Work in Collaboration with Sang-Hui Im and Min-Seok Seo October 19, 2009 18:8 World Scientiﬁc Review Volume - 9In X 6In Lepton

Implication of ACME and muon g-2 on supersymmetry

KIAS-NCTS Joint Workshop (High1 resort)! 2014. 2. 10

Department of Physics and Astronomy! Seoul National University! Hyung Do Kim work in collaboration with Sang-Hui Im and Min-Seok Seo October 19, 2009 18:8 World Scientiﬁc Review Volume - 9in x 6in lepton

2 B. Lee Roberts

was di±cult to explain by any other means.a Stern proposed an experiment to study space quantization [7] to test the Sommerfeld quantum theory, where he presented the details of what we now call the Stern–Gerlach experiment. An atomic beam of silver atoms was to be projected through a gradient magnetic ﬁeld where the net force on the magnetic dipole would separate the diÆerent magnetic quantum states. For a classical dipole the deﬂection would be continuous, since the direction of the dipole moment could 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 its ground state was one Bohr magneton [9]. Their papers made no reference to the developments in spectroscopy, and in their 1924 review article, no conclusions beyond the magnetic moment were drawn from the two-band structure. Independently, in 1925 Uhlenbeck and Goudsmit [10] proposed the

October 19,“spinning 2009 18:8 electron” to explain World the Scientific fine-structure Review Volume observed - 9in x 6in in the anomalous lepton Zeeman eÆect in atomic spectra.c The fine-structure splitting can be un- derstood as thephysics interaction of dipole of the moments magnetic dipole moment of the electron with the magnetic field produced by the nuclear motion, which in the electron’s rest frame appears to be orbiting about the electron. The electron’s Historical Introduction 7 magnetic dipole moment is along its spin and is given by q µ~ = g ~s, (1.1) moment mustmagnetic be along the spin. We2m can write an expression similar to whereEq. (1.1),q = e is the charge of the≥ particle¥ in terms of the magnitude of ± q the electron chargeelectrice, and thed~ = proportionality¥ ~s, constant g is the g(1.4)-factor for spin (which is sometimes written2mc as g ). In their second paper [11], ≥ ¥ s Uhlenbeckwhere ¥ andis a Goudsmit dimensionless conclude constant that that the isg-factor analogous for spin to g isin twice Eq. that (1.1). for orbitalWhile angular magnetic momentum, dipole moments however (MDMs) the calculated are a natural fine-structure property of charged splitting wasparticles then twice with as spin, large electric as the dipole observed moments splitting. (EDMs) Only are later forbidden in 1926, both when by Thomasparity and showed by time that reversal the factor symmetry. of 2 discrepancy between experiment and calculationThe search was a for kinematic an EDM eÆ datesect [12], back did to spin the start suggestion to become of Purcell an accepted and Ramsey [35] in 1950, well in advance of the paper by Lee and Yang [36], aIn his paper Compton acknowledges A.L. Parson (Smithsonian Misc. Collections, 1915) as firstthat proposing a measurement that the electronof the neutron was a spinning EDM ring would of charge. be a good Compton way modified to search this proposalfor parity to be violation a much smaller in the distribution nuclear force. “concentrated An experiment principally was near mounted its center.” at Compton’sOak Ridge paper [37] is almost soon thereafter unknown. which placed a limit on the neutron EDM bSee Allan Franklin,20 http://plato.stanford.edu/entries/physics-experiment/app5.html of dn < 5 10° e-cm, although the result was not published until after Stanford Encyclopedia£ of Philosophy,Appendix5,foranicediscussionputtingthe Stern–Gerlachthe discovery experiment of parity into violation. historical context. cIn theirOnce Nature parity paper violation [11] of 1926, was they established, acknowledge Landau Compton’s [38] independent and Ramsey suggestion [39] of spin.pointed out that an EDM would violate both P and T symmetries. This can be seen by examining the Hamiltonian for a spin one-half particle in the presence of both an electric and magnetic field,

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

Table 1.1. Transformation properties of the magnetic and electric ﬁelds and dipole moments.

E~ B~ µ~ or d~ P -+ + C -- - T +- -

38 The Standard Model value for the electron (muon) EDM is 10° 36 ∑ e-cm ( 2 10° e-cm), well beyond the reach of experiments (which are ∑ £ 27 19 at the 1.6 10° (1.8 10° ) e-cm level). Likewise, the Standard-Model £ £ October 19, 2009 18:8 World Scientiﬁc Review Volume - 9in x 6in lepton

Historical Introduction 7 moment must be along the spin. We can write an expression similar to October 19, 2009Eq. 18:8 (1.1), World Scientific Review Volume - 9in x 6in lepton q d~ = ¥ ~s, (1.4) 2mc ≥ ¥ where ¥ is a dimensionless constant that is analogous to g in Eq. (1.1). While magnetic dipole momentsHistorical (MDMs) Introduction are a natural property of charged 7 particles with spin, electric dipole moments (EDMs) are forbidden both by parity and by time reversal symmetry. moment must be along the spin. We can write an expression similar to The search for an EDM dates back to the suggestion of Purcell and Eq. (1.1), Ramsey [35] in 1950, well in advance of the paper by Lee and Yang [36], that a measurement of the neutron EDMq would be a good way to search d~ = ¥ ~s, (1.4) for parity violation in the nuclear force.2mc An experiment was mounted at ≥ ¥ whereOak Ridge¥ is a [37] dimensionless soon thereafter constant which placed that is a analogous limit on the to neutrong in Eq. EDM (1.1). 20 of d < 5 10° e-cm, although the result was not published until after Whilen magnetic£ dipole moments (MDMs) are a natural property of charged the discovery of parity violation. particles with spin, electric dipole moments (EDMs) are forbidden both by Once parity violation was established, Landau [38] and Ramsey [39] parity and by time reversal symmetry. pointed out that an EDM would violate both P and T symmetries. This The search for an EDM dates back to the suggestion of Purcell and can be seen by examining the Hamiltonian for a spin one-half particle in Ramseythe presence [35] in of 1950, both an well electric in advance and magnetic of the field,paper by Lee and Yang [36], that a measurement of the neutron EDM would be a good way to search for parity violation in the nuclear= µ~ force.B~ d~ AnE.~ experiment was mounted(1.5) at H ° · ° · Oak Ridge [37] soon thereafter which~ ~ placed~ a limit on the neutron EDM The transformation20 properties of E, B, µ~ and d are given in Table 1.1, and of d < 5 10° e-cm, although the result was not published until after wen see that£ while µ~ B~ is even under all three symmetries, d~ E~ is odd under the discovery of parity· violation. · both P and T. Thus the existence of an EDM implies that both P and T areOnce not good parity symmetries violation of was the established, interaction Hamiltonian, Landau [38] Eq. and (1.5). Ramsey In the [39] pointedcontext out of CPT thatsymmetry, an EDM would an EDM violate implies bothCP Pviolation.and T symmetries. This can be seen by examining the Hamiltonian for a spin one-half particle in the presence of both an electricTabledipole 1.1. and moments Transformation magnetic field, properties of the magnetic and electric= fieldsµ~ andB~ dipoled~ E.~ mo- (1.5) ments.H ° · ° · The transformation propertiesE~ of E~ , BB~~ , µ~ andµ~ ord~d~are given in Table 1.1, and we see that while µ~ B~ isP even under-+ all three symmetries, + d~ E~ is odd under · C -- - · both P and T. Thus theT existence+- of an EDM - implies that both P and T are not good symmetries of the interaction Hamiltonian, Eq. (1.5). In the context of CPT symmetry,EDM (d) an implies EDM both implies P and T violationCP violation. 38 The Standard Modeland also value CP violation for the if CPT electron symmetry (muon)holds EDM is 10° 36 ∑ e-cm ( 2 10° e-cm),Table well beyond 1.1. Transformation the reach of experiments (which are ∑ £ 27 properties19 of the magnetic and at the 1.6 10° (1.8 10° ) e-cm level). Likewise, the Standard-Model £ £electric fields and dipole moments.

E~ B~ µ~ or d~ P -+ + C -- - T +- -

Historical Introduction 5

to break the magnetic moment into two pieces: q~ g 2 µ =(1+a) where a = ° . (1.3) 2m 2 The first piece, predicted by the Dirac equation and called the Dirac moment, is 1 in units of the appropriate magneton, q~/2m. The second piece is the anomalous (Pauli) moment [23], where the dimensionless quantity a is sometimes referred to as the anomaly. The development of radio frequency engineering and microwave tech- Octobernology 19, 2009 18:8 during the Second World World Scientific WarReview Volume was - 9in quickly x 6in put to use afterward lepton in the laboratory. In 1947, motivated by measurements of the hyperfine structure in hydrogen that obtained splittings larger than expected from the Dirac theory [24–26], Schwinger [27] showed that from a theoretical Historical Introduction 5 viewpoint these “discrepancies can be accounted for by a small additional electronto break spin the magnetic magnetic moment moment” into two that pieces: arises from the lowest-order radiative i correction to the Dirac moment.q~ In his paper,g 2 Schwinger points out three µ =(1+a) where a = ° . (1.3) important features of his new2m theory. 2 The first piece, predicted by the Dirac equation and called the Dirac moment,The is 1 in new units Hamiltonian of the appropriate is superior magneton, toq the~/2m original. The second one piece in essen- is thetially anomalous three (Pauli) ways: moment it involves [23], where the the experimental dimensionless electron quantity a mass, is sometimesrather referred than the to unobservable as the anomaly. mechanical mass; an electron now Theinteracts development with of the radio radiation frequency field engineering only in and the microwave presence tech- of an ex- nologyternal during field the Second . . . the World interaction War was of quickly an electron put to with use afterward an external in thefield laboratory. is now In subject 1947, motivated to a finite by radiative measurements correction. of the hyperfine structure in hydrogen that obtained splittings larger than expected from In today’sthe Dirac language, theory [24–26], Schwinger Schwinger pointed [27] showed out that that from one a replaces theoretical the bare mass viewpoint these “discrepancies can be accounted for by a small additional andelectron charge spin with magnetic the physical moment” that (dressed) arises from mass the andlowest-order charge radiative (see Chapter 3 for additionalcorrection details). to the Dirac moment.i In his paper, Schwinger points out three Theimportant one-loop features contribution of his new theory. to a is shown diagrammatically in Fig. 1.1(b) October 19, 2009 18:8 World Scientific Review Volume - 9in x 6in lepton and has theThe value new Hamiltonianae = Æ/(2 isº superior) 0. to00116 the original, which one in essen- is independent of mass tially three ways: it involves' the experimental··· electron mass, and is therather same than for theaµ unobservableand aø . mechanical mass; an electron now October 19, 2009 18:8 World Scientific Review Volume - 9in x 6in lepton In the sameinteracts year, with the Kusch radiation and field Foley only in [29] the presence measured of an ex-ae with 4% precision, ternal field . . . the interaction of an electron with an external and found that the measured electron anomaly agreed well with Schwinger’s magnetic dipole momentfield is now of subject the to electron a finite radiative correction. prediction. They state that: “... the results can be described by g` = 1 and HistoricalIn today’s Introduction language, Schwinger pointed out that one replaces5 the bare mass g = 2(1.00119 0.00005).”j s and charge with± the physical (dressed) mass and charge (see Chapter 3 for iIn responseadditional to details). Nafe, et al. [24], Breit [28] conjectured that this discrepancy could be to break the magneticThomas moment precession into two pieces:Schwinger term explainedThe by one-loop the presence contribution of a to smalla is shown Pauli diagrammatically moment. It’s not in clearFig. 1.1(b) whether this paper Historical Introduction and has the value ae = Æ/(2º5) 0.00116 , which is independent of mass influencedq~ Schwinger’s work,g but2' in a footnote··· Schwinger states: “However, Breit has µ =(1+nota) correctlyand is thewhere drawn same for theaaµ= consequencesand°aø . . of his empirical(1.3) hypothesis.” j 2mIn the same year, Kusch2 and Foley [29] measured ae with 4% precision, The choice that g` =1andgs > 2wasguidedbytheoreticalprejudice.Themodern and found that the measured electron anomaly agreed well with Schwinger’s to break the magnetic momentThe into first two piece, pieces: predictedanomalous by magneticexperiments, the Dirac moment which equation a confine and a single called electron the in Dirac a Penning mo- trap, measure gs directly and fullyprediction. justify this They assumption. state that: “... the results can be described by g` = 1 and j ment, is 1 in units of the appropriategs = 2(1.00119 magneton,0.00005).”q~/2m. The second piece q~ g 2 ± µ =(1+ais) the anomalouswhere (Pauli)a = moment° .iIn response [23], whereto Nafe, et the al.(1.3)[24], dimensionless Breit [28] conjectured quantity that thisa discrepancy could be 2m 2 explained by the presence of a small Pauli moment. It’s not clear whether this paper is sometimes referred to as theinfluencedanomaly Schwinger’s. work, but in a footnote Schwinger states: “However, Breit has not correctly drawn the consequences of his empirical hypothesis.” The first piece, predicted by theThe Dirac development equation of and radio called frequencyj the engineering Dirac mo- and microwave tech- The choice that g` =1andgs > 2wasguidedbytheoreticalprejudice.Themodern ment, is 1 in units of the appropriatenology during magneton, the Secondq / World2mexperiments,. The War second which was confine quickly piece a single put electron to in use a Penning afterward trap, measure gs directly and ~ fully justify this assumption. is the anomalous (Pauli) momentin the [23], laboratory. where In the 1947, dimensionless motivated by quantity measurementsa of the hyperfine structure in hydrogen that obtained splittings larger than expected from is sometimes referred to as the anomaly. the Dirac theory [24–26], Schwinger [27] showed that from a theoretical The development of radioviewpoint frequency these engineering “discrepancies and can microwave be accounted tech- for by a small additional nology during the Second Worldelectron War spin was magnetic quickly moment” put that to use arises afterward from the lowest-order radiative in the laboratory. In 1947, motivatedcorrection to by the measurements Dirac moment.i In of his the paper, hyperfine Schwinger points out three structure in hydrogen that obtainedimportant splittings features of hislarger new than theory. expected from the Dirac theory [24–26], SchwingerThe [27] new showed Hamiltonian that is superiorfrom a to theoretical the original one in essen- viewpoint these “discrepancies can betially accounted three ways: for it involves by a small the experimental additional electron mass, rather than the unobservable mechanical mass; an electron now electron spin magnetic moment” thatinteracts arises with from the the radiation lowest-order field only radiative in the presence of an ex- correction to the Dirac moment.i Internal his paper, field . . . Schwinger the interaction points of an electron out three with an external important features of his new theory.field is now subject to a finite radiative correction. In today’s language, Schwinger pointed out that one replaces the bare mass The new Hamiltonian is superior to the original one in essen- and charge with the physical (dressed) mass and charge (see Chapter 3 for tially three ways: it involves the experimental electron mass, additional details). rather than the unobservable mechanical mass; an electron now The one-loop contribution to a is shown diagrammatically in Fig. 1.1(b) interacts with the radiation field only in the presence of an ex- and has the value a = Æ/(2º) 0.00116 , which is independent of mass ternal field . . . the interaction of an electrone with' an external··· and is the same for a and a . field is now subject to a finite radiative correction.µ ø In the same year, Kusch and Foley [29] measured ae with 4% precision, and found that the measured electron anomaly agreed well with Schwinger’s In today’s language, Schwinger pointed out that one replaces the bare mass prediction. They state that: “... the results can be described by g` = 1 and and charge with the physical (dressed) mass and chargej (see Chapter 3 for gs = 2(1.00119 0.00005).” additional details). ± iIn response to Nafe, et al. [24], Breit [28] conjectured that this discrepancy could be The one-loop contributionexplained to a is shownby the presence diagrammatically of a small Pauli moment. in Fig. It’s 1.1(b) not clear whether this paper and has the value a = Æ/(2ºinfluenced) 0.00116 Schwinger’s, which work, but is in independent a footnote Schwinger of mass states: “However, Breit has e not correctly drawn the consequences of his empirical hypothesis.” j ' ··· and is the same for aµ and aøThe. choice that g` =1andgs > 2wasguidedbytheoreticalprejudice.Themodern experiments, which confine a single electron in a Penning trap, measure gs directly and In the same year, Kusch andfully Foley justify this [29] assumption. measured ae with 4% precision, and found that the measured electron anomaly agreed well with Schwinger’s prediction. They state that: “... the results can be described by g` = 1 and g = 2(1.00119 0.00005).”j s ± iIn response to Nafe, et al. [24], Breit [28] conjectured that this discrepancy could be explained by the presence of a small Pauli moment. It’s not clear whether this paper influenced Schwinger’s work, but in a footnote Schwinger states: “However, Breit has not correctly drawn the consequences of his empirical hypothesis.” j The choice that g` =1andgs > 2wasguidedbytheoreticalprejudice.Themodern experiments, which confine a single electron in a Penning trap, measure gs directly and fully justify this assumption. I. INTRODUCTION

II. EXPERIMENTAL STATUS

For muon g 2[1], exp 10 aµ =11659208.9(5.4)(3.3) 10 ⇥ (1) SM 11 a =116591802(2)(42)(26) 10 µ ⇥ where the SM errors are due to the electroweak, lowest order hadronic, and higher order hadronic contributions so that

exp SM 11 a = a a =287(63)(49) 10 (2) µ µ µ ⇥ . For eEDM, recent ACME experiment reported[2]

29 d < 8.7 10 ecm (3) | e| ⇥ with 90% conﬁdence whereas the SM from CKM matrix predicts[3] 2

(2) has the dimensions of a dipole, ecm, and is determined38 by a ratio of the atomic matrix elementsde of10 the CSP ecm.and (4) de operators. Over the years, significant⇠ theoretical e↵ort has gone into computing the fd and r coe↵ecients for di↵erent molecular and atomic species; see e.g. [9–13]. W q W W equivalentfor experimental electronIf status only one species of EDM lepton is used for flavorfrom an EDM measurement,violation, refer to [4]. q the e↵ects of CSP and de cannot be separated (see e.g. W W electron-nucleon[14, 15] for recent operator discussions). Since the experimental sensitivity is usually reported as an inferred limit on the e e electron EDM, it is convenient to parametrize the e↵ect Pospelov,of CRitzSP as a contribution1311.5537 from an equivalent EDM: III. BASIS PROPERTIES OF ELECTROMAGNETICFIG. 1. Electron DIPOLE EDM de induced OPERATOR by the CKM phase via a 3 equiv closed quark loop. The contributions shown are: O(↵W ↵s) d rC . (3) 2 3 e ⌘ SP (left panel, Fig. 1a), and O(↵ ↵ ) (right panel, Fig. 1b). Taking the three leading experimental limits on the elec- In the presencetron of EDM,the supersymmetry, we list the relevant r coecients TeV [9–13], scale physicsto the conjugated gives weak an one-loop vertices. Nonzero contribution contributions to 2 20 only start at second order G , and are necessarily pro- r =1.2 10 ecm, F Tl ⇥ portional to the reduced Jarlskog/ invariant the electromagnetic dipole operator 20f such that the anomalous magnetic moment, electric rYbF =0.88 10 ecm, (4) ⇥ 2 5 D20 = s s2s3c1c2c3 sin 2.9 10 , (7) r =1.33 10 ecm. 1 dipole moment or charged leptonThO flavor⇥ violation arising fromJ can be enhanced' ⇥ compared where s and c aref the sines and cosines of the CKM an- Notice that although the f coecients for these systems i i d gles in the Kobayashi-MaskawaD basis and is the complex actually di↵er widely, the r coecients are approximately phase. The antisymmetric flavour structure of also to the Standard Modelthe same, prediction, reflecting the very similar and dynamical even can nature be of observed in upcoming experiments. leads to additional loop-level suppression of the EDMsJ of the P, T-odd perturbations to the electron Hamiltonian quarks and leptons in perturbation theory [16, 17] generated by both terms in (1). This leaves only mild As summarized in [5], we beginequiv with the general chirality-flipping electromagnetic dipole species-dependence in de . In this paper, we find that in the Standard Model A. Fundamental fermion EDMs operator, the CKM-induced CSP contribution dominates the direct contribution from de, and estimate it as In addition to the general constraints above, the EDM 18 C ( ) 10 , (5) operator for quarks and leptons breaks chiral symmetry, 1 SP J ⇠ µ⌫ 1 µ⌫ [ f ]ijfL µ⌫fR F [ † ]andijf thusR theµ⌫ coefLcientF must be at least linear in a chi- (5) where is the reduced Jarlskogi invariant.j Using the r f rality breakingi parameter.j In perturbation theory, this is J2 D 2 D coecients in (4), we can translate this into a character- generically the fermion mass mf itself. It turns out that istic CKM background to searches for the electron EDM, the antisymmetric flavour struture of actually ensures J that all 2-loop contributions to dq vanish [16], and the and let ' be the relative phaseequiv between38 Df operator and fermion mass d ( ) 10 ecm. (6) second-order weak exchanges need to be dressed with a e J ⇠ further gluonic loop. Thus, the d-quark EDM for ex- This is roughly nine orders of magnitude below the best ample arises only at 3-loop order [18, 19], and takes the current sensitivity to de, from ThO [4]. general form (6) The rest of this paper is organized' = as follows.Arg(m Inf⇤ theDf ) next section, we briefly review the CKM contributions ↵ ↵2 m m2 d(est)( ) e s W d c < 10 34ecm. (8) to fundamental fermions and other observable EDMs. In d 3 2 2 J ⇠ J (4⇡) mW mW Section 3, we turn to the CKM contribution to param- agnetic EDMs, and obtain the result (5). We2 finish with This estimates assigns ↵i/(4⇡) per corresponding loop, some concluding remarks in Section 4. and ignores additional numerical suppression or modest numerical enhancement by logarithms of quark mass ra- 2 tios. The factor of mc enters due to the flavour struc- 2. OVERVIEW OF EDMS FROM THE CKM ture of . The corresponding contribution to du is in- PHASE J 2 stead proportional to mums, and somewhat further suppressed. The most precise calculation of dq( ) can be In this section, we briefly review existing computations found in Ref. [19]. J of EDMs induced by the CKM phase. We will orga- EDMs of leptons are even further suppressed. A nize the discussion around a simple counting scheme, us- generic de diagram involves a quark loop with a minimum ing the basic symmetries to estimate the largest viable of four W -boson vertices. Such a loop can be attached to contribution to di↵erent classes of EDMs. In particular, the electron line either by two W -boson lines (Fig. 1a), CKM contributions to flavor-diagonal observables neces- at third order in the weak interaction, or via three virtual sarily vanish at first order in the weak interaction, due photons (Fig. 1b), at even higher loop order. discrepancy in anomalous magnetic moment of muon is the only evidence of BSM with a long history

from FNAL

BNL E821 from Hagiwara 2011 Advanced Cold Molecule Experiment (ACME) Harvard-Yale collaboration

Buffer Gas Beam Source Rotational Cooling Interaction Region

B E +V Electric Field Plates Magnetic Field Coils I Adjustable 298 K ThO2 Target Collimators I 50 K 4 K 16 K 690 nm +V Neon Buffer Gas PMT

-V Light Collection /4 and mirrors 20 GHz Pulsed Nd:YAG Microwave remixing

Vacuum 943 nm 1090 nm -V 1090 nm

Atmospheric Pressure Magnetic Shielding I. INTRODUCTION

II. EXPERIMENTAL STATUS

exp SM 11 a = a a =287(63)(49) 10 (2) µ µ µ ⇥ . For eEDM, recent ACME experiment reported[2]

29 d < 8.7 10 ecm (3) | e| ⇥ with 90% conﬁdence whereas the SM from CKM matrix predicts[3]

38 d 10 ecm. (4) e ⇠ for experimental status of lepton ﬂavor violation, refer to [4].

III. BASIS PROPERTIES OF ELECTROMAGNETIC DIPOLE OPERATOR

In the presence of the supersymmetry, TeV scale physics gives an one-loop contribution to the electromagnetic dipole operator such that the anomalous magnetic moment, electric Df dipole moment or charged lepton ﬂavor violation arising from can be enhanced compared Df to the Standard Model prediction, and even can be observed in upcoming experiments. BasicsAs summarized of the in anomalous [5], we begin with magnetic the general moment chirality-ﬂipping electromagnetic dipole Electrostatic properties of a charged particle: operator,Lepton electromagnetic dipole moments Charge Q,Magnetic(dipole)momentµ~,Electricdipolemoment~d 1 µ⌫ 1 µ⌫ For a spin 1/2 particle:[ f ]ijfLiµ⌫fRjF [ f† ]ijfRiµ⌫fLjF (5) e 2 D 2 D 1 µ~ = g ~s, g =2(1 + a), a = (g 2) : anomalous magnetic moment and let ' be the2m relative phase between Df operator and2 fermion mass Dirac ' = Arg(m⇤ D ) (6) Long interplay between| {z experiment} andf theory:f structure of fundamental interactions In Quantum Field Theory (with C,P2 invariance):

γ(k) µ⌫ µ 2 i k⌫ 2 = ( ie)¯u(p 0) F1(k ) + F2(k ) u(p) 2 2m 3 p p’ 6 Dirac Pauli 7 4 5 | {z } | {z } F1(0) = 1 and F2(0) = a if F2 is real 2 ae : Most precise determination of ↵ = e /4⇡. aµ: Less precisely measured than ae , but all sectors of Standard Model (SM), i.e. QED, Weak and QCD (hadronic), contribute signiﬁcantly. Sensitive to possible contributions from New Physics: 2 2 m` mµ a` 43000 more sensitive than ae [exp. precision factor 19] ⇠ mNP ) me ⇠ ! „ « „ « 3 Lepton electromagnetic dipole moments

The anomalous magnetic moment a (g 2)/2iscomingfromitsrealpart f ⌘

2 mf af = | | f cos ' (7) eQf |D | such that

e µ⌫ MDM = F2 µ⌫ F (8) L 4mf whereas the electric dipole moment is given by its imaginary part

d = sin ' (9) f |Df | such that

i = d F µ⌫. (10) LEDM 2 f µ⌫ 5 This operator is closely related to the lepton ﬂavor violation. The branching ratio of l l i ! j is directly given by

m3 (l l )= li ( [ ] 2 + [ ] 2). (11) i ! j 16⇡ | Df ji| | Df ij|

+ Moreover, l lll branching ratio can have a simple relation to l l provided the i ! j j j i ! j + photon panguin diagram which is made up of subdiagram and ll pair attached to the Df j i end of the photon line. For supersymmetry case, we have

2 Br(li 3lj) ↵ mli 11 ! = ln 2 . (12) Br(li lj) 3⇡ ml 4 ! h j i Various one loop diagrams contributing to . In [5], it was assume that Df - Small mixing between gauginos and higgsinos : From Eq. (A1), we impose m2 Z 1 (13) M 2 µ2 ⌧ 1,2 - Sizable tan and then consider dominant diagram under such conditions.

1. The Chirality Conserving Diagrams In this case, the chirality ﬂipping occurs in the external lepton mass insertion by Gordon identity.

3 Lepton electromagnetic dipole moments from supersymmetry Chirality conserving diagrams

(A) (B) ˜ ˜ W ± W3

g g g ˜ g lL 2 ⌫˜lL 2 lL lR lL 2 lL 2 lL lR

g ˜ g g ˜ g lL 1 lL 1 lL lR lL lR 1 lR 1 lR right-handed slepton in the loop Figure 1: Chirality conserving diagrams wherem ˜ corresponds to the heaviest particle’s mass in the loop. An important thing to note is that Dl by these one-loop chirality conserving diagrams cannot have a CP violating phase, in other words,

Im(Dl)=0, which means that EDM contribution from these diagrams vanishes. This is because gaugino mass insertion ﬂips the lepton chirality so that the CP phase of gaugino mass cannot enter the diagrams. Remember that physical phases of the minimal supersymmetric standard model (MSSM) in addition to the CKM phase of the standard model (SM) are given by

2 (9) Arg(MaµBµ⇤), Arg(au,d,lµBµ⇤), Arg(mfij˜ ), where m2 (i = j)denoteo↵-diagonal elements of sfermion soft supersymmetry breaking fij˜ 6 mass matrices. Then we can always choose a ﬁeld basis that additional CP phases beyond the SM occur at only M ,a ,andm2 . Therefore, if we neglect the CP phases a u,d,l fij˜ arising from o↵-diagonal elements of a and m2 , gaugino masses are the only source to l fij˜

4 The anomalous magnetic moment a (g 2)/2iscomingfromitsrealpart f ⌘

2 mf af = | | f cos ' (7) eQf |D | such that

e µ⌫ MDM = F2 µ⌫ F (8) L 4mf

whereas the electric dipole moment is given by its imaginary part

d = sin ' (9) f |Df | such that

i = d F µ⌫. (10) LEDM 2 f µ⌫ 5 This operator is closely related to the lepton ﬂavor violation. The branching ratio of l l i ! j is directly given by

m3 (l l )= li ( [ ] 2 + [ ] 2). (11) i ! j 16⇡ | Df ji| | Df ij|

2 Br(li 3lj) ↵ mli 11 ! = ln 2 . (12) Br(li lj) 3⇡ ml 4 ! h j i Various one loop diagrams contributing to . In [5], it was assume that Df - Small mixing between gauginos and higgsinos : From Eq. (A1), we impose m2 Z 1 (13) M 2 µ2 ⌧ 1,2 -LeptonSizable tan electromagnetic dipole moments and then consider dominantfrom diagram supersymmetry under such conditions.

1. The Chirality Conserving Diagrams In this case, the chirality ﬂipping occurs in the external lepton mass insertion by Gordon identity.

B˜⇤B˜ (LL, RR) : • h i 3 g2m 1 f (14) Df ⇠ 16⇡2m˜ 2

+,0 +,0 W˜ ⇤W˜ (LL) • h i g2m 2 f (15) Df ⇠ 16⇡2m˜ 2

2 2 2 Since g1/g2 =tan ✓W , This case is dominant over all the chirality conserving diagram

0 B˜⇤W˜ (LL) • h i Since bino and wino mixing is suppressed (no direct mixing; mixing through higgsinos only: see Eq. (A1)) by [m /(M 2 µ2)]2 so negligible. Z 1,2

2. The Chirality VIolating Diagrams In this case, the chirality ﬂipping takes place through the a. Slepton LR mixing and b. Higgsino-lepton-slepton coupling.

a. Slepton LR mixing

B˜B˜ (LR) • h i Since the slepton LR mixing is proportional to m (A µ tan ), for small A term or f sizable tan , it is given by

g2M µm tan 1 1 f . (16) Df ⇠ 16⇡2m˜ 4

W˜ 0B˜ • h i Since bino and wino mixing is suppressed by [m /(M 2 µ2)]2 so negligible. Z 1,2 W˜ 0W˜ 0 (LR) • h i Forbidden since wino does not couple to right handed slepton.

b. Higgsino-lepton-slepton coupling

B˜H˜ 0 • h d i 4 contribute non-vanishing CP phases to the diagrams in Fig. (1) 1, and in turn they cannot appear in the ﬁnal results because of the mentioned reason. However, there can be EDM contribution from chirality conserving diagrams at two-loop order as we shall see later. Lepton electromagnetic dipole moments B. Chirality violatingfrom diagrams supersymmetry

ylvuµ + alvd g y g ˜ ˜ g lL 2 ⌫˜lL l lR lL 1 lL lR 1 lR typically dominant diagram (N2) (N3) B˜ H˜ 0

H˜ 0 g1vu,d B˜ g1vu,d

y ˜ g g ˜ y lL l lR 1 lR lL 1 lL l lR

(N4) H˜ 0

W˜ 3 g2vu,d

g ˜ y lL 2 lL l lR

Chirality violating diagrams Figure 2: Chirality violating diagrams

In chirality violating diagrams, the lepton chirality is changed in internal propagators in the loop by Higgsino interactions or mixings by Higgs vevs. The dominant diagrams

1 Even if we introduce the right-handed neutrino sector, a CP phase in the PMNS matrix cancels in the diagram (A) in Fig. (1) because each vertex gives opposite sign of phase to each other.

5 B˜⇤B˜ (LL, RR) : • h i g2m 1 f (14) Df ⇠ 16⇡2m˜ 2

+,0 +,0 W˜ ⇤W˜ (LL) • h i g2m 2 f (15) Df ⇠ 16⇡2m˜ 2

2 2 2 Since g1/g2 =tan ✓W , This case is dominant over all the chirality conserving diagram

0 B˜⇤W˜ (LL) • h i SinceLepton bino and electromagnetic wino mixing is suppressed (no directdipole mixing; moments mixing through higgsinos only: see Eq. (A1)) by [m /(M 2 µ2)]2 so negligible. fromZ supersymmetry1,2

2. The Chirality VIolating Diagrams In this case, the chirality ﬂipping takes place through the a. Slepton LR mixing and b. Higgsino-lepton-slepton coupling.

a. Slepton LR mixing

B˜B˜ (LR) • h i Since the slepton LR mixing is proportional to m (A µ tan ), for small A term or f sizable tan , it is given by

g2M µm tan 1 1 f . (16) Df ⇠ 16⇡2m˜ 4

W˜ 0B˜ • h i Since bino and wino mixing is suppressed by [m /(M 2 µ2)]2 so negligible. Z 1,2 W˜ 0W˜ 0 (LR) • h i Forbidden since wino does not couple to right handed slepton.

b. Higgsino-lepton-slepton coupling

B˜H˜ 0 • h d i 4 B˜⇤B˜ (LL, RR) : • h i g2m 1 f (14) Df ⇠ 16⇡2m˜ 2

+,0 +,0 W˜ ⇤W˜ (LL) • h i g2m 2 f (15) Df ⇠ 16⇡2m˜ 2

2 2 2 Since g1/g2 =tan ✓W , This case is dominant over all the chirality conserving diagram

0 B˜⇤W˜ (LL) • h i Since bino and wino mixing is suppressed (no direct mixing; mixing through higgsinos only: see Eq. (A1)) by [m /(M 2 µ2)]2 so negligible. Z 1,2

2. The Chirality VIolating Diagrams In this case, the chirality ﬂipping takes place through the a. Slepton LR mixing and b. Higgsino-lepton-slepton coupling.

a. Slepton LR mixing

B˜B˜ (LR) • h i Since the slepton LR mixing is proportional to m (A µ tan ), for small A term or f sizable tan , it is given by

g2M µm tan 1 1 f . (16) Df ⇠ 16⇡2m˜ 4

W˜ 0B˜ • h i Since bino and wino mixing is suppressed by [m /(M 2 µ2)]2 so negligible. Z 1,2 W˜ 0W˜ 0 (LR) • h Leptoni electromagnetic dipole moments Forbidden since wino doesfrom not couple tosupersymmetry right handed slepton.

b. Higgsino-lepton-slepton coupling

B˜H˜ 0 • h d i There can be a simple (B˜ H˜ )mixingtermproportionalto4 m c s or g v and a d Z W 1 d (B˜ M insertion B˜ H˜ H˜ )proportionaltoM ( m s s )( µ)orM g v sin µ. 1 u d 1 Z W 1 2 For sizable tan , letter is dominant so that

g2M µm tan 1 1 f . (17) Df ⇠ 16⇡2m˜ 4

+,0 ,0 W˜ H˜ • h d i As the previous case, there can be two types of contribution. First, simple (W˜ H˜ ) d mixing term proportional to m c c or g v cos . Second, (W˜ M insertion W˜ Z W 2 2 H˜ H˜ )proportionaltoM (m s c )( µ)orM g v sin µ. For sizable tan the u d 2 Z W 2 2 latter is dominant:

g2M µm tan 2 2 f . (18) Df ⇠ 16⇡2m˜ 4

Relative Size

+,0 ,0 0 2 2 Typically, W˜ H˜ is dominant over B˜H˜ and B˜B˜ by a factor (g /g )(M /M ) • h d i h d i h i ⇠ 2 1 2 1 if M M . For instance, if M and M is the same at GUT scale, (M /M ) g2/g2. 2 & 1 1 2 2 1 ' 2 1

+ 0 0 Loop calculation shows that W˜ H˜ is larger than W˜ H˜ . Especially, for small µ • h d i h d i and large m2 , i.e. for x µ2/M 2 and y m2 /M 2 x 1,y > 1orx<1,y 1, lL ⌘ 2 ⌘ lL 2 ' ' + 0 0 the ratio (r W˜ H˜ / W˜ H˜ )betweentwographsbecomeslarger.Evenfor c ⌘h d i h d i x 1,y <1weﬁndr > 1: for y =0.1, r 3, y =0.01, r 2. For x =100,y =1, ' c c ' c ' r 4. c '

+ 2 2 2 For M M , W˜ H˜ / B˜B˜ 3(g /g )=3/ tan ✓ 10. But • 1 ' 2 h d i h i⇠ 2 1 W ' + a. For large µ, since W˜ H˜ 1/µ whereas B˜B˜ µ, B˜B˜ can be dominant. h d i/ h i/ h i 2 2 2 2 2 b. For mlL ,M2 ,µ >mlR ,M1 ,wehave

+ 2 W˜ H˜ eg mf h d i 2 tan f 64⇡2 m2 D ⇠ lL 2 (19) B˜B˜ eg1 mf µM1 h i tan f 96⇡2 m2 m2 D ⇠ lL lR

2 ˜ ˜ and so long as µM1/ml & 3/2, BB can be dominant. R h i 5 There can be a simple (B˜ H˜ )mixingtermproportionaltom c s or g v and a d Z W 1 d (B˜ M insertion B˜ H˜ H˜ )proportionaltoM ( m s s )( µ)orM g v sin µ. 1 u d 1 Z W 1 2 For sizable tan , letter is dominant so that

g2M µm tan 1 1 f . (17) Df ⇠ 16⇡2m˜ 4

Lepton electromagneticg2M µm dipoletan moments 2 2 f . (18) from supersymmetryDf ⇠ 16⇡2m˜ 4

Relative Size

+ 2 2 2 For M M , W˜ H˜ / B˜B˜ 3(g /g )=3/ tan ✓ 10. But • 1 ' 2 h d i h i⇠ 2 1 W ' In general chargino(wino-higgsino)+ diagram dominates a. For large µ, since W˜ H˜ 1/µ whereas B˜B˜ µ, B˜B˜ can be dominant. h d i/ h i/ h i 2 2 2 2 2 b. For mlL ,M2 ,µ >mlR ,M1 ,wehave

+ 2 W˜ H˜ eg mf h d i 2 tan f 64⇡2 m2 D ⇠ lL 2 (19) B˜B˜ eg1 mf µM1 h i tan f 96⇡2 m2 m2 D ⇠ lL lR

g2M µm tan 1 1 f . (17) Df ⇠ 16⇡2m˜ 4

g2M µm tan 2 2 f . (18) Df ⇠ 16⇡2m˜ 4

Relative Size

+ 0 0 Loop calculation shows that W˜ H˜ is larger than W˜ H˜ . Especially, for small µ • h d i h d i and large m2 , i.e. for x µ2/M 2 and y m2 /M 2 x 1,y > 1orx<1,y 1, lL ⌘ 2 ⌘ lL 2 ' ' Lepton electromagnetic+ 0 0 dipole moments the ratio (r W˜ H˜ / W˜ H˜ )betweentwographsbecomeslarger.Evenfor c ⌘h d i h d i x 1,y <1weﬁndrfrom> 1: for supersymmetryy =0.1, r 3, y =0.01, r 2. For x =100,y =1, ' c c ' c ' r 4. c '

+ 2 W˜ H˜ eg mf h d i 2 tan f 64⇡2 m2 D ⇠ lL 2 (19) B˜B˜ eg1 mf µM1 h i tan f 96⇡2 m2 m2 D ⇠ lL lR

2 ˜ ˜ and so long as µM1/ml & 3/2, BB can be dominant. R h i 5 For large mu or A, bino diagram dominates Lepton electromagnetic dipole moments from supersymmetry

2 2 2. Slepton universality : At leading order, mlL and mlR are proportional to identity, 2 2 2 2 respectively, say, mlL = m`LI and mlL = m`RI. The small violation from sleptoon universality can be parametrized by

2 2 (mlL )ij (mlR )ij (ij)LL 2 , (ij)RR 2 (23) ⌘ m`L ⌘ m`R

3. Proportionality : A term is universal such that LR mixing in the slepton is proportional to the lepton mass : m (A µ tan ). f

Under the assumption 2 and 3,

2 me me e µ,ae 2 aµ. (24) D ' mµ D ' mµ

Phase of Df What importantmuon in a (gis 43,0002) and eEDM,times the larger phase 'thanArg( electronm⇤ )isimportant.Such a µ ⌘ f Df phase should be expressed in tersm of basis independent one to have a physical meaning.

+ For W˜ H˜ which is the most dominant in many cases, we have h d i

g2 g2 ip i vd ip + iM2⇤ i vu⇤ iµ⇤ + p2 p2 (25) W˜ H˜ y h d i' f ⇣ (p2 ⌘ M 2)(p2⇣ µ 2) ⌘ | 2| | | where p p µ and p p µ. The ﬁrst term comes from a simple (W˜ H˜ )mixingandhas ⌘ µ ⌘ µ d a phase Arg(y v ) = Arg(m )whereasthesecondtermcomesfrom(W˜ M insertion f d f 2 W˜ H˜ H˜ ) mixing and has a phase Arg(y M ⇤µ⇤v⇤). Therefore, Arg(y⇤v⇤ )iszero u d f 2 u f dDf for the former and is given by Arg(y⇤v⇤y M ⇤µ⇤v⇤)= Arg(M µv v )= Arg(M µ(Bµ)⇤). f d f 2 u 2 u d 2 0 B˜H˜ , the phase is just the sum of real and Arg(M µ(Bµ)⇤) and assuming GUT relation h d i 1 Arg(M ) = Arg(M ), the latter is the same as Arg(M µ(Bµ)⇤). 1 2 2 For B˜B˜ (LR), we have h i

˜ ˜ i[iyf (vdA µ⇤vu⇤)]i iM1⇤ BB (LR)= 2 2 2 2 2 2 . (26) h i (p m )(p m ) p M1 lL lR | | While the second term gives the same phase as the previous example, the ﬁrst term gives

Arg(m⇤ ) = Arg(AM ⇤). f Df 1 The two loop phase coming from Barr-Zee type can be also considered[6].

7 2 2 2. Slepton universality : At leading order, mlL and mlR are proportional to identity, 2 2 2 2 respectively, say, mlL = m`LI and mlL = m`RI. The small violation from sleptoon universality can be parametrized by

2 2 (mlL )ij (mlR )ij (ij)LL 2 , (ij)RR 2 (23) ⌘ m`L ⌘ m`R

3. Proportionality : A term is universal such that LR mixing in the slepton is proportional to the lepton mass : m (A µ tan ). f LeptonUnder the assumption electromagnetic 2 and 3, dipole moments 2 me me e µ,ae 2 aµ. (24) from Dsupersymmetry' mµ D ' mµ

Phase of Df What important in (g 2) and eEDM, the phase ' Arg(m⇤ )isimportant.Such µ ⌘ f Df phase should be expressed in tersm of basis independent one to have a physical meaning.

+ For W˜ H˜ which is the most dominant in many cases, we have h d i

˜ ˜ i[iyf (vdA µ⇤vu⇤)]i iM1⇤ BB (LR)= 2 2 2 2 2 2 . (26) h i (p m )(p m ) p M1 lL lR | | While the second term gives the same phase as the previous example, the ﬁrst term gives

Arg(m⇤ ) = Arg(AM ⇤). f Df 1 The two loop phase coming from Barr-Zee type can be also considered[6].

7 selectron is heavy for electron EDM. This is what we can think of immediately if we take an order one CP violation in mind. As we found, similar spectrum was studies already in [8, 9], even though consider such spectrum for 125GeV Higgs, not electron EDM. We may add Leptonsuch attempts electromagnetic to discussion on electron EDM, but instead,dipole it would moments be also good if we are more systematic. Therefore,from we supersymmetry need to discuss more on various parameters in (N)MSSM satisfying muon g 2, electron EDM, and 125GeV Higgs. For the Barr-Zee type 2-loop diagram to be sizeable in electron EDM, too heavy selectron suppressing 1-loop diagram was suggested. To see this, we may need to more quantitative. For convenience, we saaume that the gaugino and the Higgsino are in the similar mass scale, say, M M µ M. As we discussed so far, 1-loop contribution of MSSM to 1 ⇠ 2 ⇠ ⌘ electromagnetic dipole moment for selectron is roughly given by

↵ tan e,1loop 2 me (33) D ⇠ 4⇡ sin ✓W me˜

whereas 2-loop Barr-Zee diagram gives (e.g. Eq. (12a) in [6] : Eq. (21a) says that 2 2 2 2 FH (M2 /µ ,M2µ/mH )doesnotgiveenhancemente↵ect from M2µ/mH > 1duetoits log dependence)

↵ 2 sin cos e,2loop 2 me (34) D ⇠ 4⇡ sin ✓W M ⇣ ⌘ hence we can estimate

↵ m2 De,2loop e˜ cos2 . (35) ⇠ 4⇡ sin ✓ M 2 De,1loop W Hence, 2-loop Barr-Zee dominates over 1-loop diagram for

4⇡ sin ✓ 1/2 M M m > W 30 . (36) e˜ ↵ cos ⇠ cos ⇣ ⌘ To account for muon g 2, m is roughly 100 300GeV for tan 10. In this case, selectron µ˜ ⇠ ⇠ mass need to be heavier than 300M. For small tan , say, tan =3,thesmuonmassfor muon g 2maybeslightlylargeraboutafactorof1.5-1.7.Ontheotherhand,selectron need to be larger than 90M for Barr-Zee diagram could be dominant. In a renormalization group mixing point of view, such mass hierarchy is slightly awkward. The argument below can be found in [10]. Even though it considers the possoble problem in the natural SUSY, the same can be applied here since gauge interaction rather than Yukawa

9 Lepton electromagnetic dipole moments from supersymmetry interaction is important. Neglecting Yukawa couplings (this may be apparent for lepton sector) running of the smuon soft mass is roughly given by

2 d 2 ↵a 2 ↵a 2 m = C2(µ)M + C(e)C2(µ)m (37) dt µ˜ 4⇡ a 16⇡2 e˜ a a X X and two-loop contribution becomes dominant provided

4⇡ sin ✓ 1/2 m > W 2M 60M (38) e˜ ↵ 2 ⇠ 2 ⇣ ⌘ so surely when Barr-Zee diagram is dominant over 1-loop contribution, two loop RG e↵ect becomes dominant provided such large hierarchy was maintained from high energy scale. Then smuon mass can be, not just heavier, tachyonic as the given two loop RG e↵ect makes smuon mass negative.

V. VIABLE MODELS

In this section, we discuss how electron EDM, muon g 2andleptonﬂavorviolationsuch as µ e restrict the slepton ﬂavor structure. To do this, we need to simplify expressions ! for each phenomena.

A. Slepton Mass Universality

As a simplest case, we consider the universal slepton case where m m m e˜L,R ' µ˜L,R ' ⌧˜L,R ⌘ m˜ . This can be found in many vanilla models such as mSUGRA, and from such models we can further assume thatm ˜ M . ⇠ 1,2 As discussed in Appendix B, muon g 2canbeestimatedas 2 9 tan 300GeV 10 m˜ 1 µ a =2.8 10 + µ ⇥ 20 m˜ 8 µ 10 m˜ ⇣ ⌘⇣ 2 ⌘ h i (39) 9 70GeV 10 m˜ 1 µ =2.8 10 + ⇥ m/˜ ptan 8 µ 10 m˜ ⇣ ⌘ h i + where the first term comes from W˜ H˜ whereas the second comes from B˜B˜ (LR). So h d i h i roughly, we conclude that slepton mass scale around 100GeV explains muon g 2: tan mµ˜ 100 GeV . (40) ' r 2 10 2 2 2 2 m , µ, M1,2 because the scale of (N2) is determined by 1/ max m , min(µ ,M1 ) while lR lR (C, N3, 4) are suppressed by 1/m2 in this limit and (N1) also becomes⇣ negligble by⌘ small f lL f mixing. 2 2 2 e 2 2 C where x3 = M2 /m and x4 = µ /m . These two relation is used to simplify a . In summary, we have III. EXPERIMENTAL STATUS 2 2 2 2 2 ↵m µM2 tan f (M /m ) f (µ /m ) a = µ 2 L L µ 4⇡ sin2 ✓ m2 M 2 µ2 In this section, we estimate theW magnitudeL of2 observable quantities related(B11) to the elec- 2 ⇣ 2 2 ⌘ 2 2 ↵mµµM1 tan fN (M1 /mR) fN (M1 /mL) + 2 2 2 2 2 tromagnetic dipole operators4⇡ cos in supersymmetric✓W (mR mL) modelsmR and comparemL them with current ⇣ ⌘ experimentalwhere limits. For example, at large tan & 10, dominant contributions to the Anomalous magneticx2 4x +3+2ln momentx of muon2 electromagnetic dipole operatorsf (x)= come from the,f chirality(1) = violating diagrams as we have (1 x)3 3 from supersymmetry (B12) seen. Also as explained in the previousx2 1 section,2x ln x (C)and(N11) will give dominant contri- f (x)= ,f(1) = . N (1 x)3 N 3 butions at large µ m˜,M limit. More specifically, from eq. (3), the muon anomalous l 1,2 In a special limit of mL,R M1,2 m˜ , we have a following simplified formula, magnetic moment aµ from supersymmetric' ⌘ contributions in this case becomes [? ] 2 9 tan 300GeV2 10 m˜ 1 µ aµ =2.8 109 70 GeV 10 m˜ + 1 µ (B13) a 2.8 ⇥10 20 m˜ 8 µ +10 m˜ , (13) µ ' ⇥ ⇣ m/˜ p⌘⇣tan ⌘ h 8 µ 10 im˜ which is used in [21]. ✓ Giudice◆ ✓ et al 2012 ◆ where we assume m˜ = M1,2 m˜ . In this expression, we can see the required scale lL,R ⌘ 70 GeV ofm/ ˜ ptan to explain the current deviation of aµ from the SM. From eq. (4), ⇠ Appendix C: Momentum Routing for J5. we can directly estimate theThe supersymmetric same diagram gives contribution muon EDM to the muon EDM,

Consider the one loop diagram for (g 2)µ is the one with boson and fermion internal dµ aµ 22 states and an external photon= is attachedtan to the10 bosoncm propagator.tan , For a moment, mass (14) e 2mµ ⇠ ⇥ insertion is neglected. Then we have to calculate 19 which is much smaller than the current bound on the muon EDM 10 cm [? ]. 4 d k i µ ⇠ 0 4 i(v + ia5) 2 2 2 2 ieQs(p2+ p 2k2) Assuming slepton mass(2 universality⇡) (m(p0˜ k=) mmLij, m ˜ =mRij)anda-term propor- Z Lij s Rij (C1) i i(◆k mf ) tionality (a = A y ), the electron anomalous magnetici(⇤ + i moment⇤5). and EDM are related lij l lij ⇥ (p k)2 m2 (k2 m2 ) v a s f to the muon’s by simple lepton mass scaling relations. Consider the part proportional to (vv⇤ aa)mµ part where we are interested in. Then 2 we need to consider momentum integralme of 13 ae = 2 aµ 10 , mµ ⇠ (p + p0 2k)µ (15) [(p k)2 m2][(p k)2de m2][km2 e dmµ2 ] 24 0 s =s f 10 cm tan . (p0 ke)µ mµ e ⇠ ⇥ (p k)µ = 2 2 2 2 2 2 + 2 2 2 2 2 2 . [(p0 k) ms][(p k) ms][k mf ] [(p013 k) ms][(p k) ms][k mf ] The current uncertainty related to ae is about 10 both experimentally and theoretically (C2) [? ], so the estimated value is not yet veriﬁable. On the other hand, the ACME collaboration recently announced an order of magnitude18 smaller limit for the electron EDM than

7 2 2 2 2 m , µ, M1,2 because the scale of (N2) is determined by 1/ max m , min(µ ,M1 ) while lR lR (C, N3, 4) are suppressed by 1/m2 in this limit and (N1) also becomes⇣ negligble by⌘ small f lL f mixing. e

III. EXPERIMENTAL STATUS

In this section, we estimate the magnitude of observable quantities related to the electromagnetic dipole operators in supersymmetric models and compare them with current experimental limits. For example, at large tan & 10, dominant contributions to the electromagnetic dipole operators come from the chirality violating diagrams as we have seen. Also as explained in the previous section, (C)and(N1) will give dominant contributions at large µ m˜,M limit. More speciﬁcally, from eq. (3), the muon anomalous l 1,2 magnetic moment aµ from supersymmetric contributions in this case becomes [? ] 2 9 70 GeV 10 m˜ 1 µ a 2.8 10 + , (13) µ ' ⇥ m/˜ ptan 8 µ 10 m˜ ✓ ◆ ✓ ◆ where we assume m˜ = M1,2 m˜ . In this expression, we can see the required scale lL,R ⌘ 70 GeV ofm/ ˜ ptan to explain the current deviation of a from the SM. From eq. (4), ⇠ µ we can directly estimate the supersymmetric contribution to the muon EDM,

dµ aµ 22 = tan 10 cm tan , (14) e 2mµ ⇠ ⇥ 19 which is much smaller than the current bound on the muon EDM 10 cm [? ]. ⇠ 2 2 2 2 Assuming slepton mass universality (mLij˜ = mLij, mRij˜ = mRij)anda-term proportionality (alij = Alylij), the electronSlepton anomalous universality magnetic moment and EDM are related to the muon’s by simple lepton mass scaling relations. 2 me 13 a = a 10 , e m2 µ ⇠ µ (15) d m d e e µ 29 24 the previous limit with de/e <= 8.7 1010cmcm [? ],tan which. is far smaller than the above e mµ ⇥e ⇠ ⇥ estimation. Thus assuming the simple universality13 and proportionality relations, we need The current uncertainty related to ae is about 10 both experimentally and theoretically 29 [the? ], supersymmetric so the estimatedACME value CP the phasebound is not previous on of yet electron veriﬁable. limit EDM with Onis d thee/e other < 8.7 hand,10 thecm ACME [? ], collabo- which is far smaller than the above ⇥ ration recently announcedestimation. an order of magnitude Thus assuming smaller the limit simple for the universality electron EDM and than proportionality relations, we need 4 We need , (16) the supersymmetric. 10 CP phase of for a_mu 7and ACME to be compatible to be compatible with the ACME result. 4 . 10 , (16) Breaking the slepton massSUSY CP universality, problem is more we can important get di now↵erent scaling relations for the electron electromagneticto dipole be compatible operators with with the the ACME muon’s. result. If we allow selectron mass much larger than smuon mass,Breaking the slepton mass universality, we can get di↵erent scaling relations for the

2electron2 electromagnetic2 dipole operators with the muon’s. If we allow selectron mass m˜ me m˜ 13 ae 2 2 aµ 2 10 ⇠ me˜muchmµ larger⇠ m thane˜ ⇥ smuon mass, 2 2 (17) de m˜ me dµ m˜ 24 2 2 2 29 10 mcm˜ metan

formula. me˜ 100m ˜ tan , (18) & ⇥ p wherem ˜ corresponds to the required scale in eq. (13)m to explain100m ˜ the muontan anamalous, (18) e˜ & ⇥ magnetic moment for given tan . Notice that p wherem ˜ corresponds to the required scale in eq. (13) to explain the muon anamalous tan magneticm˜ moment100 for givenGeV tan.. Notice that (19) ' r 2 tan So eq. (18) can be also expressed by m˜ 100 GeV. (19) ' r 2 1 So eq.me˜ (18)& 10 can TeV be also expressedtan tan by. (20) ⇥ r2 1 When we allow slepton mass non-universality, sleptonm flavor10 TeV changingtan o↵-diagonaltan . el- (20) e˜ & ⇥ 2 ements in slepton mass matrices will not vanish in general. These termsr can be another When we allow slepton mass non-universality, slepton flavor changing o↵-diagonal el- source of CP violating phases in supersymmety and induce sizable contributions to the ements in slepton mass matrices will not vanish in general. These terms can be another muon anomalous magnetic moment and electron EDM by sflavor mixing. These sflavor source of CP violating phases in supersymmety and induce sizable contributions to the muon anomalous magnetic moment and electron EDM by sflavor mixing. These sflavor 8

8 29 the previous limit with de/e < 8.7 10 cm [? ], which is far smaller than the above Lepton electromagnetic⇥ dipole moments estimation. Thus assuming the simple universality and proportionality relations, we need V. BRIEF SUMMARYin supersymmetry the supersymmetric CP phase of There are four possible scenarios.

4 1. Universal slepton masses . 10 , SUSY CP problem (16) 2. Heavy selectrons and light smuons to be compatible with the ACME result. 3. Light staus with the heavy sleptons in the ﬁrst two generations Breaking the slepton mass universality, we can get di↵erent scaling relations for the 4. Large a-term electron electromagnetic dipole operators with the muon’s. If we allow selectron mass We can summarize each scenario as in Table I. The contraints from radiative ﬂavor much larger than smuon mass, violating decays are given in Eq.(24, 27, 30) in terms of mixing angles. ***I will write more precisely2 for2 4a. *** 2 m˜ me m˜ 13 ae 2 2 aµ 2 10 ⇠ me˜ mµ ⇠ me˜ ⇥ Scenario (m ,m,m) (g 2) ACME µ e Higgs mass e2˜ µ˜ ⌧˜ 2 µ ! (17) de m˜ me dµ m˜ 24 29 1 (˜m, m,˜ m˜ ) Y, Eq.(13, 19) Y, < 10 4, Eq.(15, 16) easy 2 2 10 cm tanCP < 8.7 10 cm e ⇠ me˜ mµ e ⇠ me˜ ⇥ ⇥ ⇥ 2 (M,˜ m,˜ ?) Y Y, Eq.(18, 20) hard

So in this case, a3e becomes(M,˜ M,˜ m even˜ ) muchN, Eq.(29, more 31) smaller than theY current uncertaintyeasy bound, and we can interpret4a theal ACME= ylAl marginally result on Y?, theAl & electron5 TeV EDMY, byCP =0 the followingeasy more general 4b al (0,aµ, 0) formula. ⇠ Table I: Brief summary. Herem ˜ = 100 tan /2 GeV as in Eq.(19) and M˜ = 10 tan /2 TeV as in Eq.(20). p p m 100m ˜ tan , (18) e˜ & ⇥ p wherem ˜ corresponds to the required scale in eq. (13) to explain the muon anamalous magnetic moment for given tan . Notice that

[1] G. C. Cho, N. Haba, and J. Hisano, “The stau exchange contribution to muon g-2 in the tan decoupling solution,” Phys.m˜ Lett.100 B 529 (2002) 117GeV [hep-ph/0112163].. (19) ' 2 [2] G. F. Giudice and A. Romanino, “Electricr dipole moments in split supersymmetry,” Phys. So eq. (18) can beLett. also B 634 expressed(2006) 307 [hep-ph/0510197]. by

141 me˜ & 10 TeV tan tan . (20) ⇥ r2 When we allow slepton mass non-universality, slepton flavor changing o↵-diagonal elements in slepton mass matrices will not vanish in general. These terms can be another source of CP violating phases in supersymmety and induce sizable contributions to the muon anomalous magnetic moment and electron EDM by sflavor mixing. These sflavor

8 29 the previous limit with d /e < 8.7 10 cm [? ], which is far smaller than the above e ⇥ 29 the previousestimation. limit Thus with assumingd /e < 8. the7 simple10 cm universality [? ], which and is proportionality far smaller than relations, the above we need e ⇥ estimation.the supersymmetric Thus assuming CP the phase simple of universality and proportionality relations, we need the supersymmetric CP phase of 4 . 10 , (16) 4 10 , (16) to be compatible with the ACME result.. to be compatibleBreaking with the slepton the ACME mass result. universality, we can get di↵erent scaling relations for the Breakingelectron the electromagnetic slepton mass universality, dipole operators we can with get the di↵ muon’s.erent scaling If we relations allow selectron for the mass 2 2 2 2 m , µ, M1,2 because the scale of (N2) is determined by 1/ max m , min(µ ,M1 ) while electronmuch electromagnetic largerl thanR smuon dipole mass, operators with the muon’s. If we allow selectronlR mass 2 ⇣ ⌘ (C,f N3, 4) are suppressed by 1/m in this limit and (N1) also becomesf negligble by small 2 2 2 lL much larger than smuon mass,m˜ me m˜ 13 mixing.a a 10 e ⇠ m2 m2 µ ⇠ m2 ⇥e m˜ 2 m2 e˜ µ m˜ 2 e˜ e 2 2 13 (17) ae d mãmµ d m˜10 ⇠ me 2 m2 ⇠e mµ 2 ⇥ 24 29 III. EXPERIMENTALe˜ µ 2 e˜ STATUS2 10 cm tan < 8.7 10 cm e 2⇠ me˜ mµ e 2⇠ me˜ ⇥ ⇥ ⇥ (17) de m˜ me dµ m˜ 24 29 2 2 10 cm tan < 8.7 10 cm So in this case,e In⇠ thismaee˜becomes section,mµ e ⇠ we evenm estimatee˜ ⇥ much the more magnitude⇥ smaller of than observable⇥ the current quantities uncertainty related to bound, the elec- and we can interpret the ACME result on the electron EDM by the following more general So in this case, tromagneticae becomes dipole even much operators more in smaller supersymmetric than the models current and uncertainty compare them bound, with current and weformula. can interpretexperimental the ACME limits. result For on example, the electron at large EDM tan by & the10, following dominant more contributions general to the formula. electromagnetic dipole operators come from the chirality violating diagrams as we have seen. Also as explained inm thee˜ previous100m ˜ section,tan (C, )and(N1) will give dominant(18) contri- & ⇥ butions at large µ m m˜l,M1001,2 mlimit. ˜ Moretanp specifically,, from eq. (3), the muon(18) anomalous wherem ˜ corresponds to the e required˜ & scale⇥ in eq. (13) to explain the muon anamalous magnetic moment 2.a Heavyfrom supersymmetric selections contributions and light smuons in this case becomes [? ] µ p wheremagneticm ˜ corresponds moment to for the given required tan . scale Notice in eq. that (13) to explain2 the muon anamalous 9 70 GeV 10 m˜ 1 µ aµ 2.8 10 + , (13) magnetic moment for given tan . Notice' that⇥ m/˜ ptan 8 µ 10 m˜ ✓tan ◆ ✓ ◆ (19) where we assume m˜ =m˜ M1,1002 m˜ . In thisGeV expression,. we can see the required scale lL,R ' 2 tan⌘r 70 GeV ofm/ ˜ ptanm˜ to100 explain theGeV current. deviation of aµ from the SM. From(19) eq. (4), So eq. (18)⇠ can be also expressed' by 2 we can directly estimate ther supersymmetric contribution to the muon EDM, or So eq. (18) can be also expressed by dµ aµ 1 22 me˜ & 10= TeV tan tan10tancm. tan , (20)(14) e 2m⇥µ 2 ⇠ ⇥ 1 r 19 (20) which is muchm smallere˜ & 10 than TeV the currenttan bound tan on. the muon EDM 10 cm [? ]. When we allow slepton mass non-universality,⇥ 2 slepton flavor changing⇠ o↵-diagonal el- r 2 2 2 2 Assuming slepton mass universality (mLij˜ = mLij, mRij˜ = mRij)anda-term propor- Whenements we allowin slepton slepton mass mass matrices non-universality, will not vanish slepton in general. flavor changing These terms o↵-diagonal can be el- another tionality (alij = Alylij), the electron anomalous magnetic moment and EDM are related source of CP violating phases in supersymmety and induce sizable contributions to the ements in sleptonto the mass muon’s matrices by simple will not lepton vanish mass in scaling general. relations. These terms can be another muon anomalous magnetic moment and electron2 EDM by sflavor mixing. These sflavor source of CP violating phases in supersymmetyme and induce13 sizable contributions to the a = a 10 , e m2 µ ⇠ muon anomalous magnetic moment and electronµ EDM by sflavor mixing. These sflavor (15) de me dµ 24 = 8 10 cm tan . e mµ e ⇠ ⇥ 13 The current uncertainty related8 to ae is about 10 both experimentally and theoretically [? ], so the estimated value is not yet verifiable. On the other hand, the ACME collaboration recently announced an order of magnitude smaller limit for the electron EDM than

7 2. Heavy selections and light smuons

13 e) < 5.6 10 [? ]. From13 eq. (7) and eq. (21), we obtain e) < 5.6 10 [? ]. From eq. (7) and eq. (21), we obtain ⇥ ⇥ Lepton ﬂavour violation gives constraints on the ﬂavour mixing

14 2 14 2 2 2 2 2 Br(µ eBr() µ 1.85e) 101.85aµ10( ✏Lµeaµ ( ✏+Lµe ✏Rµe+ ✏Rµe), ), ! ' ! ⇥' ⇥ | | | || || | (23)(23) 3 32 2 2 2 13 13 1.45 101.45( ✏10 ( +✏Lµe✏ + ✏Rµe) < 5) .<6 5.61010. . ' ⇥' ⇥| Lµe| | | |Rµe|| | ⇥ ⇥ Therefore, Therefore, mu-e mixing should be extremely tiny

2 2 5 ✏Lµe + ✏Rµe < 2 10 . (24) 2 2 5 ✏ +| ✏ | |< 2| 10⇥ . (24) | Lµe| q | Rµe| ⇥ The electron EDM inducedq by the selectron-smuon mixing will be given dominantly from The electronthe EDM diagram induced (N1) withby the selectron-smuon mixing will be given dominantly from the diagram (N1) with 2 2 de aµ g1 M1 ˜l ˜l 30 g1 M1 L R sin 0 < 10 cm sin 0 , (25) e ⇠ 2m g2 M eµ eµ ⇥ ⇥ g2 M d a g2 M µ  2 2 g2M2 2 e µ 1 1 ˜lL ˜lR 30 1 1 2 eµ2eµ sin2 0 < 10 cm sin 0 2 , (25) wheree ⇠ again2m theg factorM (g1M1)⇥/(g2M2)indicatestherelativedi⇥ ↵gerenceM of size between µ  2 2  2 2 the diagram (C) (assumed to dominantly contribute to a ) and the diagram (N1), so 2 2 µ where again the factor (g1M1)/(g2M2)indicatestherelativedi↵erence of size between it will be absent in the case that (N1) is dominant over the other diagrams. And 0 the diagram (C) (assumed to dominantly contribute to a ) and the diagram (N1), so denotes possibly di↵erent CP phase of the diagram dueµ to the sﬂavor mixing. Anyhow it will be absentwe can in see the that case the electron that (N1) EDM is from dominant the selectron-smuon over the other mixing diagrams. is at least two And orders0 denotes possiblyof magnitude di↵erent smaller CP phase than the of ACME the diagram bound due due to to the the small sﬂavor mixing mixing. of (24). Anyhow we can see thatSimilarly the electron we can EDM estimate from the electron the selectron-smuon EDM bound induced mixing from is selectron-stau at least two mixing. orders First, selectron-stau mixing is constrained by ⌧ e decay, and the current bound on of magnitude smaller than the ACME bound due to the! small mixing of (24). 8 the mode is Br(⌧ e) < 3.3 10 [? ]. Then Similarly we can estimate! the electron⇥ EDM bound induced from selectron-stau mixing. 4 13 m˜ 2 2 2 First, selectron-stauBr(⌧ e mixing) 3.24 is10 constrained by ⌧ e decay,a andµ ( ✏ theL⌧e current+ ✏R⌧e ) bound, on ! ' ⇥ max (min(m⌧˜,m!e˜),M1,2,µ) | | | |  8 the mode is Br(⌧ e) < 3.3 10 [? ]. Then 4 4 m˜ 2 2 8 ! 2.54 ⇥10 ( ✏L⌧e + ✏R⌧e ) < 3.3 10 , ' ⇥ max (min(m⌧˜,me˜),M1,2,µ) | | | | ⇥  m˜ 4 Br(⌧ e) 3.24 1013 a 2( ✏ 2 + ✏ 2), (26) ! ' ⇥ max (min(m ,m ),M ,µ) µ | L⌧e| | R⌧e|  ⌧˜ e˜ 1,2 wherem ˜ corresponds to the value in eq. (19). This4 gives 4 m˜ 2 2 8 2.54 10 ( ✏L⌧e + ✏R⌧e ) < 3.3 10 , ' ⇥ max (min(m⌧˜,me˜),M1,2,µ) | | | | 2 ⇥ 2 2 2 max (min(m⌧˜,me˜),M1,2,µ) ✏  + ✏ < 1.14 10 . (27) | L⌧e| | R⌧e| ⇥ m˜  (26) p wherem ˜ corresponds to the value in eq. (19). This gives

10 2 2 2 2 max (min(m⌧˜,me˜),M1,2,µ) ✏ + ✏ < 1.14 10 . (27) | L⌧e| | R⌧e| ⇥ m˜  p

˜ + Hu Electron EDM from supersymmetry µ g v 2 u W˜ ˜ Barr-Zee diagramHd M2 ˜ + ˜ + g2 cos W Hu moment in the parameter region2 of loop tan estimates1. µ ⇠ g2vu ˜ h W h 2 2 ˜ d ↵ m˜ g me Hd e 2 , M2 2 2 e ⇠ ⇡ M2µ 16⇡ m˜ e e+ e 2 L g2 cos W˜ L R ↵ m˜ m e a , (34) ⇡ M µ m2 µ h ⇠ 2 µ Figure 3: Dominant two loop diagram contributing to the2 electron EDM m˜ 27 2 10 cm, ⇠ M2µ ⇥ ⇥ eL eL eR wherem ˜ is the required sparticle mass scale to explain the muon anomalous magnetic be estimated from an approximated formula below in the limit that M2,µ mZ ,mh. Figure 3: Dominant two loop diagram contributing to the electron EDM 27 moment for tan 1. Notice that the previous electron EDM bound (1.6 10 e h 2 ⇠ 2 ⇥ de ↵g2me M2 M2µ cm) before3 ACMEsin 2 sin does2F noth constrain2 , 2M2,µ, even if they are(32) both in similar size withm ˜ . be estimated from an approximatede ' 16 formula⇡ M2µ below in the limit that Mµ ,µ mmh ,m . ✓ 2 h Z◆ h 29 However, by the ACME bound de /e < 8.7 10 cm, with dhM2,mu>500↵g2m GeV to avoidM ACME2 M µ (tan beta=1,sin phi=1)⇥ e 2 e sin 2 sin F 2 , 2 , (32) e ' 16⇡3M µ 2 h µ2 m2 2 ✓ h ◆ p 2 2 1 (⇢ +1)lnM⇢2µ & ( ln20Rm ˜ ) (500 GeV) . (35) with F (⇢,R) ln R 1+ + .⇠ (33) h ⌘2 4(⇢ 1) O R ✓ ◆ 1This formula(⇢ +1)ln can be⇢ explicitlyln R checked by exact numerical simulations as in Fig. (2) in [2]. F (⇢,R) ln R 1+ + . (33) Note that the two-looph ⌘ contribution2 is4(⇢ approximately1) O R proportional to sin2, so it will be It means that even if we✓ take◆ small smuon mass to explain the muon anomalous magnetic Notesuppressed that the two-loop at large contribution tan limit.moment, is approximately However, the diagrams if proportional we consider (C, N2, to sin2 supersymmetric N3,, N4) so it inwill Fig. be models (2) will like be suppressed the by either large µ suppressedNMSSM at to large explain tan thelimit. Higgs However, mass if we with consider lower supersymmetric fine-tuning, models tan like2 the will be the relevant or large M2 as can be explicitly seen from. eq. (10-11). Furthermore, if we are interested NMSSM to explain the Higgs mass with lower fine-tuning, tan . 2 will be the relevant parameter region. In this case,in the the parameter two-loop space electron realizing EDM small can be fine-tuning sizable and in will the electroweak restrict symmetry breaking, parameter region. In this case, the two-loop electron EDM can be sizable and will restrict parameter space of the NMSSM. parameter space of the NMSSM. µ should be taken to be around the electroweak scale so that M2 should be larger than h Now let us estimate the size ofh the d /e in connection to the muon anomalous magnetic Now let us estimate the size of the1 TeV.de /e in In connection thate case, to the the muon other anomalous diagrams magnetic contributing to the muon anomalous magnetic moment at one-loop including the flavor-conserving diagrams in Fig. (1) are all suppressed 2 by large M2 and M1 assuming the gaugino mass unification relation. Therefore, the muon anomalous magnetic moment becomes hard to explain in the natural supersymmetry

scenario in the NMSSM if there exists the CP violating phase Arg(M2µBµ⇤) of the order of one. 12 ***Discussion12 on the subdominant (but negative sign) WW diagram including a light singlino in the NMSSM ...***

2 Of course, if we take M m˜ much smaller than M , the diagrams (C, D) in Fig. (1) and (N1, 2, 3) 1 ⇠ 2 in Fig. (2) may give still sizable contribution to the muon anomalous magnetic moment.

13 Now the electron EDM from the selectron-stau mixing is similarly obtained with eq. (25),

2 2 de m⌧ g1 M1 m˜ ˜l ˜l L R 00 2 aµ 2 e⌧ e⌧ sin , e ⇠ 2m g M2 max (min(m⌧˜,me˜),M1,2,µ) ⇥ µ  2  (28) 2 2 24 max (min(m⌧˜,me˜),M1,2,µ) g1 M1

11 C. Large A-term E↵ect

+ Even though W˜ H˜ is the most dominant diagram of , there are some cases where h d i Df + B˜B˜ (LR)diagrambecomesdominant.Since W˜ H˜ contains left-handed slepton only h i h d i whereas B˜B˜ (LR)containsbothleftandrighthandedsleptons,ifthereisalargeleft-right h i mixing, B˜B˜ (LR) can be enhanced by large left-right mixing insertion or, equivalently, h i lowering lightest slepton mass eigenvalue. As left-right mixing term is given by ml(A + µ tan ), we can consider the case of either large A term of large µ tan . The latter case is ﬂavor universal and studied in [15, 16]. If A term is ﬂavor universal, we expect the similar e↵ect as a large µ tan case.

Suppose m˜l m˜l . TeV. So long as slepton mass squrared are not tachyonic, either L ⇠ R 2 2 2 Aorµ tan can be large, for example, m˜ m˜ (mlA) > 0. This condition is easily lL lR satisﬁed for lepton sector by small lepton mass. For example, with pm˜ m˜ 500 GeV, lL lR ⇠ A<140 TeV. The muon g 2 using such large left-right mixing is interesting in the sense that we have an additional parameter space which looks to give too small muon g 2at ﬁrst glance but actually explain it. In this4. Large case, AB˜ B˜term(LR)diagrambecomesdominantso h i with m˜ m˜ µ MBino1,2 diagramm˜ , we can can dominate use over chargino diagram in this limit. lL ⇠ lR ⇠ ⇠ ⇠

2 BB 9 A 300GeV a =2.8 10 . (47) µ ⇥ 200m ˜ m˜ ⇣m : 100 ⌘⇣GeV, A : 2 TeV⌘ + Consider µ m˜ =150GeVandtan =2.Theusual W˜ H˜ result ⇠ h d i Instability bound is estimatedCharge in breaking Appendix minimum of [17], gives which an upper is reproduced bound on inA. Appendix D: 2 WH 2 9 tan 300GeV m˜ a1µ m=2˜ .82 10 (48) lLR ⇥< (m2 +16µ2)+m2m˜ + m2 µ. (49) 2 Hd ˜lL ˜lR 3 ml ⇣ ⌘⇣ ⌘ ⇣ ⌘ Takingsays that the it stability may give of (slightly) the false small vacuum to explain against muon quantumg 2, tunneling about half into of the account, central this value. can beThis di↵ canerent(For be supplemented the quarkA sector, bycan largebe [19]much A-term imposes larger provided thanA2 slepton+3µA2

14 A New Approach to µ-Bµ

Csaba Cs´aki,1 Adam Falkowski,2 Yasunori Nomura,3, 4 and Tomer Volansky5 1Institute for High Energy Phenomenology, Newman Laboratoryof Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA 2CERN, Theory Division, CH-1211 Geneva 23, Switzerland 3Department of Physics, University of California, Berkeley,CA94720,USA 4Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 5 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540

We present a new approach to the µ-Bµ problem of gauge mediated supersymmetry breaking. Rather than reducing the generically large contribution to Bµ we point out that acceptable elec- 2 ≪ ≪ 2 troweak symmetry breaking can be achieved with µ Bµ if at the same time Bµ mHd .This hierarchy can easily appear in models where the Higgs fields are directly coupled to the supersymmetry breaking sector. Such models can yield novel electroweak symmetry breaking vacua, can deal with the supersymmetric flavor and CP problems, allow for gauge coupling unification, and result in distinct phenomenological predictions for the spectrum of superparticles.

Introduction. Supersymmetry (SUSY) is a very supersymmetric standard model (MSSM)A New known Approach as the to µ-Bµ attractive candidate for explaining the stability of the µ problem: the SUSY preserving parameter µ is required 1 2 3, 4 5 weak scale against radiative corrections. However, it to be relatedCsaba to the Csáki, SUSYAdam breaking Falkowski, masses and,Yasunori in the Nomura, and Tomer Volansky 1Institute for High Energy Phenomenology, Newman Laboratoryof suffers from nagging problems such as the SUSY flavor absence of fine-tuning, both should be of the order2 of the problem, the SUSY CP problem, and the µ problem. weak scale. A solutionElementary to the Particle problem Physics, can Cornell arise if University,µ is Ithaca, NY 14853, USA 2CERN, Theory Division, CH-1211 Geneva 23, Switzerland The SUSY flavorwhere problemϵ 1and pointsΛH towardis the effective gauge SUSY mediated breaking scalegeneratedshould relate in conjunction the scale ΛH to with that for SUSY the gaugino, breaking. squark In the 3Department of Physics, University of California, Berkeley,CA94720,USA in the Higgs≪ sector. It is easy to see that the pattern of and slepton masses. In the following we explain how to SUSY breaking (GMSB) [1]. However, GMSB itself suf- limit of4µTheoretical=0,theMSSMhasanenhancedPeccei-Quinn Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Eq. (4) leads to a correct EWSB vacuum. The EWSB achieve these5 by coupling the MSSM Higgs fields directly fers from the variant of the µ problem2 2 called2 the µ-Bµ (PQ) symmetry.School If SUSY of Natural breaking Sciences, leads to Institute the breaking for Advanced Study, Princeton, NJ 08540 stability condition 2 µ + mHu + mHd > 2Bµ is satisfied to the SUSY breaking sector. problem [2]. The problem2 lies in|2 | the fact that generic of this (accidental) symmetry, a µ parameter of the cor- for mHd > 0, since mHd is parametrically larger than Bµ. First, recall general features for the soft gaugino masses 2 2 2 We present a new2 approach to the µ-Bµ problem of gauge mediated supersymmetry breaking. GMSB models predictConsequently, the relation Eq. (3) canB beµ solved16π withµ µ rectM magnitudea and scalar can masses bem generated.I in the GMSB While framework. this idea Be- can be ≈ ≫ lowRather we define than our reducing parameters the at generically the scale M largewhere contribution to Bµ we point out that acceptable elec- which prevents electroweak symmetrym breaking2 (EWSB) elegantly realized in the context of gravity mediation [6],2 2 Hd 1 troweak symmetry breaking can be achieved with µ ≪ Bµ if at the same time Bµ ≪ m .This tan β . (5) SUSY breaking effects are mediated to the MSSM sec- Hd if the soft masses in the Higgs sector≈ areBµ of≈ theϵ same or- it encounterstor.hierarchy In perturbative a can problem easily gauge in appear mediation, the framework in modelsM corresponds where of GMSB to the Higgs [2]. fields are directly coupled to the supersym- der as µ.Typically,solvingtheµ-Bµ problem2 is achieved2 2 2 In orderthemetry mass to scale dynamically breaking for the sector. messenger generate Such fields. modelsµ Unlessof can order there yield the is novel SUSY electroweak symmetry breaking vacua, can deal The condition for EWSB is Bµ > ( µ + mHu )( µ + 2 | | | | aspecialstructureintheSUSYbreakingsector(e.g.anwith the supersymmetric flavor and CP problems,2 allow for gauge coupling unification, and result by introducing somemHd ). additional In a typical SUSY dynamics breaking that scenario, ensures this condi-breaking masses in GMSB, the PQ symmetry must be 2 approximatein distinctR symmetry), phenomenological the soft masses predictions take the for form: the spectrum of superparticles. Bµ < µ (see e.g.tion [2, is 3, fulfilled 4]). as a result of renormalization group evolu-broken by coupling the Higgs fields directly to the SUSY tion making m2 negative. While this could also happen In∼ this letterwhere ϵ we1and proposeΛH is aHu the new eff approachective SUSY to breaking the µ-B scaleµ breakingshould relate sector. the scale SuchΛH couplings,to that for the4 however, gaugino,a squarktypically gen- in≪ our scenario, it is not necessary. Since B2 µ2m2 2 N 2 gaCI N 2 in the Higgs sector. It is easy to see that the patternµ 2 ofHd and sleptonMa masses.ga Λ In,m theI following we explainΛ how. (6) to problem. We point2 out2 that the GMSB relation B ≈ µ ≈erate both≈µ and16π2B at one≈ loop,8π leading2 16π2 to the relation m m ,bothsidesoftheaboveinequalityarepara-µ Introduction. Supersymmetryµ a=1!,2, (SUSY)3 is a very supersymmetric standard model (MSSM) known as the Eq. (4) leadsHu toHd a correct EWSB vacuum. The≫ EWSB achieve these2 by2 coupling2 the MSSM Higgs fields directly does not pose anymetrically problem of the if2 same we2 order allow of2 the magnitude, other and mass thereforeBµ 16π µ µ .Thisissaidtobeproblematicfor stability condition 2 µ + m + m > 2Bµ is satisfiedattractiveto the candidate SUSY breaking for explaining sector. the stability of the µ problem: the SUSY preserving parameter µ is required Hu Hd 2 ≈Here, ga are the≫ MSSM gauge2 couplings2 evaluated2 at M, 2 the condition|2 can| be satisfied with positive mHu . The parametersfor inm the> Higgs0, since sectorm is to parametrically also display larger a hierarchy. than Bweakµ. the scaleFirst, following against recalla general reason. radiative features If corrections.m forHu the softmH gauginod However,µ ,assuggested masses it to be related to the SUSY breaking masses and, in the Hd Hd and C are the quadratic Casimir coefficients. The quan- negative radiative corrections from top-stop loops are of I 2 ∼ ∼ In particular,Consequently, we argue Eq. that (3) the can bepattern solved with suffersnaivelyM froma tityand naggingN by scalarmeasures Eq. masses (2),problems the thenmI numberin Eq. such the of GMSB (3) as SUSY cannot the framework. breaking SUSY be satisfied flavor sector Be- withabsence of fine-tuning, both should be of the order of the course still present, but they do not have to be domi- 2 2 2 2 Blowµ fields weµ define charged.If,ontheotherhand, our under parameters the MSSM at gauge the scalem group,M andwheremΛ is Bµ, nant.2 EWSB2 is thenm driven2 by1 the dynamics thatproblem, gener- the SUSY CP problem, and the Hµu problem.Hd weak scale. A solution to the problem can arise if µ is µ m Bµ Hmd , (1) ≫ ∼ ∼ ate the boundaryHu tan β conditions,Hd Eq.. (4), rather than(5) by thesignificantSUSYthe breaking effective fine-tuning SUSY effects breaking are is mediated needed scale. In to perturbative the satisfy MSSM Eq. gauge sec- (2), since ∼ ≪ ≈≪Bµ ≈ ϵ The SUSY flavor problem points toward gauge mediated generated in conjunction with SUSY breaking. In the renormalization group (RG) evolution of the MSSM (seetor.mediation, In perturbativeΛ is the gauge SUSY mediation, breakingM masscorresponds squared split- to leads to viable EWSB. We show that this hierarchicalSUSYthe breaking experimental (GMSB) constraint [1]. However, of µ GMSB> m itselfZ then suf- requireslimit of µ =0,theMSSMhasanenhancedPeccei-Quinn The conditionalso [7] for for a EWSB related is idea).B2 > ( µ 2 + m2 )( µ 2 + theting mass divided scale for by the the SUSY messenger mass fields.for the messenger Unless there fields, is µ Hu 2 2 ∼ 2 arXiv:0809.4492v2 [hep-ph] 15 Oct 2008 pattern with the down-type Higgs soft| mass| dominat-| |fers fromm the,m variant,Bµ to of be the muchµ problem larger called than m the.Thisistheµ-B (PQ) symmetry. If SUSY breaking leads to the breaking m2 ). InLet a typical us now SUSY look at breaking the vacuum scenario, equation this (2). condi- The threeaspecialstructureintheSUSYbreakingsector(e.g.anHuF/MH.Notethat,ingeneral,therecanbed (1)Z coeffiµ- Hd O ing over Btionµ can is fulfilledcontributions be naturally as a result to obtained the of right-hand-side renormalization in GMSB (RHS) group models are evolu- allproblem of thenotoriousapproximate [2].cients The inµ the-RB problemsymmetry), RHSsµ problem, of Eq. lies the (6). in which soft For the masses general fact is considered take expressionsthat the generic form: for to be oneof this (accidental) symmetry, a µ parameter of the cor- 2 2 2 2 2 2 2 same order2 µ m m / tan β ϵ Λ . This sug- Ma and mI in GMSB, see [8]. 2 2 2 where thetion Higgs making multipletsm negative. areH directlyu WhileH thisd coupled could also toH happen theGMSBof models the most predict serious the problems relation B ofµ GMSB.16π µ µ rect magnitude can be generated. While this idea can be Hu ≈ ≈ ≈ In order to generate µ and B ,weconsiderthesuper-4 a in our scenario,gests that it the is not pattern necessary. of Eq. (4) Since couldB2 potentiallyµ2m2 lead to 2 N 2 µ gaC≈I N 2 ≫ SUSY breaking sector. Finally, we argue thatµ the pat-Hdwhich preventsBasicMa proposal.g electroweakΛ,mThe symmetry common breaking lore isΛ (EWSB) that. (6) a solutionelegantly realized in the context of gravity mediation [6], 2 2 2≈ ≈ potentiala couplings2 ofI the Higgs fields2 Hu,d 2to operators m2 m2a“fullynatural”EWSB:if,bothsidesoftheaboveinequalityarepara-ϵ ΛH mZ the electroweak ≈ 16π ≈ 8π 16π Hu Hd ≈ if the soft masses in the Higgs sectora=1!,2,3 are of the same or- it encounters a problem in the framework of GMSB [2]. tern in Eq. (1) leadssymmetry to interesting can be broken novel without phenomenology. any fine-tuning. Un-to theu,dµ-inBµ theproblem SUSY breaking must reduce sector the hierarchy between µ metrically of the same order of magnitude, and therefore O Many more detailsfortunately, of this scenario a careful analysis will be shows discussed that the in situation [5].der as isµ.Typically,solvingtheµ-Bµ problem is achieved In order to dynamically generate µ of order the SUSY 2 andHere,Bgµa.AcloseinspectionofEqs.(2,3)reveals,however,are the MSSM gauge couplings evaluated at M, the condition can be satisfied with positive mHu . The 2 The µ-Bµ problem.not that simple,Let and us begin we still bylikely reviewing need some theamountby introducing of a some= additionald θ (λuHu u dynamics+ λdHd d)+h that.c. ensures(7) breaking masses in GMSB, the PQ symmetry must be negative radiative corrections from top-stop loops are of thatand C thisI are is the notL quadratic" necessary. CasimirO To have coefficients.O a solution The quan- to Eq. (3), it cancellations in the RHS of Eq. (2) in realistic parameter< 2 2 2 2 µ-Bµ problemcourse of still GMSB. present, The but tree they level do not equations have to for be domi- theBµ isµtity su(seeffiNcientmeasures e.g. that [2, 3, the 2 4]).Bµ number 1 TeV, which2 then2 feeds in as2 a large one-loop con-can keep the generic GMSBO relation Bµ µ and, at the m m tan β m couplings λu,d dimensionless, and we adopt this conven-≫ 2 2 2 renormalizationZ tribution∼ 2 group of order (RG)Hu (500 evolution GeV)2 to ofHm thed2 .Oneshouldbe MSSM (seedoes notmediation, pose anyΛ is problem the SUSY if breaking we allow mass the squared2 other≫ split- mass Bµ 16π µ µ .Thisissaidtobeproblematicfor = µ − H,u (2) sametion time, below. generate Note4. thatLarge the our A hierarchy discussion term herem appliesB toµ atogether 2 ting divided by the SUSY mass for the messengerHd fields, ≈ ≫ 2 2 2 also2 [7] foraware− a| related of| the− fact idea). thattan if thisβ “irreducible”1 fine-tuningparameters is in2 the Higgs2 sector to also display a hierarchy.≫ the following reason. If m m µ ,assuggested withverym large classµ . of theories – the SUSY breaking sector Hu Hd Let usnot now eliminated look at the there vacuum is no compelling equation− (2). motivation The three to con-F/M.Notethat,ingeneral,therecanbeHu (1) coeffi- ∼ ∼ 2Bµ In particular,Lopsidedcan be wegauge strongly∼ argue mediation or that weakly provides the coupled, pattern an example. can contain OCsaki single et al or2008 naively by Eq. (2), then Eq. (3) cannot be satisfied with sider the “fully natural” pattern of Eq. (4). Instead, onecients in the RHSs of Eq. (6). For general expressions for sincontributions 2β = to the right-hand-side2 2 . (RHS) are all of(3) the Tomultiple be more scales, specific, and can let lead us to consider direct or theindirect following media- simple 2 2 2 2 2 2 2 2 2 2 22 2 2 Bµ µ .If,ontheotherhand,m m Bµ, same ordercan2µ considerµ m+ m moreHum general+ m/ tanHd scenariosβ ϵ Λ for. the This Higgs sug- massMa and mI in GMSB, see [8]. Hu Hd Hu Hd H patterntion of ofµ SUSY them mass breaking. parametersBµ m in, the Higgs sector:(1) ≫ ∼ ∼ | ≈| ≈ ≈ Hu Hd significant fine-tuning is needed to satisfy Eq. (2), since gests thatparameters the pattern which of Eq. still (4) lead could to realistic potentially EWSB lead without to In orderAfter to including∼ generate the≪µ and interactionsB≪µ,weconsiderthesuper- in Eq. (7), the mass Here we adopt the convention Bµ > 0. The first of these improving but not worsening2 2 the2 fine-tuning. In fact,potential couplings of the Higgs fields Hu,d to operators the experimental constraint of µ > m then requires a“fullynatural”EWSB:ifϵ ΛH mZ the electroweakleads to viableparameters EWSB. in the We Higgs2 show sector2 that receive this2 a2 direct hierarchical2 contribu-2 Z equations representstheories a problematic discussed below aspect can≈ also of lead the to minimal such a general- µ inϵ theΛH SUSY,Bµ breakingϵΛ ,m sector ϵ Λ ,m Λ , (4)2 2 ∼ 2 symmetry can be broken without any fine-tuning.arXiv:0809.4492v2 [hep-ph] 15 Oct 2008 Un-patternu,d withtion from the thedown-type SUSY breakingH HiggsH sector.u soft Assuming massH dominat-H thatd theH m ,m ,Bµ to be much larger than m .Thisisthe O ≈ ≈ ≈ ≈ Hu Hd Z fortunately,ization a careful of Eq. (4). analysis shows that the situation is SUSY breaking sector does not have a special structure notorious µ-B problem, which is considered to be one The “irreducible” fine-tuning described aboveing may over be Bµ can be naturally obtained in GMSB models µ not that simple, and we still likely need some amount of (such as= an approximated2θ (λ H PQ+ symmetry),λ H )+h the. contributionc. (7) ameliorated if we go beyond the MSSM withwhere simple the Higgs multipletsu u areu directlyd d d coupled to the of the most serious problems of GMSB. cancellations in the RHS of Eq. (2) in realistic parameter is givenLweak by" coupling to HuO strongO coupling to Hd GMSB soft terms. Improving the situation requiresSUSY ex- breaking sector. Finally, we argue that the pat- regions. The reason is that satisfying the LEP II bound Basic proposal. The common lore is that a solution tra contributions to the Higgs quartic couplingtern and/or inHere, Eq. (1)λu,d leadsare the to renormalized interestingNH couplings novel phenomenology. atN theH scale2 M. on the Higgs boson mass requires a rather heavy stop, µ λuλd ΛH ,Bµ λuλd ΛH , (8) to the µ-Bµ problem must reduce the hierarchy between µ to the scalar trilinear couplings. Such contributions mayBy rescaling≈ the operators16π2 u,d we≈ can always16π2 make the m˜ > 1 TeV, which then feeds in as a large one-loop con-Many more details of this scenarioO will be discussed in [5]. and Bµ.AcloseinspectionofEqs.(2,3)reveals,however, t appear due to direct couplings of the Higgs fields to thecouplings λu,d dimensionless, and we adopt this conven- tribution∼ of order (500 GeV)2 to m2 .OneshouldbeThe µ-Bµ problem. Let us begin by reviewing the that this is not necessary. To have a solution to Eq. (3), it SUSY breaking sector, or withHu the aid of extra singlettion below. Note thatN our discussion hereN applies to a aware of the fact that if this “irreducible”2 fine-tuning is2 m2 λ2 H Λ2 ,A λ2 H Λ , (9) 2 2 2 fields. The possibility of raising mH with Bµ µµ-Bwasµ problemvery largeHu,d of class GMSB.u,d of theories TheH tree – the levelH SUSYu,d equationsu,d breaking forH sector the is sufficient that 2Bµ 0. The the first contribution of these ameliorated if we go beyond the MSSM with simple is given by 2 2 2 2 2 2 equations represents a problematic aspect of the minimal µ ϵΛH ,Bµ ϵΛ ,m ϵ Λ ,m Λ , (4) GMSB soft terms. Improving the situation requires ex- ≈ ≈ H Hu ≈ H Hd ≈ H tra contributions to the Higgs quartic coupling and/or NH NH 2 µ λuλd ΛH ,Bµ λuλd Λ , (8) to the scalar trilinear couplings. Such contributions may ≈ 16π2 ≈ 16π2 H appear due to direct couplings of the Higgs fields to the SUSY breaking sector, or with the aid of extra singlet 2 2 2 2 NH 2 2 NH fields. The possibility of raising m with Bµ µ was m λ Λ ,AHu,d λ ΛH , (9) Hd Hu,d ≈ u,d 16π2 H ≈ u,d 16π2 considered in the context of NMSSM-type models≈ in [4]. Realization. Our approach to the µ-Bµ problem re- where NH is the effective number of messenger fields cou- quires some dynamics that naturally generates the pat- pled to Hu,d in Eq. (7). When the SUSY breaking sector tern of Eq. (4) or its variants. Moreover, a natural theory is perturbative, ΛH is F/M of these fields, and the sign V. BRIEF SUMMARY V. BRIEF SUMMARY

There are four possible scenarios. There are four possible scenarios. 1. Universal slepton masses 1. Universal slepton masses 2. Heavy selectrons and light smuons 2. Heavy selectrons and light smuons

3.3.LightLight staus staus with with the the heavy heavy sleptons sleptons in inthe the ﬁrst ﬁrst two two generations generations

4.4.LargeLargea-terma-term

WeWe can can summarize summarize each each scenario scenario as as in in Table Table I. I. The The contraints contraints from from radiative radiative ﬂavor ﬂavor violatingviolating decays decays are are given given in in Eq.(24, Eq.(24, 27, 27, 30) 30) in termsin terms of mixing of mixing angles. angles. ***I ***I will will write write moremore precisely precisely for for 4a. 4a. *** *** Summary

ScenarioScenario(m(me˜,m,mµ˜,m,m⌧˜) ) (g (g 2)µ2) ACMEACME µ µe Higgse Higgs mass mass e˜ µ˜ ⌧˜ µ ! ! 4 4 11 (˜m,(˜m,m,˜m,˜m˜m)˜ ) Y,Y, Eq.(13, Eq.(13, 19) 19) Y, Y,CP

4a al = ylAl marginally Y?, Al & 5 TeV Y, CP =0 easy 4a al = ylAl marginally Y?, Al & 5 TeV Y, CP =0 easy 4b al (0,aµ, 0) 4b a⇠l (0,aµ, 0) ⇠ Table I: Brief summary. Herem ˜ = 100 tan /2 GeV as in Eq.(19) and M˜ = 10 tan /2 TeV Table I: Brief summary. Herem ˜ = 100 tan /2 GeV as in Eq.(19) and M˜ = 10 tan /2 TeV as in Eq.(20). p p as in Eq.(20). p p

Muon g-2 and ACME can be compatible but it needs either i)1. large tan beta and vanishing SUSY CP or ii) 4a. lopsided gauge mediation or similar [1] G.(scenario C. Cho, N. Haba,2 is unattractive and J. Hisano, “The from stau exchangemu to contributione gamma to consideration muon g-2 in the [1] G. C. Cho, N. Haba, and J. Hisano, “The stau exchange contribution to muon g-2 in the decouplingand scenario solution,” Phys. 3 is Lett. already B 529 (2002) ruled 117 out [hep-ph/0112163]. by tau to mu gamma) decoupling solution,” Phys. Lett. B 529 (2002) 117 [hep-ph/0112163]. [2] G. F. Giudice and A. Romanino, “Electric dipole moments in split supersymmetry,” Phys. [2] G. F. Giudice and A. Romanino, “Electric dipole moments in split supersymmetry,” Phys. Lett. B 634 (2006) 307 [hep-ph/0510197]. Lett. B 634 (2006) 307 [hep-ph/0510197].

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