Rare Isotopes and Fundamental Symmetries: Proceedings of the Fourth Argonne INT MSU JINA FRIB Theory Workshop (Proceedings from the Institute for Nuclear Theory)

•

•

Rare Isotopes and Fundamental Symmetries

PROCEEDINGS FROM THE INSTITUTE FOR NUCLEAR THEORY Series Editors:

Wick C. Haxton (Univ. of Washington) Ernest M. Henley (Univ. of Washington)

Published

Vol. 1:

Nucleon Resonances and Nucleon Structure ed. G. A. Miller

Vol. 2:

Solar Modeling eds. A. B. Balantekin and J. N. Baheall

Vol. 3:

Phenomenology and Lattice QCD eds. G. Kileup and S. Sharpe

Vol. 4:

N* Physics eds. T.-S. H. Lee and W. Roberts

Vol. 5:

Tunneling in Complex Systems ed. S. Tomsovie

Vol. 6:

Nuclear Physics with Effective Field Theory eds. M. J. Savage, R. Seki and U van Kolek

Vol. 7:

Quarkonium Production in High-Energy Nuclear Collisions eds. B. Jaeak and X. -N. Wang

Vol. 8:

Quark Confinement and the Hadron Spectrum eds. A. Radyushkin and C. Carlson

Vol. 9:

Nuclear Physics with Effective Field Theory II eds. P. F. Bedaque, M. J. Savage, R. Seki and U van Kolek

Vol. 10:

Exclusive and Semi-Exclusive Processes at High Momentum Transfer eds. C. Carlson and A. Radyushkin

Vol. 11:

Chiral Dynamics: Theory and Experiment III eds. A. M. Bemstein, J. L. Goity and U-G. Meif!.ner

Vol. 12:

The Phenomenology of Large Nc QCD ed. R. F. Lebed

Vol. 13:

The r-Process: The Astrophysical Origin of the Heavy Elements and Related Rare Isotope Accelerator Physics eds. y.-z. Qian, E. Rehm, H. Schatz and F.-K. Thielemann

Vol. 14:

Open Issues in Core Collapse Supernova Theory eds. A. Mezzaeappa and G. M. Fuller

Vol. 15:

Opportunities with Exotic Beams eds. T. Duguet, H. Esbensen, K. M. Nollett and C. D. Roberts

Institute for

19

I~t:clcal

Theol'Y: U

of

USA

?" September 2007

Proceedings of the Fourth Argonne/INT/MSU/JINA FRIB Theory Workshop

•

•

Rare Isotopes and Fundamental Symmetries

editors

B Alex Brown Michigan State University, USA Jonathan Engel University of North Carolina, USA Wick Haxton University of Washington, USA Michael Ramsey-Musolf University of Wisconsin, USA Michael Romalis Princeton University, USA Guy Savard Argonne National Laboratory, USA

,~World Scientific NEW JERSEY. LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Proceedings from the Institute for Nuclear Theory - Vol. 16 RARE ISOTOPES AND FUNDAMENTAL SYMMETRIES

Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4271-72-1 ISBN-IO 981-4271-72-1

Printed in Singapore.

v

Series Preface The National Institute for Nuclear Theory Series

The national Institute for Nuclear Theory (INT) was established by the US Department of Energy in March, 1990. The goals of the INT include: (1) Creating a productive research environment where visiting scientists can focus their energies and exchange ideas on key issues facing the field of nuclear physics, including those crucial to the success of existing and future experimental facilities; (2) Encouraging interdisciplinary research at the intersections of nuclear physics with related subfields, including particle physics, astrophysics, atomic physics, and condensed matter; (3) Furthering the development and advancement of physicists with recent Ph.D.s; (4) Contributing to scientific education through graduate student research, INT summer schools, undergraduate summer research programs, and graduate student participation in INT workshops and programs; and (5) Strengthening international cooperation in physics research through exchanges and other interactions. While the INT strives to achieve these goals in a variety of ways, its most important efforts are the three-month programs, workshops, and schools it sponsors. These typically attract 300-400 visitors to the INT each year. In order to make selected INT workshops and summer schools available to a wider audience, the INT and World Scientific established the series of books to which this volume belongs.

In January 2004 the INT and three partners, Argonne National Laboratory, Michigan State University, and the Joint Institute for Nuclear Astrophysics, began a new workshop series to explore scientific questions that might be answered by the proposed U.S. Facility for Rare-Isotope Beams (FRIB). This volume summarizes the proceedings of the fourth workshop in this series, which was hosted by the INT in September 2007 and organized by Alex Brown, Jonathan Engel, Wick Haxton, Michael Ramsey-Musolf,

vi Michael Romalis, and Guy Savard. The organizers designed a scientific program to provide a broad overview of the potential role rare isotopes could play in tests of fundamental symmetries, such as time reversal and parity. For example, certain short-lived isotopes exhibit enhanced electric dipole polarizabilities, a phenomenon that can be exploited by experimentalists to place tighter constraints on time reversal violation. The workshop proceedings are being published so that this overview will be available to the broader nuclear, atomic, and particle physics communities, as the planning for FRIB progresses. As series editors, we would like to thank the organizers for the considerable effort they invested in designing the scientific program and in editing this volume. This volume is the 16th in the INT series. Earlier series volumes include the proceedings of the 1991 and 1993 Uehling summer schools on Nucleon Resonances and Nucleon Structure and on Phenomenology and Lattice QCD; the 1994 INT workshop on Solar Modeling; the tutorials of the spring 1997 INT program on Tunneling in Complex Systems; the 1998 and 1999 Caltech/INT workshops on Nuclear Physics with Effective Field Theory; the proceedings of the 1998 RHIC Winter Workshop on Quarkonium Production in Relativistic Nuclear Collisions; the proceeding of Nucleon Resonance Physics (1997), Confinement III (1998), Exclusive and Semi-exclusive Reactions at High Momentum (1999), Chiral Dynamics 2000, and the Phenomenology of Large-N QCD (2002), all collaborative efforts with Jefferson Laboratory; the 2004 workshop on the Astrophysical Origin of the Heavy Elements and the 2006 workshop on Opportunities with Exotic Beams, two earlier volumes from the FRIB series; and the 2004 workshop on Open Issues in Core Collapse Supernova Theory. We intend to continue publishing those proceedings of INT workshops and schools that we judge to be of broad interest to the physics community. Wick C. Haxton and Ernest Henley Seattle, Washington, April, 2008

vii

VOLUME PREFACE The Fourth Argonne/INT/MSU/JINA FRIB Theory Workshop On Rare Isotopes and Fundamental Symmetries This workshop on Rare Isotopes and Fundamental Symmetries was held September 19-22, 2007, at the INT. The fourth in a series dedicated to exploring the science important to FRIB, the proposed Facility for Rare Isotope Beams, this workshop focused on the use of radioactive ions in various symmetry tests. It is envisioned that symmetry tests would form a third leg of the FRIB experimental program, in addition to nuclear structure studies and nuclear astrophysics. At existing facilities radioactive beams in combination with atom traps and other instrumentation have opened new opportunities for such measurements. FRIBs expected intensities could help move this field further forward. The topics discussed at the workshop included: • Fermi beta decay: Nuclear systems have provided our most accurate determination of the CKM matrix element Vud that is central to tests of quark unitarity. • Electron-neutrino correlations in nuclear beta decay: Rather precise constraints on new interactions, such as scalar interactions, have already been set with trapped isotopes at facilities such as TRIUMF. • Precision mass measurements: Mass ratio measurements based on cyclotron frequency measurements in stable trap magnetic fields have enormous potential for constraining Q-values important to neutrino mass measurements (e.g., tritium beta decay), Fermi beta decay, and double beta decay. • Atomic parity violation: Atomic PNC provides our best low-energy measurement of Qweak. Atomic and nuclear (neutron skin) theoretical uncertainties are a significant issue in the associated analysis. Measurements in isotopic chains have been discussed as a possible strategy for reducing such uncertainties. • Electric dipole moments: Certain nuclei have enhanced polariz-

viii abilities due to parity near-degeneracies associated with nuclear structure phenomena such as octupole deformation. In principle enhancements of 3-4 orders of magnitude in electric dipole moments could result. Theoretical issues include the calculation of the" Schiff moment" , the residual interaction at the nucleus after atomic screening is evaluated . • Hadronic parity violation and anapole moments: The parityviolating but T-even anapole moment has been measured in a single atomic nucleus, 133Cs. As in the case of the edm, large enhancements are expected in certain nuclei due to ground-state parity doublets. In other cases, the ana pole moment arises from polarization associated with giant dipole collectivity in the nucleus. The workshop extended over 3.5 days and included presentations from 25 speakers with a mix of theory and experiment. Approximately half of the talks focused on experiments currently under development, some of which would benefit from FRIB beams. Theory talks focused on the progress in nuclear and electronic structure required to extract fundamental properties from the observations. The format allowed for considerable discussion, and included a designated end-of-the day discussion period (though this proved unnecessary, as questions were asked frequently during presentations). The participants included several locals from the atomic physics group and from CENPA. A workshop dinner was held at Ivars Salmon House. The earlier workshops in this series covered the topics of The r-Process: the Astrophysical Origin of the Heavy Elements and Related Rare Isotope Accelerator Physics; (INT, University of Washington, January 8-lO, 2004); Reaction Mechanisms for Rare Isotope Beams (Michigan State University, March 9-12,2005); and Opportunities with Exotic Beams (Argonne National Laboratory, April 4-7, 2006).

IX

ORGANIZING COMMITTEE B. Alex Brown (Chair) Jonathan Engel Wick Haxton Michael Ramsey-Musolf Michael Romalis Guy Savard

-

Michigan State University University of North Carolina University of Washington University of Wisconsin Princeton University Argonne National Laboratory

This page intentionally left blank

xi

CONTENTS Series Preface

v

Preface Experiments Searching for New Interactions in Nuclear Beta Decay Klaus P. Jungmann

1

The Beta-Neutrino Correlation in Sodium-21 and Other Nuclei P.A. Vetter, J. Abo-Shaeer, S.J. Freedman, R. Maruyama

11

Nuclear Structure and Fundamental Symmetries E. Alex Erown

21

Schiff Moments and Nuclear Structure J. Engel

31

Superallowed Nuclear Beta Decay: Recent Results and Their Impact on Vud J.C. Hardy and I.S. Towner

41

New Calculation of the Isospin-Symmetry Breaking Correlation to Superallowed Fermi Beta Decay I.S. Towner and J.C. Hardy

51

Precise Measurement of the 3H to 3He Mass Difference D.E. Pinegar, et al.

60

Limits on Scalar Currents from the 0+ to 0+ Decay of 32 Ar and Isospin Breaking in 33Cl and 32 Cl A. Garcia

67

Nuclear Constraints on the Weak Nucleon-Nucleon Interaction W.C. Haxton

75

XII

Atomic PNC Theory: Current Status and Future Prospects M.S. Safronova

80

Parity-Violating Nucleon-Nucleon Interactions: What Can We Learn from Nuclear Anapole Moments? B. Desplanques

96

Proposed Experiment for the Measurement of the Anapole Moment In Francium A. Perez Galvan, D. Sheng, L.A. Orozco, and the FRPNC Collaboration

106

The Radon-EDM Experiment Tim Chupp for the Radon-EDM collaboration

116

The Lead Radius Experiment (PREX) and Parity Violating Measurements of Neutron Densities C. l. Horowitz

126

Nuclear Structure Aspects of Schiff Moment and Search for Collective Enhancements Naftali Auerbach and Vladimir Zelevinsky

135

The Interpretation of Atomic Electric Dipole Moments: Schiff Theorem and its Corrections C.-P. Liu

150

T-Violation and the Search for a Permanent Electric Dipole Moment of the Mercury Atom M.D. Swallows, W. C. Griffith, T.H. Loftus, M. V. Romalis, B.R. Heckel, and E.N. Fortson

160

The New Concept for FRIB and its Potential for Fundamental Interactions Studies Guy Savard

170

Collinear Laser Spectroscopy and Polarized Exotic Nuclei at NSCL K. Minamisono, G. Bollen, P.F. Mantica, D.l. Morrissey and S. Schwartz

180

xiii

Environmental Dependence of Masses and Coupling Constants M. Pospelov

190

Workshop Program

201

This page intentionally left blank

1

EXPERIMENTS SEARCHING FOR NEW INTERACTIONS IN NUCLEAR ,a-DECAY KLAUS P. JUNGMANN

Kernfysisch Versneller Instituut (KVI), University of Groningen, Groningen, g147 AA, The Netherlands E-mail: [email protected] http://www.kvi.nlj trimp/trimp.html Precision measurements of ,6-decays in nuclei, muons and neutrons allow to search for non V-A contributions in weak interactions and to set limits on parameters relevant to theoretical models beyond standard theory. Novel experiments are possible in particular at presently operating stable beam facilities and at new radioactive beam facilities such as the ISAC facility at TRIUMF, the upcoming RIKEN cyclotron facility in Japan, the new proposed FRIB (RIA) facility and the newly available facility TRIp,P at KVI. EURISOL is the most powerful and versatile planned radioactive beam facility.

Keywords: ,6-decay; nuclei; muons; neutrons

1. Fundamental Interactions

The Standard Model (SM) of particle physics provides a theoretical framework which allows to describe all observations in particle physics to date. Even those recent observations in neutrino physics which suggest the existence of neutrino oscillations can be accommodated with small modifications. However, contrary to this the great success of the SM there remains a number of most intriguing questions in modern physics to which the SM can not provide further clues about the underlying physical processes, although the facts are can be described often to very high accuracy. Among those puzzling issue are the existence of exactly three generations of fundamental fermions, i.e. quarks and leptons, and the hierarchy of the masses of these fundamental particles. In addition, the electro-weak SM has a rather large number of some 27 free parameters, all extracted from experiments. 1 In all of modern physics, particularly in the SM, symmetries play an important role. Global symmetries relate to conservation laws and local symmetries give forces. Within the SM the physical origin of the observed

2

breaking of discrete symmetries in weak interactions, e.g. of parity (P), of time reversal (T) and of combined charge conjugation and parity (CP), remains unrevealed, although the experimental findings can be well described. In order to gain deeper insights into the not well understood aspects of fundamental physics, a number of speculative models beyond the present standard theory have been proposed. Those include such which involve Left-Right symmetry, fundamental fermion compositeness, new particles, leptoquarks, supersymmetry, supergravity, technicolor and many more. Interesting candidates for an all encompassing quantum field theory are string or membrane (M) theories which among other features may include supersymmetry in their low energy limit. Independent of their mathematical elegance and partial appeal all of these speculative theories will remain without status in physics unless secure experimental evidence for them being reality can be gained in future. Experimental searches for predicted unique features of those models - such as breaking of discrete symmetries are therefore essential to steer the development of theory towards a better and deeper understanding of the fundamental laws in nature. Such experiments must be carried out not only at high energy accelerators, but also in complementary approaches at lower energies. Typically the low energy experiments in this context fall into the realm of atomic physics and of high precision measurements. The advanced methods developed over the past decades in these fields are crucial for the success of this research. Precise measurements of properties of weak interactions through muon, neutron and nuclear ,6-decays are - next to searches for permanent Electric Dipole Moments (EDMs) of elementary particles, nuclei, atoms and molecules, and searches for rare lepton decays - a central subset of indispensable low energy precision particle physics experiments. 2 ,3 2. Muons

In the absence of exotic muon decays, the measurement of the muon lifetime T J-' provides the best way to determine the Fermi coupling constant C F: TJ-'

=

1927[3

-2C 5 [1

+ 8] ,

FmJ-'

where mJ-' is the muon mass and 8 < < 1 corrects for sufficiently well understood effects of virtual fields. There are three ongoing efforts to improve over the present knowledge: one at the RIKEN-RAL muon facility,4 and two at PSI, FAST 5 and MuLan. 6 As the muon mass is known from the pioneering and ground breaking precision measurements in exotic atoms,

3

i.e. from muonium spectroscopy, directly to 27 ppb, 7 the combination of all muon lifetime measurements until now yields TJL = 2.197019(21)jLs and G F = 1.166371(6) X 10- 5 GeV- 2 (5 ppm).6 Its correctness, i.e. the absence of experimental errors and of other than the 8M V-A weak interaction contributions, in particular of rare decays, is amongst other issues important for the interpretation of superallowed nuclear ,8-decays and neutron decays in terms of the unitarity of the Cabbibbo-Kobayashi-Maskawa matrix. 8

3. Nuclei 3.1. Nuclear f3-decays In standard theory the structure of weak interactions is V-A, which means there are vector (V) and axial-vector (A) currents with opposite relative sign causing a left handed structure of the interaction and parity violation. 9 ,lD Other possibilities like scalar, pseudo-scalar and tensor type interactions, which might also be possible, would be clear signatures of new physics. 80 far they have been searched for without positive result. However, the bounds on parameters are not very tight and leave room for various speculative possibilities. The double differential ,8-decay probability d 2 W/ do'edo'v is related to the electron and neutrino momenta p and if through d2 W p. - - '" 1 + a do'edo'v

E

+<J>'[A

+ < if > . [G

if + b

~+B

!+

me E

\11 - (Za)2 if+D

p~ if]

Q J + R < J>

x

!]

where me is the ,8-particle mass, E its energy, if its spin, and J is the spin of the decaying nucleus. The coefficients D and R are studied in a number of experiments at this time and they are T violating in nature. Here D is of particular interest for further restricting model parameters. It describes the correlation between the neutrino and ,8-particle momentum vectors for spin polarized nuclei. The coefficient R has a high sensitivity only within a smaller set of speculative models, since in this area of research there exist some already well established constraints, e.g., from EDM searches. 9 3.2. f3-asymmetry The ,8-asymmetry A observed first in the decay of 60Co has confirmed the maximal violation of the parity symmetry in weak interactions. This important quantity has been studied in a number of other systems confirming the

4

V-A structure of weak interactions, i.e. A=-l. Recently a 60Co experiment was performed with traditional spin polarization in 9 and 13 T magnetic fields at the university of Leuven. A preliminary value A=-0.953(22) was obtained, which presently is recheckedY

3.3. (3-neutrino correlations An efficient direct measurement of the neutrino momentum is not possible. The recoiling nucleus can be detected instead and the neutrino momentum can be reconstructed using the kinematics of the process. Since the recoil nuclei have typical energies in the few 10 eV range, precise measurements can only be performed, if the decaying isotopes are suspended in extreme shallow potential wells. Such exist in atom traps formed by laser light, where many atoms can be stored at temperatures below 1 mK. At a number of laboratories worldwide {3-v correlations are studied 10 . Recently at Berkeley the asymmetry parameter a in the {3-decay of 21 Na has been measured in optically trapped atoms. 12 The value differs from the present SM value by about 3 standard deviations. In order to explore whether this might be an indication of new physics, the {3/ ({3 + "() decay branching ratio was remeasured at Texas A&M and at KVI, because some 5 measurements existed which in part disagreed significantly. The new values of 4.74( 4)%13 and 4.85(12)%14 and agree well and do not affect significantly the SM prediction. The difference may perhaps be explained by Na dimer formation in the trap.15 The most stringent limit on scalar interactions comes from {3-v correlation measurements on the pure Fermi decay of 38mK at TRIUMF, where a was extracted to 0.5 % accuracy and in good agreement with standard theory.16 New upcoming and promissing projects are the WITCH experiment at CERN ISOLDE using ions of {3-decaying isotopes stored in a Penning trap17 and the LPCTrap experiment at GANIL, where 6He ions are trapped in a Paul (radiofrequency) trap18 and where first data were already recorded.

3.4.

Time Reversal Violation

CP violation as observed first in the neutral Kaon decays can be described with a single phase factor in the Cabbibo-Kobayashi-Maskawa formalism. Because of its possible relation to the observed matter-antimatter asymmetry in the universe, CP violation has attracted a lot of attention. a CP A. Sakharov 19 has suggested that the observed dominance of matter could be explained via CP-violation in the early universe in a state of thermal non-equilibrium and with

a

5

violation as described in the SM is however not sufficient to explain the excess of baryons. This provides a strong motivation to search for yet unknown sources CP symmetry violation. It is in particular a major driving force behind the EDM searches going on at present. With the assumption of an unbroken CPT symmetry CP violation is equivalent to T violation.

3.5. {3-neutrino correlations of spin polarized nuclei The possibilities to find T violation include certain correlation observables in nuclear ;3-decays. They hence offer excellent opportunities to find new sources of CP violation. In ;3-neutrino correlations the D coefficient 9 (for spin polarized nuclei) have a high potential to observe new interactions in a region of potential New Physics which is less accessible by EDM searches. However, the R coefficient 9 (observation of ;3-particle polarization) would explore the same areas as present EDM searches or ;3-decay asymmetry measurements. Such experiments are underway at a number of laboratories worldwide. It will be a prerequisite and crucial for the success of such measurements that nuclear polarization for trapped atoms can be effectively achieved and measured to about 10- 4 integral over an experiment.

4. Searches for Permanent Electric Dipole Moments Distinctively different precision experiments to search for an EDM are under way in many different systems. A large number of ideas for significant improvements have been made public. Still, the electron and the neutron get the largest attention of experimental groups, although besides tradition there is little which singles out these systems. Nevertheless, there is a large number of efforts in the USA and in Europe using different approaches which all have unique promising features. In composed systems, i.e. molecules, atoms or nuclei, fundamental particle dipole moments of constituents can be significantly enhanced. 21 For the electron significant enhancement factors are planned to be exploited such as those associated with the large internal electric fields in polar molecules. Recently the completeness of the Schiff moment operator, which describes such enhancements, has been questioned. 22 This may lead possibly to some modifications of presently well established enhancement factors. There is baryon number violating processes. The existence of additional sources of CP-Violation is not a necessary condition to explain the baryon asymmetry. Other viable routes could lead through CPT violation and don't need thermal non-equilibrium. 2o

6

no preferred system to search for an EDM. In fact, many systems need to be examined, because depending on the underlying processes different systems have in general quite significantly different susceptibility to acquire an EDM through a particular mechanism. An EDM may be found an 'intrinsic property' of an elementary particle as we know them, because the underlying mechanism is not accessible at present. However, it can also arise from CP-odd forces between the constituents under observation, e.g. between nucleons in nuclei or between nuclei and electrons. Such EDMs could be much higher than those expected for elementary particles originating within the usually considered models beyond the SM. This highly active field of research benefited recently from a number of novel developments. One of them concerns the Ra atom, which has rather close lying 7S7p 3 PI and 7s6d3 D2 states. Because they are of opposite parity, a significant enhancement has been predicted for an electron EDM,23,24 much higher than for any other atomic system. Further more, many Ra isotopes are in a region where (dynamic) octupole deformation occurs for the nuclei, which also may enhance the effect of a nucleon EDM substantially, i.e. by some two orders of magnitude. From a technical point of view the Ra atomic levels of interest for an experiment are well accessible spectroscopically and a variety of isotopes can be produced in nuclear reactions. The advantage of an accelerator based Ra experiment is apparent, because EDMs require isotopes with spin and all Ra isotopes with finite nuclear spin are relatively short-lived. A very novel idea was introduced recently for measuring an EDM of charged particles. The high motional electric field is exploited, which charged particles at relativistic speeds experience in a magnetic storage ring. In such an experiment the Schiff theorem can be circumvented (which had excluded charged particles from experiments due to the Lorentz force acceleration) because of the non-trivial geometry of the problem. 21 With an additional radial electric field in the storage region the spin precession due to the magnetic moment anomaly can be compensated, if the effective magnetic anomaly aef f is small, i.e. aeff < < 1. The method was first considered for muons. For longitudinally polarized muons injected into the ring an EDM would express itself as a spin rotation out of the orbital plane. This can be observed as a time dependent (to first order linear in time) change of the above/below the plane of orbit counting rate ratio. For the possible muon beams at the future J-PARC facility in Japan a sensitivity of 10- 24 e cm is expected. 25 In such an experiment the possible muon flux is a major limitation. For models with nonlinear mass scaling of EDM's such an

7

experiment would already be more sensitive to certain new physics models than the present limit on the electron EDM. The deuteron is the simplest known nucleus. An EDM could arise not only from a proton or a neutron EDM, but also from CP-odd nuclear forces. It was shown very recently26 that the deuteron can be significantly more sensitive than the neutron. Because of its rather small magnetic anomaly the deuteron is a particularly interesting candidate for a ring EDM experiment and a proposal with a sensitivity of beyond 10- 27 e cm exists. In this case scattering off a target will be used to observe a spin precession. Table 1.

Actual limits on permanent electric dipole moments. 27

Particle

Limit [e-cm]

method

e

< 1.6 < 2.8 < 3.0 < 2.1

Tl atomic beam (Berkeley) muon g-2 storage ring (Brookhaven) stored cold neutrons (Grenoble) Hg vaour cell (Seattle)

/1 n

Hg-atom

X 10- 27 19 X 10X 10- 26

x 10- 28

The highly active field of EDM searches includes at present a variety of experiments on the neutron and the electron EDM. Whereas in the neutron case basically the experiments follow the concepts of earlier measurements, novel approaches characterize the search for an electron EDM. There are continued searches in Hg and a new search in liquid Xe. Further, there are projects on molecules such as PbO, or molecular ions such as ThF+ or condensed matter such as garnets, where in all cases one relies on the huge predicted enhancements due to local fields. 5. Facilities for providing radioactive isotopes for precision experiments

At present experiments on fundamental interactions and symmetries exploiting nuclear properties are either performed in table top laboratory experiments with stable particles in several university laboratories worldwide or at a small number of accelerator laboratories where radioactive nuclids are made available. In the latter case the availability of sufficient beam time to debug precision experiments and to study systematic effects with the indicated care is a constant problem. Therefore new facilities are most welcome. The TRlfLP facility at KVI was recently commissioned. A sig28 nificant share of beamtime is allocated fundamental interaction research. An example of the achievable clean secondary beams at TRlfLP was demon-

8

strated in an experiment on the ,8-decays of A=12 isotopes in excited states of 12C, which may themselves decay into 3 a particles. Spectra obtained with beam ions implanted in a Si detector matrix are of relevance to the 12C production process in stars. 29 Future possibilities for precision experiments in a large variety of radioactive atoms may include besides FRIB the ISAC facility at TRIUMF, Canada, provided a new target will be installed, the Spiral II facility of GANIL in Caen, France, the radioactive beams at the new RIKEN cyclotrons in the Tokyo area, Japan, and to a limited extent the FAIR facility of GSI in Darmstadt, Germany. At the latter one can look especially forward to the most intense source of slow antiprotons (FLAIR at FAIR). The most intense an versatile facility will be the planned EURISOL facility. It will be based on a MW proton/deuteron driver, which opens also unique possibilities in muon and neutrino physics. 30 ,31 For the expected progress in precision measurements it will be of crucial importance that the new facilities can be operated to give a significant share of beam time to this research, because sufficient beam is not only important for collecting statistics; it is even more important to cleanly understand all possible systematic effects. 6. Neutrons Precision measurements in neutron decays offer numerous possibilities to study the very same phenomena discussed above. 32 Although it is beyond the scope of this article to discuss any details, the potential of the offered options needs to be checked in the evalution procedure before deciding on any particular experiment with nuclei (and vice versa) to assure efficient progress in the field. This holds in particular for D and R coefficient measurements. Just as examples: D coefficient measurements in the TRINE and EMIT experiments at ILL and NIST have not yet achieved the accuracy of experiments in 19Ne, however, the final state interactions are one order of magnitude better known. The FUNSPIN experiment at PSI challenges the R coefficient measurements in 8Li. Last but not least, neutron EDM searches have yielded the most stringent limit on strong CP violation. 7. Conclusions There is a variety of well motivated low energy precision experiments with radioactive nuclei. They have unique and robust discovery potentials. Novel ideas have come up in the recent past to use yet not studied systems and new

9

experimental approaches. They offer excellent opportunities to complement high energy attempts to find physics beyond the SM. The community is eagerly awaiting FRIB and other complementary facilities worldwide to rapidly proceed with the challenging physics programmes.

8. Acknowledgments The author would like to thank the organizers of the Fourth ArgonnejINT jMSU j JINA RIA Theory Workshop on Rare Isotopes and Fundamental Symmetries for providing a stimulating atmosphere with ample scientific discussion and also for their support. This work has been supported by the Dutch Stichting voor Fundamenteel Onderzoek der Materie (FOM) in the framework of the research programme 48 (TRIfLP).

References 1. W.-M. Yao et al., The Review of Particle Physics, J. of Phys. G 33, 1 (2006) 2. T. Akesson et al., Towards the European strategy for particle physics: the Briefing Book,Eur.Phys.J. C51, 421(2007) 3. K. Jungmann, Fundamental Symmetries and Interactions, Nucl.Phys. A751, 87c (2005) and K. Jungmann, Fundamental symmetries and interactions Some aspects, Eur.Phys.Jour. A 25, 677 (2005); 4. D. Tomono et al.,Precise Muon Lifetime Measurement with a Pulsed Beam at the RIKEN-RAL Muon Facility, Nucl. Phys. B, S149, 341 (2005) 5. C. Casella, FAST: A precision measurement of the muon lifetime TJL and G F, Nucl. Phys. BS150, 204 (2006) 6. D.B. Chitwood et al., Improved measurement of the positive-muon lifetime and determination of the Fermi constant, Phys. Rev. Lett. 99, 032001 (2007) and references therein 7. W. Liu et al., High precision measurements of the ground state hyperfine structure interval of muonium and of the muon magnetic moment, Phys. Rev. Lett. 82, 711 (1999) 8. A. Garcia, Unitarity of the CKM matrix, Hyperf. Int. 172, 23 (2007); H. Abele et al., Quark mixing, CKM unitarity, Eur.Phys.J. C33, 1 (2004) 9. P. Herczeg, Beta decay beyond the standard model, Prog. Part. and Nucl. Phys. 46, 413 (2001) 10. N. Severijns, M. Beck and O. Naviliat-Cuncic, Tests of the standard electroweak model in beta decay, Rev. Mod. Phys. 78, 991(2006) 11. N. Severijns, priv. com. (2007); 1. Kraev, Search for physics beyond the standard electroweak model with brute-force low temperature nuclear orientation, PhD thesis, KU Leuven, Belgium (2006) 12. N.D. Scielzo et al., Measurement of the /3-1/ Correlation using Magnetooptically Trapped 21Na, Phys.Rev.Lett. 93, 102501 (2004) 13. V.E. Iacob et al.,Branching ratios for the /3-decay of 21Na , Phys.Rev.C 74,015501,(2006)

10 14. L. Achouri, priv. com. (2006) 15. N. Scielzo, private communication (2006) 16. A. Gorelov et al.,Scalar Interaction Limits from the f3-v Correlation of Trapped Radioactive Atoms, Phys.Rev.Lett. 94, 142501 (2005) 17. V.Y Kozlov et aI, The WITCH experiment: towards weak interactions studies. Status and prospects, Hyper£. Int. 172, 15 (2006) 18. A. Mery et al., Search for tensor couplings in the weak interaction, Europ. Phys. J. Spec. Top. 150, 385 (2007) 19. A. Sakharov, Violation of CP invariance and C asymmetry and baryon asymmetry of universe, JETP Lett. USSR, 5, 24 (1967) 20. O. Bertolami et al., CPT violation and baryogenesis, Phys. Lett. B 395, 178 (1997) 21. P.G.H. Sandars, Electric dipole moments of charged particles, Contemp.Phys. 42, 97 (2001) and references therein 22. C.P. Liu et al., Atomic electric dipole moments: The Schiff theorem and its corrections, Phys. Rev. C 76, 035503 (2007) 23. J.S.M. Ginges and V.V. Flambaum, Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles, Phys.Rep. 397, 63 (2004) and references therein. 24. J. Bieron et al., Lifetime and hyperfine structure of the D-3(2) state of radium, J.Phys.B 37, L305 (2004) 25. F.J.M. Farley et al., New method of measuring electric dipole moments in storage rings, Phys.Rev.Lett. 93, 052001 (2004) 26. C.P. Liu and R.G.E. Timmermans, P- and T-odd two-nucleon interaction and the deuteron electric dipole moment, Phys.Rev. C70, 055501 (2004) 27. B.C. Regan et al., New limit on the electron electric dipole moment, Phys.Rev.Lett.88, 071805 (2002); R. McNabb et al., An Improved Limit on the Electric Dipole Moment of the Muon, hep-ex/0407008; C.A. Baker et al., An Improved Experimental Limit on the Electric Dipole Moment of the Neutron, hep-ex/0602020 (2006); M.V. Romalis eta al., A New limit on the permanent electric dipole moment of Hg-199, Phys.Rev.Lett. 86,2505( 2001) 28. G.P. Berg et al., Dual magnetic separator for TRIpP, NucLInstr.&Meth. A560, 169 (2006); K. Jungmann et al., TRIpP - trapped radiactive atoms f-licrolaboratories for fundamental physics, Physica Scripta T104, 178 (2003); E. Traykov et al., Production of Radioactive Nuclides in Inverse Reaction Kinematics, Nucl. Instr. & Meth. 572, 580 (2007); H. Wilschut et al., Status of the TRlf-lP project, Hyperfine Interactions 174, 97 (2007) 29. S.G. Pederson et al., f3-decay studies of states in 12C, Proc. of Science, NICIX, 244 (2006) 30. P.A. Butler, The first steps to EURISOL, Act. Phys. Pol. B 38, 1147 (2007) 31. A. Branyopadhyay et al., Physics at a future Neutrino Factory and superbeam facility,arXiv:0710.4947 (2007) 32. J.S. Nico and W.M. Snow, Fundamental Neutron Physics, Ann. Rev. Nucl. Part. Phys. 55,27 (2005); M. Schumann, Precision Measurements in Neutron Decay, arXiv:0705.3769v1 (2007)

11

THE BETA-NEUTRINO CORRELATION IN SODIUM-21 AND OTHER NUCLEI P.A. VETTER, J. ABO-SHAEER, S.J. FREEDMAN, R. MARUYAMA

Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94618 * E-mail: [email protected] www.lbl.gov/nsd We have measured the (3 - v correlation coefficient, af3v, in 21 Na using a laser-trapped sample. We measure the energy spectrum of the recoil nuclei by measuring their time-of-flight in coincidence with the atomic electrons shaken off in beta decay. High detection efficiency of these low-energy electrons allows good counting statistics, even with low trap density, which suppresses the photoassociation of molecular sodium, which can cause a large systematic error. Our measurement, with a 1% fractional uncertainty, agrees with the Standard Model prediction but disagrees with our previous measurement which was susceptible to error introduced by molecular sodium. We summarize precise measurements of af3v and their consequences for searches for Beyond Standard Model scalar and tensor current couplings.

1. Introduction

Some Beyond Standard Model theories predict scalar and tensor couplings of leptons and quarks which could be detected in measurements of the betaneutrino momentum correlation. 1- 3 Short-lived isotopes in neutral atom traps are an appealing source for the study of f3 decay correlations, and many experiments have been performed and proposed in such systems. 4 - 6 Ideally, recoiling daughter nuclei and f3 particles emerge from the trap with no scattering and propagate in ultra-high vacuum. Decays occur essentially at rest and are localized in a small volume. The source location and spatial distribution can be monitored optically. The nuclear polarization state can be manipulated using optical pumping. The trapped atoms are isotopically pure, and very little background activity is present when the fiducial detection region is restricted to a zone around the trap (otherwise, atoms scattered out of the trap act as background activity with poorly characterized source distribution and polarization).

12

A previous measurement of the beta-neutrino correlation in lasertrapped 21 Na found a value 3.60" smaller than that calculated from the "V minus A" current coupling of the Standard Model. 7 The results suggested a dependence on the number of atoms: at lower trap populations, a(3v was larger, and if extrapolated to zero trap population, the result agreed with the calculated value. Repeating the experiment using a more efficient detection system suggested in,s we found convincing evidence that the previous measurement of a(3v was distorted by events originating from cold, trapped molecules of 21 Na2. 2. The Beta-Neutrino Correlation in the Electroweak Standard Model In a source with no net nuclear polarization or tensor alignment, the correlation can be inferred from the f3 decay rate9 ,10

f3 - v

d3 r dEedfledfl ex F(Z, Ee)PeEe(Eo - Ee)2 v x (h(Ee) +

a(3v(Ee)~e·:: + bFierz(Ee) ; : )

.

(1)

Here (Ee,Pe) and (Ev,Pv) are the f3 and v 4-momentum, Eo is the f3 decay endpoint energy, me is the electron mass, and F(Z, Ee) is the Fermi function. In the allowed approximation, h, the f3 - v correlation coefficient (a(3v) and Fierz interference term (bFierz) are independent of Ee. Their values are determined by the weak coupling constants C i and C{ {i = scalar (5), vector (V), tensor (T), and axial-vector (An, and by the Fermi (Gamow-Teller) nuclear matrix elements, M F (MCT ). 9 For a mixed Fermi/ Gamow-Teller beta decay,

a(3v =

(IMFI2(ICvI2+ IC~12 - ICsl 2- IC~12)

-~IMCTI2(ICAI2 + IC~12 - ICTI2- IC~12))';-1

(2)

IMFI2(ICvI2+ IC~12 + ICsl2+ IC~12) +IMcTI2(ICAI2 + IC~12 + ICTI2+ IC~12),

(3)

where

~= and

bFierz

=

±2V1 - (Za)2 Re [IMFI2 (CsCy + c~qn

+ IMcTI2 (CTC A+ C~C~n] .;-1.

(4)

13

In the Standard Model (SM), C v and CA are almost purely real,a C v = C~ = 1, and CA = C~ ~ -1.27 (from experiments), and all other coupling constants are zero. Experimental limits on scalar and tensor couplings predicted by some SM extensions are model dependent, and not necessarily stringent. 1 If present, these couplings would alter af3v either through the quadratic dependence on C s and CT or through the helicitysensitive Fierz interference terms. Searching for new tree-level lepton-quark interactions in nuclear beta decay is tough because there are many different couplings to measure: Cv and CA, the opposite chirality (primed) terms, and limits on C s and CT. Absent a predicted symmetry group, we have 19 different couplings to measure. NaIvely, at tree-level, each coupling constant would be related to a new boson like l/Mfir" so that measurements with a fractional precision of 1% of C v and C A constrain physics at an energy scale ten times higher than the electroweak scale. Interpreting measurements of beta-decay correlations to test the Standard Model demands high precision auxiliary nuclear data. Several corrections alter the allowed approximation prediction of af3v and give Ee dependence to h, af3v, and bFierz at the 1% level. The input data include ground and excited state decay branching ratios, half-life, total decay energy, electron capture branching ratio, radiative corrections to order 0:, isospin symmetry breaking corrections, and magnetic moments of parent and daughter to estimate the weak magnetism contribution. Measurements of decay correlations to better than 1% precision will be limited by the precision of these auxiliary inputs. We should view these nuclear beta decay systems as a significant frontier for testing Beyond Standard Model physics, and advocate for new, high precision input data as a significant use of a potential RIA/FRIB nuclear accelerator facility. 3. (3 - v measurement technique

The experiment apparatus have been described in. 7 ,8,1l The measurement technique is at root a momentum spectrometer for the recoil nuclei. The magneto-optic trap (MOT) is located between two microchannel plates (MCPs) and several electrodes which form a focusing electric field in the region of the trap. The f3 decay leaves 21 Ne in a variety of charge states acv and C A acquire a tiny imaginary part from the complex phase in the CKM matrix in heavy quark flavor mixing. For the rest of this discussion, we assume they are purely real, and the non-Standard Model couplings for which we derive limits are also real, i.e. time-reversal invariant.

14 through shakeoff and Auger processes.u The electric field accelerates the ionized recoil nuclei to one MCP and the low-energy electrons shaken off by the 21 Ne towards the second MCP. A trigger from the electron MCP starts a time to amplitude converter, with a stop signal provided by the 21 Ne ions detected by the ion MCP. The (3 - v correlation can be inferred from the time-of-flight (TOF) spectrum since aligned lepton momenta (caused by the a(3v term in Eq. 1) result in larger nuclear recoil energies. A CCD camera acquires images of the MOT, and the trap population is inferred from the measured intensity of the fluorescence. 4. Generating Fit Templates

To interpret the time-of-flight spectra, we use Monte-Carlo simulations to generate template TOF curves for the two kinematic terms in Eq. 1 with a(3v = 0 and a(3v = 0.553. The recoil ion TOF data are fit to a linear combination of these two template spectra to determine a(3v. The Monte-Carlo generated TOF template spectra for beta decays to 21 Ne+, are shown in Fig. 1. Electron capture events are included with a 0.087% branching ratio. b Measuring the size and location of the trapped atom cloud with respect to

~

c

5

u

450

500

550 600 Ion Time-of-Flight (ns)

650

700

Fig. 1. Monte Carlo simulation of the time-of-flight spectra for 21 Ne+ given a{3v = 0.553 (calculated value) and a{3v = 0, and the difference between these two spectra.

b Although the charge state distribution for 21 Ne from electron capture is different from the positron decay mode, because of the K shell vacancy.

15

the detectors is crucial for determining a(3v. These trap parameters were determined from camera images and measured time-of-flight distributions. We measured the TOF of autoionized dimer ions (23Nat) as a function of trap position. The distance from the MOT to the ion MCP was verified by measuring the TOF of 21 Nat, and by using ,6-decay coincidence events of neutral recoils, 21 NeD. In the latter case, the rising edge of the TOF spectrum depends on the MOT /MCP distance, independent of the electric field.

5. Molecular sodium As the MOT operates, molecular sodium (Na2) is generated via photoassociation during collisions between cold trapped atoms. In sodium, the second molecular excited state manifold is autoionizing. This creates a low-energy electron and a Nat ion with very low momentum. We detect molecular sodium as coincidence events in the MCP pair with a strongly peaked TOF originating from the trap location. The observation of autoionized dimer molecules implies a population of cold, ground state molecular sodium, since both autoionization and ground state molecules are formed via the same pathways through intermediate short-range molecular states. Molecules with a net magnetic moment can be confined in the magnetic trap formed by the MOT's magnetic field gradient, implying a cold, trapped, molecular population. It is difficult to determine the absolute fraction of trapped 21 Na2 by measuring the rate of ionized dimers. This would require knowing the spontaneous emission rate and autoionization probabilities of the molecular states. We have measured the rate of autoionized 23Na2 dimers as a function of the trapped atomic population: this measures the relative molecular population of the MOT as a function of atomic population, since the formation rate of autoionized dimers must be related to the formation rate of cold, ground state, trapped dimers. We find a strong dependence of the dimer ion rate per trapped atom on the population of the MOT, shown in Fig. 2. This scaling would be very different for other laser trapped species. In our previous measurement, Ref.,7 the result for a(3v depended on the number of atoms held in the MOT, and this dependence was likely caused by counting beta decay events in which the detected recoil nucleus scattered from a molecular partner. For beta decay occuring in 21 Na2, the recoil nucleus is created near the molecular partner. The scattering potential for 21 Ne_ 21 Na is not known, and it would depend on the charge of the 21 Ne. Scattering of the nuclear recoil momentum will randomize the momentum direction, and the momentum would be shared

16

with the molecular partner.

10-5

T~~

~f ~~

10-6 4

10

5

10

6

10 Trap Population

7

10

Fig. 2. Rate of detected 23Nat per trapped atom as a function of the trap atomic population.

6. Systematic uncertainties

To address the recoil scattering from molecular partners, data were acquired with a range of trap populations. Figure 2 suggests that there should be a dependence of a(3v with trap population, as the molecular population fraction per trapped atom changes. Data were also acquired using a dark MOT technique, in which the usual trap repumping laser is not incident on the trapped atoms. The average atomic excited state population is greatly reduced, inhibiting the photoassociation process (which requires collisions involving excited state atoms) by nearly three orders of magnitude. In Fig. 3, we show the current data set and the averaged data from. 7 The perturbation on a(3v from molecular recoil scattering can be estimated by fitting these data to a curve derived from the measured rate of molecular ions (per atom) as a function of trap population, Fig. 2. This fit gives a negligible correction (0.05%) to the current data set for an extrapolation to zero trap population, while also supporting the plausibility of extrapolating the data from. 7 A component of the background in the TOF spectrum data is caused by the coincident detection of a {3+ or "( with the electron MCP and a

17

1.10

~

e 1.05

-

Scielzo 2004 Vetter 2007 Molecular scattering dependence

1.00 0.95 0.90 0.85 100

200 300 Trap population

400

500xlO

3

Fig. 3. af3v at different trap populations. Data from 7 are shown. The solid line is a fit to the dependence of a perturbation from trapped 21 Na2.

positively charged recoil 21 Ne ion. These events have a biased kinematic distribution of recoil momenta. It is difficult to accurately calculate the contribution to the TOF data from these (3+ -triggered events, since the detection efficiency of the electron MCP as a function of (3+ energy and number of incident shakeoff electrons is not precisely known. This background subtraction contributes a 0.5% uncertainty in af3v. A small correction is necessary to the measured af3v because internal conversion of the excitedstate (5/2+,350.7 kev) causes the excited state contribution in each charge state of 21 Ne to deviate from the (3 decay branching ratio. Internal conversion causes in an inner shell vacancy and Auger electron loss, giving higher charge states for excited state decays, or effectively larger decay branching ratio to the higher charge states. Since the (3 - v correlation is measured only for daughter 21 Ne that have lost ~2 electrons, the ionization process could lead to systematic effects. A calculation in 11 indicated that nuclear recoil should increase ionization for the fastest recoils, necessitating a correction of 0.6 ± 0.3% to af3v, and estimates this effect in other candidate (3 decay correlation measurement systems.

7. Conclusions

Our result, averaging the data in Fig. 3 and applying several small corrections, is af3v = 0.5502(38)(46), where the first uncertainty is statistical

18

and the second systematic. This is to be compared with the calculated value a{3v = 0.553(2), which assumes Standard Model couplings, and which has an uncertainty limited by the decay parameters (half-life, Q value, and branching ratios). In the available high-precision experiments of a{3v, the beta neutrino momentum correlation is measured essentially from the recoil energy spectrum, meaning that the observable is ~ a (5) a= m , 1

+ bFierz (E;)

with a and bFierz as in Eqs. 2 and 4, and (E{3) the mean energy of the (3±. Combining the results of several precise measurements of the betaneutrino correlation a produces a limit on the existence of non-Standard Model scalar and tensor current couplings, shown in Fig. 4. To generate this exclusion plot, we assume "normal helicity" Gs = G~ and GT = GIr, and we assume that 1m (Gs) = 1m(GT ) = 0 (i.e. time reversal invari~ ance). Each system, by virtue of the different contriubtions of Fermi and Gamow-Teller transition strengths, and different mean beta energies for the Fierz term contribution, yields a different sensitivity to possible scalar and tensor contributions. The one, two, and three a contours are derived from the combined constraints offered by the different measurements of a. For comparison, we also show (as a vertical grey bar) the allowed region of scalar coupling constants consistent with the analysis of superallowed Fermi decays Gs/Gv = -(0.00005 ± 0.00130).12 The allowed region calculated only from measurements of a{3v is roughly Gs/Gv , GT/GA < 0.01, which (naIvely) probes for new couplings at an energy scale roughly lOMw .

7.1. Outlook

fOT

atom traps

Measurements of the beta-neutrino correlation coefficient using laertrapped atoms could likely be improved to the 0.1 % level of precision, which would offer somewhat more potent limits on Beyond Standard Model physics. This would be particularly interesting in the case of 6He, which has been discussed as a candidate for experiments in several forums, because of its pure Gamow-Teller transition, which cleanly constrains tensor couplings. The decay of 18Ne would offer an opportunity to measure both pure Fermi and pure Gamow-Teller beta decays in the same nucleus, if the excited state is tagged in the detection. In our experiment, the experiment described in Ref.,4 and the work described in talks at this workshop by

19 O. 10 =--.--.----.--=

0.05

0.00

-0.05

-0.10 II:::--'--.l..-I...-L......J'---I---.L---L--L. -0.10 -0.05 0.00

0.05

0.10

Fig. 4. Amplitudes of calar and tensor conpling constants allowed by precise measnrements of The 1, 2, and 30" contours for the combined limit are shown. The vertical grey band is allowed by the limit on the Fierz term from superallowed (0+ 0+) Fermi decays.12 The systems are 6He 13 (blue), neutron 14 (pink) and 15 (orange), 21 Na (this work, red), 38mK 4 (green), and 32A r 16 (violet). A measurement for 23Ne 17 is not shown, but is used to calculate the allowed region.

Jungmann and by Behr, the achievable uncertainties are limited by some common issues. Careful attention must be paid to detector response calibration, electric field calibrations, and measurements of the trapped atom cloud distribution and location. It will be challenging to reduce the uncertainty from the absolute detection efficiencies of the MCP's for electrons and recoil ions, which relates to background subtraction, rate and position dependent detection efficiency, and more precise inelusion of electron capture .events. The neutral recoil atoms offer useful information and require detectors with good efficiency for the low-energy neutrals. We must also suppress molecular formation in laser traps, and characterize this issue for beta and other applications of laser-trapped radioactive species. This is less serious when using isotopes with shorter half-lives, in which a smaller trap population can produce a statistically useful experimental result, while still having a low atomic photoassociation rate. The uncertainty in the mo-

20

mentum dependence of the recoil ionization probability could be reduced by a more sophisticated treatment than given in RefY The uncertainty in the predicted value for af3v in 21Na (0.4%) is dominated by the uncertainties in the half-life and decay branching ratio could also soon be a limiting factor, as in other systems. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DE-AC03-76SF00098. References 1. N. Severijns and M. Beck and O. Naviliat-Cuncic, Rev. Mod. Phys. 78, 102501 (2006). 2. O. Naviliat-Cuncic and T. A. Girard and J. Deutsch and N. Severijns, J. Phys. G 17, 919 (1991). 3. F. Gluck, Nucl. Phys. A 628, 493 (1998). 4. A. Gorelov et al., Phys. Rev. Lett. 94, 142501 (2005). 5. D. Feldbaum and H. Wang and J. Weinstein and D. Vieira and X. Zhao, Phys. Rev. A 76, 051402(R) (2007). 6. H. Wilschut et al., Hyperfine Interactions 174, 97 (2007). 7. N. D. Scielzo and S. J. Freedman and B. K. Fujikawa and P. A. Vetter, Phys. Rev. Lett. 93, 102501 (2004). 8. N. D. Scielzo and S. J. Freedman and B. K. Fujikawa and 1. Kominis and R. Maruyama and P. A. Vetter and J. R. Vieregg, Nucl. Phys. A 746, 677c (2004). 9. J. D. Jackson and S. B. Treiman and H. W. Wyld, Phys. Rev. 106, 517 (1957). 10. B. R. Holstein, Rev. Mod. Phys. 46, 789 (2074). 11. N. D. Scielzo and S. J. Freedman and B. K. Fujikawa and P. A. Vetter, Phys. Rev. A 68, 022716 (2003). 12. J. C. Hardy and 1. S. Towner, Phys. Rev. C 71, 055501 (2005). 13. C. H. Johnson and F. Pleasonton and T. A. Carlson, Phys. Rev. 132, 1149 (1963). 14. C. Stratowa and R. Dobrozemsky and P. Weinzierl, Phys. Rev. D 18, 3970 (1978). 15. J. Byrne et al., J. Phys. G 28, 1325 (2002). 16. E. G. Adelberger et al., Phys. Rev. Lett. 83, 1299 (1999). 17. T.A. Carlson, Phys. Rev. 132, 2239 (1963).

21

NUCLEAR STRUCTURE AND FUNDAMENTAL SYMMETRIES B. ALEX BROWN Department of Physics and Astronomy, and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA

I discuss advances in nuclear structure theory and the implications for the study of fundamental symmetries in nuclei. Recent results for the properties of the neutron skin in heavy nuclei are summarized.

Keywords: Shell Model; Configuration Interaction; Energy Density Functionals; Neutron Skin

1. Introduction and Theoretical Methods

Much of our knowledge about the fundamental symmetries of nature comes from the experimental study of nuclei. This includes parity violation, time reversal violation, the structure of the electro-weak currents, properties of the neutrino, and tests of the unitarity of CKM matrix in the standard model. Results for many beautiful and precise experiments have been shown and proposals for new experiments that will require new facilities have been presented at this workshop. 1 The experimental results need to be related to matrix elements of the nuclear many-body wavefunctions in order to make the connection to the elementary interactions. Several broad theoretical approaches have been developed to solve the nuclear many-body problem. The most fundamental are the ab initio (AI) methods that aim for an essentially exact solution for the energies and observables starting with a given two-nucleon (NN) and three-nuclei (NNN) interaction. Progress as been made with the GFMC 2 method up to A=10 and no-core shell-model method up to A=13. 3 The coupled cluster (CC) can be applied in cases where one shell-model configuration (the reference state) dominates,4 and CC applications to the ground state of nuclei such as 16 0 and 40Ca should be a good approximation to AI. 5 In most of these approaches the nuclear wavefunctions are constrained to only nucleon de-

22

grees of freedom. The interaction must be expanded in terms of a short range cut-off to the potential using for example the methods of Effective Field Theory (EFT). 6 One must also consider three-nucleon (NNN) interactions, part of which can be derived from the EFT parameters of the NN interaction, and part of which must be constrained by nuclear data. 7 Energy Density Functional (EDF) theory as exemplified by the Skyrme Hartree-Fock approach gives a qualitative understanding of all nuclei. The basic theory is applied to a single Slater determinant, but correlations can be taken into account with, for example, the Generalized Coordinate and QRPA methods. At present the density functionals are phenomenological with parameters based upon fits to nuclear data. The present level of precision is about 700 ke V rms for the nuclear binding energies. 8 There is an coordinated effort underway in the SciDAC UNEDF program 9 to constrain and extend the functional forms based on connections to AI results for light nuclei and nuclear matter. The quark meson coupling model may also provide insight into the structure of the EDF for nucleons in the nucleus. 10 The properties of many heavy nuclei can only be treated by collective models such as deformed EDF. Correlations beyond EDF can also be considered using configuration interaction (CI) methods in a basis provided by EDF. In many cases the EDF input can be supplemented by specific experimental input such as those for the single-particle energies near the doubly magic nuclei. CI assumes an active model space for a few orbitals near the Fermi surface. The contribution of the other orbitals is treated by perturbation theory. The two-body matrix elements (TBME) for the model-space CI based on a renormalized NN (RNN) interaction are derived in two steps. First the short-range part is renormalized with the G matrix or Viowk approaches 11 to obtain TBME in a space of up to, for example, 10 major oscillator shells. This is followed by a renormalization from the 10 major oscillator shells to the small CI model space. 12 Open-source codes for the RNN TBME are available. 13 The CI results for energies can be greatly improved when linear combinations of the model-space TBME are constrained by energy data. For the sd-shell nuclei one can obtain 140 keY rms for 77 ground state binding energies and 470 excited state energies (compared to an rms of many Me V without adjustment). 14 Similar results are obtained for nuclei in the pf shell. 15 Observables such as electromagnetic matrix elements and GamowTeller decay matrix elements are well reproduced taking into account that the operators for these observables are renormalized by higher-order configuration mixing and mesonic exchange currents. 16 The rms deviation be-

23

tween the effective and RNN values for the TBME is about 200 keY. The contribution of NNN interactions to the effective model-space TBME (the interaction of two valence nucleons with nucleons in the core) needs to be evaluated. NuShellX provides a recent open-source computational advance for CI calculations. 17,18 The Hamiltonian matrix is calculated on-the-fly and J-scheme dimensions up to 108 can be considered on a PC. The structure of nuclei far from stability that will be studied with FRIB provide a crucial testing ground for the all of these nuclear structure methods. Many of these exotic nuclei will also be important for the measurements of fundamental symmetries. 2. Applications to Fundamental Symmetries

Theoretical calculations for observables related to fundamental symmetries in heavy nuclei are by necessity obtained with CI and EDF methods. Isospin-mixing corrections to the Fermi beta-decay matrix elements are based on a combination of corrections from CI for the valence nucleons together with radial overlap corrections based on EDF 19 (or the WoodsSaxon potential approximation to EDF 20). It may be possible to test these calculations by comparison to AI calculations in light nuclei - perhaps for the lOC decay. 21 But the isospin-mixing corrections for light nuclei such as lOC are much smaller than those obtained from EDF in the A=70 mass region. 19 We note that some details of the IMME related to charge-symmetry breaking in the isobaric triplets is not yet well understood in terms of the Coulomb interaction alone. 22 This may point to contributions from the poorly known CSB strong interaction. The isospin-mixing correction is enhanced in the Fermi decay of proton-rich light nuclei and nuclei with Z>28. Recent experiments for 32 Ar 23 and 62Ga 24 discussed at this meeting are consistent with the theory. For Z>28 one must also take into account weak branching to 1+ states 25 for which theoretical calculations of the GamowTeller strengths can gives some guidance. Nuclear matrix elements for double-beta decay are based on CI 26 ,27 or QRPA extensions of EDF. 28 The pf-shell model space for 48Ca is complete in terms of the spin-orbit partners required for the Gamow-Teller matrix elements in two-neutrino double-beta decay. 26 However, there are higherorder correlations that quench the strength of Gamow-Teller beta decay by 50%. 16 This has been understood in terms of second-order tensor correlations and delta-particle admixtures. 29,30 Confirmation of the origin of the quenching could come from comparison to AI results for light nuclei. For the double-beta decay of 76Ge full CI calculations in a model space

24

that contains all relevant spin-orbit partners (17/2 - 15/2 and g7/2 - g9/2) is not yet possible. For the case of 48 ca it was shown that the high-energy region of giant the Gamow-Teller strength does not contribute directly to the two-neutrino matrix element, and that matrix element is dominated by the contribution of a few low-lying intermediate states in the odd-odd nucleus. 31 Perhaps a similar mechanism holds for 76Ge and heavier nuclei, in which case CI results obtained from a model space truncated in terms of spin-orbit partners (and renormalized to single beta decay observables) may be useful. The neutrinoless double-beta decay matrix elements involve higher multi pole intermediate states. It was recently shown that the lowseniority approximation implicit in in the spherical QRPA method tends to overestimate the matrix elements for neutrino-less decay. 27 Parity non-conservation was a topic covered in several talks at this workshop. The calculations for light nuclei are based on the CI method. 32 Calculations of anapole moments for heavy nuclei are based essentially on renormalized single-particle models for the valence nucleons. 33 A combined analysis of the PNC effects in all nuclear systems shows a rather inconsistent picture in terms of the isosclar and isovector components of the two-body PNC operator (Fig. 9 in [34]). Most AI results for light nuclei are related to states associated in CI with the Op-shell configurations. It should soon be possible to obtain AI results for the non-normal parity states required for the matrix elements of parity nonconservation. Configuration mixing in 133CS for the valence orbits in between the magic number 50 and 82 will soon be possible with RNN Hamiltonians derived for this model space. 35 Perhaps after these improvements a consistent picture of parity non-conservation in nuclei will emerge. Very small limits can be set on electic-dipole moments (Schiff moments) in heavy nuclei, but they need to be interpreted in terms of nuclear wavefunctions in order to extract limits on the fundamental T-violating interactions. In the case of the Ra isotopes where collective models must be used, large nuclear-structure related enhancements are predicted due to the static and dynamic effects of octupole deformation. 36 The cases of 129Xe 37 and 199Hg38 that at present have the best experimental limits might be treatable in the future with advanced CI methods.

3. Neutron Densities The PNC interaction between electrons and the nucleus involves the exchange of the ZO boson and the associated weak charge given in the standard model (with sin20w=O.23) by 39 Qw = -N - Z(4sin20w -1) ;: : :; -N.

25

The most accurate measurement of the weak charge is obtained from atomic parity violation measurements in 133Cs. 40 The precise extraction of the weak-charge depends on a electronic form-factor that comes from largescale atomic-structure calculations. It also depends on the neutron density in the nucleus. An alternative approach involves using two or more isotopes of the same element 41 where the atomic structure part cancels out, but where the effect becomes more sensitive to the distribution of neutrons in the nucleus. 42 Thus, it becomes important to measure and calculate the neutron distributions. The most important quantity for the neutron distribution is the neutron rms radius Rn. It can be defined relative the proton rms radius Rp in terms of the skin thickness S = Rn - Rp. Proton rms radii Rp for stable nuclei are determined at a high level of accuracy from electron scattering and muonic atom data. For example, the charge rms radius obtained for 208Pb is Rch = 5.5013(7) fm, 43 which gives Rp = 5.45 fm after taking into account the finite-size effects of the protons and neutrons. 44 The neutron skin in heavy nuclei has been shown to be a unique measure of the density dependence of the neutron equation of state (EOS) near nuclear saturation density.45,46,47 Fig. 1 shows the wide range of results obtained for the neutron EOS from Skyrme interactions 45 with parameters obtained from nuclear properties. The filled circles are the results for the variational calculation of Friedman and Pandharipande. 48 However, these results depend on the three-body interaction which is not well known. 49 It is interesting to note that all models cross at a unique value near p=0.14 neutrons / fm -3, and this point could be used to constraint the NNN interaction. The linear relationship between the neutron skin and the derivative of the neutron EOS at p=O.lO neutrons/fm- 3 is shown Fig. 2 for 18 Skyrme parameter sets and for six relativistic mean-field models. 46 The densitydependent properties of the neutron EOS have a strong impact on the models of neutron stars. 47,50,51,52,53 Thus, neutron distributions in nuclei are important for physics at the smallest and largest scales. Neutron rms radii are difficult to accurately measure. A model independent method of using the parity violating asymmetry in elastic scattering of electrons from 208Pb to measure Rn to a 1% (± 0.05 fm) accuracy is proposed for the PREX experiment at JLAB. 55 There have been renewed attempts to obtain Rn from hadronic scattering data,56,57 but the error due to the many-body strong interaction effects is difficult to quantify. 58 The results of anti-protonic atom data have recently been analyzed in terms of EDF models. 59 These data are sensitive to the matter density

26

50 40

~

::2!

30

z

-LU

20 10

neutron density

Fig. 1. The neutron EOS for 18 Skyrme parameter sets. The filled circles are the Friedman-Pandharipande (FP) variational calculations and the crosses obtained with Sloe. 54

at very large radii. At these large radii the matter density is dominated by neutrons. But an extraction of an rms radius for these neutrons depends upon other features of the EDF models such as the nuclear matter incompressibility K. The charge density obtained from electron scattering is best reproduced with EDF models with K R:! 200 - 230 MeV. This eliminates many of the models which have higher values of K. Within the SHF models K is closely controlled by the power of the density-dependent potential, pex, with 0:=1 with K R:! 330 down to 0:=1/6 with K R:! 200. The anti-protonic atom data was evaluated with the help of three new Skryme forces 59 called Skxsxx with 0:=1/6 and values of xx=15, 20 and 25 representing the skin thickness of 208Pb in units of 1O- 2fm. The result of the analysis was S = 0.20(±0.04)(±0.05) fm, where ±0.04 fm is experimental error from the anti-protonic line width, and where ±0.05 fm is the theoretical error suggested from the comparison of the theoretical and experimental charge densities at large radii. The neutron-skin can also be constrained by the properties of the pygmy dipole resonance in neutron-rich nuclei. 60 Data for 132Sn suggest a value of S = 0.24(4) for 132Sn.61 Calculated neutron skins for 208Pb, 132Sn and 138Ba are shown in Fig. 3 based on 18 different Skyrme parameter sets and

27

0.4

..0

0.

co

0 N

I

I

I

I

I

0

0.3 -

-

oJ!

.....

.£

o. •

E 0.2 -

~ (j)

0.1

-

0.0 -50

"':.:

•••

,.

•

-

:-

••

-

I

I

I

I

1

0

50

100

150

200

250

derivative of the neutron EOS at p= 0.10

Fig. 2. The derivative of the neutron EOS at Po=O.lO neutrons/fm 3 (in units of MeV fm 3 /neutron) vs the S value in 208Pb for 18 Skyrme parameter sets (filled circles) and for six relativistic models (squares). The cross is the result for the Skx Skyrme Hamiltonian. 54

six different relativistic model parameter sets. 46 It shows that although there is a wide range of neutron skin values predicted with the different models, there is a strong correlation between the values in different nuclei. We can see, for example, that the value of S = 0.24(4) for 132Sn would correspond to a value of S = 0.18(4) for 208Pb. In summary, EDF models give a wide range of predictions for neutron skins of heavy nuclei and for the extrapolated neutron equation of state. It is important to find a way to accurately measure the neutron skin in heavy nuclei. Indirect and model-dependent results be must be consistent with each other. The relative neutron-skins of heavy nuclei are rather tightly constrained by EDF models. A model-independent determination from the parity-violating asymmetry of inelastic electron scattering from 208Pb can be used for calibration of other methods. FRIB will provide the means of understanding how the neutron skin evolves in nuclei far from stability. Acknowledgments

This work was supported by the National Science Foundation under Grant PHY-0555366.

28

0.4

ro 0.3

CO co

T

I

I

138Ba

C/)

+

N

;:?

.....

---E

0.1

l-

• + .t+

~ C/)

0.0 0.0

+ ++

••

•

•

rFJ§l

, ,

+ +<> ~

0.2 -

c

.E

-

~~

;:?

"cro

0

<>

I-

•

-

132Sn

-

.p

,•

I

I

I

0.1

0.2

0.3

0.4

S (fm) for 208Pb

Fig. 3. The neutron skin, S, for 208Pb vs those for 132Sn (filled circles and squares) and 138Ba (pluses and triangles) for 18 Skyrme parameter sets (filled circles and pluses) and six relativistic models (squares and triangles). The horizontal line is the Skx value for 208Pb.

References 1. http:j jwww.int.washington.edujPROGRAMSjria4.html 2. S. C. Pieper, R. B. Wiringa and J. Carlson, Phys. Rev. C 70, 054325 (2004); S. C. Pieper, Nucl. Phys. A751, 516c (2005). 3. P. Navrtil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, and A. Nogga, Phys. Rev. Lett. 99, 042501 (2007). 4. M. Horoi, J. R. Gour, M. Wloch, M. D. Lodriguito, B. A. Brown and P. Piecuch, Phys. Rev. Lett. 98, 112501 (2007). 5. G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, and A. Schwenk, Phys. Rev. C 76, 044305 (2007). 6. P. F. Bedaque and U. van Kolek, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002). 7. A. Schwenk and J.D. Holt, arXiv:0802.3741. 8. S. Gorielyand J. M. Pearson, Phys. Rev. C 77, 031301(R) (2008). 9. www.unedf.org 10. P. A. M. Guichon et al., Nucl. Phys. A772 1, 2006. 11. S. K. Bogner, T. T. S. Kuo and A. Schwenk, Phys. Rep. 386, 1 (2003). 12. M. Hjorth-Jensen, T. T. S. Kuo and E. Osnes, Physics Reports 261,

29

125 (1995). 13. www.fys.uio.no/compphys 14. B.A. Brown and W.A. Richter, Phys. Rev. C74, 034315 (2006). 15. M. Honma, T. Otsuka, B. A. Brown and T. Mizusaki, Phys. Rev. C65, 061301 (2002). 16. B. A. Brown and B. H. Wildenthal, Ann. Rev. of Nucl. Part. Sci. 38, 29 (1988). 17. NuShellX for Windows and Linux, W. D. M. Rae, 2008; knollhouse.org 18. NuShellX@MSU for Windows, B. A. Brown and W. D. M. Rae, 2008. 19. W. E. Ormand and B. A. Brown, Phys. Rev. C 52, 2455 (1995); W. E. Ormand and B. A. Brown, Phys. Rev. Lett. 62, 866 (1989). 20. 1. S. Towner and J. C. Hardy, Phys. Rev. C 77, 025501 (2008). 21. E. Cauier, P. Navratil, W. E. Ormand and J. P. Vary, Phys. Rev. C 66, 024314 (2002). 22. M. A. Bentley and S. M. Lenzi, Prog. in Part. and Nucl. Phys. 59, 497 (2007); B. A. Brown and R. Sherr, Nucl. Phys. A322, 61 (1979). 23. M. Bhattacharya et al., to be published. 24. B. Hyland et al., Phys. Rev. Lett. 97, 102501 (2006); G. F. Grinyer et al., Phys. Rev. C 77, 015501 (2008). 25. J. C. Hardy and 1. S. Towner, Phys. Rev. Lett. 88, 252501 (2002). 26. M. Horoi, S. Stoica and B. A. Brown, Phys. Rev. C 75, 034303 (2007). 27. E. Caurier, J. Menendez, F. Nowacki and A. Poves, Phys. Rev. Lett. 100, 052503 (2008). 28. V. A. Rodin, A. Faessler, F. Simkovic and P. Vogel, Nucl. Phys. A766, lO7 (2006). 29. A. Arima, K. Schimizu, W. Bentz and H. Hyuga, Adv. Nucl. Phys. 18, 1 (1987). 30. I. S. Towner, Phys. Rep. 155, 264 (1987). 31. L. Zhao, B. A. Brown and W. A. Richter, Phys. Rev. C 42, 1120 (1990). 32. M. Horoi and B. A. Brown, Phys. Rev. Lett. 74, 231 (1995). 33. N. Auerbach and B. A. Brown, Phys. Rev. C 60, 025501 (1999). 34. W. C. Haxton, C. P. Liu and M. J. Ramsey-Musolf, Phys. Rev. C 65, 045502 (2002). 35. B. A. Brown, N. J. Stone, J. R. Stone, I. S. Towner and M. HjorthJensen, Phys. Rev. C 71, 044317 (2005); erratum, Phys. Rev. C 72, 029901 (2005). 36. J. Engel, J. L. Friar and A. C. Hayes, Phys. Rev. C 61, 035502 (2000); V. V. Flambaum and V. G. Zelevinsky, Phys. Rev. C 68, 035502 (2003). 37. J. P. Jacobs et al., Phys. Rev. A 52, 3521 (1995).

30

38. M. V. Romalis et al., Phys. Rev. Lett. 86, 2505 (2001). 39. M. A. Bouchiat and C. Bouchiat, Rep. Prog. Phys. 60, 1351 (1997). 40. C. S. Wood et al., Science 75, 1759 (1997); S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2482 (1999). 41. V. A. Dzuba, V. V. Flambaum and 1. B. Kriplovich, Z. Phys. D I, 243 (1986). 42. E. N. Fortson, Y. Pang and L. Wilets, Phys. Rev. Lett. 65, 2857 (1990). 43. G. Fricke et al., Atom. Data and Nucl. Data Tables, 60, 177 (1995). 44. S. Eidelman et al. (Particle Data Group), Phys. Lett. B 592, 1 (2004). 45. B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000). 46. S. Typel and B. A. Brown, Phys. Rev. C 64, 027302 (2001). 47. C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86, 5647 (2001); Phys. Rev. C 64, 062802(R) (2001); Phys. Rev. C 66, 055803 (2003). 48. B. Friedman and V. R. Pandharipande, Nucl. Phys. A361, 502 (1981). 49. L. Tolos, B. Friman and A. Schwenk, arXiv:0711.3613; Nucl. Phys. A (2008) in press. 50. A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys. Rep. 411, 325 (2005). 51. J. Carriere, C. J. Horowitz and J. Piekarewicz, ApJ. 593,463 (2003). 52. S. F. Ban, J. Li, S. Q. Zhang, H. Y. Jia, P. Sang and J. Meng, Phys. Rev. C 69, 045805 (2004). 53. J. Meng, H. Toki, S. G. Zhou, S.Q. Zhang, W. H. Long, L. S. Geng Progress in Particle and Nuclear Physics 57, 470 (2006). 54. B. A. Brown, Phys. Rev. C58, 220 (1998). 55. R. Michaels, P. A. Souder and G. M. Urciouli, Thomas Jefferson National Accelerator Facility Proposal E-00-003, 2002, www.jlab.orgjexp_progjgeneratedjhalla.html. 56. S. Karataglidis, K. Amos, B. A. Brown and P. K. Deb, Phys. Rev. C 65, 044306 (2002). 57. B. C. Clark, L. J. Kerr and S. Hama, Phys. Rev. C 67, 054605 (2003). 58. J. Piekarewicz and S. P. Weppner, Nucl. Phys. A778, 10 (2006). 59. B. A. Brown, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C 76, 034305 (2007). 60. J. Piekarewicz, Phys. Rev. C 73, 044325 (2006). 61. A. Klimkiewicz et al., Phys. Rev. C 76, 051603(R) (2007).

31

SCHIFF MOMENTS AND NUCLEAR STRUCTURE J. ENGEL Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA E-mail: [email protected]

Measurements of atomic electric-dipole moments provide sensitive tests of CP violation and new physics. I briefly review the dependence of these moments on nuclear structure through the nuclear Schiff moment, and discuss attempts to evaluate the Schiff moment in several nuclei. After a short discussion of the uncertainty in these evaluations, I estimate the size of some previously neglected terms in the Schiff operator.

Keywords: CP violation, Schiff moments

1. Introduction Experiments with kaons and B-mesons indicate that time-reversal invariance (T) is violated at a low level. The results of these experiments can be explained by a phase in the Cabibo-Kobayashi-Maskawa (CKM) matrix of the Standard Model. But the absence of antimatter in our universe is evidence that T invariance (or more precisely, C P invariance) was badly violated long ago. The CKM phase is unable to account for so large an effect, and so theorists believe there must be another source of T violation, this one from outside the Standard Model. An atom in its ground state cannot have an electric dipole moment (EDM) without violating T. A number of experiments have searched for atomic EDMs, and the limits are tight. But because CKM T violation shows up in first order only in flavor-changing processes, it should appear in atomic experiments only after the limits are improved by 5 or 6 orders of magnitude. The same constraint does not apply, however, to T violation in extensions to the Standard-Model. The most popular extension, supersymmetry, has many flavor-conserving phases, making EDM experiments ideal for testing it. Already these experiments are putting extreme pressure on the theory.

32

Fundamental T violation can induce atomic EDMs in several ways. Here, we focus on a T-violating nucleon-nucleon interaction, which can be represented as the exchange of a pion with the T violation entering at a 7r N N vertex. The resulting interaction can be parameterized as 2

HT

= -

gm7r

{(O"I - 0"2) . (ri - is)

87rmN

X[.90TI·

T2 -

~I

(Tl z

_~1 (0\ + 52)· (rl x

+ T2z) + .92 (3TI z T2z

- r2)

- TI· T2)]

(Tlz - T2Z)}

exp( -m7r irl - r2 J) [ 1 1 1 + m7rirl - r2 i ' m7rjrl - r2j2

(1)

where the constants .90, .91, and .92 depend on the fundamental T-violating physics. This interaction leads to T-violating nuclear moments. Because of the Schiff theorem, the atom doesn't see a nuclear EDM to leading order, and the atomic effect is induced largely by the residual nuclear "Schiff moment" , which reflects finite nuclear size and relativistic kinematics. The Schiff moment is given approximately by

(Sz) = (11

0

Lep [r; - ~R~h] zp) ,

(2)

p

where R~h is the root-mean-squared nuclear charge-density radius. (We will investigate corrections to the approximations later.) The job of nuclearstructure is to determine the dependence of (Sz) on the .9'S of Eq. (1).

2. Calculations of Schiff Moments In recent years, relatively simple estimates2 have been replaced by more sophisticated many-body work. Here we quickly review some representative calculations, with apologies to any authors left out. In 199Hg, the atom that for some time has had the best limits on its EDM,l two sophisticated calculations exist, both based on mean-field theory in the even-even nucleus 198Hg, supplemented by RPA corrections from the polarization of the core by the valence neutron. Ref. 3 presents calculations that use a partially selfconsistent mean field with a Landau-Migdal residual interaction, not only in Hg, but a number of other spherical nuclei where measurements have been made or are possible. Ref. 4 contains a similar analysis in diagrammatic language, but is completely self consistent, with Skyrme interactions instead of the simpler Landau-Migdal interactions. The results the calculations in 199 Hg (each for a different Skyrme interaction, with SkO' preferred),

33

together with that of Ref. 3, appear in Tab. 1. The definition of the a coefficients, which contain all nuclear structure information, is through the relation (3) Table 1. Coefficients ai in 199Hg (defined in Eq. (3) for five different Skyrme interactions (from Ref. 4) and in Ref. 3. The units are e fm 3 .

SkM* SkP SIll SLy4 SkO' Ref. 3

ao

al

a2

0.009 0.002 0.010 0.003 0.010 0.00004

0.070 0.065 0.057 0.090 0.074 0.055

0.022 0.011 0.025 0.013 0.Q18 0.009

As first pointed out in Ref. 5, the Schiff moment can be enhanced by a few hundred in octupole-deformed nuclei. The most comprehensive estimate to date, through a self-consistent Skyrme-Hartree-Fock calculation with SkO', gives 6 aO

= -1.5,

al

= 6.0,

a2

= -4.0

(4)

in the experimentally attractive nucleus 225Ra. 3. Uncertainty

The spread in the results in Tab. I is a rough indication of the uncertainty in the nuclear-structure calculations. One way to get a reduce the uncertainty is is to examine predictions for observables that are related to the Schiff moment. One particularly useful quantity is the "isoscalar-dipole" strength. The operator that generates this strength is the isoscalar analog of the Schiff operator: (5) where the sum is over all nucleons. Figure 1 shows the corresponding strength distribution for 208Pb, calculated in the RPA with 3 different Skyrme interactions, alongside bars that indicate the peaks of the experimental distribution. The RPA-excitation of Schiff strength feeds directly into the calculation in Ref. 4 of core polarization in the ground state of

34

199Hg, so this distribution is particularly relevant. The figure indicates that the Skyrme interaction SIll does a poorer job than the other two interactions, and its predictions should be discounted accordingly.

36 SkP SkO' Sill

EX2

~

30

I

...-...

>

I I

~ 24

-

-


E

\

\

EX1

' -"

H

18 ..c ....... 0> C

Q)

'.......

CJ)

12

C')

I

0

...-

6 "

0

0

6

12

24 18 Energy (MeV)

30

36

42

Fig. 1. Isoscalar-El strength distributions for 208Pb predicted by the Skyrme interactions SkP, SkO' and SIn in self-consistent HF+RPA. The experimental bounds on the low-energy (Exl) and high-energy (E x2) peaks are also shown. (Figure taken from Ref.

4).

Besides examining the predictions of existing effective interactions, we can reduce the uncertainty in the calculations by deriving improved interactions. This task is one focus of the SciDAC UNEDF (Universal EnergyDensity Functional) collaboration. 7 It is not unrealistic to expect techniques such as the density-matrix expansion8 and coupled-cluster theory9 to produce better effective interactions within the next 5 years. Some phenomenological work in this direction has already been done. Figure 2 from Ref. 10 shows the results of tuning the Landau parameter gb, which only effects states with nonzero angular momentum and is usually

35

ignored in fits, to Gamow-Teller resonances when working with the Skyrme interaction SkO'.

1.0 0.8 +>

or 0.6 o

.............

/....

,

[/J

(:08

. . . .. •....• .., ....

0.4

.... .-••••••••

.-'-

0.2

....... .......•

• ••••

................. ..

··········t:::............. . .-............ :..............

•

.'

....

......

••••••

•••••

••••••••• ....

.208

Pb

•

124

Sn • 112Sn

0.0 ".--...

>

6 4 2

eO ................... ..........:::: : ••••• •
..

~

-2 <1 -4 -6

••••••• 0.0

..•....

0.5

••••

1.0

1.5

2.0

2.5

Fig. 2. Deviation of calculated and experimental Gamow-Teller resonance energies (lower panel) and a fraction of the Gamow· Teller strength in the resonance (upper panel) for 11 2 S n , 124Sn, and 208Pb, calculated with SkO', with the Landau parameter gb mod· ified and varied. (Figure taken from Ref. 10.)

4. Previously Neglected Terms in Schiff Operator Ref. 11 carefully rederives all the terms in Schiff operator, finding that to -+ z first order in the center of charge V == djZ, where d == 2: P =1 Tp is the

36

nuclear dipole operator and the sum is over protons a , the Schiff operator can be written more accurately than in Eq. (2) as

z

Sz

=

leO

L (f'p -

f

V

(6)

(zp - V z)

p=l

Expanding the expression to first order in

V gives

z Sz = ~ L (r;zp - r;Vz - 2(f'p. 15)zp) 10

+ ... ,

(7)

p=l

e ~ ( 2 5 2 v'21f 2 2 1) = 10 ~ rpzp - -:;/pVz + 4 --(rpY p ® V)o + ... , 3

(8)

where ® indicates angular-momentum coupling, and the last line is the form appearing in Ref. 11. Since V is itself a one-body operator, the second and third terms in both lines above contain one- and two-body operators. Prior derivations replaced r~ in the second term of Eq. (8) with its ground state expectation value ZR~h and neglected the third term (involving y2) in that equation altogether. Ref. 11 shows that these previously neglected terms are of the same order as those included for light systems such as the deuteron. Here we use sum rules to argue that in heavy nuclei the new terms can legitimately be neglected. We focus first on the operator 0 == I:p r~Vz in Eqs. (7) and (8). To the ground-state expectation value of Schiff operator, the physically important quantity, that term contributes

(0)

=

L (0\

L r;\i)(iIVz\O) =

(0)0 + (O)exc. ,

(9)

p

where the \0) is the substate of the ground-state multiplet with the largest vale of J z , the Ii) make up a complete set of intermediate states, and

(0)0 = (0\

L r;\O)(O\Vz\O) = ZR~h(Vz)

(10)

p

(O)exc.

=

L(O\ #0

L r;\i)(iIVz\O). p

r;,

The protons contribute coherently to the expectation value of I:p giving a factor of Z in (0)0. In the approximate Schiff operator used until now the sum over intermediate states is effectively replaced by the ground state aWe've neglected higher-order terms in aZ here,12 though Ref. 11 does not

37

alone, i.e. (0) is approximated by (0)0. The quality of the approximation depends in part on whether the contribution of intermediate excited states is coherent, like that of the ground state. We argue that excited states do not contribute coherently by looking at related sums. The energy-weighted sum (O)EW, which receives contributions only from excited states, is given by the expectation value of a double commutator:

p

=

~(O[ [2: r;, [H, Dzl [0).

(11)

p

Although the sum may have an imaginary part, its value is irrelevant because imaginary terms do not contribute to the unweighted sum in Eq. (9). If we neglect, as is typical, the momentum-dependence and isospin-exchange terms in the two-body interaction, we find the model-independent result (similar to that for the E1 sum): (12) where m is the nucleon mass. The coherent enhancement that gives the factor of Z in Eq. (10) is absent here. If the excited-state contribution came from a single energy, say 10 MeV, the corresponding unweighted sum (O)exc. in 199Hg would be about 4 fm 2 (D z ), a factor of about 600 less than the ground-state contribution (0)0. Of course we can't assume that the strength is so concentrated, or even that the terms in the sum contribute with the same sign, and will need to try to estimate the excited-state contribution a little differently. First, though, we look at the inverse-energy-weighted sum (O)IEw:

(13) It is straightforward to show that

(O)IEW =

-~ d~ ((DZ)H+ALpT~)

,

(14)

that is, -1/2 times the derivative with respect to A of the ground-state expectation value of D z , when A 2:: p r; is added to the Hamiltonian. If we assume that the nuclear mean field is approximately that of a spherical harmonic oscillator, the additional term just changes the strength of the

38

oscillator for the protons. (To get a qualitative estimate of the sum, we also change the neutron oscillator strength, by the same amount.) The expectation value (Dz) must be proportional to Rch, the only distance scale in the problem. Using Rch ex l/w (w is the oscillator frequency), and d/d)" = l/(mw)d/dw, we find 1

(O)IEW = - 22 (D z ) mw

.

(15)

Again there is no coherent enhancement. If all the strength were at 10 MeV, this sum rule woud again imply that the unweighted excited-state sum (O)exc. is about 4 fm 2 (Dz). What if there are many terms in the sums, with signs that fluctuate? Barring unnaturally exact cancellations, one would still expect that

(O)exc. (0) IEW ~ ~ E '

and

(16)

where E is an energy scale somewhere between 1 and 50 MeV at which, on average, states contribute to the sums. The above relations imply that E ~ V(O)EW/(O)IEW ~ V2JiW, which is about 10 MeV in a heavy nucleus. As we have seen, this value and either weighted sum rule implies that

(O)exc. «

(0)0.

(17)

The same would be the case if we chose 1 MeV or 50 Me V as the scale. Unless some unknown mechanism causes a precise cancellation in both weighted sums but not in the unweighted sum, one can neglect (O)exc .. This statement is still true when we include the third term in Eq. (7) in the definition of (0); doing so just increases (0)0 by roughly 5/3 (as reflected in Eq. (8)) and (O)EW by 5, not nearly enough to alter the conclusion above. Intermediate excited states do not contribute significantly to the expectation value of the Schiff operator in heavy nuclei. There are other ways of reaching this same conclusion. If the ground state can be well approximated by a Slater determinant, then

(0) =(O)direct =(0)0

+ (O)exchange -

1

z

L

(alr21,8)(;3Jzla),

(18)

a,f3
where a and ,8 label single-particle orbitals and the sum is over occupied levels. For large nuclei, the exchange term approaches (l/Z)(2: p r~zp), which is smaller by a factor of Z than the expectation value of the first term in Eq. (7) and therefore negligible. We are left again with (0) ~ (0)0.

39

Besides making this approximation, i.e. neglecting excited intermediate states between products of operators in Sz, calculations thus far have also ignored even the intermediate-ground-state part of the last term - the quadrupole term - in Eq. (8). Most heavy nuclei used in EDM experiments have J = 1/2, so that the intermediate-ground-state part of the quadrupole term vanishes. But even if J =I- 1/2 and it doesn't vanish, it will contribute much less than the monopole part. The largest known static quadrupole moment is about 700 fm 2 . This value implies that for large Z the groundstate part of the quadrupole term is smaller than that of the monopole term (the second term in Eq. (8)) by a factor that is at least 5 and is usually much more than that. The result of these considerations is that in heavy nuclei we need only in the second consider the first two terms in Eq. (8), and can replace term by its ground state expectation value. These simplifications yield the Schiff operator in Eq. (2) that has been used until now to calculate nuclear contributions to the dipole moments of heavy atoms. In addition, the form Eq. (6) implies that we don't need to correct the operator for centerof-mass motion, even when the new terms are included. The correction, accomplished by substituting fp - Rem for fp everywhere in the operator, doesn't cause any changes. That's not really surprising because nowhere in the derivation of the Schiff operator is a particular coordinate system assumed, so the operator must work as written in the center-of-mass frame. Nevertheless, it is good to know.

r;

References 1. M. V. Romalis, W. C. Griffith, J. P. Jacobs, and E.!N. Fortson, Phys. Rev.

Lett. 86, 2505 (2001). 2. V. V. Flambaum, I .B. Khriplovich, and O. P. Sushkov, Nucl. Phys. A 449, 750 (1986); O. P. Sushkov, V. V. Flambaum, and I .B. Khriplovich, Zh. Exp. Tear. Fiz. 87, 1521 (1984) [Sav. Phys. JETP 60, 873 (1984)]. 3. V. F. Dmitriev and R. A. Sen'kov, Phys. Atom. Nucl. 66, 1940 (2003); V. F. Dmitriev, R. A. Sen'kov, and N. Auerbach, Phys. Rev. C 71, 035501 (2005). 4. J. H. de Jesus and J. Engel, Phys. Rev. C 72,045503 (2005). 5. V. Spevak, N. Auerbach, and V. V. Flambaum, Phys. Rev. C56. 1357 (1997); N. Auerbach, V. V. Flambaum, and V. Spevak, Phys. Rev. Lett. 76, 4316 (1996). 6. J. Dobaczewski and J. Engel, Phys. Rev. Lett. 94, 232502 (2005); J. Engel, M. Bender, J. Dobaczewski, J. H. de Jesus, and P. o lbratowski , Phys. Rev. C 68025501 (2003). 7. http://vww.scidac.gov/physics/unedf.html. 8. J. W. Negele, Phys. Rev. C 1, 1260 (1970).

40 9. see, e.g., K. Kowalski, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, and P. Piechuch, Phys. Rev. Lett. 92, 132501 (2004). 10. M. Bender, J. Dobaczewski, J. Engel, and W. Nazarewicz, Phys. Rev. C 65 054322 (2002). 11. C.-P. Liu, M. J. Ramsey-Musolf, W. C. Haxton, R. G. E. Timmermans, and A. E. L. Dieperink, Phys. Rev. C 76,035503 (2007). 12. V. V. Flambaum and J. S. M. Ginges, Phys. Rev. A 65, 032113 (2002).

41

SUPERALLOWED NUCLEAR BETA DECAY: RECENT RESULTS AND THEIR IMPACT ON Vud J. C. HARDY· and I. S. TOWNER Cyclotron Institute, Texas Af3M University, College Station, Texas 77843, U.S.A . • E-mail: [email protected]

Measurements on superallowed 0+ --> 0+ nuclear beta transitions currently provide the most demanding test of the Conserved Vector Current (CVC) hypothesis and the most precise value for the up-down element, Vud , of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Both are sensitive probes for physics beyond the Standard Model. Analysis of the experimental results depends on small radiative and isospin-symmetry-breaking corrections, the validity of which is being probed by current measurements. We report on the current status of world data in light of recent improvements in both measurement and theory.

Keywords: Fermi beta decay, CVC, CKM matrix,

Vud

1. Introduction

Superallowed f3 decay between nuclear analog states of spin-parity, J7r = 0+, and isospin, T = 1, has a unique simplicity: it is a pure vector transition and is nearly independent of the nuclear structure of the parent and daughter states. The measured It value for such a transition can then be related directly to the vector coupling constant, G v , with the intervention of only a few small (rv 1%) calculated terms to account for radiative and nuclear-structure-dependent effects. Once a reliable value has been determined for G v , it is only a short step to obtain from it the value for Vud , the up-down mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix; and only another short step to the most demanding available test of the unitarity of that matrix, one of the basic precepts of the electroweak standard model. In dealing with these decays, it is convenient to combine some of the small correction terms with the measured It-value and define a "corrected"

42

Ft-value. Thus, we write 1

Ft == It(l

I

+ 6R )(1 + 6NS -

6c

)

K

= 2G~(1 + ~i{) ,

(1)

where K/(lic)6 = 27f3 liln 2j(m e c2)5 = 8120.278(4) x 10- 10 Gey- 4 s; 6c is the isospin-symmetry-breaking correction; and ~~ is the transitionindependent part of the radiative correction. The terms 6~ and 6NS comprise the transition-dependent part of the radiative correction, the former being a function only of the electron's energy and the Z of the daughter nucleus, while the latter, like 6c, depends in its evaluation on the details of nuclear structure. From this equation, it can be seen that a measurement of anyone of these superallowed transitions establishes an individual value for G v ; moreover, if the Conserved Yector Current (CYC) assertion is correct that G v is not renormalized in the nuclear medium, all such values and all the Ft-values themselves - should be identical within uncertainties, regardless of the specific nuclei involved. This assertion of CYC can be tested and a value for G v obtained with a precision considerably better than 0.1 % if experiment can meet the challenge, since the four small corrections terms only contribute to the overall uncertainty at the 0.03% level. As it turns out, experiment has exceeded that goal, leaving theory currently as the dominant contributor to the uncertainty. The It-value that characterizes any ,6-transition depends on three measured quantities: the total transition energy, QEC, the half-life, tr/2' of the parent state and the branching ratio, R, for the particular transition of interest. The QEc-value is required to determine the statistical rate function, I, while the half-life and branching ratio combine to yield the partial half-life, t. In 2005, we published a survey of world data on superallowed 0+ ----> 0+ beta decays,1 in which all previously published measurements were included, even those that were based on outdated calibrations if enough information was provided that they could be corrected to modern standards. In all, more than 125 independent measurements of comparable precision, spanning four decades, made the cut. A total of nine transitions yielded jt values with 0.1% precision or better, and three more with precision between 0.1 and 0.4%. The survey results are illustrated in Figure 1. From those results a number of important conclusions were drawn. First, the Ft values for all twelve transitions, covering the range from A=lO to A=74, formed a consistent set, which confirmed the constancy of G v to 1.3 parts in 10 4 , the tightest limit ever set. Second, the survey results set a limit on any possible contribution from scalar currents. The presence of

43 3140

3130

I

ft (1+o~) (5)

3120 "Ar

3110

rt ! ~l~·.,tII

"c 3090

if

""Aim

3080

i

"V

,.(5)

3090

UMg

3100

3100 .,.

3080

"Co

Y

3070

"Mn

c......._~~-J'-----:;~_L..---:;!:-_"--~ 3o6o'--L..----:1~0-.........--:2=0-........--:3:!:-0-......----I 10 20 30

Z of daughter

Z of daughter

Fig. 1. Results from the 2005 survey. 1 In the left panel is plotted the experimental It values corrected only for those radiative corrections that are independent of nuclear structure. In the panel on the right, the corresponding Ft values are given; they differ from the left panel simply by the inclusion of the nuclear-structure-dependent corrections, ONS and 0c. The horizontal grey band in the right panel indicates the average Ft value with its uncertainty.

ok,

a scalar current - induced or fundamental - would manifest itself as a Zdependence in the Ft values that would be most evident at low Z. There was no hint of any such curvature and a limit could be set on the scalar relative to the vector current of [Cs/C v [ ::; 0.0013, again the tightest limit ever set. Finally, with the test of CVC passed, it was possible to use the average value of G v to obtain the up-down element of the CKM matrix via the relation Vud = GV/G F , where G F is the well known 2 weak-interaction constant for purely leptonic muon decay. Since the 2005 survey closed, experiments in this field have intensified, with 10 additional measurements already published;3-12 in fact, a thirteenth precisely measured transition has been added to the set. There have also been improvements in the calculated correction terms,13,15 in large measure prompted by the new experimental results. We will briefly describe the recent advances.

2. Recent developments

The uncertainty obtained for Vud in the 2005 survey1 was dominated by the uncertainty in the theoretical correction terms. The first outcome was that it stimulated renewed interest in those calculated corrections, and before long an improved value for ~~ had been generated,13 with an uncertainty reduced by a factor of two compared with the previous result. Although

44 ~~ still remained the largest contributor to the uncertainty of Vud, it was now only slightly greater that the combined contributions of ONS and Since then, the nuclear-structure-dependent corrections, ONS and have also been reexamined 15 but that occured only after experiment had offered some hints. It might be imagined, with theory the largest contributor to the Vud uncertainty, that experiment would have no role to play until the theory could be improved. To the contrary, though, it was recognized that carefully chosen experiments could test the theory and guide it towards improvements. The approach is best illustrated by Fig. 1 and the observation that the calculated nuclear-structure-dependent corrections can be validated by their success in replacing the wide scatter of It-values (left panel of Fig. 1) with a set of statistically constant set of Ft values (right panel). Improvements in experimental precision would then test the calculations' effectiveness even more severely, as would new examples of 0+ - 0+ transitions, specifically selected for having larger calculated corrections. Although the first option, improving the precision on already well known transitions, might have seemed less appealing than the option to measure new cases, this approach has actually produced some of the most striking results in the past two years. Historically, the Q EC value of a superallowed transition has been a very challenging property to measure with sufficient precision. Since I depends on the decay energy to the fifth power, these Q values have to be determined to about ±0.01%, or ±500 eV for a 5-MeV (3 decay. Up to the time of the 2005 survey, the only measurements to meet these standards were of reaction Q-values, typically obtained from a (p,n) or eHe,t) reaction in which the {3-decay daughter nucleus was used as the target. The situation has completely changed since then. For the first time, with on-line Penning traps it has now become possible to measure parent and daughter masses individually and obtain the difference between them to a precision of a few hundred eV in light nuclei. 3,6-8 This has been, and still is, a very active research area, which almost from the start proved its value by revealing an anomaly in the result for the decay of 46V and, to a lesser extent, 42SC. In particular, with a pair of new Penning-trap measurements,3,7 the Ft value for 46V was suddenly shifted from its 2005 value of 3074.7(30)s, which appears in Fig. 1, to 3079.4(25)s, a result that disagrees with the average Ft value in that figure by over two standard deviations. The second experimental option to test the theoretical correction terms is to study new examples of superallowed transitions, particularly those

oc,

oc·

45

with large calculated corrections. The reasoning is that if the it values measured for cases with large calculated corrections also turn into corrected Ft values that are consistent with the others, then this must verify the calculations' reliability for the existing cases, which have smaller corrections. In fact, even by 2005 the cases of 22Mg, 34Ar and 74Rb were all chosen for this reason. It can be seen from Fig. 1 that all are characterized by large correction terms and all yield consistent Ft values. However it can also be seen that their error bars are still rather too large to put a serious constraint on the calculations. The reason for the large error bars in the cases of 22Mg and 34 Ar is quite different from the reason for 62Ga and 74Rb. That difference is important to appreciate. Most of the well-known superallowed emitters, like 26 AIm and 34CI, are odd-odd Tz = 0 nuclei that decay to even-even Tz = +1 daughters. For these cases, there are no 1+ states in the daughter that are energetically available to {3 decay, so the superallowed transition is the only branch that occurs with appreciable intensity: its branching ratio is essentially lOO% with negligible uncertainty. In contrast, 22Mg and 34 Ar are even-even Tz =-1 nuclei that decay to odd-odd Tz = 0 daughters, in which low-lying 1+ states do occur. For such cases, the superallowed transition is only one of several strong {3-decay branches, so its branching ratio must be measured, and the uncertainty on that measurement feeds directly into the uncertainty on its it value. Branching-ratio measurements with 0.1 % precision are extremely challenging. Even so, at Texas A&M we have taken on this challenge and have a preliminary result for the 34 Ar branching ratio at the 0.1% level. 14 We expect ultimately to reduce the it-value uncertainties to the same level for several other similar cases. The cases of 62Ga and 74Rb are quite different. They are Tz = 0 nuclei feeding even-even daughters, like the historically well-known cases. However, these being heavier nuclei, some 1+ states do appear within the {3decay energy window, and weak but important competing transitions can occur. This does introduce some complications, but recent experiments have demonstrated that these can be brought under control. l l The real difficulty lies with the nuclear models that must be used to calculate the nuclearstructure-dependent corrections. While the p, sd and h /2 shells have been reasonably well characterized by shell-model effective interactions, for nuclei in the region above A=60 experimental information is, as yet, relatively sparce, and there are no model calculations that can be regarded as reliable. Consequently, the calculated correction terms that depend on nuclear structure carry appreciably larger uncertainties for these nuclei. 15 ,16 Un-

46

fortunately what this means is that, although the measurements of their ft-values can provide a valuable check on the overall trend of the correctionterm calculations, no matter how precise the measurements are, they cannot help to reduce the uncertainties on Vud until we have a better grasp of the nuclear structure in this region of nuclei. This is undoubtedly an important task for radioactive-beam facilities in future. Thus, the situation by mid-2007 was that most new measurements were consistent with the correction terms used in the 2005 survey, with the important exception of a distinct anomaly, which had appeared at A = 46. As descri bed in detail by the previous speaker, 17 this anomaly led us first to reexamine the isospin-symmetry-breaking corrections for the 46V transition, but from what we learned there we were prompted to a more general reevaluation of the corrections for other transitions as well. 15 The outcome is that excellent self-consistency among the corrected Ft values has been restored while the value of Vud and the status of CKM unitarity has undergone a significant improvement.

3. Current status The current results from world data are illustrated in Figure 2, which compares the measured uncorrected ft values (points with error bars) with the theoretical quantity Ftj((l +ok)(l- Oc +ONS)) (grey bands), where Ft is the average value of Ft. The width of each band represents the estimated error on the theoretical corrections. The grey bands in the left panel correspond to the 2002 calculations for the correction terms,16 while in the right panel they come from the very recent reevaluation,15 which was prompted by the 46V anomaly clearly visible in the left panel. The comparison in both panels between experiment and theory tests whether the calculated structure-dependent corrections correctly match the observed nucleus-tonucleus variations in ft values. The left panel also shows the experimental improvements that have occurred in the last three years. There are grey points and error bars as well as black ones for four cases: 26mAl, 3 4Ar, 42SC and 46V. The grey points are the values from the 2005 review; 1 the black points represent the current world average, including all new measurements. 3- 12 ,14 In addition, the point for 62Ga is entirely new; in 2005, its uncertainty was too large for it to contribute. Several important observations can be made from Fig. 2. First, while results from the past three years have generally confirmed the calculated correction terms, they did create a distinct anomaly at 46V, possibly extend-

47

3080

"Ga

"Rbi i

-r~------------------------------

]"Ar

,t 3060

nM91

ft (S)

3040

Ii 10

~ "<:1

§K

;

~

\

~

"s"t}"co ~Mn

'4

IOe

f- ~C"'UI"od/

. .;

~~---------------------:~-------

"Ar t ~ / ~ tK ~ \ Calculated

"<:1

~

"SC"V

~

~ "CO

~Mn

16mAI

30200~~---1~0--~--~20~-L--~30~~--~40JLO---L--~10--~---2LO--~--3~0--~~40

Z of daughter

Z of daughter

Fig. 2. Both panels show experimental It values plotted as a function of charge on the daughter nucleus, Z. The grey bands represent the theoretical quantity Ft/«l + o~)(l - Oc + ONS))' The nuclear-structure-dependant corrections used in the left panel are those1 6 valid at the time of the 2005 review;l those used in the right panel come from the recent reevaluation. 15 The grey points and error bars in the left panel (see, for example, 46V) are results from the 2005 review; the black points and error bars are current world averages.

ing to 42SC. Secondly, the new calculations 15 have removed that anomaly but appear possibly to have created their own anomaly, with 50Mn and 54eo now disagreeing outside error bars with the calculated corrections. Rather than being a negative result, however, this possible discrepancy offers us the opportunity to use the cases of 50Mn and 54eo as a valuable test of our improved calculations. The Q EC value for each of them has been measured only twice with (claimed) high precision,18,19 and one of these references 18 also included a measurement of the QEC value for 46V, which Penning-trap measurements have recently shown3 ,6 to be low by 2 keY more than three times its originally quoted standard deviation. If, as seems likely, the problem with the 46V measurement in Ref. 18 is not limited to that measurement alone, then doubt is certainly cast on the 50Mn and 54eo QEc-value results quoted in that reference as well. New Penning-trap measurements of both Q EC values are currently in progress,20 and the question should be settled shortly. If the Q EC values in Ref. 18 prove to have been too low again, then the new Penning-trap measurements will serve to increase the Ft values for 50Mn and 54eo and could well bring them into close agreement with the average Ft value. If so, this would add strong support to our new calculations. The third important observation from Fig. 2 is that the uncertainties

48 3100

10C

22Mg 0

14

--

34Ar

42SC

26Alm "CI 38Km

SOMn 46V

"CO 62Ga 74Rb

3090

en CI)

-= n:I

3080

>

..!. ~

3070

3060

hL ~!!H t 5

10

15

20

25

30

0

35

Z of daughter Fig. 3. Current world data for superallowed Ft values. Data are from the 2005 survey 1 supplemented by more recent published data3 - 12 and the preliminary branching-ratio result for 34 Ar from Texas A&M.14

in the calculated correction terms for the cases with A ?: 70 has actually increased with the new calculations. This reflects the point already made that we have a very imperfect understanding of the nuclear structure in this region of nuclei. Unfortunately, for the time being this makes these very challenging jt-value measurements for the heavier cases much less valuable than they would otherwise be in validating the calculated correction terms or in providing useful input to the tests of CVC and CKM unitarity. As more is learned about the structure of these nuclei - something radioactive-beam facilities are ideally suited to study - and reasonable effective interactions have been derived to describe them, then the Ft values from 62Ca, 74Rb and other nuclei like them will take on the more significant role they deserve. 4. Vud and CKM U nitarity

Figure 3 presents the current results for the corrected Ft values corresponding to the data shown in the right panel of Fig. 2. Even though the values for 50Mn and 54CO are slightly low (pending remeasurement of both QEC values), all thirteen cases form a consistent set, with a resulting average Ft = 3071.4(8)s and a normalized X2 of 0.6. This leads to a value for Vud of

lVudi = 0.97418(26),

(2)

49 which compares with the value 0.97380(40) derived in our 2005 survey. 1 The reduced uncertainty of the present value comes mostly from the improvement in the calculation 13 of ~~ but also partly from recent experiments. The change in the central value comes entirely from the recent reevaluation of the nuclear-structure-dependent correction terms. 15 With the values of the other two top-row elements of the CKM matrix taken from the 2006 Particle Data Group review,2 the unitarity sum becomes

(3) in perfect agreement with the Standard Model.

Acknowledgments The work of JCH was supported by the U. S. Dept. of Energy under Grant DE-FG03-93ER40773 and by the Robert A. Welch Foundation under Grant A-1397. 1ST would like to thank the Cyclotron Institute of Texas A & M University for its hospitality during annual two-month summer visits.

References 1. J.C. Hardy and 1.S. Towner, Phys. Rev. C 71, 055501 (2005); Phys. Rev. Lett. 94, 092502 (2005). 2. W.-M. Yao et al., Journal of Physics G 33, 1 (2006). 3. G. Savard, F. Buchinger, J.A. Clark, J.E. Crawford, S. Gulick, J.C. Hardy, A.A. Hecht, J.K.P. Lee, A.F. Levand, N.D. Scielzo, H. Sharma, K.S. Sharma, 1. Tanihata, A.C.C. Villari, and Y. Wang, Phys. Rev. Lett. 95, 102501 (2005). 4. 1.S. Towner and J.C. Hardy, Phys. Rev. C 72, 055501 (2005). 5. B. Hyland, D. Melconian, G.C. Ball, J.R. Leslie, C.E. Svensson, P. Bricault, E. Cunningham, M. Dombsky, G.F. Grinyer, G. Hackman, K. Koopmans, F. Sarazin, M.A. Schumaker, H.C. Scraggs, M.B. Smith and P.M. Walker, J. Phys. G: Nucl. Part. Phys. 31, S1885 (2005). 6. T. Eronen, V. Elomaa, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, 1. Moore, H. Penttilia, S. Rahaman, S. Rinta-Antilla, A. Saastamoinen, T. Sonoda, J. Aysto, A. Bey, B. Blank, G. Canchel, C Dossat, J. Giovinazzo, I Matea, N. Adimi, Phys. Lett. B 636, 191 (2006). 7. T. Eronen, V. Elomaa, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, 1. Moore, H. Penttila, S. Rahaman, J. Rissanen, A. Saastamoinen, T. Sonoda, J. Aysto, J.C. Hardy, and V.S. Kolhinen, Phys. Rev. Lett. 97, 232501 (2006). 8. G. Bollen, D. Davies, M. Facina, J. Huikari, E. Kwan, P.A. Lofy, D.J. Morrissey, A. Prinke, R. Ringle, J. Savory, P. Schury, S. Schwarz, C. Sumithrarachchi, T. Sun, L. Weissman, Phys. Rev. Lett. 96, 152501 (2006).

50 9. P.H. Barker and A.P. Byrne, Phys. Rev. C 73, 064306 (2006). 10. V.E. Iacob, J.C. Hardy, J.F. Brinkley, C.A. Gagliardi, V.E. Mayes, N. Nica, M. Sanchez-Vega, G. Tabacaru, L. Trache, R.E. Tribble, Phys. Rev. C 74, 055502 (2006). 11. B. Hyland, C.E. Svensson, G.C. Ball, J.R. Leslie, T. Achtzehn, D. Albers, C. Andreoiu, P. Bricault, R. Churchman, D. Cross, M. Dombsky, P. Finlay, P.E. Garrett, C. Geppert, G.F. Grinyer, G. Hackman, V. Hanemaayer, J. Lassen, J.P. Lavoie, D. Melconian, A.C. Morton, C.J. Pearson, M.R. Pearson, A.A. Phillips, M.A. Schumaker, M.B. Smith, I.S. Towner, J.J. Valiente-Dob6n, K. Wendt , and E.F. Zganjar, Phys. Rev. Lett. 97, 102501 (2006). 12. J.T. Burke, P.A. Vetter, S.J. Freedman, B.K. Fujikawa and W.T. Winter Phys. Rev. C 74, 025501 (2006). 13. W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 96, 032002 (2006). 14. V.E. Iacob, to be published. 15. I.S. Towner and J.C. Hardy, arXiv:0710.3181 and to be published (2008). 16. I.S. Towner and J.C. Hardy, Phys. Rev. C 66, 035501 (2002). 17. I.S. Towner and J.C. Hardy, these proceedings. 18. H. Vonach, P. Glaessel, E. Huenges, P. Maier-Komor, H. Roesler, H.J. Scheerer, H. Paul and D. Semrad, Nucl. Phys. A278, 189 (1977). 19. V.T. Koslowsky, J.C. Hardy, E. Hagberg, R.E. Azuma, G.C. Ball, E.T.H. Clifford, W.G. Davies, H. Schmeing, U.J. Schrewe and K.S. Sharma, Nucl. Phys. A472, 419 (1987). 20. T. Eronen et al., private communication.

51

NEW CALCULATION OF THE ISOSPIN-SYMMETRY BREAKING CORRECTION TO SUPERALLOWED FERMI BETA DECAY I. S. TOWNER* and J. C. HARDY Cyclotron Institute, Texas A 8 M University, College Station, Texas 77843, U.S.A. * E-mail: [email protected]

We report new shell-model calculations of the isospin-symmetry-breaking correction, 5c, to superallowed Fermi (3 decay. The most important improvement is the inclusion of core orbitals, which are demonstrated to have a significant impact on the mismatch in the radial wave functions of the parent and daughter states. These new calculations lead to a lower average corrected Ft value and a higher value for Vud. The sum of the squares of the top-row elements of the CKM matrix agrees with unitarity. Keywords: Fermi beta decay, symmetry correction, CKM matrx,

Vud.

1. Introduction

In the most recent survey of superallowed 0+ --+ 0+ transitions, which appeared in 2005 [1], the results for all precisely measured cases were statistically consistent with one another. Since that survey, precise Penning-trap measurements [2,3] of QEC values for superallowed decays have produced the result that the 46y transition is more than two standard deviations away from the average of all other well-known transitions. This possible anomaly led us to reexamine the isospin-symmetry-breaking corrections for the 46y transition, but what we learned from that reexamination prompted us to a more general reevaluation of the corrections for other transitions as well. Our results [4] are summarized here. The situation following the Argonne QEC measurement [2] of 46y is displayed in Fig. 1. Plotted is the corrected Ft for the nine precision superallowed transitions with errors less than 0.2%. The 46y anomaly is plainly evident. The corrected Ft is defined as Ft == Jt(l

+ 6~)(1 + 6Ns -

60)

=

constant

(1)

52

and, according to the Conserved Vector Current (CVC) hypothesis, it is equal to a constant. Here the radiative correction is divided into a term that depends only on the electron's energy and the charge Z of the daughter nucleus, 8~, and a term that depends in its evaluation on the details of nuclear structure, 8N s. The isospin-symmetry-breaking correction of concern here is denoted 8c, while the experimentally determined quantity is the It value. The value of the latter depends on three measured quantities: the decay energy Q EC, the parent state half-life, and the branching ratio for the particular transition of interest.

5

9

13

17

21

25

Z of daughter Fig. 1.

Corrected Ft values for nine precision superallowed Fermi

2. Isospin-symmetry breaking correction,

f3 transitions.

~c

For weak vector interactions in hadron states, the CVC hypothesis requires the hadron state to be an exact eigenstate of SU(2) symmetry (isospin). In nuclei, SU(2) is always broken, albeit weakly, by Coulomb interactions between protons. There may be other charge-dependent effects as well. These influences shift the value of the hadron matrix element from its exact symmetry limit to a new value and this shift has to be evaluated before weakinteraction physics can be probed with hadrons. In the case of superallowed (3 decay, the hadron matrix element, M F , is written

(2)

53

where Mo is the exact-symmetry value and be is the correction we seek to evaluate. In the shell model for the cases of interest here, the A-particle wave functions representing the initial and final states for superallowed f3 decay, Ii) and 11), are states of angular momentum zero and isospin one. In a second quantisation formulation, the Fermi matrix element is written

MF

= UIT+ti) = 2)flala{3li)(aIT+If3),

(3)

a,{3

where the operator for Fermi f3 decay is the isospin ladder operator, al creates a neutron in quantum state a and a{3 annihilates a proton in quantum state f3. The single-particle matrix element, (aIT+If3), is just a radial integral

(4) If the proton and neutron radial functions R~(r) and R~(r) are identical, then the radial integral reduces to the normalization integral and has the value ra = l. Now we introduce into Eq. (3) a complete set of states for the (A - 1)particle system, 17r), by writing

(5) 7r,a

This is the essence of our model: we have allowed the radial integral to depend on the parentage expansion. Thus, we have added an additional label to ra and now write r~. If isospin is an exact symmetry, then the matrix elements of the creation and annihilation operators are related by hermiticity, (7rla a li) = Ulall7r)*. With that requirement, and with the radial integrals set to unity, the symmetry-limit matrix element is

(6) Thus we see that the breakdown of isospin symmetry can enter the evaluation of MF in one of two ways: either the matrix elements of aa and al are not related by hermiticity, or the radial integrals are not unity. Since each effect is small, we can, to first order, write the isospin-symmetry breaking correction as the sum of two terms

(7)

54

where in evaluating OC1 all radial integrals are set to unity but the matrix elements are not assumed to be related by hermiticity, while in evaluating OC2 it is assumed that (1fla a li) = Ula111f)* but the radial integrals are allowed to differ from unity. Past calculations [5,6] have indicated the radial overlap correction, OC2, is the larger of the two corrections; we will only study this term here. For the OC2 calculation, the Fermi matrix element is MF

l: IUla111f)12r~

=

n,a

(8) where Mo is the exact-symmetry value, Eq. (6), and n~ has been introduced as a radial-mismatch factor n~

= (1- r~).

(9)

With Eqs. (2) and (8), we obtain

2 OC2 ~ M

o

l: IUla~I1f)12n~

(10)

7T,G

to first order in small quantities. A large contribution to OC2 therefore requires a large spectroscopic amplitude and a significant departure of the radial integral from unity. There is an opportunity here to take guidance from experiment. The square of each spectroscopic amplitude, IUla111f)1 2, is related to the spectroscopic factor measured in neutron pick-up direct reactions. The exact relation, after inserting the isospin angular momentum couplings, is

o C2

~

'"' Tf(Tf + 1) + i - Tn (Tn + 1) ~

T (T f

n,a

f

+ 1)

ST" a,Tj

nn a

(11)

where S;"T is the spectroscopic factor for pick up of a neutron in quantum , f state ex from an A-particle state of isospin T f to an (A - 1)-particle state of isospin Tn. On setting T f = 1 and separately identfying sums to the isospindenoted 1f<, and the isospin-greater states with lesser states with Tn = Tn = ~, denoted 1f>, we obtain a very revealing formula

i,

OC2

~

l: S;; n;; - ~ l: S;; n;;. 1T< ,0::

(12)

7r> ,0::

This equation provides the key to the strategy we will use in calculating

OC2. It demonstrates that there is a cancellation between the contributions

55

of the isospin-Iesser states and the isospin-greater states. Moreover, if the orbital (X were completely full in the initial A-particle wavefunction, then the Macfarlane and French sum rules [7] for spectroscopic factors would require L7r< s;; = ~ L7r> S;; and the cancellation in Eq. (12) would be very strong. In fact, the cancellation would be complete if n;; = n;;. The key, however, is that this cancellation is not in general complete because the radial-mismatch factors for isospin-Iesser states are larger than those for isospin-greater states. Even so, cancellation is always significant, and it becomes most complete when closed-shell orbitals are involved. Furthermore, the more deeply bound the closed-shell orbital, the greater the energy spread in the spectroscopic strength and the more complete the cancellation. Thus, although the dominant contributions to OC2 come from unfilled orbitals, we conclude that closed-shell orbitals must playa role, albeit one that decreases in importance as the orbitals become more deeply bound. Based on these observations, our strategy is to use experiment to guide us in determining which closed-shell orbitals are important enough to include. Ideally, of course, one would take the spectroscopic factors determined from experiment and insert them into Eq. (12) but, especially where delicate cancellations are involved, the reliability of (forty-year-old) experimental spectroscopic factors is certainly not up to the task. Our strategy then is to use the shell model to calculate the spectroscopic amplitudes in Eq. (10) but to limit the sum over orbitals (X just to those for which large spectroscopic factors have been observed in neutron pick-up reactions.

Table 1. Illustration of the strategy used in calculating 8C2 for 46y. The measured spectroscopic factors from the 46TieHe, a)45Ti reaction [8] are shown for the states where they are largest and compared with a shell-model calculation. The contribution from each state to 8C2 is given in the last column.

45Ti Ex (keY)

0 330 1566

P';T" 7- 1

a

eHe, a) measured [8] So.

2.7(11)

0.134

3.36

0.45 0.39

O~(%)

L:"

Shell Model contribution to 8C2 (%) 5;'

2"

2"

17/2

2"

2"

d3/ 2

1.9(8)

0.157

2.45

2"

81/2

0.7(3)

0.318

1.22

0.39

0.085

2.74

-0.12

3

1

2"

1

1

7- 3

4723

2"

2"

17/2

3.6(16)

4810

2"

2"

d3/ 2

3.6(16)

0.100

4.92

-0.25

3.2(12)

0.224

2.47

-0.28

5760

3

1

2"

3

3

2"

81/2

56

We illustrate the strategy for the case of 46V. The spectroscopic factors for neutron pick up from 46Ti have been measured in the eHe, a) reaction by Borlin [8]. He identified sixteen states in 45Ti, and in Table 1 we record the six states with the largest spectroscopic factors, i.e. S > 0.5. We note that the errors on the experimental spectroscopic factors are quite large, and in two cases the quoted Sa values (column 4) exceed the Macfarlane-French sum rule [7] for pure configurations. Thus we do not use the experimental spectroscopic factor explicitly, but take them as a guide for which orbitals should be included in the shell-model calculation. In the case of 46V decay, they tell us that orbitals 17/2, d 3/ 2 and 81/2 should be included. In column five of Table 1 we give a typical value for the radial mismatch factor, n~, for the given orbital a and isospin T K • Column seven gives the contribution to 6c2 from this a and isospin TK obtained with a detailed shell-model calculation. The summed 6c2 for the shell-model calculation (the sum of all entries in column 7) is 0.58%, nearly a factor of two larger than our previous calculated value, which was published in 2002 [5]. The difference between our calculations arises as follows: In 2002 our shell-model calculations for 46V were based on the model space Up)6, with six valence nucleons occupying the pf-shell orbitals. In fact, only the 17/2 orbital contributed importantly to the 6c2 calculation so the result was 6c2 = 0.45 - 0.12 = 0.33% (see the two rows for the 17/2 orbital in Table 1). Absent from this 2002 calculation was any contribution from the core orbitals, d3 / 2 and 81/2' In our present calculations, these orbitals are included, with the d 3/ 2 orbital contributing 0.14% to 6c2 and the 81/2 contributing 0.11%. With this approach, we are now in a position to revise our earlier results [5] to include the effects of previously ignored core orbitals. Again using measured spectroscopic factors from neutron pick-up reactions, we determined that changes were required for the A = 22 and 26 cases, in which p-shell holes must contribute in addition to the original 8d-shell configurations; similarly, 8d-shell holes were required in addition to the pf-shell particles for A = 46, 50 and 54. For A = 62, 66, 70 and 74 in the upper pf-shell there are no experimental neutron pick-up reaction measurements to guide us. Our previously published calculations for these nuclei were based on (P3/2, f5/2, P1/2)n model spaces using 56Ni as a closed-shell core. It seemed prudent now for these cases at least to include the 17/2 orbital in the calculation of 6c2, and we have made this change. In the cases with A = 18 and 42, we had previously included some contribution from deeper shells; we did not need to make any changes in the former but did add the

57

81/2 and d S/ 2 shells to the latter. No additional orbitals were required for the cases with A = 10, 14, 30, 34 and 38.

.

,

2.0

-

Ell TH02

•

1.5

THOT

:

6C2(%) 1.0

0.5

•

0.0

4~

t

1m',!! Ell

Ell

Ell

• • EI

,_--L__...L.._---JI..-_....L._ _l-,_...J

L-_....L_ _......

o

10

20

30

40

Z of daughter Fig. 2. Comparison of the isospin-symmetry-breaking correction, 0C2, reported here, TH07, and that published [5] five years earlier, TH02.

3. Results and the effect on CKM unitarity The results from the new calculation are displayed in Fig. 2, where they are compared with the 2002 calculation [5]. The trend is an increase in the OC2 value, particularly for nuclei in the fp shell. With these new OC2 values, plus some modest changes to OCl, o~ and ONS discussed in ref. [4], the corrected Ft values have been recomputed. The results for the nine precision data, augmented with results for 22Mg, 34Ar, 62Ga and 74Rb, which are approaching the accuracy of the original nine, are shown in Fig. 3.

58

It is evident that 46y no longer shows any deviation from the overall average

as it did in Fig. 1. However, it is equally evident that instead the 50Mn and 54Co Ft values are now low, and by amounts that are no less statistically significant than the amount by which the 46y value was previously high. Rather than being a negative result, however, this possible discrepancy offers us the opportunity to use the cases of 50Mn and 54CO as a valuable test of our improved calculations. The Q EC value of each of them has recently been measured in a Penning trap, JYFLTRAP, by the Jyviikylii group [9] and will be published soon.

3085

Ft 3080

5

9

17

13

21

25

29

33

37

Z of daughter Fig. 3. Corrected Ft values for 13 precision super allowed Fermi f3 transitions with the isospin-symmetry-breaking correction, 0C2, discussed here.

The average corrected Ft obtained from our new analysis, 3071.4(8) s, is lower by more than one standard deviation, compared to the result obtained in our 2005 survey, 3072.7(8) s. This yields a larger value for the up-down element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix:

iVudl =

0.97418(26).

(13)

With the values of the other two top-row elements of the CKM matrix taken from the 2006 Particle Data Group review [lO], the unitarity sum becomes

iVudl 2 + iVusl 2 + iVubl 2 = 1.0000±0.001l

(14)

59

in perfect agreement with the Standard Model.

Acknowledgments The work of JCH was supported by the U. S. Dept. of Energy under Grant DE-FG03-93ER40773 and by the Robert A. Welch Foundation under Grant A-1397. 1ST would like to thank the Cyclotron Institute of Texas A & M University for its hospitality during annual two-month summer visits.

References 1. J.C. Hardy and 1.S. Towner, Phys. Rev. C 71, 055501 (2005). 2. G. Savard, F. Buchinger, J.A. Clark, J.E. Crawford, S. Gulick, J.C. Hardy, A.A. Hecht, J.K.P. Lee, A.F. Levand, N.D. Scielzo, H. Sharma, K.S. Sharma, 1. Tanihata, A.C.C. Villari, and Y. Wang, Phys. Rev. Lett. 95, 102501 (2005). 3. T. Eronen, V. Elomaa, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, 1. Moore, H. Penttilii, S. Rahaman, J. Rissanen, A. Saastamoinen, T. Sonoda, J. Aysto, J.C. Hardy, and V.S. Kolhinen, Phys. Rev. Lett. 97, 232501 (2006). 4. 1.S. Towner and J.C. Hardy, arXiv:0710.3181 and to be published (2008). 5. 1.S. Towner and J.C. Hardy, Phys. Rev. C 66, 035501 (2002). 6. W.E. Ormand and B.A. Brown, Phys. Rev. C 52,2455 (1995). 7. J.B. French and M.H. Macfarlane, Nucl. Phys. 26, 168 (1961). 8. D.D. Borlin, thesis, Washington University (1967), recorded in the nuclear data sheets of the National Nuclear Data Center: www.nndc.bnl.gov. 9. T. Eronen et al., to be published (2008). 10. W.-M. Yao et al., J. Phys. G 33, 1 (2006).

60

PRECISE MEASUREMENT OF THE 3H to 3He MASS DIFFERENCE D. B. PINEGAR*, T. P. BIESIADZINSKI,

c.

M. HOTCHKISS, R. B. WEH,

S. L. ZAFONTE and R. S. VAN DYCK, JR.

Department of Physics, University of Washington, PAB 351560, Seattle, WA 98195, USA

This article summarizes preparations at the University of Washington for a precision measurement of the mass ratio of H-3 (tritium) to He-3 with a new Penning trap mass spectrometer. This work will be continued at the MaxPlanck-Institute for Nuclear Physics in Heidelberg in the Division of Stored and Cooled Ions. Only preliminary ion observations were performed in Seattle, but the target mass uncertainty for the measurement techniques under development is 1 part in 10 11 .

Keywords: Penning-trap; Mass spectrometer; Atomic masses; Tritium; He-3

1. Introduction

Measurements of the kinematics of jJ-decay can set limits on the absolute scale of the neutrino mass eigenvalues. The KATRIN jJ-spectrometer has an anticipated sensitivity of m(ve ) '" 0.2 eV, and independent 3H - 3He mass difference measurements with 100 me V / c2 accuracy can provide an important check on systematic uncertainties. 1 In the extremely precise quadrupole electrostatic potential of a Penning trap mass spectrometer,? the axial motion of the ion approximates a 1dimensional simple harmonic oscillator. However, the electric field at the ion location is perturbed by externally applied drive voltages and inducedcharge fields from the trap electrodes. By changing the induced-charge distribution on the electrodes, the ion can modify the electrical signal between the endcaps. For the spectrometers developed at the University of Washing*Present Address: Division of Stored and Cooled Ions Max-Planck-Institut fur Kernphysik, PO Box 103980 69029 Heidelberg, Germany E-mail: [email protected]

61

ton (UW), this electrostatic coupling between the axial motion of the ion and the trap endcaps is the source of all knowledge of the state of the ion. Shifts of the axial frequency induced by cyclotron excitation allow determination of the natural cyclotron frequency. Mass ratios can be calculated because the free-space cyclotron frequency ratio of two ions in the same magnetic field is inversely proportional to their mass ratio. The mass difference.6. = meH)-meHe) can be found without knowing the individual (absolute) masses to comparable precision, because when the mass ratio C = meH)jmeHe) is measured, the uncertainty in the mass difference is

Only the term including uncertainty in C makes a significant contribution since the low energy of tritium beta-decay makes (C - 1) ;::::;; 6 X 10- 6 . The mass difference .6. has been determined using ,B-spectrometers with carefully calibrated retarding potentials. The three most precise results of this type are from the Los Alamos 2 ,3 and Mainz 4 spectrometers. At 2 18589.0(2.6) eV j c and 18590.6(2.0) eV j c2 , the 1989 and 1991 Los Alamos results are more precise than the 18591(3) eVjc 2 Mainz .6. measurement from 1993. Presently the most precise measurement of .6. comes from the SMILETRAP group, where it was determined by subtracting the two atomic masses,9 which were measured with a Penning trap time-of-flight ioncyclotron-resonance (TOF-ICR) mass spectrometer.? This destructive ICR detection method is quite different from the methods used at the UW, which makes the two methods complimentary. In particular, the need to use many single-ion cyclotron excitations to get good statistics for the TOF-ICR experiment reduces uncertainties due to contamination ions in the standard UW data runs, where each ion of interest is interrogated for many days and measurements are usually based on observations of about 10 individual single-ion data-runs. The first 3H and 3He atomic masses from a Penning trap mass spectrometer were published by the UW group in 1993. 6 They are 3016049267.25(1.54) nu and 3016029309.98(98) nu, respectively, for .6. = 18590.0(1.7) eV jc2 . However, these atomic masses, along with several other Seattle measurements from that era, such as the 1995 4He atomic mass,7 are inaccurate due to diurnal temperature variations that changed the B-field of the spectrometer, correlated with the "ion-of-interest in the afternoon, reference ion at night" data-taking procedure that was standard at that

62

time, before the data-taking was computer automated. But it appears that the resulting systematic shifts were about the same for meH) as for the meHe) data, since the 1993 UW b. result agrees quite well with the more recent SMILE TRAP results. Following suspected problems with the Seattle 4He result, m(4He) and m(3He) were measured at SMILETRAP,8 and the disagreement with the UW result of smaller stated uncertainty gave strong motivation for independently measuring meH) as well. Their 2006 values for m(3H) and m(3He) are 3016049278.7(2.5) nu and 3016029321.7(2.6) nu, respectively, 2 which gives their mass difference 18589.8(1.2) eV jc .9 This is the most precise b. determination from a single experiment, and it agrees well with 18591(I)eVjc2, the average compiled from all ;3-decay related determinations, as stated in the 2003 Atomic Mass Evaluation (AME).l0 Independent of the future 3Hj 3He mass ratio measurement with the new spectrometer, a measurement of meHe) by comparison to 12C is ongoing at the University of Washington. Early results were given in Seth Van Liew's 2004 PhD thesis,11 and extensive discussion can be found there. His original result was 3016029321.25(36) nu, but an error was recently recognized in his analysis. When it is taken into account, the atomic mass from his two 3He runs becomes 3016029321.69(8) nu,12 which is in good agreement with the less precise SMILETRAP result, as well as subsequently-acquired Seattle data. Figure 1 summarizes the atomic mass measurements discussed in this section, and also includes Lincoln Smith's RF spectrometer results, which gave a vast improvement in precision by relating masses to frequencies rather than distances and B-field strengths alone as in previous instruments. 2. Partial Demonstration of the New Technology at UW The Penning trap mass spectrometer techniques presented here were developed to measure the ratio of these particular masses at the 10- 11 precision level. They include an external ion source and beam-line to create helium and tritium ions and transport single ions to two Penning traps 4 cm apart. Electronics developed for this double Penning trap arrangement uses a single differential amplifier and extremely stable solid-state ring-electrode voltage source to measure the cyclotron frequency ratio of the two species in a stable magnetic field. 14 ,15 At the present time, all of the hardware, and most of the software for the new spectrometer has been constructed. The ion beam is transmitted to the cryogenic electrodes as expected, and

63 49290

Tritium 49280 49270

Smlh

49260 ~ 49250

+ 0 0 0 0 0

49240

~ 49230

o~1975 <')

1980

1990

HeIium-3

'"'"oj

~ 29320

.§ ~

1985

29330

29310 2930

f

. I ~

Smlh

.....

e

....

UW-PTMS 29290 29280 2.927

1975

1980

1985 1990 1995 Year of Publication

2000

2005

Fig. 1. Precise 3H and 3He Atomic Mass Measurements. The 1975 points are from Smith's RF mass spectrometerJ3 The two 1993 points? and the original result from Van Liew's thesis l l (which has an uncertainty too small to show on this scale) are from the University of Washington Penning trap mass spectrometer. The other 3 recent measurements are SMILETRAP's TOF-ICR results. 8 ,g

the ion source and trapping/transfer electronics were tested and working as expected. However, due to lack of time the fast-switching electronics and computer control software were not used to load the Penning traps from the ion beam, or transfer ions from one trap to another. Figure 2 shows a block diagram that outlines the basic relationship between the new double Penning traps, the axial ion motion detection elec-

5

64

tronics, and the way the axial frequency shift signals are sent to the ring reference system, which produces the ring voltages for the two Penning traps. Further information can be found in Ref. 16.

Room

Temperature Electronics

Ring-Bias Voltage System

Ring Correction Signals

Fig. 2. Axial motion detection and feedback electronics for two traps. This block diagram shows the use of one amplifier to observe and frequency-lock the axial ion motion in two separate traps. The room temperature electronics uses two separate mixers to monitor and lock the natural frequencies of the two ions, by monitoring the phases of the two sinusoidal ion response signals which are each present at the output of the amplifier.

It was only in the second half of the summer of 2007 that the new apparatus was used to trap ions, and efforts were focused on working with the lower Penning trap using ions loaded with the field emission point. For rough voltage scans to see different ion species, the ring electrode was biased with a manually swept power supply. With this system many ion species were observed ranging in q/m from 1 for protons at about 29 V to 1/4 for 04+ at 118 V. With the precision ring-bias system set for 80-90 V output range, C4+ (resonant at a ring potential of 88.4 V) was usually isolated, but 3He+ was also loaded by running the field emission point while leaking gas in from the 3He cylinder. When loading the trap from surface contaminants dislodged by electron bombardment, oxygen and hydrogen were the most common species found, but carbon was plentiful and nitrogen was also seen. The first ion clouds were detected when some of their axial energy (initially quite large from the loading process) was transferred to the detector circuit as the ions were brought into resonance with it during ring voltage sweeps. Next, once the ions were cooled, "noise shorting" could be seen, in which clouds of ions reduce the thermal noise of the detector circuit because their mechanical resonance reduces the impedance of the complete ion-trap-

65

detector system. Eventually, as contamination-ion elimination procedures were improved, coherently driven ion motion was seen, such as the resonance from several hundred protons shown in Fig. 3. With detector phases adjusted to obtain dispersion-shaped signals, the new ring bias system was used to lock the axial frequency of a C4+ ion cloud. However, due to a series of problems such as poor vacuum pressure, contamination ions, and eventually lack of time, clouds smaller than several hundred ions were never observed. Similarly, the new axial detector feedback loop system includes an improved compensator design that should allow higher detection bandwidth 8v than what is possible with the UW single-trap spectrometer, but because of the noise sources mentioned earlier, frequency lock-loop bandwidths above 8v rv 0.2 Hz were not obtained with the new system.

0.7

r---r------,-------,.----.---~-__,

0.6

05

'"

~

0.4

C. E

...: 03

0L--4.-11L7-9-5--4~.1-18-0---4.~11~8~5~-4~.1~18~1~0~~4~.1~18~1~5~ Frequency (M Hz)

Fig. 3. Ion cloud axial resonance. Several hundred protons in the lower hyperbolic Penning trap were observed with the new phase-sensitive detection electronics. Shifting the detection phase by 7r /2 gives the dispersion line-shape used for a control-loop errorsignal which is nulled during axial-frequency lock.

Acknowledgments

This material is supported by the National Science Foundation under Grant No. 0353712

66

References 1. E. W. Otten, J. Bonn and C. Weinheimer, International Journal of Mass Spectrometry 251, 173 (2006). 2. S. T. Staggs, et aI., Phys. Rev. C 39, 1503 (1989). 3. R. G. H. Robertson, et a!., Phys. Rev. Lett. 67, 957 (1991). 4. C. Weinheimer, et aI., Physics Letters B300, p. 210 (1993). 5. C. Carlberg, et a!., Physica Scripta 59, 196 (1995). 6. R. S. Van Dyck, Jr., D. L. Farnham and P. B. Schwinberg, Phys. Rev. Lett. 70, 2888 (1993). 7. R. S. Van Dyck, Jr., D. L. Farnham and P. B. Schwinberg, Physica Scripta T59, 134 (1995), Trapped Charged Particles and Related Fundamental Physics. Proceedings of Nobel Symposium 91. 8. T. Fritioff, C. Carlberg, G. Douysset, R. Schuch and 1. Bergstrom, The European Physical Journal D 15, 141 (2001). 9. S. Nagy, T. Fritioff, M. Bjorkhage, 1. Bergstrom and R. Schuch, Europhys. Lett. 74, 404 (2006). 10. A. H. Wapstra, G. Audi and C. Thibault, Nuclear Physics A 729, 129 (2003). 11. S. Van Liew, An ultra-precise determination of the mass of helium-3 using Penning trap mass spectrometry, PhD thesis, University of Washington, (Seattle, 2004). 12. S. Van Liew. Personal comminucation, (2008). 13. L. G. Smith and A. H. Wapstra, Phys. Rev. ell, 1392(Apr 1975). 14. R. S. Van Dyck, Jr., D. B. Pinegar, S. Van Liew and S. L. Zafonte, International Journal of Mass Spectrometry 251, 231 (2006). 15. D. B. Pinegar, S. L. Zafonte and R. S. Van Dyck, Jr., Hyperfine Interactions 174,47 (2007). 16. D. B. Pinegar, Tools for a precise tritium to helium-3 mass comparison, PhD thesis, University of Washington, (Seattle, 2007).

67

LIMITS ON SCALAR CURRENTS FROM THE 0+ ---t 0+ DECAY OF 32Ar AND ISOSPIN BREAKING IN 33Cl and 32Cl A. GARCiA Department of Physics, University of Washington Seattle, WA 98195-1560, USA * E-mail: [email protected]

We present a summary of recent experiments performed to: 1) optimize the extraction of the correlation coefficient from 32 Ar, and 2) determine isospin breaking in both 33Cl and 32Cl.

Keywords: Scalar currents, electron-neutrino correlation, isospin breaking.

1. Scalar currents

Determination of the positron-neutrino correlation in 0+ ---+ 0+ transitions allows to search for potential Scalar contributions to the weak interaction. The differential decay rate has been calculated by Jackson, Treiman and Wyld: 1

(1)

where E e , p and me are the total energy, momentum and mass of the positron; Ev the energy of the neutrino; Mi and Ef = Mf + T are the mass of the parent nucleus and the total energy of the daughter nucleus respectively, and F( - Z, p) the Fermi function for (3+ -decay. A termed called the 'Fierz interference' has been neglected but good upper limits have been placed using the log It's from nuclear decays. 2 The positron-neutrino correlation coefficient, a, depends on whether the weak currents are purely vector, or wether there is a contribution from scalar

68 currents: a

rv

[CV [2 [CV [2

+ [C~[2 - [CS[2 - [C8[2 + [C~[2 + [CS[2 + [C8[2'

(2) (3)

and the constants C represent the effective coupling constants; the subindex indicates wether they are vector or scalar and the primed (non-primed) refer to the parity violating (non-violating) components. In the absence of scalar currents the correlation coefficient should be a = 1. The 0+ -+ 0+ decay of 32 Ar presents a unique opportunity to search for scalar currents: because it is a pure Fermi transition the (3 decay is free of the complications to the correlations involved when Gamow-Teller contributions have to be considered, and because the decay is followed by proton emission, one can determine the positron-neutrino correlation simply by measuring the 'Doppler' broadenning of the proton line. 3 Fig. 1 shows what is expected for the proton line assuming pure vector or scalar currents in the beta decay. In principle, then, the challenge is to determine the shape of the superallowed peak with the best possible resolution. Because there is

-15-10 -5 0 5 10 E-Eo (keV)

15

Fig. 1. Monte-Carlo calculations of the shape of the proton line assuming pure vector (broader curve) and scalar (narrower curve) currents in the beta decay of 32 Ar. This proton line has a central energy of Ep ~ 3.3 MeV.

this special opportunity and in order to fully take advantage of it, we have worked on ancillary experiments to extract physics needed to optmize the analysis of our original 32 Ar Isolde data. A sketch of the apparatus used at Isolde to determine the positronneutrino correlation is shown in Fig. 2. Briefly, an 32 Ar beam from Isolde

69 Peltier with feed back

32Ar beam

!com

I

ISOld; L

,

+ ,.e

_~~~~::i ...... J

~beam

collimator

Fig, 2.

LN2 reservoir

3.5 Tesla

---..

LN2 intake

detector collimator

Sketch of apparatus used at Isolde.

was implanted on a thin carbon foil and protons detected at 90 degrees with respect to the direction of the incoming beam. In order to avoid systematic uncertainties related to summing of positrons and protons, the setup was submerged in a 3.5-Tesla magnetic field to impede positrons from reaching the proton detectors. The detectors were kept at a temperature of -11 °C by thermoelectric elements that held the diode temperatures constant to ±0.02 DC. This stability was crucial to obtain the best energy resolution. In order to minimize systematic uncertainties in determining the positron-neutrino correlation one needs a precise determination of the energy calibration and of the masses of the parent and daughter states. The energy calibration for the spectrum from 32 Ar was obtained from the decay of 33 Ar, which has several lines that have been identified with resonances on 32S(p, 'Y) that can be determined with precision. Recent work using the accelerator at Seattle allowed us to determine these energies to '" 0.2- -0.4 keV. 4 Fig. 3 shows the beta-delayed proton spectrum from 33 Ar where we indicate with arrows the peaks that were used for determining the energy calibration. Another important ingredient in the determination of the correlation coefficient are the masses of 32 Ar and the daughter state. The former was determined by the IsolTrap group at Isolde. 5 Further work was carried out by us to determine whether there could be signs of problems in the mass determinations by looking at the Isobaric Mass Multiplet Equation. 6 In our latest analysis we also fix the detector response function by simultaneously fitting the peaks corresponding to the superallowed transistions

70

Energy calibration 100000

!

--Data R-matrix fit

10000

> (I)

~

LO

--2 6

c

:::J

0

1000 100

()

10 500

1000

1500

2000 2500 Ep (keV)

3000

3500

Fig. 3. Beta-delayed proton spectrum from 33 Ar. The arrows indicate the lines that were used for determining the energy calibration.

from 32 Ar and 33 Ar data. Fig. 4 shows the results of the fit. One important ingredient here was to fix the width of the daughter state in 33Cl to the value determined by proton scattering. 7 ,8 We found that imposing this condition influenced the result of the correlation parameter. The values for the width obtained from two independent and qualitatively different experiments agree with each other nicely. We use r = 109 ± 9 eV in extracting uncertainties. Imposing the 'measured 33 Ar width' condition also made apparent that another small proton peak is present on the high-energy side of the peak corresponding to the superallowed transision in 33 Ar. Fig. 4 presents two fits: one in which we included only the three visible peaks and one for which we added the possibility of a fourth peak. For the case of 32 Ar we also include a small peak on the lower-energy side of the supearllowed transition that is more clearly visible than in the case of 33 Ar. Although the residuals show that the fit is significantly improved including the additional peak, the correlation coefficient extracted for 32 Ar does not depend on its presence. This yields a value for the correlation coefficient: a = 0.998(5)

(4)

assuming there are no Fierz-like contributions to scalar currents (see Ref. 3).

71

3100

3200

3300

3400

3100

3200

Fig. 4. Simultaneous fit of the superallowed transitions from 32 Ar and 33 Ar. The two fits shown for each correspond to adding/not adding an additional group at the lower-energy side of the superallowed peak for 32 Ar and at the higher-energy side of the superallowed peak for 33 Ar.

The uncertainty only reflects the statistical uncertainty. The final analysis including systematic uncertainties should be published soon. 2. Isospin Breaking

The superallowed transitions from both 33 Ar and 32 Ar can also be exploited to test models used for calculations of isospin-breaking corrections which are used for the extraction of Vud , the first element of the CKM matrix, from nuclear decays.9 Fig. 5 shows a sketch of the table of isotopes indicating the position of the nine T = 1 cases that have been used to determine Vud as well as T = 2 nuclei that follow the 'A = 4N' line, of which 32 Ar is one case. The fact that these T = 2 nuclei span about the same region of the table of isotopes as the nine well-determined cases suggests they could be used to do a systematic test of the isospin-breaking calculations. The case of 32 Ar has a particularly large predicted isospin-breaking correction: lO 82feO(mix)

= 0.4%; 82feo(RO) = 1.6(4)%;

where 8c (mix) is the part due to 'configuration mixing' and 8c(RO) is the 'radial overlap' component. So the radial overlap part, which dominates the

72

8 Fig ..5. Sketch of the t
isospin-breaking corrections in the extraction of V"d from nnclear decays, is here approximately three times larger than in the well-known cases. Consequently it provides an opportunit.y to do a stringent test of the calculations. The other part, 8c (mix) , is highly dependent on the energy of the state. Consequently it is important to determine these two contributions separately to really test the models. Fig. 6 shows how we have used our data from Isolde to extract 8c(mix). Using the positron-neutrino correlation information, as described in the previuos section, we have been able to identify one peak, in addition to the super allowed one, that corresponds to 11 Fermi decay. Our fit yields: HHALL'5

8~XP(mix) =

1.05(10)%;

which is significantly larger than the prediction.

73

10000 1000

>OJ

-'"

LO

Q

100

!l c

:J

0

u

10 3300 100

>OJ

-'"

LO

Q

~:J 0

U

10

-_..... Data ·········GT --Fermi

3540 3560 3580 3600 3620 3640

3660

Ep(keV)

Fig. 6. Top: superallowed transition. Bottom: three peaks at higher proton energy. Two fits are shown that correspond to different assumptions about the central peak: the continous line assumes a Fermi transition while the dotted line assumes a GT transition. The fit strongly favors the former.

An additional experiment at Michigan State Universityll was used to determine the 0+ - t 0+ beta-decay branch. Combining this with the lifetime and mass of the daughter state from our Isolde experiment plus the mass of 32 Ar from Blaum et al. 5 yields: a

o;';P(RO)

= 1.3(8)%;

which is consistent with the theoretical expectation, but clearly needs smaller uncertainties to be considered a stringent test of the calculations. In addition, the isospin mixing between the three J+ = 1/2+ states in 33CI have been determined and we obtain, based on the data shown in At the workshop we presented a different value, 8;';P(RO) = 0.6(7)%, but we later identified a problem with our procedure for the extraction of the branch from the MSU experiment to determine the branch. Details are presented in a paper by Bhattacharya et al.1 1

a

74

Fig. 4 (right panel)

B(F)j B(GT) ;S 0.2, B(F)j B(GT) '" 22, B(F)j B(GT) '" 0.02,

(5)

for the ratio of Fermi to GT intensity for the peaks at Ep = 3.07, 3.17, and 3.35 MeV, respectively. Acknowledgments The work described here has been done in close collaboration with Eric Adelberger and Dan Melconian. References 1. J.D. Jackson, S.B. Treiman, and H.W. Wyld Jr., Nucl. Phys. 4, 206 (1957). 2. J. C. Hardy and I. S. Towner Phys. Rev. Lett. 94,092502 (2005). 3. E.G. Adelberger, C. Ortiz, A. Garda, H. E. Swanson, M. Beck, O. Tengblad, M.J.G. Borge, I. Martel, H. Bichsel and the ISOLDE collaboration, Phys. Rev. Lett. 83, 1299 (1999). 4. S. Triambak, A. Garda, D. Melconian, M. Mella, and O. Biesel, Phys. Rev. C 74, 054306 (2006). 5. K. Blaum, G. Audi, D. Beck, G. Bollen, F. Herfurth, A. Kellerbauer, H.-J. Kluge, E. Sauvan, and S. Schwarz Phys. Rev. Lett. 91, 260801 (2003). 6. S. Triambak, A. Garda, E. G. Adelberger, G. J. P. Hodges, D. Melconian, H. E. Swanson, S. A. Hoedl, S. K. L. Sjue, A. L. Sallaska, and H. Iwamoto, Phys. Rev. C 73, 054313 (2006). 7. P.G. Ikossi et al., Phys. Rev. Lett. 36, 1357 (1976). 8. J.F. Wilkerson et al. Nucl. Phys. A549, 223 (1992). 9. J. C. Hardy and I. S. Towner Phys. Rev. C 71, 055501 (2005); Phys. Rev. Lett. 94, 092502 (2005), and this conference proceedings. 10. B. A. Brown, private communication. 11. M. Bhattacharya et al., submitted to Phys. Rev. C.

75

NUCLEAR CONSTRAINTS ON THE WEAK NUCLEON-NUCLEON INTERACTION W. C. HAXTON*

Institute for Nuclear Theory and Department of Physics University of Washington, Seattle, WA 98195 USA * E-mail: [email protected] I discuss the current status of efforts to constrain the strangeness-conserving weak hadronic interaction, which can be isolated in nuclear systems because of the associated parity violation.

Keywords: Weak interactions, parity nonconservation, anapole moments.

1. Parity Nonconservation in the NN System

In this talk I will discuss the weak nucleon-nucleon (NN) interaction: the experiments that have been done, the strategies theorists have developed to interpret measurements, and the puzzles that remain to be resolved. While the weak interaction can be observed in flavor-changing hadronic decays, the neutral current contribution to such decays is suppressed by the GIM mechanism and thus unobservable. The NN and nuclear systems are thus the only practical laboratories for studying the hadronic weak interaction in all of its aspects. 1 ,2 As the weak contribution to the NN interaction is many orders of magnitude smaller than the strong and electromagnetic contributions (the suppression relative to the strong interaction is rv 41fCm; / N N rv 10- 7 ), parity violation must be exploited to isolate weak observables. The most common observables are pseudoscalars arising from the interference of the weak and strong/electromagnetic interactions, e.g., the circular polarization of --y rays emitted from an unpolarized excited nuclear state, or the --y ray asymmetry if the nuclear state can be polarized. The observable depends on an interference between parity-conserving (PC) and parity-non-conserving (PNC) amplitudes, and the weak interaction appears linearly. Alternatively, there are processes, such as the a decay of an unnatural-parity nuclear state

g;

76

to a 0+ final state, where the amplitude is entirely weak. Such observables are proportional to the squares of weak matrix elements, and thus are not associated with a pseudoscalar.

o W~ ~

(a) Factorization

1f

± , p, W

(b) Quark Model

strong vertex

(c) Sum Rule

Fig. 1. A single-boson-exchange contribution to VPNC contains one weak vertex (left) and one strong one (right). The weak vertex is decomposed into the quark-level terms that DDH estimated, using the standard model in combination with techniques such as factorization, the quark model, and sum rules.

The range of the underlying weak interaction, mediated by Wand Z exchange, is rv 0.002 fm, much smaller than the radius of the nucleon. For this reason the nuclear weak force is often modeled as a series of meson exchanges, with one nucleon vertex strong and with the second vertex containing the weak physics, as depicted in Fig. 1. The resulting isospin of the weak meson-nucleon coupling is related to the underlying currents in an interesting way. The hadronic weak interaction has the low-energy currentcurrent form Leff

=

~

[JtvJw

+ J1Jz] + h.c.

(1)

where the charge-changing current is the sum of 6.1=1 6.S=O and Cabibbosuppressed 6.1=1/2 6.S=-l terms, ~S=O J W = cos 8c Jw

+ sm . 8c J~S=-l w .

(2)

Consequently the 6.S=O interaction has the form f _ G [ 28 JotJO L ef ~S=O - y'2 cos c w w

. 28c JltJl t] + sm w w + JzJz

(3)

where the first term, a symmetric product of 6.1=1 currents, has 6. 1=0,2, while the second term, a symmetric product of 6. 1=1/2 currents, is 6. 1=1 but Cabibbo suppressed. Consequently a 6.1=1 PNC meson-nucleon vertex

77

should be dominated by the neutral current term - a term not accessible in strangeness-changing processes. One could isolate this term by an isospin analysis of a complete set of PNC NN observables.

2. S-P Amplitudes and Meson-Exchange Potentials There are several ways to describe low-energy PNC NN interactions. Perhaps the simplest representation, the Danilov amplitudes, is an S-P partial wave description appropriate in the low-momentum limit. Table 1 lists the five partial waves. The coefficients multiplying these amplitudes can be treated as free parameters, to be determined from experiments. Once these are fixed, other low-energy PNC observables could be predicted, in virtually a model-independent way. Table 1. S-P weak PNC amplitudes and the corresponding meson-exchanges. 1 Transition

I <-+ I'

III

3S I <-+ IP1 ISO <-+ 3 Po

0<-+0 1 <-+ 1

0 0 1

3S 1 <-+ 3P1

0<-+1

n-n

x x x

2

n-p x x x x

p-p

x x x

meson exchanges

p,W p,w p,W P

1r±,p,W

A second approach expresses the interaction as a set of single meson exchanges (see Fig. I), in analogy with traditional meson-exchange treatments of the strong force, but with one of the strong vertices replaced by a weak one containing the short-range W,Z physics. The possible exchanges are constrained by symmetries, e.g., Barton's theorem excludes on-shell couplings to neutral scalar mesons. If one includes p, w, and 7r± exchanges, one has enough freedom to reproduce the five Danilov amplitudes and to model the long-range pion contribution important to higher partial waves. Much of the work that has been done in the field uses a potential developed by Donoghue, Desplanques, and Holstein (DDH)4

r,

2MV PNC(r'I =

g",NNi",

J2

- gp [h~71 -

gw

[h~

Z -

-

Tx(J+' U",

. 72 + h~Tt + h~TzzJ [(1 + /-Lv)Bx . up + B_ . v p]

+ h~TtJ [(1 + /-Ls)Bx . Uw + B_ . vw ] (4)

78 Table 2. Recommended ranges and best values for the DDH potential, along with three other parameterizations. From. 2 Coupling (x 10- 7 )

DDH Range 4

Best 4

DZ 5

FCDH 6

KM7

Jrr

0.0<-->11.4 -30.8<-->11.4 -0.4 <--> 0.0 -11.0<-->-7.6 -10.3 <--> 5.7 -1.9 <--> -0.8

4.6 -11.4 -0.2 -9.5 -1.9 -1.1 0.0

1.1

-8.4 0.4 -6.8 -3.8 -2.3

2.7 -3.8 -0.4 -6.8 -4.9 -2.3

0.2 -3.7 -0.1 -3.3 -6.2 -1.0 -2.2

hOp hIp

h2 hew hIw

h~I

where

(5) The g7rNN, gp, and gw (f7r' hp, and hw) are the strong (weak) 1f±, p, and w couplings. As noted previously, the limit m p , mw --+ 00 maps the short-

range part of this potential onto the five Danilov amplitudes. (As there are six short-range p and w couplings, there is a transformation among these couplings that leaves the S-P amplitudes unchanged. l ) The estimated parameter ranges and best values recommended by DDH are shown in Table 2, along with several other parameterizations of this potential. Such a meson-exchange treatment, by providing a model for P-D and other higher partial-waves, presumably has some validity when extended to higher momenta: analogous treatments of the strong potential are quite successful in describing intermediate-range NN interactions. A third approach, developed recently, is a fully systematic expansion in chiral perturbation theory8 in terms of m 7r / A xSB ' This allows for a consistent treatment of iterated pions, helping to define the PNC potential at intermediate ranges: the contributions, illustrated in Fig. 2, include such terms, long-range single 1f± exchange, and five contact interactions. While ten parameters arise in this approach in leading order, only five are independent in the limit of low momentum. 8,9 3. Experimental Constraints

The goal ofthe field has been to determine the weak meson-nucleon coupling strengths by fitting to experiment. If the nonperturbative strong interaction

79

Fig. 2. Contributions in a chiral perturbation theory expansion of VPNC including long-range pion exchange, the intermediate-range contribution from crossed pions, and S-P contact interactions 8

physics associated with the meson-nucleon vertices can be computed, one would then be able to connect these vertices with the underlying elementary couplings of the standard model. Ideally one would make a complete set of measurements in the NN system. However, in most cases the required sensitivity is difficult to achieve. The longitudinal analyzing power for p+ p has been measured at Bonn 10 and SIN, 11

A{+P(13.6 MeV) = (-0.93 ± 0.21) x 10- 7 (Bonn) A{+P(45 MeV) = (-1.57 ± 0.23) x 10- 7 (SIN),

(6)

constraining the 1So - 3Po ,6,1=0,1,2 amplitudes. The circular polarization of the ')'S produced in n + p radiative capture has also been measured,12

Py(n + p

--+

d + ')') = (1.8 ± 1.8) x 10- 7 .

(7)

P, depends on the 1So - 3Po ,6,1=0,2 and 3SI - 1H ,6,1=1 amplitudes. Finally, there is a upper bound on the ')'-ray asymmetry in radiative capture 13

A,(n + p

--+

d + ')')

=

(0.6 ± 2.1) x 10- 7 .

(8)

A, depends on the 3SI - 3PI ,6,1=1 amplitude. A program has begun at LANSCE and will continue at the SNS to improve this result, with a factor of 20 the long-term goal. There are also plans to measure the spin rotation of polarized neutrons passing through parahydrogen at the SNS. 14 As there are quasi-exact methods for treating few-body nuclei, PNC observables in such systems can also be interpreted reliably. The analyzing power for polarized protons scattering on 4He has been measured 15

A{+4 He (46 MeV) = (-3.3 ± 0.9) x 10- 7 (SIN).

(9)

80

This "odd proton" observable depends on a combination of isovector and isoscalar couplings quite similar to that tested in 19F, discussed below. There are also two bounds of interest,

~¢~+a(thermal) = dz

A{+d(15 MeV)

=

(8

± 14) x 10- 7 rad/m (NIST

----+

SNS)

(-0.35 ± 0.85) (LANL).

(10)

The NIST effort on the neutron spin rotation is in progress. 16 There are plans to continue the work at the SNS. Measurements in complex nuclei comprise the third class of experiments. One advantage of nuclear experiments is the opportunity, because of level degeneracies and favorable PNC/PC matrix element ratios, to significantly enhance the size of PNC observables. There are nuclear PNC effects of ~ 10%, in contrast to the 10- 7 characteristic of the NN system. One can also use isospin and other nuclear quantum numbers as a filter, isolating specific components of the PNC interaction. The disadvantage of such systems is wave function complexity, which complicates the extraction of coupling strengths from the observables. Figure 3 shows three nuclei of interest, the parity doublets in 18F, 19F, and 21 Ne. These are effectively two-level systems because the small paritydoublet splittings (39, 101, and 5.7 keY) make doublet mixing much more important than mixing with distant states. The theoretical challenge is to identify methods for calculating two-level mixing accurately. 18F is an interesting case for illustrating both the sources of PNC enhancement and the nuclear structure analysis. The circular polarization of the"( ray emitted in the decay of the 1081 keY 0-0 to the 1+0 ground state is given by

P (1081 keY) = 2 Re [(O+l!VP NC!O-O) (1 +0(g.S.)!M1!0+1)] I 39keV (1+0(g.s.)!E1!0-0)

(11)

As the typical scale of PNC nuclear mixing matrix elements is ~ leV, the first ratio in Eq. (11) is ~ 10- 5 . The second term is the ratio of a PNC M1 transition to the normal PC E1 transition. Both transition strengths are known experimentally. The E1 transition is quite suppressed: the leadingorder operator vanishes in a self-conjugate nucleus. (It corresponds to a translation of the center-of-mass.) The M1 is exceptionally strong, ~ 10.3 W.u. Thus the second ratio is ~ 110. One concludes that the expected size of P, is ~ 10- 3 , four orders of magnitude above the typical scale of PNC in the NN system. Everything is known in Eq. (11) except the sign of the NIl/ E1 ratio and the mixing matrix element.

81

3134

3662 3/2-1/2

5337 1/2+ 112

1-0

2795 1/2+1/2 2789 1/2-1/2 IMI/EII = 112 1081

0-0

1042

0+1

19F

5.7 keY

IMI/EII=296

39 keY IMI/EII=II 18F

1+0

110

1/2-1/2

21Ne 101 keY

3/2+ 1/2

1/2+ 1/2 Fig. 3.

Three sd-shell two-level nuclear systems in which PNC observables are enhanced.

Following early work by Barnes et al.,17 heroic efforts to measure Pi were made by the Queens 18 and Florence 19 groups, yielding (8 ± 39) x 10- 5 • The DDH best-value prediction is (208±49) x 10- 5 . As the mixing is purely isovector, the expected enhancement due to neutral currents was not found. First-principles calculations of PNC mixing matrix elements must address several difficulties. The underlying operator is dipole-like rv iJ . p and thus sensitive to spurious components, so that projection of the center-ofmass is important. As this operator couples opposite-parity shells, configurations in any included space are linked directly to the excluded space, leading to a sawtooth oscillation of the matrix element as new shells are added. The operator behaves under time reversal like the E1 operator, which is suppressed by correlations. VPNC is a surface operator, sensitive to the shapes of the single-particle wave functions. Most important, the down-side of exploiting parity doublets is the need to calculate a highly exclusive matrix element, one that exhausts a tiny fraction of the sum rule generated when Vp NC operates on either member of the doublet. In 18F these difficulties can be avoided by a simple trick: the doublet PNC mixing is identical, up to isospin rotation, to the exchange-current contribution to the axial-charge f3 decay transition between the 0+ 1 ground state of 18Ne (the analog of the 0+1 doublet state) and the 0-0 member of the doublet. Furthermore the ratio of the exchange current contribution to the one-body operator iJ . p L is rv 1 (both operators are of order

82

v / c) and stable: the exchange current is effectively a renormalization of the one body operator. One can use the measured (3 decay rate to determine the PNC mixing matrix element. 2o This argument, applied in a variety of nuclear structure calculations, leads to predictions of (VPNC) that are stable to about ±7%. The case of 19F is similar, though there are additional uncertainties in this case because the mixing matrix element also contains an isoscalar piece.

Fig. 4. Weak radiative corrections contributing to electron-nucleus interactions include the a) the anapole moment as well as b) terms that do not involve single photon exchange.

Another constraint 21 ,22 comes from atomic PNC experiments in which the nuclear ana pole moment generates a dependence on the nuclear spin. The anapole moment is a weak radiative correction to the electron-nucleus interaction (see Fig. 4) that acts like a contact interaction and grows as A 2 /3. In a heavy nucleus it can dominate the tree-level spin-dependent interaction from V( e )-A( nUcleus) Z exchange. There are various contributions to the anapole moment, but the most important term comes from nuclear ground-state polarization due to VPNC . As the case of interest, 133CS,23 has no ground-state parity doublet, the polarization is dominated by mixing with the collective giant resonance region. A summary of what we have learned from experiment and theory is shown in Fig. 5. To a good approximation the observables measured to date depend on two sets of couplings, one isoscalar rv -h~ - O. 7h~ and one isovector rv fir - O.12h~ - O.18h~. (The constraint from A~+P is plotted assuming the DDH value for h~.) The overall consistency of the results is not high. It appears from P,,! in 18F that the isovector coupling, where one expects the neutral current to dominate, is significantly smaller than the DDH best value. There is reasonable agreement between the odd-proton cases, 19F and p + O!, which intersect with the 18F band at a point roughly

83

30

25 20 0

~

3

...s::: tO 0

+ Q..

15 10

...s:::

' -I '

5

l~F /

/

/

/

0

o

246

8 1

10

f1l" - 0.12 hp - 0.18 hw

12

14

1

Fig. 5. Experimental PNC constraints as a function of the effective isoscalar and isovector couplings.

consistent with the DDH best value for the isoscalar coupling. However, the odd-proton constraint from 133Cs favors larger values of the couplings. (A bit of a very broad band from the upper bound on the 205Tl anapole moment is also shown: the uncertainty in this result is such that it does not impact the conclusions drawn from the Fig. 5.)

84 4. Summary

The study of hadronic PNC has proven to be a very challenging area: both the experiments and the theoretical analysis are difficult. While some reasonable consistency exists between the 18F, 19F, P+ a, and p + P results (assuming h~ is near the DDH best value), error bars are large and there is no significant redundancy among the measurements. The conclusion from 18F that the isovector coupling is small, compared to the DDH best value, may be one of the more solid results. Several 18 F experiments have placed tight upper bounds on P,,!, and the analysis, though it involves a complex nucleus, is unusually free of structure uncertainties. Consequently, we have yet to find evidence for neutral currents in ~S=O interactions. Such suppression, relative to the isovector strength, is superficially reminiscent of the enhancements embodied in the ~I=1/2 rule in flavor-changing reactions. Clearly a lot remains to be done. The ongoing effort to measure d¢ / dz in n+a is important, as the comparison with p+ 4He would allow an alternative ~I=O/ ~I=l separation to be made, checking the pattern seen in Fig. 5. The discrepancy involving the 133Cs anapole moment is troublesome. As the control of systematics in that experiment required years of effort, it is not clear when the next anapole moment measurement will be made. But, from a theoretical perspective, such a measurement in an odd-neutron system would be useful in a PNC isospin analysis. If one could tighten the constraints on the isoscalar and isovector couplings, A~+P would become an independent test of h~. While progress has been slow over the past decade, new facilities such as the SNS (with its high-intensity cold neutron beam) and FRIB (a possible source of radioactive nuclei with enhanced anapole moments) may help the field along in the next few years. Good progress has been made in theory, with the development of a more systematic expansion for the effective PNC interaction being one recent example. But the lack of redundancy among experiments puts a lot of stress on theory, requiring one to make use of constraints in NN, few-body, and nuclear systems. It is not clear whether results from NN and fewbody systems should be compared naively with those from complex nuclei. Couplings extracted from complex nuclei are necessarily effective, defined in the context of chosen shell-model spaces. We have many examples in nuclear physics (the axial-vector coupling gA being a celebrated one) where the shell-model coupling is not the underlying bare coupling. There has been some work comparing PNC calculations in small included spaces with

85

those in larger ones. There is a significant dependence on the model space, indicating that effective couplings may differ substantially from the bare values. One good exercise for theorists might be to explore this question in a light nucleus where many shells can be included, in order to test the evolution of (VPNC ) with shell number. This could provide some guidance in interpreting results from systems like 18F. Acknowledgments I thank C.-P. Liu for helpful discussions. This work was supported in part by the U.S. Department of Energy, Office of Nuclear Physics. References 1. E. G. Adelberger and W. C. Haxton, Ann. Rev. Nucl. Part. Sci. 35, 501 (1985). 2. C.-P. Liu, nucl-th/0703008/, to appear in Proc. Second Meeting of the APS Topical Group On Hadronic Physics. 3. G. S. Danilov, Phys. Lett. B 35, 579 (1971) and Phys. Lett. 18, 40 (1965). 4. B. Desplanques, J. F. Donoghue, and B. R. Holstein, Ann. Phys. 124, 449 (1980). 5. V. M. Dubovik and S. V. Zenkin, Ann. Phys. 172, 100 (1986). 6. G. B. Feldman, G. A. Crawford, J. Dubach, and B. R. Holstein, Phys. Rev. C 43, 863 (1991). 7. N. Kaiser and U. G. Meissner, Nucl. Phys. A 489, 671 (1988) and 499, 699 (1989). 8. S. L. Zhu, C. M. Maekawa, B. R. Holstein, M. J. Ramsey-Musolf, and U. van Kolek, Nucl. Phys. A 748, 435 (2005). 9. C.-P. Liu, Phys. Rev. C 75, 065501 (2007). 10. P. D. Eversheim et al., Phys. Lett. B 256, 11 (1991). 11. S. Kistryn et at., Phys. Rev. Lett. 58, 1616 (1987). 12. V. A. Knyazkov et at., Nucl. Phys. A 197, 241 (1972). 13. J. F. Cavaignac, B. Vignon, and R. Wilson, Phys. Lett. B 67, 148 (1977). 14. D. M. Markoff, J. Res. Natl. [nst. Stand. Technol. 110, 209 (2005). 15. J. Lang et al., Phys. Rev. Lett. 54, 170 (1985). 16. A. Micherdzinska, http://www.int.washington.edu/talks/WorkShops/, 2007. 17. C. A. Barnes et al., Phys. Rev. Lett. 40, 840 (1978). 18. S. A. Page et al., Phys. Rev. C 35, 1119 (1987) and Phys. Rev. Lett. 55, 791 (1985). 19. M. Bini et al., Phys. Rev. Lett. 55, 795 (1985). 20. W. C. Haxton, Phys. Rev. Lett. 46, 698 (1981). 21. W. C. Haxton, C. P. Liu, and M. J. Ramsey-Musolf, Phys. Rev. C65, 045502 (2002) and Phys. Rev. Lett. 86,5247 (2001). 22. V. V. Flambaum and D. W. Murray, Phys. Rev. C 56, 1641 (1997). 23. C. S. Wood et al., Science 275, 1759 (1997).

86

ATOMIC PNC THEORY: CURRENT STATUS AND FUTURE PROSPECTS M. S. SAFRONOVA

Department oj Physics and Astronomy, University oj Delaware, Newark, Delaware 19716, U.S.A. E-mail: [email protected]

The experimental parity-nonconservation (PNC) studies are reviewed, and the current status of theoretical calculations is described for various types of systems. The review of the relativistic all-order method and its application to PNC study is given. The current status and future prospects for improvement of the accuracy of theoretical calculations of the PNC amplitudes are discussed.

Keywords: Parity nonconservation; Anapole moment; All-order method; Perturbation theory; Configuration interaction method; Cesium

1. Current Status of Experimental and Theoretical PNC

Studies There are two distinct motivations for PNC studies in an atom: to search for new physics beyond the standard model of the electroweak interaction by precise evaluation of the weak charge Qw, and to probe parity violation in the nucleus by evaluation of the nuclear anapole moment. When an experimental study is conducted in a single isotope, both theoretical and experimental determinations of PNC amplitudes are required. The precision of theory in the evaluation of Qw for Cs (0.5%) requires further improvement to match the precision of experiment (0.35% [1]). The spin-depended PNC is a weaker effect leading (so far) to significantly lower experimental precision (14% in Cs [1]). The precision of the theory for Qw and for the anapole moment is expected to be the same, provided both effects are evaluated using the same method. It is possible to divide all atomic PNC studies into the three categories in terms of the difficulty of calculating the PNC amplitudes: (A) monovalent systems (Cs, Fr, Ra+, Ba+), (B) few valence electron systems (Yb, TI, Pb, Bi), (C) other significantly more complicated systems such as Dy or Sm.

87

We review these three cases separately below. 1.1. Monovalent systems

M. A. Bouchiat and C. Bouchiat [2,3) proposed a Stark interference scheme for measuring the ratio of the PNC dipole amplitude E pNC and the vector part of the Stark-induced amplitude f3 for transitions between states of the same nominal parity. This method led to the most precise experimental study to date, a 0.35% measurement by the Boulder group [lJ. In the same experiment, the nuclear spin-dependent contribution was determined with 14% accuracy. The most recent many-body calculation of the PNC amplitude for cesium was carried out by Dzuba et al. [4] using a method referred to as "perturbation theory in the screened Coulomb interaction" (PTSCI) in which important classes of many-body diagrams are summed to all orders. These atomic calculations led to a value in agreement with the value obtained in [5,6] using the linearized single-double coupled-cluster (SD) method and also with the precise calculation of [7J. We describe the relativistic all-order method in Sec. 3. The difference between the measured weak charge and the standard model prediction in Cs was found to be 2.5 a in Ref. [8]. Evaluation of small corrections to the PNC amplitude: Breit interaction [9-12], radiative correction arising from vacuum-polarization in the nuclear Coulomb field [13,14]' radiative correction arising from the photon vertex [14-16], and "nuclear skin" correction [17-19], reduced the difference between the measured weak charge and the standard model to 1.0 a. The most recent overview of the current status of PN C in Cs is given in [20]. A program to measure PNC in Fr has been underway for the past decade [21,22]. Isotopes from 208Fr to 221Fr have been produced and a number of parity-conserving properties has been measured with high precision. An experiment to measure the nuclear spin-dependent PNC amplitude between ground-state hyperfine levels in Fr isotopes has been proposed in [23]. The advances in the spectroscopy with trapped francium and perspectives for weak interaction studies are described in [24]. The plans for Fr on-line laser trapping at the ISAC radioactive beam facility at TRIUMF for a measurement of the nuclear anapole moment are described in [25]. Production and trapping of Fr ions have also been reported by Atutov et al. [26]. PTSCI and relativistic all-order calculations of the E pNC for the 781/2 -4881/2 transition in francium, were carried out in [6,27], respectively. While both calculations estimate accuracy at the 1% level, the discrepancy between the PTSCI and all-order (linearized coupled-cluster) result is 3%.

88 The investigation of the sources of the discrepancy in Ref. [6] noted a 0.6% difference owing to the use of different nuclear parameters in the two calculations and a 1% difference from the Breit interaction, which was omitted in the calculation of Ref. [27]. The possibility of measuring PNC in a single trapped Ba+ ion was suggested in [28]. Progress on the related spectroscopy with a single Ba+ ion is reported in [29], and precision measurements of light shifts in a single trapped Ba+ ion have been reported in [30]. The calculation of the Ba+ PNC amplitude using the relativistic coupled-cluster method has been reported in [31,32]. Parity-nonconserving 8 - d amplitudes in Cs, Fr, Ba+, and Ra+ have been calculated in Ref. [33] using a hybrid mixed-states-sumover-states. Ab initio nonperturbative calculations of the Breit correction to the parity nonconserving (PNC) amplitudes of Cs, Fr, Ba+, Ra+, and TI have been carried out in [34] and spin-depended amplitudes in these atoms have been calculated using the random-phase-approximation [35]. 1.2. Systems with few valence electrons (Yb, Tl, Pb, and

Bi). A variant of the Stark interference scheme was used in [36,37] to determine Rq, for the 6Pl/2 -+ 7Pl/2 transition in 205Tl. In another class of experiments, advantage was taken of the fact that atomic PNC induces optical activity in monatomic gas vapors. The optical rotation of linearly polarized radiation tuned to an E1-forbidden transition is measured and the ratio Rq, of the E pNC and M1 amplitudes is extracted. Precise measurements of Rq, have been carried out for the 6Pl/2 - 6P3/2 transition in 205TI in Oxford [38] and Seattle [39]; for the 6 3PO -+ 6 3P 1 transition in 208Pb in Seattle [40] and Oxford [41]; and for the 4S3/ 2 - 2D3/2 transition in 209m in Oxford [42]. Re-measurement of the E2/M1 ratio for the 6Pl/2 - 6P3/2 transition in TI [43] helped reconcile differences between the Oxford and Seattle measurements for that atom. The (68 2) ISO -+ (685d) 3D 1 PNC transition in atomic Yb is about 100 times larger than the 68 -+ 78 transition in cesium [44], being enhanced by mixing of the final state with the nearby (686p)lpl state. Measurements of PNC in ytterbium are particularly interesting since there are seven naturally occurring isotopes 168- 176 Yb giving the possibility of eliminating uncertainties arising from atomic structure calculations by comparing PNC amplitudes from different isotopes. Two of the isotopes, 171 Yb (1=1/2) and 173Yb (1=5/2), have nonvanishing nuclear spin arising from an unpaired neutron, in contrast with Cs and TI, where the unpaired nucleon is

89

a proton. Measurements of spin-dependent contributions in Vb, therefore, promise new information on the nuclear spin-dependent interaction. The PNC Yb experiment is currently underway at the University of California, Berkeley [45]. The most recent calculations of the PNC amplitude in Tl [46] was estimated to be accurate to about 3%. The best many-body calculation of the PNC amplitude in Yb [47] gives a value accurate to about 20%.

1.3. Significantly more complicated systems (such as Dy or Sm) A Stark interference experiment to detect PNC mixing between two nearly degenerate levels of opposite parity, (4fl05d6s) [10] and (4f 9 5d2 6s) [10] in dysprosium has been carried out [48]. This experiment gave a result that differed substantially from the theoretical result obtained in a multiconfiguration Dirac-Fock (CI) calculation [49]. Finally, optical rotation parameters for transitions between 7 FJ and s D J' states in the (4f6 68 2 ) multiplet of samarium were measured [50]. The upper state levels are nearly degenerate with levels of opposite parity from the (4f6 6s6p) configuration, leading to an expected enhancement of the PNC amplitude. The weak interaction matrix elements IHwl extracted from experiment ranged from 1 to 30 kHz, which was one to two orders of magnitude smaller than what was expected from semi-empirical calculations.

2. Calculation of the PNC Amplitudes Parity-nonconserving amplitude EpNc for the transition between the states v and w can be calculated using the sum-over-states approach: _ "'" (wIHpNcln)(nIHzlv) EpNc - ~ Ew - En n

"'" (vIHzln)(nIHpNcl v ) Ev - En '

+~

(1)

n

where the PNC Hamiltonian HpNc for nuclear spin-independent PNC interaction is given by HpNc =

G Qw 2V2

"'" t ~ (')'Sp(r))ij aiaj,

Hz is the electric-dipole Hamiltonian, and a! and ai are creation and annihilation operators for an electron state i. The calculations are carried out

using finite basis sets, constructed using B-splines [51]; therefore, the sum

90

in Eq. (1) is finite and ranges over allowed n states. The sum in Eq. (1) converges rapidly, and it is generally sufficient to calculate four or five terms in the above sums using accurate all-order wave functions and estimate the remainders using less accurate many-body methods such as Dirac-Fock (DF) or random-phase approximation (RPA). The calculation of the main part of spin-dependent amplitude involves a similar expressions with a different spin-dependent PNC Hamiltonian. Therefore, the calculation of the PNC amplitude reduces to the calculation of the PNC and E1 matrix elements between a few lowest states.

3. Relativistic all-order method We start from the relativistic no-pair Hamiltonian H from QED by Brown and Ravenhall [52]:

=

Ho

+ VI

obtained

Ho = LEi: aJ ai : ,

(2)

where Vijkl are two-particle matrix elements of the Coulomb interaction gijkl or Coulomb + Breit interaction gijkl + bijkl , and VHF = La (Viaja - Viaaj) is frozen-core Dirac-Fock potential. The summation index a in VHF ranges over states in the closed core. The quantity Ei in Eq. (2) is the eigenvalue of the Dirac equation h(r)¢i(r) = Ei¢i(r), where h(r) = co:·

p+ f3mc 2

-

Z

-

r

+ U(r).

(4)

Choosing the potential to be frozen-core V N -1 potential, U = VHF, greatly simplifies the calculations as the second term in Eq. (3) vanishes in this case. In the coupled-cluster method, the exact many-body wave function is represented in the form [53]

(5) where [w(O)) is the lowest-order atomic state vector. The operator 8 for an N-electron atom consists of "cluster" contributions from one-electron, twoelectron, "', N-electron excitations of the lowest-order state vector [w(O)): 8 = 8 1 + 8 2 + ... + 8 N . The exponential in Eq. (5), when expanded in terms of the n-body excitations 8 n , becomes

[w) = { 1 + 8 1 + 8 2

+ 8 3 + ~8i + 8 1 8 2 + ~8~ + ... }

[w(O)).

(6)

91

In the linearized coupled-cluster method described in detail in Refs. [54,55], all non-linear terms are omitted and the wave function takes the form

(7) Restricting the sum in Eq. (7) to single, double, and valence triple excitations yields the following expansion for the state vector of a monovalent atom in state v:

[w v ) =

[1 + L

Pmaatnaa

ma

+

'6"

+

~ L Pmnabatna~abaa + L mnab

+6

ttl Pmnvaamanaaav

mna

'6"

Pmvatna v

(8)

mcpv

t t t l ['T'(O)) Pmnrvabamanarabaaav 'l'v ,

mnrab

where the indices m, n, and r range over all possible virtual states while indices a and b range over all occupied core states. The quantities Pma, Pmv are single-excitation coefficients for core and valence electrons and Pmnab and Pmnva are double-excitation coefficients for core and valence electrons, respectively, and Pmnrvab are the triples valence excitation coefficients. To derive equations for the excitation coefficients, the state vector [w v ) is substituted into the many-body Schrodinger equation H[w v ) = E[w v ), and terms on the left- and right-hand sides are matched, based on the number and type of operators they contain, leading to the equations for the excitation coefficients given in [55]. It was shown in Refs. [20,56] that both non-linear terms and triple excitations have to be added to improve accuracy of the linearized coupled-cluster SD method. The details of the all-order method are discussed in the review [57] and references therein. The matrix elements for anyone-body operator Z = L i j Zij aI aj are obtained within the framework of the coupled-cluster method as

Zwv

=

(ww[Z[w v) , J(wv[wv)(ww[ww)

(9)

where [w v) and [ww) are given by the expansion (8). In the SD approximation, the resulting expression for the numerator of Eq. (9) consists of the sum of the DF matrix element Zwv and 20 other terms that are linear or quadratic functions of the excitation coefficients. The expression in Eq. (9) does not depend on the nature of the operator Z, only on its rank and parity. Therefore, electric and magnetic multi pole transition matrix elements, magnetic-dipole, electric-quadrupole, and magnetic-octupole hyperfine matrix elements, and nuclear spin-dependent and spin-independent PNC matrix elements, are all calculated using the same general code. The

92

all-order method yielded results for the properties of alkali-metal atoms [55] in excellent agreement with experiment.

3.1. Combining Configuration-interaction and Perturbation Theory Methods The all-order method described in the previous subsection is designed to treat core-core and core-valence correlations with high accuracy. Precision calculations for atoms with several valence electrons require an accurate treatment of the very strong valence-valence correlation; a perturbative approach leads to significant difficulties. The complexity of the all-order formalism for matrix elements also increases drastically as the number of valence electrons increases; for example, the expression for all-order matrix elements in divalent systems contains several hundred terms instead of the twenty terms in the corresponding monovalent expression. A combination of the configuration-interaction (CI) method and perturbation theory was developed in Ref. [58] and later applied to the calculation of atomic properties of various systems in a number of works (see [59-66] and references therein). The CI+MBPT method has been applied to the calculation of Tl PNC amplitude in [46]. In the CI method, the many-electron wave function is obtained as a linear combination of all distinct states of a given angular momentum J and parity [64], WJ = 2::i Ci
93

chains of such terms can be used (see, for example, Ref. [67]). The latter approach corresponds to replacing Dirac-Fock orbitals by Brueckner orbitals. The two-body Coulomb interaction term in H2 is modified by including the two-body part of core-valence interaction that represents screening of the Coulomb interaction by valence electrons; H2 ----+ H2 + ~2' where ~2 is calculated in second-order MBPT. The CI + MBPT approach is based on the Brilloiun-Wigner variant of the MBPT, rather than the RayleighSchrodinger variant. Use of the Rayleigh-Schrodinger MBPT for heavy systems with more than one valence electron leads to a non-symmetric effective Hamiltonian and to the problem of "intruder states". In the BrilloiunWigner variant of MBPT, the effective Hamiltonian is symmetric and accidentally small denominators do not arise; however, ~1 and ~2 became energy dependent. The CI + MBPT approach includes only a limited number of the corevalence excitation terms (mostly in second order) and deteriorates in accuracy for heavier, more complicated systems. In its present form, CI+MBPT also generally involves carrying out energy fitting (or scaling) of the ~ operators to reproduce experimental energies. In 2004, a modification of the effective Hamiltonian using the pair equations was proposed and tested on a "toy" 4-electron model [68]. In such an approach, both ~1 and ~2 are modified using the all-order excitation coefficients. The development of the full-scale CI + all-order method [69] may lead to significant improvement in accuracy of the PNC calculations for more complicated systems.

References 1. C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, Science 275, p. 1759 (1997). 2. M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, p. 899 (1974). 3. M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 36, p. 493 (1974). 4. V. A. Dzuba, V. V. Flambaum and J. S. M. Ginges, Phys. Rev. D 66, p. 076013 (2002). 5. S. A. Blundell, J. Sapirstein and W. R. Johnson, Phys. Rev. D 45, p. 1602 (1992). 6. M. S. Safronova and W. R. Johsnon, Phys. Rev. A 62, p. 022112 (2000). 7. M. G. Kozlov, S. G. Porsev and 1. 1. Tupitsyn, Phys. Rev. Lett. 86, p. 3260 (2001). 8. S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, p. 2484 (1999). 9. A. Derevianko, Phys. Rev. Lett. 85, p. 1618 (2000). 10. O. P. Sushkov, Phys. Rev. A 63, p. 042504 (2001). 11. V. A. Dzuba, C. Harabati, W. R. Johnson and M. S. Safronova, Phys. Rev. A 63, p. 044103 (2001).

94 12. M. G. Kozlov, S. G. Porsev and I. I. Tupitsyn, Phys. Rev. Lett. 86, p. 3260 (2001). 13. W. R. Johnson, I. Bednyakov and G. Soff, Phys. Rev. Lett. 87, p. 233001 (2001). 14. A. I. Milstein, O. P. Sushkov and I. S. Terekhov, Phys. Rev. A 67, p. 062103 (2003). 15. M. Y. Kuchiev and V. V. Flambaum, Phys. Rev. Letts. 89, p. 283002 (2002). 16. M. Y. Kuchiev and V. V. Flambaum, J. Phys. B 36, R191 (2003). 17. J. James and P. G. H. Sandars, J. Phys. B 32, p. 3295 (1999). 18. S. J. Pollock and M. C. Welliver, Phys. Lett. B 464, p. 177 (1999). 19. A. Derevianko, Phys. Rev. A 65, p. 052115 (2002). 20. A. Derevianko and S. G. Porsev, Eur. Phys. J. A 32, p. 517 (2007). 21. J. E. Simsarian, A. Ghosh, G. Gwinner, L. A. Orozco, G. D. Sprouse and P. A. Voytas, Phys. Rev. Lett. 76, p. 3522 (1996). 22. S. Aubin, E. Gomez, K. Gulyuz, L. A. Orozco, J. Sell and G. D. Sprouse, Nuclear Physics A 746, p. 459 (2004). 23. E. Gomez, S. Aubin, G. D. Sprouse, L. A. Orozco and D. P. DeMille, Phys. Rev. A 75, p. 033418 (2007). 24. E. Gomez, L. A. Orozco and G. D. Sprouse, Reports on Progress in Physics 69, p. 79 (2006). 25. G. Gwinner, E. Gomez, L. A. Orozco, A. P. Galvan, D. Sheng, Y. Zhao, G. D. Sprouse, J. A. Behr, K. P. Jackson, M. R. Pearson, S. Aubin and V. V. Flambaum, Hyperfine Interactions 172, p. 45 (2006). 26. S. N. Atutov, V. Biancalana, A. Burchianti, R. Calabrese, L. Corradi, A. Dainelli, V. Guidi, A. Khanbekyan, B. Mai, C. Marinelli, E. Mariotti, L. Moi, S. Sanguinetti, G. Stancari, L. Tomassetti and S. Veronesi, Nuclear Physics A 746, p. 421 (2004). 27. V. A. Dzuba, V. V. Flambaum and O. P. Sushkov, Phys. Rev. A 52, p. 3454 (1995). 28. N. Fortson, Phys. Rev. Lett. 70, 2383 (1993). 29. T. W. Koerber, M. H. Schacht, K. R. G. Hendrickson, W. Nagourney and E. N. Fortson, Phys. Rev. Lett. 88, p. 143002 (2002). 30. J. A. Sherman, T. W. Koerber, A. Markhotok, W. Nagourney and E. N. Fortson, Phys. Rev. Lett. 94, p. 243001 (2005). 31. B. K. Sahoo, R. K. Chaudhuri, B. P. Das and D. Mukherjee, Phys. Rev. Lett. 96, p. 163003 (2006). 32. B. K. Sahoo, B. P. Das, R. K. Chaudhuri and D. Mukherjee, Phys. Rev. A 75, p. 032507 (2007). 33. V. A. Dzuba, V. V. Flambaum and J. S. Ginges, Phys. Rev. A 6363, p. 062101 (2001). 34. V. A. Dzuba, V. V. Flambaum and M. S. Safronova, Phys. Rev. A 7373, p. 022112 (2006). 35. W. Johnson, M. Safronova and U. Safronova, Phys. Rev. A 67, p. 062106 (2003). 36. P. S. Drell and E. D. Commins, Phys. Rev. Lett. 53, p. 968 (1984). 37. P. S. Dre!! and E. D. Commins, Phys. Rev. A 32, p. 2196 (1985).

95 38. N. H. Edwards, S. J. Phipp, P. E. G. Baird and S. Nakayama, Phys. Rev. Lett. 74, 2654 (1995). 39. P. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux and E. N. Fortson, Phys. Rev. Lett. 74, 2658 (1995). 40. D. M. Meekhof, P. A. Vetter, P. K. Majumder, S. K. Lamoreaux and E. N. Fortson, Phys. Rev. A 52, 1895 (1995). 41. N. H. Edwards, P. E. G. Baird and S. Nakayama, J. Phys. B 29, 1861 (1996). 42. M. D. Macpherson, K. P. Zetie, R B. Warrington, D. N. Stacey and J. P. Hoare, Phys. Rev. Lett. 67, 2784 (1991). 43. P. K. Majumder and L. L. Tsai, Phys. Rev. A 60, p. 267 (1999). 44. D. DeMille, Phys. Rev. Lett. 74, p. 4165 (1995). 45. URL=http://socrates.berkeley.edu/ rv budkerj. 46. M. G. Kozlov, S. G. Porsev and W. R Johnson, Phys. Rev. A 64, p. 052107 (2001). 47. S. G. Porsev, Y. G. Rakhlina and M. G. Kozlov, JETP Lett. 61, 459 (1995). 48. A.-T. Nguyen, D. Budker, D. DeMille and M. Zolotorev, Phys. Rev. A 56, 3453 (1997). 49. V. A. Dzuba, V. V. Flambaum and M. G. Kozlov, Phys. Rev. A 50, p. 3812 (1994). 50. D. M. Lucas, R B. Warrington, D. N. Stacey and C. D. Thompson, Phys. Rev. A 58, 3457 (1998). 51. W. Johnson, S. Blundell and J. Sapirstein, Phys. Rev. A 37, p. 307 (1988). 52. G. E. Brown and D. G. Ravenhall, Pmc. Roy. Soc. A 208, p. 552 (1951). 53. F. Coester and H. Kiimmel, Nucl. Phys. 17, p. 477 (1960). 54. S. A. Blundell, W. R Johnson and J. Sapirstein, Phys. Rev. A 43, p. 3407 (1991). 55. M. S. Safronova, W. R Johnson and A. Derevianko, Phys. Rev. A 60, p. 4476 (1999). 56. S. G. Porsev and A. Derevianko, Phys. Rev. A 73, p. 012501 (2006). 57. M. S. Safronova and W. R Johnson, Advances in Atomic, Molecular, and Optical Physics 55 (2007). 58. V. A. Dzuba, V. V. Flambaum and M. G. Kozlov, Phys. Rev. A 54, p. 3948 (1996). 59. S. G. Porsev, M. G. Kozlov, Y. G. Rakhlina and A. Derevianko, Phys. Rev. A 64, p. 012508 (2001). 60. M. G. Kozlov and S. G. Porsev, EUT. Phys. J. D 5, p. 59 (1999). 61. M. G. Kozlov and S. G. Porsev, J. Expt. TheoT. Phys. 84, p. 461 (1997). 62. V. A. Dzuba and V. V. Flambaum, Phys. Rev. A 75, p. 052504 (2007). 63. V. A. Dzuba and V. V. Flambaum, J. Phys. B 40, p. 227 (2007). 64. 1. M. Savukov and W. R Johnson, Phys. Rev. A 65, p. 042503 (2002). 65. 1. M. Savukov, W. RJohnson and H. G. Berry, Phys. Rev. A 66, p. 052501 (2002). 66. 1. M. Savukov, J. Phys. B 36, p. 4789 (2003). 67. V. A. Dzuba and J. S. Ginges, Phys. Rev. A 73, p. 032503 (2006). 68. M. G. Kozlov, Int. J. Q. Chern. 100, p. 336 (2004). 69. M. S. Safronova, M. G. Kozlov and W. R Johnson, unpublished.

96

PARITY-VIOLATING NUCLEON-NUCLEON INTERACTIONS: WHAT CAN WE LEARN FROM NUCLEAR ANAPOLE MOMENTS? B. DESPLANQUES'

LPSC, Universite Joseph Fourier Grenoble 1, CNRS/IN2P3, INPG, F-38026 Grenoble Cedex, France • E-mail: [email protected] Knowledge about parity-violating effects both from theory and experiment is reviewed. Further information that could be obtained from measurements of nuclear anapole moments is discussed.

Keywords: Nucleon-nucleon forces, parity violation, anapole moments.

1. Introduction

In the present contribution, we are concerned with hadronic weak interactions and, more specifically, with the strangeness-conserving component. This one, contrary to the strangeness-violating one, is masked by strong (and electromagnetic) interactions. It can be disentangled through its parity-violating (pv) component, which leads to effects of order 10- 7 for low-energy N N interactions of interest here. Its experimental study is therefore expected to be difficult. It is however necessary to complement the knowledge from strangeness-violating processes. Until now, only the sector of nuclear interactions has been the object of experimental studies. Similarly to the N N strong interaction, the pv interaction is supposed to occur at the two-nucleon level in first place. Various approaches have been considered with different emphasis depending on time: more ambitious ones, based on meson exchange (see Refs. [1,2] for instance), alternating with more phenomenological ones, devoted to the consistency of effects observed at low energy [3-6]. In any case, the description of pv N N interactions involves a minimal set of 5 pieces of information corresponding to the pv elementary S - P transition amplitudes. Looking at nuclear pv effects observed up to now, it is found that, for a

97

large part, they individually agree with expectations within a factor 2 [7,8]. This is not however sufficient to get a consistent description and, thus, a reliable determination of pv N N forces. Further studies are required to provide the missing information. With this respect, we intent to review here some benchmark results, make comments partly in relation with recent works and show in what measurements of nuclear anapole moments could be useful. We largely refer to Ref. [8] for omitted details. The present paper is organized as follows. In Sec. 2, we review the description of the pv N N forces in terms of meson exchanges. This includes a discussion of the numerous uncertainties pertinent to this approach. Section 3 is devoted to the phenomenological approaches considered in the past and recently. A possible hint for the failure of the single-meson exchange picture is described in Sec. 4. Presently known information is reviewed in Sec. 5. The conclusion describes in what the knowledge of nuclear anapole moments could be useful for getting a better determination of pv N N forces. 2. Meson-exchange description of pv N N forces N

N

I V pnc N

N

8,

N, A

+ It,, '

G' Fig. 1.

,"

N

: e

,8 ...

e i1t, p, ill

N, A

e ...

...

ON,A It "

+ :It

It:

EJ

6

.

D

+ " It

..

N

D ...

"

,,

O

....

,'11:

+...

:p

E!J

....

Graphical representation of single-meson and some two-meson exchanges

Diagrams representing contributions to the pv N N forces considered up to now are shown in Fig. 1. They involve the exchange of a single meson (1f, p, w) where the pv vertex (squared box) is described by a constant to be determined while the circle represents the strong-interaction vertex, in principle known. As the weak interaction does not conserve isospin, many couplings are possible in some cases. They are usually denoted as: h~ : t::..T = 1 (long range, a priori favored),

98

h~/:

AT = 1 (short range), h~,1,2 : AT = 0,1,2 (short range), h~,l: AT = 0,1 (short range). The 7r 0 contribution is absent (Barton's theorem [9]) as well as that one due to other spin-O mesons such as rp, aO (or two pions in a S state [10]). Two-pion exchange contributions shown in Fig. 1 were considered in the 70's, within a covariant approach [10-13]. Some were recently considered at the dominant order within an effective field-theory framework. The dots in Fig. 1 stand for contributions due to heavier-meson, multi-meson or excited-baryon exchanges. These contributions, which are expected to have a quite short range, were discarded in the 70's on the basis of a large repulsion in the N N strong interaction models at these distances. This feature appears to be now a consequence of the local character of the models then used [14]. The effect is much less pronounced with non-local models such as CD-Bonn [15] or some Nijmegen ones [16]. As a consequence, the support to neglect the above short-range pv contributions is now much weaker. On the other hand, these extra contributions could involve new parameters, making more complicated the description of pv N N forces. One can only hope they are not too large. From examining N N strong interaction models, there are hints that the approximation of a single-meson exchange should be considered with caution. It has thus been shown that the combined effect of 7r and p exchange could provide some sizable repulsion [17]. In its absence, the wNN coupling has to be increased from an expected value of g~NN/47r c::: 9 to a value of about 20. On the other hand, a model like Av18 [18], which, apart from onepion exchange, is entirely phenomenological, fits experiments with a X2 of 1 per data. This shows that the relevance of a single-meson exchange is far to be established. Thus, apart from h~, pv couplings entering a single-meson exchange model based on 7r, p, w could be quite effective. Estimates for pv meson-nucleon couplings have been made by many authors. Most of them can be shown to be part of the DDH ones [1]. We therefore rely on this work for a discussion. Some detail is given in Table 1 of Ref. [8], the most important couplings being h~, h~, h~. They are obtained from an effective qq interaction, which depends on the factor K c::: 1 + g2({L) log( ~r which accounts for QCD strong-interaction effects. Part of the contributions is based on the SU(6)w symmetry and knowledge from non-Ieptonic hyperon decays. Another part is based on a factorization approximation. The original range assumed 1 S K S 6, possible SU(6)w symmetry breaking and some weight for the neutral-current contribution.

4;;2

),

99

The best-guess values were obtained by weighting the various ingredients, in particular with respect to SU(6)w symmetry breaking and qq-pair role. The update, which takes into account the fact that some ingredients are better known now, mainly affects the coupling (see Table 4 of Ref. [8] ). Results from a quite different approach based on a soliton model (KM) [2] can also be found in Table 1 of Ref. [8]. Instead, results for obtained from QCD sum rules [19-21] are not shown. Examination of various results suggests many remarks. The comparison of DDH and KM results evidences interesting similarities despite differences in the approaches. In both cases, the contribution to due to non-strange quarks is suppressed, most of the estimate involving strange quarks. The estimates for h~ and h~ differ but rough agreement could be obtained by weighting differently the various contributions in DDH (notice that KM provides couplings at q2 = 0 while DDH give them at the meson pole). On the basis of DDH estimates, the contribution of h~ has often been neglected but KM results show it should not be. On the other hand, the upper limit for this coupling in DDH has not been reproduced. Its range should be mostly negative. As for the h~ estimates from QCD sum rules [19-21], we notice that, as far as we can see, they do not correspond to any of the expected results shown in Table 4 of Ref. [8]). They are obtained from nonstrange quarks only and involve a coherent sum of two contributions while cancellations are observed in DDH and KM. It is not clear at the present time whether they represent a new contribution or whether their relation to other results has not been found yet. It has been mentioned that predictions for couplings should be corrected for various effects [22] (rescattering, vertex corrections). These ones could affect the factorization part of DDH estimates but should not change the other parts which, being based on experimental non-Ieptonic hyperon decays, already account for them. The problem in this case is whether SU(6)w symmetry breaking effects are correctly accounted for.

h;

h;

h;

3. PhenOInenological approaches to pv N N forces

For a part, phenomenological approaches have been motivated by the failure of single-meson-exchange approaches to provide a consistent description of various data. Less ambitious than the potential ones, they were originally limited to the five S - P transition amplitudes [3,4]. The important point is that their energy dependence in the few Me V range is completely determined by well-known strong-interaction properties. Thus, five parameters are required for their description [3,4]. The approach was extended to higher

100

energies, motivated by the fact that most effects known at that time were involving complex nuclei [5]. In this aim, a 6th parameter, chosen as the h~ coupling, was added. Due to its long range, the 1f-exchange force induced by this coupling indeed provides an extra sizable energy dependence for the 381 _ 3 H transition amplitude beyond a few MeV. Moreover, it contributes significant 3 P1,2 - 3 D 1 ,2 transitions in complex nuclei, with a sign opposite to the 381 - 3 P 1 one. The configuration-space expression of the interaction can be found in Refs. [5,8]. The approach was considered again recently but within the framework of an effective-field theory [6]. This one involves a chiral expansion in terms of a quantity A~l. The number of parameters to describe the pv N N interaction is the same as above. However, the part of two-pion exchange induced by the h~ coupling at the first possible order, A~\ is separated out with the motivation it is a medium-range interaction. At the same order, a specific contribution to the electromagnetic interaction appears, hence a 7th parameter in the approach. The momentum-space expression of the two-pion exchange contribution can be found in Refs. [6,23]. It only contains two terms that correspond to the local ones of a more general expression considered in the 70's [10-13]. Apart from some mistakes in the original work, the comparison with earlier works shows that [23]: i) the leading-order approximation overestimates the results at finite distances, ii) the non-local terms that appear at the next order are not negligible, iii) in comparison to a p exchange, the two-pion exchange has a longer range as expected from the two-pion tail but also a shorter range. In this case, the result is due to the fact that the two pions are exchanged in a P state. The exchange in a S state, which dominates at intermediate distances in the strong-interaction case, does not contribute here [10). For the purpose of analyzing experimental data, we introduce dimensionless quantities which are more closely related to the isospin properties of the system under consideration [5]. They are Xpp and X nn ' which involve the pp and nn forces, and xtn( +5.5h~), X;n( -5.5h~), X2n' which involve the pn force. The strengths xtn and X;n are appropriate for the description of the pn force in odd-proton- and odd-neuton-nuGleus forces while X2n does not contribute to them. The 1f-exchange contribution, is put between parentheses for the case it would be considered explicitly. We also introduce the combinations: XKr = Xpp + Xtn(+5.5h~), XlV = Xpp + X;n(-5.5h~), which determine the strengths of the proton- and neutron-nucleus forces (for a nucleus with spin and isospin equal to 0). The X parameters are closely related to the Danilov ones [5]. We stress

101

that they account for a lot of short-range unknown physics (N N shortrange correlations, heavy- and multi-meson exchange, relativity, ... ). If necessary, they can be calculated in terms of the h couplings entering a meson-exchange description of the pv N N interaction, allowing one to check the underlying model [8,24]. 4. Possible failure of the single-meson exchange model

It has been proposed that the study of pv effects in pp scattering at different energies could allow one to disentangle contributions due to p and w exchanges. There are measurements at 13.6 MeV [25] and 45 MeV [26], which mainly involve a 1 So - 3 Po transition, and at 221 Me V [27], which is sensitive to the 3 P2 - 1 D2 transition. The analysis has been performed by Carlson et al., with the results [28]: 107 h~P = -22.3, 107 h'fJ' = +5.2. The p coupling agrees with DDH expectations while the w one is at the extreme limit of the proposed range but disagrees if one notices that the range is now restricted to negative values. In short, it is found that the contribution to the 3 P2 - 1 D2 transition is too small, roughly by a factor 2. It has been thought that a longer-range force could enhance the contribution to the 3 P2 - 1 D2 transition relatively to the 1 So - 3 Po one [29]. While this is verified for the undistorted Born amplitude, it is not for the distorted one. Due to the effect of a strong short-range repulsion for the 1 So _3 Po transition amplitude, the effect is the other way round, making the problem more severe. There are different issues: i) the measurement at 221 MeV is too large (by a factor of about 2), ii) estimates of pv coupling constants miss important contributions, raising some doubt about present ones, iii) besides the single-meson exchange contribution, there are important extra contributions (multi-meson exchange, ... ). For this last issue, there would be no other choice than to adopt an effective approach, in which case, the 3 P2 _1 D2 and 1 So - 3 Po transition amplitudes are described by independent parameters. 5. Presently known information

The first information we want to consider concerns the strength of the pv pp force, Xpp. It can be obtained from measurements of the pvasymmetries AL in pp scattering at 13.6 and 45 Me V (AL = -0.10 and - 0.17Xpp respectively [30]). The fitted value [8]: Xpp ~ 0.9

X

10- 6 ,

(1)

102

provides a very good description of the measurements [25,26], leaving little space for a 3 P2 - 1 D2 transition, which is expected to be small in any case. It is stressed that the coefficients 0.10 and 0.17 are determined by well-known properties of the strong N N interaction, while the unknown physics is incorporated in the quantity Xpp. The above value provides an unambiguous benchmark for the strength of pv N N forces. The second information is obtained from the strength of the protonnucleus force, XKr, mostly determined by pv effects observed in odd-proton systems such as p - a scattering or radiative transitions in complex nuclei ( 19 F, 41K, 175Lu, 18 1Ta) [8]. While the dependence of most effects on XKr results from the underlying nuclear model used in the estimate [31], this is not so in 19 F where the estimate was based on a shell-model calculation [32]. Examination of the detailed calculations in the last case nevertheless shows that the result has the structure of a single-proton transition in an average field determined by the strength XKr, evidencing the role of two effects that were accounted for in the heavier nuclei (pairing and deformation). Another calculation [33] however suggests that correlations could affect differently isovector and isocalar contributions to XKr (see also below). A good description of the observed effects assumes the fitted value [8]:

XKr

~ 3.4 x 10- 6 .

(2)

Examining the whole fit (within experimental errors), we are inclined to think it is too good however. At this point, we simply notice that the contribution of the pp force to the proton-nucleus force is relatively small, pointing to a large contribution of the pn force which could be due either to the isoscalar part of the pv force or to the isovector one. The third available information concerns this isovector part of the force. It is obtained from the analysis of pv effects in the transition 18 F (0- -+ 1+). An extensive analysis of the effect in this process has been done [32]. It essentially involves the difference of the pv proton- and neutron-nucleus forces and results in the following upper limit:

IXKr -

X~I

::; 1.4 x

10- 6 ,

(3)

from which we can derive a range relative to the neutron-nucleus force: 2.0 (2.6) x 10- 6

::;

X~ ::; 4.2 (4.8) x 10- 6 .

(4)

Assuming that the isovector contribution is dominated by the single-pion exchange force, one would get: Ih~1 ::; 1.3 x 10- 7 . To some extent, the absence of effect in 21 N e supports the above limit, which could even be smaller if the isoscalar contribution tends to be suppressed [34]. To a lesser extent, results for 75Tc [35] lead to a similar conclusion [8].

103

Concerning the other pieces of information, Xnn and X~p, one could rely on little details in the theoretical estimates to determine them from the analysis of pv effects in various processes. The comparison of these estimates, especially in complex nuclei, however shows that these details are somewhat uncertain, preventing one to get reliable information. The strength Xnn would be best determined from measurements involving neutrons in light systems. The strength X~p does not play much role in complex nuclei [8]. The most favorable process for its determination is the measurement of the photon circular polarization in the radiative capture n + p -+ d + ,. Determining the 6th parameter, h~, introduced to get a better description of the pv N N interaction at low energy, supposes to disentangle its long-range contribution from a short-range one. As the first contribution is expected to dominate the other one however, h~ could be best determined from the measurement of the pv asymmetry in ii+p -+ d+, where the effect is maximized (see Refs. [8,23,36] and references therein). Determining the short-range part could be quite difficult in practice. Though the information is incomplete, one can nevertheless have an interesting discussion relative to the strength of the pp force, Xnn ~ 0.9 X 10- 6 and its contribution to the proton-nucleus one, XK, :::::; 3.4 x 10- 6 . The relative sign is encouraging but the relative size supposes that a large contribution to XK, comes from a pn force. This is questionable in absence of a large pion-exchange contribution, as constrained from 18 F(O- -+ 1+). In usual potentials models, the isoscalar pn contribution to XK, is at best of the order of the pp one. Thus, its strength could be larger than expected by a factor from 2 to 3. This failure could indicate that the usual potential models miss some contribution, supporting for a part findings from pp scattering. Interestingly, the analysis of pv effects in this process (see Sec. 4), with a positive wN N coupling, tends to enhance the strength of the pn force with respect to the pp one [28,29]. Another explanation supposes medium effects that could enhance the strength of the proton-nucleus pv force in heavy nuclei. Some mechanisms, in relation with an attraction in the isoscalar 0- channel [8,33,37] (RPA correlations) or relativity [38], have been discussed in the literature but the size of the effect depends on poorly known ingredients. 6. What from nuclear anapole moments? Conclusion

In first approximation, nuclear anapole moments involve the strengths of the proton- and neutron-nucleus forces, XK, and XIV [39]. Until now, there is no direct determination of the last one. Measurements of anapole mo-

104

ments in odd-neutron nuclei could therefore provide a valuable information on the strength XIV. This information could also be obtained from pv effects in n - 0: scattering (see Ref. [40] and references therein), unless there are sizable medium effects. In such a case, the two measurements will complement each other and their comparison could allow one to determine the size of these corrections for which there is some hint in odd-proton nuclei. The measurement of anapole moments in odd-proton nuclei has already been performed. In the most accurate case however 33 Gs [41]), the strength of XKr required to account for the measurement is roughly twice as much as that one given in Eq. (2), obtained from other odd-proton systems [42,43]. A factor 2 is typical of theoretical nuclear uncertainties in estimates of pv effects in nuclei but, looking at different calculations, it sounds that the uncertainty is smaller in the case of 133Gs, hence some serious concern. Noticing that the strength of the proton-nucleus force could be larger than expected on the basis of its contribution due to the pp force (see previous section), the result in 133Gs could simply be explained by an enhancement effect that is already at work in other complex nuclei. If so, one could wonder why effects in heavy nuclei have not required such an effect, especially in 41 K and 175 Lu, where previous calculations were relatively stable. An explanation could be as follows. These calculations assumed that initial and final states are described by very simple configurations with some correlations (pairing, deformation) preserving the single-particle character of the pv transition. More detailed calculations, made later on in lighter nuclei on a similar basis, have shown that the weight of such a contribution is often decreased by the consideration of further correlations [32]. In this case, some enhancement of the strength of the proton-nucleus force could also be required in 41 K and 175 Lu. Having shown that the anapole moment in 133Gs does not necessarily contradict other pv effects in nuclei, provided that some medium effect is invoked, we believe that the measurement of anapole moments in other odd-proton complex nuclei could be helpful in clarifying the present understanding of pv effects in complex nuclei. By looking at anapole moments of nuclei with different numbers of protons and neutrons, one could imagine to also determine the separate contributions to XKr (XIV) due to Xpp (Xnn) and X:n (X;n). Involving smaller contributions, this program would however suppose that both measurements and theoretical estimates are very accurate. This could be an interesting program for a much further future, once strengths, XKr and XIV, are unambiguously determined.

e

105

References 1. B. Desplanques, J. F. Donoghue and B. R. Holstein, Ann. Phys. (N. Y) 124, 449 (1980). 2. N. Kaiser and D.-G. Meissner, Nucl. Phys. A 499, 699 (1989). 3. G. S. Danilov, Sov. J. Nucl. Phys. 14, 443 (1972). 4. J. Missimer, Phys. Rev. C 14, 347 (1976). 5. B. Desplanques and J. Missimer, Nucl. Phys. A 300, 286 (1978). 6. S.-L. Zhu et al., Nucl. Phys. A 748, 435 (2005). 7. E. Adelberger and W. Haxton, Ann. Rev. Nucl. Part. Sci. 35, 501 (1985). 8. B. Desplanques, Phys. Rept. 297, 1 (1998). 9. G. Barton, Nuovo Cimento 19, 512 (1961). 10. M. Chemtob and B. Desplanques, Nucl. Phys. B 78, 139 (1974). 11. D. Pignon, Phys. Lett. 35B, 163 (1971). 12. B. Desplanques, Phys. Lett. 41B, 461 (1972). 13. H. Pirner and D. O. Riska, Phys. Lett. 44B, 151 (1973). 14. A. Amghar and B. Desplanques, Nucl. Phys. A 714, 502 (2003). 15. R. Machleidt, Phys. Rev. C 63, 024001 (2001). 16. V. G. J. Stoks et al., Phys. Rev. C 49, 2950 (1994). 17. J. W. Durso et al., Nucl. Phys. A 278, 445 (1977). 18. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995). 19. V. M. Khatsimovskii, Sov. J. Nucl. Phys. 42, 781 (1985). 20. E. M. Henley et al., Phys. Lett. B 367, 21 (1996). 21. W. H. P. Hwang, Z. fUr Physik C 75, 701 (1997). 22. S.-L. Zhu et al., Phys. Rev. D 63, 033006 (2001). 23. B. Desplanques et al., in preparation, to be posted on ArXiV soon. 24. C.-P. Liu, Phys. Rev. C 75, 065501 (2007). 25. P. D. Eversheim et al., Phys. Lett. B 256, 11 (1991). 26. S. Kistryn et al., Phys. Rev. Lett. 58, 1616 (1987). 27. A. R. Berdoz et al., Phys. Rev. Lett. 87, 272301 (2001). 28. J. Carlson et al., Phys. Rev. C 65, 035502 (2002). 29. C.-P. Liu, C. H. Hyun and B. Desplanques, Phys. Rev. C 73, 065501 (2006). 30. M. Simonius, Phys. Lett. 41B, 415 (1972), Nucl. Phys. A 220, 269 (1974). 31. B. Desplanques, Nucl. Phys. A 316, 244 (1979). 32. E. Adelberger et al., Phys. Rev. C 27, 2833 (1983). 33. M. Horoi and B. A. Brown, Phys. Rev. Lett. 74, 231 (1995). 34. B. Desplanques and O. Dumitrescu, Nucl. Phys. A 565, 818 (1993). 35. M. Hass et al., Phys. Lett. B 371, 25 (1996). 36. C. H. Hyun, S. Ando and B. Desplanques, Phys. Lett. B 651, 257 (2007). 37. V. V. Flambaum and O. K. Vorov, Phys. Rev. C 49, 1827 (1994). 38. C. J. Horowitz and O. Yilmaz, Phys. Rev. C 49, 3042 (1994). 39. V. V. Flambaum and I. B. Khriplovich, Sov. Phys. JETP 52, 835 (1980). 40. B. Desplanques, in Fundamental Physics with Pulsed Neutrons Beams (FPPNB-2000), eds. C. R. Gould et al. (World Scientific, 2000) pp. 87-96. 41. C. S. Wood et al., Science 275, 1759 (1997). 42. V. F. Dmitriev et al., Nucl. Phys. A 577, 691 (1994). 43. W. Haxton et al., Phys. Rev. C 65, 045502 (2002).

106

PROPOSED EXPERIMENT FOR THE MEASUREMENT OF THE ANAPOLE MOMENT IN FRANCIUM A. PEREZ GALVAN, D. SHENG, L. A. OROZCOt, AND THE FRPNC COLLABORATION * Joint Quantum Institute, Department of Physics, University of Maryland and NIST, College Park, MD 20742, USA t E-mail: [email protected] http://www.physics.umd.edu/rgroups/amo/oTOzco/index.html This article presents a proposal of the FRPNC collaboration for a measurement of the anapole moment of the nucleus of francium using parity non-conservation as the signature in a hyperfine transition. Keywords: Weak interaction; anapole moment; francium.

1. Introduction

Parity non conservation (PNC) is a unique signature of the weak interaction. Our current understanding of the weak interaction derives from the Standard Model of particle physics. 1 ,2 The weak interaction produces two types of PNC effects in atoms: Nuclear spin independent and nuclear spin dependent. 3 Nuclear spin dependent PNC occurs in three ways:4,5 An electron interacts weakly with a single valence nucleon (nucleon axial-vector current An Ve), the nuclear chiral current created by weak interactions between nucleons (anapole moment), and the combined action of the hyperfine interaction and the spin-independent zO exchange interaction from nucleon vector currents (VnAe).6-8 This contribution presents a program to measure the anapole moment in a chain of francium isotopes. The ultimate goal is to further our understanding of the hadronic weak interaction, this is the delicate interplay of *S. Aubin, (William and Mary), J. A. Behr (TRIUMF), G. Gomez, (San Luis Potosi), G. Gwinner (Manitoba), V. V. Fambaum (New South Wales), K. P. Jackson (TRIUMF), L. A. Orozco (Maryland), M. R. Pearson (TRIUNF), A. Perez Galvan (Maryland), D. Sheng (Maryland), G. D. Sprouse (TRIUNF), Y. Zhao (Shanxi)

107

Quantum Chromo dynamics (QCD) with the weak interaction in the nucleus. The hadronic weak interaction is richer than the one in the leptonic sector as it occurs in the presence of the strong interaction which renormalizes the axial current. Its history starts shortly after the discovery in beta decay of parity violation9- 11 and continues to date with advances both in theory and experiment that are increasing our knowledge of this subject. These developments have brought with them the need to better understand QCD at low energy. The recent review of Ramsey-Musolf and Page 12 shows the different avenues currently followed. There is an impressive theoretical development based on QCD that has produced an effective field theory (EFT) for the hadronic weak interaction. 13 The EFT relies on the important degrees of freedom of low-energy QCD. This EFT has connections with observables in a series of experiments proposed and/or currently under way. These include: rt + p --+ d + ,,(, low energy p - p scattering, low energy p - a scattering, spin rotation of polarized neutrons passing through hydrogen, measurement of spin rotation in helium, and measurement of the asymmetry in low energy photodisintegration of deuterium by polarized photons. The measurement of the seven observables form the future program for hadronic PV that is laid out in Ref.14 The program involves performing a set of few-body measurements to determine the coefficients of the operators appearing in the EFT.15 The situation of N-N parity violation, that is central to the low mass program, is not exempt from the small size of the weak interaction amplitudes relative to the strong interaction amplitudes at low energies. Theoretically the task to relate the underlying electroweak currents to low energy observables is complicated, as in this regime QCD is non-perturbative. 16 The PNC measurements in heavier nuclei, ranging from 18F to 133CS and 205Tl, provide important input in the quest for understanding the hadronhadron weak interaction. The interpretation of their results has relied on a meson exchange model that contains seven phenomenological mesonnucleon couplings, the model first proposed by Desplanques, Donoghue, and Holstein (DDH) in their seminal paperY It has not been possible to obtain a self-consistent set of values for these couplings from existing measurements. The longest-range part of the interaction is dominated by the pion-nucleon coupling constant h 7r • Values extracted from the pp and Cs anapole measurements 18 ,19 are consistent with each other, however they disagree by an order of magnitude with the value extracted from the circular polarization of the "( decay in 18F and the anapole limit in 205Tl. 20

108

Despite the inconsistency, all numbers are still within the reasonable range defined by DDH. The picture is complicated and it is important to encompass different approaches to elucidate the subtleties of the weak interaction in heavy nuclei. Erler and Ramsey-Musolf state in Ref. 15 that a study of the anapole moment provides a probe of the f).S = 0 hadronic weak interaction in nuclei. The disagreement between the results obtained with light mass systems compared to the anapole moment of Cs is puzzling. Haxton et al. suggest in Ref. 21 that strong interactions modify the isospin of weak meson-nucleon couplings in a nontrivial way. 2. Anapole measurement in francium

Zel'dovich postulated in 1957 that the weak interactions between nucleons would generate a parity violating, time reversal conserving moment called the anapole moment. 22 Flambaum and Khriplovich calculated the effect it would have in atoms. 5 Experiments in 205Tl gave a limit for its value, 20 and it was measured for the first time with an accuracy of 14% through the hyperfine dependence of PNC in 133CS.18,19 There are currently other efforts to study the anapole moment in Yt 23 and in polar molecules. 24 The measurement strategy that we propose for the nuclear anapole moment in Fr relies on PNC. It looks for the direct excitation of the microwave electric dipole (E1), parity forbidden, transition between the ground state hyper fine levels in a chain of isotopes of Fr, the heaviest alkali atom. The E1 transition between hyperfine levels is parity forbidden, but becomes allowed by the anapole-induced mixing of levels of opposite parity. The general approach has been suggested in the past. 23 ,25-31 Many atoms, optically pumped to the appropriate magnetic sublevel, would be placed inside a microwave Fabry-Perot cavity and held in a blue-detuned dipole trap (see Fig. 1). The atoms would interact with the microwave field and with a Raman field generated by a pair of laser beams, in the presence of a static magnetic field. Confinement of the atoms to the node (anti-node) of the magnetic (electric) microwave field would drive only an E1 transition between hyperfine levels. The atoms would start in the lower hyperfine level, with the signal proportional to the population of atoms in the upper hyperfine level after the excitation. Interference with a Raman transition would produce a signal linear in the El transition, which is proportional to the anapole moment of the nucleus. (See Ref. 32 for details). Parity violating atomic transitions are generated primarily by the exchange of weak neutral currents between electrons and nucleons. Assuming

109

z y

Fig. 1. Schematic setup of the proposed apparatus, adapted from Ref. 32 The dipole trap is not shon and the Raman beams (thick arrows indicate their propagation and thin [black] arrows state their polarization).

an infinitely heavy nucleon without radiative corrections, the hamiltonian

(1) where G = lO-5 is the Fermi constant, mp is the proton mass, r5 and 0: are Dirac matrices, (Tn are Pauli matrices, and "'Ii and "'nsd,i (nuclear spin dependent) with ·i p, n for a proton or a neutron are constants of the interaction. At the tree level "'nsd,i = "'2i, and in the standard model these constants are given by "'lp 1/2(1-4 sin 2 Ow), Kin -1/2, K2p -"'2n = 2 "'2 = -1/2(1 - 4 OW)7/, with sin Ow rv 0.23 the Weinberg angle, and 7] = 1.25. Kli ("'2i) represents the coupling between nucleon and electron currents when the electron (nucleon) is the axial vector. In an atom, we must add the contribution from Eq. 1 for all the nucleons. It is convenient to work in the approximation of a shell model with a single valence nucleon of unpaired spin. The second term of Eq. 1 is nuclear spin dependent and due to the pairing of nucleons its contribution has smaller dependence on Z. The result for this second term

(2)

110

where K = (I + 1/2)( _1)I+l/2~1, l is the nucleon orbital angular momentum, and I is the nuclear spin. The terms proportional to the anomalous magnetic moment of the nucleons and the electrons have been neglected. The interaction constant is given now by:34

(3) where K:2,p = K:2,n = -(1/2)1.25(1 - 4sinBw2) rv -0.05 within the tree level approximation; and we have two radiative corrections, the effective constant of the anapole moment K:a,i, and K:Qw that is generated by the nuclear spin independent part of the electron nucleon interaction together with the hyperfine interaction. Nuclear calculations give 34

(4) where a is the fine structure constant, /li and /IN are the magnetic moment of the external nucleon and of the nucleus, respectively, in nuclear magnetons, TO = 1.2 fm is the nucleon radius, A = Z + N, the weak charge Qw is approximately equal to the number of neutrons N,3 and gi gives the strength of the weak nucleon-nucleon potential with gp rv 4 for protons and 0.2 < gn < 1 for neutrons. 33 Since both K:a and K:Qw scale as A 2 / 3 , the interaction is stronger in heavier atoms. The anapole moment is the dominant contribution to the interaction in heavy atoms, for example in 209Fr, K:a,p/ K:Qw ~15. K:nsd,i = K:a,i is assumed unless stated otherwise for the rest of the proposal. The anapole moment is defined by

a

=

-1f

J

3

2

(5)

d rr J(r),

with J the nuclear current density. The ana pole moment in francium arises from the weak interaction between the valence nucleons and the core. By including weak interactions between nucleons in their calculation of the nuclear current density, Flambaum et al. 5 estimate the anapole moment from Eq. 5 of a single valence nucleon to be 1 G

a

Kj

an'

= -;; J2 j(j + 1) K:a,i = C ],

(6)

where j is the nucleon angular momentum. For the case of a single valence nucleon these values correspond to the nuclear values. Flambaun, Khriplovich and Shushkov 5,35,36 estimated the magnitude of the anapole

111 30,-~~----------------~~~ /'

25

210

Fr /' /'

/' /'

/' /'

/' /'

/' /'

/' /'

/' - - - - -Cs result /' /'

o

/' /'

/'

;{' - - -Fr20% ';-' --Fr3% ';-'

';-'

';-'

-5+---~~---r/'--------~~~~~---.--~ -2 3 8 13

Fig. 2. Range of meson coupling parameters for the expected values of the anapole moment for two Fr isotopes 209 and 210.

moments of various nuclei, demonstrating the A 2 / 3 scaling under the valence nucleon assumption, This is for a shell model with a single valence nucleon carrying all the angular momentum, Flambaum and Murray34 took the parametrization of DDH17 and found the corresponding coupling constants associated with the anapole moment of Cs under these assumptions, Figure 2 shows the possible results based on the model with a single valence nucleon or the vectorial addition of the proton and the neutron in the outer shell that induce the anapole moment, Testing the striking predictions of this simple model near closed shells, and its even-odd staggering behavior, could clarify such a phenomenological treatment to allow the extraction of the weak hadronic physics, If the prediction fails, then it would provide necessary information as input to more sophisticated shell model treatments to extract the physics. This simple approach gives results consistent with the more detailed calculations of Ref. 21 and confirms their assessment that new anapole measurements in odd-neutron nuclei would have great impact, defining a band of the weak meson-nucleon coupling plane roughly perpendicular to the Cs and TI bands , The model of Fig. 2 predicts that measuring the anapole moment on two isotopes gives an almost orthogonal crossing in the two linear combinations of the meson coupling constants from the DDH model. The values plotted in this figure are slightly different from the 133CS band in Ref. 6 as a different choice of

112

values of the "best paramenters" were used.

3. Method for the anapole moment measurement We calculate the transition amplitude for 209Fr between the hyperfine level F=4, m=O to F=5, m=-l with a microwave electric field of 476 V jcm oscillating along the x-axis and a static magnetic field of 1553 Gauss along the z-axis, and with the total anapole moment (K,a) resulting from the vectorial addition of the valence proton and neutrons in 209Fr as 0.45. We obtain32

AEdh = OEI =

01- eE· rli)/h = O.01i [476~/cm] [0~:5]

rad/s. (7)

Once the atoms are in the dipole trap we would optically pump them into a single Zeeman sublevel to prepare a coherent superposition of the hyperfine ground states with a Raman pulse of duration tR. We would drive the E1 transition with the cavity microwave field for a fixed time tEl, and then measure the population in the upper hyperfine level (normalized by the total number of atoms N) through a cycling transition. At the end of each sequence the excited state population is given by

;: ;' -_ NI Ce 12 --

~±

N sm . 2 (ORt R 2

±

OEltEl) 2'

(8)

where Ce is the upper hyperfine amplitude, OR and OEI are the respective Rabi frequencies of the Raman and E1 transition, and the sign depends on the handedness of the coordinate system defined by the external fields as explained below. For a 7r /2 Raman pulse (or a 50-50 coherent superposition) and small OEI this equation becomes 3± =

Nlc e l2

'"

N (~ ±

OE;tEl) .

(9)

The second term contains the PNC signal (E1 transition) that becomes linear through the interference with the Raman transition. We measure the population transfer for both signs and define the signal, proportional to K,a, as (10) The atoms (located at the origin as indicated in Fig. 1) are prepared in a particular Zeeman sublevel IF, m). We apply a static magnetic field B = Bz. The atoms are excited by an standing-wave microwave electric field E(t) = E cos(wmt + '¢) cos(kmy)x. The microwave frequency Wm is

113

tuned to the Zeeman-shifted hyperfine transition frequency Woo The microwave magnetic field M is deliberately aligned along B; since (for a perfect standing wave) M is out of phase with E, we thus have M(t) = M sin(wmt + 'ljJ) sin(kmy)z, with M = E in cgs units. Proper alignment of M and positioning of the standing-wave node is critical for suppressing systematic effects and line-broadening mechanisms. The Raman transition is driven by two plane-wave optical fields, ERl (t) = ERl COS(WRt + ¢R)X and E R2 (t) = ER2 COS((WR + wm)t + ¢R)Z. We assume that the Raman carrier frequency WR is detuned sufficiently far from optical resonance that only the vector part of the Raman transition amplitude (V ex iERl x E R2 ) is non-negligible. 37 The various electric and magnetic fields of the apparatus define a coordinate system related to the measured rate 3±. The transition rate 3± depends on three vectors: The polarization of the El transition, the polarization of the Raman transition (V), and the static magnetic field B which defines magnetization of the atoms. We combine these three vectors to produce the pseudoscalar i(E x (ERl xE R2 ))·B proportional to the measured quantity. We include no discussion about systematic effects in this contribution, as this is presented at great length in Ref. 32 The measurement of the upper hyperfine state population collapses the state of each atom into one of the two hyperfine levels. The collapse distributes the atoms binomially between the two hyperfine levels and leads to an uncertainty in the measured excited state fraction called projection noise. 38 The projection noise is N p = IN[c e[2(1- [ce[2). For a projection noise limited measurement, the signal-to-noise ratio is S (11) N = 2D El tRVN, p

6

With 10 francium atoms, which combined with DEI given by Eq. 7 for a field of 476 V /cm and Ka = 0.45 in tR = 1 s, Eq. 11 gives a signal-to-noise ratio of 20. Another way to state the same is that with 300 atoms we will need 104 cycles of tR duration to reach a 3% statistical uncertainty. There are other sources of noise such as the photon shot noise, that scales as IN[ce[2, or technical noise that is independent of Ceo About a 5050 initial superposition of states maximizes the signal-to-noise ratio when we include shot noise and some technical noise beyond projection noise. 4. Fr production requirements

Francium is expected to have an anapole PNC effect a factor of ten times larger than cesium. 39 Fr is, however, an unstable element and it is necessary

114

to use an accelerator to produce it in a fusion or fission reaction40 or take from the decay process of thorium. 41 The program presented in this contribution relies on the laser trapping an cooling techniques that can capture enough atoms for performing the measurement. The most efficient MOT Fr t rap 40 had an overall efficiency of a few percent; so to capture 106 in the dipole trap it would be good to have production rates in excess of 108 per second or larger with lifetimes of tens of seconds in the MOT. We expect that the developing actinite source at TRIUMF in Vancouver, Canada will be able to achieve that production rate for various isotopes both in the neutron rich and neutron deficient side. Other accelerators existing or in planning stages should also achieve comparable or better numbers. The program of PNC studies for nuclear hadronic weak interaction will greatly benefit from the Fr experiments and its success hinges on good Fr production. 5. Acknowledgments Work supported by the National Science Foundation of the USA. References 1. S. Weinberg, Phys. Rev. Lett. 19, p. 1264 (1967). 2. A. Salam, Gauge Unification of Fundamental Forces, in Elementary Particle Theory: Relativistic Groups and Analyticity (8th Nobel Symp.) , ed. N. Svartholm (Almqvist and Wicksell, Amsterdam, 1968), Amsterdam, p. 367. 3. M.-A. Bouchiat and C. Bouchiat, Rep. Prog. Phys. 60, p. 1351 (1997). 4. Y. B. Zel'dovich, Sov. Phys.-JETP 9, p. 682 (1959). 5. V. V. Flambaum, 1. B. Khriplovich and O. P. Sushkov, Phys. Lett. B 146, p. 367 (1984). 6. W. C. Haxton and C. E. Wieman, Annu. Rev. Nucl. Part. Sci. 51, p. 261 (2001). 7. W. R. Johnson, M. S. Safronova and U. 1. Safronova, Phys. Rev. A 67, p. 062106 (2003). 8. J. S. M. Ginges and V. V. Flambaum, Phys. Rep. 397, p. 63 (2004). 9. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105, p. 1413 (1957). 10. J. 1. Friedman and V. L. Telegdi, Phys. Rev. 106, p. 1681 (1957). 11. R. L. Garwin, L. M. Lederman and M. Weinrich, Phys. Rev. 105, p. 1415 (1957). 12. M. J. Ramsey-Musolf and S. A. Page, Ann. Rev. Nucl. Part. Sci. 56, p. 1 (2006). 13. S.-L. Zhu, C. Maekawa, B. Holstein, M. J. Ramsey-Musolf and U. van Kolek, Nucl. Phys. A 748, p. 435 (2005).

115 14. B. R. Holstein, Intersections of Particle and Nuclear Physics: 8th Conference, AlP Conf. Proc. 698, p. 176 (2003). 15. J. Erler and M. J. Ramsey-Musolf, Prog. Nucl. Part. Phys. 54, p. 351 (2005). 16. W. M. Snow, Eur. Phys. J. A 24 sl, p. 119 (2005). 17. B. Desplanques, J. F. Donoghue and B. R. Holstein, Ann. Phys. (N. Y.) 124, p. 449 (1980). 18. C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, Science 275, p. 1759 (1997). 19. C. S. Wood, S. C. Bennett, J. L. Roberts, D. Cho and C. E. Wieman, Can. J. Phys. 77, p. 7 (1999). 20. P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux and E. N. Fortson, Phys. Rev. Lett. 74, p. 2658 (1995). 21. W. C. Haxton, C. P. Liu and M. J. Ramsey-Musolf, Phys. Rev. C 65, p. 045502 (2002). 22. Y. B. Zel'dovich, Sov. Phys.-JETP 6, p. 1184 (1958). 23. D. Budker, Parity Noncoservation in Atoms, in WEIN 98, Physics Beyond the Standard Model, eds. P. Herczeg, C. M. Hoffman and H. V. KlapdorKliengrethaus (World Scientific, Singapore, 1998), Singapore, p. 418. 24. D. P. DeMille, S. B. Cahn, D. Murphree, D. A. Rahmlow and M. G. Kozlov, arXiv:0708.2925 (2007). 25. C. E. Loving and P. G. H. Sandars, J. Phys. B 10, p. 2755 (1977). 26. V. G. Gorshkov, V. F. Ezhov, M. G. Kozlov and A. 1. Mikhailov, Sov. J. Nucl. Phys. 48, p. 867 (1988). 27. V. E. Balakin and S. 1. Kozhemyachenko, JETP Lett. 31, p. 297 (1980). 28. V. N. Novikov and 1. B. Khriplovich, JETP Lett. 22, p. 74 (1975). 29. E. A. Hinds and V. W. Hughes, Phys. Lett. B 67, p. 487 (1977). 30. E. G. Adelberger, T. A. Traino, E. N. Fortson, T. E. Chupp, D. Holmgren, M. Z. Iqbal and H. E. Swanson, Nuc. Instr. and Meth. 179, p. 181 (1981). 31. E. N. Fortson, Phys. Rev. Lett. 70, p. 2383 (1993). 32. E. Gomez, S. Aubin, L. A. Orozco, G. D. Sprouse and D. DeMille, Phys. Rev. A 75, p. 033418 (2007). 33. 1. B. Khriplovich, Parity Non-Conservation in Atomic Phenomena (Gordon and Breach, New York, 1991). 34. V. V. Flambaum and D. W. Murray, Phys. Rev. C 56, p. 1641 (1997). 35. V. V. Flambaum and 1. B. Khriplovich, Sov. Phys. JETP 52, p. 835 (1980). 36. O. P. Sushkov, V. V. Flambaum and 1. B. Khriplovich, Sov. Phys. JETP 60, p. 873 (1984). 37. D. DeMille and M. G. Kozlov, arXiv: physics, p. 9801034 (1998). 38. W. M. Itano, J. C. Begquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen and D. J. Wineland, Phys. Rev. A 47, p. 3554 (1993). 39. V. A. Dzuba, V. V. Flambaum and O. P. Sushkov, Phys. Rev. A 51, p. 3454 (1995). 40. E. Gomez, L. A. Orozco and G. D. Sprouse, Rep. Prog. Phys. 69, p. 79 (2006). 41. Z.-T. Lu, K. L. Corwin, K. R. Vogel, C. E. Wieman, T. P. Dinneen, J. Maddi and H. Gould, Phys. Rev. Lett. 79, p. 994 (1997).

116

The Radon-EDM Experiment Tim Chupp for the Radon-EDM collaboration

University of Michigan Physics Department and FOCUS Center Ann Arbor, MI48109, USA * E-mail: [email protected] http://research. physics.lsa. umich. edu/chupp/

The Radon-EDM experiment will measure the permanent electric dipole moment of radon atoms produced at TRIUMF's ISAC facility. The choice of 223 Rn is based on the potential enhancement of sensitivity to CP violation by 2-3 orders of magnitude compared to 199Hg, due to octupole strength or deformation. We expect to measure the atomic EDM of 223Rn with precision of 10- 26 to 10- 27 e-cm and thus extend sensitivity to CP violation by one to two orders of magnitude.

Keywords: Atomic EDM; CP violation

1. Introduction

Studies of CP violating interactions impact the nature of elementary particle interactions and the origin of the predominance of matter over antimatter in the Universe. Electric dipole moment measurements provide a unique and important probe of CP violation because the signal is an unambiguous violation of CP symmetry (e.g. there are no confounding final state effects), techniques of atomic and nuclear physics provide continually improving precision, and because CP violation in the K and B meson systems is dominated by Standard Model physics. The unique impact of EDMs is evident in the tight constraints on supersymmetry set by the combination of recent results from the neutron, the electron, atoms and molecules. 1 The anticipated discovery of an EDM in one of these systems will be the first step in using CP violation to probe new physics, and measurements in several systems will, over time, fully clarify the new physics. Measurement of the EDMs provide the most sensitive available probe of flavor non-changing, CP-odd physics. In the Minimal SU(3) x SU(2) x U(l) Standard Model, CP violation enters via weak interaction flavor mixing represented by the Cabibbo-Kobayashi-Maskawa (CKM) matrix and via

117

BQCD , the vacuum expectation value of the QCD gluon field. The CKM matrix includes a single complex phase, which successfully accounts for CP mixing in the K and B mesons. In generating an EDM, the CKM phase enters twice, with opposite sign, resulting in near cancellation and EDMs much smaller than current limits. CP violating interactions would induce an atomic EDM (dA) in 223Rn that may be more than 500 times larger than in 199Hg. CP violation is also a crucial component of the Sakharov mechanism of baryogenesis,2 which could explain the dominance of matter over antimatter in the Universe. In the Sakharov mechanism, the matter-antimatter asymmetry is generated in a non-equilbirium first order phase transition by CP and baryon number violating interactions; however the phase in the CKM matrix is not sufficient to generate the observed baryon asymmetry, thus new forms of CP violation are expected. 3- 5 Most significant extensions of the Standard Model introduce additional phases that could produce the baryon asymmetry and lead to EDMs many orders of magnitude larger than the CKM values. 6 For example, supersymmetric models introduce phases that could produce the baryon asymmetry at the electroweak scale and produce EDMs of atoms or the neutron close to the current limits of sensitivity.7 In fact an electron EDM violation much smaller than the current limits could rule out electroweak baryogenesis,9 and extending the sensitivity to neutron and heavy atom EDMs would also provide strict constraints. CP violation is also a valuable observable by which to probe physics beyond the Standard Model more generally - that is, CP violation can be used to reveal a weaker interaction in the presence of the dominant strong and electroweak interactions of the Standard Model. Radon isotopes have many attractive features for advancing sensitivity to CP violation. The most important feature is enhanced sensitivity to CP violation in isotopes with oct up ole deformed nuclei. 1O- 13 For 223Rn, octupole deformation would lead to an enhancement of the observable atomic EDM by a factor estimated to be greater than 500 relative to 199HgY Radon also provides the experimental advantages of noble gas atoms, the possibilities for multiple-species experiments that directly measure the most important systematic effects, and the promise of new techniques for precision measurement with radioactive species. The many opportunities for success of this program include the potential for discovery and precision measurement of EDMs, atomic and nuclear physics of radon isotopes, new technology, and new techniques for rare isotope physics. The Radon-EDM Experiment also provides a co-magnetometer, which is widely considered

118

essential for a reliable EDM measurement. Recent theoretical advances have strengthened the case . The work of Engel and collaborators continues to clarify how octupole deformation and octupole vibrations enhance sensitivity to CP violation in the nucleus. 12- 14 The sensitivities of the atomic EDM to CP violation in the nucleus have been reevaluated. 15 and show that earlier calculations underestimated the sensitivity of noble gas atoms relative to 199Hg.

2. Experimental Approach

The 223Rn EDM will be measured in a cell with a small applied magnetic field (e.g. 1 mG) and a strong electric field parallel or anti-parallel to the magnetic field. We measure the change of the precession frequency when the electric field is flipped with respect to the magnetic field: n6.w = 4dE. A measurement cycle consists of the following: (1) Production and collection of radon. (2) Transfer to an EDM measurement cell in a highly shielded magnetic environment. (3) Polarization of the radon and other noble gas species used as comagnetometers (4) NMR pulse to initiate free precession of the noble gas spins (5) Precession rate measurements in varied field configurations.

The precession frequency w is measured by monitoring the modulated decay rate in an array of detectors oriented perpendicular to the electric and magnetic fields (see Figure 1). In the first stage measurement, the gamma rays will be detected, and the correlation of gamma-ray angular distribution with respect to the nuclear spin axis - the gamma-ray anisotropy - will lead to a modulation at 2w. The gamma ray detection rate is limited to about 15 kHz per detector. Figure 2 shows a simulation of the gamma-ray arrival times in the top panel and a histogram of the simulated the arrival times in the bottom panel. A fit to the histogram data provides the best estimate of the frequency w. The planned second stage measurement will detect betas in currentdetection mode. This is not count-rate limited and is expected to provide 100 times greater statistics in a given run time and would extend the sensitivity to CP violation an additional order of magnitude or more.

119

....

..---------

....

""",,..,..

,,

, " , ,Active Shielding

,,

~

~

,,

,, ,,

/ / /

I I

\

I

\

I I I

I I I I I I I I

I

I I I I /

I / / ~ ~ ~

....

_-------'

",

,/

Fig. 1. Schematic of the layout of eight gamma-ray detectors combined with magnetic shielding etc. for the Radon EDM experiment.

0.4

0.2 0;

~

e;

0.0

.!j -0.2

Fig. 2. simull!.tion of gfl:,m,ma ray at"rival times'over 10 setonds for an anisotropy of 0.4 and a precession frequency of 1 Hz. (bottom) Monte Carlo data histogrammed into 1/16th second bins for 60 seconds.

120 3. Recent Progress

Several techniques and components have been developed and studied in test-runs. We have developed a high efficiency transfer technique for moving radon from a catcher foil to the measurement cell described in detail in reference. 16 This provided data on the temperature dependence of diffusion rates of xenon from foils of several metals. 17 This work also led to a new measurement of the half-life of 120Xe. 18 Measurements of the polarization signal as a function of cell temperature have provided information on relaxation times and polarization. In order to study spin-exchange polarization and relaxation of radon, a 209Rn source was developed at the Stony Brook Francium Lab( see Figure 3). A beam of 160 incident on a gold target generated francium isotopes that were accelerated to 5 keV, electrostatically focussed, and implanted in a zirconium foil. The beam energy of 91 MeV optimized the 197 Au( 16 0,4n)209Fr reaction. The 209Fr decay (T1/2=50 s) has an 11% branching ratio for electron capture decay to 209Rn. Francium was implanted for about one hour (two half-lives of 209Rn). After implantation, the foil was heated and the 209Rn was transferred to a LN2 cooled cell. The optical pumping cell was approximately spherical and made of Pyrex glass. The cells were prepared by chemically cleaning with an H 2S0 4 /H 20 2 solution and baking under vacuum. A small quantity of rubidium metal was transfered into the cell under vacuum. Both coated and uncoated cells were studied. Earlier studies of the wall interactions of the spin-3/2 isotope 131 Xe 19 indicated greatrer quadrupole relaxation in coated cells. Data for the uncoated cell were taken at temperatures ranging from 150 to 220°C. Data for the octadecyltricholorosilane (OTS) coated ce1l 2o were taken at temperatures from 130 to 180°C. Up to about one million 209Rn atoms were transferred to the cell in each cycle. After transfer, the cell was quickly heated to the measurement temperature and illuminated with circularly polarized laser light at 794.7/ AA, the rubidium Dl wavelength. The cell was viewed by HPGe gamma-ray detectors oriented at 0° and 90° with respect to a 10.5 G magnetic field produced by Helmholtz coils. A gamma spectrum during implantation and after transfer of 209Rn to the measurement cell is shown in Figure 4. Gamma anisotropies, defined by the ratio of counts in a photo peak in 0° and 90° detectors with laser on and laser off, were used as a measure of the radon polarization: R

=

(noo /n90 ohON (noo / n90 LO F F 0 )

(1) '

121

where noo /n90o is the background-corrected ratio of counts in the photopeak at the 0° detector to counts in the 90° detector with laser on (LON) and with laser off (LOFF). The ratio R depends on the alignment (second moment of the Zeeman populations), which results from polarization by spin exchange combined with relaxation. Quadrupole interactions at the cell wall dominate relaxation. Data for the anisotropies over a range of temperatures are shown in Figure 5. The general theory relating alignment to spin-exchange polarization is described in references 21 and,22 and the 209Rn and 223Rn cases are an extension of these principles. 23 At low temperatures, the rubidium is almost completely polarized as optical pumping overcomes all spin destruction processes. At higher temperatures, increased spin destruction of rubidium polarization leads to a decrease in the polarization. This is parametrized by three main rates: the rubidium-radon spin-exchange rate "(SE, the radon dipole relaxation rate r 1, and the radon quadrupole relaxation rate r 2· The quadrupole wall relaxation is expected to have an Arrhenius temperature dependence:

(2) A set of six rate equations for the 209Rn sublevels can be solved to find the steady state populations, and selection rules for electron capture predict the 209 At alignment and the gamma-anisotropies. From this analysis, we can estimate the polarization relaxation time. The data of Figure 5 were fit to the model with fixed spin-exchange 2 rate constant IJSE = 2.5 X 10- 5 A calculated by Walker.? The best fits (solid lines in Figure 5) give r2' = 0.05 ± 0.01 Hz for the uncoated cell and r2' = 0.032 ± 0.009 Hz for the coated cell. 4. Conclusions

The atomic-EDM sensitivity is given by

Od a =

2~T2 JA2(1 ~ B)2 N·

(3)

where 1/T2 is the linewidth due to decay of the coherent signal (assuming RF power broadening and magnetic field inhomogeneity are negligible), A is the analyzing power for a measurement that detects a change of counts D.N = AN, and B is the fraction of N due to background. For gamma ray detection the maximum count rate allowed by germanium detectors is the fundamental limit on N. For the beta-asymmetry technique, the number

122

W D

-<

Coll!mator

Differential pumpin9

Vl

209Fr (50 s)

II~

V4~

CJ Manorneter

heating

[

~

NZ

'" U

D ~:~;]

Fig. 3.

n

197Au

Layout at the Stony Brook Francium Lab.

of decays observed is directly proportional to the total number of atoms in the cell. To estimate the potential precision of our measurements, we take T2 to be 30 seconds, consistent with the relaxation time extracted for 209Rn in our work at Stony Brook, and E= 5 kV /cm. In both cases, we assume A = 0.2 and a 100 day run. The background and count rates for the two techniques will be different. We assume a total count rate of 120 kHz for eight detectors, thus N = 1 X 10 12 for 100 days. The analyzing power A depends on the spin-sequences of each gamma-transition, and are generally not more than 0.4. Analysis of our Stony Brook data on 209Rn, shown in Figures 3 and 4 suggest anisotropies about half of that expected for maximum alignment, however this should improve with improved lasers. In our first stage measurement, the planned EDM sensitivity is 1 x 10- 26 e-cm, which would extend the sensitivity to CP violation an order of magnitude

123

beyond the 199Hg result. With the beta-detection techniques, the count rate will be limited by the radon-production rate and collection efficiency, which can be 10 MHz or higher providing an order of magnitude or greater statistical improvement for the stage two measurement.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

K.A. Olive, M. Pospelov, A. Ritz, Y. Santoso, Phys. Rev. D72, 075991 (2005). A.D. Sahkarov, JTEP Lett. 5, 24 (1967). A.D. Dolgov, (hep-ph/9707419). M. Trodden, Rev. Mod. Phys. 71, 1463 (1999). A. G. Cohen, D. B. Kaplan, and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 43, 27 (1993). M. Pospelov and A. Ritz, Phys. Rev. D63, 073015 (2001). S. Abel, S. Khalil, O. Lebedev , Nucl. Phys. B606, 151 (2001). V.V. Flambaum, V.G. Zelevinsky, Phys. Rev. C68, 035502 (2003). V. Cirigliano, S. Profumo, M.J. Ramsey-Musolf, hep-ph/0603246. N. Auerbach, V.V. Flambaum, and V. Spevak, Phys. Rev. Lett. 76, 4316 (1996). V. Spevak, N. Auerbach, and V.V. Flambaum, Phys. Rev. C56, 1357 (1997). J. Engel, J.L. Friar, and A.C. Hayes, Phys. Rev. C61, 035502 (2000). J. Engel, M. Bender, J. Dobaczewski, J.H. de Jesus, P. Olbratowski, Phys. Rev. C68, 025501 (2003). V.V. Flambaum, V.G. Zelevinsky, Phys. Rev. C68, 035502 (2003). V.A. Dzuba, V.V. Flambaum, J.S.M. Ginges, M.G. Kozlov, Phys. Rev. A66, 012111 (2002). S.R. Nuss-Warren et at., Nucl. Inst. Meth. A 533, 275 (2004). T. Warner et al., NIM A 538, 135 (2005). A.W. Phillips et al., Phys. Rev. C 74, 027302 (2006). Z. Wu, W. Happer, M. Kitano, and J. Daniels, Phys. Rev. A 42,2774 (1990). E. R. Oteiza, Ph.D. Thesis, Harvard University (unpublished). T.E. Chupp, K.P. Coulter, Phys. Rev. Lett. 55, 1074 (1985). T. E. Chupp and R. J. Hoare, Phys. Rev. Lett. 64, 2261 (1990). E.R. Tardiff et al., nucl-ex/0612006. M. E. Wagshul and T. E. Chupp, Phys. Rev. A40, 4447 (1989). E.R. Tardiff et at., in preparation. T. G. Walker, Phys. Rev. A40, 4959 (1989). Z. Wu, W. Happer, M. Kitano, and J. Daniels, Phys. Rev. A 42, 2774 (1990). Table of Isotopes, 8th edition, edited by V. S. Shirley, (John Wiley & Sons, Inc., New York, 1996), Vol. II, pp. 2598-2600.

124

:'s.->!, rFl·

'+'

i"!1!>",~",4"" .",_-1

:V~~, nil - )!llIO!2L'.!ll.-'1'_-I

160

Before transfer

140 120

100 80

60

400

300

500

600

700

800

408 keY After transfer ~ 500,00 209Rn

600

511 keY

500

400

300

337 keY 745 keY

200

100

300

400

500

600

700

800

Fig. 4. Gamma ray spectra before and after 209Rn is transferred to the optical pumping cell. The bottom axes are energy in units of keY. The 209 At levels of importance are shown at the top.

125

1

Uncoated Cell

1.05

a::

1

0.95

r

•

----------------~------------ ---------------------1-----------------• I l 1 • • I i 1

1 1

1

0.9

f

Coated Cell

1.05

a::

1

0.95

--

I

------~-----.----------------------------------------------

ill

i

r •

0.9 140

160

180

200

220

Cell Temperature eC) Fig. 5. Anisotropy data from the coated and uncoated cells for the 337 keY 209Rn gamma ray. For the uncoated cell the average of Runs 1 and 2 is displayed. R is the ratio defined in Eq. (1), and the solid curves are the fits, described in the text, from which a value for r2' is obtained.

126

The Lead Radius Experiment (PREX) and Parity Violating Measurements of Neutron Densities

c.

J. HOROWITZ

Dept. of Physics and Nuclear Theory Center, Indiana University, Bloomington, IN 47405 USA E-mail: [email protected]

Parity violating electron nucleus scattering is a clean and powerful tool for measuring the spatial distributions of neutrons in nuclei with unprecedented accuracy. Parity violation arises from the interference of electromagnetic and weak neutral amplitudes, and the ZO of the Standard Model couples primarily to neutrons at low Q2. Experiments are now feasible at existing facilities. We show that theoretical corrections are either small or well understood, which makes the interpretation clean. The Jefferson Laboratory Lead Radius Experiment (PREX) aims to measure the neutron radius of 208Pb to one %. This neutron density measurement has many implications for nuclear structure, atomic parity nonconservation experiments, and the structure of neutron stars.

1. Introduction

The size of a heavy nucleus is one of its most basic properties. However, because of a neutron skin of uncertain thickness, the size does not follow from measured charge radii and is relatively poorly known. For example, the root mean square neutron radius in 208Pb, Rn is thought to be about 0.2 fm larger then the proton radius Rp ;:::; 5.45 fm. An accurate measurement of Rn would provide the first clean observation of the neutron skin in a stable heavy nucleus. This is thought to be an important feature of all heavy nuclei. Ground state charge densities have been determined from elastic electron scattering, see for example ref. 1 Because the densities are both accurate and model independent they have had a great and lasting impact on nuclear physics. They are, quite literally, our modern picture of the nucleus. In this paper we discuss future parity violating measurements of neutron densities. These purely electro-weak experiments follow in the same tradition and can be both accurate and model independent. Neutron den-

127

sity measurements have implications for nuclear structure, atomic parity nonconservation (PNC) experiments, isovector interactions, the structure of neutron rich radioactive beams, and neutron rich matter in astrophysics. It is remarkable that a single measurement has so many applications in atomic, nuclear and astrophysics. Donnelly, Dubach and Sick2 suggested that parity violating electron scattering can measure neutron densities. This is because the Z - boson couples primarily to the neutron at low Q2. Therefore one can deduce the weak-charge density and the closely related neutron density from measurements of the parity-violating asymmetry in polarized elastic scattering. Of course the parity violating asymmetry is very small, of order a part per million. Therefore measurements are very difficult. However, a great deal of experimental progress has been made since the Donnelly et. al. suggestion, and since the early SLAC experiment. 3 This includes the Bates 12C experiment,4 Mainz gBe experiment,5 SAMPLE,6 HAPPEX,7 HAPPEXII,8 and Gog. The relative speed of the HAPPEX results and the very good helicity correlated beam properties of CEBAF show that very accurate parity violation measurements are possible. Parity violation is now an established and powerful tool. It is important to test the Standard Model at low energies with atomic parity nonconservation (PNC), see for example the Colorado measurement in CS. lO ,l1 These experiments can be sensitive to new parity violating interactions such as additional heavy Z - bosons. Furthermore, by comparing atomic PNC to higher Q2 measurements, for example at the Z pole, one can study the momentum dependence of Standard model radiative corrections. However, as the accuracy of atomic PNC experiments improves they will require increasingly precise information on neutron densities. 12 ,13 This is because the parity violating interaction is proportional to the overlap between electrons and neutrons. In the future the most precise low energy Standard Model test may involve the combination of an atomic PNC measurement and parity violating electron scattering to constrain the neutron density. There have been many measurements of neutron densities with strongly interacting probes such as pion or proton elastic scattering, see for example ref. 14 Unfortunately, all such measurements suffer from potentially serious theoretical systematic errors. As a result no hadronic measurement of neutron densities has been generally accepted by the field. Because of the uncertain systematic errors, many modern mean field interactions are fit to nuclear data without using any neutron density information, see for

128

example refs. 15 ,16 Finally, there is an important complementarity between neutron radius measurements in a finite nucleus and measurements of the neutron radius of a neutron star, see for example ref.17 Both provide information on the equation of state of dense matter. In a nucleus, Rn is sensitive to the density dependence of the symmetry energy. Likewise the neutron star radius depends also on the density dependence of the symmetry energy at normal and somewhat higher densities. In the future, we expect a number of improving radius measurements for nearby isolated neutron stars. For example, from the measured luminosity and surface temperature one can deduce an effective surface area and radius from thermodynamics. Candidate stars include Geminga 18 and RX J185635-3754.19 This paper is organized as follows. In section II we present general considerations for neutron density measurements. Section III discusses many possible theoretical corrections and shows that the interpretation of a measurement is very clean. In Section IV we discuss atomic parity nonconservation experiments. Section V describes recent progress on the Lead Radius Experiment (PREX). Finally we conclude in section IV.

2. General Considerations In this section we illustrate how parity violating electron scattering measures the neutron density. For simplicity, this section uses the plane-wave Born approximation and neglects nucleon form factors. The effects of Coulomb distortions and form factors are included in section III. These are necessary for a quantitative analysis but they do not invalidate the simple qualitative picture presented here. The effect of the parity-violating part of the weak interaction may be isolated by measuring the parity-violating asymmetry

A LR=

UR -UL ,

UR+UL

(1)

where UL(R) is the cross section for the scattering of left (right) handed electrons. In contrast to the cross section, the asymmetry is sensitive to the distribution of the neutrons in the nucleus. In Born approximation the parity-violating asymmetry is,

(2)

129

with G F the Fermi constant and Ow the weak mixing angle. The Fourier transform of the proton distribution is Fp( Q2) while that of the neutron distribution is Fn(Q2) and Q2 is the momentum transfer squared. The asymmetry is proportional to G FQ2 / a which is just the ratio of ZO to photon propagators. Since 1-4sin20w is small and Fp( Q2) is known we see that ALR directly measures Fn(Q2). Therefore, ALR provides a practical method to cleanly measure the neutron form factor and hence Rn.

3. Theoretical Corrections to the Asymmetry

In this section we document a number of corrections to the parity violating asymmetry and show that they have small uncertainties. Therefore the interpretation of a measurement should be clean. We consider coulomb distortions, parity admixtures, dispersion corrections, and meson exchange currents. Further details and additional corrections are in ref. 20 3.1. Coulomb distortions By far the largest known correction to the asymmetry comes from coulomb distortions. By coulomb distortions we mean repeated electromagnetic interactions with the nucleus remaining in its ground state. All of the Z protons in a nucleus can contribute coherently so distortion corrections are expected to be of order Za/1f. This is 20 % for 208Pb. Distortion corrections have been accurately calculated in ref.,21 see also ref. 22 Here the Dirac equation was numerically solved for an electron moving in a coulomb and axial-vector weak potentials. From the phase shifts, all of the elastic scattering observables including the asymmetry can be calculated. Finally, since the charge density is known it should be possible to "invert" the coulomb distortions and deduce from the measured asymmetry the value of a Born approximation equivalent weak form factor at the momentum transfer Q2 of the experiment. Thus, the main result of the measurement is the weak form factor Fw (Q2) which is the Fourier transform of the weak charge density pw(r),

FW(Q2)

=

J

d 3rjo(qr)pw(r).

(3)

This can be directly compared to mean field or other theoretical calculations. The weak charge density pw(r) is very closely related to the neutron density becuase neutrons carry most of the weak charge.

130

3.2. Parity Admixtures The spin zero ground state of 208Pb need not be a parity eigenstate. There is probably some small admixture of 0-. However, so long as the initial and final states are spin zero, this parity admixture can not produce a parity violating asymmetry in Born approximation. 23 A multi pole decomposition of the virtual photon has a 0+ coulomb but no 0- multipole. So long as the exchanged virtual photon is spin zero, there is no parity violating interference because there is only a single operator. This statement is true regardless of the parity of the initial or final states or if the photon coupling involves a parity violating meson exchange current. Therefore, parity admixtures should not be an issue for elastic scattering from a spin zero nucleus.

3.3. Meson Exchange Currents Meson exchange currents MEC can involve parity violating meson couplings. These are not expected to be important for a spin zero target, see the subsection on parity admixtures above. Meson exchange currents could also change the distribution of weak charge in a nucleus. However, mesons are only expected to carry weak charge over a distance much smaller then Rn. This should not lead to a significant change in the extracted neutron radius. Let r~ EC be the square of the average distance weak charge is moved by MEC. Then from ref.2o the correction to the weak radius will be of order r~Ecj Rn- This is expected to be very small because Rn is large.

3.4. Dispersion corrections By dispersion corrections we mean multiple electromagnetic or weak interactions where the nucleus is excited from the ground state in at least one intermediate state. At the low momentum transfers considered here, the elastic cross section involves a coherent sum over the Z protons and is of order Z2. In contrast, the incoherent sum of all inelastic transitions is only of order Z. Therefore we expect dispersion corrections to be of order ajZ. This is negligible.

3.5. Transverse Analyzing Power The transverse analyzing power Ay provides a direct probe of Coulomb distortion and dispersion corrections. The analyzing power is the left right asymmetry in the cross section for electrons polarized normal to the reaction plane. The analyzing power is parity conserving. However time reversal

131

symmetry insures that Ay is zero in the one photon exchange approximation. Therefore Ay provides a direct measure of two photon effects. For electron scattering from 208Pb, Ay is predicted to be relatively large, about ten times the parity violating asymmetry.24 This large value can lead to a potentially important systematic error in measuring PV asymmetries. If there is a small transverse polarization of the beam this can lead to a helicity dependence in the rate because of the nonzero A y . To control this systematic experiments should measure Ay and measure and control transverse polarizations of the beam. 3.6. Asymmetry Correction Conclusions

In this section we have discussed known corrections to the parity violating asymmetry. These corrections are either small or well known. The largest corrections are from Coulomb distortions and have been accurately calculated.

4. Relation to Atomic Parity Nonconservation Atomic parity nonconservation experiments also probe neutron densities because the parity violating amplitude involves the overlap of atomic electrons with the nuclear weak charge. Therefore atomic PNC measures the same weak form factor FW(Q2), Eq. 3, as does parity violating electron scattering. One might think that the effective Q2 probed in atomic experiments would be very low. However because of relativistic effects, atomic wave functions very rapidly at short distance and the effective Q2 is not all that much smaller than for the proposed PREX experiment. 2o One can constrain the neutron density with PV electron scattering and than use atomic PNC to test the standard model. Or one can use atomic PNC to obtain information on neutron densities. The interpretation of a measurement of PNC for a single isotope may require detailed atomic structure calculations. Alternatively, PNC measurements for a range of isotopes may directly yield information on neutron densities.

5. Progress on the Lead Radius Experiment (PREX) The Lead Radius Experiment (PREX) aims to determine the neutron radius of 208Pb to one percent by measuring the parity violating asymmetry for the elastic scattering of 850 MeV electrons at six degrees. 25 The asymmetry is of order 0.5 ppm and the goal is to measure this to 3 percent. A one week

132 test run of this Jefferson Laboratory ExperiInent was just performed in January 2008 and the full 30 day experiment may be scheduled in 2009. The target involves a thick foil (0.5 mm) of enriched 208Pb sandwiched between two thin diamond foils, see Fig. 1. The diamond, with a very high thermal conductivity, carries away heat and keeps the Pb from melting. This target was tested at high beam cnrrents for a long time and performed well. The detector concept was also tested. This involves focal plane detectors operating at very high rates (1 GHZ) in an integrating mode. In addition, tests were performed on the Compton polarimeter which aims to determine the beam polarization to one percent via electron scattering from polarized laser light. Finally tests were made of luminosity monitors that will monitor the transverse polarization of the electron beam. We eagerly await the full PREX results for the neutron radius of Pb.

Fig. 1. Assembling a Lead-diamond target. A diamond foil is being placed on a Pb foil the third opening from the right). Liquid He will be pumped through the copper frame to cool the Pb-diamond sandwich. Note that the Pb is covered with vacuum to improve thermal contact with the diamond. Image from the PREX web site.

6. Conclusion With the advent of high quality electron beam facilities such as CEBAF, experiments for accurately measuring the weak density in nuclei through

133

parity violating electron scattering (PVES) are feasible. The measurements are cleanly interpretable, analogous to electromagnetic scattering for measuring the charge distributions in elastic scattering. From parity violating asymmetry measurements, one can extract the weak charge density in nuclei and from this the neutron density. The Jefferson Laboratory PREX experiment will measure the neutron radius of 208Pb. By a direct comparison to theory, this measurement tests mean field theories and other models that predict the size and shape of nuclei and can have a fundamental and lasting impact on nuclear physics. Furthermore, PVES measurements have important implications for atomic parity nonconservation (PNC) experiments and for the properties of neutron rich matter in astrophysics. Acknowledgments I thank K. Kumar, R. Michaels, J. Piekarewicz and P. Souder for helpful discussions. This work is supported in part by DOE grant: DE-FG0287ER40365. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

B. Frois et. al., Phys. Rev. Lett. 38, 152 (1977). T.W. Donnelly, J. Dubach and Ingo Sick, Nuc. Phys. A 503 (1989) 589. C. Y. Prescott et al., Phys. Lett. 84B, 524 (1979) P. A. Souder et al., PRL65(1990) 694 W. Heil et at., Nucl. Phys. B327, 1 (1989) B. Mueller et at., PRL78(1997) 3824 K. A. Aniol et al., PRL 82 (1999) 1096 A. Acha et al., PRL 98 (2007) 032301. D. S. Armstrong et al., PRL 95 (2005) 092001. C. S. Wood et ai, Science 275, 1759 (1997). S. C. Bennett and C. E. Wieman, Phys. Rev. Lett.82, 2484 (1999). S. J. Pollock, E. N. Fortson, and L. Wilets, Phys. Rev. C 46, 2587 (1992), S.J. Pollock and M.C. Welliver, Phys. Lett. B 464, 177 (1999) P. Q. Chen and P. Vogel, Phys. Rev. C 48 1392 (1993) L.Rayand G.W.Hoffmann, Phys. Rev. C 31, 538 (1985). R.J. Furnstahl, Briand D. Serot and Hua-Bin Tang, Nucl. Phys. A615, 441 (1997); R. J. Furnstahl and Briand D. Serot, nucl-thj99n019. P. Ring et. al., Nucl. Phys. A624, 349 (1997). J. Carriere et al., ApJ. 593 (2003) 463. Patrizia A. Caraveo et. al., ApJ 461, L91 (1996); A. Golden and A. Shearer, astro-phj9812207. F. Walter, S. Wolk and R. Neuhauser, Nature 379, 233 (1996); Bennett Link, Richard 1. Epstein and James M. Lattimer, PRL 83, 3362 (1999).

134 20. C.J. Horowitz, S. Pollock, P.A. Souder and R. Michaels, Nucl-th/9912038 submitted to Phys Rev C. 21. C.J. Horowitz, Phys. Rev. C5T (1998) 3430. 22. D. Vretenar, G.A. Lalazissis, P. Ring, Nucl-th/0004018. 23. See for example, G. Feinberg, Phys. Rev. D12 (1975) 3575. 24. M. Gorchtein and C. J. Horowitz, arXiv:0801.4575, submitted to Phys. Rev. C. 25. Jefferson Laboratory Experiment E06002, P. Souder, R. Michaels, and G. Urciuoli spokespersons. See also http://hallaweb.jlab.org/parity/prex/

135

NUCLEAR STRUCTURE ASPECTS OF SCHIFF MOMENT AND SEARCH FOR COLLECTIVE ENHANCEMENTS NAFTALI AUERBACH 1,2 and VLADIMIR ZELEVINSKy2,3 ISchool of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel 2Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-1321, USA 3National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824-1321, USA

Abstract Nuclear Schiff moment may have a non-zero expectation value if both parity and time-reversal invariance are violated. The Schiff moment induces the atomic electric dipole moment currently searched by a number of experimental groups. The magnitude of the Schiff moment turns out to be sensitive to many features of complex nuclear structure; especially favorable is the combination of quadrupole and octupole deformation. We discuss these aspects, along with the new ideas for the possibility of nuclear enhancements.

1

Introduction to the Schiff moment

The measurement of the electric dipole moment (EDM) of atoms is the goal pursued by several experimental groups. The best limits obtained for the isotopes 129Xe [1] and 199Hg [2] provide an important advance on the way to information about fundamental forces violating both parity (P) and time reversal (T) invariance. Indeed, as it was argued long ago by Purcell and Ramsey [3], a non-zero expectation value (d) of the atomic EDM in a stationary state of an atom with a certain value J of angular momentum is possible only under simultaneous violation of P- and T-invariance. According to the quantum-mechanical vector model, such an expectation value is determined by the effective operator d. acting within the rotational multiplet IJ M), A

d=((d·J))

J . J(J + 1)

(1)

The pseudoscalar ((d· J)) requires parity non-conservation; on the other hand, due to rotational invariance, its value cannot depend on the angular momentum projection M and therefore should not change under time reversal (M -> -M), whereas under this transformation such a time-odd quantity changes sign. Note that these arguments forbid a non-zero expectation value of any time-even polar vector.

136

The EDM in an atom with closed electron subshells is induced by the corresponding, P, T-violating, electrostatic potential of the nucleus where it may exist because of the presence of P, T -odd forces acting between nucleons and/or quarks. The simplest consequence of such forces would be the non-vanishing dipole moment of the nucleus. However, this dipole moment is practically completely screened by atomic electrons as follows from the Schiff theorem [4, 5]. As pointed out first in Ref. [6], the operator actually inducing the atomic EDM is the so-called Schiff moment, the next vector term of the expansion of the nuclear charge distribution,

(2) (a similar operator is responsible for the isoscalar giant dipole resonance in nuclei [7]). It was recently suggested [8] that the expression (2) for the Schiff moment has to be supplemented by new terms which bring in a contribution of nucleon-nucleon correlations. An accurate rederivation of the Schiff moment [9] confirms the old result (2). Similarly to eq. (1), the expectation value of the Schiff moment in the state with a certain value of nuclear spin 1 is given by A

1

S=((S·I») 1(1+1)·

(3)

This means that we have to look at the structure of the ground state in the odd-A nucleus (1 i= 0). Various aspects of nuclear structure, single-particle and collective, and, in particular, quantum-mechanical symmetry properties emerge as decisive tools we can try to use in order to come to the best experimental candidates.

2

Microscopic calculation of the Schiff moment

The microscopic calculation of the Schiff moment requires assumptions concerning the effective P, T-violating forces; their coupling strengths are to be extracted from the experiments on the EDM (we do not discuss here the important part of the whole approach, namely the high-precision atomic calculations which are necessary for translating the observed EDM value into the unknown force parameters, see, for example, [10]). In the first order with respect to nucleon velocities, the P, T-odd forces between the nucleons a and b have the structure [6]

Wab =

v'2~m ((17abaa-17ba(h).\7aJ(ra-rb)+17~daaxab]·{(Pa-Pb),J(ra-rb)}+)' (4)

where 17ab and 17~b are to be determined by data, and G is the Fermi weak interaction constant. It is believed that the main contribution to the Schiff

137 moment comes from the coherent mean-field part of the interaction (4) and can be written as a one-body operator,

W(r)

G

=-

J2

-

1

TJ

_

- (0- . V)p(r)

2m 47r

'

(5)

where p( r) is the nuclear density. Using standard perturbation theory, we find the Schiff moment (S) of the ground state [I, M = I) == [0) of an odd-A nucleus as a sum over intermediate states In) of the same spin I =1= 0 but opposite parity admixed to the ground state by the interaction W,

(S) = 2

L n

(O[S[n)(n[W[O). Eo - En

(6)

Here it is assumed that the matrix elements of Wand S are real. In the simplest approximation, the ground state has one unpaired particle and we admix the single-particle orbitals of opposite parity. For a spherical nucleus, the mixed orbitals should have the same angular momentum, j' = j, but different orbital momenta, l' = l ± 1. In a deformed nucleus, we need to have Nilsson orbitals of opposite parity originated from such spherical levels. Rare accidental proximity can lead [11] to the small energy denominator in (6). However, in the case of single-particle mixing, this hardly can enhance the outcome since the matrix elements of W, eq. (5), are roughly proportional to those of the single-particle momentum that cancels the energy difference [6]. The single-particle properties in the nucleus are modified by the residual strong interaction. The effects of the core polarization dress the quasiparticle states and renormalize all observables. Realistic calculations performed in various versions of configuration mixing [12, 13, 14, 15] showed that the results for the Schiff moment may differ from those in the single-particle approximation by a factor of about 2.

3

Coherent mechanism: octupole deformation

For a long time it was known that, in a system of interacting particles, there exist powerful many-body mechanisms which, under certain conditions, can substantially increase effects of weak perturbations. For example, parity nonconservation is strongly enhanced in scattering of slow polarized neutrons and fission in the region of narrow neutron resonances [16, 17]. This enhancement is essentially of statistical character as can be seen from the estimates of the level spacing, D ex. N- 1 , and scaling, ex. N- 1 / 2 , of mixing matrix elements of the weak interaction between the compound states of s- and p-wave resonances; here N rv 106 is the degree of complexity of neutron resonance states (a typical number of simple shell-model components in the chaotic wave functions), see

138

reviews [18, 19] and references therein. The corresponding enhancement factor 3 is multiplied, in the case of neutron scattering, by the kinematic rv yN rv 10 coefficient related to the ratio of s- and p-wave neutron widths, rv (rs/rp)1/2, that can bring the observed effect of the longitudinal asymmetry (the difference of the total cross sections for neutrons with opposite helicity) from the typical estimate of 10-(7+8) up to 10%. The statistical mechanisms are presumably absent in the ground state structure of the nucleus. Therefore we have to search for the specific structural features which can bring closely levels of opposite parity that can have a large probability of being mixed by the interaction W. These features are related to the possible coherent mixing. The main efforts in this direction tried to use deformed nuclei as the appropriate arena for the combined action of intrinsic symmetry and weak interactions. Let us consider an axially symmetric deformed odd-A nucleus. In the usual adiabatic approximation, the nuclear rotation (which restores the proper quantum numbers of angular momentum) is adiabatic with respect to intrinsic excitation. The full wave function can be presented as the product of the rotational Wigner function Di[ K depending on the orientational angles and the intrinsic function XK. Here M is the angular momentum projection in the lab-fixed frame while K is the quantum number of the projection onto the intrinsic symmetry axis, K = (1· n), where n is the unit vector along this axis; K is the intrinsic pseudoscalar. In the frozen body-fixed frame, any polar vector, such as the Schiff moment S, can have a non-zero expectation value Sintr without any P- or T-violation. The symmetry dictates the direction of this vector along the symmetry axis,

(7) However, this intrinsic vector is averaged out by rotation because the only possible combination in the space-fixed frame is again similar to the one we have seen in eq. (3), namely proportional to the product ((n. 1)) that violates Pand T-invariance. If the P, T-violating forces create an admixture a of states of the same spin and opposite parity, the average orientation of the nuclear axis arises. In the linear approximation with respect to a,

((n· 1)) = 2aK,

(8)

and, therefore, we acquire the space-fixed Schiff moment (3) along the laboratory quantization axis, 2aKM (9) (IMISI1M) = Sintr ( ). 11+1 Now the idea is to obtain a large intrinsic Schiff moment and not to lose much in translating the result to the space-fixed frame. In order to have a significant value of the intrinsic Schiff moment, it is not sufficient to have a standard quadrupole deformation: we need a type of deformation that distinguishes two directions of the axis violating the symmetry with A

139

respect to the reflection in the equatorial plane perpendicular to the symmetry axis. The collective effect sought for may be related to the simultaneous presence of quadrupole and octupole deformation, the latter creating a pear-shape intrinsic mean field. The importance of octupole deformation for the transmission of statistical parity violation through intermediate stages of the fission process was understood long ago [20]. Now we need the oct up ole deformation in the ground state. In the phenomenological collective description of nuclear deformation in terms of the equipotential surfaces,

(10) the vector terms, l = 1, emerge, after excluding the center-of-mass displacement, through bilinear combinations of even and odd multi poles,

(27

(31

=

l+l

-y 4; {; v(2l + 1)(2l + 3) (31(31+1.

(11)

The main contribution that comes from the product of the lowest static multipoles, quadrupole and octupole, determines the collective intrinsic Schiff moment [21, 22],

(12) The collective character of the octupole moment leads to the strong enhancement of the intrinsic Schiff moment compared to the single-particle estimates. Of course, the results are sensitive to the details of the nuclear models, mean field and effective interactions, but, within a factor of about 2, the Schiff moment may be enhanced up to two to three orders of magnitude [21, 22, 24]. Such results were obtained under an assumption of close levels of opposite parity mixed by the interaction W, with the splitting 6. = IE+ -E_I:::o 50 keY. This is a real situation in 225Ra (6. = 55 keY, I = 1/2) and in 223Ra (6. = 50 keY, I = 3/2). The radium and radon isotopes seem to be promising because of clear manifestations of octupole collectivity. In addition, the large nuclear charge is favorable for the enhancement of the atomic EDM [23]. We need to note that the resulting space-fixed expectation value of the Schiff moment, according to eqs. (12) and (9), is proportional to the product aSintr and therefore to (3~. The mixing can be particularly enhanced if the admixed states are parity doublets [21, 22, 25, 26, 27]. In the presence of the octupole deformation (or for any axially symmetric shape with no reflection symmetry in the equatorial plane), the states of certain parity II are even and odd combinations of intrinsic states X±K with the quantum numbers ±K f= 0 and the intrinsic wave functions which differ just by the "right" or "left" orientation of the pear-shape

140

configuration, IIMK;II) =

V!2f+l ~ [

I

DMKXK

+II ( -

)I+K

I ] DM~KX~K'

(13)

Such doublets in fact do not even require axial symmetry; the label ±K may have a more general meaning. The intrinsic partners are time-conjugate and, according to the Kramers theorem, they are degenerate in the adiabatic approximation. In reality the doublets (13) are split by additional interactions. This can be accomplished by Coriolis forces (the body-fixed frame of the rotating nucleus is non-inertial) or by the tunneling between the two orientations. However such a splitting is not large and the closeness of intrinsic structure should help in increasing the mixing by the weak interactions. As explained in Refs. [21, 22, 25, 27, 28]' only the interaction violating both P- and T-invariance can mix the doublet partners because

The matrix elements of the pseudoscalar W change sign together with K which is possible only if the time-reversal invariance is violated, along with parity. The "normal" weak interaction is·T-invariant. Therefore it is capable of mixing the parity doublets only with the help of a mediator, a regular P, T-conserving interaction, including that one responsible for the doublet splitting. This indirect mixing of parity doublets was suggested in Ref. [27] for explaining the well known "sign problem" in 232Th (the same sign of asymmetry for all neutron resonances which display large parity non-conservation seemingly contradicts to the statistical mechanism of the enhancement). In contrast to this, the P, Tviolating interaction can mix the parity doublets directly, which is important for the enhancement of the Schiff moment.

4

Coherent mechanism: soft octupole mode

As was mentioned earlier, in a nucleus with the combination of developed quadrupole and octupole deformations, the intrinsic Schiff moment is determined by the collective octupole moment (33, whereas the Schiff moment in the space-fixed frame is proportional to its square. Obviously, the sign of the octupole moment is irrelevant. This gives rise to the idea [29, 30] that, instead of static octupole deformation, the same role of the enhancing agent can be played by the dynamic octupole deformation. The soft octupole mode (low-lying collective 3~ "one-phonon" state) is observed in many nuclei and, for a small frequency W3 of this mode, the vibrational amplitude increases, ((3~) oc 1/w3' If the Schiff moment is indeed enhanced under such conditons without static octupole deformation, this can provide a more broad choice for the experimental

141

search. Numerically, the mean square amplitude (/35) is close to the squared value (/33)2 of static octupole deformation in pear-shaped nuclei. This value can be extracted from the reduced transition probability B (E3; 0 --+ 3-). In the presence of the soft octupole mode, the octupole moment Q3jt oscillates with the low frequency, and its intrinsic component along the axis defined by the static quadrupole deformation /32 is phenomenologically given by

Q3

3

= 47l' eZR

3

(15)

/33'

This implies, eq. (12), the slowly oscillating intrinsic Schiff moment,

(16) As we have already stressed, the intrinsic Schiff moment is unrelated to the violation of fundamental symmetries. Now we need to trigger into action the mechanisms converting the intrinsic Schiff moment into observable P, T-violating effects. The description of the previous paragraph referred to the deformed even-even core. The space-fixed Schiff moment needs the non-zero nuclear spin so we proceed to the neghboring odd-A nucleus. The unpaired nucleon interacts with the octupole mode. This dynamic octupole deformation of the mean field can mix, still in the body-fixed frame, the single-particle orbitals of opposite parity. As suggested in Ref. [29], the mixing leads to the non-vanishing expectation value of the weak interaction (W) in the body-fixed frame. This process can be called "particle excitation". In a parallel process of "core excitation" [30], the octupole component of the weak P, T-violating field of the odd particle can excite the soft octupole mode in the core. The estimate of the first mechanism can be based on the octupole-octupole part of the residual nucleon interaction. The original orbital [v) acquires the octupole phonon admixture while the particle is scattered to some orbitals [v') of opposite parity, (17) [v) =} [6) = [v; 0) + av , [v'; 1),

L v'

where the number after the semicolon in the state vector indicates the number of octupole phonons. The orthogonal one-phonon state is, in the same approximation, (18) v'

The mixing amplitudes between the orbitals with energies

Ev

= /33(F3)v'v

,

bv '

Ev -

Ev' +W3

are

(19)

142 where we assume the octupole forces in the form fhF3 , the octupole collective coordinate 133 being defined by eq. (14), while F3 is operator acting on the particle and having the form - (dU j dr) Y 30 with the radial factor usually taken as a derivative of the spherical mean field potential, a reasonable approximation for realistic deformations. The quantity 133 in eq. (19) is the transition matrix element of this collective octupole coordinate between the ground and onephonon states in the even-even core. Now the states 16) and Ii) are mixed by the P, T-violating potential. This mechanism feels the coherent part of the weak interaction Wo averaged over the core nucleons. The mixing matrix element is found as

(20) In the adiabatic limit, when the oct up ole mode frequency W3 is small compared to the single-particle spacing between the orbitals of opposite parity, the weak interaction is essentially acting at a fixed octupole deformation and then it is averaged over the slowly evolving phonon wave function. Then the result practically coincides with that for the static octupole deformation discussed earlier. The only difference is the substitution of the static j3'§ by the dynamic mean square average (j3'§). In the core excitation mechanism [30], the effective part of the weak interaction Wab acts between the valence nucleon b and the paired nucleons a in the core. Because of pairing in the core, only the contribution proportional to the spin of the valence nucleon survives,

(21) We need to extract from this interaction the octupole component W3 proportional to the operator Q3 = r 3 y 30 • The result [30] depends on the specific orbital of the external nucleon and can be presented in the form

(22) (this operator has to be multiplied by the creation or annihilation operator of the 3- phonon). Here k is the numerical factor determined by the spin-orbit structure of the valence orbital; in typical cases Ikl ::::; 0.6. The matrix element of this interaction exciting an octupole phonon (that contains both proton and neutron coherent components) is given by

(23) where the coupling constant is "lb = (ZjA)"lbp is n(p) for the odd neutron (proton).

+ (NjA)"lbn,

and the subscript b

143

Using the mixing produced by the operator W3 for calculating the effective Schiff moment operator and projecting to the space-fixed frame we come to the result [30] of the same order of magnitude as in the case of the particle excitation. Compared to the static octupole deformation, the difference is, apart from numerical factors of order one, just in the substitution of static ((33)2 by the effective dynamic mean square value. Taking the limiting value in 199Hg as a current standard, we can expect the enhancement in the interval of 100 - 1000 if the energy spacing D. is of the order or less than 100 keY. The appropriate candidates are 223, 225 Ra, 223Rn, 223Fr, 225 Ac, and maybe 239pu, where the estimates of Ref. [30] are lower than in Ref. [29].

5

Coherent mechanism: soft quadrupole and octupole modes

The results of the previous consideration point out a tempting possibility of searching for the significant enhancement of the Schiff moment in a broad class of spherical nuclei where both collective modes, quadrupole and ostupole, are clearly pronounced and have low frequencies. As an example, one can mention light spherical isotopes of radium and radon. The experimental data [31] for 218,220,222Rn and for other even-even nuclei in this region show long quasivibrational bands of positive and negative parity, where energy intervals are far from the rotational rules. The phonon frequencies are quite low, and there are strong El transitions between the quadrupole and octupole bands. The softness of the modes and large phonon transition probabilities B(E2; 0 ~ 2+) and B(E3; 0 ~ 3-), along with strong dipole interband coupling, indicate that the situation might be favorable for the enhancement of the Schiff moment. The mixing of the 2+ and 3- phonons with the valence particle in a neighboring odd-A nucleus can be considered as a slow (adiabatic) process of adjustment of the valence orbitals to the oscillating mean field, as we argued in the previous section. If the particle can form states with the same spin in both types of mixing, these states should be rather close in energy and can be mixed among themselves by the weak interaction. Here we do not introduce any body-fixed frame so the angular momentum must be strictly conserved in those mixing processes. Thus, in our main eq. (6), we can have in the odd nucleus states of both parities with the same I, M quantum numbers like

II M)

~ [coa)MO" + t= C,(j' A; I)(a),QlhM ]1 0).

(24)

Here Ojm and Q>'JL are quasiparticle and phonon operators, respectively, whereas [0) represents the ground state of the even nucleus. The detailed microscopic calculations along these lines were performed in Ref. [32]. In the neutron-odd nucleus, the proton contribution needed for the

144 Schiff moment comes from the transition matrix element of the Schiff operator between the appropriate states (24) of the same spin I and opposite parity,

(I±, M

= IISzII'f, M = 1) =

L

X(jI; )")..')C2 (j)..; I±)C2 (j)..'; I'f)()..IISII)..')·

(25) where X(jI; )..N) are geometric coefficients resulting from vector coupling of angular momenta. The reduced matrix element ofthe Schiff momentum, ()..IISII)..'), is taken between the phonon states in the even-even core. Because of the strong dipole coupling between the corresponding bands in the candidate nuclei, we expect that this matrix element should not cause an additional reduction. Concrete calculations [32J used the random phase approximation (RPA) in the form of the quasiparticle-phonon model [33]. The multipole-multipole forces are fixed in even nuclei by the phonon parameters. The result for the Schiff moment can be expressed in terms of the single-particle Schiff matrix elements, (jll[S[[j2), standard pairing amplitudes, (u, v), and the RPA phonon amplitides of two-quasiparticle and two-quasihole components, (A, B),

(26) In the conventional RPA framework, the three-phonon couplings, as in eq. (26), are expressed by triangular diagrams, which come with a considerable reduction due to the combinations UI U2 - VI V2 of the pairing coherence factors. This combination is antisymmetric with respect to the single-particle Fermi surface and would vanish in the case of full symmetry around the Fermi surface. This can be understood in analogy with the well known Furry theorem of quantum electrodynamics. In that case three-photon diagrams vanish exactly because of the precise cancellation of electron and positron contributions to the loop with three photon tails. In the discrete nuclear spectrum, there is no full symmetry and the result does not vanish but still it is partially suppressed. The weak interaction was taken in the mean field form, eq. (5), -

Wb(r)

G ..j22m

_

1 dp(r) --, 47fr dr

= - - TJ(fJ· r) -

(27)

where p(r) is determined by the pairing occupancy factors in the core. There are several contributions of the interaction (27) into various parts of the complicated calculation: in the wave functions of the unpaired quasiparticle, in the matrix elements of quasiparticle-phonon coupling. in the intermediate particle and phonon propagators, and in the phonon loops. Combining these calculations with the energy denominators we come to the final results.

145 At this stage we could not find an enhancement of the nuclear Schiff moment. For example, for the 219Rn isotope the matrix element of the weak interaction equals -1.3 T/ • 10- 2 eV, and the final value of the ground state Schiff moment was 0.30 T/ .1O- 8 e·fm 3 . Typically, the reduced matrix elements (2+1813-) in the even nucleus are of the order (1-2) e·fm 3 , and the matrix elements of the Schiff operator between the ground state in the odd nucleus and its parity partner are around 0.1-0.2 e·fm 3 . Final results for the Schiff moment are of the same order as in pure single-particle models (the single-particle contribution unrelated to the soft modes [34, 13, 14] has to be added). These calculations seemingly contradict to the idea of a possible enhancement by soft collective modes. Nevertheless, a useful exercise [32] confirms that the effect indeed exists but, in the RPA framework, requires artificially low collective frequencies when the dynamic deformation amplitudes increase as (3 ex: l/w. One can consider the theoretical RPA limit of collapsing frequencies,

(28) and accurately separate the singular part of the RPA solutions. As the collective frequencies go down, the reduced matrix element (2+ 1813-) in the even nucleus, the mixing matrix element of the weak interaction in the odd nucleus and the final Schiff moment grow large. These trends are seen in the following Table. Nucleus 219Ra

~~lRa

y

1 0.1 0.01 1 0.1 0.01

(2+1813-) 1.7 20 195 2.2 23 235

m.e. W -1.3 1.1 53 0.2 -19 -253

m.e.8 -0.1 -0.2 -0.2 -0.2 -0.5 -2.7

8 0.3 -0.2 6.2 -0.1 6 560

(29)

It is clear from the Table that the matrix element (2+1813-) increases ex: l/w. Other matrix elements are also sensitive to the level spacing in the odd nucleus. Here we need to mention that the RPA results with the parameters fitted to the phonon frequencies do not produce a satisfactory description of spectra in odd nuclei. To summarize the situation, we can conclude that in the situation when the phonon-quasiparticle coupling becomes strong, the standard RPA approach that accounts for a single-phonon admixture to quasiparticle wave fuinctions, is unreliable. The effect of enhancement appears either with static deformation or in the strong coupling limit when effectively the condensate of phonons emerges that mimics the deformed field. In the exactly solvable particle-core model [35] with the soft monopole model, A = 0, the ground state of the odd-A nucleus contains a coherent phonon state with the average number of phonons defined by the coupling constant. The quasiparticle strength in this regime is strongly fragmented over many excited states. Similar effects should take place

146 for quadrupole and octupole modes [36, 37, 38] when the coherence finally leads to the phase transition to static deformation. In agreement with above arguments, the calculations [32] with artificially quenched frequencies show that the wave function of the odd nucleus becomes exceedingly fragmented. For example, in the realistic case, y = 1, for the ground state I = 7/2 in 219Ra, there exists only one large combination of amplitudes required for the mixing, namely there are particle-phonon states 7/2+ with the wave function (2g 9 / 2,2+h/2 and 7/2- with the wave function (2g9 / 2 , 3-h/2; their weights in the full RPA wave functions are 98% for negative parity but only 8% for positive parity. With quenching of frequencies, these amplitudes are getting drastically reduced, up to 2% for negative parity and 1% for positive parity. Only after the spreading of the single-particle strength reached saturation in the orbital space under consideration, one can indeed see the enhancement of the Schiff moment. Thus, the conventional RPA ansatz for the wave function of the odd nucleus as a superposition of particle-phonon components is invalid under conditions of soft collective modes. Many-phonon components take over a large fraction of the total wave function. Moreover, soft modes become mutually correlated. The correlation between soft quadrupole and octupole excitations was suggested in the global review of octupole vibrations [39]. The presence of the octupole phonon singles out the axis and triggers the spontaneous symmetry breaking with effective quadrupole condensate emerging. The predicted correlation of the two modes was confirmed by the recent experiments [40] for a chain of xenon isotopes. A similar effect of condensation is brought in by the odd particle. We gave a schematic derivation of arising correlations in Ref. [32]. Recently a model of two single-particle levels of the same large j and opposite parity with n particles interacting with quadrupole and octupole collective modes was considered using the exact diagonalization instead of the RPA [41]; the results will be reported elsewhere.

6

Conclusion

In this short review we tried to demonstrate the abundance of ideas and physical images related to the search of the effects of P, T- violating forces in atomic nuclei. Of course, there is immediate interest in measuring such effects which would lead us beyond the Standard Model, while currently we know only the upper limits. Because of extreme difficulty of such experiments and their timeconsuming nature, it is important to try to establish the most promising path and to select nuclei where we can expect the most pronounced effects. Along with that, it turns out that the wealth of physics related to the violation of fundamental symmetries in nuclei elucidates also many particular problems of nuclear structure which until now do not have definite answers. These problems are related to various manifestations of quantum-mechanical

147 symmetries in a strongly interacting self-bound many-body system as the complex nucleus. (There are also ideas in the literature of using molecular and condensed matter systems [42, 43, 44].) Parity violation is enhanced by the orders of magnitude by statistical (chaotic) properties of compound state neutron resonances. In the search for the P, Iviolation we are looking for coherent effects. The EDM of the atoms is induced by the nuclear Schiff moment through its P, I-violating potential. The best perspectives for a significant enhancement of the nuclear Schiff moment are currently seen in the nuclei with static octupole deformation in the ground state. We argued that the soft octupole mode in a combination with well developed quadrupole deformation is expected to display similar enhancement. Finally, we came to soft nuclei with slow quadrupole and octupole motion of large amplitude. Although the direct attempt in this direction did not yet bring desired results, we need to better understand nuclear physics of such nuclei where the shape is in fact ill-defined and the routine theoretical methods, such as the RPA, are probably not sufficient. This leads to new problems of structure of mesoscopic systems on the verge of shape instability. Another interesting question is that of the three-body residual forces (coming from bare three-body forces or induced by the nucleon correlations). Such forces may give stronger modemode coupling not limited by the Furry theorem discussed above. In general, the entire area of research is very promising for understanding the fundamental symmetries at work in a many-body environment.

7

Acknowledgements

The work reviewed in this article was supported by the NSF grants PHY0244453 and PHY-0555366, and by the grant from the Binational Science Foundation US-Israel. We also acknowledge support from the National Superconducting Cyclotron Laboratory at the Michigan State University and from the Institute for Nuclear Theory at the University of Washington. We thank V.V. Flambaum, R.A. Sen'kov and A. Volya for collaboration and many illuminating discussions.

References [1] J.P. Jacobs, W.M. Klipstein, S.K. Lamoreaux, B.R. Heckel, and E.N. Fortson, Phys. Rev. A 52, 3521 (1995). [2] M.V. Romalis, W.C. Griffith, J.P. Jacobs, and E.N. Fortson, Phys. Rev. Lett. 86, 2505 (2001). [3] E.M. Purcell and N.F. Ramsey, Phys. Rev. 78, 807 (1950). [4] L.I. Schiff, Phys. Rev. 132, 2194 (1963).

148 [5] P.G.H. Sandars, Phys. Rev. Lett. 19, 1396 (1967). [6] V.V. Flambaum, LB. Khriplovich, and O.P. Sushkov, Zh. Eksp. Teor. Fiz. 87, 1521 (1984) [ JETP 60, 873 (1984)]. [7] M.N. Harakeh and A.E.L. Dieperink, Phys. Rev. C 23, 2329 (1981). [8] C.-P. Liu, M.J. Ramsey-Musolf, W.C. Haxton, R.G.E. Timmermans, and A.E.L. Dieperink, Phys. Rev. C 76, 035503 (2007). [9] R.A. Sen'kov, N. Auerbach, V.V. Flambaum, and V.G. Zelevinsky, Phys. Rev. A 77, 014101 (2008). [10] V.A. Dzuba, V.V. Flambaum, J.S.M. Ginges, and M.G. Kozlov, Phys. Rev. A 66, 012111 (2002). [11] W.C. Haxton and E.M. Henley, Phys. Rev. Lett. 51, 1937 (1983). [12] V.V. Flambaum and O.K. Vorov, Phys. Rev. C 49, R1827 (1994). [13] V.F. Dmitriev and R.A. Sen'kov, Yad. Fiz. 66, 1988 (2003) [Phys. At. Nucl. 66, 1940 (2003)]. [14] V.F. Dmitriev, R.A. Sen'kov, and N. Auerbach, Phys. Rev. C 71, 035501 (2005). [15] J.H. de Jesus and J. Engel, Phys. Rev. C 72, 045503 (2005). [16] V.P. Alfimenkov et al., Usp. Fiz. Nauk 144, 361 (1984) [SOY. Phys. Usp. 27,797 (1984)]. [17] G.E. Mitchell, J.D. Bowman, S.L Pentila, and E.L Sharapov, Phys. Rep. 354, 157 (2001). [18] V.V. Flambaum and G.F. Gribakin, Prog. Part. Nucl. Phys. 35, 423 (1995). [19] V. Zelevinsky, B.A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276, 85 (1996). [20] O.P. Sushkov and V.V. Flambaum, Usp. Fiz. Nauk 136, 3 (1982) [SOY. Phys. Usp. 25, 1 (1982)]. [21] N. Auerbach, V.V. Flambaum, and V. Spevak, Phys. Rev. Lett. 76, 4316 (1996). [22] V. Spevak, N. Auerbach, and V.V. Flambaum, Phys. Rev. C 56, 1357 (1997). [23] V.V. Flambaum, Phys. Rev. A 60, R2611 (1999).

149 [24] J. Engel, M. Bender, J. Dobaczewski, J.H. de Jesus, and P. Olbratowski, Phys. Rev. C 68, 025501 (2003). [25] N. Auerbach, Nucl. Phys. A577, 443c (1994); V.Spevak and N.Auerbach, Phys.Lett. B 359, 254 (1995). [26] N. Auerbach, J.D. Bowman, and V. Spevak, Phys. Rev. Lett. 74, 2638 (1995). [27] V.V. Flambaum and V.G. Zelevinsky, Phys. Lett. B 350, 8 (1995). [28] O.P. Sushkov and V.V. Flambaum, Yad. Fiz. 31, 55 (1980) [SOy. J. Nucl. Phys. 31, 28 (1980)]. [29] J. Engel, J.L. Friar, and A.C. Hayes, Phys. Rev. C 61, 035502 (2000). [30] V.V. Flambaum and V.G. Zelevinsky, Phys. Rev. C 68, 035502 (2003). [31] J.F.C. Cocks et al., Phys. Rev. Lett. 78, 2920 (1997). [32] N. Auerbach, V.F. Dmitriev, V.V. Flambaum, A. Lisetsky, R.A. Sen'kov, and V.G. Zelevinsky, Phys. Rev. C 74, 025502 (2006). [33] V.G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons (lOP Publishing, Bristol, 1992). [34] V.V. Flambaum, I.B. Khriplovich, and O.P. Sushkov, Nucl. Phys. A449, 750 (1986). [35] S.T. Belyaev and V.G. Zelevinsky, Yad. Fiz. 2, 615 (1965) [SOy. J. Nucl. Phys. 2,442 (1966)]. [36] A. Bohr, Mat. Fys. Medd. Dan. Vidensk. Selsk. 26, #14 (1952); A. Bohr and B.R. Mottelson, ibid. 27, #16 (1953). [37] A. Abbas, N. Auerbach, N. Van Giai, and L. Zamick, Nucl. Phys. A367, 189 (1981). [38] Ch. Stoyanov and V. Zelevinsky, Phys. Rev. C 70, 014302 (2004). [39] M.P. Metlay, J.L. Johnson, J.D. Canterbury, P.D. Cottle, C.W. Nestor, S. Raman, and V.G. Zelevinsky, Phys. Rev. C 52, 1801 (1995). [40] W.F. Mueller et a1., Phys. Rev. C 73, 014316 (2006). [41] N. Auerbach, A. Volya, and V. Zelevinsky, to be published. [42] J.J. Hudson, B.E. Sauer, M.R. Tarbutt, and E.A. Hinds, Phys. Rev. Lett. 89, 023003 (2002). [43J S.K. Lamoreaux, Phys. Rev. A 66, 022109 (2002). [44J T.N. Mukhamedjanov and O.P. Sushkov, Phys. Rev. A 72, 034501 (2005).

150

LA-UR-08-1462

THE INTERPRETATION OF ATOMIC ELECTRIC DIPOLE MOMENTS: SCHIFF THEOREM AND ITS CORRECTIONS C.-P. LIU

T-16, B283, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA * Searches of permanent electric dipole moments (EDMs) using neutral systems are subject to an important screening effect which is summarized in a famous theorem by Schiff. The interpretation of actual measured EDMs thus requires this effect being properly taken care of. In this talk, we briefly outline a recent re-formulation of the Schiff theorem by Liu et al.? and discuss its differences from previous derivations with an emphasis on the so-called Schiff moment operator.

Keywords: electric dipole moments, Schiff theorem, Schiff moment

1. Schiff Screening

Permanent electric dipole moments (EDMs) are signatures of both parity (P) and time-reversal (T) violation, and serves as an alternative venue, besides high-energy colliders, to look for charge-conjugation-parity (C P) violation, assuming CPT invariance is exact. Although EDMs provide excellent testing grounds for P and T violation, their measurements, however, are a tough challenge for experiments. Except for recent developments in storage-ring techniques and ion trapping, which make EDM measurements in charged systems possible, most searches mainly involve neutral systems. Nevertheless, the measurability of the EDM of a neutral system is severely constrained by the so-called Schiff screening effect - which is first noted by Purcell and Ramsey? and is formulated rigorously later on in a theorem by Schiff.? The Schiff theorem states that: *Now at: Department of Physics, University of Wisconsin-Madison, 1150 University Avenue, Madison, WI 53706, USA.

1.51

"... for a (non-relativistic) quantum system of point, charged, electric dipoles in an external electrostatic potential of arbitrary form, there is complete shielding; i.e., there is no term in the interaction energy that is of first order in the electric dipole moments, regardless of the magnitude of the external potential." and can be easily understood by the following classical picture shown in 1.

Figure 1. The classical picture of Schiff screening: Since a neutral system can not be accelerated by au external electric field, the charge constituents of this system must re-arrange themselves so that at each of their locations, the net electric filed is zero. This figure shows an example at the center of the atom, where the atomic nucleus locates. As the external electric field is totally canceled by the internal one induced by the polarization of electron cloud, the nuclear EDM is non-detectable.

Etot=O

However, the important assumptions of the Schiff theorem are not exactly met in most situations. Take atoms as an example, they are violated, to certain degrees, because (1) the atomic electrons can be very relativistic, especially in heavy atoms (breaks the assumption of non-relativistic dynamics), (2) the atomic nucleus has a finite size (breaks the assumption of point particles, and (3) the electron--nucleus interaction contains magnetic components such as hyperfine interaction (breaks the assumption of purely electrostatic interaction). Therefore, the EDM of a neutral system being measured is in fact a residual one that evades the screening effect. These loopholes of the Schiff theorem have stimulated many discussions in the past after Schiff's original paper, with most of them focus on specific single issues such as the relativistic effect (see, e.g., Refs. ?,?,?), which for example anticipate a (Z is the atomic number, and the power can be even greater) enhancement of the electron ED M in paramagnetic atoms, or the nudear finite-size effect (see, e.g., Refs. ?,?,?,?), which for example lead to formulations of the so-called Schiff moment that is supposed to dominate EDMs of diamagnetic atoms. Given that the next-generation EDM measurements will hopefully improve the current upper limits by a few orders of magnitude and thus provide more stringent constrains on physics beyond the Standard Model, it is equally important to refine the theoretical interpretation of experimental data, which requires a series of reliable cal-

152

culations at different energy scales. The aim of Ref. ? is to carefully revisit the Schiff screening of atoms, and systematically identify all the residual terms that contribute to experimental observables. In the following section, we will outline the important steps in this re-formulation of the Schiff theorem.

2. Schiff Theorem Re-formulated As the intrinsic EDM, d, of a system must be proportional to its total angular momentum (by Wigner-Eckart theorem), the typical EDM measurement is subjecting the system to an external electric field, Eext' and looking for the change in the frequency of spin precession, i.e., the Larmor frequency which is related to the energy shift fj.E = -d . Eext' when the direction of the applied field is reversed. First, we identify the mechanisms giving rise to total energy shift that is first-order in both Eext and P-,T-violation, characterized by a superweak scale OF. , As shown in Fig. 2, they can be categorized as (i) couplings to the intrinsic electron and nuclear EDMs, de and dN, and (ii) coupling to the induced atomic EDM by the internal P-,T-violating (PVTV) electron~ nucleus interaction via electronic or nuclear polarization. In the language of perturbation, part (i) corresponds to the first order while part (ii) to the second order. Looking at panel (d) more closely, one can easily see that this diagram is much less important than panel (c): Because the typical nuclear excitation energy is of MeV scale and the atomic one is of eV scale, there is a huge suppression rv 1O~6 due to the energy denominator. (b)

+c.

C.

(d)

Figure 2. Interactions between the external field and (a) electron EDM, (b) nuclear EDM, along with external-fielddependent, induced EDMs involving (c) electronic polarization and (d) nuclear polarization.

In this spirit, the full atomic Hamiltonian is broken down as

H atom

= Ho + HI,

(1)

153

where the unperturbed Hamiltonian Ho contains the relativistic freeelectron Hamiltonian and all the P-,T-conserving (PCTC) electronelectron (ee), electron-nucleus (eN), and nuclear (nuc) interactions

z

Ho

= L (f3i me + Cti . Pi) + ~~~e) + ~~~N) + Hi':,~c,

(2)

i=l

and the perturbation HI contains (1) the interactions of electrons and nucleus with the external field, v.,~£ and Ve~), (2) the interactions of de with the external field and nucleus, v.,~2 and V;~~N), and (3) the interactions of nuclear PVTV moments (such as dN) with the external field and electrons, ~ (N) v.,xt

~ (eN)

and ~nt H

I

=

V(e) ext

+ V(N) + VUi) + V(,W) + V(N) + V(eN) ext ext lilt ext lilt·

(3)

In Fig. 3, one sees that a nuclear EDM receives contributions from (1) the intrinsic EDMs of nucleons, (2) the exchange current and (3) the wave function polarizations due to some PVTV nuclear interaction. As all these contributions are results of P-,T-violation due to either exotic charge density operators which have different transformation properties under P and T transformations from the normal one (cases (1) and (2)), or the normal charge density operator whose matrix elements are sandwiched between quantum states of broken P-,T-symmetry (case (3)), they are grouped together in Eq.3 under the realization that they appear of the same order, GF, after nuclear matrix elements are taken. However, one should note that their operators behave differently under P and T transformations.

(b)

HA" ~"" N

N

(c)

N

N

Figure 3. The contributions to a nuclear EDM from (a) nucleon EDMs, (b) PVTV exchange charges, and (c) parity admixtures induced by PVTV nuclear interactions (polarization contribution).

Second, the nuclear charge and current density operators are expanded in terms of spherical multipole operators, and the procedure relies on the

154

following expansion of the electromagnetic Greene's function

Ix

~YI ~ J~M 2'~: 1X'~H [+, +~(y- x)

(F -l)J (b)

(4)

y:t

where is the spherical harmonics. The multipole moments associated with the expansion term (a) are the usual static moments, whose interactions are extended. On the other hand, the ones associated with (b) are non-vanishing only when the field point x is inside of the source point y (indicated by the theta function B(y - x)), i.e., there is no interaction unless two electromagnetic densities overlap. These moments are thus called "local" moments. In Tab. 1, moments of these electromagnetic multipole, including Coulomb (C), transverse magnetic (M), and transverse electric (E), are classified by whether they are allowed by P- and T-invariances. According to this table, CJ=even (including Co) and MJ=odd appear in H o, while CJ=odd and even MJ=even appear in HI. All the EJ multipoles are ignored, since they are either PVTC or PCTV, which are of no interest here. Table 1. Parity and time-reversal characteristics of Coulomb (C J ), transverse magnetic (MJ ), and transverse electric (EJ) multipole moments. The script letters C, M, and £ are for the local moments, and "x" indicates no contribution transforming as indicated. Moments (C J ) , (CJ(x») (C J ) ,(CJ(x») (CJ) , (CJ(x»)

PCTC even J odd J x

PVTV odd J even J x

PVTC x x odd J

PCTV x x even J

Now we demonstrate how the screening comes out as a cancellation between Figs.2(a)+(b) and Fig.2(c). Assuming the unperturbed atomic states are formally solved: (5)

then the first- and second-order energy shifts due to various components of the atomic EDM are calculated by

155

The trick to obtain the total energy shift, of order the internal PVTV eN interaction as v(eN) mt

+ V(eN) = [6e+N 6 , mt

GF

Ii0 1+ ~V(eN) mt

Eext'

is to rewrite

+ ~V(eN) mt·

(8)

Because the commutator term in Eq. (8), which represents the internal EDM interactions, can be easily summed by applying the closure relation, the total energy shift becomes

We refer most details of the full derivation to Ref.?, and only give an example here to illustrate the screening of nuclear EDM. The internal nuclear EDM interaction is recast as

(10) where the omitted part vanishes in the point-particle limit of nucleus. After identifying 6N, one can conclude that

Z

=

L

dN

Z·E ext

i=l

(11) The Schiff screening is thus evidenced by the exact cancellation between

v.,~) and [v.,~2 , 6N], which leads to zero energy shift.

156

3. Schiff Moment Operator After the screening effect has been completely subtracted, the residual terms which contribute to the energy shift can be systematically identified from Eq. (9) using the procedure discussed above. Compared to previous derivations, the approach taken by Ref. ? differs in one crucial aspect where nuclear charge and current densities are treated as operators instead of distributions. By doing this, the whole derivation is fully quantum-mechanical with all degrees of freedoms being treated as dynamical. The matrix elements are only taken after the dynamical operators are identified; while the previous approaches require already-known nuclear density distributions which makes, for example, the identification of nuclear Schiff moment operator somewhat obscure. As discussed in the last section, there are many possibilities to break the assumptions of the Schiff theorem, and we will not elaborate them comprehensively here, but use the Schiff moment operator as an illustrative case for the differences between this new and the traditional approaches. The Schiff moment, a PVTV nuclear vector moment first identified in Ref. ?, is a measure of the leading-order, finite-size effect which evades the Schiff screening, and supposed to dominate EDMs of diamagnetic atoms. The most general form of the nuclear Schiff moment interaction, with a single valence electron at x, is given by

(12) where the symbols 129, 8, and 8 denote general (J =I=- 0), scalar, and doublescalar symmetric coupling of spherical operators, respectively; (... V) = --+ f1/2 (( ... \7) - (\7 ... )); and the subscripts "<" and ">" in local moments refer to the terms of "(X/y)2 J+l" and "I" in part (b) of the multi pole expansion, Eq. (4), respectively. The terms in the third and forth lines, which involve commutators of two nuclear multipole operators, are a result of the

157

operator formulation - they simply vanish in the approach where one starts with treating nuclear charge density as distribution. Since these commutators are non-vanishing when nuclear charge density contains velocity and/or spin dependence, these require relativistic or exchange effects that are normally small corrections to the density given by impulse approximation. The term in the fifth line describes the response of the internal nuclear degrees of freedom to the external field. As interaction with this moment is of contact nature, normally it most effectively mixes atomic s- and p-orbits. Following the treatment suggested by Flambaum and Ginges,? which uses polynomial expansions of electron wave functions to handle nuclear local moments, one can identify the local Schiff operator for atomic s-p transitions, SL, through the following relation

(p[ OSchiff Is)

=

(P[ - 47r SL . V 0(3) (x) Is) + . . . .

(13)

The omitted part" ..." corresponds to terms in the third, forth, and fifth lines in Eq. (12); it is given explicitly in Ref. ? and needs further exploration. In this presentation, we focus on SL. If further restricted to the leading order in expansions of electron wave functions, the local Schiff operator is reduced to a Schiff operator:

A 1 (Snew) = 10

~ {2 7 (YpYp) -

2

2

5 ( (dN Q9 Yp) - -5-(dN 4v'21f A)] 1/,)} 3Z Q9 YpY2(Yp

(14) that can be directly compared with the one used in literature:

(15)

The first difference one immediately observes is Snew contains a "dipoleQ9quadrupole" term. This can be easily understood since traditional derivations assume a spherical charge distribution, where no quadrupole moment exists by default. This omission be can be fixed without problem. However, the more striking difference concerns the way the second matrix elements of these two operators are evaluated. In the new form, it is a groundstate expectation value (GEV) of a composite operator (dN Q9 L: p Y;), but in the traditional form, it is a product of two GEVs, (dN) Q9 (L: p y;). Since

,

158

one way to evaluate (dN ® Lp Y~l is by inserting a complete set of states

p

n

p

the difference from (dNl ® (L p Y~l apparently depends on how big the excited states contribute to this sum. The latter difference also stems from treating nuclear density as an operator so that composite operators always surface after applying Eq. (8) to work out the Schiff screening. In a simple case of deuteron, the Schiff moment can be reliably calculated with high-quality wave functions. Assuming the PVTV two-nucleon interaction is the only source of nuclear EDM (no contributions from nucleon EDMs), the ratio of the three terms in Eq. (14) turns out to be: 1 : -5/3 : -4/3. If the latter two contributions are calculated without including the excited states, the ratio becomes: 1 : -1.03: -0.07. One can immediately see the huge difference. Whether the discrepancy becomes smaller with the increase of nuclear mass is of course an important question, as existing calculations of nuclei like Hg, Ra, etc. are all based on the traditional operator. For some qualitative discussion, we refer readers to the contribution by J. Engel in the same volume, and hopefully there will be more definite answers, especially those checked by numerical study, that resolve this question. We also note that the importance of other terms in Eq. (13) should be studied in order to have an eventual, meaningful comparison with existing calculations.

4. Summary

In this talk, we briefly outline a recent re-formulation of the Schiff theorem and highlight its several differences in comparison to the traditional approaches, in particular the form of the Schiff operator. This new work is based on more general principles of quantum mechanics, and the traditional form can be shown as a series of its approximations. Therefore, we advocate the adoption of this new framework in order to have better interpretation of experimental results, which we expect to see many improvements in next-generation runs. Whether the new framework converges to the old one without much difference for atoms of experimental interests is certainly one problem to be addressed as a high priority in this community.

159

Acknowledgments The author would like to acknowledge the support from the U.S. Department of Energy under Contract DE-AC52-06NA25396, and the discussions with J. Engel and V. V. Flambaum.

160

T-VIOLATION AND THE SEARCH FOR A PERMANENT ELECTRIC DIPOLE MOMENT OF THE MERCURY ATOM M. D. SWALLOWS·,I, W. C. GRIFFITH 2 , T. H. LOFTUS3, M. V. ROMALIS4, B. R. HECKEL3, and E. N. FORTSON 3

IJILA, 440 UCB, Boulder, CO 80309 • E-mail: [email protected] 2 NIST,

Mailcode 847.80, 325 Broadway, Boulder, CO 80305

3 Department

4 Physics

of Physics, University of Washington, Seattle, WA 98195, USA

Department, Jadwin Hall, Princeton University, Washington Road, Princeton, NJ 08544

There has been exciting progress in recent years in the search for a permanent electric dipole moment (EDM) of an atom, a molecule, or the neutron. An EDM along the axis of spin violates time-reversal (T) symmetry. Although such a dipole has not yet been detected, mainstream theories of possible new physics, such as Supersymmetry, predict the existence of EDMs within reach of modern experiments. After a brief introduction to EDM experiments and their implications for the existence of new T-violating (and hence C P-violating) interactions, we review recent work on our own EDM experiment with mercury atoms at the University of Washington, describing the newest version of this experiment and discussing current measurements. We have instituted a fixed blind offset that permits us to test for systematic errors while insuring that any cuts in the data are made objectively. Compared with our 2001 result Id(199Hg)1 < 2.1 x 10- 28 ecm, an improvement by a factor of 5 to 10 should be forthcoming, thereby probing yet further for expected new physics.

1. Introduction

Electric dipole moment (EDM) experiments have their origin in a suggestion by Purcell and Ramseyl that the assumption of the invariance of physical law with respect to spatial inversions (parity, or P-symmetry) should be subject to experimental verication. These authors proposed and carried out a search for the electric dipole moment of the neutron as a means of testing this assumption, which yielded the result Id( n) I < 5 X 10- 20 e cm. 2 Since

161

then, there have been many further searches for an EDM of the neutron, with ever increasing precision. Likewise there have been continually improved searches for an EDM of an atom or a molecule. Thus far, all EDM experiments have yielded a null result. Nevertheless, elementary particle theories that attempt to go beyond the Standard Model,3 most notably Supersymmetry,4 predict that EDMs should exist and be large enough to be detected by experiments now underway or soon to begin. 5,6 The existence of an EDM of any non-degenerate quantum system would imply a breakdown of not only P-symmetry, but also of time-reversal symmetry (T), and through the CPT theorem, a violation of CP-symmetry as well. 5,6 (C is charge conjugation, or particle/antiparticle symmetry). C Pviolation was first discovered in the decays of Ko mesons 40 years ago,7 and has recently been confirmed in B meson decays.8,9 For many years after the initial discovery, the search for a neutron EDM provided an exacting test of theories put forward to account for the Ko decays, and ruled out most of them as the experimental upper limit on the neutron moment steadily decreased to its current value. 10 Atomic and molecular EDM experiments made equally striking advances as well, starting in the 1960s 11 and leading up to recent work that includes measurements on thallium 12 and mercury,13 with a host of new experiments now planned or underway. At current accuracies, the atomic and neutron experiments set comparable and complementary bounds on Supersymmetry (SUSY) and other theories of new physics. As shown in Fig. 1, EDM predictions from SUSY models are already worrisomely large when compared to experiment. The Minimally Supersymmetric Standard Model (MSSM), with "natural" values (of order unity) for its two additional CP-violating phases, gives EDMs that are between 10 and 100 times larger than current experimental limits. Fig. 1 shows the allowed phase values in the MSSM when the neutron,15 electron (determined from the atomic thallium EDM limit12), and mercury13 EDM limits are considered. The combined limit constrains both phases to be very near zero, which indicates that the MSSM requires some degree of "fine tuning" to be a valid model. Further improvements in the precision of EDM experiments will continue to inform SUSY models, and in general can be considered a sensitive method of probing for CP-violating new physics. 3

2. EDM experiments The interaction of a particle's or atom's electric dipole moment with an applied electric field is analagous to its magnetic moment's interaction with

162

0.4

de

0.2

-0.2

'---:-;~7--::-=~~O;;-.·-05~O.-1-0-.-15-'

Fig. 1. Allowed values of CP violating phases for the MSSM, assuming a superpartner mass scale of M= 500 GeV. For SUSY to protect the guage hierarchy, M should be in the range 100 - 1000 GeV. The EDM sensitivity scales as M- 2 , so if M= 1000 GeV the angle bounds would be four times larger. The figure is adapted from Ref. 14, updated by M. Pospelov (2003).

a magnetic field. The Hamiltonian of such a particle is H = -(J.1,B +dE)· ~, where J.L is the particle's magnetic moment, d is its electric dipole moment, Band E are the magnetic and electric fields, and F is the particle's angular momentum vector. Thus, the energy levels of a particle with an intrinsic electric dipole moment will be split in an electric field, an effect that is the analog of the magnetic Zeeman effect. Most modern EDM searches exploit this analogy and measure the modication of the Zeeman splitting in a large applied electric field. Consider a spin-l/2 system with a finite EDM in a magnetic field: if the particle's spin is oriented transverse to the magnetic field, the spin will precess at a frequency determined by the energy splitting of the two spin states. If an electric field is applied, the spin precession frequency will be shifted by an amount that is linear in the electric field strength. Reversing the direction of the electric field reverses the sign of the E-induced splitting of the spin states. This behavior helps distinguish the precession due to an EDM from that due to other torques. The way T (or C P)-violation at the fundamental elementary particle level would generate an observable EDM depends upon the system under study. The neutron is sensitive almost exclusively to T -violation in the quark sector, while atoms and molecules have bound electrons and are therefore sensitive to T -violation in the lepton sector as well as the quark

163

sector. In atoms and molecules there are actually a number of ways that T-violating interactions at the particle level could give rise to an EDM, and all are enhanced considerably in heavy atoms. 6 Calculations have been made of the atomic EDM due to an EDM distribution in the nucleus, to a T -violating force between electrons and nucleons, and to an intrinsic EDM of the electron itself, corresponding respectively to hadronic (quark-quark), semi-Ieptonic (electron-quark), and purely leptonic interactions as the chief source of T -violation. Which of the possible effects will predominate in a given atom or molecule depends upon the net electronic angular momentum J. In systems with J = 0 (i.e., systems with only closed electronic shells, such as Hg, Xe, and Ra), the EDM vector points along the nuclear spin I, and the greatest sensitivity is to purely hadronic T -violation inside the nucleus. In this case, the important quantity is the nuclear Schiff Moment,3,6 which measures the part of the nuclear EDM that is not completely shielded from the outside world by the atomic electrons. Although shielding does reduce the size ofEDMs in closed shell atoms, it turns out that this loss can be more than compensated by the extra experimental EDM sensitivity attained in these atoms. Another source of an EDM along I could in principle be a tensor-pseudotensor form of electron-nucleon T -violation. 3,6 In systems with non-zero J (i.e. paramagnetic systems such as Cs, TI or open-shell molecules) the EDM has a component parallel to J, and the greatest sensitivity is to an intrinsic electron EDM, or to a scalarpseudoscalar form of electron-nucleon T -violation. 3,6 The effect of an electron EDM is actually enhanced in heavy atoms, by over a factor of 100 in cesium and considerably more in thallium and other heavier atoms. 16 An additional enhancement occurs in polar molecules due to their large internal electric fields that can couple to an EDM.17 The field axis of these molecules can generally be aligned in a relatively modest laboratory field, and their internal fields can be of order 104 -105 times available laboratory fields, yielding a corresponding increase in EDM sensitivity. 3. The

199Hg

EDM measurement in Seattle

199Hg has a 6 1So ground state electronic configuration, and a nuclear spin I = ~. Because the ground state carries no electronic angular momentum, an EDM search in mercury is primarily sensitive to T-violation associated with the quarks in the nucleus. A search for an EDM of 199Hg has been underway in our laboratory at the University of Washington for over 20 years. Our last experiment,13 which used a frequency-quadrupled semicon-

164 x4 Outer beam

tophotodetectocs .....

detector box

~

Data acquisition computer

L .. ___ ...............; :........ . to bias field coil and gradient coils x3

;---_

~,~~=\:~~ Inner beam

low-noise current sources

.... _-...... _-_ ...... _-_ ....... .

...................::- .. .

.............~

1i :::,~~; :~~t:, ...... __.t:-~~~~:~···.~~~J~-lO·kv

detector box

®

o

r7.l

LC.J

o 3-layer magnetic shields

Fig. 2.

Simplified diagram of the 199Hg EDM apparatus.

duct or laser on the 254 nm mercury absorption line to orient the 199Hg nuclear spins, yielded the result: de 99 Hg) = -(1.06 ± 0.49stat ± 0.40sysd x 10- 28 e cm, which set an upper bound on the EDM of

tde 99 Hg)t < 2.1 x 10- 28 ecm (95%confidencelevel) As shown in Fig. 1 above, the leading theoretical extension to the Standard Model, Supersymmetry, is expected to generate a 199Hg EDM comparable to our experimental limit. By increasing the precision of our result, we could provide important information about the model parameter space of Supersymmetry and other theories, or possibly observe a nonzero EDM. With such motivations in mind, upon completion of our 2001 measurement we undertook a major upgrade of the 199Hg EDM experiment. We began with a study of the spin relaxation in our vapor cells, which led us to construct new cells that on average have 1.5 times longer spin coherence times. However, the main improvement to the experiment was the construction of an apparatus that incorporates a stack of four vapor cells (see the cutaway view in Fig. 4 below). Previous versions of the experiment have all compared the spin precession frequency between two vapor cells, where the cells are in a common magnetic field and oppositely directed electric fields.

165

10

~ o :>

8

20

-10

(,0

Tim~

BO

100

12il

(sec)

Fig. 3. Pump-probe sequence showing the Larmor precession frequency expanded in the inset.

In the current experiment the two additional cells are at zero electric field and are used as magnetometers above and below the EDM sensitive cells. They help to improve our statistical sensitivity by allowing magnetic field gradient noise cancellation, and they are also used to cancel out possible magnetic systematic effects. As before, to search for an EDM, we measure the Larmor spin precession frequency of 199Hg. A common magnetic field produces Larmor precession in a vapor of spin-polarized mercury in each cell, and a strong electric field applied in opposite directions in the middle two cells modifies the precession frequency by an amount proportional to the electric dipole moment. An EDM would cause a frequency shift of 2Ed/ h, with opposite sign in the two cells; so the magnitude of the EDM is given by d = Mv/(4E), where 8v is the difference in precession frequency between the two middle cells. The current version of the experiment is shown in Fig. 2. We spinpolarize the 199Hg nuclei by optical pumping on the 253.7 nm absorption line in mercury. Since the light beam is transverse to the precession axis, the circularly-polarized pumping light is modulated at the Larmor frequency to synchronously pump the precessing spins. The optical rotation of a linearlypolarized off-resonant probe beam is used to detect the spin precession. Polarization rotation is converted to amplitude modulation using high-quality polarizers, and the resulting signals (see fig. 3) are fit to extract the Larmor frequency. The ultraviolet light for this transition is obtained by quadrupling the output of an infrared diode laser in a master oscillator, power

166

Fig. 4. Cutaway view of the EDM cell-holding vessel. High voltage (± 10 kV) is applied to the middle two cells with the ground plane in the center, so that the electric field is opposite in the two cells. The outer two cells are enclosed in the HV electrodes (with light access holes as shown here for the bottom-most cell), and are at zero electric field. A uniform magnetic field is applied in the vertical direction.

amplifier (MOPA) configuration. Our laser system produces several milliwatts of stable, tunable UV radiation with good spatial characteristics. This system has operated continuously and problem-free for several years, and requires only occasional maintenance. We lock the laser frequency to absorption lines in a separate vapor cell containing mercury at natural isotopic abundances. The cells are held as shown in Fig. 4 inside a sealed vessel filled with about 1 bar of SF 6 or N2 gas to reduce leakage currents. The vessel and electrodes are constructed of conductive polyethylene, which we found had exceptionally low magnetic impurity content. The vapor cells have been altered slightly since our last publication, containing a 100% CO buffer gas, instead of the 95% N2 / 5% CO mixture used for the 2001 measurement. Our studies of spin relaxation in mercury vapor cells 18 indicated that the wax coating on the interior of the cells could be damaged by collisions with excited metastable mercury atoms. The CO buffer gas efficiently quenches these metastable states and thus helps prevent damage to the coating. The end result is that we can achieve polarization lifetimes that are a factor of 1.5 longer than was possible with the old vapor cells. With these improve-

167

ments, we are now sensitive to spin precession frequency shifts on the lO- lO Hz scale. We have made an extensive effort to assess the noise performance, with the goal of improving the sensitivity still further. The shot noise contribution is modeled using computer simulations, which show we are within a factor of three of the shot noise limit. While the modeling also shows that further improvements to reduce the shot noise itself are possible, we must first eliminate the current extraneous noise limiting the experiment. We are pursuing these goals, while at the same time accumulating EDM data with the present sensitivity. In order to reach the 1 x 10- 29 e em level we must place tight bounds on any systematic effects in the measurement. The most dangerous effects are those which generate magnetic fields that are correlated with the direction of the applied electric field. Leakage currents which flow across the cell when high voltage is applied are one prime example. We continually monitor all leakage currents, and with careful cleaning and preparation we limit such currents to the pA level. Our measurements continue to suggest that this is below the level that could cause a problem at the present level of sensitivity. An important new safeguard is possible now that we have 4 cells. As one example, the "leak-test" combination of individual cell measurements (the sum of the outer cell frequencies minus the sum of the inner cell frequencies) is sensitive to leakage current fields while canceling any EDM effect. Another possible problem could be high voltage sparks which might change the field of a trace magnetic impurity located near the cells or electrodes (although we have constructed the apparatus from materials that are as free of such impurities as possible). Again, some combinations of cell frequencies will be sensitive to such local fields, and can reveal the presence of impurities. 3.1. Blind Analysis

Because of the occasional need to cut data (for example, when magnetic impurities do appear), we initiated a blind analysis procedure for all data taken after March 2006. The analysis program adds a fixed, HV -correlated offset to the middle cell fitted frequencies, +6/2 to the middle top cell and -6/2 to the middle bottom cell, which gives an artificial EDM-like signal of size 6, randomly generated between ±2 x 10- 28 e em (our previous upper bound). This range is large enough to insure the analysis is blind, but small enough to reveal any large spurious signals that might appear due to the changes made when the blind analysis began. Once selected, the blind offset remains fixed throughout all data, and therefore does not interfere

168

with tests for systematic effects, and of course it guards against human bias in decisions about making data cuts. We are now taking data for a new measurement of the 199Hg EDM. Thus far the accumulated statistical error is ±1.5 x 10- 29 e cm, over a factor of 10 below the upper limit of our 2001 measurement. It remains to be seen how small a systematic error will emerge from this measurement.

3.2. The

199 Hg

Stark Interference Effect

A static electric field applied to an atom with an El (electric dipole) optical transition induces M1 (magnetic dipole) and E2 (electric quadrupole) transitions. The presence of these additional transitions leads to an interference effect of a particular vector character. For a F = ~ ---+ F = ~ E1 transition, such as the one we use in the 199Hg EDM search, the fractional change in the absorptivity 0: is of the form, 80: = a(i. Es)(k x i) . it, (1) 0: where a is a factor denoting the strength of the effect, i is the direction of the electric field vector of the light driving the transition, k is the propagation direction of the light, E s is the static electric field, and it is the atomic spin polarization direction of the ground state. The factor a has been calculated to be -6.6 x 10- 8 (cm/k V) 19 for the 254 nm E1 transition in 199Hg. Along with the absorptivity shift, there is an energy shift of the ground state magnetic sublevels. This light shift is spin-dependent and linear in the strength of the applied electric field, and can therefore mimic an EDM. The effect can be measured with the present EDM apparatus with only minor modifications, and such an experiment is currently underway. It is crucial to guard against the Stark interference appearing as a systematic effect. The effect vanishes for our nominal field configuration of EsllEllB, but it is difficult to control the orientation of these vectors to the required accuracy. One way we have dealt with the problem is to probe the atoms at two different laser wavelengths where the Stark interference light shift has opposite sign, and average the results to cancel out the systematic effect. While this method is adequate, it requires us to take a significant amount of data at laser wavelengths where our measurement sensitivity is not the best it could be. A way to completely eliminate the Stark interference problem, as well as most other light shift effects, is to evaluate the Larmor frequency 'in the dark' between two probe laser pulses (which establish the Larmor phase at the beginning and end of the dark period). We are currently implementing such a scheme.

169

Acknowledgments We wish to thank Laura Kogler, Bethany Fisher, David Meyer, and our other colleagues on the mercury EDM experiment over the years. This work was supported by NSF Grant PRY 0457320 and DOE Grant DEFG02-97ER41020. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19.

E. M. Purcell and N. F. Ramsay, Physical Review 78, p. 807 (1950). J. H. Smith, E. M. Purcell and N. F. Ramsey, Phys. Rev. 108, 120 (1957). S. M. Barr, International Journal of Modern Physics A (1993). G. L. Kane, Perspectives on Supersymmetry (World Scientific, Singapore, 1998), p. xv. N. Fortson, P. Sandars and S. Barr, Physics Today 56, 33(June 2003). 1. P. Khriplovich and S. K. Lamoreaux, CP Violation Without Strangeness (Springer, Berlin, 1997). J. H. Christenson, J. W. Cronin, V. 1. Fitch and R. Turlay, Physical Review Letters 13, 138 (1964). Babar Collaboration, B. Aubert, et al., Physical Review Letters 87,091801/1 (2001). Belle Collaboration, K. Abe, et al., Physical Review Letters 87, 091802/1 (2001). C. A. Baker et al., Physical Review Letters 97, p. 131801 (2006). P. G. H. Sandars and E. Lipworth, Physical Review Letters 13,718 (1964). B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, Phys. Rev. Lett. 88 (2002). M. V. Romalis, W. C. Griffith, J. P. Jacobs and E. N. Fortson, Phys. Rev. Lett. 86, 2505 (2001). T. Falk, K. A. Olive, M. Pospelov and R. Roiban, Nuclear Physics B 560, 3 (1999). P. G. Harris, C. A. Baker, K. Green, P. Iaydjiev, S. Ivanov, D. J. R. May, J. M. Pendlebury, D. Shiers, K. F. Smith and M. van der Grinten, Phys. Rev. Lett. 85, 904 (1999). P. G. H. Sandars, Physical Review Letters 14, 194 (1964). P. G. H. Sandars, Physical Review Letters 19, 1396 (1967). M. V. Romalis and L. Lin, Journal of Chemical Physics 120, 1511 (2004). S. K. Lamoreaux and E. N. Fortson, Physical Review A 46, 7053 (1992).

170

THE NEW CONCEPT FOR FRIB AND ITS POTENTIAL FOR FUNDAMENTAL INTERACTION STUDIES

GUY SAVARD Physics Division, Argonne National Laboratory, 9700 South Cass Ave, Argonne, Illinois 60439, USA Fundamental interaction studies at low energy can benefit from the unique properties of certain radioactive ion species. The proposed FRlB facility, an evolution on the RIA concept, will be the highest intensity source for most rare isotope species and fundamental interaction studies will be an important part of its physics program. The FRIB concept will be described and its expected capabilities for fundamental interaction studies highlighted with some illustrative examples.

1. Introduction

Access to high intensity sources of radioactive isotopes offers unique opportunities for fundamental interaction studies. The broad array of atomic and nuclear properties, as well as the specific decay properties, allows one to select isotopes with the most advantageous properties to isolate or enhance a specific effect through which the manifestation of the fundamental symmetries, or their breaking, emerges experimentally. And although working with radioactive isotopes bring in some additional complications, the advantage of selecting an isotope with the optimum properties often more than compensates for these difficulties, provided that an intense source of these isotopes is available. The US nuclear physics community has been preparing for the most intense source of radioactive isotopes, the Rare Isotope Accelerator (RIA) facility [1,2], for more than a decade. Although this facility was aimed most directly at resolving issues in nuclear structure physics and nuclear astrophysics, a significant fundamental interaction project would have been possible using the beams available at this facility. A change in scope of the RIA facility, now renamed Facility for Rare Isotope Beam (FRIB), reducing the total cost by a factor of two and making it more complementary to other major investments in the rare isotope production field internationally, has changed the design and reduces somewhat the capabilities of this facility. The potential for fundamental interaction studies at FRIB is still however fascinating.

171

In the following, a brief description of the FRIB concept will be given, highlighting the main changes with respect to the initial RIA design, and the exciting capabilities that this facility will provide. Examples of fundamental interaction studies that could be carried out at this facility are also given. 2. The FRIB facility concept The physics program for FRIB tries to answer outstanding questions in four main domains: nuclear structure physics, nuclear astrophysics, fundamental interaction studies and applications of radioactive isotopes. The specifications of the facility are mainly driven by the first two topics. A key question in nuclear astrophysics has to do with understanding the nuclear physics input to explosive nucleosynthesis processes, particularly the r-process thought to be responsible for the formation of about half of the heavy elements. This process proceeds through rapid neutron-capture reactions following a path on the neutron-rich side of the valley of beta stability, starting from mid-mass nuclei and moving up all the way to the heaviest isotopes. Gaining access to these isotopes in the laboratory requires the production of very neutron-rich isotopes and their rapid and efficient extraction as a beam for essentially all atomic species, independently of their chemical properties. They also need to be made available to experiment at various energy regimes from the stopped or ion source energy beams for ground-state property (mass, lifetime, decay mode) studies, to Coulomb energy regime for single-particle structure studies, to the highest energy available for identification of the most exotic species. The required capabilities to perform this program are limited at existing facilities by chemical limitations in the ISOL type facilities which currently limit the isotope species that can be extracted as beams, and by raw production power for the fragmentation facilities that cannot produce in sufficient quantities the most exotic species. To alleviate these limitations, the FRIB concept proposes a new approach to isotope production and extraction that combines the advantages of the fragmentation and ISOL techniques with recent advances in high-power accelerator and targetry to deliver rare isotope beams of unmatched diversity and intensity at all three energy regimes. The main production technique involves the production of radioactive ions at high-energy using fragmentation on a high-power target, separation of the recoils in a large acceptance fragment separator, slowing down and cooling in a gas catcher system, followed by reacceleration in a post-accelerator to the energy of interest. The approach is

172

universal, efficient, and yields beam of ideal optical properties at the required energies. The key ingredients of this approach are shown in a cartoon layout of the FRIB facility in figure 1. Production of the rare isotopes must be optimized over the full range of nuclei accessible. This implies more than one reaction mechanism and beam target combination. To gain access to the most neutronrich isotopes in particular in the scheme proposed above requires in-flight fission of uranium and subsequent capture of the fission products in the fragment separator. The requirements of production beams of various stable species all the way to uranium at very high beam power are best met by a superconducting heavy-ion linac. The independently phased cavities allow for the acceleration of any isotopes from proton to uranium with the same accelerating structure while the large acceptance of the superconducting cavities allows for simultaneous multiple charge state acceleration, eliminating the losses at the stripping stages necessary for efficient acceleration of heavy ions. The maximum energy that can be reached with this accelerator is the major cost driver for FRIB (as it was for RIA). A cost effective maximum energy for the uranium primary beam has been determined to be 200 MeViu, half of that initially proposed for the RIA facility. The same accelerator yields higher energy for the lighter species, all the way up to 550 MeV for protons. Progress in ECR sources with the development of superconducting ECR sources [3] now yields primary beam intensities to inject into this linac that allow 400 kW beam power to be reached for all species at the reduced final energy. A detailed study of the production of radioactive isotopes with the reduced beam energy but at the same total beam power show little effect on the yield of most isotopes produced by fragmentation aside from losses due to charge state fractionation for the heaviest isotopes that are no longer fully stripped. The effect on in-flight fission products is more important since the kinematics of the fission process imparts an important transverse momentum to the fragments which increases the difficulty of separating and transporting them in the fragment separator at the lower energy. The net result is that most isotopes are produced at more than 70% of what would have been available at RIA except for the subset of isotopes coming from in-flight fission which can have losses which are in the worst case a factor of 10 compared to the full RIA. The production reaction must take place on a high-power target of preferably low-Z material capable of surviving the primary beam power density. For ion optical reasons the beam must be focused on a 1 mm diameter beam spot and no solid target material can withstand that power density. This

173

difficulty was ovcrcome by a windowless flowing liquid lithium target [4] approach developed and demonstrated at Argonne in RIA related R&D. The target is continually replenishcd by the flowing liquid hence there is no radiation damage to the target and the heat is removed via a heat exchanger in the liquid lithium circuit. The target is at the entrance of a large acceptance fragment separator that must transport the recoil ions of interest, separate thcm from thc beam and the other reaction products, and finally delivers them to a second stagc where their cnergy sprcad is reduced before they are stopped in a gas catcher. The properties of the fragment separator are determined by the energy and momentum spread of the in-flight fission products. An angular acceptance of ± 50 mrad in both the x and y directions and a momentum acceptance as large as ± 9% is desired. A high resolution, of the order of 1000, is required to minimize the energy spread in the second stage via dispersion on a solid wedge whose thickness varies so that the higher energy particles lose more energy than the lower energy one and all leave the wedge with roughly similar encrgy.

experimental areas

Figure 1. Cartoon layout of the main ingredients of the proposed FRIB facility.

These recoils then go through a final tunable degrader before entering a gas catcher where they will lose their remaining energy in ultra-pure helium gas. Helium is chosen as a stopping medium since it has a much higher electron binding energy than any other element. A recoil stopped or at the end of its range cannot therefore regain its last electron and remains ionized in the helium

174

volume. These radioactive ions can then be acted upon by DC and RF electric fields to drift and focus them towards an extraction nozzle through which they are evacuated into a series of radiofrequency quadrupole guiding devices that further cool the ions and form a beam. The whole process takes about 10-20 ms and has demonstrated efficiency in excess of 40% [5]. The ionization density into the helium gas created by the stopping recoils will be high but recent R&D work at Argonne has demonstrated successful operation under similar conditions and the currently under construction CARIBU upgrade [6] at ATLAS will operate a similar size gas catcher in battle conditions to feed the fission products from a 1 Curie fission source in the ATLAS superconducting linac for post-acceleration, providing significant operational experience in facility operation for such a device before FRIB is constructed. The beam delivered by the gas catcher system is passed through an isobar separator and can then be used directly for experiments with stopped or ionsource energy beams, or reaccelerated for experiments at astrophysical or Coulomb barrier energies. Post-acceleration will proceed via charge-state breeding to increase the charge state of the ions in an ECR or EBIT source after which the ions are accelerated by an ATLAS-like superconducting heavy-ion linac to energies from 0.3 to about 15 MeV/u. The low charge state postaccelerator plus stripping option proposed for RIA that gave higher efficiency for astrophysics energy regimes is probably outside of the reduced budget for FRIB. In addition to this main capability, FRIB can in principle also provide beams directly at high energy after the fragment separator. The intensities are still higher than at any other facilities but the reduced maximum energy will impact the purity and yield of the heaviest elements because of charge state fractionation. The heavy-ion superconducting linac can also provide enormous current of light ions such as IH and 3He at energies which are well suited for target spallation production of species for which the physics needs require the highest yield. These yields can in many cases exceed the beam currents used in stable beam experiments and, although not universally available, open great physics potential for the reduced set of isotopes that can be extracted this way. Experimental stations will be available in the three energy regimes although the bulk of the funds for those also disappeared in the transition from RIA to FRIB. There is however significant state-of-the-art equipment existing that will allow a first class physics program to start when the machine initially turns on. Large new equipment such as GRETINA and HELlOS currently under

175

construction will complement this initial set of instruments and is well suited to the core reaccelerated-beam research program of FRIB. Summarizing the FRIB facility capabilities, it will have the same 400 kW primary beam power that was expected for RIA, yield within 30% of what was predicted for RIA for most of the isotopes and lower intensity by up to a factor of 10 in the worst case for a reduced set of isotopes produced by in-flight fission. In all cases, the yield from FRIB will still exceed that of other existing or approved facilities. The maximum energy at which the radioactive ions will be available in the fast beam experimental area is reduced by a factor of2 which will affect purity for the heaviest nuclei produced by fragmentation and the size of the experimental areas and funds available for new equipment have been reduced significantly. Less notable but just as important, the multi-users capabilities that were present in RIA have been reduced significantly. This might have a very detrimental effect on the non-core programs or programs that require very long running time.

3. Opportunities for fundamental interaction studies Fundamental interaction studies at low energy often take advantage of special properties of a subset of isotopes to isolate a specific effect. In many cases, only a few isotopes will have the right properties. Since there is far more radioactive isotopes than stable isotopes, the likelihood that the best candidate is radioactive is high. The signature for a given fundamental effect in a complex nucleus or atom can easily be inextricable and the key features looked for are systems that either feature simplifications that allow one to isolate the effect or special properties that amplify the effect and make the system more sensitive to it. These searches evolve with our knowledge of the properties of these isotopes, of the implications of the symmetry breaking looked for, and the observed signals in other experiments. It is therefore difficult to predict what will come up in the decade before a full-fledged fundamental interaction program can be running at FRIB and the few representative cases described in the following are mainly highlighting programs in preparation or running that will benefit from FRIB-like beam intensities.

3.1. Electric dipole moment searches The predominance of matter over antimatter in our UnIverse requires additional CP violation than can be accommodated in the Standard Model. Searches for the sources of this additional CP violation are ongoing at all energy regime and a very powerful probe for such electroweak CP violation is the

176

7

search for an electric dipole moment (EDM) of elementary particles or quantum bound states. An EDM is odd under both parity (P) and time (T) inversion transfonnations which, via thc CPT theorem, connects its existence to CP violation. Within the Standard Model EDMs are extremely small, far below the current experimental limits in any system. However, most modifications of the Standard Model result in much larger values for EDMs which make these searches extremely sensitive to new physies.

Magneto-optical trap

EDM

Figure 2. Proposed experimental system for an atomic EDM search in octupole deformed

225Ra.

EDMs are searched for in many systems sinee the effect observed will depend on the source of the additional CP violation and searches for EDMs in the clectron, the neutron, the muon and heavy atoms such as 199Hg have all given null result, severely restricting potential extensions to the Standard Model. In the particular case of the atomic EDM, it has been demonstrated that heavy atoms with octupole nuclear deformation have a roughly WOO-fold increased sensitivity to an EDM compared to the cases that have been measured so far. This implies that for a similar experimental sensitivity experiment, a measurement on an octupole-defonned nucleus such as 225Ra, or other similar octupole-defonned nuclei in this region, could potentially be sensitive to a 1000 times smaller EDM than the best atomic measurement so far, that on 199Hg [7].

177 Large amounts of these isotopes are required at low energy for these experiments and a number of the nuclei of interest can be produced and extracted at rates of around 10 II per second using target spallation with the 400 kW proton or 3He beams that will be available at FRIB with ISOL extraction techniques. An experimental system, shown in figure 2, is being developed at Argonne for such searches using sources to produce the 225 Ra [8]. The best yield one can expect from such an approach is about 2 to 3 orders of magnitude below the rate that will be available at FRIB but should still allow a measurement exceeding in EDM sensitivity the best existing limit. With FRIB as a source, it should be able to improve EDM limits in atoms by over an order of magnitude. This approach, together with similar improvements in the neutron EDM search at the SNS [9], and possibly the electron EDM search using Francium isotopes also at FRIB, will provide multiple signals to identify the source of the CP violation if and when it is discovered.

3.2. Parity non-conservation in atoms Parity non-conservation (PNC) in atoms provides access to quantItIes extremely difficult to determine otherwise such as the nuclear anapole moment and the weak mixing angle at low momentum transfer. That field is dominated by a towering result in 133Cs [10] that obtained an experimental accuracy of about 0.5% on an alkali atom whose atomic physics is simple enough for the PNC effect to be calculated at about the 1% level. This measurement made the first clear observation of the anapole moment, made the highest precision determination of the weak mixing angle at low energy and determined one of the parameters of the weak nucleon coupling, although this result is at odds with other approaches. The accuracy with which the PNC information could be extracted from this measurement was ultimately limited by the calculations of the atomic physics effects. This dominating uncertainty can be removed by taking the difference and ratio of the PNC effect on different isotopes of the same element; the atomic uncertainties cancel out but there is an associated loss in accuracy since the net effect is reduced in size when taking the difference and ratio. But the experimental effect does increase rapidly with Z of the atom and the one heavier alkali, francium, offers just as simple an electron structure and an 18 times larger PNC effect. Francium has no stable isotopes but making measurements on two long-lived isotopes, one on the heavier production peak and the other on the lighter production peak, and taking the difference and ratio of the measured quantities could yield the PNC information at 3 times the precision, with no dependence on the atomic structure calculation uncertainties,

178

for measurements on francium isotopes of similar quality to the cesium measurement. This requires high yield on both production peaks which are best obtained by target fragmentation using thorium compound targets and the 400 kW 3He beam of FRIB. The main theoretical uncertainty remaining will come from the neutron distributions inside the francium isotopes which both FRIB and JLAB [11] experiments will help to better constrain.

3.3.

Beta-decay studies

The main properties of the weak interaction, such as parity violation, were originally determined by low-energy beta decay experiments. In fact, many of the most stringent constraints on the nature of this interaction are still obtained from such studies. One of the most successful such program involves measurements of the weak vector coupling constant through the study of 0+ to 0+ superallowed Fermi decays. These decays, when studied amongst members of the same T= 1 isospin multiplet, isolate that strength and allow its very accurate determination, essentially independently of the detailed nuclear structure which only comes in at the isospin mixing and nuclear-structure dependent radiative correction level which are typically at the percent level or below. With existing radioactive beam production facilities it has been possible to measure to very high accuracy 9 cases and add about half a dozen more in recent years at lower accuracy. These determinations involve state-of-the-art Qvalue, lifetime and branching ratio measurements best done with intense and high-purity radioactive isotope samples and have yielded the most accurate determination of the weak vector coupling constant, a powerful confirmation of the CVC hypothesis and, when combined with other measurements at higher energy, the most precise test of the unitarity of the CKM matrix [12]. The natural evolution of these studies involves measurements on heavier emitters that will allow better tests of the small nuclear structure dependent corrections. These studies however require stronger high-purity sources of these heavier emitters, particularly for the branching ratio measurements, and more nuclear structure information than currently available in the heavier N=Z region. Both require an FRIB-like facility to provide the often refractory elements in this region at the low and reaccelerated energies necessary. Beta decay is also used in searches for currents beyond the standard V-A form. Again, the availability of stronger pure sources of selected short-lived isotopes, together with the new efficient trapping techniques for both charged and neutral radioactive isotopes, yield an ideal zero-thickness source minimizing scattering and other detrimental effects for these studies (see for example [13]).

10

179

That combination of FRIB delivered high-intensity and high-purity beams with the new trapping and detection techniques should allow one to put improved limits on possible scalar or tensor currents in beta decay. 4. Conclusion The US nuclear physics community is preparing for a large investment in rare isotope science with the FRIB facility. Although the cost of this facility is roughly half of what was proposed for the RIA facility, it maintains a large fraction of the capabilities of this facility with the vast majority of the isotopes produced at more than 70% of the predicted RIA yields, in a facility of roughly half the cost. The important potential for contributions to fundamental interaction studies using rare isotopes produced at FRIB remains, although the now limited capabilities for multi-users operation might affect the amount of beam time available for such studies. Acknowledgments This work was supported by the U.S. Department of Energy, Office of Nuclear Physics, under contract No. DE-AC02-06CHI1357. References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

G. Savard, "The US Rare Isotopes Accelerator Project", Proceedings of the 2001 Particle Accelerator Conference, eds P. Lucas and S. Webber, June 18-21,2001, Chicago, Illinois, USA, p.561. B.M. Sherrill, NucI. Instr. & Meth. B204, 765 (2003). e.M. Lyneis et aI., Rev. Sci. Instrum. 75, 1389 (2004). e.B. Reed et aI., Nucl. Phys. A746, 161 (2004). G. Savard et aI., Nucl .Instr. & Meth. B204, 582 (2003). G. Savard et aI., accepted for publicaton in NucI. Instr. & Meth. B. M.V. Romalis et aI., Phys. Rev. Lett. 86, 2505 (2001). lR. Guest et aI., Phys. Rev. Lett. 98, 093001 (2007). see for example http:p25extJanI.gov/edmJedm.htmI. e.S. Wood et aI., Science 275, 1759 (1997). Jefferson Lab experiment E-OO-003, spokespersons R. Michaels, P.A. Souder, and G.M. Urciuoli. LS. Towner and J.e. Hardy, Phys. Rev. Lett 94,092501 (2003). G. Savard et aI., Phys. Rev. Lett. 95, 102501 (2005). A. Gorelov et aI., Phys. Rev. Lett. 94, 142501 (2005).

180

COLLINEAR LASER SPECTROSCOPY AND POLARIZED EXOTIC NUCLEI AT NSCL K. MINAMISONO a ,., G. BOLLENa,b, P. F. MANTICAa,c, D. J. MORRISSEya,c and S. SCHWARZ a

National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI48824, USA b Department of Physics, Michigan State University, East Lansing, MI48824, USA C Department of Chemistry, Michigan State University, East Lansing, MI48824, USA • E-mail: [email protected] a

A facility for collinear laser spectroscopy and beam polarization of exotic nuclei is being developed at NSCL. The facility will make use of thermalized rare isotope beams available at NSCL from projectile fragmentation and in-flight separation with subsequent gas stopping. This system provides access to new and unexplored territory in the nuclear chart and will be implemented at the next generation rare isotope facility. Laser spectroscopy and f3 NMR/NQR techniques will be utilized to determine nuclear charge radii and nuclear ground state electromagnetic moments as well as for fundamental interaction tests.

Keywords: laser spectroscopy; optical pumping; cooled and bunched beams;

f3 NMR; f3 NQR; mean square charge radii; electromagnetic moments; fundamental symmetries.

1. Introduction

Laser spectroscopy and optical pumping techniques have been extensively used in nuclear physics to determine the nuclear spins I, the magnetic dipole moments J-L, the spectroscopic electric quadrupole moments Q, and the mean-square charge radii (r2) of nuclear ground states and isomers [1). Atomic spectroscopy is also used in testing fundamental interactions, for example, through laser cooling and confinement of radioactive atoms [2). Most of the laser spectroscopic data has been obtained at Isotope Separator On Line (ISOL) facilities [1], where rare isotopes are extracted from thick targets bombarded by light ions. Long chains of isotopes have been systematically investigated, taking advantage of the good quality lowenergy beam (rv 60 ke V) and the high beam intensities available for many nuclides of certain elements. The number of elements for which rare iso-

181

topes can be produced with the ISOL technique is limited when long release times from the targets lead to large decay losses. These limitations have been partly overcome by the Ion Guide at an Isotope Separator On Line (IGISOL) approach, where low-energy reaction products are stopped in a gas and converted into a low-energy ion beam [3]. On the other hand, projectile fragmentation reactions and in-flight separation routinely provide high-energy beams (>50 MeV/nucleon) for isotopes of all elements lighter than uranium. The technique is universal and reaches very far from stability since decay losses are negligible. The recent conversion of these fast beams into high-quality low-energy beams via gas stopping techniques [4,5] and advanced beam manipulation techniques [6,7] has opened the door for a new range of experiments with projectile fragments, complementing and extending studies previously only possible at ISOL-type facilities. National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University was the first facility where fast rare isotope beams, produced in projectile-fragmentation, have been slowed to thermal energies with a gas stopping system [5] and used for precision experiments. An experimental program of high-precision mass measurements with a 9.4 Tesla Penning trap mass spectrometer is underway [8-10]. Extending these experimental opportunities is a logical next step. In addition to work with stopped beams, the re-acceleration of the thermalized rare isotope beams is under preparation [11]. The laser-spectroscopy activities at NSCL are expected to make a major contribution to the science program, since isotopes can be studied that are inaccessible or difficult to obtain anywhere else in the world. The activities will also set the ground work for even greater science opportunities to become available at the next-generation rare isotope facilities like the prospective Facility for Rare Isotope Beams (FRIB) in the US. NSCL has proposed the Isotope Science Facility (ISF) [12] to satisfy the technical and scientific goals of FRIB. ISF will make it possible to produce rare isotopes beams with the shortest half-lives at unprecedented intensities, using projectile-fragment reactions for rare isotope production and in-flight separation techniques. One of examples of the extended science opportunities that will be started at NSCL and then continued at ISF is the measurement of meansquare charge-radii, (r2), of short-lived very exotic nuclides. As mentioned before, most of the laser spectroscopic data to date have been obtained at ISOL facilities. However, there still is very little information on (r2) of

182 100

80 Proton

iii ..0

E 60 ::>

c c

.8 40

e

c..

20

Known laser-spectroscopy data

iiII

ISF rate sufficient for laser-spectroscopy (rate> 10 3 s-1 reaccelerated beams)

0 0

20

40

100 60 80 Neutron number

120

140

Fig. 1. Chart of nuclei depicting those isotopes studied via laser spectroscopy [1]. Previously studied cases and the reach of ISF are shown by light and dark gray squares, respectively. The solid squares are the stable nuclei. Taken from Ref. [12].

light- and medium-mass nuclei. Such shortcomings can be attributed to the difficulty of production of short-lived metallic and refractory rare isotope beams. The increased production rate at ISF will allow laser spectroscopy to be extended to isotopes far from stability. Fig. 1 shows a comparison of present laser spectroscopy data [1] and the eventual reach of laser spectroscopy at ISF. It is clear that the present data are fairly limited and the beams available at ISF will cover long isotopic chains for all elements not only in neutron-rich but also neutron-deficient regions. The increased production rate at ISF also opens up opportunities to perform laser spectroscopic experiments to test fundamental interactions using thermalized beams from projectile fragmentation.

2. Laser Spectroscopy with Thermalized Beams at NSCL An on-line laser spectroscopy and beam polarization facility is under development at NSCL, where the source of radioactive ions will be projectile fragments that are stopped in a gas stopping system [4,5,13]. After extraction and conversion into a low-energy continuous beam, the ions will be accelerated to "-' 30 ke V for transport to the laser spectroscopy facility, which is shown schematically in Fig. 2. The low-energy beam will be slowed down electrostatically before entering a linear Radio Frequency Quadrupole (RFQ) ion trap [6] filled with helium as a buffer gas at low pressure. The linear trap will reduce the beam emittance and provide short ion bunches.

183

BOB

from~ gas stopper0 '--_ _--'

Fig. 2. Schematic of the planned laser spectroscopy and beam polarization facility at NSCL. Thermalized rare isotope beams are provided by the NSCL gas stopping system. After cooling and optional buuching in a linear RFQ ion trap, a high-quality low-energy beam (pulsed or continuous) will be transported to either a laser spectroscopy or polarization beam line, where laser light will be colliuearly overlapped with the ion beam. Various techniqnes for laser spectroscopy (optical detection, ion detection) will be used as well as a bet.a-NMR and NQR setups for moment measurements.

The trap can be operated at cryogenic temperatures, a technique also pioneered at the NSCL [6,7], that is key to providing excellent beam quality. The cooled and bunched beam will then be transported to a laser spectroscopy or a laser polarizer beam line. The very short pulses with good emittance and low energy-spread will increase the detection sensitivity of laser spectroscopy experiments. For example, laser spectroscopy has been demonstrated with microsecond pulses at ion rates as low as rv 10 2 / s at JYFL [14,15]. The existing beam cooler for LEEIT at NSCL routinely proand similarly excellent beam timings vides ion pulse lengths below 100 ns properties are expected for the new beam cooler for the laser spectroscopy and beam polarization facility, all of which indicates that the sensitivity of laser spectroscopy will be drastically improved. Nuclear magnetic/quadrupole resonance techniques based on detecting !) decay CB-NMR/NQR) will be employed to beams whose nuclear spin ensemble has been polarized by optical pumping [16-18J. The ,B-NMR/NQR techniques can be applied to nuclei with production rates of rv 10 2 /s, assuming rv 10% nuclear polarization, when the NMR signal is obtained by detecting the asymmetric angular distribution from polarized nuclei (see, for example Ref. [19]). The early science program at the laser spectroscopy and beam polarization facility at NSCL will examine nuclearelectromagnetic moments and charge radii of nuclei in the sd-shell and in the light fp-shell. Some examples are discussed in the following subsections.

184

2.1. Mean-square charge radii

Changes in the nuclear charge radius determined in isotopic shift measurements are very sensitive to nuclear deformation. Such measurements have been performed for many elements along long isotopic chains [1] and revealed interesting nuclear structure effects. Light and medium mass nuclei are largely unexplored, as seen in Fig. 1. The recent measurement of the (r2) of 11 Li [20] completed at ISAC/TRIUMF serves as an excellent example of the variation of the nuclear shape at the limit of stability, and the potential reach of such measurements to the limits of the nuclear chart. At NSCL, (r2) measurements of light and medium mass nuclei, especially refractory elements, will be performed with a focus on nuclides inaccessible or difficult to extract at ISOL facilities. Techniques already developed for producing high-quality cooled and bunched beams will be key to reaching the sensitivity necessary to reach nuclides very far from stability.

2.2. Electric quadrupole moments The deviation of nuclear shape from spherical symmetry can be directly observed in the electric quadrupole moment, Q. Nonzero values of Q for nuclei far from stability serve as indicators of new regions of deformation. Most of the data for Q was obtained by laser spectroscopy and the hyperfine structures provide direct information on the sign and magnitude of Q. However, the majority of these data have been collected for heavier nuclei, and the nuclear landscape below Z = 50 remains largely unexplored [1,21]. Very limited data are available for ground state Q of light nuclei, apart from systematic studies of a few isotopic chains; B [22] and Na [18,23] isotopes. Efforts to extend such studies of neutron-rich nuclei continue. For example, measurements of the ground state Q of neutron-rich magnesium and aluminum isotopes [24] will better define the limits of the island of inversion around 32Mg. On the neutron-deficient side of the valley of stability, the nature of the proton-halo structure is one of the interesting areas to be addressed. Since the Q is defined by the proton distribution in the nucleus, Q is one of the most sensitive measures of a proton-halo structure. However, the available data are sparse. The strongest evidence for a proton halo structure is in the large Q of 8B [25], but the proton-halo structure, and its relation to Q, is still to be clarified.

2.3. Magnetic dipole moments The magnetic dipole moment, /1, is sensitive to the relative amplitudes of different orbital components of the nuclear wave function. The well known

185

form of the electromagnetic operator makes J-l an important observable for assessing nuclear structure models. This is especially true for mirror nuclei, which only differ in the exchange of proton and neutron numbers. For example, the expectation values of both spin and orbital angular momentum can be deduced from the known ground-state J-l of both mirror partners [26]. The systematics of the spin expectation value, (0-), of the isospin T = 1/2 mirror nuclei in the sd shell have been well established in the sd shell. However, (0) has only been deduced for 4 mirror partners with T = 3/2. Two anomalous results of (a) values have been reported. First, the deduced (a) for the T = 3/2, 9Li - 9C mirror system [27,28] was found to be 50% larger than the extreme single-particle expectations. It still remains a challenge to nuclear-structure theories [28,29]. Data on (a) for heavier T = 3/2 nuclei near the proton drip line may illuminate the underlying structure changes associated with loosely-bound valence protons. Secondly, the ground state J-l of 57CU, which completed the T = 1/2, A = 57 mirrormoment measurements, gives a large and negative value for (a) [30]. The result is also very different from extreme single-particle model expectations, and suggests a breaking of the 56Ni doubly magic core. Extension of J-l measurements of neutron-deficient nuclei beyond the sd shell will be critical to test the applicability of shell model interactions in medium-mass nuclei.

3. Examples of Scientific Opportunities with Laser Spectroscopy at a Next-Generation Facility like the ISF The wide range of nuclei promised at ISF naturally will allow extension of measurements of (r2) and ground-state nuclear-electromagnetic moments, as discussed in the previous sections, to nuclei at the nucleon driplines (see Fig. 1). At the same time, the overall increased production rates throughout the chart will open up possibilities to perform experiments to test fundamental symmetries, which generally require high statistics. Some of these opportunities are discussed in the following subsections. 3.1. Test of parity and time reversal symmetries The CPT theorem requires invariance under the combined application of three independent operations, namely charge conjugation (C), parity inversion (P), and time-reversal (T). Direct evidence for CP violation in the decay of the neutral kaon [31], which is now implemented in the framework of the Standard Model (SM), led immediately to searches for possible T violation. CP violation is thought to have played a crucial role in producing the excess of matter over antimatter early in the history of the universe [32].

186 The SM does not violate CP symmetry strongly enough to account for this excess. To understand baryogenesis, the physical process of generation of nucleons in the early universe, we must first discover the additional CP violation and/or equivalently the T invariance, if it indeed exists. One such effort is the determination of electric dipole moments (EDM) [33-35], which violate parity as well as time-reversal invariance. The sensitivity in observing EDMs can be enhanced by studying heavy rare isotopes. Due to relativistic effects, the measurable electron EDM in atomic systems is proportional to Z3 [36]. Therefore, measurements on the heavier Z atoms are more sensitive to the EDM. Francium is a very good candidate because of its simple atomic structure. Hadronic EDMs observed in diamagnetic atoms can experience large enhancement factors (100-1000) if the nucleus is octupole-deformed [37,38]. Such deformations exist in rare isotopes of radon, radium, and francium. New experiments with higher precision are required in a variety of atomic systems. They will help us to learn to what extent EDMs exist and may contribute to the understanding of the matter-antimatter asymmetry in our universe. Examples of rare isotopes considered for EDM studies are 211 Rn, 223, 225 Ra, and 221 Fr, which will be available at ISF. Complementary searches for CP violation can be performed in low energy ;3-decay experiments. Time-reversal violation tests via correlation experiments in ;3-decay require an odd number of spin and/or momentum vectors, which is odd in time-reversal operations. The results of the nuclear ;3-decay experiments on the neutron [39,40]' 8Li [41], 19Ne [42-44], and 56Co [45] are all consistent with T invariance and the SM. A potentially more precise time reversal violation test can be performed in a ;3-ry-ray angular correlation experiment from spin-aligned nuclei [46,47]. This type of experiment has been performed in the case of 56CO. However, it provides the poorest constraint on T-violating coefficients. One of the candidate of such experiment at ISF is 52CO. 3.2. Search for new interactions in weak nucleon current

The SM describes the ;3-decay process in terms of an exchange of charged vector bosons between the hadronic and leptonic currents. Only vector and axial-vector type interactions are allowed in the SM. However, scalar and tensor type interactions could exist and, at present, are only ruled out at a level of about 10%. Finding such new interactions would provide a signature of new physics beyond the SM, possibly requiring the exchange of leptoquarks or new charged bosons.

187

The most stringent limit on scalar currents comes from a delayed protonemission experiment on 32 Ar at ISOLDE [48] and an atom trap experiment with 38mK at ISAC [49]. The most precise experiment searching for tensor currents was carried out with 6He at Oak Ridge more than four decades ago [50]. All these experiments are still in agreement with SM predictions. In addition to the nuclei already considered, other interesting candidates, which can be produced at high rates at ISF, for electron-neutrino correlation studies in ion traps are 14 0, 26m AI, 33CI, 35 Ar, 42Sc, 46V, 50Mn, and 54Co.

3.3. Search for induced currents in weak interactions The SM predicts that (3 decays can be described by the vector-axial vector (V-A) form of the weak nucleon current. Even if scalar (S) and tensor (T) interactions in the fundamental weak quark-lepton interactions do not exist, currents can be induced by the strong interaction due to pion exchange in the nucleus [51]. In the vector current, two of the induced terms are the weak magnetism Iw and the induced scalar term Is. In the framework of the CVC theorem, where the weak currents and the isovector part of electromagnetic current form an isospin triplet, Iw and Is are exactly given. The Is should be zero and, experimentally, has been determined to be small Us < 0.0013 [52]). Theory predicts a nonzero value for Iw, which is yet to be determined experimentally with good precision. In the case of the axial vector current, two terms may be induced, a pseudo-scalar term, Ip, and a tensor term, IT. The Ip term can be determined in muon capture reactions. The fr term is known as a Second Class Current (SCC) [53,54]. According to the SM, IT should be zero. However, the small mass difference between proton and neutron (and hence up and down quarks) may result in a small but finite number, fr rv 10- 5jMeV [55]. The most stringent constraint IT < 2 X 10- 4jMe V, obtained from {3-decay angular distribution from aligned 12B and 12N [56], does not reach that level. One possibility for a precise determination of Iw in the vector current is a systematic and very precise measurement of the spectral shape, a, of the {3-decay energy spectra observed in pure Gamow-Teller transitions, where a simple interpretation of the shape factor is possible. Because the effect of Iw on the a appears in the form of rv IwEj3, high-energy {3 decays are preferred. For example, the 24m Na and 24m Al T = 1 pair would be available with required beam rates at ISF, allowing the currently very small set of test cases to be extended. An ion trap based spectrometer similar to the WITCH facility [57] at ISOLDE would be a good choice for determining the shape of the {3-decay energy spectrum with high precision and efficiency.

188

Measurement of the {3 decay angular distributions from spin-aligned mirror nuclei can be employed to search for fT in the axial vector current. The results for both nuclei takes advantage of mirror symmetry and allows systematic effects to be minimized, which is essential to extract the very small effect of fT. Polarized beams are required, and can be realized by laser optical pumping. A promising example is 13 0. The high {3-decay energy and a pure GT transition make 13 0 very sensitive to fT. The polarized 13 0 beam can be obtained by optical pumping, starting from a metastable atomic state [58]. The necessary measurement of the mirror partner 13B is now in progress at Osaka University. Systematic studies of multiple mirror partners are needed to put a reliable limit on the induced tensor term. 4. Conclusion A laser spectroscopy and nuclear polarization facility is being implemented at NSCL for experiments with thermalized projectile fragments. Highlysensitive measurements will be possible by employing laser spectroscopy techniques on cooled and bunched beams and by using (3-NMR/NQR techniques. Charge radii and electromagnetic moments of isotopes not accessible with ISOL techniques will be studied. The techniques developed will be beneficial for future studies at the next-generation facility FRIB, like ISF as proposed by MSU. The promised rare isotope beams produced by projectile fragmentation will open new opportunities to extend the limits of such studies towards the nucleon driplines and test fundamental symmetries. Acknowledgment This work is supported by the National Science Foundation, Grant PHY0606007. References 1. H. -J. Kluge and W. Nortershauser, Spectrochimica Acta B 58, 1031 (2003).

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

G. E. Sprouse and L. A. Orozco, Ann. Rev. Nucl. Part. Sci. 47, 429 (1997). P. Dendooven, Nucl. Inst. Meth. B 126, 182 (1996). M. Wada et al., Nucl. Instr. and Meth. B 204, 570 (2003). L. Weissman et al., Nucl. Instr. and Meth. A 540, 245 (2005). G. Bollen et al., Nucl. Instr. and Meth. A 532, 203 (2004). T. Sun et al., Eur. Phys. J. A 25(SI), 61 (2005). G. Bollen et al., Phys. Rev. Lett. 96, 152501 (2006). R. Ringle et al., Phys. Rev. C 75, 055503 (2007). P. Schury et al., Phys. Rev. C 75, 055801 (2007). X. Wu et al., Particle Accelerator Conference 07, http://pac07.orgj. http://www.nscl.msu.edu/future/isf.

189 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 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. 50. 51. 52. 53. 54. 55. 56. 57. 58.

A. Takamine et al., Rev. Sci. Instr. 76, 103503 (2005). A. Nieminen et al., Phys. Rev. Lett. 88, 094801 (2002). P. Campbell et al., Phys. Rev. Lett. 89, 08250 (2002). E. Arnold et al., Phys. Lett. B 197, 311 (1987). C. D. P. Levy et al., Nucl. Instr. and Meth. B 204, 689 (2003). M. Keirn et al., Eur. Phys. J. A 8, 31 (2000). H. Ogawa et al., Phys. Rev. C 67, 064308 (2003). R. Sanchez et al., Phys. Rev. Lett. 87, 033002 (2006). N. J. Stone, At. Data Nucl. Data Tables 90, 75 (2005). H. Izumi et aI, Phys. Lett. B 366, 51 (1996). G. Hubber et al., Phys. Rev. C 18, 2342 (2002). H. Ueno et al., Nucl. Phys. A 738, 211 (2004). T. Minamisono et al., Phys. Rev. Lett. 69, 2058 (1992). K. Sugimoto, Phys. Rev. 182, 1051 (1969). K. Matsuta et al., Hyperfine Int. 97/98, 519 (1996). M. Huhta et al., Phys. Rev. C 57, R2790 (1998). Y. Utsuno, Phys. Rev. C 70, 011303 (2004). K. Minamisono et al., Phys. Rev. Lett. 96, 102501 (2006). J. H. Christenson et al., Phys. Rev. Lett. 13, 138 (1964). M. Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2004). P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999). M. V. Romalis et al., Phys. Rev. Lett. 86, 2505 (2001). E. R. Commins et al., Phys. Rev. A 50, 2960 (1994). P. G. Sandars, Phys. Lett. 14, 194 (1965). V. V. Flambaum and V. G. Zelevinsky, Phys. Lett. B 350, 8 (1995). V. Spevak et al., Phys. Rev. C 56, 1357 (1997). L. J. Lising et al., Phys. Rev. C 62, 055501 (2000). T. Soldner et al., Phys. Lett. B 581, 49 (2004). R. Huber et al., Phys. Rev. Lett. 90, 202301 (2003). F. P. Calaprice et al., Phys. Rev. D 9, 519 (1974). R. M. Baltrusaitis and F. P. Calaprice, Phys. Rev. Lett. 38, 464 (1977). A. L. Hallin et al., Phys. Rev. Lett. 52, 337 (1984). F. P. Calaprice et al., Phys. Rev. C 15, 381 (1977). M. Morita and R. Saito Morita, Phys. Rev. 107, 1316 (1957). B. R. Holstein et al., Phys. Rev. C 5, 1529 (1972). E. G. Adelberger et al., Phys. Rev. Lett. 83, 1299 (1999). A. Gorelov et al., Phys. Rev. Lett. 94, 142501 (2005). C. H. Johnson et al., Phys. Rev. 132, 1149 (1963). L. Grenacs, Annu. Rev. Nucl. Sci. 35, 455 (1985). J. C. Hardy and I. S. Towner, Phys. Rev. Lett. 94, 092502 (2005). S. Weinberg, Phys. Rev. 112, 1375 (1958). D. H. Wilkinson, Eur. Phys. J. A 7, 307 (2000). H. Shiomi, Nucl. Phys. A603, 281 (1996). K. Minamisono et al., Phys. Rev. C 65, 015501 (2002). V. Y. Kozlov et al., Int. J. Mass. Spec. 251, 159 (2006). G. M. Tino et al., Phys. Rev. Lett. 64, 2999 (1990).

190

ENVIRONMENTAL DEPENDENCE OF MASSES AND COUPLING CONSTANTS

M. POSPELOV Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada E-mail: [email protected]

We construct a class of scalar field models coupled to matter that lead to the dependence of masses and coupling constants on the ambient matter density. Such models predict a deviation of couplings measured on the Earth from values determined in low-density astrophysical environments, but do not necessarily require the evolution of coupling constants with the redshift in the recent cosmological past. Additional laboratory and astrophysical tests of ~a and ~(mp/me) as functions of the ambient matter density are warranted.

1. Introduction

This talk follows the original paper by K.A. Olive and M.P. published in Phys. Rev. Dl. Perhaps the most astonishing fundamental observation of the last decade was the discovery of dark energy. So far, all cosmological data are consistent with the simplest possibility: dark energy is just a new fundamental constant of nature, which does not evolve over cosmological redshifts. On the other hand, it is intriguing to think about alternative explanations associated with this profound change in infrared physics. The most straightforward way of implementing such a change is the introduction of a new ultra-light scalar degree of freedom associated with quintessence 2. The coupling of this scalar field to matter may be the source of new cosmological phenomena such as an apparent "breakdown" of Lorentz invariance connected to the CMB frame, the existence of a "fifth force" mediated by scalar exchange, or a change of couplings and masses with time. Thus, the search for these exotic effects acquires a new actuality in the "dark energy" age. A potential hint on the difference between the laboratory values of the fine-structure constant a 3,4 and the one derived from quasar absorption spectra at high redshifts 5, t1a/ a rv -0.6 X 10- 5 (z rv 0.5 - 3), has trig-

191

gered a series of new phenomenological and theoretical studies of "changing couplings" that typically involve a quintessence-like field ¢ coupled to electromagnetic In all models of changing couplings discussed so far in the literature, the temporal change of a dominates over the possible spatial variation of a, once the constraints on the non-universal gravitational force mediated by ¢ are imposed. In Bekenstein-type models 6, the spatial variation of couplings is caused by matter inhomogeneities and thus follows the profile of the gravitational field. If the coupling of the scalar field to matter is not stronger than the matter-gravity coupling, one expects the change in cp to be less than the variation of the metric. For example, the difference between coupling constants on the surface of the Earth an in orbit is not going to exceed 10- 10 and in practice will be much smaller once fifth force constraints are imposed. In this talk, I show that such a conclusion is not generic, and there is a whole class of models where spatial variations can be more pronounced than the cosmological variations in the recent past, opening new possibilities for searching for D.a and D.m as functions of the matter density. A key to this proposal is to choose the couplings of a scalar field to matter to be much stronger than gravitational. At first glance this would appear to only worsen the problem of a fifth force induced through scalar exchange. However, this might not happen if the matter density itself leads to the effective suppression of the linear scalar field coupling to matter 7,8, and/or the range of the scalar-mediated force becomes shorter than the one needed for conventional fifth-force experiments 9. Models that escape prohibitive fifth-force constraints may predict spatial variation of a and mp/me that exceed recent temporal variations. As we will show in the remainder of this paper, such constructions can be achieved if the dynamics of the scalar field in low-density environments is determined by its selfpotential, while in regions of large overdensities, the dynamics of cp is set by its coupling to matter. In a large subclass of such models where the mattercp coupling is much stronger than gravitational, the temporal evolution happens on the time scales that are much shorter than cosmological. In these models, the global temporal evolution of the scalar field could be finished a long time ago, and on average the "cosmological" values of masses and coupling constants remain constant in space and time.

F;v'

192

2. Scalar field models of a(p) and m(p) The starting point for our analysis is the matter-gravity-scalar field action, 51> =

J 4Xv=g{ - M!l d

R

+

~*

BF~(¢) F~2F(i)/-'V + L[1j;j iNj -

- L

"

2

[Y"¢a,A) - V(¢)

B j (¢)m j 1j;j'lj;j]},

(1)

]

which can be viewed as a generalization of a scalar-tensor theory of gravity. In this expression, M p1 = (87rG N )-1/2 is the reduced Planck mass, ¢ is a dimensionless scalar field with M* being the analogue of the Planck mass in the scalar sector. The functions B Fi (¢) give the ¢-dependence to the gauge couplings in Standard Model (SM), and the sum is extended over all SM gauge groups. 'lj;j represents Standard Model fermions that are coupled to ¢ via the functions B j (¢). After performing a ¢-dependent rescaling of the matter fields, one is allowed to remove the ¢-dependence of the kinetic terms for the SM fermions 'lj;i and keep only couplings to the mass terms. Among couplings to the SM model fields, the couplings to quarks, gluons, photons and electrons are the most important. At lower energies, we can abandon the quark-gluon description in favor of an effective coupling to nucleons and reduce (1) to a more tractable form, 51> =

J 4Xv=g{ - ~~l d

B F (¢) /-,v - - 4 - F /-'v F

R

,,-.

+ ~* 2 a/-'¢a/-,¢ -

+ j=~,e['lj;jzNj

V(¢)

-

- Bj(¢)mj'lj;j'lj;j]

}

(2)

.

Since we are going to consider couplings of ¢ that are essentially much stronger than gravitational, the stability of the model will require that V(¢) and the B i (¢) functions have a minimum with respect to ¢. In what follows, we shall adopt the following ansatz, V(¢) = Ao

+ ~A2(¢ -

¢O)2

+ ... ;

Bi(¢) = 1 +

~';i(¢ -

¢i)2

+ ... ,

(3)

where ellipses stand for cubic, quartic etc. contributions around the minima. Here ';i, ¢o and ¢i are arbitrary dimensionless numbers; Ao and A2 have dimensions of [Energy]4 and we are tempted to choose Ao to be equal to the current dark energy density to "solve" the dark energy problem. A further simplification of the quadratic ansatz comes from the assumption that the proton and neutron Bp(n) functions are mostly induced by the

193

gluon B-function, and thus are approximately equal. With these simplifying assumptions, we can take ~n

C::'

~p = 1;

(4)

¢o =0.

The normalization of ~p(n) to one can be attained by rescaling M*. In principle, a negative value for ~ is also possible, but in this section we shall restrict our discussion to positive ~'s. Of course, the relations (4) are only approximate, and possible violations at the rv 1 - 10 per mill level are naturally expected due to the nonzero quark and electromagnetic content of nucleons. The choice of ¢o = 0 can always be achieved by a constant shift of ¢. The ansatz (3) and (4) is very similar to the DamourPolyakov model 8 (see also 7), where all couplings to matter fields exhibit the same minimum. In the same vein, we assume the same minimum ¢m for B F (¢) function. There are two important difference of our approach from Damour-Polyakov models: we take M* to be much smaller than the Planck mass, and introduce a self-interaction potential that has a different minimum from the minimum of Bi ( ¢) functions. In this section we disregard higher-order nonlinear corrections to v"ff, postponing their discussion to Section 4. Furthermore, we assume a region of relatively uniform matter density p. In such regions, the scalar field equation of motion takes the following form M 2 0A. *


=0 + av"ff a¢ ,

(5)

where the effective potential is given by Veff

121

= Ao + 2 A2 ¢ + 2(¢ - ¢m)

2

(6)

p.

This potential creates the minimum for the scalar field at p ¢min

=

¢m p

(7)

+ A2 '

and the physical (canonically normalized) excitation tp around this minimum has a mass

(8) By definition, the longest range for the tp-mediated force is achieved in vacuum at p = O. It is instructive to present a numerical formula for .A.eff at p» A2 : 24

.A.eff

=7x

3

M*

10- cm x 1 TeV

( 10

G:V cm- ) 3

1/2

,

(9)

194

which shows that for an extreme case with a weak-scale M* and terrestrial matter densities the range of the force falls under one millimeter. If the spatial extent of the mass distribution is much larger than the Compton wavelength of the physical excitations of ¢ the effective interaction with a "test" nucleon takes the following form,

- ( ¢;"A~ tp ¢m A2 tp2 ) Lint = -mNNN 1 + 2(A2 + p)2 - M* (A2 + p) + 2M; ,

(10)

from where we can read a p-dependent mass of a nucleon,

¢2 A2

mNeff

= mN ( 1 + 2(A:+ ~)2

)

(11)

'

and the scalar-field-corrected Newtonian interaction potential between two nucleons separated by distance r,

m'Jv ( l+exp(-me ff r )x

U(r)=G N - r-

2M~1

¢;"A~ )

M; (A2+P)2

.

(12)

Perhaps the most interesting case to consider is A2 » p for low density environments, such as e.g. the interstellar medium, and A2 « p for high density environments such as stars and planets. In that case, the change in the nucleon mass and the fine structure constant can be expressed as mNr - mNd '"

-/,2

'!.!!!:...

0:

0:

2 '

~F¢;" --2

(13)

and we assume that ¢;" and ~F¢;" are much less than one. Notice that ~F can be as large as ~F '" 0(100) without violating the assumption that ~n ~ ~p ~ 1. 3. Experimental constraints on the model

All experimental constraints on the model described by (10) can be divided into two broad categories. The constraints coming directly from the quadratic couplings of ¢ to matter to a large extent do not depend on the position of the minimum of ¢ and on whether this minimum can be reached for a realistic size of an overdensity in question. The second group of constraints follows from the linear coupling of ¢ to matter, which are very sensitive to the position of ¢ and on the size of the overdensity. Astrophysical constraints. First we discuss the astrophysical constraints on the model which employs a ¢2 coupling to photons and nucleons (the

195

linear coupling is suppressed by (A2/ PI? and is assumed to be « 1). It is clear that the quadratic coupling will be less severely constrained than a linear coupling by the thermal emission rate of ¢-quanta from the hot interiors of stars. Indeed, the overall emission rate scales as M;4 rather than f;;2 as one would routinely find in an axion-type model. As a result, instead of a lower limit to fa or order 109 - 1010 GeV, we expect to find a much more relaxed bound on M*, of the order of the electroweak scale. The emissivity of ¢ quanta due to pair annihilation of photons 1 results in an energy loss (Energy/volume/time) for a thermalized gas of photons at the level of _ 2 _ ((3)7r ~}T9 ~ f.-y,-->q,q, - n,(2wa,,-->q,q,) - - - - - 4 - - 0.06 63 M*

~}T9

X -4-.

M*

(14)

Comparing this to the typical limit on r = ExPeaTe < 1O- 14 MeV 5 that follows from the constraints on the emissivity of light particles in cores of supernovae 10, Ex

;S 10 19 erg g-1 S-1 at

PeaTe

= 3x 10 14 g cm- 3 ,

TeaTe

= 30 MeV, (15)

we obtain a typical sensitivity to the coupling of ¢ to photons,

M*~-;.1/2 ;(, 3 TeV.

(16)

Similar considerations can be applied to the bremsstrahlung-like emission process N + N -+ N + N + ¢ + ¢, where again pairs of ¢ are emitted. Skipping details of this calculation, we arrive at the constraint on M*, M* ;(, 15 TeV,

(17)

which is very similar to (16). With these constraints, we conclude that the effective range of
196 which is limited by recent searches for deviations from the gravitational1/r behavior at short distances 11. Specifying the constraints on phenomenological coefficient /33 from Ref. 12 to our model prediction (18), we arrive at G m 2 1 mm 2 V = -/33 N N - - with /33 < 1.3 X 10- 4 =?- M* > 2 TeV, (19) r r2 which is very close to the astrophysical bounds (16) and (17). We note again that the transition from a l/r to a 1/r 3 potential that may occur in our model at short distances is very similar to the transition expected in theories with two large extra dimensions. Constraints from the Yukawa part of (12) are somewhat less straightforward to implement. For a range of 0(10- 2 - 1 cm) in a medium, the constraint on its strength 12 specialized to our case with the use of (12) takes the following form (20)

r

or

16~6 x (~~r x (e~4 ~ 1012 -

13 10 ,

(21)

where we took p '::::' 10 g/cm 3 for the density of molybdenum used in experiments of Ref. 11. Perhaps an even more convenient from of the same constraint arises when we trade A;/2 / M* for 1/ Avae , the range of
~

4

2 x 10- x I (REf)) - (Rorbidl

rv

13 5 x 10-,

(23)

197

where .6.wH is the extra frequency shift of the Hydrogen maser added to the shift predicted by general relativity, and (r) is the gravitational potential at distance r from Earth's center. In our model, the difference between clock frequency on the ground and in orbit would receive an additional correction from the difference of coupling constants and masses caused by .6.¢: .6.wH _ .6.(oAm~gpm:;;l) __ (.6.¢2)

w

H

-

4

2

-1

0: megpmp

-

-2-

x (1 - 2';e

+ 4';F) ,

(24)

where for simplicity we assumed the same scaling for A QCD and quark masses with ¢ which keeps the proton g-factor gp fixed as a function of distance. The density of a medium surrounding the satellite is certainly very low, and thus it is tempting to take .6.¢2 = ¢;,. Note however, that if the in-medium range of the force is much shorter than a typical scale of a satellite, Lsat rv 1 - 10m, with its average density being Msat/ L~at rv 100kg/m3 rv O.lg/cm3 , the scalar field inside a satellite will roll back to its in-medium value, resulting in an exponential suppression of .6.¢. Therefore, we expect the clock comparison constraint to be at the level of (25) noting that the precise amount of exponential suppression would also depend on the position of clocks inside the satellite. Although very powerful for M. ;(; 108 GeV when the exponential factor is of order one, eq. (25) is not particularly constraining for an interesting range of 1Te V < M. < 100TeV, as the value of ¢-field inside the satellite would be very close to that on the surface of the Earth. Straightforward constraints on ¢;, can be deduced from the comparison of coupling constants measured in cosmological settings and in the laboratory. Interpreting the results of 5 as an upper limit [.6.0:/0:[ < 10- 5 , and converting it into parameters of our model, we obtain the constraint (26) It is also very important to stress that the Oklo constraint on .6.0: does not carry any weight in our model. Indeed, the Oklo phenomenon obviously occurred in large density environment, which means that ¢ = ¢m with good accuracy back at the time the Oklo reactor was active, as well as it is now. Cosmological constraints. Cosmology can constrain the presence of new degrees of freedom in the Universe. Big Bang NUcleosynthesis can in principle impose a constraint on a number of new relativistic degrees of freedom

198

that carries a comparable amount of entropy as photons or neutrinos. It is very easy to see, however, that even if ¢ is initially thermally excited, its decoupling occurs well before the neutrino decoupling because M;2 « G F . Since traditionally the BBN constraints are expressed in terms of the number of "new neutrino species", we can immediately conclude that ¢ contributes to this number as 4/7 or less and thus cannot be ruled out on the grounds of light element abundances. 4. Discussion: on a possibility for new tests of a(p).

One possibility to search for the environmental change of masses and couplings caused by a change in density is to try and recreate a low-density environment in the laboratory. The best quality vacuums available today achieve a density at the level of 105 particles/ cm3 , which creates a matter density comparable to eV 4 . Taking P2 »A 2 »Pl, we calculate the resulting shift of ¢ and the change in the coupling constant between the center and the walls of a spherical chamber of radius R,

a(T

= R) -

a

a(T

=

0)

~ ~F¢;" 2

{l6 ('\~J4, 1,

(27)

for R/ Avac « 1 and R/ Avac :<, 1 correspondigly. Similar changes will be experienced by masses of particles. Notice that the parametric dependence of (27) is very similar to (22), and plugging A2 and ¢m that saturate this constraint we find that

a(T = R) - a(T = 0) These shifts are extremely small, but perhaps are not so far away from the modern capabilities of frequency measurements that can be sensitive to the relative shifts as low as 10- 15 15. To put (28) in perspective, we compare this with the result obtained for L::1a/ a between two points separated vertically by rv 1m in the original Bekenstein model with a massless scalar field: Bekenstein model, (29) where 9 is again the free-fall acceleration at the Earth's surface. This is well below any detection sensitivity for the foreseeable future. The difference between (28) and (29) is an enormous factor of nine orders of magnitude

199

that can be traced back to the fact that equivalence principle is checked at macroscopic distances far better than gravity at distances under 1 mm. To conclude, we have demonstrated that a scalar coupling to matter can be much stronger than the gravitational coupling in Damour-Polyakov type models. The quadratic nature of coupling to matter allows one to escape the most prohibitive astrophysical and gravitational constraints, as only pairproduction or pair-exchange of
[hep-ph]]. 2. B. Ratra, P.J.E. Peebles, Phys. Rev. D 37, 3406 (1988); Ap. J. Lett 325, 117 (1988); C. Wetterich, Nucl. Phys. B 302, 668 (1988); R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett 80, 1582 (1998) [astro-ph/9708069]. 3. R. S. Van Dyck Jr., P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. Lett. 59, 26 (1987). 4. B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse Phys. Rev. Lett. 97, 030801 (2006); G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio and B. Odom, Phys. Rev. Lett. 97, 030802 (2006). 5. J. K. Webb et al., Phys. Rev. Lett. 87, 091301 (2001) [arXiv:astroph/0012539]; M. T. Murphy, J. K. Webb and V. V. Flambaum, Mon. Not. Roy. Astron. Soc. 345, 609 (2003) [arXiv:astro-ph/0306483]. 6. J. D. Bekenstein, Phys. Rev. D 25 (1982) 1527. 7. T. Damour and K. Nordtvedt, Phys. Rev. Lett. 70, 2217 (1993); T. Damour and K. Nordtvedt, Phys. Rev. D 48, 3436 (1993). 8. T. Damour and A.M. Polyakov, Nucl. Phys. B 423, 532 (1994) [hepth/9401069]; Gen. ReI. Grav 26, 1171 (1994) [gr-qc/9411069]. 9. J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004); J. Khoury and A. Weltman, Phys. Rev. D 69,044026 (2004); S. S. Gubser and J. Khoury, Phys. Rev. D 70, 104001 (2004). 10. G. G. Raffelt, Ann. Rev. Nucl. Part. Sci. 49, 163 (1999). 11. D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle and H. E. Swanson, Phys. Rev. Lett. 98, 021101 (2007). 12. E. G. Adelberger, B. R. Heckel, S. Hoedl, C. D. Hoyle, D. J. Kapner and A. Upadhye, Phys. Rev. Lett. 98, 131104 (2007). 13. R. F. C. Vessot et aI, Phys. Rev. Lett. 45, 2081 (1980). 14. C. M. Will, Living Rev. ReI. 4, 4 (2001). 15. T. W. Hansch, Rev. Mod. Phys. 78, 1297 (2006).

This page intentionally left blank

201

PROGRAM FOR THE WORKSHOP ON RARE ISOTOPES AND FUNDAMENTAL SYMMETRIES, INT, University of Washington September 19 - 22, 2007 • Wednesday Sept 19th morning chair (Guy Savard) - 9:45: Alex Brown Introduction and coordinates 10:00: Victor Flambaum Parity and time reversal violation in atoms and molecules and test of the Standard Model - 10:45: Klaus Jungmann Experiments searching for new interactions in Nuclear beta-decay - 11:45 Klaus Blaum High-precision Penning trap experiments for the study of fundamental symmetries • Wednesday Sept 19th morning chair (Alejandro Garcia) - 2:00: Paul Vetter The beta neutrino correlation in sodium-21 - 2:45: John Behr Decay correlations with atom traps: techniques and possibilities - 3:45: Alex Brown Progress in CI and EDF calculations for nuclear wavefunctions related to fundamental symmetries - 4:30: Jon Engel Nuclear structure physics and Schiff moments - 5:00: discussion • Thursday Sept 20th morning chair (John Behr) - 9:00: John Hardy Superallowed nuclear beta decay: recent results and their impact on Vud - Radiative and Isospin-symmetry breaking corrections in superallowedbeta decay - 11:00: David Pinegar Precise measurement of the 3H to 3He mass difference - 11:45: Alejandro Garcia Isospin breaking from T=2 0+ ----> 0+ transitions and extracting Vud from neutron beta decay • Thursday Sept 20th afternoon chair (Klaus Jungmann) - 2:00: Wick Haxton Nuclear constraints on the weak NN Potential

202

2:45: Marianna Safronova Atomic PNC theory: current status and future prospects 4:00: Bertrand Desplanqes Parity-violating nucleon-nucleon interactions: What can we learn from nuclear anapole moments? - 4:45: Luis Orozco Proposed experiment for the anapole measurement in francium 5:30: discussion 7:00: group dinner at Ivars • Friday Sept 21st morning chair (Ian Towner) 9:00: Tim Chupp The Radon EDM Experiment Atomic Parity Violation and Neutron Densities, The Jlab PREX experiment and FRIB with hadronic probes 11:00: Naftali Auerbach Nuclear structure aspects of Schiff Moments 11:45: Vladimir Zelevinsky Search for the collective enhancement of nuclear Schiff moment • Friday Sept 21st afternoon chair (Chuck Horowitz) 2:00: Peter Mueller The Ra EDM experiment at Argonne 2:45: Dmitry Budker Preliminary investigations into a possibility of a sensitive search for the Schiff moment of 207Pb in a ferroelectric - 4:00: Cheng-Pang Liu The Interpretation of Atomic Electric Dipole Moments: The Schiff Theorem and Its Corrections - 4:45: Klaus Jungmann New possibilities to search for permanent electric dipole moments in atoms and nuclei 5:00: Matt Swallows Status of the Hg EDM Experiment 5:30: Dmitry Budker Status of the Search for Parity Violation in Atomic Ytterbium • Saturday Sept 22nd morning chair (Dmitry Budker) 9:00: Guy Savard Ion Traps for Weak Interaction Studies and Prospects for FRIB 9:45: Kei Minamisono Possibilities with collinear laserpolarized atoms 11:00: Maxim Pospelov Environmental dependence of masses and coupling constants

This page intentionally left blank