Lepton flavor violation in low-scale seesaw models: SUSY and non-SUSY contributions
A. Abada, M. E. Krauss, W. Porod, F. Staub, A. Vicente, C. Weiland
LLPT-Orsay-14-43BONN-TH-14-11IFT-UAM/CSIC-14-061FTUAM-14-25August 27, 2018
Lepton flavor violation in low-scale seesaw models: SUSY andnon-SUSY contributions
A. Abada a , M. E. Krauss b , W. Porod b , F. Staub c , A. Vicente d and C. Weiland ea Laboratoire de Physique Théorique, CNRS – UMR 8627,Université de Paris-Sud 11, F-91405 Orsay Cedex, France b Institut für Theoretische Physik und Astronomie, Universität Würzburg97074 Würzburg,Germany c Physikalisches Institut der Universität Bonn, 53115 Bonn, Germany d IFPA, Dep. AGO, Université de Liège,Bat B5, Sart-Tilman B-4000 Liège 1, Belgium e Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC,Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
Abstract
Taking the supersymmetric inverse seesaw mechanism as the explanation for neutrino oscil-lation data, we investigate charged lepton flavor violation in radiative and 3-body lepton decaysas well as in neutrinoless µ − e conversion in muonic atoms. In contrast to former studies, wetake into account all possible contributions: supersymmetric as well as non-supersymmetric.We take CMSSM-like boundary conditions for the soft supersymmetry breaking parameters.We find several regions where cancellations between various contributions exist, reducing thelepton flavor violating rates by an order of magnitude compared to the case where only thedominant contribution is taken into account. This is in particular important for the correctinterpretation of existing data as well as for estimating the reach of near future experimentswhere the sensitivity will be improved by one to two orders of magnitude. Moreover, wedemonstrate that ratios like BR( τ → µ )/BR( τ → µe + e − ) can be used to determine whetherthe supersymmetric contributions dominate over the W ± and H ± contributions or vice versa. a r X i v : . [ h e p - ph ] N ov ontents (cid:96) α → (cid:96) β γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 (cid:96) − α → (cid:96) − β (cid:96) − β (cid:96) + β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 (cid:96) − α → (cid:96) − β (cid:96) − γ (cid:96) + γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 (cid:96) − α → (cid:96) + β (cid:96) − γ (cid:96) − γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 Coherent µ − e conversion in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 A.1 Mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.2.1 Fermion-Scalar vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A.2.2 Fermion-Vector vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2.3 Scalar vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.2.4 Scalar-Vector vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2.5 Vector vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
B Renormalization Group Equations 32C Loop Integrals 36D Photonic penguin contributions to LFV 38
D.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39D.2 Neutralino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40D.3 Chargino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40D.4 W + and H + contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 E Z and Higgs penguin contributions to LFV 41 E.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41E.2 Neutralino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44E.2.1 Z-penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44E.2.2 Scalar penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45E.3 Chargino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46E.3.1 Z-penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46E.3.2 Scalar penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47E.4 W + and H + contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50E.4.1 Z-penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4.2 Scalar penguins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 F Box contributions to LFV 54
F.1 Four lepton boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54F.1.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54F.1.2 Neutralino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56F.1.3 Chargino contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56F.1.4 W + and H + contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57F.2 Additional boxes for (cid:96) − α → (cid:96) − β (cid:96) + γ (cid:96) − γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 59F.2.1 Crossed neutralino contributions . . . . . . . . . . . . . . . . . . . . . . . . 59F.2.2 Crossed chargino contributions . . . . . . . . . . . . . . . . . . . . . . . . . 60F.2.3 Crossed W + and H + contributions . . . . . . . . . . . . . . . . . . . . . . . 61F.3 Two-Lepton – Two-Quark boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64F.3.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64F.3.2 Down quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65F.3.3 Up quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 G Form factors of the 4-fermion operators 69 Introduction
The recent discovery of a bosonic state at the Large Hadron Collider (LHC) [1, 2] stands as amajor breakthrough in particle physics. Although further confirmation is required, all data arecompatible with the long-awaited Higgs boson, thus completing the Standard Model (SM) particlecontent. Furthermore, the properties and decay modes of this scalar are in good agreement withthe SM expectations, making the SM picture more motivated than ever.In this context, it is crucial to keep in mind that the SM cannot be the ultimate theory.In fact, and besides theoretical arguments such as the hierarchy problem, there are very goodexperimental reasons to go beyond the SM (BSM). The best of these motivations is the existenceof non-zero neutrino masses and mixing angles, now firmly established by neutrino oscillationexperiments [3–5]. Since the SM lepton sector does not include them, one has to go beyond theSM.A generic prediction in most of these neutrino mass models is lepton flavor violation (LFV), notonly in the neutrino sector but also for the charged leptons. Depending of the exact realization ofthe neutrino mass model, the rates for the LFV processes can be very different. For instance, high-scale models typically predict small branching ratios, thus making LFV hard (if not impossible)to be discovered. In contrast, one expects measurable LFV rates if the scale of new physics isnot far from the electroweak (EW) scale. These low-scale mechanisms generating neutrino massesare thus more attractive from a phenomenological point of view, since they offer a window to newphysics thanks to their LFV promising perspectives. Moreover, they can be directly tested at theLHC via the production of new particles if these are light enough.On the experimental side, the field of LFV physics will live an era of unprecedented develop-ments in the near future, with dedicated experiments in different fronts . In the case of the muonradiative decay µ → eγ , the MEG collaboration has announced plans for future upgrades. Thesewill allow for an improvement of the current bound, BR ( µ → eγ ) < . · − [7], reaching asensitivity of about · − after 3 years of acquisition time [8]. Limits on τ radiative decays areless stringent, but they are expected to be improved at Belle II [9]. These will also search for leptonflavor violating B -meson decays. Moreover, the perspectives for the 3-body decays (cid:96) α → (cid:96) β aregood as well. The decay µ → e was searched for long ago by the SINDRUM experiment [10],setting the limit Br ( µ → e ) < . · − . The future Mu3e experiment announces a sensitivityof ∼ − [11], which would imply a 4 orders of magnitude improvement. In the case of τ decaysto three charged leptons, Belle II will again be the facility where improvements are expected [12],although recently the LHCb collaboration has reported first bounds on τ → µ [13]. The LFVprocess where the best developments are expected in the next few years is neutrinoless µ − e con-version in muonic atoms. In the near future, many different experiments will search for a positivesignal. These include Mu2e [14–16], DeeMe [17], COMET [18, 19] and PRISM/PRIME [20]. Theexpected sensitivities for the conversion rate range from a modest − in the near future to animpressive − . Finally, one can also search for LFV in high-energy experiments, such as theLHC. A popular process in this case is the Higgs boson LFV decay to a pair of charged leptons, h → (cid:96) α (cid:96) β , with α (cid:54) = β [21, 22], which has recently received some attention [23–34]. First boundson h → µτ have been reported by the CMS collaboration [35] . For other possibilities to searchfor LFV at high-energy colliders, see [36–52]. In table 1 we collect present bounds and expectednear-future sensitivities for the most popular low-energy LFV observables.With such a large variety of processes, a proper theoretical understanding of potential hierar- See [6] for a recent review. The CMS collaboration also reports an intriguing . σ excess in h → µτ leading to BR ( h → τ µ ) ∼ . µ → eγ . × − [7] × − [8] τ → eγ . × − [53] ∼ × − [9] τ → µγ . × − [53] ∼ × − [9] µ → eee . × − [10] ∼ − [11] τ → µµµ . × − [54] ∼ − [9] τ − → e − µ + µ − . × − [54] ∼ − [9] τ − → µ − e + e − . × − [54] ∼ − [9] τ → eee . × − [54] ∼ − [9] µ − , Ti → e − , Ti 4 . × − [55] ∼ − [20] µ − , Au → e − , Au 7 × − [56] µ − , Al → e − , Al 10 − − − µ − , SiC → e − , SiC 10 − [57]Table 1: Current experimental bounds and future sensitivities for some low-energy LFV observ-ables.chies or correlations in a given model becomes necessary. This goal requires detailed analyticaland numerical studies of the different contributions to the LFV processes, in order to determinethe dominant ones and to get a proper interpretation of the LFV bounds. Furthermore, the under-standing of the LFV anatomy of several models allows one to discriminate among them by usingcombinations of observables which have definite predictions [58].In this work we are interested in LFV in supersymmetric and non-supersymmetric variantsof the inverse seesaw model (ISS) [59]. This low-scale neutrino mass model constitutes a veryinteresting alternative to the usual seesaw mechanism. The suppression mechanism that guaranteesthe smallness of neutrino masses is the introduction of a slight breaking of lepton number in thesinglet sector, in the form of a small (compared to the EW scale) Majorana mass for the X singlets.This allows for large Yukawa couplings compatible with a low (TeV or even lower) mass for theseesaw mediators. With this combination, one expects a very rich phenomenology, including sizableLFV rates and additional contributions to the radiative corrections to the Higgs mass [60–62]. Inthe supersymmetric (SUSY) version of the ISS, the new singlet fermions are promoted to singletsuperfields. The appealing features of the ISS mechanism are kept also in the SUSY version.LFV in models with light right-handed (RH) neutrinos has already been studied in greatdetail. Early studies [63–66] already pointed out the existence of large enhancements in the LFVrates with respect to those found in high-scale models. More recently, there has been a revivedinterest due to the expected experimental improvements in the near future. Interestingly, dominantcontributions have been found in (non-SUSY) box diagrams induced by RH neutrinos. Thiswas first shown in [67] and later confirmed in [68–70]. In this case, the future µ − e conversionexperiments will play a major role in constraining light right-handed neutrino scenarios. Theusual photon penguin contributions get also enhanced in the presence of light RH neutrinos, seefor example [71]. Regarding the SUSY contributions, several studies have recently addressed therole of the Z -penguins. A large enhancement with respect to the usual dipole contribution wasreported in [72]. Later, this result was (qualitatively) confirmed in [70] and further exploited inseveral phenomenological studies [73–76]. However, in [77] it was shown that the results in [72](and the subsequent studies [73–76]) are incorrect, due to an inconsistency in [78]. While this hasa negligible impact in the case of high-scale seesaw models, this is not the case for low-scale seesawmodels like the supersymmetric version of the ISS.5iven that recent studies pointed out important but partial results and the upcoming ex-perimental improvements, we aim in this work for a complete calculation of the various LFVobservables taking into account all contributions at the same time. One of our results will be thatthere exist several regions in parameter space where cancellations between various contributionsoccur, changing the interpretation of existing and future experimental results. In order to do so wehave made use of FlavorKit [79], a tool that combines the analytical power of
SARAH [80–84] withthe numerical routines of
SPheno [85,86] to obtain predictions for flavor observables in a wide rangeof models. This setup makes use of
FeynArts / FormCalc [87–92] to compute generic predictionsfor the form factors of the relevant operators and thus provides an automatic computation of theflavor observables. We use this setup to compute for the first time the Higgs penguin contributionsto LFV in the inverse seesaw . In addition, we improve previous studies in others aspects as well:(i) we make use of the full 2-loop renormalization group equations (RGEs) including all flavoreffects in the SM and SUSY sectors to obtain the parameters entering the calculation, and (ii) weinclude for the first time the decays τ − → µ + e − e − and τ − → e + µ − µ − .The paper is organized as follows: in Section 2 we present the ISS model and its supersymmetricextension. The LFV observables induced by the extended particle content and the dominantcontributions are discussed in Section 3, and in Section 4 we present our numerical results. InSection 5 we draw our conclusions. In the appendices we first introduce the formulae for the massmatrices and our convention for the loop integrals before presenting the additional contributionsto the 1- and 2-loop RGEs compared to the MSSM case. More importantly, they contain thecomplete set of contributions to the LFV observables discussed in this paper. In the inverse seesaw, the Standard Model field content is extended by n R generations of right-handed neutrinos ν R and n X generations of singlet fermions X (such that n R + n X = N s ), bothwith lepton number L = +1 [59,63,94]. The corresponding Lagrangian before EWSB has the form L ISS = L SM − Y ijν ν Ri (cid:101) H † L j − M ijR ν Ri X j − µ ijX X Ci X j + h.c. , (1)where a sum over i, j = 1 , , is assumed . L SM is the SM Lagrangian, Y ν are the neutrino Yukawacouplings and M R is a complex mass matrix that generates a lepton number conserving mass termfor the fermion singlets. The complex symmetric mass matrix µ X violates lepton number by twounits and is naturally small, in the sense of ’t Hooft [99], since in the limit µ X → lepton numberis restored. This Majorana mass term also leads to a small mass splitting in the heavy neutrinosector, which then become quasi-Dirac neutrinos.After EWSB, in the basis ( ν L , ν CR , X ) , the × neutrino mass matrix is given by M ISS = m TD m D M R M TR µ X . (2) The Higgs penguin contributions to LFV processes were first considered in the context of the inverse seesawin [93]. However, our paper goes beyond this reference in two ways: by doing the computation in the mass basisand by taking into account all contributions to the Higgs penguins. The ISS requires the introduction of at least two right-handed neutrinos in order to account for the activeneutrino masses and mixings. The most minimal ISS realization [95–97] consists in the addition of two right-handedand two sterile neutrinos to the SM content. However, its minimal SUSY realization [98] requires only one pair offermionic singlets. m D = √ Y ν v and v/ √ is the vacuum expectation value (vev) of the Higgs boson.Under the assumption µ X (cid:28) m D (cid:28) M R , the mass matrix M ISS can be block-diagonalized togive an effective mass matrix for the light neutrinos [100] M light (cid:39) m TD M TR − µ X M − R m D , (3)whereas the heavy quasi-Dirac neutrinos have masses corresponding approximately to the entriesof M R .As usual, one can easily obtain a supersymmetric version of the model by promoting the corre-sponding fields to superfields (cid:98) ν Ci and (cid:98) X i ( i = 1 , , ) and including the corresponding interactionsin the superpotential. This reads W = W MSSM + ε ab Y ijν (cid:98) ν Ci (cid:98) L aj (cid:98) H bu + M R ij (cid:98) ν Ci (cid:98) X j + 12 µ X ij (cid:98) X i (cid:98) X j . (4) W MSSM is the superpotential of the MSSM given by W MSSM = (cid:15) ab Y iju (cid:98) U Ci (cid:98) Q aj (cid:98) H bu − (cid:15) ab Y ijd (cid:98) D Ci (cid:98) Q aj (cid:98) H bd − (cid:15) ab Y ije (cid:98) E Ci (cid:98) L aj (cid:98) H bd + (cid:15) ab µ (cid:98) H au (cid:98) H bd , (5)where we skipped the color indices. The corresponding soft SUSY breaking Lagrangian is givenby −L soft = −L softMSSM + (cid:101) ν Ci m (cid:101) ν Cij (cid:101) ν C ∗ j + (cid:101) X ∗ i m X ij (cid:101) X j + ( T ijν ε ab (cid:101) ν Ci (cid:101) L aj H bu + B ijM R (cid:101) ν Ci (cid:101) X j + 12 B ijµ X (cid:101) X i (cid:101) X j + (cid:101) X ∗ i m Xν Cij (cid:101) ν Cj + h . c . ) , (6)where B ijM R and B ijµ X are the new parameters involving the scalar partners of the sterile neutrinostates. Notice that while the former conserves lepton number, the latter violates lepton numberby two units. Finally, L softMSSM collects the soft SUSY breaking terms of the MSSM. −L softMSSM = (cid:16) (cid:15) ab T iju ˜ U Ci ˜ Q aj H bu − (cid:15) ab T ijd ˜ D Ci ˜ Q aj H bd − (cid:15) ab T ije ˜ E Ci ˜ L aj H bd + (cid:15) ab B µ H au H bd + h . c . (cid:17) + 12 (cid:16) M λ B λ B + δ ab M λ aW λ bW + M λ αG λ βG + h . c . (cid:17) + (cid:16) δ ab ( ˜ Q ai ) ∗ m q,ij ˜ Q bj + ( ˜ D Ci ) ∗ m d,ij ˜ D Cj + ( ˜ U Ci ) ∗ m u,ij ˜ U Cj (cid:17) + ˜ E C ∗ i m e,ij ˜ E Cj + δ ab ( ˜ L ai ) ∗ m l,ij ˜ L bj + m H d | H d | + m H u | H u | . (7)The neutrino mass matrix has the same form as in Eq. (2), just replacing v by v u , the vev of theup-type Higgs boson. The mass matrices of this model are the same as in the MSSM apart fromthe sneutrino sector. Neglecting for the moment being the soft-breaking terms which lead to asplitting between the scalar and pseudoscalar parts, the corresponding mass matrix reads m ν i = m L + v u Y Tν Y ∗ ν + D L − √ (cid:16) v d µY Tν − v u T † ν (cid:17) √ v u (cid:60) (cid:16) Y Tν M ∗ R (cid:17) − √ (cid:16) v d µY ∗ ν − v u T ν (cid:17) m (cid:101) ν C + M R M † R + v u Y ν Y † ν − M R µ ∗ S √ v u M TR Y ∗ ν − µ S M † R M TR M ∗ R + m S + µ S µ ∗ S (8)with D L = − m Z cos θ W cos 2 β . (9)The complete mass matrices including the B -parameters as well as all other mass matrices can befound in Appendix A.1. 7 Low energy observables
The fact that the LHC has not yet seen any supersymmetric particles [101, 102] implies, at leastin the specific SUSY model we consider in this work, that squarks and gluinos must be heavy.However, it could well be that sleptons, charginos and neutralinos are relatively light, thus havinglarge contributions to LFV decays. Here we will consider the processes (cid:96) α → (cid:96) β γ , (cid:96) α → (cid:96) β (cid:96) γ (cid:96) δ and µ − e conversion in nuclei. In this section we will present the effective low-energy lagrangianand the basic formulae for the observables. This will also serve to fix our notation (we stay closeto the conventions of Ref. [79]). The details for the calculations of the corresponding form factorscan be found in appendices C–G. The interaction lagrangian relevant for LFV can be written as L LFV = L (cid:96)(cid:96)γ + L (cid:96) + L (cid:96) d + L (cid:96) u . (10)with L (cid:96)(cid:96)γ = e ¯ (cid:96) β (cid:2) γ µ (cid:0) K L P L + K R P R (cid:1) + im (cid:96) α σ µν q ν (cid:0) K L P L + K R P R (cid:1)(cid:3) (cid:96) α A µ + h.c. (11) L (cid:96) = (cid:88) I = S,V,TX,Y = L,R A IXY ¯ (cid:96) β Γ I P X (cid:96) α ¯ (cid:96) δ Γ I P Y (cid:96) γ + h.c. (12) L (cid:96) d = (cid:88) I = S,V,TX,Y = L,R B IXY ¯ (cid:96) β Γ I P X (cid:96) α ¯ d γ Γ I P Y d γ + h.c. (13) L (cid:96) u = L (cid:96) d | d → u, B → C . (14)Here e is the electric charge, q the 4-momenta of the photon, P L,R = (1 ∓ γ ) are the usual chiralityprojectors and (cid:96) α and d α denote the lepton and d-quark flavors, respectively. Furthermore, wehave defined Γ S = 1 , Γ V = γ µ and Γ T = σ µν . We omit flavor indices in the form factors for thesake of simplicity. The underlying Feynman diagrams as well as the complete analytic results aregiven in appendices D–G.Whenever possible, we have compared the explicit analytical formulae for the form factorswith results already available in the literature. The supersymmetric contributions to boxes, Higgspenguins, photon penguins were found to perfectly agree with [78, 103], while the supersymmetricZ-penguins only differ from [78] via a constant term as pointed out in [77]. This constant termdoes not impact the result of [78] where a high-scale seesaw mechanism is considered but it canlead to non-physical results in low-scale seesaw models. We have also cross-checked our calculationof the non-SUSY boxes with [68], confirming their results. To the knowledge of the authors, thisis the first calculation of the non-SUSY Higgs penguins in a two Higgs doublet Model, thus nocomparison was possible. (cid:96) α → (cid:96) β γ In case of the radiative decay (cid:96) α → (cid:96) β γ , the corresponding decay width is given by [104] Γ ( (cid:96) α → (cid:96) β γ ) = α em m (cid:96) α (cid:0) | K L | + | K R | (cid:1) , (15)where the dipole form factors K L,R are defined in Eq. (11), α em being the fine structure constant.8 .3 (cid:96) − α → (cid:96) − β (cid:96) − β (cid:96) + β Next, we consider the (cid:96) − α ( p ) → (cid:96) − β ( p ) (cid:96) − β ( p ) (cid:96) + β ( p ) Γ ( (cid:96) α → (cid:96) β ) = m (cid:96) α π (cid:34) e (cid:16)(cid:12)(cid:12) K L (cid:12)(cid:12) + (cid:12)(cid:12) K R (cid:12)(cid:12) (cid:17) (cid:32)
163 log m (cid:96) α m (cid:96) β − (cid:33) (16) + 124 (cid:16)(cid:12)(cid:12) A SLL (cid:12)(cid:12) + (cid:12)(cid:12) A SRR (cid:12)(cid:12) (cid:17) + 112 (cid:16)(cid:12)(cid:12) A SLR (cid:12)(cid:12) + (cid:12)(cid:12) A SRL (cid:12)(cid:12) (cid:17) + 23 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLL (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRR (cid:12)(cid:12)(cid:12) (cid:19) + 13 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLR (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRL (cid:12)(cid:12)(cid:12) (cid:19) + 6 (cid:16)(cid:12)(cid:12) A TLL (cid:12)(cid:12) + (cid:12)(cid:12) A TRR (cid:12)(cid:12) (cid:17) + e (cid:0) K L A S ∗ RL + K R A S ∗ LR + c.c. (cid:1) − e (cid:16) K L ˆ A V ∗ RL + K R ˆ A V ∗ LR + c.c. (cid:17) − e (cid:16) K L ˆ A V ∗ RR + K R ˆ A V ∗ LL + c.c. (cid:17) − (cid:0) A SLL A T ∗ LL + A SRR A T ∗ RR + c.c. (cid:1) − (cid:16) A SLR ˆ A V ∗ LR + A SRL ˆ A V ∗ RL + c.c. (cid:17)(cid:21) . Here we have defined ˆ A VXY = A VXY + e K X ( X, Y = L, R ) . (17)The mass of the leptons in the final state has been neglected in this formula, with the exceptionof the numerical factors that multiply the K L,R contribution . Eq. (16) agrees with the one inref. [78], but includes in addition A SLR and A SRL . In [78], these contributions were absorbed in thecorresponding vector form factors, A VLR and A VRL , by means of a Fierz transformation [105]. Incontrast, A SLR and A SRL were explicitly added to the set of contributing form factors in [70]. Therelation between our coefficients and the ones of [78] is given in table 2. (cid:96) − α → (cid:96) − β (cid:96) − γ (cid:96) + γ We consider the (cid:96) − α ( p ) → (cid:96) − β ( p ) (cid:96) − γ ( p ) (cid:96) + γ ( p ) β (cid:54) = γ . The decay width is givenby Γ (cid:16) (cid:96) − α → (cid:96) − β (cid:96) − γ (cid:96) + γ (cid:17) = m (cid:96) α π (cid:20) e (cid:16)(cid:12)(cid:12) K L (cid:12)(cid:12) + (cid:12)(cid:12) K R (cid:12)(cid:12) (cid:17) (cid:18)
163 log m (cid:96) α m (cid:96) γ − (cid:19) (18) + 112 (cid:16)(cid:12)(cid:12) A SLL (cid:12)(cid:12) + (cid:12)(cid:12) A SRR (cid:12)(cid:12) (cid:17) + 112 (cid:16)(cid:12)(cid:12) A SLR (cid:12)(cid:12) + (cid:12)(cid:12) A SRL (cid:12)(cid:12) (cid:17) + 13 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLL (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRR (cid:12)(cid:12)(cid:12) (cid:19) + 13 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLR (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRL (cid:12)(cid:12)(cid:12) (cid:19) + 4 (cid:16)(cid:12)(cid:12) A TLL (cid:12)(cid:12) + (cid:12)(cid:12) A TRR (cid:12)(cid:12) (cid:17) − e (cid:16) K L ˆ A V ∗ RL + K R ˆ A V ∗ LR + K L ˆ A V ∗ RR + K R ˆ A V ∗ LL + c.c. (cid:17)(cid:21) . Here we have used the same definition as in Eq. (17). Furthermore, as for (cid:96) α → (cid:96) β , the mass ofthe leptons in the final state has been neglected in the decay width formula, with the exception ofthe dipole terms K L,R .Finally, we note that Eqs. (16) and (18) are in perfect agreement with the expressions givenin Ref. [70]. We note that the correct form for the terms proportional to K L,R was first obtained in Ref. [64]. K L,R , A L,R , A SLL e ˆ B L A SRR e ˆ B R A VLL e (cid:18) B L + F LL (cid:19) A VRR e (cid:18) B R + F RR (cid:19) A VLR − A SLR e (cid:16) ˆ B L + F LR (cid:17) A VRL − A SRL e (cid:16) ˆ B R + F RL (cid:17) A TLL e B L A TRR e B R Table 2: Relation between the form factors defined in this paper and the ones in [78]. Here ˆ B L,R = B L,R + B L,R , Higgs and ˆ B L,R = B L,R + B L,R , Higgs , and F XY = F X E Y e m Z , with E L and E R thetree-level Z -boson couplings to a pair of charged leptons (see appendix A.2). (cid:96) − α → (cid:96) + β (cid:96) − γ (cid:96) − γ Finally, we consider the (cid:96) − α ( p ) → (cid:96) + β ( p ) (cid:96) − γ ( p ) (cid:96) − γ ( p ) β (cid:54) = γ . The decay widthis given by Γ (cid:16) (cid:96) − α → (cid:96) + β (cid:96) − γ (cid:96) − γ (cid:17) = m (cid:96) α π (cid:20) (cid:16)(cid:12)(cid:12) A SLL (cid:12)(cid:12) + (cid:12)(cid:12) A SRR (cid:12)(cid:12) (cid:17) + 112 (cid:16)(cid:12)(cid:12) A SLR (cid:12)(cid:12) + (cid:12)(cid:12) A SRL (cid:12)(cid:12) (cid:17) (19) + 23 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLL (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRR (cid:12)(cid:12)(cid:12) (cid:19) + 13 (cid:18)(cid:12)(cid:12)(cid:12) ˆ A VLR (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ A VRL (cid:12)(cid:12)(cid:12) (cid:19) + 6 (cid:16)(cid:12)(cid:12) A TLL (cid:12)(cid:12) + (cid:12)(cid:12) A TRR (cid:12)(cid:12) (cid:17) − (cid:0) A SLL A T ∗ LL + A SRR A T ∗ RR + c.c. (cid:1) − (cid:16) A SLR ˆ A V ∗ LR + A SRL ˆ A V ∗ RL + c.c. (cid:17)(cid:21) . The same definitions and conventions as in the previous two observables have been used. Noticethat this process does not receive contributions from penguin diagrams, but only from boxes. µ − e conversion in nuclei We now turn to the discussion of µ − e conversion in nuclei, which will follow the conventionsand approximations described in Ref. [103, 106] (see also [107–109] for detailed works regardingthe effective lagrangian at the nucleon level, [68, 110] for a calculation including the effects ofthe atomic electric field and [111] for recent improvements on the hadronic uncertainties). The10onversion rate, relative to the the muon capture rate, can be expressed as CR( µ − e, Nucleus) = p e E e m µ G F α em Z F p π Z × (cid:26)(cid:12)(cid:12)(cid:12) ( Z + N ) (cid:16) g (0) LV + g (0) LS (cid:17) + ( Z − N ) (cid:16) g (1) LV + g (1) LS (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( Z + N ) (cid:16) g (0) RV + g (0) RS (cid:17) + ( Z − N ) (cid:16) g (1) RV + g (1) RS (cid:17)(cid:12)(cid:12)(cid:12) (cid:27) capt . (20) Z and N are the number of protons and neutrons in the nucleus and Z eff is the effective atomiccharge [112]. Similarly, G F is the Fermi constant, F p is the nuclear matrix element and Γ capt represents the total muon capture rate. p e and E e ( (cid:39) m µ in our numerical evaluation) are themomentum and energy of the electron and m µ is the muon mass. In the above, g (0) XK and g (1) XK (with X = L, R and K = S, V ) can be written in terms of effective couplings at the quark level as g (0) XK = 12 (cid:88) q = u,d,s (cid:16) g XK ( q ) G ( q,p ) K + g XK ( q ) G ( q,n ) K (cid:17) ,g (1) XK = 12 (cid:88) q = u,d,s (cid:16) g XK ( q ) G ( q,p ) K − g XK ( q ) G ( q,n ) K (cid:17) . (21)For coherent µ − e conversion in nuclei, only scalar ( S ) and vector ( V ) couplings contribute [106].Furthermore, sizable contributions are expected only from the u, d, s quark flavors. The numericalvalues of the relevant G K factors are [106, 113] G ( u,p ) V = G ( d,n ) V = 2 ; G ( d,p ) V = G ( u,n ) V = 1 ; G ( u,p ) S = G ( d,n ) S = 5 . G ( d,p ) S = G ( u,n ) S = 4 . G ( s,p ) S = G ( s,n ) S = 2 . . (22)Finally, the g XK ( q ) coefficients can be written in terms of the form factors in Eqs.(11), (13) and(14) as g LV ( q ) = √ G F (cid:20) e Q q (cid:0) K L − K R (cid:1) − (cid:0) C V LL(cid:96)(cid:96)qq + C V LR(cid:96)(cid:96)qq (cid:1)(cid:21) (23) g RV ( q ) = g LV ( q ) (cid:12)(cid:12) L → R (24) g LS ( q ) = − √ G F (cid:0) C SLL(cid:96)(cid:96)qq + C SLR(cid:96)(cid:96)qq (cid:1) (25) g RS ( q ) = g LS ( q ) (cid:12)(cid:12) L → R . (26)Here Q q is the quark electric charge ( Q d = − / , Q u = 2 / ) and C IXK(cid:96)(cid:96)qq = B KXY (cid:0) C KXY (cid:1) ford-quarks (u-quarks), with X = L, R and K = S, V . For the numerical examples we have implemented the model in the Mathematica package
SARAH [80–84], which creates the required modules for
SPheno [85, 86] to calculate the masses and mixing11atrices including the complete 1-loop corrections. In the Higgs sector we include in addition theknown 2-loop corrections to the Higgs mass from the MSSM [114–119]. However, this does notinclude 2-loop corrections stemming from the extended neutrino and sneutrino sectors, where wecan have sizable Yukawa couplings. Moreover,
SARAH calculates also the full 2-loop RGEs includingthe entire flavor structure for the model, which we have summarized in Appendix B. This will beof great importance in our numerical studies, as we use CMSSM-like boundary conditions, seebelow for their definition. In the flavor observables we include all possible contributions. Theseare calculated using the
FlavorKit interface [79]. In the context of this project we have extendedthe lists of observables implemented in
FlavorKit by (cid:96) − α → (cid:96) − β (cid:96) + β (cid:96) − γ and (cid:96) − α → (cid:96) − β (cid:96) − β (cid:96) + γ . α − em G µ . · − GeV − α S M Z m b ( m b ) m t m τ M Z unless otherwise specified.The numerical evaluation of each parameter point is performed as follows: the Y ν Yukawacouplings are calculated using a modified Casas-Ibarra parameterization [120], properly adaptedfor the inverse seesaw [121,122] (and fixing M R = 2 TeV, µ X = 10 − GeV and the lightest neutrinomass m ν = 10 − eV): Y ν = √ v u V † D √ X RD √ m ν U † PMNS . (27)Here D √ m ν = diag( √ m ν i ) , D √ X = diag( (cid:112) ˆ X i ) , ˆ X i being the eigenvalues of X = M R µ − X M TR , and V is the matrix that diagonalizes X as V XV T = ˆ X . Furthermore, we parameterize the complexorthogonal R matrix as R = θ R sin θ R − sin θ R cos θ R cos θ R θ R − sin θ R θ R cos θ R sin θ R − sin θ R cos θ R
00 0 1 . (28)Below we will set R to the unit matrix except when stated otherwise. We use the best-fit valuesfor the neutrino oscillation parameters as given in [123]: ∆ m = 7 . · − eV , ∆m = 2 . · − eV , sin θ = 0 . , sin θ = 0 . , sin θ = 0 . . (29)which are close to the ones obtained in [3–5]. We make use in our scans of the values Y ν = f · − · . − . . .
616 0 . − . .
404 1 .
78 1 . (30)fixed with f = 1 even if we vary M R . This is because one can always adjust µ X to fulfill neutrinooscillation data without affecting any of our observables. SPheno derives the SM gauge and Yukawa couplings at M Z where we take the masses andcouplings given in table 3 as input. 2-loop RGEs for the dimensionless parameters are then usedto evaluate these couplings at M GUT , defined by the requirement g = g , where g and g are the12ouplings for the U (1) Y and SU (2) L gauge groups, respectively. At M GUT the CMSSM boundaryconditions are applied m ν C = m X = m l = m e = m q = m d = m u ≡ m m H d = m H u ≡ m M = M = M ≡ M / T i ≡ A Y i i = e, d, u, ν The mixed soft-term m Xν C is set to zero at the GUT scale and is not generated via RGE effects.Moreover, the phase of µ , which is an RGE invariant, is given as input. The ratio of the Higgsvevs, tan β = v u v d , completes the list of input parameters. Then 2-loop RGEs are used to evolvethese parameters to Q EWSB = (cid:112) ˜ t ˜ t . The numerical values for superpotential terms M R and µ X ,as well as for their corresponding soft terms B M R and B µ X , are used as input at the SUSY scale. B µ and | µ | are obtained as usual from the minimization conditions of the vacuum . At M SUSY the 1-loop corrected masses are calculated before the RGEs run down to M Z to re-calculate gaugeand Yukawa couplings using the new SUSY corrections. These steps are iterated until the massspectrum has converged with a numerical precision of − . Afterwards, SPheno runs the RGEsto Q = 160 GeV for the calculation of the operators which contribute to quark flavor violatingobservables and to Q = M Z for the calculation of the operators needed for lepton flavor violatingobservables. These operators are then combined to compute the different observables using α (0) ,which includes to a large extent the effects from running the operators between M Z and the energyscale where the LFV processes take place (usually given by the mass of the decaying particle). We will use the parameter values given in table 4 as starting point for our numerical com-putations unless stated otherwise. A variation of the soft SUSY parameters is denoted by avariation of M SUSY , which actually implies a variation of three parameters at the same time M SUSY = m = M / = − A . For completeness, we note that fixing the ratio m /A usuallygives a Higgs boson mass, m h , that does not agree with the ATLAS and CMS measurements.Nevertheless, we emphasize that (1) our results depend only weakly on the value of A , and (2)contributions mediated by h itself are subdominant. Therefore, the actual Higgs boson mass is oflittle importance for our investigations here.We start with the discussion of µ decays as the bounds are strongest in this case. In Fig. 1 weshow the dependence of BR ( µ → eγ ) on M R and M SUSY as well as the individual dependence ofthe SUSY and non-SUSY contributions. The latter consist of ν - W ± and the ν - H ± contributions. In principle one could require that all B -parameters are proportional to each other, e.g. B µ : B M R : B µ X = µ : M R : µ X . However, as their actual value does not have any significant impact as long as this ratio is fulfilled up toa factor 2-3 we fix for simplicity B M R and B µ X . m M / A -1.5 TeV M R B µ X µ X B M R M R tan β
10 sign ( µ ) +Table 4: Standard values for the various parameters. M R and µ X are taken proportional to theunit matrix. 13
00 1000 2000 5000 1 ´ - - - M R = M SUSY [GeV] B R ( µ → e γ ) full contribution n o n - S U S Y c o n tr i bu t i o n SUSY contribution
100 200 500 1000 2000 5000 1 ´ - - - - - - M R [GeV] B R ( µ → e γ ) full contributionnon-SUSY contributionSUSY contribution
500 1000 2000 5000 1 ´ - - - - - M SUSY [GeV] B R ( µ → e γ ) full contributionnon-SUSY contribution SUSY contribution Figure 1: BR( µ → eγ ) as a function of M SUSY and M R . The other parameters are given in thetext. The gray area roughly corresponds to the parameter space excluded by the LHC experiments.There are two particular features: (i) if M R = M SUSY the SUSY contributions are more importantthan the non-SUSY ones and the relative importance of the SUSY contributions increases withthe scale. The reason for the latter is that the mixing between light and heavy neutrinos decreaseslike ∼ m D /M R , whereas the mixing in the sneutrino sector decreases only logarithmically withthe scale. (ii) The non-SUSY contributions can flip its sign. This is due to a sign-differencebetween the ν - H ± and the ν - W ± contributions to the coefficients K L,R . This is in contrast tothe analogous decay in the quark sector, b → sγ , where the W ± - and H ± -contributions havealways the same sign. The reason for this difference can be found in Eqs. (248)–(254), presented inappendix D, where the light neutrino masses appear instead of the mass of the heavy t -quark. Wehave checked explicitly, both numerically and analytically, that we recover the b → sγ result if wereplace the corresponding masses and Yukawa couplings. Finally, we stress that the scalar massesare functions of M SUSY , which explains why also the non-SUSY contribution actually depends onthe SUSY scale. With our specific structure of the Y ν matrices we find that M R has to be largerthan M SUSY for the sign flip to occur, which is also the reason why we do not observe it in caseof M R = M SUSY . The grey area corresponds to the part of the parameter space which is excludedin the CMSSM by the most recent ATLAS results [102]. However, we want to stress that eventhough the squark and gluino masses are essentially the same in our model as in the CMSSM, thecascade decays can be quite different due to (i) the enlarged sneutrino sector with additional lightstates and (ii) the different slepton masses. Thus, this is a rather conservative bound.In Fig. 2 we display our results for the branching ratio BR( µ → e ) as well as the various14
00 1000 2000 5000 1 ´ - - - - - M R = M SUSY [GeV] B R ( µ → e ) full contributionnon-SUSY Z penguins non-SUSY boxesSUSY Z penguins S U S Y b o x e s SUSY γ penguins n o n - S U S Y γ p e n g u i n s
100 200 500 1000 2000 5000 1 ´ - - - - - M R [GeV] B R ( µ → e ) full contributionnon-SUSYnon-SUSY boxesSUSY Z penguinsSUSY boxes Z penguinsSUSY γ penguinsnon-SUSY γ penguins
500 1000 2000 5000 1 ´ - - - - - M SUSY [GeV] B R ( µ → e ) full contributionnon-SUSY Z penguins non-SUSY boxes S U S Y Z p e n g u i n s S U S Y b o x e s S U S Y γ p e n g u i n s non-SUSY γ penguins - - - - - f B R ( µ → e ) M SUSY = 1 TeV M R = 2 TeV Figure 2: BR( µ → e ) as a function of M SUSY , M R and an overall scaling parameter f for Y ν .The other parameters are given in the text. The gray area roughly corresponds to the parameterspace excluded by the LHC experiments.contributions to this decay. Here we find that for the case M R = M SUSY the non-SUSY boxesdominate. This fact was first noted in [67] and later confirmed by [68–70]. Note that this doesnot depend on the overall strength of the Y ν couplings, which we rescale as Y ν → f Y ν . This canbe seen from the lower right plot: all contributions scale in the same way. However, the situationcan change in principle if one allows for additional flavor violation in the soft SUSY breakingparameters. Note that the sign-flip induced by the H ± contributions is not as pronounced as inthe case of µ → eγ , where it led to a change of the overall sign, as the different contributions to theoff-shell photon appear with different weights. However, it is the reason for the observed kink inthe non-SUSY γ -penguin. We also observe that we have negative interference between non-SUSY Z -penguins and the corresponding box contributions. In particular, for larger values of M R thiscan reduce BR( µ → e ) by up to an order of magnitude. Since this is precisely the region whichwill be probed by future experiments, the possible appearance of these cancellations has to betaken into account in order to interpret the experimental results properly.Similar features appear in case of µ − e conversion in nuclei, as exemplified for the case ofan aluminium ( Al ) nucleus in Fig. 3. The main difference is that there is a large part of pa-rameter space where a pronounced negative interference between the non-SUSY Z -penguin andthe corresponding box contributions can occur. Note that with the expected sensitivity of − one can probe Y ν couplings down to a few × − for M R = M SUSY = 1
TeV or, equivalently,to a mass scale of about TeV in case of Y ν as given in Eq. (30). As we found for the 3-body15
00 1000 2000 5000 1 ´ - - - - - - - M R = M SUSY [GeV] C R ( µ − e , A l ) full contributionnon-SUSY Z penguins n o n - S U S Y b o x e s S U S Y Z p e n g u i n s S U S Y b o x e s S U S Y γ p e n g u i n s non-SUSY γ penguins
100 200 500 1000 2000 5000 1 ´ - - - - - - M R [GeV] C R ( µ − e , A l ) full contributionnon-SUSY Z penguins n o n - S U S Y b o x e s SUSY Z penguinsSUSY boxesSUSY γ penguins n o n - S U S Y γ p e n g u i n s
500 1000 2000 5000 1 ´ - - - - - M SUSY [GeV] C R ( µ − e , A l ) full contributionnon-SUSY Z penguins non-SUSY boxes S U S Y Z p e n g u i n s S U S Y b o x e s S U S Y γ p e n g u i n s non-SUSY γ penguins - - - - - - f C R ( µ − e , A l ) M SUSY = 1 TeV M R = 1 TeV Figure 3: µ − e conversion on Al as a function of M SUSY , M R and the scaling parameter f for Y ν .The gray area roughly corresponds to the parameter space excluded by the LHC experiments.decays, for higher mass scales the non-SUSY Z -penguins can be as important as the correspondingbox-diagrams. The overall features are essentially element independent as can be seen in Fig. 4where we show all three observables discussed so far together and include also µ − e conversion intitanium ( Ti ). In case M R (cid:39) M SUSY , we find that µ − e conversion in nuclei is the most stringentLFV observable in our model.Turning now to the LFV τ decays, we show in Fig. 5 several branching ratios for the scenariodefined above. Unfortunately, they are too small to be observed in the near future. Below we willshow alternative scenarios (in which the R matrix is not assumed to be the unit matrix) where thisis not the case. Nevertheless, they show an interesting feature which is quite generic in this model:BR ( τ → µe + e − ) (cid:39) BR( τ → µ ) and BR ( τ → eµ + µ − ) (cid:39) BR( τ → e ) . Particularly interestingis that these branching ratios are sensitive to the relative size of the non-SUSY contributionscompared to the SUSY ones. We also stress that the various contributions contribute similarlyas in case of µ → e . For completeness we note that BR ( τ → eµ + e − ) and BR ( τ → µe + µ − ) arestrongly suppressed, at least a factor of − with respect to the other 3-body decays, as theyrequire at least one additional flavor violating vertex in the dominant contributions.It is worth stressing that the fact that the µ observables are more constraining than the τ decays is correct in large parts of the parameter space. However, there is also a substantial partwhere the opposite is true, as exemplified in Fig. 6 where we tune the parameters such that both, µ - and τ -observables can be discovered in the next generation of experiments. For this we haveadjusted the diagonal entries of µ X as well as θ R and calculated Y ν using Eq. (27). Clearly, this16
00 1000 2000 5000 1 ´ - - - - M R = M SUSY [GeV] B r a n c h i n g r a t i o s B R ( µ → e γ ) B R ( µ → e ) CR( µ − e , Al)CR( µ − e , Ti)
100 200 500 1000 2000 5000 1 ´ - - - - M R [GeV] B r a n c h i n g r a t i o s BR( µ → eγ )BR( µ → e )CR( µ − e , Al)CR( µ − e , Ti)
500 1000 2000 5000 1 ´ - - - - - - M SUSY [GeV] B r a n c h i n g r a t i o s BR( µ → eγ ) B R ( µ → e ) CR( µ − e , Al)CR( µ − e , Ti) Figure 4: BR( µ → eγ ), BR( µ → e ), µ − e conversion in Ti and Al as a function of M R and M SUSY .The gray area roughly corresponds to the parameter space excluded by the LHC experiments.part of the parameter space requires quite some hierarchy in µ X to explain neutrino data correctly.Note that even in this part of parameter space the ratios BR ( τ → µe + e − ) (cid:39) BR( τ → µ ) andBR ( τ → eµ + µ − ) (cid:39) BR( τ → e ) show the same dependence on the ratio M R /M SUSY as in theprevious case.The impact of the R matrix and the hierarchy in the µ X entries is further illustrated for thedecays (cid:96) α → (cid:96) β γ in Fig. 7. Again, we have calculated Y ν via Eq. (27), such that the resultsfrom neutrino oscillation experiments are explained correctly. One finds that, depending on theregion in the parameter space, either the µ decay or the τ decays are more important. As in caseof the 3-body decays, one finds fine-tuned combinations of the parameters where all decays canbe observed in future experiments. Note that for fixed θ R the branching ratios scale like f R /f X where f R and f X denote an overall scaling of M R and µ X , respectively. Moreover, the branchingratios scale like tan β if the SUSY contributions dominate. In case the non-SUSY contributionsdominate we find only a slight tan β dependence for very large tan β values.Finally, let us comment on the Higgs penguin contributions to the different LFV observables.In all our numerical scans they have been found to be completely negligible and that is whywe have decided not to include them in our figures. In principle, one could look for sizable Higgspenguin contributions by going to regions in parameter space with large tan β and low pseudoscalarmasses [93, 124]. This, however, would require dedicated parameter scans in order to overcomethe constraints from flavor data, as these regions are already in strong tension after the LHCbmeasurement of the B s → µ + µ − branching ratio [125]. For this reason, we have not pursued17
00 200 500 1000 2000 5000 1 ´ - - - - M R [GeV] B R ( ℓ α → ℓ β ℓ γ ℓ δ ) µ → eτ → µτ − → µ − e + e − τ → e τ − → e − µ + µ −
500 1000 2000 5000 1 ´ - - - - - M SUSY [GeV] B R ( ℓ α → ℓ β ℓ γ ℓ δ ) µ → e τ → µ τ − → µ − e + e − τ → e τ − → e − µ + µ −
100 200 500 1000 2000 5000 1 ´ M R [GeV] B R ( ℓ − α → ℓ − β ℓ + γ ℓ − γ ) / B R ( ℓ α → ℓ β ) BR( τ − → e − µ + µ − )BR( τ → τ − → µ − e + e − )BR( τ → µ )
500 1000 2000 5000 1 ´ M SUSY [GeV] B R ( ℓ − α → ℓ − β ℓ + γ ℓ − γ ) / B R ( ℓ α → ℓ β ) BR( τ − → e − µ + µ − )BR( τ → τ − → µ − e + e − )BR( τ → µ ) Figure 5: Branching ratios for τ decays as a function of M R and M SUSY . In the upper two plotsthe lines correspond to BR( µ → e ) (black solid), BR( τ → e ) (blue solid), BR( τ → µ ) (redsolid), BR( τ − → e − µ + µ − ) (blue dashed) and BR( τ − → µ − e + e − ) (red dashed). The gray arearoughly corresponds to the parameter space excluded by the LHC experiments.this goal any further. Nevertheless, we have checked that the Higgs penguins contributions to (cid:96) α → (cid:96) β (cid:96) γ (cid:96) δ and µ - e conversion in nuclei have the expected decoupling behavior for large M R and/or M SUSY scales. 18 - - - - - - - - - - µ X = µ X [GeV] B r a n c h i n g r a t i o s θ R = 0 . m = M / = 1 TeV A = − . β = 10 M R = 0 .
25 TeV µ X = diag( µ , µ , . - - - - - - - - - - µ X = µ X [GeV] B r a n c h i n g r a t i o s θ R = 0 . m = 0 . M / = 1 TeV A = − β = 20 M R = 2 TeV µ X = diag( µ , µ ,
12 eV)
Figure 6: µ - and τ -observables as a function of µ X . The underlying parameters are given in theplots. The lines correspond to BR ( τ → µγ ) (full red), BR ( τ → µ ) (dashed red), BR( τ − → µ − e + e − ) (dotted red), BR( µ → eγ ) (full black) and BR( µ → e ) (dashed black). The light gray,red, yellow and blue bands show the expected future reach of the dedicated experiments to τ → µγ , τ → µ , µ → eγ and µ → e as given in table 1. - - - - - - - - - µ X [GeV] B R ( ℓ α → ℓ β γ ) µ X = diag(10 − GeV , − GeV , µ )tan β = 3 - - - - - - - - - - µ X [GeV] B R ( ℓ α → ℓ β γ ) µ X = diag(10 − GeV , − GeV , µ )tan β = 40 - - - - - - - θ R B R ( ℓ α → ℓ β γ ) µ X = diag(10 − , − , − ) GeV M R = 4 TeV - - - - - - - θ R B R ( ℓ α → ℓ β γ ) µ X = diag(10 − , − , − ) GeV M R = 4 TeV Figure 7: Dependence of (cid:96) α → (cid:96) β γ on µ X and θ R . The lines correspond to BR( µ → eγ ) (black),BR( τ → eγ ) (blue dashed) and BR( τ → µγ ) (red dotted).19 Conclusions
This paper represents the first complete computation of selected LFV observables in scenarioswith light right-handed neutrinos. These include the radiative decays (cid:96) α → (cid:96) β γ , the 3-bodydecays (cid:96) α → (cid:96) β (cid:96) γ (cid:96) δ (in several variants) and neutrinoless µ − e conversion in nuclei. Our resultsare valid in the inverse seesaw and should also hold in low-scale type-I seesaw models with nearlyconserved lepton number, the inverse seesaw being a specific realization of these models. Comparedto previous studies, we have also included Higgs-penguins and considered non-supersymmetric aswell as supersymmetric contributions to the corresponding LFV amplitudes simultaneously.For the numerical examples we took a CMSSM inspired scenario where we also consideredthe limiting cases with either M R (cid:29) M SUSY and M SUSY (cid:29) M R . Our main conclusions can besummarized as follows: • The SUSY contributions dominate the induced photon penguins if both, M R and M SUSY ,are about the same size. For M R < ∼ M SUSY / the non-SUSY contributions start to dominatethe radiative decays (cid:96) α → (cid:96) β γ . • For low M R scales the LFV phenomenology is dominated by non-SUSY contributions. Thisholds in particular for the 3-body decays and µ - e conversion in nuclei. These are mainlygiven by boxes and Z -penguin diagrams containing right-handed neutrinos in the loop. Incontrast to the usual high-scale seesaw models, in which their contributions to LFV processesare tiny, the right-handed neutrinos can play a major role in low-scale seesaw scenarios.In what concerns the non-SUSY box contributions, our results confirm previous claims inthe literature [67–70]. Furthermore, we have highlighted the relevance of the non-SUSY Z -penguins, previously regarded as subdominant in most studies . They are particularlyrelevant for larger values of M R , where we often find a negative interference between the Z -penguins and the box contributions. This will be particularly important when the nextgeneration of experiments start to probe this mass region. • The proper decoupling of the different contributions has been checked explicitly, e.g. wehave checked that the SUSY-contributions, the ν C - X and the Higgs contributions decoupleindependently as expected. • Currently, the radiative decay µ → eγ is the most constraining LFV process. However, dueto the promising experimental prospects in the near future, the situation will change. If thecoming experiments perform as planned, µ → e will be the most relevant LFV process inthe mid term, whereas neutrinoless µ − e conversion in nuclei will set the strongest constraintsin the long term. • Ratios of τ LFV branching ratios can provide additional information about the dominant con-tributions. In particular, when the non-SUSY contributions dominate, one finds BR ( τ − → µ − e + e − ) / BR ( τ → µ ) (cid:39) BR ( τ − → e − µ + µ − ) / BR ( τ → e ) (cid:39) . − . , whereas for a SUSYdominated scenario BR ( τ − → µ − e + e − ) / BR ( τ → µ ) (cid:29) BR ( τ − → e − µ + µ − ) / BR ( τ → e ) .This can in turn be used to get a hint on the hierarchy between the seesaw and SUSY scales. Non-SUSY Z -penguins were also included in Ref. [70], where their potentially large contributions were alsoshown. cknowledgements We thank Martin Hirsch and Ernesto Arganda for fruitful discussions. A.V. acknowledges partialsupport from the EXPL/FIS-NUC/0460/2013 project financed by the Portuguese FCT. M.E.K.and W.P. have been supported by the DFG, project no. PO-1337/3-1 and the DFG researchtraining group GRK 1147. FS is supported by the BMBF PT DESY Verbundprojekt 05H2013-THEORIE ’Vergleich von LHC-Daten mit supersymmetrischen Modellen’. C.W. receives financialsupport from the Spanish CICYT through the project FPA2012-31880 and a partial support fromthe European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442) andthe Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249. A. A acknowledges support from the European Union FP7 ITN INVISIBLES (Marie CurieActions, PITN-GA-2011-289442).
A Masses and vertices
We give first our conventions for the mass matrices as well as for the corresponding rotationmatrices. These matrices are then used to express in appendix A.2 all the vertices needed tocalculate the LFV observables.
A.1 Mass matrices • Mass matrix for Neutrinos , Basis: (cid:0) ν L , ν CR , X (cid:1) m ν = √ v u Y Tν √ v u Y ν M R M TR µ X (31)This matrix is diagonalized by U V : U V, ∗ m ν U V, † = m diaν (32) • Mass matrix for CP-odd Sneutrinos , Basis: ( σ L , σ R , σ X ) m ν i = m σ L σ L m Tσ L σ R √ v u (cid:60) (cid:16) Y Tν M ∗ R (cid:17) m σ L σ R m σ R σ R (cid:60) (cid:16) B M R − M R µ ∗ X (cid:17) √ v u (cid:60) (cid:16) M TR Y ∗ ν (cid:17) (cid:60) (cid:16) B TM R − µ X M † R (cid:17) m σ X σ X (33) m σ L σ L = 12 v u (cid:60) (cid:16) Y Tν Y ∗ ν (cid:17) + (cid:60) (cid:16) m l (cid:17) + 18 (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) (34) m σ L σ R = − √ (cid:16) v d (cid:60) (cid:16) µY ∗ ν (cid:17) − v u (cid:60) (cid:16) T ν (cid:17)(cid:17) (35) m σ R σ R = (cid:60) (cid:16) m ν (cid:17) + (cid:60) (cid:16) M R M † R (cid:17) + 12 v u (cid:60) (cid:16) Y ν Y † ν (cid:17) (36) m σ X σ X = (cid:60) (cid:16) M TR M ∗ R (cid:17) + (cid:60) (cid:16) m X (cid:17) − (cid:60) (cid:16) B µ X (cid:17) + (cid:60) (cid:16) µ X µ ∗ X (cid:17) (37)This matrix is diagonalized by Z i : Z i m ν i Z i, † = m ,diaν i (38)21 Mass matrix for CP-even Sneutrinos , Basis: ( φ L , φ R , φ X ) m ν R = m φ L φ L m Tφ L φ R √ v u (cid:60) (cid:16) Y Tν M ∗ R (cid:17) m φ L φ R m φ R φ R (cid:60) (cid:16) B M R + M R µ ∗ X (cid:17) √ v u (cid:60) (cid:16) M TR Y ∗ ν (cid:17) (cid:60) (cid:16) B TM R + µ X M † R (cid:17) m φ X φ X (39) m φ L φ L = 12 v u (cid:60) (cid:16) Y Tν Y ∗ ν (cid:17) + (cid:60) (cid:16) m l (cid:17) + 18 (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) (40) m φ L φ R = − √ (cid:16) v d (cid:60) (cid:16) µY ∗ ν (cid:17) − v u (cid:60) (cid:16) T ν (cid:17)(cid:17) (41) m φ R φ R = (cid:60) (cid:16) m ν (cid:17) + (cid:60) (cid:16) M R M † R (cid:17) + 12 v u (cid:60) (cid:16) Y ν Y † ν (cid:17) (42) m φ X φ X = (cid:60) (cid:16) M TR M ∗ R (cid:17) + (cid:60) (cid:16) m X (cid:17) + (cid:60) (cid:16) B µ X (cid:17) + (cid:60) (cid:16) µ X µ ∗ X (cid:17) (43)This matrix is diagonalized by Z R : Z R m ν R Z R, † = m ,diaν R (44) • Mass matrix for Down-Squarks , Basis: (cid:16) ˜ d L,α , ˜ d R,α (cid:17) m d = m ˜ d L ˜ d ∗ L √ (cid:16) v d T † d − v u µY † d (cid:17) δ α β √ δ α β (cid:16) v d T d − v u Y d µ ∗ (cid:17) m ˜ d R ˜ d ∗ R (45) m ˜ d L ˜ d ∗ L = − (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) δ α β + 12 δ α β (cid:16) m q + v d Y † d Y d (cid:17) (46) m ˜ d R ˜ d ∗ R = 112 g (cid:16) v u − v d (cid:17) δ α β + 12 δ α β (cid:16) m d + v d Y d Y † d (cid:17) (47)This matrix is diagonalized by Z D : Z D m d Z D, † = m ,dia ˜ d (48) • Mass matrix for Up-Squarks , Basis: (˜ u L,α , ˜ u R,α ) m u = m ˜ u L ˜ u ∗ L √ (cid:16) − v d µY † u + v u T † u (cid:17) δ α β √ δ α β (cid:16) − v d Y u µ ∗ + v u T u (cid:17) m ˜ u R ˜ u ∗ R (49) m ˜ u L ˜ u ∗ L = − (cid:16) − g + g (cid:17)(cid:16) v d − v u (cid:17) δ α β + 12 δ α β (cid:16) m q + v u Y † u Y u (cid:17) (50) m ˜ u R ˜ u ∗ R = 12 δ α β (cid:16) m u + v u Y u Y † u (cid:17) + 16 g (cid:16) v d − v u (cid:17) δ α β (51)This matrix is diagonalized by Z U : Z U m u Z U, † = m ,dia ˜ u (52)22 Mass matrix for Sleptons , Basis: (˜ e L , ˜ e R ) m e = m ˜ e L ˜ e ∗ L √ (cid:16) v d T † e − v u µY † e (cid:17) √ (cid:16) v d T e − v u Y e µ ∗ (cid:17) m ˜ e R ˜ e ∗ R (53) m ˜ e L ˜ e ∗ L = 12 v d Y † e Y e + 18 (cid:16) − g + g (cid:17)(cid:16) v d − v u (cid:17) + m l (54) m ˜ e R ˜ e ∗ R = 12 v d Y e Y † e + 14 g (cid:16) v u − v d (cid:17) + m e (55)This matrix is diagonalized by Z E : Z E m e Z E, † = m ,dia ˜ e (56) • Mass matrix for CP-even Higgs , Basis: ( φ d , φ u ) m h = (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) + m H d + | µ | − (cid:16) g + g (cid:17) v d v u − (cid:60) (cid:16) B µ (cid:17) − (cid:16) g + g (cid:17) v d v u − (cid:60) (cid:16) B µ (cid:17) − (cid:16) g + g (cid:17)(cid:16) − v u + v d (cid:17) + m H u + | µ | (57)This matrix is diagonalized by Z H : Z H m h Z H, † = m ,diah (58) • Mass matrix for CP-odd Higgs , Basis: ( σ d , σ u ) m A = (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) + m H d + | µ | (cid:60) (cid:16) B µ (cid:17) (cid:60) (cid:16) B µ (cid:17) − (cid:16) g + g (cid:17)(cid:16) v d − v u (cid:17) + m H u + | µ | + ξ Z m ( Z ) (59)Gauge fixing contributions: m ( Z ) = v d (cid:16) g sin Θ W + g cos Θ W (cid:17) − v d v u (cid:16) g sin Θ W + g cos Θ W (cid:17) − v d v u (cid:16) g sin Θ W + g cos Θ W (cid:17) v u (cid:16) g sin Θ W + g cos Θ W (cid:17) (60)This matrix is diagonalized by Z A : Z A m A Z A, † = m ,diaA (61) • Mass matrix for Charged Higgs , Basis: (cid:16) H − d , H + , ∗ u (cid:17) , (cid:16) H − , ∗ d , H + u (cid:17) m H − = (cid:32) m H − d H − , ∗ d g v d v u + B ∗ µ g v d v u + B µ m H + , ∗ u H + u (cid:33) + ξ W − m ( W − ) (62) m H − d H − , ∗ d = 18 (cid:16) g (cid:16) v d − v u (cid:17) + g (cid:16) v d + v u (cid:17)(cid:17) + m H d + | µ | (63)23 H + , ∗ u H + u = 18 (cid:16) g (cid:16) − v d + v u (cid:17) + g (cid:16) v d + v u (cid:17)(cid:17) + m H u + | µ | (64)Gauge fixing contributions: m ( W − ) = (cid:18) g v d − g v d v u − g v d v u g v u (cid:19) (65)This matrix is diagonalized by Z + : Z + m H − Z + , † = m ,diaH − (66) • Mass matrix for Neutralinos , Basis: (cid:16) λ ˜ B , ˜ W , ˜ H d , ˜ H u (cid:17) m ˜ χ = M − g v d g v u M g v d − g v u − g v d g v d − µ g v u − g v u − µ (67)This matrix is diagonalized by N : N ∗ m ˜ χ N † = m dia ˜ χ (68) • Mass matrix for Charginos , Basis: (cid:16) ˜ W − , ˜ H − d (cid:17) , (cid:16) ˜ W + , ˜ H + u (cid:17) m ˜ χ − = (cid:32) M √ g v u √ g v d µ (cid:33) (69)This matrix is diagonalized by U and VU ∗ m ˜ χ − V † = m dia ˜ χ − (70) • Mass matrix for charged Leptons , Basis: ( e L ) , ( e ∗ R ) m e = (cid:16) √ v d Y Te (cid:17) (71)This matrix is diagonalized by U eL and U eR U e, ∗ L m e U e, † R = m diae (72) • Mass matrix for Down-Quarks , Basis: ( d L,α ) , (cid:16) d ∗ R,β (cid:17) m d = (cid:16) √ v d δ α β Y Td (cid:17) (73)This matrix is diagonalized by U dL and U dR U d, ∗ L m d U d, † R = m diad (74) • Mass matrix for Up-Quarks , Basis: ( u L,α ) , (cid:16) u ∗ R,β (cid:17) m u = (cid:16) √ v u δ α β Y Tu (cid:17) (75)This matrix is diagonalized by U uL and U uR U u, ∗ L m u U u, † R = m diau (76)24 .2 Vertices In this appendix we list all vertices relevant for our computations. Our conventions are as follows: • Chiral vertices are parameterized as Γ LF a F b S c P L + Γ RF a F b S c P R Γ LF a F b V µc γ µ P L + Γ RF a F b V µc γ µ P R • The momentum flow in vector and scalar-vector vertices is Γ S a S b V µc ( p µS b − p µS a )Γ V ρa V σb V µc ( g ρµ ( − p σV c + p σV a ) + g ρσ ( − p µV a + p µV c ) + g σµ ( − p ρV a + p ρV c )) Here we used polarization projectors P L,R , metric g µν and momenta p of the external fields. A.2.1 Fermion-Scalar vertices P c,Lijk = Γ L ˜ χ + i ˜ χ − j A k = − √ g (cid:16) U ∗ j V ∗ i Z Ak + U ∗ j V ∗ i Z Ak (cid:17) (77) P c,Rijk = Γ R ˜ χ + i ˜ χ − j A k = 1 √ g (cid:16) U i V j Z Ak + U i V j Z Ak (cid:17) (78) P Lijk = Γ L ˜ χ i ˜ χ j A k = 12 (cid:16) N ∗ i (cid:16) g N ∗ j − g N ∗ j (cid:17) Z Ak − g N ∗ i N ∗ j Z Ak − g N ∗ i N ∗ j Z Ak + g N ∗ i N ∗ j Z Ak + g N ∗ i N ∗ j Z Ak + g N ∗ i (cid:16) N ∗ j Z Ak − N ∗ j Z Ak (cid:17)(cid:17) (79) P Rijk = Γ R ˜ χ i ˜ χ j A k = 12 (cid:16) Z Ak (cid:16)(cid:16) − g N i + g N i (cid:17) N j + N i (cid:16) − g N j + g N j (cid:17)(cid:17) + Z Ak (cid:16)(cid:16) g N i − g N i (cid:17) N j + N i (cid:16) g N j − g N j (cid:17)(cid:17)(cid:17) (80) A d,Lijk = Γ L ¯ d iα d jβ A k = − i √ δ αβ (cid:88) b =1 U d, ∗ L,jb (cid:88) a =1 U d, ∗ R,ia Y d,ab Z Ak (81) A d,Rijk = Γ R ¯ d iα d jβ A k = i √ δ αβ (cid:88) b =1 3 (cid:88) a =1 Y ∗ d,ab U dR,ja U dL,ib Z Ak (82) A Lijk = Γ L ¯ (cid:96) i (cid:96) j A k = − i √ (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 U e, ∗ R,ia Y e,ab Z Ak (83) A Rijk = Γ R ¯ (cid:96) i (cid:96) j A k = i √ (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab U eR,ja U eL,ib Z Ak (84) A u,Lijk = Γ L ¯ u iα u jβ A k = − i √ δ αβ (cid:88) b =1 U u, ∗ L,jb (cid:88) a =1 U u, ∗ R,ia Y u,ab Z Ak (85) A u,Rijk = Γ R ¯ u iα u jβ A k = i √ δ αβ (cid:88) b =1 3 (cid:88) a =1 Y ∗ u,ab U uR,ja U uL,ib Z Ak (86) A ν,Lijk = Γ Lν i ν j A k = − i √ (cid:16) (cid:88) b =1 U V, ∗ jb (cid:88) a =1 U V, ∗ i a Y ν,ab + (cid:88) b =1 U V, ∗ ib (cid:88) a =1 U V, ∗ j a Y ν,ab (cid:17) Z Ak (87)25 ν,Rijk = Γ Rν i ν j A k = i √ (cid:16) (cid:88) b =1 3 (cid:88) a =1 Y ∗ ν,ab U Vj a U Vib + (cid:88) b =1 3 (cid:88) a =1 Y ∗ ν,ab U Vi a U Vjb (cid:17) Z Ak (88) W u,Lijk = Γ L ˜ χ − i u jβ ˜ d ∗ kγ = − δ βγ (cid:16) g U ∗ i (cid:88) a =1 U u, ∗ L,ja Z Dka − U ∗ i (cid:88) b =1 U u, ∗ L,jb (cid:88) a =1 Y d,ab Z Dk a (cid:17) (89) W u,Rijk = Γ R ˜ χ − i u jβ ˜ d ∗ kγ = δ βγ (cid:88) b =1 3 (cid:88) a =1 Y ∗ u,ab U uR,ja Z Dkb V i (90) S c,Lijk = Γ L ˜ χ + i ˜ χ − j h k = − √ g (cid:16) U ∗ j V ∗ i Z Hk + U ∗ j V ∗ i Z Hk (cid:17) (91) S c,Rijk = Γ R ˜ χ + i ˜ χ − j h k = − √ g (cid:16) U i V j Z Hk + U i V j Z Hk (cid:17) (92) S Lijk = Γ L ˜ χ i ˜ χ j h k = 12 (cid:16) N ∗ i (cid:16) g N ∗ j − g N ∗ j (cid:17) Z Hk − g N ∗ i N ∗ j Z Hk − g N ∗ i N ∗ j Z Hk + g N ∗ i N ∗ j Z Hk + g N ∗ i N ∗ j Z Hk + g N ∗ i (cid:16) N ∗ j Z Hk − N ∗ j Z Hk (cid:17)(cid:17) (93) S Rijk = Γ R ˜ χ i ˜ χ j h k = 12 (cid:16) Z Hk (cid:16)(cid:16) g N i − g N i (cid:17) N j + N i (cid:16) g N j − g N j (cid:17)(cid:17) + Z Hk (cid:16)(cid:16) − g N i + g N i (cid:17) N j + N i (cid:16) − g N j + g N j (cid:17)(cid:17)(cid:17) (94) N d,Lijk = Γ L ˜ χ i d jβ ˜ d ∗ kγ = δ βγ (cid:16) √ g N ∗ i (cid:88) a =1 U d, ∗ L,ja Z Dka − N ∗ i (cid:88) b =1 U d, ∗ L,jb (cid:88) a =1 Y d,ab Z Dk a − √ g N ∗ i (cid:88) a =1 U d, ∗ L,ja Z Dka (cid:17) (95) N d,Rijk = Γ R ˜ χ i d jβ ˜ d ∗ kγ = − δ βγ (cid:16) (cid:88) b =1 3 (cid:88) a =1 Y ∗ d,ab U dR,ja Z Dkb N i + √ g (cid:88) a =1 Z Dk a U dR,ja N i (cid:17) (96) N Lijk = Γ L ˜ χ i (cid:96) j ˜ e ∗ k = − N ∗ i (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 Y e,ab Z Ek a + 1 √ g N ∗ i (cid:88) a =1 U e, ∗ L,ja Z Eka + 1 √ g N ∗ i (cid:88) a =1 U e, ∗ L,ja Z Eka (97) N Rijk = Γ R ˜ χ i (cid:96) j ˜ e ∗ k = − (cid:16) √ g (cid:88) a =1 Z Ek a U eR,ja N i + (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab U eR,ja Z Ekb N i (cid:17) (98) N u,Lijk = Γ L ˜ χ i u jβ ˜ u ∗ kγ = − δ βγ (cid:16) √ g N ∗ i (cid:88) a =1 U u, ∗ L,ja Z Uka + N ∗ i (cid:88) b =1 U u, ∗ L,jb (cid:88) a =1 Y u,ab Z Uk a + 13 √ g N ∗ i (cid:88) a =1 U u, ∗ L,ja Z Uka (cid:17) (99) N u,Rijk = Γ R ˜ χ i u jβ ˜ u ∗ kγ = δ βγ (cid:16) √ g (cid:88) a =1 Z Uk a U uR,ja N i − (cid:88) b =1 3 (cid:88) a =1 Y ∗ u,ab U uR,ja Z Ukb N i (cid:17) (100) H d,Lijk = Γ L ¯ d iα d jβ h k = − √ δ αβ (cid:88) b =1 U d, ∗ L,jb (cid:88) a =1 U d, ∗ R,ia Y d,ab Z Hk (101) H d,Rijk = Γ R ¯ d iα d jβ h k = − √ δ αβ (cid:88) b =1 3 (cid:88) a =1 Y ∗ d,ab U dR,ja U dL,ib Z Hk (102)26 d,Lijk = Γ L ˜ χ + i d jβ ˜ u ∗ kγ = − δ βγ (cid:16) g V ∗ i (cid:88) a =1 U d, ∗ L,ja Z Uka − V ∗ i (cid:88) b =1 U d, ∗ L,jb (cid:88) a =1 Y u,ab Z Uk a (cid:17) (103) W d,Rijk = Γ R ˜ χ + i d jβ ˜ u ∗ kγ = δ βγ (cid:88) b =1 3 (cid:88) a =1 Y ∗ d,ab U dR,ja Z Ukb U i (104) V u,Lijk = Γ L ¯ u iα d jβ H + k = δ αβ (cid:88) b =1 U d, ∗ L,jb (cid:88) a =1 U u, ∗ R,ia Y u,ab Z + k (105) V u,Rijk = Γ R ¯ u iα d jβ H + k = δ αβ (cid:88) b =1 3 (cid:88) a =1 Y ∗ d,ab U dR,ja U uL,ib Z + k (106) V + ,Lijk = Γ Lν i (cid:96) j H + k = (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 U V, ∗ i a Y ν,ab Z + k (107) V + ,Rijk = Γ Rν i (cid:96) j H + k = (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab U eR,ja U Vib Z + k (108) H Lijk = Γ L ¯ (cid:96) i (cid:96) j h k = − √ (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 U e, ∗ R,ia Y e,ab Z Hk (109) H Rijk = Γ R ¯ (cid:96) i (cid:96) j h k = − √ (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab U eR,ja U eL,ib Z Hk (110) X Lijk = Γ L ˜ χ + i (cid:96) j ν ik = − i √ (cid:16) − g V ∗ i (cid:88) a =1 U e, ∗ L,ja Z i, ∗ ka + V ∗ i (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 Z i, ∗ k a Y ν,ab (cid:17) (111) X Rijk = Γ R ˜ χ + i (cid:96) j ν ik = − i √ (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Y ∗ e,ab U eR,ja U i (112) ˆ X Lijk = Γ L ˜ χ + i (cid:96) j ν Rk = − √ (cid:16) g V ∗ i (cid:88) a =1 U e, ∗ L,ja Z R, ∗ ka − V ∗ i (cid:88) b =1 U e, ∗ L,jb (cid:88) a =1 Z R, ∗ k a Y ν,ab (cid:17) (113) ˆ X Lijk = Γ R ˜ χ + i (cid:96) j ν Rk = 1 √ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Y ∗ e,ab U eR,ja U i (114) H u,Lijk = Γ L ¯ u iα u jβ h k = − √ δ αβ (cid:88) b =1 U u, ∗ L,jb (cid:88) a =1 U u, ∗ R,ia Y u,ab Z Hk (115) H u,Rijk = Γ R ¯ u iα u jβ h k = − √ δ αβ (cid:88) b =1 3 (cid:88) a =1 Y ∗ u,ab U uR,ja U uL,ib Z Hk (116) H ν,Lijk = Γ Lν i ν j h k = − √ (cid:16) (cid:88) b =1 U V, ∗ jb (cid:88) a =1 U V, ∗ i a Y ν,ab + (cid:88) b =1 U V, ∗ ib (cid:88) a =1 U V, ∗ j a Y ν,ab (cid:17) Z Hk (117) H ν,Rijk = Γ Rν i ν j h k = − √ (cid:16) (cid:88) b =1 3 (cid:88) a =1 Y ∗ ν,ab U Vj a U Vib + (cid:88) b =1 3 (cid:88) a =1 Y ∗ ν,ab U Vi a U Vjb (cid:17) Z Hk (118) F c,Lijk = Γ L ˜ χ + i ν j ˜ e k = V ∗ i (cid:88) b =1 Z E, ∗ kb (cid:88) a =1 U V, ∗ j a Y ν,ab (119)27 c,Rijk = Γ R ˜ χ + i ν j ˜ e k = − (cid:16) g (cid:88) a =1 Z E, ∗ ka U Vja U i − (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab Z E, ∗ k a U Vjb U i (cid:17) (120)In addition, we introduce ¯ W d,Lijk = ( W d,Rjik ) ∗ ¯ W d,Rijk = ( W d,Ljik ) ∗ ¯ W u,Lijk = ( W u,Rjik ) ∗ ¯ W u,Rijk = ( W u,Ljik ) ∗ (121) ¯ X Lijk = ( X Rjik ) ∗ ¯ X Rijk = ( X Ljik ) ∗ ˆ¯ X Lijk = ( ˆ X Rjik ) ∗ ˆ¯ X Rijk = ( ˆ X Ljik ) ∗ (122) ¯ N d,Lijk = ( N d,Rjik ) ∗ ¯ N d,Rijk = ( N d,Ljik ) ∗ ¯ N u,Lijk = ( N u,Rjik ) ∗ ¯ N u,Rijk = ( N u,Ljik ) ∗ (123) ¯ N Lijk = ( N Rjik ) ∗ ¯ N Rijk = ( N Ljik ) ∗ ¯ V d,Lijk = ( V u,Rjik ) ∗ ¯ V d,Rijk = ( V u,Ljik ) ∗ (124) ¯ V + ,Lijk = ( V + ,Rjik ) ∗ ¯ V + ,Rijk = ( V + ,Ljik ) ∗ (125) A.2.2 Fermion-Vector vertices F c,Lij = Γ L ˜ χ + i ˜ χ − j γ µ = F c,Rij = Γ R ˜ χ + i ˜ χ − j γ µ = eδ ij (126) C Lij = Γ L ˜ χ + i ˜ χ − j Z µ = g U ∗ j cos Θ W U i + 12 U ∗ j (cid:16) − g sin Θ W + g cos Θ W (cid:17) U i (127) C Rij = Γ R ˜ χ + i ˜ χ − j Z µ = g V ∗ i cos Θ W V j + 12 V ∗ i (cid:16) − g sin Θ W + g cos Θ W (cid:17) V j (128) M Lij = Γ L ˜ χ i ˜ χ j Z µ = − (cid:16) g sin Θ W + g cos Θ W (cid:17)(cid:16) N ∗ j N i − N ∗ j N i (cid:17) (129) M Rij = Γ R ˜ χ i ˜ χ j Z µ = 12 (cid:16) g sin Θ W + g cos Θ W (cid:17)(cid:16) N ∗ i N j − N ∗ i N j (cid:17) (130) D Lij = Γ L ¯ d iα d jβ Z µ = 16 δ αβ δ ij (cid:16) g cos Θ W + g sin Θ W (cid:17) (131) D Rij = Γ R ¯ d iα d jβ Z µ = − g δ αβ δ ij sin Θ W (132) ˆ V u,Lij = Γ L ¯ u iα d jβ W + µ = − √ g δ αβ (cid:88) a =1 U d, ∗ L,ja U uL,ia (133) ˆ V u,Rij = Γ R ¯ u iα d jβ W + µ = 0 (134) ˆ V + ,Lij = Γ Lν i (cid:96) j W + µ = − √ g (cid:88) a =1 U e, ∗ L,ja U Via (135) ˆ V + ,Rij = Γ Rν i (cid:96) j W + µ = 0 (136) E Lij = Γ L ¯ (cid:96) i (cid:96) j Z µ = 12 δ ij (cid:16) − g sin Θ W + g cos Θ W (cid:17) (137) E Rij = Γ R ¯ (cid:96) i (cid:96) j Z µ = − g δ ij sin Θ W (138) U Lij = Γ L ¯ u iα u jβ Z µ = − δ αβ δ ij (cid:16) g cos Θ W − g sin Θ W (cid:17) (139) U Rij = Γ R ¯ u iα u jβ Z µ = 23 g δ αβ δ ij sin Θ W (140) V Lij = Γ Lν i ν j Z µ = − (cid:16) g sin Θ W + g cos Θ W (cid:17) (cid:88) a =1 U V, ∗ ja U Via (141)28
Rij = Γ Rν i ν j Z µ = 12 (cid:16) g sin Θ W + g cos Θ W (cid:17) (cid:88) a =1 U V, ∗ ia U Vja (142)In addition, we introduce ˆ¯ V d,Lij = ( ˆ V u,Lji ) ∗ ˆ¯ V d,Rij = ( ˆ V u,Rji ) ∗ ˆ¯ V + ,Lij = ( ˆ V + ,Lij ) ∗ ˆ¯ V + ,Rij = ( ˆ V + ,Rij ) ∗ (143) A.2.3 Scalar vertices A hhijk = Γ A i H − j H + k = − i g (cid:16) v d Z Ai + v u Z Ai (cid:17)(cid:16) − Z + , ∗ j Z + k + Z + , ∗ j Z + k (cid:17) (144) ˜ A ijk = Γ A i ˜ e j ˜ e ∗ k = − i √ (cid:16) (cid:88) b =1 Z E, ∗ jb (cid:88) a =1 Z Ek a T e,ab Z Ai − (cid:88) b =1 3 (cid:88) a =1 Z E, ∗ j a T ∗ e,ab Z Ekb Z Ai + (cid:16) − µ (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab Z E, ∗ j a Z Ekb + µ ∗ (cid:88) b =1 Z E, ∗ jb (cid:88) a =1 Y e,ab Z Ek a (cid:17) Z Ai (cid:17) (145) ˜ A iiijk = Γ A i ν ij ν ik = 0 (146) ˜ A irijk = Γ A i ν ij ν Rk = − √ (cid:16) µ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Y ∗ ν,ab Z i, ∗ j a Z Ai − µ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Y ∗ ν,ab Z R, ∗ k a Z Ai + µ ∗ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z i, ∗ j a Y ν,ab Z Ai − µ ∗ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z R, ∗ k a Y ν,ab Z Ai + (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z i, ∗ j a T ∗ ν,ab Z Ai − (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z R, ∗ k a T ∗ ν,ab Z Ai + (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z i, ∗ j a T ν,ab Z Ai − (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z R, ∗ k a T ν,ab Z Ai + (cid:88) c =1 Z R, ∗ kc (cid:88) b =1 Z i, ∗ j b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Ai − (cid:88) c =1 Z i, ∗ jc (cid:88) b =1 Z R, ∗ k b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Ai − (cid:88) c =1 Z R, ∗ k c (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Ai + (cid:88) c =1 Z i, ∗ j c (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Ai (cid:17) (147) ˜ A rrijk = Γ A i ν Rj ν Rk = 0 (148) H hhijk = Γ h i H − j H + k = − (cid:16) Z + , ∗ j (cid:16) Z Hi (cid:16)(cid:16) g + g (cid:17) v d Z + k + g v u Z + k (cid:17) + Z Hi (cid:16)(cid:16) − g + g (cid:17) v u Z + k + g v d Z + k (cid:17)(cid:17) + Z + , ∗ j (cid:16) Z Hi (cid:16)(cid:16) − g + g (cid:17) v d Z + k + g v u Z + k (cid:17) + Z Hi (cid:16)(cid:16) g + g (cid:17) v u Z + k + g v d Z + k (cid:17)(cid:17)(cid:17) (149) ˜ H ijk = Γ h i ˜ e j ˜ e ∗ k = − (cid:16)(cid:16) − g + g (cid:17) (cid:88) a =1 Z E, ∗ ja Z Eka (cid:16) v d Z Hi − v u Z Hi (cid:17) + 2 (cid:16) √ (cid:88) b =1 Z E, ∗ jb (cid:88) a =1 Z Ek a T e,ab Z Hi + √ (cid:88) b =1 3 (cid:88) a =1 Z E, ∗ j a T ∗ e,ab Z Ekb Z Hi v d (cid:88) c =1 Z E, ∗ j c (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ca Y e,ba Z Ek b Z Hi + 2 v d (cid:88) c =1 3 (cid:88) b =1 Z E, ∗ jb (cid:88) a =1 Y ∗ e,ac Y e,ab Z Ekc Z Hi − √ µ ∗ (cid:88) b =1 Z E, ∗ jb (cid:88) a =1 Y e,ab Z Ek a Z Hi − √ µ (cid:88) b =1 3 (cid:88) a =1 Y ∗ e,ab Z E, ∗ j a Z Ekb Z Hi + g
21 3 (cid:88) a =1 Z E, ∗ j a Z Ek a (cid:16) − v d Z Hi + v u Z Hi (cid:17)(cid:17)(cid:17) (150) ˜ H iiijk = Γ h i ν ij ν ik = − (cid:16) − √ µ (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Y ∗ ν,ab Z i, ∗ j a Z Hi − √ µ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Y ∗ ν,ab Z i, ∗ k a Z Hi − √ µ ∗ (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Z i, ∗ j a Y ν,ab Z Hi − √ µ ∗ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z i, ∗ k a Y ν,ab Z Hi + √ (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Z i, ∗ j a T ∗ ν,ab Z Hi + √ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z i, ∗ k a T ∗ ν,ab Z Hi + √ (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Z i, ∗ j a T ν,ab Z Hi + √ (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Z i, ∗ k a T ν,ab Z Hi + √ (cid:88) c =1 Z i, ∗ kc (cid:88) b =1 Z i, ∗ j b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Hi + √ (cid:88) c =1 Z i, ∗ jc (cid:88) b =1 Z i, ∗ k b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Hi + √ (cid:88) c =1 Z i, ∗ k c (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Hi + √ (cid:88) c =1 Z i, ∗ j c (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z i, ∗ kc (cid:88) b =1 Z i, ∗ jb (cid:88) a =1 Y ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z i, ∗ jc (cid:88) b =1 Z i, ∗ kb (cid:88) a =1 Y ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z i, ∗ k c (cid:88) b =1 Z i, ∗ j b (cid:88) a =1 Y ∗ ν,ca Y ν,ba Z Hi + 2 v u (cid:88) c =1 Z i, ∗ j c (cid:88) b =1 Z i, ∗ k b (cid:88) a =1 Y ∗ ν,ca Y ν,ba Z Hi + (cid:16) g + g (cid:17) (cid:88) a =1 Z i, ∗ ja Z i, ∗ ka (cid:16) v d Z Hi − v u Z Hi (cid:17)(cid:17) (151) ˜ H irijk = Γ h i ν ij ν Rk = 0 (152) ˜ H rrijk = Γ h i ν Rj ν Rk = − (cid:16) − √ µ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Y ∗ ν,ab Z R, ∗ j a Z Hi − √ µ (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 Y ∗ ν,ab Z R, ∗ k a Z Hi − √ µ ∗ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z R, ∗ j a Y ν,ab Z Hi − √ µ ∗ (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 Z R, ∗ k a Y ν,ab Z Hi + √ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z R, ∗ j a T ∗ ν,ab Z Hi + √ (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 Z R, ∗ k a T ∗ ν,ab Z Hi √ (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Z R, ∗ j a T ν,ab Z Hi + √ (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 Z R, ∗ k a T ν,ab Z Hi + √ (cid:88) c =1 Z R, ∗ kc (cid:88) b =1 Z R, ∗ j b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Hi + √ (cid:88) c =1 Z R, ∗ jc (cid:88) b =1 Z R, ∗ k b (cid:88) a =1 Y ∗ ν,ac M R,ab Z Hi + √ (cid:88) c =1 Z R, ∗ k c (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Hi + √ (cid:88) c =1 Z R, ∗ j c (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 M ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z R, ∗ kc (cid:88) b =1 Z R, ∗ jb (cid:88) a =1 Y ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z R, ∗ jc (cid:88) b =1 Z R, ∗ kb (cid:88) a =1 Y ∗ ν,ac Y ν,ab Z Hi + 2 v u (cid:88) c =1 Z R, ∗ k c (cid:88) b =1 Z R, ∗ j b (cid:88) a =1 Y ∗ ν,ca Y ν,ba Z Hi + 2 v u (cid:88) c =1 Z R, ∗ j c (cid:88) b =1 Z R, ∗ k b (cid:88) a =1 Y ∗ ν,ca Y ν,ba Z Hi + (cid:16) g + g (cid:17) (cid:88) a =1 Z R, ∗ ja Z R, ∗ ka (cid:16) v d Z Hi − v u Z Hi (cid:17)(cid:17) (153) A.2.4 Scalar-Vector vertices A hwij = Γ A i H − j W + µ = i g (cid:16) Z + , ∗ j Z Ai + Z + , ∗ j Z Ai (cid:17) (154) H hwij = Γ h i H − j W + µ = 12 g (cid:16) Z + , ∗ j Z Hi − Z + , ∗ j Z Hi (cid:17) (155) F hij = Γ H − i H + j γ µ = − eδ ij (156) Z hhij = Γ H − i H + j Z µ = 12 (cid:16) g sin Θ W − g cos Θ W (cid:17) δ ij (157) F ˜ eij = Γ ˜ e i ˜ e ∗ j γ µ = − eδ ij (158) ˜ E ij = Γ ˜ e i ˜ e ∗ j Z µ = g sin Θ W (cid:88) a =1 Z E, ∗ i a Z Ej a + 12 (cid:16) g sin Θ W − g cos Θ W (cid:17) (cid:88) a =1 Z E, ∗ ia Z Eja (159) ˜ V ij = Γ ν ii ν Rj Z µ = i (cid:16) g sin Θ W + g cos Θ W (cid:17) (cid:88) a =1 Z i, ∗ ia Z R, ∗ ja (160) H wwi = Γ h i W + σ W − µ = − g (cid:16) v d Z Hi + v u Z Hi (cid:17) (161) F hwi = Γ H − i W + σ γ µ = − i W e (cid:16) v d Z + , ∗ i − v u Z + , ∗ i (cid:17) (162) Z hwi = Γ H − i W + σ Z µ = 12 g g (cid:16) v d Z + , ∗ i − v u Z + , ∗ i (cid:17) sin Θ W (163)(164)In addition, we introduce ¯ A hwij = ( A hwij ) ∗ ¯ H hwij = ( H hwij ) ∗ ¯ F hwi = ( F hwi ) ∗ ¯ Z hwi = ( Z hwi ) ∗ (165)31 .2.5 Vector vertices F w = Γ W + ρ γ σ W − µ = g sin Θ W (166) Z ww = Γ W + ρ W − σ Z µ = − g cos Θ W (167) B Renormalization Group Equations
We give in the following the 2-loop RGEs for the considered model. For parameters present in theMSSM we show only the difference with respect to the MSSM RGEs. In general, the RGEs for aparameter X are defined by ddt X = 116 π β (1) X + 1(16 π ) β (2) X (168)Here, t = log ( Q/M ) , with Q the renormalization scale and M a reference scale. Gauge Couplings ∆ β (2) g = − g Tr (cid:16) Y ν Y † ν (cid:17) (169) ∆ β (2) g = − g Tr (cid:16) Y ν Y † ν (cid:17) (170) Gaugino Mass Parameters ∆ β (2) M = − g (cid:16) M Tr (cid:16) Y ν Y † ν (cid:17) − Tr (cid:16) Y † ν T ν (cid:17)(cid:17) (171) ∆ β (2) M = 4 g (cid:16) − M Tr (cid:16) Y ν Y † ν (cid:17) + Tr (cid:16) Y † ν T ν (cid:17)(cid:17) (172) Trilinear Superpotential Parameters ∆ β (2) Y d = − Y d Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − Y d Y † u Y u Tr (cid:16) Y ν Y † ν (cid:17) (173) ∆ β (1) Y e = Y e Y † ν Y ν (174) ∆ β (2) Y e = − Y e Y † ν Y ν Y † e Y e − Y e Y † ν Y ν Y † ν Y ν − Y e Y † ν Y ν Tr (cid:16) Y u Y † u (cid:17) − Y e Y † ν Y ν Tr (cid:16) Y ν Y † ν (cid:17) − Y e Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) (175) ∆ β (1) Y u = Y u Tr (cid:16) Y ν Y † ν (cid:17) (176) ∆ β (2) Y u = − Y u Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17) − Y u Y † u Y u Tr (cid:16) Y ν Y † ν (cid:17) − Y u Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) (177) β (1) Y ν = 3 Y ν Y † ν Y ν + Y ν (cid:16) − g + 3 Tr (cid:16) Y u Y † u (cid:17) − g + Tr (cid:16) Y ν Y † ν (cid:17)(cid:17) + Y ν Y † e Y e (178) β (2) Y ν = + 65 g Y ν Y † ν Y ν + 6 g Y ν Y † ν Y ν − Y ν Y † e Y e Y † e Y e − Y ν Y † e Y e Y † ν Y ν − Y ν Y † ν Y ν Y † ν Y ν + Y ν Y † e Y e (cid:16) − Tr (cid:16) Y d Y † d (cid:17) + 65 g − Tr (cid:16) Y e Y † e (cid:17)(cid:17) − Y ν Y † ν Y ν Tr (cid:16) Y u Y † u (cid:17) − Y ν Y † ν Y ν Tr (cid:16) Y ν Y † ν (cid:17) Y ν (cid:16) g + 95 g g + 152 g + 45 (cid:16) g + g (cid:17) Tr (cid:16) Y u Y † u (cid:17) − Tr (cid:16) Y d Y † u Y u Y † d (cid:17) − Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − Tr (cid:16) Y u Y † u Y u Y † u (cid:17) − Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17)(cid:17) (179) Bilinear Superpotential Parameters ∆ β (1) µ = µ Tr (cid:16) Y ν Y † ν (cid:17) (180) ∆ β (2) µ = − µ (cid:16) Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) + 3 Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17)(cid:17) (181) β (1) µ X = β (2) µ X = 0 (182) β (1) M R = 2 Y ν Y † ν M R (183) β (2) M R = − (cid:16) Y ν Y † e Y e Y † ν M R + Y ν Y † ν Y ν Y † ν M R (cid:17) + Y ν Y † ν M R (cid:16) − Tr (cid:16) Y ν Y † ν (cid:17) + 6 g − Tr (cid:16) Y u Y † u (cid:17) + 65 g (cid:17) (184) Trilinear Soft-Breaking Parameters ∆ β (2) T d = − Y d Y † u T u Tr (cid:16) Y ν Y † ν (cid:17) − T d Y † u Y u Tr (cid:16) Y ν Y † ν (cid:17) − Y d Y † u Y u Tr (cid:16) Y † ν T ν (cid:17) − T d Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − Y d Tr (cid:16) Y e Y † ν T ν Y † e (cid:17) − Y d Tr (cid:16) Y ν Y † e T e Y † ν (cid:17) (185) ∆ β (1) T e = 2 Y e Y † ν T ν + T e Y † ν Y ν (186) ∆ β (2) T e = − Y e Y † ν Y ν Y † e T e − Y e Y † ν Y ν Y † ν T ν − Y e Y † ν T ν Y † e Y e − Y e Y † ν T ν Y † ν Y ν − T e Y † ν Y ν Y † e Y e − T e Y † ν Y ν Y † ν Y ν − Y e Y † ν T ν Tr (cid:16) Y u Y † u (cid:17) − T e Y † ν Y ν Tr (cid:16) Y u Y † u (cid:17) − Y e Y † ν T ν Tr (cid:16) Y ν Y † ν (cid:17) − T e Y † ν Y ν Tr (cid:16) Y ν Y † ν (cid:17) − Y e Y † ν Y ν Tr (cid:16) Y † u T u (cid:17) − Y e Y † ν Y ν Tr (cid:16) Y † ν T ν (cid:17) − T e Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − Y e Tr (cid:16) Y e Y † ν T ν Y † e (cid:17) − Y e Tr (cid:16) Y ν Y † e T e Y † ν (cid:17) (187) ∆ β (1) T u = 2 Y u Tr (cid:16) Y † ν T ν (cid:17) + T u Tr (cid:16) Y ν Y † ν (cid:17) (188) ∆ β (2) T u = − Y u Y † u T u Tr (cid:16) Y ν Y † ν (cid:17) − T u Y † u Y u Tr (cid:16) Y ν Y † ν (cid:17) − Y u Y † u Y u Tr (cid:16) Y † ν T ν (cid:17) − T u Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − Y u Tr (cid:16) Y e Y † ν T ν Y † e (cid:17) − Y u Tr (cid:16) Y ν Y † e T e Y † ν (cid:17) − T u Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17) − Y u Tr (cid:16) Y ν Y † ν T ν Y † ν (cid:17) (189) β (1) T ν = +2 Y ν Y † e T e + 4 Y ν Y † ν T ν + T ν Y † e Y e + 5 T ν Y † ν Y ν − g T ν − g T ν + 3 T ν Tr (cid:16) Y u Y † u (cid:17) + T ν Tr (cid:16) Y ν Y † ν (cid:17) + Y ν (cid:16) Tr (cid:16) Y † ν T ν (cid:17) + 6 g M + 6 Tr (cid:16) Y † u T u (cid:17) + 65 g M (cid:17) (190) β (2) T ν = + 125 g Y ν Y † e T e − g M Y ν Y † ν Y ν − g M Y ν Y † ν Y ν + 65 g Y ν Y † ν T ν + 6 g Y ν Y † ν T ν + 65 g T ν Y † e Y e + 125 g T ν Y † ν Y ν + 12 g T ν Y † ν Y ν Y ν Y † e Y e Y † e T e − Y ν Y † e Y e Y † ν T ν − Y ν Y † e T e Y † e Y e − Y ν Y † e T e Y † ν Y ν − Y ν Y † ν Y ν Y † ν T ν − Y ν Y † ν T ν Y † ν Y ν − T ν Y † e Y e Y † e Y e − T ν Y † e Y e Y † ν Y ν − T ν Y † ν Y ν Y † ν Y ν + 20750 g T ν + 95 g g T ν + 152 g T ν − Y ν Y † e T e Tr (cid:16) Y d Y † d (cid:17) − T ν Y † e Y e Tr (cid:16) Y d Y † d (cid:17) − Y ν Y † e T e Tr (cid:16) Y e Y † e (cid:17) − T ν Y † e Y e Tr (cid:16) Y e Y † e (cid:17) − Y ν Y † ν T ν Tr (cid:16) Y u Y † u (cid:17) − T ν Y † ν Y ν Tr (cid:16) Y u Y † u (cid:17) + 45 g T ν Tr (cid:16) Y u Y † u (cid:17) + 16 g T ν Tr (cid:16) Y u Y † u (cid:17) − Y ν Y † ν T ν Tr (cid:16) Y ν Y † ν (cid:17) − T ν Y † ν Y ν Tr (cid:16) Y ν Y † ν (cid:17) − Y ν Y † e Y e (cid:16) Tr (cid:16) Y † d T d (cid:17) + 5 Tr (cid:16) Y † e T e (cid:17) + 6 g M (cid:17) − Y ν Y † ν Y ν Tr (cid:16) Y † u T u (cid:17) − Y ν Y † ν Y ν Tr (cid:16) Y † ν T ν (cid:17) − T ν Tr (cid:16) Y d Y † u Y u Y † d (cid:17) − T ν Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) − T ν Tr (cid:16) Y u Y † u Y u Y † u (cid:17) − T ν Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17) − Y ν (cid:16) g M + 45 g g M + 45 g g M + 375 g M + 20 (cid:16) g M + g M (cid:17) Tr (cid:16) Y u Y † u (cid:17) − (cid:16) g + g (cid:17) Tr (cid:16) Y † u T u (cid:17) + 75 Tr (cid:16) Y d Y † u T u Y † d (cid:17) + 25 Tr (cid:16) Y e Y † ν T ν Y † e (cid:17) + 75 Tr (cid:16) Y u Y † d T d Y † u (cid:17) + 450 Tr (cid:16) Y u Y † u T u Y † u (cid:17) + 25 Tr (cid:16) Y ν Y † e T e Y † ν (cid:17) + 150 Tr (cid:16) Y ν Y † ν T ν Y † ν (cid:17)(cid:17) (191) Bilinear Soft-Breaking Parameters ∆ β (1) B µ = 2 µ Tr (cid:16) Y † ν T ν (cid:17) + B µ Tr (cid:16) Y ν Y † ν (cid:17) (192) ∆ β (2) B µ = − B µ (cid:16) Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) + 3 Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17)(cid:17) − µ (cid:16) Tr (cid:16) Y ν Y † ν T ν Y † ν (cid:17) + Tr (cid:16) Y e Y † ν T ν Y † e (cid:17) + Tr (cid:16) Y ν Y † e T e Y † ν (cid:17)(cid:17) (193) β (1) B µX = β (2) B µX = 0 (194) β (1) B MR = 2 (cid:16) T ν Y † ν M R + Y ν Y † ν B M R (cid:17) (195) β (2) B MR = − (cid:16) − g T ν Y † ν M R − g T ν Y † ν M R + 5 Y ν Y † e Y e Y † ν B M R + 10 Y ν Y † e T e Y † ν M R + 5 Y ν Y † ν Y ν Y † ν B M R + 10 Y ν Y † ν T ν Y † ν M R + 10 T ν Y † e Y e Y † ν M R + 10 T ν Y † ν Y ν Y † ν M R + 30 T ν Y † ν M R Tr (cid:16) Y u Y † u (cid:17) + 10 T ν Y † ν M R Tr (cid:16) Y ν Y † ν (cid:17) + Y ν Y † ν B M R (cid:16) Tr (cid:16) Y u Y † u (cid:17) − (cid:16) g + g (cid:17) + 5 Tr (cid:16) Y ν Y † ν (cid:17)(cid:17) + 2 Y ν Y † ν M R (cid:16) g M + 15 Tr (cid:16) Y † u T u (cid:17) + 3 g M + 5 Tr (cid:16) Y † ν T ν (cid:17)(cid:17)(cid:17) (196) Soft-Breaking Scalar Masses
The RGEs of the soft SUSY breaking masses are usually written in terms of a set of traces (seee.g. [126]). In the model considered here, only one changes with respect to the MSSM: ∆ σ , = 120 1 √ g (cid:16) − m H u Tr (cid:16) Y ν Y † ν (cid:17) + 30 Tr (cid:16) Y ν m ∗ l Y † ν (cid:17)(cid:17) (197)34he resulting RGEs are: ∆ β (2) m q = − (cid:16) T † u T u + 2 Y † u m u Y u + 4 m H u Y † u Y u + m q Y † u Y u + Y † u Y u m q (cid:17) Tr (cid:16) Y ν Y † ν (cid:17) − (cid:16) T † u Y u Tr (cid:16) Y † ν T ν (cid:17) + Y † u T u Tr (cid:16) T ∗ ν Y Tν (cid:17) + Y † u Y u Tr (cid:16) T ∗ ν T Tν (cid:17) + Y † u Y u Tr (cid:16) m l Y † ν Y ν (cid:17) + Y † u Y u Tr (cid:16) m ν Y ν Y † ν (cid:17)(cid:17) (198) ∆ β (1) m l = 2 m H u Y † ν Y ν + 2 T † ν T ν + 2 Y † ν m ν Y ν + m l Y † ν Y ν + Y † ν Y ν m l (199) ∆ β (2) m l = − (cid:16) T † ν T ν + 2 Y † ν m ν Y ν + 4 m H u Y † ν Y ν + m l Y † ν Y ν + Y † ν Y ν m l (cid:17) Tr (cid:16) Y u Y † u (cid:17) − (cid:16) T † ν T ν + 2 Y † ν m ν Y ν + 4 m H u Y † ν Y ν + m l Y † ν Y ν + Y † ν Y ν m l (cid:17) Tr (cid:16) Y ν Y † ν (cid:17) − (cid:16) m H u Y † ν Y ν Y † ν Y ν + 2 Y † ν Y ν T † ν T ν + 2 Y † ν T ν T † ν Y ν + 2 T † ν Y ν Y † ν T ν + 2 T † ν T ν Y † ν Y ν + m l Y † ν Y ν Y † ν Y ν + 2 Y † ν m ν Y ν Y † ν Y ν + 2 Y † ν Y ν m l Y † ν Y ν + 2 Y † ν Y ν Y † ν m ν Y ν + Y † ν Y ν Y † ν Y ν m l + 3 T † ν Y ν Tr (cid:16) Y † u T u (cid:17) + T † ν Y ν Tr (cid:16) Y † ν T ν (cid:17) + 3 Y † ν T ν Tr (cid:16) T ∗ u Y Tu (cid:17) + 3 Y † ν Y ν Tr (cid:16) T ∗ u T Tu (cid:17) + Y † ν T ν Tr (cid:16) T ∗ ν Y Tν (cid:17) + Y † ν Y ν Tr (cid:16) T ∗ ν T Tν (cid:17) + Y † ν Y ν Tr (cid:16) m l Y † ν Y ν (cid:17) + 3 Y † ν Y ν Tr (cid:16) m q Y † u Y u (cid:17) + 3 Y † ν Y ν Tr (cid:16) m u Y u Y † u (cid:17) + Y † ν Y ν Tr (cid:16) m ν Y ν Y † ν (cid:17)(cid:17) (200) ∆ β (2) m Hd = − (cid:16)(cid:16) m H d + m H u (cid:17) Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) + Tr (cid:16) Y e Y † ν T ν T † e (cid:17) + Tr (cid:16) Y e T † ν T ν Y † e (cid:17) + Tr (cid:16) Y ν Y † e T e T † ν (cid:17) + Tr (cid:16) Y ν T † e T e Y † ν (cid:17) + Tr (cid:16) m e Y e Y † ν Y ν Y † e (cid:17) + Tr (cid:16) m l Y † e Y e Y † ν Y ν (cid:17) + Tr (cid:16) m l Y † ν Y ν Y † e Y e (cid:17) + Tr (cid:16) m ν Y ν Y † e Y e Y † ν (cid:17)(cid:17) (201) ∆ β (1) m Hu = 2 (cid:16) m H u Tr (cid:16) Y ν Y † ν (cid:17) + Tr (cid:16) T ∗ ν T Tν (cid:17) + Tr (cid:16) m l Y † ν Y ν (cid:17) + Tr (cid:16) m ν Y ν Y † ν (cid:17)(cid:17) (202) ∆ β (2) m Hu = − (cid:16)(cid:16) m H d + m H u (cid:17) Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) + Tr (cid:16) Y e Y † ν T ν T † e (cid:17) + Tr (cid:16) Y e T † ν T ν Y † e (cid:17) + Tr (cid:16) Y ν Y † e T e T † ν (cid:17) + 6 m H u Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17) + 6 Tr (cid:16) Y ν Y † ν T ν T † ν (cid:17) + Tr (cid:16) Y ν T † e T e Y † ν (cid:17) + 6 Tr (cid:16) Y ν T † ν T ν Y † ν (cid:17) + Tr (cid:16) m e Y e Y † ν Y ν Y † e (cid:17) + Tr (cid:16) m l Y † e Y e Y † ν Y ν (cid:17) + Tr (cid:16) m l Y † ν Y ν Y † e Y e (cid:17) + 6 Tr (cid:16) m l Y † ν Y ν Y † ν Y ν (cid:17) + Tr (cid:16) m ν Y ν Y † e Y e Y † ν (cid:17) + 6 Tr (cid:16) m ν Y ν Y † ν Y ν Y † ν (cid:17)(cid:17) (203) ∆ β (2) m u = − (cid:16)(cid:16) T u T † u + 2 Y u m q Y † u + 4 m H u Y u Y † u + m u Y u Y † u + Y u Y † u m u (cid:17) Tr (cid:16) Y ν Y † ν (cid:17) + 2 (cid:16) Y u T † u Tr (cid:16) Y † ν T ν (cid:17) + T u Y † u Tr (cid:16) T ∗ ν Y Tν (cid:17) + Y u Y † u Tr (cid:16) T ∗ ν T Tν (cid:17) + Y u Y † u Tr (cid:16) m l Y † ν Y ν (cid:17) + Y u Y † u Tr (cid:16) m ν Y ν Y † ν (cid:17)(cid:17)(cid:17) (204) ∆ β (2) m e = − (cid:16) (cid:16) m H d + m H u (cid:17) Y e Y † ν Y ν Y † e + 2 Y e Y † ν T ν T † e + 2 Y e T † ν T ν Y † e + 2 T e Y † ν Y ν T † e + 2 T e T † ν Y ν Y † e + m e Y e Y † ν Y ν Y † e + 2 Y e m l Y † ν Y ν Y † e + 2 Y e Y † ν m ν Y ν Y † e + 2 Y e Y † ν Y ν m l Y † e + Y e Y † ν Y ν Y † e m e (cid:17) (205)35 (1) m ν = 2 (cid:16) m H u Y ν Y † ν + 2 T ν T † ν + 2 Y ν m l Y † ν + m ν Y ν Y † ν + Y ν Y † ν m ν (cid:17) (206) β (2) m ν = − (cid:16) g M ∗ T ν Y † ν + 30 g M ∗ T ν Y † ν − g T ν T † ν − g T ν T † ν − g m ν Y ν Y † ν − g m ν Y ν Y † ν − g Y ν m l Y † ν − g Y ν m l Y † ν − g Y ν Y † ν m ν − g Y ν Y † ν m ν + 10 m H d Y ν Y † e Y e Y † ν + 10 m H u Y ν Y † e Y e Y † ν + 10 Y ν Y † e T e T † ν + 20 m H u Y ν Y † ν Y ν Y † ν + 10 Y ν Y † ν T ν T † ν + 10 Y ν T † e T e Y † ν + 10 Y ν T † ν T ν Y † ν + 10 T ν Y † e Y e T † ν + 10 T ν Y † ν Y ν T † ν + 10 T ν T † e Y e Y † ν + 10 T ν T † ν Y ν Y † ν + 5 m ν Y ν Y † e Y e Y † ν + 5 m ν Y ν Y † ν Y ν Y † ν + 10 Y ν m l Y † e Y e Y † ν + 10 Y ν m l Y † ν Y ν Y † ν + 10 Y ν Y † e m e Y e Y † ν + 10 Y ν Y † e Y e m l Y † ν + 5 Y ν Y † e Y e Y † ν m ν + 10 Y ν Y † ν m ν Y ν Y † ν + 10 Y ν Y † ν Y ν m l Y † ν + 5 Y ν Y † ν Y ν Y † ν m ν + 30 T ν T † ν Tr (cid:16) Y u Y † u (cid:17) + 15 m ν Y ν Y † ν Tr (cid:16) Y u Y † u (cid:17) + 30 Y ν m l Y † ν Tr (cid:16) Y u Y † u (cid:17) + 15 Y ν Y † ν m ν Tr (cid:16) Y u Y † u (cid:17) + 10 T ν T † ν Tr (cid:16) Y ν Y † ν (cid:17) + 5 m ν Y ν Y † ν Tr (cid:16) Y ν Y † ν (cid:17) + 10 Y ν m l Y † ν Tr (cid:16) Y ν Y † ν (cid:17) + 5 Y ν Y † ν m ν Tr (cid:16) Y ν Y † ν (cid:17) + 2 Y ν T † ν (cid:16) g M + 15 Tr (cid:16) Y † u T u (cid:17) + 3 g M + 5 Tr (cid:16) Y † ν T ν (cid:17)(cid:17) + 30 T ν Y † ν Tr (cid:16) T ∗ u Y Tu (cid:17) + 10 T ν Y † ν Tr (cid:16) T ∗ ν Y Tν (cid:17)(cid:17) + 45 Y ν Y † ν (cid:16) g m H u + 15 g m H u + 6 g | M | + 30 g | M | − m H u Tr (cid:16) Y u Y † u (cid:17) − m H u Tr (cid:16) Y ν Y † ν (cid:17) − Tr (cid:16) T ∗ u T Tu (cid:17) − Tr (cid:16) T ∗ ν T Tν (cid:17) − Tr (cid:16) m l Y † ν Y ν (cid:17) − Tr (cid:16) m q Y † u Y u (cid:17) − Tr (cid:16) m u Y u Y † u (cid:17) − Tr (cid:16) m ν Y ν Y † ν (cid:17)(cid:17) (207) β (1) m X = β (2) m X = 0 (208) Vacuum expectation values ∆ β (2) v d = v d Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) (209) ∆ β (1) v u = − v u Tr (cid:16) Y ν Y † ν (cid:17) (210) ∆ β (2) v u = 3 v u Tr (cid:16) Y ν Y † ν Y ν Y † ν (cid:17) − (cid:16) g + g (cid:17) v u ξ Tr (cid:16) Y ν Y † ν (cid:17) + v u Tr (cid:16) Y e Y † ν Y ν Y † e (cid:17) (211) C Loop Integrals
The B -functions with vanishing external momenta and the arguments ( a, b ) are given by B = 1 − log (cid:18) bQ (cid:19) + 1 b − a (cid:104) a log (cid:16) ab (cid:17) (cid:105) , (212) B = −
12 + 12 log (cid:18) bQ (cid:19) − a − b ) (cid:104) a − b + 2 a log (cid:18) ba (cid:19) (cid:105) , (213)(214)36he C -functions with vanishing external momenta and the arguments ( a, b, c ) read C = − a − b )( a − c )( b − c ) × (cid:104) b ( c − a ) log (cid:18) ba (cid:19) + c ( a − b ) log (cid:16) ca (cid:17) (cid:105) (215) C = 18( a − b )( a − c )( b − c ) × (cid:104) ( c − a ) (cid:18) ( a − b )(2 log (cid:18) aQ (cid:19) − b − c ) − b log (cid:18) ba (cid:19)(cid:19) + 2 c ( b − a ) log (cid:16) ca (cid:17) (cid:105) (216) C = − a − b ) ( a − c )( b − c ) × (cid:104) c ( a − b ) log (cid:16) ca (cid:17) + b ( c − a ) (cid:18) ( b − a )( b − c ) − ( a ( b − c ) + bc ) log (cid:18) ba (cid:19)(cid:19) (cid:105) (217) C = − a − b )( a − c ) ( b − c ) × (cid:104) a ( b − c ) log (cid:18) ba (cid:19) (218) + c ( b − a ) (cid:18) ( a − c )( b − c ) + ( c ( a + b ) − ab ) log (cid:18) bc (cid:19)(cid:19) (cid:105) (219) C = − a − b ) ( a − c )( b − c ) × (cid:104) b ( a − c ) (cid:0) − (cid:0) a (cid:0) b − bc + 3 c (cid:1) + abc ( b − c ) + b c (cid:1) log (cid:18) ba (cid:19) − ( b − a )( b − c ) (cid:0) − ab + 5 ac + b − bc (cid:1) (cid:1) + 2 c ( a − b ) log (cid:16) ca (cid:17) (cid:105) (220) C = 16( a − b ) ( a − c ) ( b − c ) × (cid:104) ( a − b ) (cid:16) ( a − c )( b − c ) (cid:0) a (cid:0) b + c (cid:1) − bc ( b + c ) (cid:1) + c ( a − b )(3 ab − c ( a + 2 b )) log (cid:16) ca (cid:17) (cid:17) + b ( a − c ) ( a ( b − c ) + 2 bc ) log (cid:18) ba (cid:19) (cid:105) (221) C = 16( a − b )( a − c ) ( b − c ) × (cid:104) a ( b − c ) log (cid:18) ba (cid:19) + c ( a − b ) (cid:16) (cid:0) c (cid:0) a + ab + b (cid:1) + 3 a b − abc ( a + b ) (cid:1) log (cid:18) bc (cid:19) − ( a − c )( b − c ) (cid:0) − c ( a + b ) + 5 ab + c (cid:1) (cid:17)(cid:105) (222)In the case of external photons, often the same combinations of C -functions appear. If the argu-ments are ( a, b, b ) , these can be expressed as C + C + C = C + C + C = b ( x − (cid:0) ( x − x (2 x + 5) − − x log( x ) (cid:1) (223) C + C + C = b ( x − (cid:0) − x + 2 x log( x ) + 1 (cid:1) (224) C + 2 C − C = 2 C + 2 C − C = b ( x − (cid:0) − x ) x log( x ) + ( x − x (31 x −
26) + 7) (cid:1) (225) C + C = b ( x − (cid:0) x log( x ) + (4 − x ) x − (cid:1) (226)37nd for ( a, a, b ) we get C + C + C = − C = b ( x − ( x (( x − x + 3) + 6 x log( x ) + 2) (227) C + C + C = − C = − b ( x − (cid:0) x − x + 2 log( x ) + 3 (cid:1) (228)In the previous expressions we used x = a/b .For the photonic monopole operators we define special loop functions M SF F ( a, b ) = (( a − b ) (cid:0) a − ab + 7 b (cid:1) + 6 a (2 a − b ) log (cid:0) ba (cid:1) a − b ) (229) M F SS ( a, b ) = 6 a log (cid:0) ba (cid:1) + 11 a − a b + 9 ab − b a − b ) (230) M F SV ( a, b ) = √ a (cid:0) a + 3 a b + 6 a b log (cid:0) ba (cid:1) − ab + b (cid:1) b ( a − b ) (231) M F V S ( a, b ) = √ a (cid:0) a + 3 a b + 6 a b log (cid:0) ba (cid:1) − ab + b (cid:1) b ( a − b ) (232) M F V V ( a, b ) = 6 a ( a − b ) log (cid:0) ab (cid:1) − ( a − b ) (cid:0) a − ab + 5 b (cid:1) a − b ) (233)The necessary box functions with the arguments ( a, b, c, d ) read, in the limit of vanishingexternal momenta, D = − (cid:104) b log ba ( b − a )( b − c )( b − d ) + c log ca ( c − a )( c − b )( c − d )+ d log da ( d − a )( d − b )( d − c ) (cid:105) (234) D = − (cid:104) b log ba ( b − a )( b − c )( b − d ) + c log ca ( c − a )( c − b )( c − d )+ d log da ( d − a )( d − b )( d − c ) (cid:105) (235)In addition, we define I C D ( a, b, c, d ) = C ( a, b, c ) + dD ( a, b, c, d ) (236) D Photonic penguin contributions to LFV
In the following appendices we present our results for the form factors of the operators involvedin our computation, done in the mass basis. The flavor of the external fermions will be denotedwith Greek characters ( α , β , γ , δ ), whereas the mass eigenstates of the particles in the loops willbe denoted with Latin characters ( a , b , c , d ). A sum over repeated indices will be assumed.38 .1 Feynman diagrams ( a ) γ ˜ e c ˜ χ a ˜ e b ¯ (cid:96) β (cid:96) α ( a ) γ ˜ χ − c ν ia ˜ χ − b ¯ (cid:96) β (cid:96) α ( a ) γ ˜ χ − c ν Ra ˜ χ − b ¯ (cid:96) β (cid:96) α ( a ) γH − c ν a H − b ¯ (cid:96) β (cid:96) α ( a ) γH − c ν a W − ¯ (cid:96) β (cid:96) α ( a ) γW − ν a H − b ¯ (cid:96) β (cid:96) α ( a ) γW − ν a W − ¯ (cid:96) β (cid:96) α We give in the following the contribution of each diagram to the different operators. We indicatethe diagram by the corresponding index ( a i ) with i = 1 , . . . , .39 .2 Neutralino contributions C i = C i ( m χ a , m e c , m e b ) (237) A L ( a ) =2 F ˜ ec,b ( ¯ N Lα,a,b N Ra,β,c ( C + C + C ) m (cid:96) α + ¯ N Rα,a,b ( N La,β,c ( C + C + C ) m (cid:96) β − N Ra,β,c ( C + C + C ) m ˜ χ a )) (238) A L ( a ) = − ¯ N Rα,a,b N La,β,c F ˜ ec,b M F SS ( m χ a , m e b ) (239) D.3 Chargino contributions C i = C i ( m χ − c , m χ − b , m ν ia ) (240) A L ( a ) = − (cid:16) ¯ X Lα,b,a X Rc,β,a ( F c,Lb,c ) ∗ C m (cid:96) α − ¯ X Rα,b,a ( X Lc,β,a ( F c,Rb,c ) ∗ ( C + C + C ) m (cid:96) β + X Rc,β,a (( F c,Lb,c ) ∗ C m ˜ χ − b − ( F c,Rb,c ) ∗ ( C + C + C ) m ˜ χ − c )) (cid:17) (241) A L ( a ) = − ¯ X Rα,b,a X Lc,β,a ( F c,Rb,c ) ∗ M SF F ( m ν ia , m χ − b ) (242) C i = C i ( m χ − c , m χ − b , m ν Ra ) (243) A L ( a ) = − (cid:16) ˆ¯ X Lα,b,a ˆ X Rc,β,a ( F c,Lb,c ) ∗ C m (cid:96) α − ˆ¯ X Rα,b,a ( ˆ X Lc,β,a ( F c,Rb,c ) ∗ ( C + C + C ) m (cid:96) β + ˆ X Rc,β,a (( F c,Lb,c ) ∗ C m ˜ χ − b − ( F c,Rb,c ) ∗ ( C + C + C ) m ˜ χ − c )) (cid:17) (244) A L ( a ) = − ˆ¯ X Rα,b,a ˆ X Lc,β,a ( F c,Rb,c ) ∗ M SF F ( m ν Ra , m χ − b ) (245) D.4 W + and H + contributions C i = C i ( m ν a , m H − c , m H − b ) (246) A L ( a ) =2 F hc,b ( ¯ V + ,Lα,a,b V + ,Ra,β,c ( C + C + C ) m (cid:96) α + ¯ V + ,Rα,a,b ( V + ,La,β,c ( C + C + C ) m (cid:96) β − V + ,Ra,β,c ( C + C + C ) m ν a )) (247) A L ( a ) = − ¯ V + ,Rα,a,b V + ,La,β,c F hc,b M F SS ( m ν a , m H − b ) (248) A L ( a ) =2 ( ˆ¯ V + ,Lα,a ) ∗ V + ,Ra,β,c F hwc C ( m ν a , m H − c , m W − ) (249) A L ( a ) =( ˆ¯ V + ,Rα,a ) ∗ V + ,La,β,c F hwc M F V S ( m ν a , m W − ) (250) A L ( a ) =2 ¯ V + ,Rα,a,b ( ˆ V + ,Ra,β ) ∗ ¯ F hwb C ( m ν a , m W − , m H − b ) (251) A L ( a ) = ¯ V + ,Lα,a,b ( ˆ V + ,La,β ) ∗ ¯ F hwb M F SV ( m ν a , m H − b ) (252) C i = C i ( m ν a , m W − , m W − ) (253) A L ( a ) = − F w (cid:16) ( ˆ¯ V + ,Rα,a ) ∗ ( ˆ V + ,Ra,β ) ∗ (2 C − C + 2 C ) m (cid:96) α ( ˆ¯ V + ,Lα,a ) ∗ (( ˆ V + ,La,β ) ∗ (2 C + 2 C − C ) m (cid:96) β + 3( ˆ V + ,Ra,β ) ∗ ( C + C ) m ν a ) (cid:17) (254) A L ( a ) =( ˆ¯ V + ,Rα,a ) ∗ ( ˆ V + ,Ra,β ) ∗ F w M F V V ( m ν a , m W − ) (255)These coefficients are related to the ones used in the calculation of the flavor observables by K L = 1 e (cid:88) p A L ( a p ) (256) K L = − e m (cid:96) α (cid:88) p A L ( a p ) (257) E Z and Higgs penguin contributions to LFV E.1 Feynman diagrams
In the following B = Z, h p , A p is used. Neutralino diagrams
Self energy corrections ( n ) B ˜ χ a ˜ e b ¯ (cid:96) c ¯ (cid:96) β (cid:96) α ( n ) B ˜ e a ˜ χ b (cid:96) c ¯ (cid:96) β (cid:96) α Vertex corrections ( n ) B ˜ e c ˜ χ a ˜ e b ¯ (cid:96) β (cid:96) α ( n ) B ˜ χ c ˜ e ∗ a ˜ χ b ¯ (cid:96) β (cid:96) α hargino diagrams Self energy corrections ( c ) Bν ia ˜ χ − b ¯ (cid:96) c ¯ (cid:96) β (cid:96) α ( c ) Bν Ra ˜ χ − b ¯ (cid:96) c ¯ (cid:96) β (cid:96) α ( c ) B ˜ χ − a ν ib (cid:96) c ¯ (cid:96) β (cid:96) α ( c ) B ˜ χ − a ν Rb (cid:96) c ¯ (cid:96) β (cid:96) α Vertex corrections ( c ) B ˜ χ − c ν ia ˜ χ − b ¯ (cid:96) β (cid:96) α ( c ) B ˜ χ − c ν Ra ˜ χ − b ¯ (cid:96) β (cid:96) α ( c ) Bν ic ˜ χ + a ν Rb ¯ (cid:96) β (cid:96) α ( c ) Bν Rc ˜ χ + a ν ib ¯ (cid:96) β (cid:96) α c ) Bν Rc ˜ χ + a ν Rb ¯ (cid:96) β (cid:96) α ( c ) Bν ic ˜ χ + a ν ib ¯ (cid:96) β (cid:96) α W + and H + diagrams Self energy corrections ( w ) Bν a H − b ¯ (cid:96) c ¯ (cid:96) β (cid:96) α ( w ) Bν a W − ¯ (cid:96) c ¯ (cid:96) β (cid:96) α ( w ) BH − a ν b (cid:96) c ¯ (cid:96) β (cid:96) α ( w ) BW − ν b (cid:96) c ¯ (cid:96) β (cid:96) α ertex corrections ( w ) BH − c ν a H − b ¯ (cid:96) β (cid:96) α ( w ) BH − c ν a W − ¯ (cid:96) β (cid:96) α ( w ) BW − ν a H − b ¯ (cid:96) β (cid:96) α ( w ) BW − ν a W − ¯ (cid:96) β (cid:96) α ( w ) Bν c H + a ν b ¯ (cid:96) β (cid:96) α ( w ) Bν c W + ν b ¯ (cid:96) β (cid:96) α E.2 Neutralino contributions
E.2.1 Z-penguinsSelf-energy corrections I = B ( m χ a , m e b ) (258) I = B ( m χ a , m e b ) (259) V LLZ, ( n ,n ) =( E Lβ,c ( N La,α,b ¯ N Rc,a,b I m (cid:96) α − N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a + N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c − N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (260) V LRZ, ( n ,n ) =( E Lβ,c ( N La,α,b ¯ N Rc,a,b I m (cid:96) α − N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a + N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c − N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (261)44 ertex corrections V LLZ, ( n ) = − N La,α,b ¯ N Rβ,a,c ˜ E b,c C ( m χ a , m e c , m e b ) (262) V LRZ, ( n ) = − N La,α,b ¯ N Rβ,a,c ˜ E b,c C ( m χ a , m e c , m e b ) (263) I = B ( m χ b , m χ c ) (264) I = C ( m χ c , m χ b , m e a ) (265) I = C ( m χ c , m χ b , m e a ) (266) V LLZ, ( n ) = N Lb,α,a ¯ N Rβ,c,a ( − ( M Lc,b I m ˜ χ b m ˜ χ c ) + M Rc,b ( I − I + I m e a )) (267) V LRZ, ( n ) = N Lb,α,a ¯ N Rβ,c,a ( − ( M Lc,b I m ˜ χ b m ˜ χ c ) + M Rc,b ( I − I + I m e a )) (268) E.2.2 Scalar penguinsCP even scalarsSelf-energy corrections I = B ( m χ a , m e b ) (269) I = B ( m χ a , m e b ) (270) S LLh p , ( n ,n ) =( H Lβ,c,p ( − ( N La,α,b ¯ N Rc,a,b I m (cid:96) α ) + N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a − N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c + N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (271) S LRh p , ( n ,n ) =( H Lβ,c,p ( − ( N La,α,b ¯ N Rc,a,b I m (cid:96) α ) + N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a − N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c + N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (272) Vertex corrections S LLh p , ( n ) = N La,α,b ¯ N Lβ,a,c ˜ H p,b,c C ( m χ a , m e c , m e b ) m ˜ χ a (273) S LRh p , ( n ) = N La,α,b ¯ N Lβ,a,c ˜ H p,b,c C ( m χ a , m e c , m e b ) m ˜ χ a (274) I = B ( m χ b , m χ c ) (275) I = C ( m χ c , m χ b , m e a ) (276) S LLh p , ( n ) = N Lb,α,a ¯ N Lβ,c,a ( S Lc,b,p I m ˜ χ b m ˜ χ c + S Rc,b,p ( I + I m e a )) (277) S LRh p , ( n ) = N Lb,α,a ¯ N Lβ,c,a ( S Lc,b,p I m ˜ χ b m ˜ χ c + S Rc,b,p ( I + I m e a )) (278) CP odd scalarsSelf-energy corrections I = B ( m χ a , m e b ) (279) I = B ( m χ a , m e b ) (280)45 LLA p , ( n ,n ) =( A Lβ,c,p ( − ( N La,α,b ¯ N Rc,a,b I m (cid:96) α ) + N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a − N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c + N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (281) S LRA p , ( n ,n ) =( A Lβ,c,p ( − ( N La,α,b ¯ N Rc,a,b I m (cid:96) α ) + N Ra,α,b ¯ N Rc,a,b I m (cid:96) α m ˜ χ a − N Ra,α,b ¯ N Lc,a,b I m (cid:96) α m (cid:96) c + N La,α,b ¯ N Lc,a,b I m ˜ χ a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (282) Vertex corrections S LLA p , ( n ) = N La,α,b ¯ N Lβ,a,c ˜ A p,b,c C ( m χ a , m e c , m e b ) m ˜ χ a (283) S LRA p , ( n ) = N La,α,b ¯ N Lβ,a,c ˜ A p,b,c C ( m χ a , m e c , m e b ) m ˜ χ a (284) I = B ( m χ b , m χ c ) (285) I = C ( m χ c , m χ b , m e a ) (286) S LLA p , ( n ) = N Lb,α,a ¯ N Lβ,c,a ( P Lc,b,p I m ˜ χ b m ˜ χ c + P Rc,b,p ( I + I m e a )) (287) S LRA p , ( n ) = N Lb,α,a ¯ N Lβ,c,a ( P Lc,b,p I m ˜ χ b m ˜ χ c + P Rc,b,p ( I + I m e a )) (288) E.3 Chargino contributions
E.3.1 Z-penguinsSelf-energy corrections I = B ( m χ − b , m ν ia ) (289) I = B ( m χ − b , m ν ia ) (290) V LLZ, ( c ,c ) =( E Lβ,c ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α − W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b + W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c − W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (291) V LRZ, ( c ,c ) =( E Lβ,c ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α − W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b + W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c − W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (292) I = B ( m χ − b , m ν Ra ) (293) I = B ( m χ − b , m ν Ra ) (294) V LLZ, ( c ,c ) =( E Lβ,c ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α − ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b + ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c − ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (295) V LRZ, ( c ,c ) =( E Lβ,c ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α − ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b + ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c − ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (296) Vertex corrections I = B ( m χ − b , m χ − c ) (297)46 = C ( m χ − c , m χ − b , m ν ia ) (298) I = C ( m χ − c , m χ − b , m ν ia ) (299) V LLZ, ( c ) = W Lb,α,a ¯ X Rβ,c,a ( − ( C Lc,b I m ˜ χ − b m ˜ χ − c ) + C Rc,b ( I − I + I m ν ia )) (300) V LRZ, ( c ) = W Lb,α,a ¯ X Rβ,c,a ( − ( C Lc,b I m ˜ χ − b m ˜ χ − c ) + C Rc,b ( I − I + I m ν ia )) (301) I = B ( m χ − b , m χ − c ) (302) I = C ( m χ − c , m χ − b , m ν Ra ) (303) I = C ( m χ − c , m χ − b , m ν Ra ) (304) V LLZ, ( c ) = ˆ X Lb,α,a ˆ¯ X Rβ,c,a ( − ( C Lc,b I m ˜ χ − b m ˜ χ − c ) + C Rc,b ( I − I + I m ν Ra )) (305) V LRZ, ( c ) = ˆ X Lb,α,a ˆ¯ X Rβ,c,a ( − ( C Lc,b I m ˜ χ − b m ˜ χ − c ) + C Rc,b ( I − I + I m ν Ra )) (306) V LLZ, ( c ) = − X La,α,b ¯ X Rβ,a,c ˜ V c,b C ( m χ − a , m ν ic , m ν Rb ) (307) V LRZ, ( c ) = − X La,α,b ¯ X Rβ,a,c ˜ V c,b C ( m χ − a , m ν ic , m ν Rb ) (308) V LLZ, ( c ) = 2 W La,α,b ˆ¯ X Rβ,a,c ˜ V b,c C ( m χ − a , m ν Rc , m ν ib ) (309) V LRZ, ( c ) = 2 W La,α,b ˆ¯ X Rβ,a,c ˜ V b,c C ( m χ − a , m ν Rc , m ν ib ) (310) E.3.2 Scalar penguinsCP even scalarsSelf-energy corrections I = B ( m χ − b , m ν ia ) (311) I = B ( m χ − b , m ν ia ) (312) S LLh p , ( c ,c ) =( H Lβ,c,p ( − ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α ) + W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (313) S LRh p , ( c ,c ) =( H Lβ,c,p ( − ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α ) + W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) I = B ( m χ − b , m ν Ra ) (314) I = B ( m χ − b , m ν Ra ) (315) S LLh p , ( c ,c ) =( H Lβ,c,p ( − ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α ) + ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (316) S LRh p , ( c ,c ) =( H Lβ,c,p ( − ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α ) + ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (317) Vertex corrections I = B ( m χ − b , m χ − c ) (318) I = C ( m χ − c , m χ − b , m ν ia ) (319) S LLh p , ( c ) = W Lb,α,a ¯ X Lβ,c,a ( S c,Lc,b,p I m ˜ χ − b m ˜ χ − c + S c,Rc,b,p ( I + I m ν ia )) (320) S LRh p , ( c ) = W Lb,α,a ¯ X Lβ,c,a ( S c,Lc,b,p I m ˜ χ − b m ˜ χ − c + S c,Rc,b,p ( I + I m ν ia )) (321) I = B ( m χ − b , m χ − c ) (322) I = C ( m χ − c , m χ − b , m ν Ra ) (323) S LLh p , ( c ) = ˆ X Lb,α,a ˆ¯ X Lβ,c,a ( S c,Lc,b,p I m ˜ χ − b m ˜ χ − c + S c,Rc,b,p ( I + I m ν Ra )) (324) S LRh p , ( c ) = ˆ X Lb,α,a ˆ¯ X Lβ,c,a ( S c,Lc,b,p I m ˜ χ − b m ˜ χ − c + S c,Rc,b,p ( I + I m ν Ra )) (325) S LLh p , ( c ) = W La,α,b ¯ X Lβ,a,c ˜ H iip,c,b C ( m χ − a , m ν ic , m ν ib ) m ˜ χ − a (326) S LRh p , ( c ) = W La,α,b ¯ X Lβ,a,c ˜ H iip,c,b C ( m χ − a , m ν ic , m ν ib ) m ˜ χ − a (327) S LLh p , ( c ) = ˆ X La,α,b ¯ X Lβ,a,c ˜ H irp,c,b C ( m χ − a , m ν ic , m ν Rb ) m ˜ χ − a (328) S LRh p , ( c ) = ˆ X La,α,b ¯ X Lβ,a,c ˜ H irp,c,b C ( m χ − a , m ν ic , m ν Rb ) m ˜ χ − a (329) S LLh p , ( c ) = W La,α,b ˆ¯ X Lβ,a,c ˜ H irp,b,c C ( m χ − a , m ν Rc , m ν ib ) m ˜ χ − a (330) S LRh p , ( c ) = W La,α,b ˆ¯ X Lβ,a,c ˜ H irp,b,c C ( m χ − a , m ν Rc , m ν ib ) m ˜ χ − a (331) S LLh p , ( c ) = ˆ X La,α,b ˆ¯ X Lβ,a,c ˜ H rrp,c,b C ( m χ − a , m ν Rc , m ν Rb ) m ˜ χ − a (332) S LRh p , ( c ) = ˆ X La,α,b ˆ¯ X Lβ,a,c ˜ H rrp,c,b C ( m χ − a , m ν Rc , m ν Rb ) m ˜ χ − a (333)(334) CP odd scalarsSelf-energy corrections I = B ( m χ − b , m ν ia ) (335) I = B ( m χ − b , m ν ia ) (336) S LLA p , ( c ,c ) =( A Lβ,c,p ( − ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α ) + W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (337)48 LRA p , ( c ,c ) =( A Lβ,c,p ( − ( W Lb,α,a ¯ X Rc,b,a I m (cid:96) α ) + W Rb,α,a ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − W Rb,α,a ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + W Lb,α,a ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) I = B ( m χ − b , m ν Ra ) (338) I = B ( m χ − b , m ν Ra ) (339) S LLA p , ( c ,c ) =( A Lβ,c,p ( − ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α ) + ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (340) S LRA p , ( c ,c ) =( A Lβ,c,p ( − ( ˆ X Lb,α,a ˆ¯ X Rc,b,a I m (cid:96) α ) + ˆ X Rb,α,a ˆ¯ X Rc,b,a I m (cid:96) α m ˜ χ − b − ˆ X Rb,α,a ˆ¯ X Lc,b,a I m (cid:96) α m (cid:96) c + ˆ X Lb,α,a ˆ¯ X Lc,b,a I m ˜ χ − b m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (341) Vertex corrections I = B ( m χ − b , m χ − c ) (342) I = C ( m χ − c , m χ − b , m ν ia ) (343) S LLA p , ( c ) = W Lb,α,a ¯ X Lβ,c,a ( P c,Lc,b,p I m ˜ χ − b m ˜ χ − c + P c,Rc,b,p ( I + I m ν ia )) (344) S LRA p , ( c ) = W Lb,α,a ¯ X Lβ,c,a ( P c,Lc,b,p I m ˜ χ − b m ˜ χ − c + P c,Rc,b,p ( I + I m ν ia )) (345) I = B ( m χ − b , m χ − c ) (346) I = C ( m χ − c , m χ − b , m ν Ra ) (347) S LLA p , ( c ) = ˆ X Lb,α,a ˆ¯ X Lβ,c,a ( P c,Lc,b,p I m ˜ χ − b m ˜ χ − c + P c,Rc,b,p ( I + I m ν Ra )) (348) S LRA p , ( c ) = ˆ X Lb,α,a ˆ¯ X Lβ,c,a ( P c,Lc,b,p I m ˜ χ − b m ˜ χ − c + P c,Rc,b,p ( I + I m ν Ra )) (349) S LLA p , ( c ) = W La,α,b ¯ X Lβ,a,c ˜ A iip,c,b C ( m χ − a , m ν ic , m ν ib ) m ˜ χ − a (350) S LRA p , ( c ) = W La,α,b ¯ X Lβ,a,c ˜ A iip,c,b C ( m χ − a , m ν ic , m ν ib ) m ˜ χ − a (351) S LLA p , ( c ) = ˆ X La,α,b ¯ X Lβ,a,c ˜ A irp,c,b C ( m χ − a , m ν ic , m ν Rb ) m ˜ χ − a (352) S LRA p , ( c ) = ˆ X La,α,b ¯ X Lβ,a,c ˜ A irp,c,b C ( m χ − a , m ν ic , m ν Rb ) m ˜ χ − a (353) S LLA p , ( c ) = W La,α,b ˆ¯ X Lβ,a,c ˜ A irp,b,c C ( m χ − a , m ν Rc , m ν ib ) m ˜ χ − a (354) S LRA p , ( c ) = W La,α,b ˆ¯ X Lβ,a,c ˜ A irp,b,c C ( m χ − a , m ν Rc , m ν ib ) m ˜ χ − a (355) S LLA p , ( c ) = ˆ X La,α,b ˆ¯ X Lβ,a,c ˜ A rrp,c,b C ( m χ − a , m ν Rc , m ν Rb ) m ˜ χ − a (356) S LRA p , ( c ) = ˆ X La,α,b ˆ¯ X Lβ,a,c ˜ A rrp,c,b C ( m χ − a , m ν Rc , m ν Rb ) m ˜ χ − a (357)49 .4 W + and H + contributions E.4.1 Z-penguinsSelf-energy corrections I = B ( m ν a , m H − b ) (358) I = B ( m ν a , m H − b ) (359) V LLZ, ( w ,w ) =( E Lβ,c ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α − V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a + V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c − V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (360) V LRZ, ( w ,w ) =( E Lβ,c ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α − V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a + V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c − V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) I = B ( m ν a , m W − ) (361) I = B ( m ν a , m W − ) (362) V LLZ, ( w ,w ) =( E Lβ,c ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c )+ ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (363) V LRZ, ( w ,w ) =( E Lβ,c ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c )+ ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (364) Vertex corrections V LLZ, ( w ) = − V + ,La,α,b ¯ V + ,Rβ,a,c Z hhb,c C ( m ν a , m H − c , m H − b ) (365) V LRZ, ( w ) = − V + ,La,α,b ¯ V + ,Rβ,a,c Z hhb,c C ( m ν a , m H − c , m H − b ) (366) V LLZ, ( w ) = ˆ V + ,La,i ¯ V + ,Rβ,a,c ¯ Z hwc C ( m ν a , m H − c , m W − ) m ν a (367) V LRZ, ( w ) = ˆ V + ,La,i ¯ V + ,Rβ,a,c ¯ Z hwc C ( m ν a , m H − c , m W − ) m ν a (368) V LLZ, ( w ) = V + ,La,α,b ˆ¯ V + ,Lβ,a Z hwb C ( m ν a , m W − , m H − b ) m ν a (369) V LRZ, ( w ) = V + ,La,α,b ˆ¯ V + ,Lβ,a Z hwb C ( m ν a , m W − , m H − b ) m ν a (370) I = B ( m W − , m W − ) (371) I = C ( m ν a , m W − , m W − ) (372) I = C ( m ν a , m W − , m W − ) (373) V LLZ, ( w ) = − ˆ V + ,La,i ˆ¯ V + ,Lβ,a Z ww ( − I + 2 I + I m ν a )) (374) V LRZ, ( w ) = − ˆ V + ,La,i ˆ¯ V + ,Lβ,a Z ww ( − I + 2 I + I m ν a )) (375)50 = B ( m ν b , m ν c ) (376) I = C ( m ν c , m ν b , m H − a ) (377) I = C ( m ν c , m ν b , m H − a ) (378) V LLZ, ( w ) = V + ,Lb,α,a ¯ V + ,Rβ,c,a ( − ( V Lc,b I m ν b m ν c ) + V Rc,b ( I − I + I m H − a )) (379) V LRZ, ( w ) = V + ,Lb,α,a ¯ V + ,Rβ,c,a ( − ( V Lc,b I m ν b m ν c ) + V Rc,b ( I − I + I m H − a )) (380) I = B ( m ν b , m ν c ) (381) I = C ( m ν c , m ν b , m W − ) (382) I = C ( m ν c , m ν b , m W − ) (383) V LLZ, ( w ) = − ( ˆ V + ,Lb,i ˆ¯ V + ,Lβ,c (2 V Rc,b I m ν b m ν c + V Lc,b (1 − I − I + I m W − )))) (384) V LRZ, ( w ) = − ( ˆ V + ,Lb,i ˆ¯ V + ,Lβ,c (2 V Rc,b I m ν b m ν c + V Lc,b (1 − I − I + I m W − )))) (385) E.4.2 Scalar penguinsCP even scalarsSelf-energy corrections I = B ( m ν a , m H − b ) (386) I = B ( m ν a , m H − b ) (387) S LLh p , ( w ,w ) =( H Lβ,c,p ( − ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α ) + V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a − V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c + V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (388) S LRh p , ( w ,w ) =( H Lβ,c,p ( − ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α ) + V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a − V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c + V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) I = B ( m ν a , m W − ) (389) I = B ( m ν a , m W − ) (390) S LLh p , ( w ,w ) = − (( H Lβ,c,p ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c ) + ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c )) + ( α ↔ β ) (391) S LRh p , ( w ,w ) = − (( H Lβ,c,p ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c ) + ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c )) + ( α ↔ β ) (392) Vertex corrections S LLh p , ( w ) = V + ,La,α,b ¯ V + ,Lβ,a,c H hhp,b,c C ( m ν a , m H − c , m H − b ) m ν a (393) S LRh p , ( w ) = V + ,La,α,b ¯ V + ,Lβ,a,c H hhp,b,c C ( m ν a , m H − c , m H − b ) m ν a (394)51 = B ( m H − c , m W − ) (395) I = C ( m ν a , m H − c , m W − ) (396) S LLh p , ( w ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,c ¯ H hwp,c ( I + I m ν a ) (397) S LRh p , ( w ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,c ¯ H hwp,c ( I + I m ν a ) (398) I = B ( m H − b , m W − ) (399) I = C ( m ν a , m W − , m H − b ) (400) S LLh p , ( w ) = V + ,La,α,b ˆ¯ V + ,Rβ,a H hwp,b ( I + I m ν a ) (401) S LRh p , ( w ) = V + ,La,α,b ˆ¯ V + ,Rβ,a H hwp,b ( I + I m ν a ) (402) S LLh p , ( w ) = 4 ˆ V + ,La,i ˆ¯ V + ,Rβ,a H wwp C ( m ν a , m W − , m W − ) m ν a (403) S LRh p , ( w ) = 4 ˆ V + ,La,i ˆ¯ V + ,Rβ,a H wwp C ( m ν a , m W − , m W − ) m ν a (404) I = B ( m ν b , m ν c ) (405) I = C ( m ν c , m ν b , m H − a ) (406) S LLh p , ( w ) = V + ,Lb,α,a ¯ V + ,Lβ,c,a ( H ν,Lc,b,p I m ν b m ν c + H ν,Rc,b,p ( I + I m H − a )) (407) S LRh p , ( w ) = V + ,Lb,α,a ¯ V + ,Lβ,c,a ( H ν,Lc,b,p I m ν b m ν c + H ν,Rc,b,p ( I + I m H − a )) (408) I = B ( m ν b , m ν c ) (409) I = C ( m ν c , m ν b , m W − ) (410) S LLh p , ( w ) = 2 ˆ V + ,Lb,i ˆ¯ V + ,Rβ,c ( − H ν,Rc,b,p I m ν b m ν c + H ν,Lc,b,p (1 − I + I m W − ))) (411) S LRh p , ( w ) = 2 ˆ V + ,Lb,i ˆ¯ V + ,Rβ,c ( − H ν,Rc,b,p I m ν b m ν c + H ν,Lc,b,p (1 − I + I m W − ))) (412)(413) CP odd scalarsSelf-energy corrections I = B ( m ν a , m H − b ) (414) I = B ( m ν a , m H − b ) (415) S LLA p , ( w ,w ) =( A Lβ,c,p ( − ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α ) + V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a − V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c + V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (416) S LRA p , ( w ,w ) =( A Lβ,c,p ( − ( V + ,La,α,b ¯ V + ,Rc,a,b I m (cid:96) α ) + V + ,Ra,α,b ¯ V + ,Rc,a,b I m (cid:96) α m ν a − V + ,Ra,α,b ¯ V + ,Lc,a,b I m (cid:96) α m (cid:96) c + V + ,La,α,b ¯ V + ,Lc,a,b I m ν a m (cid:96) c )) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) I = B ( m ν a , m W − ) (417)52 = B ( m ν a , m W − ) (418) S LLA p , ( w ,w ) = − ( A Lβ,c,p ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c ) + ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (419) S LRA p , ( w ,w ) = − ( A Lβ,c,p ( ˆ V + ,Ra,i m (cid:96) α ( − V + ,Lc,a (1 − I ) m ν a + ˆ¯ V + ,Rc,a (1 + 2 I ) m (cid:96) c ) + ˆ V + ,La,i ( ˆ¯ V + ,Lc,a (1 + 2 I ) m (cid:96) α − V + ,Rc,a (1 − I ) m ν a m (cid:96) c ))) / ( m (cid:96) α − m (cid:96) c ) + ( α ↔ β ) (420) Vertex corrections S LLA p , ( w ) = V + ,La,α,b ¯ V + ,Lβ,a,c A hhp,b,c C ( m ν a , m H − c , m H − b ) m ν a (421) S LRA p , ( w ) = V + ,La,α,b ¯ V + ,Lβ,a,c A hhp,b,c C ( m ν a , m H − c , m H − b ) m ν a (422) I = B ( m H − c , m W − ) (423) I = C ( m ν a , m H − c , m W − ) (424) S LLA p , ( w ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,c ¯ A hwp,c ( I + I m ν a ) (425) S LRA p , ( w ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,c ¯ A hwp,c ( I + I m ν a ) (426) I = B ( m H − b , m W − ) (427) I = C ( m ν a , m W − , m H − b ) (428) S LLA p , ( w ) = V + ,La,α,b ˆ¯ V + ,Rβ,a A hwp,b ( I + I m ν a ) (429) S LRA p , ( w ) = V + ,La,α,b ˆ¯ V + ,Rβ,a A hwp,b ( I + I m ν a ) (430) I = B ( m ν b , m ν c ) (431) I = C ( m ν c , m ν b , m H − a ) (432) S LLA p , ( w ) = V + ,Lb,α,a ¯ V + ,Lβ,c,a ( A ν,Lc,b,p I m ν b m ν c + A ν,Rc,b,p ( I + I m H − a )) (433) S LRA p , ( w ) = V + ,Lb,α,a ¯ V + ,Lβ,c,a ( A ν,Lc,b,p I m ν b m ν c + A ν,Rc,b,p ( I + I m H − a )) (434) I = B ( m ν b , m ν c ) (435) I = C ( m ν c , m ν b , m W − ) (436) S LLA p , ( w ) = 2 ˆ V + ,Lb,i ˆ¯ V + ,Rβ,c ( − A ν,Rc,b,p I m ν b m ν c + A ν,Lc,b,p (1 − I + I m W − ))) (437) S LRA p , ( w ) = 2 ˆ V + ,Lb,i ˆ¯ V + ,Rβ,c ( − A ν,Rc,b,p I m ν b m ν c + A ν,Lc,b,p (1 − I + I m W − ))) (438)(439)53 Box contributions to LFV
F.1 Four lepton boxes
F.1.1 Feynman diagrams
Neutralino diagrams ( n l ) ˜ χ a ˜ e ∗ d ˜ e ∗ b ˜ χ c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( n l ) ˜ χ a ˜ e ∗ d ˜ e b ˜ χ c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ Chargino diagrams ( c l ) ˜ χ − a ν id ν ib ˜ χ − c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν id ν Rb ˜ χ − c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν Rd ν ib ˜ χ − c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν Rd ν Rb ˜ χ − c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν id ν ib ˜ χ + c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν id ν Rb ˜ χ + c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ c l ) ˜ χ − a ν Rd ν ib ˜ χ + c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( c l ) ˜ χ − a ν Rd ν Rb ˜ χ + c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ W + and H + diagrams ( w l ) ν a H + d H + b ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a H + d W + ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a W + H + b ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a W + W + ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a H + d H − b ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a H + d W − ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a W + H − b ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ ( w l ) ν a W + W − ν c (cid:96) α ¯ (cid:96) β ¯ (cid:96) δ (cid:96) γ .1.2 Neutralino contributions S LL ( n l ) = − N La,α,d ¯ N Lβ,a,b N Lc,γ,b ¯ N Lδ,c,d m ˜ χ a m ˜ χ c D ( m χ a , m χ c , m e d , m e b ) (440) S LR ( n l ) = − N La,α,d ¯ N Lβ,a,b N Rc,γ,b ¯ N Rδ,c,d m ˜ χ a m ˜ χ c D ( m χ a , m χ c , m e d , m e b ) (441) V LL ( n l ) = − N La,α,d ¯ N Rβ,a,b N Lc,γ,b ¯ N Rδ,c,d D ( m χ a , m χ c , m e d , m e b ) (442) V LR ( n l ) = − N La,α,d ¯ N Rβ,a,b N Rc,γ,b ¯ N Lδ,c,d D ( m χ a , m χ c , m e d , m e b ) (443) S LL ( n l ) = 12 N La,α,d N La,γ,b ¯ N Lβ,c,b ¯ N Lδ,c,d m ˜ χ a m ˜ χ c D ( m χ a , m χ c , m e d , m e b ) (444) S LR ( n l ) = − N La,α,d N Ra,γ,b ¯ N Lβ,c,b ¯ N Rδ,c,d D ( m χ a , m χ c , m e d , m e b ) (445) V LL ( n l ) = − N La,α,d N La,γ,b ¯ N Rβ,c,b ¯ N Rδ,c,d m ˜ χ a m ˜ χ c D ( m χ a , m χ c , m e d , m e b ) (446) V LR ( n l ) = − N La,α,d N Ra,γ,b ¯ N Rβ,c,b ¯ N Lδ,c,d D ( m χ a , m χ c , m e d , m e b ) (447) T LL ( n l ) = 18 N La,α,d N La,γ,b ¯ N Lβ,c,b ¯ N Lδ,c,d m ˜ χ a m ˜ χ c D ( m χ a , m χ c , m e d , m e b ) (448) F.1.3 Chargino contributions S LL ( c l ) = − W La,α,d ¯ X Lβ,a,b W Lc,γ,b ¯ X Lδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν ib ) (449) S LR ( c l ) = − W La,α,d ¯ X Lβ,a,b W Rc,γ,b ¯ X Rδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν ib ) (450) V LL ( c l ) = − W La,α,d ¯ X Rβ,a,b W Lc,γ,b ¯ X Rδ,c,d D ( m χ − a , m χ − c , m ν id , m ν ib ) (451) V LR ( c l ) = − W La,α,d ¯ X Rβ,a,b W Rc,γ,b ¯ X Lδ,c,d D ( m χ − a , m χ − c , m ν id , m ν ib ) (452) S LL ( c l ) = − W La,α,d ˆ¯ X Lβ,a,b ˆ X Lc,γ,b ¯ X Lδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν Rb ) (453) S LR ( c l ) = − W La,α,d ˆ¯ X Lβ,a,b ˆ X Rc,γ,b ¯ X Rδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν Rb ) (454) V LL ( c l ) = − W La,α,d ˆ¯ X Rβ,a,b ˆ X Lc,γ,b ¯ X Rδ,c,d D ( m χ − a , m χ − c , m ν id , m ν Rb ) (455) V LR ( c l ) = − W La,α,d ˆ¯ X Rβ,a,b ˆ X Rc,γ,b ¯ X Lδ,c,d D ( m χ − a , m χ − c , m ν id , m ν Rb ) (456) S LL ( c l ) = − ˆ X La,α,d ¯ X Lβ,a,b W Lc,γ,b ˆ¯ X Lδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (457) S LR ( c l ) = − ˆ X La,α,d ¯ X Lβ,a,b W Rc,γ,b ˆ¯ X Rδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (458) V LL ( c l ) = − ˆ X La,α,d ¯ X Rβ,a,b W Lc,γ,b ˆ¯ X Rδ,c,d D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (459) V LR ( c l ) = − ˆ X La,α,d ¯ X Rβ,a,b W Rc,γ,b ˆ¯ X Lδ,c,d D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (460) S LL ( c l ) = − ˆ X La,α,d ˆ¯ X Lβ,a,b ˆ X Lc,γ,b ˆ¯ X Lδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (461) S LR ( c l ) = − ˆ X La,α,d ˆ¯ X Lβ,a,b ˆ X Rc,γ,b ˆ¯ X Rδ,c,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (462)56 LL ( c l ) = − ˆ X La,α,d ˆ¯ X Rβ,a,b ˆ X Lc,γ,b ˆ¯ X Rδ,c,d D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (463) V LR ( c l ) = − ˆ X La,α,d ˆ¯ X Rβ,a,b ˆ X Rc,γ,b ˆ¯ X Lδ,c,d D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (464) S LL ( c l ) = − W La,α,d ¯ X Lβ,a,b ¯ X Lδ,c,b W Lc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν ib ) (465) S LR ( c l ) = − W La,α,d ¯ X Lβ,a,b ¯ X Rδ,c,b W Rc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν ib ) (466) V LL ( c l ) = W La,α,d ¯ X Rβ,a,b ¯ X Rδ,c,b W Lc,γ,d D ( m χ − a , m χ − c , m ν id , m ν ib ) (467) V LR ( c l ) = W La,α,d ¯ X Rβ,a,b ¯ X Lδ,c,b W Rc,γ,d D ( m χ − a , m χ − c , m ν id , m ν ib ) (468) S LL ( c l ) = − W La,α,d ˆ¯ X Lβ,a,b ˆ¯ X Lδ,c,b W Lc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν Rb ) (469) S LR ( c l ) = − W La,α,d ˆ¯ X Lβ,a,b ˆ¯ X Rδ,c,b W Rc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν id , m ν Rb ) (470) V LL ( c l ) = W La,α,d ˆ¯ X Rβ,a,b ˆ¯ X Rδ,c,b W Lc,γ,d D ( m χ − a , m χ − c , m ν id , m ν Rb ) (471) V LR ( c l ) = W La,α,d ˆ¯ X Rβ,a,b ˆ¯ X Lδ,c,b W Rc,γ,d D ( m χ − a , m χ − c , m ν id , m ν Rb ) (472) S LL ( c l ) = − ˆ X La,α,d ¯ X Lβ,a,b ¯ X Lδ,c,b ˆ X Lc,γ,d m ˜ χ − a , m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (473) S LR ( c l ) = − ˆ X La,α,d ¯ X Lβ,a,b ¯ X Rδ,c,b ˆ X Rc,γ,d m ˜ χ − a , m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (474) V LL ( c l ) = ˆ X La,α,d ¯ X Rβ,a,b ¯ X Rδ,c,b ˆ X Lc,γ,d D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (475) V LR ( c l ) = ˆ X La,α,d ¯ X Rβ,a,b ¯ X Lδ,c,b ˆ X Rc,γ,d D ( m χ − a , m χ − c , m ν Rd , m ν ib ) (476) S LL ( c l ) = − ˆ X La,α,d ˆ¯ X Lβ,a,b ˆ¯ X Lδ,c,b ˆ X Lc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (477) S LR ( c l ) = − ˆ X La,α,d ˆ¯ X Lβ,a,b ˆ¯ X Rδ,c,b ˆ X Rc,γ,d m ˜ χ − a m ˜ χ − c D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (478) V LL ( c l ) = ˆ X La,α,d ˆ¯ X Rβ,a,b ˆ¯ X Rδ,c,b ˆ X Lc,γ,d D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (479) V LR ( c l ) = ˆ X La,α,d ˆ¯ X Rβ,a,b ˆ¯ X Lδ,c,b ˆ X Rc,γ,d D ( m χ − a , m χ − c , m ν Rd , m ν Rb ) (480) F.1.4 W + and H + contributions S LL ( w l ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V + ,Lc,γ,b ¯ V + ,Lδ,c,d m ν a m ν c D ( m ν a , m ν c , m H − d , m H − b ) (481) S LR ( w l ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V + ,Rc,γ,b ¯ V + ,Rδ,c,d m ν a m ν c D ( m ν a , m ν c , m H − d , m H − b ) (482) V LL ( w l ) = − V + ,La,α,d ¯ V + ,Rβ,a,b V + ,Lc,γ,b ¯ V + ,Rδ,c,d D ( m ν a , m ν c , m H − d , m H − b ) (483) V LR ( w l ) = − V + ,La,α,d ¯ V + ,Rβ,a,b V + ,Rc,γ,b ¯ V + ,Lδ,c,d D ( m ν a , m ν c , m H − d , m H − b ) (484) S LL ( w l ) = 2 V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V + ,Lc,γ ¯ V + ,Lδ,c,d ( I C D ( m ν c , m W − , m H − d , m ν a ) − D ( m ν a , m ν c , m W − , m H − d )) (485)57 LR ( w l ) = 2 V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V + ,Rc,γ ¯ V + ,Rδ,c,d ( I C D ( m ν c , m W − , m H − d , m ν a ) − D ( m ν a , m ν c , m W − , m H − d )) (486) V LL ( w l ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V + ,Lc,γ ¯ V + ,Rδ,c,d m ν a m ν c D ( m ν a , m ν c , m W − , m H − d ) (487) V LR ( w l ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V + ,Rc,γ ¯ V + ,Lδ,c,d m ν a m ν c D ( m ν a , m ν c , m W − , m H − d ) (488) S LL ( w l ) = 2 ˆ V + ,La,i ¯ V + ,Lβ,a,b V + ,Lc,γ,b ˆ¯ V + ,Rδ,c ( I C D ( m ν c , m H − b , m W − , m ν a ) − D ( m ν a , m ν c , m H − b , m W − )) (489) S LR ( w l ) = 2 ˆ V + ,La,i ¯ V + ,Lβ,a,b V + ,Rc,γ,b ˆ¯ V + ,Lδ,c ( I C D ( m ν c , m H − b , m W − , m ν a ) − D ( m ν a , m ν c , m H − b , m W − )) (490) V LL ( w l ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V + ,Lc,γ,b ˆ¯ V + ,Lδ,c m ν a m ν c D ( m ν a , m ν c , m H − b , m W − ) (491) V LR ( w l ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V + ,Rc,γ,b ˆ¯ V + ,Rδ,c m ν a m ν c D ( m ν a , m ν c , m H − b , m W − ) (492) T LL ( w l ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,b V + ,Lc,γ,b ˆ¯ V + ,Rδ,c D ( m ν a , m ν c , m H − b , m W − ) (493) S LL ( w l ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V + ,Lc,γ ˆ¯ V + ,Rδ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (494) S LR ( w l ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V + ,Rc,γ ˆ¯ V + ,Lδ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (495) V LL ( w l ) = − V + ,La,i ˆ¯ V + ,Lβ,a ˆ V + ,Lc,γ ˆ¯ V + ,Lδ,c ( I C D ( m ν c , m W − , m W − , m ν a ) − D ( m ν a , m ν c , m W − , m W − )) (496) V LR ( w l ) = − V + ,La,i ˆ¯ V + ,Lβ,a ˆ V + ,Rc,γ ˆ¯ V + ,Rδ,c I C D ( m ν c , m W − , m W − , m ν a ) (497) T LL ( w l ) = ˆ V + ,La,i ˆ¯ V + ,Rβ,a ˆ V + ,Lc,γ ˆ¯ V + ,Rδ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (498) S LL ( w l ) = 12 V + ,La,α,d ¯ V + ,Lδ,c,d V + ,La,γ,b ¯ V + ,Lβ,c,b m ν a m ν c D ( m ν a , m ν c , m H − d , m H − b ) (499) S LR ( w l ) = − V + ,La,α,d ¯ V + ,Rδ,c,d V + ,Ra,γ,b ¯ V + ,Lβ,c,b D ( m ν a , m ν c , m H − d , m H − b ) (500) V LL ( w l ) = − V + ,La,α,d ¯ V + ,Rδ,c,d V + ,La,γ,b ¯ V + ,Rβ,c,b m ν a m ν c D ( m ν a , m ν c , m H − d , m H − b ) (501) V LR ( w l ) = − V + ,La,α,d ¯ V + ,Lδ,c,d V + ,Ra,γ,b ¯ V + ,Rβ,c,b D ( m ν a , m ν c , m H − d , m H − b ) (502) T LL ( w l ) = 18 V + ,La,α,d ¯ V + ,Lδ,c,d V + ,La,γ,b ¯ V + ,Lβ,c,b m ν a m ν c D ( m ν a , m ν c , m H − d , m H − b ) (503) S LL ( w l ) = − V + ,La,α,d ¯ V + ,Lδ,c,d ˆ V + ,La,γ ˆ¯ V + ,Rβ,c ( I C D ( m ν c , m W − , m H − d , m ν a ) − D ( m ν a , m ν c , m W − , m H − d )) (504) S LR ( w l ) = 2 V + ,La,α,d ¯ V + ,Rδ,c,d ˆ V + ,Ra,γ ˆ¯ V + ,Rβ,c m ν a m ν c D ( m ν a , m ν c , m W − , m H − d ) (505) V LL ( w l ) = V + ,La,α,d ¯ V + ,Rδ,c,d ˆ V + ,La,γ ˆ¯ V + ,Lβ,c ( I C D ( m ν c , m W − , m H − d , m ν a ) − D ( m ν a , m ν c , m W − , m H − d )) (506) V LR ( w l ) l = V + ,La,α,d ¯ V + ,Lδ,c,d ˆ V + ,Ra,γ ˆ¯ V + ,Lβ,c m ν a m ν c D ( m ν a , m ν c , m W − , m H − d ) (507)58 LL ( w l ) = − V + ,La,α,d ¯ V + ,Lδ,c,d ˆ V + ,La,γ ˆ¯ V + ,Rβ,c I C D ( m ν c , m W − , m H − d , m ν a ) (508) S LL ( w l ) = − ˆ V + ,La,i ˆ¯ V + ,Rδ,c V + ,La,γ,b ¯ V + ,Lβ,c,b ( I C D ( m ν c , m H − b , m W − , m ν a ) − D ( m ν a , m ν c , m H − b , m W − )) (509) S LR ( w l ) = 2 ˆ V + ,Ra,i ˆ¯ V + ,Rδ,c V + ,La,γ,b ¯ V + ,Rβ,c,b m ν a m ν c D ( m ν a , m ν c , m H − b , m W − ) (510) V LL ( w l ) = ˆ V + ,La,i ˆ¯ V + ,Lδ,c V + ,La,γ,b ¯ V + ,Rβ,c,b ( I C D ( m ν c , m H − b , m W − , m ν a ) − D ( m ν a , m ν c , m H − b , m W − )) (511) V LR ( w l ) = ˆ V + ,La,i ˆ¯ V + ,Rδ,c V + ,Ra,γ,b ¯ V + ,Rβ,c,b m ν a m ν c D ( m ν a , m ν c , m H − b , m W − ) (512) T LL ( w l ) = −
14 ˆ V + ,La,i ˆ¯ V + ,Rδ,c V + ,La,γ,b ¯ V + ,Lβ,c,b I C D ( m ν c , m H − b , m W − , m ν a ) (513) S LL ( w l ) = − V + ,La,i ˆ¯ V + ,Rδ,c ˆ V + ,La,γ ˆ¯ V + ,Rβ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (514) S LR ( w l ) = − V + ,La,i ˆ¯ V + ,Lδ,c ˆ V + ,Ra,γ ˆ¯ V + ,Rβ,c ( I C D ( m ν c , m W − , m W − , m ν a ) − D ( m ν a , m ν c , m W − , m W − )) (515) V LL ( w l ) = − V + ,La,i ˆ¯ V + ,Lδ,c ˆ V + ,La,γ ˆ¯ V + ,Lβ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (516) V LR ( w l ) = − V + ,La,i ˆ¯ V + ,Rδ,c ˆ V + ,Ra,γ ˆ¯ V + ,Lβ,c I C D ( m ν c , m W − , m W − , m ν a ) (517) T LL ( w l ) = ˆ V + ,La,i ˆ¯ V + ,Rδ,c ˆ V + ,La,γ ˆ¯ V + ,Rβ,c m ν a m ν c D ( m ν a , m ν c , m W − , m W − ) (518) F.2 Additional boxes for (cid:96) − α → (cid:96) − β (cid:96) + γ (cid:96) − γ In the case of (cid:96) − α → (cid:96) − β (cid:96) + γ (cid:96) − γ it is necessary to calculate the crossed diagrams with exchanged indices β ↔ γ explicitly. F.2.1 Crossed neutralino contributions S LL ( n l (cid:48) ) = 12 N Ld,α,a ¯ N Lβ,b,a N Lb,γ,c ¯ N Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m e c ) (519) S LR ( n l (cid:48) ) = 2 N Ld,α,a ¯ N Lβ,b,a N Rb,γ,c ¯ N Rδ,d,c D ( m χ d , m χ b , m e a , m e c ) (520) V LL ( n l (cid:48) ) = − N Ld,α,a ¯ N Rβ,b,a N Lb,γ,c ¯ N Rδ,d,c D ( m χ d , m χ b , m e a , m e c ) (521) V LR ( n l (cid:48) ) = 12 N Ld,α,a ¯ N Rβ,b,a N Rb,γ,c ¯ N Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m e c ) (522) T LL ( n l (cid:48) ) = − N Ld,α,a ¯ N Lβ,b,a N Lb,γ,c ¯ N Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m e c ) (523) S LL ( n l (cid:48) ) = 12 N Ld,α,a ¯ N Lβ,b,a ¯ N Lδ,b,c N Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m e c ) (524) S LR ( n l (cid:48) ) = 2 N Ld,α,a ¯ N Lβ,b,a ¯ N Rδ,b,c N Rd,γ,c D ( m χ b , m χ d , m e a , m e c ) (525) V LL ( n l (cid:48) ) = − N Ld,α,a ¯ N Rβ,b,a ¯ N Rδ,b,c N Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m e c ) (526) V LR ( n l (cid:48) ) = N Ld,α,a ¯ N Rβ,b,a ¯ N Lδ,b,c N Rd,γ,c D ( m χ b , m χ d , m e a , m e c ) (527)59 LL ( n l (cid:48) ) = 18 N Ld,α,a ¯ N Lβ,b,a ¯ N Lδ,b,c N Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m e c ) (528)(529) F.2.2 Crossed chargino contributions S LL ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Lδ,d,c X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (530) S LR ( c l (cid:48) ) = − X Ld,α,a ¯ X Rδ,d,c X Rb,γ,a ¯ X Lβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν ic ) (531) V LL ( c l (cid:48) ) = X Ld,α,a ¯ X Rδ,d,c X Lb,γ,a ¯ X Rβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν ic )) (532) V LR ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Lδ,d,c X Rb,γ,a ¯ X Rβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (533) T LL ( c l (cid:48) ) = − X Ld,α,a ¯ X Lδ,d,c X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (534) S LL ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Lβ,b,a ˆ X Lb,γ,c ˆ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (535) S LR ( c l (cid:48) ) = 2 X Ld,α,a ¯ X Lβ,b,a ˆ X Rb,γ,c ˆ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (536) V LL ( c l (cid:48) ) = − X Ld,α,a ¯ X Rβ,b,a ˆ X Lb,γ,c ˆ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (537) V LR ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Rβ,b,a ˆ X Rb,γ,c ˆ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (538) T LL ( c l (cid:48) ) = − X Ld,α,a ¯ X Lβ,b,a ˆ X Lb,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (539) S LL ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Lβ,b,a X Lb,γ,c ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (540) S LR ( c l (cid:48) ) = 2 ˆ X Ld,α,a ˆ¯ X Lβ,b,a X Rb,γ,c ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (541) V LL ( c l (cid:48) ) = − ˆ X Ld,α,a ˆ¯ X Rβ,b,a X Lb,γ,c ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (542) V LR ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Rβ,b,a X Rb,γ,c ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (543) T LL ( c l (cid:48) ) = −
18 ˆ X Ld,α,a ˆ¯ X Lβ,b,a X Lb,γ,c ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (544) S LL ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Lβ,b,a ˆ X Lb,γ,c ˆ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (545) S LR ( c l (cid:48) ) = 2 ˆ X Ld,α,a ˆ¯ X Lβ,b,a ˆ X Rb,γ,c ˆ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (546) V LL ( c l (cid:48) ) = − ˆ X Ld,α,a ˆ¯ X Rβ,b,a ˆ X Lb,γ,c ˆ¯ X Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (547) V LR ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Rβ,b,a ˆ X Rb,γ,c ˆ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (548) T LL ( c l (cid:48) ) = −
18 ˆ X Ld,α,a ˆ¯ X Lβ,b,a ˆ X Lb,γ,c ˆ¯ X Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (549)60 LL ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Lδ,d,c X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (550) S LR ( c l (cid:48) ) = − X Ld,α,a ¯ X Rδ,d,c X Rb,γ,a ¯ X Lβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν ic ) (551) V LL ( c l (cid:48) ) = X Ld,α,a ¯ X Rδ,d,c X Lb,γ,a ¯ X Rβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν ic ) (552) V LR ( c l (cid:48) ) = 12 X Ld,α,a ¯ X Lδ,d,c X Rb,γ,a ¯ X Rβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (553) T LL ( c l (cid:48) ) = − X Ld,α,a ¯ X Lδ,d,c X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν ic ) (554) S LL ( c l (cid:48) ) = 12 X Ld,α,a ˆ¯ X Lδ,d,c X Lb,γ,a ˆ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (555) S LR ( c l (cid:48) ) = − X Ld,α,a ˆ¯ X Rδ,d,c X Rb,γ,a ˆ¯ X Lβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (556) V LL ( c l (cid:48) ) = X Ld,α,a ˆ¯ X Rδ,d,c X Lb,γ,a ˆ¯ X Rβ,b,c D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (557) V LR ( c l (cid:48) ) = 12 X Ld,α,a ˆ¯ X Lδ,d,c X Rb,γ,a ˆ¯ X Rβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (558) T LL ( c l (cid:48) ) = − X Ld,α,a ˆ¯ X Lδ,d,c X Lb,γ,a ˆ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m ν Rc ) (559) S LL ( c l (cid:48) ) = 12 ˆ X Ld,α,a ¯ X Lδ,d,c ˆ X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (560) S LR ( c l (cid:48) ) = − X Ld,α,a ¯ X Rδ,d,c ˆ X Rb,γ,a ¯ X Lβ,b,c D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (561) V LL ( c l (cid:48) ) = ˆ X Ld,α,a ¯ X Rδ,d,c ˆ X Lb,γ,a ¯ X Rβ,b,c D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (562) V LR ( c l (cid:48) ) = 12 ˆ X Ld,α,a ¯ X Lδ,d,c ˆ X Rb,γ,a ¯ X Rβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (563) T LL ( c l (cid:48) ) = −
18 ˆ X Ld,α,a ¯ X Lδ,d,c ˆ X Lb,γ,a ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν ic ) (564) S LL ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Lδ,d,c ˆ X Lb,γ,a ˆ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (565) S LR ( c l (cid:48) ) = − X Ld,α,a ˆ¯ X Rδ,d,c ˆ X Rb,γ,a ˆ¯ X Lβ,b,c D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (566) V LL ( c l (cid:48) ) = ˆ X Ld,α,a ˆ¯ X Rδ,d,c ˆ X Lb,γ,a ˆ¯ X Rβ,b,c D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (567) V LR ( c l (cid:48) ) = 12 ˆ X Ld,α,a ˆ¯ X Lδ,d,c ˆ X Rb,γ,a ˆ¯ X Rβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (568) T LL ( c l (cid:48) ) = −
18 ˆ X Ld,α,a ˆ¯ X Lδ,d,c ˆ X Lb,γ,a ˆ¯ X Lβ,b,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m ν Rc ) (569) F.2.3 Crossed W + and H + contributions S LL ( w l (cid:48) ) = 12 V + ,Ld,α,a ¯ V + ,Lβ,b,a V + ,Lb,γ,c ¯ V + ,Lδ,d,c m ν b m ν d D ( m ν d , m ν b , m H − a , m H − c ) (570) S LR ( w l (cid:48) ) = 2 V + ,Ld,α,a ¯ V + ,Lβ,b,a V + ,Rb,γ,c ¯ V + ,Rδ,d,c D ( m ν d , m ν b , m H − a , m H − c ) (571) V LL ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Rβ,b,a V + ,Lb,γ,c ¯ V + ,Rδ,d,c D ( m ν d , m ν b , m H − a , m H − c ) (572)61 LR ( w l (cid:48) ) = 12 V + ,Ld,α,a ¯ V + ,Rβ,b,a V + ,Rb,γ,c ¯ V + ,Lδ,d,c m ν b m ν d D ( m ν d , m ν b , m H − a , m H − c ) (573) T LL ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Lβ,b,a V + ,Lb,γ,c ¯ V + ,Lδ,d,c m ν b m ν d D ( m ν d , m ν b , m H − a , m H − c ) (574) I = I C D ( m ν d , m ν b , m W − , m H − a ) (575) I = D ( m ν d , m ν b , m H − a , m W − ) (576) I = m ν b m ν d D ( m ν d , m ν b , m H − a , m W − ) (577) S LL ( w l (cid:48) ) = 14 ( − V + ,Rd,α,a ¯ V + ,Rβ,b,a ˆ V + ,Rb,γ ˆ¯ V + ,Lδ,d I − V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ V + ,Lb,γ ˆ¯ V + ,Rδ,d (4 I + 13 I )) (578) S LR ( w l (cid:48) ) = 34 ( − ( V + ,Rd,α,a ¯ V + ,Rβ,b,a ˆ V + ,Rb,γ ˆ¯ V + ,Lδ,d ) + V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ V + ,Lb,γ ˆ¯ V + ,Rδ,d ) I − V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ V + ,Rb,γ ˆ¯ V + ,Lδ,d I (579) V LL ( w l (cid:48) ) = V + ,Ld,α,a ¯ V + ,Rβ,b,a ˆ V + ,Lb,γ ˆ¯ V + ,Lδ,d I (580) V LR ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Rβ,b,a ˆ V + ,Rb,γ ˆ¯ V + ,Rδ,d ( I − I ) (581) T LL ( w l (cid:48) ) = 116 ( V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ V + ,Lb,γ ˆ¯ V + ,Rδ,d (4 I − I ) + 5 V + ,Rd,α,a ¯ V + ,Rβ,b,a ˆ V + ,Rb,γ ˆ¯ V + ,Lδ,d I ) (582) I = I C D ( m ν d , m ν b , m H − c , m W − ) (583) I = D ( m ν d , m ν b , m W − , m H − c ) (584) I = m ν b m ν d D ( m ν d , m ν b , m W − , m H − c ) (585) S LL ( w l (cid:48) ) = 14 ( − V + ,Rd,i ˆ¯ V + ,Lβ,b V + ,Rb,γ,c ¯ V + ,Rδ,d,c I − ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b V + ,Lb,γ,c ¯ V + ,Lδ,d,c (4 I + 13 I )) (586) S LR ( w l (cid:48) ) = 34 ( ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b V + ,Lb,γ,c ¯ V + ,Lδ,d,c − ˆ V + ,Rd,i ˆ¯ V + ,Lβ,b V + ,Rb,γ,c ¯ V + ,Rδ,d,c ) I − V + ,Ld,i ˆ¯ V + ,Rβ,b V + ,Rb,γ,c ¯ V + ,Rδ,d,c I (587) V LL ( w l (cid:48) ) = ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b V + ,Lb,γ,c ¯ V + ,Rδ,d,c I (588) V LR ( w l (cid:48) ) = − ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b V + ,Rb,γ,c ¯ V + ,Lδ,d,c ( I − I ) (589) T LL ( w l (cid:48) ) = 116 ( ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b V + ,Lb,γ,c ¯ V + ,Lδ,d,c (4 I − I ) + 5 ˆ V + ,Rd,i ˆ¯ V + ,Lβ,b V + ,Rb,γ,c ¯ V + ,Rδ,d,c I ) (590) S LL ( w l (cid:48) ) = 8 ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b ˆ V + ,Lb,γ ˆ¯ V + ,Rδ,d m ν b m ν d D ( m ν d , m ν b , m W − , m W − ) (591) S LR ( w l (cid:48) ) = 8 ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b ˆ V + ,Rb,γ ˆ¯ V + ,Lδ,d I C D ( m ν d , m ν b , m W − , m W − ) (592) V LL ( w l (cid:48) ) = − V + ,Ld,i ˆ¯ V + ,Lβ,b ˆ V + ,Lb,γ ˆ¯ V + ,Lδ,d ( I C D ( m ν d , m ν b , m W − , m W − ) − D ( m ν d , m ν b , m W − , m W − )) (593) V LR ( w l (cid:48) ) = 2 ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b ˆ V + ,Rb,γ ˆ¯ V + ,Rδ,d m ν b m ν d D ( m ν d , m ν b , m W − , m W − ) (594) S LL ( w l (cid:48) ) = 12 V + ,Ld,α,a ¯ V + ,Lβ,b,a ¯ V + ,Lδ,b,c V + ,Ld,γ,c m ν b m ν d D ( m ν b , m ν d , m H − a , m H − c ) (595) S LR ( w l (cid:48) ) = 2 V + ,Ld,α,a ¯ V + ,Lβ,b,a ¯ V + ,Rδ,b,c V + ,Rd,γ,c D ( m ν b , m ν d , m H − a , m H − c ) (596)62 LL ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Rβ,b,a ¯ V + ,Rδ,b,c V + ,Ld,γ,c m ν b m ν d D ( m ν b , m ν d , m H − a , m H − c ) (597) V LR ( w l (cid:48) ) = V + ,Ld,α,a ¯ V + ,Rβ,b,a ¯ V + ,Lδ,b,c V + ,Rd,γ,c D ( m ν b , m ν d , m H − a , m H − c ) (598) T LL ( w l (cid:48) ) = 18 V + ,Ld,α,a ¯ V + ,Lβ,b,a ¯ V + ,Lδ,b,c V + ,Ld,γ,c m ν b m ν d D ( m ν b , m ν d , m H − a , m H − c ) (599) S LL ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ¯ V + ,Rδ,b ˆ V + ,Ld,γ D ( m ν b , m ν d , m H − a , m W − ) (600) S LR ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Lβ,b,a ˆ¯ V + ,Lδ,b ˆ V + ,Rd,γ m ν b m ν d D ( m ν b , m ν d , m H − a , m W − ) (601) V LL ( w l (cid:48) ) = 2 V + ,Ld,α,a ¯ V + ,Rβ,b,a ˆ¯ V + ,Lδ,b ˆ V + ,Ld,γ D ( m ν b , m ν d , m H − a , m W − ) (602) V LR ( w l (cid:48) ) = − V + ,Ld,α,a ¯ V + ,Rβ,b,a ˆ¯ V + ,Rδ,b ˆ V + ,Rd,γ m ν b m ν d D ( m ν b , m ν d , m H − a , m W − ) (603) S LL ( w l (cid:48) ) = − V + ,Ld,i ˆ¯ V + ,Rβ,b ¯ V + ,Lδ,b,c V + ,Ld,γ,c D ( m ν b , m ν d , m W − , m H − c ) (604) S LR ( w l (cid:48) ) = − V + ,Ld,i ˆ¯ V + ,Rβ,b ¯ V + ,Rδ,b,c V + ,Rd,γ,c m ν b m ν d D ( m ν b , m ν d , m W − , m H − c ) (605) V LL ( w l (cid:48) ) = 2 ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b ¯ V + ,Rδ,b,c V + ,Ld,γ,c D ( m ν b , m ν d , m W − , m H − c ) (606) V LR ( w l (cid:48) ) = − ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b ¯ V + ,Lδ,b,c V + ,Rd,γ,c m ν b m ν d D ( m ν b , m ν d , m W − , m H − c ) (607) S LL ( w l (cid:48) ) = 8 ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b ˆ¯ V + ,Rδ,b ˆ V + ,Ld,γ m ν b m ν d D ( m ν b , m ν d , m W − , m W − ) (608) S LR ( w l (cid:48) ) = 32 ˆ V + ,Ld,i ˆ¯ V + ,Rβ,b ˆ¯ V + ,Lδ,b ˆ V + ,Rd,γ D ( m ν b , m ν d , m W − , m W − ) (609) V LL ( w l (cid:48) ) = − V + ,Ld,i ˆ¯ V + ,Lβ,b ˆ¯ V + ,Lδ,b ˆ V + ,Ld,γ m ν b m ν d D ( m ν b , m ν d , m W − , m W − ) (610) V LR ( w l (cid:48) ) = 4 ˆ V + ,Ld,i ˆ¯ V + ,Lβ,b ˆ¯ V + ,Rδ,b ˆ V + ,Rd,γ D ( m ν b , m ν d , m W − , m W − ) (611)63 .3 Two-Lepton – Two-Quark boxes F.3.1 Feynman diagrams
Neutralino diagrams ( n d ) ˜ e a ˜ χ d ˜ χ b ˜ d c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( n d ) ˜ e a ˜ χ d ˜ χ b ˜ d ∗ c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( n u ) ˜ e a ˜ χ d ˜ χ b ˜ u ∗ c (cid:96) α ¯ (cid:96) β ¯ u δ u γ ( n u ) ˜ e a ˜ χ d ˜ χ b ˜ u c (cid:96) α ¯ (cid:96) β ¯ u δ u γ Chargino diagrams ( c d ) ν ia ˜ χ + d ˜ χ + b ˜ u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( c d ) ν Ra ˜ χ + d ˜ χ + b ˜ u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( c u ) ν ia ˜ χ + d ˜ χ + b ˜ d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ ( c u ) ν Ra ˜ χ + d ˜ χ + b ˜ d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ + and H + diagrams ( w d ) ν a H + d H + b u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( w d ) ν a H + d W + u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( w d ) ν a W + H + b u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( w d ) ν a W + W + u c (cid:96) α ¯ (cid:96) β ¯ d δ d γ ( w u ) ν a H + d H + b d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ ( w u ) ν a H + d W + d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ ( w u ) ν a W + H + b d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ ( w u ) ν a W + W + d c (cid:96) α ¯ (cid:96) β ¯ u δ u γ F.3.2 Down quarksNeutralino S LL ( n d ) = 12 N Ld,α,a ¯ N Lβ,b,a N d,Lb,γ,c ¯ N d,Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m d c ) (612) S LR ( n d ) = 2 N Ld,α,a ¯ N Lβ,b,a N d,Rb,γ,c ¯ N d,Rδ,d,c D ( m χ d , m χ b , m e a , m d c ) (613) V LL ( n d ) = − N Ld,α,a ¯ N Rβ,b,a N d,Lb,γ,c ¯ N d,Rδ,d,c D ( m χ d , m χ b , m e a , m d c ) (614)65 LR ( n d ) = 12 N Ld,α,a ¯ N Rβ,b,a N d,Rb,γ,c ¯ N d,Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m d c ) (615) T LL ( n d ) = − N Ld,α,a ¯ N Lβ,b,a N d,Lb,γ,c ¯ N d,Lδ,d,c m ˜ χ b m ˜ χ d D ( m χ d , m χ b , m e a , m d c ) (616) S LL ( n d ) = 12 N Ld,α,a ¯ N Lβ,b,a ¯ N d,Lδ,b,c N d,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m d c ) (617) S LR ( n d ) = 2 N Ld,α,a ¯ N Lβ,b,a ¯ N d,Rδ,b,c N d,Rd,γ,c D ( m χ b , m χ d , m e a , m d c ) (618) V LL ( n d ) = − N Ld,α,a ¯ N Rβ,b,a ¯ N d,Rδ,b,c N d,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m d c ) (619) V LR ( n d ) = N Ld,α,a ¯ N Rβ,b,a ¯ N d,Lδ,b,c N d,Rd,γ,c D ( m χ b , m χ d , m e a , m d c ) (620) T LL ( n d ) = 18 N Ld,α,a ¯ N Lβ,b,a ¯ N d,Lδ,b,c N d,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m d c ) (621) Chargino S LL ( c d ) = 12 X Ld,α,a ¯ X Lβ,b,a W d,Lb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m u c ) (622) S LR ( c d ) = 2 X Ld,α,a ¯ X Lβ,b,a W d,Rb,γ,c ¯ W d,Rδ,d,c D ( m χ − d , m χ − b , m ν ia , m u c ) (623) V LL ( c d ) = − X Ld,α,a ¯ X Rβ,b,a W d,Lb,γ,c ¯ W d,Rδ,d,c D ( m χ − d , m χ − b , m ν ia , m u c ) (624) V LR ( c d ) = 12 X Ld,α,a ¯ X Rβ,b,a W d,Rb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m u c ) (625) T LL ( c d ) = − X Ld,α,a ¯ X Lβ,b,a W d,Lb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν ia , m u c ) (626) S LL ( c d ) = 12 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W d,Lb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m u c ) (627) S LR ( c d ) = 2 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W d,Rb,γ,c ¯ W d,Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m u c ) (628) V LL ( c d ) = − ˆ X Ld,α,a ˆ¯ X Rβ,b,a W d,Lb,γ,c ¯ W d,Rδ,d,c D ( m χ − d , m χ − b , m ν Ra , m u c ) (629) V LR ( c d ) = 12 ˆ X Ld,α,a ˆ¯ X Rβ,b,a W d,Rb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m u c ) (630) T LL ( c d ) = −
18 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W d,Lb,γ,c ¯ W d,Lδ,d,c m ˜ χ − b m ˜ χ − d D ( m χ − d , m χ − b , m ν Ra , m u c ) (631) W + and H + S LL ( w d ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V u,Lc,γ,b ¯ V d,Lδ,c,d m ν a m u c D ( m ν a , m u c , m H − d , m H − b ) (632) S LR ( w d ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V u,Rc,γ,b ¯ V d,Rδ,c,d m ν a m u c D ( m ν a , m u c , m H − d , m H − b ) (633) V LL ( w d ) = − V + ,La,α,d ¯ V + ,Rβ,a,b V u,Lc,γ,b ¯ V d,Rδ,c,d D ( m ν a , m u c , m H − d , m H − b ) (634) V LR ( w d ) = − V + ,La,α,d ¯ V + ,Rβ,a,b V u,Rc,γ,b ¯ V d,Lδ,c,d D ( m ν a , m u c , m H − d , m H − b ) (635) S LL ( w d ) = 2 V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Lc,γ ¯ V d,Lδ,c,d ( I C D ( m u c , m W − , m H − d , m ν a ) − D ( m ν a , m u c , m W − , m H − d )) (636)66 LR ( w d ) = 2 V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Rc,γ ¯ V d,Rδ,c,d ( I C D ( m u c , m W − , m H − d , m ν a ) − D ( m ν a , m u c , m W − , m H − d )) (637) V LL ( w d ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V u,Lc,γ ¯ V d,Rδ,c,d m ν a m u c D ( m ν a , m u c , m W − , m H − d ) (638) V LR ( w d ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V u,Rc,γ ¯ V d,Lδ,c,d m ν a m u c D ( m ν a , m u c , m W − , m H − d ) (639) T LL ( w d ) = − V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Lc,γ ¯ V d,Lδ,c,d D ( m ν a , m u c , m W − , m H − d ) (640) S LL ( w d ) = 2 ˆ V + ,La,i ¯ V + ,Lβ,a,b V u,Lc,γ,b ˆ¯ V d,Rδ,c ( I C D ( m u c , m H − b , m W − , m ν a ) − D ( m ν a , m u c , mS , m W − )) (641) S LR ( w d ) = 2 ˆ V + ,La,i ¯ V + ,Lβ,a,b V u,Rc,γ,b ˆ¯ V d,Lδ,c ( I C D ( m u c , m H − b , m W − , m ν a ) − D ( m ν a , m u c , mS , m W − )) (642) V LL ( w d ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V u,Lc,γ,b ˆ¯ V d,Lδ,c m ν a m u c D ( m ν a , m u c , m H − b , m W − ) (643) V LR ( w d ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V u,Rc,γ,b ˆ¯ V d,Rδ,c m ν a m u c D ( m ν a , m u c , m H − b , m W − ) (644) T LL ( w d ) d = − ˆ V + ,La,i ¯ V + ,Lβ,a,b V u,Lc,γ,b ˆ¯ V d,Rδ,c D ( m ν a , m u c , m H − b , m W − ) (645) S LL ( w d ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Lc,γ ˆ¯ V d,Rδ,c m ν a m u c D ( m ν a , m u c , m W − , m W − ) (646) S LR ( w d ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Rc,γ ˆ¯ V d,Lδ,c m ν a m u c D ( m ν a , m u c , m W − , m W − ) (647) V LL ( w d ) = − V + ,La,i ˆ¯ V + ,Lβ,a ˆ V u,Lc,γ ˆ¯ V d,Lδ,c ( I C D ( m u c , m W − , m W − , m ν a ) − D ( m ν a , m u c , m W − , m W − )) (648) V LR ( w d ) = − V + ,La,i ˆ¯ V + ,Lβ,a ˆ V u,Rc,γ ˆ¯ V d,Rδ,c I C D ( m u c , m W − , m W − , m ν a ) (649) T LL ( w d ) d = ˆ V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Lc,γ ˆ¯ V d,Rδ,c m ν a m u c D ( m ν a , m u c , m W − , m W − ) (650) F.3.3 Up quarksNeutralino S LL ( n u ) = 12 N Ld,α,a ¯ N Lβ,b,a N u,Lb,γ,c ¯ N u,Lδ,d,c D ( m χ d , m χ b , m e a , m u c ) (651) S LR ( n u ) = 2 N Ld,α,a ¯ N Lβ,b,a N u,Rb,γ,c ¯ N u,Rδ,d,c D ( m χ d , m χ b , m e a , m u c ) (652) V LL ( n u ) = − N Ld,α,a ¯ N Rβ,b,a N u,Lb,γ,c ¯ N u,Rδ,d,c D ( m χ d , m χ b , m e a , m u c ) (653) V LR ( n u ) = 12 N Ld,α,a ¯ N Rβ,b,a N u,Rb,γ,c ¯ N u,Lδ,d,c D ( m χ d , m χ b , m e a , m u c ) (654) S LL ( n u ) = 12 N Ld,α,a ¯ N Lβ,b,a ¯ N u,Lδ,b,c N u,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m u c ) (655) S LR ( n u ) = 2 N Ld,α,a ¯ N Lβ,b,a ¯ N u,Rδ,b,c N u,Rd,γ,c D ( m χ b , m χ d , m e a , m u c ) (656) V LL ( n u ) = − N Ld,α,a ¯ N Rβ,b,a ¯ N u,Rδ,b,c N u,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m u c ) (657)67 LR ( n u ) = N Ld,α,a ¯ N Rβ,b,a ¯ N u,Lδ,b,c N u,Rd,γ,c D ( m χ b , m χ d , m e a , m u c ) (658) T LL ( n u ) = 18 N Ld,α,a ¯ N Lβ,b,a ¯ N u,Lδ,b,c N u,Ld,γ,c m ˜ χ b m ˜ χ d D ( m χ b , m χ d , m e a , m u c ) (659) Chargino S LL ( c u ) = 12 X Ld,α,a ¯ X Lβ,b,a W u,Lb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν ia , m d c ) (660) S LR ( c u ) = 2 X Ld,α,a ¯ X Lβ,b,a W u,Rb,δ,c ¯ W u,Rd,γ,c D ( m χ − b , m χ − d , m ν ia , m d c ) (661) V LL ( c u ) = − X Ld,α,a ¯ X Rβ,b,a W u,Rb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν ia , m d c ) (662) V LR ( c u ) = X Ld,α,a ¯ X Rβ,b,a W u,Lb,δ,c ¯ W u,Rd,γ,c D ( m χ − b , m χ − d , m ν ia , m d c ) (663) T LL ( c u ) = 18 X Ld,α,a ¯ X Lβ,b,a W u,Lb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν ia , m d c ) (664) S LL ( c u ) = 12 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W u,Lb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν Ra , m d c ) (665) S LR ( c u ) = 2 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W u,Rb,δ,c ¯ W u,Rd,γ,c D ( m χ − b , m χ − d , m ν Ra , m d c ) (666) V LL ( c u ) = −
12 ˆ X Ld,α,a ˆ¯ X Rβ,b,a W u,Rb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν Ra , m d c ) (667) V LR ( c u ) = ˆ X Ld,α,a ˆ¯ X Rβ,b,a W u,Lb,δ,c ¯ W u,Rd,γ,c D ( m χ − b , m χ − d , m ν Ra , m d c ) (668) T LL ( c u ) = 18 ˆ X Ld,α,a ˆ¯ X Lβ,b,a W u,Lb,δ,c ¯ W u,Ld,γ,c m ˜ χ − b m ˜ χ − d D ( m χ − b , m χ − d , m ν Ra , m d c ) (669) W + and H + S LL ( w u ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V u,Lδ,c,b ¯ V d,Lc,γ,d m ν a m d c D ( m ν a , m d c , m H − d , m H − b ) (670) S LR ( w u ) = − V + ,La,α,d ¯ V + ,Lβ,a,b V u,Rδ,c,b ¯ V d,Rc,γ,d m ν a m d c D ( m ν a , m d c , m H − d , m H − b ) (671) V LL ( w u ) = V + ,La,α,d ¯ V + ,Rβ,a,b V u,Rδ,c,b ¯ V d,Lc,γ,d D ( m ν a , m d c , m H − d , m H − b ) (672) V LR ( w u ) = V + ,La,α,d ¯ V + ,Rβ,a,b V u,Lδ,c,b ¯ V d,Rc,γ,d D ( m ν a , m d c , m H − d , m H − b ) (673) S LL ( w u ) = − V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Rδ,c ¯ V d,Lc,γ,d D ( m ν a , m d c , m W − , m H − d ) (674) S LR ( w u ) = − V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Lδ,c ¯ V d,Rc,γ,d D ( m ν a , m d c , m W − , m H − d ) (675) V LL ( w u ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V u,Lδ,c ¯ V d,Lc,γ,d m ν a m d c D ( m ν a , m d c , m W − , m H − d ) (676) V LR ( w u ) = V + ,La,α,d ˆ¯ V + ,Lβ,a ˆ V u,Rδ,c ¯ V d,Rc,γ,d m ν a m d c D ( m ν a , m d c , m W − , m H − d ) (677) T LL ( w u ) = − V + ,La,α,d ˆ¯ V + ,Rβ,a ˆ V u,Rδ,c ¯ V d,Lc,γ,d D ( m ν a , m d c , m W − , m H − d ) (678) S LL ( w u ) = − V + ,La,i ¯ V + ,Lβ,a,b V u,Lδ,c,b ˆ¯ V d,Lc,γ D ( m ν a , m d c , m H − b , m W − ) (679)68 LR ( w u ) = − V + ,La,i ¯ V + ,Lβ,a,b V u,Rδ,c,b ˆ¯ V d,Rc,γ D ( m ν a , m d c , m H − b , m W − ) (680) V LL ( w u ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V u,Rδ,c,b ˆ¯ V d,Lc,γ m ν a m d c D ( m ν a , m d c , m H − b , m W − ) (681) V LR ( w u ) = ˆ V + ,La,i ¯ V + ,Rβ,a,b V u,Lδ,c,b ˆ¯ V d,Rc,γ m ν a m d c D ( m ν a , m d c , m H − b , m W − ) (682) T LL ( w u ) = − ˆ V + ,La,i ¯ V + ,Lβ,a,b V u,Lδ,c,b ˆ¯ V d,Lc,γ D ( m ν a , m d c , m H − b , m W − ) (683) S LL ( w u ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Rδ,c ˆ¯ V d,Lc,γ m ν a m d c D ( m ν a , m d c , m W − , m W − ) (684) S LR ( w u ) = − V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Lδ,c ˆ¯ V d,Rc,γ m ν a m d c D ( m ν a , m d c , m W − , m W − ) (685) V LL ( w u ) = 16 ˆ V + ,La,i ˆ¯ V + ,Lβ,a ˆ V u,Lδ,c ˆ¯ V d,Lc,γ D ( m ν a , m d c , m W − , m W − ) (686) V LR ( w u ) = 4 ˆ V + ,La,i ˆ¯ V + ,Lβ,a ˆ V u,Rδ,c ˆ¯ V d,Rc,γ D ( m ν a , m d c , m W − , m W − ) (687) T LL ( w u ) = − ˆ V + ,La,i ˆ¯ V + ,Rβ,a ˆ V u,Rδ,c ˆ¯ V d,Lc,γ m ν a m d c D ( m ν a , m d c , m W − , m W − ) (688) G Form factors of the 4-fermion operators
We define the sum over all penguin diagrams as V LLZ,sum = 116 π (cid:32)(cid:88) a V LLZ, ( n a ) + (cid:88) a V LLZ, ( c a ) + (cid:88) a V LLZ, ( w a ) (cid:33) (689) V LRZ,sum = 116 π (cid:32)(cid:88) a V LRZ, ( n a ) + (cid:88) a V LRZ, ( c a ) + (cid:88) a V LRZ, ( w a ) (cid:33) (690) S LLh p ,sum = 116 π (cid:32)(cid:88) a S LLh p , ( n a ) + (cid:88) a S LLh p , ( c a ) + (cid:88) a S LLh p , ( w a ) (cid:33) (691) S LRh p ,sum = 116 π (cid:32)(cid:88) a S LRh p , ( n a ) + (cid:88) a S LRh p , ( c a ) + (cid:88) a S LRh p , ( w a ) (cid:33) (692) S LLA p ,sum = 116 π (cid:32)(cid:88) a S LLA p , ( n a ) + (cid:88) a S LLA p , ( c a ) + (cid:88) a S LLA p , ( w a ) (cid:33) (693) S LRA p ,sum = 116 π (cid:32)(cid:88) a S LRA p , ( n a ) + (cid:88) a S LRA p , ( c a ) + (cid:88) a S LRA p , ( w a ) (cid:33) (694)and the sum over all boxes as X LL ( sum x ) = 116 π (cid:32)(cid:88) a X LL ( n xa ) + (cid:88) a X LL ( c xa ) + (cid:88) a X LL ( w xa ) (cid:33) (695) X LR ( sum x ) = 116 π (cid:32)(cid:88) a X LR ( n xa ) + (cid:88) a X LR ( c xa ) + (cid:88) a X LR ( w xa ) (cid:33) (696)with X = V, S, T and x = (cid:96), d, u . With these, we can finally obtain the form factors of the 4-leptonoperators as follows: A VLL = V LLZ,sum E Lγ,δ m Z + V LL ( sum (cid:96) ) (697)69 VLR = V LRZ,sum E Rγ,δ m Z + V LR ( sum (cid:96) ) (698) A SLL = (cid:88) p S LLh p ,sum H Lγ,δ,p m h p + (cid:88) p S LLA p ,sum A Lγ,δ,p m A p + S LL ( sum (cid:96) ) (699) A SLR = (cid:88) p S LRh p ,sum H Rγ,δ,p m h p + (cid:88) p S LRA p ,sum A Rγ,δ,p m A p + S LR ( sum (cid:96) ) (700) A TLL = T LL ( sum (cid:96) ) (701) A TLR = T LR ( sum (cid:96) ) (702) B VLL = V LLZ,sum D Lγ,δ m Z + V LL ( sum (cid:96) ) (703) B VLR = V LRZ,sum D Rγ,δ m Z + V LR ( sum (cid:96) ) (704) B SLL = (cid:88) p S LLh p ,sum H d,Lγ,δ,p m h p + (cid:88) p S LLA p ,sum A d,Lγ,δ,p m A p + S LL ( sum d ) (705) B SLR = (cid:88) p S LRh p ,sum H d,Rγ,δ,p m h p + (cid:88) p S LRA p ,sum A d,Rγ,δ,p m A p + S LR ( sum d ) (706) B TLL = T LL ( sum d ) (707) B TLR = T LR ( sum d ) (708) C VLL = V LLZ,sum U Lγ,δ m Z + V LL ( sum u ) (709) C VLR = V LRZ,sum U Rγ,δ m Z + V LR ( sum u ) (710) C SLL = (cid:88) p S LLh p ,sum H u,Lγ,δ,p m h p + (cid:88) p S LLA p ,sum A u,Lγ,δ,p m A p + S LL ( sum u ) (711) C SLR = (cid:88) p S LRh p ,sum H u,Rγ,δ,p m h p + (cid:88) p S LRA p ,sum A u,Rγ,δ,p m A p + S LR ( sum u ) (712) C TLL = T LL ( sum u ) (713) C TLR = T LR ( sum u ) (714)and the other chiralities are given by X WRL = X WLR ( R ↔ L ) and X WRR = X WLL ( R ↔ L ) ( X = A, B, C ; W = S, T, V ). References [1] ATLAS Collaboration, G. Aad et al., Phys.Lett.
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