37247172-lepton

LEPTON DIPOLE MOMENTS

ADVANCED SERIES ON DIRECTIONS IN HIGH ENERGY PHYSICS

PublishedVol. 1 High Energy ElectronPositron Physics (eds. A. Ali and P. Sding)Vol. 2 Hadronic Multiparticle Production (ed. P. Carruthers)Vol. 3 CP Violation (ed. C. Jarlskog)Vol. 4 ProtonAntiproton Collider Physics (eds. G. Altarelli and L. Di Lella)Vol. 5 Perturbative QCD (ed. A. Mueller)Vol. 6 QuarkGluon Plasma (ed. R. C. Hwa)Vol. 7 Quantum Electrodynamics (ed. T. Kinoshita)Vol. 9 Instrumentation in High Energy Physics (ed. F. Sauli)Vol. 10 Heavy Flavours (eds. A. J. Buras and M. Lindner)Vol. 11 Quantum Fields on the Computer (ed. M. Creutz)Vol. 12 Advances of Accelerator Physics and Technologies (ed. H. Schopper)Vol. 13 Perspectives on Higgs Physics (ed. G. L. Kane)Vol. 14 Precision Tests of the Standard Electroweak Model (ed. P. Langacker)Vol. 15 Heavy Flavours II (eds. A. J. Buras and M. Lindner)Vol. 16 Electroweak Symmetry Breaking and New Physics at the TeV Scale

(eds. T. L. Barklow, S. Dawson, H. E. Haber and J. L. Siegrist)Vol. 17 Perspectives on Higgs Physics II (ed. G. L. Kane)Vol. 18 Perspectives on Supersymmetry (ed. G. L. Kane)Vol. 19 Linear Collider Physics in the New Millennium (eds. K. Fujii, D. J. Miller

and A. Soni)

ForthcomingVol. 8 Standard Model, Hadron Phenomenology and Weak Decays on

the Lattice (ed. G. Martinelli)

LEPTON DIPOLE MOMENTS

Advanced Series onDirections in High Energy Physics Vol. 20

Editors

B Lee RobertsBoston University, USA

William J MarcianoBrookhaven National Laboratory, USA

N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

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LEPTON DIPOLE MOMENTSAdvanced Series on Directions in High Energy Physics Vol. 20

Preface

As the title suggests, lepton electromagnetic dipole moments, includinganomalous magnetic, electric, and transition moments, are the main subjectof this volume. Studies of these quantities test the Standard Model ofelementary particle physics at the level of its quantum fluctuations, andsearch for New Physics effects.

Those searches fall into two categories. The first approach entails pre-cision experimental measurements of the electron and muon anomalousmagnetic moments, which can then be compared with theoretical Standard-Model predictions of comparable accuracy. A clear discrepancy would pointto additional contributions of New Physics origin. The second approach in-volves searches for non-vanishing electric, and transition dipole moments(e.g. e). The Standard Model predicts those quantities to be unob-servably small. Hence, discovery of a non-zero value would be interpretedas direct evidence for New Physics.

The measurement and theory of the electron and muon magnetic mo-ments has a long and distinguished history. The former was intimately in-tertwined with the development of quantum electrodynamics, and the calcu-lation of the electron anomalous magnetic moment (anomaly) by Schwingerrepresented the very first quantum-loop computation. Its simple but ele-gant value is inscribed on the memorial marker located near his grave inthe Mount Auburn Cemetery in Cambridge Massachusetts.

v

vi Preface

QED calculations of the electron anomaly have become an industry,with the sixth-order (3-loop) contribution having been calculated analyt-ically by Laporta and Remiddi. The eighth- and tenth-order (4- and 5-loop) contributions have occupied a significant fraction of Kinoshitas ca-reer, and with his collaborators he continues these numerical calculationstoday. Meanwhile, the experiments by Gabrielse and his collaborators havereached the remarkable precision of 0.24 parts per billion on the electronanomaly, some 20 times more precise than independent measurements ofthe fine-structure constant . Chapters by the above-mentioned experts,along with an historical introduction by BLR and a general overview ofelectromagnetic moments by A. Czarnecki and WJM, provide an up-to-datereview of the status of the electron magnetic moment. We also include abrief discussion of the various measurements of by G. Gabrielse and anarticle by K. Pachucki and J. Sapirstein on the theory necessary to extract from helium fine structure. At present, the electron g-value along withthe QED theory provides the best measure of .

The relative sensitivity of the muon anomaly to higher mass scales com-pared to the electron goes as (m/me)2 ' 43, 000, which requires knowl-edge of the hadronic contribution arising from virtual hadrons in vacuumpolarization loops (which dominate the uncertainty on the Standard-Modelvalue of the muon anomaly), as well as the one- and two-loop contribu-tions from the weak gauge bosons, fermions and Higgs scalar. Thus, at thepresent experimental precision for the muon anomaly of 0.54 ppm, there issignificant sensitivity to the several-hundred GeV mass scale. The currentStandard-Model prediction for the muon anomalous magnetic moment andpotential effects due to New Physics are reviewed in chapters by Czarneckiand WJM; M. Davier; J. Prades, E. de Rafael and A. Vainshtein; K. Lynch;and D. Stockinger, while its experimental status is described in a chapterby J. Miller, BLR and K. Jungmann.

Dedicated searches for electric dipole moments (EDMs) date back tothe pioneering observation by Purcell and Ramsey in 1950, that a particleEDM would violate parity, but should nevertheless be searched for as a testof that symmetry. The experimental quest for an EDM of the electron, theneutron, and of atomic nuclei has become an important area in the searchfor physics beyond the Standard Model. The level of precision that has beenreached, < 1.6 1027 e-cm for the electron, < 2.9 1026 e-cm for theneutron and < 3.11029 e-cm for 199Hg, is beginning to challenge modelssuch as supersymmetry. There is substantial hope that the discovery of anEDM will come in the present generation of experiments. Reviews of all

Preface vii

of these searches, along with the related theoretical issues, are covered inthis volume by M. Pospelov and A. Ritz; E. Commins and D. DeMille; S.Lamoreaux and R. Golub; W.C. Griffith, M. Swallows and N. Fortson; allactive experts in the field. The new idea of using storage rings to search forEDMs of charged particles is covered in a chapter by BLR, J. Miller andY. Semertzidis.

The related process, the transition dipole moment that would permitlepton flavor (muon number) violation (LFV) in reactions such as N eN and + e+ are complementary to the studies of electric andmagnetic dipole moments. Since the Standard-Model predictions for suchreactions are suppressed by (m/MW )4 < 1045 and thus experimentallyunobservable, any observation of LFV in the charged sector would signalthe presence of New Physics. Charged lepton transition moments due toNew Physics and experimental searches are covered in the chapters by Y.Okada and Y. Kuno which complete the book.

The idea for this volume came about when after a seminar given atImperial College, BLR was approached by an editor from Imperial CollegePress to write a monograph on muon physics. The counter proposal was avolume dedicated to the topics covered at the series of symposia on LeptonMoments started by Klaus Jungmann in Heidelberg in 1999 and continuedby BLR on Cape Cod in 2003, 2006 and planned for 2010. We are indeedgrateful that so many of our friends and colleagues have joined with usto create this volume. We gratefully acknowledge Kevin R. Lynch for hisencyclopedic expertise in LaTeX, which he used to solve numerous issuesin putting this document together.

We dedicate this volume to Norman Ramsey, and to the memory of PaulDirac, Julian Schwinger, Polykarp Kusch and Edward Purcell, all picturedon the next page, who carried out the seminal work which began our modernjourney through the field of magnetic and electric dipole moments.

B. Lee Roberts and William J. Marciano

viii Preface

Clockwise:

Julian Schwinger,

Polykarp Kusch,

Paul Dirac,

Norman Ramsey and

Edward Purcell

Courtesy AIP Emilio

Segr Visual Archives

(full credits overleaf)

Preface ix

Photo credits: Schwinger memorial marker, photo by BLR; Schwingerphoto from AIP Emilio Segre` Visual Archives; Kusch photo from NationalArchives and Records Administration (NARA), courtesy AIP Emilio Segre`Visual Archives, Physics Today Collection, W. F. Meggers Gallery of NobelLaureates; Dirac photo from AIP Emilio Segre` Visual Archives; Ramseyphoto from AIP Emilio Segre` Visual Archives, Ramsey Collection; Purcellphoto from AIP Emilio Segre` Visual Archives, Physics Today Collection,W. F. Meggers Gallery of Nobel Laureates.

This page intentionally left blankThis page intentionally left blank

Contents

Preface v

1. Historical Introduction 1

B. Lee Roberts

2. Electromagnetic Dipole Moments and New Physics 11

Andrzej Czarnecki and William J. Marciano

3. Lepton g 2 from 1947 to Present 69Toichiro Kinoshita

4. Analytic QED Calculations of the Anomalous MagneticMoment of the Electron 119

Stefano Laporta and Ettore Remiddi

5. Measurements of the Electron Magnetic Moment 157

G. Gabrielse

6. Determining the Fine Structure Constant 195

G. Gabrielse

xi

xii Contents

7. Helium Fine Structure Theory for the Determinationof 219

Krzysztof Pachucki and Jonathan Sapirstein

8. Hadronic Vacuum Polarization and the LeptonAnomalous Magnetic Moments 273

Michel Davier

9. The Hadronic Light-by-Light Contribution to a,e 303

Joaquim Prades, Eduardo de Rafael and Arkady Vainshtein

10. General Prescriptions for One-loop Contributions to ae, 319

Kevin R. Lynch

11. Measurement of the Muon (g 2) Value 333James P. Miller, B. Lee Roberts and Klaus Jungmann

12. Muon (g 2) and Physics Beyond the Standard Model 393Dominik Stockinger

13. Probing CP Violation with Electric Dipole Moments 439

Maxim Pospelov and Adam Ritz

14. The Electric Dipole Moment of the Electron 519

Eugene D. Commins and David DeMille

15. Neutron EDM Experiments 583

Steve K. Lamoreaux and Robert Golub

16. Nuclear Electric Dipole Moments 635

W. Clark Griffith, Matthew Swallows and Norval Fortson

Contents xiii

17. EDM Measurements in Storage Rings 655

B. Lee Roberts, James P. Miller and Yannis K. Semertzidis

18. Models of Lepton Flavor Violation 683

Yasuhiro Okada

19. Search for the Charged Lepton-Flavor-ViolatingTransition Moments l l 701Yoshitaka Kuno

Epilogue 747

Subject Index 749

Chapter 1

Historical Introduction to Electric and Magnetic Moments

B. Lee Roberts

Department of Physics, Boston UniversityBoston, MA 01890 U.S.A.

[email protected]

The historical development of the discovery of spin and magnetic mo-ments is reviewed, along with the development of searches for electricdipole moments.

Contents

1.1 The Discovery of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Diracs Theory and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 The discovery of anomalous magnetic moments . . . . . . . . . . . . . 4

1.3 The Search for Electric Dipole Moments . . . . . . . . . . . . . . . . . . . . . 6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1. The Discovery of Spin

As physics developed at the beginning of the 20th century, a number ofintriguing puzzles existed that could only be explained by radically newideas. In 1911 Rutherford proposed the nuclear atom [1]. This hypothe-sis, combined with Thompsons discovery of the electron [2] and Millikansdiscovery that the electron charge is quantized [3], implied that electronswere somehow in orbit around the positive nucleus, leading to a neutralatom. Classically such a system is unstable, and in 1913 Bohr proposedhis quantum theory [4]. Of course, many conceptual problems remained,which began to be understood once Schrodingers wave equation [5] waspublished in 1926.

In 1921, two interesting proposals were published: Compton pro-posed [6] a spinning electron to explain ferromagnetism, which he realized

1

2 B. Lee Roberts

was difficult to explain by any other means.a Stern proposed an experimentto study space quantization [7] to test the Sommerfeld quantum theory,where he presented the details of what we now call the SternGerlach ex-periment. An atomic beam of silver atoms was to be projected through agradient magnetic field where the net force on the magnetic dipole wouldseparate the different magnetic quantum states. For a classical dipole thedeflection would be continuous, since the direction of the dipole momentcould have any value.b

Over the next two years the famous experiments were carried out [8],and the two-band structure observed. By 1924, Stern and Gerlach con-cluded that to within 10%, the magnetic moment of the silver atom in itsground state was one Bohr magneton [9]. Their papers made no referenceto the developments in spectroscopy, and in their 1924 review article, noconclusions beyond the magnetic moment were drawn from the two-bandstructure.

Independently, in 1925 Uhlenbeck and Goudsmit [10] proposed thespinning electron to explain the fine-structure observed in the anomalousZeeman effect in atomic spectra.c The fine-structure splitting can be un-derstood as the interaction of the magnetic dipole moment of the electronwith the magnetic field produced by the nuclear motion, which in the elec-trons rest frame appears to be orbiting about the electron. The electronsmagnetic dipole moment is along its spin and is given by

~ = g( q2m

)~s , (1.1)

where q = e is the charge of the particle in terms of the magnitude ofthe electron charge e, and the proportionality constant g is the g-factorfor spin (which is sometimes written as gs). In their second paper [11],Uhlenbeck and Goudsmit conclude that the g-factor for spin is twice that fororbital angular momentum, however the calculated fine-structure splittingwas then twice as large as the observed splitting. Only later in 1926, whenThomas showed that the factor of 2 discrepancy between experiment andcalculation was a kinematic effect [12], did spin start to become an acceptedaIn his paper Compton acknowledges A.L. Parson (Smithsonian Misc. Collections, 1915)as first proposing that the electron was a spinning ring of charge. Compton modified thisproposal to be a much smaller distribution concentrated principally near its center.Comptons paper is almost unknown.bSee Allan Franklin, http://plato.stanford.edu/entries/physics-experiment/app5.htmlStanford Encyclopedia of Philosophy, Appendix 5, for a nice discussion putting theSternGerlach experiment into historical context.cIn their Nature paper [11] of 1926, they acknowledge Comptons independent suggestionof spin.

Historical Introduction 3

theory. Thomas later wrote to Goudsmit, indicating that Kronig had alsosuggested spin [13]:

I think you and Uhlenbeck have been very lucky to get yourspinning electron published and talked about before Pauli heardof it. It appears that more than a year ago, Kronig believedin the spinning electron and worked out something; the firstperson he showed it to was Pauli. Pauli ridiculed the wholething so much that the first person became also the last and noone else heard anything of it. Which all goes to show that theinfallibility of the Deity does not extend to his self-styled vicaron earth.

Incidentally, no mention is made of the SternGerlach measurements inthe Uhlenbeck and Goudsmit papers. However, the SternGerlach resultwas noticed by Phipps and Taylor at the University of Illinois at Urbana,and they did draw the connection between the SternGerlach experimentand the electron spin proposed by Uhlenbeck and Goudsmit. They repeatedthe SternGerlach experiment with an atomic beam of hydrogen in 1926.While technically more challenging than the silver experiment, they reacheda similar conclusion, viz. that the magnetic moment of the hydrogen atomwas also one Bohr magneton [14].

Today, we understand that the magnetic moment measured in both ofthese atomic-beam experiments was that of the un-paired atomic electron.We can conclude that a magnetic moment of one Bohr magneton impliesthat the g-factor for spin is 2. Although in our undergraduate modernphysics courses we emphasize that the SternGerlach experiment showedclearly the existence of half-integer spin, historically it seems to have playeda much less important role than spectroscopy did.d In his book, The Storyof Spin, Tomonaga does not mention the SternGerlach result [16].

1.2. Diracs Theory and Beyond

It was not until Diracs famous 1928 paper [17], where he introduced hisrelativistic wave equation for the electron, that the picture became clear.Dirac pointed out that an electron in external electric and magnetic fieldshas the two extra termse

e~c(,H) + i

e~c1 (,E) , (1.2)

dThe recollections of Goudsmit agree with this assessment, see Ref. [15].eHere we use Diracs original notation.

4 B. Lee Roberts

. . . when divided by the factor 2m can be regarded as the additional poten-tial energy of the electron due to its new degree of freedom. These termsrepresent the magnetic dipole (Dirac) moment and electric dipole momentinteractions with the external magnetic and electric fields.f Dirac theorypredicts that the electron magnetic moment is one Bohr-magneton (viz.g = 2), consistent with the value measured by the experiments.g Diraclater commented: It gave just the properties that one needed for an elec-tron. That was an unexpected bonus for me, completely unexpected [18].

As an aside, Dirac had little use for the electric dipole moment (EDM),and stated The electric moment, being a pure imaginary, we should notexpect to appear in the model. It is doubtful whether the electric momenthas any physical meaning, since the Hamiltonian . . . that we started fromis real, and the imaginary part only appeared when we multiplied it up inan artificial way in order to make it resemble the Hamiltonian of previoustheories. We now understand that the presence of an electric dipole mo-ment violates both parity (P) and time reversal (T) symmetries, and CPas well if CPT holds.

1.2.1. The discovery of anomalous magnetic moments

For some years, the experimental situation remained the same. The elec-tron had g = 2, and the Dirac equation seemed to describe nature. Then asurprising and completely unexpected result was obtained. In 1933, againstthe advice of Pauli who believed that the proton was a pure Dirac parti-cle [16], Stern and his collaborators [19] showed that the g-factor of theproton was 5.5, a long way from the expected value of 2. Even moresurprising was the discovery in 1940 by Alvarez and Bloch [20] that theneutron had a large magnetic moment (see Eq. (1.1)). These two resultsremained quite mysterious for many years, and are still not perfectly un-derstood. With the advent of the quark model, one does get a 10 to 20%description of baryon magnetic moments, but given that experiments showthat very little of the proton spin is carried by the quarks, the whole spinstructure of baryons remains a topic of investigation.h It became convenient

fHowever, it appears that the Dirac complex phase is an artifact of his second-orderformalism analysis rather than a real EDM.gThe Dirac equation also predicts that the g-factor associated with orbital angularmomentum g` = 1.hA.W. Thomas claims that this crisis is resolved [21], but according to R.L. Jaffe [22]this is a minority view.


to break the magnetic moment into two pieces:

= (1 + a)q~2m

where a =g 22

. (1.3)

The first piece, predicted by the Dirac equation and called the Dirac mo-ment, is 1 in units of the appropriate magneton, q~/2m. The second pieceis the anomalous (Pauli) moment [23], where the dimensionless quantity ais sometimes referred to as the anomaly.

The development of radio frequency engineering and microwave tech-nology during the Second World War was quickly put to use afterwardin the laboratory. In 1947, motivated by measurements of the hyperfinestructure in hydrogen that obtained splittings larger than expected fromthe Dirac theory [2426], Schwinger [27] showed that from a theoreticalviewpoint these discrepancies can be accounted for by a small additionalelectron spin magnetic moment that arises from the lowest-order radiativecorrection to the Dirac moment.i In his paper, Schwinger points out threeimportant features of his new theory.

The new Hamiltonian is superior to the original one in essen-tially three ways: it involves the experimental electron mass,rather than the unobservable mechanical mass; an electron nowinteracts with the radiation field only in the presence of an ex-ternal field . . . the interaction of an electron with an externalfield is now subject to a finite radiative correction.

In todays language, Schwinger pointed out that one replaces the bare massand charge with the physical (dressed) mass and charge (see Chapter 3 foradditional details).

The one-loop contribution to a is shown diagrammatically in Fig. 1.1(b)and has the value ae = /(2pi) ' 0.00116 , which is independent of massand is the same for a and a .

In the same year, Kusch and Foley [29] measured ae with 4% precision,and found that the measured electron anomaly agreed well with Schwingersprediction. They state that: ... the results can be described by g` = 1 andgs = 2(1.00119 0.00005).jiIn response to Nafe, et al. [24], Breit [28] conjectured that this discrepancy could beexplained by the presence of a small Pauli moment. Its not clear whether this paperinfluenced Schwingers work, but in a footnote Schwinger states: However, Breit hasnot correctly drawn the consequences of his empirical hypothesis.jThe choice that g` = 1 and gs > 2 was guided by theoretical prejudice. The modernexperiments, which confine a single electron in a Penning trap, measure gs directly andfully justify this assumption.

6 B. Lee Roberts

(a) (b) (c)

e

+e

e

e

ee

SchwingerDirac

e

Fig. 1.1. The Feynman graphs for: (a) g = 2; (b) the lowest-order radiative correctionfirst calculated by Schwinger; and (c) the vacuum polarization contribution, which is oneof five fourth-order, (/pi)2, terms.

In the intervening time since the Kusch and Foley paper, many improve-ments have been made in the precision of the electron anomaly [3032], aswell as in the theory (see Chapters 3 and 4). Most recently, ae has beenmeasured to a relative precision of 0.24 ppb (parts per billion) [32], and thecomparison with theory is limited by the knowledge of the fine-structureconstant, . See Chapters 3 and 6 for the most recent theory and experi-mental values of ae.

The ability to calculate the higher-order QED contributions to theanomaly has gone well beyond what could have been imagined by the in-ventors. In response to a question about how the QED pioneers viewed thetheory Freeman Dyson said [33]:

The main point was that all of us who put QED together, in-cluding especially Feynman, considered it a jerry-built and pro-visional structure which would either collapse or be replaced bysomething more permanent within a few years. So I find itamazing that it has lasted for fifty years and still agrees withexperiments to twelve significant figures. It seems that Natureis telling us something. Perhaps she is telling us that she lovessloppiness.

The muon anomaly has been measured to a precision of 0.54 ppm [34].Naively, this level of precision would seem to limit the physics reach of themuon anomaly when compared to that of the electron. However, since therelative sensitivity of the anomaly to higher mass scales goes as (m/me)2 '43, 000, the muon anomaly has measurable sensitivity up to the severalhundred GeV scale, as discussed in the Chapter 2.

1.3. The Search for Electric Dipole Moments

Dirac [17] discovered an electric dipole moment (EDM) term in his relativis-tic electron theory. Like the magnetic dipole moment, the electric dipole


moment must be along the spin. We can write an expression similar toEq. (1.1),

~d = ( q2mc

)~s , (1.4)

where is a dimensionless constant that is analogous to g in Eq. (1.1).While magnetic dipole moments (MDMs) are a natural property of chargedparticles with spin, electric dipole moments (EDMs) are forbidden both byparity and by time reversal symmetry.

The search for an EDM dates back to the suggestion of Purcell andRamsey [35] in 1950, well in advance of the paper by Lee and Yang [36],that a measurement of the neutron EDM would be a good way to searchfor parity violation in the nuclear force. An experiment was mounted atOak Ridge [37] soon thereafter which placed a limit on the neutron EDMof dn < 5 1020 e-cm, although the result was not published until afterthe discovery of parity violation.

Once parity violation was established, Landau [38] and Ramsey [39]pointed out that an EDM would violate both P and T symmetries. Thiscan be seen by examining the Hamiltonian for a spin one-half particle inthe presence of both an electric and magnetic field,

H = ~ ~B ~d ~E. (1.5)The transformation properties of ~E, ~B, ~ and ~d are given in Table 1.1, andwe see that while ~ ~B is even under all three symmetries, ~d ~E is odd underboth P and T. Thus the existence of an EDM implies that both P and Tare not good symmetries of the interaction Hamiltonian, Eq. (1.5). In thecontext of CPT symmetry, an EDM implies CP violation.

Table 1.1. Transformationproperties of the magnetic andelectric fields and dipole mo-ments.

~E ~B ~ or ~d

P - + +C - - -T + - -

The Standard Model value for the electron (muon) EDM is 1038e-cm ( 2 1036 e-cm), well beyond the reach of experiments (which areat the 1.6 1027 (1.8 1019) e-cm level). Likewise, the Standard-Model

8 B. Lee Roberts

value for the neutron is 1032 e-cm, with the present experimental limit of2.9 1026e-cm.

Concerning these symmetries, Ramsey states [39]:

However, it should be emphasized that while such argumentsare appealing from the point of view of symmetry, they are notnecessarily valid. Ultimately the validity of all such symmetryarguments must rest on experiment.

Fortunately this advice has been followed by many experimental investiga-tors during the intervening 50 years. Today the searches for a (CP violat-ing) permanent electric dipole moment of the electron, neutron, and of anatomic nucleus have become an important part of the search for physicsbeyond the Standard Model. Since the Standard Model CP violation ob-served in the neutral kaon and B-meson systems is inadequate to explainthe predominance of matter over antimatter in the universe, the search fornew sources of CP violation beyond that embodied in the CKM formal-ism takes on a certain urgency. These searches, along with the relevanttheoretical framework, form a major portion of this volume.

References

[1] E. Rutherford, Proc. of the Manch. Lit. and Phil. Soc., IV, 55, (1911) 18,and Phil. Mag., Series 6, 21 (1911) 669.

[2] J.J. Thompson, Phil. Mag. 44 (1897) 293.[3] R.A. Millikan, Phys. Mag. XIX, 6 (1910) 209.[4] N. Bohr, Phil. Mag. 26, 1 (1913).[5] E. Schrodinger, Ann. Phys. 79 (1926) 361.[6] A.K. Compton, Jour. Franklin. Inst., 192 Aug. (1921) 145.[7] O. Stern, Z. Phys. 7, 249 (1921).[8] W. Gerlach and O. Stern, , Z. Phys. 8, 110 (1922), Z. Phys. 9 and 349(1922),

Z. Phys. 9, 353 (1924).[9] W. Gerlach and O. Stern, Ann. Phys. 74, 673 (1924).[10] G.E. Uhlenbeck and S. Goudsmit, Naturwissenschaften 47, 953 (1925).[11] G.E. Uhlenbeck and S. Goudsmit, Nature 117 (1926) 264.[12] L.H. Thomas, Nature 117, (1926) 514 and Phil. Mag. 3 (1927) 1.[13] From a letter by L.H. Thomas to Goudsmit (25 March 1926). A reproduc-

tion from a transparency shown by Goudsmit during his 1971 lecture atLeiden [15]. The original is presumably in the Goudsmit archive kept by theAmerican Institute of Physics Center for History of Physics.

[14] T.E. Phipps and J.B. Taylor, Phys. Rev. 29, 309 (1927).[15] http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.htm[16] Sin-itiro Tomonaga, The Story of Spin, translated by Takeshi Oka, U.

Chicago Press, 1997.


[17] P.A.M. Dirac, Proc. R. Soc. (London) A117, 610 (1928), and A118, 351(1928). See also, P.A.M. Dirac, The Principles of Quantum Mechanics, 4thedition, Oxford University Press, London, 1958.

[18] Abraham Pais in Paul Dirac: The Man and His Work, P. Goddard, ed.,Cambridge U. Press, New York (1998).

[19] R. Frisch and O. Stern, Z. Phys. 85, 4 (1933), and I. Estermann and O.Stern, Z. Phys. 85, 17 (1933).

[20] Luis W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940).[21] A.W. Thomas, Prog. Part. Nucl. Phys. 61, 219 (2008), (arXiv:0805.4437v1).[22] R.L. Jaffe, private communication, Nov. 2008 and

http://www.bnl.gov/gbunce/talks.asp[23] Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and

Two-Electron Atoms, Springer-Verlag, (1957), p. 51.[24] J.E. Nafe, E.B. Nelson and I.I. Rabi Phys. Rev. 71, 914(1947).[25] D.E. Nagel, R.S. Julian and J.R. Zacharias, Phys. Rev. 72, 971 (1947).[26] P. Kusch and H.M Foley, Phys. Rev 72, 1256 (1947).[27] J. Schwinger, Phys. Rev. 73, 416L (1948), and Phys. Rev. 76 790 (1949). The

former paper contains a misprint in the expression for ae that is correctedin the longer paper.

[28] G. Breit, Phys. Rev. 72 984, (1947).[29] P. Kusch and H.M Foley, Phys. Rev. 73, 250 (1948).[30] See Arthur Rich and JohnWesley, Reviews of Modern Physics 44, 250 (1972)

for a nice historical overview of the lepton g - factors.[31] R.S. Van Dyck et al., Phys. Rev. Lett., 59, 26(1987) and in Quantum Elec-

trodynamics, (Directions in High Energy Physics Vol. 7) T. Kinoshita d.,World Scientific, 1990, p. 322.

[32] D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett. 100, 120801,(2008).

[33] F. Dyson, private communication to BLR, December 2006.[34] G. Bennett, et al., (Muon (g 2) Collaboration), Phys. Rev. D73, 072003

(2006).[35] E.M. Purcell and N.F. Ramsey, Phys. Rev. 78, 807 (1950).[36] T.D. Lee and C.N. Yang, Phys. Rev. 104 (1956) 254.[37] J.H. Smith, E.M. Purcell and N.F. Ramsey, Phys. Rev. 108, 120 (1957).[38] L. Landau, Nucl. Phys. 3, 127 (1957).[39] N.F. Ramsey Phys. Rev. 109, 225 (1958).

Chapter 2

Electromagnetic Dipole Moments and New Physics

Andrzej Czarnecki

Department of Physics, University of Alberta,Edmonton, AB, Canada T6G 2G7

William J. Marciano

Physics Department, Brookhaven National Laboratory,Upton, NY 11973, USA

As an introduction to the more detailed chapters that follow, we presenta general overview of spin 1/2 fermion electromagnetic dipole momentsproduced by quantum loop effects. Standard Model predictions are givenand possible New Physics contributions are parameterized in terms ofthe mass scale responsible for anomalous magnetic, electric, and tran-sition dipole moments. Experimental measurements and bounds arediscussed. The muon anomalous magnetic moment is covered in somedetail because it may already be exhibiting signs of New Physics. Elec-tron and neutron electric dipole moments along with e transitionmoments are shown to have New Physics sensitivities extending up toO (1000 TeV) mass scales, modulo CP and flavor violation suppressions.Various other less constraining fermion dipole moments are discussed.

Contents

2.1 The Dirac Equation and Electron Dipole Moments . . . . . . . . . . . . . . . 12

2.1.1 Electron anomalous magnetic moment . . . . . . . . . . . . . . . . . . 14

2.1.2 Electron electric dipole moment . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Spin 1/2 Electromagnetic Form Factors . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Lepton anomalous magnetic and electric dipole moments . . . . . . . . 18

2.2.2 Nucleon dipole moments . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3 Complex formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.4 Transition dipole moments . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Muon Anomalous Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.2 a in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . 34

11

12 Andrzej Czarnecki and William J. Marciano

2.3.3 New Physics effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Flavor Violating Transition Dipole Moments . . . . . . . . . . . . . . . . . . 51

2.4.1 Muon flavor violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.4.2 The New Physics connection between a and e . . . . . . . . . 562.4.3 Tau flavor violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4.4 Neutrino transition dipole moments . . . . . . . . . . . . . . . . . . . . 57

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.1. The Dirac Equation and Electron Dipole Moments

The Dirac equation [1, 2],

i ( ieA (x)) (x) = me (x) , (2.1)introduced in 1928 is a cornerstone of modern physics. Using the now fa-mous four-by-four Dirac matrices, it succinctly describes a four compo-nent (spinor) electron wave function, (x), in an electromagnetic potentialA (x). That elegant equation combined quantum mechanics, special rela-tivity, spin and electromagnetic gauge invariance in one simple expressionand laid the foundation for later developments in quantum electrodynam-ics (QED). Today, it provides a basis for our SU (3)c SU (2)L U (1)YStandard Model of elementary particle physics.

The Dirac equation is primarily acclaimed for its (later realized [3])prediction of antimatter, corresponding to negative energy solutions. Sub-sequent discovery of the positron, the electrons antimatter partner, wasthus its crowning glory. However, it left us with a modern day puzzle as towhy Nature chose to populate our Universe with matter and not antimat-ter, i.e. why is it so matter-antimatter asymmetric? Resolving that puzzlewill likely require New Physics beyond Standard Model expectations. Oneof the necessary ingredients [4] is expected to be a new source of CP viola-tion that differentiates the properties of particles and antiparticles. As weshall see, a signature of that New Physics could be the existence of particleelectric dipole moments (EDMs) [5], one of the main topics of this chapterand book.

The immediate success of the Dirac equation was not, however, to pre-dict antimatter. It was the explanation [6] as to why the gyromagneticratio, ge, of the electron is equal to 2. That parameter, which expresses therelationship between the electrons magnetic moment, ~e, and its spin ~s,

~e = geQe

2me~s (2.2)

Electromagnetic Dipole Moments and New Physics 13

would be 1 if it were relating atomic orbital angular momentum and itsassociated magnetic moment.a Of course, the need for ge = 2 was alreadywell established by atomic fine structure spectroscopy before 1928. Nev-ertheless, the Dirac equation provided a natural explanation and strongunderpinning for that fundamental value.

A deviation from ge = 2 can be easily accommodated, if necessary, byadding a so-called Pauli interaction term [7, 8],

e

4meaeF(x)(x) (2.3)

F = A A, (2.4) =

i

2[, ] , (2.5)

to Eq. (2.1), where ae is called the anomalous magnetic moment because itleads to

ge = 2 (1 + ae) (2.6)

i.e. an increase in the intrinsic magnetic dipole moment by ae e2me . Such anaddition is very much required for the proton, where one finds [9] gp ' 5.59rather than 2 (see Section 2.2) due to its underlying quark substructure.However, Dirac had no need for a Pauli term, since it was known in 1928that ge = 2 with rather good certainty.

What forbids the addition of a Pauli term for an elementary spin 1/2fermion such as the electron? That term respects Lorentz covariance andlocal gauge invariance; however, it runs counter to Diracs principles ofelegance and simplicity as well as his use of minimal coupling (the replace-ment of by the covariant derivative ieA in the non-interacting Diracequation).

Today, we would automatically exclude Pauli terms at the level of ourfundamental classical interaction Lagrangian because they correspond towhat are called dimension 5 operators which are known to spoil renormal-izability. However, such dimension 5 terms can and do arise in quantumfield theories as a result of virtual loop fluctuations. In that respect, theirexistence is to be expected and they can be viewed as a window to quantumloops including effects due to heavy new particles with masses well abovedirect experimental accessibility. That feature forms the main theme ofaWe define e > 0 and Q = 1 for electrons, Q = +1 for positrons. Many field theorytexts employ e < 0 as the electron charge and express all derived results in terms of thatnegative quantity. Such an approach is a little awkward for EDMs where the unit ecmis conventionally used, since negative units can lead to sign inconsistencies.


this chapter, how measurements of various quantum induced dimension 5dipole operators can be used to provide indirect evidence for New Physicsor at least constrain speculations regarding its properties.

2.1.1. Electron anomalous magnetic moment

In 1947, small anomalous effects at about the 0.1 percent level began tobe observed in high precision atomic hyperfine spectroscopy [10, 11]. Breitsuggested [12], on empirical grounds, that such observations could be ex-plained if ge deviated slightly from 2. Schwinger then demonstrated [13]the power of QED and his own computational prowess by calculating theleading quantum contribution to ae,

ae =ge 22

=

2pi' 0.00116, (2.7)

where = e2/4pi ' 1/137 is the fine structure constant.His result agreed with experiment [14] and ushered in an era of preci-

sion measurements that tested the validity of QED to high order in andsearched for deviations that might indicate the presence of New Physics.

Today, as a result of many pioneering efforts, including the Nobel prizewinning experiments of H. Dehmelt and his collaborators [15], the electronanomalous magnetic moment has been measured with phenomenal accuracy(see Chapter 5). The most precise value, due to Hanneke, Fogwell andGabrielse [16] is currently

aexpe =ge 22

= 0.001 159 652 180 73 (28) , (2.8)

where the numbers in parenthesis represent the one sigma uncertainty in thelast two decimal places. That result is truly impressive. It can be comparedwith the four-loop QED prediction and estimated five-loop uncertainty (dueto the heroic work of many theorists, see Chapters 3, 4 and Ref. [17])

aSMe =

2pi 0.328 478 444 003

(pi

)2+ 1.181 234 016 8

(pi

)31.9144(35)

(pi

)4+ 0.0(4.6)

(pi

)5+ 1.71 1012 (2.9)

where we have truncated the two and three loop numerical coefficients at thelevel of their uncertainty (due to uncertainties in the muon and tau leptonmasses m and m ) and have included a small Standard Model correction(1.71 1012) due to hadronic loops (1.68 1012) and electroweak effects(0.03 1012).


Equations (2.8) and (2.9) can be compared in two different ways. First,assuming no New Physics, they can be equated to give the worlds mostprecise determination of the fine structure constant

1 (ae) = 137.035 999 084 (51) , (2.10)where the uncertainty comes from Eq. (2.8) and the error in Eq. (2.9).Alternatively, one can take a more direct low energy atomic physics orcondensed matter determination of and obtain a numerical predictionfrom Eq. (2.9) which can be compared with Eq. (2.8). Using the recentRydberg based value [18] (which is next best after Eq. (2.10))

1 (Rydberg) = 137.035 999 450 (620) (2.11)leads to

aSMe (Rydberg) = 0.001 159 652 177 60 (520) . (2.12)That prediction agrees with Eq. (2.8) but its error is almost 20 times larger.If New Physics is contributing to aexpe , its contribution must satisfy

|ae (New Physics)| =aexpe aSMe < 1011. (2.13)

That bound could be improved by more than an order of magnitude if were much better independently determined [19].

How large a value of ae (New Physics) might be expected from new shortdistance interactions parametrized by the mass scale ? Because anoma-lous magnetic moments change chirality (R L), we expect New Physicscontributions to ae e2me to vanish in the chiral limit me 0. Therefore, oneanticipates the quadratic dependenceb

ae (New Physics) = C(me

)2(2.14)

where C could be O (1) (see Section 2.3) or smaller, e.g. O () in weak cou-pling loop scenarios. Taking C ' 1, we find from Eq. (2.13), that


2.1.2. Electron electric dipole moment

If instead of adding the Pauli term to the Dirac equation, we were to appenda

i

2deF (x)5 (x) (2.15)

interaction, it would correspond to an electron electric dipole moment(EDM), de [20], interacting with the external electromagnetic fields F (x).Apparently, Dirac noted the possibility of EDM effects (see Chapter 1) butdismissed them as unphysical.

EDMs violate the discrete symmetries of P (parity) and T (time rever-sal) [2124]. Of course, we now know that both symmetries are violatedby weak interactions [25]; so, we should expect at some level de 6= 0 dueto Standard Model loop effects. It has been estimated, that such an effectarising from quark mixing via the CKM matrix [25, 26] (from four-looporder) is roughlydSMe ' 1038 e cm Standard Model. (2.16)In other words, dSMe is unobservably small, since current experiments probede O

(1027

)e cm and it is hard to imagine improvements in sensitivity

by more than ten orders of magnitude. However, New Physics EDM effectsthat violate P and T could arise from one or two loop order and be muchlarger than the tiny Standard Model prediction even if they stem from highmass scales.

Parameterizing the effect of New Physics (NP) on ae and de by therelationship (see Section 2.2 for a discussion)

de (New Physics) = ae (New Physics)e

2metanNPe (2.17)

with NPe a new physics model dependent phase, we can relate ae and desensitivities. Using the experimental constraint from atomic physics [27],

|de| < 2 1027 e cm (2.18)or in units of e/2me (electron Bohr magneton)

|de| < 1 1016 e2me (2.19)we find by comparing Eqs. (2.13) and (2.19) and employing Eq. (2.17)that de provides a better constraint on New Physics than ae by about105 tanNPe , i.e. it already explores scales of about 50TeV C tanNPe . If C tanNPe O (1), that represents extremely good sen-

sitivity. Even for C tanNPe ' 0.01, 5 TeV is competitive with the


scale of physics being directly explored at the LHC (Large Hadron Col-lider). See [28] for a recent example. That simple comparison suggests thatthe electron EDM is a particularly good place to look for a new sourceof P and T (CP ) violation. One that may, in fact, be linked with thematter-antimatter asymmetry of our Universe and thus responsible for ourexistence. Indeed, in some supersymmetric models, a non-zero de is oftenpredicted to be close at hand (see Chapter 13). Since the Standard Modelprediction for de is currently negligible and does not present a backgroundproblem, searches for a de 6= 0 should be pushed as far as technologicallypossible. It is expected that planned experiments will improve de sensi-tivity by more than two orders of magnitude, reaching for C tanNPe 1scales of New Physics approaching O (1000TeV). Alternatively, for lowscale New Physics scenarios with ' 200 GeV, such as supersymmetry,C tanNP as small as 4 108 will be probed.

2.2. Spin 1/2 Electromagnetic Form Factors

Having described the sensitivity of electron anomalous magnetic and elec-tric dipole moments for probing New Physics via dimension 5 induced op-erators, we now present a general field theory based analysis applicable todipole moments of arbitrary spin 1/2 fermions, elementary or composite.We also discuss flavor-changing, dimension 5, electromagnetic transitiondipole moments that allow for the decay e and related reactions.

Our discussion begins with the matrix element of the electromagneticcurrent Jem = e

f Qf ff , between initial and final states of an arbitrary

spin 1/2 fermion f , with momenta p and p respectively (so that q = pp)f (p)

Jem f (p) = uf (p) uf (p) (2.20)where uf and uf are Dirac spinor fields and has the general Lorentzstructure

= F1(q2) + iF2

(q2)q

F3(q2)q

5

+FA(q2) (q

2 2mfq)5. (2.21)

Hermiticity of Jem requires that the form factors in Eq. (2.21) be real(modulo unstable particle effects).

The three Fi(q2), i = 1, 2, 3 in Eq. (2.21) are the charge, anomalous

magnetic dipole and electric dipole form factors. FA(q2)is called the

anapole form factor. Anapole effects violate parity and are generally acomponent of electroweak loop physics. Although interesting, we will notdiscuss anapole induced interactions in this article.


The static charge and dipole moments are defined at q2 = 0,

F1 (0) = Qfe = electric charge, (2.22)

F2 (0) = afQfe

2mf= anomalous magnetic moment, (2.23)

F3 (0) = dfQf = electric dipole moment. (2.24)

The effective (quantum loop induced) Hamiltonian that gives rise to F2 andF3 interactions is

Hdipole = 12(F2f (x)f (x) + iF3f (x)5f (x)

)F (x) , (2.25)

F (x) = A (x) A (x) . (2.26)In the case of neutral spin 1/2 particles with Qf = 0, such as neutrons or(Dirac) neutrinos, F2,3(0) 6= 0 and Qf parameterization in Eqs. (2.23, 2.24)is not appropriate; so, we take Qf 1 depending on the charge of theirisospin partner, e.g. Qf 1 for the neutron and 1 for the neutrino. Inthe non-relativistic limit, the electric dipole interaction reduces to df~s ~E.That term is odd (changes sign) under P and T transformations, hence, itviolates both symmetries [21].

In modern terminology, the interactions in Eq. (2.25) are called dimen-sion 5 operators. That nomenclature stems from the fact that spinor fieldshave dimension 3/2 while the dimension of F is 2. Hence, the field prod-ucts in Eq. (2.25) have dimension 5. Since the Hamiltonian has overalldimension 4, the form factors F2 and F3 are necessarily of dimension 1(they behave like 1/M).

Dimension 5 operators are generally not allowed in fundamental classi-cal Lagrangians because they spoil renormalizability at the quantum fieldtheory level. They will, nevertheless, arise at the quantum loop level, ifno symmetry forbids them. As such, both af and df must be finite andcalculable in terms of other parameters of the theory. Unfortunately, theycan often be difficult to reliably compute because they can be due to highorders in loop perturbation theory, may be clouded by strong interactionuncertainties or, in the case of EDMs, depend on unknown model dependentphases.

2.2.1. Lepton anomalous magnetic and electricdipole moments

In Table 2.1, we list the current measured values of the electron and muonanomalous magnetic moments. Note that ae is more precisely determined


Table 2.1. Measured values and bounds for charged lepton anomalous magnetic andelectric dipole moments. EDM constraints are given in ecm as well as e/2ml magnetonunits. The muon EDM bound has recently been submitted for publication [29].

Lepton (l) al |dl|

electron 115 965 218 073(28) 1014 < 2 1027 e cm ' 1 1016 e2me

muon 116 592 080(63) 1011 < 1.8 1019 e cm ' 2 106 e2m

tau < 2 102 < 1016 e cm ' 2 102 e2m

than a roughly by a factor of 2300. However, New Physics contributionsare expected to scale as m2l for both l = e and . So, in general, a should

be about(mme

)2' 43 000 timesc more sensitive to New Physics than ae.

In addition, the Standard Model prediction for ae, as noted in Section 2.1,falls short of aexpe in precision by about a factor of 20 primarily due to theuncertainty in . So, overall a is currently about 400 times more sensitiveto New Physics than ae and probes mass scales about 20 times higher.Indeed, already a sizable difference between aexp and the Standard Modelprediction, aSM , exists and can easily be identified with various reasonableexamples of New Physics with extending into the TeV region. Thatpossible hint of New Physics will be discussed in Section 2.3 in some detail.There, we also present a rather up to date discussion of the Standard Modelprediction for a and its underlying uncertainties.

In the case of the tau anomalous magnetic moment, as well as its EDM,the current bounds on a and d in Table 2.1 come from the good agreementbetween theory and experiment for e+e +. For s = q2 4m2 , thecross section for an unexpectedly large F2 or F3 is increased to

(e+e +) = 4pi2

3s+

6

(|F2 (s)|2 + |F3 (s)|2

)+ . . . (2.27)

The agreement between (e+e +)exp at q2 ' (35 GeV)2 with ex-pectations then leads to the bounds on a and d in Table 2.1. Thoseconstraints are not competitive for probing New Physics. In fact the aconstraint is more than an order of magnitude larger than the StandardModel prediction, a ' 2pi ' 0.001. To reach even that level of sensitivitywill require a more direct experimental approach to a . For a discussion oftau dipole moments see [30].cOf course, there is always the possibility that New Physics violates electron-muon uni-versality and that simple quadratic mass relationship fails. We do not consider suchcases here.


In the case of lepton EDMs, the Standard Model predictions are farbelow experimental capabilities (e.g. dSMe ' 1038 ecm). Hence, the obser-vation of a non-vanishing lepton EDM would be heralded as a CP violatingNew Physics effect. Since the New Physics is naively expected to scale asml (or in some models as m3l ), we normally estimate

d

|de| : |d| : |d | :: 1 : 200 : 3500. (2.28)Assuming that relationship, the muon dexp sensitivity would have to reachabout 4 1025 ecm to become competitive with the existing dexpe . Asillustrated in Table 2.1, the current bound is far short of that level. Never-theless, an experiment with |d| ' 1024 ecm as its goal has been proposedat JPARC in Japan. Such an effort is extremely well motivated. It wouldbring the muon to a New Physics sensitivity similar to that of the presentelectron and neutron bounds, our current best probes.

If neutrinos are four-component Dirac particles, i.e. they have left-and right-handed components, they can have magnetic and electric dipolemoments. Those moments are expected to be very small since they arelikely to be proportional to the neutrino masses which are tiny < 1 eV.For example, if one merely adds singlet sterile right-handed neutrinos tothe Standard Model, they can give rise to Dirac neutrino masses, mi ,i = 1, 2, 3, via the Higgs mechanism as well as loop-induced magnetic dipolemoments and EDMs. The leading contribution to ai

e2me

(i.e. units ofelectron Bohr magneton) is given by [3234]

ai = 3Gmemi42pi2

= 3 1019mi (eV) (2.29)

where G = 1.166 37 105 GeV2 and the neutrino magnetic moment is~i = eme ai~si . Such small values are well below current experimentalsensitivities. However, in some New Physics scenarios, such as left-rightsymmetric models [35] or extended Higgs models, it is possible to havelarger dipole moments [36]. Therefore, it is interesting to examine whatdirect laboratory, astrophysical and cosmological bounds can be placed onneutrino dipole moments (magnetic or electric), independent of theory (see[37, 38] for a recent detailed review). In Table 2.2, we list various bounds.

Because the neutrinos considered in Table 2.2 are relativistic, we have

cited bounds for the combined moment

aie2me 2 + |di |2 rather than justaie2me as is customarily done. Of course, if one assumes |di | aie2me ,dFor examples where d is enhanced beyond Eq. (2.28) see [31].


Table 2.2. Some bounds on magnetic and electric dipole moments aie/2meand di , i = 1, 2, 3, for neutrino mass eigenstates.

Neutrino

aie2me 2 + |di |2 Source1 < 2 1010 e2me ' 4 10

21 e cm ee ee2 < 3 1010 e2me ' 6 10

21 e cm ee ee3 < 5 1010 e2me ' 3 1 10

20 e cm e ei < 3 1012 e2me ' 6 10

23 e cm Astrophysics1 < 6 1013 e2me ' 1.2 10

23 e cm Supernova 1987A2 < 8 1013 e2me ' 1.6 10

23 e cm Supernova 1987Ai < 10

16 e2me

' 2 1027 e cm Nucleosynthesis

they can be taken as simply magnetic moment bounds. Also, we apply thebounds to neutrino mass eigenstates rather than flavor states (as custom-arily done). So, for example, the best direct laboratory bounds on dipolemoments currently come from ee ee and e e scattering crosssection measurements. Those cross sections are increased by [39]

ddy

' | |2 pi2

m2e

(1y 1)

(2.30)

y =Ee me

E

if neutrinos have dipole moments e/2me,

| |2 = |a |2 +d 2mee

2 . (2.31)Converting from a flavor to a mass eigenstate basis then dilutes somewhatthe bounds. For example, using the three-by-three mixing matrix angles12, 13, and 23 (see Section 2.4), one finds (ignoring oscillations)

|e |2 = |1 |2 cos2 12 cos2 13 + |2 |2 sin2 12 cos2 13 + |3 |2 sin2 13(2.32)

so, using sin2 12 ' 0.3 and sin2 13 ' 0|e |2 = 0.7 |1 |2 + 0.3 |2 |2 . (2.33)

Similarly, using sin2 23 ' 0.5 implies 2 = 0.14 |1 |2 + 0.36 |2 |2 + 0.5 |3 |2 . (2.34)Interestingly, 3 gives the largest contribution to .

A larger than expected e e scattering cross section, particularly oneexhibiting the distinctive 1/y dependence in Eq. (2.30), would be evidence


for a non-vanishing neutrino dipole moment. Consistency of the currentee and e cross sections with Standard Model expectations, along withsolar neutrino measurements, gives the constraints [37, 38]

|e | < 6 1011, < 1.5 1010 (2.35)

used to derive the first three bounds in Table 2.2. Those rather directbounds could be improved by new dedicated low energy scattering experi-ments, perhaps by one or two orders of magnitude. It would be very difficultto do much better because of the (squared) | |2 dependence in Eq. (2.30).So, it seems unlikely that neutrino dipole moments are directly observablein scattering experiments.

Tighter constraints on (Dirac) neutrino magnetic and electric dipolemoments can be obtained from astrophysics. If neutrinos couple to photonsvia such moments, then the decay, plasmon , can occur in stellarinteriors [35, 4042]. (A plasmon is an effectively massive photon withmass p (plasma frequency) due to electron-hole excitations in a plasma.)Because neutrinos immediately escape and carry away energy, that decayprocess would speed up stellar evolution. Since evidence for such a speedup is not seen, the bound of 6 1023 ecm [43], independent of neutrinospecies, in Table 2.2 is obtained.

An independent bound on e (or 1 and 2 via Eq. (2.33)) can beobtained from the detection of e neutrinos from supernova 1987A at aboutthe expected flux level [4446]. A dipole moment would lead to the pro-duction of sterile right-handed neutrinos (left-handed antineutrinos) in thevery dense pre-supernova core. A significant flux loss would alter the su-pernova collapse dynamics and subsequent explosion. Those considerationslead to the bound |e | < 5 1013 and the slightly diluted constraints inTable 2.2.

The most constraining bound on neutrino dipole moments currentlycomes from primordial nucleosynthesis under the assumption of a back-ground magnetic field at those early times [47]. Spin precession in such afield would have resulted in sterile neutrinos and a change in the expansionrate of the Universe. The good agreement between the observed heliumabundance and the usual Big Bang prediction then leads to the bound|i | < 1016 in Table 2.2. Note that even that most stringent bound whenphrased in terms of a New Physics scale seems to suggest Cmme2 ' 1016is at best being probed. Even for C O (1), that corresponds (assumingm < 1 eV) to


Table 2.3. Comparison of the neutron and proton anomalous magnetic mo-ments (in units of e/2mN , mN = (mn +mp) /2). Also given are the currentbounds on neutron [51] and proton [52] EDMs in units of ecm and e/2mN .

N F2N (e/2mN ) |F3N |

n 1.913 042 7 (5) < 0.29 1025 ecm ' 1 1013e/2mNp +1.792 158 142 (28) < 7.9 1025 ecm ' 3 1012e/2mN

however, transition moments between different mass eigenstates which leadto i j [32] or spin-flavor precession in magnetic fields [4850]. Thedirect bounds in Table 2.2 from e scattering, solar neutrino oscillationsand the astrophysical plasmon decay constraint can be applied to suchtransition moments, but the astrophysical and cosmological constraints aresignificantly diluted (or rendered inapplicable) because the final state neu-trinos are active rather than sterile.

2.2.2. Nucleon dipole moments

The neutron and proton are known to have large anomalous magnetic dipolemoments due to their composite structure. Current values are listed inTable 2.3 in units of the nucleon Bohr magneton, e/2mN .

Since P and T symmetries are violated in Nature, it is quite likely(almost certain) that nucleons also have electric dipole moments = dN intheir spin direction. In fact, all spin 1/2 Dirac particles should have, albeittiny, EDMs. Indeed, the Standard Model predicts, on the basis of CKMquark mixing [53, 54],

|dN | ' 1032 e cm Standard Model (2.36)which (unfortunately) is more than six orders of magnitude below the ex-isting neutron EDM bound in Table 2.3 and experimentally unobservablein the foreseeable future. Of course, our inability to confirm the StandardModel prediction can be viewed as fortuitous. It means that any discoveryof a nucleon EDM with |dN | 1032 ecm is providing direct evidence forNew Physics. Furthermore, that New Physics would be an additional sourceof CP violation and might help explain the matter-antimatter asymmetryof our Universe, an exciting possibility.

We see from Table 2.3 that the nucleon magnetic dipole moments (par-ticularly the protons) are very precisely measured. In fact, the uncertaintyin the protons anomalous magnetic moment, 28 109, is only a factorof 44 worse than the muons aexp

(63 1011). If the proton were an


elementary particle and if its hadronic contributions were under theoreticalcontrol, ap because of the protons larger mass, (mp/m)

2 ' 77, wouldbe a slightly better probe than the muon a for New Physics. However,that is not the case. The proton is not elementary and there are very largehadronic uncertainties in the Standard Model prediction for ap, at least atthe several percent level, making its high precision useless as a window toNew Physics.

We can, however, learn some valuable lessons from F2p (0) and F2n (0)that may be useful in trying to anticipate and motivate possible measure-ments of F3p (0) and F3n (0), the nucleon EDMs.

We first observe that when written as isovector and isoscalar combina-tions

F(I=1)2N =

F2p F2n2

' 1.85,

F(I=0)2N =

F2p + F2n2

' 0.06, (2.37)it is clear that the isovector component dominates, i.e. F2n ' F2p.

Will the nucleon EDMs also be isovector? A priori, nothing tells us thatmust be the case. Therefore, we should strive to measure both dn and dp inorder to determine the isospin dependence and relate it to the predictionsof New Physics scenarios. Some models may predict nearly pure isovectornucleon EDMs. Others may have a relatively large isoscalar part. In otherwords, we really have to consider dn and dp as independent quantities andshould try to determine both with similar precision, if possible. Here, wemight remark that current lattice techniques are better for determining theisovector EDM combination

F(I=1)3N =

F3p F3n2

, (2.38)

rather than the isoscalar

F(I=0)3N =

F3p + F3n2

. (2.39)

Hence, an experimental determination of d(I=1)N = (dp dn) /2 will be veryimportant if we are to use lattice gauge theory to unfold the underlyingNew Physics responsible for nucleon EDMs. That requires a measurementof both dn and dp.

The values of the anomalous nucleon magnetic moments in Eq. (2.37)also illustrate a success of the SU (6) constituent quark model. (SU (6) is a


largely unsuccessful attempt to combine SU (3) flavor and SU (2) spin intoa spin-flavor symmetry.) It predicts for the full nucleon magnetic dipolemoments ~N

~n =43~d 13~u

~p =43~u 13~d (2.40)

where ~d and ~u are the constituent down and up quark dipole moments.Then employing

~u = 2~d, |~u| = 23e

2mq=

e

mN(2.41)

where mq = mN3 has been employed, one finds

F2n (0) = an = 2 e2mNF2p (0) = ap = 2

e

2mN(2.42)

which implies pure isovector anomalous nucleon magnetic moments, withvalues in very good accord with Eq. (2.37). For some things, that naivequark model approach works remarkably well and Eq. (2.42) was one of itsearly successes. As a result, the SU (6) constituent quark model is oftenused to predict nucleon EDMs from quark EDMs ~dd and ~du. FollowingEq. (2.40) one expects

~dn =43~dd 13

~du

~dp =43~du 13

~dd. (2.43)

In such a scenario, the isovector EDM will dominate if ~dd ' ~du whilethe isoscalar dominates if ~dd ' ~du. Without a specific model it is hard tofavor one over the other. Indeed, underlying calculations of quark EDMsare unlikely to satisfy simple isospin relations. On the basis of Eq. (2.43),one might, very roughly, expect

14 114.4 GeV.

2.3.3. New Physics effects

Since the anomalous magnetic moment comes from a dimension 5 operator,New Physics (i.e. beyond the Standard Model expectations) will contributeto a via induced quantum loop effects (rather than tree level). Whenevera new model or Standard Model extension is proposed, such effects areexamined and aexp aSM is often employed to constrain or rule it out.

Here we describe several examples mainly taken from our work inref. [134] of interesting New Physics probed by aexp aSM . Ratherthan attempting to be inclusive, we concentrate on two general scenar-ios: 1) Supersymmetric loop effects which can be substantial and wouldbe heralded as the most likely explanation if the deviation in aexp is con-firmed and 2) Models of radiative muon mass generation which predicta(New Physics) m2/M2 where M is the scale of New Physics. Eithercase is capable of explaining the apparent deviation in aexp aSM exhibitedin Eq. (2.85). Both examples can be cast in the form aNP ' Cm2/2,the first with C O

(pi

)and the second with C O (1). Other types of

potential New Physics contributions to a are only briefly discussed.

2.3.3.1. Supersymmetry

The supersymmetric contributions to a stem from sneutrino-chargino andsmuonneutralino loops (see Fig. 2.5). They include 2 chargino and 4 neu-tralino states and could in principle entail slepton mixing and phases. De-pending on SUSY masses, mixing and other parameters, the contributionof aSUSY can span a broad range of possibilities. Studies have been carried


out for a variety of models where the parameters are specified. Here we givea discussion primarily intended to illustrate the strong likelihood that evi-dence for supersymmetry can be inferred from aexp and may in fact be thenatural explanation for the apparent deviation from SM theory reportedby E821.

0

(a) (b)

Fig. 2.5. Supersymmetric loops contributing to the muon anomalous magnetic moment.

Early studies of the supersymmetric contributions aSUSY were carriedout in the context of the minimal SUSY Standard Model (MSSM) [135142],in an E6 string-inspired model [143, 144], and in an extension of the MSSMwith an additional singlet [145, 146]. An important observation madein [147], namely that some of the contributions are enhanced due to mix-ing by the ratio of Higgs vacuum expectation values, tan 2/1,which in some models is large (in some cases of order mt/mb 40). Inaddition, larger values of tan > 2 are generally in better accord with therecent LEP II Higgs mass bound mH > 114 GeV and, therefore, currentlyfavored. The main contribution is generally due to the chargino-sneutrinodiagram (Fig. 2.5(a)), which is enhanced by a Yukawa coupling in themuon-sneutrino-Higgsino vertex (charginos are admixtures of Winos andHiggsinos).

The leading effect from Fig. 2.5(a) is approximately given in the largetan limit byaSUSY ' (mZ)8pi sin2 W m

2

m2tan

(1 4

piln

m

m

), (2.87)

where (mZ) ' 1/128, and m = mSUSY represents a typical SUSY loopmass. SUSY mass scales are actually assumed degenerate in Eq. (2.87)[148]. (For a detailed discussion of degeneracy conditions see [149, 150].)


Also, we have included a 78% suppression factor due to leading two-loopEW effects. Like most New Physics effects, SUSY loops contribute di-rectly to the dimension 5 magnetic dipole operator. From the calculationin Ref. [124, 128, 131], one finds a leading log suppression factor

1 4pi

lnM

m(2.88)

where M is the characteristic New Physics scale. For M 200 GeV, thatfactor corresponds to about a 7% reduction. That reduction factor has thesame source as the correction given for electromagnetic transition rates inEq. (2.64). Note, Eq. (2.88) also applies to New Physics induced EDMs.

Numerically, one expects in the large tan regime (after a small negativecontribution from Fig. 2.5(b) is included, again assuming degenerate SUSYmass scales)

aSUSY ' 130 1011(100 GeVm)2

tan, (2.89)

where aSUSY generally has the same sign as the -parameter in SUSY mod-els. Eq. (2.89) represents the leading effect up to corrections of O (mW /m)and O (1/ tan).

Supersymmetric effects in a have been computed in a variety of mod-els [148, 151170]. Also two-loop effects have been determined in variousscenarios [171175]. For a detailed review of supersymmetry contributionsto a, see Chapter 12 and Ref. [149].

Rather than focusing on a specific model, we simply employ for illus-tration the large tan approximate formula in Eq. (2.89) with degenerateSUSY mass scales and the current constraint in Eq. (2.85). Then we find(for positive sgn())

tan(100 GeV

m

)2' 2.4 0.6, (2.90)

or

m ' (65 10 GeV)tan. (2.91)

(Of course, in specific models with non-degenerate SUSY mass scales, amore detailed analysis is required, but here we only want to illustrateroughly the scale of supersymmetry being probed.) Negative modelsgive the opposite sign contribution to a and are strongly disfavored bycurrent aexp aSM results.


For large tan in the range 4 40, where the approximate resultsgiven above should be valid, one finds (assuming m > 200 GeV from otherexperimental constraints and the region of Eq. (2.89) validity)

m ' 200 500 GeV (2.92)precisely the range where SUSY particles are often expected. If supersym-metry in the mass range of Eq. (2.92) with relatively large tan is respon-sible for the apparent aexp aSM difference, it will have many dramaticconsequences. Besides expanding the known symmetries of Nature and ourfundamental notion of space-time, it will impact other new exploratory ex-periments. Indeed, for m ' 200 500 GeV, one can expect a plethora ofnew SUSY particles to be discovered soon, either at the Fermilab 2 TeVpp collider or certainly at the LHC 14 TeV pp collider which is expected tostart dedicated running in 2009.

Large tan supersymmetry can also have other interesting loop-inducedlow energy consequences beyond a. For example, it can affect the b sbranching ratio. Even for the muon, New Physics in a is likely to suggestpotentially observable e, N eN and a muon electric dipolemoment, depending on the degree of flavor mixing and CP violating phases.Searches for these phenomena are now entering an exciting phase, with anew generation of experiments being proposed or constructed. The decay e is currently being searched for with 2 1013 (later to improve to21014) single event sensitivity (SES) at the Paul Scherrer Institute (PSI)[176]. The mu2e experiment at Fermilab [177] will search for the muon-electron conversion, Al eAl, with 21017 SES. A proposal has beenmade [178] to search for the muons EDM with sensitivity of about 1024

ecm. Certainly, the hint of supersymmetry suggested by aexp will providestrong additional motivation to extend such studies both theoretically andexperimentally.

2.3.3.2. Radiative muon mass models

The relatively light masses of the muon and most other known fundamen-tal fermions could suggest that they are radiatively loop induced by NewPhysics beyond the Standard Model. Although no compelling model ex-ists, the concept is very attractive as a natural way to explain the flavormass hierarchy, i.e. why most fermion masses are so much smaller than theelectroweak scale 250 GeV.

The basic idea, described in [179], is to start off with a naturally zerobare fermion mass due to an underlying chiral symmetry. The symmetry


is broken in the fermion 2-point function by quantum loop effects. Theylead to a finite calculable mass which depends on the mass scales, couplingstrengths and dynamics of the underlying symmetry breaking mechanism.In such a scenario, one generically expects for the muon

m g2

16pi2MF , (2.93)

where g is some new interaction coupling strength and MF 100 1000GeV is a heavy scale associated with chiral symmetry breaking and perhapselectroweak symmetry breaking. Of course, there may be other suppressionfactors at work in Eq. (2.93) that keep the muon mass small.

Whatever source of chiral symmetry breaking is responsible for gener-ating the muons mass will also give rise to non-Standard Model contribu-tions in a. Indeed, fermion masses and anomalous magnetic moments areintimately connected chiral symmetry breaking operators. Remarkably, insuch radiative scenarios, the additional contribution to a is quite generallygiven by [179, 180]

a(NP ) ' Cm2M2

, C ' O (1) , (2.94)where M is some physical high mass scale associated with the New Physicsand C is a model-dependent number roughly of order 1. M need not be thesame scale as MF in Eq. (2.93). In fact, M is usually a somewhat largergauge or scalar boson mass responsible for mediating the chiral symmetrybreaking interaction. The result in Eq. (2.94) is remarkably simple in thatit is largely independent of coupling strengths, dynamics, etc. Furthermore,rather than exhibiting the usual g2/16pi2 loop suppression factor, a(NP )is related to m2/M

2 by a (model dependent) constant, C, roughly of O (1),thus exhibiting the m2f/

2 possibility we discussed earlier.

Toy model example To demonstrate how the relationship in Eq. (2.94)arises, we first review a toy model example [179] for muon mass generationwhich is graphically depicted in Fig. 2.6.

If the muon is massless in lowest order (i.e. no bare m0 is possibledue to a symmetry), but couples to a heavy fermion F via scalar, S, andpseudoscalar, P , bosons with couplings g and g5 respectively, then thediagrams in Fig. (2.6) give rise to

m ' g2

16pi2MF

(M2S

M2S M2FlnM2SM2F

M2P

M2P M2FlnM2PM2F

)(2.95)

g2

16pi2MF ln

(M2SM2P

)(MS,P MF ). (2.96)


m '

S

F

+

P

F

Fig. 2.6. One-loop diagrams which can induce a finite radiative muon mass.

S, P F

F S, PF S, P

(a) (b)

Fig. 2.7. Diagrams that could potentially contribute to the anomalous magnetic mo-ment in radiative muon mass models.

Note that short-distance ultraviolet divergences have canceled and the in-duced mass vanishes in the chirally symmetric limit MS =MP .

If we attach a photon to the heavy internal fermion, F , or boson Sor P (assumed to carry fractions QF and 1 QF of the muon charge,respectively), then a new contribution to a is also induced (see Fig. 2.7).In the limit MS,P MF and QF = 1, one finds [179]

a(NP ) ' g2

8pi2mMFM2P

(M2PM2S

lnM2SM2F

lnM2P

M2F

), (2.97)

while for QF = 0

a(NP ) ' g2

8pi2mMFM2P

(1 M

2P

M2S

). (2.98)

The induced a(NP ) also vanishes in the MS =MP chiral symmetry limit.Interestingly, a(NP ) exhibits a linear rather than quadratic dependenceon m at this point.

Although Eqs. (2.96) and (2.97) both depend on unknown parameterssuch as g and MF , those quantities largely cancel when we combine both


expressions. One finds

a(NP ) ' Cm2M2P

,

C = 2[1

(1 M

2P

M2S

)lnM2SM2F

/ lnM2SM2P

]for QF = 1,

C =(1 M

2P

M2S

)/ ln

M2SM2P

for QF = 0, (2.99)

where C is very roughly O (1). It actually spans a broad range and takeon either sign, depending on the MS/MP ratio and QF . A loop produceda(NP ) effect that started out at O

(g2/16pi2

)has effectively been pro-

moted to O (1) by absorbing the couplings and MF factor into m. Alongthe way, the linear dependence on m has been replaced by a more naturalquadratic dependence.

Technicolor An alternative procedure for radiatively generating fermionmasses involves new strong dynamics, e.g. extended technicolor. In suchscenarios, technifermions acquire, via new strong dynamics, dynamical self-energies

F (p) ' mF(

2

2 p2)1 2

, (2.100)

where 0 < < 2 is an anomalous dimension, mF ' O (300 GeV), and isthe new strong interaction scale O (1 TeV).

Ordinary fermions such as the muon receive loop induced masses viathe diagram in Fig. 2.8.

F F

mF

X

Fig. 2.8. Extended technicolor-like diagram responsible for generating the muon mass.

The extended gauge boson X links and F via the non-chiral coupling

g

(a1 52

+ b1 + 52

)(2.101)


and gives rise to a mass [179, 180]

m ' g2ab

4pi2mF

(

mX

)2(2

)(1

2

), (2.102)

where (x) is the Gamma function. The possible ultraviolet divergence at = 2 corresponds to a non-dynamical mF .

If we attach a photon to one of the internal propagators of Fig. 2.8 oneobtains an anomalous magnetic moment of the form

a(New Dynamics) ' g2ab

2pi2mmFm2X

(

mX

)2F (),

(2.103)

where F () is a model dependent dynamics factor. Again, we see a lineardependence on m. However, when Eq. (2.102) and (2.103) are combined,one finds for


2.3.3.3. Other New Physics examples

Anomalous W boson properties Anomalous W boson magneticdipole and electric quadrupole moments can also lead to a deviation ina from SM expectations. Generalizing the WW coupling, the W bosonmagnetic dipole moment is given by

W =e

2mW(1 + + ) (2.106)

and electric quadrupole moment by

QW = e2mW ( ) (2.107)

where = 1 and = 0 in the Standard Model, i.e. the gyromagnetic ratiogW = + 1 = 2. For non-standard couplings, one obtains the additionalone loop contribution to a given by [189193]

a(, ) 'Gm

2

42pi2

[( 1) ln

2

m2W 13

], (2.108)

where is the high momentum cutoff required to give a finite result. Itpresumably corresponds to the onset of New Physics such as the W com-positeness scale, or new strong dynamics. Higher order electroweak loopeffects reduce that contribution by roughly the suppression in Eq. (2.88),i.e. 9%.

For ' 1 TeV, the deviation in Eq. (2.85) corresponds to 1 = 0.28 0.07. (2.109)

Such a large deviation from Standard Model expectations, = 1, is alreadyruled out by e+e W+W data at LEP II which gives [194, 195]

1 = 0.04 0.08 (LEP II). (2.110)One could reduce the requirement in Eq. (2.109) somewhat by assuminga much larger cutoff in Eq. (2.108). However, it is generally felt that 1 and should be inversely correlated. For example 1 mW /or (mW /)2. So, the rather substantial 1 needed to accommodateaexp would argue against a much larger . Similarly, the large value ofthe anomalous W electric quadrupole moment ' 4 needed to reconcileaexp aSM is also ruled out by collider data (which implies ||


We note that the existence of a W boson EDM would induce fermionEDMs in a manner very similar to the anomalous magnetic moment dis-cussion given above. Indeed, for a W EDM, dW = eW /2mW , one findsanalogous to Eq. (2.108) the fermion induced EDM [196]

df =eT3LGFmfW

42pi2

(ln

2

m2W+O (1)

)(2.111)

where T3L is the third component of the weak SU(2)L isospin of the fermionf .

New gauge bosons The local SU(3)C SU(2)L U(1)Y symmetry ofthe Standard Model can be easily expanded to a larger gauge group withadditional charged and neutral gauge bosons. Here, we consider effectsdue to a charged WR which couples to right-handed charged currents ingeneric left-right symmetric models and a neutral gauge boson, Z , whichcan naturally arise in higher rank GUT models such as SO(10) or E6. Ageneral analysis of one-loop contributions to a from extra gauge bosonshas been carried out by Leveille [197] and summarized in Chapter 10.2. Thespecific examples considered here were illustrated in [68] (see also [198]).Therefore, we will only discuss the likelihood of such bosons being thesource of the apparent aexp aSM discrepancy.

For the case of a WR coupled to R and a (very light) R with gaugecoupling gR, one finds

a(WR) ' (390 1011)g2R

g22

m2Wm2WR

. (2.112)

To accommodate the discrepancy in Eq. (2.85) requires mWR ' mW = 80.4GeV for gR ' g2, which is clearly ruled out by direct searches and precisionmeasurements which give mWR > 715 GeV. Hence, WR is not a viablecandidate for explaining the aexp discrepancy.

Extra neutral gauge bosons (with diagonal couplings) do much worsein trying to explain aexp aSM , partly because they often tend to give acontribution with opposite sign. For example, the Z of SO(10) leads to

a(Z) ' 6 1011 m2Z

m2Z. (2.113)

Given the collider constraint mZ > 720 GeV, that effect would be muchtoo small to observe in aexp . Most other Z

scenarios give similar results.An exception to the small effects from gauge bosons illustrated above

is provided by non-chiral coupled bosons which connect and a heavy


fermion F . In those cases, a ' g2

16pi2mmFM2 , where M is the gauge boson

mass. However, loop effects then give m g2mF (see the discussionin Section 2.3.3.2) and we have argued that in such scenarios a shouldactually turn out to be m2/M2. As previously pointed out in Eq. (2.105),aexp aSM then corresponds to M 2 TeV.

Many other examples of New Physics contributions to a have beenconsidered in the literature. A general analysis in terms of effective inter-actions was presented in [199]. Specific other examples include effects dueto muon compositeness [186, 200, 201], extra Higgs bosons [202206], lepto-quarks [207209], bileptons [210], two-loop pseudoscalar effects [211], com-pact and large extra dimensions [212215], extended family models [216],brane models [217219], unparticles [220], etc.

2.4. Flavor Violating Transition Dipole Moments

Searches for flavor-changing weak neutral current effects in the quark sec-tor of the Standard Model have had a rich and glorious history. The needto theoretically suppress s d transitions in decays such as KL +led to the introduction of charm and the GIM (Glashow-Iliopoulos-Maiani)mechanism [221] of loop cancellations. That mechanism was also instru-mental in suggesting that a third generation of quarks may explain CPviolation via CKM mixing. More recently, the accurate measurement ofthe b s branching ratio which occurs via transition dipole momentsconfirmed the Standard Model top quark loop prediction and has beenused [222, 223] to constrain possible New Physics effects such as in su-persymmetry. Indeed, that branching ratio currently provides sensitivityto supersymmetry [224227] competitive with and complementary to themuon anomalous magnetic dipole moment discussion in Section 2.3.

In the case of charged lepton decays, searches for flavor-changing neu-tral current effects such as e or have all, so far, proven tobe negative (see Table 2.5). Only experimental bounds on transition dipolemoments exist (see Table 2.6) in spite of the fact that we now know fromneutrino oscillation studies that lepton flavor is not conserved. Their sup-pression in charged lepton processes is accidentally due to the smallness ofneutrino masses rather than from intrinsically small flavor mixing. Indeed,we have found from oscillations that the flavor basis states e, , and produced in weak interaction reactions are related to the neutrino mass


Table 2.5. Current bounds on various flavor-changing charged lep-ton processes along with future expected or possible improvements[228].

Reaction Current bound Expected Possible

B(+ e+) < 1.2 1011 2 1013 2 1014

B(+ e+ee+) < 1.0 1012 1015

R(Au eAu) < 7 1013

R(Al eAl) 1016 1018B ( ) < 5.9 108 O (109)B ( e) < 8.5 108 O (109)

B( +) < 2.0 108 O (1010)

eigenstates 1, 2, and 3 via the mixing matrix |e||

= U |1|2|3

, where (2.114)U =

c12c13 s12c13 s13eis12c23 c12s23s13ei c12c23 s12s23s13ei s23c13s12s23 c12c23s13ei c12s23 s12c23s13ei c23c13

,cij = cos ij , sij = sin ij , i, j = 1, 2, 3

with

sin2 223 ' 1, sin2 212 ' 0.8, sin2 213 < 0.15 (2.115)and a completely undetermined phase 0 < 2pi.

The measured mixing angles 23 and 12 are quite large and give riseto some near maximal neutrino oscillation effects (such as solar e fluxreaching the earth as roughly 13e +

13 +

13 ). However, the measured

neutrino mass squared differences found from oscillations are very small,

m232 = m23 m22 ' 2.5 103 eV2

m221 = m22 m21 ' 8 105 eV2. (2.116)

Since charged lepton flavor violation in the Standard Model must vanish inthe limit where both m2ij = 0, decay amplitudes for l1 l2 + will beproportional to the m2ij and, therefore, highly suppressed. For example,in the case of , one finds for the transition dipole moments given inSection 2.2.4, the leading m232 contribution

DR = eGFm

162pi2

sin 23 cos 23 cos2 13m232m2W

,

DL ' 0. (2.117)


Table 2.6. Current experimental bounds on the transition dipole mo-ments defined in Eq. (2.62) along with possible future sensitivities. Thebounds are based on the constraints in Table 2.5 with the best future e sensitivity expected to come from Al eAl conversion.

Transition moment Current bound Possible future sensitivityFeM12 + FeE12 < 2 1026 ecm < 1 1028 ecmF M12 + F E1 2 < 5 1023 ecm < 6 1024 ecmF eM12 + F eE12 < 6 1023 ecm < 6 1024 ecm

The left-hand transition moment is non-vanishing, but rendered extremelysmall because of the m

232

m2W' 4 1025 suppression factor. Using the

branching ratio relation,

B ( ) ' 12pi2

G2Fm2

(|DR |2 + |DL |2

)B ( ) , (2.118)

leads to the prediction B ( ) ' 21054, i.e. more than 1047 belowthe current bound in Table 2.5. (A similar size B ( e) is expected.)Such tiny levels of lepton flavor violation are clearly unobservable.

So, the bad news is that we cannot measure neutrino mass and mixingin flavor-changing charged lepton decays such as e or . Thegood news is that the observation of such decays would, therefore, providedirect evidence for New Physics beyond neutrino mass effects. Also, thereare many models that predict potentially observable charged lepton flavorviolation reaction and planned experiments will probe (as we shall see)O (1000 TeV) mass scales, a significant window to New Physics.

2.4.1. Muon flavor violation

Experiments to find the rare processes e, ee+e, and N eN have been at the forefront of searches for lepton flavor violation (seeTable 2.5). With regard to flavor transition electromagnetic dipole mo-ments, the decay e provides the most direct probe and currently thebest constraint on its related New Physics [229, 230] as discussed in Chap-ters 18 and 19. Phrased in terms of the right- and left-handed transitiondipole moments in Eq. (2.65) and the total muon decay rate

( all) ' ( e) ' G2Fm

5

192pi3(2.119)


e

N N

Fig. 2.9. Coherent muon-electron conversion in the field of the nucleus N , induced bythe dipole operator coupling , e, and .

one expects a branching ratio

B ( e) ' 12pi2

G2Fm2

(|DeR |2 + |DeL |2

). (2.120)

These same dipole moments also give rise to ee+e and N eN(coherent conversion in the field of a nucleus) via virtual photon effects atq2 6= 0, but at a somewhat reduced rate [231]. One finds

B( ee+e) ' 0.006B ( e) . (2.121)

In the case of R (N eN) (N eN) / (N N ), therate for conversion is traditionally compared with ordinary weak chargedcurrent capture and both depend on the specific nucleus, N , employed. So,for example, one finds from a detailed analysis [232, 233]

R(Al eAl) ' 2.6 103B ( e) . (2.122)

However, both ee+e and N eN could turn out to be muchlarger if they are dominated by chiral conserving dimension six four fermionoperators rather than dimension five transition dipole moments. Thosedimension six operators provide a primary motivation f

37247172-lepton

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