ELECTROMAGNETIC PROBES OF FUNDAMENTAL PHYSICS
THE SCIENCE AND CULTURE SERIES
- PHYSICS
Series Editor: A. Zichichi, European Physical Socieq, Geneva, Switzerland Series Editorial Board: P. G. Bergmann, J. Collinge, V. Hughes, N. Kurti, T. D. Lee, K. M. B. Siegbahn, G. 't Hooft, P. Toubert, E. Velikhov, G. Veneziano, G. Zhou
1. Perspectives for New Detectors in Future Supercolliders, 1991 2. Data Structures for Particle Physics Experiments: Evolution or Revolution?, 1991 3. Image Processing for Future High-Energy Physics Detectors, 1992 4. GaAs Detectors and Electronics for High-Energy Physics, 1992 5. Supercolliders and Superdetectors, 1993 6. Properties of SUSY Particles, 1993 7. From Superstrings to Supergravity, 1994 8. Probing the Nuclear Paradigm with Heavy Ion Reactions, 1994 9. Quantum-Like Models and Coherent Effects, 1995 10. Quantum Gravity, 1996 11. Crystalline Beams and Related Issues, 1996 12. The Spin Structure of the Nucleon, 1997 13. Hadron Colliders at the Highest Energy and Luminosity, 1998 14. Universality Features in Multihadron Production and the Leading Effect, 1998 15. Exotic Nuclei, 1998 16. Spin in Gravity: Is It Possible to Give an Experimental Basis to Torsion?, 1998 17. New Detectors, 1999 18. Classical and Quantum Nonlocality, 2000 19. Silicides: Fundamentals and Applications, 2000 20. Superconducting Materials for High Energy Colliders, 2001 21. Deep Inelastic Scattering, 2001 22. Electromagnetic Probes of Fundamental Physics, 2003
ELECTROMAGNETIC PROBES OF FUNDAMENTAL PHYSICS
Erice, Italy
16 - 21 October 2001
Editors
W. Marciano and S. White Series Editor
A. Zichichi
L
World Scientific New Jersey. London *Singapore Hong Kong
Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224
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ELECTROMAGNETIC PROBES OF FUNDAMENTAL PHYSICS Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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xi
PREFACE
This workshop originated from a discussion with Professor Zichichi in March of 2001. The purpose of the meeting was to celebrate two developments at Brookhaven that, in both cases, are the outcome of more than a decade of preparation: the new muon g-2 measurement and the flood of data that started pouring out of RHIC. Bringing together people from these very different fields was an interesting challenge. But, at the time, when a small community of Heavy Ion physicists was beginning to define a program of electromagnetic interactions at RHIC and the LHC, it seemed logical. The result is the whitepaper “Hot Topics in Ultra-Peripheral Collisions”. In this year of Fermi’s bicentennial, his original paper, developing for the first time the Equivalent Photon Approximation, seemed to merit renewed attention. We have translated it into English from the Italian and included it in these proceedings. This project received invaluable encouragement, at various stages, from our advisory committee (Professors T.D. Lee, Bj Bjorken, Adrian Melissinos and Gigi Rolandi), as well as from Prof. Zichichi, David Lissauer, Peter Paul, Willy Chinowsky and Bradley Keister. Financial support was provided by the Ettore Majorana Foundation for Scientic Culture, The National Science Foundation and the Brookhaven National Laboratory. We would like to thank the staff of the Majorana Center, particularly Fiorella Ruggiu and Pino Aceto, who, together with our conference secretary Jackie Mooney, did a very professional job of organizing the meeting . Perry Palmedo organized the material in our “Erice cool-stuff CD’ during an internship in the BNL “Community Summer Science Program”. The cover photo is courtesy of The Amaldi Archives, Department of Physics, University “La Sapienza” , Rome. We are especially grateful to Prof. Carlos Bertulani for his preparation of the web-based material including the guidelines for preparing manuscripts.
Bill Marciano, Sebastian White Upton, New York
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xiii
CONTENTS
Preface
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
. . . . . . . .
1
. . . . . . . . . . . . . . . . . . .
15
The BNL Muon Anomalous Magnetic Moment Measurement
David W. Hertzog Hadronic Contribution to ( g - 2 ) p
Andreas Hoclcer High Precision Studies of Low Energy e+e- Annihilation into Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
S.I. Eidelman The Muon Anomalous Magnetic Moment: A Status Report
. . . . . . . . 37
William J. Marciano Strong Fields
Spontaneous Breakdownof the Vacuum by Intense Lasers . . . . . . . . . 43
Adrian C. Melassinos Electrodynamics at the Highest Energies . . . . . . . . . . . . . . . . . 53
Spencer R. Klein Fundamental Physics at an X-Ray Free Electron Laser . . . . . . . . . . . 63
A. Ringwald Parametric Gravity Wave Detector
. . . . . . . . . . . . . . . . . . . 75
G. Gemme, A. Chincarini, R. Parodi, Ph. Bernard and E. Picasso Quasi-degeneracy in Bound-State QED. Fine Structure of Heliumlike Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ingvar Lindgren, Sten Salomonson, Bjorn A s h and Ann-Marie Miirtensson-Pendrill
85
xiv
Photon Colliders in Physical Program for Large Colliders
. . . . . . . . . 97
I.F. Ginzburg and S.I. Polityko High-Energy Photon Colliders
. . . . . . . . . . . . . . . . . . . .
107
V.G. Serbo Future Determination of the Neutrino-Nucleon Cross Section at Extreme Energies . . . . . . . . . . . . . . . . . . . .
. . . . .
117
A . Kusenko Electromagnetic Tests of Lorentz and CPT Symmetry . . . . . . . . . .
121
R. Bluhm Heavy Ions
The RHIC Experimental Program: Highlights from the First Year of Operation at RHIC . . . . . . . . . . . . . . . . . . . .
. 131
J.H. Thomas Applications of the Equivalent Photon Approximation to Heavy Ion Collisions . . . . . . . . . . . . . . . . . . . . . . .
149
S. White Atomic Parity Violation in Heavy Ion Colliders . . . . . . . . . . . . .
159
Max Zolotorev and D. Budker Aspects of Coulomb Dissociation and Interference in Peripheral Nucleus-Nucleus Collisions . . . . . . . . . . . . . . . . . . . . . .
161
Joakim Nystrand, Anthony J. Baltz and Spencer R. Klein High Energy Two-Photon Interactions at the LHC
. . . . . . . . . . . 171
Krzyszt of Piotrzko wski Physics Opportunities in Ultraperipheral Heavy Ion Collisions atLHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183
G. Baur Coherent Electromagnetic Processes in Relativistic Heavy Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K. Hencken
193
xv
Relativistic Heavy Ion Excitation of Giant Resonances
. . . . . . . . . .
203
. . . . . . . .
213
. . . . . . .
223
C.A. Bertulani Heavy Ions Interpretationof Highest Energy Cosmic Rays
Steve Reucroft Two-Photon Final States in Peripheral Heavy Ion Collisions
C. G. Roldao Ultra-peripheral Collisions in STAR
. . . . . . . . . . . . . . . . . . 229
P. Yepes Supplementary Hot Topics in Ultra-peripheral Ion Collisions
. . . . . . . . . . . . . . 235
G. Bauer, C.A. Bertulani, M. Chiu, I.F. Ginaburg, K. Hencken, S.R. Klein, J. Nystrand, K. Piotrzkowski, C.G. Roldao, D. Silvennyr, J.H. Thomas, S.N. While and P. Yepes On the Theory of Collisions Between Atoms and Electrically Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . .
Enrico Fermi, 1924 Translated from the Italian by M. Gallinaro and S. White
243
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THE B N L M U O N ANOMALOUS M A G N E T I C M O M E N T M E A S U R E M E N T DAVID W. HERTZOG Department of Physics, University of Illinois at Urbana-Champaign 1110 W. Green St., Urbana, IL 61801, USA
Representing the E821 Muon g-2 Collaboration
'
The E821 experiment at Brookhaven National Laboratory is designed to measure the muon magnetic anomaly, a,, to an ultimate precision of 0.4 parts per million (ppm). Because theory can predict a, to 0.6 ppm, and ongoing efforts aim to reduce this uncertainty, the comparison represents an important and sensitive test of new physics. At the time of this Workshop, the reported experimental result from the 1999 running period achieved a,+ = 11659 202(14)(6) x lo-'' (1.3 ppm) and differed from the most precise theory evaluation by 2.6 standard deviations. Considerable additional data has already been obtained in 2000 and 2001 and the analysis of this data is proceeding well. Intense theoretical activity has also taken place ranging from suggestions of the new physics which could account for the deviation to careful reexamination of the standard model contributions themselves. Recently, a reevaluation of the pion pole contribution to the hadronic light-by-light process exposed a sign error in earlier studies used in the standard theory. With this correction incorporated, experiment and theory disagree by a modest 1.6 standard deviations.
1 Introduction
A precision measurement of the muon anomalous magnetic moment, a, = (g - 2)/2, is a sensitive test of physics beyond the standard model. Because contributions to the muon anomaly from known processes (see Fig. l),such as QED, the weak interaction, and hadronic vacuum polarization (including higher-order terms) are believed to be understood at the sub-ppm level, any significant difference between experiment and theory suggests a yet unknown, and thus not included, physical process. Conversely, agreement between theory and experiment can set tight constraints on new physics. Many standard model extensions have been postulated; of these, quite a few would manifest themselves in additional contributions to a, at the ppm level. This indirect method of probing high-mass and short-distance physics is being used by the of a,,. The experimental work is Brookhaven E821 Collaboration in a new measurement complemented by an aggressive effort by others to improve the precision of the standard model theory. In the near-term future, and generally in advance of the direct-discovery possibilities at the Tevatron or the LHC, both experiment and theory should achieve relative precision near or below 0.5 ppm. The Collaboration recently reported a 1.3 ppm precision measurement of a,, based on data obtained in the 1999 running period. A summary2 of the most up-to-date theoretical . reexaminations of the 1stinformation implied a deviation from theory by 2 . 6 ~ Subsequently, order hadronic vacuum polarization (HVP) contribution suggested some scatter in the central theory value, however the deviation remained greater than two standard deviations. Shortly 39415
1
2
(a) QED
(b) Weak
(c) Hadronic
(d) Light-by-light
Figure 1: Representative first-order Feynman diagrams for $ED, weak and hadronic vacuum polarization, and hadronic light-by-light scattering.
after this Workshop, Knecht and Nyffeler reported a new calculation of the hadronic lightby-light (LbL) diagram (see Fig. Id). This process cannot be determined from measurements and therefore must be estimated using theoretical models. It has endured a checkered past in which the sign was once reversed from positive to negative. The new finding finds a positive sign for the dominant pion pole contribution, but otherwise a similar magnitude to previous studies by others. 7,8 This finding prompted Hayakawa and Kinoshita to reexamine their own work which indeed contained a sign error-in an innocent computer algorithm-prompting a report with a new hadronic light-by-light term of a,(LbL) = +89.6(15.4) x lo-’’. Combined with the other complete hadronic LbL evaluation by Bijnens et a1.: which has also been updated to report lo a sign error, the standard model summary2 is a,(thy) = 11 659 176.7(6.7) x This adjustment reduces Aa,(exp - thy) by M 1 standard deviation and puts the discrepancy with the standard model at the modest 1.60 level. This Workshop also featured two talks by leading contributors to the 1st-order HVP evaluation. Simon Eidelmanl’ described the recent data obtained by the Novosibirsk CMD-I1 team in their experiment on e+e- + Hadrons in the important p region. These data alone, when finalized, should reduce the uncertainty to the 0.6 ppm level without the inclusion of hadronic tau decays. Andreas Hocker,12 together with Michel Davier, pioneered the inclusion of hadronic tau decay data in the evaluation of the HVP term. Despite non-negligible isospin breaking considerations, their results were responsible for a significant decrease in the overall SM theory uncertainty. But their work is not without some controversy and q~estioning?~ Additional tau data have been analyzed and the present issue discussed at this Workshop was the reliability with respect to the more direct e+e- approach; the question raised was, “Can the tau data be used with sub-percent level absolute precision?” The answer is still unclear. On the horizon experimentally is the use of radiative return at higher-energy e+e- machines which promises to add complementary precision input to the HVP database. As one has witnessed over the past year, the standard model theory of a, is very much a work in progress, just like the experiment. Recent 1st-order HVP evaluations are shown in Fig. 2. They are compared to the new E821 result and the current world average. The $ED, weak and higher-order hadronic terms, which are believed to be known to a few tenths of a ppm or better, have been subtracted from comparison the measurement of a, . This results in an effective “experiment-to-experimentll because the input to the 1st-order HVP comes from measurement. The plot demonstrates the present relative size of the uncertainties. Both experiment and “theory” uncertainties will be reduced with the analysis of additional data already obtained. Bill Marciano17 speculated on the implications of the comparison of final data with settled theory. A non-standard model result could be explained rather naturally in the context of supersymmetry. It has been recognized for
3
DH98
t-n-l
NO1
M
w
TYOl
+
BNL E821
I
,
7.0
7.5 Hadronic Vacuum Polarization
6.5
,
,
,
,
,
,
I
I
8.0 lo4
Figure 2: Recent 1st-order HVP evaluations by Davier and Ho~ker,’~ Narison l5 and de Troc6niz and YndurCn,” eaxh combining e+e- and hadronic tau decay data. The 1999 BNL result is shown with the QED, weak and higher-order hadronic terms subtracted. The solid vertical line represents the current world average experimental result.
some years that the muon anomaly scales nearly linearly with tanp, the ratio of Higgs doublet vacuum expectation values in the theory. The SUSY allowed phase space certainly has ample room for large t a n p solutions, while the lowest values, which would be exceedingly difficult to observe in up are beginning to be ruled out by direct search experiments. Of course, many other candidate explanations for non-SM up exist; many l8 have been published following the original 2 . 6 ~deviation report. The purpose of this paper is to describe the new experiment which was designed following the three pioneering CERN efforts. Francis Farley l9 presented the historical development and the key ideas incorporated in all ( g - 2) experiments. I will concentrate here on the specifics related to the E821 implementation and the 1999 data analysis. 2
The E821 Experiment
2.1 Principle The experimental goal is to measure directly ( g - 2) and not g . The leading-order contribution to up = ( g - 2)/2 is the QED “Schwinger” term whose magnitude is (a/2n) = 1.16 x This implies that to measure g directly to an equivalent sensitivity would require an experiment with nearly an 800-fold increase in precision. The muon anomaly is determined from the difference between the cyclotron and spin precession frequencies for muons contained in a magnetic storage ring, namely m
y2
-1
3.g
In principle, an additional term exists. However, it vanishes when the muon trajectory is perpendicular to the magnetic field as is the case in this experiment. Because electric quadrupoles are used to provide vertical focussing in the storage ring, the x term is necessary and illustrates the sensitivity of the spin motion to a static electric field. This term conveniently vanishes for the “magic” momentum of 3.094 GeV/c where y = 29.3. The experiment is therefore built around the principle of production and storage of muons centered at this momentum in order to minimize the electric field effect. Because of the finite momentum spread of the stored muons, a modest correction to the observed precession frequency is made to account for the muons above or below the magic momentum. Vertical betatron oscillations induced by the electric field imply
3
4
that the plane of the muon precession has a time-dependent pitch. Accounting for both of these electric-field related effects introduces a +0.81* 0.08 ppm correction to the measured precession frequency. Muons introduced into the storage ring exhibit cyclotron motion and their spins precess until the time of decay; 7-rl NN 64.4 ps. The net spin precession depends on the integrated path followed by a muon convoluted with the local magnetic field experienced along the path. Parity violation leads to a preference for the highest-energy decay electrons to be emitted in the direction of the muon spin. Therefore, a snapshot of the muon spin direction at time t after injection into the storage ring is obtained, again on average, by the selection of decay electrons in the upper part of the Lorentz-boosted Michel spectrum (Em,, M 3.1 GeV). The number of electrons above a selected energy threshold is modulated at frequency w, with a threshold-dependent asymmetry A = A(&). The decay electron distribution is described by
N ( t ) = NOexp(-t/7-rp) [ 1 + A cos(w,t
+ 4)],
where NO, the normalization, A and q5 are all dependent on the energy threshold Eth. For Eth = 2.0 GeV, A M 0.4. In summary, the experiment involves the measurement of three quantities: (1) The precession frequency, w, in Eq. 1; (2) The muon distribution in the storage ring; and, (3) The time-averaged local magnetic field. The muon anomaly is proportional to the ratio,
Term (3) above is measured using NMR in units of the free proton precession frequency, wp. Term (2) is determined from the debunching rate of the initial beam burst and from a tracking simulation. The combined denominator involves an event-weighted average of the field folded with the muon distribution. The symbol Gp represents the final average magnetic field and the muon anomaly is computed from the expression
where X is the measured” muon-to-proton magnetic moment ratio p,/pp = 3.183 345 39(10). Four independent teams evaluated w, and two studied Gp. During the analysis period, the w, and Gp teams maintained separate, secret offsets to their measured frequencies. The offsets were removed and a, was computed only after all analysis checks were complete.
3 Experiment 3.1 Storage Ring The Brookhaven muon storage ring2’ is a superferric “C”-shaped magnet, 7.11 m in radius, and open on the inside to permit the decay electrons to curl inward (Fig. 3). A 5V power supply drives a 5177 A current in the three superconducting coils. The field is designed to be vertical and uniform at a central value of 1.4513 T. Carefully shaped steel pole pieces and accompanying edge shims are separated from the main iron flux return which makes up the “C” structure. The field is further shaped by tapered iron wedges placed in the yoke/pole-piece gap and by 80 low-current surface correction coils which circumnavigate the ring on the pole piece faces. The storage volume is 9 cm in diameter and is enclosed in a vacuum chamber. A vertical slice through the storage ring illustrating some of these key features is shown in Fig. 4. Protons from the AGS strike a nickel target in 6-12 bunches separated by 33 ms within a typical 2.5-3.3 s AGS cycle. Pions created in these collisions are directed down a 72 m long
5
4
/
Figure 3: A 3D engineering rendition of the E821 muon storage ring. Muons enter the back of the storage ring through a field-free channel at approximately 10 o’clock in the figure. The three modules at approximately 2 o’clock provide the rapid current pulse which gives the muon bunch a transverse 10 mrad kick. The regularly spaced boxes on rails represent the electron detector systems.
r=4.5crn surface COrmCtlOn
&Is 1360daul
Figure 4: Slice of the storage ring magnet illustrating the key tools in the shimming kit and the relative size of the storage ring volume. The vacuum chamber, which encloses the storage volume and houses the quadrupoles and the trolley rails, is not shown.
6 beamline to the muon storage ring. Approximately half decay, x + pv, and the forward going muons, having a high degree of longitudinal polarization, remain in the channel, generally at a slightly lower momentum. The last dipoles upstream of the entrance to the storage ring are tuned to a momentum = 1.7% lower than the main channel in order to enhance the muon fraction in the beam; the result is a final p : T ratio of approximately 50 : 50 entering the ring. A superconducting inflector magnet 22 provides a field-free channel through the back of the storage ring’s iron yoke. The field created by this magnet is tuned to cancel the main storage ring field. The inflector field is then prevented from extending into the storage ring by flux trapping in a superconducting outer shield. The pulse structure of the AGS is translated to an effective muon bunch of approximately 25 ns R M S passing through the exit of the inflector into the ring. A simple circular trajectory would result in the muon bunch striking the inflector magnet 149 ns after injection, which is the cyclotron period. A pulsed “kicker” magnet provides a 10 mrad transverse deflection to the muon bunch during the first turn in the ring. In practice, three pulsed magnets, all in series, are made from current sheets and special high-voltage pulsed power supplies. The electric quadrupoles are located symmetrically at four positions occupying, in total, 43% of the ring. Immediately after particle injection, the plates are asymmetrically charged in order to scrape the beam against internal collimaters. After approximately 20 ps, the voltages are symmetrized to final values of f 2 4 kV which leads to the stored weak-focussing field index n = 0.137. The horizontal and vertical betatron frequencies are w, M 6.23 MHz and wy M 2,48MHz, respectively. Because the inflector aperture is small compared to the cross-sectional area of the storage volume, phase space is not fully occupied and the stored beam exhibits “breathing” and “swimming” motions during a fill. These motions manifest themselves as effective modulations in the acceptance function of the detectors because the acceptance is sensitive, on average, to the position of the decaying muons. The relevance of this fact becomes clear when one realizes that the difference between the cyclotron and horizontal betatron frequencies is approximately 470 kHz, which is a little greater than twice w,. These “coherent betatron oscillations’’ (CBO) are visible in the data; to account for CBO, Eq. 2 must be multiplied by a term of the type
+
C ( t )= 1 ACboe-(t/Tcbo)2 COS(Wcbot
+ 4cbo)
(5)
where 7cbo is the time constant for decay of the CBO terms having initial amplitude Acbo. 3.2
Field Measurements
The magnetic field is measured using pulsed nuclear magnetic resonance (NMR) on protons in water- or Vaseline-filled probes. The proton precession frequency is proportional to the local field strength and is measured with respect to the same clock system employed in the determination of wa. The absolute field is, in turn, determined by comparison with a precision measurement of g p in a spherical water sample23and is thus determined to a precision of better than lop7. A subset of the 360 “fixed” probes is used to continuously monitor the field during data taking. Fixed probes are embedded in machined grooves in the outer upper and lower plates of the aluminum vacuum chamber and consequently measure the field just outside of the actual storage volume. Constant field strength is maintained using 36 of the probes in a continuous feedback loop with the main magnet power supply. The determination of the field inside the storage volume is made by use of a unique nonmagnetic trolley which can travel in vacuum through the muon storage volume. The trolley carries 17 NMR probes on a grid appropriate to determine the local multipolarity of the field versus azimuth. Trolley field maps are made every few days and take several hours to complete. The deviation from the central field strength value Bo = 1.451266 T versus azimuth is shown in Fig. 5. At any given location, field measurement precision is at the 0.1 ppm level even though
7
Figure 5: Magnetic field average versus azimuth from a typical trolley run. The spike at 350” is due to a small crack in the inflector shielding which has since been repaired.
the local field varies by as much as 100 ppm from Bo. However, it is the average field which must be determined to the sub ppm level and this goal is readily achieved. A 800 ppm “spike” in the field is seen in a 1’ azimuthal segment. This small defect is due to a crack in the inflector magnet shielding. Special measurements were made to map the field in this region for the 1999 data; this inflector was replaced for the 2000 and 2001 data taking periods. A one-time systematic error of 0.2 ppm is included in the 1999 result to account for uncertainties in the procedure of measuring the errant field in this region. Comparison of the azimuthally averaged trolley field with that which is computed using the fixed probe data permits a determine of the time-dependent field. Figure 6 illustrates the excellent agreement between these two systems. The relative field is interpolated from the fixed-probe extrapolation to better than 0.15 ppm. The trolley measurements also establish the field homogeneity throughout the storage volume. Three snapshots of the azimuthally-averaged field are shown in Fig. 7. The contours represent 1 ppm deviations from the central value. The left panel illustrates the relatively uneven field achieved shortly after the ring was commissioned in 1997; here it is already better than the CERN 111field?5 An aggressive shimming program led to the improved contours for the 1998 and 1999 data taking periods (middle panel). With the inflector replaced, the field for the 2000 and 2001 data taking was again significantly improved (right panel). 4
Electron Detection and wa Analysis
The electron detection system consists of 24 lead-scintillating fiber electromagnetic calorimeters 26 located symmetrically around the inside of the storage ring and placed immediately adjacent to the vacuum chamber. The 23 cm long, radially-oriented fiber grid terminates on four lightguides which pipe the light to independent Hamamatsu R1828 2-inch PMTs, see Fig. 8. The PMT gains are carefully balanced because the four analog signals are added prior to sampling by a custom 400 MHz waveform digitizer. At least 16 digitized samples (usually 24 or more) are recorded for each decay electron event exceeding a hardware threshold of approximately 1 GeV. An example of a series of such samples is shown in Fig. 9. Offline pulse-finding and fitting establishes electron energy and muon time of decay.
8 2 -
. . __ magilet ramp down: . pa&$al-ramp.doy,n--: ......................... i ........................ :............................. j ........................... mean Gffsot for each magnet cycle subtracted PS trip i ......................................................... :........................... : ....................................................... 1 - .................. magilet :i
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Figure 6: The magnetic field strength as measured at the center of the storage ring using the NMR trolley system compared to that which was expected from data obtained using the fixed NMR probes. Each point represents an individual trolly run during the 1999 data taking period. The agreement is at the 0.15 ppm level.
-g 4
-E 4
199811999
I
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%
-5-4 -3 -2 -1 0 1 2 3 4 5
x lcml
2000
x
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x rcmi
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0 1 2 3 4 5
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Figure 7 Sequence of improving magnetic field profiles, averaged over azimuth and interpolated using a multipole expansion. The circle indicates the storage aperture. From left to right, these maps represent the field variations for the 1997 commissioning run, the 1998/1999 data taking, and finally the 2000 run. Improvements in shimming and replacement of the inflector magnet led to the two major improvements following the 1997 and 1999 runs, respectively.
9
Figure 8: Electromagnetic calorimeter system for the experiment. The figure illustrates the four lightguides from the calorimeter and the five which couple to the scintillator hodoscope.
-100 0
n
9 80 Em ’0 60 I 0
4n
40 20 0 10
20
30
40
50
60
70 80 Time (ns)
Figure 9: An “island” of waveform digitizer samples. In this representation, the constant offset has been subtracted. Two well-separated pulses are shown. The pulse-finding routines have no difficulty with such events. Special care is needed when the second pulse moves within 5 ns of the first event.
Because the full raw waveforms are stored, multi-particle pileup can be studied with sophisticated fitting algorithms which are superior to any “online,” hardware discriminator techniques. The approach taken is to fit waveforms for all occurring pulses using known detector response functions. We identify the level of pileup and remove such events, on average. By processing the data using different artificial deadtime windows around or near the driving trigger pulse, and by carefully accounting for the energy found in such windows, a pileupfree spectrum (less than 10% pileup remaining) can be created by appropriate, normalized difference spectra. Figure 10 illustrates three curves of the energy spectrum in the calorimeters. The solid curve is the “true” energy distribution found at late times where no pileup events are expected. The dashed distribution, which extends to much higher energies, is observed at early times when the rate in the detectors is high. Pileup produces effective electron energies beyond the natural maximum of 3.1 GeV. The dotted curve is the recreated pileup-free final spectrum. Its congruence with the late-time curve gives confidence in the correction procedure. Fivefold, vertically segmented scintillator hodoscopes (FSDs) are attached to the front face of many of the calorimeters. These FSDs are used to measure the rate of “lost muons” from the
10
1
2
3
4
5
6 E [GeV]
Figure 10: Energy spectrum at late times only (thin line), at all times (thick line) and after pileup subtraction (dashed line). The inset compares the average energy in a full (g - 2) cycle versus time for unsubtracted (solid circles) and subtracted (open circles) spectra. The energy distribution must be flat.
storage ring. During the initial scraping period, and extending for a period of time afterwards, errant muon trajectories will pass through one of the aperature-defining collimaters inside the storage ring. These muons lose enough energy to exit the ring, many following a path which penetrates three consecutive calorimeters and their corresponding FSDs. A triple coincidence of FSDs, together with the absence of significant energy in the calorimeters, is proportional to the lost muon rate. A representative muon loss spectra for one detector is shown in Fig. 11 At early times, the loss rate exceeds the natural muon decay rate and can be described by an exponential with an approximate 27 ps lifetime. At later times, the loss rate is essentially constant. The overall loss rate is proportional to
Integration of A ( t ) and inclusion in the fitting function is necessary to obtain a good representation of the instantaneous muon population in the ring. 5
Fitting the 1999 Data
The five adjustable parameters in Eq. 2 were sufficient to give an excellent x2 when used to fit the 1998 data! Figure 12 illustrates data and 5-parameter fit for 1999. By eye, the fit also looks excellent; however, the x2 is unacceptable because the CBO, pileup and muon losses have not been accounted for in this 20 times larger data set. The “pileup-fiee” spectrum is fit to the function f(t) = N ( t ) . C ( t ). A ( t ) (7) where N ( t ) is the ideal function (Eq. 2), C ( t )accounts for the CBO (Eq. 5), and A ( t ) characterizes the time-dependent muon lossses (Eq. 6 ) . With Eq. 7, or equivalent fitting strategies used by the other analysis teams, x2 is consistent with 1. Deviations from the ideal function of the type described are more significant at early times compared to late times. One commonly used test employed by the analysis teams is to fit the data progressively beginning at different “start” times. As shown in Fig. 13, w, only fluctuates within the expected band but is otherwise stable. The distribution of variances from neighboring fits is shown on the lower right panel of the figure. This plot has a mean consistent with zero and a normalized width consistent with 1, as expected. The bottom left panels of the figure give w, and the x2 by detector for the fixed start time of 30 p. These and other statistical tests yield satisfactory results.
11 S
0 .*
f
lo6
L.
a, P u)
c
lo5 104
1o3
lo2 10 1 XI
0
1000
2000
3000
4000
5000
o2
7000 Time (ns)
6000
Figure 11: A muon loss distribution from one detector (1999 data). Binning in this histogram is in units of the (g - 2) period, thus the normalization curve is an exponentid with lifetime of approximately 64 ps. The triples curve is proportional to the muon loss rate. After N 100 ps, the loss rate is essentially constant. At earlier times, the loss function is described by an exponential with a lifetime of approximately 27 p s . The doubles curve is similar but, with a lower coincidence requirement, contains background. The "proton" curve involves three FSD coincidences and significant energy in the calorimeter of the third station. This occurs from occasional hadronic showers. The triples and doubles curves also show proton events a t late times (= 700ps) when most muons have decayed.
I
Figure 12: The 1999 data and a simple 5-parameter fit. While the fit looks good, see the lower panels, the x2 is too high. A more complete fitting function, applied to a pileup-subtracted data set, yielded x2 consistent with 1.
12
-
229.0732
N
I
'
v
229.073
229.0728
229.0726 229.0724 229.0722 229.072
229.0718t"""""''""'""'""'""'""'""'' 0 5 10 15 20 25 30
E
-
40
u)
N
3"
35
Time (ps)
c
229.076
5
0
250
229.074 229.072
200
229.07 229.068
150 2 !L
a
4
8
6
10 12 14 16
18 20 22 24 Detector Number
1.1
100
2 N -
x
1.05
50
1
0 0
.
9
2
'
4
~
6
~
' " ~ ' ~ ~ ~ ' ~ ~ ~ ' ~ 8 10 12 14 16 18 20 22 24 Detector Number
~
~
"
~
~
'
"
~
"
~
~
Figure 13: A series of "consistency" plots from one of the wa analyses. The top plot shows the stability of the fit for ua with respect to the start time of the fit. The solid bands represent the allowed 10 deviation from the fist fit point based on the understanding that later fits are highly correlated with the starting fit. The plot on the bottom right is the deviation of fit results for adjacent start times. It should be centered at zero with a width of 1 for deviations which are purely statistical. The panels on the lower left illustrate the stability of the fit versus detector number and the x2 for each individual detector fit.
"
~
~
13 Item i;rp12n wa/2n Electric field correction Combined i;rp systematic uncertainties Combined wa systematic uncertainties Final experimental value Updated SM theory A(a,+ (exp - thy)
Value 61 791 256(25) Hz 229 072.8(3) Hz f0.81 ppm -
11659203(15) x 11 659 176.7(6.7) x 25(16) x lo-''
Relative Uncertainty 0.4 pprn 1.3 pprn 10% 0.4 ppm 0.3 ppm 1.3 ppm 0.57 ppm 1.6 u
Tabble 1: Results summary for the 1999 data analysis.
6 Results, Conclusions a n d Outlook
Table 1 summarizes the results. The experimental value from 1999 for positive muons is up+ = 11659 203(15) x (1.3 ppm. With the updated standard model theory value of .,(thy) = 11 659 176.7(6.7) x lo-'' (0.57 pprn), one finds Aa,+(exp - thy) = 25(16) x lo-''. The 2000 pel+ data is presently being analyzed and should reduce the uncertainty considerably. The 2001 ,udata is just beginning to be processed at the time of this writing. To complete the experiment, with equal positive and negative muon event samples, a final negative muon run will be necessary. The Collaboration is eager to complete this data taking phase and accompanying analysis in order to achieve our ultimate goal of 0.4 ppm precision or better in the measurement of the muon anomaly. Acknowledgments The (g - 2) experiment is supported in part by the U.S. Department of Energy, the U.S. National Science Foundation, the German Bundesminister Kr Bildung und Forschung, the Russian Minsitry of Science, and the US-Japan Agreement in High Energy Physics. The author would like to thank the organizers of the Workshop at Erice for an excellent meeting. References 1. The g-2 Collaboration for 1999: Boston U.: R.M. Carey, W. Earle, E. Efstathiadis,
2. 3. 4. 5.
E.S. Hazen, F. Krienen, I. Logashenko, J.P. Miller, J. Paley, 0. Rind, B.L. Roberts, L.R. Sulak, A. Trofimov; BNL: H.N. Brown, G. Bunce, G.T. Danby, R. Larsen, Y.Y. Lee, W. Meng, J. Mi, W.M. Morse, D. Nikas, C. Ozben, C. Pai, R. Prigl, Y.K. Semertzidis, D. Warburton; Cornell U.: Y. Orlov; Fairfield U.: D. Winn; U. Heidelberg: A. Grossmann, K. Jungmann, G. zu Putlitz; U. Illinois: P.T. Debevec, W. Deninger, F.E. Gray, D.W. Hertzog, C.J.G. Onderwater, C. Polly, S. Sedykh, M. Sossong, D. Urner; Max Planck Heidelberg: U. Haeberlen; KEK: A. Yamamoto; U. Minnesota: P. Cushman, L. Duong, S. Giron, J. Kindem, I. Kronkvist, R. McNabb, C. Timmermans, D. Zimmerman; Budker I. Novosibirsk: V.P. Druzhinin, G.V. Fedotovich, B.I. Khazin, N.M. Ryskulov, Yu.M. Shatunov, E. Solodov; Tokyo I. Tech.: M. Iwasaki, M. Kawamura; Yale U.: H. Deng, S.K. Dhawan, F.J.M. Farley, V.W. Hughes, D. Kawall, M. Grosse Perdekamp, J. Pretz, S.I. Redin, E. Sichtermann, A. Steinmetz. A. Czarnecki and W.J. Marciano, Nucl. Phys. B (Proc. Suppl.) 76, 245 (1999). R.M. Carey et al., Phys. Rev. Lett. 82, 1632 (1999). H.N. Brown et al., Muon (g - 2) Collaboration, Phys. Rev. D 62, 091101 (2000). H.N. Brown et al, Phys. Rev. Lett. 86, 2227 (2001).
14
6. M. Knecht and A. Nyffeler (2001), hep-ph/0111058 and M. Knecht, A. Nyffeler, M. Perrottet, and E. de Rafael (2001), hep-ph/0111059. 7. M. Hayakawa and T. Kinoshita, Phys. Rev. D 57,465 (1998). 8. J. Bijnens, E. Pallante, and J. Prades, Phys. Rev. Lett. 75,1447 (1995). 9. M. Hayakawa and T. Kinoshita,hep-ph/Ol12102. 10. J. Bijnens, E. Pallante, and J. Prades, hep-ph/0112255; note, the sign error here enters &om a completely different origin. 11. Simon Eidelman, these proceedings. 12. Andrew Hocker, Hadronic Contribution to (g-2),, hepph/0111243 and these proceedings. 13. Kirill Melnikov, Int. J. Mod. Phys. A 16,4591 (2001). 14. M. Davier and A. Hocker, Phys. Lett. B 435,427 (1998). 15. S. Narison, Phys. Lett. B 513,53 (ZOOl), erratum: ibid. B526, 414 (2002). 16. J.F. de Troc6niz and F.J. YndurAin, hep-ph/0106025. 17. William J. Marciano, these proceedings. 18. See, for example, the SPIRES citation search for Brown e t al., (2001) above for hundreds of interesting examples of standard model extensions. 19. A written review by Francis Farley can be found in F.J.M. Farley and E. Picasso. Adv. Ser. Direct. High Energy Phys., 7 479 (1990). 20. D.E. Groom et al., Review of Particle Physics, Eur. Phys. J. C 15,1 (2000). 21. G.T. Danby et al., Nucl. Inst. and Meth. A 457,151 (2001). 22. F. Krienen, D. Loomba, and W. Meng, Nucl. Inst. and Meth. A 283,5 (1989). 23. W.D. Phillips et al., Metrologia 13,81 (1977); X. Fei, V.W. Hughes and R. Prigl, Nucl. Inst. and Meth. A 394,349 (1997). 24. R. Prigl, U. Haeberlen, K. Jungmann, G. zu Putlitz and P. von Walter, Nucl. Inst. and Meth. A 374,118 (1996). 25. J. Bailey, et al., Nucl. Phys. B 150, 1 (1979). 26. S. Sedykh, et al., Nucl. Inst. and Meth. A 455,346 (2000).
HADRONIC CONTRIBUTION TO (9 - 2)@ Andreas Hacker Laboratoire de I'Acce'le'rateur Line'aire, IN2P3-CNRS et Universite' de Paris-Sud, BP 34, F-91898 Orsay Cedex, France The precise measurement of the muon magnetic anomaly (g - 2), at BNL constitutes a most sensitive probe of the electroweak sector of the Standard Model, provided the contribution from hadronic vacuum polarization is well enough understood. This talk summarizes the development in the evaluation of the leading order hadronic contributions. Significant improvement has been achieved in a series of analyses which is presented historically in three steps: (l), use of T spectral functions in addition to ete- cross sections, (2), extended use of perturbative QCD and (3), application of QCD sum rule techniques. The uncertainties, in particular concerning the CVC hypothesis used in step (l), and global quark-hadron duality employed in steps (2) and (3) are discussed. No new analysis results are given in these proceedings, but our previous number for the total Standard Model prediction is updated with the new contribution from hadronic light-by-light scattering.
1
Introduction
Precision measurements of electroweak observables provide powerful tests of the Standard Model. In the last 10 years significant progress has been achieved in this direction owing to the accurate and complete results from the LEP, SLC and TEVATRON colliders. These measurements yielded for the first time unique information from vacuum polarization effects in weak boson propagators which allowed the mass of the Higgs boson t o be significantly bounded. At the other end of the energy scale, the muon magnetic moment can now be measured with a precision such that new physics can be probed, provided all the contributions from the Standard Model can be under control. In this talk, I shall discuss the new precise result on (g - 2), from the BNL experiment which, when published in early 2001, exposed a significant deviation from the Standard Model prediction: this exciting situation prompted many physicists t o critically examine the status of the theoretical prediction, in particular its most delicate contribution from hadronic vacuum polarization. A mistake has been found recently in the calculation of the higher-order hadronic light-by-light (LBL) scattering diagram 2,3. Its correction turns out t o flip the (originally negative) sign of the contribution t o a,, thus greatly reducing the observed discrepancy between experiment and theory. This review will put its emphasis on the lowest order hadronic vacuum polarization which provides the dominant theoretical uncertainty. It turns out that the same physics plays an important role in the analysis of high-energy neutralcurrent data through the running of the electromagnetic coupling from q2 = 0 to Id$,relevant for limits on the Higgs mass.
15
16
Figure 1: Feynman diagrams corresponding to specific contributions to a p : first-order hadronic vacuum polarization, hadronic LBL scattering, fist-order weak interaction and possible supersymmetric contributions.
2
The muonic (g - 2)
The muon magnetic anomaly a, receives contributions from all sectors of the Standard Model,
the dominant diagrams of which are depicted in Fig. 1. The pure &ED contribution, a?ED = 116584705.7(2.9) x lo-”, has been calculated t o fourth order which represents a tour de force, only performed by one group4 (some slight change occurred recently due to computer precision problems in the original calculation ’). The fifth order term has been estimated only, but was found t o be small6. The weak contribution, u F k = 152(4) x lo-”, is known t o two-loops7. Large logarithms of ln(Mw/mf) occur, but can be resummed 8 , leading to a robust prediction. The contribution from hadrons stems mainly from vacuum polarization and will be covered in the next section. Its absolute size N 6800(160) (ca. 1995) is such that it must be known to a precision better than 1%if the experiment is to probe the level of the weak part. As it is well known, the first order correction from hadronic vacuum polarization (Fig. 1) cannot be calculated from ‘‘first principles” since most contributions arise from low-mass states, where quark confinement leads to resonances. Fortunately, the result can be expressed as a dispersion integral involving the total cross section for e+e- annihilation into hadrons, or alternatively its ratio R ( s ) to the point-like cross section,
with K ( s ) m,,/s, thus giving a large weight t o the small s region. An analog integral occurs for the running of a ( Q 2 ) where , K ( s ) = ( s - Q2)-l. A small part (N 1%) of the hadronic contribution originates from the so-called LBL scattering (see Fig. 1). These diagrams cannot be treated analogously and must be estimated through specific models for the hadron blob. As a consequence the result is not known accurately and less reliable. The effect induced by the recently reported sign flip of its contribution will be briefly discussed at the end of these proceedings. Table 2 presents estimates of the three contributions to a,, in circa 1995. Obviously, the hadronic piece must be known more accurately, by a factor of at least three, before a precise measurement can witness the effect of the weak interaction or a new physics contribution of similar magnitude, such as Supersymmetry. This is the motivation for an increased effort in the last few years t o improve the reliability and the accuracy of the hadronic contribution. N
The experimental progress on a, is chartered in Fig. 2, together with the levels of the different contributions expected in the Standard Model. While the successive CERN experiments reach enough sensitivity to uncover the expected effect of hadrons, the program underway at
17
lo-'
10
10'
lo3
Contribution to acl (x 10.'')
Figure 2: Experimental milestones on the precision of a,, and the levels of the different contributions and their present uncertainties (depicted by arrows) expected in the Standard Model.
BNL (E821) is now at the level of the weak contributions and reaches for a four times smaller sensitivity, thus demanding a corresponding improvement in the accuracy of the hadronic piece.
3
The precise BNL result
The new value recently announced by E821 lo has a precision three times higher than the previous combined CERN and BNL results"~12, up+ = 11 659 202(16) x 10-l'
.
(3)
The quoted uncertainty is dominated by statistics in muon decay counting and the major systematic errors are estimated to 3.5 x 10-l' for the precession frequency and 4.5 x for the magnetic field (NMR frequency). The E821 experimenters compare their result to the expected SM value with the hadronic contribution from vacuum polarization taken from Ref. 13, a,+ (SM) = 11 659 159.6(6.7) x
.
(4)
Averaging (3) with previous measurements yields a,(exp) - up(SM) = 43(16) x 10-l'
,
(5)
Table 1: The QED, weak and hadronic contributions to a, circa 1995. The two errors for the hadronic part correspond to vacuum polarisation and LBL scattering, respectively.
Source QED 2,W exchange Quarks and Hadrons
10l1 x a, 116584705.6 151 6800
-
10l1 x .(up) 2.9 -4 (150 3 ' 40)
-
References 4
73 9
18
where the error is dominated by the statistical experimental error (theoretical systematic errors have been added in quadrature). Using the new estimate on the hadronic LBL contribution I , a p [ L B L S ] = ($8.3 f 1.2) x 10-l’ we can rewrite Eq. (4)as
a,+ (SM) = 11 659 176.4(6.4) x 10-l’
,
(6)
,
(7)
so that the 2.6 u discrepancy. of Eq. ( 5 ) reduces to 1.6 u:
a,(exp) - a,(SM) = 26(16) x 10-l’ which is not significant.
4
New physics and (g - 2),
There exist a large number of possible contributions from new physics to a,, witnessed by the paper fiood since the publication of the BNL result (see, e.g., Refs.14J5): Supersymmetry, muon substructure, anomalous electroweak couplings, leptoquarks, lepton flavour violation, etc. The most fashionable direction is Supersymmetry and indeed contributions t o a, have been considered since the early days of SUSY phenomenology The minimal SUSY contributions are depicted in Fig. 1: they involve loops with a smuon and neutralinos, or a muon sneutrino and charginos. Introducing a generic SUSY mass scale Msusy, the contribution to a, is2’ 16,17718t19.
where t a n p is the ratio of the vacuum expectation values of the two Higgs fields. Figure 3 shows the Msusy versus t a n @ regions favored by Eq. (8) according to the “discrepancies” (5) and (7). We learn that an observed effect would have corresponded to a SUSY parameter space not excluded by LEP. More interestingly, it would correspond to masses “just around the corner” that can be directly investigated by planned accelerators.
5
H a d r o n i c v a c u u m polarization for (g - 2), - improvements in three steps
Since 1995 several improvements have been applied to the calculations of hadronic vacuum polarization in order to cope with incomplete or unprecise e+e- data. Although QCD predictions were always used a t higher energies (> 40 GeV), it became clear that reliable predictions could be made at much lower values. Let me identify the following three steps: (1) Addition of precise
7
data using CVC (see, e.g.,21.22)
(2) QCD predictions at lower energies (see, e.g.,
23,24125*26
1
(3) Constraints from QCD sum rules (see, e,g.,27y13*28) OThe interpretation of the discrepancy in terms of standard deviations is approximately valid here, since the error is dominated by experimental uncertainties from the up measurement and from the hadronic contribution, where Gaussian Bayesian priors have been used to account for the systematic experimental errors. This is common practise, not to be mixed up with a treatment of theoretical parameters which are not statistically distributed quantities, but whose uncertainties are not of dominance here.
19 1000
~
600
I 0
20
40
60
80
100
tan
Figure 3: Allowed regions for the SUSY parameter space according to the approximation (8) and using the results from Eqs. (5) and (7) as constraints.
(1)A d d i n g precise r data u n d e r CVC
The Conserved Vector Current (CVC) hypothesis expresses invariance under SU(2) of the electroweak currents. For the problem a t hand it relates the isovector vector electromagnetic and the weak hadronic currents, as occurring in e+e- annihilation and T decays. From the point of view of strong interactions this corresponds t o a factorization of the hadronic physics: hadrons (quark pairs) are created from the QCD vacuum and the probability to produce hadrons with well-defined quantum numbers a t a given mass is expressed through spectral functions. At low energy we expect spectral functions to be dominated by resonances, while QCD should provide a good description at sufficiently high energies. The corresponding energy scale must be determined from experiment. The I = 1 vector spectral function ~ ( sfor ) the two-pion channel is related t o the corresponding e+e- cross section and T branching ratio and invariant mass spectrum:
Hadronic r decays represent a clean environment to study hadron dynamics which is in many ways complementary to e+e- annihilation: The I = 1 vector spectral functions are related through CVC, while the I = 1 axial-vector r spectral function has no counterpart in the efechannel and can therefore provide additional information. It is interesting to note that r and e t e - spectral functions are experimentally determined with very different systematics, offering a way to cross check the results in the I = 1 vector sector: r data have excellent absolute normalization, because the relevant branching ratios have been measured at LEP with high statistics, large acceptance and small non-r b a c k g r o ~ n d ~ ~ . On the other hand, the shape of the spectral functions is subject to bin-to-bin corrections from resolution effects and handling of fake photons produced by hadron interactions in the electromagnetic calorimeter. Therefore, the determination of the spectral function requires an unfolding procedure which has the effect of strongly correlating the errors of adjacent bins.
20
-
2
’’
naive pakon model prediction
ALEPH -
T- -) (A-, I=1) v,
----
naive parton model prediction
m (con. ), urno (con.), KK‘ (MC) =-bar lin (MC) -
0
05
1
15
2
25
3
35
Mass’ (GeV/c’)’
0
05
1
15
2
25
3
35
Mass’ (GeV/c’)’
Figure 4: The ALEPH I = 1 vector and axial-vector spectral functions determined from 7 decays. The contributions from the different h a 1 states are indicated by the shaded areas. While strong resonance ( p , al) production dominates at low energy, the high energy behaviour is approaching the level of the quark-pair contribution.
0
efe- data have just about the opposite behaviour: the point-to-point normalization is excellent, whereas the systematic uncertainties are highly correlated among the measurements. The overall normalization is a delicate issue, because of radiative corrections and systematic errors from acceptance and luminosity.
The vector and axial-vector spectral functions have been measured at LEP by ALEPH30v31 and OPAL 32. Detailed QCD studies have been performed by both collaborations. The ALEPH results, given in Fig. 4, shows the contributions from the different final states. In the energy range made available by T decays the spectral functions are dominated by resonances, p and al in the vector and axial-vector parts, respectively. The asymptotic QCD regime is approached for the highest energies, but not yet reached. Both ALEPH and OPAL spectral functions are in good agreement with each other, but it should be noticed that their normalizations are strongly correlated as OPAL uses world-average branching ratios where the ALEPH weight is large in general.
S U ( 2 ) breaking If the T data is t o be used in the vacuum polarization calculations, that is we identify ~ J , * ~ O ( S ) with u‘+,-, (s),it is mandatory to consider in detail the amount of CVC ~ i o l a t i o n ~ Isospin ~,~~,~~. breaking is expected mainly from electromagnetic effects and it has t o be corrected for the calculation of the integral (2). The sources considered in the analysis and their quantitative effects on u F d are given in Table 2. The dominant contribution comes from short distance electroweak radiative corrections to the effective four-fermion coupling T - -+ (clc)- v,. It can be absorbed into an overall multiplicative electroweak correction SEW= 1.0194 35,36, while remaining perturbative electroweak corrections are of order a”(m,) lnn(Mz/m,) 0.3” which is safe t o ignore. The sub-leading non-logarithmic short distance correction, calculated to order O ( a ) at quark level 36, 5 a ( r n , ) / 1 2 ~ 2: 0.0010, is also small. Additional intermediate-distance corrections have been computed only for T - -+T - v, and the overall correction was found to be dominated by the leading logarithm from the short distance c ~ n t r i b u t i o n ~The ~ . electromagnetic A’ - ?yo mass splitting affects the measured cross section through phase space corrections.
21 Table 2:
Corrections for isospin violation applied to
T-
+ r-rovT.
Source
A a p ( x 10-l')
Radiative corrections to T decays SEW= 1.0194 f0.0040 35'36
-9.6
(weak final state dependence expected, verified for
T-
+ r - v 7 )37
Pion velocity: p- # Po (due to EM n--r0 mass splitting: affects cross section) Form Factor 21933: o p w interference o n--xo mass splitting (affects r,) o p--po mass splitting (affects r,) o EM p decays: ny, q y , eft-, nny (affect r,) o second class currents e.g., T - + r-qv. a (m, - md)' 10-~
-
Total correction
2.0
-7.1
+3.7 f 0.6 +3.3 0 f 0.2 -0.2 f 1.2
negligible -9.9 f 2.4
Electromagnetic corrections also affect the pion form factor, in particular the width of the p resonance(s): the p - w mixing, not present in T decays; the T* - A' and p* - po mass splitting; electromagnetic decays. The occurrence of second class currents is expected t o be proportional t o the mass splitting-squared of the light u, d quarks which is negligible. We observe that most of the effects cancel, so that the net correction applied corresponds to approximately the pure short-distance radiative correction SEW.It is important t o further investigate isospin-violating contributions t o keep the increasing precision of the T data exploitable for the purpose discussed here. The use of T data improves the precision on the evaluation of ubd by a factor of 1.621. (2) Replacing poor data by QCD prediction
The data driven analysis ' 3 shows that to improve the precision on the dispersion integral, a more accurate determination of the hadronic cross section between 2 GeV and 10 GeV is needed, where some poorly measured and sparse data points dominate the final error. Indeed, QCD analyses using T spectral function^^^-^^ revealed the excellent applicability of the Operator Product Ezpansion (OPE) at the scale of the T mass, m, P 1.8 GeV, and below. The O P E organizes perturbative and nonperturbative contributions t o a physical observable through the concept of global quark-hadron duality. Using moments of spectral functions, dimensional nonperturbative operators contributing t o the T hadronic width have been determined experimentally and found t o be small. The evolution to lower energy scales proved (to some surprise) the validity of the OPE down to about 1.1GeV. An analog analysis based on spectral moments of e+e- cross section measurements has been performed in Ref.24 (and more recently in Ref.40 - yielding compatible results). The theoretical prediction of these moments and of the total hadronic cross section in e+e- annihilation, R(so), at a given energy-squared, S O , involves the Adler D-function41, related t o the former via 38139
(see also Ref.43 for a Massless perturbative QCD predictions of D are available42to order heroic effort to go beyond this). The O P E of D also includes second order quark mass corrections far from the production threshold44 and the first order dimension D = 4 nonperturbative term
22
d
CrystalB 0
o exclusive data , , , , 1.5 2 2.5 3 ,
0.5
1
I
,
4s
+MARKr;
BbS
am ,
,
,
I
3.5
I
APLUTO
, , , ,
, , , , 4
4.5
,
/
,
,
5
(GeV)
Figure 5: The inclusive hadronic cross section ratio in ete- annihilation versus the c.m. energy 4. Shown by the cross-hatched band is the QCD prediction of the continuum contribution. The exclusive e+e- cross section measurements at low c.m. energies are taken from DM1, DM2, M2N, M3N, OLYA, CMD, ND and T data from ALEPH (seez1 for references and more detailed information).
involving the gluon condensate, ((os/7r)GG),and the quark condensates, ( m f f i q j ) ,for the quark flavours f (see, e.g., Ref. 45). The complete dimension D = 6 and D = 8 operators are parameterized phenomenologically using the saturated vacuum expectation values ( 0 6 ) and ( 0 8 ) , respectively. The nonperturbative operators (with the exception of the quark condensates, which are obtained from PCAC relations) are determined experimentally by means of a combined fit of the theoretical moments to data. It results in a very small contribution from the OPE power terms t o the lowest moment a t the scale of 1.8 GeV (repeated and confirmed at 2.1 GeV). This is in agreement with the findings from the T analyses. The calculation accounts for theoretical uncertainties, including the dependence on the choice of the renormalization scheme and scale, the uncertainty on the strong coupling, the missing r ) ~the ambiguity between contour-improved and fixed-order perturbation theory term ( ~ ~ / 7 and (see Refs. z4,31). Additional sources are the uncertainties on the running quark masses and on the nonperturbative contributions. In spite of the implicit assumption of local duality for the theoretical prediction of R , the evaluation of the dispersion integral (2) turns the duality globally, ie., remaining nonperturbative resonance oscillations are averaged over the integrated energy s p e c t r u d . The available data points together with the theoretical prediction (crossed hatched band) are shown in Fig. 5 . Good agreement is found between theory and the newest BES measurement^^^, while older data are significantly higher. The preceeding discussion justifies the application of QCD predictions for R between 1.8 GeV and the 00 production threshold, as well as from 5 GeV up to infinityz4. This yields a factor and a factor of 2.4 better accuracy on Aoyh.d(M2). of 1.3 improvement on the precision of Similar precise analyses are performed in Ref. 2 5 (applying a renormalization of experimental data on the cC resonances using QCD predictions of the continuum and assuming systematics t o be correlated) and Ref.z6 (see also Ref.47 for a recent theory driven analysis).
upd
*A systematic uncertainty is introduced through the cut at explicitly 1.8 GeV so that non-vanishing oscillations may give rise to a bias after integration. The associated (small) systematic error is estimated by means of fitting different oscillating curves to the data around the cut regionz4 .
23 Table 3: Contributions to apdfrom the different energy regions. The subscripts in the fist column give the quark flavours involved in the calculation. Energy (GeV)
(2mr - 1.8)uds (1.8 - 3.700)uds +(IS,25,3770)c (5 g.3)udsc (9.3 12)udscb
a y x 1010
+ (3.7
634.3 & 5.6,,, & 2.l:iL0 33.87 0.46theo 14-31& 0.50exp h 0.21theo 6.87& 0.11theo 1.21 & 0.05,heo 1.80 f 0.01theo
*
-
5)4sc
~
(12 - m ) u d s c b (2mt - m)t
FSO
692.4
(2mr - m ) u d s c b t
+ 5.6exp & 2.6theo
* The theoretical error accounts for uncertainties concerning the QCD prediction only. Due to the uncertainties from CVC and radiative corrections correlated average procedure applied in Ref. are folded into the systematic part of the experimental error.
*’,
(3) Improving data with QCD sum rules It was shown in Refs. 27,13 that the previous determinations can be further improved by using finite-energy QCD sum rule techniques in order t o access theoretically energy regions where perturbative QCD fails locally. In principle, the method uses no additional assumptions beyond those applied in the previous section. The idea is to reduce the data contribution to the dispersion integrals by subtracting analytical functions from the singular integration kernel in Eq. (2), and adding the subtracted part subsequently by using theory only. Two approaches have been applied in Ref.13: first, a method based on spectral moments is defined by the identity
with P,(s) = J,”dtp,(t). The regular functions pn(s) approximate the kernel K ( s ) / s in order t o reduce the contribution of the non-analytic first integral in Eq. (12), which is evaluated using experimental data. The second integral in Eq. (12) can be calculated theoretically in the framework of the OPE. The functions p,(s) are chosen in order to reduce the uncertainty of the data integral which is not necessarily equivalent to a reduction of its contribution. A second approach l3 involving local quark-hadron duality uses the dispersion relation of the Adler D-function
I
for space-like Q2 = -4’ and quark flavours f, to approximate the integration kernel. The theoretical errors of both approaches are evaluated in close analogy to the analysis presented in the previous Section. The improvement in accuracy on the dispersion integrals obtained from these constraints is weak for a p d but valuable for Aayhad(hfi).
6
Results
Table 3 shows the experimental and theoretical evaluations of a p d for the distinguished energy regions. Experimental errors between different lines are assumed to be uncorrelated, whereas theoretical errors, but those from the CC and bb thresholds which are quark mass dominated, are
24
Barkov et al. '85 Kinoshita et al. '85 Casas et al. '85 Eidelman, Jegerlehner '95 Adel, Yndurain '95 Brown, Worstell '96 Alemany, D.-H. '97 Davier, Hocker '97 Davier, Hocker '98 Jegerlehner '00 Narison '01 De Troconiz, Yndurain '01
Figure 6: Comparison of lowest order ":a evaluations and the experimental result subtracting the QED, weak and higher order hadronic contributions. Shown are the corrected LBL scattering calculation (shaded region) and the uncorrected number (dashed region). The theoretical values are taken from Refs. 49,9,21,24,13,22,47.
added linearly. The combination of the theoretical and experimental evaluations of the dispersion relation yields the final result (see Table 3 for a note concerning the theoretical error): a y [ ( Q / K ) q = (692.4f 5.6,,,
u;"
& 2.6theo) x 10-l'
,
= (11 659 176.43~5.6,,, f 3.Othe,) x 10-l' ,
dominated by the contribution from the p(770) resonance. The total u:" value contains the ) ~ (-10.0f ] contributions from non-leading order hadronic vacuum p o l a r i ~ a t i o n ~u ~; ~~ [~ (' ~ / K = 0.6) x lo-'', and from hadronic LBL scattering for which the new result from Ref.' is used. 7
Conclusions and perspectives
Much effort has been undertaken during the last years t o ameliorate the theoretical predictions on u k d . The currently most precise value obtained for the hadronic contribution is l 3 u p d = (692.4 f 6.2) x 10-lo. Figure 6 gives a chronological compilation of published results. Fairly good agreement is observed among the newest evaluations, so that the discrepancy between the BNL value and the Standard Model varies between clean compatibility and 1.6a, using the new evaluation of the hadronic LBL scattering contribution '. To maintain the sensitivity on interesting physics of the experimental improvements to be expected from BNL, more theoretical requires further studies of the following effort is needed. In particular, a better precision on items.
upd
0
0
Radiative corrections in efe-
annihilation d a t a
SU(2) breaking: let me recall that the T data not only provide precise and in many ways complementary cross section measurements, but they also constitute a powerful cross check. The current u p d evaluation being wrong would require not only the efe- data to have unaccounted systematics, but also that CVC violation is much larger than expected, since the efe- and T data are mutually (fairly) compatible.
25 0
More experimental information. In particular, complementary e+e- measurements from, e.g., new precision experiments, or analyses of radiative events using data from existing e+e- factories.
It is obvious that using the T data and CVC represents only an auxiliary remedy to cope with the lack of precision in the e+e- measurements. Certainly, the preferable scenario would be t o improve the latter well below the 1%accuracy so that one does not need t o include the former. Moreover in such a situation precise tests of CVC could be performed, providing insight into the interesting physics of possible violations. Acknowledgements I am indebted to Jacqueline Mooney, Sebastian White and Bill Marciano for organizing such a pleasant workshop with many interesting discussions around this hot topic (which was even “hotter” at the time of the workshop) and all the other fundamental questions of contemporary particle physics raised by the speakers and the audience. I gratefully acknowledge the very fruitful and pleasant collaboration with Michel Davier.
References 1. M. Knecht, A. Nyffeler, “Hadronic light-by-light corrections to the muon g-2: the pion pole contribution”, CPT-2001-P-4253, [hep-ph/0111058], November 2001 2. M. Hayakawa, T. Kinoshita, Phys. Rev. D57 (1998) 465 3. J. Bijnens, E. Pallante, J . Prades, Nucl. Phys. B474 (1996) 379 4. T. Kinoshita, B. Nizic, Y. Okamoto, Phys. Rev. D41 (1990) 593; Phys. Rev. Lett. 52 (1984) 717 5. B. Lee Roberts, private communication to M. Davier, Spring 2001 6. P.J. Mohr, B.N. Taylor, Rev. Mod. Phys. 72 (2000) 351 7. A. Czarnecki, B. Krause, W.J. Marciano, Phys. Rev. Lett. 76 (1996) 3267; Phys. Rev. D52 (1995) 2619 8. S. Peris, M. Perrottet, E. de Rafael, Phys. Lett. B355 (1995) 523 9. S. Eidelman, F. Jegerlehner, 2. Phys. C67 (1995) 585 10. Muon ( g - 2) Collaboration (H.N. Brown et al.), Phys. Rev. Lett. 86 (2001) 2227 11. CEFZN-Mainz-Daresbury Collaboration (J. Bailey et al.), Nucl. Phys. B150 (1979) 1 12. Muon ( g - 2) Collaboration (H.N. Brown et a[.), Phys. Rev. D62 (2000) 091101; Phys. Rev. Lett. 82 (1999) 1632; 13. M. Davier, A. Hocker, Phys. Lett. B435 (1998) 427 14. A. Czarnecki, W.J. Marciano, Phys. Rev. D64 (2001) 013014 15. See the entries listed a t http://phypprol.phy.bnl.gov/g2muon/new~theory.html 16. P. Fayet, ‘‘Unification of the Fundamental Particle Interactions”, eds. Ferrara, J. Ellis and P. Nieuwenhuizen (Plenum, New York, 1980), p.587 17. J.A. Grifols and A. Mendez, Phys. Rev. D26 (1982) 1809 18. J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B116 (1982) 283 19. R. Barbieri and L. Maiani, Phys. Lett. B117 (1982) 203 20. T. Moroi, Phys. Rev. D53 (1996) 6565 21. R. Alemany, M. Davier, A. Hocker, EUT. Phys. J. C2 (1998) 123 22. S. Narison, Phys. Lett. B513 (2001) 53 23. A.D. Martin, D. Zeppenfeld, Phys. Lett. B345 (1995) 558; A.D. Martin, J. Outhwaite, M.G. Ryskin, EUT.Phys. J. C19 (2001) 681 24. M. Davier, A. Hocker, Phys. Lett. B419 (1998) 419
26 25. 26. 27. 28. 29.
30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.
J.H. Kiihn, M. Steinhauser, Phys. Lett. B437 (1998) 425 J . Erler, Phys. Rev. D59 (1999) 054008 S. Groote, J.G. Korner, N.F. Nasrallah, K. Schilcher, Phys. Lett. B440 (1998) 375 G. Cvetic, T. Lee, I. Schmidt, USM-TH-108, hepph/0107069 ALEPH Collaboration (D. Buskulic et d), 2. Phys. C70 (1996) 579; OPAL Collaboration (K. Ackerstaff et ul.), EUT. Phys. J . C4 (1998) 193; L3 Collaboration (M. Acciarri et d), Phys. Lett. B345 (1995) 93 ALEPH Collaboration (R. Barate et ul.),2. Phys. C76 (1997) 15 ALEPH Collaboration (R. Barate et ul.), EUT. Phys. J. C4 (1998) 409 OPAL Collaboration (K. Ackerstaff et d), EUT. Phys. J . C7 (1999) 571 H. Czyz, J.H. Kiihn, EUT. Phys. J., C18 (2001) 497-509 V. Cirigliano, G. Ecker, H. Neufeld, Phys. Lett. B513 (2001) 361 W. Marciano, A. Sirlin, Phys. Rev. Lett. 61 (1988) 1815; Phys. Rev. Lett. 61 (1986) 1815 E. Braaten, C.S. Li, Phys. Rev. D42 (1990) 3888 R. Decker, M. Finkemeier, Nucl. Phys. B438 (1995) 17 K.G. Wilson, Phys. Rev. 179 (1969) 1499 M.A. Shifman, A.L. Vainshtein, V.I. Zakharov, Nucl. Phys. B147 (1979) 385, 448, 519 S. Menke, SLAC-PUB-8851, eConf C010430:M09, hep-ex/0106011 S. Adler, Phys. Rev. D10 (1974) 3714 L.R. Surguladze, M.A. Samuel, Phys. Rev. Lett. 66 (1991) 560; S.G. Gorishny, A.L. Kataev, S.A. Larin, I1 Nuo. Cim. 92A (1986) 119; Phys. Lett. B259 (1991) 144 P.A. Baikov, K.G. Chetyrkin, J.H. Kuhn, TTP-01-19, hep-ph/0108197 K.G. Chetyrkin, J.H. Kiihn, M. Steinhauser, Nucl. Phys. B482 (1996) 213 E. Braaten, S. Narison, A. Pich, Nucl. Phys. B373 (1992) 581 BES Collaboration (J.Z. Bai et ul,), hep-ex/0102003; Phys. Rev. Lett. 84 (2000) 594 J.F. De Troconiz, F.J. Yndurain, FTUAM-01-08, hepph/0106025; FTUAM-01-15, hepph/0107318 B. Krause, Phys. Lett. B390 (1997) 392 OLYA, CMD Collaborations (L.M. Barkov et ul.), Nucl. Phys. B256 (1985) 365; T. Kinoshita, B. NiiiC, Y. Okamoto, Phys. Rev. D31 (1985) 2108; J.A. Casas, C. Lbpez, F.J. Yndurbin, Phys. Rev. D32 (1985) 736; K. Adel, F.J. Yndurain, FTUAM-95-32, hep-ph/9509378 (1995); D.H. Brown, W.A. Worstell, Phys. Rev. D54 (1996) 3237; F. Jegerlehner, DESY-01-028, hepph/0104304 (2001)
HIGH PRECISION STUDIES OF LOW ENERGY e+e- ANNIHILATION INTO HADRONS S.I. EIDELMAN Budker Institute of Nuclear Physics, 11 Lavrentyev Ave., Novosibirsk 630090. Russia New results on the low energy e+e- annihilation into hadrons from Novosibirsk and Beijig are described. The validity of the CVC relation between e'e- and 7 decays is considered. Implications of the new measurements for the evaluation of the hadronic contribution to the muon anomalous magnetic moment are discussed.
1
Introduction
Measurement of the total cross section of e+e- annihilation into hadrons is one of the most important suppliers of the experimental information on the quark interactions. At high energy it serves as a test of perturbative QCD whereas at low enegies it provides insight into nonperturbative effects in QCD as well as valuable input to various phenomenological models describing strongly interacting particles. It became conventional to use the following dimensionless quantity characterizing the total cross section of e+e- + hadrons:
R = o(e+e-
+ hadrons)/a(e+e-+ p'p-).
(1)
R(s) is widely used for various calculations, e.g. in sum rules relating the integral of R ( s ) to fundamental physical constants or quantities determined independently. Particularly, knowledge of R ( s ) with high accuracy is required for the evaluation of a y , the hadronic contribution to the anomalous magnetic moment of the muon (g-2)/2 (see' and references therein) or the value of the fine structure constant at the Z boson mass. The ultimate goal of the experiment E8212 running in Brookhaven National Laboratory is to measure the muon anomalous magnetic moment with a relative precision of 0.35 ppm. Within the Standard Model (SM) the uncertainty of the theoretical value of the leading order a,, is dominated by the uncertainty of the hadronic contribution afEad calculated via the dispersion integral
where the QED kernel k ( s ) is a smooth function of energy varying from 0.63 at s = 4m; to 1 atstoo. The precision of the ahadcalculation depends on the approach used and varies from 1.34 ppm based on e+e- data only to 0.53 ppm if in addition 7-lepton decay data as well as perturbative
c
QCD and QCD sum rules are extensively used4. As it is clear from Eq. 2, the major contribution
27
28
to its uncertainty comes from the systematic error of the R ( s ) measurement at low energies (s < 2 GeV2), which, in turn, is dominated by the systematic error of the measured cross section e+e- -+ T+T- or pion form factor F, directly related to it. Assuming conservation of the vector current (CVC) and isospin symmetry, the spectral function of the T - + T-A'V, decay can be related to the isovector part of the pion form The detailed measurement of the spectral functions factor measured in e+e- annihilation was provided by ALEPH 6, OPAL and CLEO-11'. The comparison of the pion form factor measured at e+e- colliders with the spectral function of the T - + ~ - n ' v , decay provides a test of CVC. If CVC holds with high accuracy, T-lepton decay data can be also used to improve the accuracy of the calculations mentioned above ',lo. E821 has recently published results of the new measurement of ap in which an accuracy of 1.3 ppm was achieved. The measured value of a,, is 2.6 standard deviations higher than the SM prediction of4 '. This observation makes new high precision measurements of the cross section of e+e- + hadrms and particularly of the pion form factor extremely important.
'.
2
New Results from e+e- Colliders
2.1
Experiments at VEPP-2M
Since 1974 the e+e- collider VEPP-2M has been successfully running in the Budker Institute of Nuclear Physics in Novosibirsk in the c.m.energy range from the threshold of hadron production to 1400 MeV with a maximum luminosity of 3.103' cm-2s-1 at the q5 meson energyI3. In the last series of experiments two detectors (CMD-2 and SND) installed at VEPP-2M collected about 30 pb-' each. CMD-2 described in detail elsewhere l4 is a general purpose detector. Inside a superconducting solenoid with a field of 1 T there are a drift chamber, a proportional Z-chamber and an endcap BGO calorimeter. Outside there is a barrel CsI calorimeter and muon streamer tube chambers. The main goal of CMD-2 is to perform a high precision measurement of the exclusive cross sections of various hadronic channels and determine parameters of the low lying vector mesons. SND described in detail elsewhere l5 is a nonmagnetic detector with drift chambers for tracking and a three layer electromagnetic NaI calorimeter. Outside it there are a muon streamer tube chamber and plastic scintillators. The main goal of SND is to study p, w and q5 decays as well as the main hadronic channels. Both experiments possess some special features making high precision measurements feasible: 0
0
0
0
0
large data samples due to the high integrated luminosity and large acceptance multiple scans of the same energy ranges to avoid possible systematic effects; the step was 10 MeV for the continuum region and 2 MeV near the w and peaks absolute calibration of the beam energy using the resonance depolarization method l6 reduces to a negligible level a systematic error caused by an uncertainty in the energy measurement which can be significant for cross sections with strong energy dependence good space and energy resolution lead to small background redundancy - unstable particles are independently detected via different decay modes (w + T + T - T ' , T ' ~ ; 17 -+ ~ ~ , T + T - T '~, T ' , T + T - ~ )
aRecent progress in estimating the light-by-light scattering contribution t o a y between experiment and theory reduces t o about 1.5 standard deviations.
implies that the difference
29 Table 1: Main sources of systematic errors.
Radiative corrections Detection efficiency Fiducial volume Correction for pion losses
0.2 0.6
Total
0
0.2
detection efficiencies and calorimeter response are studied by using ”pure” experimental data samples rather than Monte Car10 events; more than 20 million 4 meson decays can be used for that purpose.
New results are available on most of the hadronic channels. We’ll briefly mention only those and CVC studies. which are relevant to the evaluation of There are new results on the process e+e- - + T + T - . This reaction has been extensively studied before17~1s~19~20. The most precise pion form factor data were obtained in late 70s - early 80s by CMD and OLYA detectors 19. Their accuracy was limited by systematic errors of the experiments, varying from 2% to 15% over the VEPP-2M energy range. In this measurement CMD-2 collected more than 2 million events of the process e+e- + T+T- from 370 to 1380 MeV. Below 600 MeV separation of Bhabha and T+T- events is performed by measuring their momentum. Above this energy the energy deposition of the final particles in the calorimeter has been used. The number of events of the reaction e+e- + j i + j i - was evaluated from QED which validity at these energies had been verified before. The systematic uncertainty of less than 0.6% was achieved in the final analysis of the data set of about 114000 events collected in the energy range 610 to 960 MeV in 1994-1995”. Table 1 lists dominant sources of the systematic error. Analysis is in progress for the rest of events and the expected systematic error ranges from 1%to 3% 22. Fig. 1 shows results of the pion form factor measurement coming from CMD-2. CMD-2 measured with high accuracy the main parameters of the w and 6, mesons using they also studied the 4 meson in its K ~ K decay L modez5. SND their decays to T+T-T’ performed a high precision measurement of three main decay modes of the 4 meson in one experiment 26. These studies allow a significant improvement in the accuracy of the leptonic widths of the w and 6,mesons. SND also studied production of three pions above the 6,and showed that the energy dependence of the cross section is consistent with the existence of at least one additional isoscalar resonance 27. These conclusions are confirmed by preliminary results from CMD-2. Both detectors observed production of four pions. CMD-2 showed that in the energy range above the 4 the a1(1260)*~Fintermediate mechanism dominates in the T+T-T+T- channel final statezs. The contribution whereas both a1(1260)*nT and WT contribute to the T+T-TOTO of other possible intermediate states is small. The collected data sample includes about 60000 events and the systematic uncertainty of the total cross sections is less than 15%. Below 1 GeV CMD-2 reliably selected about 200 events of the reaction e+e- + T+T-T+T- and measured the cross sections as low as about 50 pb near the p peak”. The measurement of the SND detector for which the data sample above the 4 was about 80000 events with the systematic uncertainty from 8 to 20% confirms the CMD-2 results on the production mechanisms 30. However, in both 4~ channels the SND cross sections are higher than those of CMD-2. The systematic uncertainties are still high and their further analysis is needed to clarify the picture. The corresponding cross
e:
23124,
30
500
7%
1000
1250
ZE,, MeV
Figure 1: New data on the pion form factor.
sections are shown in Fig. 2 and Fig. 3 together with the results of the previous measurements at VEPP-2M, DCI and ADONE (for the references see 28). Both detectors measured the cross section of the reaction e+e- --t w7ro in the w + 7r0y channel collecting several thousand events each with the systematic error of 5% for SND 31 and 8.5% for CMD-2 32. Results of both groups are consistent within systematic errors. CMD-2 reliably observed multihadronic processes e+e- --t q7r+7rP and e+e- + 7r+7r-7rTf7r-7ro with the systematuc accuracy of 15% 33. Analysis is in progress for the K+K- and KgKE final states produced above the q5 meson. Both groups measured radiative decays of the w and q5 mesons34)35. Thus, in the new experiments at the VEPP-2M collider in Novosibirsk in the c.m.energy range from 0.37 to 1.38 GeV most of the hadronic reactions contributing to R have been measured with much better accuracy than before. 2.2
R Measurement at BES
Until recently the energy range above 1.4 GeV was studied much worse (see e.g. Figs. 2,3 where cross sections of the four pion production, the dominant process above 1 GeV, are shown below the c.m.energy of 2 GeV). Despite numerous measurements of exclusive cross sections and R by various groups in Frascati, Orsay, DESY and SLAC the existing data have big scatter and large systematic uncertainties ranging from 10% to 25%. A r e d breakthrough occurred after recent experiments with the BES detector at B e i j i ~ ~ g ~ ~ in which the total cross section and R were thoroughly measured in the energy range from 2 to 5 GeV. High statistics collected in this experiment combined with the better acceptance than before and careful analysis of the systematic uncertainties provided a basis for the significant improvement of the accuracy of R ( s ) . Table 2 illustrates the progress by comparing some characteristics of the BES experiment with the R measurement by the 772 group at Frascati37.
3
Comparison t o
T
lepton decays
For the Cabibbo allowed vector part of the weak hadronic current the distribution over the mass of produced hadrons is given by
31
0
SND (VEPP-2M)
1.2
1.4
1.6
2EBE.AM
Figure 2: Cross section of the process e+e-
1.8 7
GeV + afn-2a0.
50 45
0
OLY'A(VEPP-2ZU)
40 35
30 25 20
15 10 5 0
1
1.2
1.4
1.6
Figure 3: Cross section of the process efe-
1.8
--t
2
2n+Zrr-.
32 Table 2: Comparison of yy2 and BES measurements.
Detector 6, GeV
I
772
Syst.error, %
Nhad
I
BES
I 2.0 - 3.1 I 2.0 - 3.0 5.2-8.2 920
18500
where GF is the Fermi constant, lVudl is the corresponding element of the CKM matrix, SEW is a factor taking into account electroweak radiative corrections approximately equal to 1.02 38 and v1(q2) is a spectral function:
We'll now compare recent e+e- results to those from the T decays. In the ZH channel the spectral function of CLEO' is consistent with this from ALEPH' and in general well reproduces the picture observed in efe- annihilation: the ~ ( 7 7 0 )meson peak followed by the ~(1450)and ) obvious reasons there is no p- w interference in the T decay). However, the possibly ~ ( 1 7 0 0 (for CLEO spectral function is by (3.2+1.4)% higher than that in e+e- indicating some normalization problems. We can also compare results in the W H channel. From Fig. 4 which in addition to e+edata shows recent results from CLEO 39 recalculated to the e+e- case, it is clear that in the VEPP-2M energy range (below 1400 MeV) results from the T sector are compatible with e+ewhereas above 1400 MeV the T spectral function is systematically higher than that from DM2 measurements 40. The CMD-2 analysis of intermediate mechanisms in the 47~production is consistent with the conclusions of CLEO. The model used by CMD-2 to describe their results has been successfully applied41 to describe various two pion and three pion distributions for both CLEO and ALEPH. However, there is an obvious problem with the normalization since the T - + 2 n - ~ + 7 r ~ vspectral , function is higher than that from CMD-2 in the whole energy range (Fig. 5), the difference reaching 20%. The spectral functions calculated fiom OLYA 42 and SND 30 better match the CLEO result. The ongoing analysis of the systematic uncertainties in the e+e- case can clarify the picture. We summarize the comparison in Table 3 showing the expected branching ratios for various T decays obtained by assuming that CVC is correct and averaging recent results from VEPP-2M with those from the previous measurements. T world averages are taken from43. The expected branching ratio for the H-H' hadronic state contains a -0.08% contribution coming from the p - w interference as well as a 0.06% correction for isospin breaking from 44. As seen from Table 3, the excess of the experimental value over the expected one is (0.56 f 0.31)% or 2.2% of the experimental branching ratio in agreement with the observation of CLEO The observed effect can be due to isospin breaking corrections for mn+ # m,o as well as to additional radiative corrections in T decays which according to recent estimates are not negligible and can
'.
9,44145
*This difference will probably he even larger after one takes into account that the e+e- cross section contains the vacuum polarization effects which are numerically much stronger than those for the T .
33
arto
e+e--+ Q
25
c b-
0
t
oi42 jr?cq CLEO (CESR)
0
t t
i 5
*
* * 1
1.2
1.4
1.6
2
1.8
Figure 4: Cross section of the process e+e-
0.14
1
--t WK'
CLEO(CESR)
CMDZ (VEPP-2M)
0.12
0.1 0.08
t
0.06
t
cr Y'
0.04 0.02
0
0.9
1
1.1
1.2
1.3
1.4
Figure 5: Spectral function of the decay
1.5
7-
-+
1.6
'
~K-T+K~U,
34 Table 3: Branching ratio B(T-
Hadronic State X x-r" r- 37ro 2r-7r+# WR7p-79
cvc, 2001 24.75 f 0.25 1.07 f 0.05 3.84 f 0.17 1.82 f 0.07 0.13 f 0.02
+X-v,),
World Average 25.31 f 0.18 1.08 f 0.10 4.19 f 0.23 1.92 f 0.07 0.17 f 0.02 0.24 f 0.02 30.99 f 0.31
%.
WA - CVC
0.56 f 0.31 0.01 Z t 0.11 0.35 f 0.29 0.10 f 0.10 0.04 f 0.03 -0.13 I!Z 0.11 0.83 f 0.45
reach 0.8%45>46.However, these effects still await their precise evaluation. At the moment the uncertainty of their value is a limiting factor of the CVC tests and possible improvements of the a:& calculations using T decay data in addition to e+e-. 4
New Data o n t h e P i o n Form Factor a n d a?
Let us estimate the implications of the new results on the pion form factor for a?, the contribution from the annihilation into two pions, which dominates the hadronic contribution to (g-2)/2. To this end we compare its value in the energy range 610 to 960 MeV calculated from CMD-2 data only to that based on the previous e+e- measurement^^^^^^. Table 4 presents results of the calculations performed using Eq. (2) and the direct integration of the experimental data. The method is straightforward and has been described elsewhere3. The first line of the Table 4 (Old data) gives the result based on the data of OLYA, CMD and DM1 while the second one (New data) is obtained from the CMD-2 data only. The third line (Old New) presents the weighted average of these two estimates. The assumption about the complete independence of the old and new data used in the averaging procedure seems to be well justified. For convenience, we list separately statistical and systematic uncertainties in the second column while the third one gives the total error obtained by adding them in quadrature. One can see that the estimate based on the CMD-2 data is in good agreement with that coming from the old data. It is worth noting that the statistical error of the new measurement is slightly larger than the systematic uncertainty. Because of the small systematic error of the new data, the uncertainty of the new result for a? is almost three times better than the previous one. As a result, the combined value based on both old and new data is completely dominated by the CMD-2 measurement. The example above only illustrates the importance of the improved accuracy. At the present time the analysis of the mr data as well as other hadronic final states in the whole energy range accessible to CMD-2 is in progress. Independent information is also available 36 or expected in close future from other experiments studying low energy e+e- annihilation 47. When all the above mentioned data are taken into account, one can expect a significant improvement of the overall error of a F d (by a factor of about 2) compared to the previous one based on the e+edata only '.
ay
+
5
Conclusions
Thus, new experiments in Novosibirsk and Beijing considerably improved the accuracy of R ( s ) in the energy ranges below 1.38 GeV and between 2 and 5 GeV. This should result in the and ~ ( M Z ) . significant improvement of the uncertainty of
ap
35 Table 4: Contributions of the m~channel t o (g-2)/2
Old
+ New
aKK,10374.8 3z 4.1 f 8.5 368.1 f 2.6 ik 2.2 368.9 f 2.2 f 2.3
Total error, 10-
Precise tests of the relation between the e+e- cross sections and T branching ratios will require better understanding of the effects of isospin symmetry breaking and radiative corrections. Further significant progress will become possible after new experiments planned at Beijing, Cornell and Novosibirsk. Also promising looks a possibility to study low energy efe- annihilation by the radiative return from the "(4s) or 4
6
Acknowledgments
The author is grateful to W.J. Marciano and S.N. White for an opportunity to present this talk as well as to J. Mooney and Erice colleagues for the excellent organization of the School. Special thanks are due to A. Czarnecki, G.V. Fedotovich, A. Hocker, Haiming Hu, W.J. Marciano, K.V. Melnikov and Zhengguo Zhao for numerous stimulating discussions.
References T. Kinoshita, B. NZiC and Y. Okamoto, Phys. Rev. D 31,2108 (1985). R.M. Carey e t al., Phys. Rev. Lett. 82, 1632 (1999). S. Eidelman and F. Jegerlehner, Z.Phys. C 67,585 (1995). M. Davier and A. Hocker, Phys. Lett. B 435,427 (1998). Y.S. Tsai, Phys. Rev. D 4,2821 (1971). H.B. Thacker and J.J. Sakurai, Phys. Lett. B 36,103 (1971). 6. R. Barate et al., Z.Phys. C 76,15 (1997). 7. K. Ackerstaff et al., Eur. Phys. J. C 7,571 (1999). 8. S. Anderson et al., Phys. Rev. D 61,112002 (2000). 9. R. Alemany, M. Davier and A. Hocker, Eur. Phys. J. C 2, 123 (1998). 10. S. Eidelman, Nucl. Phys. B (Proc. Suppl.) 98,281 (2001). 11. H.N. Brown et al., Phys. Rev. Lett. 86,2227 (2001). 12. M. Knecht and A. Nyffeler, hep-ph/0111058; M. Hayakawa and T. Kinoshita, hep-ph/0112102; I. Blokland, A. Czarnecki and K. Melnikov, hep-ph/0112117. 13. V.V. Anashin et al., Preprint INP 84-114, Novosibirsk, 1984. 14. E.V. Anashkin et al., ICFA Instrumentation Bulletin 5 , 18 (1988). 15. M.N. Achasov et al., Preprint BudkerINP 96-47, Novosibirsk, 1996. 16. A.P. Lysenko et al., Nucl. Instr. Meth. A 359,419 (1995). 17. A.D. Bukin et al., Phys. Lett. B 73,226 (1978). 18. A. Quenzer et al., Phys. Lett. B 76,512 (1978). 19. L.M. Barkov et al., Nucl. Phys. B 256,85 (1985). 20. D. Bisello et al., Phys. Lett. B 220,321 (1989). 21. R.R. Akhmetshin et al., Preprint BudkerINP 99-10] Novosibirsk, 1999; R.R. Akhmetshin et al., hep-ex/0112031. 22. A.E. Bondar, Talk at the Int. Conf. on High Energy Physics, Osaka, 2000. 23. R.R. Akhmetshin e t al., Phys. Lett. B 476,33 (2000). 1. 2. 3. 4. 5.
36 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.
R.R. Akhmetshin et al., Phys. Lett. B 434,426 (1998). R.R. Akhmetshin et al., Phys. Lett. B 466,385 (1999), Erratum-ibid, 508, 217 (2001). M.N. Achasov el al., Phys. Rev. D 63,072002 (2001). M.N. Achasov et al., Phys. Lett. B 462,365 (1999). R.R. Akhmetshin et al., Phys. Lett. B 466,392 (1999). R.R. Akhmetshin et al., Phys. Lett. B 475, 190 (2000). M.N. Achasov et al., Preprint BudkerINP 2001-34, Novosibirsk, 2001. M.N. Achasov et al., Phys. Lett. B 486,29 (2000). P.P. Krokovny, Master’s Thesis, Novosibirsk State University, 2000. R.R. Akhmetshin et al., Phys. Lett. B 489,125 (2000). M.N. Achasov et al., J E T P Lett. 72,282 (2000); M.N. Achasov et al., Eur. Phys. J. C 12,25 (2000). R.R. Akhmetshin et al., Phys. Lett. B 509, 217 (2001). J.Z. Bai et al., Phys. Rev. Lett. 84, 594 (2000); J.Z. Bai et al., hep-ex/0102003. C. Bacci et al., Phys. Lett. B 86,234 (1979). W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 61,1815 (1988). K.W. Edwards et al., Phys. Rev. D 61,072003 (2000). D. Bisello et al., Preprint LAL-91-64, Orsay, 1991. A.E. Bondar et al., Phys. Lett. B 466,403 (1999). L.M. Kurdadze et al., JETP Lett. 43,643 (1986); L.M. Kurdadze et al., J E T P Lett. 47,512 (1988). D.E. Groom et al., Eur. Phys. J. C 15, 1 (2000) and 2001 off-year partial update [http://pdg.lbl.gov]. H. Czyi and J.H. Kiihn, Eur. Phys. J. C 18,497 (2001). V. Cirigliano, G. Ecker and H. Neufeld, Phys. Lett. B 513,361 (2001). K. Melnikov, Int. J. Mod. Phys. A 16,4591 (2001). A. Aloisio et al., hep-ex/0107023; E. P. Solodov, hep-ex/0107027.
The Muon Anomalous Magnetic Moment: A Status Report* William J. Marciano Brookhaven National Laboratory, Upton, New York 11973 Recent improvements in the experimental determination and theoretical prediction for the muon anomalous magnetic moment, a,, are discussed. Possible reasons for the difference between experiment and Standard Model theory, Aa, = a y p - a:M, ranging from “New Physics” effects to hadronic loop uncertainties are described. A connection between Aa, and electroweak precision measurements that constrain the Higgs mass is pointed out. An outlook for the future is given.
’
Experiment E821 at Brookhaven National Laboratory has recently reported an improved determination of the muon anomalous magnetic moment, up For the p+, they find
9.
uexP II+ -
(1)
116592030(80) x
about a factor of 2 more precise than the value reported’in 2001. Additional p- data, currently under analysis, should reduce the uncertainty (assuming CPT symmetry) to about f 6 0 - 65 x 10-l’. Further running of the experiment could reduce the error to f 4 0 x or lower. However, funding is currently lacking. That is unfortunate, since the experiment has generated tremendous excitement and may be starting to see a deviation from the Standard Model. The Standard Model prediction for up naturally divides up into three parts uZM = uQED II
+
aHadrunic II
+ u,””
each of which has undergone and continues to receive thorough scrutiny. Indeed, the theoretical effort that has gone into u,”” represents one of the most impressive collective computational efforts in all of physics. Let me describe the status of each term in eq. 2. The QED contribution starts with the classic a / 2 n one loop contribution calculated by Schwinger long ago3 and extends through 5 loops. The coefficients start to grow at the 3 loop level due to a very large electron loop contribution through so-called light-by-light diagrams. For )~ loop effects have also been estimated. Currently, that reason, even the leading O ( C Y / Telectron one finds
a aQED =II 27r
+ 0.765857376(27)(z)2 + 24.05050898(44)(z)3 + 129(F)4 + 930(F)5 7T
7T
7r
(3)
with
a-’ = 137.03599877(40)
(4)
*Talk given at Erice Workshop 10/17/01,updated to include recent developments in experiment and theory.
37
38
‘.
taken from a comparison of a, theory with experiment Note, that I have not given an error estimate for the order (9)4 or ( : ) 5 terms. That is because the 0 ( 9 ) 4 coefficient is currently undergoing some revision (just as the value of 6’in eq. 4 was recently revised4). For now, I use instead a very conservative estimate of the overall error (and shift the 4 loop coefficient from its usually quoted value of 126 to 129 in anticipation of some upward movement) u$ED = 116584715(10) x
(5)
which should be reduced to about f 2 - 3 x when the revision is complete4. The added uncertainty in eq. 5 does not significantly impact the comparison of a r p and a,””. More problematic has been the computation of updrmicwhich first enters at 0($)’through vacuum polarization effects from quark (hadron) loops. Those contributions cannot (yet) be computed from first principles, although a lattice gauge theory approach may one day be competitive. Instead, one employs a dispersion relation with the measured values of cr(e+e- + hadrons) as input. Such an analysis extends from s = 4m: to 00, with the high energy region computed perturbatively. Most of the support comes from the rho resonance region where the cross-section must be measured to f l %or better and QED corrections must be applied at a similar level of accuracy. In 1998 Davier and Hocker5 carried out a thorough study of the world-wide e+e- + hudrons data and supplemented it with 7- -+ v, T-T’ data where the statistics is better. The two approaches should be equivalent for the I = 1 channel up to isospin violating corrections from Now, QED and the md - mu mass difference. They found apd(wac.pol) = 6924(62) x however, Davier et.al. find that e+e- and 7- decay data are no longer consistent, even after all known isospin violation corrections are applied to the tau. Fortunately, the e+e- data has improved and gives a pretty good prediction without being supplemented by the tau.
+
a y ( v a c . p o l ) = 6847(70) x 10-l’
(e+e- data)
(6)
Similar results have been given by Hagiwara et al’ and Jegerlehner a y ( v a c . p o l ) = 6831(59)(20) x a y ( v a c . p o l . ) = 6889(58) x Unfortunately, the of efe- data gives
T + vp
(Hagiwara et.al.)
(7)
(Jegerlehner)
(8)
+ hadrons when used where available, ,h < 1.77 GeV, in place
apd(wac.pol.)= 7046(62) x
(T
data)
(9)
I have taken the liberty of adding +27 x in eq. 9 to the tau analysis 6 . That shift is due to reduction in the electroweak radiative corrections employed i n 6 by 0.25%, due to an overestimate of the leading logsg and use of I Vud I= 0.9740 rather than the value 0.9752. After adjusting for common uncertainties, one finds that the T and e+e- based predictions differ by about 2.5 sigma. The difference in those data sets is a serious discrepancy that must be overcome. As we shall see, it impacts not only the interpretation of a r p but also precision electroweak measurements currently used to contstrain the Higgs mass or search for ”New Physics”. Until proven incorrect, I will employ the e+e- driven result in eq. 6, since the isospin violating corrections to the tau data have not been completely scrutinized and carry some inherent uncertainty. If the two data sets were to agree, the overall uncertainty in apd(vac.pol.) could be reduced by about 30 - 40% by averaging. At the 3 loop, 0(:)3,level hadron effects are of course smaller, but have a jaded history. In particular, the hadronic light-by- light loops must be computed in an effective field theory
39 approach where the T and 71 pole contributions dominate. Although not particularly large, the light-by-light contribution sign was found to be wrong last year, thus changing its value by 200% lo. (That was the third time it changed sign in the 25 year history of its evaluation.) The situation now seems stable and one finds 'OJ1 for the total 3 loop hadronic contribution, a H a d r u n i c (3
loop) = -14(35) x
(10)
Adding eq. 10 to 6, one finds
aHadronic P = 6833(70)(35) x
(11)
The electroweak contribution is smaller and much less controversial. It has been computed through 1 and 2 loops. Even the 3 loop leading logs have been evaluated 12J3 and shown to be insignificant. Also, hadronic (quark) effects have been recently scrutinized. One finds l3
a,""
= 154(1)(2) x
(12)
where the first error corresponds to hadronic uncertainties and the second primarily from the Higgs mass uncertainty. Combining the contributions in eqs. 5, 11 and 12 (in quadrature) gives the Standard Model prediction a,"" = 116591702(79) x (13) and leads to the 2.9 sigma deviation - a:"
= 328
+I
112 x
(14)
where again all errors have been combined in quadrature. The difference is large and significant. Can there be a deviation that is twice the size of electroweak contributions from W and 2 bosons? The answer is yes, it could happen. Many examples of "New Physics" have been put forward to explain eq. 14. However, by far, the leading candidate solution is supersymmetry 14. SUSY analogs of the electroweak contributions but with gauginos and sleptons replacing W , Z and leptons can give a large contribution to aP. For simplicity, I give the prediction l 5 3 l 6 for all SUSY loop masses degenerate and equal to m S U S y aSUSY P
where sgn(p) =
N
sgn(p) x 130 x
+ or - and tan(/?) =
($2)/($1).
m H
(-)100 GeV mSUSY
tan(/?)
In recent years, bounds on the Higgs mass
2 114GeV
(16)
have pushed SUSY models into the large tan(/?) region, tan(/?) 2 3 - 4, suggesting an enhancement in a$usy. So, if low energy SUSY is part of Nature's plan, it should be starting to show up in a r p . Indeed, equating eqs. 15 and 14, leads to
-
So, for tan/? 3 - 40, one finds rnSUSY N 100 - 400 GeV which is in keeping with mainstream SUSY expectations. Is the deviation in a p p - az M a Harbinger of "New Physics" or merely an indication that (vacpol) is larger than the present value in eq. 6 obtained from efe- data? Indeed, T data may be a red flag telling us that is the case. However, it is not quite that simple. The same e+edata that gives a p (vac.pol.) also plays a central role in other important tests of the Standard
apd
40
Model, the comparison of a, G, and m z with measurements of mW and sin2 t?w(mz);i?S. Such comparisons were used to predict mt before its discovery and now constrain the Higgs mass, mH. Changing efe- + hadrons data will also impact the Higgs mass prediction as well as constraints on "New Physics" from those precision tests. Let me illustrate the connection. Of particular importance for electroweak precision studies are the natural relations l7
among bare (unrenormalized) electroweak parameters. They play a role similar to a,. What I mean by that is the same relations are satisfied by renormalized (measurable) quantities, up to finite, calculable radiative corrections. The now standard prescription is to lump all radiative corrections into quantities called A r and A?, where" 7ra 1- Ar(mt,mH,Aahad ) = fiG,m&(l- m&/m2,)
2fim 1 - A?(mt, mH, Aahad ) = G,m$ sin2 2 O ~ ( m ~ ) ~
I have explicitly indicated the dependence of Ar and A? on mt,mH and hadronic vacuum polarization effects (via Aahad). The usual procedure is to employ 137.03599877(40) 1.16637(1) x 10-5GeV-2 91.1875(21)GeV = 174.3 k 5.1GeV
a-1
=
G, mz mt
= =
as input and then use mw in Ar (or sin2 t?w(mz)m in A?) to predict mH or look for a deviation from "New Physics". In those studies, a central value of
Aahad = 0.02761
(22)
has often been used19. However, more recent efe- + hadrons data will lower that value, while r + v,+ hadrons data would raise it. From the analysis of Davier et a16, I roughly findz0
Aahad= 0.02752(16)
efe- data
(23)
r data (24) Aahad = 0.02780(16) That means new e+e- data leads to larger mH predictions and bounds by about 6%, while r data suggests lower values by about 12%. Since the predictions for mH are generally low, compared to the bound in eq. (16), they suggest a Standard Model preference for the e+evalue in eq.(23). Let me be a little bit more quantitative. Using the world average mW = 80.451(33)GeV and sinzt?w(mz);i?S= 0.023085(21) from leptonic 2 pole asymmetries, one predicts2'&' mH = 5lritGeV, < 123GeV(95%CL) mH = 42fi:GeV, < 102GeV(95%CL)
e+e- data
(25)
r data
(26)
Notice that the r data already conflicts (at 95% CL) with the experimental bound on mH, while e+e- suggest a low mH, but one still consistent with experimental bounds. What does this mean? It suggests that the roughly 3 sigma deviation in a r p - a,"" may be real, since trying to alleviate it with a larger hadronic vacuum polarization effect (as in the
41
case of T data) would significantly lower an already precariously low prediction for m H . It also indicates that the e+e- - T data discrepancy must be resolved, if we want to fully utilize current and future precision measurements of mw, sin2 Bw(Mz)li;isetc. So, it seems that "New Physics" may be starting to surface in ap as well as AT and A?, with relatively light supersymmetry as the likely candidate. If that is the case, the LHC and Next Linear Collider will discover it and sort out most of its properties. However, precision measurements will continue to play an important role. For example, a r p may turn out to give our best determination of tanp. For that reason, precision studies must be pushed as far as possible and the e f e - - r discrepancy must be resolved. In the case of the latter, we should have several new measurements at operating higher energy and luminosity colliders. They can study the cross-section in the important rho resonance region via the so-called radiative return process efe- + y hadrons. The KLOE experiment at Frascoti promises results soon. Similarly, b factories should provide a wealth of r data for further scrutiny. In the case of aFp it seems clear that the current E821 effort should be pushed as far as possible. A natural place to stop would be at the level of symstematic uncertainties which could probably be reduced to f 2 0 x lo-". On the theory side, I anticipate that resolution of the e f e - - r discrepancy will lead to an overall uncertainty of about 3~50x lo-''. In the longer term, lattice gauge theory studies could perhaps reduce the error by another factor of 2, which would be well matched to experimental capabilities. If the discrepency of N t 3 0 0 x lo-'' is real, it will become a many sigma effect, and a nice constraint on whatever "New Physics" is responsible.
+
Acknowledgments Supported in part by DOE grant DE-AC03-76SF00515. References 1. G. W. Bennet et al, Phys. Rev. Lett. 89, 101804 (2002);erratum ibid 89, 129903 (2002). 2. H. N. Brown et al, Phys. Rev. Lett. 8 6 , 227 (2001). 3. J. SchwingerJhys. Rev.73, 416 (1948). 4. T. Kinoshita and M. Nio, hep-ph/0210322. 5. M. Davier and A. Hocker, Phys. Lett. B 435, 427 (1998). 6. M. Davier, S. Eidelman, A. Hocker and Z. Zhang, hep- ph/0208177. 7. K. Hagiwara, A. D. Martin, D. Nomura and T. Teubner, hep- ph/0209187. 8. F. Jegerlehner, talk on "Hadronic Contributions to the Anomalous Magnetic Moment of the Muon", Centre de Physique Theorique, Marseille, March 14- 16, 2002. 5. W. Marciano and A. Sirlin,Phys. Rev. Lett. 61, 1815 (1988). 10. M. Knecht and A. Nyffler, Phys. Rev. D 6 5 , 073034 (2002); M. Knecht, A. Nyfller, M. Perrottet and E. de Rafael, Phys. Rev. Lett. 88, 071802 (2002);M. Hayakawa and T. Kinoshita, Phys. Rev. D 6 6 , 019902 (2002);J. Bijnens, E. Pallante and J. Prades, Nucl. Phys. B 626, 410 (2002); I. Blockland, A. Czarnecki and K. Melnikov, Phys. Rev. Lett. 88, 071803 (2002). 11. B. Krause, Phys. Lett. B 390, 392 (1997). 12. G. Degrassi and G.F. Giudice,Phys. Rev. D 58, 053007 (1998). 13. A. Czarnecki, W. Marciano and A. Vainshtein, hepph/0212229. 14. D. Kosower, L. Krauss and N. Sakai, Phys. Lett. B 133, 305 (1983); T. C. Yuan et al, Z. Phys.C26, 407 (1984); T. Moroi, Phys. Rev. D 53, 6565 (1996); T. Ibrahim and P. Nath, Phys. Rev. D 62, 015004 (2002). 15. T. Moroi in ref. 14.
42 16. A. Czarnecki and W. Marciano,Phys. Rev. D 64, 013014 (2001). 17. C. Bollini, J. Giambiagi and A. Sirlin, Nuovo CimentolGA, 423 (1973); W. Marciano, NPB 84, 132 (1975). 18. A. Sirlin,Phys. Rev. D 22, 471 (1980); W. Marciano and A. Sirlin, Phys. Rev. D 22, 2695 (1980). 19. A. Ferroglia, G. Ossola, M. Passera and A. Sirlin, Phys. Rev. D 65, 113002 (2002). 20. W. Marciano, ”Precision Measurements and ”New Physics””,to be published in J. Phys.G. 28(2003). 21. G. Ossola and A. Sirlin, (private communication) find the broader range m H = 462giGeV, < 119GeV(95%CL)from 7 data, assuming an error of k0.00036 for Aahad.
SPONTANEOUS BREAKDOWN OF THE VACUUM BY INTENSE LASERS ADRIAN C. MELISSINOS Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 We review the necessary conditions for the spontaneous production of e+e- pairs from the vacuum by a high intensity time dependent electric field. We then show how this was achieved by exploiting the highly relativistic motion of 50 GeV electrons from the SLAC accelerator.
1
Introduction
Soon after the discovery of the Dirac equation1 it was noted by Klein’, Sauter3 and others that a strong static electric field could give rise spontaneously to the production of electron-positron pairs. The probability for pair production per unit volume and unit time was given by Schwinger in his 1951 paper4 as ‘w=-
n=l
Here n is the number of pairs and it suffices to consider the first term in the sum. E is the electric field and m the electron mass. The probability (l),integrated over a macroscopic space-time volume can be quite large when the exponential suppression is not extreme, that is when
In Eq.(2) we have introduced the Schwinger critical field m2c3 - 1.3 x 10l6 V/cm eti and dimensionless notation for the electric field
E,=--
T = E/Ec
(3)
(4)
Thus, as ‘Y approaches or exceeds unity, pair production will be copious. We speak of the breakdown of the vacuum. The notion of a critical field (3) and Schwinger’s result (1) can be interpreted qualitatively in simple terms. A virtual efe- pair from the vacuum has a “lifetime”
AT = A E / h = mc2/ti
43
(5)
44
During that time interval the virtual particles can gain energy from the electric field; should the energy gain equal or exceed their rest mass, the virtual particles can be found on the mass shell as real particles. Hence eE(cAr) = mc2
or
e E = m2c3/fi
(6)
as in Eq.(3). Eq.(l) is often interpreted as the tunnelling of a positron from the negative energy Dirac sea to a positive energy state. In the presence of an electric field E, the width of the barrier is Ax = 2mc2/eE. The relevant distance scale in this case is the electron's Compton wavelength A c = li/mc so that the tunnelling probability is proportional to exp
(+)
= exp
(--)
2m2C3
in qualitative agreement with Eq.(l). One also sees in this model that a magnetic field cannot give rise to pair production since the field does no work on the virtual particles. These results have fascinated theorists and experimentalists alike; not surprisingly they are also obtained as the non-perturbative limit of strong field QED. This is a rich subject that has been extensively studied [5-71. In the laboratory, strong electric fields can be obtained, for instance, by focussing intense short pulses from a laser or in heavy ion collision or when electrons axe channelled through crystal lattices l o . Applications of the Schwinger mechanism to related theoretical problems can be found in refs. [ll-131 while future experimental possibilities are discussed by A. Ringwald in these proceedings 14. Since it is not possible to create static fields of critical value in the laboratory, attention has been centered on time dependent fields such as mentioned in the previous paragraph. There are however important differences between static and time dependent fields. Most importantly, a plane wave field cannot break down the vacuum because in order to generate massive particles it must hold
E2 - B2 > 0
(7)
Instead one can consider two oppositely directed waves that interfere to produce a standing wave pattern. Then there are regions of space where the invariant condition (7) is satisfied. Other differences between static and time dependent fields have been analyzed in detail in 15. Of course, two opposing wave fields are equivalent to photon-photon scattering which leads to pair creation when Rw > mc216. But even if liw < m,c2 pairs can be produced by multiphoton interactions, if the field intensity is high enough. Multiphoton processes have been extensively studied in the atomic domain 17. They are characterized by a dimensionless (and Lorentz invariant) parameter
q is referred to as the normalized vector potential and w is the frequency of the wave. By writing q = eETmSA/mc2we see that when q = 1 the energy gained by an electron moving across one wavelength equals its rest-mass energy, mc2. Thus relativistic effects becomes important. Note that q is a classical parameter (it does not involve R ) . In the perturbative solution of multiphoton processes, q2 plays the role of an expansion parameter in place of the familiar fine structure constant, a. The probability of a process involving n photons from the incident field is then proportional to 17'". In particular when fiw < mc2 the probability that a pair is produced by the time dependent electric field is l5
45
Clearly, 2mc2/hw is the number of photons that must be absorbed to create the pair in agreement with the q2n dependence. The probability given by Eq.(9) is valid in the limit T << 1 and takes the form of Eq.(l) as T -+ 1. We note that for a static field the multiphoton parameter 77 + 03. Namely we can interpret pair production in a static field as involving the collision of an infinite number of zero energy virtual photons. The two points of view are complimentary as is so often the case. We have seen from Eqs.(3,4) that T contains li, namely it is a quantum mechanical parameter. This must be so because T measures the field strength for the onset of a purely quantum effect, the production of an e+e- pair. As written, T is not Lorentz invariant since the value of the electric field depends on the frame of reference from which it is viewed. Thus Eq.(4) should be expressed in terms of E*,the electric field in the rest-frame of the produced pair
The above observation is crucial because it shows how one can reach critical field with existing lasers by exploiting the large y-factors of high energy electron beams. Existing short pulse lasers in the visible can be focussed to near the diffraction limit and have intensity (flux) in excess of
I = 10" W/cm2
(11)
Improved intensities that could be possibly achieved in the future are given in 14. The corresponding peak field
E~ = J2;5,r
V/cm
(12)
with 20= 377 0, the impedance of free space, and I expressed in W/cm2 as in Eq.(9). We then find for I = 10" W/cm2
Ep
-
3 x 10" V/cm
(13)
-
The peak field of Eq.(13) is far from the critical field of Eq.(3). Consider however the 46.6 GeV FFTB electron beam at SLAC l8 where y 0.9 x lo5. In the electron's rest frame
E* = 2y,E
N 0.5
x 10l6 V/cm
(14)
leading to T cz 0.3. The interaction of the electrons with the laser field is in the strong field domain! The factor of 2 in Eq.(14) appears because we are dealing with a wave field. To observe the breakdown of the vacuum under such conditions we collide a high energy photon (&, 28 GeV) with the laser field (hw = 2.34 eV). If n photons are participating in the interaction, the cm energy is N
Ji
= (4E7nliw)1/2
(15)
Since it must hold f i = 2mc2, the threshold for pair production for the above &, hw is n 2 4. Clearly, a multiphoton process where four or more photons are absorbed from the laser field. It follows that
"lcm =
2
4hw
= 0.28 x 105
46 and for Ep given by Eq. (13)
T = 2ycmEP/Ec= 0.13
(17)
Such an experiment [19,20,21]was completed in 1997 and will be described in the following sections. The high energy gamma is obtained by backscattering a 46.6 GeV electron in the laser field. Because of the extremely high photon density, the gamma interacts again with the laser field leading to the production of a pair. Even though as seen in Eq.(17), T did not reach 1, the dependence of the data on l/T, clearly verifies the Schwinger result of Eq.(l). At the same time the data are also in exact agreement with the perturbative treatment of photon-photon scattering22, as they must.
2
Experimental arrangement
In it's simplest form, the experiment consists of scattering an intense laser pulse from a high energy electron beam. To study pair production, positrons are identified and detected using a magnetic spectrometer and a total absorption calorimeter. The intense flux of forward going high energy y-rays was used mainly to monitor the interaction rate. As already stated, the experiment was carried out at the FFTB l8 which delivered 5 xlOg electrons per pulse at an energy of 46.6 GeV. The repetition rate was 10 Hz. The experimental layout is shown schematically in Fig. 1. The laser crossed the electron beam at an angle of 17" and was focussed with f# fi 6 optics. It was returned to the laser room for diagnostic and monitoring purposes. The magnetic spectrometer consisted of a string of permanent magnets which also directed the beam to the dump. The location of the positron (PCAL) calorimeter is also indicated.
pair spectrometer
47 GeV e's
ECAL I dump magnets
SCAL
Figure 1: Schematic of the experimental setup: The laser pulses cross through the electron beam at the interaction point, IP1. The scattered electrons are deflected by the dump magnets into the electron calorimeter (ECAL). Positrons are deflected into the positron calorimeter (PCAL). The scattered photons are detected in a Cerenkov counter (not shown), or converted t o e+e- pairs which could be detected by the pair spectrometer.
The laser was a 0.5 Hz repetition-rate, tabletop terawatt laser, that operated at 1053 nm wavelength (IR), or at 527 nm (green) after efficient (- 45%) frequency doubling. It consisted
47
of a mode-locked Nd:YLF oscillator, Nd:glass slab amplifier. The laser system delivered up to 2.4 J in the IR at the interaction point, but typically it was operated only up to 800 mJ of IR and 500 mJ of green. Intensities above 10l8 W/cm2 at the laser focus have been produced. Important issues in the experiment are the synchronization of the laser and electron beams which must hold at the level of 1 ps or better. Fig. 2 shows a timing curve where the number of forward going Compton scattered gamma rays is plotted as a function of laser delay. The spatial overlap of the electron beam with the laser focus also poses an experimental challenge because of the tight tolerances required.
Figure 2: A “timing curve” showing the number of electrons scattered into the top row of the electron calorimeter as a function of delay of the optical pulse. The standard deviation is n = 4.3 ps.
The laser intensity could be determined from a measurement of the pulse energy U ,area A and pulse duration 7.However for highly nonlinear processes the fluctuations inherent in these measurements were too large. Instead the experiment relied on three monitors which intercepted scattered electrons N l , N2, N3 originating in one-, two-, and three-photon Compton scattering. For v2 < 1, to a good approximation
N2 = kg Ni v2 N3 = k3N1v4 (18) where the factors k z , k3 depend on the acceptance and efficiency of the monitors but are the same for all events. By making an overall fit to the data 77 could be determined for each event with a precision of & 11%and used for the analysis of the pair production data. Evidence for the production of an e+e- pair was based on the detection of a positron in the PCAL calorimeter. The momentum p , of the positron is directly related to the location of impact on the Calorimeter. The total energy, &, deposited in the calorimeter is also recorded. Thus the ratio & / p should equal unity for positrons originating from the laser focal area. This was tested by placing a thin wire at the focus so that pairs were copiously produced by the Bethe-Heitler process. A plot of E / p as measured in the calorimeter for p N 21.0 GeV using this calibration method is shown in Fig. 3. The subsidiary peaks at & / p = 2 and 3 correspond to cases when two or three positrons (at that given momentum) reach the calorimeter in the same pulse. 3
Results
A total of 175 positrons were identified with the laser on, and data were also taken with the laser off in order to measure the background. The number of positron candidates with the laser “on” and the laser “off’ normalized to the same incident flux is shown in Fig. 4(a). The positron
48
Figure 3: Distribution of the ratio Ec~u/Pc~u for calibration clusters in PCAL row 7, which spans momenta from 20.3 to 21.5 GeV.
momentum spectrum, after subtraction of the background is given in (b) of the figure. The solid line is the prediction of the simulation.
h
g20
LO 0
10
15
20
positron momentum [Ge\ rlc]
Figure 4: Number of positron candidates us. momentum for laser-on (ON) and laser-off (OFF) electron pulses. (b) Spectrum of signal positrons obtained by subtracting the laser-off from the laser-on distribution. The curve shows the expected momentum spectrum from the model calculation.
One can also plot the number of detected positrons as a function of the laser intensity expressed by the parameter q cc In Fig. 5 the laser “on” events are shown by the solid circles, while the shaded area gives the background, as deduced from the laser “ o f ’ events. A power law fit to the form
a.
Re+ cc q2n (19) gives n = 5.1 f 0.2 and is indicated by the solid line. This is in agreement with the fact that near threshold five photons must be absorbed from the laser field. One photon to create the high energy y-ray and at least four photons to create the pair as shown by Eq.(15).
49
0.2
0.1
0.3 0.4 11 at laser foeus
Figure 5: The dependence of the positron rate per laser shot on the laser field-strength parameter 1). The line shows a power law fit to the data. The shaded distribution is the 95% confidence limit on the residual background from showers of lost beam particles after subtracting the laser-off positron rate.
The data are compared with the simulation in Fig. 6. The yield is normalized to the total number of Compton scatters which is directly inferred from the monitors while the simulation is shown by the solid line. The prediction of the simulation has not been normalized and is in excellent agreement with the data. This would not have been possible without an accurate knowledge of the laser intensity for each event as discussed in Eq.(18). It is of interest, of course, to consider the positron yield as a function of the parameter T defined by Eq. (17). We would then expect in the limit 17 2 1 a dependence such as given by the Schwinger result of Eq.(l). This is confirmed by the data which are plotted in Fig. 7 as a function of l/T. A fit to the form
R,+
c(
yields A = 1.27*0.08&0.25, the first error being statistical and the second systematic. Naively one would expect A = x. However the fact that 17 < 1 introduces corrections in the case of a wave field l5 which lead to the prediction A = 1.1 in excellent agreement with the observed slope. Data from two different electron energies are combined in the plot of Fig. 7. We conclude that indeed the wave field breaks down the vacuum, albeit in the fast moving rest frame of the pair. In a complimentary fashion we can argue that multiphoton-photon scattering created an electron positron pair as excepted from strong QED calculations. 4
Discussion
It has been suggested that a strong field could also create a pair of neutral particles because of its coupling to virtual charged particles present in loop diagrams [23,24]. As an example we consider E -+ YV where the effective coupling has strength g = e G ~ r n z / ( R c=) ~3 x 10-12e
(21)
The critical field is given as in Eq.(3) but the question arises as to whether one should use the mass of the neutral particles produced (as in 23) or the mass of the lightest intermediate particle in the loop (as in 24). In the former case
50
Figure 6: The dependence of the positron rate on the laser field-strength parameter q when the rate is divided by the number of Compton scatters inferred from the EC37 monitor. The solid line is the prediction per Compton photon based on the simulation for photon-multiphoton scattering. The dashed line represents the simulation for the one-step trident process.
i
Figure 7: Number of positrons per laser shot as a function of l/T-,. The circles are the 46.6 GeV data whereas the squares are the 49.1 GeV data. The solid line is a fit to the data.
51
mzc3
E,(v)= -
gR and for my = 1 eV, E,(v)= E, opening up the possibility of an experiment. Of course, because of the smallness of g in Eq.(21), the overall production rate is highly suppressed even when the exponential is saturated. Furthermore the low efficiency for neutrino detection reduces the observable rate. It is nevertheless interesting that this rate is extremely sensitive to the neutrino mass depending on it exponentially. A word of caution however: Mikheev points out that one should use rn; instead of m: 2 5 . The breakdown of the vacuum by a strong gravitational field can also be invoked as an alternate interpretation of Hawking radiation 2 6 . Since the gravitational field couples to energy, all particles are produced and the spectrum is dominated by the particles of lowest mass, photons. They are emitted with a Planck spectrum and for a spherical star of mass M , the temperature is
where k is Boltzmann's and G the gravitational constants. Thus production of neutrinos of mass my will be suppressed with respect to photon radiation by an exponential factor
R,IR, Since for a solar mass kTH are massless.
N
-
e-
myc2f kTH
(24)
eV the ratio of Eq.(24) is vanishingly small, unless neutrinos
52
References
u,
1. 2. 3. 4. 5. 6.
P.A.M. Dirac, Proc. Roy. SOC.(London) 610 (1928). 0. Klein, Zeits f. Phys. 53, 157 (1929). F. Sauter, Zeits. f. Phys. 69, 742 (1931). J. Schwinger, Phys. Rev. 82, 664 (1951). A.I. Nikishov and V.I. Ritus, Sov. Phys. JETP 19, 1191 (1964); 20, 757 (1965). L.S. Brown and T.W.B. Kibble, Phys. Rev. 705 (1964); H.R. Reiss, J. Math. Phys. 3, 59 (1962); Phys. Rev. Lett. 26, 1072 (1971). 7. R. Bonvincini et al, Phys. Rev. Lett. 62, 2381 (1989). 8. C. Bamber et al., Laser Physics I,135 (1997). 4, (1996) and references therein. 9. R. Ganz et al., Phys. Lett. 211 (1986); B B , 561 (1988); R. Medenwalt et al., 10. A. B e h e m et al., Phys. Lett. Phys. Lett. 483 (1989). 11. H.M. Fried et al., Phys. Rev. D B , 125001 (2001). 12. E.A. Paschos, Phys. Lett. 383 (1996). 13. P. Chen and V. Telnov, Phys. Rev. Lett. 63, 1796 (1989). 14. A.Ringwald “Fundamental Physics at an X-ray Free Electron Laser”, these proceedings; Phys. Lett. 107 (2001). 15. E. Brezin and C. Itzykson, Phys. Rev. 1191 (1970). 16. G. Breit and J.A. Wheeler, Phys. Rev. 46, 1087 (1934). 2378 (1970). 17. E. Sarachik and G. Shappert, Phys. Rev. 18. V. Balakin et al., Phys. Rev. Lett. 3, 2479 (1995). 19. C. Bula et al., Phys. Rev. Lett. 16,3116 (1996). 20. D.L. Burke et al., Phys. Rev. Lett. 79, 1626 (1997). 21. C. Bamber et al., Phys. Rev. 092004 (1999). 22. N.B. Narozhny et al., Sov. Phys. JETP 20, 622 (1965). 23. J.A. Grifols, E. Masso, S. Mohanty and K. Shajesh, Phys. Rev. D X , 097701 (1999). 195 (1999). 24. A.C. Melissinos, Nuclear Physics B (Proc. Suppl.) 25. N.V. Mikheev, private communication and A.A. Gvozdev, et al., Phys. Rev. D a , 5674 (1996). 332 (1976). 26. T.Damour and R. Ruffini, Phys. Rev.
m,
m,
m, m,
m,
m,
m,
m,
m,
z,
u,
ELECTRODYNAMICS AT THE HIGHEST ENERGIES SPENCER R. KLEIN Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA At very high energies, the bremsstrahlung and pair production cross sections exhibit complex behavior due to the material in which the interactions occur. The cross sections in dense media can be dramatically different than for isolated atom. This writeup discusses these in-medium effects, emphasizing how the cross section has different energy and target density dependencies in different regimes. Data from SLAC experiment El46 will be presented to confirm the energy and density scaling. Finally, QCD analogs of the electrodynamics effects will be discussed.
1
Bremsstrahlung, Pair Production and the Formation Zone
Bremsstrahlung and pair production were described by Bethe and Heitler in 1934. In bremsstrahlung (braking radiation), an electron with energy E interacts with a target nucleus, and slows down, emitting a photon with energy k in the process. For interactions with an isolated atom, for k << E , the Bethe and Heitler bremsstralung cross section scales as
do
1
ZMZ. In pair production, a photon fluctuates to an e+e- pair. The newly created electron or positron interacts with the electromagnetic field of a target nucleus, and the pair becomes a real e+e- pair. For photon energies k >> me, the pair production cross section is independent of k . Sophisticated calculations of bremsstrahlung and pair production confirm that these energy dependencies hold within a few percent. Bethe and Heitler treated bremsstrahlung and pair production as occuring at a single point in space. With this assumption, the radiation depends on the change in electron velocity A 5 due to the scattering from the target, independent of the nature of the force that causes the velocity change. For any interaction, one can determine the expected AG distribution, and, from that, find the bremsstrahlung radiation. With this approach, it is relatively easy to generalize from isolated atoms to a dense medium. In 1953, Ter-Mikaelian and Landau and Pomeranchuk pointed out that the assumption that the interaction occurs at a single point fails when the incoming particle has a high enough energy. It is impossible to localize the reaction to a point on the projectile's trajectory. This is true classically as well as quantum mechanically. The pathlength over which the reaction can be localized is the formation length, l f . Classically, the formation length is the distance I over which the phase factor, exp(i[i.z-wt]) z - wt < 1). In a vacuum, the photon wave vector iand the photon is roughly constant (i.
53
54
+
Figure 1: Two Feynman diagrams for bremsstrahlung in a dense medium. If the nuclei are packed densely enough, the amplitudes for these two processes nearly cancel.
(&
frequency w are related by = w. The position z and time are related by the electron velocity. The electron moves more slowly than the photon, so eventually the two particles will separate by more than one photon wavelength, at which point they can be considered two separate particles. Quantum mechanically, the formation length is calculated from the momentum transfer qll from the target nucleus to the electron-photon system; l f = tL/qll. The momentum transfer is fixed by the kinematics (4momentum conservation). For E >> m with initial ( p e ) and final (p:) electron momenta,
where m and E are the electron mass and energy. This holds as long as the photon emission angle Or is smaller than the typical emission angle m / E . Because qll drops as E rises, lf can be very long for high energy reactions. For example, for a 25 GeV electron emitting a 100 MeV photon, I f = 100pm. For astrophysical energies, l f can be hundreds of meters; for E = 10'' eV, k = 5 x 10lgeV, l f = 160 m. As (for k / E fixed) E + 00 and lf -+ m, interactions with the medium as a whole determine the radiation. For pair production, the formation length is given in a manner similar to bremsstrahlung. lf = E ( E - k)/m2c3k= 2hk/(M;c3) where Mp is the invariant mass of the pair. This formula also applies to photoproduction of quark pairs (vector mesons) in hadronic environments. The effects of the formation zone have been discussed in several reviews. 4,5
2
Bremsstrahlung Suppression
Landau and Pomeranchuk3 generalized Bethe and Heitler's concept of radiation from a scattering at a single point, to radiation from the total scattering in a single formation zone. The radiation cannot be localized within the formation zone, so the entire formation zone acts as a single emitter, with radiation determined by the total scattering angle, O M S in this length. 2.1
Suppression due to multiple scattering
Bremsstrahlung from relativistic particles is reduced when the scattering angle is larger than m / E . For an isolated interaction, the usual scattering angle is less than m / E , so this is relatively unimportant. However, in a dense medium, many scatters add (in quadrature) to a single OMS, and the mean scattering angle can be much larger than m / E . When this happens, the radiation is reduced. The scattering decreases the electron forward velocity ( u z ) , making the classical z - w t ) vary more rapidly with z. phase (i. This reduction can also be explained quantum mechanically. Fig. 1 shows the radiation from two pieces of the electron trajectory, separated by a target nucleus. When the nuclei are
55
0.25
0.5
0.75
0.25
1
0.5
0.75
1
y=m
Y
Figure 2: (a) The energy weighted differential cross sections y d u / d y ( y = k / E ) for bremsstrahlung for electrons with energies E = 10 GeV, 100 GeV, 1 TeV, 10 TeV, 100 TeV, 1 PeV and 10 PeV in lead; as E increases, the cross section decreases. The units are fractional energy loss per radiation length. (b) The differential cross sections y d u / d y (Here y = E / k ) for pair production for photons with energies E = 1 TeV, 10 TeV, 100 TeV, 1 PeV, 10 PeV 100 PeV and 1 EeV in lead; as the energy increases, the cross section decreases. The units are fractional energy loss per radiation length.
close enough together (separation much less than I f ) , the amplitudes for emission from adjacent electron lines are almost completely out of phase, and so largely cancel. The target can be thought of as being divided into a small number of independent radiators, each l f thick. In a 1956 work, Migdal treated the multiple scattering using the Fokker-Planck equation, and calculated the radiation for each possible path. This approach gave accurate results for the entire k / E range. However, for pedagogical purposes, this contribution will follow the Landau and Pomeranchuk approach. Bremsstrahlung is suppressed when O M S = ( E / E s ) 2 1 f / Xis~ larger than m / E . Here E, = m c 2 m = 21.2 MeV, Xo is the material radiation length, and a x 1/137 is the fine structure constant. Defining u
ELPM=
~
m4c7Xo - 7.7 TeV/cm . X O , AEZ
(3)
we find O M S > m / E when k / E < ( E - k)/ELpM. The critical energy, ELPMranges from few TeV in solids (4.3 TeV in lead, 2.5 TeV in uranium) up to 540 TeV in water, and 234 PeV in air at sea level. For bremsstrahlung, of course, suppression is present for all E , as long as k / E < E / E L P M .This reduction holds for photons emitted at small angles ( 0 < m / E ) . Large angle photon emission ( O >> m / E ) already requires a large 411 and is much less suppressed. In the regime of strong suppression, the bremsstrahlung cross section is du dk
1
z’
(4)
the photon energy dependence changes. The change is due to the reduced coherence. The amplitudes for each bit of path-length dz in l f add in-phase, so the cross section u = I J A(z)dzI2. If a z-dependent phase reduces the coherence, the cross section is also reduced, even if IAJ is unchanged. A calculation of I f including the additional momentum transfer required due to the multiple scattering will yield Eq. (4). Fig. 2(a) shows the cross section following Migdal’s calculations. In the limit E >> E L P M suppression , is near total except for k x E . Pair production is suppressed if the produced electron and positron multiple scatter enough. This may be calculated from the bremsstahlung formulae using the crossing symmetry that relates the two processes. Fig. 2(b) shows the pair production cross sections. Small M p , symmetric pairs are suppressed the most; large M p asymmetric pairs have a naturally small l f and are less subject to suppression. Suppression is significant for M,” < km2/ELpM. Of
56 course, since Mp 2 2m, suppression requires k > ~ E L P MWhen . suppression is strong, the pair production cross section scales as
the cross section decreases as asymmetric pairs are created. 8.2
A, similar to the
bremsstrahlung case. For k >> E L P M ,very
Suppression due to photon Compton scattering
Photons from bremsstrahlung are also affected by their environment. The photons can Compton scatter off the electrons in the medium. The collective forward Compton scattering introduces a phase shift in the photons. The shift can be described classically using the dielectric constant of the medium: E(k) = 1 - (-)L P 2 (6) k where wp is the plasma frequency of the medium; tLwp is typically 40-60 eV for solids. Now, Irc'lc = &w, giving the photon an effective mass b,.This effective mass (E # 1) introduces a phase shift, reducing l f . The reduction is significant when k < $up.This inequality is satisfied when k / E < tLwp/mc2 M lop4. The latter number applies for solids. For smaller k, the photon mass dominates, and l f = 2ck/hw; and the cross section scales as u x k.
(7)
The k-dependence is drastically changed, and the radiation disappears as k + 0. This is sometimes known as the longitudinal density effect or as dielectric suppression. Migdal allowed for dielectric suppression in his calculations, allowing the combined effect to be calculated As k + 0, qll increases and l f decreases, actually increasing the localization. Baier and Katkov pointed out that, for sufficiently small k / E , the interaction may be localized to within the atomic nucleus, and the nuclear form factor affects the emission. Because of this suppression, the infrared divergence completely disappears! The medium eliminates the need for artificial cutoffs. The total bremsstrahlung cross section is then finite. For example, a 25 GeV electron in lead emits about 10 photons per radiation length.
2.3 Mutual Suppression of Bremsstmhlung and Pair Production At sufficiently high energies, l f > XO. Then, a nascent photon from a bremsstrahlung interaction will convert to an e+e- pair before it is fully formed. The pair production is localized (to within it's own l f ) , and can be used to better locate the bremsstrahlung. This localization reduces the formation length. In the simplest approach one can limit the formation length to Xo. Then U/UBH = xo/lfand du - = ko dk i.e. daldk is independent of k. After accounting for LPM and dielectric suppression, this effect appears for E > Ep = X o w p E B / f i c ,with Ep about 24 TeV in lead, rising to 205 TeV in aluminum, 540 TeV in water and 15 PeV in sea level air. It appears for intermediate energy photons, as is shown in Fig. 3. A similar effect occurs for pair production. One of the produced leptons will emit a bremsstrahlung photon, thereby localizing the pair production and reducing l f . A more sophisticated calculation would treat the bremsstrahlung and pair production as a single Feynman diagram, finding the amplitude for the combined interaction. As the incident energy rises, the amplitudes for more and more generations of the electromagnetic shower become intertwined, and the calculation become intractable.
57
-
Pair ko
-yI
4
b a
k Figure 3: Schematic view of bremsstrahlung d u / d k with several suppression mechanisms, showing the different k ranges. For E < E,, the pair creation suppression disappears and LPM suppression connects with dielectric suppression.
8.4
Finite Targets and Surface Radiation
The calculations discussed so far only apply to infinite targets. Finite thickness targets have surfaces which can affect the radiation. The simplest case is a very thin target, with thickness T less than I f . Then, the target interacts as a whole, and the radiation depends on the total scattering in the target: 1720T zsln() - 1. (9) dk XO Here, dN/dk is the radiation from the entire target. The radiation depends only logarithmically on the target thickness! It is important to accurately model for the Coulomb scattering. The usual Gaussian approximation underestimates the number of large angle scatters, and hence the radiaton. Eq. (9) is based on an an accurate distribution. The radiation will vary from electron to electron, depending on the scattering. Electrons that scatter a lot will emit more radiation than those that scatter less. For thicker targets, it is conceptually simple to consider the radiation as coming from a bulk target plus the entrance and exit surfaces. This surface radiation is naturally treated as a type of transition radiation. As long as absolute rigor is not required, this is a fruitful approach. Conventional transition radiation is due to the adjustment that an electrons electromagnetic fields make when they enter a medium with a different dielectric constant. The fields must make a comparable adjustment when the electron enters a medium with a different amount of multiple scattering per unit length; the multiple scattering causes the electron to jitter and the electron drags the fields with it. Ter-Mikaelian pointed out that both types of transition radiation depend on the difference in formation length between the two media:
2
*
dN dk
- x (If
- I>)?
In the 1960's, several calculations of this transition radiation appeared. Although the basic methods were similar, the numerical results were not. Newer calculations have treated the target as an integrated whole, with good results. These calculations incorporated many advances; a fuller description and references is given in Ref. 5. Almost all of the calculations allowed for electrons traversing arbitrary density profiles. The path is considered as a whole, including interference between radiation from each pair of points on the trajectory. Most of the calculations used accurate models of the Coulomb potential instead of the Gaussian scattering model. Both elastic and inelastic (electron-electron) scattering were included.
58
Figure 4: A block diagram of SLAC Experiment E l 4 6 in End Station A. Electrons traversed a thin target and were bend downward into a set of wire chambers and lead glass blocks. Bremsstrahlung photons travelled 50 m downstream into a BGO calorimeter.
2.5 Experimental Tests Shortly after Migdals work appeared, several groups tried to measure LPM suppression of pair conversion from very high-energy photons in cosmic rays. These early experiments all suffered from very limited statistics, typically 1-50 events. The first accelerator experiment, at Protvino in 1976,used 40 GeV electrons interacting in solid targets. lo Their data indicated that there was some suppression. Later, Kasahara studied the development of xl00 TeV showers in lead/emulsion chambers, and found a significant elongation, matching the LPM predictions. l1 This study is of special interest because it is the only experiment to probe the region E M E L P M . The first accurate measurement was by SLAC experiment E-146,l2>l3in 1993. The collabe ration studied bremsstrahlung of 200 keV to 500 MeV photons from 8 and 25 GeV electrons in 7 different target materials. At least two different thicknesses were used for each target material. The target thicknesses were optimized to trade off between multiple interactions in thick targets vs. transition radiation in thin targets. As Fig. 4 shows, electrons interacted in a thin target and were magnetically bent downward into a set of momentum-measuring wire chambers and electron-counting lead glass blocks. A 45 crystal array (7 by 7, minus the 4 corners), 18 X o thick BGO calorimeter detected the emitted photons The calorimeter segmentation was used to determine the photon position. The collaboration took great care to minimize systematic errors. The experiment took data at 120 Hz,and only bunches containing a single electron were used for the analysis. The electron beamline upstream of the target was kept in vacuum to minimize bremsstrahlung from air. The calorimeter was calibrated by two independent methods. Semi-independent analyses were performed for k > 5 MeV and k < 5 MeV photons. For k > 5 MeV, most photon interactions in the calorimeter were pair conversions. At lower energies, Compton scattering dominated. Also, for the 25 GeV beams, synchrotron radiation from the bending magnets was significant for k < 1 MeV. Most of this radiation was from the downward bend by the spectrometer magnet, with a much smaller contribution from the beam-line steering magnets (not shown). The contribution from the spectrometer magnet was largely removed by dividing the calorimeter into diagonal quadrants. Data from the bottom quadrant, where most of the synchrotron radiation hit, was not used. For k > 5 MeV, the systematic error was 5%. Below 5 MeV, the systematic error was 9% for the 8 GeV electrons, rising to 15% for the 25 GeV electrons. The increase is because of the synchrotron radiation removal cut, because the cut efficiency depends on how well the beam is centered. Some of the €3-146 cross section (do/&) data is shown in Figs. 5 -7. The histogram bin widths are proportional to ln(k). With logarithmic bins, the Bethe-Heitler da/dk x l/k
59 0 15
(a) 2% Xo Carbon
(b) 6% Xo Carbon
01
OM OM 0
0
0 15
0 12 01
01
OW
005
OM
B
8
3
B
OM 0
0 02
1
10
k (MeV)
100
m 0 2
1
10
loo
5M
k (MeV)
Figure 5: SLACEl46 data for 200 keV to 500 MeV photons from 8 and 25 GeV electrons passing through carbon targets. The crws sections are given as dN/d(lnk)/Xo where N is the number of events per photon energy bin per incident electron, for (a) 2% XO carbon and (b) 6% XOcarbon targets in 25 GeV electron beams, while (c) shows the 2% XOcarbon and (d) the 6% XOcarbon target in an 8 GeV beam. Three simulations are shown. The solid histogram shows LPM and dielectric suppression of bremsstrahlung, plus conventional transition radiation. Also shown are the Bethe-Heitler plus transition radiation (dashed histogram) and LPM suppression only plus transition radiation (dotted histogram) simulations.
becomes dg/dln ( k ) which does not vary with k . The Bethe-Heitler simulations (dashed lines) in Fig. 5 are sloped because a single electron traversing a target may interact more than once; the multiple interaction probability increases with the square of the target thickness. Figure 5 shows 3 simulations: one based on the Bethe-Heitler cross section, one with LPM suppression alone, and the last with LPM and dielectric suppression. Both mechanisms are required to explain the data. For the 25 GeV electrons, the upturn in the simulations for k < 500 keV is due to conventional transition radiation. The data has a similar, but larger upturn. This may be due to some remaining synchrotron radiation from the electron bending magnets. For the 25 GeV carbon data, the simulation and the data are not in complete agreement for k < 15 MeV. This discrepancy is puzzling. The amorphous graphite target had a micw crystalline structure, with non-uniform density at a length scale comparable to I f . However, if this were a problem, it should also have appeared in the 8 GeV data. For heavier targets, the suppression is much larger and the agreement is better. For example, Fig. 6 compares the E-146 uranium target data with an LPM plus dielectric suppression simulation (solid line). For k > 5 MeV, the agreement is excellent for both targets at both 8 and 25 GeV. At lower energies, the data is above the simulations. The difference could be due to the additional transition radiation discussed earlier. The other curves in the figure show different predictions of transition radiation; none fit the data. The other targets are believed to be homogenous, with the possible exception of thin oxide layers on the surfaces. The collaboration also studied thin targets. Figure 7 shows the data for a 0.7% X O (3.2pm)
60 0.15
z
(a) 3% Xo Uranium
(b) 5% Xo Uranium
0.1
008
B
006
ow
0.05
B
B
2
OM 0 0 15
0
0 12
8 GeV beam 01
F
01
008
B
006
p a
ow
0 05
02
10
1
1w
50002
1
k (MeV)
10
1w
5w
k (MeV)
Figure 6 SLAC E l 4 6 measurements and simulations for 3% X O and 5% XO uranium targets in 8 and 25 GeV electron beams. The solid histogram shows a simulation with LPM and dielectric suppression and conventional transition radiation. The other simulations use different calculations of transition radiation due to multiple scattering. The solid line in panel (a) is based on Eq. (9).
'*
02
8
(b) 0.7% Xo Gold
02
0 15
0 15
01
01
005
0 05
0
0 02
1
10 k (MeV)
loo
50002
1
10
100
sw
k (MeV)
Figure 7: SLAC E l 4 6 data on 8 and 25 GeV electrons hitting a 0.7% XO gold target. Shown are calculations by Blankenbecler and Drell (dashed line), and using Eq. (9) (dot-dashed line). The solid histogram is a simulation assuming Migdal plus dielectric suppression.
61
thick gold target. For 25 GeV electrons in this target, l f > T for 1 MeV < k < 7 MeV. In this range the data is considerably above the LPM plus dielectric prediction. However, calculations using the thin-target limit, Eq. (9), or based on whole-target approaches work well; the dotdashed line in the figure is based on Eq. (9) while the dashed line is a whole-target calculation by Blankenbecler and Drell. In short, for E << E L P M the , LPM effect seems to be understood to the 10% level in a variety of regimes. For higher energies, the effect is clearly there, but good measurements are needed to confirm the formulae.
3
QCD Analogs
The problem of hadrons traversing a hadronic medium is closely related to that of electrons traversing a charged medium. Reactions involving small momentum transfer are poorly localized, and the overall environment must be considered. l5 This is true at both the hadronic and partonic level. Here, we will consider two very diverse examples: a quark or gluon traversing a nuclear environment, and vector meson photoproduction in a nucleus. Vector meson photoproduction is a fairly well studied process, with many similarities with e+e- pair conversion. l4 A photon fluctuates to a virtual qij pair. The q or ij interacts elastically with the target nucleus, and the pair emerges as a real vector meson. The elastic scattering can be modelled by Pomeron exchange. The details are unimportant here; our interest is in the target atomic number (A) dependence. The nuclear radius R A M l.2A1l3 fm and l f = 2hk/M$, where MV is the vector meson mass. For p production in lead, l f > RA for k > 20 GeV, and the nucleus interacts as a whole. In this regime, one can further differentiate two regimes. For vector mesons where the qq scattering cross section is small ( J / $ or T),the pair will likely undergo a single interaction in the nucleus. Then, the interaction amplitudes from the different nucleons add coherently, and u x A2. If the q?j dipole is large (the p ) , then a single pair can interact with many nucleons. As with the LPM effect, the interference is destructive, and the effects of the interior nucleons disappear. In effect, the qij pair sees only the front surface of the nucleus, with area A2f3,and D x A4/3. At lower k , where l f << RA, the interactions are independent, and u M A. So, simple kinematics considerations lead to 3 different A-scaling regimes, independent of the details of the interaction. In contrast, quark or gluon (parton) energy loss in dense media is still not well understood. A parton produced in a relativistic heavy ion collision will interact and lose energy to the surrounding medium, be it hot nuclear matter or a quark gluon plasma. l6 For most of the energy-loss interactions, l f > RA and interference between multiple interactions is important. The previously discussed 'thin target' approach applies: the energy loss should be calculate based on the total scattering angle. The created partons are bare, without an accompanying colored fields. Because of the retarded potentials, newly created partons (or electrons) will take some time to grow accompanying colored (or electromagnetic) fields. Until these fields are fully formed, the interaction cross sections will be reduced. Because of these effects, the total energy loss grows as L 2 , where L is the distance traversed in the medium. One key question concerns where the lost energy goes. Experiments reconstruct jets by measuring energy deposited in a cone. If the energy lost by the parton is transfered to soft particles which remain in the jet cone, the measured jet energy will be unaffected. The energy loss can only be measured by studying the energy of the individual particles in the jet. However, if the soft particles scatter outside the jet cone, the measured jet energy will be unchanged. Bremsstrahlung suppression is angular; photons emitted with angles Or >> m / E are much less suppressed. So, in the strong suppression limit, 0, >> m / E . Because the angular dependence is entirely due to kinematics, the same effect should apply in QCD. When suppression is strong,
62
radiated gluons are more likely to be outside the jet cone. Unfortunately, although there is broad agreement on many elements of energy loss calculations, different numerical results may vary widely. However, despite the uncertainty, simple kinematic arguments give significant information about the dependence on L and density. 4
Conclusions
High-energy electrodynamic reactions are highly sensitive to their environment. In-medium effects can drastically change the cross sections of bremsstrahlung and pair production. Individual target atoms are indistinguishable, and the electron or photon projectiles interact with the medium as a whole. At the highest energies, it becomes impossible to separate electromagnetic showers into individual reactions, and current theoretical approaches break down. Similar effects occur for hadronic reactions in dense media such as heavy ions. In some cases, kinematics-derived scaling laws can explain the in-medium effects, while in others, more work is needed.
Acknowledgments The author would like to thank Bill Marciano and Sebastian White for organizing an enjoyable conference. This work was supported by the U.S. D.O.E. under contract DE-AC03-76SF00098.
References 1. H. A. Bethe and W. Heitler, Proc. R. SOC.London, Ser. A 146, 83 (1934) 2. E. L. Feinberg, Priroda Vol 1, pg. 30 (1994). 3. L. D. Landau and I. J. Pomeranchuk, Dokl. Akad. Nauk SSSR 92, 535 (1953); L. D. Landau and I. J. Pomeranchuk, Dokl. Akad. Nauk SSSR 92, 735 (1953). 4. M. L. Ter-Mikaelian, High Energy Electromagnetic Processes i n Condensed Media, John Wiley, New York, 1972. 5. S. Klein, Rev. Mod. Phys. 71, 1501 (1999). 6. A. B. Migdal, Phys. Rev. 103, 1811 (1956). 7. V. N. Baier and V. M. Katkov, Phys. Rev. D57, 3146 (1998). 8. V. M. Galitsky and I. I. Gurevitch, Nuovo Cim. 32, 396 (1964). 9. N. F. Shul’ga and S. P. Fomin, JETP Lett. 63, 837 (1996). 10. A. Varfolomeev et al., Sov. Phys. JETP 42, 218 (1976). 11. K. Kasahara, Phys. Rev. D31, 2377 (1985). 12. P. Anthony et al., Phys. Rev. Lett. 75, 1949 (1995); P. Anthony et al., Phys. Rev. D56, 1373 (1997). 13. P. Anthony et al., Phys. Rev. Lett. 76, 3550 (1996). 14. T. H. Bauer, R. D. Spital, D. R. Yennie and F. M. Pipkin, Rev. Mod. Phys. 50, 261 (1978). 15. S. Klein, hep-ph/9808235, in Proc. IVth Wlcshp. on Quantum Chromodynamics, June 1-6, 1998, Paris, France, ed. H. M. Fried and B. Muller, World Scientific, 1999. 16. R. Baier, D. Schiff and B. G. Zakharov, Ann Rev. Nucl. Parti. Sci. 50, 37 (2000).
FUNDAMENTAL PHYSICS AT AN X-RAY FREE ELECTRON LASER A. RINGWALD Deutsches Elektronen-Synchrotron DESY, Notkestraje 85, 0-22607 Hamburg, Germany X-ray free electron lasers (FELs) have been proposed to be constructed both at SLAC in the form of the swx&d Linac Coherent Light Source as well as a t DESY, where the sw called XFEL laboratory is part of the design of the electron-positron linear collider TESLA. In addition to the immediate applications in condensed matter physics, chemistry, material science, and structural biology, X-ray FELs may be employed also to study some physics issues of fundamental nature. In this context, one may mention the boiling of the vacuum (Schwinger pair creation in an external field), horizon physics (Unruh effect), and axion production. We review these X-ray FEL opportunities of fundamental physics and discuss the necessary technological improvements in order to achieve these goals.
1 Introduction
There are definite plans for the construction of free electron lasers (FELs) in the X-ray band, both at the Standord Linear Accelerator Center (SLAC), where the so-called Linac Coherent Light Source (LCLS) has been proposed l , as well as at DESY, where the so-called XFEL laboratory is part of the design of the electron-positron ( e f e - ) linear collider TESLA (TeVEnergy Superconducting Linear Accelerator) 2,3. X-ray free electron lasers will give us new insights into natural and life sciences. X-rays play a crucial role when the structural and electronic properties of matter are to be studied on an atomic scale. The spectral characteristics of the planned X-ray FELs, with their high power, short pulse length, narrow bandwidth, spatial coherence, and a tunable wavelength, make them ideally suited for applications in atomic and molecular physics, plasma physics, condensed matter physics, material science, chemistry, and structural biology In addition to these immediate applications, X-ray FELs may be employed also to study some physics issues of fundamental nature4. In this context, one may mention the boiling of the vacuum (Schwinger pair creation in an external field), horizon physics lo,ll (Unruh effect), and axion production 12913. In this contribution, I shall concentrate on these particle physics opportunities of X-ray FELs. I shall also discuss the necessary improvements in X-ray FEL technology in order to achieve these goals. 1,293.
5,617,8,9
2
X-Ray Free Electron Lasers
Before we enter the discussion of particle physics issues, let us briefly review the principle of X-ray free electron lasers. aInvited talk a t the Workshop on Electromagnetic Probes of Fundamental Physics, Erice, Italy, October 2001.
63
64
L -
Undulator
'
beam dump
distance Figure 1: Principle of a singlepass X-ray free electron laser in the self amplified spontaneous emission mode3
Conventional lasers yield radiation typically in the optical band. The reason is that in these devices the gain comes from stimulated emission from electrons bound t o atoms, either in a crystal, liquid dye, or a gas. The amplification medium of free electron lasers 14, on the other hand, is free (unbounded) electrons in bunches accelerated to relativistic velocities with a characteristic longitudinal charge density modulation (cf. Fig. 1). The radiation emitted by an FEL can be tuned over a wide range of wavelengths, which is a very important advantage over conventional lasers. The basic principle of a single-pass free electron laser operating in the self amplified spontaneous emission (SASE) mode l5 is as follows. It functions by passing an electron beam pulse of energy E, of small cross section and high peak current through a long periodic magnetic structure (undulator) (cf. Fig. 1). The interaction of the emitted synchrotron radiation, with opening angle
l/y = mec2/Ee= 2 . (25 GeVIE,) , (1) where me is the electron mass, with the electron beam pulse within the undulator leads to the buildup of a longitudinal charge density modulation (micro bunching), if a resonance condition,
is met. Here, X is the wavelength of the emitted radiation, XU is the length of the magnetic period of the undulator, and KU is the undulator parameter,
which gives the ratio between the average deflection angle of the electrons in the undulator magnetic field BU from the forward direction and the typical opening cone of the synchrotron radiation. The undulator parameter should be of order one on resonance. The electrons in the developing micro bunches eventually radiate coherently - the gain in radiation power P ,
P
c(
e2 N," B$y2,
(4)
over the one from incoherent spontaneous synchrotron radiation ( P c( N,) being proportional to the number N, 2 lo9 of electrons in a bunch (cf. Fig. 2 (left)) - and the number of emitted
65
10'
lo2
lo3
lo4
lo5
lo6
Energy [eV] Figure 2: Left: Spectral peak brilliance of X-ray FELs and undulators for spontaneous radiation at TESLA, together with that of third generation synchrotron radiation sources3. For comparison, the spontaneous spectrum of an X-ray FEL undulator is shown. Right: Schematic view of the TESLA XFEL electron beam transport (top) and the XFEL laboratory (bottom) '.
photons grows exponentially until saturation is reached. The radiation has a high power, short pulse length, narrow bandwidth, is fully polarized, transversely coherent, and has a tunable wavelength. The concept of using a high energy electron linear accelerator for building an X-ray FEL The feasability of a single-pass FEL was first proposed for the Stanford Linear Accelerator operating in the SASE mode has recently been demonstrated down t o a wavelength of 80 nm using electron bunches of high charge density and low emittance from the linear accelerator at the TESLA test facility (TTF) at DESY. An X-ray FEL laboratory is planned as an integral part of TESLA 2,3 (cf. Fig. 2 (right)). Some characteristics of the radiation from the planned X-ray FELs at TESLA are listed in Table 1.
'.
Table 1: Typical photon beam properties of the SASE FELs at TESLA3
unit wavelength peak power average power numb. photons per pulse bandwidth (FWHM) pulse duration (FWHM)
nm
11
SASE1
I
)I 0.1 + 0.5 I
SASE2 0.085+0.133 110 t 200 610 + 1100
0.07
fs
100
100
0.08 100
I
0.08 100
I
0.29 + 0.7 100
66
3
Applications in Particle Physics
The spectral characteristics of X-ray free electron lasers suggest immediate applications in condensed matter physics, chemistry, material science, and structural biology, which are reviewed in the conceptual L2 and technical design reports of the planned X-ray FEL facilities. In this section, we want to emphasize that X-ray FELs may be employed also to study some particle physics issues. In this context, one may mention the boiling of the vacuum (Schwinger pair creation in an external field), horizon (Unruh effect), and axion p r o d ~ c t i o n ' ~ ~ ~ ~ . Whereas the last application mainly requires large average radiation power ( P ) , the first two applications require very large electric fields and thus high peak power densities P / ( n g 2 ) , where g is the laser spot radius. Here, one could make use of the possibility to focus the X-ray beam to a spot of small radius, hopefully down to the diffraction limit, 0 2 X N O(O.1) nm. In this way, one may obtain very large electric fields and accelerations, 516!738,9
a =
e&_ - 1.9'
P 'I2 f (lTw)
me
(-)0.1nm
,
much larger than those obtainable with any optical laser of the same power. 3.1 Boiling the Vacuum Spontaneous particle creation from vacuum induced by an external field, fmt put forth to examine the production of e+e- pairs in a static, spatially uniform electric field and often referred to as the Schwinger mechanism, ranks among the most intriguing nonlinear phenomena in quantum field theory. Its consideration is theoretically important, since it requires one to go beyond perturbation theory, and its experimental observation would verify the validity of the theory in the domain of strong fields. Moreover, this mechanism has been applied to many problems in contemporary physics, ranging from black hole quantum evaporation l8 to particle production in hadronic collisions and in the early universe 20, to mention only a few. One may consult the monographs 21 for a review of further applications, concrete calculations and a detailed bibliography. It is known since a long time that in the background of a static, spatially uniform electric field the vacuum in quantum electrodynamics (QED) is unstable and, in principle, sparks with spontaneous emission of e+e- pairs17. However, a sizeable rate for spontaneous pair production requires extraordinary strong electric field strengths & of order or above the critical value m c2
m2c3
& e1.3 x 1Ol8 V/m. c - = A = eXe eA
(7)
Otherwise, for & << &, the work of the field on a unit charge e over the Compton wavelength of the electron he = A/(m,c) is much smaller than the rest energy 2m,c2 of the produced efepair, the process can occur only via quantum tunneling, and its rate i s exponentially suppressed, d4 n,+,d3x dt
-N-
C
4 n3Xe
exp [-n
21
.
Unfortunately, it seems inconceivable to produce macroscopic static fields with electric field strengths of the order of the Schwinger critical field (7) in the laboratory. In view of this
67 Table 2: Laser parameters and derived quantities relevant for estimates of the rate of spontaneous e’e- pair production ’. The column labeled “Optical” lists parameters which are typical for a petawatt-class (1 PW = 1015 W) optical laser 35, focused to the diffraction limit, u = A. The column labeled “Design” displays design parameters of the planned XFEL at DESY (“SASE-5’’ in Ref. and Table 1). Similar values apply for LCLS The column labeled “Focus: Available” shows typical values which can be achieved with present day methods of X-ray It assumes that the XFEL X-ray beam can be focused to a rms spot radius of u N 21 nm with an energy extraction efficiency of 1 % 3 6 . The column labeled “Focus: Goal” shows parameters which are theoretically possible by increasing the energy extraction of LCLS (by the tapered undulator technique) and by a yet unspecified method of diffraction-limited focusing of X-rays ‘.
’.
Wavelength Photon energy Peak power Spot radius (rms) Coherent spike length (rms)
x
Peak power density Peak electric field Peak electric field/critical field Photon energy/e rest energy Adiabaticity parameter
s=$
Opticals5 Focus: Diffraction limit 1w 1.2 eV 1 PW 1 Ilm 500 fs i 20 ps
f i w = !E
P U
At
e
=
3
m
.=+ &/,?
eE
3 x 1026 4 x 1014 % 3 x 10-4 2 x 10-6 9 x 10-3
Design
‘
0.4 nm 3.1 keV 110 GW 26 pm 0.04 fs
X-ray FEL Focus: Available 36 0.4 nm
3.1 keV 1.1 GW 21 nm 0.04 fs
5 x 1019
3
8 x loz3
1 x 10”
g
z x 1013 5
1 x 10-7 0.006 6 x lo4
Focus: Goal 0.15 nm 8.3 keV 5 TW 0.15 nm 0.08 ps
3
7 x 1031 2 x 1017
1 x 10-5
0.006 5 x 102
3 2
0.1 0.02 ,
0.1
difficulty, in the early 1970’s the question was raised whether intense optical lasers could be employed to study the Schwinger mechanism24s25. Yet, it was found that all available and conceivable optical lasers did not have enough power density to allow for a sizeable pair creation rate 24,25,26,27,28,29,30,31,32,33,34 With the possibility of X-ray lasers at the horizon, this question has been addressed recently As a quasi-realistic picture of the electromagnetic field of a laser, a pure electric again field oscillating with a frequency w = 2 m / X was considered, under the assumption that the field amplitude & is much smaller than the Schwinger critical field, and the photon energy is much smaller than the rest energy of the electron, 5,617,819.
E
<< E, ,
t u << ~ mec2 ;
(9)
conditions which are well satisfied at realistic X-ray lasers (c. f. Table 2). Under these conditions, it is possible to compute the rate of efe- pair production in a semiclassical manner, using generalized WKB or imaginary-time methods 25,26,28,30,33,34.Here, the ratio q of the energy of the laser photons over the work of the field on a unit charge e over the Compton wavelength of the electron.
plays the role of an adiabaticity parameter. As long as q << 1, i.e. in the strong-field, lowfrequency limit, the non-perturbative Schwinger result (8) for a static uniform field applies. On *At about the same time, the thorough investigation of the question started whether the necessary superstrong fields around E, can be generated microscopically and transiently in the Coulomb field of colliding heavy ions with Z1 Zz > Z, % 170”. At the present time, clear experimental signals for spontaneous positron creation in heavy ion collisions are still missing and could only be expected from collisions with a prolonged lifetime 23. ‘Such a field may be created in an antinode of the standing wave produced by a superposition of two laser beams. In Ref. 34, pair creation in the overlap region of two lasers, whose beams make a fixed angle to each other, was considered.
+
68
”-
g lo+ Y
.-5, U
v)
S
$ lo-“ tj Q ES
s lo-”
Figure 3: Time evolution of the number density of produced e+e- pairs at the focus of an X-ray laser
the other hand, for large 71, i. e. in the low-field, high-frequency limit, the essentially perturbative result
---
d4n,+,d3xdt
c
eeE&
--
4 ~ 3 % ( 4 ~AW~
)
4meczlfw
is obtained for the rate of pair production. It corresponds to the n-th order perturbation theory, n being the minimum number of quanta required to create an e+e- pair: n_>2m,c2/(A~) >> 1. For an X-ray laser, with AW l t l O keV, the adiabatic, nonperturbative, strong field regime, 17 ,< 1, starts to apply for & _> fL~E,/(rn,c~) V/m (c.f. Eq. (10)). An inspection of the tunneling rate (8) leads then to the conclusion that one needs an electric field of about 0.1 &, N 1017 V/m in order to get an appreciable amount of spontaneously produced e+e- pairs at an X-ray laser7. Under such conditions the production rate is timedependent, with repeated cycles of particle production and annihilation in tune with the laser frequency (cf. Fig. 3), but the peak particle number is independent of the laser frequency: up to lo3 pairs may be produced in the spot volumes. In Table 2 we have summarized the relevant parameters for the planned X-ray FELs. We conclude that the power densities and electric fields which can be reached with presently available technique (column labeled “Focused: Available” in Table 2) are far too small for a sizeable effect. On the other hand, if the energy extraction can be improved considerably, such that the power of the planned X-ray FELs can be increased to the terawatt region, and if X-ray optics can be improved 37 to approach the diffraction limit of focusing, leading to a spot size in the 0.1 nanometer range, then there is ample room (c. f. column labeled “Focus: Goal” in Table 2) for an investigation of the Schwinger pair production mechanism at X-ray FELs.
-
3.2
-
Unruh Effect
Black hole evaporation, the so-called Hawking effect and the similar Unruh effect 38 are two fundamental effects in present-day theoretical physics. Both are thermal-like effects involving microscopic degrees of freedom of quantum fields which are not causally connected (event horizons). Experimental detection of Hawking radiation from real, massive (Mbh) general-relativistic black holes seems impossible, since the corresponding temperature, as seen by an outside sta-
69
Polarization
Filter Detector
Figure 4: Schematic diagram of an experiment to detect the Unruh effect at an X-ray free electron laser44.
tionary observer, is tiny, THawking =
hn = 6 . lo-’ K 2xk
~
Ma ( z , ) 1
where K. is the surface gravity of the black hole, k is the Boltzmann constant, and Ma denotes the solar mass. Furthermore, the proposed detection 39 of Hawking radiation from primordial mini black holes, which are relics from the big bang, is rather indirect ‘. Even the detection of condensed matter analogues of Hawking radiation, while more accessible than that from real black holes, is currently far from laboratory realization (see e.g. Ref. 43 and references therein). Under these circumstances, it seems worthwhile to look more closely onto the Unruh effect. It implies that a particle uniformly accelerated by an acceleration a will find itself surrounded by a thermal heat bath at a temperature TUnruh =
ha -= 4. 2xck
We see that enormous accelerations are required to produce a measurable effect. Here, the X-ray lasers come into play: very large accelerations might be available at their focus (cf. Eq. (6)). A scheme how the Unruh effect could be detected at the focus of an X-ray laser ‘‘J1is shown in Fig. 4. In the spot of a standing laser wave, where only an electric field exists, a single electron is accelerated with a N loz6 m/s2, a value which is possible with state-of-the-art means of focusing (cf. Eq. (6) and Table 2, column labeled “FOCUS: Available”). The acceleration of the electron through the vacuum causes a jitter in the electron’s motion, in addition to the usual zero point fluctuations. This jitter modifies the radiation emitted by the electron -over and above the classical Larmor radiation. The additional, acceleration-related radiation has a characteristic a dependence (a distorted thermal spectrum) and angular dependence (cf. Fig. 4). In particular, there is a blind spot in the angular dependence of the classical Larmor radiation. Any radiation in this blind spot should be traceable t o the distortion of the zero-point fluctuations. Whether ultimately one will call this a verification of the Unruh effect or just basic quantum field theory (QED) is a matter of taste or linguistics l l . After all, the Unruh temperature itself will not be measured. Nevertheless, it seems worthwhile to pursue this type of experiment. models of TeV-scale quantum gravity with extra dimensions 40, mini black holes may be generated and their evaporation studied at near-future collider 41 and existing cosmic ray facilities 42.
70
mA (ev)
Figure 5: Exclusion region in mass m~ vs. axion-photon coupling gay for various current experiments (adapted from Ref. 47, where also the corresponding references can be found). Also shown in this figure, and labeled with “SASEB”, is the projected sensitivity l3 of a photon regeneration experiment using the SASE5 XFEL (cf. Table l), as well as the one of a hypothetical XFEL with average power ( P )= 10 GW (“10 GW XFEL”).
3.3 Axion Production An axion (Ao)45 is a hypothetical, very light, weakly coupled (pseudo-)scalar particle. It arises from a natural solution to the strong C P problem: why the effective ¶meter in the Lagrangian of quantum chromodynamics (QCD),
is so small, Oefi ,< lo-’, as required by the current limits on the neutron electric dipole mo1 is perfectly allowed by QCD gauge invariance? Here, a, is the ment, even though Beff are the (dual) gluon field strength tensors. The strong fine-structure constant, and GPUa(GPVa) axion appears as a pseudo Nambu-Goldstone boson of a spontaneously brocen Peccei-Quinn symmetry46,whose scale f A determines the mass,
-
mA = 0.62.
eV x
(15) (lo1;AGeV)
The original axion model, with f A v = 247 GeV being of the order of the scale of electroweak symmetry breaking, is experimentally excluded (see e.g. Ref. 47 and references therein), however so-called invisible axion r n o d e l ~ ~where ~ , ~ f~A , >> v, are still allowed. The interactions of axions with Standard Model particles are model dependent. Of particular importance is the axion-photon coupling gAy, N
where Fclv ( F P ” ) is the (dual) electromagnetic field strength tensor. The quantity E / N is the ratio of electromagnetic over color anomalies, a model-dependent ratio of order one. It is noteworthy however, that two quite distinct models, namely the so-called KSVZ48 (or hadronic)
71
Figure 6: Schematic view of axion production through photon conversion in a magnetic field (left) and subsequent detection through photon regeneration (right) 5 2 .
and the so-called DFSZ4’ (or grand unified) one, lead to quite similar axion-photon couplings, as shown in Fig. 5, which displays the axion-photon coupling (16) , in terms of its mass (15). Also shown in Fig. 5 are the exclusion regions arising from astrophysical considerations and various experiment$. Apart from the laser experiments quoted in Fig. 5, all the others rely on axion production in cosmological or astrophysical environments and aim only at their detection: The microwave cavity experiments assume that axions are the galactic dark matter, the telescope search looks for axions thermally produced in galaxy clusters, and the solar-magnetic and solarGermanium experiments search for axions from the sun. In connection with the X-ray laser, a photon regeneration experiment seems to be appropriate and possible 13. It may be based on the idea 50 to send a laser beam along a superconducting dipole magnet (with E 11 B), in whose magnetic field the photons may convert via a Primakoff process (cf. Eq. (16)) into axions (cf. Fig. 6). If another dipole magnet is set up in line with the first magnet, with a sufficiently thick wall between them, then photons may be regenerated from the pure axion beam in the second magnet and detected. For light axions, with
<<
r n ~
=
1.6. lop3 eV
();l’z -
I
where e is the length of the magnetic field, the axion beam produced is collinear and coherent with the photon beam, and the overall rate for production and subsequent detection is 1 4 (P) rate c( - (gAy B e ) -, 16 W
where ( P ) is the average laser power. A pilot experiment 51,52 with an optical laser, X = 514 nm, of power ( P ) = 3 W, and with a magnet with B = 3.7 T and e = 4.4 m, excluded axion-like pseudescalars with mass mA < lop3 eV and axion-photon couplings QA-( > 6 . 7 . lo-’ GeV-’. The overall envelope of current limits from laser-based experiments is shown in Fig. 5 (“Laser Experiments”). Also shown in this figure, and labeled with “SASE-5”, is the projected sensitivity l3 (cf. Eqs. (17) and (18)) of a possible photon regeneration experiment using the SASE-5 XFEL (cf. Table l), along with a state-of-the-art magnetic field of B = 10 T over e = 10 m. Clearly, in order to reach a sensitivity comparable to the one arising from stellar evolutionf, which excludes gA-, ,> 6 . GeV-’ and is labeled “HB Stars” in Fig. 5, additional efforts are necessary. For example, one might envisage an XFEL with ( P ) = 10 GW, and a magnetic field of B = 40 T over e = 40 m, to obtain a projected sensitivity of gA-( ,> 8 . lo-’’ GeV-’, for mA ,< 10W2 eV. eFor a discussion and references, see Ref. 47. j T h e Cern Axion Solar Telescope 53 (CAST) has a designed sensitivity of ga-, compete with the stellar evolution limit.
,> 5 . lo-”
GeV-l and will
72 4
Conclusions
We have considered several particle physics applications of X-ray free electron lasers, such as spontaneous e+e- pair creation from vacuum, the Unruh effect, and axion production. We have seen that for all these applications still some improvement in X-ray FEL technology over the presently considered design parameters is necessary. But these opportunities appear to be very well worth the effort. In addition, I should point out that the subject of fundamental physics at an X-ray FEL is still in its infancy and in a development phase. No doubt, there will be unpredecedented opportunities to use these intense X-rays in order to explore even more issues of fundamental physics that have eluded man’s probing so far. Acknowledgments I would like to thank H. Mais, A. Melissinos, G. Raf€elt, and J. Roilbach for fruitful discussions and a careful reading of the manuscript. Furthermore, I would like to congratulate Bill Marciano and Sebastian White for organizing such a remarkable workshop. References 1. J. Arthur et al. [LCLS Design Study Group Collaboration], SLAC-R-0521 (1998). 2. R. Brinkmann, G. Materlik, J. Rossbach, and A. Wagner, Hamburg, Germany: DESY (1997) 1183 p. Hamburg DESY - DESY-97-048 (97/05,recSep.) 1183 p. (ECFA 97-182). 3. G. Materlik and T . Tschentscher, TESLA: The superconducting electron positron linear collider with an integrated X-ray laser laboratory, Technical design report, Pt. 5: The X ray free electron laser, DESY-01-011, DESY-2001-011, DESY-01-011E, DESY-2001-011E, DESY-TESLA-2001-23, DESY-TESLA-FEL-2001-05, ECFA-2001-209. 4. T . Tajima, “Fundamental Physics with an X-Ray Free Electron Laser,” subm. to Comments Plasma Phys. Contr. Fusion; T. Tajima and G. Mourou, arXiv:physics/0111091. 5. A. C. Melissinos, in Quantum Aspects of Beam Physics, Proc. 15th Advanced ICFA Beam Dynamics Workshop, Monterey, Calif., 4-9 Jan 1998, ed. P. Chen (World Scientific, Singapore, 1998), p. 564. 6. P. Chen and C. Pellegrini, in Quantum Aspects of Beam Physics, Proc. 15th Advanced ICFA Beam Dynamics Workshop, Monterey, Calif., 4-9 Jan 1998, ed. P. Chen (World Scientific, Singapore, 1998), p. 571. 7. A. Ringwald, Phys. Lett. B 510,107 (2001). 8. R. Alkofer, M. B. Hecht, C. D. Roberts, S. M. Schmidt, and D. V. Vinnik, Phys. Rev. Lett. 87,193902 (2001). 9. V. S . Popov, Pisma Zh. Eksp. Teor. Fiz. 74, 151 (2001) [JETP Lett. 74, 133 (ZOOl)]. 10. P. Chen and T. Tajima, Phys. Rev. Lett. 83,265 (1999). 11. H. C. Rosu, Phys. World 10, 21 (1999); M. Visser, Matters Gruv. 17,4 (2001) [grqc/O102044]. 12. A. C. Melissinos, Nucl. Phys. Proc. Suppl. 72, 195 (1999). 13. A. Ringwald, in preparation. 14. J. M. Madey, J . Appl. Phys. 42, 1906 (1971). 15. A. M. Kondratenko and E. L. Saldin, Part. Accel. 10, 207 (1980); R. Bonifacio, C. Pellegrini, and L. M. Narducci, Opt. Commun. 50, 373 (1984). 16. J. Andruszkow et al. [TESLA Collaboration], Phys. Rev. Lett. 85,3825 (2000). 17. F. Sauter, 2.Phys. 69,742 (1931); W. Heisenberg and H. Euler, Z. Phys. 98,714 (1936); J. Schwinger, Phys. Rev. 82,664 (1951). 18. S. W. Hawking, Nature 248,30 (1974); Commun. Math. Phys. 43,199 (1975); T.Damour
73 and R. Rufki, Phys. Rev. D 14,332 (1976); G. W. Gibbons and M. J. Perry, Proc. Roy. Soc. Lond. A 358,467 (1978); S. P. Gavrilov and D. M. Gitman, Phys. Rev. D 53,7162 (1996); M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85,5042 (2000). 19. A. Casher, H. Neuberger, and S. Nussinov, Phys. Rev. D 20, 179 (1979); B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys. Rept. 97,31 (1983); T.S. Biro, H. B. Nielsen, and J. Knoll, Nucl. Phys. B 245,449 (1984). 20. L. Parker, Phys. Rev. 183,1057 (1969); N. D. Birrell and P. C. Davies, Quantum Fields in Curved Space (Cambridge University Press, 1982). 21. W. Greiner, B. Miiller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer-Verlag, Berlin, 1985); A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Vacuum Quantum Effects in Strong Fields (Atomizdat, Moscow, 1988; Friedmann Laboratory Publishing, St. Petersburg, 1994); E. S. Fradkin, D. M. Gitman, and Sh. M. Shvartsman, Quantum Electrodynamics with Unstable Vacuum (Springer-Verlag, Berlin, 1991). 22. Ya. B. Zel’dovich and V. S. Popov, Usp. Fiz. Nauk. 105,403 (1971) [Sow. Phys. Usp. 14,673 (1972)l; B. Miiller, J. Rafelski, and W. Greiner, 2.Phys. 257,62 (1972); ibid. 257,183 (1972). 23. W. Greiner and J. Reinhardt, in Quantum Aspects of Beam Physics, Proc. 15th Advanced ICFA Beam Dynamics Workshop, Monterey, Calif., 4-9 Jan 1998, ed. P. Chen (World Scientific, Singapore, 1998), p. 438. 24. F. V. Bunkin and I. I. Tugov, Dokl. Akad. Nauk Ser. Fiz. 187,541 (1969) [Sow. Phys. DokZ. 14,678 (1970)l. 25. E. Brezin and C. Itzykson, Phys. Rev. D 2,1191 (1970). 26. V. S. Popov, Pisma Zh. Eksp. Teor. Fiz. 13,261 (1971) [ J E T P Lett. 13,185 (1971)]; V. S. Popov, Zh. Eksp. Teor. Fiz. 61,1334 (1971) [Sow. Phys. J E T P 34,709 (1972)l. 27. G. J. Troup and H. S. Perlman, Phys. Rev. D 6,2299 (1972); V. S. Popov, Zh. Eksp. Teor. Fiz. 62,1248 (1972) [Sow. Phys. JETP 35,659 (1972)]; V. S. Popov and M. S. Marinov, Yad. Fiz. 16,809 (1972) [Sow. J. Nucl. Phys. 16,449 (1973)]; N. B. Narozhnyi and A. I. Nikishov, Zh. Eksp. Teor. Fiz. 65,862 (1973) [Sow. Phys. JETP 38,427 (1974)]. 28. V. S. Popov, Pisma Zh. Eksp. Teor. Fiz. 18,435 (1973) [ J E T P Lett. 18,255 (1974)]. 29. V. M. Mostepanenko and V. M. Frolov, Yad. Fiz. 19,885 (1974) [Sow. J . Nucl. Phys. 19,451 (1974)]. 30. V. S. Popov, Yad. Fzz. 19,1140 (1974) [Sow. J. Nucl. Phys. 19,584 (1974)j. 31. M. S. Marinov and V. S. Popov, Fortsch. Phys. 25,373 (1977). 32. J. I. Katz, Astrophys. J. Supp. 127,371 (2000) [physics/9806027]. 33. G. D u n e and T. Hall, Phys. Rev. D 58,105022 (1998). 34. H. M. Fried, Y. Gabellini, B. H. McKellar, and J. Avan, Phys. Rev. D 63,125001 (2001). 35. M. Perry and G. Mourou, Science 264,917 (1994). 36. W. Graeff, private communication. 37. J. B. Hastings et al., “X-ray Laser Physics”, in LCLS - The First Experiments (September 2000). 38. W. G. Unruh, Phys. Rev. D 14,870 (1976). 39. F. Halzen, E. Zas, J. H. MacGibbon, and T. C. Weekes, Nature 353,807 (1991). 40. N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 429, 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Lett. B 436, 257 (1998); N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali, Phys. Rev. D 59,086004 (1999); L. J. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). 41. S. B. Giddings and S. Thomas, hep-ph/0106219; S. Dimopoulos and G. Landsberg, Phys. Rev. Lett. 87,161602 (2001); S. Hossenfelder, S. Hofmann, M. Bleicher, and H. Stocker, hepph/0109085; K. Cheung, hep-ph/0110163. 42. J. L. Feng and A. D. Shapere, hep-ph/0109106; L. Anchordoqui and H. Goldberg, hep-
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43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.
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PARAMETRIC GRAVITY WAVE DETECTOR G . GEMME ', A. CHINCARINI l, R. PARODI', PH. BERNARD and E. PICASSO
'Istituto Nazaonale di Fisica Nucleare, via Dodecaneso, 33, I-1 6146, Genova, Italy 2CERN, CH-1211, Geneva 23, Switzerland 3Scuola Normale Superiore, piazza dei Cavalieri, 7, I-56126, Pisa, Italy
Since 1978 superconducting coupled cavities have been proposed as a sensitive detector of gravitational waves. The interaction of the gravitational wave with the cavity walls, and the resulting motion, induces the transition of some energy from an initially excited cavity mode to an empty one. The energy transfer is maximum when the frequency of the wave is equal to the frequency difference of the two cavity modes. In 1984 Rewe, Reiner and Melissinos built a detector of the type proposed, and used it as a transducer of harmonic mechanical motion, achieving a sensitivity to fractional deformations of the order 6x/x G lo-''. In this paper the working principles of the detector are discussed and the last experimental results summarized. New ideas for the development of a realistic gravitational waves detector are considered; the outline of a possible detector design and its expected sensitivity are also shown.
1
Introduction
In this paper we shall discuss the mechanism of the interaction of a gravitational wave with a detector based on two coupled electromagnetic cavities. In previous works this issue was discussed using the concept of a dielectric tensor associated with the gravitational wave: The interaction was analyzed in the reference frame where the resonator walls were at rest even in presence of a gravitational perturbation. Here we shall analyze the interaction in the proper reference frame attached to the detector and we shall therefore consider both the coupling between the wave and the mechanical structure of the detector and the perturbation induced on the field stored inside the resonator due to the time-varying boundary conditions. The proposed detector exploits the energy transfer induced by the gravitational wave between two levels of an electromagnetic resonator, whose frequencies w1 and w2 are both much larger than the characteristic angular frequency R of the g.w. and satisfy the resonance condition - w1 = 0.The interaction between the g.w. and the detector is characterized by a transfer of energy and of angular momentum. Since the elicity of the g.w. (i.e. the angular momentum along the direction of propagation) is 2, it can induce a transition between the two levels provided their angular momenta differ by 2; this can be achieved by putting the two cavities at right angle or by a suitable polarization of the electromagnetic field axis inside the resonator. In the scheme suggested by Bernard et al. the two levels are obtained by coupling two identical high frequency ~avities.29~ The angular frequency w1 is the frequency of the level symmetrical in the fields of the two cavities, and w2 is that of the antisymmetrical one. The frequency difference between the symmetric and the antisymmetric level is determined by the coupling and can be adjusted by a careful resonator design. Since the detector sensitivity is proportional to the square of the resonator quality factor, superconducting cavities must be used for maximum sensitivity.
75
76 The power transfer between the levels of a resonator made up of two pill-box cavities, mounted end-to-end and coupled by a small circular aperture in their common end wall, was checked in a series of experiments by Melissinos et al., where the perturbation of the resonator volume was induced by a piezoelectric crystal?^^ Recently the experiment was repeated by our group with an improved experimental set-up? we obtained an order of magnitude sensitivity to fractional deformations of the resonator length as small as a!/! M lo-'' In this paper we shall discuss the detector's working principles and briefly review the last experimental results obtained by our group on the first prototype. Finally a possible detector design, based on two coupled spherical cavities, is discussed and its expected sensitivity is shown. 2
Fundamental principles
When the resonator's boundary is deformed by an external force the local displacement veco t,: C(F,t), can be expresyd as a superposition of the mechanical undamped normal modes {&(q}:C(.',t) = Caqa(t)&(q. qa(t) is the generalized coordinate of the at), mode, obeying the dynamical equation of motion:
and the modes are normalized according to the relation
being p(Z) the mass density and Ma the reduced mass of the ath mode. For a homogeneous system Ma = M , where M is the mass of the system. In eq. 1 an empirical damping term, proportional to the velocity, has been included. f o ( t ) is the generalized force, given by
where f(F,t) is the external force density acting on the system. For a plane g.w travelling along the z axis the force density, in the proper reference frame attached to the detector, has the form:
2 2 ,h12 - h21. where h:(t), is the adimensional amplitude of the wave, and h\ = 4 To study the mechanism of the energy transfer between the two levels of an electromagnetic resonator perturbed by a gravitational wave we shall make use of the fact that the electromagnetic field insideJhe resonator can be expanded over the fields of the normalized, orthogonal normal modes {En(q}and {fin(r3}:7
For simplicity we shall assume that in the frequency range of interest only two e.m. modes give a significant contribution ( n = 1,2), and that the external force couples strongly only to one mechanical mode (a = rn). If we now introduce a perturbation of the resonator boundary and assume that the perturbation is small i.e. that we can still expand the fields inside the
77 perturbed volume over the normal modes of the unperturbed resonator, we obtain the following set of equations for the magnetic field expansion coefficients:
where we have defined the time-dependent expansion coefficients as &i(t) 'Mi(t)z l? . f i i dV and the coupling coefficients as:
fisv
6sv l?.l?i dV ,
The integral in eq. 9 is made on the unperturbed surface of the resonator. The electromagnetic quality factor &, = G, fRs, takes into account the dissipation arising from the finite conductivity of the walls. R, is the material-dependent surface resistance of the walls, and the geometric factor G, of the nth e.m. mode is given by:
In the following we shall assume &I = &2 5 &. The term in the r.h.s. of eq. 8, describes the deformation of the walls induced by the stored e.m. fields, i.e. a bacl-action effect of the e.m. field on the detector's boundary: it is well known that in a resonant cavity the stored magnetic field interacts with the rf wall current, resulting in a Lorenz force which causes a deformation of the cavity shape?i8 The radiation pressure is given by: 1 PL(7,t ) = 2 (pol?(?, t)2 - &(7, t ) Z ) (11)
fgl
Expanding again the fields 2 and l? in terms of the normal modes (eq. 5) we get:
P L ( F ] t )=
Since exp(jwlt), ;Ftz(t) exp(jwzt), &(t) exp(jw1t) and &(t) exp(jw2t) only the cross-product terms will give a significant contribution at the resonance frequency R = wq - w1. The other, rapidly oscillating terms, will just give an average deformation of the detector's walls, determining a static frequency shift of the resonant modes, which can easily be compensated by an external tuning device! The generalized back-action force (cfr. eqs. 3 and 8), acting on the mth mechanical mode of the structure, will be given by: N
N
N
N
78 where we have introduced the coefficients A?$ Js (Bl . I&) frn.dS and f38= Js (31.gz) frn . ds’. The analysis of the system of differential equations 6-8 can be simplified if we neglect the small perturbation on the initially excited e.m. mode (say mode l ) , just taking into account the effects on the initially empty mode. Furthermore we shall consider the coupling between two TE modes of a resonator: for these modes we have vanishing electric field on the resonator surface and, as can readily be calculated, C g = 0 and A; M C$/w2. With this assumptions we can recast the coupled system of equations in the following form:
The solution of quations 14-15 is straightforward when the back-action term is switched off in eq. 15 and if we assume % 1 ( t ) = R(Aexp(iw1t)). In this case we have the following asymptotic solution (in the frequency domain):
and
The field amplitude will be maximum when w
NN w2.
If we design our detector so that
Eq. 18 shows that the amplitude of field in the initially empty mode is proportional to the amplitude of the field in the excited mode A, and to the electromagnetic (Q) and mechanic (Qm) quality factor of the system. A detailed dicussion of the derivation of equations 6-8 and of the noise sources affecting the performance of the detector will be given elsewhere. 3
Experimental results
The electromagnetic properties of a prototype detector, made up of two pill-box cavities, mounted end-to-end, and coupled trough an iris on the axis, were measured in a vertical cry* stat after careful tuning of the two cells frequencies. In order to get maximum sensitivity we need to have two identical coupled e.m. resonators, or, in other words, a flat field distribution between the two cavities. The symmetric mode frequency was measured at 3.03 GHz and the mode separation was 1.38 MHz. In order to suppress the noise coming from the symmetric mode at the detection frequency, the transmission detection scheme, with two magic-tees, was used, as described elsewhere! In figure 2 the signal from the A port of the output magic-tee is shown for an input power Pi = 1 W and no adjustments made on the phase and amplitudes of the rf signal entering and leaving the cavity. The overall attenuation of the symmetric mode is R x -48 dB.
79
Figure 1: Superconducting cavity mounted on the test cryostat RL
--10,0dBm
CENTER
l O u l /
3.0351350Hz
J.C3345E?5OHZ
SPAN
3.000MHZ
Figure 2: Transmission of the symmetric mode (no optimization) measured at the A port of the output magic-tee.
After balancing the arms of the two magic-tees in order to launch the symmetrical mode at the cavity input and to pick up the antisymmetrical one at the cavity output, with 1 W (30 dBm) of power at the C port of the first magic-tee, 6.3 x W (-112 dBm) were detected at the A port of the second one, giving an overall attenuation of the symmetric mode of R M -140 dB (see figure 3). At a detection frequency of R / 2 r M 1 MHz the sensitivity of the system is quite independent from the value of R, because of the high cavity Q. Nevertheless for lower frequencies, in a range R 5 10 kHz, where astrophysical sources of gravitational waves are expected to exist, this noise source can become dominant and the achieved rejection is fundamental in order to pursue the design of a working g.w. detector in the 1-10 kHz frequency range. The cavity loaded quality factor was QL = 1 x lo9 at 1.8 K, and the energy stored in the cavity with 10 W input power was approximately 1.8 J (limited by the maximum power delivered by the rf amplifier), with both the input and output ports critically coupled (PI M Pz M 1). To excite the antisymmetric mode a piezoelectric crystal (Physik Instrument PIC 140, with longitudinal piezoelectric coefficient /q = 2 x 10-l' m/V) was k e d to one cavity wall. The
80 RL
-6O.OdQrn
CENTER
1OdE/
3.035335GHz
3.034485GHz
S P A N
3.000MHz
Figure 3: Transmission of the symmetric mode in the optimized system, measured at the A port of the output magic-tee. Measurement taken with 1 kHz resolution bandwidth.
driving signal to the crystal was provided by a synthesized oscillator with a power output in the range 2-20 mW (3-13 dBm). The oscillator output was further attenuated to reduce the voltage applied to the piezo by a series of fixed attenuators and a variable attenuator (10 dB step). The oscillator frequency was carefully tuned to maximize the energy transfer between the cavity modes. The signal emerging from the A port of the output magic-tee was amplified by the LNA (48 dB gain) and fed into a spectrum analyzer. In figure 2 an example of the parametric conversion process is shown. The minimum detected noise signal level at the antisymmetric mode frequency, with no excitation coming from the piezo, was Potlt(w2)= 5 x lop1' W in a bandwidth Sf = 100 Hz, W/Hz; the main contribution to this giving a noise power spectral density Potlt(wz)= 5 x signal was the johnson noise of the rf amplifier used to amplify the signal picked from the A port of the output magic-tee. Taking into account the input and output coupling coefficients the sensitivity if the system is given by hmin M 3 x 10-20(Hz)-1'2 4
Future perspectives
The second phase of the R&D program is focused on the development of a detector based on two spherical coupled cavities (see figure 4). In order to approach the interesting frequency range for g.w. detection, the mode splitting (i.e. the detection frequency) will be w2 - w1 M 10 kHz. The internal radius of the spherical cavity will be T M 100 mm, corresponding to a frequency of the TEoll mode w = 2 GHz. The overall system mass and length will be M M 5 kg and L M 0.8 m. The choice of these frequencies for the resonator and mode splitting will be also useful in order to test the feasibility of a detector working at M 200 MHz and at a detection frequency of M 1 KHz. A tuning cell, or a superconducting bellow, will be inserted in the coupling tube between the two cavities, allowing to tune the coupling strength (i.e. the detection frequency) in a narrow range around the design value. The choice of spherical cells depends on several factors: From the point of view of the electromagnetic design the spherical cell has the highest
81
Figure 4: Artistic view of the coupled spherical cavities with the central tuning cell.
geometrical factor, and so the highest quality factor, for a given surface resistance.
For the TEoll mode of a sphere the geometric factor G has a value G = 850 0, while for a standard elliptical accelerating cavity the TMolo mode has a value of G = 250 R. Looking at the best reported values of quality factor of accelerating cavities, which typically are in the range lolo - l o l l , we can extrapolate that the quality factor of the TEoll mode of a spherical cavity can exceed Q % l o l l . From the mechanical point of view it is well know that a sphere has the highest interaction cross-section with a g.w. and that only a few mechanical modes of the sphere do interact with a gravitational perturbation (the quadrupolar ones)? The mechanical design is highly simplified if the spherical geometry is used since the deformation of the sphere is given by the superposition of just one or two normal modes of vibration and thus can be easily modeled. In fact the proposed detector acts essentially as a standard g.w. resonant bar detector: the gravitational perturbation interacts with the mechanical structure of the resonator, deforming it. The e.m. field stored inside the resonator is affected by the timevarying boundary conditions and a small quantity of energy is transferred from the initially excited e.m. mode to the initially empty one, provided the g.w. frequency equals the frequency difference of the two modes. A possible design of the detector makes use of both the mechanical resonance of the resonator structure, and the e.m. resonance. This can be accomplished if the detector is designed in order to have the mechanical mode frequency equal to the e.m. modes frequency difference w m = w2 - w1. In particular, for the detector designed to work in the 10 kHz frequency range, the two lowest quadrupolar modes frequency will be approximately at 4 and 17 kHz. The expected sensitivities of the detector for w2 - w1 = 4 kHz and w2 - w1 = 10 kHz are shown in figures 5 and 6. In the calculation of the above curves the brownian motion contribution to the detector noise as well as the noise coming from the detection electronics has been taken into account. Note that, also when w2 - w1 # w m (fig. 6) the sensitivity of the system is fairly good. a DActuallythe sensitivity of the system at 10 KHz is better than the sensitivity at 4 KHz. This is essentially due to the lower value of the brownian noise at higher frequency.
82 hmh = 1 2424s023
10."
3900
39M
3940
3Sa,
3980 40W 1020 FWU-Y Mzl
4040
4080
4080
41m
Figure 5: Calculated system sensitivity for a periodic source after 1 year integration time (urn= u kHz, Q = lo", Qm = 5 x lo3, T = 1.8 K).
0
2
-WI = 4
The spherical cells can be esily deformed in order to remove the unwanted e.m. modes degeneracy and to induce the field polarization suitable for g.w. detection.
The interaction between the stored e.m. field and the time-varying boundary conditions is not trivial and depends both on how the boundary is deformed by the external perturbation and on the spatial distribution of the fields inside the resonator. It has been calculated that the optimal field spatial distribution is with the field axis of the two cavities orthogonal to each other. Different spatial distributions (e.g. with the field axis along the resonators' axis) give a smaller effect or no effect at all. The spherical shape can be easily used to investigate whether the niobium-on- copper technique could be useful for the detector final design. The choice between bulk niobium or niobium-on-copper for the final detector design has not yet been made and is still under investigation. Both techniques present in principle advantages and drawbacks. A prototype of two coupled spherical cavities in bulk niobium will be built at CERN in 2002. A single cell, seamless, copper spherical cavity has been built at INFN-LNL by E. Palmieri and will be sputter coated at CERN. 5
Conclusions
A first prototype of the detector, made up of two pill-box cavities, mounted end-to- end, has been built and successfully tested. A detector based on two coupled spherical cavities is now being designed, and preliminar tests on nomal conducting prototypes are being made. The planned timeline is as follows: 0
0
In 2002 a bulk niobium detector (two spherical cavities, w = 2 GHz, R = 10 kHz, fixed coupling) will be built at CERN; In 2003 a variable coupling detector will be built and tested.
If experimental results will be encouraging, by the end of 2003 a proposal for the realization of a g.w. detector, based on superconducting rf cavities will be made.
83
Figure 6: Calculated system sensitivity for a periodic source after 1 year integration time (urn= 4 kHz, u2 - W I = 10 kHz, 2 ' = lo", Qm = 5 x lo3,T = 1.8 K).
Acknowledgements Several people gave a significant contribution to this work. In particular we wish to thank Prof. C.M. Becchi for his useful suggestions and Prof. A.C. Melissinos for his constant interest in our work and for the fruitful discussions that took place in Erice.
References 1. F. Pegoraro and L.A. Radicati. Dielectric tensor and magnetic permeability in the weak field approximation of general relativity. Journal of Physics A , 13:2411-2421, 1980. 2. F. Pegoraro, L.A. Radicati, Ph. Bernard, and E. Picasso. Electromagnetic detector for gravitational waves. Physics Letters, 68A(2):165-168, 1978. 3. F. Pegoraro, E. Picasso, and L.A. Radicati. On the operation of a tunable electromagnetic detector for gravitational waves. Journal of Physics A , 11(10):1949-1962, 1978. cm harmonic 4. C.E. Reece, P.J. Reiner, and A.C. Melissinos. Observation of 4 x displacement using a 10 GHz superconducting parametric converter. Physics Letters, 104A(6,7):341, 1984. 5. C.E. Reece, P.J. Reiner, and A.C. Melissinos. Parametric converters for detection of small harmonic displacements. Nuclear Instruments and Methods, A245:299-315, 1986. 6. Ph. Bernard, G. Gemme, R. Parodi, and E. Picasso. A detector of small harmonic displacements based on two coupled microwave cavities. Review of Scientific Instruments, 72( 5) ~2428-2437, 2001. gr-q~/0103006. 7. J.C. Slater. Microwave Electronics. D. Van Nostrand Company, Inc., New York, 1950. 8. H. Padamsee, J. Knobloch, and T. Hays. Rf superconductivity for accelerators. John Wiley & Sons, New York, 1998. 9. J.A. Lobo. What can we learn about gw physics with an elatic spherical antenna? Physical Review D, 52:591, 1995.
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QUASI-DEGENERACY IN BOUND-STATE QED. FINE STRUCTURE OF HELIUMLIKE IONS INGVAR LINDGREN, STEN SALOMONSON, BJORN ASEN, and ANN-MARIE MARTENSSON-PENDRILL
Department of Physics, Chalmers University of Technology and Goteborg University, SE 41296 Goteborg, Sweden A new procedure for bound-state QED, based upon a covariant form of the time-evolution operator is reported. This procedure is applicable also t o systems with closely spaced or quasidegenerate levels, where the standard S-matrix formalism can generally not be used due to quasi-singuladies. We have applied the procedure t o the heliumlike ions of neon and argon, and produced the first numerical &ED results for a complete fine-structure splitting. Good agreement is obtained with recent experimental data. The new procedure is closely related t o standard many-body perturbation theory (MBPT), which may open up possibilities to combine QED and MBPT in a more systematic way.
1
Introduction
There is presently a renewed interest in the study of the fine structure of helium and heliumlike ions, the main reason being that accurate comparison of experimental and theoretical data can lead to an independent determination of the fine-structure constant a Quite recently a very accurate measurement of the fine-structure separation 2 3P1 - 2 3P0 in neutral helium has been reported with an accuracy of 30 ppb Compared with accurate analytical calculations this yields a value of the fine-structure constant, 1/137.0359864(31), which differs from the accepted value by four standard deviations. Quite accurate values exist also for the separations in heliumlike ions, some of which are shown in Table 1 The most accurate QED calculations on heliumlike ions are performed by means of the analytical Za expansion, which works well for low Z . For higher 2 the convergence rate decreases drastically, and a rapidly increasing number of terms is needed to reach high accuracy. For high Z , where Z a approaches unity, the power expansion cannot be used, and it is necessary to utilize numerical methods, using a Furry-like picture, which corresponds to an expansion to all orders in Z a . The standard numerical technique for bound-state QED is based upon the S-matrix formulation with the Gell-Mann-Low-Sucher procedure 12,13. Such calculations have been performed and also to some excited states of such ions 18,19. for the ground state of heliumlike ions 14,15,16,17 Generally, however, this technique is not applicable for excited states with multiplet structure, where the states can be very closely spaced or quasi-degenerate. Presently, only two procedures are available for dealing with such problems in its full generality, namely the two-times and the covariant Green’s-function procedure, developed by Shabaev and coworkers evolution-operator procedure, recently developed by us 24,25. 192,314.
’.
6,718,9,10,11.
20921,22,23
85
86 Table 1: Experimental data for some fine-structure separations of He-like ions in the ls2p multiplet. (The values for Z=2 and 3 are in MHz, the remaining ones in cm-')
a
George et al. Myers e t al.
f
Storry et al. ', Castilla et al. Curdt et al. 9 Myers et al. lo
Riis et al. Kukla et al.
Thompson et al.
Another approach to deal with the structure of heliumlike ions is to use many-body procedures of various kind, where the electron correlation can be treated essentially to all orders, and corrected for QED effects by using the analytical expansion. The shortcoming of such an a p proach is that it is difficult to go beyond the leading order for the QED effects. It would therefore be desirable to be able to combine MBPT and the numerical QED approach in some systematic way, which seems to be necessary to achieve sufficient accuracy for light and medium-heavy elements. The present work represents one step in that direction. The atomic fine-structure is entirely due to relativity and QED, with leading orders of (2a)'and ( Z C Y respectively, )~, relative to the non-relativistic energies. For that reason the hestructure is a good candidate for testing the theory, when very accurate experimental results are becoming available, particularly for heliumlike ions. For high 2 a comparison between theory and experiment will test QED at strong electric fields and for low 2 this can be used to determine the fine-structure constant, a, provided the theory of QED is trusted. 2
Time-independent perturbation theory. Quasi-degeneracy
In perturbation theory the Hamiltonian is normally partitioned into a zeroth-order Hamiltonian and a perturbation
H = Ho + HI.
(1)
Working with a single reference or model function, a, the standard (Rayleigh-Schrodinger) perturbation theory gives the first-order contribution to the wave function
The model function, @, is in this case an eigenfunction of HO and forms together with the remaining eigenfunctions the basis functions,
H o @= Eo@ and H o @ = Ei@a. This can easily be generalized to the case with several degenerate model functions, all of which are then excluded from the summation in (2). In the case of quasi-degeneracy (very closely lying states), some of the energy denominators in (2) can be very small, which may cause serious convergence problems. In relativistic manybody perturbation theory, applied to the fine structure in the ls2p multiplet of He-like systems,
87 the states ls2p1/2 and ls2p312 are very close in energy and hence strongly mixed for light elements. This has led to serious problems using a single model function26, problems which can be remedied by using an extended model space We consider now a number of eigenfunctions of the Hamiltonian 27928329.
H'Pa = E a V
((Y=
1,2,...d),
(3)
referred to as the target functions. The relation between the target functions and the corresponding model functions is given by a wave operator,
!Pa = R!Pg
((Y
= 1,2...d).
(4)
In the intermediate normalization we apply here, the model functions are projections of the corresponding target functions on the model space, = 1 , 2 . . . d ).
(5)
[R, Ho]P = ( H ' R - 0PH'R)P.
(6)
'Pg = P!Pa The wave operator satisfies the Bloch equation
((Y
27330328
Expanding the wave operator order by order, leads instead of ( 2 ) to the first-order contribution
where m/i represent basis functions inside/outside the model space. The sum is here restricted to states in the complementary space (Q), outside the model space ( P ) . The Bloch equation can also be used to derive the linked-diagramexpansion for an extended model space, which can be expressed 27!28 [R,Ho]P= ( H ' R - R P H ' R ) L ~ ~ ~ ~ , ~ P .
(8)
The last term represents what is known as folded diagrams. In the general case the model functions are not eigenfunctions of Ho. Instead they are solutions of the secular equation
Hetf is matrix operator, acting in the model space and referred to as the effective Hamiltonian. The eigenvectors of the effectiveHamiltonian are the model functions (4) and the eigenvalues are the corresponding exact energies (3). In intermediate normalization the effective Hamiltonian has the form He# = PHRP = PHoP
+ PH'RP.
(10)
In the MBPT procedure with an extended model space, the states of the model space are excluded from the summations of the type (7), thus eliminating not only degenerate but also quasi-degenerate states. The contribution due to (quasi)degenerate states is included to all orders of perturbation theory by diagonalizing the effective Hamiltonian. This procedure has been shown to improve the convergence rate drastically in the case of quasi-degeneracy 29. In the case of the ls2p multiplet of heliumlike ions the two states (sl12p l / 2 ) J=l and (sl/2 ~ 3 1 have the same symmetry, and for light elements they are closely spaced and hence strongly mixed.
2
)
88 By including these states in the model space their mixture will be appropriately taken care of, which might be quite difficult, using the standard procedure with single reference function. The main question is now: Can a similar procedure be applied in bound-state QED? A major obstacle is here that the standard procedure for bound-state QED, the Smatrix formulation, requires energy conservation between the initial and h a 1 states and hence cannot be used to evaluate the elements of the effective Hamiltonian (9) non-diagonal in energy, needed to treat quasi-degeneracy using an extended model space. Therefore, the procedure has to be modified, as we shall demonstrate below. 3
Time-dependent perturbation theory
In order to find a procedure applicable also to a quasi-degenerate situation, we shall consider time-dependent perturbation theory, although the problems we shall study usually are time independent. The basic tool is here the time-evolution operator, U , which regulates the development of the wave function in time
*(t)
U ( t ,t o ) * ' ( t o )
(t > t o ) .
We work in the interaction picture, where the operators and wave functions are related to those in the conventional Schrodinger picture by a
OI(t)= eiHot0 s e-iHot;
Q I ( t ) = eiHotQs(t).
In this picture the time-dependent Schrodinger equation takes the form
d dt
i-QI(t)
= Hf QI(t),
which leads to the expansion 31
where T is the time-ordering operator. Due to the adiabatic damping, the perturbation vanishes in the limits t + +m and the exact wave function becomes equal to the unperturbed function in these limits, Q, = *(hm). The wave function in time-independent theory corresponds to the time-dependent function at t = 0, Q = *(O). Gell-Mann and Low l2 have shown that for a single reference function (closed-shell case), Q,, the function
is a normalized eigenfunction of the Hamiltonian (provided the limit exists). The evolution operator has singularities due to unlinked diagrams, which are eliminated by the denominator. When several reference or model functions are used (multi-dimensional model space), the Gell-Mann-Low procedure has to be modified 32,33,
"We use here relativistic units:
c = m = t2. = eo =
1; e2 = 4 ~ a .
89
6"are here certain 'parent' functions in the model space. In this case the evolution operator has singularities or quasi-singularities also for linked diagrams, when a model-space state appears as an intermediate state. This corresponds to the folded diagrams in time-independent MBPT, mentioned above. In order to treat the (quasi)singularities, we rewrite the evolution operator, operating on the model space, as 25
+ U ( t ,-00)PU(O,
U ( t ,-00) = 1
-03)'
(13)
leaving out the subscript y. This leads to the expansion
-
u
= l+U+VPU+UPUPU+...
U
= u-1-VPU-UPUPU
U is referred to as the reduced evolution operator and can be shown to be regular, i.e.,fre_e from (quasi)singularities. The (quasi)singularities are eliminated by the counterterms (-UPU etc.) in (14). For t = 0 the definition (13) becomes U ( 0 ,-00) = 1
+ U(0,- m ) P U ( O , -m),
which leads to
U ( 0 ,-00) = ( 1 iQ U(0, -oo))PU(O, -m). Inserted in the generalized Gell-Mann-Low relation (12) this gives Qa = Qa(0) = [1+ Q U(0, -m)] qg,
where
This we can compare with the MBPT equation above ( 4 ) , showing that the wave operator becomes
I0
=
1 + Q U(0,-m).
I
The effective interaction is defined
H& = he^
~
PHoP
and this can be expressed by means of the reduced evolution operator as34,25
4
4.1
Bound-state QED
S-matrix
The S-matrix is related to the time-evolution operator by S = U ( m , -00). The S-matrix for single-photon exchange between electrons, represented by the Feynman diagram in Fig. 1, is given by
90
Figure 1: The S-matrix for single-photon exchange
Here b ,
4(z)= cj (bj(z);
Gt(z) = c; (b;(z)
(18)
are the electron-field operators in the interaction picture, y is a n adiabatic-damping factor, which eventually goes t o zero, y + +O, and 1(z2,21) = e2c$~2V~ ~ ~ ~ ( z 2 , z l )
(19)
represents the electron-electron interaction, DF~,,(Q,2 1 ) being the photon propagator. Identification with the second-quantized expression,
yields the 'matrix element'
where
The relation between the S-matrix and the energy shift is given by the Sucher formula 13,31
bWe use here the Furry pzcture, where the single-electron orbitals are solutions of the Dirac equation in the field of the nucleus (and possibly other electrons), V ( T ) , hD
'$3
= [P' Q
+ P + v(T)]
'$3
= € 3 63.
c,'/c3 are the corresponding creation/annihilation operators. We apply the summation convention with summation over repeated indices.
91
+ t = t’
t = t’
r
;
r
l
a
t = t’ Figure 2: The covariant evolution operator for single-photon exchange.
which in the lowest order ( n = 2) becomes
In the present case this gives
The Kronecker delta factor implies here that the result is nonvanishing only for q + q’ = 0 or E, &b = E, + E ~ which , means that energy must be conserved between the initial and final states. This makes the method inapplicable for treating quasi-degeneracy, using the method of extended model space.
+
4.2
Covariant form of the evolution operator
We shall now demonstrate how the evolution operator, discussed in Sec. 3, can be used to treat quasi-degeneracies also in QED problems. In this case, however, it is necessary to allow time to run also in the backward direction in order to include the contribution from positron states. Therefore, we have to generalize the standard procedure, as illustrated for single-photon exchange in Fig. 2. This corresponds to the e x p r e s s i ~ n ~ ~ ~ ~ ~
where 0 is the Heaviside step function. The time integrations are here performed over all times, making this operator covariant - like the S operator but in contrast to the evolution operator for forward evolution only (11). The operator (26) can also be expressed by means of the electron propagator as
92 which yields the matrix element
where y, has the same sign as E,.
Using the Feynman gauge we obtain
e2 f ( k ) = -(1- a1 . cq) sin(krl2) 47r2 r12 The contributions to the first-order wave operator (15) and effective interaction (16) then become
In the first case the state ITS) lies in the complementary space (Q) (outside the model space, P ) and in the second case inside the model space. The effective interaction agrees with the S-matrix result, when the initial and final energies are the same. But the important point is here that the evolution-operator result is valid also when these energies are different, making the procedure applicable also to an extended model space with several unperturbed energies. Another important advantage of the evolution-operator method is that - in contrast to the S-matrix formulation (23) - no limiting procedure of the adiabatic damping factor y is needed. (This factor is here needed only to indicate the position of possible poles.) This implies that the integrations can be performed independently with ezuct energy conservation at each vertex. This simplifies considerably the treatment of (quasi)singularities, as we shall demonstrate with the two-photon exchange below.
4.3
Two-photon exchange
There are two Feynman diagrams representing the two-photon exchange between the electrons, the ’ladder diagram’ and the ’crossed-photon diagram’, illustrated in Fig. 3. The covariant evolution operator for the ladder diagram is given by
93 s
u
2
-; r- L-3 t-
4
I
s
4
I
1
r
h
s
a
r
l
s
- ;- r- -
a
t
t = t‘
h
I
-
- a-
1
~
1
t
= t’
Figure 3: Twephoton exchange diagrams, the ladder diagram (left) and the crossed-photon diagram (right)
The ladder diagram has a (quasi)singularity, when the intermediate state (tu) is (quasi)degenerate with the initial state (ab). In order to study this (quasi)singularity, we first separate the ladder into a separable and a non-separable part. A diagram is said to be separable, if the two photons do not overlap in tame, or in other words that there exists an intermediate time (t = t”), for which there is no free photon, as indicated in the leftmost diagram in Fig. 4. If the intermediate state of a separable diagram lies in the model space, the diagram is said to be reducible. It can be shown that the separable two-photon diagram is given by 25
where q = E, - E ~ q’; = Eb - E,; p = E, - E t ; p’ = Eb - E, and v ( q , q ’ ) is given by (30). The corresponding contribution to the effective Hamiltonian is then, using (16),
which is (quasi)singular, when the intermediate state (tu) is (quasi)degenerate with the initial state (ab) ( p p’ FZ 0 ) . In the expression for the effective interaction (16), the reduced evolution operator (11) appears, which in the present case is given by
+
G(4) = u(4)-
u(Z)pu(2).
(36)
The counterterm, - U ( 2 ) P U ( 2 ) is , here
+
+
) AE =p If we introduce W ( E 0 )= V ( q p’, q’ p ) = V(E0 - E~ - E,, EO- E, - ~ t with EO- E t - E,, the contribution to the effective Hamiltonian can be expressed
This expression is regular
-
free from (quasi)singularities. When A E
-+
+ p’
=
0, the result becomes
which identical to the S-matrix result. The result (38), however, is more general and (asthe single-photon result) valid for a model space that is not necessarily degenerate. A similar result can be derived for the wave operator, which is also regular.
94
T
l
T
i
t
a
T
l
T
t
a
= t’ t = t” t = t’ t=O Figure 4: The reduced evolution operator for the two-photon exchange. The second diagram represents the counterterm U ( 2 ) P U ( 2in ) (36)
t
When a separable two-photon diagram is irreducible (not reducible), then there is no counterterm, and the contribution to the effective Hamiltonian is given by (35). The summation is then performed only over intermediate states in the complementary space (Q), and there is no (quasi)degeneracy. This is then quite similar to the standard expression of second-order perturbation theory (2), which is entirely contained in the separable two-photon diagram. In addition, the separable two-photon diagram contains QED effects - beyond standard MBPT due to negative energy states and retardation. The remaining, non-separable part of the ladder diagram is always regular, and this is also the case of the crossed two-photon diagram, which has no reducible part. These parts can be evaluated in a straightforward manner. The non-separable diagrams represent pure $ED effects - with no analogy in MBPT. The above illustrates the close analogy between standard MBPT and bound-state QED using the covariant-evolution-operatormethod - a fact which may open up possibilities of combining &ED and MBPT in a more systematic fasion than has previously been possible. 5
Rcsults and discussion
We have applied the covariant-evolution-operatormethod to evaluate the fine-structure separation of the ls2p multiplet of the He-like ions of Ne and Ar, and the results are reproduced in Table 2. Our calculations include the exchange of one and two photons between the electrons, as described above, which contains relativistic many-body effects to second order in addition to the QED effects. Furthermore, we have included relativistic electron correlation effects beyond second order from a separate many-body calculation. The remaining QED efects, not included in the diagrams evaluated, namely the self-energy and vacuum-polarization contributions, are estimated by means of the analytical Za expansion. Our numerical results are compared with those of Drake 35 and Plante et al. 26, as well as with available experimental results ll,’. Our calculation represents the first application of bound-state QED to a quasi-degenerate he-structure separation, using a numerical technique to all OTdeTS of Za. The results of our calculations agree very well with available experimental data. Our results agree also well with those of Drake and Plante et al. The calculations of Drake are performed using highly correlated non-relativistic wave functions of Hylleraas type with relativistic and QED effects estimated from the analytical expansion. In the work of Plante et al. a relativistic MBPT procedure is used, with the QED correction separately added. For the elements presented here the difference between the theoretical results is hardly significant compared to the experimental and theoretical uncertainties. (For the argon ion the result of Drake differs from the experimental result by 2-3 standard deviations, which is likely due to inaccuracy in the relativistic correction.) When higher accuracy is needed, however,
95 Table 2: Comparison between experiment and theory for the fine-structure separations the ls2p multiplet of some He-like ions (in /I Hartree N 27,2 /I eV).
I z 1 3 ~ 1 - 3 ~ I 0 3 ~ z - 3 ~ 0 jP2 -3
1
Pl
Drake 35 PIante et al.
II 1 18
I
I
23600(60) 23690 23792
8464 124960(30) Expt'l 124810(60)' Drake 3s 124942 Plante et al. 124938
26
higher-order QED effects will be important. Already for argon we have found that the effects beyond the leading (Z(11)3order are at least comparable to that of the leading term. As can be seen from Table 1, quite accurate experimental data are available for neutral helium as well as for the ions of lithium and fluorine. For very low Z the numerical problems increase, due to slow convergence of the partial-wave expansion. In addition, the electron correlation plays here a relatively more important role, and two-photon exchange may not be sufficient. For these reasons, it will be difficult at present to compete with the analytical results for He and Lif. In order to achieve higher accuracy with the numerical technique, it would be necessary to combine the QED and MBPT procedures in some way, and developments along these lines are in progress at our laboratory. A good test of the present procedure described here, however, would be to try to reproduce the experimental result of the fluorine ion, which has an accuracy of the order of one ppm. Here high-order QED effects will certainly be significant. Such an accurate numerical result could then be used to test the analytical expansion and possible to estimate higher-order non-calculated terms of that expansion. In this way the numerical results could contribute to the accuracy of the theoretical result also for very low Z and possibly to the evaluation of the he-structure constant. Acknowledgments
This work has been supported by the Swedish Natural Science Research Council and the Alexander von Humboldt Foundation. References
1. F. Minardi, G. Bianchini, P. C. Pastor, G. Guisfredi, F. S. Pavone, and M. Inguscio. Phys. Rev. Lett., 82:1112, 1999. 2. C. H. Storry, M. C. George, and E. A. Hessels. Phys. Rev. Lett., 84:3274, 2000. 3. J. Castillega, D. Livingston, A. Sanders, and D. Shiner. Phys. Rev. Lett., 84:4321, 2000. 4. K. Pachucki and J. Sapirstein. J. Phys. B, 33:5297, 2000. 5. M. C. George, L. D. Lombardi, and E. A. Hessels. Phys. Rev. Lett., 87:173002, 2001. 6. E. Riis, A. G. Sinclair, 0. Poulsen, G. W. F. Drake, W. R. C. Rowley, and A. P. Levick. Phys. Rev. A , 49:207, 1994. 7. J. K. Thompson, D. J. H. Howie, and E. G. Myers. Phys. Rev. A , 57:180, 1998. 8. E. G. Myers, H. S. Margolis, J. K. Thompson, M. A. Farmer, J. D. Silver, and M. R. Tarbutt. Phys. Rev. Lett., 82:4200, 1999. 9. W. Curdt, E. Landi, K. Wilhelm, and U. Feldman. Phys. Rev. A , 62:022502, 2000.
96
10. E. G. Myers and M. R. Tarbutt. Phys. Rev. A , 61:010501R, 1999. 11. K. W. Kukla, A. E. Livingston, J. Suleiman, H. G. Berry, R. W. Dunford, D. S. Gemmel, E. P. Kantor, S. Cheng, and L. J. Curtis. Phys. Rev. A , 51:1905, 1995. 12. M. Gell-Mann and F. Low. Phys. Rev., 84:350, 1951. 13. J. Sucher. Phys. Rev., 107:1448, 1957. 14. S. Blundell, P. J. Mohr, W. R. Johnson, and J. Sapirstein. Phys. Rev. A , 48:2615, 1993. 15. I. Lindgren, H. Persson, S. Salomonson, and L. Labzowsky. Phys. Rev. A, 51:1167, 1995. 16. H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgren. Phys. Rev. Lett., 76:204, 1996. 17. V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, and G. Soff. Phys. Lett. A , 234:361, 1997. 18. P. J. Mohr and J. Sapirstein. Phys. Rev. A, 62:052501, 2000. 19. B. &n, S. Salomonson, and I. Lindgren. Phys. Rev. A , acceptcd, 2002. 20. V. M. Shabaev. J . Phys. B, 26:4703, 1993. 21. V. M. Shabaev. Phys. Rev. A , 50:4521, 1994. 22. V. M. Shabaev and I. G. Fokeeva. Phys. Rev. A , 49:4489, 1994. 23. V. A. YeroWlin, A. N. Artemyev, T. Beier, G. Plunien, V. M. Shabaev, and G. Soff. Phys. Rev. A , 60:3522, 1999. 24. I. Lindgren. Mol. Phys., 98:1159, 2000. 25. I. Lindgren, B. a s h , S. Salomonson, and A.-M. Mktensson-Pendrill. Phys. Rev. A , 64:062505, 2001. 26. D. R. Plante, W. R. Johnson, and J. Sapirstein. Phys. Rev. A, 49:3519, 1994. 27. I. Lindgren. J. Phys. B, 7:2441, 1974. 28. I. Lindgren and J. Morrison. Atomic Many-Body Theory. Springer-Verlag, Berlin, 1986. 29. A.-M. Mirtensson-Pendrill, I. Lindgren, E. Lindroth, S. Salomonson, and D. S. Staudte. Phys. Rev. A, 51:3630, 1995. 30. V. KvasniEka. Adv. Chern. Phys., 36:345, 1977. 31. S. S. Schweber. A n Introduction to Relativistic Quantum Field Theory. Harper and Row, N.Y., 1961. 32. T. T. S. Kuo, S. Y. Lee, and K. F. Ratcliff. Nucl. Phys. A , 176:65, 1971. 33. G. Oberlechner, F. Owono-N’-Guema, and J. Richert. Nouvo Cimento B, 68:23, 1970. 34. R. W. Jones and F. Mohling. Nucl. Phys. A , 151:420, 1970. 35. G. Drake. Can. J. Phys., 66:586, 1988.
PHOTON COLLIDERS IN PHYSICAL PROGRAM FOR LARGE COLLIDERS Talk given at Workshop on Electromagnetic Probes of Fundamental Physics, Eriche, Sicily, 16-21 October 2001 I.F. Ginzburg', S.I. Polityko2 Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia; Irkutsk State University, Irkutsk, Russia In this report our main attention is focused on the problems which can be solved at Photon Colliders naturally but are hardly solved at LHC and e'e- Linear Collider: The New Physics new particles, new interactions, supersymmet y, Dirac monopole, extra dimensions.. . SM and the case of a SM-like scenario - EWSB, Higgs bosom, anomalies in intenctiom of gauge bosom, QCD, Pomeron and odderon.. . By-product: Production of light Goldstone particles from region of e + y conversion. ~
1 Introduction
Photon Collider will be a specific mode of Linear Collider (LC) obtained with the aid of laser light backscattering on the accelerated electrons in Linear Collider near the collision point Let us enumerate its main characteristics in the frame of TESLA project '. 1. Characteristic photon energy E, = 0.8E ( E - energy of electron in the basic e+e- collider). 2. For high energy peak, E , ~ J > 0.7E7,,, (separated well from low energy part of spectrum) 0 Luminosity L ,, FZ L,,/3, L,, = L,,/4 with L,,dt, S L,,dt = 200 + 150 fb-'/year. 0 Mean energy spread < AE, > X O.O7E, (by factor 2 + 3 worse than in efemode considering beamstrahlung and ISR). 0 Mean photon helicity < A, >= 0.95, with easily variable sign. One can also get the linear polarization. (In the e+e- mode only longitudinal polarization is relevant.) 3. The e + y conversion region is ey collider with , & N 1.2 MeV and L 0.1 fb-'/sec! 4. The total additional cost is estimated as about 10% from that of ete- LC. '9'.
s
N
The Standard Model (SM) is verified now with high precision except for mechanism of electroweak symmetry breaking (EWSB). The new large colliders are constructed with the dream to find New Physics and understand EWSB mechanism. We discuss different physical pictures obtained after running of large colliders of next generation (Tevatron, LHC and LC): 1. Clear signals of New Physics (new particles, strong deviations from SM) will be found. 2. The physical picture will coincide with that expected in SM within experimental precision - SM-like scenario, determined for the time of observations: 0 No new particles and interactions will be discovered except for single Higgs boson. The couplings of Higgs boson to W , 2 and quarks will coincide with those predicted in S M within the experimental precision.
97
98
We discuss the potential of Photon Collider in both these scenarios separately. For more details on some mentioned points see refs. 2,4. 2
Hunting for New Physics
2.1 Discoveq of new particles
The discovery of a new particle will be a clean signal of some definite form of a new theory. In the discussion below we denote the kinematically allowed discovery bound for its mass as Mb. The production of pair of charged particles in yy collisions near the threshold is described by QED with reasonable accuracy. The corrections due to other (even strong) interactions can be neglected in the estimates of opportunity to discover the particle. Therefore, the real discovery limit of new particle in yy channel is close to Mb FZ 0.8E, it is lower than that in e+e- mode. The ey collisions provide us with final states which cannot be produced with similar intensity other ways. In reactions like ey -+ B A with light particle A and new particle B the kinematical discovery bound Mb can be much higher than in other reactions, Mb 5 1.8E. However, the cross section of such process depends on new coupling constants like eAB. Therefore - in contrast to yy collisions - the absence of such signals can be explained by absence of new particle B as well as by its "electrofobic" nature (very small eAB coupling). W The ey mode provides the best opportunities for discovering a number of new particles. The discovery limits Mb of some new states are presented in Table 1. For the excited
1 e* u,'
W' S U
W
2
name excited e excited v new W
1
reaction ey 3 e* e y + Wu,' e y +vW'
I
M < Mh 1.8E 1.8E - MW 1.8E
ey + W x 1.8E - M , ey - + Z E 1.8E - M; ey +Ex 1.8E - M , Y - LSP. lightest suDerDarticle
wino zino selectron
1
I
1
observed e y or eZ eW wz. W r
1
1
Table 1: Some discovery limits for ey mode
electron the expected cross section is high enough to observe it even with the weak enough coupling constant.
The yy mode. Pair production. The cross section of the pair production yy + P f P ( P = S - scalar, P = F - fermion, P = W - gauge boson) not far fr_oom_the threshold is given by QED as (Xi - circular, t i - linear polarization of photons, 4 = L(!l,'2))
(with functions f:, g$ and g$ written e.g. in ref. 15). These f : (for the unpolarized photons) are shown in Fig. 1 ( W 2= s). The yy cross sections are evidently higher than the corresponding e f e - collisions, which are also shown here a. "The observable e'e+ P+P- cross section includes also 2' contribution, dependent on other quantum numbers besides charge and spin. Typically, at s >> M$ we have u(e+e- + P f P - ) < 1.3u(e+e- + y' +
P+P-).
99 0.14
0.7
0.12
0.6
0.1
yy
+ s*s-
0.5
0.0s
0.4
0.06
0.3
0.04
0.2
0.02
0.1
0
20
10
0 x =W2/4M2
Figure 1: The yy + P'P- cross sections, divided by (mx2/M;), nonpolarized photons, C, = 1. The sections e'e- + y' + P'P- are also shown.
CTOSS
With expected luminosities the first discovery of new particle is preferable in efe- mode. After that, the key problem will be to study the nature of the discovered particle. In this respect, the yy production provides essential advantages compared to e+e- collisions: These cross sections decrease slow with energy growth. Therefore, one can study these processes relatively far from the threshold where the decay products don't overlap. Near the threshold fp 0: (1f X l X 2 & cos 24) with sign for P = S (scalar) and - sign for P = F (fermion). This polarization dependence provides the opportunity to know spin of produced particle independent on its charge. (This problem arises, e.g., at the discovery of SUSY particles since spin of invisible neutral is unknown). The possible CP violation in the P y interaction can be seen as a variation of cross section with changing the sign of photon helicity.
+
H The discovery of some other particles in yy mode. T h e leptoquark ( l q ) can be discovered in reactions like yy -+ t ( l t )with h/r, = 1.5E7. T h e scalar o r tensor resonances R appearing due t o the strong interaction in Higgs sector can be discovered in process yy R with Mb = 2E,. (via quark loop) '. The maximal value T h e gluino 9 can be produced in process yy of this cross section is (a2a~/Mg)ln(M~/M at ~2M3 ) < J"rr < 2M*. For example, = 1.5 TeV. this cross section is about 1 pb at M,j = 0.5 TeV, M3 = 0.25 TeV for If the stop squark is not too heavy, the very narrow atom-like (scalar) stoponium with mass 200-400 GeV should also exist (such states cannot be observed at hadron collider). It can be clearly seen at yy collider with cross section averaged over photon spectrum < 0 >= 10 - 50 fb and clear enough signature6.
e+ +
---f
--f
N
2.2
Some problems with new interactions
Due to high values of the basic cross sections for the pair production of charged particles, Photon Collider would be an excellent place for observation of (even small) possible flavor changing neutral currents (FCNC), for example, with superparticles. In the 2HDM or MSSM in the CP conserving case the masses of heavy Higgs scalar H and pseudoscalar A can be close to each other or they are mixed (in the CP violated - (23' - case). The observations of decay products at LHC and e f e - LC cannot resolve these opportunities due to low resolution for these bosons. The polarization asymmetries in Higgs boson production at Photon Collider can resolve these variants, i.e. establish, whether CP parity in Higgs sector is violated or not.
100
Using the yy -+ 77, etc. processes for the search of effects from extra dimensions or heavy point-like Dirac monopole. In both cases the process is considered strongly below new mass scale M or particle production threshold 2M. In both cases the cross section can be written as
with specific polarization dependence and angular distribution ( S and D waves, roughly isotropic). This wide angle elastic yy scattering has very clear signature and small QED background. The observation of strong elastic yy scattering raising quickly with energy will be the signal of one of these mechanisms. The study of polarization and angular dependence at photon collider and some similar processes can discriminate what mechanism is relevant. 0 Effects of extra dimensions are considered in the scenario where gravity propagates in the (4 n)-dimensional bulk of space-time, while gauge and matter fields are confined to the (3+l)-dimensional world volume of a brane configuration. The extra n dimensions are compactified with scale R what produces the Kaluza-Klein excitations having masses 71-nlR. The corresponding scale in our world is assumed to be M N few TeV. The particles of our world interact (as AA + BB) via the set of Kaluza-Klein excitations having spin 2 or 0 as e.g. TP”TP”/M4, where T P ” is stress-energy tensor. The coefficients are accumulated in the definition of M (with A = 1). The yy initial state has numerical advantage as compared to e+e- one. The yy final state has the best signature and the lowest SM background. The interference (with SM) effect enhances this anomaly for yy + WW process (simultaneously with enhancement of background). 0 Point-like Dirac monopoles. This monopole existence would explain mysterious quantization of the electric charge. There is no place for it in modern theories of our world but there are no convincing reasons against its existence. At s << M2 the electrodynamics of monopoles is expected to be similar to the standard QED. At g&‘/(47rM) < 1 (with g = n / ( 2 e ) ) the effect is described by monopole loop, and one can use coefficient A cx gs, calculated within QED. Both A and details of angular and polarization dependence depend strongly on spin of monopole J , e.g., A ( J = l ) / A ( J = 0) N” 1900. Effect can be seen at TESLA500 at M < 4 - 10 TeV (depending on monopole spin). Modern limitation (Tevatron) is about 10 times lower.
+
3
SM and SM-like scenario
If after experiments at LHC and e+e- LC a SM-like scenario (defined in the beginning of report) will be realized, the main problems for study at Colliders become the following 0 The study of EWSB mechanism. 0 Discovery of signals of New Physics via (small) deviations from SM predictions. 0 Description of observed phenomena in SM, especially QCD: 0 Total cross sections, Pomeron and odderon, minijets. 0 Photon structure function ’). 0 Some problems with heavy quarks. Photon Colliders provide new (sometimes unique) keys for solving all these problems. 3.1
The study of E WSB
Assuming Higgs mechanism for EWSB, one should consider 3 main opportunities: SM with one Higgs doublet and Mh < 400 t 700 GeV - standard SM case. 0 More complex Higgs sector, the simplest variant - Two Doublet Higgs Model (2HDM). In these cases Higgs boson will be discovered at the Tevatron or LHC, its spin and couplings to W, 2 and fermions will be precisely measured at e+e- LC.
101 0 SM with one Higgs doublet but with strong Higgs self-interaction. In this case standard Higgs particle does not exist and strong interaction in Higgs sector will manifest itself as the strong interaction of gauge bosons W and Z (their longitudinal components). The measuring of hyy and h Z y couplings is an excellent tool for these problems. Indeed, 0 In the SM these couplings appear only at the loop level. Therefore, the S/B for new signals is better than that for the processes allowed at tree level. 0 All fundamental charged particles contribute to these effective couplings. 0 The expected accuracy in the measurement of the two-photon width is N 2% at the luminosity integral 30 fb-' and Mh 5 150 GeV lo. This uncertainty can be reduced with the expected 3-year luminosity integral about 500 fb-'. .The anomalous interactions of Higgs boson with light l4 (CP conserving and nonconserving) can be summarized in an effective interaction
Here FPU and 2"" are standard field strengths for the electromagnetic and Z field, fibu = ~ i ' ~ ~ @ F , pv/ 2=, ( G ~ f i ) - ' / ~ v.e.v. of Higgs field. Finally, Ei are the phases of couplings, in general not equal to 0 or 7r. ~
1 100
I -50
200
250
300
M,. GeV
-0.2
i
1 0
Figure 2: The longztudinal asymmetry in yy
7z
+h
5-
ZX
production due to anomalous interactzons (3).
The CP conserving anomalies give the deviation of measured yy + h and ey + eh cross sections from SM prediction. The (22' anomalies give the polarization asymmetries - variation of cross sections with change of sign of photon helicities (longitudinal), or with the angle between directions of linear polarization of colliding photons (transverse). One example - the longitudinal asymmetry - is shown in Fig. 1. We see that the effects can be seen at reasonable values of anomaly scales hi and phases &, Distinguishing SM/2HDM in the SM-like scenario 'I. The SM-like scenario can be realized both in the SM and in other models. The simplest extension of Higgs sector is the 2HDM with the Model I1 for the Yukawa coupling (the same is realized in MSSM). It contains 2 Higgs doublet fields $1 and $2 with v.e.v.'s v c o s p and wsinp. The physical sector contain charged scalars H* and three neutral scalars hi with no definite CP parity, in the CP conserving case these h, become scalars h and H (with Mh < M H ) and pseudoscalar A . Generally, 2HDM permits to have relatively strong clp and large FCNC. To make these effects naturally weak, the terms in Higgs potential, giving ($1, $2) mixing should be relatively
102
small, and the properties of the observed Higgs boson are close to those of h or H . In this case masses M H , M A and MH+ are 5 3 TeV due to perturbativity constraint 12. The SM-like scenario means that the squared coupling constants (not coupling constants themselves) are close to the SM value. In the 2HDM it can be realized in many ways even in the CP conserving case (Table 2). The widely discussed decoupling limit corresponds to solution Ah+ supplemented with a demand of unnatural strong (41, $2) mixing (giving heavy H , A and H" with MH = MA = MHi without perturbativity limitation).
Table 2: Allowed realizations of SM-like scenario in the 2HDM (11)
In these SM-like cases the observed Higgs boson can be either h or H . The pseudoscalar A and other scalar, H or h, are almost decoupled to gauge bosons and cannot be seen at e+eLC in the standard processes. If mass of any of these elusive Higgs bosons is below 350 GeV and tanp << 1, it can be seen in yy yy process (for h - using the low energy part of photon spectrum) (and in e+et f H at 2E > 2Mt M H ) . In any case, one can distinguish models via measurement of twephoton width of observed SM-like Higgs bosonll. For MH* = 800 GeV the ratios of the twephoton Higgs width and the of cross sections for ey eh process (for latter reaction - at ,h = 1.5 TeV for the left hand polarized photons) to their SM values are shown in Fig. 3 for the natural set of parameters of 2HDM. The bands around central line represent possible difference of Higgs-fermion and HiggsW couplings (squared) from their SM values in the SM-like scenario according to anticipated uncertainty of future measurements The deviations from SM, shown in Fig. 3 for solutions A and Bd, are about 10% (one may compare with anticipated 2% accuracy). These deviations are given by contribution of heavy charged E g g s bosons. For solutions Bu changing of relative sign of contributions of tloop and W-loop increases the observable cross section more than twice in comparison with SM. Therefore, measurement of two-photon width at Photon Collider can resolve these cases reliably. --f
--f
+
--f
'.
-
The possible strong interaction in the Higgs sector should also be the strong interaction of longitudinal W's. It is expected to be observed in the process yy + W W . Based on the experience with process yy -+ T+T- the following picture is expected: The strong interaction modifies weakly the cross section near the threshold in comparison with its SM value (but the phase of amplitude reproduces that of strong interacting WLWLscattering and it can
103 a,mr.a'-,muu.
ZHDM(II)/SM
- Solution A
1.05
0.8
I """""""""""""" 120 140
160
180 M,, 200 [GeVI 220 240
L
Ye4
m
I54
UhG&
Figure 3: Solutions A and B++d. The ratio of quantities in 2HDM to their SM values. The two-photon Higgs width - left panel; the cmss section uL(ey + e h ) - right panel.
be significant). It makes strong interaction in the Higgs sector hardly observable in the cross sections below the energies, given by the masses of resonances, 1.5-2 TeV. The charge asymmetry in the process e y -+ eW+W- is sensitive to this phase of amplitude even at relatively low energy of TESLA (0.8-1 TeV) 15, considerably below possible resonance production. Indeed, the essential contribution to this asymmetry is given by interference of twephoton production subprocess of WW pair (in C-ven state) and bremsstrahlung (onephoton) production subprocess (in C-odd state) like for the process e+e- + e+e- T+W- 16. (The interference with axial part of Z exchange contributes additionally to this asymmetry). 3.2
Anomalous interactions of gauge bosons
At relatively low energies the New Physics cannot manifest itself via new heavy particles. Thus, it reveals itself as certain anomalies in the interactions of known particles. The goal of corresponding studies is to find and discriminate these anomalies. The correlation between coefficients of different anomalies will be the key for understanding what is the nature of New Physics. The practically unique process under interest in the e+e- mode is e+e- + W+W- . With appropriate electron polarization the neutrino exchange contribution disappears, and residual cross section (described by photon and 2 boson exchange) has maximum about 2 pb within LEP operation interval and further decreases with energy. The cross sections of other processes
200 GeV, what provides about lo7 W's per year. Due to high value of these basic cross sections, many processes of 3-rd and 4-th order have large enough cross sections: e y + eWW, yy + ZWW, e y + vWZ, yy + W W W W , ..., see Fig. 4. Large variety of these processes allows to discover and separate well anomalies in specific processes and (or) distributions. The cross section ey + vW c( (1- 2X,), it is switched on or off with variation of electron helicity A,. It gives very precise test of absence of right handed currents in the interaction of W with the matter. The modern simulations of process ey + vW- with only lepton decay modes for W decay show that the sensitivity of this reaction to the quadruple momentum of W is twice higher than it can be reached at e+e- LC 13. ---f
The two-loop radiative corrections to yy --+ W + W - and ey 4 v W - processes should be considered. They are measurable and sensitive to the (unsolved) problems of 0 construction of S-matrix in the theory with unstable particles; gluon corrections (like Pomeron exchange) between quark components of W's.
104
w
and ye processes (uL0,, treelevel)
CompHEPv2 4
:-
=
. . . . . . . . . . . . . . . . . . . . . . . . x1 . . . . . . . . . . . . . . . . . . . . .
,,
,*
,
”” _._..._..._._.._..__--..-.....-----___---,,
10‘
“21 .
tb
0
200
4W
000
0W
1000
1200
1400
I000 1800
2000
Figure 4: The 2-nd and 3 - 4 order processes at Photon Collider
3.3
QCD and Hadron Physics, t-quarks, etc.
All the problems studied at HERA and LEP will be studied at Photon Collider but in the wider interval of parameters and with much higher accuracy. Among them we list those which look most interesting now. 0 Nature of growth of total cross sections. The widespread concepts assume standard Regge type factorization and universal energy behavior for different processes. With Photon Colliders, - for the first time in particle physics at high enough energies - one can have the set of mass shell cross sections of very high energy processes, appropriate for the testing of factorization or its violation. That are upp,measured at Tevatron and LHC, arp,measured at HERA, THERA and urr, measurable at Photon Collider. The preliminary stage of operations with low luminosity can be used to observe cross sections at small scattering angles. 0 The structure function of photon is a unique QCD object calculable completely without phenomenology impacts at high enough photon virtuality Q2 and moderate z9.In modern data phenomenological hadronic component of photon dominates and accuracy of data is low. The experiments at ey collider would increase the obtainable region of Q2 significantly and improve accuracy 2 . 0 The study of charge asymmetry of produced hadrons in yy collisions will give quite new information about QCD at small distances. The charge asymmetry of the produced hadrons in the ey collisions with transverse momentum of scattered electron p l 2 30 GeV will show in explicit form the relation between hadron states produced by vector and axial currents 17. 0 In addition to the usually discussed problems related to the t quarks, the specific one is the study of axial anomaly in the process e y -+ e t f existing even in the SM. At small transverse momenta of scattered electrons p l -+ 0 the cross section of subprocess yZ, -+ t f with longitudinally polarized Z diverges as M?/p: (in contrast to a ( y y ~ tending to 0) 18. 0 The study of charge asymmetry in e y -+ ebb can help to discover S and D wave resonances in b&. -+
4
The using of conversion region
The e -+ y conversion region is eycollider with c.m.s. energy about 1.2 MeV and with huge luminosity about 0.1 fb-l/sec! It will be a unique source of light Goldstone particles (axions,
105 majorons, etc. - LGP, a) 19 , weakly interacting with the matter. They are expected to exist in numerous schemes20. The production processes are e + 70 —> e + a and 770 —> a (here 70 denotes laser photon). Denoting x = 4EujQ/m% and a = m^/m 2 , we obtain that the LGP energy in these reactions is limited from above as
+
E
E
*-*'" £+-*-***••»—• - -2h^ <»™-°- <« The angular spread of LGP is very narrow, in practice, it is given by angular spread of incident electrons within beam (~ 10~5). Note that due to high density of laser photon in the conversion region the nonlinear processes like e + n-jo —> e + a (n = 2,3,..), etc., also become possible. The upper limit of LGP energy in these reactions is higher than that given by eq. (4). Some numerical estimates for LGP being axion or arion with mass about 10 KeV and electron beam energy 250 GeV are presented in Table 3 (the production rate of LGP's familon and majoron is negligibely low). 9a.ee
Standard axion "Invisible" axion Arion
6
2 • 103 • 10~8 2 - 10-6
Cross-section a(cm 2 ) 2.2 • 10~35 4.9 • 10-39 2 • 10~35
The number of produced LGP's per year 2 • 101U 5-10 b 2 • 101U
Table 3: Estimates of typical coupling constants, production cross sections and the number of LGP's, produced per year.
To observe these LGP's, the special detector is proposed. It should be some pin-type lead rod with radius about 2 cm and length 300-500 m, placed in vacuum behind a shield to get rid of the background (Fig. 5). The LGP interacts with lead nuclei like pion but with much lower coupling constant, a + Pb —> h (hadrons). The produced hadron jets with total energy ~ ta and characteristic transverse momentum p± ~ 300 MeV/c should be recorded in the round scinillator and calorimeter with diameter in 1-3 m in the end of the device.
Figure 5: Scheme of LGP detection
The main background is given by neutrinos produced by photons in the shield (reactions like 7 + p —> TT + ..., TT —^ p,v). The proposed scheme allows to reduce this background strongly. Indeed, the energy of the main part of such neutrinos is much lower than the upper limit (4). Besides, the produced neutrinos are spread over the angular interval which is about 3-4 orders of magnitude wider than that for LGP's, reducing relative flux of neutrino in the lead rod by factor 10~6 - 10~8.
106
Acknowledgments We are thankful D. Anipko, V.Ilyin, I. Ivanov, M. Krawczyk, P. Olsen, A. Pak, M. Vychugin for collaboration related different parts of paper and A. Djouadi, V. Serbo, M. Spira, V. Telnov, P.Zerwas for useful discussions. This work was supported by RFBR grants 99-02-17211 and 0015-96691 and INTAS grant 00-00679. I.F.G. is also thankful by organizers of Eriche Workshop for invitation and support. References 1. I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Nucl. Instrum. Methods 205 (1983) 47; I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, V.I. Telnov, Nucl. Instrum. Methods A 219 (1984) 5; Zeroth-order Design Report for the NLC, SLAC Report 474 (1996); R. Brinkmann et al., Nucl. Instrum. Methods 406 (1998) 13. 2. B.Badelek et al. TESLA Technical Design Report, p. VI, chap.1 DESY 2001-011, TESLA Report 2001-23, TESLA FEL 2001-05 (2001) hepex/0108012, p.1-98 3. R.D. Heuer et al. TESLA Technical Design Report, p. I11 DESY 2001-011, TESLA Report 2001-23, TESLA FEL 2001-05 (2001) 192p. hep-ph/0106315 4. E. Boos, A. De Roek, I.F. Ginzburg, K. Hagiwara, R.D. Heuer, G. Jikia, J. Kwiecinski, D.J. Miller, T . Takahashi, V.I. Telnov, T. Rizzo, I. Watanabe, P.M. Zerwas, Goldplated processes at photon collider, Nucl. Instrum. Methods A 472 (2001) 100-120; hep-ph/0103090 5. S.P. Li, H.C. Liu, D. Silverman Phys. Rev. D 31 (1987) 1736. 6. D.S. Gorbunov, V.A. Ilyin, hep-ph/0004092; D.S. Gorbunov, V.A. Ilyin, V.I. Telnov, hepph/0012175 7. See e.g. H. Davoudiasi J . Mod. Phys. A 15 (2000) 2613, K. Cheung, Phys. Rev. D 61 (2000) 0105015; T. Rizzo, Phys. Rev. D 60 (1999) 115010, hep-ph/0008037 8. I.F. Ginzburg, S.L. Panfil, Sov. Yud. Fzz. 36 (1982) 850; I.F. Ginzburg, A. Schiller, Phys. Rev. D 60 (1999) 075016. 9. E. Witten, Nucl. Phys. B 1 2 (1977) 189. 10. G. Jikia, S. Soldner-Rembold, Nucl. Phys. B (Proc. Suppl.) 82 (2000) 373; M. Melles, W.J. Stirling, V.A. Khoze,Phys. Rev. D61(2000) 054015. 11. I.F. Ginzburg, M. Krawczyk, P.Osland, hep-ph/9909455; hep-ph/0101229; hepph/0101331. 12. I.F. Ginzburg, M. Krawczyk, P.Osland, in preparation; I.F. Ginzburg, M.V. Vychugin, hep-ph/0201117. 13. D.A. Anipko, I.F. Ginzburg, A.V. Pak, hep-ph/0201072. 14. A.T. Banin, I.F. Ginzburg, I.P. Ivanov, Phys. Rev. B59 (1999) 115001; I.F. Ginzburg, I.P. Ivanov, Eur. Phys. Journ. C 22 (2001) 411-421, hep-ph/0004069; E. Gabrieli, V.A. Ilyin, B. Mele, Phys. Rev. D 60 (1999) 113005 15. I.F. Ginzburg, Proc. 9th Int. Workshop on Photon - Photon Collisions, San Diego (1992) 474-501, World Sc. Singapore. 16. V.L. Chernyak, V.G. Serbo, Nucl. Phys. B 67 (1973) 464; I.F. Ginzburg, A. Schiller, V.G. Serbo, Eur. Phys. Journ. C18 (2001) 731. 17. I.F. Ginzburg, to be published in Proc. PHOTON'2001 18. I.F. Ginzburg, V.A. Ilyin, in preparation. 19. S.I. Polityko, Sou. Yad. Fiz. 43 (1986) 146; 56 (1993) 144; J.Phys.G: Nucl.Part.Phys. 27 (2001) 2421. 20. See e.g., P.Sikivie, hep-ph/0002154
HIGH-ENERGY PHOTON COLLIDERS V.G. SERB0 Novosibirsk Slate University, 630090, Novosibirsk, Russia This talk is devoted to yy collisions including methods of creating the colliding yy beams of high energy. It is a new and promising area in the high energy physics connected with the fundamental problems of strong and electroweak interactions.
1 1.1
Introduction The subject
Our knowledge about elementary particles and their interactions is mainly obtaidd from particle collisions. Most of fundamental results in particle physics has been obtained from experiments at the pp, pp, e+e- and ep colliders. Principal characteristics of colliders are: the energy in the center-of-mass system (c.m.s) Ecm; luminosity of a collider L which determines collision rate N of events with the cross section u by relation N = L a; types of colliding particles. The progress on high-energy colliders can be seen from Table 1. Up to now and in the nearest future, the pp and pp colliders are the machines with the highest energy. That is why such ephochal discoveries as W and 2 bosons (responsible for weak interaction) and t quark had been performed at the SppS and the TEVATRON, respectively. Table 1: High energy colliders
Collider
Ecm,
TeV
Start date
SPPS TEVATRON LHC
0.6 2 14
1981 1987 2005
HEM
0.31
1992
SLC LEP-I LEP-I1
0.1 0.1 0.2
1989 1989 1999
Linear collider Photon collider
0.5 0.4
2010? 2010+?
Muon collider
0.li 3
107
??
108
For detail study of new phenomena, it is important not only the energy but also types of colliding particles. The e+e- colliders, being less energetic then p p colliders, have some advantages over proton colliders due to much lower background and simpler initial state. Well known example - the study of 2 boson. About thirty years ago a new field of particle physics - photon-photon interactions has appeared. Since that time the two-photon physics is actively investigated in a number of accelerators. This talk about two-photon colliders is based on reviews ’, and on the materials of the TESLA Technical Design Report3. The physical problems which can be solved or clarified in such collisions are reviewed in Ginzburg’s talk.
1.2 Interaction of Photons in the Mmwell Theory and in QED In the classical Maxwell theory of electromagnetism, photons do not interact with each other. In quantum electrodynamics (QED) photons can interact via virtual e+e- pairs. For example, an elastic yy scattering is described by Feynman diagrams of Fig. 1. At low energies, w << mec2
1
1
+ crossed
Figure 1: Feynman diagrams for the elastic yy scattering in QED.
where w is the photon c.m.s. energy, this cross section is very small 6
-
-
For example, for visible light, w 1 eV, the cross section is a,, cm2 . It is too small to be measured even with the most powerful modern lasers, though there were such attempts4. At energies w > mc2, two photons can produce a pair of charged particles. The cross section of the characteristic process yy + p+p- is equal to
It is larger than the (‘standard’’ cross section for the production of the same pair in the e+ecollisions 47rr,2 mEc4 ae+e-,p+p-= -(3) 3 s 2
Collisions of Equivalent Photons at e+e- Storage R i n g s
Unfortunately, there are no sources of intense high-energy photon beams (like lasers at low energies). However, there is indirect way to get such beams - to use equivalent photons which accompanied fast charged particles. Namely this methods was used during last three decades for investigation of two-photon physics at eie- storage rings. The essence of the equivalent photon approach can be explained in the following way5i6 (see also’ $99). The electromagnetic field of
109
an ultra-relativistic electron is similar to the field of a light wave. Therefore, this field can be described as a flux of the equivalent photons with energy distribution dn,ldw. The number of these photons per one electron with the energy E is
dn,
212 E dw - In- -
N
(4)
w w
7r
or approximately dn,
dw
N
0.03 -.
(5)
w
At the e+e- colliders the equivalent photons also collide and can produce some system of particles
Figure 2: Production of system X a) by two equivalent photons y' with 4-momenta (energies) q1 q2 (wz) emitted by an electron and a positron and b) in the annihilation process e+e- + X .
(WI)
and
X (see Fig. 2, y* denotes the equivalent photon) e+e-
+ e+e--y*-y*+ e + e - X .
(6)
Thus, this process is directly connected with the subprocess y*y*-+ X . The cross section for twc+photon production of e+e- in collisions of two fast particles with charges Z1e and Zze, i.e. for the 212~ + ZlZze+e- process, was calculated by Landau and Lifshitz in 1934 (see7 $100). At first sight, the cross sections of the two-photon processes at e+e- colliders (Fig. 2a) are very small since they are the 4-order processes: Ctwo-phot cx c y 4 , while for the annihilation processes of Fig. 2b the cross sections Uannih K a'. However, the annihilation cross sections decrease with increase of the energy (compare with (3))
while the two-photon cross sections increase Ctwo-phot
ff4
"
~
Inns.
m:harc2
Here n = 3 + 4 depending on the process, and the characteristic mass m,-har is constant (for example, mchar m,, for X = p+p- and m,har m, for X = hadrons). As a result, already at f i > 2 GeV N
N
ueie-+eie-,,i,,-> ue+e-+,,ip-.
(9)
Another example, at the LEP-I1 electron-positron collider with the energy & = 200 GeV, the number of events for two-photon production of hadrons with the invariant mass W,, > 2 GeV is by a three order of magnitude larger than that in the annihilation channel.
110
At e+e- storage rings the two-photon processes efe- 4 e+e-efe- had been observed for the first time in 1970 ( Novosibirsk’). The importance of two-photon processes for the lepton and hadron production at efe- storage rings had been emphasized in the papers Arteage-Romero, Jaccarini, Kessler and Parisi Balakin, Budnev and Ginzburg lo and Brodsky, Kinoshita and Terazawa l l . In the papers lo it was shown that e+e- colliding beam experiments can give information about a new fundamental process y*y* + hadrons and the necessary formulae and estimations were obtained. At that time there were a lot of theoretical investigations of various aspects of two-photon physics , but only a few experimental results have been obtained related mainly to the processes yy + e+e-, y7 + p + p - . This period of two-photon physics was summarized in review by Budnev, Ginzburg, Meledin and Serbol. A few years later (approximately from 1977) it was shown in a number of theoretical papers that the two-photon processes are very convenient for the test and detailed study of the Quantum Chromodynamics (QCD) including investigation of a photon structure function (Witten; Zerwas); a jet production in the yy collisions (Llewelyn Smith; Brodsky, De Grand, Gunion and Weis; Baier, Kuraev and Fadin); the yy + CECEprocess and the problem of the perturbative Pomeron (Balitsky and Lipatov). A new wave of experimental activity in this field was initiated by the experiment at SLACI2 which demonstrated that two-photon processes can be successfully studied without detection of the scattered electrons and positrons. After that there was a flow of experimental data from almost all detectors at the e+e- storage rings. Later on several detectors (for example, MD-1 and PEP-9) were equipped with a special system at small angles for detection of the scattered electrons and positrons which allowed to measure the invariant mass of two photons and their virtualities. This period was reviewed by Kolanoski 13.
’,
3
Results O b t a i n e d in Virtual y*y* Collisions
In experiments at efe- storage rings a lot of interesting results about y*y*collisions have been obtained (see reviews I3J4 and Proceedings of Workshops on Photon-Photon Collisions), among them: production of C-even resonances in y*y* collisions, such as and measurement of their yy width; 0
0
T’,
7, q’,
f2,
a2,
qc, xc, ...
measurement of the total yy + hadrons cross section up to c.m.s. energy W,, about 150 GeV; measurement of the total y*y* + hadrons cross section with large values of W& and -4; N 10 (GeV/c)’; photon virtualities -qf N
0
0
0
a number of exclusive reactions: y*y*+ T T , K K , p p , p p , pw, etc.; investigation of the photon structure function in the collision of almost real photon and highly virtual photon with -q2 up t o 600 (GeV/c)’; jet production in yy collisions.
Unfortunately, the number of equivalent photons per one electron is rather small, and correspondingly the y*y*luminosity is about 3 t 4 orders of magnitude smaller than that in e+ecollisions. Therefore, it is not surprising that the most important results at e+e- storage rings were obtained in the efe- annihilation.
111 4
4.1
Photon Colliders on a Base of the Linear eke- Colliders
Linear e+e- Colliders
New opportunities for twephoton physics are connected with future linear e&e- colliders. Projects of such accelerators are now under development in several laboratories. Since 1988 this field is developed in a very tight international collaboration of physicists from many countries. In 1996-97 three projects NLC (North America) 15, JLC (Asia) and TESLA (Europe) l7 have published their Conceptual Design Reports of the linear colliders in the energy range from a few hundred GeV to about one TeV; in 2001 the TESLA Technical Design Report has been published. One team at CERN is working now on the conception of multi-TeV Compact Linear Collider (CLIC). A linear collider consists of several main systems: electron injectors, pre-accelerators, a positron source, two damping rings, bunch compressors, main linacs, interaction regions, a beam dump. Current parameters of these projects are presented in Table 2. Parameters of the projects NLC and JLC are presented in one column since their teams had a tight collaboration and developed a common set of the collider parameters. Note the very small transverse bunch sizes.
=
Table 2: Some parameters of the linear colliders NLC/JLC, TESLA and CLIC
C.m.s. energy Luminosity (with pinch) Repetition rate No. bunchltrain No. particleslbunch Collision rate Bunch spacing Accel. gradient Linac length Beams power Norm.transv.emittance IP beta-function R.m.s. beam size at IP R.m.s. beam length
VLC/JLC I'ESLA CLIC __ __ 0.5 0.5 3 3 10 2.2 120 5 100 190 2820 154 2 0.75 0.4 14.1 22.8 15.4 0.67 337 1.4 25 150 50 10 20 20 22.5 14 30 36013.5 100013 7012 810.1 1510.4 310.1: 24512.7 55515 4311 110 300 25 __ N
So, it is likely that a first linear collider will have energy about 500 GeV with some possible extension (TESLA up to 0.8 TeV, JLC/NLC up to 1.5 TeV). Compared to the SLC, these colliders of the so called next generation are designed on one order higher energy and four orders of magnitude higher luminosity !
4.2 Idea of High-energy yy and y e Colliders with Real Photons Unlike the situation in storage rings, in linear colliders each ek bunch i s used only once. It makes possible to L'convert'' electrons to high-energy photons and to obtain the yy or ye colliding beams with approximately the same energy and luminositya as in the basic eke~
"This can not be done at usual e+e- storage rings where a high luminosity is provided by large number of collisions (- lo9 - l o l l ) of the same e+ and e- bunches. Conversion of the electron and positron bunches into the y bunches at the storage ring gives only a single collision of gamma bunches. The resulting luminosity will be very low because obtaining of new e+ and e- bunches at storage rings takes long time.
112
collisions. Moreover, yy luminosity may be even larger due to absence of some collisions effects. This idea was put forward by Novosibirsk group in 1981-1983l8 and was further developed in details. Among various methods of e + y conversion (bremsstrahlung, ondulator radiation, beamstrahlung and so on), the best one is the Compton scattering of laser light on high-energy electrons. In this method a laser photon is scattered backward taking from the high-energy electron a large fraction of its energy. The scattered photon travels along the direction of the initial electron with an additional angular spread 0 l/ye where ye = E / ( m e c 2 )is the Lorentz factor of the electron. This method was known long time ago l9 and has been realized in a number of experiments. However, the conversion coefficient of electrons to high-energy photons k = Nr/Ne was very small in all these experiments. For example, in the SLAC experiment it
-
-
-
In our papers lS it was shown that at future linear e*e- colliders it will be possible to g e t k 1 at a quite reasonable laser flash energy of a few Joules. Therefore, two principal facts, which make possible a photon collider, are: (i) linear colliders are single-pass accelerators, the electron beams are used here only once; (ii) obtaining of conversion coefficient k 1 is technically feasible. It should be noted that positrons are not necessary for photon colliders, it is sufficient and much easier to use the electron-electron colliding beams. The yy and y e option at the linear e+e- colliders has been included in Conceptual 15J7J6 and Technical3 Designs of linear colliders. All these projects foresee the second interaction regions for the yy and y e collisions.
-
4.3 Scheme of a Photon Collider To create a yy or y e collider with parameters comparable to those in e f e - colliders, the following requirements should be fulfilled: (i) the photon energy w M EOfrom 100 GeV to several TeV; (ii) the number of high-energy photons N7 N, 10"; (iii) photon beams should be focused on the spot with transverse sizes close to those which electron bunches would have at the interaction point a, x cry cm X I O - ~ cm. The best solution for this task is to use a linear e*e- collider as a basis and convert the e* beams into y beams by means of the backward Compton scattering ". The principal scheme is shown in Fig. 3. An electron beam after the final focus system is traveling towards the interaction point IP. At the distance b -0.1 t 1 cm from the interaction point, the electrons collide with the focused laser beam in the conversion region C. The scattered high-energy photons follow along the initial electron trajectories (with small additional angular spread l/ye), hence, they are also focused at the interaction point IP. The produced y beam collides downstream with the oncoming electron or a similar y beam. It is very important that modern laser technology allows to convert most of electrons to high-energy photons. This means that the yy luminosity will be close to the luminosity of the basic e*e- beams.
- -
-
-
4.4 Cornpton Scattering
-
-
In the conversion region a laser photon with energy wo 1 eV is scattered on an electron with energy EO 100 GeV at a small collision angle a0 (Fig. 4) and produced a final photon with the energy w and the emission angle 0. Kinematics of the backward Compton scattering is 'To reduce background, the "used" electrons can be deflected from the interaction point by an external magnetic field. In the scheme without magnetic deflection background is somewhat higher, but such scheme is simpler and allows t o get higher luminosity.
113
Figure 3: A principal scheme of a photon collider. High-energy electrons scatter on laser photons and produce high-energy y beam which collides with similar y or e beam at the interaction point IP.
characterized by two dimensionless variables x and y
The maximum energy of the scattered photon w, and the maximum value of the parameter y is determined as
For example, in collision of the photon with wo = 1.17 eV (the region of the most powerful solid
Figure 4: Kinematics of the backward Compton scattering
state lasers) and the electron with Eo = 250 GeV, the parameter z = 4.5 and the maximum photon energy w, = 0.82Eo = 205 GeV is close enough to the initial electron energy Eo. It worth to note that z M 5 is a good choice for photon colliders because at x > 4.8 the produced high-energy photons disappear from the beam due to production e+e- pairs in collisions with the laser photons, and as result the yy luminosity is reduced. The total c.m. energy of a laser photon and an incident electron is W o e= m e c 2 m . For the typical value of z 5 this energy is not large, 2.5 mec2. Therefore, the transverse momentum of the scattered photon is small, k l 5 mec, and its emission angle 0 (relative to the motion of an incident electron - see Fig. 4 ) is of the order of l/ye = mec2/Eo. The total Compton cross section is N
N
114 7, =
5[(1+:)1n(x+1)--+---- 5 X 2
1 X + l
2 ( XI+ 1 ) 12 ’
2
a0 = T
T ~ = T
( L )= 2.5. lopz5cm2, m,c2
where Xio’ is the mean helicity of the initial electron (IXio’I 5 l/2) and P, is that of a laser photon (IPcl 5 1). Note, that polarizations of initial beams influence the total cross section (as well as spectrum) only if both their helicities are nonzero, i.e. at $’PC # 0. In the region of interest 5 = 1 + 5, p p o l - (1.5 t 0.7) uo, < 0.1 , (13)
bduyPol
i.e. the total cross section is large enough and only slightly depends on the polarization. On the contrary, the energy spectrum does essentially depend on the value of X$”P,. The energy spectrum of scattered photons is defined by the differential Compton cross section:
where
T
= y/[(l - y)z]
5 1 and
T -+
1 at y
+
ym. The “quality” of the photon beam, i.e.
Figure 5: Energy spectrum of scattering photons at
2
= 4.8.
the relative number of hard photons, is improved when one uses beams with the negative value of ALo’P,. For 2Xio’Pc = -1 the peak at w = wm nearly doubles improving significantly the monochromaticity of the y beam (cf. curves a and b in Fig. 5)
2 dUc(X,ym, 2XL0’Pc = -l)/dy dUc(X,ym72XL0’Pc = O)/dy 1+ ( X +
‘
It is not difficult to obtain that the full width of spectrum at the half of maximum is
for unpolarized beams and even smaller at ALo’Pc< 0. The colliding photons with the energies w1 and w2 have invariant mass W,, = G. The distribution of the yy luminosity over W,, is broad with a pronounced peak near the high-energy edge W,, = 2wm. This peak has a typical width at half maximum about 15%.
115
The energy spectrum of high-energy photons becomes more peaked if the initial electrons are longitudinally polarized and laser photons are circularly polarized. This gives almost a factor 3 + 4 increase of the luminosity in the high-energy peak. The polarization of high-energy photons depends on the polarization of laser photons and initial electrons. If the laser photons are polarized circularly (or linearly), the high-energy photons will be highly polarized circularly (or linearly) as well, that is very important for many experiments. 5
Conclusion
-
Physical problems, which have been already studied in the yy collisions, were mainly connected with physics of strong interaction at the large h/(m,c) and moderate small distances f i / p l , where p l 5 10 GeV/c. In future it will be a continuation of the present day experiments plus physics of gauge (W*, 2)bosons and Higgs H bosons, i.e. it will be problems of the electroweak interactions, Standard Model and beyond, search of new particles and new interactions. Physical potential of such yy and y e colliders will be on the same level with future e+e- and p p colliders. Moreover, there is a number of problems in which photon colliders are beyond competition. Photon collider makes it possible to investigate both problems of new physics and of “classical” hadron physics and QCD. Since photon couple directly to all fundamental charged particles - leptons, quarks, W bosons, supersymmetric particles, etc. - a photon collider can provide a possibility to test every aspect of the Standard Model and beyond. Besides, photons can couple to neutral particles (gluons, 2 bosons, Higgs bosons, etc.) through charged particles box diagrams. On the other hand, in a number of aspects photons are similar to hadrons, but with simpler initial state. Therefor, a photon collider will be perfect in studying of QCD and other problems of hadron physics. The problems of the yy and ye colliders were discussed on many conferences: Photon-Photon Collisions, Linear Colliders, and dedicated yy Workshops. Very rich physics, potentially higher than in e+e- collisions luminosity, simplification of the collider (positrons are not required) are all attractive to physicists. Progress in development of linear e+e- colliders and high power lasers (both conventional and free-electron lasers) makes it possible to consider photon colliders as very perspective machines for investigation of elementary particles. N
Acknowledgments
I am very grateful to Organizing Committee and especially to Sebastian White for the invitation to this Conference and for the financial support. This work is partially supported by the Russian Ministry of Education (grant E00-3.3-146). References 1. V.M. Budnev, I.F. Ginzburg, G.V. Meledin, V.G. Serbo, Phys. Rep. 15 C, 181 (1975). 2. I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, photon Collisions and Colliders (Springer, in preparation). 3. TESLA Design Report, DESY 2001-011, ECFA 2001-209; TESLA TDR, Part VI,Chapter 1: The Photon Collider: hep-ex/0108012. 4. D. Bernard, Nucl. Phys. Proc. Suppl. 82, 439 (2000). 5. E. Fermi, Zeit. Physik 29, 315 (1924). 6. E.J. Williams, Kgl. Danske Vidensk. Selskab. Mat.-Fiz. Medd. 13, 4 (1935); K.F. von Weizsacker, Zeit. Physik 88, 612 (1934).
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7. V.B. Berestetskii, E.M. Lifshitz, L.P Pitamskii, Quantum Electrodynamics (Nauka, Moscow, 1989). 8. V.E. Balakin et al., Phys. Lett. 34 B, 663 (1971). 9. A. Jaccarini, N. Arteaga-Romero, J. Parisi, P. Kessler, Compt. Rend. 269 B, 153, 1129 (1969); Nuovo Cim. 4, 933 (1970). 10. V.E. Balakin, V.M. Budnev, I.F. Ginzburg. JETP Lett. 11, 388 (1970); V.M. Budnev, I.F. Ginzburg, Phys. Lett. 37 B, 320 (1971) 320. 11. S. Brodsky, T. Kinoshita, H. Terazawa. Phys. Rev. D 4, 1532 (1971). 12. G. Abrams et al., Phys. Rev. Lett. 43, 477 (1979). 13. H.Kolanoski Two-Photon Physics at e f e - Storage Rings. Springer Tracts in Modern Physics, v. 105 (Springer Verlag, 1984). 14. D. Morgan, M.R. Pennington and M.R. Whalley, J. Phys. G20, A1 (1994). 15. Zeroth-Order Design Report for the Next Linear Collider, LBNL-PUB-5424, SLAC Report 474 (May 1996). 16. JLC Design Study, KEK-REP-97-1 (April 1997). 1.Watanabe et. al., KEK Report 97-17. 17. Conceptual Design of a 500 GeV Electron Positron Linear Collader with Integrated X-Ray Laser Facility, DESY 97-048, ECFA-97-182. R.Brinkmann et al., Nucl. Instr. Methods A 406, 13 (1998). 18. I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov, Pis’ma ZhETF 34, 514 (1981); Nucl. Instr. Methods 205, 47 (1983). 19. F.R. Arutyunian and V.A. Tumanian, Phys. Lett. 4, 176 (1963); R.H. Milburn, Phys. Rev. Lett. 10, 75 (1963). 20. J. Ballam et al., Phys. Rev. Lett. 23, 498 (1969).
Future d e t e r m i n a t i o n of the neutrino-nucleon cross section at extreme energies Alexander Kusenko Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547 RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, N Y 11973 Future detectors of cosmic rays, such as EUSO and OWL, can test the Standard Model predictions for the neutrino interactions at energies well beyond the reach of any terrestrial experiment. The relative rates of horizontal and upgoing air showers, combined with the angular distribution of upgoing air showers will allow one to measure the neutrino-nucleon cross section at 4 lo5 GeV or higher.
-
Detection of ultrahigh-energy (UHE) neutrinos will mark the advent of UHE neutrino astronomy, allowing the mapping on the sky of the most energetic and most distant sources in the Universe. In addition, the prospects for detection of the neutrino-induced upgoing air showers (UAS) by the ground-level fluorescence detectors such as HiRes, Telescope Array, and Pierre Auger2, as well as the future orbiting detector^^,^, present an opportunity t o conduct a particle physics experiment5 and measure the neutrino cross section uvNat an unprecedented center-ofmass energy lo5 - lo6 GeV. The relative rates of the horizontal air showers (HAS) and UAS initiated by neutrinos depend on uWNin such a way that the cross section can be determined without a precise knowledge of the incident neutrino flux. Moreover, the angular distribution of UAS provides an additional and independent information about the cross section. The first question is, of course, whether there is a sufficient flux of neutrinos t o detect. Observations of ultrahigh-energy cosmic rays (UHECR) imply the existence of a related flux of ultrahigh-energy neutrinos generated in the interactions of UHECR with cosmic microwave background radiation '. In addition, active galactic nuclei 7, gamma-ray bursts ', and other astrophysical objects can produce a large flux of neutrinos lo. Finally, if the solution t o the puzzle of UHECR involves Z-bursts ', there is a strong additional flux of ultrahigh-energy neutrinos 11,12. The flux of UHE neutrinos at energies lo1' - lo2' eV is uncertain. However, as discussed below, the proposed measurement of the neutrino cross section is not sensitive to these uncertainties '. Calculations of the neutrino-nucleon cross section g W Nat lo2' eV necessarily use an e x t r a p olation of parton distribution functions and Standard Model parameters far beyond the reach of present experimental data. The resulting cross section l3 at lo2' eV is 10-31cm2. It is of great interest t o compare this prediction with experiment t o test the small-2 behavior of QCD, as well as the possible contributions of new physics beyond the electroweak scale. For the purposes of such a measurement, we assume the cross section t o be a free parameter bounded from below by the value 2 x 10-34~m2measured at HERA at ,/Z = 314 GeV. (This corresponds t o a laboratory energy E, = 5.2 x 1013 eV of an incident neutrino.) UHE neutrinos are expected t o arise from pion and muon decays. The subsequent oscillations generate a roughly equal fraction of each neutrino flavor. Tau neutrinos interacting below the surface of the Earth can create an energetic 7-lepton, whose decay in the atmosphere produces an UAS. It is clear that, for smaller values of the cross section, the Earth is more transparent for neutrinos, so that more of them can interact just below the surface and produce a T that can
',
N
N
117
118
cm2 Figure I.: The air shower probability per incident tau neutrino as a function of the neutrino cross section. The incident neutrino energy is 10’’ eV and the assumed energy threshold for detection of UAS is E t h = 10’*eV for curve 1 and lO”eV for curve 2.
come out into the atmosphere. As long as the mean free path X u is smaller than the radius of the Earth, the rates of UAS increase with A, cc l/u,,,. The rates of HAS, however, are proportional t o u,,; they decrease for a smaller cross section. The comparison of the two rates, shown in Fig. 1, can allow a measurement of the cross section which is practically independent of the uncertainties in the incident neutrino flux. In addition, the angular distribution of UAS alone can be used as an independent measurement of the cross section. The peak of the angular distribution of UAS occurs when cosOpeakx Xu/2Re, which depends on the cross section.
It is comforting t o know that the program of UHE neutrino astronomy, which is one of the goals of EUSO and OWL, is not at risk, regardless of any theoretical uncertainties in the neutrino cross section. For a larger cross section, HAS are more frequent than HAS, while for a smaller value UAS dominate. Nevertheless, the total rates of combined events remain roughly constant for a wide range of uuNras shown in Fig. 1. On the other hand, some of the reported bounds on the neutrino flux are directly affected by the uncertainties in the neutrino-nucleon cross section. For example, the reported bounds on the UHE neutrino flux due t o the non-observation of neutrino-initiated HAS l4 and of radio signals produced by neutrino interactions near the surface of the moon are weaker if the cross section is smaller. To conclude, the future neutrino cosmic-ray experiments can determine the neutrino-nucleon cross section at energies as high as 10l1 GeV, or higher, by comparing the rates of UAS with those of HAS; or by measuring the angular distribution of UAS events. Hence, there is an exciting opportunity do a particle physics experiment using a natural “beam” of cosmic UHE neutrinos in the near future. In addition, the overall prospects for UHE neutrino astronomy are not marred by possible theoretical uncertainties in the value of the neutrino-nucleon cross section: the total number of horizontal and upgoing events remains sufficient for a wide range of u,,.
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Acknowledgments This work was supported in part by the DOE grant DE-FG03-91ER40662. References 1. J. L. Feng, P. Fisher, F. Wilczek and T. M. Yu, hepph/0105067. 2. X. Bertou, P. Billoir, 0. Deligny, C. Lachaud and A. Letessier-Selvon, arXiv:astroph/0104452. 3. G. Domokos and S. Kovesi-Domokos, arXiv:hepph/9805221. 4. D. Fargion, arXiv:astro-ph/0002453. 5. A. Kusenko and T. Weiler, arXiv:hepph/0106071. 6. For reviews, see, e.g., P. Bhattacharjee and G. Sigl, Phys. Rept. 327, 109 (2000); M. Nagano and A. A. Watson, Rev. Mod. Phys. 72, 689 (2000); T. J. Weiler, arXiv:hepph/0103023; P. Biermann and G. Sigl, arXiv:astro-ph/0202425. 7. F. W. Stecker, C. Done, M. H. Salamon and P. Sommers, Given at High-energy Neutrino Astrophysics Workshop on Astrophysics of High-energy Neutrinos: Particle Physics, Sources, Production Mechanisms and Detection Prospects, Honolulu, Hawaii, 23-26 Mar 19gq K. Mannheim, Astropart. Phys. 3, 295 (1995); R. J. Protheroe, arXiv:astro-ph/9607165; F. Halzen and E. Zas, Astrophys. J. 488, 669 (1997) [arXiv:astroph/9702193]. 8. E. Waxman, Phys. Scripta T85, 117 (2000) [arXiv:astr~-ph/9911395]; Nucl. Phys. Proc. Suppl. 87, 345 (2000) [arXiv:astro-ph/0002243]. 9. T. Weiler, Phys. Rev. Lett. 49, 234 (1982); Astropart. Phys. 11, 303 (1999); D. Fargion, B. Mele and A. Salis, Astrophys. J. 517, 725 (1999). 10. For a recent review, see, e.g., E. Waxman, Nucl. Phys. Proc. Suppl. 100, 314 (2001). 11. R.J. Protheroe, astro-ph/9809144; G. Gelmini and A. Kusenko, Phys. Rev. Lett. 82,5202 (1999); Phys. Rev. Lett. 84, 1378 (2000); S. Yoshida, G. Sigl and S.-J. Lee, Phys. Rev. Lett. 81, 5505, (1998); K. Mannheim, R.J. Protheroe, J.P. Rachen, Phys. Rev. D63, 023003 (2001); J.L. Crooks, J.O. Dunn, and P.H. F'rampton, astro-ph/0002089; 2. Fodor, S. D. Katz and A. Ringwald, [hepph/0105064]; 12. G. B. Gelmini, hepph/0005263; J. Alvarez-Muniz and F. Halzen, arXiv:astro-ph/0102106; 0. E. Kalashev, V. A. Kuzmin, D. V. Semikoz and G. Sigl, arXiv:hepph/0112351; G. Gelmini and G. Varieschi, hepph/0201273. 13. G.M. Frichter, D.W. McKay and J.P. Ralston, Phys. Rev. Lett. 74, 1508 (1995); Phys. Rev. Lett. 77E, 4107 (1996); R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Astropart. Phys. 5, 81 (1996); R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Phys. Rev. D 58, 093009 (1998) 14. R.M. Baltrusaitis, et al. (Fly's Eye Collaboration), Phys. Rev. D31, 2192 (1985). 15. P. Gorham, K. Liewer, and C. Naudet, [astreph/9906504] and Proc. 26th Int. CR Conf., Salt Lake City, Utah, Aug. 1999; Alvarez-Muniz and E. Zas, astro-ph/0102173; P. W. Gorham, K. M. Liewer, C. J. Naudet, D. P. Saltzberg and D. R. Williams, astroph/0102435.
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ELECTROMAGNETIC TESTS OF LORENTZ AND CPT SYMMETRY R. BLUHM Physics Department, Colby College, Waterville, ME 04901, USA A review is presented of some recent Lorentz and CPT tests in atomic and particle systems where the predominant interactions are described by quantum electrodynamics. A theoretical framework extending QED in the context of the standard model is used to analyze these systems. Experimental signatures of possible Lorentz and CPT violation are investigated, and bounds are discussed.
1
Introduction
In recent years, there has been a growing interest in testing Lorentz and C P T symmetry! This is due t o both theoretical developments as well as improved experimental capabilities. For example, it has been shown that string theory can lead t o violations of C P T and Lorentz symmetry? This is because strings are nonpointlike and have nonlocal interactions. They can therefore evade the C P T theorem. In particular, there are mechanisms in string theory that can induce spontaneous breaking of Lorentz and C P T symmetry. It has also been shown that geometries with noncommutative coordinates can arise naturally in string theor? and that Lorentz violation is intrinsic t o noncommutative field theories? Experimental signals due t o effects in these kinds of theories are expected at the Planck scale, Mp1 = N 10’’ GeV, where particle physics meets up with gravity. This energy scaIe is inaccessible in accelerator experiments. However, a promising apprach h a s been t o adopt Lorentz and CPT violation as a candidate signal of new physics originating from the Planck scale. The idea is to search for effects that are heavily suppressed at ordinary energies, e.g., with suppression factors proportional to the ratio of a low-energy scale t o the Planck scale. Normally, such signals would be unobservable. However, with a unique signal such as Lorentz or C P T violation (which cannot be mimicked in conventional physics) the opportunity arises t o search for effects originating from the Planck scale. This approach t o testing Planck-scale physics has been aided by the development of a consistent theoretical framework incorporating Lorentz and C P T violation in an extension of the standard model of particle physics? In the context of this framework, it is possible t o look for new signatures of Lorentz and CPT violation in atomic and particle systems that might otherwise be overlooked. Experiments in QED systems are particularly well suited t o this approach since they are often sensitive to extremely low energies. Experiments in atomic physics are routinely sensitive to small frequency shifts at the level of 1 mHz or less. Expressing this as an energy shift in GeV corresponds to a sensitivity of approximately 4 x lo-’’ GeV. Such a sensitivity is well within the range of energy one might associate with suppression factors originating from the Planck scale. For example, the fraction m,/Mpl multiplying the proton mass yields an energy of approximately 1O-l’ GeV, while for the electron the fraction m,/Mpl times the electron m a s
121
122
is about 2.5 x GeV. Some examples of QED systems that are highly sensitive to Lorentz and C P T violation include experiments with p h o t o n ~ , 6electron^^^^^^^^^'^^'^ ~~~~ muons:4,15,16 protons,17J8 and neutron^!^ These examples include some of the classic tests of Lorentz and C P T symmetry, such as g - 2 experiments in Penning traps?' and atomic-clock comparisons - the so-called Hughes-Drever experiments?1~22In addition to these examples involving leptons and baryons, there are other experiments that provide bounds on meson^?^!^^ In the next section, I begin with a brief review of Lorentz and C P T symmetry. This includes a discussion of some of the theoretical ideas that have been put forward over the years for ways in which Lorentz symmetry and C P T might be violated in nature. Different theoretical approaches t o searching for Lorentz violation are also described. I then briefly review the standard-model extension. I t is the QED sector of this theory that is used t o investigate recent electromagnetic tests of Lorentz and CPT symmetry. These tests are described in the subsequent sections, with the photon and fermion sectors treated separately. Several new Lorentz and C P T bounds are summarized. Lastly, some recent ideas involving possible tests of Lorentz and C P T symmetry in a space satellite are pre~ented.2~ 2
L o r e n t z and CPT Symmetry
It appears that nature is invariant under Lorentz symmetry and C P T ? All physical interactions seem to be invariant under continuous Lorentz transformations consisting of boosts and rotations and under the combined discrete symmetry CPT formed from the product of charge conjugation C , parity P, and time reversal T. The C P T theorem links these ~ y m m e t r i e s ?It~states that (under mild technical assumptions) all local relativistic field theories of point particles are symmetric under CPT. A consequence of the C P T theorem is that particles and antiparticles should have exactly equal lifetimes, masses, and magnetic moments. Numerous experiments confirm Lorentz and CPT symmetry to extremely high precision. The Hughes-Drever type experiments are widely considered the best tests of Lorentz symmetry. These experiments place very stringent bounds on spatially anisotropic interactions?l The best C P T test listed by the Particle Data Grou$8 compares the masses of neutral KO mesons with their antiparticles and obtains a bound o n their difference of a few parts in 2.1
Experimental Tests in QED Systems
Many of the sharpest tests of C P T and Lorentz symmetry are made in particle and atomic systems where the predominant interactions are described by QED. For example, the Hughes-Drever type experiments typically compare two clocks or high-precision magnetometers consisting of different atomic species. The best C P T tests for leptons and baryons cited by the Particle Data Group are made by atomic physicists working with Penning traps. These experiments have obtained bounds of order 2 x on the relative difference in the g-factors of electrons and positrons and of order 9 x 10-l' on the relative difference in the charge-to-mass ratios of protons and antiprotons. In addition to these, two proposed experiments a t CERN intend to make high-precision spectroscopic comparisons of trapped hydrogen and a n t i h y d r ~ g e n ?One ~ possibility is t o compare 1s-2s transitions in hydrogen and antihydrogen. These are forbidden transitions and can only occur as two-photon transitions. They have a small relative linewidth of approximately High precision comparisons of these and other transitions in hydrogen and antihydrogen will yield sharp new CPT bounds. It is interesting to note that of all the experiments testing Lorentz and C P T in matter it is the atomic experiments which have the highest experimental precisions (as opposed to sensitivity). For example, in neutral meson experiments quantities are measured with precisions
123 of approximately while in atomic experiments frequencies are typically measured with precisions of or better. Nonetheless, the C P T bound from the neutral meson experiments is many orders of magnitude better than those from the atomic experiments. It would therefore be desirable to understand the atomic experiments better and t o gain greater insight into their sensitivity. Part of the difficulty in doing this stems from the fact that these experiments all compare different physical quantities, such as masses, g factors, charge-to-mass ratios, and frequencies. One way t o find a more meaningful approach t o making cross comparisons would be to work within a common theoretical framework.
2.2 Ideas for Violation A number of different ideas for violation of Lorentz or CPT symmetry have been put forward over the years. In order t o evade the C P T theorem one or more of the assumptions in the proof of the theorem must be disobeyed. A sampling of some of the theoretical ideas that have been put forward include the following: nonlocal interactions:' infinite component fields:' a breakdown of quantum mechanics in gravity:' and spontaneous Lorentz and C P T violation occuring in the context of string theory? It has also recently been shown that Lorentz violation is intrinsic to noncommutative field theories? To investigate some of the experimental consequences of possible Lorentz or C P T violation, a common approach has been to introduce phenomenological parameters. Examples include the anisotropic inertial mass parameters in the model of Cocconi and S a l ~ e t e r : ~ the 6 parameter used in kaon physicsB4 and the T H t p model which couples gravity and electromagneti~m?~ Another approach is t o introduce specific types of lagrangian terms that violate Lorentz or C P T symmetry? These approaches are straightforward and are largely model independent. However, they also have the disadvantage that their predictive ability across different experiments is limited. To make further progress, one would want a consistent fundamental theory with Lorentz and C P T violation. This would permit the calculation of phenomenological parameters and the prediction of signals indicating symmetry violation. No such realistic fundamental theory is known at t h i s time. However, a candidate extension of the standard model incorporating C P T and Lorentz violation does exist. 2.3 The Standard-Model Extension An extension of the standard model incorporating Lorentz and C P T violation h a s been developed by Kosteleck? and collaborators. It provides a consistent theoretical framework that includes the standard model (and SU(3)xSU(2) xU(1) gauge invariance) and which allows for small violations of Lorentz and C P T symmetry! I t is motivated in part from string theory and includes any low-energy effective theory that could arise from spontaneous breaking of Lorentz symmetry? The idea in this context is to assume the existence of a fundamental theory such as string theory in which Lorentz and C P T symmetry hold exactly but are spontaneously broken at low energy. As in any theory with spontaneous symmetry breaking, the symmetries become hidden at low energy. The effective low-energy theory contains the standard model as well as additional terms that could arise through the symmetry breaking process. A viable realistic fundamental theory is not known at this time, though higher dimensional theories such as string or M theory are promising candidates. A mechanism for spontaneous symmetry breaking can be realized in string theory because suitable Lorentz-tensor interactions can arise which destabilize the vacuum and generate nonzero tensor vacuum expectation values. It has been shown that any realistic noncommutative field theory is equivalent t o a subset of the standard-model extension? Colladay and Kosteleck? have written down a general extension of the standard model that could arise from spontaneous Lorentz symmetry breaking of a more fundamental theory, which maintains SU(3)xSU(2) xU(1) gauge invariance, and is power-counting renormalizable!
124
They have shown that the theory maintains many of the other usual properties of the standard model besides Lorentz and CPT symmetry, such as electroweak breaking, energy-momentum conservation, the spin-statistics connection, and observer Lorentz covariance. Issues related t o the stability and causality of the standard-model extension have been investigated as well? 3
&ED Extension
To consider experiments involving electromagnetic interactions it suffices t o restrict the standardmodel extension to its &ED sector. The lagrangian describing electromagnetic interactions of a fermion field $ of mass m and charge q = - [ e l with photons A , can be written as
L = 130 f
Lphoton
f lfermion
.
(1)
Here, LO is the usual QED lagrangian in the absence of Lorentz and C P T violation, 1
LO= i$ypD,$ - $m$ - -F,,FFU 4
,
(2)
where ZD, = ia, - qA,, F,,, = a,,A,, - &A,, and natural units with fi = c = 1 are used. The Lorentz and C P T violating terms are Lphoton
1 1 =- ( ~ A F ) ~ ~ ~ X - -~ ( ,k ,~A) n ~ ~Fp ~v F~ ~ ~ F ’ ~ 2 4
(3)
for the photon sector and Germion
= -a,&p+
-
+ ic,,,&YD”$ + i d , , , h w p D u +
bp$ysyIL$ - iH,Y$ffPLy$
(4)
for the fermion sector. Each of the additional terms involves a constant parameter. The terms involving the effective , C P T , while the terms with H,,, c,,,, d,,,, and coupling constants a,, b,, and ( ~ A F ) violate ( k ~ ) , + ~preserve C P T . All seven terms break Lorentz ~ y r n m e t r y ?The ~ renormalizability of this theory has recently been shown to hold t o one loop? The QED extension has also been used to study scattering cross sections of electrons and positrons in the presence of C P T and Lorentz violation!’ In the following sections, the photon and fermion sectors will be discussed separately. 4
Photon Sector
The extra terms in ,&hoton lead to modifications of Maxwell’s equations and the energy density and dispersion relations for photons. A thorough discussion of these modifications is given by Colladay and Kosteleckf! Here, I will briefly summarize some of the theoretical issues and experimental bounds for these terms. In many respects, the theory for the photon sector is analogous t o electromagnetism in certain types of macroscopic media, such as a crystal. This results in the Lorentz and C P T violation causing effects such as photon birefringence. Bounds ca.n therefore be obtained from experiments looking a t photons originating from cosmological sources. The CPT-odd term involving ( ~ A F ) ,gives rise to negative-energy contributions,6 which would cause instabilities in the theory. However, this term is expected to vanish for theoretical reasons! It can be set t o zero at tree level, and then the question arises as to whether it acquires radiative corrections from quantum loop corrections. Remarkably, the structure of the standardmodel extension leads t o an anomaly cancelation mechanism that preserves the vanishing of ( ~ A F ) ,at , the one-loop level?,’ In addition to these theoretical constraints, very sharp bounds on
125
( I C A F ) , at the level of GeV can be obtained from cosmological birefringence experiments! For these reasons, ( I C A F ) , will be assumed t o vanish in the following sections. The CPT-even term involving ( I C F ) ~ ~ leads ,~ t o positive energy contributions. There are no theoretical reasons to expect that it vanishes. This term has 19 independent real components. Their contributions have been shown t o result in a wavelength dependence in the relative phase difference between the photon polarizations.8 This gives rise t o a new method of extracting bounds in spectropolarimetry of cosmological sources. A recent survey of different source8 results in bounds at the level of on many of the contributions from the (k,),,,, term. 5
Fermion Sector
The effects of Lorentz and C P T violation in matter are controlled by the term in the lagrangian. It involves the five effective coupling constants a,, b,, H,,, c,,,, and d,,,, which are all assumed to be small. It is these terms that cause leading-order corrections in &ED systems involving matter. They effectively give rise t o a modified structure for the mass and gamma matrices in the Dirac equation, (irpq, - ~ ) q =>o , (5) where and
Additional interactions involving the photon Lorentz-violation parameters ( ( I C A F ) ~ or ( I C F ) ~ ~ , , , ) coupling to matter through the photon propagator will be of sub-leading order and can be ignored. The leading-order corrections can then be found using relativistic quantum mechanics in a perturbative treatment. In the last several years, a number of experiments in &ED systems have been performed that have resulted in sharp new bounds on Lorentz and C P T violation. These bounds are typically expressed in terms of the parameters a,, b,, c,,,, d,,, and H,,,. This permits a straightforward way of making comparisons across different types of experiments and avoids problems that can arise when different physical quantities (9 factors, charge-to-mass ratios, masses, frequencies, etc.) are used in different experiments. A thorough investigation of possible C P T and Lorentz violation must look at as many different particle sectors as possible, since each different particle sector in the QED extension has a set of Lorentz-violating parameters that are independent. The parameters of the different sectors are distinguished in the following using superscript labels. Recent experiments have obtained bounds o n parameters for the e l e c t r ~ n p ~ ’ ~ m ,u~~~n, , ~’ ~ , ’ ~ , ~ ~ pro tor^^^^^' and n e ~ t r 0 n . l ~ Before discussing these recent experiments individually, it is useful t o examine some of the more general results that have emerged from these investigations. First, the sharp distinction between what are considered Lorentz tests and CPT tests has been greatly diminished. Experiments traditionally viewed as C P T tests are also sensitive to Lorentz symmetry and vice versa. In particular, it has been demonstrated that it is possible to test for C P T violation in experiments with particles alone. This has opened up a whole new arena of C P T tests. A second general feature is that the sensitivity t o C P T and Lorentz violation in these experiments stems primarily from their ability to detect very small anomalous energy shifts. While many of the experiments were originally designed t o measure specific quantities, such as differences in g factors or charge-to-mass ratios of particles and antiparticles, it is now recognized that they are most effective as CPT and Lorentz tests when all of the energy levels in the system are investigated for possible anomalous shifts. Because of this, several new signatures of C P T and Lorentz violation have been investigated in recent years that were previously overlooked. Examples are
126
given in t h e following sections. Finally, another common feature of these experiments is t h a t they all have sensitivity to t h e Planck scale.
5.1
Penning-Trap Experiments
T h e aim of the original experiments with Penning traps was to make high-precision comparisons of the g factors and charge-to-mass ratios of particles and antiparticles confined within the trap?” This was obtained through measurements of the anomaly frequency w, and the cyclotron frequency w,. For example, g - 2 = 2wa/wc. T h e frequencies were typically measured to lo-’ for t h e electron, which determines g to lo-’’. In computing these ratios it was not necessary to keep track of the times when w, and w, were measured. More recently, however, additional signals of possible C P T and Lorentz violation in this system have been found, which has led to two new tests being performed. T h e first was a reanalysis performed by Dehmelt’s group of existing d a t a for electrons and positrons in a Penning t r a p ? T h e goal was t o search for an instantaneous difference in the anomaly frequencies of electrons and positrons, which can be nonzero when C P T and Lorentz symmetry are broken. (In contrast t h e leading-order instantaneous cyclotron frequencies remain equal). T h e original measurements of g - 2 did not involve looking for possible instantaneous variations in w,. Instead, t h e ratio w,/wc was obtained using averaged values. T h e new analysis is especially relevant because it can be shown t h a t the CPT-violating corrections t o the anomaly frequency w, can occur even though the g factor remains unchanged. T h e new bound found by Dehmelt’s group can be expressed in terms of t h e parameter bz, which is the component of b; along the quantization axis in the laboratory frame. They obtained lb5l <, 3 x GeV. A second new test of C P T and Lorentz violation in the electron sector has been made using only d a t a for the electron!” Here, the idea is t h a t the Lorentz and CPT-violating interactions depend on the orientation of the quantization axis in the laboratory frame, which changes as the Earth turns on its axis. As a result, both t h e cyclotron and anomaly frequencies have small corrections which cause them to exhibit sidereal time variations. Such a signal can be measured using just electrons, which eliminates t h e need for comparison with positrons. T h e bounds in this case are given with respect to a nonrotating coordinate system such as celestial equatorial coordinates. T h e interactions involve a combination of laboratory-frame components t h a t couple to the electron spin. T h e combination is denoted as bz s bz - md& - H f 2 . T h e bound can be expressed in terms of components X , Y , 2 in the nonrotating frame. It is given as <, 5 x 10-”GeV for J = X , Y. N
N
161
5.2 Clock-Comparison Experiments T h e Hughes-Drever experiments are classic tests of Lorentz invariance?l These experiments look for relative changes between two atomic “clock” frequencies as t h e Earth rotates. The “clock” frequencies are typically atomic hyperfine or Zeeman transitions. In a 1995 experiment, very sharp bounds at leading-order for the-proton, neutron :nd electron were obtaLned in the GeV, b‘j N lop3” GeV, and b; experiment of Berglund et al. These were b; ‘v GeV for J = X , Y . More recently, several new clock-comparison tests have been performed or are in the planning stages. For example, Bear et a / . have used a two-species noble-gas maser to test for Lorentz and CPT violation in the neutron sector!’ They obtained a bound l&’jl <, 10p31GeV for J = X , Y . This is currently the best bound for the neutron sector. As sharp as these bounds are, however, it should be kept in mind t h a t certain assumptions about the nuclear configurations must be made t o obtain them. For this reason, these bounds should be viewed as accurate t o within one or two orders of magnitude. To obtain cleaner bounds it is necessary to consider simpler atoms or to perform more precise nuclear modeling.
=
127
5.3 Hydrogen-Antihydrogen Experiments Hydrogen atoms have t h e simplest nuclear structure. Two experiments are being planned at CERN which will make high-precision spectroscopic measurements of the 1s-2s transitions in hydrogen and antihydrogen. These are forbidden two-photon transitions with a relative linewidth of approximately T h e magnetic field plays an important role in the sensitivity of these transition to Lorentz and C P T breaking. For example, in free hydrogen in the absence of a magnetic field, the 1s and 2 5 levels shift by the same amount at leading order, and there are no leading-order corrections to the 1s-2s transition. However, in a magnetic t r a p there are fields t h a t mix the different spin states in the four hyperfine levels. Since the Lorentz-violating couplings are spin-dependent, there will be sensitivity at leading order to Lorentz and C P T violation in comparisons of 1s-2s transitions in trapped hydrogen and antihydrogen. An alternative to 1s-2s transitions is to consider measurements of ground-state Zeeman hyperfine transitions in hydrogen alone. It has been shown t h a t these transitions in a hydrogen maser are sensitive to leading-order Lorentz-violating effects. Measurements of these transitions have recently been made using a double-resonance t e ~ h n i q u e ! They ~ yield new bounds for the electron and proton. T h e bound for the proton is &I\; <, GeV. Due t o the simplicity of the hydrogen nucleus, this is an extremely clean bound. It is currently the best Lorentz and C P T symmetry for the proton.
5.4
Spin-Polarized Matter
A recent experiment at t h e University of Washington uses a spin-polarized torsion pendulum to achieve high sensitivity t o Lorentz violation in the electron ~ e c t o r ! Its ~ sensitivity stems from the combined effect of a large number of aligned electron spins. T h e experiment uses stacked toroidal magnets t h a t have a net electron spin S = 8 x loz2, but which have a negligible magnetic field. T h e pendulum is suspended on a turntable and a time-varying harmonic signal is sought. An analysis of this system reveals t h a t in addition to a signal with the period of t h e rotating turntable, the effects of Lorentz and C P T violation induce additional time variations with a sidereal period caused by Earth’s rotation. T h e group at t h e University of Washington has analyzed their d a t a and has obtained a bound on the electron parameters equal to lb;l <, GeV for J = X , Y and 1b51 <, lo-’’ GeVJ3 These are currently the best, Lorentz and C P T bounds for the electron.
5.5
hluon Experiments
Experiments with muons involve second-generation leptons. They provide independent Lorentz and C P T tests. There are several different kinds of experiments with muons t h a t are currently being conducted, including muonium experimentk4 and g - 2 experiments with muons at Brookhaven!’ In muonium, the experiments measuring t h e frequencies of ground-state Zeeman hyperfine transitions in a strong magnetic field have the best sensitivity to Lorentz and CPT violation. A recent analysis has looked for sidereal time variations in these transitions. A bound at a level of I&;l 5 5 x lo-” GeV has been ~ b t a i n e d ! T ~ h e g - 2 experiments with positive muons are relativistic with “magic” boost parameter 6 = 29.3. Bounds on Lorentz-violation parameters should be attainable in these experiments at a level of GeV. These experiments are currently underway at Brookhaven and their results should be forthcoming in t h e near future.
128
I
Expt
Muon g - 2
Sector
I Params ( J = X , Y ) I Bound (GeV)
muon
&$
5x
10-25
(estimated) Table 1: Summary of leading-order bounds
6
Clock-Comparison Experiments in Space
In summary, five new sets of Lorentz and C P T bounds have been obtained in recent years for the electron, proton, neutron, and muon. T h e leading-order bounds are summarized in Table 1. All of these limits are within t h e range of sensitivity associated with suppression factors arising from the Planck scale. However, as sharp as these bounds are, there is still room for improvement, and i t is likely tha.t t h e next few years will continue t o provide increasingly sharp new tests of Lorentz and C P T symmetry in QED systems. In particular, it should be possible to obtain bounds on many of t h e parameters t h a t do not appear in Table 1. One promising approach is to conduct atomic clock-comparison tests in a space ~ a t e l l i t e . 2 ~ These will have several advantages over traditional ground-based experiments, which are t y p ically insensitive to the direction 2 of Earth’s axis and ignore boost effects associated with timelike directions. For example, a clock-comparison experiment conducted aboard the International Space Station (ISS) would be in a laboratory frame t h a t is both rotating and boosted. It would therefore immediately gain sensitivity to both the Z and timelike directions. This would more than triple the number of Lorentz-violation parameters t h a t are accessible in a clockcomparison experirncnt. Since there are several missions already planned for the ISS which will compare Cs and R b atomic clocks and H masers, the opportunity to perform these new Lorentz and CPT tests is quite real. Another advantage of an experiment aboard the ISS is t h a t t h e time needed to acquire d a t a would be greatly reduced (by approximately a factor of 16). In addition, new types of signals would emerge t h a t have no analogue in traditional Earth-based experiments. T h e combination of these advantages should result in substantially improved limits on Lorentz and CPT violation.
129
Acknowledgments
I would like to acknowledge my collaborators Alan Kosteleckji, Charles Lane, and Neil Russell. This work is supported in part by t h e National Science Foundation under grant number PHY9801869. References
1. See, for example, V.A. Kosteleckji, ed., CPT and Lorentz Symmetry 11 (World Scientific, Singapore, 2002). 2. V.A. Kosteleckji and S. Samuel, Phys. Rev. D 39 (1989)683;ibid. 40 (1989) 1886;Phys. Rev. Lett. 63 (1989) 224; ibid. 66 (1991) 1811;V.A. Kosteleckji and R. Potting, Nucl. Phys. B 359 (1991) 545;Phys. Lett. B 381 (1996) 89. 3. A. Connes, M. Douglas, and A. Schwarz, J H E P 02 (1998)003. 4. S.M. Carroll, J.A. Harvey, V.A. Kosteleckji, C.D. Lane, and T. Okamoto, Phys. Rev. Lett. 87,141601 (2001). 5. D. Colladay and V.A. Kosteleckji, Phys. Rev. D 55 (1997)6760;Phys. Rev. D 58,116002 (1998). 6. S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D 41 (1990) 1231. 7. R. Jackiw and V.A. Kosteleck?, Phys. Rev. Lett. 82 (1999) 3572. 8. V.A. Kosteleck? and M. Mewes, Phys. Rev. Lett. 87,251304 (2001). 9. H.G. Dehmelt et al., Phys. Rev. Lett. 83 (1999)4694. 10. R . Mittleman et al., Phys. Rev. Lett. 83 (1999)2116. 11. R . Bluhm, V.A. Kost,eleckji and N . Russell, Phys. Rev. Lett. 79 (1997) 1432;Phys. Rev. D 57 (1998) 3932. 12. R. Bluhm and V.A. Kosteleckf, Phys. Rev. Lett. 84 (2000) 1381. 13. B.R. Heckel et al., to appea+. 14. V.W. Hughes et al., Phys. Rev. Lett. 87,111804 (2001). 15. H.N. Brown et al., Phys. Rev. Lett. 86 (2001)2227. 16. R . Bluhm, V.A. Kosteleckji and C.D. Lane, Phys. Rev. Lett. 84 (2000)1098. 17. D.F. Phillips e t al., Phys. Rev. D 63,111101 (2001). 18. R . Bluhm, V.A. Kosteleckji and N. Russell, Phys. Rev. Lett. 82 (1999)2254. 19. D . Bear et al., Phys. Rev. Lett. 85 (2000) 5038. 20. P.B. Schwinberg, R.S. Van Dyck, Jr., and H.G. Dehmelt, Phys. Rev. D 34 (1986) 722; R.S. Van Dyck, Jr., P.B. Schwinberg, and H.G. Dehmelt, Phys. Rev. Lett. 59 (1987)26; G . Gabrielse et al., Phys. Rev. Lett. 82 (1999)3198. 21. V.W. Hughes, H.G. Robinson, and V. Beltran-Lopez, Phys. Rev. Lett. 4 (1960) 342; R.W.P. Drever, Philos. Mag. 6 (1961) 683; J.D. Prestage et al., Phys. Rev. Lett. 54 (1985) 2387; S.K. Lamoreaux et al., Phys. Rev. A 39 (1989) 1082;T.E. Chupp et al., Phys. Rev. Lett. 63 (1989) 1541;C.J. Berglund e t al., Phys. Rev. Lett. 75 (1995)1879. 22. V.A. Kosteleckji and C.D. Lane, Phys. Rev. D 60,116010 (1999). 23. V.A. Kostelecki and R . Potting, Phys. Rev. D 51 (1995) 3923;D. Colladay and V.A. Kosteleckji, Phys. Lett. B 344 (1995)259;Phys. Rev. D 52 (1995)6224;V.A. Kosteleckji and R . Van Kooten, Phys. Rev. D 54 (1996) 5585;V.A. Kosteleckji, Phys. Rev. Lett. 80 (1998) 1818;Phys. Rev. D 61, 016002 (2000);64,076001 (2001). N. Isgur e t al., Phys. Lett. B 515,333 (2001). 24. B. Schwingenheuer et al., Phys. Rev. Lett. 74 (1995)4376;L.K. Gibbons et al., Phys. Rev. D 55 (1997) 6625;OPAL Collaboration, R. Ackerstaff et al., Z. Phys. C 76 (1997)401; DELPHI Collaboration, M.Feindt et al., preprint DELPHI 97-98C O N F 80 (July 1997). 25. R. Bluhm, V.A. Kosteleckji, C.D. Lane, and N. Russell, to appear (hep-ph/Ollll41).
130 26. For reviews of C P T , see R.F. Streater and A S . Wightman, PC T, Spin, and Statistics and All That, Benjamin Cummings, Reading, 1964; R. G. Sachs, The Physics of Time Reversal, University of Chicago, Chicago, 1987. 27. J. Schwinger, Phys. Rev. 82 (1951)914;J.S. Bell, Birmingham University thesis (1954); Proc. Roy. SOC. (London) A 231 (1955) 479; G. Liiders, Det. Kong. Danske Videnskabernes Selskab Mat.fysiske Meddelelser 28,No. 5 (1954);Ann. Phys. (N.Y.) 2 (1957) 1; W.Pauli, in W. Pauli, ed., Neils Bohr and the Development of Physics, McGraw-Hill, New York, 1955,p. 30. 28. J. Bartels e t al., Review of Particle Properties, Eur. Phys. J. C 15 (2000)1. 29. B. Brown et al., Nucl. Phys. B (Proc. Suppl.) 56A (1997) 326;M.H. Holzscheiter et al., Nucl. Phys. B (Proc. Suppl.) 56A (1997)336. 30. P. Carruthers, Phys. Lett. B 26 (1968) 158. 31. A.I. Oksak and I.T. Todorov, Commun. Math. Phys. 11 (1968)125. 32. S. Hawking, Commun. Math. Phys. 87 (1982) 395. 33. G.Cocconi and E. Salpeter, Nuovo Cimento 10 (1958) 646. 34. See for example, T.D. Lee and C.S. Wu, Annu. Rev. Nucl. Sci 16 (1966)511. 35. A.P. Lightman and D.L. Lee, Phys. Rev. D 8 (1973)364. 36. V.A. Kosteleckjr and R. Lehnert, Phys. Rev. D 63 (2001) 065008. 37. Additional fermion terms invloving parameters e,, f p , and g+, are ignored. Gauge invariance excludes these terms in the standard-model extension. However, they could be generated effectively due to strong binding in a nucleus and are included in a more general discussion ?* 38. V.A. Kosteleckjr, C.D. Lane and A.G.M. Pickering, Phys. Rev. D, in press (hepth/Ollll23). 39. D. Colladay and V.A. Kostelecki, Phys. Lett. B 511,209 (2001).
The RHIC Experimental Program: Highlights from the First Year of Operation at RHIC Jim Thomas Lawrence Berkeley National Laboratov, Berkeley, CA
1 Introduction
The Relativistic Heavy Ion Collider (RHIC) is located at Brookhaven National Laboratory on Long Island, New York. The collider is 3.83 km in circumference and it accelerates a variety of heavy ion beams; from protons to gold. The top energy is 250 GeV per beam for protons and 100 GeVlamu per beam for gold. So, the collision energies are Js = 500 GeV and Js = 200 GeVIamu, respectively. The design luminosity for heavy ions is 2x1OZ6cm-2 set.' and the beam lifetime will eventually be greater than 10 hours. RHIC is also capable of running polarized protons and it will have a full complement of Siberian Snakes. The detectors have been designed to measure scattering asymmetries and so a robust program of polarized physics will be possible, in the future. RHIC has completed two run cycles. The first run ended in 2000 and the second run ended in 2002. Au+Au beams were run at Js = 130 GeVIamu for the first cycle and then the beam energy was raised to Js = 200 GeVIamu for the second cycle of operation. There was also a brief p+p run at 200 GeV in 2002 to test the capabilities of the polarized beam facilities. I will report on the physics results that have been achieved with the heavy ion data in the 2000 run. The motivation for building RHIC was to study nuclear matter under extreme conditions, at high temperature, and at densities greater than 10 times normal nuclear matter density. Under these conditions, we expect quark and gluon degrees of freedom to become important and the underlying dynamics should change as the nuclear system makes the transition from cold matter to extremely hot and dense matter. In fact, it was predicted that nuclear matter will undergo a phase transition into a Quark Gluon Plasma (QGP) at a critical temperature near the rest mass of the pion and at about 10 times the density of normal nuclear matter. This prediction has been explored with lattice gauge calculations and these calculations have routinely shown that there is a large jump in the energy density for two and three flavor systems at a critical temperature, T,, of 160 MeV[l]; see Figure 1. The lattice calculations have a difficult time dealing with the mass of quarks as they appear in nature and so we do not know if the phase transition is first order, second order, or whether there
131
132
14.0 12.0 10.0
8.0 6.0
4.0 2.0 0.0
I.o
1.5
2.0
2.5
3.0
3.5
4.0
Fig. 1. Lattice gauge calculation of the energy density in a system of quarks with 2 or 3 flavors. The fi gure is from reference [ 13. The arrows on the right hand side of the fi gure indicate the Stephan Boltzman limit for a free Quark Gluon gas.
is a tri-critical point on the phase diagram. So, experimentally we are hoping to learn more about hot, dense, matter and to see if nuclear systems undergo a first or second order phase transition into a QGP or perhaps a cross over from one side of the diagram to the other without exhibiting any critical behavior at the cross-over point.
2 The Detectors Four experiments took data in the 2000 run cycle. They are STAR[2], PHENIX[3], BlZAHMS[4] and PHOBOS[S]. STAR and PHENIX are the large experiments and PHOBOS and BRAHMS are smaller experiments; each with a more specialized physics agenda. STAR is the hadronic signals experiment. It uses the world’s largest time projection chamber (TPC) as its primary tracking detector[6]. See Figure 2. The TPC is 4.2 meters long and 4 meters in diameter. It has full azimuthal coverage and records tracks in the range from -1.8 units to 1.8 units of rapidity and it sits in a solenoidal magnetic field of 0.5 Tesla. Charged particle tracking close to the interaction region is accomplished by a Silicon Vertex Tracker (SVT) consisting of 216 silicon dnft detectors arranged in 3 layers. The TPC and SVT allow for identification of secondary vertices from weak decays of, for example, A, E,and 0’s. For the 2000 and 2002 runs, the TPC was supplemented by a ring imaging cherenkov detector to extend the particle ID to high transverse momentum. And in the fu-
133
Coils
/
Magnei
Silicon Vertex , Tracker
L- Forward Time Projection Chamber
Fig. 2. The STAR experiment measures hadronic observables using a large acceptance Time Projection Chamber, a Silicon Vertex Tracker, a Ring Imaging Cherenkov Detector and other detectors. In the future, STAR will have a full barrel Electromagnetic Calorimeter.
ture, STAR will have a full barrel calorimeter, and additional detectors. Only the TPC and the RICH were operational in the 2000 run. PHENIX is a multi-purpose experiment that includes central rapidity hadron and electron spectrometers as well as muon arms at forward rapidities. See Figure 3. The central arms each cover 90 degrees in 4 and they cover the range from -0.35 units to 0.35 units of rapidity. The central arms include tracking chambers and a time of flight wall (TOF) for charged particle measurements plus a RICH backed up with calorimeters for electron, photon and neutral particle energy measurements. In the future, PHENIX will have forward and backward facing muon spectrometers which will provide excellent tracking and particle ID for muons in the range from 1.1 to 2.5 units of rapidity. Only the central arm spectrometers were operational in the 2000 run. The BRAHMS experiment has two spectrometers. Each arm is designed to be versatile and to cover a wide rapidity interval, albeit with a small aperture. The arms rotate about a common axis and the mid-rapidity spectrometer covers the angular range from 95 to 30 degrees. The Forward spectrometer extends this coverage from 30 degrees to 2.3 degrees with respect to the beam. This allows BRAHMS to achieve the widest rapidity coverage of all the experiments at RHIC; from 0 5 q 5 4. See Figure 4. The PHOBOS experiment is a table-top two arm spectrometer. It includes a strong field magnet with many layers of Silicon micro-strip detectors surrounding the strong field region to do the spectroscopy of low pt charged hadrons. It also has additional silicon detectors to measure charged particle multiplicities into 47r solid angle and two time-of-flight
1 34
Fig. 3. The PHENIX experiment measures leptons, photons and hadrons in selected solid angles using tracking chambers, EM Calorimeters, and Ring Imaging Cherenkov detectors. In the future, PHENIX will have muon arms in the forward and backward directions. The figure is provided courtesy of the PHENIX collaboration. walls (TOF) to improve the particle identification capabilities in specific rapidity intervals. See Figure 5. Finally, there are a set of zero degree calorimeters[7] (ZCALs) installed in all four experiments and they are used to classify the events seen by each experiment. They are located k17 meters from the interaction vertex and they are located behind the first bend magnet in each beamline. The primary purpose of the the ZCALs is to collect the neutral particles that are moving in the forward and backward directions. The response of the ZCALs is non-linear because there is an abundance of neutral particles in mid-peripheral collisions but a deficit in very peripheral collisions, due to a lack of interacting particles, and a deficit in central collisions because the colliding nuclei have evaporated. Therefore, the ZCAL response is not unique and charge particle multiplicity counters are used to break the ambiguity. See Figure 6. The ZCALs provide a common triggering and analysis tool for each experiment. Thus, it is possible to quantitatively compare data from all four experiments according to whether the data came from the most central collisions (e.g. top 5% of the interaction cross-section) or from more peripheral collisions. But perhaps the most satisfying part of the detector program is that they work. The first events were recorded by the STAR collaboration on June 25th, 2000 and all four detectors observed Au-Au collisions during the first run of the RHIC accelerator. A beautiful example
135 H2,T5
RICH D4
T4
Forw a rd Spectrometer 2.3 < 0 < 30
T1 '1
TOFW
Multiplicity
Beam Beam counters
Fig. 4. The BRAHMS experiment measures inclusive hadronic particle production over a large rapidity and a large pt range. It uses two moving spectrometer arms. The fi gure is provided courtesy of the BRAHMS collaboration.
of an event as seen in the STAR TPC, and displayed by the online level three trigger, is shown in Figure 7. Figure 7 is an end-view of an event seen in the STAR TPC. The detector is azimuthally symmetric and it has an active volume starting at 50 cm from the beam axis and extending to 200 cm from the beam axis. This is a low multiplicity event, which was chosen so you can see the individual tracks in a projected view. However, event multiplicities up to several thousand per unity rapidity are routinely seen, recorded, and analyzed by the TPC.
3 Physics Results from the First Year of Operation at RHIC
The RHIC experimental program finished it's first cycle of operation in September, 2000, and its second cycle of operation in January, 2002. I will concentrate on published data from the fist cycle of operation because the data from the second cycle has not been fully
136
Rinc
Fig. 5. The PHOBOS experiment measures low pt charged hadrons into 4n azimuth. The fi gure is provided courtesy of the PHOBOS collaboration.
analyzed yet . . .with one exception. The PHOBOS collaboration has measured the multiplicity of particles produced in RHIC collisions at 56, 130, and 200 GeV per nucleon[8], [9]. The 56 and 130 GeV data were taken during the first cycle of operation while the 200 GeV data point is a very fast analysis that was done with data taken in the Znd cycle of operation. Figure 8 shows the PHOBOS data in comparison to the HiJing model[ 101. The multiplicity of charged particles, scaled by the number of participating nucleons (Npart),is shown versus the number of participating nucleons. The data are interesting because the maximum multiplicities at RHIC are large but not as large as expected from simple ideas about a first order phase transition. Naive extrapolations, made before these data were available, suggested that a first order jump from 3 pionic degrees of freedom to 37 gluonic degrees of freedom in a QGP would lead to a huge increase in particle production in the h a 1 state. Clearly, this does not happen at 130 or 200 GeV. But rather than suggest that we have not seen the QGP, the data are telling us that hot and dense matter produced in RHIC collisions is more complex than we had anticipated. Many models and many predictions were made before the data were available and almost all of them fail to describe the data. The HiJing model is the rare exception. It was available before the data were taken and, at least, it is a good description of particle multiplicities at RHIC. The interesting thing about HiJing is that it was created as a description of hard processes in nuclear collisions, such as jets and mini-jets, and it was not tuned to, and is not a good description of, the soft physics that is dominant at lower energies and yet is still important at RHIC; especially during the final state hadronization phase.
137
PaddkslBBC
Central Multiplicity
Fig. 6. The Zero Degree Calorimeters measure neutrons moving in the forward and backward directions. The ZDC response is double-valued because the mid-peripheral collisions yield a large number of beam-like neutrons, peripheral collisions yield very few neutrons, and central collisions also yield very few beam-like neutrons. The labels identify the top 5% of the interaction cross-section, top lo%, and so forth. The fi gure is provided courtesy of the PHENIX collaboration.
Fig. 7. A low multiplicity event seen in the STAR TPC. The beam line goes into the page and through the center of the fi gure. The colors represent different rates of energy loss for each track.
138
~i
0.5
n '
Fig. 8. The PHOBOS collaboration made the first measurements of maximum multiplicities at RHIC. The multiplicities are large but not as large as expected. The figure is from Ref. [lo] but it has been modifi ed to include the 200 GeV data point from Ref. [9]. The HiJing calculations are shown as a histogram with a solid line to guide your eye.
The BRAHMS collaboration has data that can extend our comparison of the data and theoretical models in very interesting ways because their spectrometer can be rotated through a wide range of laboratory angles. Thus, they can collect data over a very broad rapidity interval. See Figure 9. The figure and the data are from reference [l 11, while the model calculations are from references [lo], [121, and [131. UrQMD[12] is an elegant example of a model that was built to simulate ultra-relativistic heavy ion collisions. It follows the course of a reaction, collision by collision, and it was built using a set of known physical processes and cross-sections. So it is a surprise to find that while UrQMD does an excellent job of simulating final state hadronization, it is not a good description of RHIC multiplicity data. See Figure 9. The AMPT model[ 131 is built upon the initial parton momentum distributions generated by the HIJING model and then it adds its own unique hadronization scheme in the final state. T h s leads to modifications of the HiJing predictions which are generally supported by the data. Another important observation from the BRAHMS data shown in Figure 9 is that the multiplicity of particles is invariant with respect to pseudo-rapidity over an interval of about 2 units. It is not invariant over the full range of RHIC multiplicities. Thus, the simple idea of a Bjorken tube in which hadronization is invariant is an approximation that is not fully born out by the data. This behavior can also be seen in the BRAHMS p-bar to
139
Fig. 9. The BRAHMS collaboration has made charged particle multiplicity measurements over a very wide rapidity interval. The data help discriminate between models which are used to describe heavy ion collisions. p ratios[l4] as a function of rapidity. Here too, the ratio is not constant but falls off after 1.5 units of pseudo-rapidity. This indicates that hadronization and baryon production is not invariant across pseudo-rapidity as suggested by a simple Bjorken model. So, it is clear that the PHOBOS and BRAHMS data discard a wide class of models that were expected to be a good description of the data and, in particular, models based on soft processes at lower energies appear to be inaccurate at RHIC energies. Instead, hard processes such as mini-jet production are beginning to play an important role. In addition, final state interactions seem to be scaling according to the initial state gluon densities.
Kharzeev and Nardi[lS] have started to explore this idea. They propose a simple separation of hard and soft processes according to whether the observed process scales as the number of participating nucleons in a collision (soft process) or the number of binary collisions (hard process). In the perturbative picture, the “hard” component of multiplicity is proportional to the mini-jet production cross section.
The PHOBOS collaboration has fit their multiplicity data[ 161 with this equation and they find that the hard process coefficient is z M 0.15. This simple scaling rule for soft and hard processes quantifies how important the hard processes are in the production of charged particle multiplicity. This result is incompatible with final state saturation models but it is consistent with initial state gluon saturation models.
140
Kharzeev and Levin[l7] have used an initial state gluon saturation model to predict the total charged particle multiplicity and the rapidity dependence of charged particle production at 200 GeV before it was measured. They predicted an increase of 10-14% between 130 and 200 GeV and PHOBOS measured 14f6%. The Kharzeev and Nardi prediction is based on the RHIC 130 GeV data as well as CERN and HERA data at lower energies. The essential idea behind their calculations is that particle multiplicitiesare proportional to the number of gluons in the initial state. But the initial state in ultra-relativisticheavy ion reactions is Lorentz contracted and the low momentum gluons interact coherently with the entire Nucleus. The running of the coupling constant ensures that the coupling is large for the low momentum gluons and thus the cross-section for combining low momentum gluons into higher momentum gluons is large . ..so large that the initial state is saturated and p a = 1. ( p is the density of gluons and (7 is the cross-section for glue-glue interactions.) The saturation condition reduces the flux of low momentum gluons and reduces the multiplicity of particles in the final state relative to what you might expect from a simple extrapolation of gluon number based on the increase in Js. This explains, in part, why the early models of RHIC collisions were wrong. They did not correctly include the suppression of particle yield due to gluon saturation in the initial state. Thus, the data suggest that final state particle multiplicities depend on the properties of the initial state in ways that we did not expect. RHIC data are full of surprises. The Lorentz properties of the initial state are important. We haven’t found any evidence for a simple first order phase transition from ordinary hadronic matter to a quark gluon plasma; and the final state particle multiplicities are lower than we expected. So the early ideas that were used to motivate the program are showing their humble origins. However, we have discovered some things that are unusual and which cannot be fully explained. Flow is an example; both radial and elliptic flow. Radial flow is a measure of the collective motion of particles away from their point of interaction while elliptic flow is a measure of the asymmetry of this particle distribution in momentum space. The radial expansion of the fireball, created during a nucleus-nucleus collision, has thermal characteristics and so a plot of the differential multiplicity versus the transverse mass, mT, is exponential. The inverse slope of the exponential is loosely referred to as the temperature, however, the temperature is different for different particle species and strongly dependent on the particle mass. Xu[181 has proposed that the mass dependence has a simple explanationbecause the total kinetic energy of a particle has two components; a thermal component and a kinematical component.
E
=
3
-T
2
+
1
-mu2
2
Using equation 2, Xu proposed a universal freeze out temperature that is common to all particle species once the mass dependent kinematical term is removed from the data. Xu and Kaneta[191 have performed this analysis for pion, kaon, and proton spectra that were
141
RE
Fig. 10. Radial flow measurements suggest explosive radial expansion and high pressure in nuclear collisions at RHIC. For example, the radial expansion velocity is 55% of the speed of light at 130 GeV and it is apparently climbing with 4s. On the other hand, the freeze out temperature is saturated and does not appear to be climbing with Js.
measured by the STAR collaboration. They find that there is a well defined freeze-out temperature at RHIC and it is 120f 15 MeV. The data also suggests that the average radial flow velocity, p, of these particles is 0.55f0.1 c at Js = 130 GeV. This is very surprising. The freeze-out temperatures at RHIC are essentially the same as at CERN and the AGS while the radial expansion velocity at RHIC (130 GeV) is a substantial increase over the CERN and AGS values. See Figure 10. Now let's turn to elliptic flow. Elliptic flow is a measure of the asymmetry of the emission pattern of particles in momentum space following a peripheral nucleus-nucleus collision. It is the result of the almond shaped overlap zone between two nuclei when they collide, and the flow value is a measure of the efficiency with which this coordinate space asymmetry can be translated into momentum space. One consequence of the geometric asymmetry in the overlap zone is that it is easiest for particles from the center of the collision to emerge along the short axis of the almond. Or, equivalently, one can think of the surfaces of the almond as emitters of particles and there is more surface area facing to the side than to the top (if the long axis of the almond is vertical). Either thought leads to the idea that a coordinate space asymmetry can be translated into a momentum space asymmetry in the emission of particles after a heavy ion collision. It is traditional to perform a Fourier decomposition of the momentum space particle distributions in the x-y plane and v2 is the
142 "*,
0. I
2,0.09 a. 0.08
0.07
0.0s 0.04 0.03 0.02 0.01
'0
0.1
0.2
0.3 0-4 0.5 0,Ci U.7 0.8 0.9 I P IGeVW
Fig. 11. Elliptical flow is large at FWIC and the mass dependence of the signal is in good agreement with hydro-dynamical calculations. These data are from the STAR collaboration[211. Pion and proton data are shown. The solid lines are the predictions of a hydro-dynamical model (modifi ed blast wave).
coefficient of the 2nd harmonic term. In other words: 212
=
(c os(24) )
where
=
atan
(z)
(3)
and pz and py are the momenta of the individual particles. The average is performed over all particles in the event. The RHIC data show that v2is large at Js = 130 GeV. The peak value, integrated over all pT, is 6% for peripheral collisions[20]. The same quantity measured at the AGS is 2% and the value measured at the SPS is 3.5%. Hydrodynamical models predict that elliptic flow will always be large and these models consistently over-predict the amount of flow at lower energies but they seem to get it right at RHIC energies. See Figure 11. In contrast, Cascade models such as UrQMD successhlly predict the amount of flow observed at the SPS but they fail to predict the amount of flow at RHIC. v2 as a function of pT is shown in Figure 11 [Zl]. The interesting feature in Figure 11 is that flow increases with PT as predicted by hydrodynamics and the model does a good job of describing the mass dependence of the data. This is a surprise because hydrodynamics assumes thermal equilibrium and therefore the data is suggesting that the system evolves very rapidly towards an equilibrium state; much more rapidly than at the SPS and more rapidly than was anticipated for RHIC. The hydrodynamical model predicts that flow persists to very high transverse momentum. However, the data do not support this prediction. The data[22], [23] show a pronounced saturation effect and the amount of elliptic flow levels off at momenta above 2 GeV/c. See Figure 12. The data suggest that the process which generates flow is being attenuated at high pT and some kind of quenching is going on in the fire ball. This is a surprise and was
143
a 0.3
v
N
2-
0.25 0.2
0.15 0.1
0.m O0
0’5
1
1,s
2
2.5
3
3.5
4
4.5
5
P t I CeV/el Fig. 12. The elliptical flow signal for pions persists to high momentum and saturates above 2 GeV/c. These data, from the STAR collaboration, suggest that the flow signal is preferentially quenched at high PT.
not anticipated at RHIC. The initial attempts at describing the data with a cascade model required unreasonably high initial gluon densities or extreme elastic parton cross-sections. More recently, Gyulassy, Vitev, and Wang[24] have approximately reproduced the data by dividing the problem into a soft non-perturbative component, which includes hydrodynamic elliptic flow, and a perturbative QCD hard component. The hard component includes jet quenching and energy loss that is a function of the initial gluon density achieved in the collision. See Figure 12.
A related phenomena can be seen by comparing A-A and p-p scattering cross-sections at high PT. The cross-section for scatteringAu on Au is a steeply falling exponential function. Similarly, the cross-section for scattering protons on protons is a steeply falling exponential function and so we might expect that the Au-Au cross-section is a simple super-position of p-p interactions. For example, the ratio between the Au-Au cross-section and the pp cross-section at low PT should be proportional to the number of wounded nucleons in the collision[25], and the ratio at high pT should be proportional to the number of binary encounters in the collision. In between these two limits, we should see a smooth transition from soft to hard physics. Figure 13 shows the ratio of inclusive spectra for charged hadrons at 130 GeV compared to the UA1 p-bar p spectrum interpolated to 130 GeV[23]. The binary collision limit and the wounded nucleon limit are shown on the figure. The interesting feature is that the ratio does not rise from the wounded nucleon scaling limit to the binary collision scaling limit, as expected, but instead it saturates and perhaps decreases at transverse momenta above 2 GeVIc. The PHENIX collaboration has seen a similar phenomenon in their charged hadron data[26] at Js = 130 GeV. They plot the yield of charged hadrons in central collisions
144
Fig. 13. The yield of high pt particles, at mid-rapidity, does not look like a simple binary scaling of proton-proton collisions. This may be evidence for radiative parton energy loss or other energy loss mechanisms due to the high gluon densities in the initial state. (scaled per binary collision) compared to the yield of charged hadrons in peripheral collisions (scaled per binary collision) and show that it is suppressed at high pt. They have also investigated the neutral noproduction yield. The data are shown in Figure 14 as a ratio RAA where:
The figure suggests that the r0data are even more suppressed at high PT than the charged particle data shown in Figure 13. Figure 14 also shows nodata taken by the WA98 collaboration[27],[28] at Js = 17.3 GeV. The striking feature of the WA98 data is that it is not suppressed and it climbs to the binary collision scaling limit and even surpasses the limit. Presumably, the reason why the CERN data rises above 1.0 is that the Cronin effect comes into play and multiple scattering in the initial state causes the incoming partons to have a modest pT kick and this distorts the spectrum. The Cronin effect should also be at work in the RHIC data, trying to increase the ratio above 1.O, and so the WA98 data underscores the fact that the RHIC data is highly suppressed. One possible explanation for the suppression effect is that a new energy loss mechanism is
145 a
ce 2.5
2
1.5
1
0.5
"0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
pr (GeVlc)
Fig. 14, The PHENIX collaboration has measured the relative yield of charged and neutral particles at RHIC. The WA98 collaboration has made similar measurements at the CERN SPS. They are very different. The RHIC data are highly suppressed, perhaps by radiative parton energy loss mechanisms.
coming into play for the partons as they traverse the hot dense matter that is created in a nucleus-nucleus collision. Wang[29] has studied the data and he has successfully described the RHIC results using a parameterization of the Cronin effect, nuclear shadowing, and radiative parton energy loss. He estimates that the average parton dE/dx rate in Au-Au at RHIC energies is approximately 0.25 GeV/fm; but this effective value has been diluted by the rapid expansion of the fireball and he estimates that the rate of energy loss in a static system with a gluon density appropriate for the initial stages of the expanding system would be 12 GeV/fm. This is about 40 times higher than the rate of energy loss in cold, static, nuclear matter (0.3 GeV/fm), and since parton energy loss is proportional to gluon density, this suggests that the gluon density in the initial stages of a Au-Au collision is 40 times higher than in cold nuclear matter.
Finally, the PHENIX collaboration has measured the Bjorken energy density in Au-Au collisions at RHIC[30]. They use a large electromagnetic calorimeter to measure the total transverse energy in each collision. Then they apply the Bjorken formula for thermalized energy density to the top 2% most central collisions. Using a characteristic time of 1 fm/c, they find that Bjorken energy density is 4.6 GeV/fm3. This is a factor of 1.6 larger than at CERN energies (JS = 17.3 GeV) and 30 times higher than in cold nuclear matter.
146
4
Conclusions
RHIC data are not what we had expected. Attempts to describe the data by extrapolating low energy cross-sections and low energy dynamics appear to be wrong, or incomplete. The best cascade model predictions at RHIC are made by HiJing and it has a lot of input from perturbative QCD and very little input from soft physics. The initial state is very important. The Lorentz contraction of the incoming nucleus and the running of the coupling constant leads to gluon saturation. This effect suppresses the large multiplicity of particles that we expected based on simple Js scaling of low-energy data. The initial state is also very dense. The energy density is high and the gluon density is high; both are 30 or 40 times higher than found in cold nuclear matter. The high value for the glum density may cause unusually high radiative parton energy loss at high PT and this could explain the suppression of elliptic flow at high pT. The initial state also allows multiple scattering and the Cronin effect which are important in a complete description of the high pT suppression of particle yields at RHIC. RHIC and CERN data are different. Elliptic flow measurements at RHIC are in good agreement with the predictions of models based on hydrodynamics. This is a surprise because hydro models assume thermodynamic equilibrium and thermodynamic equilibrium is not a good description of CERN results, but yet the elliptic flow data suggest that RHIC collisions evolve very rapidly towards thermal equilibrium. In addition, the radial flow expansion velocities at RHIC are large; 55% of the speed of light. The RHIC and CERN data also differ by a factor of 1.5 to 2 in energy density, and the initial gluon densities are very different as can be seen by the dramatic suppression of high p~ particle yields at RHIC. The suppression is not seen in the CERN data. Have we seen a new state of matter? It is hard to say; but the data do suggest that we seen a very unusual state of matter at RHIC.
References [l] F. Karsch, hep-lat/Ol06019, hep-pW0103314, and Nucl. Phys. A698, 199 (2002). [2] http://www.star.bnl.gov and NIM A, RHIC Special Volume, in press. [3] http://www.phenix.bnl.gov and NIM A, RHIC Special Volume, in press. [4] http://www.rhic.bnl.gov/brahms [5] http://www.phobos.bnl.gov and NIM A, RHIC Special Volume, in press.
[6] H. Wieman et al., IEEE Trans. Nuc. Sci. 44, 671 (1997) and J. Thomas, et al., Nucl. Instrum. & Meth. A478, 166 (2002). See also: http://www.star.bnl.gov + Group Documents + TPC +Hardware.
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[7] C. Adler, A. Denisov, E. Garcia, M. Murray, H. Strobele, and S. White, Nucl. Instrum. & Meth. A470,488 (2001). [8] B.B. Back et al., Phys. Rev. Lett. 85,3100 (2000) and nucl-ex/O007036. [9] B.B. Back et al., Phys. Rev. C, in press, and nucl-ex/0201005. [lo] X.N. Wang and M. Gyulassy, Phys. Rev. Lett. 86,3496 (2001) and nucl-tW0008014. [ l l ] I.G. Bearden et al., Phys. Lett. B523,227 (2001) and nucl-ex/0108016. [12] S.A. Bass et al., Prog. Part. Nucl. Phys. 41, 255 (1998) and nucl-tW9803035. [13] B. Zhang, C.M. KO,B.A. Li, and Z. Lin, Phys. Rev. C 61,067901 (2001) and nucl-th/9907017. [14] I.G. Bearden et al., Phys. Rev. Lett. 87, 112305 (2001). [15] D. Kharzeev and M. Nardi, Phys. Lett. B507, 121 (2001) and nucl-th/0012025. [16] B.B. Backet al., Phys. Rev. C65,031901 (2002) and nucl-ex/0105011. [17] D. Kharzeev and E. Levin, Phys. Lett. B523,79 (2001) and nucl-th/0108006. [18] N. Xu, Nucl. Phys. A 610, 175c (1996). [19] N. Xu and M. Kaneta, Nucl. Phys. A698,306 (2002) and nucl-ex 0104021. [20] K.H. Ackermann et al., Phys. Rev. Lett. 86,402 (2001) and nucl-ex/0009011. [21] C. Adler et al., Phys. Rev. Lett. 87, 182301 (2001) and nucl-ex/Ol07003. [22] C. Adler et al., submitted to Phys. Rev. Lett. (2002) and nucl-ex/0206011. [23] J.C. Dunlop, Nucl. Phys. A698,515 (2002). [24] Gyulassy, Vitev, and Wang, Phys. Rev. Lett. 86,2537 (2001) and nucl-th/ 0012092. [25] A. Bialas, M. Bleszynski, and W. Czyz, Nucl. Phys. B111, (1976) 461. [26] K. Adcox et al., Phys. Rev. Lett. 88,022301 (2002) and nucl-ex/OlO9003. [27] K. Reygers et al., nucl-ex/0202018. [28] M.M. Aggarwal et al., ELKPhys. J. C23,225 (2002) and nucl-ex/Ol08006. [29] X.N. Wang hep-pW0111404. [30] K. Adcox et al., Phys. Rev. Lett. 87,052301 (2001) and nucl-ex/0104015.
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Applications of the Equivalent Photon Approximation to Heavy Ion Collisions Sebastian N. White Brookhaven National Laboratory, Upton, New York 119’73 Two of the main research tools in High Energy Physics- electron and proton colliders are often described as having complementary roles. But their roles have changed in the last 30 years. For example the term “electron collider physics” connotes a broader topic today than the original purpose- to study the electromagnetic structure of the proton. The Relativistic Heavy Ion Collider (RHIC) which started operation a year ago will be joined by the Large Hadron Collider (LHC) in 2007 with a 30-fold increase in center-of-mass energy. In many respects these instruments are complementary. The heavy ion community anticipates a full program for both machines which focuses on the study hot hadronic matter and search for the quark-gluon plasma. Nevertheless, it is likely that the term “Heavy Ion Collider Physics” will come to include other areas of research as techniques with the colliders evolve- particularly in those areas where the energy increase from RHIC to the LHC are most important.
1
Introduction
In this paper I discuss Heavy Ion Colliders as laboratories for the highest energy photon hadron collisions on the planet. This was one of the themes of this workshop. I discuss briefly the Equivalent Photon Approximation(EPA) which motivates some of the experiments. This is similar to the role of the parton model, which motivates many experiments in hard probe physics. There are already several experimental paper&3 from RHIC dealing with electromagnetic interactions of Heavy Ions. How this field might evolve at the LHC is discussed elsewhere in these proceedings’. I emphasize the experimental details of nuclear coulomb dissociation measurements. That the measurements are clean and precise is, in itself, a significant result and affects the discussions about the program at LHC.
1.1
the Equivalent Photon Approximation
Fermi When first introduced by Fermi in 1924 4 , the Equivalent Photon Approximation(EPA) was perhaps most significant for establishing a correspondence between the interaction of charged particles to those of x- and y -rays with atoms. Fermi treated the interaction of non-relativistic electrons and ions (Radium-C Q ’s) with Hg vapor and Helium gas and used available results on photc+ionization and excitation as input to his calculation. The method is still widely used in spite of the development of qed. It is often more convenient to make use of experimental input from y interactions than to calculate the full interaction diagrams. Fermi’s expression for the photon “flux factor” from a charged particle with velocity , v , at impact parameter, b, has a spectral distribution:
where KOand K1 are the Bessel functions, is essentially identical to the expression we use today for the relativistic case. The expression usually reduces to a single term in the relativistic case since, in Fermi’s decomposition, the term due to the longitudinal Electric field component decreases with velocity.
149
150
Weizdcker-Williams In 1933, about a year after the discovery of the positron in cosmic rays, Williams visited Copenhagen where he discussed the calculation of electron-positron production by high energy charged particles. Bohr encouraged Weizsiicker to work on this problem also. The papers of Weizsacker and Williams extend Fermi’s method to the ultra-relativistic case. The line of work that this initiated illustrates the fact that charged particle beams are often the most practical tool for studying the highest energy photon interactions. The method of equivalent quanta has since been extended to other problems besides the calculation of the equivalent photon flux from charged particles. For example, in the ’60s and ’70s the Equivalent Electron Approximation was introduced The Equivalent W Approximation was used to calculate Higgs particle production ’, for example. The method is used to describe coherent interactions of particle beams with stationary magnetic fields ’. Also between particle beams in colliders s, where the transverse beam dimensions have also to be considered in the discussion of the equivalent photon spectrum.
‘.
extension to qcd Several authors pointed out that the method could be used to calculate analogous processes such as q + qg in qcd” taking into account the different coupling constant(a + a s ) ,etc. A similar description was used to calculate diffractive jet production”. Diffractive scattering is sometimes described using an equivalent Pomeron “flux factor” and the Pomeron, itself, is then equivalent to a constituent parton flux. The analogy between photon exchange and Pomeron exchange interactions of hadrons occurs naturally because of the similarity of these pictures and their event characteristics. However the breakdown of the Pomeron exchange picture (factorization) in high energy HERA and Tevatron data should remind us of the basic difference between these processes as discussed in refJ5. Before leaving the topic of hadronic diffraction it is worth mentioning that hadronic diffraction (“Pomeron exchange” ) and electromagnetic interactions at the LHC are inevitably part of the same experimental program. For example, the electromagnetic two photon collision event and the double Pomeron event are indistinguishable. Both are characterized by leading particles in both beam directions and central particle production. Yet they have very different cross section dependence on nuclear species (ie Z4 vs . The latter is a very serious background to 2-photon collisions when LHC runs pp colliding beams but can be neglected in Pb-Pb collider running at the L H e 2 . On the other hand, all of the techniques used to select rapidity gaps and study diffractive interactions by the ”rapidity gap method” at the Tevatron are directly applicable in Heavy Ion running at the LHC.
applications Early applications to Nuclear Physics were based on measurements with electron beams and nuclear targets. For some time, photo-absorption cross-sections on many nuclear targets have been available for incident photon energies, E7,of N 6 MeV up to E7 > 1 GeV. The dominant feature at low energies is the “giant dipole” resonance (GDR) which results in the emission of 1 or more neutrons (in heavy nuclei the coulomb barrier suppresses proton emission)- ie y AuT& -+ n A& and has a threshold of E7 > 8.1 MeV. Photo-absorption on nuclei with E7 > 1 GeV has been cal~ulated’~, scaling available high energy y - p data by target mass(AT) and correcting for nuclear shadowing by a factor of order 60% . Available data are of adequate quality for calculation of charged particle interactions (using the EPA) for Heavy Ion measurements discussed below. Because strong interaction
+
+
151
cross-sections are typically much larger than the corresponding electromagnetic ones , the first applications naturally used electron beams. (In the present discussion we leave out lepton pair production (the original motivation for Weizsacker and Williams’ calculations) but this is treated elsewhere in these proceedings 16.) Since the dominant feature of low energy photo-absorption on nuclei is neutron emission, early tests used radiochemical techniques to measure the activation of foils exposed to an electron beam. Systematic measurements with electron beams on nuclear targets were performed by Wilson, Panofsky and co-workers13v14in the 1950’s. In these experiments foils of different thickness were exposed to a 35-500 MeV electron beam. For significant target thickness the dominant photoinduced reactions are due to bremsstrahlung with subsequent photo-nuclear absorption. In these experiments, data from exposures with several thickness were used to extrapolate to the “no radiator” limit- ie the intrinsic photon content of the electron as envisioned by Fermi. 1.2 experiments with nuclear beams
Fermi’s paper dealt with electromagnetic interactions of both electrons and ions with atoms (and equally good data were available for both). It has taken 70 years for ion beams to become a viable alternative to electron beam measurements . The reasons for this are obvious. The strong interaction cross-sections of heavy nuclei are large ( barns) but approximately energy independent. The electromagnetic cross-sections, on the other hand, increase with energy and are comparable to them at RHIC. The second reason is that these two types of interaction are experimentally distinguishable at RHIC. We illustrate this below with data taken during the first RHIC run.
Single Coulomb Dissociation Coulomb Dissociation refers to the breakup of one nucleus by the Electromagnetic field of the other. As noted above, electro-disintegration by an ion beam is more difficult to isolate than the same process induced with an electron beam- particularly at low energy. Experiments at the Bevalac started in the 70’s first observed this process. Using beams with different atomic number (Z), these experiment found a roughly quadratic Z dependence thus confirming a The calculated Coulomb Dissociation cross-se~tionl~ takes the form:
Wthresh
The large ( Z 2 )factor due to the nuclear charge of the beam particles is effective so long as the nucleus can act as a coherent source. For this reason, Coulomb Dissociation is predominantly due to quasi-real photons, ie q2 < I . / R i . Several features of the above expression distinguish ion coulomb interactions from from those of electrons with hadrons:
+
1. bmin: this a natural cutoff parameter. Experimentally, when b < (RA R B ) ( when the nuclei A and B overlap) strong interaction will occur between the nuclei, changing the characteristics of events. In any case, for b < bminr the point charge approximation is no longer valid.
2. urnas: there is a natural cutoff in the y energy spectrum directly related to the source size. The cutoff due to the logarithm above is derived in ref?g. Coulomb Dissociation has been observed at the CERN SPS with various beams and targed4 using radiochemical techniques. At the LHC and RHIC the cross-sections are 220 and 95 barns,
152
re~pectivel$~.This cross-section is so large that it is the dominant source of beam loss (and consequently determines the magnet quench limit) for operation of the LHC with Heavy I o d 6 . M u t u a l Coulomb Dissociation In spite of its large cross-section, the Single Coulomb Dissociation process is not intrinsically a clean process at a Heavy Ion Collider. For example, interaction of the beam with residual gas in the accelerator vacuum chamber could cause similar breakup of beam ions. For this reason, a variant on the single Coulomb Dissociation process was investigatedzO~zz~zl. In Mutual Coulomb Dissociation(MCD), both ions interact with each-other’s photon field. There is, in fact, a correlation, since the photon spectral distributions depend on impact parameter. The calculation proceeds along the same lines as eq’n (2) while noting that the order of integration (over b and w ) can be interchangea2 we have:
where a @ )refers to the first-order (ie not unitarity corrected) calculation. 2
The Measurements at RHIC
In order to trigger effectively on this cross-section at RHIC, we built Zero Degree Calorimeters to capture and measure the neutrons from Nuclear breakup very close to the beam directionz8. During the year-1 run at RHIC we collected data with m = 1 3 0 GeV (with a beam energy of A A x~65 GeV). The ZDCs are small transverse-area hadron calorimeters with an angular acceptance of (01 < 2 mrad with respect to the beam axis. They are located in the beam line behind the beam bending magnets at the four RHIC interaction regions and measure the energy of unbound neutrons in small forward cones around the beam. Charged particles are deflected out of the ZDC acceptance by the beam bending magnet as seen in Fig. 1. An interaction trigger required an energy deposit above 12 GeV in the left and right ZDCs. The energy scale was determined from fits to the one and two neutron peaks, which are clearly seen in Fig. 2. The energy resolution at the one neutron peak is approximately 21%, consistent with test beam results z8. The following discussion is based on data from the PHENIX setup. Similar results from the PHOBOS and BRAHMS locations were also presented in ref?. Events were selected by a ZDC coincidence corresponding to a distance of 5 20 cm from the nominal interaction point (calculated using the time-of-flight to the ZDCs) yielding a total of 160,601 events. The PHENIX Beam Beam Counters (BBCsY’ are arrays of quartz Cerenkov detectors which measure relativistic charged particles produced in cones around each beam (3.05 < 1771 < 3.85, with 27~azimuthal coverage). There are 64 photomultipliers (PMTs) in each BBC arm. A coincidence of the 2 BBC arms was used to tag hadronic collisions for the ZDC trigger sample. In Fig. l(b) we plot the neutron multiplicity measured in the ZDCs (i.e. total amplitude relative to the single neutron peak) for 2 classes of events, according to the number of hit BBC photomultipliers above threshold, ngBC
1. “Hadronic” events for which nggC > 1 in each arm. 2. “Coulomb” events with nggC
5 1 in at least one arm.
The BBC trigger efficiencies for Coulomb and hadronic collisions were calculated from simulations of particle production for each class. We find that 99% of Coulomb interactions satisfy the Coulomb cut criteria since only 3% of the events have a charged particle in either arm z5. However PHENIX calculated that only 92 & 2% of hadronic interactions passed the “Hadronic”
153 L"
'
'
"'
'
'
"
'
'
" "
" "
'
'
"
" '
'
'
'A
Meters A-A
1 "
"
1
"
'
I
"
'
"
I
I". 0.
-10. -Z".
-10.
0.
10.
EO.
30.
4".
CentimeLers
0
5
10
15
20
25
30
35
Neutron multiplicity Figure 1: (1eft)ZDC location: (top) plan view and (bottom)position and spread of neutrons, Au ions and protons at the ZDC location. (right,b) Single arm ZDC neutron spectrum for the Coulomb and hadronic events.
":;
/ \ # I
_""."..." ".""..-.I..-.. -.-...-
100
0
Figure 2: Left ZDC spectrum with "Coulomb" selection for events in which there is one neutron in the right ZDC.
cut defined above 30. The other events are misidentified and represent a 14% contamination of the "Coulomb" sample. Diffractive hadronic interactions are not treated in our simulation and these are a potential background to our "Coulomb" sample. However this is a negligible correction since, for example, the high energy p+W diffraction dissociation cross-section is 20 mb32, which is a small fraction of the total p+W inelastic cross-section of 2.7b. Background from accidental coincidences between single beam interactions was calculated from the individual ZDC rates and found to be negligible. The ZDC multiplicity spectra are very different for the 2 classes of events as can be seen from Fig. l(b). The "Coulomb" events tend to have low total neutron multiplicity whereas the fraction of "Hadronic" events with 1 or 2 neutrons is very small. Hadronic interactions which fail the "Hadronic" cut are peripheral events corresponding to a few elementary N-N collisions so events with low BBC multiplicity were used to model the shape of the 14% background for "Coulomb" events. The corrected L'Coulomb'' shape is identical to the ZDC multiplicity spectrum obtained by measuring the left ZDC distribution for events with one neutron in the right one. This cut strongly suppresses hadronic collisions. The ZDC trigger efficiency depends on the geometrical acceptance for neutrons in the ZDC. The angular distribution of neutrons about the beam direction is expected to be different for hadronic and Coulomb events so we discuss the two cases separately. In low energy photopre
154
ui
measured(PHEN1X)
calculate&’
25
gtot
-
10.8 f0.5
11.2
u g e a
-
7.1
7.3
“seom
.661 +.014
.67
.659
&
.117 f .004 .345 .012
.125 .329
,139
-
,327
H
* .345 * .014
-
Table 1: Ratios of cross sections for experiment and theory. The values of utotand ugeomare in barns.
duction experiments the neutron multiplicity increased with photon energy and their maximum was 120 MeV/c 33,34. For 65GeV neutrons this corresponds to a maximum opening angle of less than 2 mrad and therefore an efficiency of 100%. However -20% of the MCD cross section is due to equivalent photons with energy 2 2 GeV for which no photoproduction data are available. Based on the trend of lower energy data we expect these events to have a mean neutron multiplicity larger than 1. Even if the p~ distribution broadens at higher photon energies the ZDC inefficiency is expected to be small since it is unlikely that every neutron will miss the detector. Therefore, no ZDC acceptance correction is applied to the measured Coulomb rates. The ZDC acceptance correction for hadronic collisions was measured using an independent data sample in which we required only a BBC coincidence trigger. BBC trigger events were scanned for interactions in which the ZDC trigger condition failed. The measured efficiency is 98 2% and this correction is applied to the calculated hadronic rates. The acceptance correction was also independently estimated assuming a neutron p~ spectrum characteristic of a Fermi distribution 35. The calculated geometrical acceptance per neutron in the ZDC detector is 75% at JsivN=130 GeV. Multiple neutrons are detected for most nuclear collisions and so the calculated ZDC detection efficiency per event is close to unity, consistent with the measurement. We corrected our measured ratio of “Hadronic” events to total events to account for the (experiment dependent) BBC trigger efficiency, E B B C . In PHENIX, EBBC was found to be [92 f2(syst)]% 30. The trigger efficiency is very weakly dependent on input assumptions of the Glauber model, such as the radius parameter. Note that it is only this efficiency and not the Glauber cross-section that is used in the current analysis. The BBC trigger had a background contamination of [I f ~ ( s y s t ) ] % ~ ~ . Table 1 presents our measured value of the ratio of ggem to the total ZDC cross section, utot. This ratio agrees well with calculations in To compare to specific channels calculated for Mutual Coulomb Dissociation, we fit the measured ZDC energy spectra to neutron multiplicity distributions, convoluted with the energy resolution. We constrained the fits such that u ~ ( 2 n=) fi x a ~ ( l n )etc. , This is justifiable from first principles. However the fitted areas can vary significantly if this constraint on the relative peak widths is removed and this variation dominates our systematic errors. Figure 2 shows the energy spectrum obtained in one ZDC (ZDC left) for the “Coulomb” event selection when the other ZDC pulse height is consistent with 1 neutron. The total number of events in Fig. 2, after background subtraction, corresponds to the crosssection for the (1,X) topology in which one neutron is observed in the right beam direction. Table 1 lists the sum of 1 neutron in left and right topologies as (1,X). Of course, any number of protons could also be emitted in the reaction but these are not detected. The decay topology (1,l)corresponds to exactly 1 neutron in ZDC left and 1 neutron in ZDC right. Similarly the fits to specific neutron multiplicities in the ZDCs such as (2,l) are derived from fits to the left ZDC spectrum, for events where we have cut on the pulse height in the right ZDC. The errors
+
22925.
155
PHOBOS BRAHMS Aumann et a1
Y
0.05
lo2
10
1
dS,
(GeV)
Figure 3: Ratio of 2n to In cross-sections vs.
fi,
events
lo00
...... 0-1
-0.8 -0.6 -0.4 0 . 2
-0
0.2
0.4
0.6
0.8
1
Energy asymmetry Figure 4: Asymmetry spectra for Coulomb and hadronic events.
shown include both statistical and systematic errors in the fit procedure. In Fig. 3 we compare our measured 2n to I n ratio with lower energy data on single beam dissociationz4 from AGS and SPS. The ratio tends to increase with energy and the values are in reasonable agreement with calculations found in 2z,25. Since the calculation of U M C D is sensitive to the choice of minimum impact parameter bminr ref!g proposed initially measuringutot instead and obtained utot = 10.8 zt 0.5b at = 130 GeV. Since this result is more precise than current measurements of age- we use it to calculate UMCD,
~ M C D= atot
- u g e a = utot . (1 - u g e o m / u t o t )
(4)
and the error on UMCD has contributions from the measured ratios and the theoretical error on utot. Using a weighted mean of the measurements of ugea/utot by the 3 experiments in Eqn. 4 gives U M C D = 3.67 f 0.26 barns. In Coulomb events each nucleus interacts independently with the field of the other. we therefore expect only a weak correlation between fragment multiplicities of the left and right
156
Figure 5: The differential photon luminosity scaled by the beam luminosity a t LHC,RHIC and the eIC. The energy w scale refers to the equivalent energy in the rest frame of the nuclear (Pb or Au) target.
going nuclei. For hadronic collisions both nuclei have the same average number of “wounded nucleons” and therefore the left and right energy in each event should be symmetric. We define the asymmetry
where ELeft and ERight are the total ZDC energy signal in the Left and Right ZDCs respectively. Note we require > 98GeV in at least 1 of the ZDCs to overcome the correlation inherent in the trigger. The ZDC asymmetry will be a useful tool in isolating electromagnetic interaction events at RHIC and the LHC. 3
Beyond RHIC
A tremendous increase in energy scale of collisions occurred between the BEVALAC and RHIC The additional increase at the LHC will yield new data on photon-hadron interactions and information about the partonic structure of nuclei. Many measurements that have been carried out with the HERA e-p collider have interesting counterparts in the LHC program. Although we can’t directly compare RHIC and the LHC to e-hadron colliders, it is interesting to compare the photon fluxes in the two cases. To do so we integrate over one variable- q 2 . In the (coherent) Heavy Ion case we are concerned with “quasi-real” photons so we integrate the corresponding electron beam distribution to q2 < (2me)’. This is really an arbitrary cutoff. In this case: dn(u) = --In(-) 2a yefj w(6) dw T mew In both cases the effective lorentz factor y e f f is calculated from the equivalent beam energy for a stationary target (ie yeff = 2y&, - 1).We also take into account the different luminosities for RHIC and the electron-ion collider and LHC designs. The event rate is proportional to the product of the beam luminosity and the gamma flux per beam particle. Fig. 5 reflects the fact
157 that the US nuclear physics community is discussing an electron-Ion collider (eRHICY6 with 6000 times RHIC luminosity and the LHC will have about 10 times current RHIC luminosity. 4
Acknowledgements
I would like to thank Tony Baltz, Valery Serbo and Stan Brodsky for many useful discussions and Sally Dawson for a discussion which led to the calculation in fig. 5. I also thank Alexei Denisov and Mickey Chiu, my collaborators on the analysis of ZDC data taken in the PHENIX experiental area. The US Department of Energy supported this work. References
G.Baur et al., Hot Topics in Ultra-Peripheral Collisions, these proceedings. M.Chiu et al., Phys. Rev. Lett. 89, 012302 (2002) F. Meissner (STAR Collaboration) nucl-ex/0112008 E. Fermi, these proceedings von Weizsacker and Williams, Z. Physik 88 (1934) 612; E.J. Williams, Phys. Rev. 45 (1934) 729 6. P. Kessler, Nuovo Cim. 17 (1960) 809 7. S. Dawson et al., Ch. 3 of The Higgs Hunter’s Guide , Perseus Publishing, Cambridge, Ma. and S. Dawson Nucl. Phys. B249 (1985) 42. 8. I.F. Ginzburg, V.G. Serbo, Phys. Rev. D 49 (1994) 2623 9. V.Serbo, private communication 10. V.M. Baier, V.S. Fadin, V.A. Khoze, Nucl. Phys. B 65 (1973) 381 11. Ghgelman and P. Schlein, Phys.Lett. B152 (1985) 256 12. Engel, Braun, Pajares and Ranft, Z. Phys. C 74 (1997) 687 13. K.L. Brown and R. Wilson, Phys. Rev 9 3 (1954) 443 14. Panofsky, Newton and Yodh, Phys. Rev. 98 (1955) 751 15. Collins, Frankfurt and Strikman, Phys.Lett. B307 (1993) 161 16. K. Hencken, these proceedings 17. C.A. Bertulani, these proceedings 18. G.Baur, K. Hencken, D.Trautmann, J. Phys. G 24 (1998) 1657. 19. A. J. Baltz, M. J. Rhoades-Brown, J. Weneser, Phys. Rev. E 54, 4233 (1996). 20. A. J. Baltz, S. N. White, BNL-67127 (1996) 21. S. N. White, Nucl. Instrum. Meth. A409, 618 (1998). 22. A. J. Baltz, C. Chasman and S. N. White, Nucl. Instrum. Meth. A417, 1 (1998). 23. H.H. Heckman and P.J. Lindstrom, Phys.Rev.Lett.37 (1976) 56 24. T. Aumann et al., Phys. Rev. C 47, 1728 (1993) and proceedings of “Heavy Ions at the AGS”, 286 (1996) 25. I. A. Pshenichnov, J. P. Bondorf, I. N. Mishustin, A. Ventura, and S. Masetti, Phys.Rev. C 64, 024903 (2001) and I. A. Pshenichnov, private comm. 26. D. Brandt, LHC Project report 450 27. G. Baur hep-ph/0112239 28. C. Adler et al., Nucl. Instrum. Meth. A 470, 488(2001). 29. K. Ikematsu et al., Nucl. Instrum. Meth. A 411, 238(1998). 30. K. Adcox, et al., Phys. Rev. Letters 86 3500 (2001). 1. 2. 3. 4. 5.
158 31. This plot differs somewhat from the corresponding one in ref 30, where the BBC PMTs were run at high gain for good timing but suffered some pulse height saturation. 32. T. Akesson et al., Z.Phys. C 49, 355-366 (1991) 33. F. Tagliabue, J. Goldemberg, Nucl.Phys. 23, 144 (1961). 34. A. Veyssiere et al., Nucl. Phys. A159,561 (1970). 35. J. Barette et al., Phys. Rev. C 45, 819 (1992), and references therein. 36. “The Electron-Ion Collider, white paper submitted to the NSAC long range planning meeting, March, 2001.
Atomic Parity Violation in Heavy Ion Colliders Max Zolotorev, D. Budker LBNL, Berkeley, CA USA It has long been a dream of atomic physicists to measure PNC in hydrogen (for a review of hydrogen experiments, see I ) , or a hydrogenic system. It appears that recent developments in relativistic ion colliders, high-brightness ion sources, and laser cooling methods of ions in storage rings, may open such a possibility for relatively light (Z-10) hydrogenic ions ’. Due to their simple atomic structure, high precision theoretical calculations can be carried out in these ions. In addition, neutron distribution uncertainties will not present a serious problem in relatively light ions considered here, both because the structure of light nuclei is much better understood than that of the heavy nuclei, and because the electron wavefunction gradient at the nucleus is relatively small for light nuclei.
Consider the conceptually simplest variant of a PNC experiment in a hydrogenic system: circular dichroism (i.e. the difference in transition rates for right- and left-circularly polarized light) on the 1s-2s transition in the absence of external electric and magnetic fields. Dichroism arises due to interference between the A41 and the PNC-induced Elamplitudes of the transition. Due to the PNC iiiteraction, the 2s state acquires an admixt,ure of the 2P1/2 state; the magnitude of the PNC admixture is probed by tuning the laser in resonance with the highly forbidden 1s-2s kfl-transition and observing circular dichroism. Tracing Z-dependences of various atomic paraniet,ers, one finds that while t.he weak interaction matrix element increases cxZ5, the PNC asymmetry, i.e. the relative difference in absorption for the two circular polarizations, decreases cxZ-’, while the statistical sensitivity increases cxZ4 ’. The frequencies of the 1S+2S transitions for the hydrogenic ions lie outside the range directly accessibly to laser sources. This problem can be solved by using relativistic Doppler tuning. For an ion wit,h relativistic factor 7 = 1 >> 1 colliding head-on with a photon of frequency
JiT
Wl&
the frequency in the ion’s rest frame is given by: % o n f r a m e = Y (1
+ P ) Wlob
2YWlab.
(1)
In order to tune to t,he 1S+2S resonance for a hydrogenic ion, it is necessary to satisfy the condition:
Using the Relat,ivist,ic Heavy Ion Collider (RHIC, y -loo), with visible and near-UV lasers, it is possible to access 1S-+2St,ransitions for ions with Z up to -11 (Na).
159
160
Evaluating the statistical sensitivity of the experiment, one arrives at the following expression:
9
Here r D = (w . Ap)ion zs w,,,frame . is the Doppler width, Nimsis the average number of ions entering the interaction region per unit time, T is the overall measurement time, and bH, = d E i s a dimensionless saturation paramet,er. This shows t,hat for an opt,inially desigiied PNC experiment, the stat,istical sensit,ivity is completely determined by the t,otal number of available ions and by the transition widths. In order to obtain a certain sensitivky to weak interaction paraniet,ers, e.g. t,o sin2Qw,where Q,is t,he Weinberg angle, it is necessary tsohave exposure NT which can be represented in units of particle-Amperesx year:
using Ne ions (Z=lO) in RHIC, and substit,utiug As an example, for &sin2ell) = 6 . 10-2(which is found to be an optimal value liniit,ed by laser photoionization), one obt,aiiis the necessaxy running time 1 week. In t,his estimat,e we assumed Ay/y=10-6, which is possible to achieve using laser cooling 3,2. Many technical problems would have to be addressed before a PNC experiment could be carried out,. This includes development, of a hydrogenic ion source for the accelerator, implernentatioii of laser cooling, design of an efficient detect,ion scheme for ions excited to the 2 s state, etc. However, all of these problems appear, at least in principle, tractable, arid the proposed technique may offer sensitivity sufficient for test.ing physics beyond the standard model. Another promising t,echnique involving heavy (Z up to 92) helium-like ions has been discussed in4. , y ~ 1=
N
References
E. Hinds, in The Spectiurn of A t o m i c Hydrogen: Advances, ed. G. W . Series (World Scientific, Singapore, 1988). 2. M. Zolotorev and D. Budker, Phys. Rev. Lett. 78(25), 4717 (1997); Prospects for Measuring Parity Nonconservation in Hydrogenic Ions Using High-Energy Accelerators, in: “Parity Violation in At,oms and Elect,ron Scattering” B. Frois arid M. A. Bouchiat, eds., World Scient,ific, 1999, p. 3G4. 3. D. Habs, V. Balykin, M. Grieser, R. Grimm, E. Jaeschke, M. Music, W. Pet,rich, D. Scliwalm, A. Wolf, G. Huber, and R.. Neuriiann, in Electron Cooling and New Cooking Techniques, R. Calahrese and L. Tecchio, eds, World Scient,ific, Singapore, 1991. 4. R. W . Dunford, Phys. Rev. A 54(5), 3820 (1996). 1.
Aspects of Coulomb Dissociation and Interference in Peripheral Nucleus-Nucleus Collisions Joakim Nystrand' , Anthony J. Baltz' and Spencer R. Klein3 'Department of Physics, Lund University, SE-221 00 Lund, Sweden Brookhaven National Labomtory, Upton, N Y 11973, U.S.A. 3Luwrence Berkeley National Laboratory, Berkeley, CA 94720, U S A . Presented at the Workshop on Electromagnetic Probes of Fundamental Physics, Erice, Italy, 16-21 October, 2001. Coherent vector meson production in peripheral nucleus-nucleus collisions is discussed. These interactions may occur for impact parameten much larger than the sum of the nuclear radii. Since the vector meson production is always localized to one of the nuclei, the system acts as a two-source interferometer in the transverse plane. By tagging the outgoing nuclei for Coulomb dissociation it is possible to obtain a measure of the impact parameter and thus the source separation in the interferometer. This is of particular interest since the lifetime of the vector mesons are generally much shorter than the impact parameten of the collisions.
1
Coherent Peripheral Nuclear Collisions
This presentation will discuss some aspects of nucleus-nucleus collisions without physical overlap, i.e. collisions with impact parameters, b, larger than the sum of the nuclear radii, R, i.e. b > 2R. Particles can be produced in these collisions through an interaction of the fields of the nuclei. The interactions can involve both the electromagnetic and nuclear fields, but because of the short range of the nuclear force, purely nuclear processes are suppressed for b > 2R. If the momentum transfers ftom the nuclei are small enough (Q < hc/R), the fields couple coherently to all nucleons. This enhances the cross sections and gives the events an unique signature, which can be used for identification. The restrictions on the momentum transfer do not prevent the production of heavy systems, however, in high-energy collisions. States with masses up to about 27hc/R, where -y is the Lorentz factor of the beams in the laboratory system, can be produced in a heavy-ion collider; this corresponds to masses of a few GeV/c2 at the Relativistic HeavyIon Collider (RHIC) at Brookhaven National Laboratory and about 100 GeV/c2 at the Large Hadron Collider (LHC) at CERN. The electromagnetic field of a relativistic nucleus can be treated as an equivalent flux of photons. This was first realized by Fermi and the idea was further developed by Weizstkker and Williams2. The method is now known as the Weizsacker-Williams method. For relativistic nuclei, the impact parameter is a well-defined variable, and the photon flux should be evaluated in impact parameter space3. The density of photons with energy k at a perpendicular distance
',
"presenting author
161
162
b ( b > R) from the center of a nucleus is (in units where h = c = l y n(k,b) =
* --(F) K'(y) dk&b
=
kb
a22 1 7r2
kb2
Here, a is the fine structure constant, Z the charge of the ion, and K1 a modified Bessel function. The total number of photons with energy k , n ( k ) , is obtained by integrating the photon density over all impact parameters for which there are no hadronic interactions (roughly b > 2R). The photons from one of the nuclei may interact with the other nucleus either electromagnetically or hadronically. Electromagnetic interactions include collective excitation of the nucleus into a Giant Dipole Resonance (GDR) and particle production through two-photon interactions. The hadronic interactions are usually divided the following 3 categories: Vector meson dominance, direct, and anomalous5. The latter two involve interactions with a parton in the nucleus and do not lead to a rapidity gap between the produced particle and the target nucleus. The coherent photonuclear interactions that will be considered here will be treated in the framework of the vector meson dominance model. According to vector meson dominance, the photon interacts hadronically by first fluctuating into a vector meson, or more generally a @-pair, carrying the same quantum numbers (Jpc=l--) as the photon. The scattering amplitude for the photonuclear interaction is
4
is the momentum transfer from the target where fv is the photon vector meson coupling, and nucleus. The sum is over all applicable vector meson states. The photon vector meson couplings are constrained from data on the semi-leptonic decay widths, l'V+e+e-,
where MV is the vector meson mass. The coherent reaction 7 A + V A thus corresponds to elastic scattering, V A -+ V A if the cross terms are neglected (i.e. cases where the photon fluctuates into a state V and then scatters off the target into a state V'). The nuclear momentum transfer can be treated as exchange of a meson or a Pomeron, and the vector meson is produced through 7-Pomeron (or 7-meson) fusion. The Pomeron is the colorless exchange particle of the strong force and carries the same quantum numbers as the vacuum (Jpc=O++). The nuclear transverse momentum transfer is determined by the nuclear form factor, F ( t ) , and the photonuclear cross section is
where tmin = ( M $ / 4 7 k ) 2 is the minimum momentum transfer needed to produce a vector meson. The form factor vanishes for t >> (1/R)2 and the cross section will thus go to zero for small photon energies k < M;R/47. Convoluting the photonuclear cross section with the photon spectrum gives the total cross section for vector meson production in nucleus-nucleus collisions:
o ( A + A --f A
+A +V )=2
Jom
n(k)U,A+vA(k)dk
.
(5)
The integral goes from 0 to 00; in practice it is cut-off at high k by the exponential fall-off in the photon spectrum for k > 7 / R and at low k by the nuclear form factor of the target. The factor of 2 takes into account that the nuclei can act both as target and photon emitter.
163 Table 1: Cross sections for exclusive vector meson production in Au+Au at RHIC and Pb+Pb at the LHC.
Vector Meson
u fmbl RHIC Au+A; LHC Pb+Pb
5200
590 59 39 0.29
Po W
4
JI Q
490 460 32
The reaction ~p + V p has been studied at HERA and in fixed target experiments’. The cross sections in nucleus-nucleus collisions have been calculated in *, using the WeizsiickerWilliams photon spectrum and a Glauber for the photonuclear cross sections. The Glauber model calculations use the experimentally determined photon-proton cross sections as input. The results for gold at RHIC and lead at the LHC are given in Table 1. The cross sections are very large, roughly 10% and 50% of the hadronic cross section for po production with gold at RHIC and lead at LHC, respectively. The production is centered around mid-rapidity with a width determined by the mass of the vector meson. The rapidity, y, is related to the photon energy, k, through
2k
y = In(-)
Mv
.
The shape of the rapidity distribution is determined by the photon spectrum and the energy variation of the photonuclear cross section and hence by the nuclear form factor. Three examples of vector meson rapidity distributions in Au+Au interactions at RHIC are shown in Fig. 1. F’urther details about the vector meson production calculations are given in Fkf.8.
2
A two-source interferometer
The two nuclei can act both as target and photon emitter in nucleus-nucleus collisions. This is different from e A collisions where the electron emits the photon and the vector meson production is localized to the target nucleus. The transverse plane in a nuclear collision at impact parameter b is illustrated in Fig. 2. The impact parameter is not measured in the interactions. The cross section is obtained by integrating over all possible impact parameters, b > 2R, as was discussed in the previous section. The impact parameter dependence comes from the photon spectrum (Eq. 1). The calculations in the previous section assumed that the total cross section is given by the sum of the cross sections for production from the two sources (Eq. 5). This is a reasonable assumption in most cases. For transverse momenta, p ~ of, the produced meson smaller than l / b , it is however not possible to distinguish which nucleus the meson came from, as can be understood from Fig. 2. One then has to add the amplitudes.’ The of the vector meson is the sum of the photon and Pomeron transverse momenta. The coherent couplings of the fields ensures that the pi- will be of the order of pi- < 1/R. The transverse momentum distribution is given by the convolution of the photon, f1, and Pomeron, f 2 , distributions,
dn = f1,2(pT) = dPr
1
h&)f2&
-
The transverse momentum distribution of virtual photons of energy k is given by lo
(7)
164
P 0
50 ’1.
O
- 4 - 2
0
2
4
-4
0
2
4
Zj 100 0” 75 I
2
~~
50
25
0
-2
Y Figure 1: Rapidity distributions of vector mesons in Au+Au interactions at RHIC. The dashed histograms show the rapidity distributions when the photon is emitted by the nucleus with positive rapidity
Similarly, the distribution of the nuclear transverse momentum transfer, q1, is determined by the form factor
+
d‘(yA) &l a ~ ~ ( t , i ,q:)l2ql.
(9)
The transverse momentum distribution of the photon is generally narrower than that of the Pomeron, 50 the vector meson a distribution is dominated by the nuclear form factor. The differential cross section, duldydp?., for a vector meson with rapidity, y, and transverse momentum, p ~ is, the product of the photon density and the photonuclear cross section, i.e.
This can be written
do
where the A1 and A2 correspond to the amplitudes for production off each of the two nuclei. This is only valid for b >> l/m. The appropriate form for all impact parameten is instead
At mid-rapidity, the amplitudes will be of equal magnitude because of symmetry, and by comparing Eq. 10 with l3q. 11, one sees that
165
I
Figure 2: The transverse plane for p" production. The production is always localized to one of the nuclei because of the short range of the nuclear force. When the transverse momentum of the po-mewn is smaller than l / b , it is not possible to distinguish at which nucleus it was produced.
Production at mid-rapidity is the only case that will be considered here; the general case y # 0 is discussed elsewhere. To properly calculate the differential cross section from Eq. 12, one has to determine the relative phases of A1 and A2. The vector meson has negative parity so A1 and A2 have opposite signs; exchanging the positions of the two nuclei is equivalent to a reflection of the spatial coordinates, Z -+ -Z. To a distant observer at the position F, a vector meson produced at the position of nucleus 1, Z1, will have a different phase from one produced at the position of nucleus 2, &, because of the different path lengths. It is a reasonable assumption here, and it simplifies the calculations considerably, that the emission is from two point sources. The size of the nuclei is thus neglected. This is justified, since the nuclear dimensions (- 7 fm) are much smaller than the typical impact parameters. It is furthermore assumed that the vector mesons can be treated as plane waves. The amplitude for a single source is then
where A0 is the magnitude of Ai. The sum of the amplitudes is then
and the magnitude squared is
The impact parameter is b' = 21 - 22; This is the same interference pattern as from a twwource interferometer with slit separation lbl, albeit with the opposite sign (destructive interference). This system is particularly interesting because the vector mesons, except for the J/4,are short-lived in comparison with the source separation divided by the speed of light; the lifetimes are po 1.3 fm/c, w 23 fm/c, $ 44 fm/c, and J / 4 2300 fm/c. The interference will thus involve the decay products, which are in an entangled state, rather than the vector meson itself. Observing the interference would thus be a proof of entanglement and hence an example of the Einstein-Podolsky-Rmen paradox 9. Using Eq. 16, the integral in Eq. 12 can be written
For large values of m,the term containing cosw. g) will go through several oscillations in the integration over b, and the net contribution to the integral will go to zero. For small transverse
166
momenta however, pr << l/(b), 3.6- 0 for all relevant impact parameters, and the interference term will lead to a vanishing cross section in this region. Since the median impact parameters for light vector mesons are around 40 fm at RHIC and about 200 fm at the LHC, one expects the interference to be significant for pr < 5 MeV/c at RHIC and p~ < 1 MeV/c at the LHC. The transverse momentum distribution calculated with and without interference is shown in Fig. 4, but before the results are discussed further, the consequences of vector meson production in coincidence with nuclear Coulomb break-up will be considered. 3
Electromagnetic dissociation
The photon spectrum (Eq. 1) is proportional to Z 2 and inversely proportional to k and b2. The density of low-energy photons is thus high for small b. This may lead to violation of unitarity, as can be seen by rewriting the cross section as an interaction probability for a given impact parameter.
The superscript '1' on P indicates that it is the first order probability. For photonuclear processa with low thresholds and/or high cross sections P'(b) may exceed one for small impact parameters. This is the case for a general photonuclear excitation followed by nuclear break-up, A + B + A + r + B + X B, in gold and lead interactions at RHIC and the LHC. Clearly, Eq. 18 cannot be interpreted as a probability under such circumstances. Instead, P'(b) should, in the general case, be interpreted as a first-order amplitude. Unitarity can then be restored by accounting for multiple excitations, where the probability of having exactly N excitations is given by a Poisson distribution.
+
The cross section for a photonuclear excitation of a single nucleus in Au+Au interactions at RHIC is very large, about 95 barns. The largest contribution to this (65 barns) is excitation of the target nucleus into a Giant Dipole Resonance (GDR). The GDR decays by emitting a single neutron in about 83% of the events. l 2 The cross section for mutual Coulomb excitation of both ions was calculated assuming that the two excitations are independent. Two cases were treated: a general excitation into a state which results in the emission of an arbitrary number of neutrons (Xn), and an excitation into a GDR which results in the emission of a single neutron (In). The total cross section for mutual excitation was found to be 11 barns and of this roughly 7 barns are from hadronic interactions and about 4 barns from Coulomb dissociation. The cross section for mutual GDR excitation followed by the emission of exactly one neutron in both directions was calculated to be 0.4 barn. This Coulomb cross section is of importance for the trigger in experiments at RHIC and the LHC. All RHIC experiments are equipped with identical ZereDegree Calorimeters (ZDCs) for triggering and event selection. The calorimeters, situated on both sides of the experiments approximately 18 m from the center of the interaction region, detect free neutrons in the forward direction from the fragmentation of the beam nuclei. Calculations indicate that about 1/3 of the ZDC coincidence triggers are from Coulomb interactions. Results from the first year of data taking at RHIC are in good agreement with the calculations'4, which indicates that the assumption of independent excitations is a reasonable one. These large cross sections, and hence probabilities, suggest that it should be possible to observe vector meson production in coincidence with mutual dissociation. This has indeed been observed by the STAR collaboration 15. If the Coulomb break-up and the vector meson
167 Table 2: Crass sections for exclusive vector meson production in Au+Au at nuclear Coulomb dissociation.
Vector Meson
RHIC and Pb+Pb at the LHC with
0 [mbl RHIC Au+Au LHC Pb+Pb In,ln Xn,Xn ln,ln Xn,Xn 3.7 42 14 190 0.34 3.9 1.3 17 1.3 17 3.1 0.27 0.16 2.2 0.044 0.0036
I
Po W
4 J/J
I
production are independent, the probabilities factorize, and the cross sections can be calculated along the same lines as for mutual Coulomb excitation. The validity of the assumption of factorization is hard to prove rigorously. It is reasonable if one considers the different time scales involved. The lifetime of a GDR in a heavy nucleus is about 50 fm/c in the rest frame of whereas the typical hadronic formation time is of the order of 1 fm/c in the center the of mass 17. The nucleus, if it is excited into a GDR, will thus remain intact during and long after the production of the vector meson. The fact that the protons and neutrons are in a state of collective oscillations is not expected to affect the coherent vector meson nucleus scattering significantly. Vector meson production in coincidence with Coulomb excitation leading to a final state with (Xn,Xn) and (ln,ln) neutrons will be considered. The probability for mutual Coulomb excitation and no hadronic interaction at an impact parameter b is
~ c ( b=) (1 - 2e-PA(b) + e-ZPA(b))
,-PA(’)
(20)
where P&) is calculated from Eq. 18 with u ( y A ) ,the total photonuclear cross section, taken from a parameterization of experimental data 13. PA(b) is the first-order hadronic interaction probability, which is calculated from the Glauber model p i ( b ) = / T ~ ( F - @ ( l-exp(-up~,~~T~(fl)d~fl,
(21)
where UNN is the total nucleon-nucleon cross section and TA,Bis the nuclear thickness functionI8. Pc(b) has a maximum of about 35% at b = 15 fm in Au+Au interactions at RHIC. Similarly, the probability for mutual Coulomb excitation leading to the emission of single neutrons is 2 PC(ln,Jn)( b ) = (PA(In) ( 6 , ) e-2pA(b)-pH(b) (22) where P&,,)(b) is also calculated from Eq. 18 but with o(yA) being the cross section for GDR excitation followed by single neutron emission. The distribution of impact parameters will thus be altered in interactions where the nuclei dissociate. The integral over b (Eq. 12) has to be weighted with the Coulomb probabilities Pc(b) or &(ln,ln)(b):
The total vector meson cross sections, integrated over y and p ~ at, RHIC and the LHC are presented in Table 2. Requiring coincidence with a general photonuclear interaction (Xn,Xn) reduces the cross sections with roughly a factor of 10 at RHIC and up to almost a factor of 30 at the LHC. The cross section with emission of a single neutron in each direction is about 10% of the cross section in coincidence with a general photonuclear interaction.
168
g 120
Au+Au RHIC '
'
'-6'
'
'
I
'
'
a
I
' ' '
I
'
' '
I
' ' '
I
'
' '
Y
>
q 100 b U
80
60 40
20 -4
-2
0
2
4
6 Y
Figure 3 Rapidity distributions of pO-mesons in Au+Au interactions at FlHIC with (dashed curve) and without (solid curve) nuclear breakup. The curve with breakup has been multiplied by a factor of 10.
The modified impact parameter distribution also affects the photon spectrum, as can be understood from Eq. 10. The increased relative abundance of small impact parameters in interactions with breakup hardens the photon spectrum. This modification of the photon spectrum leads to a corresponding change in the rapidity distribution, favouring production at central rapidities. This is illustrated in Fig. 3, where the rapidity distribution of po mesons is calculated with and without break-up. The altered impact parameter distribution is, furthermore, of importance for the transverse momentum spectrum and the interference discussed previously. The two nuclei act as a twosource interferometer in the transverse plane. By requiring mutual Coulomb dissociation of the nuclei it is possible to obtain a handle on the source separation. The median impact parameters are approximately a factor of two smaller in interactions with Coulomb breakup compared with exclusive production. The interference is thus expected to be significant at correspondingly higher transverse momenta. The calculated rn- spectrum for po mesons produced in Au+Au interactions at RHIC is shown in Fig. 4 for interference in exclusive production, and interference in reactions with Coulomb dissociation (and, for comparison, without interference). The plot shows dN/dpT, as opposed to dN/d&, so all curves go to zero at low p~ because of vanishing phase space. As expected, the interference is stronger and reaches larger pr in interactions with Coulomb breakup.
4
Conclusions
The consequences of vector meson production in coincidence with nuclear Coulomb dissociation in peripheral heavy-ion interactions at RHIC and the LHC have been studied. Production in coincidence with nuclear breakup alters the phase space distribution of the vector mesons compared with exclusive production. The breakup requirement enables one to get an independent handle of the impact parameter of the interactions. This will benefit the understanding of the predicted interference phenomenon.
169
h
%
0.04
E
p. iI
0.03
'..'
, Au+AuRHlC -
P
0.02
-
0.01
-
0
;
,
,
,
,
I
.
,
,
,
I
pr [GeVlc] Figure 4: Transverse momentum distributions of p" mesons in Au+Au interactions at RHIC. The solid curve is without interference, the dashed with interference in exclusive production, the dot-dashed with interference and Coulomb break-up (Xn,Xn).
Acknowledgments
This work was supported by the Swedish Research Council (VR). References
1. E. Fermi Z. Phys. 29, 315 (1924). 2. E.J. Williams Proc. Roy. Society London A 139, 163 (1933); Kgl. Danske Videnskap. Mat.-fys. Medd. 13, No. 4 (1935); C. Weizsaer Z. Phys. 88, 612 (1934). 3. R.N. Cahn and J.D. Jackson, Phys. Rev. D 42 3690 (1990); G. Baur and L.G. Ferreira Filho, Nucl. Phys. A 518, 786 (1990). 4. J.D. Jackson Classical Electrodynamics 2nd Ed., John Wiley & Sons, New York (1975), Eq. 15.52. 5. G.A. Schuler and T. Sjostrand, Phys. Lett B 300, 169 (1993), Nucl. Phys. B 407, 539 (1993). 6. A. Donnachie and G. Shaw, in Electromagnetic Zntemctions of Hadrons, Eds. A. Donnachie and G. Shaw (Plenum Press, 1978), Vol. 2, p. 169. 7. J.A. Crittenden Ezclusive Production of Neutral Vector Mesons at the Electron-Proton Collider HERA (Springer-Verlag, 1997), and references therein. 8. S.R. Klein and J. Nystrand Phys. Rev. C 60, 014903 (1999). 9. S.R. Klein and J. Nystrand Phys. Rev. Lett. 84, 2330 (2000). 10. M. Grabiak, B. Miiller, W. Greiner, G. Soff, P. Koch J. Phys. G 15, L25 (1989); M. Vidovic, M . Greiner, C. Best, G. Soff Phys. Rev. C 47, 2308 (1993). 11. A.J. Baltz, M.J. Rhoades-Brown, J. Weneser, Phys. Rev. E 54, 4233 (1996). 12. A. Veyssiere et al., Nucl. Phys. A159, 561 (1970). 13. A.J. Baltz, C. Chasman, S.N. White, Nucl. Inst. Meth. A 417, l(1998). 14. M. Chiu et al., nucl-ex/0109018. 15. S. Klein (STAR Collaboration), nucl-ex/0104016, nucl-ex/0108018; F. Meissner, nuclex/Ol12008; P. Yepes, these proceedings.
170 16. G.F. Bertsch, P.F. Bortignon, R.A. Broglia, Rev. Mod. Phys. 55, 287 (1983). 17. J.D. Bjorken, Phys. Rev. D 27, 140 (1983). 18. R.J. Glauber in Lectures in Theoretical Physics, Eds. W.E. Brittin and L.G. Dunham (Interscience, New York, 1959); C.Y. Wong, Introduction to High-Energy Heavy-Ion Cob lisions (World Scientific, 1994).
HIGH ENERGY TWO-PHOTON INTERACTIONS AT THE LHC
KRZYSZTOF PIOTRZKOWSKI Universitg catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve E-mail: [email protected]
Two-photon events at the LHC are characterized by the protons scattered at very small angles and the particles centrally produced via the yy fusion. To select these events from the huge samples of generic p p interactions a detection of the scattered protons, or tagging twophoton interactions is necessary. It requires installation of the high-resolution positionsensitive detectors close to the proton beam and far from the interaction point. Efficient measurement of the forward-scattered protons will open a new field of studying high-energy photon-photon interactions at remarkable luminosity, reaching 1% of that in p p collisions. In this paper a few aspects of tagging two-photon interactions as well as several most exciting topics in the high-energy two-photon physics at the LHC are presented.
171
172
Introduction Recently, a method of tagging two-photon interactions at the LHC has been proposed '). These interactions, to a good approximation, proceed in two steps: first, the photons are emitted by incoming protons, and then the photons collide producing a system X. The method is based on the measurement of the forward scattered protons using detectors similar to those planned for measurements of the elastic p p scattering at the LHC *). The final state X will be detected in the central detectors as in the CMS experiment. Tagging is essential for extraction of high-energy two-photon interactions, at the yy center of mass energy W even beyond 1 TeV. A significant luminosity of the tagged two-photon interactions will allow for new, complementary physics studies at the LHC, in particular for searches for new phenomena. In general two-photon events are cleaner than the pp ones and usually the final state is fully contained within acceptance of the central detectors. Particularly interesting are those events where only two or one heavy particle is exclusively produced via the yy fusion as in the yy 4 If case, for example. The event reconstruction can usually be more precise than for the generic p p collisions, and it corresponds more to the experimental conditions at the e'e- colliders. In fact, also the considered physics topics in highenergy yy collisions are closely related to the subjects studied at the e'e- colliders - in particular to the physics studies planned for the future yy collider '). One should note that of course the photon collider offers much higher luminosity, but a lower energy reach than that available in the two-photon collisions at the LHC. Last but not least, the two-photon physics will be done parasitically to the mainstream physics program, and will complement it, at a very limited incremental cost. The content of this paper has two major ingredients. First, after a short introduction of the tagging technique several experimental issues are discussed, in particular those relevant for estimates of the effective yy luminosity. Secondly, the most interesting physics topics are discussed, with emphasis on the exclusive production of the Higgs boson and pairs of the charged SUSY particles.
173
Tagging Aspects In two-photon processes protons are scattered at angles comparable to the beam angular divergence at the interaction point (IP). The scattered protons can however be measured when a fraction of the initial proton energy, x, carried away by a photon is significant. In such a case these protons are more strongly deflected by the beam-line magnets and can be detected in the so-called Roman pots installed far away from the IP and close to the proton beam (see Fig. 1). The detectors are capable of measuring the distance of the scattered proton and its momentum direction with respect to the proton beam at a given location.
Figure 1. Sketch of the LHC beam-line with the proposed localization of the Roman pot detectors for tagging two-photon events; Q and D symbols stand for the machine quadrupoles and dipoles.
At some 240 m from the IP, the relations between the detector variables and the scattered proton energy and angle at the IP, are particularly simple. The distance in the horizontal plane Ax measures then the proton energy loss, hence the tagged photon energy. Other measured variables, direction of the proton momentum in the horizontal plane 0, and distance from the beam axis in the vertical plane Ay are used to reconstruct the proton scattering angle at the TP (from its two projections, €I,* and
174
0,*) as schematically presented is in Tab. 1. The beam dispersion at this location, 0, is of about 100 mm. Variable at the IP
Detector variable @ 240 m
OX*
30,
@Y*
100 [yrad/mm] Ay
X
Ax/D
Table 1. Relations between ‘true’ variables at the IF’ and the detector variables according to the LHC beam optics v6.0, where D -100 mm.
The photon virtuality @ can then be calculated using @ z (l-~)E~[(0,*)~ +(0,*)’], where E is the beam energy, and the photon energy is equal to xE. The tagging efficiency is determined by minimum distance between the detector sensitive edge and the proton beam. For small beam widths, a 1 mm minimum detector approach is usually required to ensure enough space for the beam steering ’). At the recently advocated detector location at about 240 m
1,4)
the horizontal beam
width is small and the 1 mm distance corresponds to a minimum tagged photon energy of 70 GeV, that is to 1% of the beam energy. If the maximum tagged energy of 700 GeV and, the maximum virtuality, QZmax = 2 GeV’, of the colliding photons are assumed, one obtains the tagged effective luminosity spectra as a function of W see Fig. 2. The double tagging corresponds to the case when the two scattered protons are detected, whereas the single tagging occurs when only one proton is detected. In such a case, however also those two-photon events are tagged where one proton dissociates and does not survive the interaction. In fact, these inelastic twophoton events have even higher effective luminosity than the nominal, elastic events, and the total luminosity available for the tagged two-photon collisions is significant, reaching 1% of the p p luminosity for W > 100 GeV. One should also note that the luminosity spectrum extends to a very large W, even beyond 1 TeV. The detector approach of 1 mm assumed above in fact requires an edgeless detector, i.e. a detector sensitive right from its mechanical edge. This might be considered too optimistic and
175
instead one can conservatively consider a 2 mm distance between the detector sensitive edge and the beam.
S,/G&
IdWS, 70
-5
70
70 -6
70
70 -3
70
200
I Y
250
400
500
600
800
750
7OOO
1000 W ( GeV)
250
250
500
500
750
750
7qoo
1000
WdGeV)
Figure 2. Tagged photon-photon luminosity spectrum Syy and its integral 1," dW Syy, assuming double tags (shaded histograms) and single tags, for all events (solid line) and for elastic events (dashed line), above for a 1 mm and below for a 2 mm approach
In Fig. 2 the luminosity spectra and their integrals (equal to the probability of yy collisions in a p p single collision) are shown also for that case, demonstrating that the single-tagged spectrum is modestly affected, whereas the low Wpart of the double-
176
tagged spectrum is as expected suppressed. This shows that the tagging efficiency does not critically depend on the closest approach to the proton beam. S,/GeV
J d WS, 10-2
250
750
500
1000 W (GeV)
Figure 3. Tagged photon-photon luminosity spectrum
IL
k\
250
w>w.
500
750 1000 1250
W o w 9 and its integral 5
d W w , assuming double tags
~ 0
(shaded histograms) and single tags, for all events (solid line) and for elastic events (dashed line); all for a 1 mm approach but assuming for the inelastic events MN < 7000 GeV.
So far, following Ref. MN
'),
for the inelastic production the maximum dissociation mass
of 20 GeV has been assumed. Allowing for the interactions more inelastic and
MN, for example, as large as 7000 GeV significantly increases the effective
'(y
luminosity, see Fig. 3. Virtuality of the photon coupled to the inelastic vertex also increases - for the additional events (with MN > 20 GeV) the average @ is about 150 GeV2,resulting in a significant p T of the system X.Reconstruction of these events is more difficult and requires detailed studies. One should also note that for the high-@ events the contribution from the Z exchange is not negligible. Two-photon interactions in ion collisions are enhanced owing to the coherence effects '). At low W the enhancement scales as Z4 but with increasing Wit becomes weaker resulting in incoherent production (-Z2) at large W. Tagging '(y interactions is
177
however not very practical in this case. It is then restricted to a very large W domain where the coherent interactions are much suppressed and yy luminosity is small.
100 200 Figure 4. Tagged luminosity spectrum S, and its integral assuming elastic production and a 1 mm approach.
300
5 W,(Ge\
400
dW S, for pAr collisions and single tagging,
Situation is different for proton-ion collisions, where the scattered proton can be detected in the same manner as in the p p case, and the single-tagged luminosity is high, especially below W = 100 GeV where the enhancement is still significant. For example, almost 50% of the proton-Argon luminosity LpAris available for yy collisions at W > 50 GeV (see Fig. 4), and LpAritself might reach values close to cm-2s-'. The luminosity of the tagged yy events at medium W might be therefore comparable to that in the p p collisions. In addition, tagging in PA collisions can be used to check yy event selection in the AA collisions. The untagged yy luminosity in the ArAr collisions is high and competitive for W < 100 GeV '). Finally, two-photon exclusive production of lepton pairs will be an excellent monitoring tool for the tagging efficiency and its energy scale. These events can be selected using a standard CMS di-muon trigger and be used off-line for a number of systematic studies, including the luminosity normalization and contribution of the inelastic production, or the accidental tagging.
178
Physics Highlights Exclusive production of heavy particles will possibly be the most exciting subject in two-photon research at the LHC. In particular, the Higgs boson might be observed in
yy collisions at the LHC - as can be seen in Fig. 5 , it will be statistically limited and will not be a discovery channel, it will however bring new, relevant information. The events are very central, usually within the acceptance of the CMS tracking (1~-1<2.5), therefore high overall detection efficiency is expected. The 'irreducible' background due to the b-quark, W and Z pair production, for the respective decay channels, is comparable to the signal for a few GeV window in the 'yy center-of-mass energy.
-5
-4
-3
-2
-1
0
1
2
3
4
5 7)
Figure 5. (Above) Number of the Standard Model Higgs boson events as a function of its mass, exclusively y collisions for the integrated pp, P A and& luminosity of 30 fb-',300 and 30 pb-', produced in y respectively; the event distributions for the major decay modes,?I + bb, W'W and ZZ - all branching ratios and widths obtained with the HDECAY 2 . 0 program 6, . (Below) The pseudo-rapidity distribution of the b-quark from the Higgs boson decays for MI, = 120 GeV.
179
The W and 2 boson pair production in the
yy collisions at the LHC is interesting
on its own right. The number of the produced Wpairs will be similar to that at LEP I1 but at much higher center-of-mass energy - for example, about 3000 W-pairs will be tagged at W > 500 GeV for the integratedpp luminosity of 30 fb-’.This will allow for precision tests of the yWW coupling and for new physics searches, as for example signatures of Large Extra Dimensions ’). The 2 pair two-photon production is much suppressed in the Standard Model but an interesting search for the anomalous y . 2 coupling can also be performed. The exclusive
two-photon production will be statistically limited - only about
one hundred events will be tagged for the ‘canonical’ luminosity. The observed crosssection, which is proportional to the fourth power of the particle charge, will however directly and precisely measure the top charge. It has been recently argued 8, that a measurement of the two-photon production of photon pairs at very high transverse momenta at the LHC can be used for searching for the Dirac monopoles. Tagging two-photon events will certainly improve the reconstruction of these events, decrease systematic uncertainties and will help to suppress the backgrounds. If supersymmetric particles are found at the LHC then the sparticle pairs produced in yy collisions can be used to test the structure and parameters of the underlying theory. The two-photon production of pairs of the charged sparticles is particularly simple - it is a pure QED process. In Fig. 6, the number of the tagged spairs is plotted as a function of the sparticle mass, assuming a simplified scenario of the degenerate sleptons, i.e. the equal masses for the left- and right-handed sleptons ’). It shows that the charginos might be detected up to the masses of about 300 GeV, and the sleptons and charged Higgs bosons up to the masses of about 200 GeV, covering a significant fraction of benchmark spectra of the minimal SUSY
lo).
Studies of the tagged two-
photon production will be complementary and less model-dependant with respect to the nominal analyses of the complex final states and decay chains. In addition, the photon energy measurement will improve the reconstruction of event kinematics.
180
M(,GeV) Figure 6 . Number of pairs of the charginos, sleptons and charged Higgs bosons produced in the single tagged two-photon interactions, plotted against the mass of the produced sparticle.
In conclusion, a few new aspects of tagging two-photon production at the LHC discussed in this paper provide a further strong motivation for work towards a technical realization of this proposal. The initial survey of topics in high-energy twophoton physics shows that this research program will be a significant and complementary extension of the nominal physics program at the LHC. Finally, the same tagging technique can also be utilized to select 'yp,or 'yq and 'yg, interactions at the LHC, for which the energy reach and the effective luminosity are even higher than for the yy case. The initial study of this experimental opportunity is a subject of the publication in preparation.
181
Acknowledgements
The author would like to thank the workshop organizers for support and invitation to this very interesting and inspiring conference. References 1.
K. Piotrzkowski, Phys. Rev. D 63 (2001) 071502(R).
2. TOTEM Collaboration, Technical Proposal, CERN/LHCC/99-07. 3. T E S U Technical Design Report, Appendix, The Photon Collider at TESLA, B. Badelek et al.,
DESY 2001-01 1, TESLA Report 2001-23, March 2001. A. Faus-Golfe et al., hep-ex/0102011. 5. For example, see G. Baur etal., hep-ph/0112211. 6. A. Djouadi, J. Kalinowski and M. Spira, Comp. Phys. Comm. 108 (1997) 56. 7. T. G. Rizzo, Phys. Rev. D 60 (1999) 1 15010. 8. L.F. Ginzburg and A. Schiller, Phys. Rev. D 57 (1998) 6599. 9. J. Ohnemus et al., Phys. Lett. B 328 (1994) 369. 10. M. Battaglia etal., Eur. Phys. J. C22 (2001) 535. 4.
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PHYSICS OPPORTUNITIES IN ULTRAPERIPHERAL HEAVY ION COLLISIONS AT LHC G.Baur Institut fur Kernphysik Forschungszentrmm Juelich, Germany Due to coherence, there are strong electromagnetic fields of short duration in very peripheral heavy ion collisions. They give rise to photon-photon and photon-nucleus collisions with high flux up to an invariant mass region hitherto unexplored experimentally. Photon-photon and photon-hadron physics at various invariant mass scales are discussed. Due to the very strong electromagnetic fields there are interesting many-photon exchange processes like double giant dipole resonance excitation in nuclei, multiple e+e- pair production and vector meson pair production. Maximum equivalent photon energies in the lab-system(col1ider frame) are typically of the order of 3 GeV for RHIC and 100 GeV for LHC. Vector meson production(coherent as well as incoherent) and photon-gluon fusion leading to heavy quark jets are processes of great physics interest. The high photon-photon luminosities in very peripheral collisions a t the LHC will allow to study interesting physics channels in the invariant mass region up to about 100 GeV. It is hoped that this will bridge a gap to the physics a t the future photon colliders. In pp collisions this maximum energy is even higher, and their is also the experimental possibility to tag on these photons.
1
Introduction
In ultraperipheral (UPC) heavy ion collisions nuclei interact with each other through their electromagnetic fields. In these collisions with impact parameter b larger than the sum of the nuclear radii a new range of physical parameters enters. Electromagnetic fields are very strong and of short duration. The interaction can be very well described by the equivalent photon method. Fluxes are very high and extend up to energies hitherto unexplored experimentally. A review article for Physics Reports on coherent yy and y A interactions in relativistic ion colliders has just been completed '. It is the aim of this talk to discuss the physics of UPC. Emphasis is put on the LHC and RHIC energy regions. Detectors at RHIC as well as at LHC are built with the aim to study central heavy ion or pp collisions. The question of how to detect such UPC events at the colliders is not dealt with here. An important question is the triggering on UPC events. The STAR group at RHIC led by Spencer Klein has shown how to trigger on these events and how to study them experimentally. I refer to the discussion in these proceedings Since the electromagnetic fields are very strong, many photons can be exchanged in these collisions. In section 2 processes where more than one photon is emitted from a nucleus are discussed. There are a few important and interesting processes of this type. Generally, the one photon exchange mechanism is sufficient for inelastic processes. A very useful view on the electromagnetic processes i s the parton picture with the photons as the partons. In this picture the scattering is described as an incoherent superposition of the 'i3.
183
184
-
Figure 1: One photon exchange in inelastic ultraperipheral heavy ion collision. The exchanged photon is almost 0. red, i.e. Q2 = -q2
scattering of the various constituents. For example, nuclei consist of nucleons which in turn consist of quarks and gluons, photons consist of lepton pairs, electrons consist of photons, etc.. Relativistic nuclei have photons as an important constituent. This is due to the coherent action of all the charges in the nucleus: for these conditions the wavelength of the photon is larger than the size of the nucleus, therefore it does not resolve the individual nucleons but sees the coherent action of all of them. The coherence condition limits the virtuality Q2 = -q2 of the photon to very low values Q2
< 1/R2,
where the radius of a nucleus is approximately R = 1.2 fm with A the nucleon number. This is due to the rapid decrease of the nuclear electromagnetic form factor for high Q2 values. For most purposes these photons can therefore be considered as real (“quasireal”),see Fig.1. From the kinematics of the process one has a photon four-momentum of qp = ( w , q i , q 3 = w/v), where w and q 1 are energy and transverse momentum of the quasireal photon in a given fiame, where the projectile moves with velocity v. This leads to an invariant four-momentum transfer of 2-
Q
where the Lorentz factor is 7 = E/rn = l energy of the quasireal photon to w
w2
- - +v272 q2
1,
/ d m . The condition Eq. 7
< W m a z = R’
(2)
(1) limits the maximum
(3)
and the perpendicular component of its momentum to
At LHC energies (y = 3000) this means a maximum photon energy of about 100 GeV in the laboratory system, at RHIC (y = 100) this number i s about 3 GeV. In section 3 7-A interactions are discussed. 77 processes are a (small) subset of such processes. They are very important in general and they allow to study very interesting physics in a clean way. This is discussed in section 4. Conclusions and an outlook are given in section 5. 2
Strong Field Effects, Exchange of Many Photons
The Sommerfeld parameter
185
Figure 2: a) Classical picture of heavy ion scattering. This is also used by Fermi in his celebrated paper on the equivalent photon approximation '. He considered the nonrelativistic case v << c. The relativistic case is discussed in a very pedagogical manner in '. b) For 7 >> 1 many photons are exchanged in elastic heavy ion scattering. A typical graf is shown.
characterizes the strength of the electromagnetic interaction between the ions with charges Z1 and Z2 respectively, see Fig. 2a. This parameter 11 can be >> 1, e.g.' for Pb-Pb collisions we have 11 = 49. According to classical arguments given by N.Bohr elastic scattering can be considered as classical for 11 >> 1. For relativistic heavy ions the classical trajectory is almost a straight line with an impact parameter b. The momentum transfer Ap is essentially perpendicular to the beam direction and is given by
This momentum transfer is built up from the exchange of many photons6, see Fig. 2b. Due to the large value of 11 the integral over the impact parameter b in the Glauber amplitude can be evaluated using the stationary phase (or saddle point) approximation. This phase is given by
q5 = -qb
+ 211 ln(kb)
(7)
where k is the wave number of the projectile nucleus. The condition #(b) = 0 leads to b = i.e. Ap = hq. Thus the momentum transfer q(or scattering angle 0 = f ) is related to the classical impact parameter. This is in contrast to e.g. p-p scattering where 11 = 1/137 << 1 and the Born approximation (one photon exchange) is sufficient. For the heavy ions the strong interactions for where h i n is the sum of the nuclear radii, lead to complete absorption collisions with b < Gzn, (black disk approximation). This disregards the diffuseness of the nucleus, but it is quite a good approximation for many purposes. In the case of strong Coulomb coupling(7 >> 1) the elastic scattering cross section is given by a Fresnel diffraction pattern(rather than a Fraunhofer one for 11 < 1). This is explained in Ref. ', see also Sect 5.3.5 of Ref. a. The electric charge of the relativistic ion gives rise to an electromagnetic potential, the Lienard-Wiechert potential Ap(F,t ) . This potential interacts with a target current j,. This target current describes e.g. nuclear states, vector mesons or e+e- pairs in the field of a nucleus. This defines a time dependent interaction V ( t )as( see e.g. 9,
V ( t )= / d 3 r A p ( F 7 t ) j p ( l ; l The dependence of V ( t )on the impact parameter is not shown explicitly. Coupled equations for the excitation amplitudes a,@) for certain states n can be set up. The solution of these equations is greatly facilitated if the sudden approximation can be applied, see e.g. Ref?'. This is the case if the collision time is much smaller than the nuclear excitation time. This condition is fulfilled in many interesting cases and we assume now that the sudden approximation can be used. This "frozen nucleus"-approximation is also used in Glauber theory. The relation between
186
GDR
GDR,~'.)
Figure 3: Grafs contributing to the simultaneous excitation of the GDR and p a production
the semiclassical approach and the (quantal) Glauber (or eikona1)approximation is explained in Ref?O. This is done for the non-relativistic as well as the relativistic case. The excitation amplitude is given by a,(t + co) =< n1 exp(iR)10 > (9) where R = - j-'," V(t)dt( we put h=l). The operator R is a direct sum of operators in the space of nuclear states, the space of the nucleus- vector meson system, the nucleus-e+e- system,etc.. We can expand the exponential in eq. 9. Terms linear in R give e.g. the excitation of nuclear states,like the collective giant dipole resonance(GDR), vector meson production or e+e-pair production. Terms quadratic in R give e.g. contributions to double phonon GDR-excitation , double vector meson production , two e+e- pair production . It also describes e.g. vector meson production and GDR excitation in a single collision. A contribution to the second order amplitude a(2)is e.g. a(2)= - < GDR(R(0>< poJRJO> (10) where 10 > denotes the ground state of the nucleus. The factor 1/2! in the expansion of e z p ( i R ) , see eq. 9, is compensated by the two possibilities in the time ordering of the GDR excitation and the vector meson production respectively, see Fig. 3. For three independent processes, say GDR-excitation, vector meson- and e+e- pair production , there are 6 different time orderings, which compensate the $ factor in the expansion of eq. 9, and so on. In this formalism it is clearly seen that these processes are independent and the elementary amplitudes factorize, as one would have intuitively expected. This property is used e.g. in the experimental analysis of vector meson production with simultaneous GDR excitation. The neutrons from the GDR decay serve as a trigger on UPC 2,3. The ion motion is not disturbed by the excitation process. The reason is that the kinetic energy of the ion is much larger than the excitation energy. 2.1 Exchange of Many Photons in Multi-Phonon Giant Resonance Excitation An especially simple and important case is the excitation of a harmonic oscillator. In terms of the corresponding creation and destruction operators at and a the Hamiltonian of the system is given by 1 H = hw(ata -) (11)
+2
where w denotes the energy of the oscillator. We have the boson commutation rule [a,at] = 1. Only one mode is shown explicitly, in general one has to sum (integrate) over all the possible modes. The excitation operator is assumed t o be linear in the destruction and creation operators
R = -(ua+u*a+)
187
where u is a c-number which characterizes the excitation process (the matrixelement of R between the ground state and the one-phonon-state). This leads to the excitation of a so-called coherent state, see ll. For the excitation of multiphonon states this is explicitly shown in 12. One has
where the operator identity eA+B = eAeBe-i[A,B]was used, which is valid for two operators A(= -iu*at) and B(= -iua) for which the commutator is a c-number. Electromagnetic excitation of nuclear states, especially the collective giant multipole resonances was discussed at this workshop by Carlos Bertulani 13. The possibility to excite multiphonon GDR states is discussed in 14, where also its main properties like decay widths are discussed. The parameter which describes the probability @ of GDR excitation isI5
where mN denotes the nucleon mass, the neutron- proton-, and mass-number of the excited nucleus are given by N2,&, and A2 respectively. The excitation probability is inversely proportional to the energy w(- 80MeVA-1/3) of the GDR state. Thus soft modes are more easily excited, as one may have expected. In this (rather accurate) estimate, it was assumed that the classical dipole sum rule (Thomas-Reiche Kuhn sum rule) is exhausted to 100 percent. For the excitation of an N-phonon state, a Poisson distribution is obtained. For the heavy systems @ is of the order of $ for close collisions ( b R1 &). Quite similarly , double po production was studied in Ref. 16. In addition to the label m for the magnetic substates of the GDR, one has a continuous label (the momenta) in the case of vector meson production. The probability to produce a vector meson in a close collision is of the order of one to three percent for the heavy systems.
-
+
2.2 Production of Multiple Electron-Positron Pairs
In Ref. l7 it was shown that multiple e+e--pairs can be produced in relativistic heavy ion collisions. In this work the sudden (or Glauber) approximation and a quasiboson approximation for e+e- pairs was assumed. This will be well fulfilled in practice. Using a QED calculation (including Coulomb corrections in the Bethe-Maximon approach) for one pair production as an input, a Poisson distribution is obtained for multiple pair production. This is quite natural, since this problem is now reduced to the excitation of a harmonic oscillator (the modes are labelled by the spins and momenta of the e+e- pairs) , see above. d where m is the The characteristic dimensionless parameter for this problem is E = iZ1Zza2 electron mass and b > l/m. For impact parameters b- l / m and heavy systems like Pb-Pb or Au-Au the parameter E is of the order of unity. In a series of papers by K.Hencken et al. (see Is) the impact parameter dependence of e+e- pair production was studied numerically in lowest order QED. Only recently an approximate analytical formula for the total pair production probability in lowest order P(')was found. In an impact parameter range of l / m < b < -y/m it is given by l9
-
P(') = 9.rr2 Z G ( 2 1 n y 2- 3ln(mb))ln(mb)
-
(15)
The N pair-production probability decreases strongly with increasing impact parameter b (approximately like b-2N). Therefore the probabilities P(l)(b)should be known accurately for an impact parameter range of 0 < b <several l / m . Muon pair production is also of interest. In close collisions, the pair production probability is of the order of 1, as can be seen kom eq. 15. For a more exact evaluation, form factor effects
188
-
would have to be taken into account. Due to the much smaller Compton wave length of the 40000. muon, total cross sections are smaller than about a factor of (m,/m,)’ 3
yA interactions
It seems appropriate at this point to recall various interesting results which have been obtained References to these experiments in the past decade in UPC experiments at lower energies can be found e.g. in ”. At beam energies in the GeV region, the equivalent photon spectrum = 10 - 20MeV. With such a soft spectrum of photons up to extends up to w, the GDR region, many interesting topis in nuclear structure physics and astrophysics could be investigated. The excitation of the DGDR was already briefly discussed above in subsection 2.1: strong (quasirea1)photon sources are essential for such investigations. Another application is the study of radiative capture processes: they are related to photodissociation via time reversal 2oiz1. E.g. the astrophysical S-factor for the 7Be(p,y)8B reaction could be studied with the electromagnetic dissociation of ‘ B rare isotope beams at RIKEN(Japan),GSI(Darmstadt)and Michigan State University. This is of great interest for the solar neutrino problem: most of the neutrinos detected in the Kamiokande or SNO detectors originate from the weak decay of 8B. This nucleus is solely produced in the 7Be(p,y) capture reaction, thus this astrophysical S-factor directly determines the (high energy) neutrino flux from the sun. While GDR excitation and subsequent particle(main1y neutron) decay is a source of beam loss, a means to trigger on UPC as well as a tool for a luminosity monitor zz (”yesterday’s sensation is today’s calibration”), the physics interest concentrates on the higher equivalent photon energies available at FiHIC(up to about 6OOGeV) and LHC (up to about 600 TeV), as viewed from the nucleus rest frame. The physics is quite similar to what is done at HERA, with essentially two differences: now the photons are only quasireal(Q2 0) and the hadron is now a nucleus instead of a proton. There is (diffractive) coherent and incoherent vector meson production with many interesting applications. Indeed, relativistic heavy ion colliders are vector meson factories This opens the way for interesting experimental studies, like the interference effect in exclusive vector meson production 2 5 . As explained in this refererence, this provides also an example of the Einstein-Podolsky-Rosen paradoxon. Double po production 16wasalready mentioned above and the prospects for vector meson spectroscopy were discussed inz3. Certainly, the forthcoming experimental results will spur further theoretical developments. Another interesting topic is photon-gluon fusion into 44. The two jets can be identified and the kinematics resconstructed. These processes were first studied in Ref. z6 and Ref. 27. Quite recently, it was suggested to investigate the colour glass condensate with this r n e t h ~ d ~ ~ , ~ ’ . In this way, the low-x gluon distribution can be directly studied experimentally . Top quark production will also be possible3’. The points above are most important also for the future experiments at RHIC and LHC. I only refer here toz3 and’. Further discussions and calculations are also reported at this workshop in3’.
’.
197e,yfmy N
-
23124.
4
yy physics
At LHC ultraperipheral collisions provide equivalent photons with energies up to 100 GeV in the c.m. system. yy-collisions have been investigated at LEP in the invariant mass region up to 185 GeV. The effective yy luminosity for medium mass heavy ions exceeds the one achieved at LEP, see e.g. Fig.6 of the recent review A review of yy physics is also given there, so I will only mention a few considerations here, see also the talk of Kai Hencken 31. These very high effective yy luminosities will allow QCD studies like the study of 77 widths of C = +1 (heavy)
’.
189
Figure 4: Cross section per GeV for fermion pair production at the LHC for Ca-Ca collisions. A luminosity of Lcoca = 4~1030cm-2s-' was assumed. The process yy -+ hadrons is also shown. A "(heavy ion) year" corresponds to lo6 s. For further details see
mesons and the total cross section for yy -+ hadrons,see Fig.4. Also the study of new physics will be possible. One can speculate about new particles with strong coupling to the yy-channel. Large rrrwidths will directly lead to large yy production cross-sections. The two-photon width of quarkonia is proportional to the wave function squared in the center of the system. Thus we can expect, that if a system is very tighty bound it should have a large two-photon width due to the factor 1Q(0)l2 which is large in these cases. Examples for such tightly bound systems were discussed in the eighties e.g. in Refs. 32,33. Composite scalar bosons at W,, M 50 GeV are expected to have y-y-widths of several MeV 32,33. The search for such kind of resonances in the yy-production channel will be possible at LHC. In the TESLA Report part3 p.110 (Ref. 34) the reader can find some remarks about the "agnostic" approach to compositeness. From eq. 68 and figure 14 of Ref. one can directly obtain a value for the production cross-sections of such states and the corresponding rates in the various collider modes. Due to the high flux of equivalent photons such searches seem worth-while, and a possible discovery would be quite interesting. Certainly one will have to wait for the future yy colliders in order to study yy- physics in the region of several 100 GeV. However, it may be possible to have a glimpse into this region with the (u1tra)peripheral collisions at LHC which will be working (taking data) in a few years from now. With their high luminosities pp collisions are also a very interesting source of equivalent photons. Their energy spectrum extends up to very high photon energies beyond 200 GeV. In Refs. 35,36 yy-processes at pp colliders are studied. It is observed there that non-strongly interacting supersymmetric particles(sleptons, charginos, neutralinos and charged Higgs bosons) are difficult to detect in hadronic collisions at LHC. The possibility of producing such particles in -yy interactions is examined. Clean events can be generated which should should compensate for the small production number. This is even more important since there is a new experimental approach to tagging in pp coillisions by measuring very forward proton scattering Particularly exciting is the possibility to detect Higgs boson production via the yy fusion37. In3' it was proposed to search for heavy magnetic monopoles in yy collisions at hadron colliders like the Tevatron and LHC. The idea is that photon-photon scattering below the monopole 37138.
190
production threshold is enhanced due to the strong coupling of magnetic monopoles to photons. where 7~ = x t l , 3 ~ 2.... , Since = 1/137 The magnetic coupling strength g is given by g = the magnetic coupling strength is indeed quite large. In this reference differential cross sections for yy scattering via the monopole loop are calculated for energies below the monopole production threshold. The result depends strongly on the assumed value of the spin of the monopole. With this elementary cross section as an input, the cross section for the process pp + yyX is calculated. Elastic(i.e.X = pp ) and inelastic contributions are taken into account. The signature of such a process is the production of two photons where the transverse momentum of the pair is much smaller than the transverse momentum of the individual photons. At the Tevatron such a search was performed. They looked at a pair of photons with high transverse energies. No excess of events above background was found4'. Thus a lower limit on the mass of the magnetic monopole could be given. A mass of 610, 870, or 1580 GeV/c2 was obtained, for the assumed values of a monopole spin of 0, 1/2, or 1 respectively. For further discussions see also41. Such kind of searches could also be performed at the heavy ion colliders: the flux is enhanced by a factor of Z 4 ,on the other hand, the maximum equivalent photon energy is lower. Some exploratory theoretical calculations would be useful.
%
5
&
Conclusions. Outlook
Ultraperipheral heavy ion collisions provide a strong source of equivalent photons up to very high energies. This offers the unique possibility to study photon-hadron(nuc1eus) and photon-photon processes in hitherto inaccessible regions. Some of these comparatively silent (as compared to the violent central collisions) events are very interesting. With the forthcoming experimental results from RHIC the field of ultraperipheral collisions is rapidly expanding. These results will also be helpful for the planning of UPC experiments at LHC. This kind of physics is of great interest and potential. Acknowledgments I wish to thank Sebastian White and Bill Marciano for inviting me to come to Erice to a very stimulating workshop in a most pleasant atmosphere. References
1. G.Baur, K.Hencken, D.Trautmann, S. Sadovsky, Y. Kharlov, Coherentyy and yA interactions in very peripheral ion collisions, Physics Reports, in press, hep-ph/0112211 2. P.Yepes, these proceedings 3. Joakim Nystrand,these proceedings 4. E.Fermi, Z.Phys. 29(1924) 315 5. J.D. Jackson, Classical Electrodynamics (Wiley,New York,1975) 6. F.Hayot and C.Itzykson Phys. Lett. 39B(1972) 521 7. W.E. Frahn Ann. Phys.(N.Y.)72(1972) 524 8. W. Norenberg and H.A. Weidenmuller, Introduction to the Theory of Heavy-Ion Collisions, Second Enlarged Edition, Springer-Verlag, Berlin Heidelberg New York 1980 9. C.A.Bertulani and G.Baur Phys.Reports 163(1988)299 10. G.Baur Nucl.Phys.A531(1991)685 11. R.Glauber Phys.Rev. 131(1963)2766 12. G.Baur, C.A. Bertulani Nucl Phys.A482(1988)313~ 13. C.A. Bertulani, these proceedings
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14. G.Baur and C.A. Bertulani, Proc. Int. School of Heavy Ion Physics, 12-220ctober 1986 Erice, Italy, Plenum Press New York 1988 , ed. by R.A. Broglia and G.Bertsch, p.343 15. G.Baur and C.A. Bertulani Phys.Lett.B 174(1986)23 16. S.Klein and J.Nystrand Phys.Rev.C60(1999) 014903 17. G.Baur Phys.Rev.A 42(1990)5736 18. K.Hencken ,D.Trautmann and G.Baur, Phys. Rev.A51(1995) 998 19. R.N.Lee, A.I.Milstein and V.G.Serbo hep-ph/0108014 20. G.Baur, K.Hencken, D.Trautmann, S.Typel and H.H. Wolter Erice Summer School in Nuclear Physics, Progress in Particle and Nuclear Physics 46(2001)99 21. G.Baur, C.A. Bertulani and H-Rebel Nucl.Phys.A458(1986)188 22. M.Chiu et al. hep-ex/0109018 23. S.Klein physics/OO12021, Talk presented at the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics, October 15-20, 2000,Capri,Italy,2000 24. S.Klein, these proceedings 25. S.R.Klein and J.Nystrand, Phys.Rev. Lett. 84(2000)2330 26. Ch.Hoffman,G. Soff, A.Schafer and W. Greiner , Phys.Let t .B262(1991)210 27. N.Baron and G.Baur, Phys.Rev.C48( 1993)1999 28. F.Gelis, A.Peshier, hep-ph/0107142 29. F.Gelis, APeshier, hep-ph/0111227 30. S.R.Klein, J.Nystrand, and R.Vogt Eur.Phys.J C21(2001)583 31. K.Hencken, these proceedings 32. F.M.Renard, Phys.Lett.B126(183)59 33. U.Baur, H.Fritzsch, and H.Faissner, Phys.Lett.B135(1984)313 34. Tesla Technical Design Report, edited by F.Richard et al. (DESY,Hamburg,2001) No.DESY 2001-011 35. J.Ohnemus, T.F. Walsh, and P.M. Zerwas, Phys.Lett.B328(1994)369 36. M.Drees et al. Phys.Rev.D50(1994)2335 37. K.Piotrzkowski Phys.Rev.D63(2000)071502 38. K.Piotrzkowski, these proceedings 39. 1.F.Ginzburg and A. Schiller Phys.Rev.D57(1998) R6599 40. B.Abbott et al. Phys.Rev.Lett. 81(1998) 524 41. K.A.Milton et al. hep-ph/0111062
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COHERENT ELECTROMAGNETIC PROCESSES IN RELATIVISTIC HEAVY ION COLLISIONS
K. HENCKEN
Departement fur Physik und Astronomie, Institvt fur Physak, Universatat Basel, Klingelbergstr. 88, 4056 Basel, Switzerland Using the strong electromagnetic fields in peripheral heavy ion collisions gives rise t o a number of interesting possibilities of applications in both photon-photon and photon-hadron physics. We look at the theoretical foundations of the equivalent photon approximation and the specific problems in the heavy ion case. The interesting physics processes that can be studied in this way are outlined. Electron positron pair production plays a special role. We look at multiple pair production and Coulomb corrections as typical strong field effects. But electron positron pair production is also an important loss process and has some practical applications.
1
Introduction
Central collisions are the reason, why heavy ion colliders have been or are currently built. But as soon as these colliders are available it is of interest to look also at very peripheral (“ultraperipheral”) collisions, which are characterized by the condition that the impact parameter b is larger than the sum of the nuclear radii. The two ions do not interact directly via the nuclear interaction in this case and only the electromagnetic field or the long range part of the nuclear interaction (described by Pomeron- or meson-exchange) is still able to interact. Peripheral collisions are quite complementary to the central ones, both in terms of the characteristics of the events, but also concerning the physics of interest. Peripheral collisions can be characterized as being “silent” compared to the “violent’ central interaction with their large multiplicities. Whereas in central collisions one is looking for ‘Lmutter under extreme conditions” the focus of peripheral collisions is more towards the “elementary interaction of photons”. Peripheral collisions can in many cases study properties which cannot be easily extracted from the central collisions. The coherent action of all the protons within the nucleus gives an enhancement factor of Z 2 or Z4, which is the reason for the often very large cross sections and rates. The strong electromagnetic fields have already found a number of applications in atomic physics and in nuclear physics. In the past years their use at higher energies for both photon-photon and photon-hadron physics has been studied theoretically. Some reviews of the field can be found the most recent one is4. With the recent measurements at the STAR detector at RHIC this field has now left the purely theoretical stage and has come into reality.
193
194
JbiPI Figure 1: The strong electromagnetic fields surrounding the heavy ions in relativistic collisions are an intense source of (quasireal) photons. They can be used in very peripheral collisions for photon-photon and photonnucleus collisions.
2
The Equivalent Photon Approximation in the heavy ion case
Lepton (electron) colliders have been traditionally used to study yy physics. The equivalent photon approximation is one of the main theoretical tools there. This method was originally developed by Fermi5 from the observation that the Coulomb field of a fast moving object resembles the flux of a spectrum of real photons and later extended to the relativistic case by Weizsacker6 and Williams7. Its application for photon-photon physics is discussed in detail in8. The main idea behind the quantum mechanical derivation of the equivalent photon approximation is the fact that photons with a small virtuality q2 are dominant due to the singular behavior of the photon propagator. The cross section for virtual photons can then be replaced by the one for real photons and the cross section is written as
where n ( w ) is the equivalent photon number in the photon-photon case. More recently photon-photon physics was also investigated for ep collisions or e A collisions at HERA In ion-ion collisions there are current measurements at STAR and both CMS and ALICE have plans to measure photon-photon and photon-hadron processes as well. In contrast to the point-like structure-less electrons with a small charge and only electromagnetic interaction the situation with ions is more complex. A number of additional processes, which occur in the photon-photon case, are shown in Fig. 2. Similar diagrams exist as well in the photon-ion case. Let us address the different processes in more detail. The finite size of the ions can be taken care of by using the elastic form factor. The inelastic excitation will contribute also to the photon spectrum. This was studied for the excitation of the most important excited state, the giant dipole resonance (GDR), in lo. It was found that this is only a rather small contribution, whereas, e.g., the PA transition gives an additional effect of about 10%. At even higher q2 one also has incoherent contributions. For the proton it was found that the photon emission from individual quarks gives a contribution that is larger than the elastic one l 1 2 l 2 . In the heavy ion case such a contribution is expected to be much smaller for two reasons: First the charge of the ‘Lpartons’’(quarks for the proton, protons for the heavy ion) is small compared to the total charge for ions, whereas they are of the same order for the proton. Therefore the loss of the coherence enhancement will be larger here. In addition the exclusion of the initial/final state interaction (see below) will remove a large part of this contribution. The virtuality of the photon in the inelastic case needs to be rather large, q2 >> 1/R2; their range will therefore be rather small. On the other hand the exclusion of the initial/final state interaction eliminates most of the small impact parameters.
’.
195
Figure 2: Due to the more complex nature of the ions corrections occur in photon-photon processes in the heavy ion case. Besides the elastic photon emission, which is governed by the elastic form factor (a), the photon can also be emitted inelastically (b). The two ions of course do interact with each other either via the Coulomb interaction or due to their nuclear interaction for small impact parameter (c) (“initial state interaction”). Finally if a hadronic find state is produced it can again interact with one of the ions (d) (“final state interaction”).
The exclusion of the nuclear interaction at small impact parameter (see Fig. 2(c)) is not easy to do in the plane wave approach. Of course photon-photon processes will occur in this case as well but they are completely dominated by the particles produced in the nuclear interactions; therefore these collisions are not useful and one should exclude them also from luminosity calculations. A possible approach for the inclusion of the strong interaction is a calculation starting from the eikonal approximation, see also the discussion in 13. The strong Coulomb interaction between the two ions, characterized by the large Sommerfeld parameter 17 = 2122c-u >> 1, on the other hand allows to use the semiclassical approximation, see also 13. At high energies the heavy ions move essentially along a straight line and it is possible t o define an impact parameter dependent photon number. In the case of a point-like particle this number is well known to be
”‘
’I
’
Y
In such a semiclassical pictures the exclusion of the nuclear interaction can be done by excluding all impact parameters with b < R1+ R2. Alternatively one can instead introduce the probability for no hadronic interaction given, e.g., from Glauber theory and get the photon-photon flux by integrating the product of the equivalent photon numbers and this probability over all impact parameter. In general the difference between this two approaches is not very large. Another type of “initial state interaction” are additional electromagnetic processes occurring together with the photon-photon or photon-hadron interaction (Fig. 3). In the semiclassical approximation this process will be described by the product of the individual probabilities, which is discussed also in 13. The main contribution is expected t o be from the GDR, but excitation to higher energies contribute as well. For the heaviest ion the probability for GDR excitation is rather large and approaches one. Therefore the excitation of higher states (DGDR), see 14, need to be taken into account. It is found that about 95% of all Pb-Pb collisions are accompanied by such an excitation, but only about 1%of the Ca-Ca collisions. As the equivalent photon number N ( w ,b) falls off as l / b 2 , a very convenient parameterization of this probability is P ( A * ,b) = 1 - exp [ - ( ~ o n s t / b ) ~ ] (3) The importance of such an effect depends on the trigger condition used in the experiment. If one triggers for no breakup of the ions at all, one would loose this part of the luminosity. On the other hand this effect is currently used a t RHIC to trigger for peripheral collisions by looking for the neutrons coming from the decay of the GDR in both ions, which are measures in both ZDCs 15. Due t o their much larger ion luminosity, it was found that medium heavy ions (like Ca-Ca) give in the end a much larger total luminosity (given by the product of ion luminosity and photon-photon luminosity). Therefore medium heavy ion beams are often more preferable. In
196 A
A
A
A
A
A’
A
A’
Figure 3: The strong fields allow for the excitation of the ions in addition t o the photon-photon process. For the heaviest ion the probability for this is rather large. The semiclassical approximation allows for a very easy description of these processes in terms of the products of the individual probabilities.
addition the ions have a smaller size and therefore the maximum photon energy wmax = Y / R will be larger in this case as well. Apart from the proton-proton case (see also 16) it will probably be impossible to measure the scattered ions, therefore only “untagged events” will be possible. These are characterized by a small total transverse momentum of the final state smaller than about 2/R. In terms of the rapidity the photon luminosity peaks at Y = 0 and has a range of about
3
Physics Potential i n 77 and yA collisions
The available invariant mass range in the photon-photon case is given by the maximum photon energy umax N Y/R and therefore the maximum invariant mass is 2 y l R ’. The invariant mass range for RHIC extends up to a few GeV and therefore makes this of interest for meson spectroscopy in this range. The luminosities are comparable to what has been available at LEP. Both meson production and meson pair production can be studied. Detailed studies of these processes have been done by the “Ultraperipheral Collision Group” at STAR, see also The suppression of the two-photon production of a meson would also be an indication for its gluonic nature, making it possible to look, for example, for glueballs. For the LHC the invariant mass range goes up to a few 100 GeV and therefore extends the possibilities of LEP both in mass range and in luminosity. Meson spectroscopy is possible there, especially for mesons with b and c quarks, see Fig. 4(a). In addition the total cross section 77 +hadrons has been measured at LEP. Deviations from the Pomeron universality have been found there. Photon-Photon processes at the LHC have the potential to look for this process at higher energies. In addition it is worthwile to study whether the detectors there will be able to detect also in the more forward direction, where most of the events will occur and which up to now are undetectable and need to be modeled. The possibility to find new particles at the LHC has attracted some interest in the past. Especially Higgs boson production and the search for supersymmetric particles have been studied in detail. Unfortunately the production rates (for ion-ion collisions) seem to be rather small (few per year) in these cases and for realistic parameters. Photon-ion collisions are possible over a large range of energies. Please remember that the maximum photon energy in this case is given by wmaX = yi,,,,/R with the Lorentz factor 7ion= - 1 in the rest frame of one of the ions. Therefore the maximum photon energy is about 300 GeV for RHIC and about 500 TeV for the LHC. The total photon-ion cross section is dominated by the nuclear excitation mainly of the GDR (the so called “Weizsacker-Williams 1771s.
“Of course these “maximum energies” should not be treated as absolute maxima. Photon with higher energies do occur but their spectrum decreases exponentially.
197
Id
lo2
Id
M [GeVl
Figure 4: The cross sections and rates for different mesons in yy collisions are shown for meson production in PbPb collisions at the LHC in (a). Figure (h) shows the same properties for the difhactive vector meson production as well as the total hadronic cross section for a hypothetical collision of Ph on protons for LHC energies. The results for Pb-Pb collisions can be obtained by a multiplication of a factor describing the coherent (or incoherent) production. For the rates a ion-ion luminosity of 1026cm-2s-’ was assumed.
process”) at the lowest energies. This is an important loss process as the excited nuclei decay mostly via neutron emission. But it is also used at RHIC as a luminosity monitor, see 19. Going to higher energies coherent diffractive processes are of interest. At the LHC the coherent diffractive vector meson production on a nucleus up t o the Upsilon can be studied. The expected rates for the hypothetical collisions of photons from Pb on a proton are shown in Fig. 4(b). For the heavy ion case they need of course t o be multiplied by a factor depending on the details of the coherent production on the nuclei. In this way one might be able t o see the transition from the “soft” t o the ‘‘hard” Pomeron with increasing meson mass. Looking for vector mesons with a larger transverse momenta one will also be able t o study the inelastic vector meson production. Again Fig. 4(b) will be helpful t o estimate the rates. Finally it has been proposed to study the photon-gluon fusion process and with it to be able t o measure the in-medium gluon distribution function. Such a measurement would be very interesting as some models predict a saturation of the gluon density at small Bjorken 2. 4
Lepton Pair Production and QED of Strong Fields
An interesting subject in itself is the production of light lepton pairs. Electron-positron pair production (and t o some extend also muon pair production) plays a special role in peripheral heavy ion collisions. Due t o their small mass they are produced quite easily. It is interesting to note that the work of Landau and Lifschitz in 1934 ’O, followed by Racah 1937 ”, are probably the first calculation of an electromagnetic process in relativistic ion collisions. The cross section for e+e- pair production is quite large. Using, e.g., the formula derived by Racah’’ (J
=z4a4 - 28 [ln3 Km2
27
-&,- 2.191n2 T2~ ~+, . . .] ,
(5)
one finds cross sections of about 30kbarn for Au-Au collisions at RHIC and 200kbarn for Pb-Pb collisions a t LHC, leading, e.g., t o lo7 pairs/sec produced in Pb-Pb collisions at LHC. This cross section increases with in3 $-m, which is rather fast and will eventually lead t o a violation of the Froissart bound. Therefore new phenomena will occur a t high energies. Most electrons and positrons are produced with rather modest energies (of the order of several me) and into the very forward direction. Therefore they will remain unobserved. Still the cross section for large angles and large energies is quite sizable. Finally we want t o point out that the equivalent photon approximation can only be used with great care for e+e- pair
198
Figure 5: The electromagnetic fields in relativistic heavy ion collisions are so strong that higher order processes will occur. Multiple pair production in a single collision is possible as shown in (a). Coulomb corrections lead to the distortion of the wave functions of electron and positrons (b). Finally due to the strong fields also multipleparticle processes can occur (c). The impact parameter dependent probability for the production of up to four pairs in lowest order is shown in (d) for Pb-Pb collisions a t the LHC.
production. This is due to the smallness of m, compared to 1/R. The total cross section can be calculated within EPA by using as cutoff parameter me instead of 1/R or restricting the impact parameters to be larger than A, = l / m , = 400fm. Calculations using only a restricted part of the phase space or for small impact parameter on the other hand need to be done either with a more refined analysis of this cutoff parameter or using a full calculation. 4.1
Multiple Pair Production and Coulomb Corrections
Electron-positron pair production has attracted some interest in recent years due to the observation that the impact parameter dependent probability calculated in perturbation theory can exceed unity already for RHIC and for impact parameters up to the Compton wavelength of the electron (A, % 400fm) ”. That the pair production cross section rises too fast with collision energy was already observed by Heitlerz3, but was thought to be an “academic problem” by him. In a series of papers starting from 22 it was found that the probability larger than one means that more than one pair will be produced on the average within a single collisions (Fig. 5(a)). Neglecting the antisymmetrization of the final state (and therefore treating the pair as a “quasiboson”) the probability P ( N ,b) for the N pair production was found to follow a Poisson distribution P ( N ,b) = P N( b ) exp(-P(b)) (6) N! with P ( b ) the probability for a single lepton line. P ( b ) , which is the probability calculated, e.g., in lowest order perturbation theory, is naturally interpreted as the average multiplicity of pairs 24125
~
Corrections to the Poisson distribution, as well as multiple-particle effects, have been discussed in24,where also the earlier work is discussed. The impact parameter dependent probability was calculated in lowest order z6 and it was found that on the average three to four pairs will be produced for small impact parameter. Cross section for the production of up to five pairs have been given as wellz5 (see Fig. 5(d)). The results for impact parameter of the order of a few A, have been recently confirmed 27. As the fields are rather strong (the effective coupling constant Z a = 0.6 is not small) one might doubt the results of the lowest order calculations, expecting that higher order processes
199
(on the single pair production process) should be important. Two different types of higher order corrections can be loosely distinguished 28: The distortion of the electron and positron wavefunctions after their production are normally subsumed under the term “Coulomb corrections”, see also Fig. 5(b). A second class deals with the fact that more than one pair can be present at an intermediate stage before the pairs annihilate crosswise (Fig. 5(c)). Such “multiple-particle” processes24are unique to the situation of heavy ion collisions with two strong fields. A calculation for small impact parameter 24 found them to be rather small. A classical result for Coulomb corrections in the pair production cross section are the BetheMaximon ~ o r r e c t i o n sdescribing ~ ~ ~ ~ ~ ,the production of an electron-positron pair through a single (real) photon interacting with a strong Coulomb fieldb. In the Bethe-Heitler formula 2w 109 In----f(Za) m e 42
the Coulomb corrections are contained in the correction term
with the Euler constant y x 0.57721 and $ the Psi (or Digamma) function. Using an analysis of the different orders in lny of the contribution with either one or multiple photon emissions from each ion, the Coulomb corrections to the total cross section are calculated in31. Rather large reductions compared to the Born cross sections (about 25% and 14% for RHIC and LHC) are given. Such an analysis is unfortunately not helpful for the calculation of multiple pair production as this requires the exchange of multiple photons from each ion. A different approach was proposed by a number of authors 32,33,34 making use of the form of the interaction in the high energy limit y + a.Using retarded boundary conditions it is found that only a certain class of diagrams is dominant in the high energy limit. The sum to all orders can be related to the one in lowest order approximation by a replacement of the photon propagator 1
1
A calculation of the impact parameter dependent probability was done in35, where it was found that the Coulomb corrections lead to a substantial reduction of the probabilities at small impact parameter and therefore also of the multiple-pair production cross sections. that this high energy limit leads to the interesting result It was already pointed out in that the cross section (but not P(b) itself) including all higher orders is identical to that in lowest order. This is of course in disagreement to the Bethe-Maximon corrections as discussed above. Different solutions for this have been proposed in the mean time (for a discussion see4). That a too simplified use of the eikonal approximation will lead to a result in disagreement with Bethe-Maximon theory was already pointed out in45. Using a more refined analysis37 the Bethe-Maximon corrections were found to be present here as well. Still the Coulomb corrections for the multiple pair production process have not been calculated up to now. 33136
4.2 Bound-Free Pair Production The electron from the produced pair has a tiny probability to be produced not as a free particle but into the bound state of one of the ions
A +A
+ A + (Ae-)K,L,... + e+
*This coincides with the limit 21+ 0 in the heavy ion case.
200
Figure 6: Processes with one muon emitted under a large angle and a jet in the opposite direction can be thought of to occur from the deep inelastically scattering of an equivalent muon (a). They are characterized by one muon scattered into one direction and a jet in the opposite one (b). The spectrum of the muons as a function of their energy is shown for both Pb-Pb collisions at LHC and Au-Au collisions at RHIC (c).
Even though this probability is small in combination with the large cross section for the pair production makes for a cross section of the order of 100 barn. As the Z/A ration changes to (2- 1)/A in this process, the ion is lost from the beam. Together with the electromagnetic excitation of the ions themself this is the dominant loss process. But whereas the excited ions, which subsequently decay mostly by neutron emissions have a large momentum spread, the momentum transfer coming from the electron capture is small and a very narrow Pbal+ beam emerges after the interaction region. Recently it was shown38 that this narrow beam of ions will hit the wall hundreds of meters after the interaction region in a very narrow spot depositing an energy of the order of tens of watts in a very small spot and the maximum beam luminosity of the P b beam at the LHC is limited due to the magnet cooling in this region. Recently a full calculation of the bound-free pair production in first order in the target interaction was made not only for production into the 1s-state but also to higher s and p states39. The results were also compared with other calculations up to now. In general a good agreement was found, therefore one expects that the size of this process seems to be well under control. 4.3 Equivalent Leptons: Heavy Ions as a source of high energetic leptons
The process of lepton (in the following mainly muon) production with a large transverse momentum of one of the muons can be best described within the picture of the “equivalent lepton approximation” 40,41. The photon emitted from one of the ions contains muons as partons, which can scatter deep inelastically with the other ion (see Fig. 6(a)). Using this approach the spectrum of the equivalent muons (or electrons) can be written as
With this the cross section can be calculated, see Eq. (144) of4. The equivalent muon spectrum for RHIC and LHC are given in Fig. 6(c). The deep inelastic scattering process will give information about the quark distribution function within the nucleus. In contrast to a lepton beam, which is mono-energetic, the muon here has a continuous spectrum. Nevertheless by measuring the scattered muon and the energy and longitudinal momentum of the jet going into the opposite direction (see Fig. 6(b)) one is
20 1
F
Figure 7: The emission of photons from the created electrons and positrons is the main source of bremsstrahlung at “high” photon energies and large angles. In the IR limit diagrams (a) and (b) give the main contribution. The result of a calculation using three different approximation schemes are shown for the conditions of Pb-Pb collisions at the LHC in (c).
able to reconstruct the energy of the initial muon. Of course these cross section are small. Nevertheless it is worthwhile to study how these processes can be used and work on this is in progress.
4.4 Bremsstrahlung from produced pairs: The “glowing of heavy ion collisions” The emission of bremsstrahlung photons from the ions itself is highly suppressed due to their large mass 42. On the other hand the pair production cross section is rather large and due to their small mass bremsstrahlungsemission is possible easily. It is well known that in the case of lepton colliders this process can become the dominant source for photons at large angles 43. A calculation of the bremsstrahlungsspectrum was done using different approximations 44. The results for different energies and angles are shown in Fig. 7(c). It is an open question whether these processes could be used, eg., in order to “see” the interaction vertex of the two ion bunches directly by looking at these photons.
5
Summary
The strong electromagnetic fields can be used for both photon-photon and photon-ion processes. These processes are measurable in very peripheral collisions, as only the long range electromagnetic interaction is present in this case. They are interesting processes to be studied at both RHIC and LHC. The theoretical description of these processes is well understood and corrections to the dominant coherent photon emission have been studied already. Electron-positron pair production plays a special role in these collisions. Besides the theoretical interest connected to the strong field effects, they are also of practical importance as a possible loss process or even as a possibility to measure the parton distributions inside the ions. As can also be seen from the other contributions to this workshop, the beautiful results of the STAR detector at RHIC have shown that we have definitely left the regime of theoreticans dreams and hopes and are entering the aera of experimental realities and results.
Acknowledgments The author would like to thank Sebastian White and Bill Marciano for the organization of this workshop and the possibility to participate. The wonderful atmosphere has lead to a number of most interesting discussions, which will be surely very fruitful for the future of this kind of physics.
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References
1. C. A. Bertulani and G. Baur, Physics Reports 163,299 (1988). 2. F.Krauss, M. Greiner, and G. Soff, Prog. Part. Nucl. Phys. 39,503 (1997). 3. G.Baur, K. Hencken, and D. Trautmann, Topical Review J. Phys. G 24,1657 (1998) 4.G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, and Yu. Kharlov, to appear in Physics Reports, 2002,e-print hep-ph/Ol122 11. 5. E. Fcrmi, 2. Phys. 29,315 (1924). 6. C. F.Weizsacker, 2. Phys. 88, 612 (1934). 7. E. J. Williams,Phys. Rev. 45,729 (1945). 8. V. M. Budnev, I. F. Ginzburg, G. V. Meledin, and V. G. Scrbo, Physics Reports 15,181 (1975). 9. G. Ingelman, A. De Roeck, and R. Klanner, eds., Future Physics at HERA, 1996. 10. K. Hencken, D. Trautmann, and G. Baur, Phys. Rev. C 53,2532 (1996). 11. J. Ohnemus, T.F. Walsh, and P. M. Zerwas, Phys. Lett. B 328,369 (1994). 12. M. Drees and D. Zeppenfeld, Phys. Rev. D 39, 2536 (1989). 13. G. Baur, this proceedings. 14. C. A. Bertulani, this proceedings. 15. S. R. Klein and the STAR collaboration, LBNL-47723,e-print nucl-ex/0104016. 16. K. Piotrzkowski, this proceedings. 17. J. Nystrand, this proceedings. 18. P. Ycpes, this proceedings. 19. S. White, this proceedings. 20. L. D. Landau and E. M. Lifshitz, Phys. 2. Sowjet., 6,244 (1934). 21. G Ebcah, Nuovo Cimento 14,93 (1937). 22. G. Baur, Phys. Rev. A 42,5736 (1990). 23. W. Heitler, The Quantum Theory of Radiation, Oxford University Press, London, 1954. 24. K. Hencken, D. Trautmann, and G. Baur. Phys. Rev. A 51,998 (1995). 25. A. Alscher, K. Hencken, D. Trautmann, and G. Baur, Phys. Rev. A 55,396 (1997). 26. K.Hencken, D. Trautmann, and G. Baur, Phys. Rev. A 51,1874 (1995). 27. R.N. Lee, A. I. Milstein, and V. G. Serbo, e-print hep-ph/0108014, 2001. 28. A. Aste, G. Baur, K. Hencken, and D. Trautmann, c-print hep-ph/0112193,2001. 29. H. A. Bethe and L. C. Maximon. Phys. Rev. 93,768 (1954). 30. H. Davies, H. A. Bethe, and L. C. Maximon. Phys. Rev. 93,788 (1954). 31. D. Yu. Ivanov, A. Schiller, and V. G. Serbo. Phys. Lett. B 454,155 (1999). 32. B. Segev and J. C. Wells, Phys. Rev. A 57, 1849 (1998). 33. A. J. Baltz and L. McLerran, Phys. Rev. C 58, 1679 (1998). 34. U.Eichmann, J. Rcinhardt, S. Schramm, and W. Greiner, Phys. Rev. A 59, 1223 (1999). 35. K. Hencken, D. Trautmann, and G. Baur, Phys. Rev. C 59, 841 (1999). 36. B. Scgev and J. C. Wells, Phys. Rev. C 59,2753 (1999). 37. R. N.Lee and A. I. Milstein, Phys. Rev. A 61,032103 (2000). 38. S.R. Klein, Nucl. Instrum. Methods A 59,51 (2001). 39. H. Meier, Z. Halabuka, K. Hencken, D. Trautmann, and G. Baur, Phys. Rev. A 63, 032713 (2001). 40. M. Chen and P. Zerwas, Phys. Rev. D 12,187 (1975). 41. V.N. Baier, V. S. Fadin, and V. H. Khose, Nucl. Phys. B 65,381 (1973). 42. H. Meier, K.Hencken, D. Trautmann, and G. Baur, E. Phys. J. C 2,741 1998). 43. V. S.Fadin and V. A. Khoze, Sov. Phys.-JEW 17,313 (1973). 44.K. Hencken, D. Trautmann, and G. Baur, Phys. Rev. C 60,34901 (1999). 45. R. Blankenbecler and S.D. Drell, Phys. Rev. D 36, 2846 (1987).
RELATIVISTIC HEAVY ION EXCITATION O F GIANT RESONANCES C.A. BERTULANI NSCL, Michigan State University, East Lansing, MI 48824-1321, USA Giant resonances and giant resonances built on other giant resonances in nuclei are observed with very large cross sections in relativistic heavy ion collisions. A theoretical effort is underway to understand the reaction mechanism which leads t o this process, as well as a better understanding of the microscopic properties of multiphonon states, e.g., their strength, energy centroids, widths and anharmonicities.
1 Giant Resonances 1.1
Single giant resonances
Giant resonances in nuclei were first observed in 1937 by Bothe and Gentner who obtained an unexpectedly large absorption of 17.6 MeV photons (from the 7Li(p,y) reaction) in some targets. These observations were later confirmed by Baldwin and Klaiber (1947) with photons from a betatron. In 1948, Goldhaber and Teller interpreted these resonances (called isovector giant dipole resonances (IVGDP)) with a hydrodynamical model in which rigid proton and neutron fluids vibrate against each other, the restoring force resulting from the surface energy. Steinwendel and Jensen later developed the model, considering compressible neutron and proton fluids vibrating in opposite phase in a common fixed sphere, the restoring force resulting from the volume symmetry energy. The standard microscopic basis for the description of giant resonances is the random phase approximation ( W A ) in which giant resonances appear as coherent superpositions of one-particle one-hole (lplh) excitations in closed shell nuclei or two quasiparticle excitations in open shell nuclei (for a review of these techniques, see, for example, ref. 4). The isoscalar quadrupole resonances were discovered in inelastic electron scattering by Pitthan and Walcher (1971) and in proton scattering by Lewis and Bertrand (1972). Giant monopole resonances were found later and their properties are closely related to the compression modulus of nuclear matter. Following these, other resonances of higher multipolarities and giant magnetic resonances were investigated. Typical probes for giant resonance studies are (a) 7's and electrons for the excitation of IVGDR, (b) a-particles and electrons for the excitation of isoscalar giant monopole resonance (ISGMR) and giant quadrupole resonance (ISGQR), and (c) (p, n), or (3He, t), for Gamow-Teller resonances, respectively. 1.2 Multiphonon resonances
Inelastic scattering studies with heavy ion beams have opened new possibilities in the field (for a review of the recent developments, see ref. 5 ) . A striking feature was observed when either the beam energy was increased, or heavier projectiles were used, or both6. This is displayed in figure 1, where the excitation of the GDR in 'OsPb was observed in the inelastic scattering of 170at
203
204
\
0
'
I
I
I
I
I
10 20 30 Excitation Energy [MeV]
I
40
Figure 1: Experimental cross sections in arbitrary units for the excitation of ''Pb targets by I7O (22.A MeV and 84.A MeV) and by 36Ar (95.A MeV), as a function of the excitation energy.
22 MeV/nucleon and 84 MeV/nucleon, respectively, and 36Ar at 95 MeVfnucleon What one clearly sees is that the 'bump' corresponding to the GDR at 13.5 MeV is appreciably enhanced. This feature is solely due to one agent: the electromagnetic interaction between the nuclei. This interaction is more effective at higher energies, and for increasing charge of the projectile. In ref. it was shown that the excitation probabilities of the GDR in heavy ion collisions approach unity at grazing impact parameters. It was also obtained that, if the DGDR (double GDR), or GDR2 (i.e.) a GDR excited on a GDR state), exists then their excitation cross sections in heavy ion collisions at relativistic energies are of the order of hundreds of millibarns. The calculation was based on the semiclassical approach, appropriate for heavy ion scattering at high incident energies, and the harmonic oscillator model for the giant resonances. The semiclassical model treats the relative motion between the nuclei classically while quantum mechanics is used for the internal degrees of freedom. In the harmonic picture for the internal degrees of freedom the GDR is the first excited state in a harmonic well, the DGDR is the second state, and so on. In ref. lo it was shown that the excitation probabilities and cross sections are directly proportional to the photonuclear cross sections for a given electric (E) and magnetic (M) multipolarity. For an impact parameter b, excitation energy E , and a multipolarity T X (n= E or M, X = 1, 2, 3, . . .) the excitation probabilities are given by 718.
where a,"'(E) is the photonuclear cross sections for the photon E and multipolarity n X . The total photonuclear cross section is u7(E)= C,xu$X(E).In the semiclassical approach, the eqviualent photon numbers N,x(E, b) are given analytically lo. A quantum mechanical derivation of the excitation amplitudes in relativistic Coulomb excitation shows that equation (1)can also be obtained by using the saddle-point approximation in the DWBA integrals l l . The total Coulomb excitation cross sections are then obtained by an integration of equation (1) over the impact parameter b, including a factor, T ( b ) , which accounts for the strong absorption at small impact parameters: a,x(E) = 2 n J d b b T ( b ) P , x ( E , b). The total number of equivalent photons n,x(E) = 2 n J d b b N,x(E, b) is given in The cross section for the excitation of a giant resonance is obtained from these expressions, by using the experimental photonuclear absorption cross section for u,"'(E) in equation (1). One problem with this procedure is that the experimental photonuclear cross section includes all multipolarities with the same
205
weight: uFP(E)= ETA U ~ ~ ( while E ) , the calculation based on equation (1) needs the isolation of .,"'(I?). This can only be done marginally, except in some exclusive measurements. Generally, one finds in the literature the (7, n), (7, 2n), and (r, 3n) cross sections, which include the contribution of all multipolarities in the giant resonance energy region. A separation of the different multipolarities can be obtained roughly by use of sum rules, or some theoretical model for the nuclear response to a photoexcitation. Assuming that one has u,"'(E) somehow (either from experiments, or from theory), a simple harmonic model based on the Axel-Brink hypotheses can be formulated to obtain the probability to access a multiphonon state of order n. In the harmonic oscillator model the inclusion of the coupling between all multiphonon states One of the basic changes is that the excitation probabilities can be performed analytically calculated to first-order, P,'it(E, b), are modified to include the flux of probability to the other states. That is, for the first harmonic state,
'.
where P,'f(b) is the integral of P,'it(E, b) over the excitation energy E. In general, the probability to reach a multiphonon state with the energy E(") from the ground state, with energy Ido),is obtained by an integral over all intermediate energies
The character and spin assignment of the multipolarity A* depends on how the intermediate states couple with the electromagnetic transition operators. For example, in the case of the DGDR (GDR2 ), assuming a O+ ground state and excluding isospin impurities, the final state has either spin and parity O+ or 2+, respectively. A simpler reaction model than the one above can be obtained by assuming that all states can be approximated by a single isolated state. For example, we can assume that the photoabsorption cross sections in the range of the GDR is due to a single state with energy equal to the centroid energy of the GDR exhausting the whole excitation strength. Then the multiphonon states are sharp and equidistant. Eq. (3) becomes
The above relation was used to calculate the cross sections for the excitation of the GDR, GDR2, GDR3, ISGQR and IVGQR in 13'Xe, respectively, for collisions with P b nuclei as a function of the bombarding energy, as shown in figure 2. Each resonance is considered to be a single state exhausting 100% of the respective sum rule. Also shown in the figure is the geometrical cross fm2 . The cross sections for the excitation of the GDR2 is section, (GC), u 4 A Y 3 large, of the order of hundreds of mb. Much of the interest in looking for multiphonon resonances relies on the possibility of finding exotic particle decay of these states. For example, in ref. l5 a hydrodynamical model was used to predict the proton and neutron dynamical densities in a multiphonon state of a nucleus. Large proton and neutron excesses at the surface are present in a multiphonon state. Thus, the emission of exotic clusters from the decay of these states are a natural possibility. A more classical point of view is that the Lorentz contracted Coulomb field in a peripheral relativistic heavy ion collision acts as a hammer on the protons of the nuclei lo. This (collective) motion of the protons seems only to be probed in relativistic Coulomb excitation. Despite all the advantages of relativistic Coulomb excitation, the DGDR was first found in pion scattering at the Los Alamos Pion Facility 12. In pion scattering off nuclei the DGDR can be described as a two-step mechanism induced by the pion-nucleus interaction. Only about
-
+
206
Bombarding Energy [ A . GeV] Figure 2: Theoretical cross sections for the excitation of the GDR,ISGQR, IVGQR, GDR' and the GDR3 , in the reaction '''Pb + z08Pb,as a function of the bombarding energy.
five years later, the first Coulomb excitation experiments for the excitation of the DGDR were performed at the GSI facility in Darmstadt/Germany 13914. One of these experiments observed the neutron decay channels of giant resonances excited by relativistic projectiles. The excitation spectrum of relativistic 136Xe projectiles incident on P b were compared with the spectrum obtained in C targets. A comparison of the two spectra immediately proved that the nuclear contribution to the excitation is very small. Another experiment l4 dealt with the photon decay of the double giant resonance. The advantages of relativistic Coulomb excitation of heavy ions over other probes (pions, nuclear excitation, etc.) was clearly demonstrated in several other GSI experiments 13114J6,17. 2
Energy, width and strength of the DGDR
Experimentally 18, it was found that the energy of the DGDR agrees with the harmonic prediction, i.e., that its energy should be twice the energy of the GDR, although small departures from this prediction were seen, especially in pion and nuclear excitation experiments. The width of the DGDR seems to be fi times that of the GDR, although a value equal to 2 is not ruled out completely. An unexpected result was obtained for the ratio between the measured cross sections and the calculated ones. This seems to be strongly dependent on the experimental probe. The largest values of ueZp/uthcome from pion experiments, yielding up to a value of 5 for it. In Coulomb excitation experiments this ratio was initially 13J4 of the order of 2.
2.1
Width of the DGDR
In a microscopic approach, the GDR is described as a coherent superposition of one-particle one-hole states. One of the many such states is pushed up by the residual interaction to the experimentally observed position of the GDR. This state carries practically all of the E l strength. This situation is simply realized in a model with a separable residual interaction. We write the GDR state as (one phonon with angular momentum 1M) I1,lM >= Ai,IO >, where A!, is a proper superposition of particle-hole creation operators. Applying the quasi-boson approximation we can use the boson commutation relations and construct the multiphonon states (N-phonon states). An N-phonon state will be a coherent superposition of N-particle Nhole states. The width of the GDR is essentially due to the spreading width, i.e., to the coupling
207
to more complicated quasi-bound configurations. The escape width only plays a minor role. Let us take a simple model for the strength function1'. A state la > (i.e., a GDR state) is coupled by some mechanism to more complicated states la >. For simplicity we assume a constant coupling matrix element V, =< alVla >=< culVla >= u. With an equal spacing of D of the levels I C Y > one obtains a width r = 2.rru2/D for the state la >. We assume the same mechanism to be responsible for the width of the N-phonon state: one of the N-independent phonons decays into the more complicated states la > while the other (N - 1)-phonons remain spectators. We write the coupling interaction in terms of creation (destruction) operators ch (c,) of the complicated states / a > as V = u(AIMca +AIM&). (5) For the coupling matrix elements U N , which connects an N-phonon state IN IN - 1, cy > (N-1 spectator phonons) one obtains
UN=< N
-
1,aIVlN >= v
-
llAlMlN
>= vd%,
> to the state (6)
i.e., one obtains for the width r n of the N-phonon state
Thus, the factor N in (7) arises naturally from the bosonic character of the collective states. For the DGDR this would mean r2 = 2r1. The data points seem to favor a lower multiplicative factor. We can also give a qualitative explanation for a smaller rz/I'l value. First we note that the value l?z/I'l = 2 can also be obtained from a folding procedure, as given in equation (3). If the sequential excitations are described by Breit-Wigner (BW) functions P,x(E) with the centroid & and the width r, the convolution (3) yields a BW shape with the centroid at 2& for the DGDR and the total width of 2r1 . However, if one uses Gaussian functions (instead of BW) for the shape of one-phonon states, it is easy to show that one also obtains a Gaussian for the N-phonon shape, but with the width given by f l F 1 . The latter assumption seems inconsistent since the experimentalists use BW fits for the shape of giant resonances, which are in good agreement with the experimental data. However, one can easily understand that the result fir, is not restricted to a Gaussian fit. For an arbitrary sequence of two excitation processes we have < E >=< El E2 > and < E2 >=< (El E2)' >; for uncorrelated steps it results in the addition in quadrature (AE)' = (A&)' (AE2)'. Identifying these fluctuations with the widths up to a common factor, we obtain for identical phonons r2 = fir,. The same conclusion will be valid for any distribution function which, as the Gaussian one, has a finite second moment, in contrast to the BW or Lorentzian ones with second moment diverging. We may conclude that, in physical terms, the difference between l?z/J?l = 2 and r2/r1 = fi is due to the different treatment of the wings of the distribution functions which reflect small admixtures of remote states.
+
+
+
2.2 Strength of the DGDR Microscopically, the harmonic picture is accomplished within the RPA approximation. The excited states of the nucleus are described as superpositions of particle-hole configurations with respect to the ground state. The multiphonon resonances are built using products of the 1resonance states, yielding O+ and 2+ double phonon states. The interaction with the projectile is described in terms of a linear combination of particle-hole operators weighted by the timedependent field for a given multipolarity of the interaction. Since the time-dependent Coulomb field of a nucleus does not carry monopole multipolarity, the DGDR states can be reached via
208 1
'
1
'
1
'
1.4
- Coupled-ChannelP
12l-
-----
Perturbtie
1
Figure 3: Occupation probability of the ground state, laoI2, and of the GDR, [all', for the reaction '"Pb + zosPb at 640.A MeV, as a function of the reaction time. The reaction time is given in terms of the adimensional quantity Twt/b, with b equal to the impact parameter in the collision. The broken curves are the predictions of perturbation theory, while the full curves are the predictions of coupled-channels calculations.
two-step E l transitions and direct E2 transitions (for a O+ ground state). Early calculations failed to explain the experimental data. There seems to be two possible reasons for uezp/uth# 1; (a) either the Coulomb excitation mechanism is not well described, or (b) the response of the nucleus to two-phonon excitations is not well known. Many authors studied the effects of the excitation mechanism of the DGDR. In ref. 2o the cross sections were calculated using second-order perturbation theory. It was found that the theoretical values were smaller than the experimental ones by a factor of 1.3 - 2. However, it was suggested that second-order perturbation theory is not adequate for relativistic Coulomb excitation of giant resonances with heavy ions and that it is necessary to perform a coupled channels calculation. We can see this more clearly in figure 3, taken from", where a coupledchannels study of multiphonon excitation by the nuclear and Coulomb interactions in relativistic heavy ion collisions was performed. The figure shows the probability amplitude to excite the GDR in '08Pb, la1I2, and the occupation probability of the ground state, la01~,for a grazing collision of '08Pb '08Pb at 640 MeV/nucleon. The broken curves are the predictions of the first-order perturbation theory. We see that the asymptotic excitation probability of the GDR is quite large ( 40%). In first-order perturbation theory the occupation probability of the ground state is kept constant, equal to unity. Obviously, one greatly violates the unitarity condition in this case. An appropriatc coupled-channcls calculation (full curves) shows that the ground-state occupation probability has to decrease to meet the unitarity requirements, while the excitation probability of the GDR is also reduced slightly for the same reason. In ref. 22 it was shown that a good coupled-channels calculation does not need to account for the exact coupling equations in all channels. The strongest coupling, responsible for the effect observed in figure 3 is the coupling between the ground state and the GDR states. This has to be treated exactly within a coupled-channels calculation. The coupling between the GDR and the other states (including the DGDR, IVGQR, ISGQR, etc.) can be treated perturbatively. The results ofz2 showed appreciable dependence of the excitation cross sections of the DGDR on the width of both the GDR and the DGDR for "'Pb '08Pb at 640 MeV/nucleon. It was also shown that the most favorable energies for the measurement of the DGDR corresponds to the SIS energies at the GSI/Darmstadt facility.
+
+
209
P'' b (7,n)
16
18
20
22
24
Figure 4: Photoneutron cross section for "'Pb. Experimental data (dots with experimental errors) are from ref. [28]. The long-broken curve is the high-energy tail of the GDR, the short-broken curve is the IVGQR and the curve with squares is their sum. The contribution of two-phonon states is plotted by a curve with triangles. The full curve is the total calculated cross section.
2.3 Anharmonicities Another possible effect arises from a shift of the energy centroid of the DGDR due to anharmonic effects 23. In ref. 22 one obtained U D G D R = 620, 299 and 199 mb for the centroid energies of EDGDR= 20, 24 and 27 MeV, respectively. This shows that anharmonic effects can play a big role in the Coulomb excitation cross sections of the DGDR, depending on the size of the shift of EDGDR.However, in ref. 2o the source for anharmonic effects were discussed and it was suggested that it should be very s m d , i.e., A(')E = E D G D R - ~ E G D rr,R 0. The anharmonic behavior of the giant resonances as a possibility to explain the increase of the Coulomb excitation cross sections has been studied by several authors23p24(see also ref. 25, and references therein). It was found that the effect is indeed negligible and it could be estimated25as A(2)E< E G D R / ( ~ O A )AP4l3 MeV. Recent studies26 of the reaction mechanism with anharmonic effects support the idea that these are indeed very small. N
2.4
Other routes to the DGDR
From the above discussion we see that the magnitude of the Coulomb excitation cross sections of the DGDR can be affected due to uncertainties in: (a) strength, (b) width, (c) energies, or (d) reaction mechanism. Cases (a) and (c) are the basis of the Axel-Brink hypothesis and we have seen that a modification of their values would only be considered seriously if anharmonic effects were large, which does not seem to be the case. Case (b) is an open question. Microscopic calculations 24 have shown that, taking into account the Landau damping, the collective state splits into a set of different 1: states distributed over an energy interval, where i is the order number of each state. A further fragmentation of the 1; states into thousands of closed packed states, is obtained by the coupling of one- and two-phonon states. This leads to a good estimate of the spreading width of the GDR. However, the DGDR states were obtained by a folding procedure: I[li8 l;]Jr=0+,1+,2+ >M' (1rnlm'IJM)ll; > m 11; >m' . (8)
c
m.m'
The width of the DGDR is thus fixed by the width of the GDR. It is therefore impossible
210
b
20
30
25
35
Figure 5: The contribution for the excitation of two-phonon 1- states (long-broken curve) in first-order perturbation theory, and of two-phonon'0 and 2' DGDR states to second-order (short-broken curve). The total cross section (for 20BPb(640.AMeV) "'Pb) is shown by the full curve.
+
to make any quantitative prediction for the width of the DGDR, other than saying that 4 5 r D G D R / r G D R 5 2. Thus, we return to the discussion of the reaction mechanism, and how it could affect the magnitude of the cross sections. The nuclear excitation of giant resonances is very small in magnitude compared with Coulomb excitation in collisions with heavy ions at relativistic energies ". In ref. 22 it was shown that the nuclear-Coulomb interference i s also a small effect. In ref. 27 the contribution of non-natural parity 1+two-phonon states were investigated in a coupled-channels calculation. The diagonal components [ l i€3 lill+are forbidden by symmetry properties but non-diagonal ones [li€3 lJ1+ , may be excited in the two-step process bringing some "extra strength" in the DGDR region. Consequently, the role of these non-diagonal components depends on how strong the Landau damping is. A coupled-channels calculation found that the contribution of the 1+ states to the total cross section is small. The reason for this i s better explained in second-order perturbation theory. For any route to a final magnetic substate M , the second-order amplitude will be proportional to (OO1pllp)V'lp,~+l- x (lplpIlM)Vg1,,1-,1+ ( p +) p ' ) , where V'lP,i+f is the p-component of the interaction potential (for a spin-zero ground state, p is also the angular momentum projection of the intermediate state). Assuming that the phases and the products of the reduced matrix elements for the two sequential excitations are equal, we obtain vElp,0+1- x VEI,,!,~-+~+= V ~ ~ ~ t , ox +V~E-~ ~ , ~ - - ,Thus, ~ + . under these circumstances, and since (OOlpllp) = 1, we get an identically zero result for the excitation amplitude of the 1+ DGDR state as a consequence of (1plpIlM) = - ( l p ' l p ~ l M ) . We note that multiphonon states can be obtained by coupling all kinds of phonons. Each configuration [Ayl @A?] can be obtained theoretically from a sum over several two-phonon states made of phonons and of complicated states with a given spin and parity A?, A? , and different RPA root numbers i l , is of their constituents. The cross sections can be obtained accordingly:
+
As an example, in ref. 28 the total number of two-phonon 1- states generated in this way was about lo5. The absolute value of the photoexcitation of any two-phonon state under consideration is negligibly small but altogether they produce a sizeable cross section. The 1- two-phonon were used to calculate their contribution to the (7,n) cross section states obtained in ref.
'*
21 1
Figure 6: Deviation of the experimental results from the harmonic oscillator prediction for the energy, width, cross section, ratio between the decay by emission of two gammas, and of the one-neutron decay width, respectively.
in '08Pb, via direct E l excitations. This is shown in figure 4. Experimental data (dots with experimental errors) are from29. The long-broken curve is the high-energy tail of the GDR, the short-broken curve is the IVGQR and the curve with squares is their sum. The contribution of two-phonon 1- states is plotted by a curve with triangles. The full curve is the total calculated cross section. Thus, already at the level of photonuclear data the contribution of two-phonon 1states is of relevance. Here they are not reached via two-step processes, but in direct excitations. Since the energy region of these states overlap with that of the DGDR, in Coulomb excitation experiments they should also contribute appreciably. In fact, it was shownz8 that their contri'08Pb (640.A MeV) in the DGDR region is of the bution to the total cross section for '"Pb order of 15%. In figure 5 we show the contribution of the excitation of two-phonon 1- states (long-dashed curve) in first order perturbation theory, and for two-phonon O+ and 2+ DGDR states in second-order (short-dashed curve). The total cross section (for '08Pb (640.A MeV) z08Pb)is shown by the solid curve.
+
+
3
Present situation and perspectives
Presently, experiments tell us that the harmonic model reproduces the cross section for the GDR quite well, but it gives smaller values than the measured cross sections by as much as 30%. In figure 6 the present situation on our knowledge of the energy, width, excitation cross section, branching ratio for gamma to neutron emission, and the neutron emission width, respectively, is shown in comparison with calculations based on the simple harmonic picture. We see that the theory-experiment agreement is much better than that obtained in the pioneer experiments. As we have seen in this short review there are several effects which compete in the excitation of double giant resonances in relativistic Coulomb excitation. These effects were discovered in part by the motivation to explain discrepancies between the harmonic picture of the giant resonances and the recent experimental data. We cannot say at the moment how much we have progressed towards a better understanding of these nuclear structures, as some controversies still remain in the literature (see, for example, 30). Recent studies of giant resonances in ultra-relativistic collisions have been performed at CERN 31 and Brookhaven 32. Since the nuclei fragment after the excitation to a giant resonance, this process can be used for beam monitoring as well3'. The field is just in its infancy and important experimental and theoretical progress will occur in the
212
near future.
References W. Bothe and W. Gentner, Z. Phys. 106, 236 (1937) M. Goldhaber and E. Teller, Phys. Rev. 74, 1046 (1948) H. Steinwedel and J.H.D. Jensen, Z. Naturf. 5a, 413 (1950) J. Speth and J. Wambach, Int. Review of Nuclear and Particle Physics vol. 7, ed. J Speth (Singapore, World Scientific), 1991 5 . C.A. Bertulani and V.P. Ponomarev, Phys. Rep. 321, 139 (1999) 6. J. Barrette et. al., Phys. Lett. B209, 182 (1988) 7. J. Beene et. al., Phys. Rev. C41, 920 (1990) 8. J. Beene, Int. Nuclear Physics Conf. on Giant Resonances (Gull Lake, 1993) (1994 Nucl. Phys. A569 163c 9. G. Baur and C.A. Bertulani, Phys. Lett. B174, 23 (1986); Phys. Rev. C34, 1654 (1986) 10. C.A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988) 11. C A. Bertulani and A.M. Nathan, Nucl. Phys. A554, 158 (1993) 12. Y. Moidechai et. al., Phys. Rev. Lett. 61, 531 (1988) 13. R. Schmidt et. al., Phys. Rev. Lett. 70, 1767 (1993) 14. J.L. Ritman et. al., Phys. Rev. Lett. 70, 2659 (1993) 15. A.C. Vasconcellos-Gomes and C.A. Bertulani, Nucl. Phys. A517, 639 (1990) 16. T. Aumann et. al. Phys. Rev. C47, 1728 (1993) 17. K. Boretzky et. al., Phys. Lett. B384,30 (1996) 18. Ph. Chomaz and N. Frascaria, Phys. Rep. 252, 275 (1995) 19. G. Baur and C.A. Bertulani, Proc. Int. School of Heavy Ion Physics (Erice, Italy, October 1986), ed. R.A. Broglia and G.F. Bertsch (New York: Plenum), p. 331 20. C.A. Bertulani and V. Zelevinsky, Phys. Rev. Lett. 71, 967 (1993); Nucl. Phys. A568, 931 (1993) 21. L.F. Canto, A. Romanelli, M.S. Hussein and A.F.R. de Toledo Piza, Phys. Rev. Lett. 72, 2147 (1994) 22. C.A. Bertulani, L.F. Canto, M.S. Hussein and A.F.R. de Toledo Piza, Phys. Rev. C53, 334 (1996) 23. C. Volpe et. al., Nucl. Phys. A589, 521 (1995) E. Lanza et. al., Nucl. Phys. A613, 445 (1997) 24. V. Ponomarev et. al., Phys. Rev. Lett. 72, 1168 (1994); Z. Phys. A356, 251 (1996) 25. G.F. Bertsch and H. Feldmeier, Phys. Rev. C56, 839 (1997) 26. D.T. de Paula, T. Aumann, L.F. Canto, B.V. Carlson, H. Emling and M.S. Hussein, Phys. Rev. C64, 064605 (2001); C.A. Bertulani, Comput. Phys. Commun. 116, 345 (1999) 27. C.A. Bertulani, V. Ponomarev and V.V. Voronov, Phys. Lett. B388, 457 (1996) 28. V. Ponomarev and C.A. Bertulani, Phys. Rev. Lett. 79, 3853 (1997) 29. S.N. Belyaev et. al., Phys. At. Nucl. 58, 1833 (1995) 30. C.A. Bertulani, P.F. Bortignon, V.Yu. Ponomarev and V.V. Voronov, Phys. Rev. Lett. 87, 269201 (2001) 31. S. Datz et. al., Phys. Rev. Lett. 79, 3355 (1997) 32. S. White, private communication 1. 2. 3. 4.
HEAVY IONS INTERPRETATION O F HIGHEST ENERGY COSMIC RAYS STEVE REUCROFT Department of Physics, Northeastern University,
Boston, M A 02115, USA A brief review of the energy spectrum of primary cosmic rays above 10'' eV and the measurement techniques used t o investigate the ultra high energy ones is given. This is followed by a discussion of the atmospheric shower profile of the highest energy event observed using the Fly's Eye detector. Finally, cosmic ray simulation tools are discussed and used t o investigate the heavy ion interpretation of the highest energy primary. The best way to contribute further to the understanding of this issue is by the collection of new and better data.
1 Cosmic Ray Energy Spectrum
The cosmic ray (CR) spectrum above lo1' eV (where the Sun's magnetic field is no longer a concern) can be described by a series of power laws with the flux falling about 3 orders of magnitude for each decade increase in energy,' see Fig. 1. Above l O I 4 eV, the flux becomes so low that direct measurements using sophisticated equipment on satellites or high altitude balloons are limited in detector area and in exposure time. Ground-based experiments with large apertures make such a low flux observable after a magnification effect in the upper atmosphere: the incident cosmic radiation interacts with atomic nuclei of the air molecules and produces extensive air showers which spread out over large areas. Continuously running using ingenious installations has raised the maximum observed primary particle energy to higher than lo2' eV? While theoretical subtleties surrounding CR acceleration provide ample material for discussion, the debate about the origin of CRS up to the knee (- 1015.5eV) has reached a consensus that they are produced in supernova explosions! The change of the spectral index (from -2.7 to -3.0) near the knee, presumably reflects a change in origin and the takeover of another, yet unclear type of source. The spectrum steepens further to -3.3 above 1017.7 eV (the dip) and then flattens to an index of -2.7 at 10'8.5 eV (the ankle). A very widely held interpretation of the modulation features is that above the ankle a new population of CRS with extragalactic origin begins to dominate the more steeply falling Galactic population! The origin of the extragalactic channel is somewhat mysterious. CRS do not travel unhindered through intergalactic space, as there are several processes that can degrade the particles' energy. In particular, the thermal photon background becomes highly blue shifted for ultrarelativistic protons. The reaction sequence m -+ A+ + 7r'p effectively degrades the primary proton energy providing a strong constraint on the proximity of CR-sources, a phenomenon known as the Greisen-Zatsepin-Kuz'min (GZK) cutoff5 The energy attenuation length of proton& is shown in Fig. 2. A heavy nucleus undergoes photodisintegration in the microwave and infra-red backgrounds; as a result, iron nuclei do not survive fragmentation over N
N
"This hypothesis is supported by AGASA data?
213
214 . .. . . .
.. .
Fluxes of Cosmic Roys
x
=
10
E
F L b
‘1.
t- ( 1
particle p e r m ’ - s ~ c n d )
-,7
10
-
-,i 10 )
C
-
Figure 1: Cosmic ray energy spectrum.
comparable distances: Ultra high energy gamma rays would travel even shorter paths due to pair production on radio photons8 Therefore, if the CR sources are all at cosmological distances, the observed spectrum must virtually end with the GZK cutoff at E M 8 x lo1’ eV. The spectral cutoff is less sharp for nearby sources (within 50 Mpc or so)? The arrival directions of the transGZK events are distributed widely over the sky, with no plausible optical counterparts (such as sources in the Galactic Plane or in the Local Supercluster). Furthermore, the data are consistent with an isotropic distribution of sources in sharp constrast to the anisotropic distribution of light within 50 Mpc!’ The difficulties encountered by conventional acceleration mechanisms in accelerating particles to the highest observed energies have motivated suggestions that the underlying production mechanism could be of non-acceleration nature. Namely, charged and neutral primaries, mainly light mesons (pions) together with a small fraction (3%) of nucleons, might be produced at extremely high energy by decay of supermassive elementary X particles (mx eV)!l However, if this were the case, the observed spectrum should be dominated by gamma rays and neutrinos, in contrast to current observation! l2 Alternative explanations involve undiscovered neutral hadrons with mmses above a few GeV!3 neutrinos producing nucleons and photons via or else neutrinos attaining cross resonant 2-production with the relic neutrino backgro~nd;~ sections in the millibarn range above the electroweak s~a1e.l~ If neutrinos/neutral-hadrons are primaries, they should point back to their sources, thereby enabling point source astronomy for the most energetic sources of flux at and above the GZK energy. However, the current CR-sample does not show significant angular correlation with powerful high-redshift sources!‘ N
215
Figure 2: Energy attenuation length of nucleons in the intergalactic medium. Notice that independently of the initial energy of the nucleon, the mean energy values approach to 100 EeV after a distance of approx 100 Mpc.
2
Measurement Techniques
There are two major techniques used in the detection of ultra high energy CRS. The first one is to build an array of sensors (scintillators, water Cerenkov tanks, muon detectors) spread over a large area. The detectors count the particle densities at any given moment, thus sampling the shower particles hitting the ground. By means of the ground lateral distributions of the shower one can deduce the direction, the energy and possibly the identity of the primary CR. The Akeno Giant Air Shower Array (AGASA) is the largest array so far constructed to measure extensive air showers. This array comprise 111 scintillation detectors each of 2.2 m2 spread over a grid of 100 km2 with 1 km spacing.17 The array detectors are connected and controlled through a sophisticated optical fibre network. The array also contains a number of shielded scintillation detectors which provide information of the muon content of the showers. The second technique, pioneered by the University of Utah’s Fly’s Eye detector,18 consists in studying the longitudinal development of extensive air showers by detecting the fluorescence light produced by the interactions of the charged secondaries. Measurement of atmospheric fluorescence is possible only on clear, dark nights. Since mid-1998 a successor to the Fly’s Eye instrument, called Hi-Res, has started taken data at the Dugway site!g In its final form it is expected to have a time-average aperture of 340 km2 sr at 1019 eV and 1000 km2 at 10’’ eV. The Auger experiment has been conceived to measure the properties of cosmic rays above lo1’ eV with unprecedented statistical precision?’ The Observatory will consist of two sites (in the Northern and Southern hemispheres) designed to work in a hybrid detection mode, each covering an area of 3000 km2 with 1600 particle detectors overviewed by 4 fluorescence detectors. Surface array stations are water Cerenkov detectors (a cylindrical tank of 10 m2 top surface and 1.2 m height, filled with filtered water and lined with a highly reflective material, the Cerenkov radiation is detected by 3 photomultiplier-tubes installed at the top) spaced 1.5 km from each other in an hexagonal grid. These stations will operate on battery-backed solar power and will communicate with a central station by using wireless local area network radio links. Event timing will be provided through global positioning system (GPS) recivers. Of course, the hybrid reconstruction (see Fig. 3) will be extremely valuable for energy calibration, but such “golden” events are expected to be less than 10% of the total event rate.
216
Figure 3: Schematic representation of the operation of a hybrid air shower detector.
It is worth mentioning that an array of ground-based detectors is ideally suited for involving school students and their teachers in cosmic ray research. This has been proposed several times (for example, SCROD 21,22). 3
Shower Profile of Highest Energy Fly’s Eye Event
The Fly’s Eye observes an air shower as a nitrogen fluorescence light source which moves at the speed of light along the path of a high energy particle traversing the atmosphere!8 In other words, it directly detects the longitudinal development of the air cascade. The simulation of the shower evolution depends sensitively on the first few interactions, necessarily related to the quality of our “understandying” of hadronic collisions. The well known codes QGSJET 23 and SIBYLLZ4 represent two of the best simulation tools to model hadronic interactions at the highest energies. The underlying idea behind SIBYLL is that the increase in the cross section is driven b,y the production of mini jet^?^ The probability distribution for obtaining N jet pairs (with > p p , where is a sharp threshold on the transverse momentum below which hard interactions are neglected) in a collision at energy & is computed regarding elastic p p or pp scattering as a dikactive shadow scattering associated with inelastic processes. The algorithms are tuned to reproduce the central and fragmentation regions data up to P p collider energies, and with no further adjustments they are extrapolated several orders of magnitude. On the other hand, in QGSJET the theory is formulated entirely in terms of Pomeron exchanges. The basic idea is to replace the soft Pomeron by a so-called “semihard Pomeron”, which is defined to be an ordinary soft Pomeron with the middle piece replaced by a QCD parton ladder. Thus, minijets will emerge as a part of the “semihard Pomeron”, which is itself the controlling mechanism for the whole intera~tion?~ The different approaches used in both codes to model the underlying physics show clear differences in cross sections (see Fig. 4) and multiplicity predictions which increase with rising energy?5 As can be seen in Fig. 5, for proton-induced showers, the differences become washed out as the shower front gets closer to the ground. Nevertheless, the footprints of the first hadronic collisions are still present in the longitudinal
pF
217
z E
800
I
i 700
600
500
400
300
Figure 4: p a i r cross sections of SIBYLL (dashed line), QGSJET (dots), and AIRES (solid line) superimpossed on data obtained from collider experiments and cosmic ray experiments. 25
2
3
1.
Sea level
10
10
10
10
1
1
10
10
-2
2
3
10
2
3
Figure 5: Comparison of muon lateral distributions at different atmospheric depths. Ep = 10 EeV. 26
218
V Flys Eye doto A proton 0 gold 0 dust-grain
t
i
-0 G
-0
Depth ( g / cm’)
Figure 6: Longitudinal development of 300 EkV showers from different primary species together with the data of the highest energy event recorded by Fly’s Eye.
’’
development. In Fig. 6 we show the atmospheric shower profile of 300 EeV showers induced by a proton, a gold-nucleus and a dust g r a d obtained after Monte Carlo simulation with the program AIRES (version 2.2.1)?* SIBYLL was used to generate hadronic interactions above 200 GeV. The simulated results are superimposed over the experimental data of the highest energy Fly’s Eye event?O It is clearly seen that the gold-nucleus longitudinal development better reproduces the data than a proton or a dust grain. However, this is not the case when the hadronic collisions are modelled with QGSJET. As can be seen in Fig. 7, in this case a medium mass nucleus likely fits the Fly’s Eye data. All in all, the primary chemical composition remains hidden by the hadronic interaction model. In light of this, in the next section we examine in more detail the speculative case of superheavy nuclei. 4
Superheavy Nuclei
It has been generally thought that 56Fe is a significant end product of stellar evolution and higher mass nuclei are rare in the cosmic radiation. Strictly speaking, the atomic abundances of middleweight (60 5 A < 100) and heavy-weight ( A > 100) elements are approximately 3 and 5 orders of magnitude lower, respectively, than that of the iron gr0up.3~The synthesis of the stable super-heavy nuclides is classically ascribed to three different stellar mechanisms referred to as the s-, r-, and p-processes. The s-process results from the production of neutrons and their capture by pre-existing seed nuclei on time scales longer than most P-decay lifetimes. There is observational evidence that such a kind of process is presently at work in a variety of chemically peculiar Red Giants 33 and in special objects like FG Sagittae 34 or SN1987A?5 The abundance of well developed nuclides peaks at mass numbers A = 138 and A = 208. The neutron-rich (or r-nuclides) are synthesized when seed nuclei are subjected to a very intense neutron flux so that P-decays near the line of stability are far too slow to compete with the neutron capture. It has long been thought that appropriate r-process conditions could be found in the hot (7’2 10l°K) and dense ( p N 1O1O - 10l1 g/cm3) neutron-rich (neutronized) material located behind the b A very massive particle containing lo7 nucleon^?^
21 9
00
Depth (g/ cm')
Figure 7 Atmospheric cascade development of 300 EeV calcium-nucleus together with Fly's Eye data. 31
outgoing shock in a type I1 supernova event?6 Its abundance distribution peaks at A = 130 and A = 195. The neutron-deficient (or p-nuclides) are 100-1000 times less abundant than the corresponding more neutron rich isobars, while their distribution roughly parallels the s- and r-nuclides abundance curve. It is quite clear that these nuclides cannot be made by neutron capture processes. It is generally believed that they are produced from existing seed nuclei of the s- or r-type by addition of protons (radiative proton captures), or by removal of neutrons (neutron photodisintegration). The explosion of the H-rich envelopes of type I1 supernovae has long been held responsible for the synthesis of these nuclides? Putting all this together, starbursts appear as the natural sources able to produce relativistic super-heavy nuclei. These astrophysical environments are supposed to comprise a considerable population of 0 and Red Giant stars:7 and we believe the supernovae rate? is as high as 0.2-0.3 yr-'. It was recently put forward that within this type of galaxies, iron nuclei can be accelerated to extremely high energies if a two step process is invoked!' In a first stage, ions are diffusively accelerated up to a few PeV at single supernova shock waves in the nuclear region of the galaxy! Since the cosmic ray outflow is convection dominated, the typical residence time of the nuclei in the starburst results in t 1 x 10" s. Thus, the total path traveled is substantially shorter than the mean free path (which scales as AP2l3) of a super-heavy nucleus. Those which are able to escape from the central region without suffering catastrophic interactions could be eventually re-accelerated to superhigh energies at the terminal shocks of galactic superwinds generated by the starburst. The mechanism efficiently improves as the charge number 2 of the particle is increased. For this second step in the acceleration process, the photon field energy density drops to values of the order of the cosmic background radiation (we are now far from the starburst region). The dominant mechanism for energy losses in the bath of the universal cosmic radiation is the photodisintegration process. The disintegration rate R (in the system of reference where the microwave background radiation is at 2.73K) of an extremely high energy nucleus with Lorentz ) factor I?, propagating through an isotropic soft photon background n ( ~reads:'
-
where a stands for the total photon absortion cross section. Primed quantities refer to the
220 Table 1: Giant dipole resonance parameters e0 [MeV]
uo [mb]
ro [MeV]
13.15 13.90
255 365
2.9 4.0
Figure 8: Energy attenuation length of gold (solid) and iron (dashed) nuclei in the intergalactic medium
rest frame of the nucleus. The total photon absortion cross section is characterized by a broad maximum, designated as the giant resonance, located at an energy of 12-20 MeV depending on the nucleus under consideration. For the medium and heavy nuclei, A 2 50, the cross section can be well represented by a single, or in the case of the deformed nuclei, by the superposition of two Lorentzian curves of the form
In order to make some estimates] hereafter we refer our calculations to a gold nucleus (the resonance parameters are listed in table I)?' Figure 8 shows the energy attenuation length of a gold nucleus due to interactions with the infrared and microwave backgrounds?' For comparison, we also show the energy attenuation length of an iron nucleus. We can conclude that despite the fact the abundances of superheavy nuclei are around 4 orders of magnitude lower respective to the iron group, the volume for the potential sources of extremely high energy superheavy nuclei ( E z 300 EeV) increases substantially. 5
Conclusions
The energy spectrum of primary cosmic rays above lolo eV has several compelling features. Perhaps the most impressive is the apparent occurrence of events above the GZK cutoff. This issue has been studied using simulation techniques and leads to a plausible interpretation of the highest energy cosmic ray event as due to the atmospheric interaction of an energetic heavy ion. The only reliable way to contribute further to the understanding of this issue is by the collection of new and better data in future experiments.
Acknowledgments
221
I acknowledge crucial help from my friend and colleague Luis Anchordoqui; he provided me with all the figures and most of the text! I thank Bill Marciano and Sebastian White for organizing such an interesting and eclectic workshop and I am in awe of Antonino Zichichi for providing Science with such an amazing resource as the one at Erice. Finally, I am grateful to the US National Science Foundation for financial support. References 1. For comprehensive reviews the reader is referred to, J. Cronin, T. K. Gaisser and S. P. Swordy, Sci. Amer. 276 44 (1997); S. Yoshida, and H. Dai, J. Phys. G 24, 905 (1998); M. Nagano and A. A. Watson, Rev. Mod. Phys. 72,689 (2000). 2. J. Linsley, Phys. Rev. Lett. 10, 146 (1963); M. A. Lawrence, R. J. 0. Reid and A. A. Watson, J. Phys. G 17,773 (1991); M. Takeda et al., Phys. Rev. Lett. 81,1163 (1998); D. J. Bird et al., Phys. Rev. Lett. 71, 3491 (1993); E. Antonov et al., JETP Lett. 69, 650 (1999). 3. P. 0. Lagage and C. J. Cesarsky, Astron. Astrophys. 118,223 (1983); P. L. Biermann, Astron. Astrophys. 271,649 (1993). 4. N. Hayashida et al., Astropart. Phys. 10, 303 (1999). 5. K. Greisen , Phys Rev. Lett. 16, 748 (1966); G.T. Zatsepin and V.A. Kuz’min, Pis’ma Zh. Eksp. Teor. Fiz. 4, 114 (1966) [JETP Lett. 4,78 (1966)]. 6. L.A. Anchordoqui, M.T. Dova, L.N. Epele, and J.D. Swain Phys. Rev. D 55,7356 (1997). 7. J. L. Puget , F. W. Stecker and J. H. Bredekamp, Astrophys. J. 205,638 (1976). Updated in, L. A. Anchordoqui, M. T. Dova, L. N. Epele, and J. D. Swain, Phys. Rev. D 57, 7103 (1998); L. N. Epele and E. Roulet, JHEP 10,009 (1998); F. W. Stecker and M. H. Salamon, Astrophys. J. 512,521 (1999). 8. R. J. Protheroe and P. Johnson, Astropart. Phys. 4,253 (1996). 9. G. R. Farrar, T. Piran, Phys. Rev. Lett. 84, 3527 (2000); P.L. Biermann, E.J. Ahn, G. Medina Tanco, T. Stanev Nucl. Phys. B (Proc.Supp1) 87,417 (2000); L. A. Anchordoqui, H. Goldberg, T. J. Weiler, Phys. Rev. Lett. 87, 081101 (2001); L. Anchordoqui, H. Goldberg, S. Reucroft, and J. Swain, Phys. Rev. D 64,123004 (2001). 10. E. Waxman, K. B. Fisher and T. Piran, Astrophys. J. 483, 1 (1997); M. Hillas, Nature 395, 15 (1998). 11. P. Bhattacharjee and G. Sigl, Phys. Rep. 327 109 (2000). 12. M. Ave, J. A. Hinton, R. A. Vazquez and A. A. Watson, Phys. Rev. Lett. 85, 2244 (2000). 13. G. R. Farrar,Phys. Rev. Lett. 76,4111 (1996); D. J. H. Chung, G. R. Farrar and E. W. Kolb, Phys. Rev. D57, 4696 (1998). 14. T. J. Weiler, Phys. Rev. Lett. 49,234 (1982); Astrophys. J 285,495 (1984); Astropart. Phys. 11, 303 (1999); D.Fargion, B. Mele and A. Salis, Astrophys. J. 517,725 (1999). 15. S. Nussinov and R. Shrock, Phys. Rev. D 59,105002 (1999); G. Domokos and S. KovesiDomokos, Phys. Rev. Lett. 82, 1366 (1999); P. Jain, D. W. McKay, S. Panda, and J. P. Ralston, Phys. Lett. B 484,267 (2000); C. Tyler, A. Olinto and G. Sigl, Phys. Rev. D 63,055001 (2001); L. Anchordoqui, H. Goldberg, T. McCauley, T. Paul, S. Reucroft and J. Swain [hepph/0011097]; A. Jain, P. Jain, D. W. McKay and J. P. Ralston [hepph/0011310]. 16 G. R. Farrar, P. L. Biermann, Phys. Rev. Lett. 81,3579 (1998); G. Sigl et al., Phys. Rev. D 63,081302 (2001); A. Virmani, S. Bhattacharya, P. Jain, S. Razzaque, J. P. Ralston, and D. W. McKay, [astro-ph/0010235].
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17. N. Chiba et al., Nucl. Instrum. Meth. A 311, 338 (1992); H. Ohoka, S. Yoshida and M. Takeda [AGASA Collaboration], Nucl. Instrum. Meth. A 385,268 (1997). 18. R. M. Baltrusaitis et al., Nucl. Inst. Methods Phys. Res., Sect. A 240,410 (1985). 19. P. Sokolsky, AIP Conference Proceedings 433,65 (1998). 20. http://uvu.auger.org/admin/ 21. http://hepnt.physics.neu.edu/scrod/index.html 22. L. A. Anchordoqui, J. Cook, M. Gabour, N. Kirsch, 3. MacLeod, T . P. McCauley, Y. Musienko, T. C. Paul, S. Reucroft, J. D. Swain, R. Terry, [hep-ex/0106002]. 23. N. N. Kalmykov, S. S. Ostapchenko, A. I. Pavlov, Nucl. Phys. B (Proc. Supp.) B52, 17 (1997). 24. R. S. Fletcher, T. K. Gaisser, P. Lipari and T. Stanev, Phys. Rev. D 50, 5710 (1994). 25. L. A. Anchordoqui, M. T. Dova, L. N. Epele and S. J. Sciutto, Phys. Rev. D 59,094003 (1999). 26. L. A. Anchordoqui, [astreph/9812445]. 27. L. A. Anchordoqui, Phys. Rev. D 61,087302 (2000). 28. S. J. Sciutto, in Proc. X X V I International Cosmic Ray Conference, (Edts. D. Kieda, M. Salamon, and B. Dingus, Salt Lake City, Utah, 1999) vol.1, p.411. 29. L. A. Aslchordoqui, M. T. Dova, T. P. McCauley, T. Paul, S. Reucroft, and J. D. Swain, Nucl. Phys. B (Proc. Suppl.) 97,203 (2001). 30. D. J. Bird et al., Astrophys. J. 441, 144 (1995). 31. L. A. Anchordoqui, M. Kirasirova, T. P. McCauley, S. Reucroft, J. D. Swain, Phys. Lett. B 492, 237 (2000). 32. E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. Hoyle, Rev. Mod. Phys. 29,547 (1957). 33. V. V. Smith, AIP Conference Proceedings 183,200 (1989). 34. G. E. Langer, R. P. Kraft, K. S. Anderson, Astrophys. J. 189,509 (1974). 35. P. A. Mazzali, L. B. Lucy, K. Butler, Astron. Astrophys. 258,399 (1992). 36. M. D. Delano and A. G. W. Cameron, Astrophys. Space. Sci. 10, 203 (1971); W. Hillebrandt, K. Takahashi and T. Kodama, Astron. Astrophys. 52,63 (1976). 37. S. Sakai and B. Madore, [astro-ph/9906484]. 38. See for instance, T. W. B. Muxlow et al., Mont. Not. Roc. Astron. SOC.266,455 (1994); J. S. Ulvestad and R. J. Antonucci, Astrophys. J. 448,621 (1997); D. A. Forbes et al., Astrophys. J. 406,L11 (1993); R. de Grijs et al., [astro-ph/9909044]. 39. L. A. Anchordoqui, G. E. Romero, and J. A. Combi, Phys. Rev. D 60,10300 (1999). 40. F. W. Stecker, Phys. Rev. D 180, 1264 (1969). 41. E. G. Fuller and M. S. Weiss, Phys. Rev. 112,560 (1958). 42. L. A. Anchordoqui, M. T. Dova, T. P. McCauley, S. Reucroft, and J. D. Swain, Phys. Lett. B 482,343 (2000).
Two-photon final states in peripheral heavy ion collisions C . G. Roldlo Instituto de Fisica Tedn'ca, Universidade Estadual Paulista, SCo Paulo, Brazil We discuss processes leading to two photon final states in peripheral heavy ion collisions at RNIC. Due to the large photon luminosity we show that the continuum subprocess yy --t ~y can be observed with a large number of events. We study this reaction when it is intermediated by a resonance made of quarks or gluons and discuss its interplay with the continuum process, verifying that in several cases the resonant process ovewhelms the continuum one. It is also investigated the possibility of observing a scalar resonance (the u meson) in this process. Assuming for the u the mass and total decay width values recently reported by the E791 Collaboration we show that RHIC may detect this particle in its two photon decay mode if its partial photonic decay width is of the order of the ones discussed in the literature.
1 Introduction
In the Relativistic Heavy-Ion Collider (RHIC) operating at the Brookhaven Laboratory beams of heavy-ions are colliding with the main interest in the search of a quark-gluon plasma in central nuclear reactions. In addition to this important feature of heavy-ion colliders, peripheral collisions may give rise to a huge luminosity of photons opening the possibilities of studying two-photon and other interactions as discussed by several authors In order to study peripheral y - y heavy-ion collisions we have to pay attention to two important facts, which are fundamental to determine the kind of physics that is truly accessible in these interactions. The first one is to avoid events where hadronic particle production overshadows the y - 7 interaction, i.e., events where the nuclei physically collide (with impact parameter b < ~ R ARA , being the nuclear radius) are excluded from calculations of the usable luminosit?. The second point to remember is that the photons will carry only a small fraction of the ion momentum, favoring low mass final states. Therefore, hadronic resonances with masses up to a few GeV can be produced at large rates 3,4, as well as any two-photon process not leading to extremely heavy final states. One of the purposes of our work was motivated by the possibility of measurement of the continuum process 77 -+ yy3, as well as the resonant one yy + R -+ yy which was predicted in Ref. as a clear signal for resonances made of quarks or gluons. The continuum 77 -+ yy reaction is interesting per se but is also a background for the resonant process. This last one is quite important to be observed because it involves only the electromagnetic coupling of the resonance. Its knowledge with high precision is very useful, for instance, to unravel the possible amount of mixing in some glueball candidates 5 , complementing the information obtained through the observation of hadronic decays. Another interesting study is the possible production of a light scalar meson (c).
'.
223
224
2
2.1
The reaction y
+y + y +y
The continuum reaction
We compute the yy -+ yy subprocess, for the Au ion at RHIC energies, f i = 200 GeV/nucleon. The photon distribution in the nucleus can be described using the equivalent-photon or WeizsackerWilliams approximation in the impact parameter space. We use the photon distribution obtained by Cahn and Jackson 6, wich takes into account the condition for realistic peripheral collisions (for details see Ref!). The subprocess yy -+ yy up to energies of a few GeV is dominated by the continuous fermion box diagram, and is a background for the resonant yy -+ R 3 77 process. It was first calculated exactly by Karplus and Neumans and De Tollisg. The helicity amplitudes are discussed in Ref. The electron gives the major contribution to the continuous 77 + yy subprocess. The muon and u contributions are smaller roughly by one order of magnitude, and the d and s quark contributions are even smaller. The process is proportional to (q;)4 where q f is their charge, and this is the main reason for the suppression of the quarks contribution7. The cross section is strongly restricted when we impose a cut in the scattering angle equal to 1 cosO( = 0.57. This cut is conservative, but it will make possible t o compare the cross section of the box diagram with rival processes, that will be discussed in the following sections, as well as it is enough to eliminate the effect of double bremsstrahlung (which dominates the region of I cos 01 KZ 1). Finally, this kind of cut is totally consistent with the requirements proposed in Ref. lo. Just to give one idea of the number of events, with a luminosity of 2.0 x 1026cm-2s-’ lo and integrating in a bin of energy 700 f 100 MeV (which of any strong resonance decaying into two-photons), we have 1532 events/year assuming 100% efficiency in tagging of the ions and photon detection ’.
’.
2.2 Resonances decaying into two-photons Photon pair production via the box diagram is a background to yy -+ R 3 yy process (or vice versa), both have the same initial and final states, and for this reason they can interfere one in another. Normally the interference between a resonance and a continuum process is unimportant, because on resonance the two are out of phase. The total cross section for the elementary subprocess yy + R 3 yy is described by a Breit-Wigner profile. We are going to discuss only J = 0 resonances made of quarks as well of gluons. In Fig.(l) we show the result for some resonance production (7,v’,7(1440), f0(1710)), whose invariant mass of the produced photon pair is between 500 MeV and 2000 MeV. For comparison we also show the curve of the continuum process. It is possible to see in that figure the well pronounced peaks of the resonances 7 and q‘. We assumed for their masses the values of 547.3 MeV and 957.78 MeV, respectively, the q total decay width is equal to 1.18 keV and the 7’one is equal to 0.203 MeV. Their partial decay width into photons are 0.46 keV (7) and 4.06 keV (7’).In the same figure we can see the predicted cross section for the glueball candidates production in peripheral heavy ion collisions by double photon fusion. We restrict the analysis to the J = 0 glueballs candidates q(1440) and fo(1710). For the = 56 ~ ( 1 4 4 0we ) used the mass and total decay width values of Ref. 11, mR = 1405 MeV, rtotUl MeV, for the decay width into photons we use the value given in Ref. 12, rrr = 5.4 keV. We see in Fig.(l) that the peak for this resonance is of the same order of the continuum process. For the other glueball candidate, f0(1710), the peak is clearly above the background. For this one we assumed the values listed in Ref. l1 of mass and total width, mR = 1715 MeV rtotul = 125 MeV, and for the two-photon decay width we adopted the value encountered by the ALEPH Collab. 13, rrr = 21.25 keV. In all these cases the resonances can be easily studied in peripheral heavy ion collisions.
225
(a)
rm
8m SmlmD
u iw
Figure 1: Invariant mass distribution of photon production (with the cut I cosBl < 0.5). The solid curve is for the box diagram, the dashed curves are due to the process yy --t R --f yy,where R are the pseudoscalars resonances q and q’ and the glueballs candidates q(1410) and fo(1710).
’
In Ref. we discuss other possible backgrounds for final states, for example, the photonuclear production of vector resonances which decay into three photons and one of the photons is missed by the detectors may fake the signal that we are discussing. We have respectively for q , q‘ and fo(1710) the following number of events above background 7 . 4 4 lo5, ~ 2.67~ lo4 and 42, assuming a luminosity of 2 . 0 ~ cmP2s-l and on year of operation.
2.3 Can we observe the u meson in peripheral collisions? Recently the E791 Collaboration at Fermilab found a strong experimental evidence for a light and broad scalar resonance, that is, the sigma, in the D+ + n-r+n+ decay 14. The resonant amplitudes present in this decay were analyzed using the relativistic Breit-Wigner. The values f 17 of mass and total decay width found by the collaboration with this procedure are 478’;; MeV and 3242;; f 21 MeV, respectively.
J uo
sm
“,“.“I
aa
1m
Un %nIW
Figure 2: Invariant mass distribution of photon pair production. The solid curve is due to box diagram only, the dashed one is due to the process -yy--t o + -yyin the Breit-Wigner approximation, the dash-dotted is the scalar contribution computed with the effective lagrangian, and the dotted one is due to the total process, computed with box amplitudes plus the scalar boson amplitudes. In all cases r7-,= 4.7 keV, and the angular cut is equal to -0.5 < case < 0.5.
We calculated the cross section for sigma production in the Breit-Wigner approximation for different values of r(u 4 yy), the results can be found in Ref. ’. These amplitudes will be computed with the help of an effective Lagrangian for the scalar interaction with photons, which is given by gsFpVFpV@,, where gs is the coupling of the photons to the scalar field and FP” is the electromagnetic field four-tensor. The helicity amplitudes are presented in Ref. In Fig.(2) we show the full cross section for the process yy + 77 due to the box diagram and the existence of the scalar resonance u with amplitudes given by effective Lagrangian computed and with rrr = 4.7 keV 15. Contrarily to the previous cases the u is a quite large LLresonance”
’.
226
due to its large decay width off resonance the interference is not negligible. In Fig.(2) we can see the continuous curve which is due only to the box diagram, the dashed one is the cross section of a scalar resonance computed with a Breit-Wigner profile, the dash-dotted curve is due only to the scalar effective lagrangian contribution and the dotted one is obtained when we consider the total cross section for two photon production in peripheral heavy ion collisions, i.e., the box diagram amplitudes plus the helicity amplitudes of the effective lagrangian. The figure shows clearly that the above mentioned interference is destructive. The effective Lagrangian model used to compute the u contribution to the photon pair production gives a larger cross section than the calculation with the Breit-Wigner approximation at energies above M M 600 MeV. It is dominated by the s channel contribution. We consider the Breit-Wigner result as the best signal representation for the resonant process because we are using the E791 data and this one was fitted by a Breit-Wigner profile. The effective Lagrangian gives a nonunitary amplitude that overestimates the sigma production above 600 MeV and shows the model dependence in the u analysis. The Breit-Wigner profile is not a bad approximation as long as we stay above the two pions threshold and in the following we assume that the signal is giving by it (the dashed curve of Fig.(2)) and the background is giving by the box diagram result (the solid curve of Fig.(2)). We also computed the possible background of double Pomeron exchange for this reaction and verified that it turned out to be very small. We changed the values of the u mass and total width around the central ones reported by the E791 Collaboration. We do not observed large variations in our result, but noticed that it is quite sensitive to variations of the partial decay width into photons. It is interesting to and characterizes the look at the values of the significance which is written as L u s i g n a l / d G statistical deviation of the number of the observed events from the predicted background. The significance as a function of the two photons decay width of the sigma meson, with mass equal to 478 MeV and total decay width of 324 MeV, is shown in Fig.(3), were we used a luminosity of L = 2.0 x loz6 cm-2s-1 at RHIC and assumed one year of operation. The significance is above 20 95% confidence level limit for two photon decay width greater than 4.7 keV, while for a 5u discovery criteria can be obtained with rrr > 7.5 keV.
'~~
a; I
._..." ....."
, I
1
J
1
5
6
c YY)
1
8
e
10
Figure 3: Significance as a function of decay width into two photons, rT7, for a sigma meson with mass equal to 478 MeV and total decay width of 324 MeV. The solid curve was obtained integrating the cross sections in the interval 438 < M < 519 MeV, the dashed one in the interval 300 < M < 800 MeV.
3
Conclusions
The continuum yy -+ yy subprocess has an interesting interplay with the one resulting from the exchange of a resonance. We discuss the resonance production and decay into a photon pair. This is a nice interaction to observe because it involves only the electromagnetic couplings of the resonance. Therefore, we may say that it is a clean signal of resonances made of quarks (or gluons) and its measurement is important because it complements the information obtained through the observation of purely hadronic decays. It may also unravel the possible amount of
227
mixing in some glueball candidates ’. We discuss the interference between these process and compute the number of events for some specific cases. The possibility of observing resonances that couple weakly to the photons is exemplified with the o meson case. This meson, whose existence has been for many years contradictory, gives a small signal in the reaction 7y + o + yy. However, its effects may be seen after one year of data acquisition, providing some clue about this elusive resonance. Using values of mass, total and partial widths currently assumed in the literature, we compute the full cross section within a specific model and discuss the significance of the events. Our work shows the importance of the complete simulation of the signal and background of these processes including an analysis of possible systematic errors, indicating that two photon final states in peripheral collisions can be observed and may provide a large amount of information about the electromagnetic coupling of hadrons.
Acknowledgments This work has been done in collaboration with A. A. Natale and J. P. V. Carneiro. I would like to thank I. Bediaga, C. Dib, R. Rosenfeld and A. Zimerman for many valuable discussions. This research was supported by FundacBo de Amparo 8. Pesquisa do Estado de SBo Paulo (FAPESP).
References 1. G. Baur, K. Hencken, D. Trautmann, S. Sadovsky and Y. Kharlov, hep-ph/0112211 Phys. Rep., i n press; C. A. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988); G. Baur, J. Phys. G24, 1657 (1998); S. Klein and E. Scannapieco, hepph/9706358 (LBNL-40457); J. Nystrand and S. Klein, hep-ex/9711021 (LBNL-42524); C. A. Bertulani, nucl-th/0011065, nucl-th/0104059. 2. G. Baur, in Proc. SBPF Intern. Workshop on Relativistic Aspects of Nuclear Physics, Rio de Janeiro, 1989, edited by T. Kodama et al. (World Scientific, Singapore, 1990). 3. G. Baur and C. A. Bertulani, Nucl. Phys. A505 835 (1989). 4. A. A. Natale, Phys. Lett. B362 177 (1995); Mod. Phys. Lett. A9 2075 (1994). 5. F. E. Close and A. Kirk, (Eur. Phys. J. C21 531 (2001). 6. R. N. Cahn and J. D. Jackson, Phys. Rev. D42, 3690 (1990). 7. A. A. Natale, C. G. RoldLo and J. P. V. Carneiro, Phys. Rev. C65 014902 (2002). 8. R. Karplus and M. Neuman, Phys. Rev. 83, 776 (1951). 9. B. De Tollis, Nuovo Cimento 32, 757 (1964); 35, 1182 (1965). 10. J. Nystrand and S. Klein (LBNL-41111) nucl-ex/9811007, in Proc. Workshop on Photon Interactions and the Photon Structure eds. G. Jarlskog and T. Sjostrand, Lund, Sweden, Sept., 1998. 11. D. E. Groom et al. (Particle Date Group), Eur. Phys. J. C15 1. (2000). 12. F. E. Close, G. R. Farrar and Z. Li, Phys. Rev. D55 5749 (1997). 13. ALEPH Collaboration, R. Barate et al., Phys. Lett. B472 189 (2000). 14. E791 Collaboration, E. M. Aitala et al., Phys. Rev. Lett. 86 770 (2001). 15. M. Boglione and M. R. Pennington, Eur. Phys. J. C9 11 (1999).
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Ultra-Peripheral Collisions in STAR P. Yepes for the STAR Collaboration Rice University
M S 315, 6100 Main Street, Houston, T X 77005 E-mail: [email protected] Ultra-peripheral heavy ion collisions involve long range electromagnetic interactions a t impact parameters larger than twice the nuclear radius, where no hadronic nucleon-nucleon collisions occur. The first observation of coherent po production with and without accompanying nuclear breakup, along with the observation of e+e- pair production are reported by the STAR collaboration.
1
Introduction
We refer to ultra-peripheral heavy ion collisions, as those nuclear interactions with a impact parameter, b, larger than twice the nuclear radius RA, where no hadronic nucleon-nucleon collisions occur The large charge of a relativistic heavy nucleus is a strong source of quasi-real can then be described by the vector photons. Exclusive po meson production AUAU.+AUAU~' meson dominance model 2: a photon emitted by one nucleus fluctuates to a virtual quark-anti quark pair; this intermediate state scatters diffractively from the other nucleus, emerging as a vector meson. The diagram for this process is shown in Fig. la. Here, the gold nuclei remain in their ground state. Additional photon exchange can yield nuclear excitation and the subsequent emission of single or multiple neutrons, as shown in Fig. l b for the process AuAu + Au*Au*p0. Photon and Pomeron can couple coherently to the spatially extended electric and nuclear charge of the gold nuclei. Requiring a coherent interaction of the whole nucleus imposes a maximum momentum transfer of h/d, where d is the nuclear dimension. In the transverse direction along which nuclei are not Lorentz contracted, d is of the order of the nucleus radius, and therefore p r < R/RA (- 100 MeV/c for RA 7fm). In the longitudinal direction nuclei are contracted by a factor y, therefore the maximum longitudinal momentum is y times larger: pll< h y / R ~ (w 6 GeV/c at RHIC).
'.
N
A"
-
..-
I
a)
Au
b)
Figure 1: Feynman diagrams for (a) po production, and (b) pa production with nuclear excitation. Factorization is indicated by the dashed line; nuclear excitation may occur by the exchange of a single or multiple photons.
229
230 The coupling strength of the photon is proportional to the square of the charge Z 2 (6241 for AuAu);the strength of the Pomeron coupling lies between A4/3 for surface coupling to A2 in the bulk limit (lo3 to lo4 for AuAu). For gold collisions at fi= 130 GeV those large couplings translate into a p production cross section of about 400 mb, or e 5 % of the total hadronic cross section. In the case of p production, it is not possible to determine which nucleus is the photon source and which is the target. Therefore the amplitudes for po production from both ions interfere. Since the po has negative parity, this interference is destructive. The short-lived p o decay before they travel the distance of the impact parameter b, and the interference is believed to be sensitive to the post-decay wave function6.
Figure 2: End and side views of a typical p candidate event in the STAR TPC. The candidate tracks are almost back-to-back radially, but boosted longitudinally.
The p meson decays into m~with an approximately 100% branching ration. However a m final state may be produced directly or through the p. The amplitudes for p production, A , and direct m r production, B, interfere an&
where the p width is corrected for the increasing phase space as rn,, increases. The p component undergoes a 180' phase shift at Mp, so the interference skews the p peak shape, enhancing production for M,, < M p and suppressing the spectrum for MTX> M p .
2
STAR Detector
m=
In the year 2000, RHIC collided gold nuclei at a center-of-mass energy of 130 GeV/nucleon. The STAR detector consists of a 4.2 m long cylindrical time projection chamber (TPC) of 2 m radius. In 2000 the TPC was operated in a 0.25 T solenoidal magnetic field. Particles are identified by their energy loss in the TPC. A central trigger barrel of scintillators surrounds the TPC, providing fast signaling for online event selection. Two zero degree calorimeters (ZDC) at f 18m from the interaction point are sensitive to the neutral remnants of nuclear breakup.
231
3 Analysis Production of the po meson in ultra-peripheral collisions has a clear experimental signature, especially when compared with the typical high multiplicity AuAu interaction. It is characterized by a 7riT+rpair in an otherwise 'empty' spectrometer. Fig. 1 shows a typical event candidate; the tracks are approximately back-to-back in the transverse plane due to the small p~ of the pair. Two data sets are used for the analysis: the 'minimum bias' and the 'topology' samples.
Figure 3: Reconstruction efficiency, including ac- ~i~~~~ 4: reconstruction efficiency, inCePtance, = a function of the rapidity, for the in- cluding acceptance, as a function of transverse moclusive pa production. mentum, before the cut in the opening angle is a p plied. Only pa mesons with 19) <1 were considered.
3.1
Minimum Bias Sample
The former includes 800k events triggered by requiring simultaneous signal in both ZDC's. The analysis of the 'minimum bias' sample, on which this paper focuses, is optimized to select the following reaction: AuAu + Au*Au*p'. Events are required to have two tracks of opposite charge coming from a common vertex. In addition the vertex should be within 15cm from the beam axis, and contained in the TPC in the longitudinal direction. Moreover the opening angle in the transverse plane is required to be larger than 2.7 radians. A sample of lOOK Monte Carlo AuAu -+ AuAup' events, generated with STARLIGHT, and run through the STAR simulation and reconstruction chain was utilized to estimate the reconstruction efficiency for po inclusive production. The efficiency, including geometrical acceptance, as a function of the po rapidity is depicted in Fig. 3. As can be seen the efficiency peaks at 70% at mid rapidity. The dip at rapidity zero is due to the poor vertex resolution for those events. This is due to the fact that it is not possible to determine a common vertex for two tracks perfectly back-to-back. Fig. 4 depicts the efficiency as a function of transverse momentum, before the cut on the angle in the transverse plane is applied. That cut affects the high end of the distribution, but has little effect in the overall acceptance, since p o p~ distribution peaks below 100 MeV/c. Fig. 5.a shows the uncorrected transverse momentum spectrum for the selected events, along with estimated background from hadronic processes and dielectron events. A clear peak, the signature for coherent coupling, can be observed at p~ < 100 MeV/c. Those events are compatible with coherently produced po candidates. The hadronic background is estimated from like-sign combination pairs, which is normalized to the signal for p~ > 200 MeV/c. As can be noted, the hadronic background does not have a peak at low p ~ Di-electron . background is estimated
232
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+Had. Backgrounc
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"0 0.05 0.1 0.15 0.2 0.25 0.3 Transverse Momentum (GeV)
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Figure 5: a) The uncorrected p~ spectrum for po candidates in the minimum bias sample, along with estimates for the hadronic and di-electron backgrounds. b) The p~ spectrum for po candidates after background subtraction and correcting for acceptance, reconstruction efficiency, and smearing effects.
by selecting electrons in the kinematic zone where the energy loss dE/dx in the TPC provides good separation between electron and pion as explained below. Fig. 5.b shows the corrected transverse momentum after background subtraction, and correction for acceptance, detection efficiency, and smearing effects. Fig. 6.a depicts the T ~ invariant T mass distribution for the events with a total p~ smaller than 150 MeV/c, along with estimates for hadronic and di-electron backgrounds. As can be observed, a clear peak is shown at the p o mass, while backgrounds are concentrated at low masses. The spectrum corrected for background, acceptance, reconstruction efficiency and smearing is shown in Fig. 6.b. The corrected spectrum is well fit by Eq. 1,with IBIAJsimilar to that found by the ZEUS collaboration for y p + PO$. 3.2 Topology Sample
For the analysis of the process AuAu + AUAU~'a data sample selected with a low-multiplicity topology trigger was utilized. In the level 0 trigger, the central trigger barrel was divided into 16 coarse pixels. These pixels cover half a unit in rapidity, and 90' in the transverse plane. Hits in opposite pixels were required, while pixels in the top and the bottom acted as a veto to supress cosmic rays. No requirement on the ZDC signals was imposed. A high level triggep further removed background. The STAR collaboration collected about 30k events in 7 hours, with this configuration. The po candidates from this data set have a transverse momentum
233
I -+-Cor. signal
_ _ l _ _ _ _ _ _ _
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#
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40
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0.40.50.60.70.80.9 1 1.1 1.21.31.4 Invariant Mass (GeV)
Figure 6: a) The uncorrected m,, spectrum of p o candidates with p~ < 150 MeV/c, along with estimates for hadronic and di-electron backgrounds. b) The corrected msn spectrum of po candidates with p~ < 150 MeV/c along with the results from a fit to equatiion(1).
and an invariant mass distributions similar to the ones already shown in Figures 5 and 6. Therefore, they are also characterized by a PT distribution peaked below 100 MeV/c, and and events, around 300, clustered around the po mass. In contrast to the minimum bias data, the topology triggered data had almost no energy deposition in the ZDC consistent with the two gold nuclei remaining in their ground state. 3.3 Electron-Positron pair production
Two-photon interactions include the purely electromagnetic process of electron-positron pair production as well as single and multiple meson production. The coupling Z a (0.6 for Au) is large, hence e+e- pair production is an important probe of quantum electrodynamics in strong fields At momenta below 140 MeV/c, efe- pairs are identified by their energy loss in the TPC as shown for the minimum bias data sample in Fig. 7a. Fig. 7b shows the p~ spectrum for identified efe- pairs; a clear peak at p~ < 50 MeV/c identifies the process AuAu + Au*Au*efe-.
',
4
Summary
In summary, for the first time, exclusive po production AUAU+AUAU~O and po production with and without nuclear breakup was observed in ultra-peripheral heavy ion collisions. The po are
234 I
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dEldx vs. P (130 G e V Au + Au) L
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+ A")
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Figure 7: (a) Energy loss d E / d x of tracks in the 2-track, minimum bias data; triangles indicate events where both particles are identified as electrons. (b) The PT spectrum for identified e'e-pairs.
produced at small perpendicular momentum, showing their coherent coupling to both nuclei. In addition, the coherent electromagnetic process AuAu -+ Au*Au*e+e- was observed. 1. G. Baur, K. Hencken and D. Trautmann, J. Phys. G24, 1657 (1998); C. A. Bertulani and G. Baur, Phys. Rep. 163,299 (1988). 2. J.J. Sakurai, Ann. Phys. 11 (1960) 1, and Phys. Rev. Lett. 22 981 (1969), T.H. Bauer et al., Rev. Mod. Phys. 50 261 (1978). 3. S. Klein and J. Nystrand, Phys. Rev. C60,014903 (1999). 4. P. Soding, Phys. Lett. 19,702 (1966). 5. S. Klein and J. Nystrand, private communication. 6. S. Klein and J. Nystrand, Phys. Rev. Lett. 84,2330 (2000). 7. J. Breitweg et al., Eur. Phys. J C2, 247 (1998). 8. J. S. Lange et al., Nucl. Instrum. Meth. A453,397 (2000).
HOT TOPICS IN ULTRA-PERIPHERAL ION COLLISIONS G. BAUR”, C. A. BERTULANIb, M. CHIU‘, I. F. GINZBURGd, K. HENCKEN‘, S. R. KLEINf, J. NYSTRAND~,K.PIOTRZKOWSKI~c . G. ROLDAO~D. SILVERMYR~,J. H. THOMAS!, S. N. WHITEj and P. YEPESk a Forschungzentrum Julich, Julich, Germany Natl. Supercond?LctingCyclotron Lab., Michigan State University, East Lansing, MI, 48824 USA Columbia University, New York, NY, 10027 USA Sobolev Institute of Mathematics, SB RAN, 630090, Novosibirsk, Russia Uniuersit6t Basel, 4056 Basel, Switzerland f Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA 9 Department of Physics, Lund University, Lund SE-22100, Sweden Departement de Physique, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Instituto de Fisica Teorica, Uniuersidade Estadual Paulista, Sao Paolo, Brazil j Brookhaven National Laboratory, Upton, NY, 11973, USA Physics and Astronomy Dept., Rice University, Houston, TX, 77005, USA Ultra-peripheral collisions of relativistic heavy ions involve long-ranged electromagnetic interactions at impact parameters too large for hadronic interactions to occur. The nuclear charges are large; with the coherent enhancement, the cross sections are also large. Many types of photonuclear and purely electromagnetic interactions are possible. We present here an introduction to ultra-peripheral collisions, and present four of the most compelling physics topics. This note developed from a discussion at a workshop on “Electromagnetic Probes of Fundamental Physics,” in Erice, Italy, Oct. 16-21, 2001.
1 What are Ultra-Peripheral Collisions?
Ultra-peripheral collisions are interactions that occur at impact parameters, b large enough , RA is the nuclear that no hadronic interactions can occur. In simple terms, b > ~ R Awhere radius. Only electromagnetic interactions are possible; they can be purely electromagnetic (‘twophoton’) or photonuclear(yA). Ultra-peripheral collisions of heavy ions are interesting because the large ion charge (2) gives rise to extremely intense short-duration electromagnetic fields. Following the WeizsackerWilliams method, these fields are usually described as a spectrum of almost real (quasireal) photons. When the photon wavelength is larger than the nucleus, the emission is coherent and the flux is proportional to Z 2 . For relativistic particles with Lorentz boosts y (to the lab . the target nucleus rest frame, frame), this occurs for lab-frame photon energies k < y f i c / R ~ In , 500 GeV at photons from the other nucleus have a maximum energy of (2y2 - l ) h c / R ~about the Relativistic Heavy Ion Collider (RHIC) at BNL, and 1 PeV at the Large Hadron Collider (LHC) at CERN. RHIC and LHC are high-luminosity r A colliders; LHC has an energy reach far beyond other existing or planned machines. Figure 1shows the ratio of the yA to A A luminosity for heavy ions at RHIC and the LHC; yp collisions may also be of interest. Also shown, for comparison, is the ratio for the proposed eRHIC electron-ion collider.
235
236
Figure 1: The relative 7 A : A Aluminosity at RHIC and the LHC, compared with that expected at the proposed eRHIC. The eRHIC curve has been multiplied by 6000, the ratio of the planned eRHIC luminosity t o the RHIC design luminosity. The photon energy w is that in the target nucleus rest frame.
The maximum yy collision energy W,, is 2hCy/R~,about 6 GeV at RHIC and 200 GeV at the LHC. Figure 2 compares the yy flux at the LHC with the completed LEP I1 e3-e- collider at CERN. The LHC will have a significant energy and luminosity reach beyond LEPII, and could be a bridge to yy collisions at a future e+e- linear collider. These cross section enhancements extend the physics reach of heavy ion colliders. For example, the coherent p production rate at RHIC is 120s-1 with gold, rising to 230,000s-' with calcium at the LHC. The LHC is a vector meson factory. The cross section for efe- production reaches 200 kb for lead at the LHC. Many processes are unique to heavy ion colliders, either because of the high-2 beam particles or because of interference between yy and coherent yA processes. Ultra-peripheral processes have striking signatures. Two-photon and photon-Pomeron processes lead to final states with a small number of centrally produced particles, with rapidity gaps (regions of phase space containing no particles) separating the central final state from both beams. In addition, neither nucleus should be broken up. For coherent photon emission, the photon perpendicular momentum p~ is less than l i / R ~ .If the coupling to both nuclei is coherent, the transverse momentum p~ of the final state is less than 2 h / R ~= 100 MeV/c. Even incoherent interactions (with respect to one of the nuclei) such as photoproduction of heavy quarks will be characterized by a single rapidity gap and intact nucleus. We present four compelling physics topics for ultra-peripheral collisions: gluon shadowing in nuclei, Pomeron coupling to nuclei, interferometry with short-lived particles and searches for new physics, most notably by studying the yWW vertex. Numerous other interesting subjects, many unique to heavy ion collisions, are not included, either because they require specialized apparatus to study, or because the physics is not yet clearly defined. For example, electromagnetic production of e+e- pairs tests QED in the non-perturbative, strong-field regime created by the high-2 sources, but requires a dedicated experiment. Production of lepton or hadron pairs accompanied by capture of the negatively charged particle also fall into this category. Interference between identical final states (like r+7rT- or e+e-) produced via yy and yA channels is sensitive to the relative phases of many processes. This interference is very interesting, but we don't know exactly what to measure yet. Additional details and other peripheral collisions physics are presented in several recent reviews. 2
What is the most interesting physics?
These four topics presented are highlighted for their strong physics interest and their feasibility with existing and planned detectors (possibly with slight modifications).
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2.1
Gluon Shadowing in Nuclei
The parton distributions (structure functions) of heavy nuclei is of interest because the low-3: gluon density is high, and, in fact, gluon saturation should be present. These strong fields could be a colored glass condensate, with new and unexpected properties. Even in the absence of a condensate, accurate measurements of the parton distributions are important in understanding the initial states of relativistic heavy ion collisions. To date, most measurements of parton shadowing in nuclei have relied on lepton deep inelastic scattering (DIS). DIS is only sensitive to the quark content of the nuclei; the gluon content is inferred from the Q2 evolution of the structure functions. Furthermore, the only data on nuclear structure functions is from fixed target experiments, limited to fractional momenta 3: > lop3 and low momentum transfer. The gluon distributions can be measured at heavy ion colliders by studying photoproduction of heavy (charm and bottom; 4,5 top is also observable at the LHC 6, quarks; this reaction occurs primarily by photon-gluon fusion. RHIC will be able to extend the previous measurements to higher Q 2 , while the LHC will be able to reach much smaller 3: and higher Q 2 . The LHC measurements will probe regions where many models predict that gluon saturation will be observed. For example, in a colored glass model, photoproduction of charm and bottom at the LHC is much less than is expected in conventional parton distributions. In photoproduction, the photons are almost real, but the high mass and p~ of the produced heavy quarks introduces a high Q2 scale. These events can be separated from heavy quark production in grazing hadronic interactions by requiring one rapidity gap in the event, along with one undisturbed nucleus, with a charged particle multiplicity cut to eliminate more central hadronic collisions. The heavy quark production cross sections are sensitive to the heavy quark mass and to higher-order corrections. Despite this, RHIC and the LHC can offer a ‘first look’ at as yet unexplored regions of z and Q 2 . More importantly, by comparing the heavy quark production in AA (effectively YA) and p A (effectively y p ) collisions, the shadowing of nuclear particle distributions can be measured. In the ratio, the QCD uncertainties cancel, and the achievable accuracy should be sufficient for a meaningful measurement of shadowing. It is difficult to obtain these structure functions from other methods. The leading alternative, studies of jets, direct photons and the like in p A collisions at RHIC and the LHC suffers from large systematic uncertainties.
2.2 Pomeron Couplings to Nuclei Many hadronic interactions do not involve color exchange. Examples include proton-proton elastic scattering, photoproduction of vector mesons, and other diffractive phenomena, including jet and W* production. These processes are generally described in terms of Pomeron exchange. The Pomeron has a long and complex history. It is generally agreed that it involves the strong force, but is colorless, and that it has the quantum numbers of the vacuum (Jpc = 0++). Beyond that, there is a great diversity of views as to its structure. There are two classes of Pomeron models corresponding to hard and soft interactions. The soft Pomeron represents the absorptive part of the scattering cross section. Often, this is dressed up in Regge theory. It describes much data, including elastic scattering and photoproduction of vector mesons, but lacks internal detail, and a connection to QCD. In photoproduction of vector mesons, a photon fluctuates to a qij pair. This pair then scatters elastically (via Pomeron exchange) from the nucleus, emerging as a vector meson. The hard Pomeron model usually applies for higher momentum transfers, where QCD is expected to be reasonably perturbative. The hard Pomeron is usually modeled as two gluons, with additional structure, such as a gluon ladder.
238 Id= Id' Id0
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Figure 2: The effective yy luminosity as a function of the yy energy W,, at the LHC, compared with that at LEP 11. The A A and e+e- luminosities are ATAT: 5.2 x 1 0 2 9 n - 2 s - 1 , PbPb: 4.2 x 1026m-2s-', pp: 1 0 3 3 n - 2 s - ' (corresponding to about 1 collision/ beam crossing; the LHC p p design luminosity is 10 times higher), and e+e-: 5 x 1 0 3 ' n - 2 s - ' at an e+e- center of mass energy of 200 GeV.
One test of these models is their coupling to nuclei. The soft Pomeron couples equally to all nucleons. For heavy states like J / $ and Y,the cross section for coherent photonuclear production of vector mesons scales as A2. For lighter mesons, a single qii state can interact with multiple Pomerons; this can be calculated accurately in the Glauber model. In the 'black-disk' limit (very large q?j states), the cross section scales as A4/3. However, if the Pomeron is made of gluons, the situation changes. The gluon distributions of nucleons change when they are placed in nuclei (the EMC effect/shadowing). If Pomerons are made of gluons, and if the gluon content of nuclei is different from that in independent nucleons, then the Pomeron coupling should change. These couplings can be studied systematically by comparing production rates of different mesons at different p~ from different targets. For p~ < TL/R*, the mesons couple coherently to the entire nucleus, while at higher p ~ the , coupling is to individual nucleons. The two types of interactions have different properties, and will provide useful cross-checks. Measurements at LHC will extend the photoproduction cross section measurements to energies significantly above the HERA regime. HERA data for the J / + finds u = W0,'. This rapid increase with yp center-of-mass collision energy W cannot continue indefinitely or the J / + cross section will exceed the total photoproduction cross section. A similar rise is seen for the Y,albeit with lower statistics. lo The y p cross sections can be measured with pA collisions; LHC can reach an order of magnitude beyond HERA in W . Heavy meson production in AA (yA) collisions is of interest because of its sensitivity to gluon saturation. l1 The LHC can study nuclear vector meson photoproduction at previously unavailable energies. The cross section is a probe of the gluon density in very low-2, high-A region where many models predict that saturation should be observed. The meson mass and p~ variation will cover the ranges usually described by the hard and soft Pomerons.
'
2.3 Interferometry with Short-Lived Particles For coherent yA processes, the two nuclei act as an interferometer, with two nuclei serving as sources. Because the electromagnetic field has a long range, and the hadronic scattering a short range, vector meson production is well localized at the two sources, especially compared with the typical impact parameters of 20-60 fm at RHIC, rising to 20-250 fm at the LHC. The sign of the interference depends on the final state parity. Because most of the vector mesons have lifetimes far shorter than the time needed to travel
239
Figure 3: The PT spectrum of twetrack events observed by the STAR detector. The large peak for p~ < 100 MeV/c is expected for coherent production of p mesons.
between the two nuclei, amplitudes from the two sources must decay independently. This seems to preclude interference; consider, for example, one source decaying J / $ -+ efe- and the other J/$J -+ r+r-7ro. Yet, quantum mechanics requires that they interfere. The resolution to this paradox is that the post-decay wave functions must contain amplitudes for all possible decay modes: branching ratios, angular distributions, and even decay times. Because this requires a spatially distributed (non-factorizable) wave function, the system is an example of the Einstein-Podolsky-Rosen paradox. It is a test of quantum mechanics for unstable particles. The interference is destructive for p ' ~. < fi, so the vector meson production is reduced for PT < A / ( b ) . As the data from the STAR detector in Fig. 3 shows, in this region, a clean sample of mesons with p~ < f i / R can ~ be selected with simple cuts. l3
''
Multiple vector meson production is also observable in the strong fields. The probability of p production in grazing collisions p is about 1%with gold at RHIC, rising to 3% with lead at the LHC. l4 If each meson is treated independently, then the probability of producing n p at RHIC and LHC are P ( n ) = exp ( - p ) p n / n ! , or, for n = 2, (1%)'/2 and (3%)'/2, for n = 3, (1%)3/6 and (3%)3/6. The total rates are obtained by integrating P(b) over impact parameter; the doublet rates are large (106/year for popo at both RHIC and the LHC), while the triplets are still observable. These samples can be used to study quantum correlations; the final state is considerably richer than for single mesons. The two source ions can each emit single or multiple photons. Prcduction correlations should produce Hanbury-Brown Twiss (HBT)-like enhancements at small relative momentum. However, because the vector mesons can be produced literally on top of each other, many new phenomena may be observable. For example, the vector mesons could interact with each other, leading to new final states such as p+p-. Finally, since for rclative mocoherence will be maintained until the mesons decay, correlated (stimulated) menta, Sp < ti/., decays will be possible. Vector meson lasing occurs for p 2 1, depending on the source geometry. Unfortunately, p = 1 seems unreachable. However, even superradiant behavior should be visible even at the expected p = 0.01 - 0.03. As the previous paragraph shows, this leads to very interesting phenomenology. The strong field sources and unique geometry (two identical meson sources) produce a new configuration - an interferometer for short-lived particles. The system is new, and much theoretical and experimental work is required to exploit it. Experimentally, clean samples of coherent vector meson production have been extracted, and initial measurements of interference are not far off.
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2.4
Searches for New Physics
Two-photon physics at the LHC allows for a wide range of searches for new physics; as Fig. 2 shows, the LHC will have an energy/luminosity reach considerably beyond LEP2. The triple gauge coupling yWW can be studied using the copious two-photon W pair production. This is believed to be the most sensitive process to the new physics of large extra dimensions. l5 Twophoton production of 2 pairs or the Higgs boson is less copious, but could complement more conventional studies at the LHC, and test for deviations from the Standard Model predictions. In addition, two photon production rates of the supersymmetric charginos and sleptons are significant. l6 Because of the high center of mass energies, luminosity and running time, searches for new physics with yy interactions will be most competitive with p p mode at the LHC (see Fig. 2). The hadronic backgrounds can be reduced by requiring that there be rapidity gaps in the event and tagging the forward scattered protons. l7 The LHC magnets will act as magnetic spectrometers for the scattered protons, allowing the reconstruction of the photon momenta. This will give an unbiased estimate of the photon-photon collision energy. It will also allow for significant background rejection. Tagging will also detect double-Pomeron interactions. Two-photon interactions can be separated from these double-Pomeron interactions because the Pomerons have a larger average transverse momentum than the photons. Depending on the final state under consideration, either single or double tagging may be used. The anticipated luminosity of the tagged two-photon collisions at W,, > 100 GeV reaches then almost 1% of the p p luminosity, even for rather conservative assumptions on the tagging efficiency. l6 Especially at RHIC, tagged double Pomeron interactions are of strong interest to study meson spectroscopy. For reasons that are poorly understood, when the transverse momentum transfer from the protons are in the same direction, conventional qq mesons are produced, while when the transverse momentum transfers are in opposite directions, many suspected exotic (non qq) mesons are preferentially produced. l8 Single tagging is also possible for p A collisions. The W,, reach is lower, but up W,, N 100 GeV the pA two-photon luminosity is very competitive with the p p case. This channel can also be used to cross-check two-photon studies using p p and AA. 3
Conclusions
We have presented four compelling examples of the physics accessible in ultra-peripheral collisions. They address a variety of new physics, including relatively straightforward, but important measurements of nuclear shadowing, studies of the nature of the Pomeron, and tests of quantum mechanics and searches for new physics. Many other hadronic measurements are possible with existing detectors. If experiments requiring new hardware were considered, many other processes would compete for spaces on the list. Many interesting atomic physics experiments are possible. For example, near-threshold single and multiple efe- pair production probes &ED in a nonperturbative regime. Pair production with e - capture is important because it limits the maximum luminosity achievable with heavy ions at the LHC.
References 1. G. Baur, K. Hencken, D. Trautmann, S. Sadovsky and Y. Kharlov, hep-ph/0112211, to appear in Phys. Rep.; G. Baur, K. Hencken and D. Trautmann, J. Phys. G24, 1657 (1998); F. Krauss, M. Greiner and G. Soff, Prog. Part. Nucl. Phys 39, 503 (1997); C. A.
24 1
Bertulani and G. Baur, Phys. Rep. 163, 299 (1988). 2. “The Electron-Ion Collider, white paper submitted to the NSAC long range planning meeting, March, 2001. 3. L. McLerran, hep-ph/0104285. 4. M. Greiner, M. Vidovic, Ch. Hofmann, A. Schafer and G. Soff, Phys. Rev. C51, 911 (1995). 5. N. Baron and G. Baur, Phys. Rev. C48, 1999 (1993). 6. S. Klein, 3. Nystrand and R. Vogt, Eur. Phys. J. C21, 563 (2001). 7. F. Gelis and A. Peshier, hep-ph/0107142. 8. H. Abramowicz and A. Caldwell, Rev. Mod. Phys. 71, 1275 (1999); N. Cartigilia, hepph/9703245. 9. J. Breitweg et al., Zeit. Phys. 75, 215 (1997). 10. J. Breitweg et al., Phys. Lett. B437, 432 (1998). 11. L. Frankfurt, M. Strikman and M. Zhalov, hep-ph/0111221. 12. S. Klein and J. Nystrand, Phys. Rev. Lett. 84, 2330 (2000). 13. S. Klein, nucl-ex/0104016. 14. S. Klein and J. Nystrand, Phys. Rev. C60, 014903 (1999). 15. T. Rizzo, Phys. Rev. D60, 115010 (1999). 16. K. Piotrzkowski, these proceedings, hep-ex/0201027. 17. K. Piotrzkowski, Phys. Rev. D63, 071502 (2001). 18. CERN experiments WA-76, WA-91 and and WA-102 have many papers showing how double-Pomeron interactions are an excellent place to search for exotic mesons. See D. Barberis et al. Phys. Lett. B507, 14 (2001); Phys. Lett. B488, 225 (2000); Phys. Lett. B484, 198 (2000) and previous works.
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243 Appeared in Nuovo Cimento 2, pp. 143-158 (1925) (translated from Italian by Michele Gallinaro and Sebastian W h i t e , New York 2001)
Translators’ Note: In the fall of 1924, Enrico Fermi visited Paul Ehrenfest at Leyden on a 3-month fellowship from the International Education Board (IEB). Fermi was 23 years old. In his trip report to the IEB, Fermi says he learned a lot about cryogenics and worked on two scientific papers, including the following one. It was submitted in German to Zeitschrift fur Physik. The German version was known to Weizsacker and Williams and cited in the papers (10 years) later in which they extended Fermi’s method to the Ultra-Relativistic case. The German version was subsequently translated into a Russian version and perhaps other languages. Fermi’s Italian version (printed in Nuovo Cimento) is less widely known and does not appear in the “Collected Works”. Nevertheless, Persico remarks that this was one of Fermi’s favorite ideas and that he often used it in later life. So, we would like to think of this as a late looth birthday present to the Italian Navigator. We would like to thank Professor T.D. Lee for his encouragement of this project and for interesting discussions about Fermi. Also Tom Rosenblum at the Rockefeller Archives for bringing Fermi’s correspondence to our attention and Bonnie Sherwood for typing the original manuscript.
“On the Theory of Collisions between Atoms and Electrically Charged Particles” Note by: Enrico Fermi, 1924
I. When an atom, in its normal state, is illuminated with light of the appropriate frequency, it can be excited, which is to say that it goes to a quantum state of higher energy, absorbing a quantum of light. If the quantum of the exciting light is greater than the energy needed to ionize the atom, it can be ionized losing, depending on the light frequency, an electron of either the superficial layers or those deep within the atom. During the phase of relaxation of the atom there is emission of light and, depending on the circumstances, it is possible to obtain the phenomena of optical resonance or fluorescence. Some natural phenomena, very similar to these, occur also during excitation by collision. In fact, if the atoms in a gas are bombarded with electrons of high enough velocity, they can be excited or ionized. They can also lose, if the velocity of the bombarding electrons is very large, some electrons belonging to the innermost layers of the atom. The scope of the present work is to further specify the analogies (existing) between these two classes of phenomena, and to quantitatively derive the phenomena of excitation by collision from those of optical absorption. Therefore, we consider that when a charged particle passes near a point, it produces, at that point, a variable electric field. If we decompose this field, via a Fourier transform, into its harmonic components we find that it is equivalent to the electric field at the same point if it were struck by light with an appropriate continuous distribution of frequencies. If we now imagine that, at that point, there is an atom, the hypothesis occurs quite naturally, that the electric field of the particle produces on the atom those same effects of excitation or ionization that the equivalent light would produce. Let us suppose that we know the absorption coefficient for light by the atom, as a function of frequency. We would then have a way to calculate the probability that a charged particle, passing with a given velocity at a given distance from an atom, will ionize it. It seems, on the other hand absolutely necessary to introduce a limitation to this correspondence between the electric field of the light and that of a charged particle. We note, in fact, that a particle of velocity, v, cannot produce phenomena, by collision, which require an energy greater than its own kinetic energy. If, instead, we perform a harmonic decomposition of the electric field it produces, we see that it will include all possible frequencies, including the highest ones. Therefore we must accept that all frequencies whose quantum, hv, is greater than the particle’s kinetic energy can have no effect (since not enough energy is available to supply an entire quantum). We have applied our hypothesis to three phenomena that allow experimental verification. They are: a) Excitation of the line 2537 (a resonance line) of mercury. The optical absorption of this line is known with sufficient precision. Experiments also exist that give, at least to an order of magnitude, the probability that atoms of mercury bombarded with slow electrons are excited. Our theory gives for this probability the right order of magnitude. Naturally, given the limited precision of the experimental data, a precise test of the theory through these phenomena is not possible, but the case nonetheless is of some interest since no other theory of the excitation, by collision, of a quantum state exists, only theories of ionization by collision. b) The number of ion pairs produced per cm of path length by a-particles from Radium-C in Helium. We chose Helium because, being a monoatomic gas, and possessing only k-shell electrons, it’s possible to know with some precision its absorption coefficient as a function of
244
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frequency. The agreement between experiment and theory is very good. The previous theories of ionization by collision a are all based on a principle different from the present theory. The essential point of those theories can be summarized in the following way: when a charged particle passes within the vicinity of an atom it transfers to the electrons of the atom a fraction of the particle’s energy. The energy transfer is calculated assuming, typically, that the electrons are free within the atom. One then allows the electron to be detached from the atom every time that the energy transferred to it via this mechanism is greater than the ionization energy of the atom. c) The path length in Helium of a’s from Radium-C. The mechanism of this phenomenon is, according to our theory, the following: the a particle creates, along its path, a variable electric field whose energy is absorbed by the surrounding atoms, as if it were the electric field of a light wave. The energy, which is thus absorbed, has to obviously be subtracted from the kinetic energy of the atparticle which is thus decelerated. Also, in this case, the agreement between the theoretical and experimental values is excellent. Previously, the deceleration of a particles in matter had been studied theoretically by Thomson and by Bohr The former interpreted it as due to the fact that the a particle, attracting electrons of the surrounding atoms along its path, transfers to them a fraction of its kinetic energy. Since, on the other hand, when one calculates this energy as if the electrons were free, one finds an infinitely large deceleration, Bohr redid the calculation assuming that the electrons inside the atoms were no longer free, but bound by quasi-elastic forces. In this way he finds a good agreement with experiment, taking for the frequency of this binding, as the case may be, optical or x-ray (Rontgen) frequencies. In the second section of this work we establish some general formulas for the calculation of the “field of light” that produces an electric field equivalent to that of the particle. In the third, fourth and fifth sections we discuss the application to a), b) and c). 11. Let’s calculate, first of all, the spectral distributions corresponding to those of the electric field created by a particle with electric charge, E , passing with velocity, ‘u, at a minimum distance, b, from a point, P. The components of the electric force, at P , parallel and perpendicular to the motion of the particle, are
‘.
E.b E2 = ~ ~ 8’ ~ ) ; (b2 +‘u282)i
Eve
El = (b2
+
(1) ‘
Where 8 is the time-elapsed starting from the passage of the particle at the minimum distance from P. Taking now T to be a time that we will later extend to infinity, we can expand E l and E2 in a Fourier series for all values of 0 between -(1/2)T and +(1/2)T. We then find El =
c
2irn T
a, sin -8,
E2 =
c
2irn b, cos -8
T
where
The electric field, a, sin(2irnelT)is now equal to the electric field of a light wave with intensity (c/4x) . ( a i / 2 ) and frequency u = n/T. We now let I(v)du be the light intensity in a frequency interval du, multiplied by the duration, T , of its action. We find “J.J. Thomson - Phil. Mag. 23, p. 449, 1912. N. Bohr - Phil. Mag. 30, p. 581, 1915. S. Rosseland - Phil. Mag. 45, p. 65, 1923.
*N.Bohr - Phil. Mag. 25, p. 10; 1913, Phil. Mag. 30, p.581; 1915.
246
I
- c z2 27T
( u )du =
{ (7 b2
m
lim _cTaZ+ b i Tdu = T--+oo47~ 2
cos 27~uOdO~)
+
(b2
u2
+ w202)T
(7-
O sin 2xuOdO) (b2
+ u202):
’1
du.
The integrals that appear in this expression can be represented through the modified Bessel functions of the second kind of order zero and one. Explicitly we have
m
J
-m
x sin wxdx 3
= -2wK, (w)
(1+x2)3
With these formulas we immediately find
where we substitute for brevity
B ( w ) = KO’( w )
+ K,2 ( w )
and we then find
I(u) =
8 7 r c ~ ~ ’ u ~( 2 y b )
-
~
21
(5)
Since all the frequencies whose quantum is larger than the kinetic energy of the particle cannot, according to our hypothesis, have any effect, we find finally that the charged particle’s passage is equivalent to the following spectral distribution
I (v) =
81r‘$”2
B
( F ),
hu 5 $nu2 , hu > ;mu2
111. We now want to show, using a simple example, the application of the general method: therefore we will study the excitation by collision of spectral lines. For example, we consider an atom of mercury, and take a(.) to be its absorption coefficient at the frequency u. This means that if the atom is struck by light of intensity I(u)dv it absorbs, on average, the energy a(v)I(u)duand therefore the absorption probability of a quantum hu is a ( u ) I ( u )du (7) hu Since the absorption at the resonance line corresponds to the excitation of the atom to the 2p state, we fmd that (7) represents the excitation probability if only the frequencies in the interval ‘See, for example, Gray, Mathew and Macrobert - Bessel Functions - London, 1922, where also numerical tables of the values of these functions can be found. For the Fourier series expansions of these expressions, see also Bohr, 1.e.
247
d v would be effective. The probability II(b) for the excitation to be caused by the passage of the particle at a distance, b, is naturally smaller than the sum of all the probabilities obtained from the individual frequency intervals dv when one treats these as independent. One easily finds that
Since, in the case of the resonance, a ( v )has a value different from zero only in a very narrow band near the resonance frequency vg, we can write
where we use a=
1
a (v)dv.
Equation (8) can therefore be written, in this case as n ( b ) = 1- e - G 1 ( ” ) .
(8’)
To derive from this an observable directly accessible to experiment, we calculate the effective radius, p , of the atom for resonant excitation. That is to say the equivalent radius needed to obtain the same overall excitation probability if the excitation probability where unity within this radius. This radius is evidently given by xp2 = 27r
J
b dbII ( b ) .
We therefore find, keeping in minde Eq. (6),that for imv2 > hv
The integral can be evaluated approximately f and onc finds dEffectively the probability that the atom doesn’t absorb any quantum is
I ( u )a ( u ) du in which the product must be extended over all the infinitesimal intervals du Taking the logarithms of both sides we find in the limit
from which we obtain Eq. (8). “Effectively an electron, which passes with low velocity near an atom, will be strongly scattered by it. Since we are only after the order of magnitude, we’ll apply Eq. (6) anyway. ’To find an approximate expression for
248
Experimentally the excitation by collision of the resonance line 2537 of mercury was studied by Miss Sponerg. Unfortunately this work gives us only the order of magnitude of the excitation probability. In fact, Sponer reached the conclusion that in the collisions of electrons having velocity not much greater than 4.9 volts, with atoms of mercury, inelastic collisions amount to a few percent of all collisions. To calculate p using Eq. (9’) we take for v a velocity corresponding to a potential of 8 volts; the value of (Y can be obtained from a work of Fuchtbauer ‘. This author finds, in fact, that when a mercury atom is illuminated with an amount of light having the spectral distribution I(v), the absorption probability of a quantum of “resonance light” is PI(v0) where P = 8 x lo-’. Evidently cy = Phvo, and therefore cy = 6 x lop4. Equation (9’) gives us then p = 0.8 x This value is considerably larger than that found by Sponer. If, in fact, we take the fraction of inelastic collisions to be as large as 9% we still find, for p, the value 0.4 x lo-* which is about half as large as what we calculated. On the other hand, it is easy to understand the reason for this discrepancy. When, an electron of velocity equivalent to a few volts passes at a distance of the order of magnitude of 10-8cms from an electron, it is already strongly scattered in such a way that its distance of closest approach is considerably larger than if it had remained on its original trajectory. One then understands that the error made in neglecting the deflection leads to an overestimate of p. Since, on the other hand, the uncertainty in the measurements excludes anyway a precise test of the theory, and with the present hypotheses, we obtained the right order of magnitude for p, it seems superfluous to attempt a more precise calculation which certainly would be more complicated. Ionization phenomena by collision can be explained in a very similar way. It is known that all atoms beyond the limit of their principal series exhibit, both in absorption and emission, a continuous spectrum, corresponding to the transition of the valence electron to the state in
for a << 1, we observe that when x one can set
< 0.4 one has with
sufficient accuracy B ( z )= l/xz. For a
1 - e-a’B(=) = a B ( z )
< 1 and z > 0.4
.
The numerical calculation gives us for the second integral 0.973 . a. The first can be calculated easily using the asymptotic expressions for the logarithmic integral and one finds that it has the value a - - log - 0 . 7 0 5 ~ ~ . 2 We then obtain finally I ( a ) = (0.268- Z 1l o g a ) a .
gHertha Sponer, ZS. f. Phys. 7, p.185, 1921. hFiicthbauer - Phys. 2s. 21, pp. 322, 694, 1920.
249
which it is ionized and furthermore possesses some kinetic energy. The spectrum consists of a reasonably broad band. It has a sharp cutoff towards the red due to the limit of the principal lines and is blurred towards the violet. If, now, the velocity of the colliding electron is such that the frequency rnv2/2h lies beyond the absorption band, it is still possible, as a rough approximation, to apply Eq. (9') also for the calculation of the equivalent radius of the atom for ionization effects; if instead the limiting frequency falls within the absorption band the radius will naturally be smaller than found with Eq. (9') and will finally go to zero when the frequency mv2/2h coincides with or falls below the limit of the principal series. Qualitatively this behavior of p as a function of v is confirmed by the experiment Unfortunately it seems impossible to pursue it quantitatively because precise data on the intensity of the continuous absorption spectrum are not available. IV. Instead, it is considerably simpler to calculate the ionization produced by a particles. In fact, a-particles, because of their considerable mass, are hardly deflected and one can, to a good approximation, use Eq. (6). The empirical results on absorption in the region of Rontgen rays can be summarized with the following formulaj
'.
where a(.) represents the atomic absorption coefficient, D is the term of the absorption due to [Compton] scattering, 2 is the atomic number, and K is a coefficient, which exhibits some discontinuities at the edge of the series. For values of v larger than the threshold, vo, of the k-series we have K = 0.6 x lo3', while for v < vo, K = 0.1 x lo3'. From here one derives that the contribution to the absorption due to ionization from the k-shell, is
H represents, naturally, the discontinuity of the K coefficient when we cross the threshold for the k series. One has therefore H = 0.5 x lo3' The probability II(b), for an a-particle to ionize the k-shell can be calculated with Eq. (8). Since the mass of the a-particle is very large, we can substitute the upper limit mv2/2h,with 03. Thus we find 8xc2HZ4 hv4
n(b)= 1 - e
SfB(39) "0
The equivalent radius can be calculated with the formula
0
We find then
'Nettleton - Proc. Nat. Acad. of Sci, 10, p. 140, 1924. 3See, for example, H.A. Kramers - Phil. Mag. 46, p. 863, 1923.
250
where for brevity one puts a=
87rE2HZ4
bow4
.
The integrals in (13) can be calculated approximately
for a
<< 1, and one finds (13’)
We want to apply this formula to calculate the ionization of Helium by a-particles from Radium-C. Since Helium possesses k-shell electrons, and [the k-shell] is complete, Eq. (13’) will be applicable, always taking H = 0.5 x lo3’. Corresponding to the ionization potential of 24.5 volts, we have uo = 6.0 x and for a-particles from Radium-C, u = 1.98 x lo9. We therefore find a = 0.0091; with this value one finds p2 = 0.56 x
p = 0.75 x lo-*
The number of ion pairs produced per cm by the particle is naturally x p 2 n where n = 2.6 x lo1’ represents the number of atoms per cm3 at 15°C. This number is then 4800. Instead, experimentally, one finds 4600. The agreement between theory and experiment is quite satisfactory; in fact, we observe that by chance the agreement is better than could be foreseen given the precision of the data. In fact the large uncertainty, especially in the coefficient H , could easily account for a discrepancy of 20% and perhaps more. V. An additional experimental confirmation of our theory can be found in the treatment of deceleration of a-particles in matter. We’ll also apply this method to Helium for which, as was seen earlier, it is possible to find a reasonable value for the absorption coefficient. First of all we want to calculate the average energy loss that an a-particle undergoes passing at a distance, b, from an atom. Let n(b)be the probability that the particle passing at a
When a < 0.1 one can break up I into an integral from 0 to 0.4 and another from 0.4 to infinity. The first can be calculated observing that for E < 0.4 we can substitute with reasonable approximation, J$l?(u)
=1 -0.64 3E3
c
and then the asymptotic expressions for the logarithmic integral can be applied. The second integral can instead be approximately written as m
m
0.4 2 The coefficient, a, calculated numerically, is 0.28. Thus we find
251
distance, b, ionizes the atom, and P(b,u)du be the probability that would exist for the atom to be ionized, if only frequencies in the interval, dv were effective. The probability that the ionization takes place with the absorption of a quantum of frequency u is then
This ionization corresponds to an energy loss hu. The average energy loss of the particle is therefore m J P (b, v ) hvdv n(b)
Om
s0 p ( b ,).
dv
If n represents the number of atoms per unit volume, the particle passes, during path 1, within a distance of b t o b db from 2 x n b d b atoms. If T is its kinetic energy, we have
+
_ dT -- 2 m dx
7P (b,
s
bdbn(b)
u ) hvdu
Om
s0 p ( 4
0
dv
now we have
Equation (15) now becomes, with a simple transformation, keeping in mind Eq. (14)
When a is very small these expressions can be evaluated approximately and one finds 'To calculate this integral we need to use (for f
Jdu E
< 0.4)
1
z,zB (ti) = 7- 0.64, 35
the approximate expressions
7:
--B
e
1
(u) = - - 0.64 2P
.
Furthermore it is convenient to divide the integral into a term from 0 to 0.4 which can be easily calculated using the asymptotic expressions for the logarithmic integral. To calculate the second term from 0.4 to infinity, one needs only to observe that for 5 > 0.4 and Q very small we can take
One needs now to calculate numerically
and the result of the overall integral is 0.24
from which, applying Eq. (15), one derives Eq. (17).
1
- -log 4
Q
252
dT dx
cx2HZ4n
0.96 - log
8 n c 2H 2 4 hvov4
and since T = (1/2)mv2, we find then
dv dx
C E ~ H Z ~ ~ 0.96 +log
(17')
From this expression we can easily derive a formula which gives us the path-length traveled by the particles while their velocity is reduced from its initial value, V O , to the final value, v. Explicitly one finds vo
X =
du log 21
mu: v3dv C E ~ H Z0.96 ~ ~ log & ~d
+
Naturally this formula is valid only when both limits are large with respect to unity, since otherwise Eq. (17'), from which we derived Eq. (18) is not applicable, and we would then need to calculate (16) exactly also for large values of a. However, we can calculate the path-length in Helium of a-particles from Radium-C when their velocity is reduced by half. In fact, substituting n = 2.6 x lo1'; E = 2 x 4.77 x lo-'' = 9.54 x in Eq. (18) m = 6.6 x lopz4;vo = 6.0 x vo = 1.98 x lo9, we find 288
x = 0.56
J
du
= 32
18
Since the range [English word used in the original] of a-particles of velocity vo/2 is equal to about 1/8th of the range of particles with velocity VO, we immediately deduce for the latter 37cm. Experimentally we find, in good agreement with this theoretical value, a range of approximately 33 cm.
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