Treating the MSSM as an effective theory, we study the implications of having
dimension five operators in the superpotential for flavor and CP-violating
processes, exploiting the linear decoupling of observable effects with respect
to the new threshold scale \Lambda. We show that the assumption of weak scale
supersymmetry, when combined with the stringent limits on electric dipole
moments and lepton flavor-violating processes, provides sensitivity to \Lambda
as high as 10^7-10^9 GeV, while the next generation of experiments could
directly probe the high-energy scales suggested by neutrino physics.Comment: 4 pages, 1 figur

Adam Ritz

I. B. Khriplovich

Maxim Pospelov

Yudi Santoso

English

arXiv

Pospelov, M

Ritz, A

Santoso, Y

CERN Document Server

PRL 96, 091801 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006Flavor- and CP-Violating Physics from New Supersymmetric ThresholdsMaxim Pospelov,1,2 Adam Ritz,2,3 and Yudi Santoso1,21Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada2Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada3Theoretical Division, Department of Physics, CERN, Geneva 23, CH-1211 Switzerland(Received 25 October 2005; published 7 March 2006)0031-9007=Treating the minimal supersymmetric standard model as an effective theory, we study the implicationsof having dimension-five operators in the superpotential for flavor- and CP-violating processes, exploitingthe linear decoupling of observable effects with respect to the new threshold scale . We show that theassumption of weak-scale supersymmetry, when combined with the stringent limits on electric dipolemoments and lepton-flavor-violating processes, provides sensitivity to  as high as 107–109 GeV (and upto 1017 GeV through the  term), while the next generation of experiments could directly probe the high-energy scales suggested by neutrino physics.DOI: 10.1103/PhysRevLett.96.091801 PACS numbers: 12.60.Jv, 12.10.Kt, 11.30.Er, 11.30.HvWeak-scale supersymmetry (SUSY) is a theoreticalframework that helps to soften the so-called gauge hier-archy problem by removing the powerlike ultraviolet sen-sitivity of the dimensionful parameters in the Higgspotential. It also has other advantages, notably an improve-ment in gauge coupling unification and a natural darkmatter candidate, which have made it the standard para-digm for physics beyond the standard model (SM).However, the simplest scenario—the minimal supersym-metric standard model (MSSM)—suffers from a numberof well-known tuning problems, due in part to the largearray of possible parameters responsible for soft SUSYbreaking [1] and, consequently, the possibility of cata-strophically large flavor- and CP-violating amplitudes.The absence of new flavor structures and order-one sourcesof CP violation in the soft-breaking sector, as evidenced,respectively, by the perfect accord of the observed K and Bmeson mixing and decay with the predictions of the SM [2]and the null results of electric dipole moment (EDM)searches [3–5], motivates continuing work on the specificsof SUSY breaking.In the present Letter, we will instead ask: Given asolution to the flavor and CP problems in the soft-breakingsector, what sensitivity do we have to new high-scalesources of flavor and CP violation? Such effects wouldarise through SUSY-preserving higher-dimensional opera-tors generated at a new threshold   MW . Such thresh-olds are indeed expected due to various completions of theMSSM, e.g., via mechanisms for SUSY breaking andmediation, the breaking of flavor symmetries, and more-over via the physics generating neutrino masses and mix-ings. Intermediate scales are also suggested by the axionsolution to the strong CP problem, SUSY leptogenesisscenarios, and more entertainingly as a lowered grandunified theory/string scale arising from large compactifi-cation radii of extra dimensions. In contrast to nonuniver-sal or complex soft-breaking terms, the flavor- andCP-violating observables induced by such operators willscale as mSUSY1, and, thus, the constraints on non-06=96(9)=091801(4)$23.00 09180minimal flavor or CP translate directly into sensitivity to far above the scale of the superpartner masses mSUSY.At dimension five, there are several well-knownR-parityconserving operators associated with neutrino masses,HuLHuL, and baryon- and lepton-number violation,UUDE, QQQL [6]. The constraints on proton decay putsevere restrictions on the size of the latter baryon- andlepton-number-violating operators, b > 1024 GeV,where 1=b is the overall normalization scale. The ‘‘super-seesaw’’ operator HuLHuL is a welcome addition to theMSSM superpotential, as it generates Majorana massesand mixing for neutrinos, which imply  1014–1016 GeV. Note that, in the seesaw scenario, theactual scale of right-handed neutrinos, MR, is lower than, since 1  Y2M1R with a small Y, as is also favoredby SUSY leptogenesis.In what follows, we analyze in detail the remainingoperators allowed in the R-parity conserving MSSM atdimension-five level [6]. We write the superpotential asW WMSSM yhhHdHuHdHuYqeijklqeUiQjEkLlYqqijklqqUiQjDkQl~YqqijklqqUitAQjDktAQl; (1)where yh, Yqe, Yqq, and ~Yqq are dimensionless coefficients,the latter three being tensors in flavor space. The paren-theses in (1) denote a contraction of color indices. Notethat, since we will consider only supersymmetric thresh-olds, the superfield equations of motion can be used toeliminate all dimension-five corrections to the Ka¨hler po-tential, e.g., K5  cuQUHyd , absorbing them in W 5 andthe Yukawa terms, and slightly modifying the soft-breaking sector. A renormalizable realization of (1) caneasily be obtained, e.g., the MSSM extended by a singlet N(the NMSSM) or an extra pair of heavy Higgs bosons.The full Lagrangian descending from (1) is rather cum-bersome, and we will focus our attention here on those1-1 © 2006 The American Physical SocietyPRL 96, 091801 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006dimension-five operators which are of potential phenome-nological interest, specifically those that involve two SMfermions and two sfermions. We then proceed to integrateout the sfermions to obtain operators composed from theSM fields (or, more precisely, those of a type II two-Higgs-boson doublet model). We will impose the requirements offlavor triviality and CP conservation in the soft-breakingsector. Thus, all dimension  4 coefficients in the Higgspotential, trilinear terms Ai, gaugino masses Mi, and the parameter, will be taken real. We will also make thesimplifying assumption of universal sfermion masses, de-noted msq, msl, which we will take, along with , Mi, to besomewhat larger than MW . Deferring the full details [7], wequote the relevant results below:Correction to the SM fermion masses.—The SM opera-tors of lowest dimension that are of phenomenologicalinterest are the fermion mass operators. From the diagramsin Fig. 1(a), we obtain the following corrections:MeijYqeklijM0u 	kl3lnqe=msq82qeA	ucot;MdijKqqklijM0u 	kllnqq=msq42qqA	ucot;(2)with a similar correction to Mu. The notation impliessummation over the repeated flavor indices, and we havedefined the combination Kqq  Yqq  2 ~Yqq=3. M0e;d;udenote unperturbed mass matrices arising fromdimension-four terms in the superpotential. Note that thecorrections proportional to Au directly break SUSY, whilethose proportional to  arise from corrections to the Ka¨hlerpotential.Dipole operators.—At dimension five, dipole operatorsfirst arise at two-loop order, as in Fig. 1(b). In the chargedlepton sector, they result inFIG. 1. Several representative loop corrections to: (a) SM fer-mion masses; (b) dipole amplitudes contributing to EDMs (cf.the supersymmetric Barr-Zee diagrams [18]),  ! e	, b ! s	,g 2; and (c),(d) dimension-six four-fermion operators. Thecrossed vertex descends from dimension-five terms in the super-potential (1).09180L eAucotqem2sqe123Mu	klYqeklij EiFPLEjH:c:;(3)where we treated LR squark mixing as a mass insertion andused PL  1 	5=2 and F  F. In the quarksector, the corresponding results are more cumbersome dueto a large number of possible diagrams.Jumping an additional dimension, we now considerdimension-six four-fermion operators generated by variousterms in (1). Two representative diagrams are shown inFigs. 1(c) and 1(d).Semileptonic operators.—Integrating out gauginos andsfermions as in Fig. 1(c), we find the following semilep-tonic operators, sourced by QULE:L qe  1qemSUSYs3Yqeijkl UiQj EkLl  H:c:: (4)Here m1SUSY denotes a combination of superpartnermasses folded with a loop function F: m1SUSY M3m2sq FM23=m2sl, and Fa  2f1 a a lna=1a2g, with F1  1 (see [8] for the unequal mass case). In(4) we have retained only the gluino-squark contribution,which is expected to dominate unless there are additionalhierarchies between the masses of sleptons and squarks.Four-quark operators.—Integrating out gluinos andsquarks as in Fig. 1(c), we arrive at the following four-quark effective operators:L qq  1qqmSUSYs12Kqq83 UQ DQ   UtAQ  DtAQ H:c:; (5)where the summation over flavor is carried out exactly as in(1). The largest down-type F  2 operator arises insteadfrom Fig. 1(d),L dd 1qqmSUSY1162Y	uimY	dnjKqqijkl13 QmDn DkQl QmtADn DktAQlH:c:; (6)which inevitably contains additional Yukawa suppressionoriginating from the Higgsino-fermion-sfermion vertices.Here mSUSY is a combination of SUSY masses as in (4) and(5) with M3 replaced by .We will now turn to the phenomenological consequen-ces and the sensitivity to qe and qq in various experi-mental channels. Of course, one of the most importantissues is the flavor structure of the new couplings constantsYqe, Yqq, and ~Yqq. We will assume that these coefficientsare of order one and do not factorize: Yqe  YuYe. Withthis assumption, we should first determine the natural scalefor  such that the corrections to SM fermion masses donot exceed their measured values.Particle masses and  term.—Taking MuAukl MuAu33 mtAt  175 300 GeV in (2), and assuming1-2PRL 96, 091801 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006a maximal Yqe3311 O1, we arrive at the estimate,me 3mtAtYqe3311 lnqe=msq82qe 1 MeV107 GeVqe: (7)Equation (7) clearly implies that the natural scale for newphysics encoded in the semileptonic operators in the super-potential is qe  107 GeV, while the corresponding scalein the quark sector is slightly lower.A strikingly high naturalness scale emerges from con-sideration of the effective shift of  due to the masscorrections (2). Assuming uncorrelated phases betweenYqq and the eigenvalues of Yu and Yd, we find  Immdmd ImKqq3311mtAt lnqq=msq42mdqq 107 GeVqq: (8)Equation (8) translates directly to an extremely strongbound on qq in scenarios where  ’ 0 is engineered byhand, either by using discrete symmetries at high energies[9] or by imposing an approximate global U1 symmetryat tree level to ensure m0u  0. In these cases, the experi-mental bound on the neutron EDM, jdnj< 6 1026e cm[5] (soon to be updated [10]), combined with standardestimates for dn  [11] implies remarkable sensitivity toscales qq  1017 GeV. Future progress in EDM searches(for both neutrons and heavy atoms) can bring this up to thePlanck scale and beyond. In contrast, no constraints from(8) ensue within the axion scenario.Electric dipole moments from four-fermion operators.—EDMs of neutrons and heavy atoms and molecules are theprimary probes for sources of flavor-neutral CP violation[11]. In addition to dn, the strongest constraints onCP-violating parameters arise from the atomic EDMs ofthallium, jdTlj< 9 1025e cm [3], and mercury, jdHgj<2 1028e cm [4].Assuming that  is removed by an appropriate symme-try, EDMs are mediated by higher-dimensional operators,and both (4) and (5) are capable of inducing atomic ornuclear EDMs if the overall coefficients contain an extraphase relative to the quark masses. Restricting Eq. (4) tothe first generation, we find the following CP-odd opera-tors (with real me, mu):L CP   s ImYqe11116qemSUSY uu ei	5e  ui	5u ee: (9)Accounting for QCD running from the SUSY scale to1 GeV, and using the hadronic matrix elements over nu-cleon states hNj uu dd=2jNi ’ 4 NN andhnj ui	5ujni ’ 0:4mN=mu ni	5n, we determine the in-duced corrections to the CP-odd electron-nucleonLagrangian, L  CS NN ei	5e CP Ni	5N ee,CS  2 1041 GeVqe ; CP 4 1031 GeVqe ; (10)using maximal ImYqe and taking mSUSY  300 GeV.09180Comparing (10) to the limits on CS and CP deducedfrom the Tl and Hg EDM bounds [11], we obtain thefollowing sensitivity:qe * 3 108 GeV from Tl EDM; (11)qe * 1:5 108 GeV from HgEDM; (12)qq * 3 107 GeV from HgEDM: (13)The last relation results from sensitivity to theCP-violating operators  di	5d uu from (5), leading tothe Schiff nuclear moment and the Hg EDM. These areremarkably large scales and, indeed, not far below thescales suggested by neutrino physics. In fact, the nextgeneration of atomic or molecular EDM experiments[12] may reach sensitivities sufficient to push qe intoregions close to the suggested scale of right-handedneutrinos.Semileptonic operators involving heavy quark super-fields are, in turn, strongly constrained via two-loop cor-rections (3) to the dipole amplitudes. The bound on dTlimplies jdej & 1:6 1027e cm, which for maximalImYqe1133 implies:qe * 1:3 108 GeV: (14)Results analogous to (3) apply for the quark EDMs andcolor EDMs, furnishing a similar sensitivity to qq.Lepton-flavor violation.—Searches for lepton-flavorviolation (LFV), such as  ! e	 decay and  ! e con-version in nuclei, have resulted in stringent upper boundson the corresponding branching ratio Br ! e	< 1:21011 [13] and the rate of conversion normalized on cap-ture rate R ! e onTi< 4:3 1012 [14], with fur-ther improvement anticipated. The latter bound implies aparticularly high sensitivity to the semileptonic operatorsin (1). The conversion is mediated by  uu ei	5 and uu e and involves the same matrix elements as CS.Using bounds on such scalar operators derived elsewhere(see, e.g., [15]), we conclude that  ! e conversionprobes energy scales as high asqe * 1 108 GeV from  ! e on Ti: (15)The constraint on  ! e	 probes similar, but slightlylower, scales as it requires a two-loop diagram as inFig. 1(b). Disregarding an O1 factor between (11) and(15), we conclude that searches for EDMs and LFV probethese extensions of the MSSM up to comparable energyscales of 108 GeV.Hadronic flavor constraints.—Often, the most con-straining piece of experimental information comes fromthe contribution of new physics to the mixing of neutralmesons K and B. However, in the present case, there isnecessarily a significant loop and Yukawa suppressionarising from (6), and the sensitivity is correspondinglyweakened. Taking mKexp’3:5106 eV [16], we findqq * tan=50  200 GeV [7]. mB exhibits a similarsensitivity, while K is about 3 orders of magnitude more1-3TABLE I. Sensitivity to the threshold scale. The naturalnessbound on ImYqq does not apply to the axionic solution of thestrong CP problem, the best sensitivity to Imyh is achieved atmaximal tan, and the Hg EDM constraint on ImYqq applieswhen at least one pair of quarks belongs to the 1st generation.Operator Sensitivity to  (GeV) SourceYqe3311 107 Naturalness of meImYqq3311 1017 Naturalness of , dnImYqeii11 107–109 Tl, Hg EDMsYqe1112, Yqe1121 107–108  ! e conversionImYqq 107–108 Hg EDMImyh 103–108 de from Tl EDMPRL 96, 091801 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending10 MARCH 2006sensitive but still well below the scales probed by EDMsand LFV. In contrast, it is clear that these observablesprovide much better sensitivity to SUSY dimension-sixoperators, which impose no additional suppression factors.Denoting the corresponding scale as 0, we find 0 * 8106 GeV, while K is sensitive to scales 108 GeV.Two-loop contributions to b ! s	 [as in Fig. 1(b)] arenot Yukawa suppressed and, with the current precisionBrB ! Xs	  104 [16], are somewhat more sensitive.We find qq * 103–104 GeV (for Yqq3233  1), still wellbelow the sensitivity in other channels.Constraints on the Higgs operator.—The high sensitiv-ity to QULE and QUQD arises primarily because they canflip the light fermion chirality without Yukawa suppres-sion. It would then come as no surprise if HuHdHuHd wereto have little implication for CP- and flavor-violating ob-servables; the operator will, of course, provide correctionsto the sfermion and neutralino mass matrices and caninduce CP-odd mixing between A and h, H, but theseeffects do not lead to high sensitivity to h.Remarkably enough, it turns out that EDMs do exhibit ahigh sensitivity to HuHdHuHd at large tan through cor-rections to the Higgs potential and, in particular, the effec-tive shift of the m212 parameterm212HuHd!m212effHuHdm212yhv2SMhHuHd:(16)Crucially, a complex phase in m212eff , due to Imyh, isenhanced at large tan because m212 ’ m2A= tan. The re-sulting phase affects the one-loop SUSY EDM diagrams(see, e.g., [17]):de  eme tan162m2SUSY5g2224 g2124sinArgM2m212eff: (17)Expanding to leading order in 1=h, using (16), and im-posing the present limit on de discussed earlier, one findsimpressive sensitivity for large tan,09180h *2107 GeVtan502300GeVmSUSY300GeVmA2: (18)In conclusion, we have examined new flavor- andCP-violating effects mediated by dimension-five superpo-tential operators and have shown that the sensitivity tothese operators extends far beyond the weak scale (assummarized in Table I). The semileptonic operators thatmediate flavor violation in the leptonic sector and/or breakCP could be detectable even if the scale of new physics isas high as 109 GeV and well above the naturalness scale.Our results can be translated into constraints on CP andflavor violation in specific models leading to (1), e.g., theNMSSM or the MSSM with an extra pair of Higgs bosons.Moreover, the sensitivity quoted in (11) and (15) is robust,having a mild dependence on the SUSY threshold. Finally,since these effects decouple linearly, an increase in sensi-tivity by just 2 orders of magnitude would already startprobing scales relevant for neutrino physics. Our resultsmotivate further searches for EDMs and LFV in the SUSYframework even if the soft-breaking sector provides nonew sources, as happens, e.g., in models with low scaleSUSY breaking.1-4[1] S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150(1981); for reviews, see, e.g., S. P. Martin, hep-ph/9709356; M. A. Luty, hep-th/0509029.[2] See, e.g., M. Bona et al., hep-ph/0509219.[3] B. C. Regan et al., Phys. Rev. Lett. 88, 071805 (2002).[4] M. V. Romalis et al., Phys. Rev. Lett. 86, 2505 (2001).[5] P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999).[6] S. Weinberg, Phys. Rev. D 26, 287 (1982); N. Sakai andT. Yanagida, Nucl. Phys. B197, 533 (1982).[7] M. Pospelov, A. Ritz, and Y. Santoso (to be published).[8] L. J. Hall, R. Rattazzi, and U. Sarid, Phys. Rev. D 50, 7048(1994).[9] R. Kuchimanchi, Phys. Rev. Lett. 76, 3486 (1996); R. N.Mohapatra and A. Rasin, Phys. Rev. Lett. 76, R3490(1996); G. Hiller and M. Schmaltz, Phys. Lett. B 514,263 (2001).[10] P. G. Harris, in The 13th International Conference onSupersymmetry and Unification of Fundamental Inter-actions, Durham, United Kingdom, 2005 (unpublished).[11] See, e.g., I. B. Khriplovich and S. K. Lamoreaux, CPViolation Without Strangeness (Springer, New York,1997); J. S. M. Ginges and V. V. Flambaum, Phys. Rep.397, 63 (2004); M. Pospelov and A. Ritz, Ann. Phys.(N.Y.) 318, 119 (2005).[12] See, e.g., D. DeMille et al., Phys. Rev. A 61, 052507(2000); J. J. Hudson et al., Phys. Rev. Lett. 89, 023003(2002).[13] M. L. Brooks et al., Phys. Rev. Lett. 83, 1521 (1999).[14] J. Kaulard et al., Phys. Lett. B 422, 334 (1998).[15] A. Faessler et al., Nucl. Phys. B587, 25 (2000).[16] S. Eidelman et al., Phys. Lett. B 592, 1 (2004).[17] K. A. Olive et al., Phys. Rev. D 72, 075001 (2005).[18] D. Chang, W. Y. Keung, and A. Pilaftsis, Phys. Rev. Lett.82, 900 (1999); 83, 3972(E) (1999).

Flavor- and CP-violating physics from new supersymmetric thresholds

Treating the MSSM as an effective theory, we study the implications of having dimension five operators in the superpotential for flavor and CP-violating processes, exploiting the linear decoupling of observable effects with respect to the new threshold scale \Lambda. We show that the assumption of weak scale supersymmetry, when combined with the stringent limits on electric dipole moments and lepton flavor-violating processes, provides sensitivity to \Lambda as high as 10^7-10^9 GeV, while the next generation of experiments could directly probe the high-energy scales suggested by neutrino physics

Flavor and CP violating physics from new supersymmetric thresholds

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Flavor and CP violating physics from new supersymmetric thresholds

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