The dominant MSSM effects in the rare K decays, K+ --> pi+ nu nu bar, KL -->
pi0 nu nu bar, KL --> pi0 e+ e- and KL --> pi0 mu+ mu-, are discussed both
within and without the minimal flavor violation hypothesis, at moderate and
large tangent beta. In each case, the sensitivities to MSSM soft-breaking terms
are compared, laying emphasis on possible correlations among observables. In
most scenarios, rare K decays offer unique windows into the Delta S=1 sector of
the soft-breaking terms. Therefore, together with B-physics and collider
observables, these modes will be essential for reconstructing the still elusive
SUSY-breaking mechanism.Comment: Talk given at the KAON07 International Conference, May 21-25 2007,
  Frascati, Ital

Smith, Christopher

English

arXiv

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PoS(KAON)058Recent progress on supersymmetric effectsin rare K decaysChristopher Smith∗†Institut für Theoretische Physik, Universität Bern, CH-3012 Bern, SwitzerlandE-mail: chsmith@itp.unibe.chThe dominant MSSM effects in the rare K decays K+ → pi+ν ¯ν , KL → pi0ν ¯ν , KL → pi0e+e− andKL → pi0µ+µ−, are discussed both within and without the minimal flavor violation hypothesis,at moderate and large tanβ . In each case, the sensitivities to MSSM soft-breaking terms arecompared, laying emphasis on possible correlations among observables. In most scenarios, rareK decays offer unique windows into the ∆S = 1 sector of the soft-breaking terms. Therefore,together with B-physics and collider observables, these modes will be essential for reconstructingthe still elusive SUSY-breaking mechanism.KAON International ConferenceMay 21-25 2007Laboratori Nazionali di Frascati dell’INFN, Rome, Italy∗Speaker.†Work supported by the Schweizerischer Nationalfonds.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(KAON)058SUSY and rare K decays Christopher Smith1. IntroductionThe FCNC-induced decays, K+ → pi+ν ¯ν , KL → pi0ν ¯ν , KL → pi0e+e− and KL → pi0µ+µ−,are very suppressed in the Standard Model (SM), where they can be predicted very accurately[1].Therefore, these modes are ideal for probing possible New Physics effects[2]. In the present talk,the signatures of supersymmetry (SUSY), in its simplest realization as the MSSM, are reviewed.As is well-known, SUSY unifies matter (fermions) and interactions (bosons), and has a number ofdesirable features, e.g. it provides a dark matter candidate, helps unify the gauge couplings at high-energy and stabilizes the electroweak scale[3]. Even though the minimal supersymmetrizationof the SM requires one super-partner for each SM particle (and two Higgs doublets), it is veryconstrained and involves only a few free parameters. The problem, however, is that SUSY must bebroken, and the precise mechanism still eludes us. Therefore, in practice, an effective descriptionis adopted, introducing all possible explicit soft-breaking terms allowed by the gauge symmetries.In the squark sector, there are LL and RR mass-terms and trilinear couplings giving rise to LRmass-terms after the Higgses acquire their VEV’s, 〈H0u,d〉= vu,d:LLL,RRso f t =− ˜Q†m2Q ˜Q− ˜Um2U ˜U†− ˜Dm2D ˜D†, L LRso f t =− ˜UAU ˜QHu + ˜DAD ˜QHd ,with ˜Q = (u˜L, ˜dL)T , ˜U = u˜†R, ˜D = ˜d†R. Obviously, m2Q,U,D and AU,D, which are 3×3 matricesin flavor-space, generate a very rich flavor-breaking sector as squark mass eigenstates can differsubstantially from their gauge eigenstates.What to expect from SUSY in rare K decays: In the SM, the Z-penguin is the dominant con-tribution, and is tuned by λt = V ∗tsVtd (Fig.1a). The four MSSM corrections depicted in Figs.1b− e(together with box diagrams), represent the dominant corrections, and are thus the only MSSMeffects for which rare K decays can be sensitive probes. Let us briefly describe each of them.First, there is the charged Higgs contribution to the Z-penguin (Fig.1b), which is, at moderatetanβ = vu/vd , aligned with the SM one (∼ λt). Then, there is the supersymmetrized version ofFigs.1a−b, with charginos – up-squarks in place of W±/H± – up-quarks in the loop (Fig.1c), andwhich is sensitive to the mixings among the six up-squarks (ZU ), a priori not aligned with the CKMmixings. Another purely supersymmetric contribution, relevant only for charged lepton modes, isthe gluino electromagnetic penguin (Fig.1d), sensitive to down-squark mixings (ZD). The last classof effects consists of neutral Higgs FCNC (Fig.1e), and arises at large tanβ ≈mt/mb ≈ 50. Indeed,the 2HDM-II structure of the Higgs couplings to quarks, required by SUSY, is not preserved be-yond leading order due to Lso f t , and the “wrong Higgs”, Hu, gets coupled to down-type quarks,Le f f ⊃ ¯diRY ikd (H0d + εY †u YuH0†u )k jdjL. Clearly, once the Higgses acquire their VEV’s, there is amismatch between quark mass eigenstates and Higgs couplings; both are no longer diagonalizedsimultaneously and Higgs FCNC are generated[4].Bottom-up approach and Minimal Flavor Violation: There are obviously too many para-meters in Lso f t to have any hope to fix them all from rare K decays. At the same time, however,observed FCNC transitions and CP-violation seem to indicate that new physics induces only smalldepartures with respect to the SM. Therefore, one starts from a lowest-order basis in which theflavor-breakings due to m2Q,U,D and AU,D are minimal. This can take the form of mSUGRA, align-ment of squarks with quarks or the Minimal Flavor Violation hypothesis (MFV). In a second stage,2PoS(KAON)058SUSY and rare K decays Christopher Smiths dL LZW ±VVs dL LZH ±VVs dL LZχ±UZ UZs dL,R R,LDZ DZg~uLi uRidL,Ri~uL,Ri~a) b) c)d)s dL,R R,Le)γ h ,H ,A0 0 0s dL LZW ±VVs dL Lχ±g)f )VLRδtm tmULRδU*V **Zs dR RZH ±uLiud dLj ΑUHuu0*χ±R LRi~ ~ d~d ui ju~(δ  )D ijg~RRR LR LH ±h)** **Figure 1: a− e) Dominant MSSM contributions to rare K decays. f − g) Dominant sources of SU(2)L-breaking in the Z-penguin. h) Schematic representation of the H± contribution to the Z-penguin at largetanβ .one probes the possible signatures of departures from this minimal setting. The goal being, ulti-mately, to constrain SUSY-breaking models, which imply specific soft-breaking structures. At thatstage, information from rare K decays, colliders and B-physics must of course be combined.Here we adopt MFV as the lowest order basis, i.e. we impose that the SM Yukawas Yu,d arethe only sources of flavor-breaking[5]. In practice, this means that Lso f t terms can be expanded as(ai,bi ∼ O(1), A0 and m0 setting the overall mass-scale as in mSUGRA)m2Q = m20(a11+b1Y†uYu +b2Y†dYd +b3(Y†dYdY†uYu +Y†uYuY†dYd)),m2U = m20(a21+b4YuY†u),m2D = m20(a31+b5YdY†d),AU = A0Yu(a41+b6Y†dYd),AD = A0Yd(a51+b7Y†uYu) ,such that all FCNC’s and CP-violation are still essentially tuned by the CKM matrix. For example,the dominant contributions to the Z-penguin are those breaking the SU(2)L gauge-symmetry[6, 7].In the SM, this breaking is achieved through a double top-quark mass insertion (Fig.1 f ). Similarly,in the MSSM, it is the double t˜L − t˜R mixing via the AU trilinear terms which plays the dominantrole (Fig.1g in the sCKM basis)[8]. Within MFV, this gives a factor m2t λt |a4− cotβ µ∗|2 [9], stillenhanced by m2t and tuned by λt .2. Supersymmetric effects in K → piν ¯νSUSY effects in the SM-like operators, (s¯d)V±A( ¯νν)V−A, cannot be distinguished since only(s¯d)V ( ¯νν)V−A contributes to the K → piν ¯ν matrix-element. All MSSM effects are thus encodedinto a single complex number, Xν ≡ yνL + yνR [7]:He f f = yνL (s¯d)V−A ( ¯νν)V−A + yνR (s¯d)V+A ( ¯νν)V−A → (yνL + yνR)(s¯d)V ( ¯νν)V−A .At moderate tanβ , the dominant MSSM contribution comes from chargino penguins becauseof their quadratic sensitivity to up-squark mass-insertions (Figs.1c,1g). Within MFV, this means,given the mt enhancement present in the δULR sector, that K → piν ¯ν are particularly sensitive. Still,a significant enhancement would require a very light stop and chargino[9], mostly because of theconstraint from ∆ρ[10]. Any enhancement & 5% would thus falsify MFV if sparticles are found3PoS(KAON)058SUSY and rare K decays Christopher Smiths dL,R R,Ld~d dj id~(δ  )D ijg~LLRL Rh ,H ,A0 0 0Hu0*|Re  | < 90  yV,A onlySEEWPSM | Im  | < 35  yPLK     piµµ  +0LK e    pi   e+01111Lbound on yK     µ µ+ S,Pb) c)a)Figure 2: a) Sensitivity of K+ → pi+ν ¯ν to AU terms, compared to B-physics observables. b) Schematicrepresentation of the neutral Higgs FCNC beyond MFV, at large tanβ . c) Impacts of dim-6 FCNC operatorsin the B(KL → pi0µ+µ−) vs. B(KL → pi0e+e−) plane.above∼ 200GeV , and if tanβ & 5 (to get rid of the H± contribution). Turning on generic AU terms,the largest deviations arise in K → piν ¯ν , see Fig.2a[9]. Further, the decoupling is slower than forobservables sensitive to chargino boxes like εK . All in all, given that K+ → pi+ν ¯ν has alreadybeen seen, how large the effect could be for KL → pi0ν ¯ν? By an extensive, adaptive scanning overthe MSSM parameter space, it has been shown[11] that it is possible to saturate the GN model-independent bound[12], which represents a factor ∼ 30 enhancement of B(KL → pi0ν ¯ν) over theSM.At large tanβ , the chargino contributions decouple, both within and without MFV, while theHiggs FCNC obviously does not contribute (Fig.1e). However, higher order effects in the H±contribution to the Z-penguin (Fig.1h), sensitive to δ DRR, can become sizeable beyond MFV[13].Further, this contribution is slowly decoupling as MH increases compared to tree-level neutral Higgsexchanges, as for example in Bs,d → µ+µ−.SUSY effects in other dimension-six operators, (s¯d)( ¯ν(1,γ5)ν) and (s¯σµνd)( ¯νσ µν(1,γ5)ν),require active right-handed neutrinos and will not be discussed here[14]. Another possible class ofoperators, since the neutrino flavors are not detected, are (s¯ΓAd)( ¯ν iΓBν j) with i 6= j and ΓA,B someDirac structures. In the MSSM, such lepton flavor violating operators arise only from suppressedbox diagrams, and cannot lead to significant effects[15]. However, they could be sizeable in thepresence of R-parity violating terms[15, 16].3. Supersymmetric effects in KL → pi0`+`−Though the SM predictions for these modes are less accurate than for K → piν ¯ν , they aresensitive to more types of New Physics operators[17]. Indeed, the final-state leptons are nowcharged and massive. Therefore, besides electromagnetic effects, common to both the muon andelectron modes, the relatively large muon mass opens the possibility to probe a whole class ofhelicity-suppressed effects.SUSY effects in the QCD operators, i.e. in the chromomagnetic s¯σµνdGµν or four-quarkoperators, have no direct impact on KL → pi0`+`−. Indeed, the two-photon CP-conserving con-tribution is fixed entirely in terms of the measured K → pipipi , piγγ modes[18], while the indirect4PoS(KAON)058SUSY and rare K decays Christopher SmithK → piν ¯ν KL → pi0`+`−MFV, tanβ ≈ 2 Best sensitivity, but max. Less sensitive, but preciselyenhancement < 20-25% correlated with K → piν ¯νMFV, tanβ ≈ 50 Negligible effectsGeneral, tanβ ≈ 2 Best probes of δULR δULR : correlated with K → piν ¯ν(quadratic dependence in δULR) δ DLR : correlated with ε ′/ε (but cleaner)General, tanβ ≈ 50 Good probes of δ DRR Good probes of δ DRR,LL,(slow decoupling as MH → ∞) corr. with KL → µ+µ− (but cleaner)Table 1: Sensitivity of rare K decays to MSSM effects, within and without the MFV hypothesis, and withmoderate and large tanβ . The dominant contributions can come from single, (δ ij)12, and/or double (e.g.(δ ij)∗32(δ ij)31) mass insertions, see text for the precise dependences.CP-violating contribution is fixed from the measured εK and B(KS → pi0`+`−)[19]. New physicscan thus explicitly enter through short-distance semi-leptonic FCNC operators only.SUSY effects in the SM operators, which are the vector and axial-vector operatorsHe f f = y7V (s¯d)V(¯``)V + y7A (s¯d)V(¯``)A ,can in principle be disentangled thanks to the different sensitivities of the two modes to the axial-vector current (it also produces `+`− in a helicity-suppressed 0−+ state). Various MSSM con-tributions can enter in y7A and y7V . First, chargino contributions to the Z-penguin (Fig.1c) enteras y7A,y7V ∼ (δURL)∗32(δURL)31, and are thus directly correlated to the corresponding contribution toK → piν ¯ν discussed previously[9, 20]. Within MFV, the maximal effect for KL → pi0`+`− is aboutone third of the one for KL → pi0ν ¯ν , hence may be inaccessible due to theoretical uncertainties.Secondly, gluino contributions to the electromagnetic operator s¯σµνdFµν (Fig.1d) can be absorbedinto y7V ∼ (δ DRL)12. Even if directly correlated with ε ′/ε , sizeable effects in KL → pi0`+`− are stillpossible[21]. Finally, H± contributions arise at large tanβ (Fig.1h), with y7A,y7V ∼ (δ DRR)12, andare directly correlated with those for K → piν ¯ν[13].SUSY effects in the (pseudo-)scalar operators, which can be helicity-suppressed (i.e., y ∼m`) or not:He f f = yS (s¯d)(¯``)+ yP (s¯d)(¯`γ5`)+ y′S (s¯γ5d)(¯``)+ y′P (s¯γ5d)(¯`γ5`).The first (last) two operators contribute to KL → pi0`+`− (KL → `+`−). In the MSSM at large tanβ ,they arise from Higgs FCNC[22], and are thus helicity-suppressed (Fig.2b). Sizeable effects for themuon mode are possible beyond MFV, where they are sensitive to (δ DRR,LL)12 and (δ DRR)23(δ DLL)31mass-insertions. Also, even if this contribution is correlated to the one for KL → µ+µ−, given thelarge theoretical uncertainties for this mode, a factor ∼ 4 enhancement is still allowed (Fig.2c)[17].On the other hand, helicity-allowed contributions to these operators do not arise in the MSSM, butcould be generated from R-parity violating couplings. Still, a precise fine-tuning of these couplingswould be needed to have an effect for KL → pi0`+`− without overproducing Bexp(KL → e+e−) =9+6−4 ·10−12[17].5PoS(KAON)058SUSY and rare K decays Christopher SmithSUSY effects in the (pseudo-)tensor operators, (s¯σµνd)( ¯`σ µν(1,γ5)`), the last possibledimension-six semi-leptonic FCNC operators, are helicity-suppressed in the MSSM[23] and, beingalso phase-space suppressed, do not lead to any significant effect[17]. Further, they cannot arisefrom R-parity violating couplings.4. ConclusionThe four rare K decay modes, K+ → pi+ν ¯ν , KL → pi0ν ¯ν , KL → pi0e+e− and KL → pi0µ+µ−,are the only theoretically clean windows into the ∆S = 1 sector. If SUSY is discovered, the patternof deviations they could exhibit with respect to the SM (see Table 1) will be essential to constrainthe MSSM parameter-space, and hopefully unveil the nature of the SUSY-breaking mechanism.References[1] U. Haisch, these proceedings.[2] For a review of new physics signals, see C. Tarantino, these proceedings, [hep-ph]0706.3436.[3] For an introduction, see for example S. P. Martin, hep-ph/9709356.[4] K. S. Babu, C. Kolda, Phys. Rev. Lett. 84 (2000) 228.[5] L. J. Hall, L. Randall, Phys. Rev. Lett. 65 (1990) 2939;G. D’Ambrosio, G.F. Giudice, G. Isidori, A. Strumia, Nucl. Phys. B645 (2002) 155.[6] Y. Nir, M.P. Worah, Phys. Lett. B423 (1998) 319.[7] A.J. Buras, A. Romanino, L. Silvestrini, Nucl. Phys. B520 (1998) 3.[8] G. Colangelo, G. Isidori, JHEP 09 (1998) 009.[9] G. Isidori et al., JHEP 08 (2006) 064.[10] A. J. Buras et al., Nucl. Phys. B592 (2001) 55.[11] A. J. Buras, T. Ewerth, S. Jager, J. Rosiek, Nucl. Phys. B714 (2005) 103.[12] Y. Grossman, Y. Nir, Phys. Lett. B398 (1997) 163.[13] G. Isidori, P. Paradisi, Phys. Rev. D73 (2006) 055017.[14] Y. Kiyo et al., Prog. Theor. Phys. 101 (1999) 671; G. Perez, JHEP 9909 (1999) 019.[15] Y. Grossman, G. Isidori, H. Murayama, Phys. Lett. B588 (2004) 74.[16] N. G. Deshpande, D. K. Ghosh, X. G. He, Phys. Rev. D70 (2004) 093003;A. Deandrea, J. Welzel, M. Oertel, JHEP 0410 (2004) 038.[17] F. Mescia, C. Smith, S. Trine, JHEP 08 (2006) 088.[18] G. Buchalla, G. D’Ambrosio, G. Isidori, Nucl. Phys. B672 (2003) 387;G. Isidori, C. Smith, R. Unterdorfer, Eur. Phys. J. C36 (2004) 57.[19] G. D’Ambrosio, G. Ecker, G. Isidori, J. Portoles, JHEP 08 (1998) 004.[20] P. L. Cho, M. Misiak, D. Wyler, Phys. Rev. D54 (1996) 3329.[21] A. J. Buras et al., Nucl. Phys. B566 (2000) 3.[22] G. Isidori, A. 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Recent progress on supersymmetric effects in rare K decays

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Recent progress on supersymmetric effects in rare K decays

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