Lepton flavour violation in the MSSM: exact diagonalization vs mass expansion

Journal of High Energy Physics, Jun 2018

Abstract The forthcoming precision data on lepton flavour violating (LFV) decays require precise and efficient calculations in New Physics models. In this article lepton flavour violating processes within the Minimal Supersymmetric Standard Model (MSSM) are calculated using the method based on the Flavour Expansion Theorem, a recently developed technique performing a purely algebraic mass-insertion expansion of the amplitudes. The expansion in both flavour-violating and flavour-conserving off-diagonal terms of sfermion and supersymmetric fermion mass matrices is considered. In this way the relevant processes are expressed directly in terms of the parameters of the MSSM Lagrangian. We also study the decoupling properties of the amplitudes. The results are compared to the corresponding calculations in the mass eigenbasis (i.e. using the exact diagonalization of the mass matrices). Using these methods, we consider the following processes: ℓ → ℓ ′ γ, ℓ→3ℓ ′ , ℓ→2ℓ ′ ℓ ′′ , h→ℓℓ ′ as well as μ→e conversion in nuclei. In the numerical analysis we update the bounds on the flavour changing parameters of the MSSM and examine the sensitivity to the forthcoming experimental results. We find that flavour violating muon decays provide the most stringent bounds on supersymmetric effects and will continue to do so in the future. Radiative ℓ → ℓ ′ γ decays and leptonic three-body decays ℓ → 3ℓ ′ show an interesting complementarity in eliminating “blind spots” in the parameter space. In our analysis we also include the effects of non-holomorphic A-terms which are important for the study of LFV Higgs decays.

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Lepton flavour violation in the MSSM: exact diagonalization vs mass expansion

Received: March Lepton avour violation in the MSSM: exact diagonalization vs mass expansion Andreas Crivellin 2 3 Zo a Fabisiewicz 0 2 3 Weronika Materkowska 0 2 3 Ulrich Nierste 1 2 3 Stefan Pokorski 0 2 3 Janusz Rosiek 0 2 3 Paul Scherrer Institut 2 3 Villigen PSI 2 3 Switzerland 2 3 0 Faculty of Physics, University of Warsaw 1 Institut fur Theoretische Teilchenphysik, Karlsruhe Institute of Technology 2 Pasteura 5 , 02-093 Warsaw , Poland 3 76128 Karlsruhe , Germany The forthcoming precision data on lepton avour violating (LFV) decays require precise and e cient calculations in New Physics models. In this article lepton avour violating processes within the Minimal Supersymmetric Standard Model (MSSM) are calculated using the method based on the Flavour Expansion Theorem, a recently developed technique performing a purely algebraic mass-insertion expansion of the amplitudes. The expansion in both avour-violating and avour-conserving o -diagonal terms of sfermion and supersymmetric fermion mass matrices is considered. In this way the relevant processes are expressed directly in terms of the parameters of the MSSM Lagrangian. We also study the decoupling properties of the amplitudes. The results are compared to the corresponding calculations in the mass eigenbasis (i.e. using the exact diagonalization of the mass matrices). Using these methods, we consider the following processes: ` ! `0 , ` ! 3`0, ` ! 2`0`00, h ! ``0 as well as ! e conversion in nuclei. In the numerical analysis we update the bounds on the avour changing parameters of the MSSM and examine the sensitivity to the forthcoming experimental results. We nd that avour violating muon decays provide the most stringent bounds on supersymmetric e ects and will continue to do so in the future. Radiative ` ! `0 decays and leptonic three-body decays ` ! 3`0 show an interesting complementarity in eliminating \blind spots" in the parameter space. In our analysis we also include the e ects of non-holomorphic A-terms which are important for the study of LFV Higgs decays - 1 Introduction 2 E ective LFV interactions 2.1 2.3 2.4 2.2 Z ` `0 interactions ` `0 interactions LFV Higgs interactions Box contributions 2.4.1 2.4.2 2.4.3 2.4.4 Operators with two leptons and two quarks 3.1 3.2 3.4 5.1 5.2 5.3 5.4 3 Observables Radiative lepton decays: `I ! ` J h(H) ! `I `J decays 3.3 `I ! `J `K `L decays ! e conversion in nuclei 4 Mass eigenstates vs. mass insertions calculations 5 Phenomenological analysis Generic bounds on LFV parameters Dependence on the mass splitting Correlations between LFV processes Non-decoupling e ects in LFV Higgs decays 6 Conclusions A MSSM Lagrangian and vertices B Loop integrals C Divided di erences D Box diagrams in the mass eigenstates basis E E ective lepton couplings in the leading MI order E.1 Lepton-photon vertex E.1.1 E.1.2 Tensor (magnetic) couplings Vector couplings E.2 Lepton-Z0 vertex E.3 CP-even Higgs-lepton vertex E.4 CP-odd Higgs-lepton vertex E.5 4-lepton box diagrams { i { Introduction conversion in nuclei and ` ! `0 + or ` ! `0e+e So far, the LHC did not observe any particles beyond those of the Standard Model (SM). Complementary to direct high energy searches at the LHC, there is a continuous e ort in indirect searches for new physics (NP). In this respect, a promising approach is the search for processes which are absent | or extremely suppressed | in the SM such as lepton avour violation (LFV) which is forbidden in the SM in the limit of vanishing neutrino masses. The experimental sensitivity for rare LFV processes such as ` ! `0 , ! e will improve signi cantly in the near future, probing scales well beyond those accessible at foreseeable colliders. Furthermore, the discovery of the 125 GeV Higgs boson h [1, 2] has triggered an enormous experimental e ort in measuring its properties, including studies of its LFV decays. The most recent experimental limits on the LFV processes are given in table 2 in section 5. Many studies of LFV processes within the MSSM (and possible extensions of it) exist (see e.g. refs. [3{29] and ref. [30] for a recent review). In this article we revisit this subject in the light of the new calculational methods which have been recently developed [31, 32]. These methods allow for a systematic expansion of the amplitudes of the LFV processes in terms of mass insertions (MI), i.e. in terms of o -diagonal elements of the mass matrices. We show that a transparent qualitative behaviour of the amplitudes of the LFV processes is obtained by expanding them not only in the avour-violating o -diagonal terms in the sfermion mass matrices but also in the avour conserving but chirality violating entries related to the tri-linear A-terms as well as in the o -diagonal terms of the gaugino and higgsino mass matrices. This procedure is useful because in the MI approximation we work directly with the parameters of the Lagrangian and can therefore easily put experimental bounds on them. We compare the results of the calculations performed in the mass eigenbasis (i.e. using a numerical diagonalization of the slepton mass matrices) with those obtained at leading non-vanishing order of the MI approximation, in di erent regions of the supersymmetric parameter space and considering various decoupling limits. Of course, the MI approximation [33, 34] has already been explored for many years as a very useful tool in avour physics. However, a detailed comparison between the full calculation and the MI approximation is still lacking, partly because a fully systematic discussion of the MI approximation [31] to any order and the technical tools facilitating it [32] have not been available until recently. Concerning the phenomenology, we summarise and update the bounds on the avour violating SUSY parameters, show their complementarity and examine the impact of the anticipated increase in the experimental sensitivity. We investigate in detail the decay h ! showing the results in various decoupling limits and analyse the role of the socalled non-holomorphic A-terms [35{42], which are usually neglected in literature. We also avoid simplifying assumptions on the sparticle spectrum and assume neither degeneracies nor hierarchies among the supersymmetric particles. This article is structured as follows: in section 2 we establish our conventions and present the results for the 2-point, 3-point, and 4-point functions related to avour violating charged lepton interactions in the mass eigenbasis, i.e. expressed in terms of rotation { 1 { matrices and physical masses. Section 3 contains the formula for the decay rates of the processes under investigation. In section 4 we discuss the MI expansion and summarise important properties of the decoupling limits MSUSY ! 1 and MA ! 1. In section 5 we present the numerical bounds on LFV parameters obtained from current experimental measurements and discuss the dependence of the results on the SUSY spectrum. We also discuss the correlations between the radiative decays and the 3-body decays of charged lepton as well as the non-decoupling e ects in LFV neutral Higgs decays. Finally we conclude in section 6. All required Feynman rules used in our calculations are collected in appendix A. The de nitions of loop integrals can be found in appendix B. In appendix C we explain the notation for the \divided di erences" of the loop functions used in the expanded form interactions generated at the one-loop level.1 We use the notation and conventions for the MSSM as given in refs. [43, 44].2 In our analysis, we include the so-called non-holomorphic trilinear soft SUSY breakA0IJ Hi2?LiI RJ + A0dIJ Hi2?QiI DJ + A0uIJ Hi1?QiI U J + H:c: ; l (2.1) ing terms: Lnh = 3 X 2 X I;J=1 i=1 letters i = 1; 2 are SU(2)L indices. 2.1 ` `0 interactions photons as which couple up(down)-sfermions to the down(up)-type Higgs doublets. Here, as throughout the rest of the paper, capital letters I; J = 1; 2; 3 denote avour indices and the small We de ne the e ective Lagrangian for avour violating couplings of leptons to on-shell L ` = e X I;J F JI `J PL`I + F IJ `J PR`I F ; The SM contribution to F JI is suppressed by powers of m2=M W2 and thus completely negligible. In the mass eigenbasis the supersymmetric contributions to F JI come from the diagrams displayed in gure 1. Let us decompose F in the following way F JI = F JAI mJ F JLIB mI F JRIB ; be included and renormalization is required. notation, up to the minor di erences summarised in the appendix A. 1Note that these expressions are not valid in the avour conserving case where additional terms should 2The conventions of [43, 44] are very similar to the later introduced and now widely accepted SLHA2 [ 45 ] (2.2) (2.3) { 2 { ? Lk ? Lk Nn Cn Cn ??K interaction (mirror-re ected self-energy diagram not shown). with HJEP06(218)3 (4 )2F JAI = X V`J~KC;nR V`I~KC;nL mCn C11(mCn ; m~K ) X V`JL~kNn;RV`IL~kNn;L mNn C12(mL~k ; mNn ) ; (4 )2F JLIB = X V`J~KC;nL V`I~KC;nL C23(mCn ; m~K ) + X V`JL~kNn;LV`IL~kNn;LC23(mL~k ; mNn ) : where q = pI and reads Here, V abbreviates the tree-level lepton-slepton-neutrino and lepton-sneutrino-chargino vertices, i.e. the subscripts of V stand for the interacting particles and the chirality of the lepton involved. The super-scripts refer to the lepton or slepton avour as well as to the chargino and neutralino involved. The speci c form of the chargino and neutralino vertices VL(R) is de ned in appendix A and the 3-point loop functions Cij are given in appendix B. F A (F LB) denotes the parts of the amplitude which is (not) proportional to the masses of fermions exchanged in the loop. F RB can be obtained from F LB by exchanging L $ R on the r.h.s. of eq. (2.4). Gauge invariance requires that LFV (axial) vectorial photon couplings vanish for onshell external particles. However, o -shell photon contributions are necessary to calculate three body decays of charged leptons. The vectorial part of the amplitude for the ``0 vertex can be written as iAJI = ieq2uJ (pJ ) JLIPL + JRIPR uI (pI ) ; pJ and JLI is at the leading order in p2=MS2USY momentum independent Ni(Ci) L?k(??K ) Ni(Ci) Ni(Ci) l J l I l J mirror-re ected self-energy diagram not shown). portant and have to be included [47{52]. HJEP06(218)3 2.2 Z ` `0 interactions In order to calculate the three body decays of charged leptons as are considered in section 3.3 it is su cient to calculate the e ective Z ` `0 interactions in the limit of vanishing external momenta. The Wilson coe cients of the e ective Lagrangian for the Z coupling to charged leptons are generated at one-loop level by the diagrams shown in gure 2 and can be written as with L`JZI = FZJLI`J PL`I + FZJRI`J PR` I Z ; FZJLI = FZJRI = JI ZL JZIR + e(1 esW cW ZL(R) denote the contribution originating from the one-particle irreducible (1PI) vertex diagram and V L(R) is the left-(right-)handed part of the lepton self-energy dened as JI (p2) = JVIL(p2) p= PL + JVIR(p2) p= PR + JmIL(p2) PL + JmIR(p2) PR : (2.9) Contrary to the left- and right-handed magnetic photon-lepton couplings, which change chirality, the Z`I `J coupling is chirality conserving. Therefore, the Wilson coe cients of the left-handed and right-handed couplings are not related to each other but rather satisfy FZIJL(R) = FZJLI(R). In the mass eigenbasis the vectorial part of the lepton self-energy and { 4 { X V`IL~jNi;LV`L~N;LB1(p; mLj ; mNi ) ; Jji (4 ) 2 JZIL = X V`I~KC;iLV`~C;L JKj VCijCZ;LC2(m~K ; mCi ; mCj ) 2VCijCZ;RmCi mCj C0(m~K ; mCi ; mCj ) X V`I~KC;iLV`J~KC;iL C2(m~K ; m~K ; mCi ) V`IL~jNi;LV`L~N;L Jjk VNikNZ;LC2(mLj ; mNi ; mNk ) 2VNikNZ;RmNi mNk C0(mLj ; mNi ; mNk ) X V`IL~jNi;LV`JL~kNi ;L VLjLkZ C2(mLj ; mLk ; mNi ) ; (2.11) 2 X 3 i=1 K=1 4 + X (4 ) at vanishing external momenta with obvious replacements L $ R for JVIR; JZIR. 2.3 LFV Higgs interactions To compactify the notation, we denote the CP-even Higgs boson decays by H0K ! `I `J , where, following again the notation of [43, 44], H CP-odd neutral Higgs boson by A0. H01; h H02. As usual, we denote In order to study h ! ``0 decays precisely, we keep the terms depending on the external Higgs mass. Therefore, we assume the following e ective action governing the LFV Higgslepton interaction: AHe = `J (kJ )(FhJ`IK (kJ ; kI )PL + FhI`JK (kJ ; kI )PR)`I (kI )H0K (kI ` + `J (kJ )(FAJ`I (kJ ; kI )PL + FAI J` (kJ ; kI )PR)`I (kI )A0(kI kJ ) kJ ) : (2.12) In addition, to calculate the ! e conversion rate one needs to include the e ective Higgsquark couplings. For this purpose, one can set all external momenta to zero and consider the e ective Lagrangian q LHe = uJ (FhJuIK PL + FhIuJK PR)uI H0K + dJ (FhJdIK PL + FhIdJK PR)dI H0K : (2.13) However, in this article we consider only the lepton sector and therefore do not give the explicit forms of Higgs quark couplings. The relevant 1-loop expressions in the same notation as used in the current paper are given in ref. [53] and the formulae that take into account also non-decoupling chirally enhanced corrections and 2-loop QCD corrections in the general MSSM can be found in refs. [54{56].3 3Earlier accounts on chiral resummation can be found in refs. [57{65]. { 5 { q Li l l J the MSSM (the mirror-re ected self-energy diagram is omitted). At the 1-loop level there are eight diagrams contributing to the e ective lepton Yukawa couplings. The ones with slepton and neutralino exchange are displayed in gure 3, while diagrams with the chargino exchange can be obtained by the obvious replacements N ! C; L ! ~. The expressions for Fh and FA are obtained from 1PI triangle diagrams and the scalar part of lepton self-energies (see eq. (2.9)) while the chirality conserving parts of the selfenergies are absorbed by a eld rotation required to go to the physical basis with a diagonal lepton mass matrix. Therefore, F JIK (kJ ; kI ) = h JIK (kJ ; kI ) h FAJI (kJ ; kI ) = JAI (kJ ; kI ) ZR1K v1 i sin v1 JmIL(0) ; JmIL(0) ; where the ZR denotes the CP-even Higgs mixing matrix (see appendix A) and the scalar self-energy contributions are evaluated at zero momentum transfer and given by: (4 ) 2 JmIL(0) = mCi V`I~LCi;LV`J~LCi;R B0 (0; m~L ; mCi ) X mNi V`L~N;LV`L~N;R B0 (0; mLj ; mNi ) Iji Jji The neutralino-slepton contributions to the 1PI vertex diagrams can be written as (the symbols in square brackets denote common arguments of the 3-point functions)4 (4 ) 2 JIK (kJ ; kI ) = h V`JL~mNn;LV`IL~lNn;LVHKL~lmL~mNn C0[kJ ; kI kJ ; mNn ; mL~m ; mL~l ] 4 X 6 l;n=1 m=1 X V`JL~nNm;RV`IL~nNl ;L(VNlKHmN;RC2 +VNlKHmN;LmNl mNm C0)[kJ ; kI kJ ; mL~n ; mNm ; mNl ] ; 4As we shall see later using MI expanded formulae (see appendix E.3), due to strong cancellations the leading order terms in eqs. (2.15), (2.16) are suppressed by the ratios of m`=MW or A0l=MSUSY. Additional terms linear in m`=MW , not included in eq. (2.16), appear in 1PI vertex diagrams when external lepton masses are not neglected. We calculated such terms and proved explicitly that after performing the MI expansion they were suppressed by additional powers of v2=MS2USY and therefore, a posteriori, negligible. Thus, we do not display such terms in eq. (2.16). 2 X 3 X i=1 L=1 4 + X 6 X 6 X K (qK ) l V`JL~mNn;LV`IL~lNn;LVA1Ll~mL~ mNn C0[kJ ; kI kJ ; mNn ; mL~m ; mL~l ] 4 l;n=1 m=1 X V`JL~nNm;RV`IL~nNl ;L(VNl1AmN;RC2 +VNl1AmN;LmNl mNm C0)[kJ ; kI kJ ; mL~n ; mNm ; mNl ] ; while the chargino-sneutrino triangle diagram is obtained by replacing L~ ! ~; N ! C and adjusting the summation limits appropriately in vertex factors V:::::: (see appendix A). 2.4 Box contributions 4-fermion interactions are also generated by box diagrams. The corresponding conventions for incoming and outgoing particles are shown in gure 4. We calculate all box diagrams in the approximation of vanishing external momenta. The e ective Lagrangian for the 4-lepton interactions involves the quadrilinear operators OVJIXKYL = (`J OSJXIKYL = (`J PX `I ) OTJIXKL = (`J PX `I ) (`K PY `L) ; (`K PY `L) ; `I ) (`K PX `L) ; or BTJIXKL. where X; Y stands for the chirality L or R.5 The Wilson coe cients of these operators are calculated from the box diagrams in gure 4 and are denoted by BNJIXKYL with N = V ,S, The operator basis in eq. (2.17) is redundant. First, we note that ONJIXKYL = ONKYLJXI OTJIXKL = OTKXLJI : for N = V; S; OVJIXKXL = OVKXIJXL; OVJIXKYL = 2 OSKXIJYL for X 6= Y; { 7 { (2.16) (2.17) (2.18) Second, there are Fierz relations among di erent operators: 5Note that the upper L index in box formfactors denotes the sfermion avor while the lower L subscript denotes its chirality, even if both symbols are identical. Also, recall that (`J PL`I ) (`K PR`L) = 0. Furthermore, we have OTJIXKL = OSJXIKXL = 2 OSKXIJXL 1 8 OTKXIJL: OVJIXKYL y = OVIJXLYK ; OSJLIKRL y = OSIJRLLK ; OSJLIKLL y = OSIJRLRK ; OTJILKL y = OTJIRLK : Eqs. (2.18) to (2.20) must be taken into account when deriving the e ective Lagrangian. decays. We can therefore specify to I = 3 for the e ective Lagrangian. Furthermore, we can choose either (J; K) = (1; 2) or (J; K) = (2; 1) without the need to sum over both cases: the Fierz identities in eq. (2.19) permit to bring all operators into the form (e : : : ) ( : : : `) (corresponding to the case (J; K) = (1; 2)) or into an alternative form with e interchanged with . Thus we have 2 6 6 X L=1;2 4 X N=V;S X;Y =L;R L4J`3KL = with J 6= K and J; K; L 2; X X=L;R BNJ3XKYLONJ3XKYL + BTJ3XKLOTJ3XKL77 + h.c. 3 5 describes + decays. as the four-lepton interaction in the Lagrangian. Note that the \+h.c." piece of L4J`K The Wilson coe cients BNJ3XKYL and BTJ3XKL in eq. (2.21) are simply identical to the results of the sum of all contributing box diagrams to the decay amplitude. The latter is given in eq. (3.7) with the coe cients of the spinor structure in the right column of table 1. The relation to the analytic expressions in eqs. (D.3) to (D.6) is BNJIXKYL = BAJINKXLY + BBJINKXLY + BCJINKXLY + BDJINKXLY ; for N = V; S (2.22) and an analogous expression for BTJIXKL. 2.4.2 Leptonic operators with J = K and I 6= L The case J = K occurs for the decays ! e e e and ! ` ` ` 0 with `; `0 = e; . Thanks to the Fierz identities in eq. (2.19) we may restrict the operator basis to OVJIXJXL ; OVJIXJYL = 2OSJXIJYL; OSJXIJXL = 1 12 OTJIXJL; with X; Y = L; R and X 6= Y: (2.23) { 8 { (2.19) (2.20) (2.21) The four-lepton piece of the e ective Lagrangian for the decay `I L4J`IJL = with L; J < I: 2 4 X X L=1;2 X;Y =L;R X X=L;R CeVJIXJYLOVJIXJYL + CeSJXIJXLOSJXIJXL5 + h.c. ! `J `J `L reads: 3 (2.24) For the matching calculation it is useful to quote the tree-level matrix elements of the operators: h lJ (pJ ; sJ )lJ (p0J ; s0J )lL+(pL; sJ )jOVJIXJXL jlI (pI ; sI )i lJ (pJ ; sJ )lJ (p0J ; s0J )lL+(pL; sJ )jOVJIXJYLjlI (pI ; sI )i = [u(pJ ; sJ ) PX u(pI ; sI )][u(p0J ; s0J ) PX v(pL; sL)] [u(p0J ; s0J ) PX u(pI ; sI )][u(pJ ; sJ ) PX v(pL; sL)] = 2 [u(pJ ; sJ ) PX u(pI ; sI )][u(p0J ; s0J ) PX v(pL; sL)] = [u(pJ ; sJ ) PX u(pI ; sI )][u(p0J ; s0J ) PY v(pL; sL)] [u(p0J ; s0J ) PX u(pI ; sI )][u(pJ ; sJ ) PY v(pL; sL)] = [u(pJ ; sJ ) PX u(pI ; sI )][u(p0J ; s0J ) PY v(pL; sL)] h lJ (pJ ; sJ )lJ (p0J ; s0J )lL+(pL; sJ )jOSJXIJXLjlI (pI ; sI )i 2 [u(pJ ; sJ )PX u(pI ; sI )][u(p0J ; s0J )PY v(pL; sL)]; = [u(pJ ; sJ )PX u(pI ; sI )][u(p0J ; s0J )PX v(pL; sL)] [u(p0J ; s0J )PX u(pI ; sI )][u(pJ ; sJ )PX v(pL; sL)] = [u(pJ ; sJ )PX u(pI ; sI )][u(p0J ; s0J )PX v(pL; sL)] [u(pJ ; sJ ) PX u(pI ; sI )][u(p0J ; s0J ) PX v(pL; sL)] (2.25) Here we have used the Fierz transform to group the spinors into the canonical order [u(pJ ; : : :) : : : u(pI ; : : :)][u(p0J ; : : :) : : : v(pL; : : :)]. This allows us to use the same formula for spin-summed squared matrix elements as in the case of J 6= K of section 2.4.1. To quote the Wilson coe cients CeNJIXJYL, N = V; S in terms of the box diagrams BNJIXJYL in eq. (2.22) we must compare the results of the MSSM decay amplitude in eq. (3.6) with the matrix elements in eq. (2.25) and read o coe cients of the various Dirac structures. The result is The Fierz identities further imply the equalities CeVJIXJXL = CeVJIXJYL = BVJIXJYL CeSJXIJXL = 2 BSJXIJXL: 2 1 BVJIXJXL ; BSJXIJYL = BTJIXJL = 2BVJIXJYL 41 BSJXIJXL : { 9 { These operators do not appear in lepton decays, but trigger muonium-antimuonium transitions and describe muon or tau pair production in e {e collisions at energies far below MSUSY. Their Wilson coe cients are tiny in the MSSM. Operators with two leptons and two quarks The analogous Lagrangian for the 2-lepton-2-quark interactions reads where X N;X;Y LI2`J2Kq L = BqIJNKXLY OqJNIKXLY OqIJVKXLY = (`I OqIJSKXLY = (`I PX `J ) OqIJTKXL = (`I PX `J ) (qK PY qL) ; (qLPY qK ) ; `J ) (qK PX qL) : (2.28) Again, we consider only purely leptonic contributions here in detail and do not give explicit expressions for the 2-lepton-2-quark box diagrams. The relevant expressions in the mass eigenbasis can be found using formulae of appendix D and inserting proper quark vertices from refs. [43, 44] into these. 3 Observables In this section we collect the formulae for the LFV observables in terms of the e ective interactions de ned in section 2. All the processes listed here will be included in the future version of the SUSY FLAVOR numerical library calculating an extensive set of avour and CP-violating observables both in the quark and leptonic sectors [66{68]. 3.1 Radiative lepton decays: `I ! ` J The branching ratios for the radiative lepton decays `I ! ` J are given by for the hadronic decay modes of the G2F mI5=(192 3) for the tree-level leptonic decay width and the 0:0005 [69] are introduced to account Even though in our numerical analyses we restrict ourselves to LFV processes, we remind the reader that the expressions for the anomalous magnetic moments and electric dipole moments of the charged leptons can be also calculated in term of the quantities de ned in eq. (2.4) and read: aI = dlI = 4mI Re F IAI 2e Im F IAI mI F ILIB + F IRIB ; (3.1) (3.2) (3.3) ? L ?K ? J J ?K The decay branching ratios for the CP-even and CP-odd Higgs bosons read: Br(H0K ! `I+`J ) = Br(A0 ! `I+`J ) = 16 16 mH0K mA H0K A F IJK 2 h + F JIK 2 h FAIJ 2 + FAJI 2 (3.4) h A with F IJK ; F IJ de ned in eq. (2.14). Note that summing over lepton charges in the nal state, `I+`J and `J+`I , would produce an additional factor of 2. 3.3 ` I ! `J `K `L decays The LFV decays of charged lepton into three lighter ones can be divided into 3 classes, depending on the avours in the nal state: (A) ` ! `0`0`0: three leptons of the same avour, i.e. ! e e+e , ! e e+e and , with a pair of opposite charged leptons. ! + ! e (B) ` (C) ` i.e. as `, i.e. + ! e and ! e+e . and ! e e . ! `0 `00+`00 : three distinguishable leptons with `0 carrying the same charge as `, ! `0 `00+`00 : three distinguishable leptons with `0 carrying the opposite charge Class (C), representing a L = 2 processes, is tiny within the MSSM: it could only be generated at 1-loop level by box diagrams suppressed by double avour changes, or at the 2-loop level by double penguin diagrams involving two LFV vertices. Therefore, we will not consider these processes in our numerical analysis. In order to calculate Br(`I ! `J `K `L) we decompose the corresponding amplitude A as A = A0 + A : (3.5) The relevant diagrams are displayed in gure 5. A0 contains contributions from 4-lepton box diagrams and from penguin diagrams (including vector-like o -shell photon couplings, see eq. (2.5)) which in the limit of vanishing external momenta can be represented as the 4-fermion contact interactions. A is the on-shell photon contribution originating from the magnetic operator (see eq. (2.2)) which has to be treated separately with more care as the photon propagator becomes singular in the limit of vanishing external momenta. We further decompose A0 for the two cases (A) and (B) according to its Lorentz structure: 0 A(A) = 0 A(B) = X with X; Y = L; R. Note that the amplitude A(A) in general contains a second term which is obtained from the one given in eq. (3.6) by replacing (pJ $ p0J ). However, one can use Fierz identities to reduce it to the structure given in eq. (3.6). The basis of Dirac quadrilinears Q is the same as the one used to decompose 4-lepton box diagrams in eq. (2.17): (3.6) (3.7) (3.8) (3.10) (3.11) pJ )2 [u(pJ )i pJ )2 [u(pJ )i (C LPL + C RPR)(pI pJ ) u(pI )][u(p0J ) v(pL)] (C LPL + C RPR)(pI pJ ) u(pI )][u(pK ) v(pL)]: (3.9) S = 1 ; V = ; T = ; and 0Q is obtained from Q by lowering the Lorentz indices. The amplitudes originating from on-shell photon exchange are given by The full form of the coe cients CN(A;B); C is displayed in table 1, where we compacti ed the expressions by using the following abbreviations for the Higgs penguin contributions:6 VHJI = 2 X N=1 ZR1N m2H0N F JIN ; h VAJI = i sin m2A0 FAJI : Note that in eq. (3.7) and eq. (3.9) we do not explicitly display avour indices, but they are speci ed in table 1. Neglecting the lighter lepton masses whenever possible, the expression for the branching ratios can be written down as (for comparison see [23]): Br(`I ! `J `K `L) = NcBr(`I ! e ) 32G2F 4 jCV LLj2 + jCV RRj2 + jCV LRj2 + jCV RLj 2 + jCSLLj2 + jCSRRj2 + jCSLRj2 + jCSRLj 2 + 48 jCT Lj2 + jCT Rj 2 + X where Nc = 1=2 if two of the nal state leptons are identical (decays (A)), Nc = 1 for decays (B) and X denotes the contribution to matrix element from the photon penguin A , 6Note that we de ne lepton Yukawa coupling appearing in table 1 to be negative, YlI = p2mlI =v1. CSLL CSRR CSLR CSRL CT L CT R ! e conversion in nuclei The full 1-loop expressions for the ! e conversion in Nuclei depend on both the squark and slepton SUSY breaking terms. Thus, in principle the resulting upper bounds on the slepton mass insertions to some extent depend on the squark masses. Therefore, we do not include ! e conversion in nuclei in our numerical analysis.7 However, for completeness we collect here the complete set of formulae required to calculate the rate of this process. 7Recent discussion of interplay between the bounds on MI's in the slepton and squark sectors can be found in ref. [70]. of eq. (3.7) and eq. (3.9) for decay types (A) and (B). BQXY ,BT X denote the irreducible box diagram contributions (see eq. (2.21)), the terms with FZ stem from the Z penguin Lagrangian (eq. (2.7)), V is the sum of the vector-like photon contributions (eq. (2.5)), Higgs contributions are de ned in eq. (3.10) and the coe cients F of the magnetic operator are de ned in eq. (2.2). including also its interference with the A0 part of the amplitude (m denotes the mass of the heaviest nal state lepton) X(A) = Re (2CV LL + CV LR 1 2 CSLR) C?R + (2CV RR + CV RL 1 2 CSRL) C?L + + 16e m`I 64e2 m`2I 16e m`I 32e2 m`2I log log m`2I m2 m`2I m2 3 (jC Lj2 + jC Rj2) : X(B) = Re (CV LL + CV LR) C?R + (CV RR + CV RL) C?L ! e conversion in nuclei is produced by the dipole, the vector, and the scalar operators already at the tree level [71]. Following the discussion of ref. [72] we use the e ective Lagrangian where N = V; S and X; Y = L; R with the operators de ned as L !e = X N;X;Y CqNI qXIY ONqI qXI Y + CXggOXgg OqI qI V XY = (e OqI qI S XY = (ePX ) (qI PY qI ) PX ) (qI PY qI ) OXgg = s m GF (ePX ) Ga Ga (3.13) (3.14) Using the notation introduced in previous sections, the corresponding Wilson coe cients can be expressed as CVdI XdIL = Cd1`2VIIXL CVdI XdIR = Cd1`2VIIXR + CVuI XuIL = Cu1`2IVIXL + CVuI XuIR = Cu1`2IVIXR CSdILdXI = Cd1`2SIILX + CSuILuXI = Cu1`2ISILX + CSdIRdIX = Cd1`2SIIRX + CSuIRuXI = Cu1`2ISIRX + 1 1 1 m2Z 2sW cW m2Z 3sW cW m2Z 2sW cW e e e 1 1 s2W FZ12X For this process, a Lagrangian involving only quark, lepton and photon elds is not su cient. Instead, an e ective Lagrangian at the nucleon level containing proton and neutron elds is required. It can be obtained in two steps. First, heavy quarks are integrated out. This results in a rede nition of the Wilson coe cient of the gluonic operator [73] Cgg L ! C~Lgg = Cgg L 1 12 X q=c;b CSqqLL + CSqqLR GF m mq with an analogous equation for CgRg. Second, the resulting Lagrangian is matched at the scale of n = 1 GeV to an e ective Lagrangian at the nucleon level. Following [74] the transition rate N!e = ( N ! e N ) can then be written as N !e = m5 4 + (L $ R); e CLD F 12=m + 4 GF m mpC~S(pL)SN(p) + C~V(pR) VN(p) + (p ! n) (3.15) (3.16) 2 (3.17) where p and n denote the proton and the neutron, respectively. The e ective couplings in eq. (3.17) can be expressed in terms of our Wilson coe cients as C~V(pR=n) = C~S(pL=n) = X q=u;d;s X q=u;d;s Cqq V RL + Cqq V RR fV(qp)=n ; CSqqLL + CSqqLR m mqGF fS(qp)=n + C~Lgg fGp=n with analogous relations for L $ R. The Wilson coe cients in eqs. (3.18) and (3.19) are to be evaluated at the scale n. The nucleon form factors for vector operators are xed by vector-current conservation, i.e. fV(up) = 2, fV(un) = 1, fV(dp) = 1, fV(dn) = 2, fV(sp) = 0, fV(sn) = 0. Hence, the sum in eq. (3.18) is in fact only over q = u; d. The calculation of the scalar form factors are more involving. The values of the up- and down-quark scalar couplings fS(up==dn) (based on the two- avour chiral perturbation theory framework of [75]) can be found in refs. [76, 77], while the values of the s-quark scalar couplings fS(sp)=n can be borrowed from a lattice calculation [78].8 In summary, one has fS(up) = (20:8 fS(dp) = (41:1 fS(sp) = fS(sn) = (53 1:5) 2:8) The form factor for the gluonic operator can be obtained from a sum rule. In our normalisation fGp=n = 8 9 1 X q=u;d;s fS(qp)=n : The quantities DN , S(p=n), and V (p=n) in eq. (3.17) are related to the overlap integrals [81] N N between the lepton wave functions and the nucleon densities. They depend on the nature of the target N . Their numerical values can be found in ref. [71]: DAu = 0:189; SA(pu) = 0:0614; VA(pu) = 0:0974; SA(nu) = 0:0918; VA(nu) = 0:146; DAl = 0:0362; SA(pl) = 0:0155; VA(pl) = 0:0161; SA(nl) = 0:0167; VA(nl) = 0:0173; (3.22) for gold and aluminium, respectively. by the capture rate, the latter given in ref. [82]: Finally, the branching ratio is de ned as the transition rate, (see eq. (3.17)), divided cAaupt = 8:7 10 15 MeV; Al capt = 4:6 10 16 MeV : (3.23) 8For earlier determinations of the pion-nucleon sigma terms see [79, 80]. (3.18) (3.19) (3.20) (3.21) For each process, we have given the exact one-loop expressions calculated in the mass eigenbasis (ME). These formulae are compact and well suited for numerical computations, however, do not allow for an easy understanding of the qualitative behaviour of the LFV amplitudes for various choices of the MSSM parameters. Therefore, in this section we expand the Wilson coe cients in terms of the \mass insertions", de ned as the o -diagonal elements (both avour violating and avour conserving) of the mass matrices. Such an expansion allows us to: parameters. Recover the direct analytical dependence of the results on the MSSM Lagrangian HJEP06(218)3 Prove analytically the expected decoupling features of the amplitudes in the limit of a heavy SUSY spectrum. In the case of Higgs boson decays, we also identify explicitly the terms decoupling only with the heavy CP-odd Higgs mass MA (which also determines the heavy CP even and the charged Higgs masses). The decoupling properties also serve as an important cross-check of the correctness of our calculations. Test the dependence of the results on the pattern of the MSSM spectrum and the size of the mass splitting between SUSY particles. Better understand the possible cancellations between various types of contributions and correlations between di erent LFV processes. The mass insertion expansion in avour o -diagonal terms has been used for a long time in numerous articles on the subject. However, often various simplifying assumptions have been made, i.e. some terms have been neglected or a simpli ed pattern of the slepton spectrum was considered. This is understandable as a consistent MI expansion of the amplitudes for the LFV processes in the MSSM, mediated by the virtual chargino and neutralino exchanges, is technically challenging. The standard approach used in literature is to calculate diagrammatically the LFV amplitudes with the \mass insertions" treated as the new interaction vertices. We follow the common practice and normalise such slepton mass insertions to dimensionless \ -parameters":9 ILJL = ILJR = q (ML2L)IJ (ML2L)II (ML2L)JJ ; AIJ l (ML2L)II (MR2 R)JJ 1=4 ; IRJR = 0LIRJ = (MR2 R)IJ q (MR2 R)II (MR2 R)JJ A0IJ l (ML2L)II (MR2 R)JJ 1=4 ; ; (4.1) where ML2L; MR2 R; Al; A0l are the slepton soft mass matrices and trilinear terms. As lepton avour violation is already strongly constrained experimentally, it is su cient to expand the amplitudes up to the rst order in avour-violating 's. For instance, the 9We assume that trilinear Al, A0l terms scale linearly with the slepton mass scale. e ective vertices listed in section 3 take the schematic form: 1 F IJ = (4 )2 FLILJ ILJL + FRIJR JRIR + FAIJLR JLIR + FBIJLR ILJR + F A0ILJR 0LJRI + F B0ILJR LR 0IJ : (4.2) The MSSM contributions to FLL; : : : ; F B0LR can be classi ed according to their decoupling behaviour, distinguishing the following types (M denotes the average SUSY mass scale): 1. E ects related to the diagonal trilinear slepton soft terms or to the o -diagonal elements of supersymmetric fermion mass matrices, decoupling as v2=M 2. 2. E ects related to the external momenta of the (on-shell) Higgs or Z0 bosons, decoupling as Mh2=M 2 or MZ2 =M 2 (we did not include the MZ dependence as it is not necessary for the considered processes). 3. Non-decoupling e ects related to the 2HDM structure of the MSSM. Such contributions are constant in the limit of a heavy SUSY scale M but, in case of the SM-like Higgs boson h, decouple with the CP-odd Higgs mass like v2=MA2 (the effective couplings of heavier H; A bosons do not exhibit such a suppression). They are proportional either to the lepton Yukawa couplings or to the non-holomorphic A0l terms. The structure of the box diagrams is more complicated as they carry 4 avour indices. Their MI expansion is given in appendix E.5. All box diagram contributions decouple at least as v2=M 2. Calculating consistently the quantities FLL; : : : ; F B0LR to the order v2=M 2 is not trivial for chargino and neutralino contributions. If the MI expansion is used only for the sfermion mass matrices but the calculations for the supersymmetric fermions are done in the mass eigenbasis, the direct dependence on the Lagrangian parameters is hidden and the decoupling properties of the amplitude cannot be seen directly. However, one can also treat the o -diagonal entries of the chargino and neutralino mass matrices as \mass insertions". With such an approach, the nal result is expressed explicitly in terms of Lagrangian parameters, but the computations can get very complicated. At the order v2=M 2 one needs to include diagrams with all combinations of two fermionic mass insertions (each providing one power of v=M1, v=M2 or v= ) or avour diagonal slepton terms originating from trilinear A-terms (providing powers of vAl=M 2, vA0l=M 2). Thus, to obtain an expansion of the F 's in eq. (4.2), one needs to formally go to the 3rd order of MI expansion, adding all diagrams with up to two avour conserving and one avour violating mass insertion. Therefore, the number of diagrams grows quickly with the order of the expansion and such a method is tedious and prone to calculational mistakes. In our paper, we employ a recently developed technique using a purely algebraic MI expansion of the ME amplitudes listed in section 3, without the need for direct diagrammatic MI calculations (\FET theorem") [31], automatised in the specialised MassToMI Mathematica package [32, 83]. The use of this package and full automation of the calculations allows us to perform the required 3rd order MI expansion for a completely general SUSY mass spectrum, without making any simplifying assumptions. Such a result would be very di cult to obtain diagrammatically, as in the intermediate steps of the calculations (before accounting for the cancellations and simpli cations between various contributions) the expressions may contain up to tens of thousand terms, even if the nal results collected in appendix E are again relatively compact. In detail: We perform the expansion always up to the lowest non-vanishing order in the slepton LFV terms, taking into account the possible cancellations. Compared to previous analyses, we consider the non-holomorphic trilinear soft terms as well. In the MI expanded expressions we include all terms decreasing with the SUSY mass scale as v2=MS2USY (or slower), where MSUSY denotes any of the relevant mass parameters in the MSSM Lagrangian (apart from the soft Higgs mass terms): diagonal soft slepton masses, gaugino masses M1; M2 or the parameter. We do not assume degeneracy or any speci c hierarchy for the sleptons, sneutrinos or supersymmetric fermion masses. In calculating the LFV Higgs decays we keep the leading terms in the external Higgs boson mass (m2h=MS2USY). The full set of the expanded expressions in the MI approximation for the photon, Z0 and CP-even Higgs leptonic penguins and for the 4-lepton box diagrams is collected in appendix E. We illustrate the accuracy of the derived MI formulae in gure 6. The plots show the ratio of the MI expanded couplings over the ones obtained in the mass eigenbasis with exact diagonalization. For this purpose, we start from the following setup where all mass parameters are given in GeV: (4.3) F 322 h (4.4) tan = 5 M1 = 150 M2 = 300 Next, to see the decoupling e ects we scale this spectrum uniformly up to slepton masses of 2 TeV. For each of the six penguin Wilson coe cients describing the transition between 2nd and 3rd generation, F 2L3(R) (eq. (2.2)), F 23 ZL(R) (eq. (2.7)) and Fh2L3 Fh232, Fh2R3 (eq. (2.13)) we plot the quantity as a function of the average slepton mass. The accuracy of left-handed (right-handed) Wilson coe cients is illustrated with red(blue) lines. As can be seen from gure 6, the accuracy of MI expanded amplitudes is very good even for light SUSY particles and for MSUSY > 500 GeV always better than 95%. F = FMI FME 1 ; (YlC )2v1ZR1K (ZLCi?ZLCl + Z L (C+3)i? (C+3)l 1 Z L 4s2 2s2 W ) ZLCi?Z(C+3)l L + Y C l ZLClZL(C+3)i? ) ZR1K (AlCD?ZLClZ(D+3)i? L + ACDZLCi?Z(D+3)l l L ) ZR2K (A0lCD?ZLClZ(D+3)i? L + A0CDZLCi?Z(D+3)l l L ) ; v1ZR1K v2ZR2K il + W ZLCi?ZLCl V V V ij ij ij CCZ;L CCZ;R NNZ;L = = = e 3) CP-even-Higgs-slepton and CP-even-Higgs-sneutrino vertices: (A.11) (A.12) (A.13) (A.14) VHKLilL = 3 X C=1 e 2 2c2 W ZR2K p 2 1 p 2 (YlC 3 X C;D=1 e 2 4s2W c2W i cos p 2 3 X C;D=1 VA1LilL = VA1L~~I = 0 : e VHK~L~I = (v1ZR1K v2ZR2K ) LI : 4) CP-odd-Higgs-slepton and CP-odd-Higgs-sneutrino vertices: (AlCD tan + A0CD l Y C l CD )ZLCi?Z(D+3)j L L ; (AlCD tan + A0CD l Y C l CD )Z Z L Cj (D+3)i? 5) CP-even-Higgs-neutralino and CP-even-Higgs-chargino vertices: VNiKHlN;L = VNiKHlN;R = (ZR1K ZN3l ZR2K ZN4l)(ZN1isW ZN2icW ) + (ZR1K ZN3i ZR2K ZN4i)(ZN1lsW ZN2lcW ) ; VCiKHlC;L = VCiKHlC;R = ZR1K Z2iZ+1l + ZR2K Z1iZ+2l : 6) CP-odd-Higgs-neutralino and CP-odd-Higgs-chargino vertices: VNi1Al N;L = VNi1Al N;R = (v2ZN 3j 4j v1ZN )(ZN1isW ZN2icW ) + (v2ZN3i VCi1AlC;L = VCi1AlC;R = 2j ZN cW ) ; (v2Z2iZ+1j + v1Z1iZ+2j) : VLijLZ = e We de ne the following loop integrals for 2-point and 3-point functions with non-vanishing = Z Z Z Z d4k d4k d4k (2 )4 (k2 (2 )4 (k2 (2 )4 (k2 d4k (2 )4 (k2 1 k m21)((k m21)((k p)2 p)2 m22) ; m22) ; (k2)n m21)((k + p)2 m22)((k + p + q)2 k m21)((k + p)2 m22)((k + p + q)2 (4 )2 (p C11(p; q; m1; m2; m3) + q C12(p; q; m1; m2; m3)) In our expanded results we need only the integrals above, their derivatives and higher point 1-loop integrals calculated at vanishing external momenta. Let us de ne (A.15) (B.1) m23) ; m23) : (B.2) (B.3) (B.4) (B.5) B Loop integrals external momenta p and q: i i i i (4 )2 (4 )2 B0(p; m1; m2) = p B1(p; m1; m2) = (4 )2 C2n(p; q; m1; m2; m3) = (4 )2 Li2n(m1; : : : ; mi) = Z d4k (k2)n (2 )4 i Q (k2 j=1 m2) j : In common notation L23n = C2n; L24n = D2n; L25n = E2n etc. For i 3 one has: Li0(m1; : : : ; mi) = Li2(m1; : : : ; mi) = i X j=2 i X j=2 mj2 log mj21 m2 i Q k=1;k6=j mj4 log mj21 m2 (mj2 m2) k ; ; i Q k=1;k6=j (mj2 m2) k (with the exception of L2 3 C2 having also an in nite part, which however is always cancelled out in avour violating processes and is thus not given here explicitly). To simplify our formulae, we use the relation 2Li0(m1; m2; : : : ; mi) = Li2+1(m1; m1; m2; : : : ; mi) + Li2+1(m1; m2; m2; : : : ; mi) + : : : + Li2+1(m1; : : : ; mi 1; mi; mi) ; which can be obtained by di erentiating with respect to the integral form of the homogeneity property L0( m1; : : : ; mi) = i 4 2iLi0(m1; : : : ; mi) ; and using the relation (k = 1; : : : ; i) In addition, we de ne the following integrals: C00(m1; m2; m3) = 2(m21 m22)(m21 p=q=0 The expansion of the amplitudes given in the mass eigenbasis in terms of mass insertions can be naturally expressed [31] by the so-called divided di erences of the loop functions. In case of a function of a single argument, f (x), divided di erences are de ned recursively as: f [0](x) = f (x) ; f [1](x; y) = f [2](x; y; z) = f [0](x) x f [0](y) y ; f [1](x; y) f [1](x; z) y z ; : : : : m2kLi0+1(m1; : : : ; mk; mk; : : : ; mi) = Li2+1(m1; : : : ; mk; mk; : : : ; mi) Li0(m1; : : : ; mk; : : : ; mi) : (B.6) As can be easily checked, a divided di erence of order n is symmetric under permutation of any subset of its arguments. It also has a smooth limit for degenerate arguments: lim fx0;:::;xmg!f ;:::; g f [k](x0; : : : ; xk) = f [k m]( ; xm+1 : : : ; xk) : (C.2) (B.7) HJEP06(218)3 2m21m22 + m22m23) log mm2322 ; m23)2(m22 m23)3 (B.8) (B.9) (B.10) (B.11) (B.12) (C.1) To compactify the formulae for functions of many arguments, we use the notation f [k](x0; : : : ; xk) f (fx0; : : : ; xkg) ; where the order of the divided di erence is de ned by the number of arguments inside curly brackets. Then, for example a divided di erence of the 1st order in the 1st argument and of the 3rd order in the 2nd argument for the function of 3 variables, g(x; y; z), can be For the loop functions de ned in appendix B one should note that their natural arguments are squares of masses. However, we use mi's instead of mi2's to compactify the notation. Thus, for loop functions we write divided di erences as (C.3) (C.4) L(m1; : : : ; fmi; m0ig; : : : ; mn) = m2 i m02 i L(m1; : : : ; mi; : : : ; mn) L(m1; : : : ; m0i; : : : ; mn) ; (C.5) with squared masses in the denominator. The FET expansion works for any transition amplitude, also in the case of nonvanishing external momenta or for multi-loop calculations. However, it is particularly e ective for 1-loop functions with vanishing external momenta, due to the fact that the notion of the divided di erences is naturally encoded in the structure of such functions: a divided di erence of a n-point scalar 1-loop function is a (n + 1)-point function (see eq. 3.13 in ref. [31] for generalisation to the case of non-vanishing external momenta). Thus, for example one has B0(m1; fm2; m3g) = B0(fm1; m2g; m3) = C0(m1; m2; m3) B0(m1; fm2; m3; m4g) = C0(m1; m2; fm3; m4g) = D0(m1; m2; m3; m4) (C.6) : : : We use such relations extensively to nd cancellations between various terms and to identify the lowest non-vanishing order of mass insertion expansion for a given process. D Box diagrams in the mass eigenstates basis There are four types of box diagrams with four external leptons involving slepton (sneutrinos) and neutralinos (charginos) in the loop, displayed in gure 13. Both chargino-sneutrino and neutralino-slepton pairs contribute to diagrams A) and B), while only neutralinos (Majorana fermions) can be exchanged in the \crossed" diagrams C) and D). Using whenever necessary Fierz identities, the amplitudes describing each of the diagrams N = A; B; C; D can be brought into the form iAJNIKL = i X BNJIQKXLY [u(pJ ) QPX u(pI )][u(pK ) QPY v(pL)] (D.1) with V = , S = 1 and T = . Note that for T only the case X = Y is non vanishing. Assuming that the generic couplings for an incoming lepton `I - an incoming l I Sk Sk fj l K A) fi fj l K C) Sl Sl l I l I fi fj Sl B) Sk Sl D) l K l K fi fj l J l L l J l L scalar particle Sk and an outgoing fermion fi takes the form the contribution from diagram A) in gure 13) to the Wilson coe cients BQXY can be written down as: iV`ISkfi = i A`ISkif PL + B`ISkfi PR ; (D.2) (4 )2BAJIVKLLL = A`ISkif A`JSlif A (4 )2BAJIVKRLR = B`ISkfi B`JSlif B (4 )2BAJIVKLLR = A`ISkif A`JSlif B (4 )2BAJIVKRLL = B`ISkfi B`JSlif A (4 )2BAJISKLLL = A`ISkif B`JSlif B (4 )2BAJISKRLR = B`ISkfi A`JSlif A (4 )2BAJISKLLR = A`ISkif B`JSlif A (4 )2BAJISKRLL = B`ISkfi A`JSlif B Kkj `Sf Kkj `Sf Kkj `Sf Kkj `Sf (4 )2BAJITKLL = 0 ; (4 )2BAJITKRL = 0 ; Kkj `Sf Kkj `Sf Kkj `Sf Kkj `Sf A Llj A Llj Llj A `Sf mfi mfj D0 ; B `Sf mfi mfj D0 ; B A Llj `Sf mfi mfj D0 ; Llj `Sf mfi mfj D0 ; (D.3) where D0; D2 above are the abbreviations for 4-point loop functions with respective mass arguments, D0 = D0(mfi ; mfj ; mSk ; mSl ); D2 = D2(mfi ; mfj ; mSk ; mSl ) (see appendix B). Using the same notation, the contributions from diagram B), C), D) are: (4 )2BBJIVKLLL = (4 )2BBJIVKRLR = (4 )2BBJIVKLLR = (4 )2BBJIVKRLL = (4 )2BBJISKLLL = (4 )2BBJISKRLR = (4 )2BBJISKLLR = (4 )2BBJISKRLL = (4 )2BBJITKLL = (4 )2BBJITKRL = A`ISkif A Jkj `Sf A`KSlfi A Llj B`ISkfi B `Sf B`KSlfi B Llj A`ISkif A `Sf B`KSlfi B Llj B`ISkfi B A`ISkif B B`ISkfi A A`ISkif B B`ISkfi A A`ISkif B B`ISkfi A Jkj `Sf Jkj `Sf Jkj `Sf Jkj `Sf Jkj `Sf Jkj `Sf Jkj `Sf A`KSlfi A Llj B`KSlfi A B`KSlfi A B`KSlfi A A`KSlfi B `Sf mfi mfj D0 ; (D.4) (D.5) (4 )2BDJIVKLLR = B`ISkfi A`LSlif B Jkj `Sf A Klj `Sf D2 ; (4 )2BCJIVKLLL = A`ISkif A`LSlif A (4 )2BCJIVKRLR = B`ISkfi B`LSlif B (4 )2BCJIVKLLR = B`ISkfi A`LSlif B (4 )2BCJIVKRLL = A`ISkif B`LSlif A Jlj `Sf Jlj `Sf Jlj `Sf Jlj `Sf A Kkj `Sf B A Kkj `Sf Kkj `Sf B Kkj `Sf mfi mfj D0 ; mfi mfj D0 ; D2 ; D2 ; (4 )2BCJISKLLL = (4 )2BCJISKRLR = A`ISkif A`LSlif B B`ISkfi B`LSlif A Jlj `Sf Jlj `Sf B Kkj `Sf A Kkj `Sf mfi mfj D0 ; mfi mfj D0 ; (4 )2BCJISKLLR = B`ISkfi A`LSlif A (4 )2BCJISKRLL = A`ISkif B`LSlif B (4 )2BCJITKLL = A`ISkif A`LSlif B (4 )2BCJITKRL = B`ISkfi B`LSlif A (4 )2BDJIVKLLL = A`ISkif A`LSlif A (4 )2BDJIVKRLR = B`ISkfi B`LSlif B Jlj `Sf Jlj `Sf Jlj `Sf Jlj `Sf Jkj `Sf Jkj `Sf B Kkj `Sf A B Kkj `Sf Kkj `Sf A Kkj `Sf A Klj `Sf B Klj `Sf D2 ; D2 ; mfi mfj D0 ; mfi mfj D0 ; mfi mfj D0 ; mfi mfj D0 ; (4 )2BDJIVKRLL = (4 )2BDJISKLLL = (4 )2BDJISKRLR = (4 )2BDJISKLLR = (4 )2BDJISKRLL = (4 )2BDJITKLL = (4 )2BDJITKRL = 12 A`ISkif A`LSlif B`JSkfj B`KSlfj mfi mfj D0 ; 12 B`ISkfi B`LSlif A`JSkfj A`KSlfj mfi mfj D0 ; 12 B`ISkfi A`LSlif A`JSkfj B`KSlfj D2 ; 12 A`ISkif B`LSlif B`JSkfj A`KSlfj D2 ; 18 A`ISkif A`LSlif B`JSkfj B`KSlfj mfi mfj D0 ; 18 B`ISkfi B`LSlif A`JSkfj A`KSlfj mfi mfj D0 : (D.6) L~; A`Sf ! V`L~N;L; B`Sf ! V`L~N;R. To obtain the actual MSSM contributions to the 4-lepton operators, one should add terms from eqs. (D.3), (D.4) with replacements f ! C; S ! ~; A`Sf ! V`~C;L; B`Sf ! V`~C;R and f ! N; S ! L~; A`Sf ! V`L~N;L; B`Sf ! V`L~N;R (summing over repeated indices of loop particles) and terms from eqs. (D.5), (D.6), substituting there only f ! N; S ! The contributions to 2-quark 2-lepton operators can be obtained from diagrams A) and C) by replacing `K and `L with qK and qL as de ned in eq. (2.28). Therefore, the expressions for Bq QXY can be obtained replacing vertices of leptons `K and ` L by the relevant quark-squark vertices. Such vertices are not listed in appendix A but can be found in refs. [43, 44]. The explicit form of ``dd box amplitudes can be also found in appendix A.3 of ref. [112]. E E ective lepton couplings in the leading MI order We list below the MI expanded expressions for the leptonic penguin and box diagram amplitudes. For penguins we follow the decomposition of eq. (4.2), with FXY denoting functions of avour diagonal SUSY parameters multiplying the respective slepton mass insertions: F XIJ = 1 (4 )2 F XIJLL To compactify the notation, we also introduce the abbreviation q where X; Y = L or R. E.1 E.1.1 Lepton-photon vertex Tensor (magnetic) couplings After performing MI expansion, one can see that terms coming from F A in eq. (2.4) are always suppressed by the powers of lepton Yukawa couplings or lepton masses, and may add to or cancel terms generated from F LB; F RB. Thus, in the expressions below we give the sum of both types of contributions. The chargino contributions contain only terms proportional to LL slepton mass insertions (see appendix C for the notation of divided di erences and curly brackets around the function arguments) (F LL)JCI = 2p2s2W e2v1YLJ MLIJL C11(jM2j ; fm~I ; m~J g) + C11(j j ; fm~I ; m~J g) C23(jM2j ; fm~I ; m~J g) + j j2 + jM2j2 + 2 M2 tan C11(fj j ; jM2jg; fm~I ; m~J g) (E.3) HJEP06(218)3 The non-vanishing neutralino contributions are: (F LL)JNI = M1?C12(fme~LI ; me~LJ ; me~RJ g; jM1j) ML2R JJ 2 2c2W MLIJL 2vp12 YLJ 2 c s W W2 (C12(fme~LI ; me~LJ g; j j) C23(fme~LI ; me~LJ g; jM2j)) 2 c s W C12(fme~LI ; me~LJ g; j j) C23(fme~LI ; me~LJ g; jM1j) + jM2j2 + ?M2? tan W2 C12(fme~LI ; me~LJ g; fj j; jM2jg) jM1j2 + ?M1? tan C12(fme~LJ ; me~LI g; fj j; jM1jg) (E.4) (F RR)JNI = 2 2c2W M RIJR M1?C12 (fme~LI ; me~RI ; me~RJ g; jM1j) ML2R II pv12 YLI (C12 (fme~RI ; me~RJ g; j j) 2C23(fme~RI ; me~RJ g; jM1j) + jM1j2 + ?M1? tan C12(fme~RI ; me~RJ g; f ; jM1jg) (F ALR)JNI = v2 v1 F 0 ALR N JI = e2v1 2p2c2W MLIJR M1?C12(fme~LI ; me~RJ g; jM1j) E.1.2 Vector couplings Loop functions C01 and C02 appearing in eq. (2.6) scale with the inverse of the squared SUSY scale M 2. Thus, only LL and RR terms contribute to the MI expanded expressions at the v2=M 2 order, as LR mass insertions always come with additional v=M powers. The non-vanishing chargino and neutralino contributions are: e 2 s W e 2 2e2 c W (V L LL)JCI = 2 MLIJL C01(jM2j; fm~I ; m~J g) (V L LL)JNI = (V R RR)JNI = 2s2W c2W MLIJL (c2W C02(jM2j; fme~LI ; me~LJ g) + s2W C02(jM1j; fme~LI ; me~LJ g) 2 M RIJR C02(jM1j; fme~RI ; me~RJ g) (E.5) (E.6) (FZL LL)JCI = MLIJL v22 D0(jM2j; j j; m~I ; m~J ) v22)E2(jM2j; jM2j; j j; m~I ; m~J ) jv2M2 + v1 j2 F2(jM2j; jM2j; j j; j j; m~I ; m~J ) Neutralino contributions have a more complicated form. They can be written down as: Lepton-Z0 vertex The leading v2=MS2USY terms in the e ective Z`I `J vertex de ned in eq. (2.7), expanded to the 1st order in LFV mass insertions, depend on divided di erences of scalar C0 and C2 3point functions. They can be expressed as higher point 1-loop functions (see appendices B and C). We give here the expressions using explicitly scalar 4-, 5- and 6-point functions D, E and F . The only non-negligible chargino contribution to Z``0 vertex read: v2 v1 (F Z0L ALR)JNI = v2 v1 (FZL BLR)INJ v2 v1 (F Z0R ALR)JNI = v2 v1 (FZR BLR)INJ 5 4s5W cW + (v12 + 1 2 e3p2 e3p2 e3p2 e3p2 where we de ned XZJINL1 = p2(s2W E2(jM1j; me~LJ ; me~LI ; me~RJ ; me~RJ ) (FZL LL)JNI = (FZR LL)JNI = (FZL RR)JNI = (FZR RR)JNI = 16s3W c3W MLIJL (XZJINL4 + XZJINL5 + XZIJNL5) 8sW c3W MLIJL (XZIJNR4 + XZJINR5 + XZIJNR5) 16s3W c3W M RIJR (XZJINL2 + XZJINL3 + XZIJNL3) 8sW c3W M RIJR (XZJINR2 + XZJINR3 + XZIJNR3) (FZL ALR)JNI = (FZL BLR)INJ = (FZR ALR)JNI = (FZR BLR)INJ = = = e3v1 4sW c3W q MLIJR XZJINL1 MLIJR XZJINR1 (E.7) (E.8) (E.9) (E.10) (E.11) + c2W E2(jM2j; me~LJ ; me~LI ; me~RJ ; me~RJ ))(ML2R)JJ + YlJ 2v1(M1 s2W D0(jM1j; j j; me~LI ; me~RJ ) c2W M2 D0(jM2j; j j; me~LI ; me~RJ )) s2W (v1M1 + v2 )(E2(jM1j; j j; me~LI ; me~RJ ; me~RJ ) + E2(jM1j; j j; j j; me~LI ; me~RJ )) + c2W (v1M2 + v2 )(E2(jM2j; j j; me~LI ; me~RJ ; me~RJ ) + E2(jM2j; j j; j j; me~LI ; me~RJ ))) + F2(jM1j; me~LJ ; me~LI; me~RJ ; me~RI; me~RI)) + c2W (F2(jM2j; me~LJ ; me~LI; me~RJ ; me~RJ ; me~RI) + F2(jM2j; me~LJ ; me~LI; me~RJ ; me~RI; me~RI))) XZJINL3 = YlI (ML2R)JJ 2v1(M1 s2W E0(jM1j; j j; me~LJ ; me~RJ ; me~RI) c2W M2 E0(jM2j; j j; me~LJ ; me~RJ ; me~RI)) + F2(jM1j; j j; me~LJ ; me~RJ ; me~RI; me~RI) + F2(jM1j; j j; j j; me~LJ ; me~RJ ; me~RI)) + F2(jM2j; j j; me~LJ ; me~RJ ; me~RI; me~RI) + F2(jM2j; j j; j j; me~LJ ; me~RJ ; me~RI))) XZJINL4 = p2s2W c2W 2 v2) s4W D0(jM1j; j j; me~LJ ; me~LI) + c4W D0(jM2j; j j; me~LJ ; me~LI) + 2s2W c2W Re (M1M2 )E0(jM1j; jM2j; j j; me~LJ ; me~LI) s4W (E2(jM1j; jM1j; j j; me~LJ ; me~LI) + E2(jM1j; j j; j j; me~LJ ; me~LI)) c4W (E2(jM2j; jM2j; j j; me~LJ ; me~LI) + E2(jM2j; j j; j j; me~LJ ; me~LI)) + 2s2W c2W E2(jM1j; jM2j; j j; me~LJ ; me~LI) + 12s4W (j j2 jM1j2)F2(jM1j; jM1j; j j; j j; me~LJ ; me~LI) + 12c4W (j j2 jM2j2)F2(jM2j; jM2j; j j; j j; me~LJ ; me~LI) + s2W c2W (j j2 Re (M1M2 ))F2(jM1j; jM2j; j j; j j; me~LJ ; me~LI) (E.14) XZJINL5 = YlI (ML2R)II 2v1(s2W M1 E0(jM1j; j j; me~LJ ; me~LI; me~RI) c2W M2 E0(jM2j; j j; me~LJ ; me~LI; me~RI)) s2W (v1M1 + v2 )(F2(jM1j; j j; me~LJ ; me~LI; me~RI; me~RI) + F2(jM1j; j j; j j; me~LJ ; me~LI; me~RI)) + c2W (v1M2 + v2 )(F2(jM2j; j j; me~LJ ; me~LI; me~RI; me~RI) + F2(jM2j; j j; j j; me~LJ ; me~LI; me~RI))) + p2 (s2W F2(jM1j; me~LJ ; me~LI; me~LI; me~RI; me~RI) + c2W F2(jM2j; me~LJ ; me~LI; me~LI; me~RI; me~RI)) (ML2R)II 2 XZJINR1 = YlI (2v1M1 D0(jM1j; j j; me~LI; me~RJ ) (v1M1 + v2 )(E2(jM1j; j j; me~LI; me~LI; me~RJ ) + E2(jM1j; j j; j j; me~LI; me~RJ ))) p 2 2 E2(jM1j; me~LI; me~LI; me~RJ ; me~RI)(ML2R)II (E.12) (E.13) (E.15) (E.16) XZJINR2 = jM1j2E0(jM1j; j j; j j; me~RJ ; me~RI ) 2 (jM1j2 + E2(jM1j; jM1j; j j; me~RJ ; me~RI ) + F2(jM1j; j j; j j; me~RJ ; me~RI ; me~RI ) j j2)(F2(jM1j; j j; j j; me~RJ ; me~RJ ; me~RI ) p XZJINR3 = YlI (ML2R)II (2v1M1 E0(jM1j; j j; me~LI ; me~RJ ; me~RI ) (v1M1 + v2 )(F2(jM1j; j j; me~LI ; me~LI ; me~RJ ; me~RI ) 2 2 F2(jM1j; me~LI ; me~LI ; me~RJ ; me~RI ; me~RI ) (ML2R)II 2 XZJINR4 = 2 2 (F2(jM1j; me~LJ ; me~LJ ; me~LI ; me~RJ ; me~RI ) + F2(jM1j; me~LJ ; me~LI ; me~LI ; me~RJ ; me~RI ))(ML2R)JJ (ML2R)II XZJINR5 = YlI (ML2R)JJ (2 v2E0(jM1j; j j; me~LJ ; me~LI ; me~RJ ) (v1M1 + v2 )(F2(jM1j; jM1j; j j; me~LJ ; me~LI ; me~RJ ) + F2(jM1j; j j; me~LJ ; me~LI ; me~RJ ; me~RJ ))) E.3 CP-even Higgs-lepton vertex (E.17) (E.18) (E.19) (E.20) (E.21) HJEP06(218)3 The dominant MI terms in the e ective CP-even Higgs - lepton couplings (see eq. (2.13)) can be split into four classes, 1 F IJK = h (4 )2 FhInJdK + FhIYJK + FhIdJeKc + FhImJK ; de ned as (below we give the sum of neutralino and chargino contributions, the latter appearing only as single term depending on sneutrino masses in eq. (E.23) and follow notation of eq. (4.2)): v: 1. Contributions proportional to non-holomorphic A0l trilinear terms,10 non-decoupling for (Fh0 nd ALR)IJK = (Fhnd LL)IJK = (Fhnd RR)IJK = e2(v1ZR2K v2ZR1K ) q e2(v1ZR2K e2(v1ZR2K p2c2W v1 p2c2W v1 p2c2W v1 v2ZR1K ) v2ZR1K ) MLIJR M1? C0(jM1j ; me~LI ; me~RJ ) (E.22) MLIJL M1?D0(jM1j ; me~LI ; me~LJ ; me~RJ )A0LJJ M RIJR M1? D0(jM1j ; me~LI ; me~RI ; me~RJ )A0LII (v1ZR2K v2ZR1K)=v1 = ( sin( cos( )= cos )= cos for K = 1 for K = 2 . 10For comparison with commonly used notation of the Higgs mixing angles, note that 2p2v1c2W s2W e 2 (FhY RR)IJK = v1ZR2K v2ZR1K M1? ?(D0(jM1j ; me~LI ; me~RI ; me~RJ ) D0(jM1j ; j j ; me~RI ; me~RJ )) M RIJR YLI (E.23) 3. Contributions decoupling as v2=MS2USY. We neglect here terms proportional to LL, RR, 0 LR as they are dominated by non-decoupling contributions listed in points 1) and 2). Only the terms proportional to ILJR and JRIL are generated starting at order v2=MS2USY. To simplify the expressions, below we also neglect terms additionally suppressed by lepton Yukawa couplings (this approximation becomes inaccurate for large and tan 30, when the diagonal LR elements of the slepton mass matrix proportional to Yl become imv2ZR2K )M1 (2s2W D0(jM1j; me~LI ; me~RJ ; me~RJ ) 2. Contributions suppressed by the lepton Yukawa couplings, also non-decoupling for (FhY LL)IJK = v1ZR2K v2ZR1K + 2D0(jM1j ; me~LI ; me~LJ ; me~RJ )) + 2D0(jM2j ; j j ; m~I ; m~J ))) MLIJL YLJ c2W M2? ?(D0(jM2j ; j j ; me~LI ; me~LJ ) 1)D0(jM1j; me~LI ; me~LI ; me~RJ )) + 2(v1ZR1K + v2ZR2K )(c2W (M1 + M2 )E2(jM1j; jM2j; j j; me~LI ; me~RJ ) 2s2W M1 E2(jM1j; jM1j; j j; me~LI ; me~RJ )) + 2(v2ZR1K + v1ZR2K ) M1 (c2W M2 E0(jM1j; jM2j; j j; me~LI ; me~RJ ) s2W M1 E0(jM1j; jM1j; j j; me~LI ; me~RJ )) + (c2W E2(jM1j; jM2j; j j; me~LI ; me~RJ ) s2W E2(jM1j; jM1j; j j; me~LI ; me~RJ ))) 2 2 pe2vc2W1 ZR1K M1 AII 2 l + AJJ 2 l E0(jM1j; me~LJ ; me~LI ; me~RJ ; me~RJ ) E0(jM1j; me~LI ; me~LI ; me~RJ ; me~RI ) ! q MLIJR (E.24) (Fhdec BLR)IJK = MLJRI 4. Contributions decoupling as Mh2(H)=MS2USY. Here, we do not show numerically small terms suppressed by lepton Yukawa couplings or avour-diagonal A terms: e2M 2 p2c2W (Fhm ALR)IJK = H0K ZR1K M1 C00(jM1j; me~RJ ; me~LI ) q MLIJR (E.25) (see eq. (B.7)). { 46 { For the processes considered in this article, the contribution from the LFV CP-odd Higgslepton vertex can become important only in the case of the three body charged lepton decays and only in the limit of MSUSY v, when photon, Z0 and box contributions decouple. Thus, we give here only the dominant non-decoupling terms for this vertex. FAIJ = (4 )2 FAIJnd + FAIJY + FAIJm : (E.26) As for CP-odd Higgs vertices, we give the sum of the neutralino and chargino contributions, the latter appearing only as single term depending on sneutrino masses in eq. (E.28): p2c2W cos (FAY LL)IJ = ie2 2p2c2W s2W cos (FAY RR)IJ = p2c2W cos ie2 1 + 2D0(jM1j ; me~LI ; me~LJ ; me~RJ )) + 2D0(jM2j ; j j ; m~I ; m~J ))) MLIJL YLJ D0(jM1j ; j j ; me~RI ; me~RJ )) M RIJR YLI M1? ?(D0(jM1j ; me~LI ; me~RI ; me~RJ ) 2. Contributions suppressed by lepton Yukawa couplings: (E.28) (E.29) s2W M1? ?(D0(jM1j ; j j ; me~LI ; me~LJ ) c2W M2? ?(D0(jM2j ; j j ; me~LI ; me~LJ ) 3. Contributions proportional to MA2 =MS2USY (see eq. (B.7) for the de nition of C00). As in eq. (E.25) we do not show numerically small terms suppressed by lepton Yukawa couplings or avour-diagonal A terms: (FAm ALR)IJK = ie2MA2 sin p2c2W M1 C00(jM1j; me~RJ ; me~LI ) MLIJR E.5 4-lepton box diagrams All genuine box diagram contributions listed in eqs. (D.3){(D.6) have negative mass dimension and without any cancellations explicitly decouple like v2=MS2USY. Thus, it is su cient to expand them only in the lowest order in chargino and neutralino mass insertions. Also the LR slepton mass insertions are always associated with additional factors of v=MSUSY. Thus in the leading v2=MS2USY order only LL and RR slepton mass insertion can contribute to formulae for box diagrams. Expressions listed below are valid only for tions of indices I = J; K = L or I = K; J = L - for these one would also take into account avour conserving diagrams. As mentioned in section 3.3, we do not consider MI expanded expressions for exotic L = 2 processes. The chargino diagrams contribute signi cantly only to the BV LL, all other contributions are at least double Yukawa suppressed and very small. The BV LL term is: (4 )2BVJILKLLC = E2(jM2j; jM2j; m~I ; m~J ; m~K ) KL JLILMLIJL + JL LKLI MLIKL + E2(jM2j; jM2j; m~J ; m~K ; m~L ) IK JLLLMLJLL + IJ LKLLMLKLL (E.30) HJEP06(218)3 (4 )2BVJILKRLN = Contributions arising from neutralino box diagrams, both normal and crossed added together, are listed below in eqs. (E.31){(E.36). We do not give here formulae for the neutralino contributions to BSLL; BSRR; BT L and BT R, as they are also double Yukawa suppressed and small. 4 16s4W c4W (4 )2BVJILKLLN = KL JLILMLIJL + IK JLLLMLJLL (3c4W E2(jM2j; jM2j; me~LI ; me~LJ ; me~LL ) + 3s4W E2(jM1j; jM1j; me~LI ; me~LJ ; me~LL ) 2c4W D0(jM2j; me~LI ; me~LJ ; me~LL ) 2s4W D0(jM1j; me~LI ; me~LJ ; me~LL ) + 4s2W c2W Re (M1M2 )E0(jM1j; jM2j; me~LI ; me~LJ ; me~LL ) + 2s2W c2W E2(jM1j; jM2j; me~LI ; me~LJ ; me~LL )) LKLLMLKLL (3c4W E2(jM2j; jM2j; me~LI ; me~LK ; me~LL ) + 3s4W E2(jM1j; jM1j; me~LI ; me~LK ; me~LL ) 2c4W D0(jM2j; me~LI ; me~LK ; me~LL ) 2s4W D0(jM1j; me~LI ; me~LK ; me~LL ) + 4s2W c2W Re (M1M2 )E0(jM1j; jM2j; me~LI ; me~LK ; me~LL ) + 2s2W c2W E2(jM1j; jM2j; me~LI ; me~LK ; me~LL )) (4 )2BVJIRKRLN = KL IRJRM RIJR + IK LRJRMRJRL (2D0(jM1j; me~RI ; me~RJ ; me~RL ) 3E2(jM1j; jM1j; me~RI ; me~RJ ; me~RL )) 3E2(jM1j; jM1j; me~RI ; me~RK ; me~RL ))) LRKR MRKRL (2D0(jM1j; me~RI ; me~RK ; me~RL ) 4c4W ( KL JLILMLIJL(2D0(jM1j; me~LI ; me~LJ ; me~RL ) 3E2(jM1j; jM1j; me~LI ; me~LJ ; me~RL )) + IJ LRKR MRKRL(2D0(jM1j; me~LI ; me~RK ; me~RL ) 3E0(jM1j; jM1j; me~LI ; me~RK ; me~RL )) (E.31) (E.32) (E.33) 2c4W ( JL LKLI MLIKL (2D0(jM1j; me~LI ; me~LK ; me~RL ) (4 )2BVJIRKLLN = (4 )2BSJRIKLNL = 4c4W ( KL IRJRM RIJR(2D0(jM1j; me~RI ; me~RJ ; me~LL ) 3E2(jM1j; jM1j; me~RI ; me~RJ ; me~LL )) + IJ LKLLMLKLL(2D0(jM1j; me~RI ; me~LK ; me~LL ) 3E0(jM1j; jM1j; me~RI ; me~LK ; me~LL )) 3E2(jM1j; jM1j; me~LI ; me~LK ; me~RL )) + IK LRJRMRLRJ(2D0(jM1j; me~LI ; me~RJ ; me~RL ) 3E2(jM1j; jM1j; me~LI ; me~RJ ; me~RL ))) 2c4W ( JL IRKRM RIKR(2D0(jM1j; me~RI ; me~RK ; me~LL ) 3E2(jM1j; jM1j; me~RI ; me~RK ; me~LL )) + IK JLLLMLLLJ (2D0(jM1j; me~RI ; me~LJ ; me~LL ) 3E2(jM1j; jM1j; me~RI ; me~LJ ; me~LL ))) (E.34) (E.35) (E.36) Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Andreas Crivellin, Zofia Fabisiewicz, Weronika Materkowska, Ulrich Nierste, Stefan Pokorski, Janusz Rosiek. Lepton flavour violation in the MSSM: exact diagonalization vs mass expansion, Journal of High Energy Physics, 2018, 3, DOI: 10.1007/JHEP06(2018)003