributions of quarks in proton (antiquarks in antiprotons are the same). This was pointed out in refs. [61,58].

The transverse spin structure of the nucleon

313

Infact, the double transverse spin asymmetry AT T ≡

(44)

dσ ↑↑ − dσ ↑↓ dσ ↑↑ + dσ ↑↓

has, in the collinear parton model, without any need of TMDs, the simple expression % (45)

AT T ≡ a ˆT T

q

e2q [h1q (x1 ) h1q (x2 ) + h1¯q (x1 ) h1¯q (x2 )] % 2 ¯(x1 ) q¯(x2 )] q eq [q(x1 ) q(x2 ) + q

where all distributions refer to a proton and (46)

a ˆT T ≡

σ ↑↓ dˆ σ ↑↑ − dˆ sin2 θ cos 2ϕ. = dˆ σ ↑↑ + dˆ σ ↑↓ 1 + cos2 θ

By appropriately choosing the energy range, one could select to study AT T in a region where the cross-section is large and one explores the valence quark x values, where the transversity distributions are expected (and have been ﬁrst measured to be [38]) large. This ideal domain would be oﬀered by the p¯ p asymmetric collider option at PAX [58]. We also notice that QCD corrections, which are known to be very large for Drell-Yan cross sections, almost do not aﬀect the value of AT T , being such corrections essentially a multiplicative K-factor, spin independent, which cancels out in the expression of the spin asymmetry [62, 63]. We conclude by noticing that other approaches to transversity, rather than the coupling to the Collins function and the (hopefully) future DY processes with polarized antiprotons, have been proposed in the literature. Also, we have not presented important recent progress in the description of the transverse space structure of the proton. Some of these points are discussed in the lectures by D. Hasch. 6. – Conclusions We have discussed the issue of the transverse structure of the proton. It concerns not only transverse spin, but also the parton transverse motion, and its coupling with spin. It is an interesting and blooming ﬁeld, which oﬀers new insights into the inner composition of the nucleons. The transverse degrees of freedom remain unchanged under Lorentz boosts and by looking at them, one can access real intrinsic properties, like the possible orbital motion of quarks. One has to work in the framework of QCD, coupling perturbative and nonperturbative features; the boundary between the two domains is not always so clear and well cut, and one has to resort to models which combine the two features into an operative formalisms. The language we have used here is that of the transverse momentum dependent partonic distributions and fragmentation functions (TMDs). It is not only a simple generalization of the usual collinear formalism, where the transverse degrees of freedom have been integrated over, but it reﬂects new genuine properties of the partons.

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Several experimental cases in which TMDs play a crucial role have been discussed, both in the unpolarized case, the Cahn eﬀect, and the polarized one, the SSAs. The latter have been known for a long time and keep being discovered by new experiments. The special feature about them is that they are absent in elementary QCD and QED dynamics and must then originate from non-perturbative QCD properties. The status of TMDs, their extraction from data and phenomenological analysis is rather well established in SIDIS experiments, where a generally accepted formalism based on a factorization theorem is used. Many dedicated experiments are collecting or plan to collect information on TMDs, thus obtaining basic information on the nucleon structure and on the hadronization properties of quarks. A similar situation holds, in principle, for Drell-Yan processes, which are, however, much more diﬃcult to measure and would require a dedicated machine. The situation is more complicated for inclusive hadron production in hadronic interactions, for which good data exist and more are being collected, but where the theoretical framework is less established; the factorization scheme with TMDs is still under discussion, although it appears to be phenomenologically successful. Problems like the universality of TMDs and their QCD evolution and are still uncertain. Many progresses have been recently achieved and there seems to be a unifying interpretation of diﬀerent approaches [64]. But a lot of work remains to be done. ∗ ∗ ∗ I would like to thank the Societ` a Italiana di Fisica, the leturers, the students and the people organizing life and work at the Villa for creating a unique atmosphere.

REFERENCES [1] See, for example, Anselmino M., Efremov A. and Leader E., Phys. Rep., 261 (1995) 1. [2] Leader E., this volume, p. 263. [3] Barone V., Drago A. and Ratcliffe P. G., Phys. Rep., 359 (2002) 1; Barone V. and Ratcliffe P. G., Transverse Spin Physics (World Scientiﬁc) 2003. [4] Mulders P. J. and Tangerman R. D., Nucl. Phys. B, 461 (1996) 197; 484 (1997) 538(Erratum). [5] Boer D. and Mulders P. J., Phys. Rev. D, 57 (1998) 5780. [6] Boer D., Mulders P. J. and Pijlman F., Nucl. Phys. B, 667 (2003) 201. [7] Boglione M., this volume, p. 431. [8] Bourrelly C., Leader E. and Soffer J., Phys. Rep., 59 (1980) 95. [9] Dharmaratna W. G. D. and Goldstein G., Phys. Rev. D, 53 (1996) 1073. [10] Kane L. G., Pumplin J. and Repko W., Phys. Rev. Lett., 41 (1978) 1689. [11] Krueger K. et al., Phys. Lett. B, 459 (1999) 412. [12] E704 Collaboration (Adams D. L. et al.), Phys. Lett. B, 264 (1991) 462. [13] STAR Collaboration (Nogach L.), AIP Conf. Proc., 915 (2007) 543; hep-ex/0612030. [14] STAR Collaboration (Simon F.), AIP Conf. Proc., 870 (2006) 428; hep-ex/0608050.

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[15] For a review of data see, e.g., Heller K., Proceedings of Spin 96, edited by de Jager C. W., Ketel T. J. and Mulders P. J. (World Scientiﬁc) 1997; or Panagiotou A. D., Int. J. Mod. Phys. A, 5 (1990) 1197. [16] Hasch D., these proceedings. [17] HERMES Collaboration (Diefenthaler M.), AIP Conf. Proc., 792 (2005) 933; hepex/0507013. [18] Bacchetta A., D’Alesio U., Diehl M. and Miller C. A., Phys. Rev. D, 70 (2004) 117504. [19] Ji X., Ma J.-P. and Yuan F., Phys. Rev. D, 71 (2005) 034005; Phys. Lett. B, 597 (2004) 299. [20] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. and Prokudin A., Phys. Rev. D, 71 (2005) 074006. [21] Cahn R. N., Phys. Lett. B, 78 (1978) 269; 40 (1989) 3107. ¨rk C., Eur. Phys. J. A, 31 (2007) [22] Anselmino M., Boglione M., Prokudin A. and Tu 373. [23] Sivers D. W., Phys. Rev. D, 41 (1990) 83; 43 (1993) 261. [24] Boer D., Phys. Rev. D, 60 (1999) 014012. [25] Collins J. C., Nucl. Phys. B, 396 (1993) 161. [26] Anselmino M., Boer D., D’Alesio U. and Murgia F., Phys. Rev. D, 63 (2001) 054029. [27] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. and Prokudin A., Phys. Rev. D, 72 (2005) 094007. [28] Vogelsang W. and Yuan F., Phys. Rev. D, 72 (2005) 054028. [29] Collins J. C., Efremov A. V., Goeke K., Menzel S., Metz A. and Schweitzer P., Phys. Rev. D, 73 (2006) 014021. [30] Anselmino M. et al., Transversity 2005 (World Scientiﬁc, Singapore) 2006; hepph/0511017. [31] Brodsky S. J., Hwang D. S. and Schmidt I., Phys. Lett. B, 530 (2002) 99. [32] Brodsky S. J., Hwang D. S. and Schmidt I., Nucl. Phys. B, 642 (2002) 344. [33] Collins J. C., Phys. Lett. B, 536 (2002) 43. [34] Belitsky A. V., Ji X. D. and Yuan F., Nucl. Phys. B, 656 (2003) 165; Ji X. D. and Yuan F., Phys. Lett. B, 543 (2002) 66. [35] Boer D., Mulders P. J. and Pijlman F., Nucl. Phys. B, 667 (2003) 201. [36] Anselmino M. and Murgia F., Phys. Lett. B, 483 (2000) 74. [37] Efremov A. V., Goeke K. and Schweitzer P., Phys. Rev. D, 73 (2006) 094025. [38] Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F., Prokudin A. ¨rk C., Phys. Rev. D, 75 (2007) 054032. and Tu [39] COMPASS Collaboration (Alexakhin V. Y. et al.), Phys. Rev. Lett., 94 (2005) 202002; COMPASS Collaboration (Ageev E. S. et al.), Nucl. Phys. B, 765 (2007) 31. [40] Burkardt M., Phys. Rev. D, 69 (2004) 091501. [41] D’Alesio U., private communication. [42] Brodsky S. J. and Gardner S., Phys. Lett. B, 643 (2006) 22. [43] Burkardt M. and Schnell G., Phys. Rev. D, 74 (2006) 013002. [44] Lu V. and Schmidt I., Phys. Rev. D, 75 (2007) 073008. [45] Sivers D., Phys. Rev. D, 74 (2006) 094008; e-Print: arXiv:0704.1791 [hep-ph]. [46] Jaffe M. and Ji X. D., Phys. Rev. Lett., 71 (1993) 2547. [47] Anselmino M., Boglione M., D’Alesio U., Leader E., Melis S. and Murgia F., Phys. Rev. D, 73 (2006) 014020. [48] Kotzinian A., Nucl. Phys. B, 441 (1995) 234.

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[49] Bacchetta A., Mulders P. J. and Pijlman F., Phys. Lett. B, 595 (2004) 309; Bacchetta A., Diehl M., Goeke K., Metz A., Mulders P. J. and Schlegel P., JHEP, 0702 (2007) 093. [50] Collins J. C. and Soper D. E., Phys. Rev. D, 16 (1977) 2219. [51] Vogelsang W., this volume, p. 349. [52] D’Alesio U. and Murgia F., e-Print: arXiv:0712.4328 [hep-ph]. [53] Anselmino M., Boglione M. and Murgia F., Phys. Lett. B, 362 (1995) 164; Phys. Rev. D, 60 (1999) 054027; Anselmino M. and Murgia F., Phys. Lett. B, 442 (1998) 470. [54] Anselmino M., Boglione M., D’Alesio U., Leader E. and Murgia F., Phys. Rev. D, 71 (2005) 014002. [55] D’Alesio U. and Murgia F., Phys. Rev. D, 70 (2004) 074009. [56] STAR Collaboration (Adams J. et al.), Phys. Rev. Lett., 92 (2004) 171801. [57] Boglione E., D’Alesio U. and Murgia F., e-Print: arXiv:0712.4240 [hep-ph]. [58] PAX Collaboration (Barone V. et al.), e-Print: hep-ex/0505054. [59] Gamberg L. P. and Goldstein G. R., Phys. Lett. B, 650 (2007) 362. [60] Anselmino M., D’Alesio U. and Murgia F., Phys. Rev. D, 67 (2003) 074010. [61] Anselmino M., Barone V., Drago A. and Nikolaev N. N., Phys. Lett. B, 594 (2004) 97. [62] Shimizu H., Sterman G., Vogelsang W. and Yokoya H., Phys. Rev. D, 71 (2005) 114007. ´ C. and Ratcliffe P. G., Phys. Lett. [63] Guzzi M., Barone V., Cafarella A., Corian o B, 639 (2006) 483. [64] Ji X., Qiu J. W., Vogelsang W. and Yuan F., Phys. Rev. Lett., 97 (2006) 082002; Phys. Rev. D, 73 (2006) 094017; Phys. Lett. B, 638 (2006) 178.

Transverse spin phenomena in DIS—Experiments D. Hasch INFN, Laboratori Nazionali di Frascati - Via E. Fermi 40, I-00044 Frascati, Italy

Summary. — This lecture reviews the strategies for measuring transverse spin phenomena in deep-inelastic scattering. Understanding such transverse polarisation phenomena in hadronic physics is a long-standing and intriguing problem. Despite of a wealth of experimental data on transverse spin asymmetries there is no measurement yet of the quark tranversity distribution which, together with the unpolarised and the helicity distribution, completes the three basic distributions to be measured in order to obtain a full description of the quark structure of the nucleon at leading twist. Modern developments in hadron physics emphasize the role of correlations of transverse momentum of partons and spin. Such spin-orbit correlations are described by a new class of transverse-momentum–dependent distribution and fragmentation functions (TMDs), which generalise the standard parton distributions. The information on spin-orbit correlations together with independent measurements related to the intrinsic motion of quarks will be the key to construct a complete picture of the internal structure of the nucleon going beyond the collinear approximation. Experiments that aim at pinning down various TMDs are currently running at Cern (Compass Collaboration), Desy (Hermes Collaboration), JLab, Kek and Rhic. Here, we will concentrate on the studies of transverse spin phenomena at the Hermes and Compass experiments in semi-inclusive deep-inelastic scattering on transversely polarised nucleon targets. After an overview of the analysis techniques we will present the ﬁrst exciting results and give an attempt to interpret them.

1. – Preface This lecture is part of the CLXVII Course of the Enrico Fermi School on “Strangeness and Spin in Fundamental Physics”, Varenna, June 2007. Detailed presentations of the theory of the description of hadrons in QCD, in particular of deep-inelastic scattering c Societ` a Italiana di Fisica

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and the new concept of transverse-momentum–dependent distribution and fragmentation functions (TMDs) is given in the lectures of M. Anselmino, R. Jaﬀe, E. Leader and W. Vogelsang. Here we will give a brief introduction only as far as it is needed for an understanding of the analysis of the data, the obtained results and their discussion. A global analysis of the data presented here with the aim to extract the transversity distributions and the Collins fragmentation function is given in E. Boglione’s lecture. There are many diﬀerent processes that can be used to access transversity and TMDs. All these processes involve at least two hadrons of which at least one is transversely polarized. The processes can be divided into two groups: hadron-hadron scattering and lepton-hadron scattering. The ﬁrst group is discussed in the lecture of N. Saito. We will concentrate on the second group and on measuring transverse spin phenomena in lepton-hadron scattering oﬀ transversely polarised nucleons at the Hermes and Compass experiments. The spin physics program and the prospects for accessing TMDs at JLab are discussed in the lecture by H. Avakian. 2. – A brief introduction to transversity and its friends A very successful tool to gain information about the inner structure of hadrons is Deep-Inelastic Scattering (DIS) of leptons, i.e. the absorption by a quark of a spacelike virtual photon with large 4-momentum Q2 (1 ). Inclusive deep-inelastic scattering was the ﬁrst process in which pointlike partons were identiﬁed inside hadrons. The physics is most transparent in the frame in which the nucleon target moves contrary to the photon with “inﬁnite” momentum. In this frame the transverse motion of the partons is “frozen” during the interaction time, while their transverse momenta are unchanged; all particles move collinear. The structure of hadrons in terms of quark and gluon degrees of freedom is parametrised by parton distribution functions. While the x-dependence of these functions has to be measured in experiments, the Q2 -dependence can be computed from pertubative QCD (for a deﬁnition of the kinematic variables see table II). . 2 1. Parton distribution functions. – The hadronic tensor appearing in the DIS crosssection contains the quark-quark correlation matrix describing the conﬁnement of the quarks inside the nucleon. When averaging over the intrinsic transverse momentum pT , this correlation matrix contains exactly three terms: a vector, axial-vector and tensor component, which correspond to three fundamental quark distributions having a probabilistic interpretation. Two of these have been experimentally explored in some detail—the unpolarised density q(x) representing the probability of ﬁnding a quark with → → fraction x of the nucleon momentum, and the helicity density Δq(x) ≡ q ⇒ (x) − q ⇐ (x) reﬂecting the probability of ﬁnding the helicity of the quark to be the same as that of the target nucleon. Viewed in the same helicity basis, the third distribution known as (1 ) A detailed description of the DIS process, the concept of factorisation, deﬁnition of twist, etc. can be found in [1, 2].

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Fig. 1. – Leading twist transverse-momentum–dependent quark Distribution Functions (DFs). All DFs depend on x and pT . For clarity the index q of the quark ﬂavour is omitted unless it contributes to the name of the DF. Nucleon and quark are represented by light and dark grey circles, respectively. Their spin orientations with respect to the incident virtual photon which is thought of entering the ﬁgure from the left side, are indicated by arrows. Chiral-odd DFs are related to scattering amplitudes involving a helicity ﬂip. T-odd functions are odd under naive time reversal, which is time reversal without interchange of initial and ﬁnal state.

transversity [3-6], δq or alternatively hq1 or ΔT q, is related to a forward-scattering amplitude involving a helicity ﬂip of both quark and target nucleon (N ⇒ q ← → N ⇐ q → ) and has no probabilistic interpretation in this basis. However, it retains a probabilistic interpretation in a basis of transverse spin eigenstates: δq = q ↑⇑ − q ↑⇓ . The transversity and helicity densities diﬀer because quarks inside the nucleon move relativistically, hence boosts and rotations do not commute. They are therefore independent quantities which probe diﬀerent QCD properties. In a more general framework where the parton distributions are kept diﬀerential in transverse momentum, eight diﬀerent pT -dependent quark distributions [7] appear at leading twist(2 ). In ﬁg. 1 they are classiﬁed according to their chirality and their behaviour under time reversal. Their probabilistic interpretations are illustrated too. Chiral-odd distribution functions are related to scattering amplitudes involving a helicity ﬂip. Time reversal odd (T-odd) functions are odd under naive time reversal, which is time reversal without interchange of initial and ﬁnal state. Time reversal changes the nucleon state |P, S into | − P, −S while naive time reversal just interchanges the initial state to |P, −S . Therefore these functions are not completely T-odd and can in general be nonzero despite of the T-invariance of the strong interaction. The most prominent transverse-momentum–dependent distributions are the Sivers (2 ) Leading twist (twist-2) is deﬁned as a term of leading order in a 1/Q expansion following [1].

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D1q→h

H1⊥q→h

Fig. 2. – Leading twist transverse-momentum–dependent quark Fragmentation Functions (FF). Both FFs depend on z and kT . The struck quark (produced hadron) is indicated as a dark (light) grey circle.

⊥ f1T and the Boer-Mulders h⊥ 1 functions. They can be related to the interference of wave functions for diﬀerent orbital momentum states. As such they parametrise the correlations between the transverse momentum of quarks and the spin of a transversely polarised hadron or the transverse spin of the quark, respectively. The information on these spin-orbit correlations will be a key to construct a complete picture of the internal structure of hadrons beyond the collinear approximation. Integrating the distribution functions over the transverse momentum of the quark, most of the functions vanish except of the aforementioned three basic ones q(x), Δq(x) and the transversity δq(x). Unlike the other two basic distributions, transversity cannot be measured in inclusive DIS due to its chiral-odd nature, but only in a process in which it combines with another chiral-odd quantity. The most direct approach is to measure double transverse-spin asymmetries in polarised Drell-Yan processes which couples two transversity distributions. This approach is however experimentally not yet feasible (the Pax experiment at the future Fair facility at Gsi plans to measure double-spin asymmetries in Drell-Yan processes from polarised proton antiproton scattering.). Another possibility is the semi-inclusive DIS process where fragmentation functions enter the cross-section in conjunction with the distribution functions.

. 2 2. Fragmentation functions. – In a semi-inclusive DIS measurement at least one produced hadron h is detected in addition to the scattered lepton (see table II for the relevant kinematic variables). The hadronic tensor contains now an additional quarkquark correlation function describing the way the struck quark evolves into a hadronic ﬁnal state of which the hadron h is detected. This process is called hadronisation or fragmentation of the quark. Like distribution functions, also fragmentation functions have a probabilistic interpretation at leading twist. The unpolarised fragmentation function D1 (z, kT ) is the probability distribution that a struck quark of ﬂavour q with transverse momentum kT fragments into a certain hadron of type h with energy fraction z. An example for a spin-dependent fragmentation function is the Collins function ⊥ H1 (z, kT ) which represents the diﬀerence of the probability densities for quarks with transverse spin state ↑ and ↓ fragmenting into a hadron h as illustrated in ﬁg. 2. The Collins function [8] describes a correlation of the transverse polarisation of the struck quark and the transverse momentum of the produced hadron orthogonal to the virtual photon direction thereby inﬂuencing the distribution in the azimuthal angle φ (for the

321

Transverse spin phenomena in DIS—Experiments

y

x k

k

⊥ S q

φS

Ph⊥

z

Ph

φ

Fig. 3. – The deﬁnitions of the azimuthal angles of the hadron production plane (in grey) and ⊥ of the target spin, relative to the lepton scattering plane the axis of the relevant component S h ⊥ ⊥ h × k· q ×P × k· q ×S q × k·P q × k·S −1 q (in white). Explicitly, φ = |q×k·P | cos−1 |qq× | and φS = | | cos | , where k|| q ×P q × k·S | q × k|| q ×S h

h

0 < cos−1 < π. The deﬁnitions follow the Trento convention [16].

⊥

⊥

deﬁnition of the angle see ﬁg. 3). The Collins fragmentation function is chiral-odd and also T-odd. It appears naturally in combination with the transversity distribution, hence oﬀering access to transversity in semi-inclusive DIS processes. This is the way for gaining information on transversity we will follow in this lecture. Diﬀerent phenomenological models have been developed to describe the unpolarised fragmentation process. A very successful model for the description of experimental data is the Lund string fragmentation model [9]. In this model, the colour ﬁeld connecting the initial quarks is assumed to possess a constant ﬁeld energy density. This results in a potential which increases linearly with the distance between the quarks. After one of the quarks is struck and moves away, the energy stored in the colour string rises. As soon as it exceeds the rest mass of a quark antiquark pair the string may break up and create such a pair. The partners of this pair are then connected to the initial quarks by two new strings. They continue to break independently until a string-connected quark-antiquark pair is close to the mass shell of a colour singlet hadron. The number of independent fragmentation functions for quarks of ﬂavour q fragmenting into hadrons of type h, decreases signiﬁcantly when charge conjugation and isospin symmetry is applied, which is valid for a strong interaction process like fragmentation. In case of three diﬀerent quark ﬂavours which hadronise into pions, only three independent unpolarised fragmentation functions remain: (1)

+

−

¯

+

−

−

+

¯

−

+

D1,fav (z) = D1u→π (z) = D1u¯→π (z) = D1d→π (z) = D1d→π (z), D1,dis (z) = D1u→π (z) = D1u¯→π (z) = D1d→π (z) = D1d→π (z), +

−

+

−

D1,s (z) = D1s→π (z) = D1s¯→π (z) = D1s¯→π (z) = D1s→π (z). They are called favoured, disfavoured and strange fragmentation function. The fragmentation function for neutral pions is obtained by averaging that of charged pions for up and down quarks and is the same for strange quarks. Applying the same symmetry

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Table I. – Accessing transversity and TMDs by measuring single-spin asymmetries (SSA) or polarisation transfer in semi-inclusive DIS. The observable of interest appears as a convolution or a product of a quark distribution function with a Fragmentation Function (FF). Experiments given in brackets are planning such measurements in near future. Reaction ↑

Observable

Experiments

lN →lhX

SSA related to a convolution of transversity Hermes [11], Compass [12] and Collins–FF / Sivers–fct. and upolarised FF (Hall-A [13])

l N↑ → l h h X

SSA related to the product of transversity and Di-hadron FF

polarisation transfer related to the product of l N ↑ → l Λ↑ X l N ↑ → l hJ=1 X transversity and a spin-dependent FF

Hermes [14], Compass [15] (Compass)

constraints to the Collins function yields again three independent functions: the favoured ⊥ ⊥ ⊥ H1,fav , the disfavoured H1,dis , and the strange H1,s Collins function. . 2 3. Universality. – An important part of the concept of factorisation is the universality of distribution and fragmentation functions, i.e. they are the same in all processes for which factorisation holds. Hence, parton distribution functions measured in DIS can be used to calculate hadron-hadron scattering processes and fragmentation functions measured in e+ e− scattering processes are applied to describe semi-inclusive DIS. Generally, there are very few processes for which a full factorisation proof in all orders of pertubative QCD has been obtained, as, e.g., Drell-Yan processes, collinear DIS and deeply virtual compton scattering. Often one ﬁnds “good arguments” that factorisation should hold. Universality has been demonstrated for all six T-even distribution functions in ﬁg. 1. A very exciting ﬁnding is the extra sign for the T-odd Sivers and Boer-Mulders functions when viewing it in diﬀerent processes like DIS and Drell-Yan [10]: (2)

⊥ ⊥ f1T,DIS = −f1T,DY ,

⊥ h⊥ 1,DIS = −h1,DY .

The experimental veriﬁcation of this sign change of the T-odd functions when being measured in semi-inclusive DIS or in Drell-Yan processes constitutes a crucial test of the current understanding of single-spin asymmetries in terms of pertubative QCD. The currently most accessible way to obtain information about transversity and TMDs is indeed to measure spin and azimuthal asymmetries, and in particular single-spin asymmetries in semi-inclusive DIS. Table I presents an overview of DIS reactions that allow to access transversity and the experiments that measure or aim to measure those reactions. In this lecture we will describe in detail measurements of the ﬁrst two reactions and their potential for gaining information on transversity and transverse-momentum–dependent distribution functions.

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Table II. – Kinematic variables in deep-inelastic scattering. k = (E, k), k = (E , k )

4-momenta of initial and ﬁnal state leptons

θ, φ

polar and azimuthal angle of the scattered lepton

θγ ∗ h

angle between the direction of the virtual photon and the hadron

lab

P = (M, 0)

4-momentum of the initial target nucleon

q=k−k

4-momentum transfer to the target 2 lab

s = (P + k) ≈ M 2 + 2M E

squared centre-of-mass energy

θ 2 lab W 2 = (P + q)2 = M 2 + 2M ν − Q2 lab

Q2 = −q2 ≈ 4EE sin2

P · q lab = E − E M Q2 lab Q2 x= = 2P · q 2M ν P · q lab ν = y= P·k E ν=

squared invariant mass of the virtual photon squared mass of the ﬁnal state energy transfer to the target Bjorken scaling variable fractional energy transfer to the target

) Ph = (Eh , p

4-momentum of a hadron in the ﬁnal state

Ph⊥

component of Ph perpendicular to the virtual photon direction

z=

P · Ph lab Eh = P·q ν

longitudinal momentum fraction of the hadron

3. – How to measure tranversity and its friends in DIS? In the deep-inelastic scattering process a lepton scatters oﬀ a nucleon via the exchange of a virtual boson; breaking up the nucleon which forms a ﬁnal hadronic state X. The √ maximum possible momentum transfer is determined by the centre-of-mass energy s which is about 7 GeV at the Hermes and 18 GeV at the Compass experiment. In inclusive DIS only the scattered lepton is detected whereas additional hadrons of the ﬁnal state X are detected in semi-inclusive DIS processes. The relevant kinematic variables of the process are summarised in table II. . 3 1. The generalised semi-inclusive DIS cross-section. – The generalised description of the semi-inclusive DIS cross-section also includes the transverse momentum of the hadrons Ph⊥ which in combination with the azimuthal angles φ and φS deﬁned in ﬁg. 3 adds three more degrees of freedom resulting in a 6-fold diﬀerential cross-section (3)

d6 σ =: dσ. dx dy dz dφS dφ dPh⊥

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It is convenient to split the cross-section into unpolarised and polarised terms: (4)

dσ = dσUU + dσLU + dσUL + dσLL + dσUT + dσLT ,

where we use two subscripts to indicate beam and target polarisation, respectively: U means unpolarised, L longitudinally polarised, and T transversely polarised. Here and in the following, the diﬀerential kinematic variables are omitted for clarity. The unpolarised and polarised terms of the semi-inclusive DIS cross-section in eq. (4) with their diﬀerent azimuthal modulations are constructed as (5)

dσbeam target =

2α2 · factor · modulation · e2q I[W · DF · F F ]. 2 sxy q,¯ q

Our object of interest, the product of the distribution and fragmentation function [DF · F F ] appears in a convolution integral(3 ) multiplied by a weight W dependent on Ph⊥ , pT , kT , x and z. The integral cannot therefore be factorised and any extraction of the distribution function of interest will depend on a model for solving the integral (see the lecture by M. Anselmino and E. Boglione for details). This model dependence has to be kept in mind when comparing the extracted amplitudes with predictions. In the following schematic equation we omit the kinematic factors (which generally are diﬀerent for the diﬀerent terms), the summation and weights in order to just illustrate the relations between speciﬁc modulations of the polarised or unpolarised cross-section and certain combinations of distribution and fragmentation functions at leading twist (the full expressions can be found in [7] involving all leading twist functions) (6)

⊥ ⊥ ⊥ dσ ∝ [qD1 ] + cos 2φ h⊥ 1 H1 + |SL | sin 2φ h1L H1 ! ⊥ ⊥ + |ST | sin (φ + φS ) δqH1⊥ + sin (φ − φS ) f1T D1 + sin (3φ − φS ) h⊥ 1T H1 + λ|SL | ([ΔqD1 ] + cos (φ − φS ) [g1T D1 ]) .

Here, λ is the beam helicity and SL and ST are the longitudinal and transverse target spin components with respect to the virtual photon. The ﬁrst two terms are therefore σUU terms, SL and ST correspond to the single polarised σUL and σUT terms, and λSL to the double-polarised σLL term. In eq. (6) we ﬁnd all leading twist functions illustrated in ﬁg. 1. In the unpolarised part we ﬁnd back the momentum distribution q(x) and in the double-polarised part the helicity distribution Δq(x). Both quark distributions can be measured more easily in inclusive DIS without involving the unpolarised fragmentation function. All other functions are accessible only in semi-inclusive DIS. Transversity and the Sivers function appear as sin(φ + φS ) and sin(φ − φS ) modulations of the σUT crosssection term, respectively. It becomes immediately clear how to measure them: weighting the cross-section with the respective modulations sin(φ ± φS ) will project out the terms (3 ) I[W · DF · F F ] ≡

R

d2 p T d2kT δ( pT −

h⊥ P z

− kT )W · DF · F F .

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Transverse spin phenomena in DIS—Experiments

Table III. – Characteristics of the Hermes and Compass experiments and requirements on kinematic variables relevant for the presented results. Compass has also taken data on a transversely polarised proton target (N H3 ) this year.

beam type and energy eﬀective target type target type target dilution factor spin reversal each average target polarisation kinematic requirements

resulting kinematic ranges

Hermes

Compass

electrons of 27.6 GeV p atomar hydrogen 1 1–3 min 74% ± 8%

muons of 160 GeV d solid state 6 LiD 0.38 ≈ 5 days ≈ 47% ± 5%

Q2 > 1 GeV2 W 2 > 10 GeV2 y < 0.95 0.2 < z < 0.7 θγ ∗ h > 0.02 rad

Q2 > 1 GeV2 W 2 > 25 GeV2 0.1 < y < 0.9 0.2 < z < 1.0 –

0.023 < x < 0.4 0.05 < Ph⊥ < 1.3 GeV

0.003 < x < 0.4 0.1 < Ph⊥ < 3.0 GeV

containing the transversity and Sivers functions. In order to measure these functions, we consequently need a transversely polarised target and a good hadron identiﬁcation. 4. – Experimental techniques Most recently the Hermes experiment at Desy and the Compass experiment at Cern have provided results on azimuthal single-spin asymmetries measured with transversely polarised targets. The main characteristics of the two experiments as well as the requirements on kinematic variables for the presented results are summarised in table III. ˇ Both experiments use the information of a ring-imaging Cerenkov counter for the sepa± ± ration of π , K and p over the available momentum ranges. A detailed description of the experiments is given in [17] for Hermes and [18] for Compass. Both experiments measure asymmetries for opposite orientations of the target spin. This technique has the advantage that detector ineﬃciencies cancel as long as the detector performance is stable of the period of two spin orientations. At Hermes which uses a polarised gas target the nuclear polarisation of the atoms was ﬂipped very frequently at 1–3 min time interval. At Compass which uses a solid-state target in a 0.42 T holding ﬁeld a polarisation reversal takes 2 days and the polarisation was ﬂipped only once per week. However, the two up and down stream target cells were polarised in opposite ways thereby taking data simultaneously in the two opposite spin conﬁgurations. . 4 1. Instrumental eﬀects. – Instrumental eﬀects like the acceptance of the spectrometer can bias the extracted observables and produce fake asymmetries. The analysis meth-

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ods have to be chosen such that instrumental eﬀects cancel or are corrected for. Small eﬀects (small compared to the statistical precision) and uncertainties in the methods or corrections are included in the systematic uncertainty for the extracted asymmetries. Two examples of acceptance eﬀects are shown in ﬁg. 4 for Compass and Hermes. Compass utilises simultaneously two target cells polarised in opposite direction in order to cancel instrumental eﬀects to a large extend. However, this causes a dependence of the event distributions on the geometrical acceptance going from the upstream to the downstream end of the target (see ﬁg. 4-left). This geometrical acceptance eﬀect is minimised by extracting the asymmetries in a so-called “ratio product method” from the number of events in the upstream (u) and downstream (d) target cells for the opposite spin orientations (+/−)

(7)

Aj (Φj ) =

+ (Φj ) Nj,u

+ (Φj ) Nj,d

· − , − Nj,u (Φj ) Nj,d (Φj )

j = C(Collins angle), S(Sivers angle).

As long as the spectrometer response is stable over the data-taking period including a spin ﬂip in the target cells, which is here in the order of 10 days, geometrical acceptance eﬀects cancel with this method. The Hermes spectrometer consists of two separate parts above and below the beam lines. This geometry is reﬂected as a clear gap in the distribution of the angle φS (see ﬁg. 4-right) which causes cross contaminations of diﬀerent amplitudes by the acceptance.

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In order to account for such correlations, the azimuthal amplitudes are extracted simultaneously from a ﬁt to the 2-dimensional event distributions in the angles φ and φS as described in more detail in the following section. . 4 2. Asymmetry amplitudes extraction. – The presented Compass asymmetry amplitudes are obtained from 1-dimensional ﬁts of the form p0 · (1 + Acoll · sin(ΦC )) and p0 · (1 + Asiv · sin(ΦS )) to the event distributions of eq. (7) for the Collins- and Sivers-type combination of the φ and φS angles. Note that Compass uses a deﬁnitions for φC which diﬀers by a factor π from the combination of (φ + φS ) when using the deﬁnitions of ﬁg. 3. This results in an opposite sign for the extracted Collins asymmetries when compared to the Hermes ones. At Hermes, the cross-section asymmetry with respect to the target polarization is evaluated for the 2-dimensional event distributions in φ and φS AhUT (φ, φS ) =

(8)

1 Nh+ (φ, φS ) − Nh− (φ, φS ) . Pz Nh+ (φ, φS ) + Nh− (φ, φS )

+/−

Here, Nh represents the semi-inclusive yield for opposite target spin states +/− for a hadron type h and Pz is the average target polarisation value. The corresponding azimuthal amplitudes are evaluated using maximum-likelihood ﬁts. The Collins sin (φ + φS ) hUT and the Sivers sin (φ − φS ) hUT amplitudes are extracted simultaneously to avoid cross contaminations. All possible cross-section modulations of the σUT cross-section term up to subleading twist are taking into account by adding sin (2φ − φS ) and sin φS terms in addition to those given in eq. (6) in the probability density function F , with F = f (2 sin (φ ± φS ) hUT , . . . , φ, φS ) (9)

F =

1 h 1 + Pz 2 sin (φ + φS ) UT · sin (φ + φS ) + 2 h +2 sin (φ − φS ) UT · sin (φ − φS ) + h

+2 sin (3φ − φS ) UT · sin (3φ − φS ) + h

+2 sin (2φ − φS ) UT · sin (2φ − φS ) + ! h +2 sin φS UT · sin φS . Monte Carlo studies have shown that this method fully accounts for possible cross contaminations of the diﬀerent amplitudes in the σUT cross-section term due to the experimental acceptance. . 4 3. Charge conjugation and isospin symmetry. – Considering both charge conjugation and isospin symmetry, the isotriplet of the π-mesons is reﬂected in the following relation for the asymmetry amplitudes: (10)

+

−

0

sin(φ ± φS ) πUT + C · sin(φ ± φS ) πUT − (1 + C) · sin(φ ± φS ) πUT = 0.

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Here, C represents the unpolarized DIS cross-section ratio for π − and π + production π− π+ (C = σUU /σUU ). This relation is valid for any single-spin or double-spin asymmetry in semi-inclusive DIS at both twist-2 and twist-3 level and in leading and next-to-leading order in αs . The relation (10) constitutes an important validity check for the extraction of any pion asymmetries. . 4 4. Lepton beam and virtual photon asymmetries. – Theoretically, the asymmetry amplitudes are deﬁned for particles polarised with respect to the virtual photon direction. Experimentally however, the particle spin is oriented with respect to the lepton beam while the virtual photon direction changes in every scattering event. A nucleon transversely polarised with respect to the incoming lepton beam has a transverse and longitudinal spin component [19] (11)

ST =

cos θγ ∗ 1 − sin2 θγ ∗ sin2 φS

,

SL =

sin θγ ∗ cos φS

,

1 − sin2 θγ ∗ sin2 φS

where θγ ∗ is the polar angle between the momenta of incoming lepton and the virtual photon. For suﬃciently high beam momenta, the angle between the incoming lepton and the virtual photon is small and sin2 θγ ∗ can be neglected. The components can then be approximated by ST ≈ cos θγ ∗ and SL ≈ sin θγ ∗ cos φS where the longitudinal component can be up to 15% at Hermes kinematics. When integrated over φS the longitudinal spin component vanishes and the resulting longitudinal polarisation for all scattering events is zero. However, the polarised cross-section dσUL contains a sin φ modulation which couples to the cos φS modulation of the longitudinal spin component since sin φ cos φS = 21 [sin(φ + φS ) + sin(φ − φS )]. Hence, the longitudinal twist-3 crosssection term contributes to the measured Collins and Sivers amplitudes (12)

sin(φ ± φS ) lUT = cos θγ ∗ sin(φ ± φS ) qUT +

1 sin θγ ∗ sin φ qUL , 2

where the subscripts l and q indicate the amplitudes deﬁned with respect to the lepton beam and the virtual photon direction, respectively. In the following, lepton beam asymmetries are presented. The contribution from the longitudinal component has been checked to be negligible compared to the experimental uncertainties of the measurements. . 4 5. Contribution from exclusive channels. – Although formally, the contribution to the asymmetries of π-mesons and K-mesons from the decay of exclusively produced vector mesons is a part of the semi-inclusive DIS cross-section, a too large contribution might contradict the assumptions of factorisation, i.e. summation over a larger number of contributing channels. As an indication, the simulated fraction of π-mesons and charged K-mesons originating from exclusive vector meson production and decay is shown in ﬁg. 5 for Hermes kinematics. A dominant contribution of these channels appears only

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at very large z > 0.7. This kinematic region is therefore not considered in the Hermes analyses and no correction is applied for the remaining contributions shown in the ﬁgure. Compass has not yet evaluated such contributions. However, the fractions are expected to be smaller because of the higher Q2 values at Compass (the exclusive cross-sections behave like 1/Q6 compared to 1/Q4 for the DIS cross-section). 5. – The Collins asymmetries and how to interpret them The amplitude of interest is sin(φ + φS ) , i.e. the so-called Collins amplitude, which is sensitive to the transversity distribution function δq(x) and the Collins fragmentation function H1⊥ (z). The Hermes results [11] for the Collins amplitudes extracted according to eq. (9) are shown in ﬁg. 6 for pions (left) and for charged kaons (right). These results are milestones in the ﬁeld as the signiﬁcant non-zero asymmetries for charged pions demonstrated for the ﬁrst time unambiguously that both the transversity distribution and the Collins fragmentation function exist and are non-zero! Moreover, the results for the Collins amplitudes of the charged pions are quite surprising. The measured amplitudes are positive for π + and negative for π − which is expected if the transversity distributions resemble the helicity distributions to the extent that δu is positive and δd is negative and smaller in magnitude as most of the models predict. However, the magnitude of the negative π − amplitudes appears to be even larger than that for π + , which is indeed unexpected. The azimuthal amplitudes relate to the following combinations of distribution and fragmentation functions (with fav = favoured and dis = disfavoured) (13)

+

sin(φ + φS ) π : 4δuH1⊥fav + δdH1⊥dis , −

sin(φ + φS ) π : 4δuH1⊥dis + δdH1⊥fav . As mentioned, most models predict |δu(x)| > |δd(x)| which is analogous to the behaviour of the helicity distributions. Adding u-quark dominance in a proton target and an extra enhancement by the squared quark charges one expects the π − Collins amplitudes to be much smaller than the π + amplitudes. An explanation of the larger negative π − azimuthal amplitudes could be a substantial magnitude with opposite sign for the dis-

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Fig. 6. – Most recent Hermes results for the Collins amplitudes for π-mesons (left ﬁgure) and for charged kaons compared to charged pions (right ﬁgure) as a function of x, z and Ph⊥ . The strikingly non-zero amplitudes for charged pions demonstrated for the ﬁrst time that both transversity and the Collins fragmentation function exist and are sizeable.

favoured Collins function: H1⊥dis ≈ −H1⊥fav . While unpolarised fragmentation functions give just the probability for the fragmentation of a quark q into a hadron h those functions exhibit only positive values. In contrast, the Collins fragmentation function introduces an orientation into the fragmentation process and allows for a sign of the function. How can the opposite signs of the favoured and disfavoured Collins functions be understood? A possible explanation can be found in the description of the Collins fragmen. tation process in the light of the string model of fragmentation (see also subsect. 2 2) which was suggested by Artru [4, 20]. The ideas are illustrated in ﬁg. 7. The struck quark ﬂips —on average— its transverse spin and is kicked out of the nucleon (ﬁg. 7(a)). The quark-antiquark pair created in a string break is expected to preserve the vacuum quantum numbers J P = 0+ . Since the positive parity of this state requires aligned spins of quark and antiquark, an orbital angular momentum of L = 1 has to compensate the spins as indicated in ﬁg. 7(b). This means that the quarks are rotating around each other which creates additional transverse momentum when the antiquark will eventually form a pseudoscalar meson with the struck quark. Since the transverse spin of the struck quark has a deﬁnite orientation, the antiquark has to have the opposite orientation in order to form a pseudoscalar meson. Therefore the transverse momentum the formed meson has gained will have a deﬁnite direction too. Hence, the outgoing meson is deﬂected with respect to the virtual photon direction, indicated by the open arrow in ﬁg. 7(c). Following these ideas, the remaining quark from the ﬁrst string break has already a transverse momentum in the opposite direction as the ﬁrst produced meson and will gain

Transverse spin phenomena in DIS—Experiments

331

Fig. 7. – Collins eﬀect in the string fragmentation model by Artru [20] for a transversely polarised nucleon with its spin orientation in (left panels) and perpendicular to (right panels) the lepton scattering plane (see text).

even more transverse momentum in that direction in the next string break by picking up the orbital angular momentum of the antiquark from the next quark-antiquark pair with which it forms the second meson. This formation corresponds now to a negative Collins function. Accordingly, the favoured and disfavoured Collins functions within this particular model are always of opposite sign. To some extent this reﬂects just momentum conservation as the transverse momentum of all the created ﬁnal states has to vanish. Let us have a closer look to the speciﬁc case of a u quark in a proton with δu > 0, i.e. most of the u quarks have their spins aligned with the proton spin. The target polarisation vector lays in the scattering plane (within the page): φS = 0, as shown in ﬁg. 7(a). The spin of the u quark is ﬂipped when absorbing the virtual photon. The creation of a d d¯ pair yields a π + pseudoscalar meson which is deﬂected upwards with respect to the page, i.e. φ = π/2 (see chart (c)). For a target spin orientation perpendicular to the scattering plane (φS = π/2, ﬁg. 7(d)) the spin of the u-quark does not ﬂip. The produced pion is deﬂected to the left-hand side of the target spin when looking at the direction of the virtual photon (within the page), so that φ = 0, as illustrated in charts (e) and (f). For both target spin orientations, the azimuthal modulation of the Collins eﬀect sin(φ + φS ) = sin(π/2) is therefore positive, just as the measured Collins amplitudes for π + , which are dominated by scattering oﬀ u-quarks. The Artru model yields the correct sign for the π + Collins amplitudes under the condition of a positive transversity distribution for u-quarks which is predicted so far by

332

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Fig. 8. – Compass results for the Collins amplitudes for charged pions (upper panel) and for charged kaons (lower panel) as function of x, z and pt . The zero asymmetries can be understood as a cancellation of the contributions from up and down quarks when measuring on an isoscalar target.

all models for transversity. Coming back to the measured data, the π 0 Collins asymmetries represent an im. portant check of isospin symmetry in the π-meson triplet as discussed in subsect. 4 3. 0 + − Following eq. (10), the π amplitudes are expected to lay between the π and π amplitudes which is nicely seen in ﬁg. 6. Numerical evaluations conﬁrm that the Collins amplitudes measured by Hermes fulﬁll eq. (10) for the π-meson triplet. Finally, ﬁg. 6 also shows the Collins amplitudes for charged kaons which are in the right hand side compared to the charged pion amplitudes. The amplitudes are compatible within statistical accuracy for positive pions and kaons, while they are of opposite sign for negative pions and kaons. However, there is no reason to expect similar amplitudes for K − and π − , being K − : (¯ us) a fully sea object. . 5 1. Collins asymmetries from a deuterium target. – The Compass asymmetries [12] for charged pions and kaons shown in ﬁg. 8 are small, compatible with zero. Such small asymmetries can be understood as a cancellation of the contributions from up and down quarks when measuring on an isoscalar target. As discussed earlier, the up and down quark transversity distributions are expected to have opposite sign if they resemble to a

Transverse spin phenomena in DIS—Experiments

333

Fig. 9. – Deﬁnition of the azimuthal angles for two-hadron production in electron-positron annihilation.

certain extent the helicity distributions. This behaviour is also predicted by most of the models. While the Hermes asymmetries measured on a proton target mainly constrain the u-quark transversity distribution, the Compass deuteron data provide additional information on the d-quark distribution. Very valuable, complementary information on the d-quark transversity distribution is expected from future HallA (JLab) measurements with a transversely polarised neutron target. . 5 2. What about the Collins fragmentation function? – In order to extract any information on the transversity distribution or on the Collins fragmentation function from the measured Collins asymmetries, one of the two functions has to be modelled or taken from additional experimental information. As shown earlier, the Hermes data alone restrict the ratio of the disfavoured and the favoured Collins functions. However, they do not allow to separate the transversity distributions. Fragmentation functions are most commonly measured in electron-positron annihilation with the production of a quark-antiquark pair that fragments into hadrons. The study of correlated production of two hadrons may provide information on the spindependent Collins function. Since there is a nonzero probability that quarks have their spins aligned transversely to the lepton beams, the Collins function causes the correlation of the transverse momenta of the produced hadrons. This leads to an azimuthal modulation in the cross-section of inclusive two-hadron production. This modulation can be expressed either by one azimuthal angle φ0 , deﬁned with respect to the momentum of one of the produced hadrons, or by two azimuthal angles φ1 and φ2 , deﬁned with respect to the measured thrust axis which is an approximation of the quark-antiquark axis. These two deﬁnitions are depicted in ﬁg. 9. In both reference systems, the azimuthal asymmetry contains a cosine modulation: cos 2φ0 or cos(φ1 + φ2 ), respectively. The amplitudes of these modulations are proportional to the product of the quark and antiquark Collins

334 A0

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Fig. 10. – Belle results for the asymmetry Acos 2φ0 deﬁned in eq. (14) as function of z1 for 4 diﬀerent intervals of z2 . The asymmetries are sensitive to the Collins fragmentation function. The triangles and squares correspond to two diﬀerent methods for the extractions which are not supposed to yield the same magnitudes (see [22] for details).

functions [21] 2

(14)

Acos 2φ0 =

1 sin θ0 1 + cos2 θ0 M1 M2 ⊥[1]q

Acos(φ1 +φ2 ) =

F H1⊥q (z1 )H1⊥¯q (z2 ) D1q (z1 )D1q¯(z2 )

,

⊥[1]¯ q

(z1 )H1 (z2 ) sin2 θ1 H1 , 1 + cos2 θ1 D1q (z1 )D1q¯(z2 )

where F is a convolution integral over transverse quark and hadron momenta, M1 and M2 are the masses of the two hadrons, and z1 and z2 are their energy fractions. The polar angle θ0 (θ1 ) is deﬁned between the hadron momentum (the thrust axis) and the electron momentum (see ﬁg. 9) and determines the transverse polarisation of the quark-antiquark pair: sin2 θ0/1 /(1 + cos2 θ0/1 ). A ﬁrst measurement of azimuthal asymmetries in the production of charged-pion pairs in electron-positron annihilation was performed with the Belle detector at the asymmetric Kek electron-positron collider. The most recent, high-statistics result from Belle [22] is shown in ﬁg. 10. Signiﬁcant asymmetries are measured, increasing with increasing values of z. Yet, the extraction of the Collins function from these promising data is only possible with assumptions on the ratio of the disfavoured to favoured Collins function. The restriction on this ratio derived from the Hermes data provides helpful information for a ﬁrst attempt of the Collins function extraction. In addition, future

Transverse spin phenomena in DIS—Experiments

335

measurements for the production of neutral-pion pairs will allow the determination of this ratio. . 5 3. A brief look at model predictions for transversity. – Transversity has been calculated in various diﬀerent models like the chiral quark-soliton model, quark-diquark models or in a pertubative QCD approach (see also the lecture of M. Anselmino and E. Boglione). In addition, there is a basic constraint on the transversity distribution resulting from positivity (15)

|δq(x)| ≤

1 (q(x) + Δq(x)) 2

which is the so-called Soﬀer bound as it was ﬁrst pointed out by Soﬀer [23]. This bound is more restrictive as the obvious positivity limit |δq(x)| ≤ q(x). We will illustrate the use of the available data for gaining information on transversity and on the Collins function by giving only one example before coming to the ﬁrst extraction of transversity in a global analysis. Figure 11 shows the results of a simultaneous analysis of Hermes and Belle data [39]. For the x-dependence of the amplitudes the Hermes data are ﬁtted using a chiral Quark-Soliton Model (cQSM) parametrisation of the transversity distribution and a Gaussian ansatz for the transverse-momentum dependence of both the transversity distribution and the Collins function. The dark shaded area indicates the uncertainty associated with the ﬁt. This ﬁt is used to extract the Collins fragmentation function from the Hermes data. Within the model assumptions, indeed opposite values are obtain for the favoured and disfavoured Collins fragmentation function integrated over z: (16)

⊥(1/2)fav

2Bgauss H1

= (3.5 ± 0.8)% ,

⊥(1/2)dis

2Bgauss H1

= −(3.8 ± 0.7)% ,

where Bgauss is a factor depending on the widths used in the Gaussian ansatzes, and ⊥(1/2) H1 is the (1/2)-moment of the Collins function. In contrast, for the z-dependence the curves are predictions for the azimuthal amplitudes, based on an extraction of the Collins function from a ﬁt to the Belle data and the same cQSM for the transversity distribution. Here, the dark shaded area indicates the uncertainty related to the ﬁt to the Belle data and the light shaded area indicates the sensitivity of the asymmetry ⊥ to the unknown ratio of the Gaussian width of δq(x) and H1T (z). The curves for the z-dependence indicate that within the model assumptions the azimuthal asymmetries measured by Hermes and Belle are consistent. One can therefore go a step further and take them as support that these measurements are indeed related to the same eﬀect, i.e. the Collins eﬀect, hence their combined analysis can be used to extract transversity. Although such an extraction was not performed by the authors, it was concluded that δu(x) is positive and within 30% of the Soﬀer bound while δd(x) is practically unconstrained by the data and varies within the limits of the Soﬀer bound. This analysis did

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Fig. 11. – The Collins amplitudes for charged pions measured by Hermes compared to model predictions [24]. For the x-dependence of the amplitudes the Hermes data are ﬁtted using a chiral Quark-Soliton Model (cQSM) for transversity to obtain a value for the favoured and disfavoured Collins fragmentation function. In contrast, for the z-dependence the curves are based on a ﬁt for the Collins function to the Belle data and the same cQSM is used for transversity. The light shaded area indicates the sensitivity of the asymmetry to the unknown ratio of the Gaussian width of δq(x) and H1⊥ (z). The dark shaded area indicates the uncertainty associated with the ﬁts.

not yet use the new, high-statistic Belle and Hermes data shown here but earlier data sets with much reduced statistics. . 5 4. First glimpse of transversity. – Using the measurements of Hermes, Compass and Belle in a recent global analysis [25] the transversity distribution for up and down quarks have been extracted for the ﬁrst time. These results place another milestone in the ﬁeld of transverse spin phenomena. A detailed description of this analysis is given in the lecture of E. Boglione and we only show the obtained results in ﬁg. 12 in order to complete the picture obtained so far about transversity and the Collins function. The extracted Collins fragmentation functions are consistent with the values found in and discussed before [39]. This is not the case for the transversity distributions. The up-quark transversity found here is considerably smaller than the Soﬀer bound (indicated by the thick line in ﬁg. 12). While the shape of the distribution functions is determined from the Hermes data, the relative size of the functions is mainly constrained by the Compass data. Again, this analysis

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does not yet use the new, high-statistic data sets from Belle, Compass and Hermes shown here. We therefore look forward to an update of this extraction. 6. – Transversity from two-hadron production A very promising, alternative process to access transversity is the semi-inclusive production of two hadrons in DIS. In this process, the transverse spin of the quark inside a transversely polarised nucleon can be correlated to the transverse momentum of the hadron pair, instead of the transverse momentum of a single hadron. This allows to probe transversity without the inclusion of partonic transverse momenta. The latter is a very important point as the transversity distribution appears therefore in a direct product with the dihadron fragmentation function and is not embedded in a convolution integral as in the single hadron case. Two-hadron DIS production is therefore considered as a very clean process to access transversity. Of course, one needs again independent information on the unknown dihadron fragmentation function which in principle can be obtained from the Belle data, too. The obvious disadvantage is a signiﬁcantly smaller count number of hadron pairs compared to single hadron events; about ﬁve times less π + π − pairs than single π + events can be identiﬁed in the Hermes data on a transversely polarised target. Moreover, the two-hadron semi-inclusive cross-section is more complicated as it depends on 9 kinematic variables instead of 6. Luckily, the fact that one can integrate the cross-section over Ph⊥ (and still remain sensitive to transversity) reduces the dependence from 9 to 7 variables.

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Fig. 13. – Deﬁnition of the azimuthal angle φR⊥ (left) and the polar angle θ (right) for twohadron production in DIS. The latter is deﬁned in the two-hadron centre-of-mass system.

The study of such two-hadron fragmentation processes was already proposed in 1993 [26] and 1998 [27]. The diﬀerential cross-section for semi-inclusive two-hadron production in DIS on a transversely polarised target, integrated over the transverse momentum of the hadron pair, is [28] (17)

d7 σUT = 2 dx dy dz dφS dφR⊥ dcos θ dMππ α2 2 T | |R| sin(φR⊥ + φS ) sin θ − eq B(y)|S 2 2πsxy q,¯q Mhh sp 2 2 ·δq(x) H1 (z, Mhh ) + cos θH1pp (z, Mhh ) ,

where z is the energy fraction of the hadron pair and Mhh its invariant mass. For two = 1 M 2 − 4M 2 . The azimuthal angle φR⊥ and the hadrons with equal mass Mh , |R| hh h 2 polar angle θ are deﬁned in ﬁg. 13. The transversity distribution appears in conjunction with a combination of two-hadron fragmentation functions that describe the interference of diﬀerent production channels of the hadron pair. The chiral-odd and T-odd functions 2 2 H1sp (z, Mhh ) and H1pp (z, Mhh ) are related to the interference between the s-wave and p-wave channels and between two p-wave channels, respectively. The azimuthal asymmetry for semi-inclusive π + π − -pair production has been obtained from the 2002–2004 Hermes data (note, this is not yet the full Hermes statistics like shown for the Hermes single-hadron results which include the full 2002–2005 transverse target data set). The azimuthal amplitude of interest is extracted by a 2-dimensional ﬁt dependent on φR⊥ + φS and θ and found to be signiﬁcantly positive [14]: (18)

sin(φR⊥ +φS ) sin θ

AUT

= 0.040 ± 0.009 (stat.) ± 0.003 (syst.).

The amplitude is positive over the measured Mππ range as shown in ﬁg. 14. This result is clearly inconsistent with a sign change at the mass of the ρ0 vector meson, predicted in

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Transverse spin phenomena in DIS—Experiments

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Fig. 14. – Hermes result for the asymmetry amplitude AUT R⊥ S for semi-inclusive π + π − -pair production as a function of the invariant mass Mππ of the pion pair.

a 2-meson phase shift analysis [27]. The asymmetry moments rather show a maximum around the ρ0 mass, as predicted, e.g., in [29]. Also Compass has measured asymmetries in two-hadron production using all combinations of charged pion and kaon pairs or pion-kaon combinations. The obtained asymmetries are all compatible with zero and not shown here. As for the single hadron case, the small asymmetries can again be understood as cancellation of contributions from up and down quarks when measuring on an isoscalar target. 7. – The Sivers asymmetries and how to interpret them The amplitude of interest here is sin(φh − φS ) , i.e. the so-called Sivers amplitude, ⊥ which is sensitive to the Sivers distribution function f1T (x) and the usual unpolarised fragmentation function D1 (z). The Hermes results [11] for the Sivers amplitudes extracted according to eq. (9) are shown in ﬁg. 15 for pions (left) and for charged kaons (right). Also these results constitute milestones in the ﬁeld as the signiﬁcant non-zero asymmetries for π + , π 0 and K + demonstrated for the ﬁrst time that T-odd distribution functions indeed exist in DIS. Moreover, the existence of such functions depending on transverse momentum of the quarks inside the nucleon implies that these quarks also carry non-vanishing orbital angular momentum which is one of the still missing pieces in the spin puzzle. A direct relation, however, between the Sivers function or other similar functions that describe spin-orbit correlations and the angular momentum contribution of the quarks to the nucleon spin could not yet be established. First, promising attempts of ﬁnding such, still model dependent, relations have been made in [30] and [31]. Following the ideas of [30] a descriptive picture of the relation between the orbital angular momentum of the quarks and the Sivers function can be obtained when the

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distribution function is expressed in impact parameter space: (19)

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φS = π/2 u FSI π+ φ=π Fig. 17. – Illustration of the scattering process oﬀ a u quark in the semi-classical picture with the production of a π + meson. The attractive FSI deﬂects the active quark towards the centre of momentum.

with the impact parameter bT = (bx , by ). The reference point for the impact parameter is the transverse centre of the longitudinal momentum, i.e. the sum over the transverse positions rT,i of all quarks, antiquarks and gluons in the target weighted by their momentum fractions xi . The impact parameter dependent distribution function q(x, bT ) of unpolarised quarks is axial symmetric for unpolarised and for longitudinally polarised nucleons. In case of transversely polarised nucleons, the distribution of unpolarised quarks qX (x, bT ) is distorted perpendicular to the spin and the momentum of the nucleon. This distortion vanishes when there is no quark orbital angular momentum parallel to the nucleon spin. Even though the mathematical description of the distortion is model independent, models for the impact parameter dependences have to be used in order to visualise these distortions. An example for the distributions of unpolarised u and d quarks in a transversely polarised nucleon is shown in the left two panels of ﬁg. 16 for a momentum fraction of x = 0.3. The signs of the distortions in ﬁg. 16 are ﬁxed by the signs of the anomalous magnetic moments of the proton and the neutron. In a semiclassical picture, the superposition of translation and orbital motion of the quarks can be identiﬁed as the cause of the distortion of the distribution function. For quarks with an orbital angular momentum parallel to the nucleon spin in the x-direction, the orbital momentum adds to the quark momentum in the top and subtracts in the bottom. Hence, a quark with a given momentum fraction xq is probed by the virtual photon at a higher momentum fraction xobs > xq in the top and a smaller fraction xobs < xq in the bottom. In the top the unpolarised distribution is therefore shifted towards higher x values while in the bottom it is shifted to smaller x values as shown in the right panel of ﬁg. 16. Since the unpolarised distribution function decreases with increasing values of x in the valence region, the increase of the momentum on one side of the nucleon spin results in a larger number of quarks for a certain observed momentum fraction xobs at this side. At the opposite side, less quarks are observed at xobs due to the decrease of the quark momentum, resulting in a distortion of the distribution at xobs towards the top. For quarks with antialigned orbital angular momentum, the distribution is distorted towards

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the bottom. This semi-classical picture thus yields a positive orbital angular momentum for u quarks and a negative orbital angular momentum for d quarks. In ﬁg. 17 the scattering process is schematically illustrated for a nucleon spin orientation perpendicular to the scattering plane, i.e. φS = π/2. For a positive orbital angular momentum of the u quarks, the u quark density is enhanced in the left hemisphere of the nucleon when looking along the virtual-photon direction so that it will be absorbed more likely by a u quark in that region. After the absorption, Final-State Interactions (FSI) bend the quark towards the centre. The FSI are attractive since the struck quark and the spectators form a colour antisymmetric state. The outgoing positive pion that contains the struck quark is therefore observed on the right-hand side of the nucleon spin, i.e. φ = π. Thus, the description of the quark distributions in the impact parameter space yields a positive Sivers amplitude sin(φ − φS ) = sin π > 0 for u quarks fragmenting into π + . This is exactly the result obtained by Hermes for the π + Sivers amplitudes which are dominated by the scattering oﬀ u quarks. Coming back again to the measured data, very interesting are the observed large Sivers amplitudes for K + which are roughly twice as large as that for π + . Since the quark content of these two mesons diﬀers only on the antiquark involved, this observation suggests a signiﬁcant Sivers function for antiquarks in the proton. Going even a step further, it might also suggest that antiquarks carry signiﬁcant orbital angular momentum in the nucleon. . And ﬁnally, isospin symmetry for the π-meson triplet as discussed in subsect. 4 3 is fulﬁlled for the Sivers asymmetries measured at Hermes. . 7 1. Sivers asymmetries from a deuterium target. – The Sivers asymmetries measured by Compass on a deuteron target [12] are shown in ﬁg. 18 for charged pions and charged kaons. Also these asymmetries are all compatible with zero. This yields to an equivalent picture as for the Collins asymmetries, that the Sivers functions for up and down quarks are of similar size and opposite sign. In fact, predictions for the Sivers function from ⊥u ⊥d the QCD limit of a large number of colours Nc [32], state that f1T = −f1T up to 1/Nc -corrections. . 7 2. A brief look at models and extractions of the Sivers function. – The Sivers function couples to the unpolarised fragmentation function which is reasonably well known from electron-positron annihilation processes. Figure 19 shows the extraction of the Sivers up and down quark distributions using three diﬀerent models for the Sivers function [33-35]. The extraction [33] (full line) is based on a combined ﬁt to the Hermes and Compass data, while the other two ﬁt the Hermes data only but describe well the Compass data when using the obtained parameters to calculate the asymmetries for Compass kinematics. All three extractions use the Kretzer parametrisations [36] for the unpolarised fragmentation function. The two curves of each set indicate the 1-sigma regions of the various parametrisations. An uncertainty due to the diﬀerent models for the parton transverse momenta is not included in the shown range of the curves. The three approaches describe equally well the Hermes Sivers asymmetries. The diﬀerences

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in size and shape of the extracted Sivers up and down quark distributions reﬂect therefore the model dependence of the ﬁt results. The parametrisation [35] imposes the constraint ⊥u ⊥d from the limit of a large number of colours Nc : f1T = −f1T modulo 1/Nc corrections, which results in the symmetric extraction for the up and down Sivers distributions shown in the left panel of ﬁg. 19 with dashed lines. None of the models involve parametrisations for the sea quarks as they could not be constraint by the data used for the ﬁts. Therefore the model [33] which describes well the Hermes pion asymmetries (see ﬁg. 19, left) fails to describe the Hermes K + asymmetries (see ﬁg. 19, right). A combined ﬁt to the new pion and kaon Sivers asymmetries from Hermes and Compass should allow to constrain also sea quark parameterisations. It is interesting to also look at diﬀerent parametrisations of the unpolarised fragmentation function. The most recent parametrisation [37], which also includes the pion and kaon multiplicities measured at Hermes in the extraction, constrains much better the separation between favoured and disfavoured fragmentation functions. In fact, using these new parametrisations of the unpolarised fragmentation function and including the Hermes kaon amplitudes in the ﬁt, yields to a reasonably well description of the Hermes K + Sivers amplitudes with the ansatz from [33, 38]. None of the above extractions made use of the new high-statistic Compass and Hermes data shown here. These new data together with the new parametrisations of the unpolarised fragmentation function should constrain much better diﬀerent models for the Sivers up and down quark distributions. . As discussed in subsect. 2 3, of particular interest are studies of the Sivers function in Drell-Yan processes in order to test their unusual universality property to appear with opposite signs in DIS and in Drell-Yan processes. The experimental check of this prediction is planned by Compass when running in the hadron-beam mode and by the proposed Pax experiment at the future Fair facility at Gsi [39]. 8. – What should come next? We are currently in a very exciting situation for the ﬁeld of studying transverse spin phenomena. New, data of much improved precision are available from the Compass and Hermes experiments for the Collins and Sivers asymmetries for pions and kaons and from Belle for the Collins fragmentation function. We await eagerly the new results from Compass from the data taken with a proton target this year and from the hadron run planned for the next year. In future, JLab at 11 GeV and the possible option of a transversely polarised target at the Clas experiment will provide access to the high x-region where the transversity distribution is expected to be large. The most crucial test of our current understanding of azimuthal single-spin asymmetries in terms of pertubative QCD will be the experimental veriﬁcation of the predicted sign change of T-odd distribution functions, like the Sivers function, when being measured in DIS or in Drell-Yan processes. First, very promising global analyses of available data from DIS and electron-positron annihilation processes have been performed by theory groups. The ultimate goal will

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be to perform such global analyses on all available data from DIS, hadron-hadron and electron-positron scattering that provide information on transverse momentum dependent distribution and fragmentation functions (TMDs) in various diﬀerent observables. Theorist groups are preparing for this task. We are looking forward to all projects that aim in pinning down various TMDs at future facilities such as the Fair-project at Gsi, J-Parc and the proposed Eic. There are exciting ideas for studying these new functions at the Lhc and more ideas will certainly come up. ∗ ∗ ∗ We would like to thank the three former P.h.D. students R. Seidl, U. Elschenbroich and P. van der Nat for their pioneering work in analysing the Hermes data taken with a transversely polarised target. The interested reader will ﬁnd a wealth of useful information for analysing azimuthal single-spin asymmetries in DIS in their outstandingly written theses [40]. In particular, ﬁgures 7, 16 and 17 and their description we took from U. Elschenbroich’s thesis.

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REFERENCES [1] Jaffe R. L., hep-ph/9602236; Spin, twist and hadron structure in deep inelastic processes, Talk given at Ettore Majorana International School of Nucleon Structure: 1st Course: The Spin Structure of the Nucleon, Erice, Italy, 3-10 Aug 1995. [2] Anselmino M., Jaffe R., Leader E. and Vogelsang W., lectures at this school. [3] Ralston J. P. and Soper D. E., Nucl. Phys. B, 152 (1979) 109. [4] Artru X. and Mekhfi M., Z. Phys. C, 45 (1990) 669. [5] Jaffe R. L. and Ji X., Nucl. Phys. B, 375 (1992) 527. [6] Barone V., Drago A. and Ratcliffe P. G., Phys. Rep., 359 (2002) 1; Barone V. and Ratcliffe P. G., Transverse Spin Physics (World Scientiﬁc) 2003. [7] Bacchetta A. et al., JHEP, 0702 (2007) 93. [8] Collins J. C., Nucl. Phys. B, 396 (1993) 161. [9] Andersson Bo., Gustafson G., Ingelman G. and Sjostrand T. J. C., Phys. Rep., 97 (1983) 31. [10] Collins J. C., Phys. Lett. B, 536 (2002) 43. [11] Hermes Collaboration (Airapetian A. et al.), Phys. Rev. Lett., 94 (2005) 12002; for the most recent preliminary results shown here see: Diefenthaler M. for the Hermes Collaboration, e-Print: arXiv:0706.2242 [hep-ex], Proceedings of 15th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2007), Munich, Germany, 16-20 Apr 2007. [12] Compass Collaboration (Alexakhin V. et al.), Phys. Rev. Lett., 94 (2005) 202002; Compass Collaboration (Ageev E. et al.), Nucl. Phys. B, 765 (2007) 31; for recent preliminary results see also: Martin A. for the Compass Collaboration, e-Print: hepex/0702002. [13] HallA Collaboration, The E06-010 and E06-011 experiments (approved) http:// hallaweb.jlab.org/experiment/transversity (2006). [14] van der Nat P. for the Hermes Collaboration, First measurement of interference fragmentation on a transversely polarized hydrogen target, prepared for the 13th International Workshop on Deep-inelastic Scattering (DIS 2005), Madison, Wisconsin, USA, 27 Apr-1 May 2005. [15] Schill C., for the Compass Collaboration, Proceedings of 15th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS2007), Munich, Germany, 16-20 Apr 2007. [16] Bacchetta A. et al., Phys. Rev. D, 70 (2004) 117504. [17] HERMES Collaboration (Ackerstaff K. et al.), Nucl. Instrum. Methods A, 417 (1998) 230. [18] COMPASS Collaboration (Abbon P. et al.), Nucl. Instrum. Methods A, 577 (2007) 455. [19] Diehl M. and Sapeta S., Eur. Phys. J. C, 41 (2005) 515. [20] Artru X., hep-ph/9310323; Proposals for measuring transversity distributions in deep inelastic electron scattering and a model for E-704 asymmetries, prepared for the 5th International Workshop on High-energy Spin Physics, Protvino, Russia, Sep 20-24, 1993. [21] Boer D., Jakob R. and Mulders P., Nucl. Phys. B, 504 (1997) 345. [22] Ogawa A., Grosse-Perdekamp M. and Seidl R., Fragmentation function measurements at Belle, Proceedings of the “17th International Spin Physics Symposium” (Spin2006), Kyoto, Japan, 2-7 Oktober 2006. [23] Soffer J., Phys. Rev. Lett., 74 (1995) 1292. [24] Efremov A., Goeke K. and Schweitzer P., Phys. Rev. D, 73 (2006) 094025. [25] Anselmino M. et al., Phys. Rev. D, 75 (2007) 54032.

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[26] [27] [28] [29]

[30] [31] [32] [33] [34] [35] [36] [37] [38]

[39] [40]

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Collins J. et al., Nucl. Phys. B, 420 (1994) 565. Jaffe R. et al., Phys. Rev. Lett., 80 (1998) 1166. Bacchetta A. and Radici M., hep-ph/0407345; and Phys. Rev. D, 74 (2006) 114007. Radici M., Alternative approaches to transversity: How convenient and feasible are they?, prepared for the International Workshop on Transverse Polarisation Phenomena in Hard Processes (Transversity 2005), Como, Italy, 7-10 Sep 2005. Burkardt M., Phys. Rev. D, 66 (2002) 114005. Meissner S., Metz A. and Goeke K., Phys. Rev. D, 76 (2007) 034002. Pobylitsa P., hep-ph/0301236. Anselmino M. et al., Phys. Rev. D, 72 (2005) 94007, hep-ph/0511017. Vogelsang W. and Yang F., Phys. Rev. D, 72 (2005) 507266, hep-ph/0511017. Collins J. C. et al., Phys. Rev. D, 73 (2006) 014021, hep-ph/0511017. Kretzer S., Phys. Rev. D, 62 (2000) 54001. de Florian D., Sassot R. and Stratmann M., Phys. Rev. D, 75 (2007) 114010. Anselmino M., presentation at the 6th Circum-Pan-Paciﬁc Symposium on High Energy Spin Physics (PACSPIN07) Jul 30 - Aug 2, 2007, Vancouver, Canada, www.triumf.info/ hosted/pacspin07/. Efremov A. et al., Phys. Lett. B, 612 (2005) 233. Seidl R., PhD Thesis (University of Erlangen); Elschenbroich U., PhD Thesis (University of Gent); van der Nat P., PhD Thesis (University of Amsterdam) www-hermes.desy.de/notes/pub/.

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QCD spin physics in hadronic interactions W. Vogelsang Physics Department, Brookhaven National Laboratory - Upton, NY 11973

Summary. — We discuss spin phenomena in high-energy hadronic scattering, with a particular emphasis on the spin physics program now underway at the ﬁrst polarized proton-proton collider, RHIC. Experiments at RHIC unravel the spin structure of the nucleon in new ways. Prime goals are to determine the contribution of gluon spins to the proton spin, to elucidate the ﬂavor structure of quark and antiquark polarizations in the nucleon, and to help clarify the origin of transverse-spin phenomena in QCD. These lectures describe some aspects of this program and of the associated physics.

1. – Introduction For many years now, spin has played a very prominent role in QCD. The ﬁeld of QCD spin physics has been driven by the hugely successful experimental program of polarized deeply-inelastic lepton-nucleon scattering (DIS) [1]. One of the most important results of this program has been the ﬁnding that the quark and anti-quark spins (summed over all ﬂavors) provide only about a quarter of the nucleon’s spin, ΔΣ ≈ 0.25 in the proton helicity sum rule [2] (1)

1 1 = ΔΣ(Q2 ) + ΔG(Q2 ) + Lq (Q2 ) + Lg (Q2 ), 2 2

c Societ` a Italiana di Fisica

349

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W. Vogelsang

implying that sizable contributions to the nucleon spin should come from the gluon spin contribution ΔG(Q2 ), or from orbital angular momenta Lq,g (Q2 ) of partons. Here, Q is the resolution scale at which one probes the nucleon. To determine the gluon spin contribution on the right-hand side of eq. (1) has become a major focus of the ﬁeld. Like ΔΣ, it can be probed in polarized high-energy scattering. Several current experiments are dedicated to a direct determination of the spin-dependent gluon distribution Δg(x, Q2 ), Δg(x, Q2 ) ≡ g + (x, Q2 ) − g − (x, Q2 ),

(2)

where g + (g − ) denotes the number density of gluons in a longitudinally polarized proton with same (opposite) sign of helicity as the proton’s, and where x is the gluon’s light-cone momentum fraction. The ﬁeld-theoretic deﬁnition of Δg is (3)

Δg(x, Q2 ) =

i 4π x P +

& + ˜ +ν (λn)|P, S && dλ eiλxP P, S|G+ν (0) G

Q2

,

˜ μν its dual. The written in A+ = 0 gauge. Gμν is the QCD ﬁeld strength tensor, and G 2 integral of Δg(x, Q ) over all momentum fractions x becomes a local operator only in A+ = 0 gauge and then coincides with ΔG(Q2 ) [2-4]. The COMPASS experiment at CERN and the HERMES experiment at DESY attempt to access Δg(x, Q2 ) in charmor high-pT hadron ﬁnal states in photon-gluon fusion γ ∗ g → q q¯ [5, 6]. A new milestone has been reached with the advent of the ﬁrst polarized proton-proton collider, RHIC at BNL [7-9]. By colliding longitudinally polarized protons at high energies, RHIC will provide precise and detailed information on Δg, over a wide range of x and Q2 , and from a variety of probes. There are also important spin phenomena in QCD associated with transversely polarized high-energy nucleons [10, 11]. Single-transverse Spin Asymmetries (SSAs), in particular, play an important role for our understanding of QCD and of nucleon structure. They have a long history, starting from the 1970s and 1980s when surprisingly large SSAs were observed in hadronic reactions such as p↑ p → πX at forward angles of the produced pion [12]. The last few years have seen a renaissance in the experimental studies of SSAs. The HERMES collaboration at DESY, SMC and COMPASS at CERN, and the CLAS collaboration at the Jeﬀerson Laboratory have investigated SSAs in semi-inclusive hadron production eN ↑ → eπX in deep-inelastic scattering [13]. For proton targets, remarkably large asymmetries were found. With the advent of RHIC, there are new possibilities for extending the studies of SSAs in hadronic scattering into a regime where the use of QCD perturbation theory in the analysis of the data appears to be justiﬁed. These lectures discuss some of the most important aspects of high-energy polarized pp scattering at RHIC. We will start out with a brief synopsis of perturbative QCD and its applications in hadronic scattering. We will then discuss separately the physics associated with longitudinally and transversely polarized pp collisions.

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QCD spin physics in hadronic interactions

2. – QCD perturbation theory and its applications The prerequisite for many of our eﬀorts to study the inner structure of the nucleon is the “asymptotic freedom” of QCD [14]. This term describes the property of the strongcoupling constant αs to decrease with increasing momentum transfer (or towards shorter distances): (4)

αs (μ2 )

αs (Q2 ) = 1+

(μ2 )

αs 12π

(33 − 2nf ) log

Q2 μ2

,

where nf is the number of quark and anti-quark ﬂavors. Because of asymptotic freedom, it should be possible to use perturbation theory in terms of the quarks and gluons of QCD when the momentum transfer is suﬃciently high. Indeed, despite the fact that ultimately all strong-interaction phenomena will also involve hadronic mass scales and hence the conﬁnement regime of QCD, QCD perturbation theory has turned out to be highly useful and successful for quantities that are either – insensitive to details of long-distance physics at leading power in momentum transfer (infrared-safe), or – for which short-distance and (non-perturbative) long-distance phenomena may be separated (“factorized”) from one another. In the following, let us discuss examples of both. . 2 1. Perturbative QCD in e+ e− annihilation. – The simplest QCD observable in e+ e− annihilation is the fully inclusive total cross-section for e+ e− → hadrons. At suﬃciently high center-of-mass energy s, a ﬁrst-order approximation to this cross-section is given by that for e+ e− → q q¯, with massless quarks, which is (5)

σe+ e− →qq¯ ≡ σ0 =

4πα2 Nc 3s

e2q ,

q=u,d,s,...

where Nc = 3 is the number of colors in QCD and eq denotes the quark fractional electromagnetic charge. Once the q and q¯ have been produced in the collision, they will hadronize, which is certainly a non-perturbative process. However, the cross-section we are considering is totally inclusive, that is, we sum over all hadronic ﬁnal states. The total probability for the q and q¯ to turn into some hadronic ﬁnal state is unity, which is the reason why σ0 is a useful ﬁrst-order approximation. What happens at higher orders in perturbation theory? At the next order, O(αs ), one needs to consider virtual loop corrections to e+ e− → q q¯ (interfering with the lowest-order process), as well as real-gluon emission from one of the outgoing quark legs. Both these contributions are separately inﬁnite. For example, the real-emission diagrams become singular when the

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W. Vogelsang

gluon is radiated parallel to one of the outgoing quark lines, or when it becomes very soft. The same happens in the virtual diagrams. These singularities signal the onset of long-distance physics. However, the sum of real and virtual diagrams is ﬁnite, giving rise to a moderate correction to the perturbative cross-section: (6)

αs σe+ e− → hadrons = σ0 1 + . π

Let us take stock. We have found a perturbative way of calculating the cross-section for e+ e− → hadrons. We do this by calculating “e+ e− → strongly interacting particles” at the level of the quarks and gluons of QCD. The reason why this can be done is precisely that we have considered a cross-section that is insensitive to details of long-distance physics. We have not asked for the cross-section for producing, say, ﬁve low-energy pions at certain angles, but rather a cross-section that collects all hadronic ﬁnal states, no matter how they were eventually generated in the hadronization mechanism. For such long-distance-insensitive observables QCD perturbation theory is useful. At the level of quarks and gluons, insensitivity to long-distance physics means that the observable is deﬁned in such a way that it does not change if a ﬁnal-state quark or gluon collinearly splits into a pair, or if a ﬁnal-state particle becomes very soft. Such observables are referred to as “infrared-safe” (IR-safe). The perturbative cross-section for e+ e− → hadrons is (7)

σe+ e− → hadrons = σ0 1 + αs (s) c1 + αs2 (s) c2 + . . .

with coeﬃcients ci . Corrections to this formula as such are of non-perturbative nature and suppressed by inverse powers of s. That such “power-corrections” occur may for example be seen by keeping a ﬁnite quark mass mq in the calculation of the lowest-order cross-section σ0 , in which case (8)

( ) 2m2q 4m2q 1+ ≈ 1− s s q=u,d,s,... * ( )+

m4q 4πα2 2 ≈ Nc eq 1 + O . 3s s2

4πα2 Nc σ0 = 3s

'

e2q

q=u,d,s,...

If mq ΛQCD , it will set a non-perturbative mass scale. The power suppression of nonperturbative eﬀects by two powers (or more) of s is actually generic in e+ e− → hadrons. The total cross-section for e+ e− → hadrons is the simplest example of an “infraredsafe” (IR-safe) observable. It was discovered that there are many other IR-safe observables in e+ e− annihilation. A classic IR safe observable is the “Sterman-Weinberg jets” [15]. An event in e+ e− annihilation qualiﬁes as a Sterman-Weinberg jet event if one can ﬁnd two oppositely directed cones of half-opening δ so that all hadronic energy √ except a fraction s is inside these cones. This observable has been the progenitor of all jet observables in QCD.

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QCD spin physics in hadronic interactions

. 2 2. Factorized deep-inelastic scattering. – For our purposes, factorized cross-sections are particularly interesting. These are not per se IR-safe, but allow a separation of longdistance from short-distance phenomena. The classic example is the cross-section (or structure function) for deep-inelastic scattering (DIS). At large momentum transfer Q2 , a DIS structure function may be written in the simple parton model as

1

FDIS =

(9)

x

dξ q(ξ) Fˆ (x/ξ), ξ

where q is the quark parton distribution (we assume here that we have only one quark ﬂavor) and Fˆ is the partonic structure function describing the scattering of the virtual DIS photon oﬀ the quark. The parton model has the notion of a separation of longdistance (q) from short-distance (Fˆ ) eﬀects. How does this hold up in QCD? One ﬁnds that this separation can be proven systematically to all orders, resulting schematically in (10)

FDIS

Q λ

=q

2 Q κ , αs (μ) ⊗ Fˆ , αs (μ) + O , λ μ Q2

μ

where ⊗ denotes the integral convolution in x given in (9). Here λ denotes generically a non-perturbative mass scale. All dependence on λ is in the quark distribution function, whereas all dependence on the hard scale Q resides in Fˆ . However, this factorization implies the presence of a “factorization scale” μ. This scale may be thought of as the scale that distinguishes when contributions are to be absorbed into the quark distribution or into the partonic structure function, respectively. Corrections to the factorized form in eq. (10) are suppressed by inverse powers of Q2 and become negligible at high Q2 . Physics of course should be independent of the choice of μ. This requirement leads to the scale evolution of parton distribution functions. For the helicity distributions, for example, the evolution equations [16, 17] read (11)

d d ln μ2

Δq Δg

(x, μ2 ) =

ΔPqq (αs (μ), x) ΔPgq (αs (μ), x)

ΔPqg (αs (μ), x) ΔPgg (αs (μ), x)

⊗

Δq Δg

! x, μ2 ,

where the ΔPij are known as “splitting functions” [17-19] and are evaluated in QCD perturbation theory. Evolution resums collinear logarithms of the form αsk ln(Q/Q0 )m (m ≤ k) between the initial scale Q0 for the parton distributions and the hard scale Q. . 2 3. Factorized pp scattering. – Factorization is the basic concept that underlies most of RHIC spin physics [20]. Like for DIS, cross-sections for large-momentum-transfer reactions in pp scattering may be factorized into long- and short-distance contributions. The long-distance pieces contain information on the structure of the nucleon in terms of its distributions of constituents, “partons”. The short-distance parts describe the hard interactions of these partons and can be calculated from ﬁrst principles in QCD perturbation theory. While the parton distributions describe universal properties of the nucleon, that is, are the same in each reaction, the short-distance parts carry the process

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W. Vogelsang

dependence and have to be calculated for each reaction considered. The factorization just described for the case of DIS is the prototype. As an explicit example in pp scattering, we consider the cross-section for the reaction pp → π(pT )X, where the pion is at high transverse momentum pT , ensuring large momentum transfer. X denotes an arbitrary hadronic ﬁnal state, which is summed over. The statement of the factorization theorem is then: (12)

dσ =

dxa

dxb

dzc fa (xa , μ) fb (xb , μ) Dcπ (zc , μ)

a,b,c c × dˆ σab (xa PA , xb PB , Pπ /zc , μ),

c where the sum is over all contributing partonic channels a + b → c + X, with dˆ σab the associated partonic cross-section. The fa,b describe the distributions of partons in the nucleon. In this particular example, the fact that we are observing a speciﬁc hadron in the reaction requires the introduction of additional long-distance functions, the parton-to-pion fragmentation functions Dcπ . These functions have been determined with some accuracy by observing leading pions in e+ e− collisions and in DIS [21]. As we discussed above, a factorization of a physical quantity into contributions associated with diﬀerent length scales will rely on a “factorization” scale that deﬁnes the boundary between what is referred to as “short-distance” and “long-distance”. The dependence on the value of μ decreases order by order in perturbation theory. This is a reason why knowledge of higher orders in the perturbative expansion of the partonic cross-sections is important. We recall that eq. (12) is of course not an exact statement. There are corrections to eq. (12) that are down by inverse powers of the momentum transfer, the so-called “power corrections”. These corrections may become relevant towards lower pT . As we shall see below, comparisons of RHIC data for unpolarized cross-sections with theoretical calculations based on eq. (12) do not suggest that power corrections play a very signiﬁcant role in the RHIC kinematic regime, even down to fairly low pT . Figure 1 oﬀers a graphic illustration of QCD factorization. Thanks to factorization, one can study nucleon structure, represented by the parton densities fa,b (x, μ), through a measurement of dσ, hand in hand with a theoretical calculation of dˆ σ . The latter may be evaluated in QCD perturbation theory. Schematically, they can be expanded as

(13) c,(0)

c,(0)

c = dˆ σab dˆ σab

+

αs c,(1) dˆ σab + . . . . π

dˆ σab is the Leading-Order (LO) approximation to the partonic cross-section. The lowest order can generally only serve to give a rough description of the reaction under study. It merely captures the main features, but does not usually provide a quantitative understanding. The ﬁrst-order (“Next-to-Leading Order” (NLO)) corrections are usually indispensable in order to arrive at a ﬁrmer theoretical prediction for hadronic cross-sections.

355

QCD spin physics in hadronic interactions

p

fa fa

h

h

D

f

f

V fb p

X’

fb

Fig. 1. – Factorization of pp → π 0 X in terms of parton densities, partonic hard-scattering cross-sections, and fragmentation functions.

We emphasize that the factorization in eq. (12) involves integrations over the partons’ momentum fractions xa,b , zc only, which appear in the relations between the partonic and hadron momenta, pa = xa Pa in the initial state and pc = Pπ /zc in the ﬁnal state. Such a factorization is known as “collinear” factorization. It applies in particular to singleinclusive reactions such as pp → πX. 3. – Longitudinally polarized pp collisions at RHIC . 3 1. Results from unpolarized pp scattering at RHIC . – There have already been results from RHIC that demonstrate that the NLO framework is very successful. This is shown by ﬁg. 2 which presents comparisons of data from PHENIX and STAR for mid-rapidity π 0 [22] and jet [23], forward π 0 [24], and mid-rapidity direct-photon [25] production with NLO calculations [26-31]. This provides the basis for extending this type of analysis to polarized reactions. In addition, each of the cross-sections shown are strongly dominated by partonic scatterings with initial gluons [8]. As an example, in ﬁg. 3 we decompose the NLO mid-rapidity π 0 cross-section into the relative contributions from the various two-parton initial states. It is evident that qg and gg scattering dominate. . 3 2. Probing the spin structure of the nucleon in polarized pp collisions. – The measured quantities in spin physics experiments at RHIC are spin asymmetries. For collisions of longitudinally polarized proton beams, one deﬁnes a double-spin asymmetry for a given process by (14)

ALL =

dΔσ dσ(++) − dσ(+−) ≡ , dσ(++) + dσ(+−) dσ

W. Vogelsang

1

10 10 10 10 10

(Data-QCD)/QCD

'V/ V (%)

10

-2

-3

PHENIX Data

-4

KKP FF

-5

Kretzer FF

8

10

(a)

107

-6

-7

-8

40 20 0 -20 -40

b)

4

c)

STAR p+p g jet + X s=200 GeV midpoint-cone rcone =0.4 0.2<η<0.8

6

10

5

10

104

2

10

a)

-1

1/2π dσ/(dηdpT) [pb/GeV]

10

3

10

102

Combined MB

10

2

Combined HT NLO QCD (Vogelsang)

1

0 4

d)

2 0 0

5

10

15

Systematic Uncertainty Theory Scale Uncertainty

1.8 1.4 1.0 0.6 0.2 0

10

-2 -3

Ed V /dp [pb GeV c ]

pT (GeV/c)

data / theory

3

3

-2

E*d V/dp (mbGeV c3)

356

20

30

(a)

NLO pQCD (by W.Vogelsang) CTEQ 6M PDF BFGII FF P=1/2pT,pT,2pT

10

3

3

3

10

40 50 pT [GeV/c]

PHENIX Data

4

10

(b)

2

10

(Data-Theory) Theory

1

2

(b)

1 0

-1

0

2

4

6

8

10

12

14 16 J pT [GeV/c]

√ Fig. 2. – Data for the cross-section for single-inclusive π 0 production pp → π 0 X at s = 200 GeV at mid-rapidity from PHENIX (upper left) [22] and at forward rapidities from STAR (lower left) [24], for mid-rapidity jet production from STAR (upper right) [23], and for mid-rapidity prompt-photon production from PHENIX (lower right) [25]. The lines show the results of the corresponding next-to-leading order calculations [28, 29, 31].

357

QCD spin physics in hadronic interactions

0.8

0.6

qg

0.4

qq + qq + ... 0.2

gg 0 0

5

10

15 p

T

[GeV]

Fig. 3. – Relative contributions to the mid-rapidity NLO cross-section for pp → π 0 X at 200 GeV from gg, qg, and qq initial states [8].

√

s=

where the signs indicate the helicities of the incident protons. The basic concepts laid out so far for unpolarized inelastic pp scattering carry over to the case of polarized collisions: spin-dependent inelastic pp cross-sections factorize into “products” of polarized parton distribution functions of the proton and hard-scattering cross-sections describing spindependent interactions of partons. As in the unpolarized case, the latter are calculable in QCD perturbation theory since they are characterized by large momentum transfer. Schematically, one has for the numerator of the spin asymmetry: (15)

dΔσ =

Δfa ⊗ Δfb ⊗ dΔˆ σab ,

a,b=q,¯ q ,g

where ⊗ denotes a convolution and where the sum is over all contributing partonic channels a + b → c + X producing the desired high-pT or large-invariant mass ﬁnal state. dΔˆ σab is the associated perturbative spin-dependent partonic cross-section, deﬁned as (16)

dΔˆ σab =

1 [dˆ σab (++) − dˆ σab (+−)] , 2

the signs now denoting the helicities of the initial partons a, b. The sensitivity with which one can probe the polarized parton densities will foremost depend on the weights with which they enter the cross-section. Good measures for this are the so-called partonic “analyzing powers”. The latter are just the spin asymmetries (17)

a ˆLL =

σab (+−) dˆ σab (++) − dˆ dˆ σab (++) + dˆ σab (+−)

358

a^LL

W. Vogelsang 1

C

A

0.75

B

0.5

D

0.25 0

A ggogg D qqoqq B qqoqq E ggoqq C qq’oqq’ E qqogg C qq’oqq’ E qqogJ C qgoqg E qqoq’q’ C qgoqJ E qqoll

-0.25 -0.5 -0.75

E -1 -0.8

-0.4

0

0.4

0.8

cosT Fig. 4. – Helicity asymmetries for the most important partonic reactions at RHIC at lowest order in QCD.

for the individual partonic subprocesses. Figure 4 shows these analyzing powers at LO for all partonic reactions. One can see that they are usually very substantial. As an example, consider q q¯ annihilation processes into any number of bosons. Because of helicity conservation at the Standard-Model fermion-boson vertices (for massless quarks), the annihilating quark and anti-quark need to have opposite helicities in order for the reaction to take place, regardless of how many ﬁnal-state bosons couple to them. Thus all reactions of the type q q¯ → γ ∗ , q q¯ → gg, . . ., have a partonic analyzing power −1. The same is true for the s-channel process q q¯ → q q¯ , but not for the process q q¯ → q q¯ which also has a t-channel contribution. Since the partonic cross-sections are calculable from ﬁrst principles in QCD, eq. (15) may be used to determine the polarized parton distribution functions from measurements of the spin-dependent pp cross-section on the left-hand side. The crucial point here is that, as discussed in the previous section, the parton distributions are universal. They are the same in all inelastic processes, not only in pp scattering, but also for example in deeply-inelastic lepton nucleon scattering which up to now has mostly been used to learn about nucleon spin structure. This means that inelastic processes with polarization have the very attractive feature that they probe fundamental and universal spin structure of the nucleon. In eﬀect, one is using the asymptotically free regime of QCD to probe the deep structure of the nucleon. At RHIC, there are a number of sensitive and measurable processes. The key ones are listed in table I, where we also give the dominant underlying partonic reactions and the aspect of nucleon spin structure they probe. We emphasize that, even though we have only shown LO results in ﬁg. 4, the NLO corrections are available for each process

359

QCD spin physics in hadronic interactions

Table I. – Key processes at RHIC for the determination of the parton distributions of the longitudinally polarized proton, along with the dominant contributing subprocesses, the parton distribution predominantly probed, and representative leading-order Feynman diagrams. From [8]. Reaction p p →π+X

Dominant partonic process

Probes

gg → gg

Δg

LO Feynman diagram

q g → qg

p p → jet(s) + X

g g → gg q g → qg

Δg

p p →γ+X p p → γ + jet + X

q g → γq q g → γq

Δg Δg

p p → γγ + X

q q¯ → γγ

Δq, Δ¯ q

g g → c¯ c, b¯b

Δg

q q¯ → γ ∗ → μ+ μ−

Δq, Δ¯ q

p p → DX, BX

p p → μ+ μ− X (Drell-Yan) p p → (Z 0 , W ± )X p p → (Z 0 , W ± )X

q¯ → W ± q q¯ → Z 0 , q ± q q¯ → W , q q¯ → W ±

(as above)

Δq, Δ¯ q

relevant for RHIC-Spin, thanks to considerable eﬀorts made over the past decade or so [28-33]. These calculations bring the theoretical calculations for RHIC-Spin to the same level that has been so successful in the unpolarized case. For each of the processes in table I the parton densities enter with diﬀerent weights, so that each has its own role in helping to determine the polarized parton distributions. Some will allow a clean determination of gluon polarizations, others are more sensitive to quarks and antiquarks. Eventually, when data from RHIC become available for most or all processes, a “global” analysis of the data, along with information from lepton scattering, will be performed . which then determines the Δq, Δ¯ q , Δg. For further details, see subsects. 3 5. We will now discuss some of the recent results from RHIC on Δg, along with the associated theoretical predictions.

360 ALL

W. Vogelsang 0.06

Δ G=G

0.05 0.04 0.03

Δ G=-G

0.02

std

0.01

Δ G=0

0 -0.01 0

1

2

3

4

5

6

7

8

9

p (GeV/c) T

Fig. 5. – Data for the double-spin asymmetry ALL for mid-rapidity single-inclusive π 0 production √ at s = 200 GeV from PHENIX (left) [35] and for jet production from STAR (right) [36].

. 3 3. Access to Δg in polarized proton-proton scattering at RHIC . – The measurement of gluon polarization in the proton is a major focus and strength of RHIC [7, 8]. As we saw, several diﬀerent processes will be investigated at RHIC [7, 8] that are very sensitive to gluon polarization: high-pT prompt photons pp → γX, jet or hadron production ¯ pp → jet X, pp → hX, and heavy-ﬂavor production pp → (QQ)X. An important role for the determination of Δg will also be played by measurements of two-particle, jet-jet (or hadron-hadron) and photon-jet correlations. For these, at the leading-order approximation, the hard-scattering subprocess kinematics can be calculated directly on an event-by-event basis, giving an estimate of the gluon momentum fraction [34]. In √ √ addition, besides the current s = 200 GeV, also s = 500 GeV will be available at RHIC at a later stage. All this will allow to determine Δg(x, Q2 ) in various regions of x, and at diﬀerent scales. Results for ALL in pp → πX are now available from PHENIX [35], and ALL for singleinclusive jet production has been measured by STAR [36] (see also [9]). The results are shown in ﬁg. 5. The curves shown in ﬁg. 5 represent the ALL values calculated at NLO for a range of gluon distributions from [37], from a suggested very large positive gluon polarization (“GRSV-max”) with an integral ΔG = 1.9 at scale Q = 1 GeV, to a “maximally” negative gluon polarization, (“Δg = −g”), for which ΔG(1 GeV2 ) = −1.8. The curves labeled “GRSV-std” represent the best ﬁt of [37] to the polarized DIS data, which has a more “natural” ΔG(1 GeV2 ) of about 0.4, and the results for “Δg = 0” correspond to very little gluon polarization, ΔG(1 GeV2 ) = 0.1. One can see that the RHIC data are already discriminating between the various Δg distributions. For each of the channels studied so far, the results appear to rule out a very large gluon polarization, with either positive or (to a lesser extent) negative gluon polarization. The possibility that ΔG 0 was suggested [38] when the DIS experiments ﬁrst discovered that the quarks (and anti-quarks) carry only very little of the proton spin. A large positive gluon polarization could mask a “bare” quark polarization [1]. At

361

QCD spin physics in hadronic interactions

this point one cannot distinguish, however, between the gluon carrying 70% of the proton spin or carrying none of the proton spin, or determine the sign of the gluon polarization. . 3 4. Weak boson production. – Within the standard model, W bosons are produced through pure V − A interaction. Thus, in hadronic scattering, the helicity of the participating quark and antiquark are ﬁxed in the reaction, making W production an ideal tool to study the spin structure of the nucleon [39]. To leading order, W s are produced through ud¯ → W + , for example. One can deﬁne a parity-violating single-longitudinal spin asymmetry as the diﬀerence of left-handed and right-handed production of W s, divided by the sum: AW L =

(18)

dσ(−) − dσ(+) . dσ(−) + dσ(+)

In the parton model, for the case of the process ud¯ → W + , one will have: (19)

+

= AW L

¯ 2 ) − Δd(x ¯ 1 )u(x2 ) Δu(x1 )d(x ¯ 2 ) + d(x ¯ 1 )u(x2 ) . u(x1 )d(x

To obtain the asymmetry for W − , one interchanges u and d. By identifying the rapidity of the W , yW , relative to the polarized proton, we can obtain direct measures of the quark and antiquark polarizations, separated by quark ﬂavor. The momentum fractions carried by the quarks and antiquarks, x1 and x2 , can at LO be determined from yW : (20)

MW x1 = √ eyW , s

MW x2 = √ e−yW . s

+

From this one can easily see that AW approaches Δu/u in the limit yW 0, whereas L ¯ d. ¯ Expected sensitivities for measurements of for yW 0 the asymmetry becomes −Δd/ ¯ ¯ Δu/u, Δd/d, Δ¯ u/¯ u, Δd/d at RHIC are shown in ﬁg. 6. The experimental diﬃculty at RHIC is that the W is observed through its leptonic decay W → lν, and only the charged lepton is observed. Since none of the detectors at RHIC is hermetic, measurement of missing transverse momentum is not available. It has been shown [40] that distributions in the rapidity of the charged lepton will also be powerful tools for measuring Δu/u, ¯ d. ¯ Δd/d, Δ¯ u/¯ u, Δd/ . 3 5. Global analysis. – The eventual determination of quark and gluon polarizations will require consideration of all existing data through a “global analysis” that makes simultaneous use of results for all probes, from RHIC and from lepton scattering. The technique is to optimize the agreement between measured spin asymmetries, relative to the accuracy of the data, and the theoretical spin asymmetries, by minimizing the associated χ2 function through variation of the shapes of the polarized parton distributions. The advantages of such a full-ﬂedged global analysis program are manifold:

362

W. Vogelsang

1.0

'f/f

RHIC pp s = 500 GeV ³L dt = 800 pb1

AL (W +) _ AL (W )

0.5

'u/u 'd/d 0

'd/d

'u/u

0.5

2 Q 2= MW

GS95LO(A) BS('g=0) 1.0 2 10

1

10

x

Fig. 6. – Expected sensitivity for the ﬂavor-decomposed quark and antiquark polarization at RHIC. Darker points and error bars refer to the sensitivity from AL (W + ) measurements, and lighter ones correspond to AL (W − ).

1) The information from the various reaction channels is all combined into a single result for Δg(x), Δq(x), Δ¯ q (x). 2) The global analysis eﬀectively deconvolutes the experimental information, which in its raw form is smeared over the fractional gluon momentum x, and ﬁxes the gluon distribution at deﬁnite values of x. Figure 7 highlights the importance of this. The ﬁgure shows [41] the contributions of the various regions in gluon momentum fraction to the mid-rapidity spin-dependent cross-section for pp → π 0 X at RHIC, for six diﬀerent sets of polarized parton distributions [37] mostly diﬀering in the gluon distribution. The pion’s transverse momentum was chosen to be 2.5 GeV. One can see that the distributions are very broad, and that the x-region that is mostly probed depends itself on the size and form of the polarized gluon distribution. This makes it very diﬃcult to assign a good estimate of the gluon momentum fraction to a data point at a given pion transverse momentum. The global analysis solves this problem. The further advantages of a global analysis are: 3) State-of-the-art (NLO) theoretical calculations can be used without approximations. 4) It provides a framework to determine errors on the quark and gluon polarizations. 5) Correlations with other experiments, to be included in χ2 and sensitive to degrees of freedom diﬀerent from Δg, are automatically respected. Global analyses of this type have been developed very successfully over many years for unpolarized parton densities. Examples of early work on global analyses of RHIC-Spin and polarized DIS data in terms of polarized parton distributions are [42-44].

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QCD spin physics in hadronic interactions

d'V / dpT dlog10x

pp o SX

3000

40000

'G(P)=1.24 30000

x|0.03y 0.19

7500

x|0.04y0.23

2000

5000

10000

1000

2500 -2

-1 log x 10

0

-2

'G(P)=0

400

x|0.04y 0.27

200

-2

-1 log x 10

0

-1 log x 10

2000

0

'G(P)=0.24

'G(P)=0.60

10000

x|0.03y 0.16

20000

0

pT = 2.5 GeV

-2

-1 log x 10

25000

1500

'G(P)=-0.45

20000

'G(P)=-1.24

1000

x|0.008y 0.08

15000

x|0.03y 0.13

500

10000

0

5000

-500

-2

-1 log x 10

0

-2

-1

log10x

Fig. 7. – NLO dΔσ/dpT d log10 (x) (arbitrary normalization) for the reaction pp → π 0 X at RHIC, for pT = 2.5 GeV and six diﬀerent values for ΔG(μ2 ) at μ ≈ 0.4 GeV [37]. The shaded areas denote in each case the x-range dominantly contributing to dΔσ. From [41].

4. – Collisions of transversely polarized protons . 4 1. Transversity. – Besides the unpolarized and the helicity-dependent densities, there is a third set of leading-twist parton distributions, transversity [45]. They measure the net number (parallel minus antiparallel) of partons with transverse polarization in a transversely polarized nucleon: (21)

δf (x, Q2 ) = f ↑ (x, Q2 ) − f ↓ (x, Q2 ).

In a helicity basis, one ﬁnds that transversity corresponds to a helicity-ﬂip structure, which precludes a gluon transversity distribution at leading twist [46]. It also makes transversity a probe of chiral symmetry breaking in QCD [47]: perturbative-QCD interactions preserve chirality, and so the helicity-ﬂip required to make transversity non-zero must primarily come from soft non-perturbative interactions for which chiral symmetry is broken.

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In contrast to the distributions f and Δf , we have essentially no knowledge from experiment so far about the transversity distributions δf . Again the fact that perturbative interactions in the Standard Model do not change chirality means that inclusive DIS is not useful. Collins, however, showed [47] that properties of fragmentation might be exploited to obtain a “transversity polarimeter”: a pion produced in fragmentation will have some transverse momentum kT with respect to the momentum of the transversely polarized fragmenting parent quark. There may then be a correlation of the form T · (Pπ × k⊥ ). The fragmentation function associated with this correlation is the Collins S function. It makes a leading-power [47] contribution to the single-spin asymmetry A⊥ in the reaction ep↑ → eπX: (22)

T | sin(φ + φS ) A⊥ ∝ |S

e2q δq(x)H1⊥,q (z),

q

where φ (φS ) is the angle between the lepton plane and the γ ∗ π-plane (and the transverse target spin). HERMES has reported clear signs of a nonvanishing Collins asymmetry in ep↑ scattering [13]. Recently, ﬁrst independent information on the Collins functions has come from BELLE measurements in e+ e− annihilation [48]. This has allowed to obtain a ﬁrst “glimpse” of transversity from a combined analysis of single-transverse spin asymmetries in Semi-Inclusive Deep-Inelastic Scattering (SIDIS) and the BELLE data [49]. Clean and direct information on transversity might be gathered from polarized proton-proton collisions at RHIC, using the Drell-Yan process [7, 8]. In pp collisions, however, the Drell-Yan process probes products of valence quark and sea antiquark distributions. It is possible that antiquarks in the nucleon carry only little transverse polarization since the perturbative generation of transversity sea quarks from g → q q¯ splitting is missing. Also, at RHIC the partonic momentum fractions are fairly small, so that the denominator of ATT is large. NLO studies [50] estimate the size of ATT at RHIC to be at most a few per cent. It has recently been proposed to add polarization to planned pp ¯ experiments at the GSI-FAIR facility, and to extract transversity from measurements of ATT for the DrellYan process [51, 52]. Initially, experiments could be performed in a ﬁxed-target mode, using the 15 GeV antiproton beam. At later stages of operations, there are plans for an asymmetric p¯p collider, with an additional proton beam of energy 3.5 GeV. The results from such measurements would be complementary to what can be obtained from RHIC or SIDIS. In p¯p collisions the Drell-Yan process mainly probes products of two quark densities, δq × δq, since the distribution of antiquarks in antiprotons equals that of quarks in the proton. In addition, kinematics in the proposed experiments are such that rather large partonic momentum fractions, x ∼ 0.5, are probed. One therefore accesses the valence region of the nucleon, where the polarization of partons is expected to be large. Estimates [52] for the GSI PAX and ASSIA experiments show that the expected spin asymmetry ATT should indeed be very large, of order 30% or more. The theoretical framework for GSI kinematics is somewhat more involved than for RHIC, since in the region to be accessed higher-order corrections to the partonic cross-

365

QCD spin physics in hadronic interactions

NLO NNLO full resummed 1st order expansion 2nd 3rd 4th 6th 8th

2

V / VLO

10

3

10

V / VLO

10

10

1

NLO NNLO full resummed 1st order expansion 2nd 3rd 4th 6th 8th

1

0

0

10 2.5

3

4 3.5 M (GeV)

4.5

5

10 2

4

8

6

10

12

M (GeV)

Fig. 8. – “K-factors” for the Drell-Yan cross-section in ﬁxed-target p¯p collisions at s = 30 GeV2 (left) and for an asymmetric collider mode with s = 210 GeV2 (right), as functions of the lepton pair invariant mass M . For details, see [55].

sections are particularly important. The NLO corrections for the transversely polarized Drell-Yan cross-section have been calculated in [53]. In the region of interest here, however, certain logarithmic terms in the partonic cross-section become important to all orders in perturbation theory and need to be resummed. Such a “threshold resummation” is a well-established technique in QCD [54]. In the case of ATT for the Drell-Yan process, it has been addressed in detail recently in [55]. Figure 8 shows results of [55] for the K (= σ higher order /σ LO ) factors for the unpolarized Drell-Yan cross-section at s = 30 GeV2 (left) and s = 210 GeV2 (right), at NLO, NNLO, and for the Next-to-Leading Logarithmic (NLL) resummed case, along with various higher-order expansions of the resummed result. As can be seen, the corrections are very large, in particular in the lower-energy case. Figure 9 shows the corresponding spin asymmetries ATT . ATT turns out to be extremely robust and remarkably insensitive to higher-order corrections. Perturbative corrections thus make the cross-sections larger

0.4

0.4

0.35

ATT

ATT

0.3 0.3

0.2 LO NLO res. (P0=0) res. (P0=0.3 GeV)

0.25

0.2

2.5

3

4 3.5 M (GeV)

4.5

5

LO NLO res. (P0=0) res. (P0=0.3GeV)

0.1 2

4

6

8

10

12

M (GeV)

Fig. 9. – Corresponding spin asymmetries ATT (φ = 0) at LO, NLO and for the NLL resummed case.

366

W. Vogelsang

independently of spin. They would therefore make easier the study of spin asymmetries, and ultimately transversity distributions. For further details, including a discussion on the role of nonperturbative eﬀects, see [55]. We ﬁnally note that it has recently also been proposed [56] to study the spin asymmetry ATT for the reaction p¯p → π 0 X, which would give additional information on transversity. . 4 2. Single-transverse spin asymmetries. – As we mentioned in the introduction, there have been observations of striking single-transverse-spin eﬀects in hadronic scattering. Large single-spin asymmetries in single-inclusive pion production persist to RHIC energies, as seen by STAR [57] and BRAHMS [58]. At PHENIX [59], the asymmetry was studied in the mid-rapidity regime, where it was found to be small. The observed large size of SSAs in hadronic scattering has presented a challenge for QCD theorists [45, 60]. Two mechanisms have been discussed [61-64] and extensively applied [63, 65-67] in phenomenological studies. The ﬁrst relies on the use of transversemomentum–dependent parton distributions for the transversely polarized proton. For these distributions, known as Sivers functions [61], the parton transverse momentum is assumed to be correlated with the proton spin vector, so that spin asymmetries naturally arise from the directional preference expressed by that correlation. The other mechanism (referred to as Efremov-Teryaev-Qiu-Sterman (ETQS) mechanism) is formulated in terms of the collinear factorization approach and twist-three transverse-spin-dependent quarkgluon correlation functions of the proton [62-64]. A concept common to both mechanisms is the factorization of the spin-dependent cross-section into functions describing the distributions of quarks and gluons in the polarized proton, and partonic hard-scattering cross-sections, calculated in QCD perturbation theory. The question of which mechanism should be used in the analysis of a single-spin asymmetry is therefore primarily tied to the factorization theorem that applies for the single-spin observable under consideration. For the single-inclusive process p↑ p → πX, there is only one hard scale, the transverse momentum ⊥ of the produced pion, and the SSA is power-suppressed (“higher-twist”) by 1/⊥ . In this case, one can prove a collinear factorization theorem in terms of the quark-gluon correlation functions [63, 64], and the ETQS mechanism applies. On the other hand, the observables typically investigated in deep-inelastic lepton scattering are characterized by a large-scale Q (the virtuality of the DIS photon) and by the much smaller, and also measured, transverse momentum q⊥ of the produced hadron. In this two-scale case, single-spin asymmetries may arise at leading twist, i.e. not suppressed by 1/Q. The relevant factorization theorem is formulated in terms of transverse-momentum–dependent (TMD) functions [68-71], in particular the Sivers functions. In the following, we will describe both types of observables, the single-spin asymmetry in single-inclusive scattering and the asymmetry in a “two-scale” situation. We will also discuss recent work [72] that has connected the two mechanisms in the case of the singlespin asymmetry for the Drell-Yan process.

QCD spin physics in hadronic interactions

367

. 4 3. Single transverse-spin asymmetry in high-⊥ pion production in pp collisions. – The single-transverse spin asymmetry in the process pp → πX is among the simplest spin observables in hadronic scattering. One scatters a beam of transversely polarized protons oﬀ unpolarized protons and measures the numbers of pions produced to either the left or the right of the plane spanned by the momentum and spin directions of the initial polarized protons. This deﬁnes a “left-right” asymmetry. Equivalently, the asymmetry may be obtained by ﬂipping the spins of the initial polarized protons. This gives rise to the customary deﬁnition (23)

AN (, sT ) ≡

Δσ(, sT ) σ(, sT ) − σ(, −sT ) ≡ , σ(, sT ) + σ(, −sT ) σ()

where sT denotes the transverse spin vector and the four-momentum of the produced pion. We assume the pions to be produced at large transverse momentum ⊥ . As we mentioned above, measurements of single-spin asymmetries in hadronic scattering experiments over the past three decades have shown spectacular results. Large asymmetries of up to several tens of per cents were observed at forward (with respect to the polarized initial beam) angles of the produced pion. Until a few years ago, all these experiments were done with a polarized beam impeding on a ﬁxed target [12]. These experiments necessarily had a relatively limited kinematic reach, in particular in ⊥ . Now, with RHIC, it has become possible to investigate AN at higher energies [57-59], in a kinematic regime where the theoretical description is bound to be under better control. Indeed, as we saw in ﬁg. 2, at RHIC also the unpolarized pion production cross-section has been measured, in the same kinematic regimes as covered by the measurements of the single-spin asymmetries, and is well described by the NLO perturbative calculations based on collinear factorization. Despite the conceptual simplicity of AN , the theoretical analysis of single-spin asymmetries in hadronic scattering is remarkably complex. The reason for this is that the asymmetry for a single-inclusive reaction like p↑ p → πX (the symbol ↑ denoting from now on the polarization of the proton) is power-suppressed as 1/⊥ in the hard scale set by the observed large pion transverse momentum. This is in contrast to typical double (longitudinal or transverse) spin asymmetries that usually scale for large ⊥ . In essence, the leading-twist part cancels in the diﬀerence σ(, sT ) − σ(, −sT ) in the numerator of AN . That AN must be power-suppressed is easy to see: the only leading-power distribution function in the proton associated with transverse polarization is transversity. For transversity to contribute, the corresponding partonic hard-scattering functions need to involve a transversely polarized quark scattering oﬀ an unpolarized one. Cross-sections for such reactions vanish in perturbative QCD for massless quarks because they require a helicity-ﬂip for the polarized quark, which the perturbative q q¯g vertex does not allow. In addition, a non-vanishing single-spin asymmetry requires the presence of a relative interaction phase between the interfering amplitudes for the diﬀerent helicities. At leading twist this phase can only arise through a loop correction, which is of higher order in the strong-coupling constant and hence leads to a further suppression. These arguments are,

368

W. Vogelsang

P, ST

P, ST ⊗ P

=

σ, B pc ⊗

P

⊗ π()

Fig. 10. – Generic Feynman diagram contributing to the single transverse-spin asymmetry for inclusive pion production in proton-proton scattering at leading twist (twist-three). The polarized cross-section can be factorized into convolutions of the following terms: twist-three quark-gluon correlation functions for the transversely polarized proton, parton distributions for the unpolarized proton, pion fragmentation functions, and hard-scattering functions calculable in QCD perturbation theory.

in fact, more than 30 years old [73] and led to the general expectation that single-spin asymmetries should be very small, in striking contrast with the experimental results. Power-suppressed contributions to hard-scattering processes are generally much harder to describe in QCD than leading-twist ones. In the case of the single-spin asymmetry in pp → πX, a complete and consistent framework could be developed, however [63]. It is based on a collinear factorization theorem at non-leading twist that relates the singlespin cross-section to convolutions of twist-three quark-gluon correlation functions for the polarized proton with the usual parton distributions for the unpolarized proton and the pion fragmentation functions, and with hard-scattering functions calculated from an interference of two partonic scattering amplitudes: one with a two-parton initial state and the other with a three-parton initial state [62, 63]. A generic Feynman diagram for the scattering process is shown in ﬁg. 10, along with its factorization just described. Typical Feynman diagrams for the hard-scattering in case of quark-gluon scattering are shown in ﬁg. 11. As was shown in [63], the phase needed to generate a single-spin asymmetry arises naturally in the hard-scattering functions, even at tree level, thanks to its pole structure. Imaginary parts arise from the scattering amplitude with an extra initial-state gluon when its momentum integral is evaluated by the residues of unpinched poles of the propagators indicated by the bars in ﬁg. 11. The on-shell condition associated with any such pole ﬁxes the momentum fraction of the extra initial-state gluon. Roughly speaking, all of the diagrams in ﬁg. 11 provide an unpinched pole at x1 = x2 [63]. After a collinear expansion, one ﬁnds the following expression for the spin-dependent cross-section: (24)

E

d3 Δσ(sT ) ∝ Dc→h (z) ⊗ fb (x ) ⊗ 3 d a,b,c

d Ta,F (x, x) ⊗ Hab→c (ˆ ⊗ Ta,F (x, x) − x s, tˆ, u ˆ), dx

369

QCD spin physics in hadronic interactions

x2 P + k⊥

x1 P

x1 P

x2 P + k⊥

(x2 − x1 )P + k⊥

(x2 − x1 )P + k⊥

x P

x P

x P

x P

(a)

x1 P

(x2 − x1 )P + k⊥

x2 P + k⊥

x P

x P

(x2 − x1 )P + k⊥

x1 P

x P

x 2 P + k⊥

x P

(b)

Fig. 11. – Speciﬁc examples of twist-three diagrams for generic (a) initial-state and (b) ﬁnal-state interactions.

where sˆ, tˆ, u ˆ are the partonic Mandelstam variables, fb (x ) and Dc→h (z) are the usual unpolarized parton distribution and fragmentation functions, respectively, and the twistthree functions Ta,F are deﬁned as

(25)

Ta,F (x1 , x2 ) =

dy1− dy2− ix1 P + y1− +i(x2 −x1 )P + y2− e 4π × P, sT |ψ¯a (0)γ + sT σn¯n Fσ+ (y2− ) ψa (y1− )|P, sT .

The sum in eq. (25) runs over all ﬂavors, with Hab→c the associated hard-scattering functions, calculated from diagrams like the ones shown in ﬁg. 11. The Hab→c are power-suppressed with respect to their unpolarized counterparts, and they are also down √ by αs due to the additional gluon that couples in the diagrams. As one can see, a speciﬁc combination of the Ta,F occurs in eq. (25), involving the derivative of Ta,F . Such derivative terms arise in the collinear expansion, because there are terms proportional to the initial partons’ transverse momenta in the δ-functions ﬁxing the light-cone momentum fractions [63]. The other terms in eq. (25) that involve Ta,F (x, x) without a derivative, were recently derived in [67]. Based on eq. (25), one can perform some phenomenology for the single-spin asymmetry, examining the salient features of the new RHIC data and of the earlier E704 ﬁxed-target pion production data. In [67] a simple model ansatz for the twist-three

370

W. Vogelsang AN

AN

p p at s=200GeV n

0.2

S0

p p at s=200GeV

S+

S-

+

K

0.1

n

0

-0.1

0.1

K

-

0.1

0 0 -0.1

-0.1 0.2

0.3

0.4

0.5

0.6

xF

0.25

0.3

xF

0.35

0.25

0.3

xF

0.35

Fig. 12. – Comparison of the single-spin asymmetries AN using the ﬁt results of [67] to the RHIC data by the STAR [57] (left) and BRAHMS [58] (right) collaborations. The lower dotted line in the ﬁgure shows the contribution to AN by the “non-derivative” terms alone.

¯ s, s¯) was made, relating them quark-gluon correlation functions Ta,F (x, x) (a = u, u ¯, d, d, to their unpolarized leading-twist counterparts: (26)

Ta,F (x, x) = Na xαa (1 − x)βa fa (x).

The parameters in this ansatz were determined through a “global” ﬁt to the experimental data for AN as functions of Feynman-xF . As one example, ﬁg. 12 compares the ﬁt results to the experimental data from RHIC. The overall quality of the ﬁt is relatively poor, but the basic trends of the data are reproduced. With new precise experimental information expected to arrive from RHIC, however, we will be entering an era where detailed global analyses of the data on AN will become possible. . 4 4. Single transverse-spin asymmetries in two-scale situations. – As we discussed earlier, further exciting single-spin phenomena may occur when one has observables characterized by two very separate momentum scales, a hard scale Q, and a much lower measured transverse momentum q⊥ . For such observables one may have a factorization in terms of transverse-momentum–dependent (TMD) functions. Among these are the Sivers functions [61], which are TMD parton distributions. They represent distributions of unpolarized quarks of ﬂavor a in a transversely polarized nucleon, through a correlation between the quark’s transverse momentum k⊥ and the nucleon polarization vector sT :

(27)

1 sT · (P × k⊥ ) , fˆa (x, k⊥ , sT ) = fa (x, k⊥ ) + ΔN fa (x, k⊥ ) 2 |sT | |P | |k⊥ |

371

QCD spin physics in hadronic interactions

where the function fa (x, k⊥ ) with k⊥ = |k⊥ | is the unpolarized TMD parton distribution, and ΔN fa denotes the Sivers function. The latter is also sometimes written as (28)

ΔN fa (x, k⊥ ) ≡ −

2k⊥ ⊥a f (x, k⊥ ). MN 1T

For the Sivers function to exist, ﬁnal/initial-state interactions are required, as well as an interference between diﬀerent helicity Fock states of the nucleon. In the absence of interactions, the Sivers function would vanish by time-reversal invariance of QCD, hence it is often referred to as a “naively time-reversal-odd” distribution. As was shown in [74-76], the interactions are represented in a natural way by the gauge link that is required for a gauge-invariant deﬁnition of a TMD parton distribution. Interference between diﬀerent helicity Fock states implies the presence of orbital angular momentum [74, 77]. The Sivers functions have also been linked to spatial distributions of partons in the proton [77-80]. These properties motivate the study of this function. The Sivers function will contribute to the target SSA in semi-inclusive DIS, but also to SSAs in polarized pp scattering processes such as the Drell-Yan process and di-jet or jet-photon correlations. A particularly interesting aspect is that the Sivers functions are not universal in the usual sense, i.e. they are not the same in each hard-scattering process. This might at ﬁrst sight appear to make the study of these functions less interesting. However, the non-universality has in fact a clear physical origin, and its closer investigation has turned out to be an extremely important and productive development in QCD. We will only describe it qualitatively here. We have already mentioned that, in order to be non-zero, the Sivers functions require an additional ﬁnal/initial-state interaction, represented by the gauge link that makes the function gauge invariant. This may be viewed as a rescattering of the parton in the color ﬁeld of the nucleon remnant. Depending on the process, the associated color Lorentz forces will act in diﬀerent ways on the parton. In DIS, so far explored experimentally, the ﬁnal-state interaction between the struck parton and the nucleon remnant is attractive. In contrast, for the Drell-Yan process it is repulsive. Therefore, the Sivers functions contribute with opposite signs to the single-spin asymmetries for these two processes [74-76]. This is a remarkable and fundamental prediction that really tests all concepts we know of for analyzing hard-scattering reactions in strong interactions. The veriﬁcation of the predicted non-universality of the Sivers functions is an outstanding challenge in strong-interaction physics. Let us give a simple QED example [81] that captures the essential physics. In ﬁg. 13(a) we consider a “toy” DIS process. A transversely polarized charge-less “hadron”, consisting of particles with electric charges +1 and −1, is probed by a highly virtual photon. In order not to be forced to vanish by time-reversal invariance, a single-spin asymmetry for the process requires the presence of an interaction phase. Such a phase may be generated by a rescattering of the struck “parton” in the ﬁeld of the “hadron remnant”, by exchange of a photon as shown in the ﬁgure. The amplitude with the additional exchanged photon interferes with that without the photon. More precisely, two diﬀerent phases appear, the S- and P -wave Coulomb phases. The diﬀerence of these phases is infrared-ﬁnite and

372

W. Vogelsang

+

γ∗

γ∗

−

+

+ −

−

(a)

(b) γ∗

γ∗

(gb)

r

r

r

attractive

repulsive

(c)

(d)

Fig. 13. – (a), (b) Simple QED example for process-dependence of the Sivers functions in DIS and the Drell-Yan process. (c), (d) Same for QCD.

generates the single-spin asymmetry [74]. As the electric charges of the two interacting particles are opposite, this ﬁnal-state interaction is attractive. Now consider a similar model for the Drell-Yan process in ﬁg. 13(b). “Partons” of opposite charge annihilate to produce a highly virtual photon. The interaction generating the phase in this case is “initial-state” and is between the remnant of the transversely polarized “hadron” and the initial parton from the other, unpolarized, “hadron”. These necessarily have identical charges, and the interaction is repulsive. As a result, the spineﬀect in this case needs to be of opposite sign as that in DIS. These simple models are readily generalized to true hadronic scattering in QCD. In DIS, the ﬁnal-state interaction is through a gluon exchanged between the 3 and ¯ 3 states of the struck quark and the nucleon remnant, which is attractive, as indicated in ﬁg. 13(c). In the Drell-Yan process, the interaction is between the 3 and 3 states (or ¯ 3 and ¯ 3) and therefore repulsive, as shown in ﬁg. 13(d). This is the essence of the—by now widely quoted—result that the Sivers functions contributing to DIS and to the Drell-Yan process have opposite sign [74-76, 82]: (29)

& & ⊥a f1T (x, k⊥ )&

DY

& & ⊥a = −f1T (x, k⊥ )&

. DIS

In the full gauge theory, the phases generated by the additional (ﬁnal-state or initialstate) interactions can be summed to all orders into a “gauge-link”, which is a pathordered exponential of the gluon ﬁeld and makes the Sivers functions gauge-invariant.

QCD spin physics in hadronic interactions

373

Fig. 14. – “Asymmetric jet correlation”. The proton beams run perpendicular to the drawing.

The non-universality of the Sivers functions is then reﬂected in a process-dependence of the space-time direction of the gauge-link. The crucial role played by the gauge link has given rise to intuitive model interpretations of single-spin asymmetries in terms of spatial deformations of parton distributions in a transversely polarized nucleon [79]. The process-dependence of the Sivers functions can also be tested in more complicated QCD hard scattering. An example is the single-spin asymmetry in di-jet angular correlations [83, 84]. The basic idea is very simple and presented in ﬁg. 14. As we discussed above, the Sivers function represents a correlation of the form sT · (P × k⊥ ) between the transverse proton polarization vector, its momentum, and the transverse momentum of the parton relative to the proton direction. In other words, if there is a Sivers-type correlation, then there will be a preference for partons to have a component of intrinsic transverse momentum to one side, perpendicular to both sT and P . Suppose now for simplicity that one observes a jet in the direction of the proton polarization vector, as shown in ﬁg. 14. A “left-right” imbalance in k⊥ of the parton will then aﬀect the Δφ distribution of jets nearly opposite to the ﬁrst jet and give the cross-section an asymmetric piece around Δφ = π. Tremendous progress has been made recently in our understanding of the gauge links for this observable [85]. The more involved color structure of the hard-scattering functions has profound consequences on the gauge links. It was found that if all interactions are summed up, the resulting gauge link for the TMD parton distributions takes a much more complicated form. It does not involve just the color charge of the relevant parton, but has in general knowledge about the full hard-scattering process and its color structure. As such, diﬀerent correlators were found to appear in diﬀerent partonic channels, even if the parton type entering from the polarized proton is the same. This observation makes the non-universality of the TMD parton distributions much more dramatic than previously indicated by their sign diﬀerence between the SIDIS and Drell-Yan processes. Reference [86] stressed that the non-trivial gauge link structure found in [85] implies

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that a “standard” factorization in terms of universal TMD parton distributions cannot hold for this process. Standard factorization still holds to ﬁrst order in perturbation theory [87] but breaks down beyond [88]. The STAR collaboration has now published ﬁrst experimental data for the single-spin asymmetry in di-jet correlations [89], which show an asymmetry consistent with zero. Clearly, the ﬁeld is moving ahead at a remarkable pace! . 4 5. Relation between mechanisms for single-spin asymmetries. – ¿From what we discussed so far, the two mechanisms for single-spin asymmetries, twist-three quarkgluon correlation functions on the one hand and TMD distributions on the other, might appear to be essentially unrelated. However, one can make an argument that a consistent theoretical description of the SSA for a hard process over its full kinematical regime requires both mechanisms to be present and to contain the same physics in the region where they both apply. To give a speciﬁc example, let us consider the SSA for the DrellYan process when the invariant mass Q of the pair as well as its transverse momentum q⊥ are measured [72]. At relatively large pair transverse momentum, q⊥ ∼ Q, there is only one large scale, and the SSA will be power-suppressed in that scale. This directs us to use the ETQS mechanism with its collinear factorization involving the twist-three quark-gluon correlation functions and corresponding hard-scattering functions calculated at lowest order . from partonic 3 → 2 processes, as described in subsect. 4 3. We can next investigate what happens in this case when we make the ratio q⊥ /Q small, keeping however both scales perturbative, Q q⊥ ΛQCD . We refer to q⊥ in this regime as “moderate” transverse momentum. The ETQS mechanism will still apply here (even though the hard-scattering functions will develop large logarithms of the ratio q⊥ /Q that will eventually need to be resummed to all orders in the strong coupling). At the same time, however, the factorization in terms of TMD distributions applies now [68-70], which involves the Sivers functions. If both mechanisms are internally consistent, they must describe the same physics in this region.

Fig. 15. – Cartoon of diﬀerent kinematic regions q⊥ ∼ Q and q⊥ ΛQCD relevant for the single-spin asymmetry in the Drell-Yan process. In the region of overlap, Q q⊥ ΛQCD both mechanisms describe the same physics [72].

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In recent publications [72], it was demonstrated that the two mechanisms indeed provide the same description of the single-spin asymmetry for the Drell-Yan process in the regime ΛQCD q⊥ Q, and that there is a direct correspondence between the Sivers functions and the twist-three quark-gluon correlation functions. The key observation is that, at moderate transverse momentum, the Sivers function may be calculated perturbatively, using the twist-three quark-gluon correlation functions. In other words, the ETQS mechanism generates a non-vanishing Sivers function in this kinematic regime. These results may in some sense be viewed as establishing a uniﬁcation of the two mechanisms. At large q⊥ , the ETQS mechanism applies. At moderate transverse momentum, a smooth transition from the ETQS mechanism to the one based on TMD factorization occurs, with the two approaches containing the same physics. At yet lower q⊥ (∼ ΛQCD ), the TMD factorization still applies, containing in a natural way the transition from perturbative to non-perturbative physics. A cartoon of this connection between the two mechanisms is given in ﬁg. 15. This uniﬁed picture should prove to be the best approach to phenomenological studies of single-spin asymmetries. ∗ ∗ ∗ I thank M. Anselmino for the kind invitation to the International School of Physics Enrico Fermi “Strangeness and Spin in Fundamental Physics”. I am grateful to M. Stratmann and F. Yuan for useful discussions. I am also grateful to the US Department of Energy (contract number DE-AC02-98CH10886) for providing the facilities essential for the completion of his work.

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Spin structure of the nucleon: RHIC results and prospects for J-PARC N. Saito High Energy Accelerator Research Organization (KEK) 1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan

Summary. — This lecture is delivered to provide an up-to-date picture of the spin structure of the nucleon basing on the recent results from RHIC experiments. A future prospect of the possible experiments at J-PARC is also given.

1. – Introduction Spin is one of the most important concepts in the development of modern physics. The concept appears at many diﬀerent levels including very large-scale phenomena such as spiral galaxies and very microscopic levels such as the space-time structure described in spin networks. In particle/nuclear physics the concept is also very important, since it couples to the angular-momentum conservation originated in the rotational symmetry of space. The statistical characteristic of an elementary particle is also determined by the spin. On the other hand, the structure of the nucleon has been investigated for many years especially by using the lepton scattering. Such experiments provided the basis of the quantum chromodynamics through the discovery of the asymptotic freedom. The discovery potential of hadron colliders would have never been so promising without a c Societ` a Italiana di Fisica

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detailed knowledge of the nucleon structure. This knowledge is also very fundamental, since more than 99% of the visible universe consists of nucleons. Given these backgrounds, it is understandable that the phenomenon called “proton spin crisis” was considered with great interest. The quark-spin contribution to the proton spin was measured to be small in lepton scattering experiments [1]. However, the nucleon already had gone through so many crises. For example, the mass of the nucleon cannot be explained from the bare quark mass. Instead, most of the mass is dynamically produced. The second example is the momentum of the proton. When the fractional momentum, x, carried by quarks is integrated over 0 ≤ x ≤ 1, it yields only ∼ 50% of the total momentum. The rest of the momentum is carried by the gluons. This is referred to as momentum sum rule. Similarly there is a spin sum rule to explain the proton spin from quark spin, gluon spin and their orbital motion: (1)

1 proton 1 = ΔΣ + Δg + Lq + Lg . 2 2

The fractional quark-spin contribution ΔΣ is obtained to be 0.1–0.3 from lepton scattering data combined with the β-decay constants of octet baryons, which is signiﬁcantly smaller than naive expectation, and called “proton spin crisis”. The gluon-spin contribution Δg and quark and gluon orbital contributions, Lq and Lg , respectively, remain unmeasured. These components are the 1st moment of the corresponding Bjorken x-dependent functions at a certain energy scale, Q2 , e.g. (2)

! Δg Q2 =

1

! g x, Q2 dx.

0

An example of Q2 evolution of the 1st moments is displayed in ﬁg. 1. There is a theory guideline for the separation of the proton spin described in the equation, by Ji, Tang, and Hoodboy [2]: (3)

1 1 3Nf ΔΣ + Lq = ; 2 2 3Nf + 16

Δg + Lq =

16 1 . 2 3Nf + 16

Each corresponds to 0.18–0.26 and 0.32–0.24, respectively, depending on the number of ﬂavors Nf , three through six. Once Δg is measured to a reasonable precision, then we will know roughly how the spin of the proton is distributed to each component. The above spin sum rule is for the longitudinal spin structure of the proton. Triggered by the “spin crisis” and also very interesting experimental data such as the unexpectedly large AN for forward pion production [3, 4], there has been a lot of progress in understanding the transverse-spin structure, too. There are several good reviews [5] on this subject. Readers should refer to those reviews for a more comprehensive picture.

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Fig. 1. – An example of the Q2 evolution of the proton spin components in leading order is displayed.

2. – Polarized parton distribution functions In ﬁg. 2, we schematically show the quark and gluon distributions in the proton. The parton distributions are deﬁned as the momentum distribution of quarks and gluons,

Fig. 2. – Deep inelastic scattering and parton distribution functions are depicted.

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Fig. 3. – Spin-independent and -dependent parton distributions are depicted. There is no transverse gluon distribution that can be deﬁned for the proton, which is spin 1/2.

where the momentum is denoted by x, the momentum fraction of the proton carried by the partons. There are three independent quark distributions deﬁned at the leading twist as shown in ﬁg. 3. In the helicty distributions, the diﬀerence of momentum distributions of the same and opposite helicities is measured. In the transversity distributions, transversely polarized quarks in the transversely polarized proton are measured. As for the gluon distributions, there is no transverse distribution deﬁned at the leading twist; while transverse distribution requires the spin ﬂip, the spin ﬂip of the gluon (spin=1) cannot be absorbed by the proton (spin=1/2). In ﬁg. 4, we show the current understanding of polarized parton distributions obtained through next-to-leading order analysis of data from lepton scattering experiments [6]. The valence quark distributions, Δuv (x) and Δdv (x), are determined to a reasonable precision. The gluon distribution Δg(x) and sea-quark distribution Δqsea (x) remain to be determined better. Uncertainties of gluon polarization will be discussed in some details later. Sea-quark polarization is much improved by the Hermes data on semi-inclusive DIS. However, the majority of the global analysis groups assumes SU (3)ﬂavor -symmetric sea due to the limited precision obtained so far. To go beyond the current picture of SU (3)symmetric sea, more ﬂavor-sensitive measurements are necessary. 3. – Experimental eﬀorts There are many experimental eﬀorts triggered by the “spin crisis”. Ongoing and future experiments are summarized in table I. Experimental data to determine the spin structure of the nucleon so far is largely dominated by the lepton scattering data with ﬁxed target. The eﬀorts are being extended to cover various reactions including pp and

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Fig. 4. – Polarized parton distribution functions extracted from inclusive DIS data and π 0 production in pp collisions at RHIC through next-to-leading order Q2 evolution.

Table I. – Current and future spin physics facilities. Experiment

Reaction

Beam energies

Status

Hermes at DESY Compass at CERN RHIC-Spin at BNL J-Lab

e± p, d μp, d pp e− N

completed in 2007 continuing continuing continuing

eRHIC at BNL 12 GeV upgrade at J-Lab ELIC at J-Lab J-PARC GSI-FAIR

e− p e− N e− p pp, pA p¯p

Ee = 27 GeV ﬁxed target Eμ = 160 GeV ﬁxed target √ s = 200, 500 GeV collider Ee ∼ 5 GeV ﬁxed target √ s = 100 GeV collider Ee = 12 GeV ﬁxed target √ s = 20–65 GeV collider Ep = 50 GeV ﬁxed target √ s ∼ 15 GeV collider

planned planned planned under construction planned

FINeSSE

νN elastic

Eν = 1 GeV ﬁxed target

proposed

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ep colliders with the successful operation of the ﬁrst-ever-built polarized pp collider, RHIC [7]. Future facilities will cover an extended x-range, and also the elastic scattering νN → νN [8, 9] which could provide the 1st moment of the polarized strange quark Δs. These experimental facilities utilize diﬀerent probes to pin down the spin structure. Each probes has its advantages and disadvantages. As we are going to see below, it is important to use all probes to obtain a comprehensive picture of the spin structure. . 3 1. Electromagnetic interaction. – The classical, however, still leading probe in the study of structures is the electromagnetic interaction. A simple lepton scattering interaction is well understood and precisely calculable. There are many advantages in this reaction including the clear deﬁnition of the kinematics which requires only the four momenta of the incoming and outgoing lepton. It is directly sensitive to an electric-charge squared, which results in two diﬃculties: i) separation of the quark and anti-quark contribution, and ii) identiﬁcation of the gluon contribution. Gluon would come into play only in the subleading contribution. This type of measurement has been done at CERN (Compass), DESY (Hermes), and J-Lab. Unfortunately one of the successful experiment Hermes has been terminated as of Summer 2007. Higher-energy polarized ep colliders are planned at J-Lab (ELIC) and at BNL (eRHIC). Drell-Yan production of lepton pairs also originates in the QED subprocess, which will be achievable at RHIC. Luminosity developments are underway by improving the accelerator every year. An experiment, PAX, at GSI, is planned to use p¯↑ p↑ collisions to measure transversity distributions in the nucleon. . 3 2. Strong interaction. – Until recently the gluon contribution in the spin structure of the nucleon has been poorly known. By using the strong interaction, we can obtain a high sensitivity to the gluon contribution, since gluon-related processes are the leading contribution. Therefore, the direct measurement of the gluon polarization using the polarized pp collider, RHIC, has been longed for. Jet production is one of the most promising processes due to its abundance. The leading hadron can be used as a jet surrogate, too. In both cases the leading processes are gg, gq and qq scattering and the gg and gq dominate in the lower-pT region where statistics is high. The Star experiment at RHIC presented its recent results on ALL for √ jet production in pp collision at s = 200 GeV from Run-3 (2003) [10]. The Phenix experiment has published the ALL for π 0 production in Run-5 [11]. They now obtained new results from Run-6, as shown in ﬁg. 5, which will be discussed later in some details. The gold-plated mode for the gluon polarization measurement is still prompt photon production, which is dominated by the gluon Compton process, gq → γq, which is displayed in ﬁg. 6. In a sense, this is a half strong and a half electromagnetic, since its leading contribution starts from O(αS αEM ). Similarly it goes for photoproduction of charmed mesons or hadron pairs in the photon-hadron interaction, which is being explored in Hermes and Compass. Here the real/virtual photon and gluon fuse into a

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Fig. 5. – Asymmetry ALL for π 0 production.

q q¯ pair, thus it is referred to as photon-gluon fusion. Current experimental data points to constrain gluon polarization Δg/g(x) solely from this process. . 3 3. Weak interaction. – However, the still missing information is the ﬂavor separation, which can be ﬁnally achievable using a weak interaction. W production in pp collisions is a pure V − A process, where only the left-handed quark and the right-handed anti-quark can contribute and it is an ideal reaction to study the spin structure. W couples to the weak charge, which is highly correlated with the ﬂavor, as shown in ﬁg. 7. Therefore, it is also suitable in ﬂavor structure studies.

Fig. 6. – Left: an example of gluon Compton process, which is the dominant subprocess for prompt photon production in pp collisions. Right: projection for the statistical precision for ALL measurements for prompt photon production.

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Fig. 7. – The helicty structure of W production in polarized pp collisions is shown in (a) and (b). On the right panel, statistical projection for the spin-ﬂavour studies using W production in pp √ collisions at s = 500 GeV is displayed.

√ Such measurement will be feasible at RHIC when it reaches its highest energy s = √ 500 GeV. In 2005, we have achieved s = 410 GeV. We plan to commission the machine at 500 GeV in 2008. Physics production at 500 GeV is expected to start in 2009. Flavour studies are being done by using another class of lepton scattering processes called semi-inclusive DIS, in which an additional hadron is required in the ﬁnal states to select the probed ﬂavor as shown in ﬁg. 8. This attempt has been rather successful and is reported by the Hermes experiment. In the near future, high-intensity ν beams will be available at J-PARC and Fermilab. There are some eﬀorts to realize the measurement of elastic scattering νN → νN . The measurement is useful in the determination of Δs [12], which is currently obtained to be

Fig. 8. – Diagrammatic representation of semi-inclusive deep-inelastic scattering of leptons from the nucleon.

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Fig. 9. – Gluon polarization obtained through the next-leading order QCD analysis using inclusive deep-inelastic scattering of leptons from the nucleon and inclusive π 0 production in polarized pp collisions at RHIC.

negative. To determine the components appearing in eq. (1), we need to integrate the corresponding parton distributions over 0 ≤ x ≤ 1. On the other hand, the elastic scattering will provide the integrated value as a whole in the limit of Q2 = 0 GeV2 . A previous experiment at BNL [13] provided the cross-section for the elastic scattering. However, the determination was rather limited due to a rather high-Q2 cut (Q2 ≥ 0.4 GeV2 ). In addition, most of the events were from the bound proton in carbon which may be subject to substantial nuclear eﬀects. Therefore, it is desirable to have a pure hydrogen target and go to the lowest possible Q2 . Such measurements are being studied either at Fermilab or J-PARC. 4. – Gluon polarization There have been signiﬁcant eﬀorts to measure gluon polarization in the proton. The current constraints are summarized in ﬁg. 9. There are several direct measurements as summarized in table II. Data points are available from Hermes, SMC, and Compass experiments. Phenomenologically the gluon polarization Δg/g(x) is determined from the scaling violation of DIS data. Two typical results [6, 14] are shown with the range of uncertainties. They are both consistent with the direct measurements and there is still room for either positive or negative gluon polarization. A theoretical prediction by Brodsky and Schmidt [15], Δg/g(x) ∼ x, is also shown in the ﬁgure, which is consistent with the GRSV curve. This is a remarkable agreement, given these are obtained through

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Table II. – Direct measurements of Δg. In E704, Phenix, and Star experiments, x-range and Q2 are not speciﬁed. Δg/g(x)

μ2

Experiment

subprocess

x-range

Ref.

E704 (¯ pp, pp → π 0 X) Hermes (γp → h+ h− X) SMC(γp → h+ h− X) Compass (γp → DX) Compass (γp → h+ h− X) Q2 < 1 GeV2 Q2 > 1 GeV2 * Phenix (pp → π 0 X) Star (pp → jet + X)

gg, qg scat. γg → q q¯ γg → q q¯ γg → c¯ c

− 0.17 0.07 0.15

“Large” Δg rejected 0.41 ± 0.18 ± 0.03 2.1 GeV2 −0.20 ± 0.076 ± 0.010 > 2.5 GeV2 −1.08 ± 0.78 −

[21] [22] [24] [25]

γg → q q¯ γg → q q¯ gg, qg scat. gg, qg scat.

0.095 0.15 − −

0.024 ± 0.089 ± 0.057 3 GeV2 0.06 ± 0.31 ± 0.06 3 GeV2 “GRSV-max” Δg rejected “GRSV-max” Δg disfavored

[26] [25] [11] [10]

completely diﬀerent approaches. We discuss what has been expected, and measured, and what is going to be measured in a near future below. . 4 1. Expectation. – There are two reasons (at least phenomenologically) to expect a rather large Δg. One reason is to ﬁll the gap between the expected value of the fractional ˜ and the measured-extracted value (ΔΣ) contribution of quark spin to the proton (ΔΣ) through the axial anomaly: (4)

˜ − Nf αs Δg. ΔΣ = ΔΣ 2π

Since αs ∼ 0.5 at Q2 = 1 GeV2 , Δg should be ∼ 2 to explain the diﬀerence. Another reason to expect a large Δg is to compensate the spin sum rule shown in eq. (1). Since ΔΣ is extracted to be 0.1–0.3, Δg ∼ 0.40 is already enough. There is enough range of variation in the model predictions to cover these naive expectations. The bag-model calculation gives the value Δg ∼ −0.4 [16]. The QCD sum-rule calculation gives the “upper limit” Δg ∼ 2 ± 1 [17]. Diﬀerently from the quark-spin contribution, it seems rather hard to evaluate it in numerical simulations using the lattice gauge theory [18]. There has been a substantial discussion on the gauge dependence of gluon polarization [19]. However it should be noted that Δg is highly scale dependent. It is predicted that αs Δg should remain constant. Since αs runs as ln Q2 , Δg should run as 1/ ln Q2 . Since αs (1 GeV2 ) ∼ 0.5 and αs (MZ2 ) ∼ 0.1, Δg(1 GeV2 ) = 2 means Δg(MZ2 ) ∼ 10. In a sense, the proton spin crisis is over-corrected! The total quark-gluon contribution will be reduced back to 0.5 especially by the negative contribution from Lg [20]. In any case, a naive expectation may hold only at a certain scale and if we go to a diﬀerent energy scale, we need to adapt ourselves to a deviation from natural expectation.

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. 4 2. Measurements. – Experimentally there have been extensive eﬀorts to measure, or constrain at least Δg(x) in either lepton or hadron scattering. Direct constraints from pp collision were obtained in Fermilab E704 [21] and by Hermes [22] in the ’90s. Indirect constraints have been obtained through scaling violation of g1p,n (x) by many phenomenological analysis groups. As in the case of any structure function measurements, one experiment can cover only a limited x range. Obviously we are trying to answer the question of the 1st moment, which is the integral of the structure function at 0 ≤ x ≤ 1. Diﬀerently from the quark measurement there are no direct measurements of the 1st moment in gluon polarization. Therefore it is very important to have a) several experiments to cover a reasonable range of x to extrapolate the measurements to x → 0 and x → 1 reliably, b) a reasonable measure of the kinematical values x and Q2 covered by the experiments. The latter part is not trivial as was in the quark case. A major process to measure Δg(x) in lepton scattering experiments is the photongluon fusion process. Reconstruction of the parton level kinematics is not possible, unless entire jet fragments are detected in the ﬁnal state. Then the QCD event generator is often employed to estimate the momentum fraction, x, carried by the initial state gluon. The technique is further extended to evaluate the background contribution such as the resolved photon process. The hadron collision is more complicated because two initial partons are involved. Even with the complete detection of the ﬁnal states, the ambiguity of assignment of the reconstructed x to the initial state remains. In principle, the QCD event generator can be used to reconstruct the parton level kinematics, to estimate the background, and to evaluate the eﬀect of misassignment. Instead, the so far adopted approach is to compare with the full ﬂedged theoretical calculation taking some models on Δg(x) in the market. However, this approach cannot answer an important question, namely what is the x region measured. Ultimately a global QCD analysis has to be done including the hadron collision data. Such eﬀorts are underway. For an experimentalist to obtain some idea on the gluon polarization, we made the following crude approximation. The asymmetry ALL can be calculated in full next-to-leading order [23]. The asymmetry is essentially a function of Δg/g(x): (5)

ALL =

Ed3 Δσ/dp3 . Ed3 σ/dp3

The cross-section is a convolution of the polarized parton distributions Δfi (x), where i, j = g, q, and spin-dependent cross-section at the parton level, dΔσij /dt, and the fragmentation function, Dπ0 /i (z) integrated in the appropriate phase space ΔΩ: (6)

Ed3 Δσ (pT ) = Σ(i,j)=(g,q) dp3

Δfi (x1 )Δfj (x2 ) ΔΩ

dΔσij Dπ0 /i (z)dΩij . dt

Now we are going to apply this formula to the Phenix π 0 measurement. Its acceptance

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is limited in the central rapidity region, therefore the contribution is dominated by the x1 = x2 region [23]. When we apply the mean-value theorem for multiple integration, there should exist an x value, ξ, to represent the integral: (7)

ALL (pT ) = α

2 Δg Δg (ξ) + β (ξ) + γ. g g

The assumption made here is that relevant x values for the gg scattering part and the qg scattering one are the same. We tested this method by using a couple of Δg(x) models, and it works with a reasonable precision (∼ 10%). We performed χ2 -minimization to obtain a constraint on Δg/g(x). Encouraged by the agreement between the Brodsky-Schmidt model and the GRSV curve, Δg/g(x) ∼ a · x is employed. As expected from the quadratic dependence on Δg/g(x) in eq. (7), two possible solutions are obtained and shown with an uncertainty band, which is signiﬁcantly underestimated because of the choice of the functional form and of other assumptions made in this approach. In combining these results with the lepton scattering data, we can see some preference for positive gluon polarization, although further precision data is desirable. If we make another stretch to calculate the 1st moment with the obtained positive solution, we obtain Δg ∼ 0.3 at Q2 = 1 GeV2 . Even with an underestimated error, the solution is still consistent with zero within 2σ. Obviously we need more precision before we conclude on the size of the gluon-spin contribution to the proton spin. . 4 3. Prospects. – We expect that the Compass experiment will accumulate more data on the helicity distribution. The Hermes collaboration is preparing the new results on Δg/g(x) from the hadron pair production. The Star experiment has shown preliminary data in this conference, and higher statistics data being analyzed from Run-5. At RHIC, a more signiﬁcant improvement in both luminosity and polarization is expected by the new snake magnet in its injector, AGS. Within a few years, the determination of Δg/g(x) around x ∼ 0.1 will be improved to a reasonable precision, so that we can conclude about the gluon-spin contribution to the proton spin. A smaller-x region can be constrained √ from the s = 500 GeV run at RHIC in a few years. At the end of this section, it would be worth mentioning the importance of the unpolarized distributions. All measurements mentioned above are sensitive to the gluon polarization, Δg/g(x), in certain x ranges. To answer the question of Δg, we need to multiply it by the unpolarized gluon distribution g(x), which is poorly determined in large-x and small-x regions. A smaller-x region can be measured at future facilities such as eRHIC/ELIC, and a larger-x region can be measured at J-PARC. In the next decade, our knowledge on the gluon section of the proton structure will be signiﬁcantly enhanced. 5. – Transverse-spin phenomena There have been a substantial number of measurements of single transverse-spin asymmetry, AN , which is essentially a left-right asymmetry with the upwardly polarized proton

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391

Fig. 10. – Left: single transverse-spin asymmetry AN for pion production in pp collisions at √ s = 19.4 GeV, and right: pictorial presentation of the possible origins of the asymmetry, Sivers eﬀect and Collins eﬀect.

beam for hadron productions, since the early ’80s. While theoretical predictions for the asymmetries are essentially zero for the hard processes, because non-zero asymmetry, especially the experiment at Fermilab using the polarized proton beam from Λ decay, E704, has measured “mirror-symmetric asymmetries” for charged-pion production as shown in ﬁg. 10. At the phenomenological level, the asymmetries can be explained in either Sivers eﬀect or Collins eﬀect, or a combination of both. Sivers eﬀect refers to the transversemomentum asymmetry in the initial quark emission from the polarized proton. Since the cross-section decreases rapidly with respect to the transverse momentum, even a small diﬀerence between the left-emitted quark and the right-emitted quark would result in a signiﬁcant asymmetry in the ﬁnal-state particle, e.g. the pion. On the other hand, Collins eﬀect refers to the transverse-momentum asymmetry in the ﬁnal-state particle emission. The initial-state proton polarization can be transferred to the quark, and the polarization could survive at a certain probability, then the ﬁnal-state particle emission could be aﬀected by the quark polarization. At RHIC, Star was the ﬁrst experiment which conﬁrmed that the asymmetry persists even at the collider energy. The asymmetry AN for forward production of π 0 was measured to be non-zero, and similar in size to the E704 π 0 asymmetry. Then Brahms experiment has provided a series of beautiful measurments on AN . Those data are overplotted together with the E704 data in ﬁg. 11. Even with the order of magnitude √ diﬀerence in s, the data seem to have a similar trend as a function of xF suggesting that partonic description would be appropriate.

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Fig. 11. – Single transverse-spin asymmetry AN for inclusive pion production is plotted for Star and Brahms experiment with E704 data as a function of xF .

In addition to these pp scattering data, the Hermes experiment has discovered signiﬁcant transverse spin asymmetries in light meson production such as pion and kaon. The data are analyzed to extract either Collins eﬀect or Sivers eﬀects. Non-zero asymmetries are observed in either cases. Furthermore a spin-dependent fragmentation function, which represents Collins eﬀect has been measured in the Belle experiment at the KEKB factory. They have found non-zero eﬀects in the azimuthal correlation of the pion production in e+ e− collisions. In the future, one of the crucial measurements to understand the transverse-spin phenomena would be that of AN for Drell-Yan production of lepton pairs. In this case, there is no room for Collins eﬀect to participate. Such measurement will be possible in the future at RHIC or at J-PARC with the polarized beam option. Recently the connection between transverse-spin phenomena and orbital angular momentum or generalized parton distributions is discussed. Exclusive processes, such as deeply virtual compton scattering or exclusive meson production have been studied especially to constrain the GPDs of the nucleon. Burkhart has suggested the profound picture of the proton structure, impact-parameter–dependent parton distribution functions. Similarly transverse-momentum–dependent parton distributions have been developed by Anselmino et al. to explain the non-zero AN . Being stimulated by the recent experimental data, a more universal picture of the spin structure of the nucleon is emerging.

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393

Fig. 12. – Production of primary, secondary, and tertiary beams at J-PARC to serve for a variety of scientiﬁc programs.

6. – J-PARC: high-intensity proton beam facility J-PARC is the acronym for Japan Proton Accelerator Research Complex, being constructed by KEK (National High Energy Accelerator Research Organization) and JAEA (Japan Atomic Energy Agency). The facility is constructed in the Tokai campus of JAEA, which is located at about 100 km north of the Tsukuba campus of KEK. This new proton accelerator is targeted at a wide range of ﬁelds, nuclear and particle physics, materials science, biology and nuclear engineering. Such scientiﬁc program will be pursued using its primary proton, secondary (π, K, n, etc.), and tertiary (μ and ν) beams. These beams will be created by bombarding proton beams on nuclei at rest, as illustrated in ﬁg. 12.

Fig. 13. – Schematic diagram of the J-PARC facility. A plan of the project is shown separately for its Phase-1 and Phase-2.

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N. Saito

Fig. 14. – Aerial view of the J-PARC facility as of November, 2006.

J-PARC comprises the following components: a) a 400 MeV (181 MeV on Day-1) proton Linac as an injector, b) a 25 Hz, 3 GeV proton synchrotron with 1 MW power (RCS, Rapid Cycle Synchrotron), and c) a 50 GeV proton synchrotron with slow and fast extraction capabilities for nuclear-particle physics experiments. The schematic diagram of the facility is shown in ﬁg. 13. The project is approved in two phases. In the ﬁrst phase of the project, Phase-1, the Linac up to 181 MeV, the RCS, and the 50 GeV proton synchrotron, together with an experimental hall for material and life science, a hadron experimental facility, and a neutrino beam line will be constructed. In the second phase, Phase-2, the Linac energy will be recovered to the original design value, 400 MeV, the MLF beamlines will be added and the Hadron Experimental Hall will be extended to accommodate a greater variety of scientiﬁc programs. The aerial view of the facility is shown in ﬁg. 14. As of October 2007, commissioning of the RCS is about to start(1 ). With further commissioning in the RCS and in the 50 GeV MR in spring 2008, the entire facility will be ready for users in late 2008. . 6 1. Day-1 experiments with 50 GeV proton synchrotron. – There are two major day-1 experiments; a neutrino oscillation experiment, T2K with the fast extracted beam and nuclear physics experiments using kaons with the slow extracted beam to the Hadron Experimental Hall. The neutrino experiment will be performed with the narrow-band neutrino beam heading to the SuperKamiokande. Following the discovery of the oscillation by the K2K experiment through the disappearance of νμ , T2K will further accumulate data (1 ) In the end of October, the ﬁrst beam was successfully accelerated to 3 GeV at the RCS.

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395

to observe the appearance of νe at SuperKamiokande. Consequently, the mixing angle between the νe and ντ , θ13 , will be measured. The experiment utilizes the re-furbished SuperKamiokande and the near detector at 280 m from the target. T2K is also proposing another detector at 2 km, especially because the energy spectrum at 2 km is very similar to the one at the SuperKamiokande, while the spectrum slightly diﬀers at 280 m. The 2 km detector is required for further precision measurement of the mixing angle. The ﬁrst set of experiments at the Hadron Experimental Hall consists of nuclear physics experiments focused on Ξ-hypernuclei and K − bound states of nuclei, to search for a new state of nuclear matter by implanting hadrons other than protons and neutrons. Especially in the case of the strangeness implantation, such as Λ-baryon or K-mesons, the spatial size of the hadron is expected to be shrunk signiﬁcantly. Therefore, we can expect the formation of high-density nuclear matter, which is expected to be realized in the neutron star in the universe. This kind of experiments serves as a unique method to explore the phase diagram of quantum chromodynamics (QCD) which is the fundamental theory of strong interactions. There are many other experiments proposed at the Hadron Experimental Hall. Those experiments include the kaon rare-decay experiments, K 0 → π 0 ν ν¯, a search for T -violation using the μ polarization in the Kμ3 channel, modiﬁcation of vector mesons in nuclear matter, and Drell-Yan production of lepton pairs. The beam lines for these experiments will be built step by step. . 6 2. Possible spin physics at J-PARC . – There is a great interest in the spin physics subjects at J-PARC. One example is the T -violation experiment utilizing the transverse polarization of muons in K + → π 0 μ+ νμ . The experiment aims to search for a possible violation of the time-reversal symmetry beyond the Standard Model of particle physics. They are trying to improve the current limit set by the same group (KEK-E246) by a factor of 20 especially with the detector upgrades. There is a letter of intent submitted for measurements of muon g-2, EDM and deuteron EDM by utilizing the precession of the spin polarization in the storage ring. These measurements also represent how spin is a powerful tool in fundamental physics. In addition, there is an attempt to accelerate a polarized proton in the entire J-PARC facility. There is a study group to pursue this possibility and one solution is identiﬁed; namely by adding two Siberian snakes in the main ring, signiﬁcant polarization of the proton can be maintained throughout the acceleration. This option would open a possibility to study the spin structure of the nucleon in the large-x region. Some accelerator studies as well as theoretical predictions were performed and they should provide a unique ground for the spin structure studies in the next decade. 7. – Summary There has been a signiﬁcant improvement in the knowledge of the spin structure of the nucleon. We reviewed the progress in the gluon sector, and showed the possible size of the gluon-spin contribution, which turns out to be suﬃcient to fulﬁll the spin

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sum rule, but unlikely to be large enough to explain the small quark-spin contribution through the axial anomaly. Further clariﬁcation is necessary before we conclude on the gluon polarization in the nucleon. Transverse-spin phenomena are also described. Recent data and theoretical developments provide a hint for a more comprehensive picture of the spin structure of the nucleon including the orbital-angular-momentum contribution. Near-future measurements at RHIC, J-PARC and other facilities will further explore the spin structure of the nucleon. ∗ ∗ ∗ We would like to thank the entire RHIC Spin Collaboration, Hermes, Compass experiments, S. Kumano, M. Hirai, M. Stratmann, and W. Vogelsang for providing the materials for the lecture and this manuscript. Special thank is addressed to the patient editor, Mrs. C. Vasini. This work was supported by Grant-in-Aid for Creative Scientiﬁc Research (18GS0210) of Japan Society for the Promotion of Science (JSPS).

REFERENCES [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

European Muon Collaboration (Ashman J. et al.), Phys. Lett. B, 206 (1988) 364. Ji X., Tang J. and Hoodbhoy P., Phys. Rev. Lett., 76 (1996) 740. Fermilab E704 Collaboration (Adams D. L. et al.), Phys. Lett. B, 264 (1991) 462. STAR Collaboration (Adams J. et al.), Phys. Rev. Lett., 92 (2004) 171801. Bass S. D., Rev. Mod. Phys., 77 (2005) 1257 and references therein. Asymmetry Analysis Collaboration (Hirai M. et al.), Phys. Rev. D, 69 (2004) 054021. For accelerator details, see Alekseev I. et al., Nucl. Instrum. Methods A, 499 (2003) 392; for physics program, see Bunce G., Saito N., Soffer J. and Vogelsang W., Ann. Rev. Nucl. Part. Sci., 50 (2000) 525; for updates in the plan, see http://spin.riken.bnl.gov/rsc/report/masterspin.pdf. FINeSSE Collaboration (Bugel L. et al.), Fermilab Proposal hep-ex/042007. http://www.nucl.phys.titech.ac.jp/∼neuspin/. Kiryluk J. (for the Star collaboration) this volume, p. , hep-ex/0512040. Phenix Collaboration (Adler S. S. et al.), Phys. Rev. Lett., 93 (2004) 202002. Garvey G. T., Louis W. C. and White D. H., Phys. Rev. C, 48 (1993) 761. Ahrens L. A. et al., Phys. Rev. D, 35 (1987) 785. Gluck M., Reya E., Stratmann M. and Vogelsang W., Phys. Rev. D, 63 (2001) 094005. Brodsky S. J. and Schmidt I., Phys. Lett. B, 234 (1990) 144. Jaffe R. L., Phys. Lett. B, 365 (1996) 359. Mankiewicz L., Piller G. and Saalfeld A., Phys. Lett. B, 395 (1997) 318. “Comments on Lattics Calculations of Proton Spin Components” Keh-Fei Liu, Report for the Workshop on Future Physics at HERA, Hamburg, Germany, 25-26 (1995), hep-lat/9510046. Bashinsky S. V. and Jaffe R. L., Nucl. Phys. B, 536 (1998) 303. ¨gler P. and Scha ¨ fer A., Phys. Lett. B, 448 (1999) 99. Martin O., Ha E581/E704 Collaboration (Adams D. L. et al.), Phys. Lett. B, 261 (1991) 197.

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[22] Hermes Collaboration (Airapetian A. et al.), Phys. Rev. Lett., 84 (2000) 2584. ¨ger B., Stratmann M., Kretzer S. and Vogelsang W., Phys. Rev. Lett., 92 (2004) [23] Ja 121803. [24] Spin Muon Collaboration (Adeva B. et al.), Phys. Rev. D, 70 (2004) 012002. [25] Bedfer Y. (for the Compass collaboration), Proceedings of 11th International Workshop on High Energy Spin Physics Dubna-SPIN-05, Dubna, Russia, 27 Sep - 1 Oct 2005, hep-ex/0601034. [26] Compass Collaboration (Ageev E. S. et al.), Phys. Lett. B, 633 (2006) 25.

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The nature of spin(∗ ) R. Bertini Dipartimento di Fisica “A. Avogadro” and INFN - Torino, Italy

Summary. — The concept of spin has been revisited, in a simple way and in the framework of Lorentz symmetry and quantum theory. The basic proof of the existence of spin, given by the results of Stern-Gerlach experiments, is also discussed.

1. – Introduction The concept of spin was ﬁrst introduced, in 1925 [1], by two 25 years old Ph.D. students: G. E. Uhlenbeck and S. A. Goudsmith. This idea was suggested to explain some experimental results: the fact that an atomic beam splits into an even number of separate beams in a Stern-Gerlach [2] experiment requires 2l + 1 to be even. This indicates that the orbital angular momentum of the electron in an atom should be a half-integer in order to make 2l + 1 even. But it is impossible for the angular momentum to be half integer with integer quantized orbits. To solve this dilemma, Uhlenbeck and Goudsmith made the bold suggestion that the electron is not a pointlike charge and in addition to its orbital angular momentum around the nucleus, an electron rotates like a (∗ ) Note added by the editors. Due to personal reasons, this text does not cover the full set of lectures delivered by Prof. Bertini, but is limited to a basic and pedagogical introduction to spin. c Societ` a Italiana di Fisica

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Fig. 1. – A sphere that ﬂies by an observer in a straight line is seen from diﬀerent directions, labelled 1 to 4, as time goes by. The surface coordinate grid shows that this straight line ﬂy by is accompanied by an apparent rotation.

top. They called this new intrinsic motion spin and the spin angular momentum is (1a)

Ls =

s(s + 1) · ¯h,

s=

1 . 2

This new concept of spin, so revolutionary for that time that famous physicists hardly accepted it, appears now fully embedded in the present description of extremely small objects given by quantum mechanics and relativistic theory. To illustrate, in this framework, the concept of spin consider the example, given in ﬁg. 1, where the appearance of a spherical object past an observer is shown. We notice that the sphere seems to rotate as it ﬂies by, even though it does not rotate at all with respect to its own direction of motion. This apparent rotation relative to the observer is called orbital spin. Now suppose that the sphere rotates on an axis as it moves. Clearly, this rotation will add to the orbital spin, to produce a compound rotation. But can we separate these two eﬀects if we see such a particle ﬂy by? In classical mechanics we can, but in quantum relativity we cannot, because of the impossibility to deﬁne a time order of the events. Therefore we would say now: Any particle, even if not moving with respect to some observer, can behave as if it were rotating. The Lorentz symmetry of space-time, in

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The nature of spin

Fig. 2. – Even number of nodes (left side). Odd number of nodes (right side).

conjunction with quantum uncertainty, endows each particle with a property, called spin, which behaves like a rotation. The spin of a particle is an internal degree of freedom, an overt consequence of the underlying Lorentz symmetry of quanta in space-time. The spin physics is now a well-developed discipline with dedicated tools to study it and opening a new opportunity to understand the intimate structure of the nucleon. It will be the object of these lectures to underline the reasons why speciﬁc values of the spin quantum number are assumed by indivual particles and its relationship to Lorentz symmetry of space-time and quantum mechanics rules. 2. – SPIN and polarisation Spin is quantized: it appears only like a whole multiple of a ﬁxed unit. Consider a little rotating ball. Because of the quantum behaviour of small objects, there are wave properties associated with the ball: in particular the wave has a certain phase. Because of interference, the quantum path around the equator of the ball must be in phase with itself. A spinning quantum particle can rotate only in certain special ways: spin is quantized. To see what the quantum of spin is, let consider a closed quantum path that is locked in phase, so that the nodes coincide after the path has wrapped itself around the spinning particle equator. Clearly this requires that the path closes after a whole number n of revolutions. Let λ be de Broglie’s wavelength related to the particle momentum mv and R the radius of the path. We then have (2a) (2b)

2πR = nλ

n = 0, 1, 2 . . . ,

λ = 2π¯h/mv,

(2c)

2πR = n2π¯h/mv,

(2d )

mvR = n¯h = S.

We recall that mvR is the angular momentum. Because we are dealing here with the rotation of the particle itself, we speak of spin angular momentum or simply spin indicated by S and equal to n¯ h. The appearance of the number n, that is an integer, is an illustration of a quantum number indicating an internal degree of freedom of the particle. The number n can be odd or even and represents the number of revolutions of

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Fig. 3. – Twisting of the topology of a boson phase loop to obtain a doubled-over fermion loop.

the particle (see ﬁg. 2). Let ﬁrst n be even, for example n = 4. The third node coincides then with the ﬁrst and the second with the fourth. We have then a superposition of two indistinguishable paths, each with n/2 nodes. But even so n/2 = m is a whole number and S = m¯ h, where m is an integer. This is a Bose-Einstein particle, a boson. Let be n an odd integer: 1, 3, 5, . . . . Then the nodes do not coincide in pairs anymore. Each point in the second loop is exactly midway between two points in the ﬁrst loop. The two loops are not the same and we obtain 2πR = nλ/2. We obtain a class of particles with spin quantisation S = n¯ h/2. These are Fermi-Dirac particles or fermions. To change a boson into a fermion, we must change the topology of its phase path (see ﬁg. 3). We have to deforme it from a circular (boson) shape into a double-loop (fermion) conﬁguration. This is impossible unless we lift half a loop oﬀ the table, twist it in space and put it back on the other half, thus, unless we can step out of space into another dimension, we never can change a boson into a fermion or the other way around. Unless our Universe has extra freedom beyond space-time, spin can only change in whole units of h ¯. An extension of the Lorentz symmetry of space-time to a similar symmetry of a space with more than four dimensions, like is done in some theories, allows us to change angular momentum by half a unit, so that Δs = ±1/2 rather than Δs = ±1 that was mandatory until now. By demanding the right to step out into another direction, we can twist and untwist the phase paths of bosons and fermions. The extended form of the Lorentz group is called the super-Poincar´e and the corresponding symmetry supersymmetry. An extended discussion of all these topics concerning spin can be found in ref. [3]. Spin can have a certain orientation in space called POLARISATION. If you measure the spin of a particle in a ﬁxed, but otherwise arbitrary direction (z direction), either you observe a whole multiple of h ¯ /2 or nothing at all. For S = s¯h, where either s = 0, 1, 2, . . . or s = 1/2, 3/2, 5/2 . . . . Being spin the oﬀspring of relativity, we expect that the interactions among particles with spin will be Lorentz symmetric. We recall that the number of degrees of freedom of a Lorentz symmetric object is a power of 4, because space-time has four dimensions. Therefore we can have 40 = 1, 41 = 4, 42 = 16 and so on. Thus, a Lorentz symmetric object is either one number, or a string of four numbers, or an array of 4 × 4 numbers, and so forth. Let inspect the more usual cases:

The nature of spin

403

– s = 0: scalar particle: in the absence of spin, one number describes the particle orientation. 40 – s = 1/2, together with its antiparticle, forms a four-components state vector called a spinor. 41 – s = 1 is a vector particle or vector boson. 41 . Two longitudinal (Left + Right) and trasverse (Up + Down). – s = 2 is a tensor particle with a 4 × 4 array numbers. The choice of a special direction to measure polarisation is called ﬁxing a gauge. If the chosen direction is the direction of motion, the way the particle spins is called handedness or helicity: we can have right (R), left (L) or zero helicity. If the spinning axis is perpendicular to the direction of motion, we have transversity. We can change the handedness of a massive particle, simply by moving so fast that we can overtake it: with respect to us, the particle will then move in the other direction, so that its handedness is reversed. Lorentz symmetry implies that the Minkowski distance x2 −c2 t2 in space-time remains constant. Then also E 2 − c2 p2 must be invariant, and in particular E 2 − c2 p2 = m2 c4 . There are two special cases. One if the particle momentum is negligible compared to mc. In that case we have E 2 = m2 c4 , which implies E = mc2 or E = −mc2 . The ﬁrst possibility applies to particles, the second to antiparticles. The other special case occurs if the particle momentum is much larger than mc to obtain approximately E = cp. Thus, if we keep the momentum ﬁxed and let the mass go to zero, we still have a possible connection between energy and momentum. Apply now de Broglie’s prescriptions: E = 2π¯ hν and p = 2π¯ h/λ. But this is precisely the requirement for a wave that propagates with the speed of light. Therefore zero-mass particles can exist, provided they move with the speed of light. What about their spin? Suppose that the particle is transversely polarised; if we look along its spin axis, one side of it would be moving in the direction of the particle’s motion, whereas the other side would be moving in the opposite direction. But if the particle is massless this cannot be. The whole should move with speed of light and the diﬀerence of motion of the two sides must be zero. So it cannot be rotating: its angular momentum in the transverse direction is zero. A particle with zero rest mass can only have two degrees of spin freedom. It cannot change its handedness, for massless particles must move with the speed of light, that is maximal and can never be overtaken. Example of such a particle is the photon. 3. – Magnetic moments, Stern-Gerlach and polarised source We know, from classical electromagnetism, that a magnetic moment μ is associated with a small current as such generated by the electron revolving around the nucleus or by the electron spinning on its axis. From the motion of a single electron inside the atom we

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R. Bertini

Fig. 4. – Correlation of the magnetic moments and the angular momenta of the electron.

can expect then two magnetic moments: one related to the orbital angular momentum: (3a)

μl = − l(l + 1) · ¯h ·

e . 2me c

μs = − s(s + 1) · ¯h ·

e . 2me c

Another to the spin: (4a)

These two magnetic moments combine as illustrated in ﬁg. 4, for a single electron. The resulting magnetic moment μ may be decomposed into two components: one along the direction of the total angular momentum J = L + S, called μj ; the other perpendicular to J, which has not eﬀect on the total on the average, because of its rotational motion around J. So it is μj that should be taken into account as total magnetic moment of an electron. The interaction of the magnetic moment μ with an external magnetic ﬁeld B causes a precession of the magnetic moment around the direction of the magnetic ﬁeld B. Similarly to the case discussed in ﬁg. 4, here the magnetic moment μ may be decomposed into two components: one along the direction of B, called μz , assuming z along B; the other perpendicular to B, which has not eﬀect on the total on the average, because of

405

The nature of spin

Fig. 5. – The set-up for a Stern-Gerlach experiment.

its rotational motion around B. The component of the force along z coming from the interaction of μ with B and acting on the single electron will be (5a)

Fz = −

∂Bx ∂By ∂Bz ∂(μ · B) = μx + μy + μz . ∂z ∂z ∂z ∂z

∂Bz z Therefore the choice of a magnetic ﬁeld with ∂B ∂x = ∂y = 0 will give a force Fz , that ∂Bz depends only on μz and ∂z . This is a key feature of the Stern-Gerlach experiment: create a magnetic ﬁeld rapidly changing along z but not along x or y. As shown in ﬁg. 5(b), this can be done with an appropriate shaping of the magnet poles. We want to make a Stern-Gerlach experiment on an atom with a single electron in the most external shell. The most simple example of such a case being the hydrogen atom, we encounter an additional problem: the hydrogen molecule is diatomic with the spin of the electrons of the two atoms coupling to a total spin zero. We have therefore to break the hydrogen molecule to deal only with single hydrogen atoms. This is done heating at a temperatureT in the oven O, from where the atoms escape through a small hole with a velocity v = 3kT /m, where k is Boltzmann’s constant and m the mass of the atom. The trajectories of the hydrogen atoms, directed

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R. Bertini

along the horizontal axis x by the slits S1 and S2 , are bended into the vertical direction z when they reach the inhomogeneous magnetic ﬁeld active for a distance d. They ﬁnally hit the sensitive plate P, where they leave a mark at the position (6a)

z2 = μz

∂Bz dD ∂Bz dD ∂Bz dD = −mj gj μB = ±μB , ∂z mv 2 ∂z mv 2 ∂z mv 2

where D is the distance between screen P and the middle of the magnetic-ﬁeld region, mj = j, j − 1, . . . , −j, so that there are 2 j + 1 values of mj . As shown in ﬁg. 5(c), the number of lines on plate P corresponds to the number of values of z2 . Therefore from the number of lines we may calculate the value of j and then the values of mj . From the interval distance between two neighboring lines, we may calculate mj gj . In our case, hydrogen atoms in the ground state: j = 1/2; mj = ±1/2; gj = 2; mj gj = ±1. In conclusion the Stern-Gerlach experiment shows that the spatial quantization of angular momentum is an experimental fact, the correctness of the hypthesis of electron spin, s = 1/2 and ﬁnally the correctness of the values given for the spin magnetic moment of the electron μs,z = ±μB , gs = 2. In addition to a convenient choice of slits, one may choose only atoms with spin +1/2 for positive values of z2 , and with spin −1/2 for negative values of z2 . We have therefore a source of polarised beams. For all these reasons the Stern-Gerlach experiment is probably the most important experiment in the hystory of spin. The method they introduced is still used in modern machines to produce polarised beams. REFERENCES [1] Uhlenbeck G. E. and Goudsmith S. A., Naturwiss., 13 (1925) 953; Nature, 117 (1926) 264. [2] Gerlach W. and Stern O., Z. Phys., 9 (1922) 349. [3] Icke V., The Force of Symmetry (Cambridge University Press) 1999.

Studies of semi-inclusive and hard exclusive processes at JLab H. Avakian Jeﬀerson Lab - 12000 Jeﬀerson Avenue, Newport News, VA 23606, USA

Summary. — The main goal of experiments proposed for the CLAS12 detector in conjunction with the 12 GeV CEBAF accelerator is the study of the nucleon through hard exclusive, semi-inclusive, and inclusive processes. This will provide new insights into nucleon dynamics at the elementary quark and gluon level. In this contribution we provide an overview of ongoing studies of the structure of nucleon in terms of quark and gluon degrees of freedom and future physics program planned with CLAS and CLAS12.

1. – SIDIS and the transverse structure of the nucleon The spin structure of the nucleon has been of particular interest since the EMC [1] measurements implied that the helicity of the constituent quarks account for only a fraction of the nucleon spin. The so-called “spin puzzle” was subsequently conﬁrmed by a number of other experiments at CERN [2], SLAC [3, 4], HERA [5, 6], and JLab [7]. Possible interpretations of this result include the contribution of the orbital momentum of quarks and signiﬁcant polarization of either the strange sea (negatively polarized) or gluons (positively polarized). The contributions to the sum rule for the total helicity of c Societ` a Italiana di Fisica

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H. Avakian

the nucleon include the following: (1)

1

1 sea glue = (Δq val + Δq sea ) + Lval + ΔG, z + Lz + Lz 2 2 q

where Δq, Lz , and ΔG are, respectively, the quark helicity, the orbital angular momentum of all partons, and the gluon helicity. Present knowledge about the spin structure of the nucleon, described by parton distribution functions comes mainly from polarized Deep Inelastic Scattering (DIS). The polarization of the individual ﬂavors and anti-ﬂavors were mainly studied using ﬁts to the inclusive data. Inclusive DIS is sensitive to only the squared charges of the partons, and requires additional assumptions (e.g., an SU (3) symmetric sea), which leads to certain ambiguities. Semi-Inclusive Deep Inelastic Scattering (SIDIS) studies, when a hadron is detected in coincidence with the scattered lepton that allows so-called “ﬂavor tagging”, provide more direct access to contributions from various quarks. In addition, they give access to the transverse momentum distributions of quarks, not accessible in inclusive scattering. Azimuthal distributions of ﬁnal-state particles in semi-inclusive deep-inelastic scattering provide access to the orbital motion of quarks and play an important role in the study of transverse-momentum distributions of quarks in the nucleon. Signiﬁcant progress has been made recently in understanding the role of partonic initial- and ﬁnal-state interactions [8-10]. The interaction between the active parton in the hadron and the spectators leads to gauge-invariant Transverse Momentum Dependent (TMD) parton distributions [8-12]. Furthermore, QCD factorization for semi-inclusive deep inelastic scattering at low transverse momentum in the current-fragmentation region has been established in refs. [13, 14]. This new framework provides a rigorous basis to study the TMD parton distributions from SIDIS data using diﬀerent spin-dependent and independent observables. TMD distributions (see table I) describe transitions of a nucleon with one polarization in the initial state to a quark with another polarization in the ﬁnal state. The diagonal elements of the table are the momentum, longitudinal and transverse spin distributions of partons, and represent well-known parton distribution functions related to the square of the leading-twist, light cone wave functions. Oﬀ-diagonal elements

Table I. – Leading-twist transverse-momentum–dependent distribution functions. U , L, and T stand for transitions of unpolarized, longitudinally polarized, and transversely polarized nucleons (rows) to corresponding quarks (columns). N/q

U

U

f1

L T

⊥ f1T

L

T h⊥ 1

g1

h⊥ 1L

g1T

h 1 h⊥ 1T

Studies of semi-inclusive and hard exclusive processes at JLab

409

require non-zero orbital angular momentum and are related to the wave function overlap ⊥ of L = 0 and L = 1 Fock states of the nucleon [15]. The TMDs f1T and h⊥ 1 , which are related to the imaginary part of the interference of wave functions for diﬀerent orbital momentum states and are known as the Sivers and Boer-Mulders functions [16-18, 9-11], describe unpolarized quarks in the transversely polarized nucleon and transversely polarized quarks in the unpolarized nucleon, respectively. They vanish at tree-level in a T -reversal invariant model (T -odd) and can only be non-zero when initial- or ﬁnal-state interactions cause an interference between diﬀerent helicity states. These functions parameterize the correlations between the transverse momentum of quarks and the spin of a transversely polarized target or the transverse spin of the quark, respectively. They require both orbital angular momentum, as well as non-trivial phases from the ﬁnal-state interaction, that survive in the Bjorken limit. Experimental results on the Sivers functions for up and down quarks so far are consistent with a heuristic model of up and down quarks orbiting the nucleon in opposite directions. The most simple mechanism that can lead to a Boer-Mulders function is a correlation between the spin of the quarks and their orbital angular momentum. In combination with a ﬁnal-state interaction that is on average attractive, already a measurement of the sign of the Boer-Mulders function, would thus reveal the correlation between orbital angular momentum and spin of the quarks. The impact parameter space displacement of transversely polarized quark distributions in an unpolarized target is described by a chirally odd GPD [19], and has recently been calculated for the ﬁrst time in lattice QCD [20, 21]. The resulting transverse ﬂavor dipole moment for transversely polarized quarks in an unpolarized nucleon suggests that the Boer-Mulders functions are signiﬁcantly larger than the Sivers functions. Moreover, consistent with large NC predictions, the displacement of transversely polarized u- and d-quarks was found to be in the same direction, indicating the same sign for the BoerMulders functions for u- and d-quarks, and suggesting a further enhancement of the SIDIS asymmetry from the d-quark contribution. Similar quantities arise in the hadronization process. One particular case is the Collins T -odd fragmentation function H1⊥ [22] describing fragmentation of transversely polarized quarks into unpolarized hadrons. Parton model analyses [23-26] of sub-leading singlespin asymmetries observed at HERMES [27, 28] and CLAS [29] led to the introduction of new twist-3 T -odd distribution functions [14, 12]. The oﬀ-diagonal TMD distributions arise from interference between amplitudes with left- and right-handed polarization states, and only exist because of chiral symmetry breaking in the nucleon wave function in QCD. Their study therefore provides a new avenue for probing the chiral nature of the partonic structure of hadrons. The universality of the TMD correlation functions has been proven, resulting in a sign change for two T -odd TMD distributions between Drell-Yan and DIS [9, 14], an exciting prediction that has to be conﬁrmed by future experiments. The CLAS12 set-up will provide capabilities for exploring nucleon structure in the valence quark region, allowing studies of semi-inclusive processes both in the target and current fragmentation regions.

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. 1 1. Present experimental results on spin-azimuthal asymmetries. – In recent years, Semi-Inclusive Deep Inelastic Scattering (SIDIS) has emerged as a powerful tool to probe nucleon structure through measurements of Single Spin Asymmetries (SSAs) [27, 30, 31]. In contrast to inclusive deep-inelastic lepton-nucleon scattering where transversemomentum is integrated out, these processes are sensitive to transverse-momentum scales on the order of the intrinsic quark momentum PT ∼ k⊥ . Measurements of SSAs in SIDIS provide access to a list of novel physics observables including transversity (h1 ) [32, 33], ⊥ and the time-reversal odd Sivers distribution function (f1T ) [16, 17, 8-10]. . 1 1.1. Transversely polarized target. For transversely polarized targets, several azimuthal asymmetries already arise at leading order. Four contributions related to the corresponding distribution functions were investigated in refs. [22, 34, 35, 8, 10, 36]: (2)

cos φ σLT ∝ λe ST y(1 − y/2) cos(φ − φS )

(3)

sin φ σU T

∝ ST (1 − y) sin(φ + φS )

q,¯ q

e2q xh1 (x)H1⊥q (z)

q,¯ q

+ST (1 − y + y /2) sin(φ − φS ) 2

+ST (1 − y) sin(3φ − φS )

q e2q xg1T (x)D1q (z),

⊥q e2q xf1T (x)D1q (z)

q,¯ q ⊥q e2q xh⊥q 1T (x)H1 (z),

q,¯ q

where φ and φS are the azimuthal angles of the hadron and transverse spin in the photon frame, x, y, z deﬁne the fractions of the proton momentum carried by the struck quark, electron momentum carried by the virtual photon. and the virtual photon momentum carried by the ﬁnal-state hadron, respectively. D1q (z) and H1⊥q (z) are the spinindependent and spin-dependent fragmentation functions. The leading-twist transversity distribution h1 [32,33] and its ﬁrst moment, the tensor charge, are as fundamental for understanding of the spin structure of the nucleon as are the helicity distribution g1 and the axial vector charge. The transversity distribution h1 is charge conjugation odd. It does not mix with gluons and for non-relativistic quarks it is equal to the helicity distribution g1 . Thus, it probes the relativistic nature of quarks and it has a very diﬀerent Q2 evolution than g1 . The tensor charge is reliably calculable % 1 ¯ in lattice QCD with δΣ = f 0 dx(hf1 − hf1 ) = 0.562 ± 0.088 at Q2 = 2 GeV2 , which is twice as large as the value of axial charge [38]. A similar quantity (δΣ ≈ 0.6) was obtained in the eﬀective chiral quark soliton model [39]. A detailed study of the Q2 and xB dependences as a function of the azimuthal angle φ will allow the separation of contributions from diﬀerent mechanisms. During the last few years, ﬁrst results on transverse SSAs have become available [30,31]. HERMES measurements for the ﬁrst time directly indicated signiﬁcant azimuthal moments generated both by Collins (ﬁg. 1) and Sivers (ﬁg. 2) eﬀects.

411

Studies of semi-inclusive and hard exclusive processes at JLab

π

AUT (Collins)

0.05

+

π

-

π

0

0.025 0 -0.025 -0.05 HERMES

-0.075 -0.1

CLAS12

0

0.5 0

0.5

0

0.5

x sin(φ+φ )

S Fig. 1. – Projected transverse spin asymmetry from the Collins eﬀect (AU T ) in single π production with CLAS at 11 GeV. Curves represent predictions by Anselmino and collaborators [37].

. 1 1.2. Longitudinally polarized target. Spin-orbit correlations are also accessible in SIDIS with a longitudinally polarized target, where they give rise to the Mulders leadingtwist distribution function h⊥ 1L . It is related to the real part of the interference of wave functions for diﬀerent orbital momentum states, and describes transversely polarized quarks in the longitudinally polarized nucleon. For a longitudinally polarized target the only azimuthal asymmetry arising in leading order is the sin 2φ moment, sin 2φ σU ∝ SL 2(1 − y) sin 2φ L

(4)

⊥q e2q xh⊥q 1L (x)H1 (z).

q,¯ q

The physics of σU L , which involves the Collins fragmentation function H1⊥ and Mulders distribution function h⊥ 1L , was ﬁrst discussed by Kotzinian and Mulders in

A UT (Sivers)

0.04

π+

π-

π0

HERMES

0.03

CLAS12

0.02 0.01 0

-0.01 0

0.5

0

0.5

0

0.5

x sin(φ−φS )

Fig. 2. – Projected transverse spin asymmetry from the Sivers eﬀect (AU T production with CLAS12 at 11 GeV.

) in single π

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1996 [34-36]. The same distribution function is accessible in double-polarized Drell-Yan, where it gives rise to the cos 2φ azimuthal moment in the cross-section [40]. Measurements of the sin 2φ SSA [36], thus allow the study of the Collins eﬀect with no contamination from other mechanisms. A recent measurement of the sin 2φ moment of σU L by HERMES [27] is consistent with zero. A measurably large asymmetry has been predicted only at large x (x > 0.2), a region well covered by JLab [23]. The kinematic dependence of the SSA for π + , measured from the CLAS EG1 data set at 6 GeV [41] is consistent with predictions [23]. The π + SSA is dominated by the u-quarks; therefore with some assumption about the ratio of unfavored to favored Collins fragmentation functions, it can provide a ﬁrst glimpse of the twist-2 Mulders TMD + function. The distribution function h⊥ target SSA [41], 1L was extracted using the π which is less sensitive to the unknown ratio of unfavored (d-quark fragmenting to π + ) to favored (u-quark fragmenting to π + ) polarized fragmentation functions (see ﬁg. 5). The curve is the result of the calculation by Efremov et al. [23], using h⊥ 1L from the chiral 2 2 quark soliton model evolved to Q = 1.5 GeV . The extraction, however, suﬀers from low statistics and has a signiﬁcant systematic error from the unknown ratio of the Collins favored and unfavored fragmentation functions, the unknown ratio of hd1L /hu1L , as well as from background from exclusive vector mesons. Current statistical errors for π − , and in particular π 0 , which is relatively free of possible higher twist contributions [42], are large and do not allow strong conclusions from the measured SSAs. More data are required for a statistically signiﬁcant measurement of the sin 2φ moment. . 1 1.3. Unpolarized target. The only leading-twist contribution to the unpolarized target cross section depending on the azimuthal angle has a term with the Boer-Mulders function coupling to the Collins function [22]: (5)

cos 2φ σU ∝ 2(1 − y) cos 2φ U

⊥q e2q xh⊥q 1 (x)H1 (z).

q,¯ q

The physics of which involves the Collins fragmentation function H1⊥ and the Boer-Mulders distribution function h⊥ 1 , was ﬁrst discussed by Boer and Mulders in 1998 [43]. In recent years, the cos 2φ asymmetry in leptoproduction was phenomenologically studied using diﬀerent approximations for the Boer-Mulders function [44-46]. 2 Independent information on the Boer-Mulders function h⊥ 1 (x, kT ) can be obtained from the study of the cos 2φ azimuthal asymmetry in unpolarized Drell-Yan processes, which has been measured in πN collisions [47, 48]. In refs. [49, 50] this asymmetry was estimated by computing the h⊥ 1 distribution of the pion and of the nucleon in a quark spectator model [51,52]. The cos 2φ azimuthal asymmetry in SIDIS was computed assuming that the π + production is dominated by u-quarks and using the same distributions 2 2 h⊥ 1 (x, kT ) and f1 (x, kT ) used in ref. [50]. The calculation of the cos 2φ asymmetry appeared to be in rather good agreement for low values of PT (up to 0.5 GeV) with SIDIS data coming from the ZEUS experiment [53] at large Q2 values (0.01 < x < 0.1, 0.2 < y < 0.8, 0.2 < z < 1, Q2 > 180 GeV2 ) where the higher-twist contributions are not expected to be relevant. cos 2φ σU U ,

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413

It is important to note that both π + and π − azimuthal moments may have signiﬁcant contributions from exclusive vector meson production. The fraction of π + in the single pion sample, coming from exclusive ρ0 decays, is somewhat less but still signiﬁcant at large z and in particular for small x. The two-pion data from CLAS12 will allow us to extract exclusive two-pion asymmetries and estimate their contribution to the single-pion SSA. . 1 2. TMD Measurements with JLab at 12 GeV . – Projections for target single-spin asymmetry measurements with CLAS12 at 11 GeV are plotted in ﬁgs. 1, 2. The projected error bars have been calculated assuming a luminosity of 2× 1034 cm−2 s−1 , with an N H3 target polarization of 85% and a dilution factor of 0.14, with 2000 hours of data taking. The asymmetry is integrated over all hadron transverse momenta. The target single-spin asymmetry from polarized quark fragmentation extracted for CLAS12 kinematics at 11 GeV is plotted in ﬁg. 1. The estimate was done assuming h1 ≈ g1 and an approximation for the Collins fragmentation function from ref. [23]. Additional cuts were applied on z (z > 0.5) and the missing mass of the e π + system (MX (π + ) > 1.3 GeV). φ The extraction of the transversity from Asin U T could be performed using parameteriza¯ and certain approximations tions for the unpolarized distribution functions u(x) and d(x) ⊥ for the polarized Collins fragmentation function H1 . The measurement of transversity is complicated by the presence of an essentially unknown Collins function. Recently, the Collins function for pions was calculated in a chiral-invariant approach at a low scale [54] and it was shown that at large z the function rises much faster than previously predicted [23, 55] in the analysis using the HERMES data on target SSA. It was also pointed out that the ratio of polarized and unpolarized fragmentation functions is almost scale independent [54]. Signiﬁcant asymmetry was measured by Belle [56] indicating that the Collins function is indeed large. The transverse asymmetry measurements were performed at HERMES [57] and COMPASS [31, 58]. The ﬁrst extraction of the transversity distribution has been carried out recently [37] combining e+ e− and semiinclusive DIS data [30]. The statistics, however, are not enough to make statistically signiﬁcant predictions in the valence region, where the eﬀects are large. Signiﬁcantly higher statistics from CLAS12 data, especially in the large-x region, will enable the extraction of the x and Q2 dependencies for diﬀerent azimuthal moments in a wide kinematic range allowing the source of the observed SSA to be revealed and will allow extraction of the underlying distribution functions. The measurement of the transverse asymmetry from the Sivers eﬀect (see ﬁg. 3) with π 0 will provide a model-independent extraction of the Sivers function. Furthermore, measurements with proton and neutron targets will provide model-independent information on ﬂavor partners of the Sivers function. The transversely polarized target measurements also provide access to the leadingq twist TMD g1T (x) appearing in convolution with the unpolarized fragmentation function q D1 (z) in a cos φ moment of the cross-section. Signiﬁcant asymmetries were predicted re-

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0.02 0 -0.02 F1T

φ-φs )P⊥/M A sin( UT

0.015

0.01

0.005

0

0

-0.04 -0.06

0.25

0.5

-0.08 x

0.2

0.4

0.6 x

φ−φS Fig. 3. – Projected transverse spin asymmetry (Asin ) in single π 0 production with CLAS12 UT at 11 GeV. The curves are calculated using models for the Sivers function from Efremov et P 2 ⊥q e f . al. [59], F1T = u,d q 1T q

q cently for CLAS12 [60] providing access also to g1T (x), describing longitudinally polarized quarks in the transversely polarized nucleon. The eﬀects related to orbital motion of quarks and in particular correlations of spin and transverse momentum of quarks play more important role in the valence region. It was shown that spin-orbit correlations may lead to signiﬁcant contribution to partonic momentum and helicity distributions [61] in large-x limit. Measurements of transverse momenta of ﬁnal-state hadrons in SIDIS with longitudinally polarized targets will provide complementary to transverse target information, probing the longitudinal nucleon structure beyond the collinear approximation. The P⊥ -dependence of the double-spin asymmetry, measured for diﬀerent bins in z and x will provide a test of the factorization hypothesis and probe the transition from the non-perturbative to perturbative description. At large PT (ΛQCD PT Q) the asymmetry is expected to be independent of P⊥ [13]. There are indications that the double-spin asymmetry (see ﬁg. 4) at small PT tends to increase for π − and decrease for π + . A possible interpretation of the PT -dependence of the double-spin asymmetry may involve diﬀerent widths of transversemomentum distributions of quarks with diﬀerent ﬂavor and polarization [62] resulting from a diﬀerent orbital structure of quarks polarized in the direction of the proton spin and opposite to it [63,64]. This interpretation may demand a diﬀerent width for d-quarks than for u-quarks, consistent with observation from lattice QCD studies of a diﬀerent spread in transverse distances for d-quarks compared to u-quarks [20]. The same eﬀect may be responsible for the relatively large cos φ moment of the double-spin asymmetry (see ﬁg. 4, right panel). Detailed measurements of ALL and its cos φ moment as a function of PT in diﬀerent bins in x, z, Q2 combined with measurements of azimuthal moments of the unpolarized cross-section proposed for CLAS12 will allow study of the ﬂavor dependence of transversemomentum distributions.

415

Studies of semi-inclusive and hard exclusive processes at JLab ep→e′ π+X

ep→e′ π+X +

-

π

π

0

π

π

CLAS 5.7 GeV CLAS12

0.5

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0 -0.1 0

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PT

Fig. 4. – The double-spin asymmetry ALL (left) and its cos φ moment (right) as a function of the transverse momentum of hadrons, PT , averaged in the 0.4 < z < 0.7 range.

Projections for the resulting kinematic dependence of the leading-twist SSA are shown in ﬁg. 5. Calculations were done using h⊥ 1L from the chiral quark soliton model evolved to Q2 = 1.5 GeV2 [23], f1 from GRV95 [65], and D1 from Kretzer, Leader, and Chris+ − tova [66]. Three diﬀerent curves correspond to H1⊥u→π /H1⊥u→π = 0, −1.2, −5 [67]. Corresponding projected error bars for the Mulders TMD parton distribution are shown in ﬁg. 5. An important ingredient for the estimates are so-called “Lorentz-invariance relations” that connect h⊥ 1L with h1 [34]. Meanwhile these relations are known not to be valid exactly [68, 69]. It is of importance to ﬁnd out experimentally to which extent such relations can provide useful approximations, or whether they are badly violated, since there is little theoretical intuition on that point [70].

π+

π-

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a) 0.5 0

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Fig. 5. – (Left) The projected x-dependence of the target SSA at 11 GeV. The triangles illustrate the expected statistical accuracy. The open squares and triangles show the existing measurements of the Mulders TMD from HERMES and the CLAS 5.7 GeV EG1 data sets, respectively. The curves are calculated using ref. [67]. (Right) Projection for the Mulders distribution function for the u-quark from the π + SSA from CLAS12 (predicted) compared with the CLAS EG1 data set at 5.7 GeV.

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A UU

cos2 φ

0.1

e p → e′ π X

0.05

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0

0

-0.05

-0.05

-0.1 10 5 2 2 Q (GeV )

π+ ππ0

0.1

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1 PT (GeV)

2

Fig. 6. – The cos 2φ moment (Boer-Mulders asymmetry) for pions as a function of x and PT for Q2 > 2 GeV2 (right) with CLAS12 at 11 GeV from 2000 hours of running. Values are calculated + − assuming H1⊥u→π = −H1⊥u→π .

Proposed measurements of SSAs in SIDIS will pin down the corresponding TMD distributions and will constrain the ratio of favored to unfavored polarized fragmentation functions. The new data will also allow a more precise test of the factorization ansatz and the investigation of the Q2 -dependence of sin 2φ, sin φ, and cos φ asymmetries. This will enable us to study the leading-twist and higher-twist nature of the corresponding observables [71, 33, 55, 24-26, 14]. The Boer-Mulders contribution, being leading twist, is expected to survive at higher 2 Q and that can be tested at the large Q2 accessible with CLAS12. At large transverse momentum, i.e. Ph⊥ ΛQCD , the transverse-momentum dependence of the various factors in the factorization formula [13] may be calculated from perturbative QCD. Following the similar arguments in Ji-Qiu-Vogelsang-Yuan [72], the cos 2φ azimuthal asymmetry has the following behavior at ΛQCD Ph⊥ Q: (6)

cos 2φ |Ph⊥ ΛQCD ∝

1 2 . Ph⊥

When the transverse momentum is compatible with the large-scale Q, the above result will be modiﬁed, because the gluon radiation from the pQCD diagram will dominate, and contribute to the azimuthal asymmetry not being suppressed by any hard scale. At Q2 values accessible at JLab (1.0 < Q2 < 10 GeV2 ), however, the reduction of the Boer-Mulders asymmetry due to Sudakov form factors arising from soft-gluon contributions [73] is not expected to be signiﬁcant. Measurement of the PT dependence of the Boer-Mulders asymmetry (see ﬁg. 6) will allow for checking of the predictions of a uniﬁed description of SSA by Ji and collabo-

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rators [13, 72] and for study of the transition from a non-perturbative to a perturbative description. The cos 2φ asymmetry for semi-inclusive deep inelastic scattering in the kinematic regions of CLAS12 is predicted to be signiﬁcant (a few percent on average) and tends to be larger in the small-x and large-z region. The preliminary data from CLAS at 6 GeV indeed indicate large azimuthal moments both for cos φ and cos 2φ. The combined analysis of the future CLAS12 data on cos 2φ and of the previous ZEUS measurements in the high-Q2 domain (where higher-twist eﬀects are negligible) will provide information on the Boer-Mulders function, shedding light on the correlations between transverse spin and transverse momenta of quarks. Signiﬁcantly increasing the kinematic coverage at large Q2 and PT , CLAS12 (see ﬁg. 6) will map the quark TMDs in the valence region allowing study of the transition from a non-perturbative description at small PT to a perturbative description at large PT . Measured single- and double-spin asymmetries for all pions in a large range of kinematic variables (xB , Q2 , z, P⊥ , and φ) combined with measurements with unpolarized targets will provide detailed information on the ﬂavor and polarization dependence of the transverse-momentum distributions of quarks in the valence region, and in particular, on the xB , z, and P⊥ dependence of the leading TMD parton distribution functions of u and d-quarks. Such measurements across a wide range of x, Q2 , and PT would allow for detailed tests of QCD dynamics in the valence region complementing the information obtained from inclusive DIS. They would also serve as novel tools for exploring nuclear structure in terms of the quark and gluon degrees of freedom of QCD. With upgraded energy and luminosity, CLAS12 can study single- and double-spin asymmetries, involving essentially unexplored chiral-odd and time-odd distributions functions, including transversity [32, 33], Sivers [16, 8-10], Boer-Mulers [43], and Collins [22] functions, providing detailed information on the quark transverse momentum and spin correlations [22, 35, 34, 18, 74]. 2. – GPDs and quark distributions in transverse space Recently it was realized that parton distribution functions depending on the fraction of longitudinal momentum of the nucleon carried by the quark, x, represent special cases of a more general, much more powerful, way to characterize the structure of the nucleon, the Generalized Parton Distributions (GPDs) [75-78]. They depend on the fractions of the nucleon momentum carried by the quark before and after the process, x + ξ and x − ξ (ξ deﬁnes the longitudinal momentum transfer to the nucleon), as well as on the transverse-momentum transfer to the nucleon, Δ⊥ . The presence of spin—both of the nucleon and the quark—as well as quark ﬂavors (u, d, s) lead to the appearance of various independent spin/ﬂavor components of the GPDs. Together, they provide a comprehensive description of the quark structure of the nucleon. The GPDs are the Wigner quantum phase space distribution of quarks in the nucleon—functions describing the simultaneous distribution of particles with respect to both position and momentum in a quantum-mechanical system, representing the closest analogue to a classical phase space density allowed by the uncertainty principle. In addi-

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tion to the information about the spatial density (form factors) and momentum density (parton distributions), these functions reveal the correlation of the spatial and momentum distributions, i.e. how the spatial shape of the nucleon changes when probing quarks and gluons of diﬀerent wavelengths. The concept of GPDs has led to completely new methods of “spatial imaging” of the nucleon, either in the form of two-dimensional tomographic images (analogous to CT scans in medical imaging), or in the form of genuine three-dimensional images (Wigner distributions). GPDs also allow us to quantify how the orbital motion of quarks in the nucleon contributes to the nucleon spin—a question of crucial importance for our understanding of the “mechanics” underlying nucleon structure. The spatial view of the nucleon enabled by the GPDs provides us with new ways to test dynamical models of nucleon structure. The GPDs describe the correlation of the quark longitudinal momenta (x + ξ, x − ξ) with the transverse-momentum transfer to the nucleon (Δ⊥ ). This information permits a simple interpretation in terms of a spatial distribution of quarks in the nucleon [79]. For ξ = 0, the two-dimensional Fourier transform of the GPD with respect to Δ⊥ describes the distribution of quarks with longitudinal momentum fraction x over the transverse distance, b, from the center of the nucleon (impact parameter). The integral of this spatial distribution over b gives the total parton density at a longitudinal momentum fraction x. This (1 + 2)-dimensional “mixed” momentum and coordinate representation corresponds to a set of “tomographic images” of the quark distribution in the nucleon at ﬁxed longitudinal momentum, x. Further motivation for the study of GPDs comes from the fact that certain moments of the GPDs—integrals over the quark momentum fractions—are related to fundamental static properties of the nucleon that cannot directly be accessed experimentally otherwise. In particular, the second moment of the GPDs gives the fraction of the nucleon spin carried by the quarks, including their spin and orbital angular momentum. Starting with the historic measurements by the EMC collaboration 20 years ago, determining how the nucleon’s spin is composed from the spins of the quarks and gluons and their orbital motion has been the central goal of polarized deep-inelastic scattering experiments. Measurements of the GPDs give access to the quark orbital angular momentum, thus providing information about another crucial piece of the nucleon “spin puzzle”. The momentum transfer, Q2 , in eN scattering deﬁnes the spatial resolution of the probe. The description of hard exclusive processes in terms of GPDs applies to the limit of large Q2 , where the reaction is dominated by the scattering from a single, quasi-free quark (“leading-twist approximation”). At lower Q2 , eﬀects related to the interaction of quarks during the hard-scattering process, or coherent scattering involving more than one quark, become important (“higher-twist eﬀects”). The minimum value of Q2 required for the GPD description to be applicable in practice depends on the ﬁnal state, and can ultimately only be determined experimentally. For DVCS (see ﬁg. 7), the experience with inclusive DIS and other two-photon processes such as γ ∗ γ → π 0 (measured in e+ e− annihilation) suggests that the leading-twist approximation should be reliable already at Q2 ∼ few GeV2 . Thus, DVCS can be used to extract information about GPDs at

Studies of semi-inclusive and hard exclusive processes at JLab

γ *(q)

419

γ (q')

x+ξ

x-ξ

GPD

p

p'=p+Δ

Fig. 7. – Reactions in eN scattering probing the generalized parton distributions. Deeplyvirtual Compton scattering and meson production probe the GPDs at non-zero longitudinal and transverse momentum transfer, ξ = 0, Δ⊥ = 0. Diﬀerent mesons (ρ, π, K) select diﬀerent spin-ﬂavor components of the GPDs.

the momentum transfers accessible in ﬁxed-target experiments. For meson production, the experience with the pion form factor at high Q2 and available meson electroproduction data suggest that higher-twist eﬀects are signiﬁcant up to momentum transfers of Q2 ∼ 10–20 GeV2 . While such eﬀects can be reduced by forming ratios of observables, or can be accounted for in phenomenological models, it seems likely that the use of meson production data for a quantitative extraction of GPDs requires measurements at signiﬁcantly higher momentum transfers than in DVCS. In the eN → eN γ cross-section, the DVCS amplitude interferes with the known amplitude of the Bethe-Heitler (BH) process, in which the ﬁnal-state photon is emitted from the electron (see ﬁg. 8). The total cross-section is given by [80]

(7)

α3 xB y dσ ep→epγ √ = dxB dy dΔ2 dϕ 16π 2 Q2 1 + 2

& &2 &T & & & , & e3 &

where = 2xB M/Q, y is the fraction of the electron energy lost in the nucleon rest frame and ϕ is the angle between the leptonic plane (e, e ) and the photonic plane (γ ∗ , γ).

DVCS

BH

+ (a)

+ (b)

(c)

Fig. 8. – Diagrams contributing to the electroproduction of a real photon. The DVCS process (a) is shown along with the interfering Bethe-Heitler diagrams (b) and (c).

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The total amplitude T is the superposition of the BH and DVCS amplitudes: 2

2

|T |2 = |TBH | + |TDVCS | + I,

(8)

∗ ∗ I = TDVCS TBH + TDVCS TBH ,

(9)

where TDVCS and TBH are the amplitudes for the DVCS and Bethe-Heitler processes, and I denotes the interference between these amplitudes. The individual contributions to the total ep → epγ cross-section can be written (up to twist-3 contributions) [80] as (10)

ΓBH (xB , Q2 , t) |TBH | = P1 (ϕ)P2 (ϕ)

2

2

cBH 0

|TDVCS | = ΓDVCS (xB , Q , t) 2

(11)

ΓI (xB , Q , t) I= P1 (ϕ)P2 (ϕ)

+

cDVCS 0

cos(nϕ) +

sBH 1

sin ϕ ,

2

DVCS DVCS cn cos(nϕ) + sn sin(nϕ) , + n=1

cI0

cBH n

n=1

2

2

+

3

cIn

cos(nϕ) +

sIn

sin(nϕ) ,

n=1

where P1 (ϕ) and P2 (ϕ) are the BH electron propagators and ci , si are azimuthal moments in the corresponding cross-section contributions. Depending on whether the beam helicity or target spin is ﬂipped, diﬀerent GPD contributions enter the cross section azimuthal moments (σLU , σU L ). In practice, cross-section asymmetries are experimentally easier to determine (12)

A=

dσ ← − dσ → dσ ← + dσ → ΓA (xB , Q2 , t)

sI1 sin ϕ + sI2 sin 2ϕ , I DVCS + (cBH + cI + Γ cDVCS ) cos ϕ cBH D 0 0 + c0 + ΓD c0 1 1

where ΓA , ΓD are known kinematical prefactors. DVCS measurements thus allow one to separate the imaginary and real parts of the DVCS amplitude (cf. ﬁg. 7) by measuring combinations of cross-sections and asymmetries with respect to the beam spin (helicity), beam charge (e+ /e− ), and/or target or recoil polarization. The imaginary part of the amplitude probes the GPDs at x = ξ = x/2, the real part probes a certain integral over the quark momentum fractions. The diﬀerent nucleon spin components of the GPDs can be extracted by measuring target spin asymmetries. Measurements of the t (Δ⊥ ) dependence provide the information necessary for transverse nucleon imaging. Information about the ﬂavor decomposition requires measurements with both protons and neutrons. Additional information about the spin/ﬂavor separation can come from meson production data. Studies of DVCS and meson production processes will require a combination of high-energy and high beam intensity, and are generally much more challenging than traditional inclusive scattering experiments.

421

Q2 (GeV2)

Studies of semi-inclusive and hard exclusive processes at JLab

One single (x ,Q2) bin B

One (x ,Q2,t) bin out of five B

4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

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ALU 0

0.1 0.2 0.3 0.4 0.5 xB

90

180 270 360

φ (deg)

0.3 1.5
90

180 270 360

φ (deg)

Fig. 9. – Measurements of the beam spin asymmetry, ALU , of the eN → e N γ cross-section from CLAS at 6 GeV. The plot shows the kinematic coverage in x and Q2 (left) and the azimuthal angle dependence of ALU in a typical (x, Q2 ) bin, both integrated over t (center) and in a sub-bin in t (right). The sin φ dependence is characteristic of the BH-DVCS interference cross-section. The magnitude of the asymmetry at φ = 90o can be related to a linear combination of the GPDs at x = ξ.

. 2 1. Present experimental results on hard exclusive processes. – Measurements of exclusive processes at large momentum transfers have been carried out in eN scattering experiments with existing ﬁxed-target facilities (HERMES at DESY, JLab with 6 GeV beam energy) and the HERA collider. These studies have demonstrated the basic feasibility of such measurements, and have provided crucial evidence for the applicability of a handbag-based description of such processes. They are also providing the ﬁrst useful constraints for GPD phenomenology. Experiments at ﬁxed-target facilities aim to extract the interference terms between the DVCS and the Bethe-Heitler (BH) amplitudes in the eN → e N γ cross-section. The interference terms are experimentally accessible from combinations of measurements of the spin-dependent and independent cross-sections and relative asymmetries, as well as from measurements of the beam charge dependence (e+ /e− ) of the cross-section. In kinematic regions where the BH amplitude is much larger than the DVCS amplitude, the interference with the BH amplitude acts as a natural “ampliﬁer and ﬁlter” for the DVCS amplitude, boosting it to comfortably measurable levels. Measurements of the beam spin asymmetry in eN → e N γ have been performed by HERMES (0.02 < x < 0.3) [81], CLAS (0.15 < x < 0.55) [82, 83] and Hall A [84, 85]. Figure 9 shows the kinematic coverage of the CLAS detector at 5.7 GeV, as well as the results for the asymmetry in a typical (x, Q2 ) bin. A new inner calorimeter was used to detect photons at small scattering angles. The azimuthal angle dependence of the asymmetry clearly exhibits the approximate sin φ behavior characteristic of the leading-twist BH-DVCS interference. This asymmetry can be related to a linear combination of GPDs at x = ξ; the experimental results are consistent with the predictions of present GPD models. An important point is that with the CLAS detector data in all (x, Q2 , t) bins are taken simultaneously, making it possible to extract information about the GPDs over a wide kinematic range.

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While the unpolarized GPD H dominates the beam spin asymmetry at small t [80], " 1 ), the target single-spin asymmetry at small t has a signiﬁcant σLU ∼ F1 H − ξF2 H( " " [80]: σU L ∼ F1 H−ξF contribution from the polarized GPD H 2 H. Therefore, a combined analysis of the beam and target single-spin asymmetries will allow for separation of H " and H. Recently published results by the CLAS Collaboration on the ﬁrst measurements of the longitudinal target spin asymmetry [86] conﬁrmed again that the factorization is likely to be applicable at Q2 values as low as 2 GeV2 . These measurements eventually will allow " E) " nucleon one to separate the contributions from unpolarized (H, E) and polarized (H, GPDs. An order of magnitude more data are expected from JLab during the next two years, which would allow for more accurate extraction of GPD parameters. The measurements of the DVCS cross-sections and beam spin asymmetries carried out by JLab with 6 GeV beam energy support the theoretical expectation of dominance of the single-quark reaction mechanism (leading-twist approximation) for DVCS for momentum transfers Q2 of a few GeV2 , essential for the GPD interpretation of the eN → e N γ data. They also demonstrate the feasibility of accurate diﬀerential measurements of the t-dependence of the cross-sections needed for the GPD-based reconstruction of the spatial images of the nucleon. The HERMES Collaboration measured for the ﬁrst time the beam charge asymmetry of the cross-section, which probes a dispersive integral of the GPDs over the quark momentum fractions [87]. DVCS cross-sections at high Q2 were measured at HERA [88, 89], including its t-dependence; the results are well described by leading-order (LO) and Nextto-Leading Order (NLO) QCD calculations incorporating QCD evolution of the GPDs, thus fully conﬁrming the applicability of QCD factorization to exclusive processes at high energies. . 2 2. GPD measurements with Jeﬀerson Laboratory at 12 GeV . – CLAS12 will provide a unique combination of high beam intensity (luminosity), high energy, and largeacceptance detectors, which will enable studies of exclusive processes such as DVCS and meson production in the valence quark region. Detection of the photon in the CLAS inner calorimeter, in addition to the recoil proton and scattered electron, provides the exclusivity condition crucial for complete control over diﬀerent background processes. Using the epX sample has its own advantages with regard to background suppression (π 0 ) and azimuthal angle coverage. The two samples probe ep → e γp in regions of diﬀerent relative magnitudes of the DVCS and Bethe-Heitler amplitudes. In this way, GPDs can be extracted using both absolute cross-section and polarization asymmetry data. The possibility to use both methods in experiments at a single facility represents a crucial advantage of CLAS12. To separate the diﬀerent spin components of the GPDs, measurements of a variety of polarization observables (beam and target spin) will be performed. The longitudinal e Ee correspond to respective Compton form factors [80]. (1 ) H, E, H,

423

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0.5

0.4

0.45

0.3

0.4

0.2

0.35

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0.25

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360 0

0.2

0.4

0.6

0.8

1 2

-t (GeV )

0.2

0.3

0.4

0.5

xB

0 0.6

Fig. 10. – On all ﬁgures: points represent expected 12 GeV data from CLAS12 with statistical error bars. Lines are models with diﬀerent input parameters, none of which include twist-3 contributions: The full line is a model with a Regge-type t-dependence and D-term. The dotted line includes the Regge-type t-dependence but has no D-term. The dash-dotted line has the D-term but the t dependence only comes from form factors. Left ﬁgure: sample Beam Spin Asymmetry (BSA) as a function of ϕ for xB = 0.2, Q2 = 3.3 GeV2 , and −t = 0.45 GeV2 . Middle ﬁgure: BSA as a function of −t for xB = 0.2, Q2 = 3.3 GeV2 , and ϕ = 90◦ . Right ﬁgure: BSA as a function of xB for t = 0.45 GeV2 , Q2 = 3.3 GeV2 and ϕ = 90◦ .

beam single-spin asymmetry, ALU (see ﬁg. 10), will access mainly the unpolarized Dirac GPD, H. Combined analysis of the DVCS data on a longitudinally polarized target single spin asymmetry (see ﬁg. 11) with the beam single-spin asymmetry will provide separation of contributions from unpolarized and polarized GPDs. The double-spin asymmetry for longitudinal target polarization, ALL , provides information on the real part of the corresponding GPDs, complementary to beam charge asymmetries. The results of these measurements can directly be translated into transverse spatial images of the nucleon. As an example, ﬁg. 12 shows the projected results for the Dirac GPD, H, as a function of x and t, and its corresponding transverse spatial representation. Complementary information can be obtained about integrals of the GPDs over the quark momentum fraction. With the help of GPD parameterizations, this information can be used to construct 2-dimensional tomographic images of the nucleon. Additional information about the ﬂavor structure of GPDs will come from ratios of meson production cross-sections in channels with the same spin/parity quantum numbers, such as η/π 0 and K ∗ /ρ+ . These channels will be measured simultaneously with DVCS, not requiring any extra beam time. Measurements of exclusive meson production in non-diﬀractive channels in CLAS12 would allow for detailed studies of the spin, ﬂavor, and spatial distributions of quarks in the nucleon in the valence region, complementing the information from DVCS measurements. Very interesting information can already be gained by comparing observables for diﬀerent mesonic channels, without detailed modeling of the GPDs. For example, a comparison of π 0 and η provides model-independent information about the ratio of the quark spin distributions Δu and Δd and their spatial distributions. Comparison between π +

424

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sinϕ

a)

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AUL

A UL

H. Avakian

b)

c)

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0.5

xB

-t (GeV2)

Fig. 11. – (a) CLAS12 expectations for target spin asymmetry vs. ϕ for Q2 = 4.1 GeV2 , xB = 0.36, and −t = 0.52 GeV2 . The points show the values from ref. [90] using CTEQ6 PDFs (thin curve) with the estimated errors from the proposed measurement. The thick solid curve is using e = 0, and for the dashed curve is H e is also set to zero. (b) sin ϕ MRST02 PDFs with E = E moments of the target spin asymmetry vs. −t at Q2 = 4.1 GeV2 and xB = 0.36, and (c) vs. xB at Q2 = 4.1 GeV2 and −t = 0.52 GeV2 . The projected error bars represent the statistical uncertainties only.

x = 0.45

H (x = ξ, t) / F1 (t)

15

x = 0.25 0.35 0.45

10

3

2

3

2

1

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0.5

1

0

1

0

3

x = 0.35

0

0.5

1

0

x

0

0.5

1

b [fm]

Fig. 12. – Left: projected results for the Dirac GPD of the proton, H(x = ξ, t), as a function of x and t, as extracted from the DVCS beam spin asymmetry, ALU , measured at JLab at 12 GeV. Shown is the ratio of the GPD to the the proton’s Dirac form factor, F1 (t). Right: transverse spatial image of the proton obtained by Fourier-transforming the measured GPD. The errors were estimated assuming a dipole-like t-dependence.

Studies of semi-inclusive and hard exclusive processes at JLab

425

and K + production, as well as between ρ+ and K ∗+ , allows one to study SU (3) ﬂavor symmetry breaking in the nucleon’s quark distributions in diﬀerent spin/parity channels. More information about the spatial distribution of quarks can be obtained from the GPD analysis of absolute cross-sections (σL ) in these channels. Separation of the various response functions (L, T , etc.) would provide a crucial test of the dominance of the single-quark reaction mechanism at large Q2 . . 2 3. GPD studies with a transversely polarized target. – Asymmetries in the exclusive production of photons and vector mesons with a transversely polarized target were identiﬁed as the most sensitive observables providing access to the total orbital angular momentum. Eight observables, namely the ﬁrst harmonics cos φ and sin φ of the interference term, are accessible in polarized beam and target experiments [80]. Thus, experiments with both longitudinally and transversely polarized targets can measure all eight Fourier coeﬃcients cI1,Λ and sI1,Λ and with Λ = {unp, LP, TPx , TPy }. The DVCS Single-Spin Asymmetry (SSA) for a transversely polarized target is the most sensitive observable to the elusive GPD E, providing access to the orbital angular momentum. First results on DVCS single-spin asymmetries from the HERMES Collaboration for transverse target polarizations [91,92] indeed indicate great sensitivity of target single spin asymmetries to the contribution of u-quarks to the total angular momentum. The most sensitive to the GPD-E asymmetry appeared to be the cos φ moment of the spin-dependent contribution σU T [80], (13)

σU T ∼

1−x t t F2 H + (2 − x)F1 E. 2 2−xM 4M 2

Transverse target DVCS SSA measurements in addition to unpolarized SSA and longitudinally polarized SSA measurements will provide the full set of data needed for the extraction of Compton form factors and corresponding GPDs. AU T is especially sensitive to the GPD E, and as such will constrain any extraction of the angular momentum J. The projection curves for CLAS12 running with a transversely polarized target have been calculated assuming a luminosity of 2 × 1034 cm−2 s−1 , with an N H3 target polarization of 85% and a dilution factor 0.14, with 2000 hours of data taking (see ﬁg. 13). The theoretical uncertainty in the factorization procedure on the amplitude level for the meson sector is translated into large variations of the physical cross section. However, in the single-spin asymmetry, given by the ratio of the Fourier coeﬃcients of the cross section, the ambiguities approximately cancel [94]. Thus, the perturbative predictions for this quantity are rather stable. The NLO eﬀects result in +7% −18% corrections to the LO prediction for 0.1 < x < 0.5. The handbag-based calculations were performed for the case when the incoming virtual photon is longitudinally polarized. The cross section for the transversely polarized photons is suppressed by a power of Q [95], but at JLab energies it may still be signiﬁcant. Insensitivity to the higher-order corrections make single spin asymmetries appropriate quantities for experimental study at JLab, and will provide an important test of the applicability of handbag-based predictions.

426 AUT

H. Avakian

0.4

0.4 -t = 0.5 GeV

0.3

2

0.2

AUTsin(φ- φs)cosφ

Q2=2.6,xB=0.25 0.6 0.5

Ju = 0.1 Jd = 0

0.2

HERMES CLAS12

0.4 0.3

0

Ju=0.12

0.2 Ju=0.34

0.1 0

-0.2

-0.1

Ju=0.50

HERMES (preliminary) JLab 12 GeV (projected)

-0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

0.1

0.2

0.3

0.4

0.5

x

φ Fig. 13. – Projected transverse spin asymmetry (Asin U T ) in exclusive photon production at 11 GeV. All points correspond to diﬀerent values of Ju calculated for the bin with Q2 = 2.6 GeV2 and x = 0.25 (left). Projections for the transverse target asymmetry for exclusive ρ0 production from a hydrogen target (ﬁlled squares) using CLAS12 are shown compared to preliminary HERMES data [93].

Even though the power corrections for the absolute cross-section of exclusive meson electroproduction analyzed in terms of generalized parton distributions are expected to be large, there are indications of a precocious scaling in the ratios of observables [94]. The measurement of spin asymmetries could therefore become a major tool for studying GPDs in the Q2 domain of a few GeV2 . Projections for CLAS12 for measurements of transverse asymmetries for vector mesons are shown in ﬁg. 13. The transverse asymmetries for ρ mesons (neutral and charged) are widely accepted as an important source of independent information on the GPD E. SSA measurements in hard exclusive processes will allow mapping of the underlying GPDs and will provide access to the orbital angular momentum of quarks. The quark angular momentum in the nucleon, Jq , can be estimated if one uses the results of measurements of DVCS and meson production observables to constrain GPD parameterizations, which incorporate information about GPDs obtained from other processes (inclusive DIS, form factors). These parameterizations allow one to estimate the second moment of the GPDs, based on the information about the GPDs at x1 = x2 and the momentum fraction integrals probed in the observables. Figure 13 shows the constraints on Ju and Jd from DVCS and ρ asymmetries, which is particularly sensitive to the Pauli form factor-type GPD, E. One sees that accurate measurements of the asymmetries will be able to constrain Jq in this way. While not fully model independent, this method of extracting Jq will become more and more accurate as amplitude calculations and GPD parameterizations become more reﬁned as a result of measurements of a variety of other DVCS and meson production observables. High-statistics data will allow us to constrain the quark angular momentum in the proton.

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The data from CLAS12 with a transversely polarized target focussing on hard exclusive photon and meson production, combined with data from unpolarized and longitudinally polarized targets, will provide a complete set of measurements required for the separation of all four leading-twist, chiral-even GPDs, and in particular, will provide a constraint on the quark orbital angular momentum. GPDs unify the momentum-space parton densities measured in inclusive deepinelastic eN scattering with the spatial densities (form factors) measured in eN elastic scattering. They describe correlations between the momentum and spatial distributions of quarks, which are revealed in exclusive processes in eN scattering at large momentum transfer (deeply virtual Compton scattering, meson production). A full program to extract GPDs from measurements requires coverage of a large kinematic range in ξ, t, and Q2 , along with measurements of several ﬁnal states together with the use of polarized beam and polarized targets (both longitudinal and transverse polarizations). The 12 GeV upgrade of the electron accelerator and of the equipment required for the GPD program will provide the tools and kinematic coverage needed for a broad program of DVCS and meson production measurements. The JLab 12 GeV upgrade will allow us to map the nucleon GPDs in the valence quark region. 3. – Conclusions Understanding of spin-orbit correlations, together with independent measurements related to the spin and orbital angular momentum of the quarks, will help to construct a more complete picture of the nucleon in terms of elementary quarks and gluons going beyond the simple collinear partonic representation. The JLab 12 GeV upgrade will provide the unique combination of wide kinematic coverage, high beam intensity (luminosity), high energy, high polarization, and advanced detection capabilities necessary to study nucleon 3D parton distributions in hard processes. Measurements of semi-inclusive processes combined with inclusive and exclusive measurements with an upgraded JLab will allow us to study the quark structure of the nucleon with unprecedented detail. ∗ ∗ ∗ We thank A. Kotzinian and C. Weiss for many related discussions. This work was supported by the United States Department of Energy. Jeﬀerson Science Associates (JSA) operates the Thomas Jeﬀerson National Accelerator Facility for the U.S. DOE under contract DE-AC05-060R23177. REFERENCES [1] [2] [3] [4] [5]

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Global ﬁt for the simultaneous determination of the transversity distribution and the Collins fragmentation functions M. Boglione INFN, Sezione di Torino - Via P. Giuria 1, I-10125 Torino, Italy

Summary. — The ﬁrst part of this lecture is a general introduction to distribution and fragmentation functions, the “soft” functions which imbed the full information on the polarization structure of the nucleon in terms of its elementary constituents. I then concentrate on the description of two of these objects, yet unknown: the transversity distribution and the Collins fragmentation functions. The determination of these two soft functions will be the focal point of the last part of my lecture, where a global analysis of the experimental data on azimuthal asymmetries in SemiInclusive Deep Inelastic Scattering (SIDIS), from the HERMES and COMPASS Collaborations, and in e+ e− → h1 h2 X processes, from the Belle Collaboration, will be presented.

1. – Distribution and fragmentation functions One of the fundamental challenges of hadron physics is the mapping of the nucleon spin structure in terms of its elementary constituents, quarks and gluons. The basic packages of information on a polarized nucleon are contained inside three independent Parton Distribution Functions (PDFs): c Societ` a Italiana di Fisica

431

432

M. Boglione

Fig. 1. – Pictorial illustration of the unpolarized, helicity and transversity distribution functions.

– The unpolarized distribution function, q(x, Q2 )[g(x, Q2 )], which gives the number density of unpolarized quarks[gluons] carrying a light cone fraction x of the parent unpolarized-proton momentum, at a given scale Q2 . – The helicity distribution function Δq(x, Q2 )[Δg(x, Q2 )], related to the probabilities of ﬁnding the helicity of the parton to be either the same or opposite to that of the longitudinal polarized parent nucleon. – The transversity distribution function ΔT q(x, Q2 ), which gives the diﬀerence in number density of quarks transversely polarized in a direction parallel or antiparallel to that of the transversely polarized parent nucleon. Notice that helicity conservation of QCD and QED interactions implies that there is no gluonic counterpart to ΔT q (see refs. [1] and [2]). The quark distribution functions (1) (2) (3)

+ − (x, Q2 ) − q+ (x, Q2 ), Δq(x, Q2 ) = q+

ΔT q(x, Q2 ) = q↑↑ (x, Q2 ) − q↑↓ (x, Q2 ), + − q(x, Q2 ) = q+ (x, Q2 ) + q+ (x, Q2 ) = q↑↑ (x, Q2 ) + q↑↓ (x, Q2 )

± is the number density of quarks are given a pictorial illustration in ﬁg. 1. Notice that q+ with ± helicity inside a proton with positive helicity; similarly, q↑↑↓ is the number density of spin-up quarks inside a proton with up transverse polarization. The gluon helicity and unpolarized distribution functions are deﬁned in an analogous way, and are described in detail in ref. [3]. Many diﬀerent notation schemes exist in the literature: q(x, Q2 ) and Δq(x, Q2 ) are often denoted f1q (x, Q2 ) and g1q (x, Q2 ), respectively, while ΔT q(x, Q2 ) can also be called hq1 (x, Q2 ) or δq(x, Q2 ) by diﬀerent authors.

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Fig. 2. – A parton inside its parent nucleon has, in general, an intrinsic transverse momentum k⊥ which makes that parton non-collinear to its parent nucleon. There are crucial correlations between parton intrinsic transverse momenta and spin which cannot be neglected when studying the spin structure of the nucleon in terms of its elementary constituents.

These PDFs are integrated distribution functions as they only depend on the lightcone fraction x and on the scale Q2 , while the parton intrinsic transverse-momentum degrees of freedom (see ﬁg. 2) are integrated over. Indeed they are perfectly suitable for a collinear treatment of the process under study; however, there are crucial correlations between parton intrinsic transverse momenta and spin which cannot be neglected when studying the spin structure of the nucleon in terms of its elementary constituents. When the intrinsic transverse momentum is not integrated out, one ﬁnds a set of eight independent parton distribution functions, see ﬁg. 3, and new conventions for the notation need to be introduced. In what follows we will explicitly indicate all polarization indices: the suﬃx q or p will signal an unpolarized quark or proton, while p+ and p↑

Fig. 3. – Transverse momentum dependent distribution functions.

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M. Boglione

indices will indicate longitudinally and transversely polarized protons, with respect to their direction of motion. The quark polarizations will be denoted by sx , sy and sz and refer to the quark own helicity frame, which does not coincide with the parent proton helicity frame due to the presence of the intrinsic transverse momentum. This delicate issue is discussed in great detail in ref. [3]. Indeed, sx and sy are the two components of the quark transverse polarization vector, while sz is its longitudinal component. From now on, for simplicity, we will omit the Q2 dependence of the structure functions, which will be implicitly understood. Figure 3 gives a pictorial illustration of the eight Transverse Momentum Dependent (TMD) distribution functions: – In the center we have the unpolarized TMD distribution function, fq/p (x, k⊥ ), and on its left the helicity TMD distribution functions, Δfsz /p,+ (x, k⊥ ), deﬁned in such a way that (4) fq/p (x) ≡ q(x) = d2 k⊥ fq/p (x, k⊥ ), Δfsz /p+ (x) ≡ Δq(x) = d2 k⊥ Δfsz /p+ (x, k⊥ ). (5) – Below the TMD helicity distribution, a “new” function, Δfsx /p+ , which corresponds to the probability of ﬁnding a quark transversely polarized along x inside a longitudinally polarized proton. – On the right side of the diagram we ﬁnd the two components, Δfsx /p↑ and Δfsy /p↑ , of the transversity quark distribution function. When suitably combined and integrated over k⊥ , they give back the usual transversity function, ΔT q(x), see ref. [3]. – Above them, another “new” TMD function, Δfsz /p↑ , which gives the number density of longitudinally polarized quarks inside a transversely polarized proton. – On the top of the diagram, two TMD distributions which have been very debated lately: the Sivers and the Boer-Mulders distribution functions, somehow complementary to each other, giving the number density of unpolarized quarks in a transversely polarized proton and the number density of transversely polarized quarks inside an unpolarized proton. Notice that the Boer-Mulders function turns out to be equal to Δfsy /p+ , the number density of quarks transversely polarized along the y direction inside a longitudinally polarized proton. For a detailed description of the Sivers distribution function see ref. [1]. The integrated unpolarized parton distribution functions, q(x, Q2 ) and g(x, Q2 ), and the u and d quark helicity distribution functions, Δq(x, Q2 ) are well known over a wide range of x and Q2 , see for example refs. [1] and [4]. For detailed discussions on the sea contributions to the helicity distribution function see refs. [5] and [6]. The gluon helicity distribution function, Δg(x), was largely undetermined (even in sign) until very recently, when statistical analyses of PHENIX and STAR experimental data on the longitudinal

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435

double spin asymmetry ALL for neutral pion production in proton-proton scattering showed that the hypothesis Δg 0 is favoured over the alternative possibilities of a large positive or negative Δg (see refs. [7] and, for example, [8]). The k⊥ distributions of the unpolarized and helicity distribution functions, fq,g/p (x, k⊥ ) and Δfsq,g + (x, k⊥ ), z /p are unmeasured. However, very recent lattice QCD calculations [9] have shown that the k⊥ dependence of the unpolarized distribution function is compatible with a Gaussian distribution. For modelling purposes we assume a factorized Gaussian form, suitable to describe non-perturbative eﬀects at small PT values and simple enough to allow analytical integration over the intrinsic transverse momenta: 2

(6)

2

e−k⊥ /k⊥ fq,g/p (x, k⊥ ) = fq,g/p (x) , 2 π k⊥ 2

(7)

Δfsq,g (x, k⊥ ) = Δfsq,g (x) z /p z /p

2

e−k⊥ /k⊥ L , 2 π k⊥ L

2 2 where k⊥

and k⊥

L are free parameters to be determined by ﬁtting sensitive experi2 mental data (see ref. [10] for the determination of k⊥

and [8] for a discussion on the sensitivity of the ALL proton-proton scattering longitudinal double spin asymmetry to 2 various choices of the parameter k⊥

L ). In eq. (7), fq/p (x) ≡ q(x), fg/p (x) ≡ g(x), q g Δfsz /p (x) ≡ Δq(x) and Δfsz /p (x) ≡ Δg(x) are the usual unpolarized and helicity densities which we take from the literature; in particular we refer to refs. [11, 12]. On the contrary, the distribution of transversely polarized quarks in a transversely polarized nucleon, ΔT q(x, Q2 ), is so far completely unknown. The reason is that, being related to the expectation value of a chiral-odd quark operator, it appears in physical processes which require a quark helicity ﬂip: this cannot be achieved in the usual inclusive DIS, due to the helicity conservation of perturbative QED and QCD processes, see ﬁg. 4. Detailed descriptions of the transversity distribution function are also given in refs. [1, 2, 4]. The problem of measuring the transversity distribution has been largely discussed in the literature [13]. The most promising approach is considered the double transverse spin asymmetry AT T in Drell-Yan processes in p¯ p interactions at a squared c.m. energy of the order of 200 GeV2 , which has been proposed by the PAX Collaboration [14-17]. However, this requires the availability of polarized antiprotons, which is an interesting, but formidable task in itself. An alternative method to extract the transversity distribution function combining SIDIS and e+ e− → h1 h2 X data will be the focus of this lecture and will be extensively discussed in what follows. As for the unpolarized and helicity densities, the dependence of the transversity distribution function is unknown, while simple positivity constraints require its x distribution to be limited by the so-called Soﬀer bound [18]

(8)

|ΔT q(x)| ≤

1 [q(x) + Δq(x)], 2

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M. Boglione

Fig. 4. – Hand-bag diagrams representing the structure of the three integrated distribution functions, fq/p (x), Δq(x) and ΔT q(x). The “up” and “down” spin states are expressed in terms √ of the helicity ± states by | ↑↓ = 1/ 2(|+ ± i |−). In the helicity basis, the chiral odd nature of the transversity distribution function is explicitly visible. Notice that ΔT q(x) decouples from DIS, where helicity ﬂips are forbidden by helicity conservation.

see ref. [2] for a nice pictorial explanation. Therefore, we devise a parametrization which automatically satisﬁes all the required properties of ΔT q(x): 2

(9)

ΔT q(x, k⊥ ) =

2

1 T e−k⊥ /k⊥ T Nq (x) [fq/p (x) + Δq(x)] , 2 2 π k⊥ T

with (10)

NqT (x) = NqT xα (1 − x)β

(α + β)(α+β) , αα β β

2 2

T = k⊥

, but from our ﬁts we learn that present experiand |NqT | ≤ 1. In general k⊥ 2 2 mental data are insensitive to such a diﬀerence, therefore we simply assume k⊥

T = k⊥

. (α+β)

assures that the function NqT (x) ≤ 1. The expoNotice that the coeﬃcient (α+β) αα β β 2 , nents α and β are taken to be ﬂavor independent. Finally, the average value of k⊥ assumed to be constant and ﬂavor independent, is taken from ref. [10], where it was

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437

Fig. 5. – Transverse momentum dependent fragmentation functions.

obtained by ﬁtting the azimuthal dependence of SIDIS unpolarized cross-section: (11)

2

= 0.25 GeV2 . k⊥

This value is in perfect agreement with the recent lattice calculation of ref. [9], which 2 = 0.5 GeV. quotes k⊥ When considering semi-inclusive hadron production, one has to take into account the so-called fragmentation process, i.e. the hadronization of the parton (resulting from the elementary interaction) into the observed ﬁnal state, h. Often a collinear conﬁguration is assumed, in which the hadron h retains the same direction of the parent parton. Thus, for unpolarized ﬁnal hadrons, the fragmentation function Dh/q,g (z) is associated to the probability of parton q or g to generate the detected hadron h, carrying a light-cone fraction z of the parent parton momentum. However, once again, when studying spin-dependent observables, the general, non-collinear conﬁguration has to be taken into account and intrinsic transverse degrees of freedom cannot be integrated out. When only spinless ﬁnal hadrons are considered (pions and kaons, for instance), the TMD fragmentation set is made of only two objects: the unpolarized fragmentation function Dh/q (z, p⊥ ) and the so-called Collins fragmentation function, ΔNDh/q↑ (z, p⊥ ), which is related to the probability for a transversely polarized quark to fragment into a spinless hadron. Similar deﬁnitions hold for gluons [3]. Notice that this function, likewise the transversity distribution function, is chirally odd. A pictorial representation of the unpolarized and Collins quark fragmentation functions is given in ﬁg. 5. The Collins fragmentation function imbeds the correlation between the p⊥ of the produced hadron h and the transverse spin of the parent quark q, (12)

Dh/q,s (z, p⊥ ) = Dh/q (z, p⊥ ) +

1 N Δ Dh/q↑ (z, p⊥ ) sˆ · (pˆq × pˆ⊥ ), 2

where pq is the momentum of the fragmenting quark, along the z-axis in its own helicity frame, while the hadron transverse momentum is p⊥ = |p⊥ |(cos ϕ, sin ϕ, 0) in that reference frame; Sq is the spin vector of the parent quark, as showed in ﬁg. 6.

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M. Boglione

Fig. 6. – The Collins fragmentation function imbeds the correlation between the p⊥ of the produced hadron h and the transverse spin Sq of the parent quark q.

The integrated unpolarized fragmentation function Dh/q (z) is quite well known, as it can be determined by ﬁtting a huge amount of experimental data, covering large ranges in z and Q2 ; some recent parametrizations can be found in [19-21], for example. For its p⊥ dependence we have, up to know, no experimental information. However, the BELLE Collaboration is presently analysing a set of high-statistics e+ e− → h1 h2 X data, and might soon be able to give some valuable information on the shape and size of Dh/q (z, p⊥ ) as a function of p⊥ , in the z range covered by their experiment. For the time being, we use a Gaussian smearing assuming full factorization between z and p⊥ dependence 2

(13)

Dh/q (z, p⊥ ) = Dh/q (z)

2

e−p⊥ /p⊥ , π p2⊥

normalized so that (14)

Dh/q (z) =

dp⊥ Dh/q (z, p⊥ ),

and where p2⊥ is a free parameter to be determined according to the experimental data. For our calculation we use the functions Dh/q (z) given in ref. [19]. The Collins function ΔN Dh/q↑ (z, p⊥ ), often denoted as [22] (15)

ΔN Dh/q↑ (z, p⊥ ) =

2p⊥ ⊥q H (z, p⊥ ), zmh 1

was unknown until very recently, when the BELLE Collaboration published their data on the azimuthal modulations of the e+ e− → h1 h2 X asymmetry. The ﬁtting procedure to extract the z dependence of the Collins function from these data will be discussed in great detail in sect. 3; an early attempt to extract ΔN Dh/q↑ using the BELLE experimental data can be found in ref. [23]. For its p⊥ dependence we will again choose a Gaussian factorized form, multiplied by an extra term, h(p⊥ ), which imbeds the requirement that the Collins function should vanish at small values of p⊥ , and allows for slight modiﬁcations of the Gaussian width (with respect to that of the unpolarized fragmentation function)

Global fit for the simultaneous determination of the transversity etc.

439

through the parameter M 2 : 2

(16)

2

e−p⊥ /p⊥ Δ Dh/q↑ (z, p⊥ ) = 2 Nq (z) Dh/q (z) h(p⊥ ) , π p2⊥ N

C

with (17)

NqC (z) = NqC z γ (1 − z)δ

(18)

h(p⊥ ) =

√

2e

(γ + δ)(γ+δ) , γ γ δδ

p⊥ −p2⊥ /M 2 e , M

with |NqC | < 1 and where the Gaussian width p2⊥ is the same as that used for the unpolarized fragmentation function, eq. (13), while M is an extra free parameter. Similarly to what we did for the transversity distribution function, the parametrization of the Collins function is devised in such a way that the required positivity bound (19)

|ΔN Dh/q↑ (z, p⊥ )| ≤ 2Dh/q (z, p⊥ )

is automatically satisﬁed, since NqC (z) and h(p⊥ ) are normalized to be smaller than 1 in size for any value of z and p⊥ , respectively. Finally, the average value of p2⊥ is taken 2 from ref. [10] where it was obtained, together with k⊥

, by ﬁtting experimental data on the Cahn eﬀect in SIDIS unpolarized cross-section: (20)

p2⊥ = 0.20 GeV2 .

Such value is assumed to be constant and ﬂavor independent. As mentioned previously, the complexity of measuring the transversity distribution function is intimately related to its chiral odd nature. A consequence of this “oddness”, is that the transversity function always appears coupled to another chiral-odd function. While waiting for precise Drell-Yan experimental data, p¯ p → l+ l− X, which will give us direct access to the “square” of the transversity function, the most accessible channel, which involves the convolution of the transversity distribution with the Collins fragmensin(φ +φ ) tation function [24], is the azimuthal asymmetry AU T S h in SIDIS processes, namely p↑ → π X. This is the strategy being pursued by HERMES, COMPASS and JLab Collaborations. A crucial improvement, towards the success of this strategy, has been recently achieved thanks to the independent measurement of the Collins function (or rather, of the convolution of two Collins functions), in e+ e− → h1 h2 X unpolarized processes by Belle Collaboration at KEK [25]. By combining the SIDIS experimental data from HERMES [26, 27] and COMPASS [28], with the Belle data, we have, for the ﬁrst time, a large enough set of data points as to attempt a global ﬁt which involves, as unknown functions, both the transversity distributions and the Collins fragmentation functions of u and d quarks. As we will

440

M. Boglione

l´

l k⊥ k´

p

PT

Ph

φh

ϕ

⊥

P y

S

φS

x z y x z

Fig. 7. – Three-dimensional kinematics of the SIDIS process, according to Trento conventions [22]. The photon and the proton collide along the zˆ-axis, while the leptonic plane deﬁnes the xz ˆ plane. The fragmenting quark and the ﬁnal hadron h are emitted at azimuthal angles ϕ and φh , and the proton transverse spin direction is identiﬁed by φS .

describe in detail in what follows, with this strategy we will be able to extract simultaneously the Collins fragmentation function ΔN Dπ/q↑ (z, p⊥ ) and the transversity distribution function ΔT q(x) for q = u, d. We can then use the transversity distributions and the Collins functions so determined, to give predictions for forthcoming experiments at JLab and CERN-COMPASS. 2. – Transversity and Collins functions from SIDIS processes The role of transversity distribution and Collins fragmentation functions was presented in many of the lectures of this School. For detailed discussions on this and related issues I will therefore refer the reader to the contributions of M. Anselmino, D. Hash, E. Leader and N. Saito to these proceedings. In this section I will simply recall the deﬁnitions and formulae which are strictly needed to the purposes of this lecture. The exact kinematics for SIDIS p → h X processes in the γ ∗ − p c.m. frame, which we will call S, including all intrinsic motions, was extensively discussed in ref. [10], and is schematically represented in ﬁg. 7. We take the virtual photon and the proton colliding along the zˆ-axis with momenta q and P , respectively, and the leptonic plane to coincide with the xz ˆ plane. The ﬁnal hadron h is produced, out of this plane, with threemomentum Ph = (PT , Phz ). We work in the kinematic regime in which PT ΛQCD k⊥ , where k⊥ is the magnitude of the intrinsic transverse momentum k⊥ of the initial quark with respect to the parent proton and PT = |PT | is the magnitude of the ﬁnal hadron transverse momentum. We neglect second-order corrections in the k⊥ /Q expansion: in this approximation, the transverse momentum p⊥ of the observed hadron h with respect

Global fit for the simultaneous determination of the transversity etc.

441

to the direction of the fragmenting quark is related to k⊥ and PT by the simple expression p⊥ = PT − zk⊥ ; in addition, the light cone momentum fractions x and z coincide with the usual measurable SIDIS variables, z = zh = (P ·Ph )/(P ·q) and x = xB = Q2 /(2P ·q). In this region factorization holds [29,30], leading-order q → q elementary processes are dominating and the soft PT of the detected hadron is mainly originating from intrinsic motions. The transverse Single-Spin Asymmetry (SSA) for this process is deﬁned as ↑

(21)

AU T =

↓

d6 σ p → hX − d6 σ p → hX dσ ↑ − dσ ↓ , ↑ → hX ↓ → hX ≡ 6 p 6 p dσ ↑ + dσ ↓ d σ +d σ

↑,↓

↑,↓

where d6 σ p →hX is a shorthand notation for (d6 σ p →hX )/(dxB dy dzh d2 PT dφS ). It will often happen, in comparing with data or giving measurable predictions, that the numerator and denominator of eq. (21) will be integrated over some of the variables, according to the kinematical coverage of the experiments. ↑ and ↓ refer, respectively, to polarization vectors S and −S, see ﬁg. 7. A full study of eq. (21), with all contributions at all orders in k⊥ /Q, will be presented in a forthcoming paper [31]. We consider here, at O(k⊥ /Q), the sin(φS + φh ) weighted asymmetry, (22)

sin(φS +φh )

AU T

=2

dφS dφh [dσ ↑ − dσ ↓ ] sin(φS + φh ) , dφS dφh [dσ ↑ + dσ ↓ ]

measured by the HERMES [26, 27] and COMPASS [28] Collaborations. This asymmetry singles out the spin-dependent part of the fragmentation function of a transversely polarized quark with spin polarization sˆ and three-momentum pq , see eq. (12), resulting in S +φh ) (23) Asin(φ = UT

d(Δˆ σ) N Δ Dh/q↑ (z, p⊥ ) sin(φS +ϕ+φhq ) sin(φS + φh ) e2q dφS dφh d2 k⊥ ΔT q(x, k⊥ ) dy q ·

dˆ σ Dh/q (z, p⊥ ) e2q dφS dφh d2 k⊥ fq/p (x, k⊥ ) dy q

In the above equation fq/p (x, k⊥ ) is the unpolarized TMD distribution function of a quark q inside the parent proton p, eq. (6), while Dh/q (z, p⊥ ) is the unpolarized TMD fragmentation function of quark q into the ﬁnal hadron h of eq. (13), ΔT q(x, k⊥ ) is the TMD transversity distribution, eq. (9), while ΔN Dh/q↑ (z, p⊥ ) is the Collins fragmentation function of eq. (16). dˆ σ /dy is the planar unpolarized elementary cross-section (24)

2πα2 dˆ σ = [1 + (1 − y)2 ] dy sxy 2

442

M. Boglione S’

ha dr on

y’

x’ z’

p Ph y

S

PT

z

z

k’ photon

Ph

k q

x

proton

k

P

k Fig. 8. – Kinematics of the fragmentation process.

and ↑

(25)

↑

↑

↓

dˆ σ q →q dˆ σ q →q 4πα2 d(Δˆ σ) = − = (1 − y). dy dy dy sxy 2

The sin(φS + ϕ + φhq ) azimuthal dependence in eq. (23) arises from the combination of the phase factors in the transversity distribution function, in the non-planar q → q elementary scattering amplitudes, and in the Collins fragmentation function; φS and ϕ identify the directions of the proton spin S and of the quark intrinsic transverse momentum k⊥ , see ﬁg. 7; φhq is the azimuthal angle of the ﬁnal hadron h, as deﬁned in the helicity frame, S , of the fragmenting quark, in which the hadron h moves along the z -axis (see ﬁg. 8). It is worth concentrating this angle, as it plays a pivotal role in the hadronization process, in both SIDIS and e+ e− scattering, as in any other hadron production reaction. In the S reference frame, φhq coincides with the azimuthal angle which identiﬁes the hadron transverse momentum p⊥ . This vector can be written in terms of the momentum vectors of the ﬁnal hadron, Ph = (PT , Phz ), and of the fragmenting quark, k = k + q = (k⊥ , kz ) ref. [10], see ﬁg. 8: (26)

(27)

ˆ ) k ˆ = p⊥ = Ph − (Ph · k PT · k⊥ + Phz k z PT · k⊥ + Phz k z z z = PT − k , P − k ⊥ h |k |2 |k |2 2 k⊥ = PT − zh k⊥ + O , Q2

Global fit for the simultaneous determination of the transversity etc.

443

with [10] Phz = (zh W )/2 − PT2 /(2zh W ), 1−x W x k2 k z = + B ⊥2 , 2 1 − xB x Q

(28)

2 + (k z )2 . |k |2 = k⊥

ˆ To reach the helicity frame S of the fragmenting quark, moving along the direction k as deﬁned in the γ ∗ − p reference frame, S (see ﬁg. 8), we have to perform two rotations, as described in Appendix D of ref. [3]. The S -axes are then given by

ˆ , z = k

(29)

ˆ⊥ , y = z × k ˆ⊥) × k ˆ , x = y × z = (z × k where (30)

ˆ = 1 k⊥ cos φ⊥ , k⊥ sin φ⊥ , k z , k |k |

ˆ ⊥ = (cos φ⊥ , sin φ⊥ , 0). k

By replacing eqs. (28) and (30) inside eq. (29), we ﬁnd the expression of the S axes in terms of the fragmenting quark transverse-momentum components in the γ ∗ −p reference frame: (31)

1 z cos φ , k sin φ , k , k ⊥ ⊥ ⊥ ⊥ |k | y = (− sin φ⊥ , cos φ⊥ , 0), 1 z z x = k cos φ⊥ , k sin φ⊥ , −k⊥ . |k | z =

In the helicity frame of the fragmenting quark S , the azimuthal angle φhq is simply deﬁned as (32)

cos φhq = pˆ⊥ · x ,

sin φhq = pˆ⊥ · y .

By replacing eqs. (26) and (31) inside eqs. (32), we obtain, in general, (33)

1 z z , P k cos(φ − φ ) − P k T h ⊥ ⊥ h |k | |p⊥ | 1 sin φhq = PT sin(φh − φ⊥ ) . |p⊥ |

cos φhq =

444

M. Boglione

2 2 Neglecting O(k⊥ /Q2 ), O(k⊥ /W 2 ) and O(PT2 /Q2 ) terms, one simply ﬁnds

PT k⊥ cos(φh − ϕ) − z , p⊥ p⊥ PT sin(φh − ϕ). sin φhq = p⊥

cos φhq =

(34)

Notice that these relations do not coincide with the usual approximation φhq φh − ϕ commonly used in the literature, not even to ﬁrst order in k⊥ /Q! A full study of eq. (22), taking into account intrinsic motions with all contributions at all orders, following the general approach of ref. [3], will be presented in a forthcoming paper [31]. Here, in agreement with all papers on the Collins eﬀect in SIDIS so far appeared in the literature, we work at O(k⊥ /Q) and use eqs. (23) and (34). By insertion of the above expressions into eq. (23), we obtain, in agreement with refs. [32, 33], S +φh ) (35) Asin(φ = UT 2 2 PT 1 − y √ p2⊥ 2C e−PT /PT C 2 T eq Nq (x) fq/p (x) + Δq(x) NqC (z) Dh/q (z) 2e 2 M sxy 2 p⊥ PT2 2C q 2 2 e−PT /PT [1 + (1 − y)2 ] 2 eq fq/p (x) Dh/q (z) PT2 sxy 2 q

,

where (36)

p2⊥ C =

M 2 p2⊥ , M 2 + p2⊥

2 PT2 = p2⊥ + z 2 k⊥

,

2 PT2 C = p2⊥ C + z 2 k⊥

.

S +φh ) Equation (35) expresses Asin(φ in terms of the parameters α, β, γ, δ, NqT , NqC UT and M . In sect. 4 we shall ﬁx them by performing a best ﬁt of the measurements of HERMES, COMPASS and Belle Collaborations. Actually, we shall consider, following S +φh ) the experimental data, Asin(φ as a function of one variable at a time, by properly UT integrating the numerator and denominator of eq. (35): the integration over x and z S +φh ) gives the PT distribution of Asin(φ , whereas the integrations over PT and z or PT UT and x yield the x and z distributions. Notice that, with our approximations, x = xB and z = zh .

3. – Collins functions from e+ e− processes The kinematics corresponding to the e+ e− → h1 h2 X process is schematically represented in ﬁg. 9: the two detected hadrons h1 and h2 are the fragmentation products of a quark and an antiquark originating from e+ e− collisions. We choose the reference frame so that the e+ e− → q q¯ scattering occurs in the xz ˆ plane, with the back-to-back quark and antiquark moving along the zˆ-axis. This choice requires, experimentally, the

Global fit for the simultaneous determination of the transversity etc. p

⊥2

P2

e

445

P1

q

p

⊥1

θ e+

q

y

x z

Fig. 9. – Three dimensional kinematics of the e+ e− → h1 h2 X process, in the q q¯ c.m. frame. In this conﬁguration the reconstructed thrust axis identiﬁes the zˆ-direction, the lepton-quark scattering plane deﬁnes the xz ˆ plane.

reconstruction of the jet thrust axis, but it involves a very simple kinematics and a direct contribution of the Collins functions, as we shall see. A diﬀerent choice, originally suggested in the literature [34], is discussed at the end of this section. In the conﬁguration of ﬁg. 9, the four-momenta of the e+ , e− (k + , k− ) and of the q, q¯ (q1 , q2 ) are (37)

√ √ s s (1, 0, 0, 1), q2 = (1, 0, 0, −1), 2 2 √ √ s s + (1, − sin θ, 0, cos θ), k = (1, sin θ, 0, − cos θ). = 2 2

q1 = k−

The ﬁnal hadrons h1 and h2 carry light cone momentum fractions z1 and z2 and have intrinsic transverse momenta p⊥1 and p⊥2 with respect to the direction of fragmenting quarks, p⊥1 = p⊥1 (cos ϕ1 , sin ϕ1 , 0),

(38)

p⊥2 = p⊥2 (cos ϕ2 , sin ϕ2 , 0), so that their four-momenta can be expressed as √ √ p2⊥1 p2⊥1 s s √ , p⊥1 cos ϕ1 , p⊥1 sin ϕ1 , z1 √ , + − P1 = z 1 (39) 2 2 2z1 s 2z1 s √ √ p2⊥2 p2⊥2 s s √ , p⊥2 cos ϕ2 , p⊥2 sin ϕ2 , −z2 √ . + + P2 = z 2 (40) 2 2 2z2 s 2z2 s At large c.m. energies and not too small values of z, one can neglect second-order √ corrections in the p⊥ /(z s) expansion, to work with the much simpler kinematics: (41) (42)

√ √ s s , p⊥1 cos ϕ1 , p⊥1 sin ϕ1 , z1 z1 , 2 2 √ √ s s , p⊥2 cos ϕ2 , p⊥2 sin ϕ2 , −z2 . P2 = z 2 2 2 P1 =

446

M. Boglione

Notice also that in this limit the lightcone momentum fractions z coincide with the observable energy fractions zh , √ p2 zh = 2Eh / s = z + ⊥ z. zs

(43)

The cross-section corresponding to this process, with unpolarized leptons, can be written as + −

(44)

dσ e e →h1 h2 X = dz1 dz2 d2 p⊥1 d2 p⊥2 d cos θ 3 1 ˆ h1 /q h2 /¯ q ˆ ∗ Mλq λq¯ ;λ+ λ− M λq λq¯ ;λ+ λ− Dλq λq (z1 , p⊥1 ) Dλq¯λq¯ (z2 , p⊥2 ), 32πs q 4 {λ}

ˆ λ λ ;λ λ are the helicity amplitudes corresponding to the elementary scattering where M q q ¯ + − ¯ s, s¯ (neglecting heavy ﬂavors) and % e+ (λ+ )e− (λ− ) → q(λq )¯ q (λq¯), q = u, u ¯, d, d, {λ} indicates a sum over all helicity indices. In this case there are only two non-zero, independent amplitudes: ˆ +−;+− = M ˆ −+;−+ = e2 eq (1 + cos θ), M ˆ −+;+− = M ˆ +−;−+ = e2 eq (1 − cos θ). M

(45)

h /¯ q

h /q

The functions Dλq1 λ (z1 , p⊥1 ) and Dλq2¯λ (z2 , p⊥2 ) are the probability densities which q q ¯ describe the fragmentation of quarks and antiquarks into the physical hadrons h1 and h2 , respectively (see Section II.C of ref. [3] for detailed explanations). In particular, the h/q h/q diagonal elements D++ (z, p⊥ ) and D−− (z, p⊥ ) correspond to the transverse-momentum– dependent unpolarized fragmentation function Dh/q (z, p⊥ ), h/q

h/q

D++ (z, p⊥ ) = D−− (z, p⊥ ) = Dh/q (z, p⊥ ),

(46)

whereas the non-diagonal elements (47)

h/q

h/q

D+− (z, p⊥ ) = D+− (z, p⊥ ) eiϕ , D−+ (z, p⊥ ) = D−+ (z, p⊥ ) e−iϕ = −D+− (z, p⊥ ) e−iϕ h/q

h/q

h/q

are related to the Collins fragmentation function ΔN Dh/q↑ (z, p⊥ ) [10] by (48)

h/q

h/q

ΔN Dh/q↑ (z, p⊥ ) = −2i D+− (z, p⊥ ) = 2i D−+ (z, p⊥ ).

The angle ϕ in eq. (47) is the azimuthal angle identifying the direction of the observed hadron h in the helicity frame of the fragmenting quark q. Similar relations hold for the antiquark fragmentation functions, where one has to take into account a sign diﬀerence in

Global fit for the simultaneous determination of the transversity etc.

447

ϕ originating from the fact that the antiquark is chosen to move along the −ˆ z direction. Finally, inserting eqs. (45)-(48) into eq. (44) and performing the sum over the quark helicities one obtains + −

(49)

dσ e e →h1 h2 X = dz1 dz2 d2 p⊥1 d2 p⊥2 d cos θ 3πα2 2 eq (1 + cos2 θ) Dh1 /q (z1 , p⊥1 ) Dh2 /¯q (z2 , p⊥2 ) 2s q +

1 sin2 θ ΔNDh1 /q↑ (z1 , p⊥1 ) ΔNDh2 /¯q↑ (z2 , p⊥2 ) cos(ϕ1 + ϕ2 ) . 4

Equation (49) shows that the study of the correlated production of two hadrons (one for each jet) in unpolarized e+ e− collisions oﬀers a direct access to the Collins functions, both regarding their z and p⊥ dependences. So far, only data on the z dependence are available. Notice that by integrating over the intrinsic transverse momenta p⊥1 and p⊥2 one recovers the usual unpolarized cross section, + −

3πα2 dσ e e →h1 h2 X = (1 + cos2 θ) e2q Dh1 /q (z1 ) Dh2 /¯q (z2 ), dz1 dz2 d cos θ 2s q

(50) having used

d2 p⊥ Dh/q (z, p⊥ ) = Dh/q (z).

(51)

To construct the physical observable measured by the Belle Collaboration, we now perform a change of angular variables from (ϕ1 , ϕ2 ) to (ϕ1 , ϕ1 + ϕ2 ) and then integrate over the moduli of the intrinsic transverse momenta, p⊥1 and p⊥2 , and over the azimuthal angle ϕ1 . This leads to (52)

+ − 3α2 2 dσ e e →h1 h2 X = e (1 + cos2 θ) Dh1 /q (z1 ) Dh2 /¯q (z2 ) dz1 dz2 d cos θ d(ϕ1 + ϕ2 ) 4s q q

+

1 sin2 θ cos(ϕ1 + ϕ2 ) ΔNDh1 /q↑ (z1 ) ΔNDh2 /¯q↑ (z2 ) , 4

where we have deﬁned d2 p⊥ ΔNDh/q↑ (z, p⊥ ) ≡ ΔNDh/q↑ (z).

(53)

By normalizing eq. (52) to the azimuthal averaged cross-section, (54)

dσ ≡

+ − 3α2 2 1 dσ e e →h1 h2 X = e (1 + cos2 θ) Dh1 /q (z1 ) Dh2 /¯q (z2 ), 2π dz1 dz2 d cos θ 4s q q

448

M. Boglione

one has + −

(55)

dσ e e →h1 h2 X 1 dσ dz1 dz2 d cos θ d(ϕ1 + ϕ2 ) % 2 N N 2 1 sin θ q ↑ (z2 ) q eq Δ Dh1 /q ↑ (z1 ) Δ Dh2 /¯ % =1+ cos(ϕ · + ϕ ) 1 2 2D 4 1 + cos2 θ e (z ) D (z h2 /¯ q 2) q q h1 /q 1

A(z1 , z2 , θ, ϕ1 + ϕ2 ) ≡

Actually, Belle data are collected over a range of θ values, according to the acceptance of the detector (see eq. (83)). Thus, eqs. (52) and (54) are integrated over the covered θ range resulting in some speciﬁc sin2 θ and 1 + cos2 θ values. Finally, to eliminate false asymmetries, the Belle Collaboration considers the ratio of unlike-sign to like-sign pion pair production, AU and AL , given by 2

sin θ 1 + 14 cos(ϕ1 + ϕ2 ) 1+cos 2 θ PU AU R≡ = 2 sin θ AL 1 + 14 cos(ϕ1 + ϕ2 ) 1+cos 2 θ PL

(56)

sin2 θ 1 cos(ϕ1 + ϕ2 ) (PU − PL ) 4 1 + cos2 θ ≡ 1 + cos(ϕ1 + ϕ2 ) A12 (z1 , z2 ),

−1 +

with % q

(57) PU = % (58) PL =

q

e2q [ΔNDπ+ /q↑ (z1 ) ΔNDπ− /¯q↑ (z2 ) + ΔNDπ− /q↑ (z1 ) ΔNDπ+ /¯q↑ (z2 )] % 2 , q (z2 ) + Dπ − /q (z1 ) Dπ + /¯ q (z2 )] q eq [Dπ + /q (z1 ) Dπ − /¯ e2q [ΔNDπ+ /q↑ (z1 ) ΔNDπ+ /¯q↑ (z2 ) + ΔNDπ− /q↑ (z1 ) ΔNDπ− /¯q↑ (z2 )] % 2 , q (z2 ) + Dπ − /q (z1 ) Dπ − /¯ q (z2 )] q eq [Dπ + /q (z1 ) Dπ + /¯

(59) A12 (z1 , z2 ) =

1 sin2 θ (PU − PL ). 4 1 + cos2 θ

For ﬁtting purposes, it is convenient to re-express PU and PL in terms of favoured and unfavoured fragmentation functions, (60)

Dπ+ /u = Dπ+ /d¯ = Dπ− /d = Dπ− /¯u ≡ Dfav ,

(61)

Dπ+ /d = Dπ+ /¯u = Dπ− /u = Dπ− /d¯ = Dπ± /s = Dπ± /¯s ≡ Dunf ,

and similarly for the ΔN D, obtaining (62)

PU =

[5 ΔNDfav (z1 ) ΔNDfav (z2 ) + 7 ΔNDunf (z1 ) ΔNDunf (z2 )] , [5 Dfav (z1 ) Dfav (z2 ) + 7 Dunf (z1 ) Dunf (z2 )]

(63) PL = [5 ΔNDfav (z1 ) ΔNDunf (z2 ) + 5 ΔNDunf (z1 ) ΔNDfav (z2 ) + 2 ΔNDunf (z1 ) ΔNDunf (z2 )] , [5 Dfav (z1 ) Dunf (z2 ) + 5 Dunf (z1 ) Dfav (z2 ) + 2 Dunf (z1 ) Dunf (z2 )]

Global fit for the simultaneous determination of the transversity etc.

449

P1T

θ2 e

P1

P2 e+

φ1 y

x z

Fig. 10. – Three-dimensional kinematics of the e+ e− → h1 h2 X process. In this conﬁguration the zˆ direction is identiﬁed by the momentum of the ﬁnal hadron h2 , while h1 is emitted at an ˆ plane. azimuthal angle φ1 with respect to the lepton-h2 plane, deﬁned as the xz

having neglected heavy quark contributions. PU and PL are the same as in ref. [23], remembering eq. (15) and noticing that ΔN Dh/q↑ (z) = d2 p⊥ ΔN Dh/q↑ (z, p⊥ ) (64) 2p⊥ ⊥q ⊥(1/2)q H (z, p⊥ ) = 4 H1 (z). = d2 p⊥ zmh 1 In addition, the Belle Collaboration presents a second set of data, analysed in a diﬀerent reference frame: following ref. [34], one can ﬁx the zˆ-axis as given by the direction of the observed hadron h2 and the xz ˆ plane as determined by the lepton and the h2 directions. There will then be another relevant plane, determined by zˆ and the direction of the other observed hadron h1 , at an angle φ1 with respect to the xz ˆ plane. This kinematical conﬁguration is shown in ﬁg. 10; it has the advantage that it does not require the reconstruction of the quark direction. √ However, in this case the kinematics is more complicated. At ﬁrst order in p⊥ /(z s) one has (65) (66) (67) (68) (69)

P2 = |P2 | (1, 0, 0, −1), √ √ s p⊥2 s p⊥2 q2 = ,− cos ϕ2 , − sin ϕ2 , − , 2 z2 z2 2 √ √ s p⊥2 s p⊥2 , cos ϕ2 , sin ϕ2 , , q1 = 2 z2 z2 2 √ s , P1 = P1T cos φ1 , P1T sin φ1 , z1 2 z1 z1 p⊥2 cos ϕ2 , P1T sin φ1 − p⊥2 sin ϕ2 , 0 . p⊥1 = P1T cos φ1 − z2 z2

Moreover, the elementary process e+ e− → q q¯ does not occur in general in the xz ˆ plane,

450

M. Boglione

and thus the helicity scattering amplitudes involve an azimuthal phase ϕ2 . One can still perform an exact calculation, using the general approach discussed in ref. [3]. A detailed description will be presented in a forthcoming paper [31]. We give here only the results √ valid at O(p⊥ /z s). The analogue of eq. (49) now reads + −

(70)

dσ e e →h1 h2 X = dz1 dz2 d2 p⊥1 d2 p⊥2 d cos θ2 3πα2 2 eq (1 + cos2 θ2 ) Dh1 /q (z1 , p⊥1 ) Dh2 /¯q (z2 , p⊥2 ) 2s q +

1 sin2 θ2 ΔNDh1 /q↑ (z1 , p⊥1 ) ΔNDh2 /¯q↑ (z2 , p⊥2 ) cos(2ϕ2 + φhq 1 − φhq¯2 ) , 4

where φhq 1 and φqh¯2 are the azimuthal angles of the detected hadrons h1 and h2 around the direction of the parent fragmenting quarks, q and q¯, respectively. Technically, φhq 1 is the azimuthal angle of p⊥1 in the helicity frame of q, while φqh¯2 is the azimuthal angle of p⊥2 in the helicity frame of q¯. They can be expressed in terms of the integration variables we are using, p⊥2 and P1T , by a procedure analogous to that used for the SIDIS √ fragmentation process. In this case we will limit the study to lowest order in p⊥ /(z s); the helicity frame of the fragmenting quark q, S1 , can be reached by performing the following transformations: (71)

√ s px⊥2 py⊥2 , , , z2 z 2 2 1 z 1 × q1 − py⊥2 , px⊥2 , 0 , = y1 = z1 × pˆ⊥1 = |z1 × q1 | p⊥2 x y p p 2p⊥2 ⊥2 ⊥2 x1 = y1 × z1 = , ,− √ p⊥2 p⊥2 z2 s 2 z1 = qˆ1 = √ s

and (72)

√ px⊥2 py⊥2 s , , , z2 z2 2 1 y z 2 × q2 = p⊥2 , −px⊥2 , 0 , y2 = z2 × pˆ⊥2 = |z2 × q2 | p⊥2 x y p⊥2 p⊥2 2p⊥2 x2 = y2 × z2 = , ,− √ , p⊥2 p⊥2 z2 s z2

2 = qˆ2 = − √ s

from which we obtain P1T z1 p⊥2 cos(φ1 − ϕ2 ) − , p⊥1 z2 p⊥1 P1T = p⊥1 · y1 = sin(φ1 − ϕ2 ), p⊥1

(73)

cos φhq 1 = p⊥1 · x1 =

(74)

sin φhq 1

451

Global fit for the simultaneous determination of the transversity etc.

and (75)

cos φhq¯2 = p⊥2 · x2 = 1,

(76)

sin φqh¯2 = p⊥2 · y2 = 0.

Finally we have (77)

! cos 2ϕ2 + φhq 1 − φhq¯2 =

1 p⊥1 p⊥2

P1T p⊥2 cos(φ1 + ϕ2) −

z1 cos 2ϕ2 , z2

to be replaced in the last line of eq. (70). Integrating eq. (70) over p⊥2 and P1T , but not over φ1 , and normalizing to the azimuthal averaged unpolarized cross-section (54), we obtain the analogue of eq. (55), (78)

A(z1 , z2 , θ2 , φ1 ) = sin2 θ2 1 z1 z2 cos(2 φ1 ) 1+ π z12 + z22 1 + cos2 θ2

% q

e2q ΔNDh1 /q↑ (z1 ) ΔNDh2 /¯q↑ (z2 ) % 2 , q (z2 ) q eq Dh1 /q (z1 ) Dh2 /¯

in agreement with ref. [23] taking into account the diﬀerent notations. Finally, eq. (56) becomes in this conﬁguration (79)

R 1 + cos(2 φ1 )A0 (z1 , z2 ),

with (80)

A0 (z1 , z2 ) =

sin2 θ2 1 z1 z2 (PU − PL ), 2 2 π z1 + z2 1 + cos2 θ2

where PU and PL are the same as deﬁned in eqs. (62) and (63). 4. – Transversity and Collins functions from a global ﬁt We may now pursue our strategy of simultaneously gathering information on the transversity distribution function ΔT q(x, k⊥ ) and the Collins fragmentation function ΔNDh/q↑ (z, p⊥ ). To such a purpose we perform a global best-ﬁt analysis of experimental data involving these functions, namely the data from the SIDIS measurements by the HERMES [26, 27] and COMPASS [28] Collaborations, and the data from unpolarized e+ e− → h1 h2 X processes by the Belle Collaboration [25]. ΔT q(x, k⊥ ) and ΔNDh/q↑ (z, p⊥ ) are parametrized as shown in eqs. (9)-(18). Considering the scarcity of data, in order to minimize the number of parameters, we assume ﬂavor-independent values of α and β and, similarly, we assume that γ and δ are the same for favored and unfavored Collins fragmentation functions, eqs. (60) and (61); we then remain with a total number of 9 parameters. Their values, as determined through our global best ﬁt are shown in tables I and II, together with the errors estimated by MINUIT.

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Table I. – Best values of the free parameters for the u and d transversity distribution functions and for the favored and unfavored Collins fragmentation functions, eqs. (9)–(18), as obtained sin(φ +φ ) by simultaneously ﬁtting HERMES and COMPASS data on the AU T S h asymmetry and the Belle data on the A12 asymmetry, eq. (59), proportional to cos(ϕ1 + ϕ2 ). Notice that the errors generated by MINUIT are strongly correlated, and should not be taken at face value. The signiﬁcant ﬂuctuations in our results are shown by the shaded areas in ﬁgs. 11, 12 and 13, as 2 2 T = k⊥ and p2⊥ are ﬁxed, according to eqs. (11), (20). explained in the text. The values of k⊥ χ2 /d.o.f. = 0.81

Fit I (A12 ) Transversity distribution function

NuT α 2 T k⊥

= = =

0.48 ± 0.09 1.14 ± 0.68 0.25 GeV2

NdT β

= =

−0.62 ± 0.30 4.74 ± 5.45

Collins fragmentation function

C Nfav γ p2⊥

= = =

0.35 ± 0.16 1.14 ± 0.38 0.20 GeV2

C Nunf δ M2

= = =

−0.85 ± 0.36 0.14 ± 0.36 0.70 ± 0.65 GeV2

As the two diﬀerent sets of Belle data are based on a diﬀerent analysis of the same experimental events, they are strongly correlated. Therefore, we have treated them separately in our combined analysis of the HERMES, COMPASS and Belle data; the best-ﬁt values of table I are obtained by ﬁtting the SIDIS results together with the Belle data on the cos(ϕ1 + ϕ2 ) dependence, eq. (56), while the values in table II originate from the Belle data on the cos(2φ1 ) dependence, eq. (79). We notice that the two sets of resulting best-ﬁt parameters are in full agreement within the uncertainties; this gives a good check of the consistency of the measurements and the stability of our analysis. In the sequel we shall present results and predictions based on the values of table I; the

Table II. – Best values of the free parameters for the u and d transversity distribution functions and for the favored and unfavored Collins fragmentation functions, eqs. (9)–(18), as obtained sin(φ +φ ) by simultaneously ﬁtting HERMES and COMPASS data on the AU T S h asymmetry and the Belle data on the A0 asymmetry, eq. (80), proportional to cos(2φ1 ). Notice that the errors generated by MINUIT are strongly correlated, and should not be taken at face value. The signiﬁcant ﬂuctuations in our results are shown by the shaded areas in ﬁgs. 11, 12 and 13, as explained in 2 2 T = k⊥ and p2⊥ are ﬁxed, according to eqs. (11), (20). the text. The values of k⊥ χ2 /d.o.f. = 0.77

Fit II (A0 ) Transversity distribution function

NuT α 2 T k⊥

= = =

0.42 ± 0.09 1.20 ± 0.83 0.25 GeV2

NdT β

= =

−0.53 ± 0.28 5.09 ± 5.87

Collins fragmentation function

C Nfav γ p2⊥

= = =

0.41 ± 0.10 0.81 ± 0.40 0.20 GeV2

C Nunf δ M2

= = =

−0.99 ± 1.24 0.02 ± 0.37 0.88 ± 1.15 GeV2

Global fit for the simultaneous determination of the transversity etc.

453

0.12

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S +φh ) for Fig. 11. – HERMES experimental data [26, 27] on the azimuthal asymmetry Asin(φ UT ± π production are compared to the curves obtained from eq. (35) with the parametrizations of eqs. (9)-(18), and the parameter values, determined through our global best ﬁt, given in table I. The shaded area corresponds to the theoretical uncertainty on the parameters, as explained in the text.

corresponding results based on the values of table II are hardly distinguishable (examples of this are shown explicitly in ﬁg. 13 and in ﬁg. 15, right panel). Our best ﬁts of the experimental data from HERMES, COMPASS and Belle are shown in ﬁgs. 11, 12 and 13, respectively. The central curves correspond to the central values of the parameters in table I, while the shaded areas correspond to one-sigma deviation at 90% CL and are calculated using the errors and the parameter correlation matrix generated by MINUIT, minimizing and maximizing the function under consideration, in a 9-dimensional parameter space hyper-volume corresponding to one-sigma deviation. The transversity distribution functions ΔT u(x, k⊥ ) and ΔT d(x, k⊥ ) as resulting from our best ﬁt —eqs. (9)-(18) and table I— are plotted as a function of x and k⊥ in ﬁg. 14; for comparison, the Soﬀer bound is also shown, as a bold line. The solid central line corresponds to the central values in table I and the shaded area corresponds to the uncertainty in the parameter values, as explained above. Similarly, the resulting Collins functions ΔN Dfav (z, p⊥ ) and ΔN Dunf (z, p⊥ ) are plotted as a function of z —integrated over d2 p⊥ , and normalized to twice the unpolarized fragmentation functions— and as a function of p⊥ in ﬁg. 15; for comparison, we also show the Collins functions from refs. [23] and [35], respectively as dashed and dotted lines (left panels), and the corresponding positivity bound. The dashed lines in the right panels show the results corresponding to the parameters of table II. A few comments are in order. – In ﬁg. 14 we show the extracted transversity distribution for u and d quarks. The x dependence is based on the simple parametrization assumed in eqs. (9) and (10),

454

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S +φh ) , for the production of positively and negatively Fig. 12. – The measurements of Asin(φ UT charged hadrons, from the COMPASS experiment operating on a deuterium target [28] are compared to the curves obtained from eq. (35) with the parametrizations of eqs. (9)-(18), and the parameter values, determined through our global best ﬁt, given in table I. The shaded area corresponds to the theoretical uncertainty on the parameters, as explained in the text. Notice the extra π phase in addition to φS + φh in the ﬁgure label, to keep into account the diﬀerent choice of the Collins angle, with respect to Trento [22] and HERMES conventions, adopted by COMPASS Collaboration.

0.2

0.2 < z1 < 0.3

0.2

0.3 < z1 < 0.5

A12(z1, z2)

A0(z1, z2)

0.15 0.1 0.05 0

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A0(z1, z2)

0.1 0.05 0 -0.05

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Fig. 13. – The experimental data on two diﬀerent azimuthal correlations in unpolarized e e → h1 h2 X processes, as measured by Belle Collaboration [25], are compared to the curves obtained from eqs. (59) [A12 ] and (80) [A0 ] with the parametrizations of eqs. (16), (17) and (18). The solid lines correspond to the parameters given in table I, obtained by ﬁtting the A12 asymmetry; the shaded area corresponds to the theoretical uncertainty on these parameters, as explained in the text. The dashed lines correspond to the parameters given in table II obtained by ﬁtting the A0 asymmetry. The agreement between the results obtained from the two ﬁts shows the consistency between the two sets of Belle data and the solidity of our analysis.

455

Global fit for the simultaneous determination of the transversity etc.

x 'T u(x, k )

0.5

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0.4 0.3 0.2 0.1 0

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0.4

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k (GeV)

Fig. 14. – The transversity distribution functions for u and d quarks as determined through our global best ﬁt. In the left panel, x ΔT u(x) (upper plot) and x ΔT d(x) (lower plot), see eq. (9), are shown as functions of x and Q2 = 2.4 GeV2 . The Soﬀer bound [18] is also shown for comparison (bold blue line). In the right panel we present the unintegrated transversity distributions, x ΔT u(x, k⊥ ) (upper plot) and x ΔT d(x, k⊥ ) (lower plot), as deﬁned in eq. (9), as functions of k⊥ at a ﬁxed value of x. Notice that this k⊥ dependence is not obtained from the ﬁt, but it has been chosen to be the same as that of the unpolarized distribution functions: we plot it in order to show its uncertainty (shaded area), due to the uncertainty in the determination of the free parameters.

which contain NqT , α and β as free parameters; our result represents the ﬁrst extraction ever of the transversity distributions ΔT u(x) and ΔT d(x). – The k⊥ dependence has been assumed to be the same as for the unpolarized distributions. The ﬂavor dependence is contained in the coeﬃcients NqT and in the + proportionality of ΔT q(x) to [q(x) + Δq(x)]/2 = q+ (x), the number density of quarks with positive helicity inside a positive-helicity proton. – Our results show that the transversity distribution is positive for u quarks and negative for d quarks; the magnitude of ΔT u is larger than that of ΔT d, while they are both signiﬁcantly smaller than the corresponding Soﬀer bound. – The shaded regions in ﬁg. 14 show that both ΔT u(x, k⊥ ) and ΔT d(x, k⊥ ) are, considering the limited amount of data, already well determined. It is worth noticing that while the HERMES data alone tightly constrain the transversity distribution of u quarks, the addition of COMPASS data to the ﬁt allows to better constrain the transversity distribution function of d quarks. We have checked that ﬁtting only HERMES and Belle data, ignoring the COMPASS results, still leads to a

456

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Q = 2.4 GeV z = 0.36

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-' N Dunf (z)/2Dunf (z)

' N Dfav(z)/2Dfav(z)

M. Boglione

Q = 2.4 GeV z = 0.36

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Fig. 15. – Favored and unfavored Collins fragmentation functions as determined through our global best ﬁt. In the left panel we show the z dependence of the p⊥ integrated Collins functions deﬁned in eq. (53) and normalized to twice the corresponding unpolarized fragmentation functions; we compare them to the results of refs. [23] (dashed line) and [35] (dotted line). In the right panel we show the p⊥ dependence of the Collins functions deﬁned in eq. (16), at a ﬁxed value of z. The Q2 value is 2.4 GeV2 , having assumed that the Q2 evolution of ΔN D is the same as that of D. The solid lines show the results based on the parameters of table I, while the dashed ones show the results corresponding to the parameters of table II. In all cases we also show the positivity bound (upper lines).

similar good χ2 /d.o.f.; the resulting functions would give a slightly worse descripS +φh ) tion —when compared to the global ﬁt— of the x dependence of Asin(φ , as UT measured by COMPASS. This is mainly related to a less stringent determination of ΔT d(x, k⊥ ) in the absence of deuteron target data. Although their measured azimuthal asymmetry is very small, the inclusion of COMPASS data signiﬁcantly contributes to the extraction of the transversity distributions. Diﬀerent ﬁtting pro+ cedures were earlier attempted, for example by ﬁxing ΔT q = Δq or ΔT q = q+ [36]: they lead to a slightly worse description of Belle data. – The extracted Collins functions are shown in ﬁg. 15; they agree with similar extractions previously obtained in the literature [23, 35]. The shaded areas indicate well constrained Collins functions for u and d quarks in the large (valence) z region, much smaller than their corresponding positivity bound. – We note once more that, in analyzing SIDIS data, we have neglected the contributions of the sea quarks and antiquarks (assuming the corresponding transversity distributions in a proton to vanish), taking into account only u and d ﬂavors. In analyzing Belle data and introducing the favored and unfavored Collins fragmen-

Global fit for the simultaneous determination of the transversity etc.

457

tation functions, we have considered the contributions of u, d and s quarks, all √ abundantly produced in the e+ e− annihilation at s 10 GeV. – The partonic distribution and fragmentation functions are taken from refs. [11, 12] and [19]. The QCD evolution is taken into account in the unpolarized distributions, in the unpolarized fragmentation functions and, following ref. [37], for the transversity distributions. Finally, we explicitly list, for clarity and completeness, the kinematical cuts we have imposed in numerical integrations, according to the set-up of the HERMES experiment: (81)

0.2 ≤ zh ≤ 0.7,

0.023 ≤ xB ≤ 0.4,

0.1 ≤ y ≤ 0.85,

Q2 ≥ 1 GeV2 ,

W 2 ≥ 10 GeV2 ,

2 ≤ Eh ≤ 15 GeV,

the COMPASS experiment: 0.2 ≤ zh ≤ 1,

(82)

Q ≥ 1 GeV , 2

2

0.1 ≤ y ≤ 0.9, W 2 ≥ 25 GeV2 ,

and the Belle experiment (83)

−0.6 ≤ cos θlab ≤ 0.9,

QT ≤ 3.5 GeV,

where θlab is the polar production angle in the laboratory frame (related to the scattering angles θ and θ2 used in this paper) and QT is the transverse momentum of the virtual photon from the e+ e− annihilation in the rest frame of the hadron pair [34]. 5. – Predictions for ongoing and future experiments We can now use the transversity distributions and the Collins functions we have obtained from ﬁtting the available HERMES, COMPASS and Belle data, see table I, to give predictions for new measurements planned by COMPASS and JLab Collaborations. S +φh ) The transverse single spin asymmetry Asin(φ will be measured by the COMPASS UT experiment operating with a polarized hydrogen target (rather than a deuterium one). In ﬁg. 16 we show our predictions, obtained by adopting the same experimental cuts which were used for the deuterium target, see eq. (82). Notice that this asymmetry is found to be sizeable, up to 5% in size. S +φh ) for pion production oﬀ transversely The JLab experiments will measure Asin(φ UT polarized proton and neutron targets, at incident beam energies of either 6 or 12 GeV. The kinematical region spanned by these experiments is very interesting, as it will enable to explore the behavior of the transversity distribution function at large values of x, up

458

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S +φh ) as it will be measured by the Fig. 16. – Predictions for the single-spin asymmetry Asin(φ UT COMPASS experiment operating with a transversely polarized hydrogen target. For the extra π phase in the ﬁgure label see the caption of ﬁg. 5.

to x ∼ 0.6. The adopted experimental cuts for JLab operating on a proton target at 6 GeV are the following: (84)

0.4 ≤ zh ≤ 0.7,

0.02 ≤ PT ≤ 1 GeV,

0.1 ≤ xB ≤ 0.6,

0.4 ≤ y ≤ 0.85,

Q ≥ 1 GeV , 2

W 2 ≥ 4 GeV2 ,

2

1 ≤ Eh ≤ 4 GeV, whereas for a beam energy of 12 GeV they are (85)

0.4 ≤ zh ≤ 0.7,

0.02 ≤ PT ≤ 1.4 GeV,

0.05 ≤ xB ≤ 0.7,

0.2 ≤ y ≤ 0.85,

Q ≥ 1 GeV , 2

W 2 ≥ 4 GeV2 ,

2

1 ≤ Eh ≤ 7 GeV. For a neutron target at 6 GeV the cuts read (86)

0.46 ≤ zh ≤ 0.59, 0.68 ≤ y ≤ 0.86,

0.13 ≤ xB ≤ 0.40, 1.3 ≤ Q2 ≤ 3.1 GeV2 ,

5.4 ≤ W 2 ≤ 9.3 GeV2 ,

2.385 ≤ Eh ≤ 2.404 GeV,

459

Global fit for the simultaneous determination of the transversity etc.

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S +φh ) as it will be measured at JLab Fig. 17. – Predictions for the single spin asymmetry Asin(φ UT operating on polarized hydrogen (proton, upper plot) and He3 (neutron, lower plot) targets at a beam energy of 6 GeV.

whereas for an incident beam energy of 12 GeV they are (87)

0.3 ≤ zh ≤ 0.7,

0.05 ≤ xB ≤ 0.55,

0.34 ≤ y ≤ 0.9,

Q2 ≥ 1 GeV2 ,

W 2 ≥ 2.3 GeV2 . Our corresponding predictions, according to eq. (35) and our extracted transversity and Collins functions, are shown in ﬁgs. 17 and 18. It is important to stress that, as the large x region is not covered by the HERMES S +φh ) are very and COMPASS experiments, our predictions for the x dependence of Asin(φ UT sensitive to the few available data points from HERMES and COMPASS at moderately large x values. As a consequence, the predictions for the JLab experiments may vary drastically in the region 0.4 ≤ xB ≤ 0.6, as indicated by the large shaded area in ﬁgs. 17 and 18. On the contrary, the results on the PT and zh dependences are more stable, as they only depend on the transversity distribution function integrated over x.

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Fig. 19. – Our results, based on the extracted transversity and Collins functions, for the azS +φh ) for K ± production, compared with the HERMES experimental imuthal asymmetry Asin(φ UT data [26, 27].

S +φh ) Finally, we compute the azimuthal asymmetry Asin(φ for the production of K UT mesons and compare it with existing HERMES results [26,27]. These data have not been included in our best ﬁt, as they might involve the transversity distribution of strange quarks in the nucleon, which we have neglected for SIDIS data on π production. We show our results in ﬁg. 19, obtained using the extracted u and d transversity distributions. Again, we have used favored (ΔN DK + /u↑ ) and unfavored (ΔN DK − /u↑ , ΔN DK ± /d↑ ) Collins functions, as in eqs. (16), (17) and (18). For these we have used the same parameters NqC , γ, δ and M of table I, with the appropriate unpolarized fragmentation functions DK ± /q [19]. We notice that our computations are in fair agreement with data concerning the K + production, which is presumably dominated by u quarks; instead, there seem to be discrepancies for the K − asymmetry, for which the role of s quarks might be relevant. New data on the azimuthal asymmetry for K production, possible from COMPASS and JLab experiments, might be very helpful in sorting out the eventual importance of the sea quark transversity distributions in a nucleon.

6. – Comments and conclusions We have performed a combined analysis of all experimental data on spin azimuthal asymmetries which involve the transversity distributions of u and d quarks and the Collins fragmentation functions, classiﬁed as favored (when the fragmenting quark is a valence quark for the ﬁnal hadron) and unfavored (when the fragmenting quark is not a valence quark for the ﬁnal hadron). We have ﬁxed the total number of 9 parameters by best ﬁtting the HERMES, COMPASS and Belle data.

Global fit for the simultaneous determination of the transversity etc.

461

All data can be accurately described, leading to the extraction of the favored and unfavored Collins functions, in agreement with similar results previously obtained in the literature [23, 35]. In addition, we have obtained, for the ﬁrst time, an extraction of the so far unknown transversity distributions for u and d quarks, ΔT u(x) and ΔT d(x). They turn out to be opposite in sign, with |ΔT d(x)| smaller than |ΔT u(x)|, and both smaller than their Soﬀer bound [18]. The knowledge of the transversity distributions and the Collins fragmentation funcS +φh ) tions allows to compute the azimuthal asymmetry AUsin(φ for any SIDIS process; T we have then presented several predictions for incoming measurements from COMPASS and JLab experiments. They will provide further important tests of our complete understanding of the partonic properties which are at the origin of SSA. Data on K production will help in disentangling the role of sea quarks. Further expected data from Belle will allow to study in detail not only the z dependence of the Collins functions, but also their p⊥ dependence. The combination of data from SIDIS and e+ e− → h1 h2 X processes opens the way to a new phenomenological approach to the study of the nucleon structure and of fundamental QCD properties, to be further pursued. ∗ ∗ ∗ This lecture is based on some work performed with M. Anselmino, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin and C. Turk, whom I warmly thank for their long and fruitful collaboration. It is a pleasure to thank all the organizers of this School for their kind hospitality.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Anselmino M., this volume, p. 287. Vogelsang W., this volume, p. 349. Anselmino M. et al., Phys. Rev. D, 73 (2006) 014020. Hasch D., this volume, p. 317. Leader E., this volume, p. 263. Jaffe R., contribution to this School. Saito N., this volume, p. 379. Anselmino M. et al., arXiv:0710.1569 [hep-ph]. Musch B. U. et al., PoS LAT2007 (Sissa, Trieste) 2007, p. 155, arXiv:0710.4423 [hep-lat]. Anselmino M., Boglione M., D’Alesio U., Kotzinian A., Murgia F. and Prokudin A., Phys. Rev. D, 71 (2005) 074006. Gluck M., Reya E. and Vogt A., Eur. Phys. J. C, 5 (1998) 461. Gluck M., Reya E., Stratmann M. and Vogelsang W., Phys. Rev. D, 63 (2001) 094005. Barone V., Drago A. and Ratcliffe P. G., Phys. Rep., 359 (2002) 1. Barone V. et al., hep-ex/0505054. Anselmino M., Barone V., Drago A. and Nikolaev N. N., Phys. Lett. B, 594 (2004) 97.

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Efremov A. V., Goeke K. and Schweitzer P., Eur. Phys. J. C, 35 (2004) 207. Pasquini B., Pincetti M. and Boffi S., Phys. Rev. D, 76 (2007) 034020. Soffer J., Phys. Rev. Lett., 74 (1995) 1292. Kretzer S., Phys. Rev. D, 62 (2000) 054001. Kniehl B. A., Kramer G. and Potter B., Nucl. Phys. B, 582 (2000) 514. de Florian D., Sassot R. and Stratmann M., Phys. Rev. D, 75 (2007) 114010, arXiv:0708.0769 [hep-ph]. Bacchetta A., D’Alesio U., Diehl M. and Miller C. A., Phys. Rev. D, 70 (2004) 117504. Efremov A. V., Goeke K. and Schweitzer P., Phys. Rev. D, 73 (2006) 094025. Collins J. C., Nucl. Phys. B, 396 (1993) 161. Seidl R. et al., Phys. Rev. Lett., 96 (2006) 232002. Airapetian A. et al., Phys. Rev. Lett., 94 (2005) 012002. Pappalardo L. et al., in the proceedings of the XIV International Workshop on Deep Inelastic Scattering, Tsukuba city, Japan, April 20th - April 24th. (2006). Ageev E. S. et al., Nucl. Phys. B, 765 (2007) 31. Ji X.-d., Ma J.-P. and Yuan F., Phys. Lett. B, 597 (2004) 299. Ji X.-d., Ma J.-p. and Yuan F., Phys. Rev. D, 71 (2005) 034005. Anselmino M., Boglione M., D’Alesio U., Murgia F. and Prokudin A., in preparation (2008). Kotzinian A., Nucl. Phys. B, 441 (1995) 234. Mulders P. J. and Tangerman R. D., Nucl. Phys. B, 461 (1996) 197. Boer D., Jakob R. and Mulders P. J., Nucl. Phys. B, 504 (1997) 345. Vogelsang W. and Yuan F., Phys. Rev. D, 72 (2005) 054028. Prokudin A. and Turk C., AIP Conf. Proc., 915 (2007) 584. Martin O., Schafer A., Stratmann M. and Vogelsang W., Phys. Rev. D, 57 (1998) 3084.

International School of Physics “Enrico Fermi” Villa Monastero, Varenna Course CLXVII 19–29 June 2007

“Strangeness and Spin in Fundamental Physics” Directors Mauro ANSELMINO Dipartimento di Fisica Teorica Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707227 Fax: ++39-011-6707214 [email protected] Tullio BRESSANI Dipartimento di Fisica Sperimentale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707322 Fax: ++39-011-6707324 [email protected]

Scientiﬁc Secretaries Alessandro FELICIELLO INFN, Sezione di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707320 Fax: ++39-011-2367320 [email protected] c Societ` a Italiana di Fisica

Philip G. RATCLIFFE Dipartimento di Fisica e Matematica Universit` a dell’Insubria Via Valleggio 11 22100 Como Italy Tel.: ++39-031-2386231 Fax: ++39-031-2386209 [email protected]

Lecturers Raimondo BERTINI Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707423 Fax: ++39-011-6707423 [email protected] Avraham GAL Racah Institute of Physics University of The Hebrew 91904 Jerusalem Israel Tel.: ++972-26584930 Fax: ++972-25611519 [email protected] 463

464

Delia HASCH INFN - LNF Via E. Fermi 40 00044 Frascati (RM) Italy Tel.: ++39-06-94032310 Fax: ++39-06-94032559 [email protected]

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Tomofumi NAGAE High Energy Accelerator 1-1 Oho, Tsukuba 305-0801 Ibaraki Japan Tel.: ++81-298645677 Fax: ++81-298645258 [email protected]

Eugenio NAPPI INFN, Sezione di Bari Via G. Amendola 173 70125 Bari Italy Tel.: ++39-080-5443199 Fax: ++39-080-5534938 [email protected]

Elenco dei partecipanti Naohito SAITO KEK High Energy Accelerator Research Org Tsukuba-shi 305-0801 Ibaraki-ken Japan [email protected] Stefan SCHRAMM Center for Scientiﬁc Computing Max von Laue Straße 1 60438 Frankfurt am Main Germany Tel.: ++49-69-79847352 Fax: ++49-69-79847360 [email protected] Werner VOGELSANG BNL Nuclear Theory University of Brookhaven Nat. Lab. Building 510 Upton, NY 11973 USA Tel.: ++1-631-3442172 Fax: ++1-631-3444067 [email protected]

Seminar Speakers

Harut AVAGYAN Jeﬀerson Lab. 12000 Jeﬀerson Avenue Newport News, VA 23606 USA Tel.: ++1-757-2697764 Fax: ++1-757-5800 [email protected]

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Elenco dei partecipanti Giorgio BENDISCIOLI Dipartimento di Fisica Nucleare e Teorica Universit` a di Pavia Via U. Bassi 6 27100 Pavia Italy Tel.: ++39-0382-987351 [email protected]

Maria Elena BOGLIONE Dipartimento di Fisica Teorica Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707223 Fax: ++39-011-6707214 [email protected]

Carlo GUARALDO INFN - LNL Via E. Fermi 40 00044 Frascati (RM) Italy Tel.: ++39-06-94032318 Fax: ++39-06-94032559 [email protected]

Robert L. JAFFE Center for Theoretical Physics University of Massachusetts Inst. 77 Massachusetts Avenue Cambridge, MA 02139 USA Tel.: ++1-617-2534858 Fax: ++1-617-2538674 [email protected]

Students Anastasia BOLSHAKOVA JINR Joliot-Curie st. 6 141980 Dubna Russia Tel.: ++7-49621-67568 Fax: ++7-49621-66666 [email protected] Stefania BUFALINO Dipartimwento di Fisica Sperimentale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707321 Fax: ++39-011-6707324 [email protected] Barbara DALENA Dipartimento di Fisica e INFN Universit` a di Bari Via G. Amendola 173 70126 Bari Italy Tel.: ++39-080-5443187 Fax: ++39-080-5443151 [email protected] Diego FASO Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707266 [email protected]

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Hiroyuki FUJIOKA KEK - IPNS Saito Lab. 1-1 Oho, Tsukuba-shi #305-0801 Tokyo Japan Tel.: +81-29-879-6041 Fax: +81-29-864-7831 [email protected]

Francesca GIORDANO Dipartimento di Fisica e INFN Universit` a di Ferrara Via G. Saragat 1 44100 Ferrara Italy Tel.: ++39-0532-974311 [email protected]

Elenco dei partecipanti Teppei KATORI MS 309, Fermilab P.O. Box 500 Batavia, IL 60510-0500 USA Tel.: ++1-630-855-2077 [email protected]

´ Alejandro LOPEZ RUIZ Department of Subatomic and Radiation Physics University of Gent Proeftuinstraat 86 9000 Gent Belgium Tel.: ++494089984674 Fax: ++494089984034 [email protected]

Ivan GNESI Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707049 Fax: ++39–011-6707269 [email protected]

Xiaorui LU Physics Department Tokyo Institute of Technology O-okayama 2-12-1 152-8551 Tokyo Japan Tel.: ++0081-3-5734-2369 Fax: ++0081-3-5734-2742 [email protected]

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Annalisa MASTROSERIO Dipartimento di Fisica Universit` a di Bari Via G. Amendola 173 70126 Bari Italy Tel.: ++39-080-5443203 Fax: ++39-080-5442531 [email protected]

467

Elenco dei partecipanti Stefano MELIS Dipartimento di Fisica Teorica Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707235 [email protected]

Kei MORIYA Physics Department University of Carnegie Mellon 5000 Forbes Ave Pittsburgh, PA 15213 USA Tel.: ++1-412-370-4547 Fax: ++1-412-618-0648 [email protected]

Sergio Anefalos PEREIRA INFN - LNF Via E. Fermi 40 00044 Frascati (RM) Italy Tel.: ++39-06-94032569 Fax: ++39-06-94032559 [email protected]

Sergey PETROCHENKOV Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10129 Torino Italy Tel.: ++39-011-6707477 [email protected]

Luciano Libero PAPPALARDO Dipartimento di Fisica Universit` a di Ferrara Via G. Saragat 1 44100 Ferrara Italy Tel.: ++39-0532-974321 [email protected]

Manuel PINCETTI Dipartimento di Fisica Nucleare e Teorica Universit` a di Pavia Via A. Bassi 6 27100 Pavia Italy Tel.: ++39-0382-987445 Fax: ++39-0382-526938 [email protected]

Bakur PARSAMYAN Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707284 [email protected]

Alexey PROKUDIN Dipartimento di Fisica Teorica Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707235 Fax: ++39-011-6707214 [email protected]

468

Oleg B. SAMOYLOV JINR Joliot-Curie st. 6 141980 Dubna Russia Tel.: ++7-49621-67568 Fax: ++749621-66666 [email protected]

Sabyasachi SARKAR Matrivani Insitute 6A, Seven Tanks Lane Kolkata India Tel.: ++91-3325578107 [email protected]

George SERBANUT Dipartimento di Fisica Generale Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707060 Fax: ++39-011-6707269 [email protected]

Giuseppe SIMONETTI Dipartimento di Fisica e INFN Universit` a di Bari Via Orabona 4 70126 Bari Italy Tel.: ++39-080-5443187 Fax: ++39-080-5443151 [email protected]

Elenco dei partecipanti Katarzyna SZYMANSKA Dipartimento di Fisica Politecnico di Torino C.so Duca degli Abruzzi 10129 Torino Italy Tel.: ++39-011-5647384 Fax: ++39-011-5647399 [email protected]

Tomonori TAKAHASHI KEK INPS 1-1 Oho, Tsukuba-shi #305-0801 Tokyo Japan Tel.: ++81-29-879-6040 Fax: ++81-29-864-7831 [email protected]

¨ Christian TURK Dipartimento di Fisica Teorica Universit` a di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-011-6707066 Fax: ++39-011-6707214 [email protected]

Elena ZEMLYANICHKINA JINR Joliot-Curie st. 6 141980 Dubna Russia Tel.: ++7-49621-66205 Fax: ++7-49621-65767 [email protected]

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Elenco dei partecipanti

Observers Daniela CALVO INFN, Sezione di Torino Via P. Giuria 1 10125 Torino Italy Tel.: ++39-3474356971 Fax: ++39-011-6707324 [email protected]

Patrizia ROSSI INFN - LNF Via E. Fermi 40 00044 Frascati (RM) Italy Tel.: ++39-06-94032549 Fax: ++39-06-94032559 [email protected]

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PROCEEDINGS OF THE INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

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Course CXVII (1991) Semiconductor Superlattices and Interfaces edited by A. Stella Course CXVIII (1991) Laser Manipulation of Atoms and Ions edited by E. Arimondo, W. D. Phillips and F. Strumia

Course CXXXI (1995) Coherent and Collective Interactions of Particles and Radiation Beams edited by A. Aspect, W. Barletta and R. Bonifacio Course CXXXII (1995) Dark Matter in the Universe edited by S. Bonometto and J. Primack

Course CXIX (1991) Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CXXXIII (1996) Past and Present Variability of the SolarTerrestrial System: Measurement, Data Analysis and Theoretical Models edited by G. Cini Castagnoli and A. Provenzale

Course CXX (1992) Frontiers in Laser Spectroscopy ¨nsch and M. Inguscio edited by T. W. Ha

Course CXXXIV (1996) The Physics of Complex Systems edited by F. Mallamace and H. E. Stanley

Course CXXI (1992) Perspectives in Many-Particle Physics edited by R. A. Broglia, J. R. Schrieffer and P. F. Bortignon

Course CXXXV (1996) The Physics of Diamond edited by A. Paoletti and A. Tucciarone

Course CXXII (1992) Galaxy Formation edited by J. Silk and N. Vittorio

Course CXXXVI (1997) Models and Phenomenology for Conventional and High-Temperature Superconductivity edited by G. Iadonisi, J. R. Schrieffer and M. L. Chiofalo

Course CXXIII (1992) Nuclear Magnetic Double Resonsonance edited by B. Maraviglia Course CXXIV (1993) Diagnostic Tools in Atmospheric Physics edited by G. Fiocco and G. Visconti Course CXXV (1993) Positron Spectroscopy of Solids edited by A. Dupasquier and A. P. Mills jr. Course CXXVI (1993) Nonlinear Optical Materials: Principles and Applications edited by V. Degiorgio and C. Flytzanis Course CXXVII (1994) Quantum Groups and their Applications in Physics edited by L. Castellani and J. Wess Course CXXVIII (1994) Biomedical Applications of Synchrotron Radiation edited by E. Burattini and A. Balerna 1 This

Course CXXXVII (1997) Heavy Flavour Physics: a Probe of Nature’s Grand Design edited by I. Bigi and L. Moroni Course CXXXVIII (1997) Unfolding the Matter of Nuclei edited by A. Molinari and R. A. Ricci Course CXXXIX (1998) Magnetic Resonance and Brain Function: Approaches from Physics edited by B. Maraviglia Course CXL (1998) Bose-Einstein Condensation in Atomic Gases edited by M. Inguscio, S. Stringari and C. E. Wieman Course CXLI (1998) Silicon-Based Microphotonics: from Basics to Applications edited by O. Bisi, S. U. Campisano, L. Pavesi and F. Priolo

course belongs to the NATO ASI Series C, Vol. 460 (Kluwer Academic Publishers).

Course CXLII (1999) Plasmas in the Universe edited by B. Coppi, A. Ferrari and E. Sindoni

Course CLIV (2003) Physics Methods in Archaeometry edited by M. Martini, M. Milazzo and M. Piacentini

Course CXLIII (1999) New Directions in Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CLV (2003) The Physics of Complex Systems (New Advances and Perspectives) edited by F. Mallamace and H. E. Stanley

Course CXLIV (2000) Nanometer Scale Science and Technology edited by M. Allegrini, N. Garc´ıa and O. Marti

Course CLVI (2003) Research on Physics Education edited by E.F. Redish and M. Vicentini

Course CXLV (2000) Protein Folding, Evolution and Design edited by R. A. Broglia, E. I. Shakhnovich and G. Tiana

Course CLVII (2003) The Electron Liquid Model in Condensed Matter Physics edited by G. F. Giuliani and G. Vignale

Course CXLVI (2000) Recent Advances in Metrology and Fundamental Constants edited by T. J. Quinn, S. Leschiutta and P. Tavella

Course CLVIII (2004) Hadron Physics edited by T. Bressani, U. Wiedner and A. Filippi

Course CXLVII (2001) High Pressure Phenomena edited by R. J. Hemley, G. L. Chiarotti, M. Bernasconi and L. Ulivi

Course CLIX (2004) Background Microwave Radiation and Intracluster Cosmology edited by F. Melchiorri and Y. Rephaeli

Course CXLVIII (2001) Experimental Quantum Computation and Information edited by F. De Martini and C. Monroe

Course CLX (2004) From Nanostructures to Nanosensing Applications edited by A. D’Amico, G. Balestrino and A. Paoletti

Course CXLIX (2001) Organic Nanostructures: Science and Applications edited by V. M. Agranovich and G. C. La Rocca Course CL (2002) Electron and Photon Conﬁnement in Semiconductor Nanostructures ´dran, A. Quatedited by B. Deveaud-Ple tropani and P. Schwendimann Course CLI (2002) Quantum Phenomena in Mesoscopic Systems edited by B. Altshuler, A. Tagliacozzo and V. Tognetti Course CLII (2002) Neutrino Physics edited by E. Bellotti, Y. Declais and P. Strolin Course CLIII (2002) From Nuclei and their Constituents to Stars edited by A. Molinari, L. Riccati, W. M. Alberico and M. Morando

Course CLXI (2005) Polarons in Bulk Materials and Systems with Reduced Dimensionality edited by G. Iadonisi, J. Ranninger and G. De Filippis Course CLXII (2005) Quantum Computers, Algorithms and Chaos edited by G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti Course CLXIII (2005) CP Violation: From Quarks to Leptons edited by M. Giorgi, I. Mannelli, A. I. Sanda, F. Costantini and M. S. Sozzi Course CLXIV (2006) Ultra-Cold Fermi Gases edited by M. Inguscio, W. Ketterle and C. Salomon Course CLXV (2006) Protein Folding and Drug Design edited by R. A. Broglia, L. Serrano and G. Tiana

Course CLXVI (2006) Metrology and Fundamental Constants ¨nsch, S. Leschiutta, A. edited by T. W. Ha J. Wallard and M. L. Rastello

Course CLXIX (2007) Nuclear Structure far from Stability: New Physics and New Technology edited by A. Covello, F. Iachello, R. A. Ricci and G. Maino