In Memory Of Vernon Willard Hughes: Proceedings Of The Memorial Symposium In Honor Of Vernon Willard Hughes, Yale University, USA 14 - 15 November 2003

Vernon W. Hughes

Proceedings of the Memorial Symposium in Honor of Vernon Willard Hughes

Yale University, USA

14 - 15 November 2003

editors

Emlyn Willard Hughes California Institute of Technology, USA

Francesco Iachello Yale University, USA

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World Scientific

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IN MEMORY OF VERNON WILLARD HUGHES Proceedings of the Memorial Symposium in Honor of Vernon Willard Hughes Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying. recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface On November 14-15, 2003, the Vernon Willard Hughes Memorial Symposium was held at Yale University in New Haven, Connecticut. This volume contains the Proceedings of that Symposium. The Symposium was organized by a Committee composed of Charles Baltay, David DeMille, Paul Fleury, Emlyn Hughes and Francesco Iachello (Chair). About 100 scientists attended the Symposium from the international community, Yale, surrounding universities and the country as a whole. The Symposium commenced with a welcoming address by Susan Hochfield, Provost of Yale University. Talks were presented by scientists from several countries on topics related to Vernon’s work, in particular his discovery of muonium, his major contributions to the spin structure of the proton and to the muon (g-2) experiment. Other subjects were also discussed. A Symposium banquet was held on Friday evening with D. Allan Bromley presiding. Gisbert zu Putlitz was unable to be present, but D. Allan Bromley read his remarks. Daniel Kleppner and Nicholas Samios said some nice words. John Marburger Jr., Science Adviser to the President, was also present and made remarks. We are grateful to all the banquet speakers for their words. The biographical memoir of Vernon, written by Robert K. Adair for the National Academy of Sciences, is included in this volume as an important contribution to Vernon’s legacy. For future reference, we have also included in this volume the complete publication list of Vernon Willard Hughes comprising over 400 articles, a true account of major developments in atomic, nuclear and particle physics in the years 1950-2003. We are also grateful to the Provost Office, the Physics Department through its Chairman R. Shankar and the School of Engineering through its Chairman Paul Fleury for grants to support outside speakers. We owe a great deal of gratitude to the Physics Department for providing the infrastructure and technical services, and to the Conference Secretary, Diane Altschuler, and the staff of the Physics Department, Laurelyn Celone and Marguerite Scalesse, for their help. Without them, this important event, commemorating the contributions of one of the leading world figures in 20th Century Physics would not have been possible.

Emlyn Hughes Prancesco Iachello V

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Contents

Preface

V

Vernon Hughes and the Early Years of Molecular Beam Resonance Norman F. Ramsey, Higgins Professor of Physics Emeritus, Harvard University

1

The 46 Years of Muon g-2 Francis Farley, Visiting Senior Research Scientist, Yale University

8

Muonium - The Early Experiments Richard Prepost, Professor of Physics, University of Wisconsin Muonium Lifetime and Heavy Quark Decays (Lessons Learned from Muonium) William J. Marciano, Senior Scientist, Brookhaven National Laboratory

26

42

Recent Developments of the Theory of Muon and Electron g-2 Toichiro Kinoshita, G. Smith Professor of Physics Emeritus, Cornell University

58

Vernon Hughes and the Quest for the Proton’s Spin Robert L. Jaffe, Morningstar Professor of Physics, Massachusetts Institute of Technology

78

The Spin Structure of the Nucleon: A Hughes Legacy Gordon D. Cates, Professor of Physics, University of Virginia

96

vii

viii

Muon g-2: The Last Word? Ernst Sichtermann, Division Fellow, Lawrence Berkeley National Laboratory

116

Past, Present and Future of Muonium Klaus P. Jungmann, Professor of Physics, University of Groningen

134

Parity Nonconservation in Electron-Electron Scattering Emlyn W. Hughes, Professor of Physics, California Institute of Technology

154

Exploring the Nucleon Spin: The Next Decade Abhay L. Deshpande, RIKEN Fellow, Brookhaven National Laboratory

171

Remarks at the Symposium Banquet Honoring Vernon Hughes D. Allan Brornley, Presiding, Sterling Professor of the Sciences, Yale University

191

Banquet Talk in Honor of Vernon W. Hughes Gisbert zu Putlitz, Professor of Physics, University of Heidelberg

193

Tests of CP T for Muons Vernon W. Hughes, Yale University

196

Vernon Willard Hughes, 1921-2003 (A Biographical Memoir) Robert K. Adair, Professor of Physics, Yale University

204

Publication List of Vernon W. Hughes

223

VERNON HUGHES AND THE EARLY YEARS OF MOLECULAR BEAM RESONANCE NORMAN F. RAMSEY Faculty of Arts and Sciences, Harvard University Cambridge MA 02138

The first phase of molecular beam resonance studies began with 1.1. Rabi's invention [ 11 of the molecular beam magnetic resonance method in September of 1937. This invention was stimulated by Rabi's brilliant theoretical paper [a] entitled "Space Quantization in a Gyrating Magnetic Field," by C.J. Gorter's publication [a] entitled "Negative Result of an Attempt to Detect Nuclear Magnetic Spins" and by a visit of Gorter to Rabi's Columbia laboratory. Imme,diately after the invention, two of Rabi's research groups modified their apparatus [I] as shown in Fig. 1 to detect the resonance frequencies at which transitions occurred.

Y

v 0 magnet

A magnet

Fig. 1 . Schematic diagram showing the principle of the first molecular beam magnetic resonance experiments. The two solid curves indicate two paths of molecules having different orientations that are not changed during passage through the apparatus. The two dashed curves in the region of the B magnet indicate two paths of molecules whose orientation has been changed in the C region so the refocusing is lost due to the change in the component along the direction of the magnetic field.

1

2

With these apparatuses, Rabi, S. Millman, P. Kusch, J.R. Zacharias, J.M.B. Kellogg and N.F. Ramsey during the next few years measured a number of spins and magnetic moments of nuclei including the proton, the deuteron, and the nuclei of 'Li and many other atoms [a]. Kellogg, Rabi, Ramsey and Zacharias [a] discovered that the deuteron had a quadrupole moment which implied the existence of a nuclear tensor force. Although the magnetic resonance method was originally developed to study nuclear magnetic moments in nonparamagnetic molecules, Kusch, Millman and Rabi [a] in 1940 applied it to the paramagnetic atoms Li, K, Na, Rb and Cs. They measured both the atomic moments and the hyperfine separations.

The first publication with the molecular beam magnetic resonance method was in 1938, but by 1941, after only three years of great productivity, research in the field was exclusively at Columbia University. This research was coming to an end, however, as the scientists and the laboratories became involved in World War I1 defense-related research. Vernon Hughes was too young to have been involved in this first short but productive period of molecular beam resonance research. However he was deeply involved in the next phase.

In the fall of 1945, Rabi and I returned from our war time research work to revive the Columbia molecular beam laboratory. It was a difficult but exciting period. Since the ONR program for supporting basic research had not yet been established, there was little financial support, in marked contrast to the generous funding for our wartime research. Although two molecular beam apparatuses had been left more or less intact, the remainder had been hastily dismantled with many of the components dumped into attic trashcans so the research rooms could be used for defense research. I spent many hours searching the attic and identifying useable components.

After a few months Vernon and other graduate students began to arrive. They were a great group of students since many of them had already spent one to five years doing research in defense laboratories. Vernon, for example, worked in the MIT Radiation Laboratory group developing accurate timing circuits. As a result he was much more skillful with electronics than either Rabi or 1. In some respects it was embarrassing to have Vernon as a student be more

3

knowledgeable about many research techniques than we on staff, but the skill and knowledge he brought to the lab more than made up for the embarrassment. The conditions at the time were rather unfair to Vernon. I had received one of the last Ph.D's granted before the war and was working as a tenured associate professor whereas Vernon, who had almost as much research experience, was officially a graduate student.

Since initially we could not afford to build a major new apparatus, Vernon started with an atomic beam electric resonance apparatus built before and during the War by Harold Hughes [a] and further improved by John Trischka and Vernon. A schematic diagram of the apparatus is shown in Figure 2.

etec tor-

7 "1.0

0

Fig. 2. Schematic diagram of electric-resonance apparatus showing paths of molecules. The transverse scale of the drawing is much larger than the longitudinal scale.

The vertical scale in this diagram is tremendously magnified. The molecular states in the diagram are identified by giving the values (J, m J). Vernon [2-51 and his fellow student, Lou Grabner measured "RbF, s7RbFand 3 k F in the first rotational and zero'th vibrational states. Their values for the electric quadrupole interaction parameters e2qQ/h in MHz were -70.31 f 0.06, -34.00 f 0.06 and 7.938 f 0.040 for 85RbF,"RbF and 39KFrespectively. They also showed that for 85RbFand 87RbFthe value of e2qQ/h decreased by 1.1% in going from one

4

vibrational state to the next higher one whereas for 39KFthe corresponding decrease was 1.3%.

As this work progressed, Vernon and Lou [2,3,5] observed that they were getting a resonance transition not only at the frequency for the quadrupole moment interaction but at half that frequency as well. They attributed this to a two-photon transition, which had never before been seen in spectroscopy. They confirmed that the transition was not due to a second harmonic from the oscillator by using two different frequencies and finding the two-photon resonance when the sum of the two frequencies was correct. Vernon at this time also developed the theory for a two-photon transition in a microwave resonance spectrum [5,7].

Vernon's interest in simple fundamental experiments showed up at this time by his undertaking, at Rabi's suggestion, an experiment [9] to see if a molecule could be significantly deflected by a uniform electrostatic field. ~ Since then Vernon and others His negative result showed that 1q e +q ~ l < l O - ' e. [lO,l 11 have lowered the limit to (qe+q p(17) along with the lppm rP+ experiment mentioned b e f ~ r eSo, . ~ in my opinion, the g p puzzle currently provides the main near term motivation for high precision muon lifetime measurements. The PSI experiments will stop both the p+ and p- and then basically count the decays as a function of time. The p- stopping material (hydrogen) will reduce the lifetime due to the capture mode, but what about modifications of the ordinary decay mode p- .+ e-FevP due to nuclear binding effects. Similarly, will bound state effects due to muonium M = p+eformation in the stopping material affect the p+ lifetime at lppm? These issues have been addressed in the l i t e r a t ~ r e In . ~the ~ ~next ~ ~ ~two sections I summarize what are rather surprising and interesting results from those studies. 3. Muonium (p+e-) Decay

Measurement of the p+ lifetime (at rest) requires stopping the p+ in matter and counting the number of outgoing muons as a function of time. At the lppm, one might worry about environmental effects on the lifetime. For example, will electron screening or muonium (p+e-) formation modify the decay rate? The easiest case to examine is muonium, a simple bound state with very well defined properties.lg

48

Modifications of r, for the 1s bound state of muonium, due to Coulombic interactions, can be expressed as an expansion in terms of the two small dimensionless parameters a = 1/137 and m,/m, N 1/207. Such effects must vanish as a -, 0 or m,/m, --t 0. Hence, one expects corrections of the form an(me/m,)mwhere n and m are integers. Before considering the leading corrections, let me review some basic properties of muonium’s ground state. The binding energy, average potential and kinetic energy, as well as average electron and muon velocities are given by

E = &&rne N -13.5 eV

(1la)

( V )= -a2m, N -27 eV ( T )= ia2m, 2 13.5 eV

(1lb)

(Be) 2

(114

(P,)

21

(114

(1le)

The muon rest frame and lab frame differ because (p,) # 0. That effect gives some spectral distortion due to Doppler smearing of the positron energy by terms of O(am,/m,). However, Lorentz invariance tells us that there are no corrections to the lifetime linear in velocity. Instead, that small muon lifetime velocity only gives rise to a very small time dilation increase

which is about 6 x a negligible increase. The next-to-leading spectral distortion will entail effects of order - ( V ) / m , = a2me/m,. Naively, one might expect a potential energy shift in the outgoing positron energy to give rise to a decay rate reduction due to the phase space change rn;

-+

(m, i- ( v ) 4 ) ~mE(1- 5a2m,/m,)

(13)

That small 21 1 x reduction would, if real, be about the same size as the expected PSI experiment’s ~ensitivity.~ However, it has been shown3 that final state e+e- interactions give rise to an equal but opposite sign effect that cancels the shift in eq. (13). In fact, electromagnetic gauge :~ are no anme/m, invariance, can be used to prove a general t h e ~ r e mThere corrections! The absence of such l / m , corrections is very well known to people who work on b quark physics where the operator product expansion

49

is used to show that there are no l/mb correctionssI2' t o the b quark lifetime when placed in different hadronic bound state environments, i.e. Bd, B,, A b etc. In other words, through first order in l/mb all b hadrons should have the same lifetime, a somewhat anti-intuitive feature. I return to this point in section 5. Note also that the cancellation of phase space effects due t o the electron screening potential with final state interactions also occurs for nuclear beta decays.21 but is usually disguised because the standard ,@decay formalism employs Q values for atoms rather than fully ionized nuclei. That approach hides the phase space effect; so only the final state interaction is corrected for. It has also been observed for the decay of the p - while in a muonic atom bound state. Indeed, the cancellation is a universal phenomenon. A simple illustration of the above cancellation is provided by a static sphere model in which one thinks of muonium as a p + surrounded by a fixed thin electron sphere with radius Bohr radius. The effect of that sphere gives rise to a potential V = -a2me which shifts the entire positron decay spectrum by V . The fully integrated decay rate r ( p + --+ e+vP) for a p + in the sphere is modified such that

-

There is an apparent change in phase space (limits of integration) which is cancelled by the final state interaction of the e+ with the electron sphere which shifts the differential decay rate. Overall, the shifts in eq. (14) correspond t o a simple change of variables with no net effect. So, we have learned that the very small time dilation shift in eq. (12) is in fact the leading correction and it represents a totally negligible effect for lppm muon lifetime measurements. A similar argument can be made for electron screening effects in metals which collectively give rise t o potentials similar to muonium. What are some of the higher order (in a and me/m,) corrections t o the muonium lifetime? That question is not of phenomenological importance for the muon lifetime issue, but muonium is such a simple system that we should be able to learn some useful lessons from such a study. One such correction is due to the muon-electron hyperfine interaction which modifies the lifetime by terms of order a4m2/mi. The QCD analog of that effect provides a leading l/mg correction t o b hadron lifetimes (along with time dilation), see section 5.

50

What are the leading l/mE corrections? Muonium exhibits two such effects. The first is the availability of a capture mode M --+ vep, in muonium3

Although that effect is tiny 6.6 x it demonstrates an important feature. The 2 vs 3 body final state gives rise to a very large 4 8 enhance~ ment fact. This type of capture effect can be quite important for bhadron lifetimes where 2 body annihilation or scattering can play a significant role. A final order (arr~,/m,)~ muonium decay correction is well illustrated by the simple example in eq. (14). Classically, the entire range of integration down to me-a2me is allowed. However, the real lowest energy state positronium has only 3V. So, part of the lower range of the integration in eq (14) are actually not allowed. That effect is the real quantum phase space reduction. It suppresses the muon decay rate in muonium by a tiny correction -16(am,/m,)3 I I-7 x Although very small, it also exhibits a large factor N 16 enhancement factor. Overall, one finds the following lessons learned from the relationship between the muon and muonium total decay rates3 N

N

+

+ 4 8 ~ ( ~ ~ m , / m-, 1 ) ~6 ( ~ t m , / m ~ )* ~* .

where the 4 corrections exhibited correspond to time dilation, hyperfine effects, annihilation and quantum phase space reduction respectively. Similar types of corrections will occur for other bound states, as we now illustrate by several examples. 4. p- Decay in Atomic Orbit

Muonic atoms have much larger Coulomb interactions than muonium because of their small radii and tight binding to nuclei. The average bound state potential and muon velocity are

51

For high Z nuclei, pfl will give rise to large spectral distortions. Again, the naive phase space reduction due to ( V )cancels with the electron-nucleus final state interaction, as observed by Uberall in 1960.7The leading remaining effect is a time dilation increase of the p- in orbit lifetime by a 1++Z2a2 factor which can be important for high 2 as well as a lOppm p- lifetime measurement in 2 = 1 hydrogen. There are also capture proce~ses'~ which provide a new decay mode which grows significantly with 2,

hcapture N 1000(Za)31'(p evD) -+

(18)

and again has the large 2 body enhancement factor. (It actually grows as Z4 modulo final state Pauli exclusion.) There is also a quantum state reduction of O ( Z 3 a 3 )due to the suppression" of decays with Ee- < Z2a2m,. m2 +m:

due to the posAnother interesting effect is decays with Ee- > sibility of nuclear recoil. In fact, the e- spectrum will have a tail extending all the way to m,(l - Z2a2). All such corrections are calculable;18 so they should not cloud the interpretation of the p- lifetime in hydrogen. Nevertheless, for a lOppm experiment they must be closely scrutinized.

,&

5 . b-Hadron Lifetimes

A very nice illustration of the lessons learned from muonium is provided by the lifetimes of b mesons and baryons. QCD rather than QED bound state potentials are involved, but the physics is universal. To a good first approximation all b-hadrons should have the same ( b quark) lifetime. That is a remarkable feature when one considers the broad range of b-hadron masses and the complexity of final state decay interactions. Nevertheless, the cancellation of naive phase space effects and QCD final state interactions will occur due to QCD gauge invariance. As a result, there are no l/mb corrections to the b quark lifetime induced by its hadronic environment. That property is well known to b physics workers. It is usually proven by using the operator product expansion20i22to show the leading lifetime corrections are of order l/m;. There are O(l/m;) time dilation corrections (due to the b quark Fermi motion) as well as QCD hyperfine interactions that vary from one b hadron to another. However, those corrections are expected to represent only few percent effects. Of course, the l/m: corrections to lifetimes can be potentially important. Two body processes b u -+ c d in b baryons and b d -+ c fi in the B d meson are relatively suppressed by l/m: but

+

+

+

+

52 can have large 2 body phase-space enhancements similar to the 48r factor observed for M 4 ueDp in section 3. But, are those corrections enough to bring all b-hadron lifetimes into accord with one another? To illustrate the current situation, I give in table 3 some b hadron properties along with their lifetimes12 Table 3. bhadron properties and lifetime ratios. State

Mass (MeV)

Lifetime (ps)

Lifetime/TBo

Bi = bd

5279 5279 5370 5624

1.542(16) 1.674(18) 1.461(57) 1.229(80)

1 1.083(17) 0.947(38) 0.797/53)

B l = bii B, = bB Ah = bud

How do those lifetime ratios compare with theoretical expectations, after time dilation, hyperfine interactions, and 2 body “capture” interactions are taken into account?20i22The predicted ratios are12 TB-/TB;

=1

+ O.O5(f~/200MeV)2

(19a)

TB,/TB; = 1f 0.01

(19b)

TAb/TB: = 0.9

(194

Those theoretical expectations seem to be in some disagreement with the lifetimes observed for B, and Ab. In fact, the direction of the disagreement appears to be correlated with their larger masses, i.e. looks like a phase space effect. The difference between theory and experiment, particularly for the Ab is sometimes called the b lifetime puzzle. How will it be resolved? and T? will change. Maybe the O(l/mi) 2 body rates are Perhaps T? larger than theory estimates. Large O(l/m:) effects may be the source.23 Here, I would like to suggest that a lesson learned from muonium may be the cause. The quantum phase-space reduction may be giving additional different l / m i corrections for the various 6-hadrons. (Sometimes called a breakdown of quark-hadron duality by b physics workers.) It is a tiny effect for muonium but could be a few percent for the b-hadron system. How this b lifetime puzzle issue is resolved will be interesting to watch. 6. The K + + nOe+vevs KO + n-e+ve Decay Puzzle

Another interesting puzzle has recently surfaced in Ke3(K + rev) decays. That special decay is important because it is traditionally used to obtain

53

the CKM quark mixing matrix element lVusl = sin8cabibbo. The neutral K L -+ r+e--Fe K L -+ r-e+ue decay rates together give rise to12

+

r ( K L 4 r e v ) = 0.494(5) x

MeV

(20)

On the other hand, a recent measurement by the E865 c ~ l l a b o r a t i o nat~ ~ Brookhaven found

r ( K + -+ roe+ve)= 0.273(5) x MeV (21) If isospin were a perfect symmetry, one would expect the ratio of those two rates to be 2 (because 2 modes are included in eq. (20)). However, one currently finds r ( K L -+ r*ev) = 1.81(2)(3) r ( K + 3 roev) a significant 9.5%deviation from the isospin limit. That difference gives rise (even after isospin violating corrections) to quite different determination^^^ of IV,, I from neutral and charged Ke3 decays, an unacceptable situation. What does the above problem have to do with lessons from muonium? Well, there are 2 sources of isospin violation in eq. (22), the md - mu mass difference and QED (electromagnetic) effects. The first of those gives rise to a factor ofL6

3 md-mu 1+21 1.034 (23) 2 m, - (md mu)/2 correction (from T-77 mixing) that is accounted for in extractions of lVusl from K A . It is not enough to account for the 9.5% difference on its own. In fact, it is essentially cancelled by the mKL - mK+ mass difference effect which is primarily (but not totally) due to md - mu. That gives rise to a

+

(m~+/m~ N 0.96 ~)' (24) compensating correction. The 9.5% difference must be due to electromagnetic effects or experimental error(s). Now is where the lesson from muonium comes in; specifically, the cancellation between electromagnetic phase space effects and final state interactions. Of the kaon and pion mass differences mKL - m K + , m,+ - m,o 12 involved in the Ke3 decays

54 mK+ = 493.65 m,o

MeV

= 134.97 MeV

m K L = 497.67 MEV

m,+ = 139.57 MeV

the latter is primarily of electromagnetic origin. It gives rise to a phasespace reduction of the expected ratio of 2 in eq. (22) by26 0.1561 0.9726 (26) 0.1605 That 2.6% change goes in the right direction. However, there is a fairly large final state electromagnetic (Coulombic) interaction (FSI) between the x+e- (or r e + ) which enhances the neutral Ke3 relative to the charged Ke3 by Pion Mass Phase Space reduction

FSI

N

+

1 xcr

N

11 -=

1.023

(27)

Multiplying eq. (27) times eq. (26) demonstrates the near cancellation as expected from the general theorem found for muonium, beta-decay, b decays etc. It is again a consequence of electromagnetic gauge invariance ; albeit a more subtle demonstration. In a sense, the cancellation confirms the claim made above that the pion mass difference is primarily of electromagnetic origin. Where does that leave us? The product of the 4 isospin violating corrections give rise to an overall factor of (1.034 * 0.96)/(0.9726. 1.023) = 0.998

(28)

The 9.5% deficit remains and leads to about a 4.7% difference in the Valextracted from charged and neutral Ke3 results. Something ues of /Vusl is wrong and it doesn’t appear to be the up-down mass difference or the electromagnetic effects (since the theorem works). There are various possibilities or some combination of them: 1) One of the Ke3 results is incorrect. It could be an actual branching ratio measurement or the use of an incorrect K* or K L lifetime employed to convert t o the partial decay rates in eqs. (20) or (21). In the case of both K* and K L , their properties are obtained using overall fits to all kaon data. New dedicated measurements would be welcome. 2) Perhaps the up-down quark mass difference is a factor of 2 or more larger than assumed in eq. (23). That would favor the

55

smaller value of lVusl5 0.220 extracted from neutral Ke3 decays, a value that unfortunately does not seem to respect CKM ~ n i t a r i t y 3) . ~It~ ~ ~ ~ ~ is possible that the form factors and used in the extraction26 of lVus/differ by more than the m d - mu correction effect in eq. (23). How will the Ke3 puzzle be resolved? We will have to wait and see. Note Added: After the symposium, a more detailed study of radiative and chiral corrections to Ke3 decays appeared.2g It found an increase of about +1.5% due to isospin violation, thus, also suggesting that the 9.5% in eq. (22) difference is of experimental origin and perhaps more likely in the neutral kaon system. Also, a new Ke3 result by the KTeV Collaboration at Fermilab? finds a 5% increase in that branching ratio, reducing the 9.5% difference t o 4.51

fto f?'

7. Concluding Remarks

I have tried to describe how lessons learned from our study of the muon lifetime in muonium can by analogy teach us something interesting about other bound state decay effects. Naive phase space decay rate reduction was shown to be canceled by final state interactions. Other effects such as p+e- capture and quantum state reductions, although tiny for muonium, can have significant analogous implications for muonic atoms and b-hadrons. The examples discussed have interesting puzzles associated with them. The induced pseudoscalax coupling, g p , in muon capture on hydrogen is too large. The Ab lifetime is too short. The charged and neutral Ke3 decay rates appear t o be in disagreement. Which one, if either, gives the right value of IVusl? Those types of puzzles are very healthy for physics. They stimulate new experiments and new ideas. Their resolution can lead to new discoveries and scientific or technological advances. It is the excitement of those discoveries and satisfaction of our intellectual appetites they provide that draws us to scientific research. On a personal note, I would like to end by expressing my gratitude for the work and discoveries of Vernon Hughes. His prized discovery, muonium, provided me with insights into bound state physics. He introduced me to the muon g - 2. I had the pleasure to work on some theory related to his famous polarized eD experiment at SLAC and a number of his other adventurous discoveries. He was the champion of muon physics. Hopefully, those of us who were inspired by his devotion to science will continue his tradition of excellence.

56 References 1. V.W. Hughes et al. Phys. Rev. Lett. 5,63 (1960); Phys. Rev. A l , 595 (1970); V.W. Hughes, Ann, Rev. Nucl. Sci. 16,445 (1966) 2. see Introductory Muon Science by K. Nagamine, Cambridge U. Press, 2003. 3. A. Czarnecki, G.P. Lepage and W. Marciano, Phys. Rev. D61,073001 (2000). 4. R. Carey and D. Hertzog et al, “A precision measurement of the positive muon lifetime using a pulsed muon beam and pLan detector”, PSI Proposal R-99-07. 5. V.W. Hughes, Int. J . Mod. Phys. A1851,215 (2003); G.W. Bennett et al., Phys. Rev. Lett. 89,101802 (2002); hep-ex/0401008. 6. R. Serber and H.S. Snyder, Phys. Rev. 87,152 (1952). 7. H. Uberall, Phys. Rev. 119,365 (1960). 8. 1.1. Bigi, N.G. Uraltsev and A.I. Vainshtein, Phys. Lett. B293,430 (1992). 9. W. Marciano, J . Phys. G: Nucl. Part. Phys. 29,23 (2003). 10. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959); S. Berman, Phys. Rev. 112,267 (1958). 11. T. van Ritbergen and R. Stuart, Phys. Rev. Lett. 82,488 (1999). 12. K. Hagiwara et al. (Particle Data Group), Phys. Rev. D66, 010001 (2002). 13. A. Sirlin, Phys. Rev. D22, 471 (1980); W. Marciano and A. Sirlin, Phys. Rev. D22,2695 (1980). 14. W. Marciano, Phys. Rev. D60,093006 (1999). 15. T. Gorringe and H. Fearing, Rev. Mod. Phys. 76,31 (2004); nucl-th/0206039. 16. A. Adamczak et al., PSI Proposal R-97-05. 17. G. Bardin et al., Nucl. Phys. A352,365 (1981); T.Suzuki, D. Measday and J. Roalsvig, Phys. Rev. C35,2212 (1987). 18. V. Gilinsky and J. Mathews, Phys. Rev. 120,1450 (1960); R. Huff, Annals of Phys. 16,288 (1961); P. Hanggi, R. Viollier, U. Roff and K. Alder, Phys. Lett. 51B,119 (1974); F. Herzog and K. Alder, Helvetica Physica Acta 53, 53 (1980); 0.Shanker, Phys. Rev. D25, 1847 (1982). 19. V.W. Hughes and G. zu Putlitz in Quantum Electrodynamics, ed. T. Kinoshita (World Scientific, Singapore, 1990) p. 882. 20. N. Uraltsev, hep-ph/9804275; B. Blok and M. Shifman, Nucl. Phys. B399, 441 and 459 (1993). 21. M. Rose, Phys. Rev. 49,727 (1936). 22. M. Voloshin, Phys. Rev. D61, 074026 (2000); hep-ph/9908455; Phys. Rept. 320,275 (1999); 1.1. Bigi, hep-ph/0001003. 23. F. Gabbeani, A. Onishchenko and A. Petrov, Phys. Rev. D68,114006 (2003); hepph/0303235. 24. A. Sher et al., Phys. Rev. Lett. 91,261802 (2003). 25. W. Marciano, in Kaon Phys., eds. J. Rosner and B. Winstein, Chicago 1999, p. 603; hep-ph/9911381. 26. H. Leutwyler and M. Roos, Zeit fiir Physilc C25,91 (1984). 27. Quark Mixing, CKM Unitarity, eds. H. Abele and D. Mund (2002) Mattes Verlag, Heidelberg. 28. W. Marciano, hep-ph/0402299.

57 29. V. Cirigliano, H. Neufeld and H. Pichl, hepph/0401173. 30. T. Alexopoulos et al., (2004), hep-ex/0406001

RECENT DEVELOPMENTS OF THE THEORY OF MUON AND ELECTRON G - 2 *

TOICHIRO KINOSHITA Laboratory for Elementary Particle Physics Cornell University Ithaca, N Y 14853, USA E-mail: tk0hepth.wrnell.edu

This paper is dedicated to Vernon Hughes to honor his fundamental contributions to high precision measurements in atomic and particle physics throughout his long and fruitful research career.

1. Introduction

In 1986 Hughes and I submitted a proposal of high precision muon g - 2 experiment at Brookhaven National Laboratory, which was approved in 1987. The goal of experiment E821 was to improve the precision of the last CERN experiment by a factor 20 to about 4 x 10-l'. This will test the Standard Model (SM) prediction of electroweak effect and significantly enhance the sensitivity to new physics. After years of painstaking preparation and test it has finally approached the designed precision and begun to produce exciting physics. The latest measured value of the anomalous magnetic moment of negative muon is N

a,- (ezp) = 11 659 214 (8)(3) x 10-l'

(0.7 ppm),

(1)

where up = 51 (9, - 2) and the numerals 8 and 3 in parentheses represent the statistical and systematic uncertainties in the last digits of measurement. The world average value of a, obtained by combining this and earlier measurements is 233*435

a,(ezp) = 11 659 208 (6) x 10-l'

(0.5 ppm).

(2)

'This article is based upon the work supported by the National Science Foundation under Grant No. PHY-0098631.

58

59

This result provides the most stringent test available thus far of the Standard Model, which is written traditionally as a sum of three parts:

+ a,(had) + a,(weak),

a,(SM) = a,(QED)

(3)

although such partition is unambiguous only in the lowest order. The contribution of the three terms are roughly 100 %, 60 ppm, and 1.3 ppm, respectively. Unfortunately, the hadronic contribution a, (had) has a large uncertainty (- 1 ppm) and prevents us from carrying out the test of the Standard Model to the extent achieved by the measurements of a,. The lowest order hadronic vacuum-polarization effect has been determined in two ways: (i) e+e- annihilation cross section, and (ii) hadronic T decay. Several recent evaluations of the effect are 6*79899

~ t ) ( h ~ d . l=O6963 ) (62)e,p(36),,d x a ~ ) ( h a d . l O= ) 6948 (86) x ~ F ) ( h a d . l O= ) 6924 (59)ezp(24)radx a f ) ( h ~ d . L O=) 6996 (85),,p(19),,d(20)pr.,

x

~P( ~ ~ ) ( h ~= d 7110 . l O (50),,,(8),,d(28)~~(2) ) X

(4)

Together with the terms given later in Eqs. (6), (8), (9), and higher-order hadronic vacuum-polarization term 8Jo, the values in (4) lead to predictions which deviate from the measurement (2) by 2.70, 2.60, 3.30, 2.10, and 1.40, resp. Differences among the first four lines are due to different interpretations and treatments of basically identical data. However they all agree that the measurement of the e+e- annihilation cross section must be improved, in particular in the region below the p w resonances. Such efforts are underway at several laboratories. Particularly interesting and promising is the new radiative-return measurements 11J2: e+e-

+ y + hadrons.

(5)

A new theoretical development is an attempt to calculate the hadronic vacuum-polarization effect on muon g - 2 by lattice QCD13. The contribution of hadronic light-by-light scattering to a, is harder to evaluate reliably because it cannot utilize any experimental information and must rely solely on theory. After correction of a sign error, it seems to have settled down to around 14315316317118919

a,(had.ZZ)

-

80 (40) x

(6)

60 More recently, however, a considerably different value was reported a,(had.ZZ)

-

20:

136 (25) x

(7)

In view of the fact that this moves the prediction of the Standard Model closer to the experiment, it is important that it is checked by an inde pendent calculation. A first principle calculation in lattice QCD would be particularly welcome. The weak interaction contribution is known to the 2-loop order 21i22

a,(weak) = 152 (1) x a,(weak) = 154 (1) (2) x

(8)

where the first error on the second line is from theory and the second error is from Higgs mass uncertainty. The numerical difference between these two values is insignificant for comparison with experiment. Nevertheless it is desirable to have it resolved in one way or another. The best value of a,(QED) quoted previously was 23

uJP’~)(QED) = 116 584 705.7 (1.8) x

(9)

Terms of order a, a2 and a3 are known exactly The a4 term contributes 3.3 ppm, which is larger than the weak term (8) and the experimental error (2). Thus it must be known accurately and reliably for a meaningful test of the Standard Model. Unfortunately, the QED term a,(QED) was mostly evaluated by only one group and in only one way until recently. This is not a desirable situation, in particular, in view of the recent discovery of program error in eighteen a* diagrams 28. An independent evaluation of remaining a4 terms is urgently needed to assure the validity of this complicated calculation. Another concern about a,(QED) is of computational nature: The uncertainty given in (9) ( which is about (- 0.016 ppm)) was estimated by the integration routine VEGAS 29 assuming that they are purely statistical. This assumption is not exactly valid since any numerical work dealing with a finite number of digits suffers to varying degrees from non-statistical errors caused by rounding off of digits. Indeed we have found that some of our integrals suffer from sizable non-statistical errors. Thus this digitdeficiency (or d-d) error must be sorted out and controlled. Of course all this painstaking work may eventually become unnecessary when integration is carried out completely analytically. (A promising development in the analytic integration of multi-loop Feynman integrals was reported recently

-

24125126,27.

61 30.) For some years to come, however, the numerical integration method will be the only practical approach available.

In this talk I report the latest work on a,(QED) and ae(QED): (a) All a4 terms contributing to a, - a, have been verified by at least two independent formulations. (b) The d-d problem has been reduced to a manageable level. (c) The uncertainty of a,(QED) in the a5 term is being examined. (d) New (tentative) values of a, (electron g - 2) and cr(a,) are obtained. The term a,(QED) can be written in the general form:

ap(QED) = A1 + A2(m,/me) +Az(m,/m,) where

+ A3(m,/me, mp/m,),

(10)

A1 is calculated to order a4. (See Sec. 4 for the values of A?), A?), A?), and A?).) A?), A?), A f ) have been evaluated by numerical integration, analytic integration, asymptotic expansion in mp/me, or power series expansion in m,/m,. Thus they are known “exactly” 24825926927:

Ap)(mp/m,)= 1.094 258 282 7(104), A?’(mp/mr) = 7.8059 (25) x AP)(mp/me)= 22.868 379 36(22), A r ) ( m , / m , ) = 36.054(21) x low5, Ar)(mp/meymp/m,) = 0.527 63 (17) x lov3.

(12) The errors come only from measurement uncertainties of a,m, and m,. On the other hand, the analytical values of A?), A?), and A?’) are not yet known. Numerical approach has been the only available means to evaluate these terms. The best reported value of A?) was 31932:

Ar)(mp/me)= 127.50(41).

(13)

A$ was given a crude numerical estimate 31: Ar)(mp/me,mp/m,)= 0.079(3). (14) A P ) was also given a very crude estimate based on the renormalization group argument and a lot of guesswork 31,33: A$l”(m,/m,) = 930 (170).

(15)

62

Thus the theoretical uncertainty in a, (QED) comes primarily from AP) and A t o ) . Clearly there is still a lot of room for improvement in the context of numerical approach. Let us first discuss the improvement of A,(8 1.

2. Evaluation of AF)(mr/m,)

Ar)(m,/m,) consists of 469 Feynman diagrams. For the sake of analysis it is convenient to classify them in four (gaugeinvariant) sets. Group I. Second-order muon vertex whose virtual photon line has insertion of lepton vacuum-polarization (v-p)loops. This group consists of 49 Feynman diagrams. It is further subdivided into gauge invariant subsets I(a), W), I(c), I(d). Group 11. Fourth-order muon vertices with lepton v-p loops. This group contains 90 Feynman diagrams. Group 111. Sixth-order muon vertices with electron v-p loop of 2nd order. This group consists of 150 diagrams. Group IV.Muon vertex containing light-by-light scattering subdiagram with further radiative corrections. This group has 180 diagrams. It is further classified into gauge invariant subsets IV(a), IV(b), IV(c), IV(d).

Figure 1. Typical diagrams of Groups I(a) and I(b). In this and subsequent figures fermions propagate in the external magnetic field. I(a) has 7 diagrams. I(b) has 18 diagrams.

Groups I and I1 had been evaluated by more than one method, numerical, analytic, or others 32334935.The best results obtained are u y ) = 16.720 359 (20),

(16)

u g ) = -16.674 591 (68).

(17)

63

Figure 2.

Typical diagrams of Group I(c), which consists of 9 diagrams.

P6a

Figure 3.

P6b

P6c

P6d

Diagrams of Group I(d). It has 15 diagrams.

Figure 4. Typical diagrams of Group 11, which has 90 diagrams.

Groups I11 and N ( a ) had been evaluated only numerically. But their codes are fully tested since they are obtained from the sixth-order codes

64

Figure 5.

Some of 150 diagrams of Group 111.

Figure 6. Some of 54 diagrams of Group IV(a).

which give results identical with those of the analytic approach. We find 35 111 -

10.793 43 (414),

(18)

= 116.759 18 (30),

M,$ilp2 =

2.697 44 (15),

=

4.328 89 (30),

(19)

where the first and second members of the superscripts such as (e,e) refer to 1-1 and w p loops and e and p indicate that the loop is made of electron and muon, respectively. The remaining diagrams of Groups IV consist of three subgroups: IV(b) (LLA, LLB, LLC, LLD: 60 diagrams), IV(c) (LLE, LLF, LLG, LLH, LLI: 48 diagrams), and IV(d) (LLJ, LLK, LLL: 18 diagrams). Group IV integrands consist of several thousand terms of complicated rational functions. Because of their size, our initial effort was focused on making FORTRAN code as small and efficient as possible. This was achieved with the help of the Ward-Takahashi identity: Q P A Y P ,a) = -c(P

a + C ( p - i), + 5)

where AP is the sum of vertex diagrams generated from the self-energylike diagram C by insertion of an external magnetic field in all muon and electron propagators in C. Differentiating both sides of (20) with respect to qu we obtain

65

LLA

LLB

LLC

LLD

LLI

LLJ

LLK

LLL

Figure 7. Qpical diagrams of Group IV(b), W ( c ) ,rV(d).

The magnetic projection of the right hand side (RHS) is more complex than that of the left hand side (LHS) but gives smaller code. The a3term was evaluated using both RHS (Version A ) and LHS (Version B) 36. Since a4 term is so huge only Version A was used initially. The momentum integration was carried out analytically, initially by SCHOONSCHIF' 37, originally written by Veltman, and more recently by its successor FORM 38. This leads to exact integrals of up to 11 Feynman parameters, all generated from a small number of templates. Their renormalization terms are related exactly to the 6th- and lower-order integrals, which are known analytically. This enabled us to thoroughly cross check all diagrams of IV(b) and IV(c). For 18 diagrams of group IV(d), however, UV terms could not be related to known lower-order diagrams, making the cross-checking less effective. Initially IV(d) was evaluated in Version A only. Not satisfied by the insufficient verification of its codes we decided to check IV(d) by formulating it also in Version B. Comparison of the two versions revealed that the template for IV(d) needed additional terms not present in IV(b) and IV(c). Let us now discuss briefly the d-d problem encountered in the numerical integration. Our renormalization is a point-wise procedure, which requires in particular subtraction of 00 from 00 at singular points. This procedure is analytically well-defined but numerically dangerous, and is the major cause

66

of our d-d problem. Numerical integration has been carried out by an adaptive-iterative Monte-Carlo integration routine VEGAS, written by Lepage 29. The FORTRAN codes of Group IV are very large and require enormous amount of computing time. Furthermore, the complication caused by the dd problem forced us frequently to go to quadruple precision, which slowed down the computation by w 20, making it difficult to accumulate large statistics. Before 1990 it was simply not practical, if not impossible, to evaluate them with large enough statistics. The discovery of an error in IV(d) prompted us to check IV(b) and IV(c) by evaluating them also in Version B. This is no longer a problem because of vastly increased computing power available now. Integrands of two versions look very different, even the numbers of integration variables are different. But numerical evaluation of two versions give identical results within the uncertainties generated by VEGAS. Having carried out an exhaustive check, we are sure that all diagrams of Group IV and hence all terms contributing to AF)(m,/me)are now free from analytic and numerical error. The new results of numerical integration are 28135

/p) Iv(a)- -0.417

04 (375),

= 2.907 22 (444), aIv(c) (8) a?$(,) = -4.432 43 (58).

(22)

Comments: The values of some individual integrals are larger than 100. In each gauge-invariant set IV(b) or IV(c), however, the integrals nearly cancel out when combined. Meanwhile their errors add up in quadrature. Thus small errors in individual terms can lead to a large relative error for the sum over a gauge-invariant set. The d-d effect is thus amplified strongly in gauge-invariant sets. The new total value is 35

Ay)(mp/me)= 132.682 3 (72).

(23)

The old value was31,32

The difference comes mostly from IV(b) and IV(c). While the size of A,(8)(m,/me) is determined by a ~ v ( ~ its) error , is dominated by a ~ v ( a ) ,

a ~ v ( , )This . is why the change in A f ) ( m , / m , ) was less than 3 % in spite of the analytical and numerical problems with IV(b), IV(c), and IV(d).

67

A?) has also been reevaluated

35:

A?)(mp/me,mp/mT)= 0.0376(1).

(25) Its difference with the old value 0.079(3) 31 comes mostly from the diagrams not included in the old estimate. The relative contributions of various QED terms are shown in Table 1. Table 1. Relative contributions of the QED terms to the muon g -2. Term

Contribution in ppm

Error in p p m

994623.88 ppm 5063.86 ppm 0.36 ppm

0.0073 ppm 0.48 x

ppm

0.12x 1 0 - ~ ppm

245.82 ppm

0.24 x

ppm

0.0039 ppm

0.23 x

ppm

0.0057 ppm

0.19 x 1 0 - ~ ppm

3.31 ppm 0.94 x

0.18 x

ppm

ppm

0.21 x 1 0 - ~ ppm

0.054 ppm

0.0099 ppm

Collecting all results of orders a4 and a5 we find

a,(QED) = 116 584 719.4 (1.5) x

(26) In conclusion we found that the improvement of the a4 term did not significantly affect the comparison of theory and experiment of a,. The net effect of our calculation is to increase the value of the QED prediction by 13.7 x and eliminate an important source of uncertainty in a,,. It is seen from Table 1 that, as far as QED is concerned, the a5 term is now the most important source of uncertainty in Q ~ The . a5 term will be examined in the next section. The overall theoretical uncertainty of the Standard Model remains dominated by that of the hadronic vacuum-polarization effect and the hadronic light-by-light scattering effect.

3. Improving ~ F ’ ) ( m , / m , ) Previously, AFo)(m,/me)was estimated to be 930 (170), which contributes only 0.054 ppm to a,, well within the current experimental uncertainty. But

68

it will become a significant source of error in the future when the accuracy improves in the next generation of a, measurements. Better estimates of AY'(m,/m,) will then be needed. At this point, however, it is more out of curiosity than necessity that I began to look into this problem. The first step is to find the number of Feynman diagrams contributing to AFo)(m,/m,). It turns out to be 9080, a very discouraging number indeed ! Nevertheless, let us go ahead and classify them into several gaugeinvariant sets. The result is shown graphically by Figs. 8, 9, 10, 11, 12, and 13.

10)

Figure 8. Some diagrams of Set I. It is built from a second-order vertex. 498 diagrams contribute to A ~ o ) ( m p / m e ) .

Fortunately, it is not difficult to identify the diagrams that may give large contributions. They are characterized by some of the following criteria: (a) Diagrams containing a light-by-light scattering (l-Z-scattering) subdiagram in which one of the photon lines represents the external magnetic field,

69

Figure 9. Some diagrams of Set 11. It is built from fourth-order proper vertices. 1176 diagrams contribute to AFO)(mp/me).

Figure 10. Some diagrams of Set 111. It is built from sixth-order proper vertices. 1740 diagrams contribute to AgO)(mp/me).

Figure 11. Some diagrams of Set IV.It is built from eighth-order proper vertices. 2072 diagrams contribute to AFo)(m,/me).

(b) Diagrams containing a vacuum-polarization (v-p) subdiagram, (c) Diagrams containing several v-p subdiagrams. Diagrams of types (a) and (b) are both sources of ln(mp/me). The presence of ln(mp/me) of type (a) in the diagrams containing a light-by-light scattering subdiagram was initially discovered by numerical calculation of the sixth-order muon g - 2 39. What makes this term really large, however, is the presence of a large coefficient r2. This was given a nice physical interpretation by Elkhovskii 40.

70

Figure 12. Some diagrams of Set V. It consists of 10th-order proper vertices with no closed lepton loop. There are 6354 diagrams in this set. But none contributes t o Aro)(m,/me).

Figure 13. Some diagrams of Set VI. Each one of W(a) - VI(k) represents a gaugeinvariant subset that consists of diagrams containing various light-by-light scnttering subdiagrams. 3594 contribute to A!jlo)(m,/ m e ) .

The logarithm of type (b) is a consequence of charge renormalization. The structure resulting from the charge renormalization procedure gives rise to the renormalization group equation, which enables us to determine

71

several coefficients of descending powers of ln(m, / m e ) (sometimes even down to the constant term) by a purely algebraic manipulation in terms of known constants of lower-order diagrams It has been applied to obtain some leading terms of the a5 term, too 43. An easy but very crude way to estimate the effect of v - p insertion is to examine the structure of the renormalized photon propagator: 24,41342.

D r ( q ) = -i---[~ gpv + -(a! 1 ln(q2/m,) 2 -5 q2 lr3 9

+ ---)I.

It is seen from this that the v - p insertion is roughly equivalent to multiplying a, with a factor (a!/x)KVwhere

Here q is a fudge factor of order 1. KV N 3 for q = 1. This enables us to make crude order-of-magnitude estimates of individual integrals. Applying it to the sum over a gauge-invariant set requires some caution, however: Since member diagrams of the set tend to have strong cancelation, simplistic application of (28) can lead to a value badly off the mark unless individual integrals are known very precisely. The case.(c) is actually a part of (b) but mentioned separately to emphasize that insertion of v-p loops in various photon lines tends to contribute with the same sign and thus the size of contribution increases roughly in proportion to the number of such insertions. Based on these criteria one may conclude that the most important diagrams are those of Set VI(a) [252 diagrams] of Fig. 13 which contain a light-by-light scattering subdiagram and two vacuum-polarization subdiagrams. A somewhat smaller but still significant contribution may come from the Set VI(b) [162 diagrams] of Fig. 13. We have thus far evaluated the contributions of several subsets of the set VI, including VI(a) and VI(b) 44: Az[Vl(a)] = 629.141 (12), Az[Vl(b)] = 181.129 ( 5), Az[Vl(c)] = -36.057 (321), Az[Vl(e)] = -4.261 (214), Az[Vl(f)] = -38.335 (281), Az[Vl(i)] = -27.337 (115).

(29)

72

Note that the contributions from the subsets VI(c), VI(f), and VI(i) are sizable and negative so that they reduce considerably the positive contributions of VI(a) and VI(b). Other sets computed thus far are 44:

A2[I(a)]= 22.566 973 (3)*, Az[I(b)]= 30.667 091 (3)*, A~[I(c= ) ] 5.141 395 (1)*, A2[I(d)]= 8.8921 (ll), Az[I(e)]= -1.219 20 (71), Az[II(a)]= -70.4716 (105)*, Az[II(b)]= -34.7718 ( 29)*, A z [ I I ( f ) ]= -77.5224 (414). (31) Parts of data with * agree with the analytic results 45. A part of I(d) was also evaluated using an exact spectral function 46. Parts of I(c), I(d), and I(e) are in approximate agreement with the leading terms obtained by the renormalization group method 43. The rest of the results will be useful in fixing the unknown constants in the renormalization group analysis. All values in (29), (30), and (31) have been obtained by FORTRAN codes that can also be used to evaluate corresponding a4 integrals by a trivial change of parameters. Since a4 codes had been fully verified, these values may be regarded as firmly established. Nevertheless we still regard them as preliminary since we want to carry out few more checks. The (partial) sum of terms evaluated thus far is 44 A2Cpart.~um]= 587.55 (50). (32) It is plausible that the largest remaining contribution comes from the diagrams of the set VI(k) [120 diagrams], which has no lower-order analogue. This was crudely estimated to be 185 (85) 33 using the method developed in 40. Another non-negligible contribution might arise from VI(j). This set has a ln(mp/m,) term coming from one of the light-by-light subdiagram according to the criterion (a), while the second light-by-light subdiagram does not generate a logarithmic term since it is not attached to any external photon line. Short of direct numerical calculation, however, it is difficult to estimate its size or sign. It was given only its likely error range 0 f 40 33. To reduce the uncertainty coming from some other diagrams, we will evaluate the contributions of the sets I(f), I(g), I(h), III(a), III(b), IV in

73

the near future. According to the criteria (a), (b), and (c) these terms will not be large and their uncertainties will be smaller than those of VI(j) and VI(k). Further reduction of uncertainties by explicit calculation of these terms is crucial for obtaining a good and reliable estimate of A Y ) ( m , / m , ) . 4. Electron g

-2

The term ae(QED) can be written in the general form:

ae(QED) = A1 + A2(me/mp)+ A2(me/m,)

+ A3(me/mp,me/m,), (33)

where

The first four coefficients of A1 are the following:

A?) = 0.5, A?) = -0.328 478 965 A?) = 1.181 241 456 A?) = -1.726 0 (50).

. . ., .. ., (35)

A?) and A?) are known analytically. A y ) is known by both numerical integration 47 and analytical calculation 48. A?) is a newly revised (still tentative) value. A2,A3 and weak and hadronic contributions to a, are very small and known with sufficient accuracy for comparison with experiment. The correction of an error in Group IV(d) caused sizable shifts in a, and also in ~ ( a ,which ) is determined from the theory and experiment of the electron g - 2 2 8 . But the largest uncertainty in a, comes from 518 diagrams of Group V: diagrams which have no closed fermion loop. The internal consistency of codes for these diagrams was checked thoroughly in Version A. Unfortunately, it has not been checked by Version B or other means thus far since it requires an enormous amount of additional work. I should like to emphasize, however, that the complete verification of Group IV reinforces our confidencein Group V, which has actually gone through a more extensive check than Group IV.The numerical work on A?) is almost finished. The latest value is shown in (35). Note that the new uncertainty is 7 times smaller than the old one. As a byproduct of the calculation of Apo)(m,/me)in progress, many terms from the sets I, 11, and VI that contribute to the mass-independent

74

term A?') are being evaluated 44. All terms evaluated thus far are relatively small in size. However, work on more sets is needed t o obtain a better assessment of the size of the a5 term of a,. 5 . New values of a

For years the biggest obstacle in testing QED using a, was the unavailability of a with high enough precision. Recently the situation has been improved significantly by the atom interferometry m e a s ~ r e r n e n t : ~ ~

a - ' ( h / M c S )= 137.036 000 3 (10)

(7.4 p p b ) .

(36)

This leads to

ae(h/Mcs) = 1 159 652 175.7 (8.5)(0.2) x

(37)

where the first uncertainty comes from a of (36) and the second uncertainty is that of QED.This leads to ae(exp)- U e ( h / M c s ) = 12.6 (9.5) x

(38)

assuming that A(1o) has a value within the range (-3, 3). An alternative (and more sensible) test of QED is to calculate a from the theory and measurement of a,. This leads to the (still tentative) value of a(ae):

&-'(a,) = 137.035 998 84 (1.8) (2.4) (50) = 137.035 998 84 (50)

(3.7 ppb).

(39)

where the uncertainties on the first line are from the a4 and a5 terms and the experiment 5 0 , respectively. Note that the uncertainty in the a4term is smaller than the guesstimated uncertainty of the a5term. Until a reliable (even if crude) estimate of the a5 term is obtained, further reduction of uncertainty in A?) cannot improve theory significantly. This is why the a5 term must be examined. A(,'') has contributions from 12672 Feynman diagrams, in which the Set 5 (6354 diagrams) is the most difficult to evaluate. For comparison most precise values of 01-l available at present are shown in Fig. 14 51,52353,5435: If the uncertainty of a ( h / M c s )shrinks to 3.1 ppb, which Wicht et al. are trying to achieve 49, it will become more precise than the current best a(a,). Then we would have U e ( h / M c s )= 1

159 652 175.2 (3.6)(0.2) x

(40)

75

( a-'-137.036)x

lo7

I

CODATA 1998 l*l

I

I........................

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

ac Josephgon I

Cesium de Broglie

I-..lt.-l I

Neutron de Broilie

I............... .............. {

I

Electron 9-2 Muonium h.f.s.

1.1

I....................................

{-

I

I

-100

-200

0

Figure 14. Comparison of various

100

0-l.

and

a-'(h/Mc8) - Q-'(u,) = 150 (66) x lo-', or, about 2.3 s. d. discrepancy in two a's. Meanwhile, a new measurement of a, is making a good progress 5 6 . It is expected that it leads to a(a,) with the precision of 0.4 ppb or better, bringing the test of QED (and SM) to a higher level of rigor. If the discrepancy such as (41) persists even with the new measurement, it might indicate either an unexpectedly large asterm (about - 2 O O ( a / 7 ~ ) ~ ) or a possible breakdown of the Standard Model that cannot be attributed to short distance physics. This would be really exciting. Could it possibly be the first hint that QED is not entirely seamless after all ? Acknowledgment

I thank M. Nio for her assistance in preparation of the paper.

76 References G. W. Bennett et al., arXiv:hep-ex/0401008. G. W. Bennett et al., Phys. Rev. Lett. 89,101804 (2002). H. N. Brown et al., Phys. Rev. Lett. 86,2227 (2001). H. N. Brown et al., Phys. Rev. D 62,091101 (2000). J. Bailey et al., Phys. Lett. 68B,191 (1977); F. J. M. Farley and E. Picasso, in Quantum Electrodynamics, edited by T. Kinoshita (World Scientific, Singapore, 1990), pp. 479 - 559. 6. M. Davier, S. Eidelman, A. Hocker, and Z. Zhang, Euro. Phys. J. C 31, 503 (2003) [arXiv:hep-ph/0308213]. 7. S. Ghozzi and F. Jegerlehner, Phys. Lett. B585,222 (2004). 8. K. Hagiwara et al., arXiv:hep-~h/0312250. 9. V. V. Ezhela, S. B. Lugovsky, and 0. V. Zenin, arXiv:hep-ph/0312114. 10. B. Krause, Phys. Lett. B390,392 (1997). 11. A. Aloisio et al., KLOE Collaboration, arXiv:hep-ex/0307051. 12. M. Davier, talk given at the Cape Cod symposium, May 2003, http://g2pcl.bu.edu/ieptonmom. 13. T. Blum, Phys. Rev. Lett. 91, 052001 (2003); T. Blum, arXiv:heplat /0310064. 14. M. Knecht and A. NyReler, Phys. Rev. D 65,073034 (2002). 15. M. Knecht and A. NyfFeler, M. Perrottet, and E. de Rafael, Phys. Rev. Lett. 88, 071802 (2002). 16. M. Hayakawa and T. Kinoshita, Phys. Rev. D 66,019902 (2002) [arXiv:hepph/0112102]. 17. J. Bijnens, E. Pallante, and J. Prades, Nucl. Phys. B626,410 (2002). 18. I. Blockland, A. Czarnecki, and K. Melnikov, Phys. Rev. Lett. 88, 071803 (2002). 19. M. J. Ramsey-Musolf and M. B. Wise, Phys. Rev. Lett. 89, 041601 (2002). 20. K. Melnikov and A. Vainshtein, arXiv:hep-ph/0312226. 21. M. Knecht, S. Peris, M. Perrottet,, and E. de Rafael, JHEP 11,003 (2002) [arXiv:hep-ph/0205102]. 22. A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D 67,073006 (2003) [arXiv:hep-ph/0212229]. 23. V. W. Hughes, and T. Kinoshita, Rev. Mod. Phys. 71,S133 (1999). 24. T. Kinoshita, Nuovo Cim. B 51,140 (1967). 25. S. Laporta, Nuovo Cim. B 106,675 (1993). 26. S. Laporta and E. Remiddi, Phys. Lett. B 301,440 (1993). 27. S. Czarnecki and M. Skrzypek, Phys. Lett. B 449,354 (1999). 28. T. Kinoshita and M. Nio, Phys. Rev. Lett. 90,021803 (2003). 29. G. P. Lepage, J. Comput. Phys. 27, 192 (1978). 30. V. A. Smirnov and M. Steinhauser, Nucl. Phys. B672,199 (2003). 31. T. Kinoshita and W. J. Marciano, in Quantum Electrodynamics, edited by T. Kinoshita (World Scientific, Singapore, 1990), pp. 419 - 478. 32. P. A. Baikov and D. J. Broadhurst, in New Computing Techniques in Physics Research IV. International Workshop on Software Engineering and Artificial

1. 2. 3. 4. 5.

77 Intelligence for High Energy and Nuclear Physics, edited by B. Denby and D. Perret-Gallix (World Scientific, Singapore, 1995), pp. 167-172; arXiv:hepph/9504398. 33. S. Karshenboim, Yad. Phys. 56, 252 (1993) [Phys. At. Nucl. 56, 857 (1993)]. 34. S. Laporta, Phys. Lett. B 312, 495 (1993) . 35. T. Kinoshita and M. Nio, arXiv:hep-ph/0402206. 36. P. Cvitanovic and T. Kinoshita, Phys. Rev. D 10, 4007 (1974). 37. H. Strubbe, Compt. Phys. Commun. 8, 1 (1974); 18, 1 (1979). 38. J. A. M. Vermaseren, FORM ver. 2.3 (1998). 39. J. Aldins, S. Brodsky, A. Dufner, and T. Kinoshita, Phys. Rev. Lett. 23, 441 (1969); Phys. Rev. D 1,2378 (1970). 40. A. S. Elkhovskii, Yad. Fiz. 49, 1056 (1989)[Sov. J. Nucl. Phys. 49, 654 (1989)]. 41. B. Lautrup and E. de Rafael, NucI. Phys. B 70, 317 (1974). 42. T. Kinoshita, H. Kawai, and Y. Okamoto, Phys. Lett. B 254, 235 (1991); H. Kawai, T. Kinoshita, and Y. Okamoto, Phys. Lett. B 260, 193 (1991). 43. A. L. Kataev, Phys. Lett. B 284, 401 (1992). 44. T. Kinoshita and M. Nio, paper on the tenth-order QED contribution to a,,, in preparation. 45. S. Laporta, Phys. Lett. B 328, 522 (1994). 46. T. Kinoshita and M. Nio, Phys. Rev. Lett. 82, 3240 (1999); Phys. Rev. D 60, 053008 (1999). 47. T. Kinoshita, Phys. Rev. Lett. 75, 4728 (1995). 48. S. Laporta and E. Remiddi, Phys. Lett. B 379, 283 (1996). 49. A. Wicht et 01. in Proc. of 6th Symp. on Req. Standards and Metrology (World Scientific, Singapore, 2002), pp. 193 - 212. 50. R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. Lett. 59, 26 (1987). 51. A. Jeffery et al., IEEE Trans. Instrum. Meas. 46, 264 (1997); Metrologia 35, 83 (1998). 52. P. Mohr and B. Taylor, Rev. Mod. Phys. 72, 351 (2000). 53. E. R. Williams et al., IEEE !iTans. Instrum. Meas. 38, 233 (1989). 54. E. Kriiger, W. Nistler, and W. Weirauch, Metrologia 36, 147 (1999). 55. W. Liu, Phys. Rev. Lett. 82, 711 (1999). 56. G. Gabrielse, Cape Cod Symposium, May 2003; G. Gabrielse, Cornell physics colloquium, Dec. 1, 2003.

VERNON HUGHES AND THE QUEST FOR THE PROTON’S SPIN

ROBERT L. JAFFE Center for Theoretical Physics Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, M A 02139 E-mail: jaffe @mit.edu Vernon Hughes dedicated much of the latter part of his career to the question “What carries the spin of the proton?” The question remains unanswered and near the top of the list of fascinating questions in QCD. I present a perspective on the question and Vernon’s pursuit of an answer.

1. Introduction

We all know that the spin of the proton is $ti. The question is: How do the contributions of quarks and gluons add up. Vernon Hughes loved this subject. There is a famous old photo which alleges to show a group of physicists discussing spin. [Figure 1.1 I believe it captures the intensity and excitement that Vernon brought to his work on spin in QCD. Vernon was a tough cookie. He pursued terrific goals with a singlemindedness that often drove his friends and collaborators to distraction. He was, however, remarkably patient with theorists, especially young ones and ones who shared his love of spin. Falling in both those categories back when spin in QCD first attracted his attention, I was lucky to have shared with Vernon twenty five years of interest in the spin of the proton. Among the high points were workshops here at Yale in the 1970’s and 1980’s when QCD spin physics was out of favor and Vernon was on a mission to stimulate interest among theorists and experimenters. Vernon managed to badger us into recognizing the importance of spin in deep inelastic phenomena, and stimulated much good work in theory as well as experiment. Looking back at his career, it is clear that Vernon had a taste for an elegant experiment that could decide a complex issue by measuring one or two numbers. His pursuit of muonium and the muon’s anomalous magnetic 78

79

Figure 1. “Two high energy physicists discussing the spin of the proton.”

moment are cases in point. In the spin substructure of the nucleon he identified a similar problem in QCD, where they are hard to come by. Once he settled on measuring the spin sum rules, he pursued the goal with characteristic intensity. The pursuit took him from SLAC to CERN by way of Fermilab, and lasted from the early 1970’s until the end of his life. He achieved his goal, but to his and everyone else’s surprise, the measurement of the nucleon’s spin sum rules raised new, pressing questions that have given birth to a new generation of QCD spin physics experiments, led by several of Vernon’s younger students and colleagues. Abhay Deshpande has described them here. Vernon’s achievements in QCD spin physics are important and easy to summarize: 0

0

0

Vernon realized that the Bjorken Sum Rule1 is fundamental to QCD. He realized that the nucleon spin sum rules2 could provide finely tuned information about how nucleons are put together. He developed and led an experimental program spanning 25 years

80

0

that culminated in precise measurements of both. He inspired and prodded theorists to respond to these experiments, especially to define the components of the nucleon spin.

A measure of his impact can be seen in the ‘&stateof the art” summary of measurements of g:p shown in Fig. 2.

10 3

--

X d . 0 2 5 ( x 512)

10

--

10

X.O.125 ( x 32)

t

Figure 2.

A recent summary of measurements of gyp as a function of

I

2

and

Q2

In this talk I will give a theorist’s perspective on the problem of the nucleon’s spin in QCD and Vernon’s contributions to it. 2. What is the issue and why should we care?

QCD is the theory of matter. More than 99% of the visible mass in the Universe is made of quarks and gluons. There is of course great current excitement about dark matter and dark energy. While it is essential to find

81

out what they are, it is almost certain that all of what actually happens in the Universe - change, structure, complexity, etc. - plays out in terms of particles made of quarks, gluons and a dash of electrons. QCD is also the only non-trivial theory that we are certain describes Nature. The electroweak sector of the Standard Model is mere perturbation theory. The world beyond the Standard Model is fascinating but the relevance of any particular suite of ideas to Nature is speculative at best. Furthermore QCD incorporates all the features one expects to encounter in future deeper, more unified theories: the interactions follow from local symmetries alone; there are no free parameters (at least in the light quark sector); mass emerges dynamically; although simple at the fundamental level, the theory is capable of generating rich, non-perturbative structure including dynamical breaking of chiral and conformal symmetries; the ground state is a mystery to us; finally, there are fascinating regularities (eg. vector dominance, the constituent quark model, diquark and instanton dynamics) that do not follow trivially from the underlying Lagrangian. String theorists do well to study QCD to see what sort of problems are in store for them when they finally figure out what string theory is! QCD is hard for the same reasons it is elegant. For light quarks there are no parameters. The coupling runs with distance scale. It is small at short distances, so we can probe hadrons in deep inelastic processes. However hadrons form at precisely the scale where the coupling is of order unity. None of the clever attacks on light-quark QCD - chiral dynamics, large N,, instanton models, QCD sum rules, constituent quark and bag models, etc. - provides a progressively improvable approach to the structure and properties of the nucleon. Each has added insight: for example, we have a qualitative understanding of the nucleon’s magnetic moment from quark models, and of the NAT system from combinations of N , 4 00 and chiral dynamics. Lattice QCD, although capable in principle of answering many important questions about the nucleon, has proved unable to shed light on important and well-posed questions like “What is the gluon contribution to the proton spin?” or “What is the quark orbital angular momentum in the proton?” Certainly no one was able to anticipate the almost complete violation of naive expectations observed when Vernon and his colleagues first measured the nucleon spin sum rule.

82 3. Why the nucleon’s spin?

What attracted Vernon to the nucleon spin problem seems t o have been first the possibility of testing QCD precisely by checking Bjorken’s Sum Rule, and second, the possibility of making a precise measurement of the quark contribution to the nucleon’s spin by measuring the separate proton and neutron spin sum rules. Bjorken’s Sum Rule dates from the Dark Ages of QCD, when quarks were not yet reputable, and field theory was still in eclipse. In his enormously influential 1966 paper on quarks at short distances Bjorken wrote down his famous sum rule (converted to modern notation),

where gI”(x,Q2)and g:”(x, Q 2 ) are the proton’s and neutron’s polarized deep inelastic structure functions. At the time Bjorken was most interested in the fact that the right hand side was independent of Q2 - evidence for scaling. The possibility of measuring g:psenwas so remote that Bj followed his result with a now famous quote that set the stage for Vernon’s work on this subject: “Something may be salvaged from this worthless equation by constructing an inequality. . . ” It is worth examining the theoretical basis of the sum rule. Bjorken assume that ‘the commutators of electromagnetic current operators behaved as in free field theory,

where the operators O(Z) are linear combinations of the unrenormalized currents themselves. At the time this was heresy: Chewian “nuclear democracy” was dominant and hadrons were supposed to be composites only of one another4. Now we know that asymptotic freedom in QCD validates Bjorken’s assumption up to calculable logarithmic corrections. The sum rule follows from broad general properties of QCD plus isospin invariance alone. It relates physics at totally different distance scales: QA is the nucleon’s axial charge measured in P-decay, effectively at zero momentum Q 2 )are measured at asymptotically high momentum transfer, and gIPSen(z, transfer. Such sum rules should give pause to advocates of the modern “effective” approach to field theory. While QCD can be formulated in terms of

83 effective operators defined a t a hadronic scale, you won't discover relations like eq. (1) that way. Bjorken's Sum Rule continues to occupy a special place in QCD. It is now possible to compute corrections t o the sum rule in a threefold series: a) perturbative QCD corrections - a power series in a s ( Q 2 ) 5 1b) target mass corrections - a power series in M 2 / Q 2 , and c) higher twist corrections - a power series in ( F ) / Q 2where , (F)are the matrix elements of more complex operators measuring quark and quark-gluon correlations in the nucleon6. Years of work by theorists is summarized by the 2004 version of the sum rule:

1'

as - -2 43a2 1- (xlQ2)= -6l g gvA 7r 127r2

{

dx gyp-en

+

"1

1

Q2

{9

dxx2 2gyp-en(x, Q 2 )

+ 6g2ep-en(xl

Q2)}

- --Fu-d(Q2) 1 4

(3)

Q 2 27

Sum rules are interesting if both the left and the right hand sides are directly related to experiment and if both have important theoretical significance. Bjorken's Sum Rule relates two strikingly different ways of measuring the (isospin weighted) up and down quark contributions to the nucleon's spin. The parton model provides the simplest interpretation of the left hand side:* In the parton picture the polarized structure function gI(x, Q 2 ) measures the helicity weighted momentum distribution of quarks in a nucleon a t infinite momentum. When an electron scatters from a nucleon with four momentum P , transferring four momentum q , the nucleon structure information is encoded in structure functions that depend on the Lorentz invariants Q 2 = -q2 and x = Q 2 / 2 P . q. The contributions to g1 a t large Q 2 are given by gyP(xc1 Q2) =

e i (Aqa(xl Q 2 )

+ &(x1

Q2))

(4)

a

where Aqa(xlQ 2 ) is the helicity weighted probability to find a quark of type a and momentum fraction x in the polarized proton. (Aq is the same for antiquarks.) These probabilities evolve slowly as the probe resolution, *Subtleties can always be resolved by recourse to the operator product expansion and perturbative QCD.

a4

Q 2 , is increased. When integrated over x we obtain

where Aqa(Q2) has the interpretation of the total contribution of the helicity of quarks and antiquarks of flavor a to the helicity of the nucleon at infinite momentum. The proton-neutron difference that enters the sum rule is proportional to the difference of u and d quark helicities. The right hand side of the sum rule arises more formally. The cross section for deep inelastic electron scattering is proportional to the product of two electromagnetic currents acting at points separated by a light-like interval. This product has an expansion similar to (2), and the particular operator, 0 ,singled out by a) letting Q2 -+ 00, b) taking the p n difference, c) looking for nucleon polarization dependence, and d) integrating over IC, is the isospin weighted axial vector current,

A$ = i i y ” 7 5 ~- &”75d

(6)

So the dominant term on the right hand side of (3) is given by the matrix element of this operator - a very fortunate result: not only is A: the operator that mediates the Gamow-Teller contribution to neutron P-decay, but it is also the (isospin weighted) quark spin contribution to the angular momentum operator. This is actually a slightly subtle subject to which I’ll return at the end of the talk. So, if we define,

AQa ( ~ ~ 1=s ”( ~ s l ~ d ‘ 7lp2 5 ~IPS) a

(7) then we can interpret AQa as the contribution of the spin of quarks and antiquarks of flavor a to the spin of the proton. The renormalization scale p 2 appears because the individual quark axial currents are not conserved and therefore not renormalization group invariant. The isovector combination, A U - A D however, is conserved and is proportional to the nucleon’s axial vector charge, AU

- AD =

( N s l i i ~ ” 7-5 ~d ; ( ” ~ ~ , d l N ~ ) / sg”A = 1.2670 f 0.0030 (8)

Identifying the isospin weighted helicity sum in the infinite momentum frame with the isospin weighted spin contribution in the rest frame, we get Bjorken’s sum rule. This derivation invites one to speculate about the possibility of measuring other flavor weighted quark spin contributions - a possibility that ~ > led ~ to the nucleon spin sum rules was considered in the early 1 9 7 0 ’ ~and often referred to as the Ellis-Jaffe Sum Rules2.

85

The total light quark spin contribution to the proton’s spin can be obtained by adding the contributions from the u, d , and s quarks:+ AX(p2) = AU(p2)

+ AD($) + AS(p2)

(9)

There are serious problems with such a generalization of Bjorken’s Sum Rule: first, this combination cannot be extracted from baryon P-decay data, and second, the famous Adler-Bell-Jackiw triangle anomaly ruins the conservation of the associated flavor-singlet axial-vector currentg>lo, rendering A E renormalization scale (and scheme) dependent as indicated by the p dependence in (9) l. The first problem actually spurred the experimental efforts on polarized deep inelastic scattering in the 1980’s and 1990’s. It certainly attracted Vernon’s interest. First consider baryon P-decay. The operators that mediate weak semileptonic decays of baryons are all flavor changing, either d -+ u or s u, and therefore have no SU(3)flavor-singletcomponents. Thus by taking suitable combinations of nucleon and hyperon @-decays,and using SU(3)flaVorsymmetry, one can measure the flavor non-singlet combinations AU - A D and AU A D - 2AS, but there is no sensitivity to AX. In the standard language of SU(3)flavor,12 ---f

+

(AU - AD) (AU

+ AD - 2AS)

+

F D 3 g A = 1.267 f 0.011 3 F - D = 0.585 f 0.025

t 10)

Neutral weak currents which contribute to elastic neutrino scattering or parity violating electron scattering are sensitive to AX13114,but so far no one has succeeded to use those processes to extract AE. Enter polarized deep inelastic scattering: The integrals over g p ( z ,Q2) and gp“(z, Q2) measure the charge-squared weighted sum of quark axia,l charges, which can be re-expressed in terms of the F, D, and AX,

1 18

= -(3F

Jd

co

1 + D + 2AX(Q2)) = (9F - D + 6AS(Q2)) 18

d z g y ( z , 0’) = 1 18

= -(-20

+ 2AX(Q2))=

1

(6F - 4 0 + 6AS(Q2)) (11)

+The small contributions of heavy quarks can be computed using QCD perturbation theory.13

86

Thus polarized deep inelastic scattering measures AE(Q2), the total quark spin contribution to the spin of the nucleon, or equivalently if you prefer, AS(Q2), the fraction of the nucleon’s spin carried by strange quarks and antiquarks. Back in 1973 Ellis and I speculated that the nucleon contained no polarized strange quarks, and estimated the integrals of gyp and gTn by setting A S = 0. This was before asymptotic freedom, which taught us among other things, that A S is Q2-dependent, so if it were to vanish at one Q2, it could not be zero at a higher scale. Still, the assumption gave experimenters like Vernon something to shoot at. When we set A S to zero we obtained AE = 0.60 f0.05. So even in those benighted times it was clear that quarks’ spin did not account for 100% of the nucleon’s spin. Parenthetically this was the same phenomenon responsible for dropping the value of g A from its non-relativistic quark model value of 5/3 down to its experimental value closer to 4/315. As early as 1974 Sehgal suggested that quark orbital angular momentum made up the other 40%16. The other difficulty with the separate proton and neutron spin sum rules is the fact that the integrals are Q2 dependent. This was not recognized until quite late”. In the absence of the triangle anomaly all the axial currents are conserved up to quark mass terms. This is enough to prove that their matrix elements are renormalization group invariant. So originally the spin sum rules were thought to be on the same theoretical foundation as Bjorken’s Sum Rule. In 1978 Kodaira et a117718showed that the anomaly gave rise to a small (two loop) anomalous dimension and an associated weak Q2 dependence for the flavor singlet axial current and therefore for AX. This allows the quark spin fraction measured in the deep inelastic domain to differ both in value and interpretation from the quark spin “measured” in quark models. This subtlety has spawned hundreds of theory papers, and is still controversial. It won’t be settled soon either, because quark models are not sufficiently well defined to allow one to assign renormalization scale or scheme dependence to the numbers extracted from them. Now, however, is not the time or place to follow this thread of QCD spin physics history further.

4. Testing the spin sum rules

Vernon’s long standing interest in spin physics led him to propose an experiment to measure polarized deep inelastic scattering at SLAC. Others at this meeting have described the how the polarized target and beam were developed and how Vernon fought for the physics, so I will be brief. The

87 first experiment, E80, too crude to test any sum rule, was notable because it demonstrated that polarization persisted in the deep inelastic domain as Bjorken had predictedlg. The first data on polarized DIS is shown in Fig. (3), Despite the quality of the data and the fact that only proton targets

0.8

-

0.4

4

e “

0

-0.4

-0.8

Figure 3. Left: first data on g1/Fl for e p scattering. The old scaling variable w = 1/x was still in use. Right: First attempt to compare polarized DIS with theoretical models.

had been studied, Vernon and collaborators quickly attacked the Bjorken Sum Rule and tried to distinguish among theoretical models. In a 1978 paper entitled “Deep Inelastic e-p Asymmetry Measurements and Comparison with the Bjorken Sum Rule and Models of Proton Spin Structure””, they attempted the first comparison with theory. Their results are shown in Fig. (3) where they compared with some of the models of the day. E80 was followed by a much higher precision experiment, E130, which did allow for a meaningful comparison with theoryz1. The results, which were much discussed at the time, are shown in Fig. (4). The curve was the prediction of a model by Carlitz and Kaur22,consistent with both the Bjorken and Ellis-Jaffe Sum Rules. Clearly a meaningful test of the sum rules required still higher precision data, and in the case of Bjorken’s Sum Rule, data from a “neutron”, ie. deuteron or 3He target. Even as El30 was in process, SLAC was redirecting its polarized electron program to measure parity violating deep inelastic scattering and to the now famous test of the Standard Vernon’s attempts to obtain approval for a followup to E80 and El30 were rejected by the SLAC PAC. After an unsuccessful attempt to interest Fermilab in a polarized muon deep inelastic scattering program Vernon and his group migrated to Europe, where they joined the

88 0.7

0.6 0.5

:0.4 h

0,

0.3 0.2

0.I 0

0 I.*.'

0.2

0.4

0.6 X

0.8

1.0 *"*>A.

Fig 4 Figure 4. Data from El30 compared t o the Carlitz-Kaur model, which was consistent with the Bjorken and Ellis-Jaffe Sum Rules.

European Muon Collaboration, renamed the Spin Muon Collaboration, and helped lead its attack on polarized DIS. The result was confirmation of the Bjorken Sum Rule and a clear signal that something other than quark spin carries a majority of the spin of the proton. The SMC final word on the Bjorken Sum Rule was 0.174f0.005f0.010 measured for the left hand side at Q2 = 5 GeV2, and 0.181 f 0.003 for the right hand side computed from &decay, and perturbative QCDZ4. The world's data on the integrand for the Bjorken Sum Rule are shown in Fig. (5). The SMC data on the proton spin sum rule, shown in Fig. (6), came as quite a surprise to uninitiated, who incorrectly expected 100% of the nucleon spin to be on the quarks' spin, and to the initiated, who expected approximately 60%. Instead the number has settled down somewhere around

89

(World's data in 1998)

Figure 5. The world's data on the integrand of the Bjorken Sum Rule.

25-35%,25

AE(Qg) AU(Qi) AD(Qg) AS(Qg)

= 0.28 f 0.16 = 0.82 f 0.05 = -0.44

f 0.05 = -0.10 f 0.05

(12)

at QE = 5 GeV2. Notice that a relatively small polarized strange quark contribution (-0.10 f0.05) corresponds to a shift of 3 x -0.10 = -0.30 in AX and moves the result from the expected value of NN 0.60 to the observed value of M 0.30. The situation was quite chaotic in 1988 and many theorists wrote things about the spin content of the proton that they would perhaps sooner forget today26. The relation between the quark spin and quark axial charge was not generally understood. Some thought that the quark spin plus the gluon spin had to add up to the nucleon spin, forgetting about orbital angular momentum. The operator description of gluon spin, was worked out by Manohar in 199128,generalizing work by Collins and Soper many years before, which had been forgottenz9. During this period Vernon persistently

90

lo-’

s,’

Figure 6. Left: the SMC data on g;p(z) and d d g 7 P ( z ’ ) . Right: The proton spin sum rules measured by different experiments at different Q 2 .

asked “What is the sum rule for the spin of the proton?”, ie. What are the components of the total angular momentum in QCD, and how can they be measured? His persistent questioning stimulated a series of papers in which the components of the angular momentum were defined and the possibility of measuring them was considered. Perhaps the most notable fallout from this was Ji’s formulation of generalized parton distributions and the elucidation of their role in the spin puzzle. Most of these issues have now been settled, and the principal focus of the community has shifted to the measurement of other components of the nucleon spin, especially the contribution of the spin of the gluons. Groups at RHIC (STAR and PHENIX), at CERN (COMPASS), and at HERA (Hermes) are attempting to measure the gluon contribution, AG(Q2). Abhay Deshpande has described some of these efforts in his talk. It suffices to say that we don’t know if AG contributes significantly. Indeed we don’t even know if it is positive - although initial estimates seem to suggest so. Measuring AG is now the highest priority for the field. 5. So what is the sum rule for the spin of the proton? Vernon’s question: “What is the sum rule for the proton’s spin?” is answered by2?, 1 1 -2 = -AE(Q2) 2 WQ2)L , ( Q ~ ) L , ( Q ~ )

+

+

+

which looks rather obvious: the sum of the spin (AX and AG) and orbital ( L , and L,) contributions of quarks and gluons. However, there is more here than meets the eye. In fact there are two different versions of (13), neither entirely satisfactory.

91

In order to make sense of the total angular momentum one must first recognize that it is not a vector, but instead a rank-2 antisymmetric tensor. In fact it is most fruitfully regarded as the integral over space of the time component of a rank-3 angular momentum tensor density, J k = t k i j d3xMoij, where27

s

MP”‘

=

- z”D’)$

z&cl(x”D”

2

+ ;€PuAu4yuy5$

+ 2Tr {Fpa(z’Du - x”D’)A,} + n(FP’A” F ~ ~ A ~ } + terms that do not contribute to J”’ -

(14)

This conserved tensor (dPMP”’ = 0) is a variant$ of the Noether current associated with Lorentz invariance of which rotations are, of course, a subgroup. The four terms shown are the quark and gluon orbital and spin contributions respectively. The quark spin contribution is the simplest: the components of the rank-3 tensor are proportional to components of the axial current - a great simplification. Sum rules for the proton’s spin follow from considering either the time ( p = 0) or light-cone-time ( p = +) component of MPVX. Here I would like to distinguish between a sum rule and an operator relation. A sum rule expresses the expectation value of a local operator in a state as an integral (or sum) over a distribution measured in an inelastic production process involving the same state. This is the traditional definition of a sum rule, dating back to the Thomas, Reiche, Kuhn Sum Rule of atomic spectroscopy. They are especially powerful because the distribution which is integrated has a simple, heuristic interpretation as the momentum (Bjorken-z) distribution of the observable associated with the local operator. Bjorken’s Sum Rule is a classic example as discussed above. Another, less powerful but still interesting type of relation-sometimes called a sum rule in the QCD literature-arises simply because an operator can be written as the sum of two (or more) other operators, 0 = 01 0 2 . If the expectation values of all three operators can be measured, then this relation, and the assumptions underlying it, can be tested. Such a relation exists for the contributions to the nucleon’s angular momentum 27,30,

+

1 2

-

- = L,

1 + -C + J, 2

$It has been constructed from the symmetrized, “Belinfante” stress tensor.

92

where the three terms can be interpreted as the quark orbital angular momentum, the quark spin, and the total angular momentum on the gluons. Ji has shown how, in principal, to measure the various terms in this relation 30. A sum rule of the classic type also exists for the contributions to the nucleon's angular momentum 31,32733,

2 =

1'

{

dx L,@, Q2)

+ -12A c ( z , Q2) + Lg(x,Q2) + AG(x,Q2)

where the four terms are precisely the x-distributions of the quark orbital angular momentum, quark spin, gluon orbital angular momentum, and gluon spin. However it appears that the distributions L q ( x , Q 2 )and Lg(x, Q2) are not experimentally accessible. So the value of the sum rule is obscure. To extract both the relation (15) and the sum rule (16) polarize the nucleon along the 3-direction in its rest frame and set v = 1,X = 2 in order to select rotations about the 3-direction. The matrix element of M0l2 is normalized in terms of the nucleon's momentum (P= ( M ,0, 0,O)) and spin (SP = (O,O, 0, M ) ) 27. First consider the time component:' M'l2,

i 1 M'l2 = - q t ( Z x f i ) 3 q + - q t c 7 3 q + 2 T r E j ( Z ~ i f i ) 3 A j + T r ( E ~ x(17) )3. 2 2 The four terms look like the generators of rotations (about the 3-axis) for quark orbital, quark spin, gluon orbital, and gluon spin angular momentum respectively. Taking the matrix element in a nucleon state at rest, one obtains 1 1 - = L, -C Lg + A G . (18) 2 2

+

+

There are problems, however. There are no parton representations for i g , Lq, or AG, so it is not a sum rule in the classic sense. We know from the nucleon spin sum rules how to write C as an integral of the helicity weighted quark distribution, but AG is not presented as an integral of the helicity weighted gluon distribution. Interactions prevent a clean separation into quark and gluon contributions. And worse still, Lg and AG are not separately gauge invariant, so only the sum j g = Lg AG is physically meaningful. The most important feature of the relation, eq. (18), is the result derived by Ji, that j , = L,+ic and j gcan, in principle, be measured in deeply virtual Compton scattering 30.

+

93 Turning to the +-component sum rule, we find a much simpler form, 1 z8)3q++ - q l y s q + + 2 T r F + j ( Z x l8)Aj + T r ~ + - ~ j F + z A j 2 (19) in A+ = 0 gauge.§ The four terms in M+12 correspond respectively to quark orbital angular momentum, quark spin, gluon orbital angular momentum, and gluon spin, all about the 3-axis. Each is separately gauge invariantq and involves only the “good”, i.e., dynamically independent, degrees of freedom, q+ and Each is a generator of the appropriate symmetry transformation in light-front field theory. The resulting sum rule, M+12

1

= &(ZX

Al.

1 2

- = L,

1 + -C + L, + AG 2

is a classic deep inelastic sum rule. It can be written as an integral over z-distributions as given in (16). Each term is an interaction independent, gauge invariant, integral over a partonic density associated with the appropriate symmetry generator. AX is the same quark spin contribution that we have seen before. AG is the gluon spin helicity distribution that will be measured over the next decade. However the parton distributions of quark and gluon orbital angular momentum have so far eluded us. We do not know any experiment that can access them. So the experimental answer to Vernon’s question still awaits us. We might be lucky and find that AG together with AX M 0.3 saturate the angular momentum sum rule. More likely, however, orbital angular momentum is important and the challenge of measuring it or relating it to other measurable or calculable quantities remains for Vernon’s descendents, both figuratively - the next generation of experimentalists and theorists studying QCD - and literally, since one of the leaders in this endeavor is another Hughes! References 1. J.D. Bjorken, Phys. Rev. 148,1467 (1966). 2. J. Ellis and R.L. Jaffe, Phys. Rev. D 9, 1444 (1974), erratum 10,1669 (1974).

§This gauge condition must be supplemented by the additional condition that the gauge fields vanish fast enough at infinity. VNote however, that in any gauge other than A+ = 0, the operators are nonlocal and appear to be interaction dependent. The same happens to the simple operators involved in the momentum sum rule, eq. (20)

94 3. N. Makins, Talk given at 8th International Workshop on Deep Inelastic Scattering and QCD (DIS 2000), Liverpool, England, 25-30 Apr 2000. Published in “Liverpool 2000, Deep inelastic scattering”. 4. M. Gell-Mann, in Proceedings of the 13th International Conference on High Energy Physics (University of California Press, Berkeley, 1967). 5. S. A. Larin and J. A. Vermaseren, Phys. Lett. B B259,345 (1991). 6. E. V. Shuryak and A. I. Vainshtein, Nucl. Phys. B B201,141 (1982). 7. A. J . G. Hey and J. E. Mandula, Phys. Rev. D 5,2610 (1972). 8. M. Gourdin, Nucl. Phys. B 38 (1972) 418. 9. S. L. Adler, Phys. Rev. 177 (1969) 2426. 10. J. S. Bell and R. Jackiw, Nuovo Cim. A 60 (1969) 47. 11. R. L. Jaffe, Phys. Lett. B 193,101 (1987). 12. M. Hirai, S. Kumano and N. Saito [Asymmetry Analysis Collaboration], Phys. Rev. D 69,054021 (2004) [arXiv:hep-ph/O312112]. 13. D. B. Kaplan and A. Manohar, Nucl. Phys. B 310,527 (1988). 14. R. D. Mckeown, Phys. Lett. B 219,140 (1989). 15. A. Chodos, R. L. Jaffe, K. Johnson and C. B. Thorn, Phys. Rev. D 10,2599 (1974). 16. L. M. Sehgal, Phys. Rev. D 10, 1663 (1974) [Erratum-ibid. D 11, 2016 (1975)]. 17. J. Kodaira, S. Matsuda, T. Muta, K . Sasaki and T . Uematsu, Phys. Rev. D 20 (1979) 627. 18. J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu, Nucl. Phys. B 159,99 (1979). 19. M. J. Alguard et al., Phys. Rev. Lett. 37,1261 (1976). 20. M. J. Alguard et al., Phys. Rev. Lett. 41,70 (1978). 21. G. Baum et al., Phys. Rev. Lett. 51, 1135 (1983). 22. R. D. Carlitz and J. Kaur, Phys. Rev. Lett. 38,673 (1977) [Erratum-ibid. 38,1102 (1977)l. 23. Experiments in which Vernon and the Yale group were active participants, C. Y. Prescott et al., Phys. Lett. B 77,347 (1978); C. Y. Prescott et al., Phys. Lett. B 84,524 (1979). 24. B. Adeva et al. [Spin Muon Collaboration], Phys. Rev. D 58,112001 (1998); Phys. Rev. D 58,112002 (1998). 25. These are the final numbers from the SMC. The analysis of spin dependent parton distributions continues to be a subject of considerable interest in anticipation of the new experiments at COMPASS and RHIC. For a recent analysis with qualitatively similar results, see J. Blumlein and H. Bottcher, Nucl. Phys. B 636,225 (2002) [arXiv:hep-ph/0203155]. 26. For a review and critique of early work see Ref. 27. 27. R. L. Jaffe and A. Manohar, Nucl. Phys. B 337,509 (1990). 28. A. V. Manohar, Phys. Lett. B 255,579 (1991). 29. J. C. Collins and D. E. Soper, Nucl. Phys. B 194,445 (1982). 30. X. D. Ji, Phys. Rev. Lett. 78,610 (1997) [arXiv:hepph/9603249]. 31. P. Hagler and A. Schafer, Phys. Lett. B 430, 179 (1998) [arXiv:hepph/9802362].

95 32. S. V. Bashinsky and R. L. Jaffe, Nucl. Phys. B 536,303 (1998) [arXiv:hepph/9804397]. 33. A. Harindranath and R. Kundu, Phys. Rev. D 59, 116013 (1999) [arXiv:hepph/9802406].

THE SPIN STRUCTURE OF THE NUCLEON: A HUGHES LEGACY

GORDON D. CATES Department of Physics, University of Virginia, Charlottesville, VA 22903 More than any other individual, Vernon Hughes can be pointed to as the father of the experimental investigation of nucleon spin structure. Even theoretical development in this area was spurred on by Vernon’s pioneering efforts to make the control of spin degrees of freedom an experimental reality. This talk traces some of Vernon’s work in this area, as well as examining, briefly and not in a complete fashion, some of the other work that can be looked upon as Vernon’s legacy.

1. Introduction

More than any other individual, Vernon Hughes was responsible for initiating and leading the experimental investigation of the spin structure of the nucleon. Vernon embraced the importance of utilizing spin degrees of freedom as a means for testing our understanding of matter. Realizing that such experiments would require a suitable source of polarized electrons, he began work on a prototype in the early 196O’s1i2.As the results of deep inelastic scattering of unpolarized electrons began t o unfold in the late 1960’s, Vernon was poised to begin exploring the underlying spin structure of the nucleon. It is amusing to trace the influence of Vernon’s work in the literature. In 1966, Bjorken wrote the famous paper in which he derived the Bjorken sum rule using current algebra3. In this paper, referring t o spin-polarized cross sections, he states that “ It will be a long time before these cross sections are measured.” Later, in the same paper, he refers to what we now call the Bjorken sum rule as “ ... this worthless equation ...”. Because of Vernon’s work, however, it soon became clear that the study of spin dependent cross sections was not such a far fetched goal. In a paper written four years later titled “Inelastic Scattering of Polarized Leptons from Polarized nucleon^"^ Bjorken starts out in the introduction: “Some time ago, a high-energy sum rule involving electromagnetic scat96

97

tering of longitudinally polarized leptons from polarized protons and neutrons was derived and then dismissed as ‘worthless’. However, it turns out to be interesting to reconsider that negative conclusion in light of the present experimental and theoretical situation.” Later Bjorken states that: “It appears to be possible to produce electron or muon polarized beams which have nearly 100% longitudinal polarization.” and he specifically references a paper by Hughes, Lubell, Posner, and Raith5. In just four years a measurement that had seemed completely impractical had become something worth contemplating quite seriously. It seems unlikely that this change of attitude would have taken place if Vernon had not demonstrated that a source of polarized electrons could indeed be built. Vernon, in his usual way, brought to nucleon spin structure a love for that which is fundamental. He emphasized the value of studying sum rules because by doing so, one could glean the most precise information about the problem. The Bjorken sum rule in its simplest form can be written

where gy and g i are the spin structure functions of the proton and neutron respectively, and they are integrated over the full range of the Bjorken scaling parameter x. The constants g~ and g v are the axial-vector and vector couplings that characterize &decay of the neutron. As Bob Jaffe discusses in these proceedings, the Bjorken sum rule can also be written to include both perturbative and non-perturbative corrections. I often have felt that Vernon was attracted to the Bjorken sum rule because he wanted to launch QCD onto a path of increasingly accurate measurements in much the same way as has been the case in QED. I say this, however, recognizing that Vernon always considered the proton an unfortunately complicated object. Ellis and Jaffe recognized that one could also construct sum rules for the proton and neutron individually6. Here some care needs to be taken because the quantities one encounters are affected by the axial anomaly reference and are sensitive to the factorization scheme and renormalization scale. Still, the formalisms explored by Ellis and Jaffe made it possible, within the naive quark-parton model, to deduce the extent to which the spin of the nucleon comes from quark contributions. To do so requires a measurement of the so-called first moment of the spin structure function of either the proton or the neutron, :?I = J ; gydx or I?; = gydx. I will say more on this later.

Jt

98

2. The early SLAG program

In 1970, Vernon proposed to measure the spin asymmetry in the scattering of polarized electrons from a polarized proton target, an experiment that came to be known as E80. The proposal came at a time when the quark-parton model was still in its nascent period, and was designed to see certain gross features of spin structure that one would associate with the quark-parton model. E80 presented large technical challenges, requiring what would be the first high energy polarized electron beam, and a polarized proton target that went well beyond what had been accomplished previously. The polarized electron source was based on the aforementioned prototype developed at Yale in the 1960’s. Eventually known as PEGGY, it utilized an atomic beam of 6Li that was state selected using hextapole magnets, and photoionized using a pulsed source of ultraviolet light.7. Depicted in Fig. 1, at the time it was built, PEGGY was overwhelmingly the most intense source of polarized electrons ever constructed. During E80 it produced around lo9 electrons/pulse at 120 pulses/sec with a polarization of around 0.5. Following E80 it was established that the polarization was limited by a multistep ionization process. With the elimination of this process, the electron polarization of PEGGY was increased to the impressive level of 0.85.

IWechanical Chopper

Longitudinally polarizing

Figure 1. Illustrated are the major subsystems of PEGGY, the first polarized electron source t o be used on a high-energy accelerator, built by Vernon’s group at Yale.

99

E80 used a polarized nuclear target based on dynamic nuclear polarizatioq (DNP). While E80 was not the first high energy experiment to use a DNP based target, the E80 target broke new ground in certain performance characteristics. It was probably the first such target to utilize a 5 T magnet, something that enabled the use of a 4He refrigerator while still achieving good polarization. With a 4He refrigerator, the E80 target was more tolerant of beam intensity than otherwise would have been the case. The E80 target used butanol beads as a target material, doped with 1.4% porphyrexide. Beam rastering and regular target annealing were used to deal with the effects of radiation damage. Some of the subsystems of the E80 target are shown schematically in Figure 2'.

Figure 2. Illustrated are some of the major subsystems of the Yale/SLAC polarized proton target that was built for use in E80.

The goal of E80 was to determine the spin asymmetry

Al =

*l/2

- *3/2

*l/2

+ *3/2

where ( ~ 3 1 2 )is the total virtual photoabsorption cross section for the nucleon for the case where the total angular momentum of the proton and

100

the virtual photon, when projected onto the direction of the virtual p h e ton, is 1/2 (3/2). In order to determine AI, E80 recorded experimental asymmetries of the form

where dot1 (doTT) is the differential cross section for scattering when a longitudinally polarized electron beam is antiparallel (parallel) to a longitudinally polarized target. The experimental asymmetry Aexpis related to the physics asymmetries A1 and the transverse physics asymmetry A2 by Aexp= D(Al +qAa) where q is a kinematic factor and D , known as the depolarization factor, represents the depolarization of the virtual photon with respect to the polarization of the incident polarized electrong. For the kinematics studied during E80, and indeed most deep inelastic spin-structure experiments using a longitudinally polarized target, Aezp M D A l . The results of E80, the first of their kind, are shown in Fig. 41°. The asymmetries were large and positive, a result that favored the quark-parton model according to a prediction by Bjorken based on his sum rule4.

0.8 0.4

a

t f o \

Q

-0.4

-0.8

Figure 3. Shown are the first spin asymmetries from the scattering of polarized electrons from polarized protons obtained during the SLAC experiment E80.

101

The SLAC program continued with E130, in which several improvements were incorporated1l?l2.The electron polarization was increased substantially to 0.85 from 0.50, and increases in the target polarization were also achieved. For some of the El30 running a new spectrometer was constructed with substantially larger acceptance. With these improvements and additional running time, data on the spin structure of the proton were mapped out over a range of Bjorken z of 0.10 < z < 0.64. This permitted the first crude test of the Ellis-Jaffe sum rule, and under the assumption that the neutron asymmetries would be negligibly small (based on simple quark models), a first look at the Bjorken sum rule.

--

--0

0.2

0.6

0.4

0.8

1.0

X Figure 4. The final results of E80 (open diamonds) and El30 (closed squares) together with several theoretical predictiond2.

The early SLAC spin-structure experiments provided the first information on the spin structure of the proton, and started a new field, but stopped short of making definitive measurements of the Bjorken and Ellis-Jaffe sum rules. If Vernon had had his way, however, the early SLAC program would have gotten quite a bit further. In Fig. 5, the projected errors of a second generation of experiments are shown. Known alternatively as “Son of E130” or E138, the proposal described an experiment whose statistical accuracy is not so bad when compared to the modern experiments that have actually been carried out. Sadly, particularly given the great surprises that were later seen in nucleon spin structure, El38 was not approved. In the late 1970’s SLAC was in hot competition with CERN working to bring the SLAC linear collider, SLC, online. This effort was taxing the lab, and

102

perhaps made the spin-structure program seem like a distraction. In retrospect, however, it is hard not to see El38 as quite a missed opportunity. Luckily, however, there was no way that Vernon was going to drop his dogged pursuit of the physics!

PROTON ASYMMETRY

-0.2 0

0.2

0.4

0.6

X

DEUTERON ASYMMETRY

0.8

1.0

-0.2

0

’

‘

0.2

’

’

1

0.4

0.6

’

I

0.8

’

1.0

X

Figure 5. The projected errors on A1 from the proton and the deuteron from SLAG E138, an experiment known as “Son of E130” that was proposed by Vernon as a follow-on to E80 and El30 but was not allowed to run.

3. The CERN program

With the option of further studies at SLAG cut off, Vernon sought an alternative means to pursue his study of nucleon spin structure. He first joined Fermilab experiment E665 which was studying unpolarized deep inelastic muon-nucleon scattering, hoping to stimulate interest in studying polarized muon-nucleon scattering. While this did not work out, Vernon was subsequently invited to join the European Muon Collaboration (EMC) at CERN, where interest in polarized muon-nucleon scattering was building, and a large volume polarized target was under development. The spinstructure program at CERN proved to be a huge success, with the efforts of EMC being followed by the Spin Muon Collaboration (SMC) of which Vernon was spokesperson. The CERN program produced seminal data that triggered explosive growth in spin-structure studies. To this day the EMC and SMC results have the best coverage in the important kinematic regime of low Bjorken IC.

103

3.1. The EMC spin-structure experiment Using the M2 muon beam at the CERN SPS accelerator, the EMC experiment collected data on the scattering of polarized muons off what was then the largest polarized proton target in existence. With incident muon energies up to 200 GeV, the EMC collected more than a million events over a range of Bjorken x spanning 0.01 < x < 0.7. While the flux of muons was modest, up to 4 x lo7 per pulse every 14 seconds, the polarized target had two cells each of which was 360 mm in length, resulting in a reasonable event rate. The proton polarization was typically between 0.75 and 0.80 and the muon polarization was roughly 0.8, both of which are quite high and contributed to the quality of the data.

Figure 6. Results from EMC on x g1 of the proton as a function of Bjorken x, and the integral of gy as a function of the lower limit of integration.

When the EMC published their results in 198813>14, they pushed our knowledge of spin structure to values of x that were an order of magnitude smaller than had previously been the case in the early SLAC experiments, greatly improving the accuracy with which the first moments of the spin structure functions could be evaluated. In Fig. 6 the EMC results for the spin structure function gy of the proton are shown. The improved

104

coverage at low x is readily apparent. Also shown is the integral of gy as a function of the lower limit of integration. While the full integral requires an extrapolation to x = 0, the plot makes a convincing argument that the integral is converging to a value well below the prediction of the Ellis-Jaffe sum rule, which is also shown on the figure. The EMC result, as presented in their first paper13, was that the integral of gy 1

F ? = J d g?(x)dX = 0.114 f 0.012(st~t.)f 0.026(syst.)

,

(4)

a result that was in strong disagreement with the Ellis-Jaffe (EJ) Sum Rule6, which was quoted in the same paper as predicting that a =

Jd

1

gf(x)dx = 0.189 f0.005 .

(5)

The Ellis-Jaffe sum rule was derived using SU(3) current algebra under the assumption that the strange quark sea was unpolarized. Conversely, it was recognized that if the Ellis-Jaffe sum rule was violated, one explanation was that the strange quark sea is highly polarized. Within the naive quark-parton model (QPM), the first moment of gy has a particularly simple form in terms of Au, Ad and As, where Aqi = (4: - q/)dz is the probability difference for the momentum distributions corresponding to a quark that is aligned or anti-aligned with the nucleon spin. Again within the naive QPM, Aqi is the fraction of the nucleon’s spin carried by the quark helicity of flavor i. Writing the first moment I?; and the Bjorken sum rule out in terms of the Aqi’s, we have

Jt

4 9

1 9

-AU + -Ad

1 + -AS = 2 I?; 9

(6)

A u - A d = - gA gv A u + A ~ - ~ A=s 3 F - D where the third expression follows from SU(3) flavor symmetry. A more theoretical discussion of these matters can be found in Bob Jaffe’s paper in this proceedings. Within this framework, including also first-order QCD corrections, the EMC computed AX = Au Ad As, the fraction of spin carried by all the quarks, to be 0.120 f0.116 f0.234. That is, the fraction

+

+

*A more modern value for the EJ Sum Rule, evolved to a Q2 of 5GeV2, corrected to order a:, and using updated values for the SU(3) couplings F and D would be ry = 0.163 f0.00429.

105

of spin carried collectively by all the quarks is quite small and consistent with zero! The EMC result touched off what at the time was called the “proton spin crisis”, and what some would now call the proton spin puzzle. There was great surprise that so little of the spin of the nucleon was carried by quark spin. The EMC result certainly changed prevailing views of nucleon spin structure, and provided strong motivation for further studies. Experimental efforts were launched at several major laboratorie~l~. At CERN the Spin Muon Collaboration, or SMC, was formed as a follow-on to the EMC with Vernon as spokesman. At SLAC, the very laboratory that had shut down Vernon’s original program, a new set of experiments were undertaken, some of which were led by Vernon’s son Emlyn. And at DESY, the HERMES experiment was formed, a program that continues taking data to this day. In addition to experimental activity, there has also been a huge amount of theoretical progress. Indeed, it is probably fair to say that the effort that went into understanding the EMC result set the stage for many of the more topical subjects in QCD today. At the time of this writing, the paper announcing the EMC result^'^ has been cited roughly 1000 times and nucleon spin structure has grown into a large and rich field.

3.2. The Spin Muon Collaboration (SMC) With the enormous impact of the EMC results, the motivation to continue studying nucleon spin structure at CERN was very high. The Spin Muon Collaboration (SMC) was formed under Vernon’s leadership, and proceeded to collect data from 1992 until 1996 on both the proton and the deuteron. For a Q2 > 1 (GeV/c)2, the SMC collected data down to a value of Bjorken z = 0.003, roughly three times smaller than was the case for the EMC. To improve their statistics, the SMC used a proton/deuteron target comprising two cells, each 60 cm in length, a truly huge polarized target. They also collected data over an impressive length of time. In the end SMC presented data comprising 15.6 million events on the proton and 19.0 million events on the deuteron (after cuts), an impressive increase over the 1.2 million events collected by the EMC. The final papers written by the SMC, published in 1998, included a paper detailing the SMC’s final experimental results16, and a next-to-leading order (NLO) perturbative QCD analysis of the world spin structure data such as it was at that time including the final SMC results17. Then, as is still the case, the SMC experiment had the best coverage at low Bjorken

106

x of any experiment to study spin structure. And despite a limited muon flux, their statistics were also impressive. The SMC experiment gave Vernon a vehicle for pursuing scientific interests that were cut short at SLAC. SMC also formed a vibrant intellectual center for the study of spin structure that helped spur worldwide activity both in experiment and theory. 4. The second generation SLAC nucleon spin-structure

program

When the EMC results were released, which dealt strictly with the proton, it was clear that studies were needed of the neutron. The EMC collaboration itself emphasi~ed’~ “ ... it is of crucial importance to measure the asymmetries from a target containing polarised neutrons ...” In 1989 the first of a new generation of spin-structure experiments was proposed at SLAC to address this need. Dubbed E142, the experiment proposed t o measure the scattering of polarized electrons from a polarized 3He target18t9. Organized and led by Emlyn Hughes, there was amusing irony that it was a Hughes that was bringing the study of spin structure back to SLAC. El42 was a legacy of Vernon’s in more ways than one! While some accelerator-based experiments had previously used gaseous 3He targets, El42 brought the practice to a new level. The target was much larger than those that preceded it, and provided a luminosity approaching 1036Cm-2~-1. El42 was the first of five experiments in a new generation of SLAC spin-structure experiments. It was followed by E143, an experiment that used polarized solid targets containing first hydrogen and then deuteriumlg. El42 and El43 were both run at energies of 29 GeV or less. The energy was limited not by the accelerator, but by the beam lines that transported the beam into End Station A, the area in which the experiments were performed. With an upgrade of the beam line making an energy near 50 GeV possible, a second set of experiments was performed including El54 that again used polarized 3He20921y22, and E15523 and E 1 5 5 ~that ~ ~ ,again used solid hydrogen and deuterium polarized targets. In addition to higher energy, E154, E155, and E155x benefitted from additional improvements, including substantially higher electron polarization due to the use of a “strained” GaAs photocathode. The new generation of SLAC spin-structure experiments, both before and after the 50 GeV beam-line upgrade, were characterized by superb

107

statistics, but took place at significantly lower energies than were available at CERN and consequently had more modest coverage in terms of Bjorken x. The precision of the later SLAC results, however, ensures that they are weighted heavily in any type of global fit of spin-structure data. Because I am mostly familiar with the polarized 3He experiments, I will restrict my detailed comments on the second generation of SLAC experiments to El42 and E154.

4.1. SLAC E l 4 2 El42 was designed to provide a high luminosity study of the spin structure of the neutron at a time when virtually no data on the neutron existed. Polarized electrons were scattered from a polarized 3He target at energies of 19.4 GeV, 22.7 GeV, and 25.5 GeV. Two spectrometers were used at 4.5" and 7.0". The average Q2 of the data was 2GeV2, and data were taken at values of Bjorken x as low as 0.03. Average beam currents were in the range of 1 - 4pA, with an average polarization of 0.38 f 0.02. The target polarization was 0.36 f 0.02 Approximately 300 million events were used in determining gy from E142. While this is substantially more than the 1.2 million events collected by the EMC, the advantage was less pronounced than one might naively conclude. The target and beam polarizations during El42 were about a factor of two lower than in EMC, and the dilution factor (about .11) was about .65 the dilution factor of EMC (about .17). The statistical errors on the two experiments were thus not that dissimilar. And of course, EMC had the clear advantage that their lowest x bin was 0.015 versus 0.030 for E142. Still, the quality of the information that El42 provided was impressive, and at the time, there were no other precision data for the neutron. It is interesting to compare the El42 results for g i to the EMC results for the proton and the early SMC results for the deuteron25 that were being published at roughly the same time. Such a comparison is shown in Fig. 7 taken from the thesis of one of the El42 students26. It is clear that the El42 data were quite precise by any standard, and completely transformed the experimental situation for the neutron. Critical to the success of El42 was the construction of a suitable polarized 3He target. The target was based on the technique of spin-exchange optical p ~ m p i n g ~a ~two-step l ~ ~ , process in which 1) rubidium atoms are polarized by optical pumping, and 2) spin is transferred from the Rb valence electrons to the nuclei of 3He atoms by the hyperfine interaction during

108

.............. 1................ 4.................................................................. L

4.05

I

0

-

1........................................................................

+

9.05

o.05

x&

O

9.05

CC

I

L E-142 Neutron

o.05

xgy

: : :I

a

'

-: : : f 1 SMC Deuteron

@

I

I

+ +

*

w/ :

1:-f. ......................................................................... + + '

"

+

*$ ....... ........ 1

-

............-

I

Figure 7. Shown are data from EMC, SLAC E142, and SMC on the nucleon spin structure function g1(z) such as existed around the time the El42 results were first published.

collisions. At the time it was proposed, it appeared that a polarized 3He target could be built with a luminosity approaching cm-2 s-'. Achieving such a luminosity, however, meant scaling the volume of polarized gas by a large amount over anything that had been done previously. It was not clear to what extent unanticipated problems would be encountered. Spin-exchange optical pumping typically takes place in a sealed glass cell, containing up to around 10 atmospheres of 3He, about 70 Torr of N2, and on the order of 100 milligrams of metallic Rb. The magnitude of the challenge that faced El42 is illustrated in Fig.8 which shows two El42 target cells, ready to be filled with 3He, together with a cell just over 2 cm in diameter which was more typical of samples used for spinexchange optical pumping at that time. The SLAC cells have volumes of around 150 cm3, whereas the smaller cell has a volume around 10 cm3. Targets for use at TRIUMF and Bates with volumes as large as 35cm3 were under development at the time that El42 was proposed, but their

PerP 5. SLAC E154

ormannce had not yetormannce been established. had not yet been established. ormannce had not yetormannce been established. had not yet been established.

ormannce had not yet been established. ormannce had not yet ormannce been established. had not yet been established. 60 1

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also used two new spectrometers, one at 2.75", and one at 5.5". The impact of the experimental improvements is readily apparent in the comparison of El54 data with that of El42 shown in Fig. 9. 0.03

xg: 0.02

0.01

0

-0.01

-0.02

-0.03 lo-'

1

X

Figure 9. Plotted is a comparison of the El54 data with the El42 data on the product of Bjorken x with the longitudinal neutron spin structure function 91".

To this day, El54 provides the most precise data on the spin structure of the neutron over the kinematic range it covered. As a result, the El54 results are heavily weighted in world next-to-leading order perturbative QCD analyses of spin-structure data. With Emlyn at its helm and many of Vernon's former colleagues in other leadership positions, El54 represented an excellent example of Vernon's legacy. 6. Next-to-leading order perturbative QCD analyses

While sum rules involving integrals of the spin structure functions provide a means for accessing remarkably fundamental information, it is a practical reality that the full range of the Bjorken scaling variable, 0 5 z 5 1, is not experimentally accessible. Of particular importance is the region from the lowest value of x at which data are available down to x = 0, since this is a regime in which sea quarks and gluons become increasingly prevalent.

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Probably the most accepted way of coping with this problem is a nextto-leading order (NLO) perturbative QCD analysis. In this approach, the spin structure function is parameterized at low Q2 in a manner that can describe well both the low-x and high-x data. The spin structure function is then evolved to the Q2of interest, and the parameters are iterated to fit the data. In this way a physically reasonable spin structure function with a well defined analytical form is generated that can be integrated over the full range

ormannce had not yet been established. lo-* X

Figure 10. Shown are data on gy from El54 and SMC for the region z < 0.1, together with several fits, as indicated on the plot.

The urgency of needing a well defined prescription is illustrated in Fig. 10 which shows the El54 data on the neutron as well as SMC data on the neutron (from considering the difference of proton and deuteron data) such as existed at the time El54 was published. The data are inconsistent with the simplest Regge theory interpretation that gy is constant at low x. The El54 collaboration considered several alternative possibilities. One was a Regge theory extrapolation with a constrained power-law fit in which it was assumed that 9;" x - ~ ,and -0.5 < Q 5 0. This results in a determination of the first moment of gy of ry = -0.041 f0.004f 0.006, but only fits well the lowest three points. Another was to consider an unconstrained

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power law fit, fitting the data for x < 0.1. It was found that Q = 0.9 f0.2 resulting in a determination of r;l.= -0.2, but it is not possible to quote an error on I’;l. because if a = 1 the integral of g1 diverges. A Pomerontype fit was also considered but was not particularly successful. From these considerations, it is clear that the El54 data, taken by themselves, are not sufficiently constraining to yield a reliable low-2 extrapolation, and hence, a determination of r;l.. By employing an NLO pQCD analysis, there is at least a well defined prescription within which to analyze the world body of spin structure data. The evolution of the structure functions with Q2 is handled well, making it possible to combine data from many different kinematic conditions. I worry that there is a certain arbitrariness to the parameterizations that are chosen, but I will leave the consideration of such points to the experts. Many NLO pQCD analyses have been performed. The last such analysis performed by the SMC was published in 199817. There was also an NLO pQCD analysis performed by the El54 collaboration22. Many others exist as well, and some of their results are summarized nicely by Filippone and Ji29. The most recent of which I am aware, which includes the El55 proton data, was published in 200430. One of the quantities that comes out of an NLO pQCD analysis is a value for the first moment of the singlet quark distribution, AX. The analysis must be performed within a particular factorization scheme, which among other things, affects whether or not AC contains a contribution from the gluon spin, AG. In the quark-parton model, AX is the fraction of the nucleon’s spin carried by quark spin. In the MS factorization scheme, in which AX does not contain a contribution from AG, AX is found t o be constrained to the range 0.05 - 0.28 depending on the particular NLO pQCD analysis in question. The original discovery that started the proton spin crisis has certainly held up. It appears that rather little of the spin of the nucleon is carried by quark spin. NLO pQCD fits of the world’s inclusive deep inelastic scattering data are much less definitive regarding AG. While most fits seem to indicate that AG is positive, a reliable determination will need to await new experiments.

7. A large and growing legacy One equation that can be used to discuss Vernon’s legacy with regard to spin structure is the following: -1= - A1X + A G + L , + L g . (9) 2 2

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On the left is the spin of the nucleon, and on the right are the various sources of angular momentum that in principle can contribute. Here the first term represents the angular momentum carried by the quark spin, the second term represents the angular momentum carried by the gluon spin, and the last two terms represent the angular momentum carried by orbital angular momentum of quarks and gluons respectively. The focus of this paper has been measurements of g1 through inclusive deep inelastic scattering. Such measurements predominantly provide information on the first term pertaining to quark helicity. There are many other types of measurements, however, which I cannot hope to enumerate within the scope of this paper. I have essentially ignored A2 and g2, which are important for many reasons. Nor have I discussed semi-inclusive measurements where one detects a hadron in the final state. Such measurements potentially provide more direct measurements of flavor specific spin distributions such as Au, Ad, and As,and have been the focus of some of the current measurements at HERMES. I have also ignored what is arguably the next large push in spin-structure studies, a determination of the second term AG. The COMPASS experiment at CERN and the RHIC Spin program at Brookhaven are both seeking to quantify the degree to which the polarization of gluons contribute to the spin of the nucleon. The contributions to the spin of the nucleon from orbital angular momentum is a fascinating subject in which there have been some interesting developments in recent years. Within the context of Generalized Parton Distributions (GPD’s), it has been suggested that information regarding L, can be gotten from deeply virtual Compton scattering31. Such experiments appear very challenging, and I believe it is still unclear how this will unfold. Before being skeptical, however, it is important to remember the degree to which polarized deep inelastic scattering looked impractical in the late 1960’s. At Jefferson Laboratory there may already be indirect evidence concerning the role of orbital angular momentum. Measurements of the ratio of the electric to the magnetic form factors of the proton have shown a dramatic decrease with Q2, where the naive expectation was that the ratio would be roughly constant32. Explanations of this phenomena have tended to include a non-zero component of angular momentum associated with the quark wave functions33. Several other experiments, including a measurement of A? at high 234,have also seen effects that can be interpreted as evidence of orbital angular momentum. Despite these interesting developments, it is clear that any type of thorough understanding of L, is still at the earliest stage. Nevertheless, the theoretical and experimental

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activity in this area is yet another sign of the richness of the evolving field of spin structure. During my talk, I could not resist pointing out that Vernon’s leadership has indirectly lead to useful technological spin-offs. The EMC results on the proton created a compelling need for better polarized neutron targets. The polarized 3He target that was used for El42 and El54 answered that need, and also represented a large step forward in the use of spin-exchange optical pumping for the polarization of large quantities of noble gas. In a series of experiments that took place immediately following E142, it was demonstrated that polarized 129Xeand 3He could be used for a new type of magnetic resonance imaging35i36. The gases are inhaled, and provide a source of signal for MR images of the gas space of the lungs. A comparison of a 3He image of human lungs with a nuclear medicine scan, the current state-of-the-art, is shown in Fig. 11. The technology that was developed for El42 lead quite directly into MR imaging with noble gases. This is a nice example of an unanticipated spin-off from basic research, basic research that came in part from a field that began with Vernon’s leadership.

Figure 11. Shown are two images off the gas space of human lungs (from different subjects). At left is a traditional ventilation scan in which the subject inhales radioactive gas and an image is made using a gamma camera. At right is an MRI in which the signal source is inhaled nuclear-polarized 3He. Both images were made at UVa.

Fortunately there is another talk in this proceedings that discusses the future of spin-structure studies. Not only are there upcoming experiments such as COMPASS and RHIC Spin, there is also discussion of constructing a polarized electron-ion collider. What started as a field with a handful of people has expanded into a substantial community. Vernon has left us a wonderful legacy, and will be sorely missed.

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References 1. R. L. Long, Jr., W. Raith and V. W. Hughes, Phys Rev. Lett 15, 1 (1965). 2. V. W. Hughes, R. L. Long, Jr., M. S. Lubell, M. Posner and W. Raith, Phys. Rev. A 5 , 195 (1972). 3. J. D. Bjorken, Phys. Rev. 148, 1467 (1966). 4. J. D. Bjorken, Phys. Rev. D 1,465 (1970); Phys. Rev. D 1,1376 (1970). 5. V. Hughes, M. Lubell, M. Posner, and W. Raith, in Proceedings of the Sixth International Conference on High-Energy Accelerators (unpublished). 6. J. Ellis, R. L. Jaffe, Phys. Rev. D 9, 1444 (1974); 10, 1669 (1974). 7. M. J. Alguard et al., Nucl. Instr. Meth. 163, 29 (1979). 8. W.W. Ash in High Energy Physics with Polarized Beams and Targets, ed. M.L. Marshak, Am. Inst. Phys., (New York, 1976), p.485. 9. P.L. Anthony et al. (the SLAC E-142 Collab.), Phys. Rev. D 54, 6620 (1996). 10. M.J. Alguard et al., Phys. Rev. Lett. 37, 1261 (1976). 11. M.J. Alguard et al., Phys. Rev. Lett. 41, 70 (1978). 12. G. Baum et al., Phys. Rev. Lett. 51, 1135 (1983). 13. J. Ashman et al., Phys. Lett B 206, 364 (1988). 14. J. Ashman et al., Nucl. Phys. B328, 1 (1989). 15. E.W. Hughes and R. Voss, Annu. Rev. Nucl. part. Sci. 49, 303 (1999). 16. B. Adeva et al. (The SMC collab.), Phys. Rev. D 58, 112001 (1998). 17. B. Adeva et al. (The SMC collab.), Phys. Rev. D 5 8 , 112002 (1998). 18. P.L. Anthony et al. (the SLAC El42 collab.), Phys. Rev. Lett. 71,959 (1993). 19. K. Abe et al. (the SLAC El43 collab.), Phys. Rev. D 58, 112003 (1998). 20. K. Abe et al. (the SLAC E-154 collab.), Phys. Rev. Lett. 79, 26 (1997). 21. K. Abe et al. (the SLAC El54 collab.), Phys. Lett. B 404 (1997). 22. K. Abe et al. (the SLAC El54 collab.), Phys. Lett. B 405, 180-190 (1997). 23. P.L. Anthony et al. (the SLAC El55 collab.), Phys. Lett. B 463, 339 (1999); Phys. Lett. B 493, 19 (2000). 24. P.L. Anthony et al. (the SLAC E155x collab.), Phys. Lett. B 553, 18 (2003). 25. B. Adeva et al., Phys. Lett. B302, 553 (1993). 26. H. Middleton, Ph.D. thesis, Princeton University, 1994. 27. T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997). 28. A. Ben-Amar Baranga, S. Appelt, M.V. Romalis, C.J. Erickson, A.R. Young, G.D. Cates and W. Happer, Phys. Rev. Lett. 80, 2801 (1998). 29. B. W. Filippone and X. Ji, Adv. in Nucl. Phys. 26, 1 (2001). 30. M. Hirai, S. Kumano, and N. Saito (Asymmetry Analysis Collaboration), Phys. Rev. D 69, 054021 (2004). 31. X.D. Ji, Phys. Rev. Lett. 78, 610 (1997). 32. M.K. Jones et al., Phys. Rev. Lett. 84, 1398 (2000). 33. See for instance A.V. Belitsky, X. Ji, and F. Yuan, Phys. Rev. Lett. 91, 092003 (2003) or G. A. Miller, Phys. Rev. C 66, 032201(R) (2002). 34. X. Zheng et al. (JLab Hall A Collab.), Phys. Rev. Lett. 92, 012004 (2004). 35. M. S. Albert, G. D. Cates, B. Driehuys, W. Happer, B. S a m , C. S. Springer Jr., and A. Wishnia, Nature 370, 199 (1994). 36. J. R. MacFall et al., Radiology 200, 553 (1996).

MUON g - 2: THE LAST WORD?

ERNST P. SICHTERMANN, representing the muon g - 2 Collaboration Yale Unversity

P.O. Box 208121 New Haven, CT 06520, USA and Lawrence Berkeley National Laboratory 1 Cyclotron Road Berkeley, CA 94720, USA E-mail: EPSichtermannOlbl.gov In the early 1980's, Vernon W. Hughes initiated a fourth generation of muon g - 2 measurements aiming at an uncertainty well below 1ppm, over an order of magnitude more precise than the results from the famous measurements at CERN. The new experiment has measured the anomalous g values of the positive and negative muon, each t o a precision of 0.7 parts per million (ppm), at the Brookhaven Alternating Gradient Synchrotron. The final results, a,+ = 11659 203(6)(5) X lo-'' and a,- = 11659 214(8)(3) x lo-'' are consistent with the previous measurements. Their average is a,(exp) = 11659208(6) x lo-'' (0.5ppm).

1. Introduction

The anomalous g values, a = (g - 2)/2, of leptons arise from quantum mechanical effects. Their precise measurement has historically played an important role in the development of particle theory. The anomalous magnetic g value of the electron, a,, has been measured to within about four parts per billion (ppb)2,and is among the most accurately known quantities in physics. Its value is described in terms of Standard Model field interactions, with nearly all of the measured value contributed by QED processes involving virtual photons, electrons, and positrons3. Heavier particles contribute to a, only at the level of the present experimental uncertainty. The anomalous magnetic g value of the muon, a,, is more sensitive than a, to processes involving particles more massive than the electron, characteristically by a factor (rn,/m,)2 4.104.4A series of three experiment^^>^ at CERN measured a, to within 7 parts per million (ppm), an uncertainty which is predominantly of statistical origin. The CERN generation of ex-

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117

-

periments thus tested electron-muon universality and established the existence of a hadronic contribution to a, with a relative size of 59ppm. Electroweak processes are expected to contribute at the level of 1.3ppm, as are many speculative extensions of the Standard Model7. Theoretical evaluations of the a , were attributed to have a 8ppm uncertainty at the time of the last CERN muon g - 2 experiment, and arose principally from the uncertainty coming from hadronic contributions. As R.W. Williams emphasized in his talk, “Muon g - 2 - the Last Word”’, the theoretical and experimental uncertainties were at the same level and would be hard to improve. Vernon W. Hughes considered pursuing an improved measurement because of the importance of having a precise knowledge of the muon g - 2, despite these observations. In the year 1984 he organized a workshop at Brookhaven National Laboratory (BNL) to initiate a new g - 2 measurement and to work out the general parameters of the experiment (Fig. 1). A letter of intent was submitted to BNL, followed by a proposal in 1985’. N

Figure 1. At Brookhaven National Laboratory, summer 1984. Standing, from left: Gordon Danby, John Field, Francis Farley, Emilio Picasso, and Frank Krienen; kneeling from left: John Bailey, Vernon Hughes, and Fred Combley.

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The goal was an uncertainty well below the electroweak contribution, more than an order of magnitude improvement over the CERN measurements, using a similar concept as the last CERN muon g - 2 experiment and a new superconducting storage ring. The first muon data were collected in 19971°. The long stretch between the letter of intent and the first data reflects on the difficulty of the measurement, as well as on difficulties to secure adequate funds. The new measurements reached an uncertainty of 1.3ppm in a,+ for the positive muon from data collected in the year 199911, followed by 0.7ppm uncertainty from p+ data collected in 200012. Shortly after the Memorial Symposium in honor of Vernon Willard Hughes the collaboration finalized its analysis of a,- of the negative muon once more achieving a 0.7ppm uncertainty13. The focus in the sections below is on some of the many aspects of these last measurements. Vernon W. Hughes had a keen interest in theoretical evaluations of the muon anomalous magnetic g value, and in particular its hadronic contribution. Close relations with the Budker Institute for Nuclear Physics in Novosibirsk were established early on. The Novosibirsk measurements of the hadron production cross section a(e+e- --+ hadrons) have played a lead role in providing improved knowledge of the hadronic contribution to the anomalous moment. At present, the Standard Model expectation for up is known about an order of magnitude more precisely than it was in 1984.

2. Experiment

The concept of the present experiment is similar to that of the last of the CERN experiments5i6and involves the study of the orbital and spin motions of polarized muons in a magnetic storage ring. Protons with energies of 24 GeV from the AGS were directed onto a rotating, water-cooled nickel target. Pions with energies of 3.1 GeV emitted from the target were captured into a 72 m straight section of focusing-defocusing magnetic quadrupoles, which transported the parent beam and naturally polarized muons from forward pion decays. For most of the data taking periods, longitudinally polarized muons of slightly lower energies were injected into a 14.2m diameter storage ring magnet14 through a field-free inflector15 region in the magnet yoke. A fast non-ferric kicker16 located at approximately one quarter turn from the inflector region produced a 10mrad deflection which placed the muons onto stored orbits. Pulsed electrostatic quadrupoles17 provided vertical focusing. The magnetic dipole field of about 1.45T was measured with an nuclear magnetic resonance (NMR) s y ~ t e m relative ' ~ ~ ~ to ~ the free

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proton NMR frequency wp over most of the 9 cm diameter circular storage aperture. Twenty-four electromagnetic calorimeters2' read out by 400 MHz custom waveform digitizers (WFD) were used on the open, inner side of the C-shaped ring magnet to measure the decay positrons and electrons. The WFD and NMR clocks were phase-locked to the same LORAN-C21 frequency signal. Muon decay violates parity, which in the laboratory frame results in a modulation of the number of decay electrons (positrons),

produced with energies above a threshold E. Here, No is a normalization, 64ps is the muon lifetime in the laboratory frame, A 0.4 is an asymmetry factor, 4 is a phase, and w, is the angular difference frequency of muon spin precession and momentum rotation. The muon anomalous magnetic g value is evaluated from the ratio of the measured frequencies, R = w,/wp, according to:

YT

N

N

R

x

a, = - R ' in which X = p,/pp is the ratio the muon and proton magnetic moments. The value with smallest stated uncertainty, X = 3.183 345 39(10)22, results from measurements of the microwave spectrum of ground state m u o n i ~ m and~ theory25. ~ ~ ~ ~ 3. Data Analysis

The proton NMR frequency wp and the muon spin precession frequency w, were analyzed independently by several groups within the collaboration. The values of R = w,/wp and a, were evaluated only after each of the frequency analyses had been finalized; at no earlier stage were the absolute values of both frequencies, wp and w,, known to any of the collaborators. 3.1. The magnetic field frequency The measurement of the magnetic field is based on proton NMR in water. A field trolley with 17 NMR probes was moved typically 2-3 times per week throughout the entire muon storage region, thus measuring the field in 17 x 6 . lo3 locations along the azimuth. The trolley probes were calibrated in dedicated measurements taken before, during, and after the muon data collection periods. In these calibration measurements, the field in the storage region was tuned to very good homogeneity at specific calibration

120

Figure 2. Top view of the g - 2 apparatus. The beam of longitudinally polarized muons enters the superferric storage ring magnet through a superconducting inflector magnet located at 9 o’clock and circulates clockwise after being placed onto stored orbit with three pulsed kickers modules in the 12 o’clock region. Twenty-four lead scintillating-fiber calorimeters on the inner, open side of the C-shaped ring magnet are used to measure muon decay positrons and electrons. The central platform supports the power supplies for the four electrostatic quadrupoles and the kicker modules.

locations. The field was then measured with the NMR probes mounted in the trolley shell, as well as with a single probe plunged into the storage vacuum and positioned to measure the field values in the corresponding locations. Drifts of the field during the calibration measurements were determined by remeasuring the field with the trolley after the measurements with the plunging probe were completed, and in addition by interpolation of the readings from nearby NMR probes in the outer top and bottom walls of the vacuum chamber. The difference of the trolley and plunging probe readings forms an inter-calibration of the trolley probes with respect to the plunging probe, and hence with respect to each other. The plunging probe, as well as a subset of the trolley probes, were calibrated with respect to a standard probe with a l c m diameter spherical H2O sample in a similar sequence of measurements in the storage region, which was opened to air for that purpose. The standard probe is the same as the one used in the muonium measurements that determine the ratio X of muon to proton magnetic moment^^^?^^. The leading uncertainties in the calibration procedure

121

result from the residual inhomogeneity of the field at the calibration locations, and from position uncertainties in the active volumes of the NMR probes. The ring magnet design14, the inflector design15, and extensive shimming contributed to the overall uniformity of the field throughout the storage ring. Figure 3 shows one of the magnetic field measurements with the center NMR probe in the trolley in the year 2000. A uniformity of flOOppm in the center of the storage region was achieved for the full azimuthal range, in particular also in the region where the inflector magnet is located. Between the data taking periods in 2000 and 2001 the polarity

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azimuthal positron [deg] Figure 3. The NMR frequency measured with the center trolley probe relative to a 61.74 MHz reference versus the azimuthal position in the storage ring for one of the 22 measurements with the field trolley during the data collection in the year 2000. The continuous vertical lines mark the boundaries of the 12 yoke pieces of the storage ring. The dashed vertical lines indicate the boundaries of the pole pieces.

of the ring, inflector, and beamline magnets was reversed. After several ramping cycles the field was found t o be of equal uniformity. Figure 4 shows a two-dimensional multipole expansion of the azimuthal average of the field in the muon storage region from a typical trolley measurement in 2001. Since the average field is uniform to within 1.5ppm over the storage aperture, the field integral encountered by the (analyzed) muons is rather insensitive to the precise location and profile of the beam. The measurements with the field trolley were used to relate the readings of about 370 NMR fixed probes in the outer top and bottom walls of the

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storage vacuum chamber t o the field values in the beam region. The fixed NMR probes were read out continually. Their readings were used t o interpolate the field when the field trolley was “parked” in the storage vacuum just outside the beam region and muons circulated in the storage ring.

Multipoles [ppm]

Normal Skew Quad -0.28

0.11

%XI -0.72

-0.45

Octu

0.09

0.01

Decu

1.04

0.38

-4 -3 -2 -1 0 1 2 3 4 radial distance [cm] Figure 4. A 2-dimensional multipole expansion of the azimuthal average of the field measured with trolley probes with respect to the central field value of 1.451 269 T. The multipole amplitudes are given at the storage ring aperture, which has a 4.5cm radius as indicated by the circle.

For the data collection in the year 2001 the field frequency wp weighted by the muon distribution was found to be, wp/(27r) = 61 791 400(11) Hz (0.2ppm)13.

(3)

The uncertainty has a leading contribution from the calibration of the trolley probes and is thus predominantly systematic. The result was confirmed by a second, largely independent analysis, which made use of additional calibration data, a different selection of fmed NMR probes, and a different method to relate the trolley and fked probe readings. The history of systematic uncertainties in the field measurements since 1998 is given in Table 1. The uncertainty in the field measurement was improved by a factor of three over the course of the experiment. A new superconducting inflector magnet and shield, installed between the data collection periods in 1999 and 2000, improved the field homogeneity and further relaxed the demands on the knowledge of the muon beam distribution. Other significant improvements resulted from refinements in the calibration measurements and in the data analysis. The technique is not

123 Table 1. Systematic uncertainties for the wp analysis. The uncertainty "Others" groups uncertainties caused by higher multipoles, the trolley frequency, temperature, and voltage response, eddy currents from the kickers, and time-varying stray fields. Source of errors Absolute calibration of standard probe Calibration of trolley probe Trolley measurements of Bo Interpolation with fixed probes Inflector fringe field Uncertainty from muon distribution Others Total svstematic error on wm

Size [ppm] 1998 0.05 0.3

1999 0.05

0.1

0.10 0.15 0.20 0.12 0.15

0.3 0.2 0.1

0.5

0.20

0.4

2000 0.05 0.15 0.10 0.10

0.03 0.10 0.24

2001 0.05 0.09 0.05 0.07 0.03 0.10 0.17

yet fully exhausted; modest further improvements may result from better measurement of the residual field from kicker eddy currents, and from further refinement in the calibration and analysis. 3.2. T h e muon s p i n precession frequency

The muon frequency w, was determined by fitting the spectrum of arrival times of the decay electrons (positrons) measured with the lead scintillatingfiber calorimeters on the inner side of the storage ring. The calorimeters were read out with waveform digitizers (WFD) which sampled the photomultiplier signals every 2.5ns. The WFD traces were fitted off-line with average pulse-shapes, which were determined for each calorimeter individually from a selection of about lo4 pulses in the energy range 1-3 GeV. The selection was made so as to ensure that transient detector effects had faded away and the traces consisted of detector responses to single electrons (positrons). Two independently determined sets of pulse-shapes were used, as well as two independent implementations of the pulse-finding algorithm. A fraction of several percent of the recordings was found to contain multiple electron (positron) pulses per WFD trace. Extensive studies of the algorithm showed that in such cases each of the pulses was identified and measured correctly, provided that the pulse separation exceeded -3 ns and the pulse energy was larger than -0.3 GeV. Pulses with lower energies escaped reconstruction and pulses separated by less than -3 ns were reconstructed as a single pulse, so called pile-up. Pileup distorts the electron (positron) time spectrum because of miscounting of the number of pulses and misidentification of the energies and times. Since the phase 4 in Eq. 1 depends on the electron (positron) energy and

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correlates strongly with the frequency w, in fits, pile-up potentially causes a sizable error in the fitted value of w,. It is thus advantageous to apply a correction to the data prior to the fitting. The availability of the WFD trace, as well as the fact that the 3ns interval is smaller than the other time-scales in the experiment, allowed us to do so in a way that is based on the data itself and that is self-normalizing1'. The data collection in the year 2000 resulted in a sample of about 4 . lo9 reconstructed positrons with energies greater than 2 GeV and times between 50 p s and 600 p s following beam injection. A slightly smaller sample of electrons was available for analysis from the data collection in 2001. Figure 5 shows their time spectrum after corrections for pile-up and for the bunched time structure of the beam and had been applied.

Figure 5. The time spectrum of analyzed electrons collected in the year 2001, after corrections for pileup and for the bunched time structure of the injected beam had been made.

The main characteristics of the spectrum were that of muon decay and spin precession (Eq. 1). However, additional effects needed to be considered. These effects included detector gain and time instability, muon losses, and oscillations of the beam as a whole, so-called coherent betatron oscillations

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(CBO). The latter were caused by the injection of the beam through the relatively narrow 18(w) x 57(h)mm2 aperture of the 1.7m long inflector channel into the 90mm diameter aperture of the storage region. Their frequencies are determined by the field focusing index n of the storage ring, which is proportional to the electric field gradient, and have been observed directly with fiber harp monitors that were plunged into the beam region for this purpose. Numerically most important to the determination of w, were CBO in the horizontal plane. In the year 2000 the weak focusing storage ring was operated with a field focusing index n = 0.137, a historical setting which is well away from beam and spin resonances. This setting nevertheless formed a considerable complication since it caused the horizontal CBO frequency Wcbo,h 2 7 ~. 466 kHz to be numerically close to twice the frequency w, N 27~.229 kHz. The interference frequency W,-bo,h - w, was thus close to w,, and since the calorimeter acceptances varied with the muon decay position in the storage ring and with the momentum of the decay positron, a small modulation of the time and energy spectra of the observed positrons resulted. This affected the observed asymmetry and phase, and caused a systematic uncertainty in the fitted frequency w, at the level of 0.2ppm. New working points, n = 0.122 and n = 0.142 corresponding to Wcbo,h 2n 419 kHz and Wcbo,h 2 7 ~.491 kHz, were chosen for the data collection in 2001 to move the interference frequency away from w,. The uncertainty was reduced. The muon frequency value for the data collection in 2001 is, N

N

N

w a / ( 2 n ) = 229073.59(15)(5) Hz (0.7ppm)13,

(4)

where the first uncertainty is statistical and the second is systematic. The value was obtained from five largely independent analyses of mostly the same data. The analyses differed in approach and thus gave somewhat different sensitivity to systematic effects. Two of the analyses fitted directly the time spectrum of electrons in the energy range of 1.8-3.4GeV, using slightly different parametrizations. The third analysis was a likelyhood analysis, which used the observed dependence of the asymmetry on the electron energy to maximize the statistical power and to extend the energy range of the analyzed electrons down to 1.5GeV. The fourth and fifth analyses made use of a cleverly devised data-transformation of the time spectrum, which virtually eliminated the dependency on muon decay and on other effects with time scales larger than 7, 4.4p.9. The results of all five analyses agreed to within the expected statistical fluctuations N

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caused by differences in data selection and in weighting. The frequency in Eq. 4 includes a correction of +0.77(6) ppm for the net contribution to the muon spin precession and momentum rotation caused by vertical beam oscillations and, for muons with y # 29.3, by horizontal electric Table 2. Systematic uncertainties for the wa analysis. The uncertainty "Others" groups uncertainties caused by the Efield and pitch corrections, by beam debunching/randomization, by the fitting procedure and binning, by timing shifts, and for 2000 and 2001 by residual AGS background. Source of errors Coherent betatron oscillations Pileup Gain changes Lost muons AGS background Others Total systematic error on wa Total statistical error on wn.

Size [ppm] 1998 0.2