and the Anomalous Anomalous of the

Jonathan L. Fenga and Konstantin T. Matchevb a Center for Theoretical , Massachusetts Institute of Technology Cambridge, MA 02139, U.S.A.

b Theory Division, CERN, CH–1211 Geneva 23, Switzerland The recently reported measurement of the muon’s anomalous magnetic moment differs from the prediction by 2.6σ. We examine the implications of this discrepancy for supersym- metry. Deviations of the reported magnitude are generic in supersymmetric theories. Based on the new result, we derive model-independent upper bounds on the of observable supersymmetric particles. We also examine several model frameworks. The sign of the reported deviation is as predicted in many simple models, but disfavors -mediated supersymmetry breaking.

14.80.Ly, 12.60.Jv, 14.60.Ef, 13.40.Em hep-ph/0102146 MIT-CTP-3083, CERN-TH/2001-039 Measurements of magnetic dipole moments have a The standard model prediction has been greatly re- rich history as harbingers of profound progress in particle fined in recent years. The current theoretical status is physics. In the leptonic sector, the ’s gyromag- reviewed in Ref. [5] and summarized in Table II. The un- netic ratio ge 2 pointed the way toward Dirac’s theory certainty is dominated by the hadronic polariza- of the electron.≈ Later, the electron’s anomalous magnetic tion contribution that enters at 2-loops. This is expected + moment ae (ge 2)/2 α/2π played an important to be reduced by recent measurements of σ(e e− role in the development≡ − of ≈ electrodynamics and hadrons) at center-of- energies √s 1GeV.Thus,→ . Since then, increasingly precise mea- although the statistical significance of the∼ present devi- surements have become sensitive both to very high order ation leaves open the possibility of agreement between effects in quantum electrodynamics and to hadronic pro- experiment and the standard model, the prospects for a cesses, and the consistency of experiment and theory has definitive resolution are bright. If the current deviation stringently tested these sectors of the standard model. remains after close scrutiny and the expected improve- Very recently, the Muon (g 2) Collaboration has re- ments, the anomalous value of aµ will become unambigu- ported a measurement of the muon’s− anomalous magnetic ous. moment, which, for the first time, is sensitive to contri- In this study, we consider the recent measurement of aµ butions comparable to those of the weak [1]. to be a signal of physics beyond the standard model. In (See Tables I and II.) The new Brookhaven E821 result particular, we consider its implications for supersymmet- is ric theories. Supersymmetry is motivated by many inde- pendent considerations, ranging from the gauge hierarchy exp 10 a = 11 659 202 (14) (6) 10− (1.3 ppm) , (1) problem to the unification of gauge couplings to the ne- µ × cessity of non-baryonic dark , all of which require where the first uncertainty is statistical and the second supersymmetric particles to have weak scale masses. De- systematic. Combining experimental and theoretical un- viations in aµ with the reported magnitude are therefore certainties in quadrature, the new world average differs generic in supersymmetry, as we will see. In addition, from the standard model prediction by 2.6σ [1]: in contrast to other low-energy probes, aµ is both flavor- exp SM 10 and CP-conserving. Thus, while the impact of supersym- a a =(42 16) 10− . (2) µ − µ ± × metry on other observables can be highly suppressed by Although of unprecedented precision, the new result scalar degeneracy or small CP-violating phases in sim- is based on a well-tested method used in several previ- ple models, supersymmetric contributions to aµ cannot ous measurements. Polarized positive are circu- be. In this sense, the anomalous magnetic moment of lated in a uniform magnetic field. They then decay to the muon is a uniquely robust probe of supersymmetry, , which are emitted preferentially in the direc- and an anomaly in aµ is a natural place for the effects of tion of the muon’s spin. By analyzing the number of en- supersymmetry to appear. ergetic positrons detected at positions around the storage The anomalous magnetic moment of the muon is e mn the coefficient of the aµ µσ¯ µFmn,where ring, the muon’s spin precession frequency and anoma- 4mµ mn i m n lous magnetic moment are determined. The new result σ = 2 [γ ,γ ]. The supersymmetric contribution, SUSY is based solely on 1999 data. Analysis of the 2000 data aµ , is dominated by well-known neutralino-smuon 10 is underway, with an expected error of 7 10− (0.6 and chargino-sneutrino diagrams [7]. In the absence ∼ × 10 ppm), and the final goal is an uncertainty of 4 10− of significant slepton flavor violation, these diagrams (0.35 ppm) [6]. × are completely determined by only seven supersymme-

1 10 TABLE I. Recent measurements of aµ 10 and the cu- × searches, and the upper limit follows from mulative world average. requiring a perturbative bottom quark Yukawa Data Set Result World Average up to 1016 GeV. Supersymmetric contributions may ∼ CERN77 [2] 11 659 230 (85) (7 ppm) — therefore be greatly enhanced by large tan β. SUSY BNL97 [3] 11 659 250 (150) (13 ppm) 11 659 235 (73) To determine the possible values of aµ without SUSY BNL98 [4] 11 659 191 (59) (5 ppm) 11 659 205 (46) model-dependent biases, we have calculated aµ in a BNL99 [1] 11 659 202 (14) (6) (1.3 ppm) 11 659 203 (15) series of high statistics scans of parameter space. We SUSY use exact mass eigenstate expressions for aµ .Our calculations agree with Refs. [9–11] and cancel the cor- TABLE II. Contributions to the standard model prediction 10 responding standard model diagrams in the supersym- for aµ 10 . (See Ref. [5] and references therein.) × metric limit [12]. We require chargino masses above 103 Standard model source Contribution GeV and smuon masses above 95 GeV [13]. We also as- QED (up to 5-loops) 11 658 470.6(0.3) sume that the lightest supersymmetric particle (LSP) is Hadronic vac. polarization (2-loop piece) 692.4(6.2) stable, as in -mediated theories, and require it to Hadronic vac. polarization (3-loop piece) 10.0(0.6) be neutral. Finally, we record the mass and identity of Hadronic -by-light − 8.5(2.5) − the lightest observable supersymmetric particle (LOSP) Weak interactions (up to 2-loops) 15.2(0.4) for each scan point. Given the assumption of a stable Total 11 659 159.7(6.7) LSP, the LOSP is the 2nd lightest supersymmetric par- ticle, or the 3rd if the two lightest are a neutralino and try parameters: M1, M2, µ,tanβ, mµ˜L , mµ˜R ,and the sneutrino. Aµ. The first four enter through the chargino and neu- We begin by scanning over the parameters M2, µ, tralino masses: M , M ,andµ are the U(1) gaugino, 1 2 mµ˜L ,andmµ˜R , assuming gaugino mass unification M1 = SU(2) gaugino, and Higgsino mass parameters, respec- M2/2, Aµ =0,andtanβ = 50. The free parameters 0 0 tively, and tan β = Hu / Hd governs gaugino-Higgsino take values up to 2.5 TeV. The resulting values in the mixing. The lasth fivei determineh i the slepton masses, SUSY (MLOSP,aµ ) plane are given by the points in Fig. 1. where mµ˜L and mµ˜R are the SU(2) doublet and sin- We then relax the gaugino mass relation and consider glet slepton masses, respectively, and the combination both positive and negative values of M2/M1. The result- mµ(Aµ µ tan β) mixes left- and right-handed smuons. ing values are bounded by the solid curve. As can be seen, − Our sign conventions are as in Ref. [8]. and as verified by high statistics sampling targeting the The qualitative features of the supersymmetric contri- border area, the assumption of gaugino mass unification butions are most transparent in the mass insertion ap- has no appreciable impact on the envelope curve. Finally, proximation. The structure of the magnetic dipole mo- we allow any Aµ in the interval [ 100 TeV, 100 TeV]. The ment operator requires a left-right transition along the resulting sample is extremely model-independent,− and is -slepton line. In the basis, this transi- bounded by the dashed contour of Fig. 1. Note that the tion may occur through a mass insertion in an external envelope contours scale linearly with tan β to excellent muon line, at a Higgsino vertex, or through a left-right approximation. mass insertion in the smuon . The last two From Fig. 1 we see that the measured deviation in aµ contributions are proportional to the muon Yukawa cou- is in the range accessible to supersymmetric theories and pling and so may be enhanced by tan β. For large and is easily explained by supersymmetric effects. moderate tan β, it is not hard to show that the supersym- The anomaly in aµ also has strong implications for metric contributions in the mass insertion approximation the superpartner spectrum. Among the most impor- are all of the form tant is that at least two superpartners cannot decouple g2 if supersymmetry is to explain the deviation, and one i m2 µM tan βF , (3) of these must be charged and so observable at colliders. 16π2 µ i This is reflected in the upper bound on MLOSP for non- SUSY where i =1, 2, and F is a function of superparticle vanishing aµ . The large value of tan β is chosen to 4 masses, with F MSUSY− in the large mass limit [9]. allow the largest possible MLOSP. The solid contour is ∝ SUSY SUSY 2 2 Equation (3) implies aµ /ae mµ/me 4 parametrized by 4 ∼ ≈ × 10 ; aµ is therefore far more sensitive to supersymmet- ric effects than a , despite the fact that the latter is 350 aSUSY tan β 390 GeV 2 e µ = . (4) times better measured. Also, for M /M > 0, although 10 max 2 1 42 10− 50 MLOSP the contributions of Eq. (3) may destructively interfere, ×   SUSY SUSY typically sign(aµ )=sign(µM1,2); we have found ex- If aµ is required to be within 1σ (2σ) of the measured ceptions only rarely in highly model-independent scans. deviation, at least one observable superpartner must be Finally, the parameter tan β is expected to be in the lighter than 490 GeV (800 GeV). range 2.5 < tan β < 50, where the lower limit is from In Fig. 2 we repeat the above analysis, but for the ∼ ∼ 2 FIG. 1. Allowed values of MLOSP, the mass of the lightest SUSY FIG. 2. As in Fig. 1, but assuming a visibly decaying LSP. observable supersymmetric particle, and aµ from a scan of parameter space with M2 =2M1, Aµ =0,andtanβ = 50. scale parameters through two-loop Green (red) points have smuons (charginos/neutralinos) as equations [16] with one-loop threshold corrections and the LOSP. The 1σ and 2σ allowed aSUSY ranges are indicated. µ calculate all superpartner masses to one-loop [17]. Elec- The allowed aSUSY scale linearly with tan β.Relaxingthe µ troweak is broken radiatively with a full one- relation M2 =2M1 leads to the solid envelope curve, and loop analysis, which determines µ . further allowing arbitrary Aµ leads to the dashed curve. A | | stable LSP is assumed. In minimal , many potential low-energy effects are eliminated by scalar degeneracy. However, case where the LSP decays visibly in collider detectors, SUSY aµ is not suppressed in this way and may be large. as in models with low-scale supersymmetry breaking or SUSY In this framework, sign(aµ )=sign(µM1,2). As is R- violating interactions. In this case, the LOSP well-known, however, the sign of µ also enters in the su- is the LSP. We relax the requirement of a neutral LSP, persymmetric contributions to B Xsγ. Current con- and require slepton masses above 95 GeV and neutralino → straints on B Xsγ require µM3 > 0iftanβ is large. masses above 99 GeV [13]. The results are given in Fig. 2. In minimal supergravity,→ then, gaugino mass unification For this case, the solid envelope curve is parametrized by implies that a large discrepancy in aµ is only possible for aSUSY > 0, in accord with the new measurement. aSUSY tan β 300 GeV 2 230 GeV 4 µ µ = + , (5) In Fig. 3, the 2σ allowed region for aSUSY is plotted for 42 10 10 50 M max M max µ × − " LOSP   LOSP  # µ>0. Several important constraints are also included: bounds on the neutralino relic density, the Higgs boson and the 1σ (2σ) bound is M < 410 GeV (640 GeV). LOSP mass limit mh > 113.5GeV,andthe2σ constraint 2.18 These model-independent upper bounds have many 4 4 × 10−

3 FIG. 4. Contours of aSUSY 1010 in the minimal µ × SUSY anomaly-mediated model, for tan β =10andµ>0. The FIG. 3. The 2σ allowed region for aµ (hatched) in min- green region is excluded by mτ˜ > 72 GeV, and the red region imal supergravity, for A0 =0,µ>0, and two representative 4 is excluded at 2σ by B Xsγ<4.10 10− . values of tan β. The green regions are excluded by the require- → × ment of a neutral LSP (left) and by the chargino mass limit [1]H.N.Brownet al. [Muon (g 2) Collaboration], hep- of 103 GeV (bottom and right), and the blue (yellow) region ex/0102017. − has LSP relic density 0.1 Ωh2 0.3(0.025 Ωh2 1). [2] J. Bailey et al. [CERN-Mainz-Daresbury Collaboration], The area below the solid≤ (dashed)≤ contour is≤ excluded≤ by Nucl. Phys. B150, 1 (1979). B X γ (the Higgs boson mass), and the regions probed s [3]R.M.Careyet al., Phys. Rev. Lett. 82, 1632 (1999). by→ the tri-lepton search at Tevatron Run II are below the [4]H.N.Brownet al. [Muon (g 2) Collaboration], Phys. dotted black contours. (See text.) − Rev. D62, 091101 (2000) [hep-ex/0009029]. contrast to the case of conventional supergravity theo- [5] A. Czarnecki and W. J. Marciano, hep-ph/0010194. [6] R. M. Carey, talk given at ICHEP00, Osaka, Japan, 27 ries [8,15]. SUSY July – 2 August 2000; R. Prigl [Muon (g 2) Collabora- In Fig. 4 we investigate how large a positive aµ may tion], hep-ex/0101042. − be in the minimal anomaly-mediated model. This model [7] For references to the early literature, see Ref. [5]. is parametrized by Maux, m0,tanβ,andsign(µ), where [8] J. L. Feng and T. Moroi, Phys. Rev. D61, 095004 (2000) Maux determines the scale of the anomaly-mediated soft [hep-ph/9907319]. terms, and m0 is a universal scalar mass introduced to [9] T. Moroi, Phys. Rev. D53, 6565 (1996) [hep- SUSY remove tachyonic sleptons. To get aµ > 0, we choose ph/9512396]; Erratum, D56, 4424 (1997). µM1,2 > 0. We see, however, that the constraint from [10] M. Carena, G. F. Giudice and C. E. Wagner, Phys. Lett. B390, 234 (1997) [hep-ph/9610233]. B Xsγ is severe, as this sign of µ implies a construc- → [11] T. Blazek, hep-ph/9912460. tive contribution from charginos to B Xsγ in anomaly mediation. Even allowing a 1σ deviation→ in a ,wehave [12] S. Ferrara and E. Remiddi, Phys. Lett. B53, 347 (1974). µ [13] See, e.g., LEP experiments’ presentations at LEP Fest, checked that for all tan β, it is barely possible to ob- October 10-11, 2000, CERN, http://delphiwww.cern.ch/ tain 2σ consistency with the B Xsγ constraint, and offline/physics links/lepc.html. → ∼ minimal anomaly mediation is therefore disfavored. The [14] E. Gabrielli and U. Sarid, Phys. Rev. Lett. 79, 4752 dependence of this argument on the characteristic gaug- (1997) [hep-ph/9707546]; K. T. Mahanthappa and S. Oh, ino mass relations of anomaly mediation suggests that Phys. Rev. D62, 015012 (2000) [hep-ph/9908531]. similar conclusions will remain valid beyond the minimal [15] U. Chattopadhyay, D. K. Ghosh and S. Roy, Phys. Rev. model. D62, 115001 (2000) [hep-ph/0006049]. In conclusion, the recently reported deviation in [16] I. Jack, D. R. Jones, S. P. Martin, M. T. Vaughn and Y. Yamada, Phys. Rev. D50, 5481 (1994) [hep- aµ is easily accommodated in supersymmetric models. Its value provides model-independent upper bounds on ph/9407291]. masses of observable superpartners and already serves to [17] D. M. Pierce, J. A. Bagger, K. Matchev and R. Zhang, Nucl. Phys. B491, 3 (1997) [hep-ph/9606211]. discriminate between well-motivated models. We await [18] A. B. Lahanas and D. V. Nanopoulos, Phys. Rept. 145, the expected improved measurements with great antici- 1 (1987). pation. [19] M. Schmaltz and W. Skiba, Phys. Rev. D 62, 095005 Acknowledgments — This work was supported in part (2000) [hep-ph/0001172]. by the U. S. Department of Energy under cooperative [20] J. L. Feng, K. T. Matchev and F. Wilczek, Phys. Lett. research agreement DF–FC02–94ER40818. B 482, 388 (2000) [hep-ph/0004043]; Phys. Rev. D 63, 045024 (2001) [astro-ph/0008115]. [21] K. T. Matchev and D. M. Pierce, Phys. Rev. D60, 075004 (1999) [hep-ph/9904282]. [22] L. Randall and R. Sundrum, Nucl. Phys. B 557,79 (1999) [hep-th/9810155].