arXiv:0704.2394v1 [hep-ex] 18 Apr 2007 yee ihjpsj2.cls with Typeset n acltdmksi infiatiptprmtrin constants. parameter fundamental input of significant determination a the it makes calculated and ieao sivreypootoa oteobtn par- orbiting the ( to mass proportional inversely ticle’s is like atr hr yrgnlk tm r omd iha with formed, of are charge in nuclear -like where the , of role the on focus physics. will I view, atrsses h ihpeiint hc h muo- the which to precision condensed ( high of nium The properties systems. magnetic nuclear matter as a interaction, well as weak as it semileptonic use properties, the to study and to properties, tool its of measurements cision ob tbe h undcy hog h ekforce. weak the through decays muon appears the which stable, be the to Unlike forces. electromag- weak the and through netic other and with the interact electrodynamics, Like can quantum electron’s. obeys the muon times the 207 lep- electron, about Today generation mass second force. a a strong with is the ton, muon of the weakly carrier that too but understand the we interacted particle, be called, to a was matter such it with predicted as had “mesotron” force this nuclear the of atceo neti aue”I 96Adro n Ned- and Anderson 1936 dermeyer In nature.” uncertain of particle e yKunze by ber Introduction 1. ncsi as hc a ofimdi 97b Street by 1937 in confirmed Stevenson, was ” which and than rays, penetrating cosmic more in but than sive yCusr n Leprince-Ringuet. and Crussard by oetqatmnmeso high- of numbers quantum lowest ulu.Te2 The nucleus. uni elisd fteaoi lcrncod with cloud, 1 electron the atomic of the radius of Bohr inside the well is muon ∗ -alades [email protected] address: E-mail emo eaiemoscnb ruh ors in rest to brought be can negative of beam A h unlftm f2 of lifetime muon The h unwsfis bevdi isncodcham- cloud Wilson a in observed first was muon The µ + 2 e − eotdtepeec f“atce esmas- less “particles of presence the reported EWRS un ekdcy uncpue antcmoment, magnetic capture, muon decay, weak muon, KEYWORDS: tm yefiesrcuecnb measured be can structure hyperfine atom) 1 r teraction, beifraino h oicto ftewa urn yth by current weak nuclear the from obtained of that at modification to the the complementary on on capture information Its able theory. standard-model and experiment eodtesadr tnadmdl hr tpeetteei there re present in at where role model, important standard an standard played the has beyond moment magnetic anomalous tnadmdlpril ob icvrd n t ea has decay t its was and muon discovered, The matter. be ( of to theory particle sub-atomic standard-model the understand to n 3 P ector V n13,weei a eotdt e“a be to reported was it where 1933, in ic t icvr nte13s h unhspae nimpor an played has muon the 1930s, the in discovery its Since ihn,Tkuh n Ichimiya, and Tekeuchi Nishina, [ = → Z h orrdu o hydrogen- a for radius Bohr The . < n 1 2 S ver.1.2 − ~ G c . S -a nrisaesitdbe- shifted are energies x-ray 2 unPyis ilro h tnadModel Standard the of Pillar A Physics: Muon F ] / xa ector V Axial n h osraino etnnme flvr nmo decay muon in (flavor) number of conservation the and , µ tmcsaewl niethe inside well state atomic [ mc emt n omk pre- make to one permits s > 2 Zα Z 5 ,s htfrthe for that so ), ] h uaatheory Yukawa The uncaos the atoms, muonic tutr ftewa neato,tesrnt fteweak the of strength the interaction, weak the of structure ) 6 Rcie coe ,2018) 5, October (Received otn A025 USA 02215, MA Boston, nti re- this In eateto Physics of Department otnUniversity Boston .LeRoberts Lee B. 4 and 1 β rvd rcs esrmn of should measurement experiment This precise radius. a charge provide rms the is hr h uncharge muon the where aino ula otma-qaecag ai.The infor- radii. ∆ hydrogen, provided charge muonic in have shift root-mean-square rays Lamb in- nuclear x potential on these Coulomb and mation the nucleus, of the modification side the of cause esrda ttePu cerrIstt(S) sgiven is (PSI), Insitut Scherrer Paul by the at at measured uno h tmcncesfloigtecsaet the to the cascade the of following 1 (OMC), nucleus atomic capture the on muon muon ordinary called capture, neato ytehdoi matter. hadronic weak the the by of modification interaction the on information provides and undcyi ueylpoi rcs,adtedomi- the and process, Thus is leptonic mode . decay purely nant and a , is decay electron, muon the into decay a of capture weak the to found in,adtu un,i h aoaoy h energeti- The make laboratory. decay to the favorable in possible cally muons, became thus it and 1950s, , the In . os(id)o etio r itntparticles. distinct are fla- neutrinos two the of that (kinds) and separately, vors conserved electron is number, muon, lepton individual and the that us tells decay erhdfrwstenurnls atr fa of capture nucleus, neutrinoless atomic the was for searched the antcmmn ie by given moment magnetic a of sensitivities to of reaching object studies 10 the planned be to and continue present and flavor,” lepton “violate ttelvlof level the at decay. S ~µ h unms of mass muon The h un ieteeeto,i pn1 spin a is electron, the like muon, The − 7 s rudstate, ground 14 µ =  − ∗ 8 209 n 10 and g and oarltv rnhn ai of ratio branching relative a to s ( . 2 974(6) q m e mcnceshspoie valu- provided has nucleus omic − 3.4 a s − ) etnflvrviolation flavor lepton ~s 16 rvddifraino the on information provided r prils n the and “particles” are ; togitrcinwihis which interaction strong e ∼ µ titn hoiso physics of theories stricting respectively. , − − µ efis second-generation first he 5 − 5 + σ µ . × atrl norquest our in role tant 226 µ + − → N µ ieec ewe the between difference ∼ 10 Z 1+ (1 = + → R 0 e etit h unto muon the restricts MeV 106 K → N q − → p 2 e 4 seleeto ytenucleus, the by electron -shell = e uhpoessaesi to said are processes Such . − 0 + eγ − + h muon’s The . ± Z + a . a erhdfradnot and for searched was − ) ν 036 e N 2 µ 1 and , q E m ~ R ¯ + hc a o found not was which , N R 2 p ; P ν p 3 h eknuclear weak The . + − e

µ . 2 g < ν + in- S hsthree-body This e,where meV, s µ / hc sbeing is which , h Land´e the , 2 a and steanalog the is , etn with lepton, 2 ulPaper Full × ≡ 10 e ( µ g + − s − 10 r the are 5 2 − Also . nan on Here 2) (1) R g ; p - 2 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts factor is slightly greater than the Dirac value of 2. The middle equation above is useful from a theoretical point of view, as it separates the magnetic moment into two pieces: the Dirac moment which is unity in units of the appropriate magneton, e~/2m, and is predicted by the Dirac equation; and the anomalous (Pauli) moment, (a) (b) (c) where the dimensionless quantity a is referred to as the anomaly. The muon anomaly, like the electron’s, arises Fig. 1. Muon decay in (a) the ; (b) the Fermi from radiative corrections that are discussed below. theory; (c) radiative corrections to the Fermi theory. When the muon was discovered, it was an unexpected surprise. Looking at this from our 21st century perspec- 14 tive, it is easy to forget how we reached what is now called electroweak gauge coupling g is related to GF by the “standard model” of subatomic physics, which incor- 2 GF g porates three generations of , e, µ and τ and their = 2 (1 + ∆r) (2) neutrinos; three generations of ; the electro-weak √2 8MW gauge , γ W and Z; and the that carry the where ∆r represents the weak mediated tree-level strong force. When this author joined the field as a grad- process and its radiative corrections.15 In the standard uate student in the mid 1960s, none of this was clear. model, the Fermi constant is related to the vacuum ex- 2 Quarks were viewed by many as a mathematical device, pectation value of the Higgs field by GF =1/(√2 v ). not as constituent particles. Even after quarks were in- While the Fermi theory is nonrenormalizeable, the ferred from deep inelastic electron scattering off the pro- QED radiative corrections are finite to first order in GF , ton, we only knew of the existence of three of them. While and to all orders in the fine-structure constant α. This 14 the V A structure of the weak interaction was first in- gives the relationship between GF and the muon life- − ferred from nuclear β decay, the study of muon decay has time, τµ, provided a useful laboratory in which to study the purely 1 G2 mµ5 leptonic weak interaction, to search for physics beyond F = 3 (1 + ∆q) (3) the standard model, such as additional terms in the in- τµ 192π teraction besides the standard-model V A structure, as where ∆q is the sum of phase space, and QED and − well as looking for standard model forbidden decays like hadronic radiative corrections. More properly one should µ eγ. For many years, the experimental value of the write G since new physics contributions could make G → µ muon’s anomalous magnetic moment has served to con- different for the three leptons.16 strain physics beyond the standard model, and continues The MuLan experiment at PSI has recently re- that role today. ported a new measurement of the muon lifetime 2.197 013(21)(11) µs ( 11 parts per million (ppm)),17 G 2. Muon Decay and F to be compared with± the previous world average − − 18 The muon decay µ e νµν¯e is purely leptonic. 2.197 03(4) µs (19 ppm). The new world average → Since mµ << MW , muon decay can be described by muon lifetime of 2.197 019(21) µs gives, assuming a local four- (contact) interaction. While non- only standard-model physics in muon decay, GF = renormalizeable, at low energies it provides an excel- 1.166 371(6) 10−5 GeV2 (5 ppm). This experiment × lent approximation to the full electroweak theory. The should eventually reach a precision of 1 ppm on τµ. weak Lagrangian is written as a current-current interac- tion, where the leptonic current is of the (V A) form, 3. Nuclear Muon Capture − uγ¯ λ(1 γ )u. The weak capture of a muon on a proton has much in − 5 Michel11 first wrote down a parameterization of muon common with nuclear β decay. As for other low-energy decay, defining five parameters, ρ, η, ξ, δ and h, which weak processes, the interaction can be described as a are combinations of the different possible couplings al- current-current interaction with the (V A) leptonic cur- λ − lowed by Lorentz invariance in muon decay. The standard rent given byu ¯νµ γ (1 γ5)uµ. Because the strong inter- model has clear predictions for these parameters and action can induce additional− couplings,19 the hadronic they have been measured repeatedly over the intervening current is more complicated. The most general form of years to search for physics beyond the standard model. the vector current allowed by Lorentz invariance is20 This tradition continues today, with the TWIST experi- 2 2 ′ g (q ) g (q ) ment at TRIUMF, which is mid-way through a program u¯ (p ) g (q2)γλ + i M σλν q + S qλ u (p). n V 2m ν m p to improve on the precision of the Michel parameters by  N µ  (4) an order of magnitude.12 While there are some scenarios The corresponding form of the axial-vector current is in which new physics would conspire to leave the Michel 13 2 2 parameters at their standard model value, a variance ′ 2 λ gP (q ) λ gT (q ) λν u¯n(p ) gA(q )γ γ γ q i σ qν γ up(p), from the standard-model values would be a clear sign of − 5 − m 5 − 2m 5  µ N  new physics at work. (5) The muon lifetime, see Fig. 1 is directly related to the where mµ and mN are the muon and masses re- strength of the weak interaction, which in Fermi theory spectively; the g(q2) are the induced form factors: vector, is described by the constant GF . The standard-model weak magnetism, scalar, axial-vector, pseudoscalar and J. Phys. Soc. Jpn. Full Paper B. Lee Roberts 3 tensor. The scalar and tensor terms are called “second time in this protonium target to the the free µ+ life- class currents” because of their transformation proper- time, gives the capture rate and determines gp. The Mu- ties under G-parity, and in the standard model are ex- Cap experiment has recently reported a first result,26 19 2 2 pected to be quite small. It is traditional to set these gp(q = 0.88mµ)=7.3 1.1, consistent with the ex- second-class currents equal to zero. pectation− from chiral perturbation± theory. They have a Nuclear β decay is sensitive to the vector, axial-vector factor of four more data which are being analyzed. While and weak-magnetism form factors, but in muon cap- it is not clear what is wrong with the previous ordi- ture the the capture rate has a measurable contribution nary muon capture and radiative capture experiments, from the induced pseudoscalar interaction, the least well the MuCap result seems to indicate that a modern ex- known of the weak nucleon form factors. Radiative muon periment, with a gaseous target and information from − capture (RMC), µ + p n + γ + νµ, should in prin- the TPC, has settled the long-standing discrepancy. ciple be more sensitive to→ the induced pseudoscalar cou- pling than OMC, since with the three-body final state, 4. The Magnetic and Electric Dipole Moments q2 can get closer to the pole than is possible in or- The electric and magnetic dipole moments have been dinary muon capture, which was pointed out many years an integral part of relativistic electron (lepton) theory ago.21, 22 The interested reader is referred to the review since Dirac’s famous 1928 paper, where he pointed out by Gorringe and Fearing for further discussion.19 that an electron in external electric and magnetic fields In the past, current algebra and the Goldberger- has “the two extra terms Trieman relation expressed gP in terms of gA. With the eh eh (σ, H)+ i ρ (σ, E) , (6) development of quantum chromodynamics (QCD), and c c 1 chiral perturbation theory, new interest has developed in . . . when divided by the factor 2m can be regarded as the value of g .19 The presently accepted theory value is P the additional potential energy of the electron due to g (q2 = 0.88m2 )=8.26 0.23.19 P µ its new degree of freedom.27” These terms represent the The experimental− history± is rather interesting. For magnetic dipole (Dirac) moment and electric dipole mo- many years the radiative capture reaction µ− + p ment interactions with the external magnetic and electric n + γ + ν was thought to be the “golden” channel→ to µ fields. study. However, this experiment is extremely difficult, In modern notation, the magnetic dipole moment and it was first observed experimentally in the 1990s.23, 24 (MDM) interaction becomes To achieve an adequate muon stopping rate, it was nec- essary to stop the muons a liquid hydrogen target. The ie u¯ eF (q2)γ + F (q2)σ qδ u (7) value obtained for g by this experiment was in disagree- µ 1 β 2m 2 βδ µ p  µ  ment with both the ordinary muon capture experiment where F (0) = 1, and F (0) = a . The electric dipole in liquid hydrogen, and with the theoretical expectation. 1 2 µ moment (EDM) interaction is The muon chemistry in hydrogen, especially liquid, is rather complicated, and may be the source of these dis- ie 2 2 ν − u¯µ F (q ) F (q )γ σβδq uµ (8) crepancies. The incident µ first forms an atom with a 2m 2 − 3 5  µ  proton in the hydrogen target, but then quickly picks up where F (0) = a , F (0) = d , with a second proton to form a pµp with the pro- 2 µ 3 µ ~ tons in the ortho (J = 1) state. The ortho state of this η e −14 dµ = η 4.7 10 e cm. (9) molecule can decay to the J = 0 (para) state. The ortho 2 2mc ≃ × × and para proton states have different admixtures of the     (This η, which is the EDM analogy to g for the MDM, muon-proton spin: ortho has 3/4 singlet and 1/4 triplet should not be confused with the Michel parameter η.) µ p and para has 1/4 singlet and 3/4 triplet. Since the − The existence of an EDM implies that both P and T muon capture rate is 40 times greater from the singlet 28, 29 − are violated. This can be seen by considering the µ p state than from the triplet, it is essential that Λ , OP non-relativistic Hamiltonian for a spin one-half particle the ortho to para transition rate in the pµp molecule, in the presence of both an electric and magnetic field: is known in order to extract g from the measured cap- p = ~µ B~ d~ E~ . The transformation properties of ture rate in liquid hydrogen.19 Even after a recent mea- EH~ , B~ ,−~µ and· d~−are· given in the Table I, and we see that surement25 of the transition rate Λ , the difficulty in OP while ~µ B~ is even under all three, d~ E~ is odd under accommodating previous results on ordinary and radia- both P and· T. While parity violation has· been observed tive muon capture results in hydrogen continues. The in many weak processes, direct T violation has only been complications of muon chemistry in liquid hydrogen can observed in the neutral system.30 In the context of be avoided by using a 10 bar ultra-pure hydrogen tar- CPT symmetry, an EDM implies CP violation, which is get, which has a density 1.16% that of liquid hydrogen. allowed by the standard model for decays in the neutral At this lower density, the sensitivity to Λ is greatly OP kaon and B- sectors. reduced. It is this approach that the recent MuCap ex- Observation of a non-zero electron or muon EDM periment at PSI has used. − would be a clear signal for new physics. To date no per- The MuCap experiment stops µ in a gaseous hydro- manent EDM has been observed for the electron, the gen target that functions as a time projection cham- , or an , with the experimental ber (TPC), making it possible to determine where the − limits given in Table II. It is interesting to note that muon stops in the target. A comparison of the µ life- 4 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts

are shown in Fig. 2. Thus Table I. Transformation properties of the magnetic and electric (SM) (QED) (hadronic) (weak) fields and dipole moments. ae,µ = ae,µ + ae,µ + ae,µ . (10) E~ B~ ~µ or d~ P - + + The dominant contribution from quantum electrody- C --- namics (QED), called the Schwinger term,36 a(2) = T +- - α/2π, and is shown diagrammatically in Fig. 2(a). The QED contributions have been calculated through four loops, with the leading five-loop contributions calcu- 37 in his original paper27 Dirac stated “The electric mo- lated. Examples of the hadronic and weak contribu- ment, being a pure imaginary, we should not expect to tions are given in Fig. 2(b)-(e). appear in the model. It is doubtful whether the electric The hadronic contribution cannot be calculated di- moment has any physical meaning, since the Hamilto- rectly from QCD, since the energy scale is very low 2 nian . . . that we started from is real, and the imaginary (mµc ), although Blum has performed a proof of prin- 44 part only appeared when we multiplied it up in an artifi- ciple calculation on the lattice. Fortunately dispersion 38 cial way in order to make it resemble the Hamiltonian of theory gives a relationship between the vacuum polar- + − previous theories.” Even in the 4th edition of his quan- ization loop and the cross section for e e , ∞ → tum mechanics book from 1958, well after the suggestion αmµ 2 ds 31 aµ(Had; 1) = ( ) 2 K(s)R(s); (11) of Purcell and Ramsey that one should search for a 3π m2 s Z4 π permanent EDM, Dirac held fast to this point of view. + − + − + − CP R σtot(e e hadrons) / σtot(e e µ µ ) , While violation is widely invoked to explain the ≡ { → } { 38, 39→ } -antibaryon asymmetry of the universe, the CP and experimental data are used as input violation observed to date in the neutral kaon, and in the sectors is too small to explain it. This γ γ γ γ γ CP deficit has motived a broad program of searches for ∗ ∗ µ µ ∗ µ∗ h WW µ∗ µ∗ µ π− EDMs in a range of systems. Many extensions to the γ µ γ µ µ ν µ µ γ + γ µ Z standard model, such as , do not forbid π  new sources of CP-violation, and the failure to observe (a) (b) (c) (d) (e) it has placed severe restrictions on many models. Fig. 2. The Feynman graphs for: (a) Lowest-order QED (Schwinger) term; (b)Lowest-order hadronic contribution; (c) hadronic light-by-light contribution; (d)-(e) the lowest order elec- Table II. Measured limits on electric dipole moments, and their troweak W and Z contributions. The * emphasizes that in the standard model values loop the muon is off-shell. With the present limits on mh, the Particle Present EDM Standard Model contribution from the single Higgs loop is negligible. Limit (e cm) Value (e cm) n 2.9 × 10−26 (90%CL)32 10−31 − −27 33 −38 e ∼ 1.6 × 10 (90%CL) 10 The muon anomaly is sensitive to a number of poten- −18 34 −35 µ < 10 (CERN) 10 tial candidates for physics beyond the standard model:41 ∼ 10−19 † (E821) 199Hg 2.1 × 10−28 (95%CL)35 (1) muon substructure, where the contribution depends †Estimated on the substructure scale Λ as 2 mµ δaµ(Λµ) 2 , (12) ≃ Λµ The magnetic dipole moment can differ from its Dirac value (g = 2) for several reasons. Recall that the proton’s (2) W -boson substructure. g-value is 5.6 (ap = 1.79), a manifestation of its - (3) new particles that couple to the muon, such as the internal structure. On the other hand, the leptons supersymmetric partners of the weak gauge bosons, appear to have no internal structure, and the MDMs are (4) extra dimensions thought to arise from radiative corrections, i.e. from vir- The potential contribution from supersymmetry has tual particles that couple to the lepton. We would empha- generated a lot of attention,42, 43 the relevant diagrams size that these radiative corrections need not be limited are shown in Fig. 3 below. A simple model with equal to the standard-model particles, but rather the physical masses41 gives values of the lepton anomalies represent a sum-rule over 2 all virtual particles in nature that can couple to the lep- (SUSY) α(MZ ) mµ 4α m˜ aµ 2 2 tan β 1 ln (13) ton, or to the through vacuum polarization loops. ≃ 8π sin θ m˜ − π mµ W   The standard model value of a lepton’s anomaly, aℓ, 2 has contributions from three different sets of radiative − 100 GeV (sgnµ) 13 10 10 tan β (14) processes: quantum electrodynamics (QED)– with loops ≃ × m˜   containing leptons (e,µ,τ) and photons; hadronic – with where tan β is the ratio of the two vacuum expectation hadrons in vacuum polarization loops; and weak – with values of the two Higgs fields. If the SUSY mass scale loops involving the bosons W, Z, and Higgs. Examples (SUSY) were known, then aµ would provide a clean way to determine tan β. J. Phys. Soc. Jpn. Full Paper B. Lee Roberts 5 ∼ 0 ν χ When a single energy threshold is placed on the decay µ µ µ∼ ∼ electrons, the number of high-energy electrons is modu- µ − − µ + µ χ χ lated by the spin precession frequency, Eq 17, producing γ γ the time distribution −t N(t, Eth)= N0(Eth)e γτ [1 + A(Eth) cos(ωat + φ(Eth))]. Fig. 3. The lowest-order supersymmetric contributions to the (18) muon anomaly. The χ are the of the standard- 40 as shown in Fig. 4. The value of ωa is obtained from a model gauge bosons. least-squares fit to these data. The five-parameter func- tion (Eq. 18) is used as a starting point, but many addi- tional small effects must be taken into account.40 4.1 Measurement of the Anomalous Magnetic Dipole Moment Measurement of the magnetic anomaly uses the spin motion in a magnetic field. For a muon moving in a mag- netic field, the spin and momentum rotate with the fre- 10 quencies: 1 qgB~ qB~ qB~ ~ωS = (1 γ); ~ωC = . (15)

− 2m − γm − −mγ -1 10

The spin precession relative to the momentum occurs at Million Events per 149.2ns the difference frequency, ωa, between the spin and cy- -2 clotron frequencies, Equation 15, 10

~ ~ -3 g 2 qB qB 10 ~ωa = ~ωS ~ωC = − = aµ . (16) − − 2 m − m 0 20 40 60 80 100   Time modulo 100µs [µs] The magnetic field in Eq. 16 is the average field seen by the ensemble of muons. This technique has been used in Fig. 4. The time spectrum of 3.6 × 109 electrons with energy 45 all but the first experiments by Garwin, et al., which greater than 1.8 GeV from the 2001 data set. The diagonal “wig- used stopping muons, to measure the anomaly. After gles” are displayed modulo 100 µs. Garwin, et al., made a 12% measurement of the anomaly, a series of three beautiful experiments at CERN culmi- 34 + − nated with a 7.3 ppm measure of aµ. In E821, both µ and µ were measured, and as- In the third CERN experiment, a new technique was suming CPT invariance, the final result obtained by 40 exp −11 developed based on the observation that electrostatic E821, aµ = 116592080 (63) 10 , is shown in quadrupoles could be used for vertical focusing. With Fig. 5, along with the individual measurements× and the the velocity transverse to the magnetic field (β~ B~ = 0), standard-model value. The present standard-model value · 38 SM(06) −11 the spin precession formula becomes is aµ = 116591785 (61) 10 , and one finds −11 × ∆aµ = 295(88) 10 ,a3.4 σ difference. q 1 β~ E~ × ~ωa = aµB~ aµ × . (17) − m − − γ2 1 c "  −  # µ+ For γmagic = 29.3, (pmagic = 3.09 GeV/c), the second (9.4 ppm) CERN − (10 ppm) CERN µ term vanishes; one is left with the simpler result of Equa- + (13 ppm) E821 (97) µ tion 16, hence the name “magic”, and the electric field + (5 ppm) E821 (98) µ + does not contribute to the spin precession relative to the (1.3 ppm) E821 (99) µ + (0.7 ppm) E821 (00) µ − momentum. There are two major advantages of using the (0.7 ppm) E821 (01) µ magic γ and a uniform magnetic field: (i) the knowledge World Average needed on the muon trajectories to determine the aver- age magnetic field is much less than when gradients are −11 present, and the more uniform field permits NMR tech- S−M Theory X 10 116 590 000 116 591 000 116 592 000 116 593 000 niques to realize their full accuracy, thus increasing the 116 594 000 116 595 000 a µ knowledge of the B-field. The spin precession is deter- mined almost completely by Eq. 16, which is independent Fig. 5. Measurements of the muon anomaly, indicating the value, of muon momentum; all muons precess at the same rate. as well as the muon’s sign. As indicated in the text, to obtain the 40 CPT This technique was used also in experiment E821 at the value of aµ− and the world average invariance is assumed. Brookhaven National Laboratory Alternating Gradient The theory value is taken from Ref. [38], which uses electron- Synchrotron (AGS). The reader is referred to Ref. [38] annihilation to determine the hadronic contribution. for a discussion of muon decay relevant to E821, and to Ref. [40] for details of E821. Muons are stored in a storage ring,40 and the ar- One candidate for the cosmic dark matter is the light- 0 rival time and energy of the decay electrons is measured. est supersymmetric partner, the , χ in Fig. 3. 6 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts

In the context of a constrained minimal supersymmet- systematic effects. The limit obtained by this method in 34 ric model (CMSSM), (g 2)µ provides an orthogonal the CERN experiment, and likely to be obtained by constraint on dark matter−46 from that provided by the E821, cannot directly exclude the possibility that the WMAP survey, as can be seen in Fig. 6. entire difference between the measured and standard- model values of aµ could be caused by a muon EDM. Such a scenario would imply that the EDM would be tan β = 10 , µ > 0 −19 8 800 dµ = 2.4(0.4) 10 e-cm, a factor 10 larger than the current limit× on the electron EDM.≈ While this would 700 mh = 114 GeV be a very exciting result, it is orders of magnitude larger 600 mχ± = 104 GeV than that expected from even the most speculative mod- 500 els47–50 (GeV) 0

m 400 To reduce systematic errors in the muon EDM mea- WMAP surement, a “frozen spin” technique has been proposed51 300 g-2 restrictions which uses a radial electric field in a muon storage ring, 1V 200 ± 2V operating at γ << γmagic to cancel the (g 2) preces- 100 ± −

scalar mass scalar sion. The EDM term, Eq. 19, would then cause the spin

0 to steadily move out of the plane of the storage ring. 100 200 300 400 500 600 700 800 900 1000 Electron detectors above and below the storage region mass m (GeV) 1/2 would detect a time-dependent up-down asymmetry that increased with time. As in the (g 2) experiments, de- − Fig. 6. Limits on dark matter placed by various inputs in tectors placed in the plane of the beam would be used, CMSSM, with tan β = 10. The ∆ between experiment and in this case to make sure that the radial-E-field can- standard-model theory is from Ref. [38], see text. The brown cels the normal spin precession exactly. Adelmann and wedge on the lower right is excluded by the requirement the 52 dark matter be neutral. Direct limits on the Higgs and Kirsh have proposed that one could reach a sensitivity −23 ± of 5 10 e cm with a small storage ring at PSI. A let- χ masses are indicated by vertical lines. Restrictions from the × − WMAP satellite data are shown as a light-blue line. The (g − 2) ter of intent at J-PARC53 suggested that one could reach 1 and 2-standard deviation boundaries are shown in purple. The < 10−24 e cm there. The ultimate sensitivity would need − region “allowed” by WMAP and (g 2) is indicated by the el- an even more− intense muon source, such as a lipse, which is further restricted by the limit on Mh. (Figure courtesy of K. Olive) factory. 5. The Search for Lepton Flavor Violation With the apparent 3.4 σ difference between theory and The standard-model gauge bosons do not permit lep- experiment, a new experiment to improve the error by tons to mix with each other, unlike the quark sector a factor of 2 to 2.5 has been proposed to Brookhaven where mixing has been known for many years. Quark 54 laboratory, but at present it is not funded. The theoret- mixing was first proposed by Cabibbo, and extended to 55 ical value will continue to be improved, both with the three generations by Kobayashi and Maskawa, which is + − described by a mixing 3 3 matrix now universally called expected availability of additional data on e e anni- × hilation to hadrons, and with additional work on the the CKM matrix. With the discovery of neutrino mass, hadronic light-by-light contribution.38 we know that lepton flavor violation (LFV) certainly ex- ists in the neutral lepton sector, with the determination 4.2 The Search for a Muon Electric Dipole Moment of the mixing matrix for the three neutrino flavors having With an EDM present, the spin precession frequency become a world-wide effort. relative to the momentum must be modified. The total While the mixing observed in neutrinos does predict some level of charged lepton mixing, it is many orders frequency becomes ~ω = ~ωa + ~ωη, where of magnitude below present experimental limits.13 New q η E~ dynamics,57–66 e.g. supersymmetry, do permit leptons to ~ωη = + β~ B~ , (19) −m " 2 c × !# mix, and the observation of standard-model forbidden processes such as with η defined by Eq. 9, and ωa by Eq. 17. The spin + + + + + − − − motion resulting from the motional electric field, β~ B~ µ e γ; µ e e e ; µ N e N; (21) × → → → is the dominant effect, so ωη is transverse to B~ . An EDM + − − + − + ′ would have two effects on the precession, there would µ e µ e ; µ + e + (22) → N → N be a slight tipping of the precession plane, which would would clearly signify the presence of new physics. The cause a vertical oscillation of the centroid of the decay present limits on lepton flavor violation are shown in electrons that out of phase with the ωa precession; and Fig. 7. the observed frequency ω would be larger, If lepton mixing occurs via supersymmetry, there will 2 be a mixing between the supersymmetric leptons (slep- qηβB ω = ω2 + . (20) tons) which would also be described by a 3 3 mixing a 2m × s   matrix. The schematic connection between lepton flavor The muon limit in Table II, placed by the non- violations and the dipole moments is shown in Fig. 8, and observation of the vertical oscillation, is dominated by there are models that try to connect these processes.67 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts 7

Searches for Lepton Number Violation Liq. Xe Scintillation Liq. Xe Scintillation Detector Detector -1 10

-3 10 Thin Superconducting Coil J Stopping Target -5 Muon Beam J 10 e+ Timing Counter e+ -7 Drift Chamber 10

-9 10 Drift Chamber µ → e γ -11 1m 10 µ- N→ e- N Projected µ+ -→ µ- + -13 e e µ → + = 180° 10 e e e TeJ → π+ µ -15 KL e 10 K → µ e L + + → π0 µ Proposed, phase 1 e -17 KL e P J 10 E = E = 52.8 MeV -19 e J

UL Branching Ratio (Conversion Probability) 10 1950 1960 1970 1980 1990 2000 2010 Year Fig. 9. The side and end views of the MEG experiment. Since Fig. 7. Historical development of the 90% C.L. upper limits on the muon is at rest, the photon and positron are at a relative branching ratios respectively conversion probabilities of muon- angle of 180◦. The positron is tracked in a magnetic field which number violating processes which involve muons and . Also produces a constant bending radius, independent of angle. shown is the projected goal of the MEG (µ+ → e+γ) experiment which is underway at PSI, and the projected sensitivity of the recent letter of intent to J-PARC for muon-electron conversion. perimentally more challenging. It is the coherent muon to (Figure from Ref. [13]) electron conversion, where with adequate energy resolu- tion, the conversion electron can be resolved from back- MDM ground, that with adequate muon flux can be pushed to µ − − e EDM the 10 18 or 10 19 sensitivity. Such a program has been ~µ 2 2 2 ~µ ~µ 71 ~e m˜˜ ∆m˜˜ ∆m˜˜ proposed for J-PARC.  ee eµ eτ  µ ∆ 2 2 ∆ 2 µ µ e mµ˜e˜ mµ˜µ˜ mµ˜τ˜ The to antimuonium conversion (left-hand ~  ∆ 2 ∆ 2 2  ~ B  mτ˜e˜ mτ˜µ˜ mτ˜τ˜  B process in Eq.22) represents a change of two units of lepton number, analogous to K0 K¯ 0 oscillations. This 72 Fig. 8. The supersymmetric contributions to the anomaly, and to process was originally proposed by Pontecorvo. An µ → e conversion, showing the relevant slepton mixing matrix experiment at PSI73 obtained a single event sensitiv- −11 elements. The MDM and EDM give the real and imaginary parts ity of PMM¯ = 8.2 10 which implies a coupling of the matrix element respectively. −3 × GMM¯ 3 10 GF at 90% C.L., where GF is the Fermi coupling≤ × constant. A broad range of speculative theories such as left-right symmetry, R-parity violating LFV In a large class of models, if the ∆ℓ =1 decay goes supersymmetry, etc.,74 could permit such an oscillation. through the transition magnetic moment, one finds13 6. Summary and Conclusions B(µN eN) − → =2 10 3B(A, Z), (23) B(µ eγ) × Since its discovery, the muon has provided an impor- → tant tool to study the standard model, and to constrain where B(A, Z) is a coefficient of order 1 for nuclei heav- its extensions. Experiments in the planning stage for ier than aluminum.68 For other models, these two rates 13 (g 2), the search for an electric dipole moment and can be the same, so in the design of new experiments, lepton− flavor violation in muon decay or conversion will the reach in single event sensitivity for the coherent muon continue this tradition. Research and development for conversion experiments needs to be several orders of mag- new more intense muon sources, such as the muon ion- nitude smaller than for µ eγ to probe the former class 75 → ization cooling experiment (MICE), will further propel of models with equal sensitivity. Detailed calculations of increases in sensitivity. Muon experiments form an im- µ e conversion rates as a function of atomic number − 69 portant part of the precision frontier in , have also been carried out, and if observed, measure- which will continue to provide vital information comple- ments should be carried out in several nuclei. mentary to that from the highest energy . From the experimental side, the next generation µ I wish to acknowledge A. Czarnecki, T. Gorringe, D. eγ experiment, MEG, is now under way at PSI,70 with→ a −13 −14 Hertzog, P. Kammel, K. Jungmann, Y. Kuno, W. Mar- sensitivity goal of 10 10 . Since the decay occurs ciano, J.P. Miller, Y. Okada, and E. de Rafael, for help- at rest, the photon and− positron are back-to-back, and 2 ful conversations, and TG, DH, KJ, PK and JM for their share equally the energy mµc . This experiment makes excellent suggestions on this manuscript. This work was use of a unique “COBRA” magnet which produces a con- + supported in part by the U.S. National Science Founda- stant bending radius for the mono-energetic e , indepen- tion and U.S. Department of Energy. dent of its angle. The photon is detected by a large liquid Xe scintillation detector as shown in Fig. 9. Of the various lepton-flavor violating reactions, only 1) See Figure 5 in Paul Kunze, Z. Phys. 83, 1 (1933). coherent muon conversion does not require coincidence 2) Carl D. Anderson and Seth H. Neddermeyer, Phys. Rev. 50 measurements. The decay µ 3e, while theoretically (1936) 263, and Seth H. Neddermeyer and Carl D. Anderson, appealing, requires a triple coincidence→ and sensitivity Phys. Rev. 51 (1937) 844. 52 to the whole phase space of the decay, and thereby is ex- 3) J.C. Street, E.C. Stevenson, Phys. Rev. (1937) 1003. 8 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts

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