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 , 02093 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 massinsertion expansion of the amplitudes. The expansion in both avourviolating and avourconserving 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 threebody 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 nonholomorphic Aterms 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
Nondecoupling 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 Leptonphoton vertex
E.1.1
E.1.2
Tensor (magnetic) couplings
Vector couplings
E.2 LeptonZ0 vertex
E.3 CPeven Higgslepton vertex
E.4
CPodd Higgslepton vertex E.5 4lepton 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 avourviolating o diagonal terms in the
sfermion mass matrices but also in the
avour conserving but chirality violating entries
related to the trilinear Aterms 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 nonvanishing 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 nonholomorphic Aterms [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 2point, 3point, and 4point 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 3body decays of charged lepton as
well as the nondecoupling 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 oneloop level.1 We use the notation and conventions for the
MSSM as given in refs. [43, 44].2
In our analysis, we include the socalled nonholomorphic 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 onshell
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 (mirrorre ected selfenergy 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 treelevel leptonsleptonneutrino and leptonsneutrinochargino
vertices, i.e. the subscripts of V stand for the interacting particles and the chirality of the
lepton involved. The superscripts 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 3point 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
mirrorre ected selfenergy 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 oneloop 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 oneparticle irreducible (1PI)
vertex diagram and
V L(R) is the left(right)handed part of the lepton selfenergy
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 righthanded magnetic photonlepton couplings, which change
chirality, the Z`I `J coupling is chirality conserving. Therefore, the Wilson coe cients of
the lefthanded and righthanded couplings are not related to each other but rather satisfy
FZIJL(R) = FZJLI(R). In the mass eigenbasis the vectorial part of the lepton selfenergy 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 CPeven Higgs boson decays by H0K
! `I `J ,
where, following again the notation of [43, 44], H
CPodd 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 1loop expressions in the same
notation as used in the current paper are given in ref. [53] and the formulae that take into
account also nondecoupling chirally enhanced corrections and 2loop 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 mirrorre ected selfenergy diagram is omitted).
At the 1loop 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 selfenergies (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 CPeven Higgs mixing matrix (see appendix A) and the scalar
selfenergy 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 neutralinoslepton contributions to the 1PI vertex diagrams can be written as (the
symbols in square brackets denote common arguments of the 3point 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 charginosneutrino 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
4fermion 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
4lepton 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 fourlepton 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 fourlepton 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 treelevel 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 spinsummed 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 muoniumantimuonium
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 2lepton2quark 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 2lepton2quark 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
CPviolating 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 treelevel 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 CPeven and CPodd 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 1loop level by box diagrams suppressed by double avour changes, or at the
2loop 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 4lepton
box diagrams and from penguin diagrams (including vectorlike o shell photon couplings,
see eq. (2.5)) which in the limit of vanishing external momenta can be represented as the
4fermion contact interactions. A is the onshell 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 4lepton 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 onshell 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 1loop 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 vectorlike 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 vectorcurrent 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 downquark 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 squark 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 pionnucleon sigma terms see [79, 80].
(3.18)
(3.19)
(3.20)
(3.21)
For each process, we have given the exact oneloop 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 CPodd Higgs mass MA (which
also determines the heavy CP even and the charged Higgs masses). The decoupling
properties also serve as an important crosscheck 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 avourviolating
'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 (onshell) 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. Nondecoupling 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
SMlike Higgs boson h, decouple with the CPodd 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 nonholomorphic
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 Aterms (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 nonvanishing order in the slepton
LFV terms, taking into account the possible cancellations. Compared to previous
analyses, we consider the nonholomorphic 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 CPeven Higgs leptonic penguins and for the 4lepton 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 lefthanded (righthanded)
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) CPevenHiggsslepton and CPevenHiggssneutrino 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) CPoddHiggsslepton and CPoddHiggssneutrino 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) CPevenHiggsneutralino and CPevenHiggschargino 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) CPoddHiggsneutralino and CPoddHiggschargino 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 2point and 3point functions with nonvanishing
=
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 1loop 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 socalled 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 multiloop calculations.
However, it is particularly
e ective for 1loop 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 npoint scalar 1loop function is a (n + 1)point function (see eq. 3.13
in ref. [31] for generalisation to the case of nonvanishing 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 nonvanishing 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 charginosneutrino
and neutralinoslepton 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 4point 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 4lepton 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 2quark 2lepton 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 quarksquark 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
Leptonphoton 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 nonvanishing 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
nonvanishing 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:
LeptonZ0 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 1loop functions (see appendices B
and C). We give here the expressions using explicitly scalar 4, 5 and 6point functions D,
E and F .
The only nonnegligible 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
CPeven Higgslepton vertex
(E.17)
(E.18)
(E.19)
(E.20)
(E.21)
HJEP06(218)3
The dominant MI terms in the e ective CPeven 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 nonholomorphic A0l trilinear terms,10 nondecoupling 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 nondecoupling 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 nondecoupling 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 avourdiagonal 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 CPodd
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 nondecoupling terms for this vertex.
FAIJ =
(4 )2 FAIJnd + FAIJY + FAIJm :
(E.26)
As for CPodd 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 avourdiagonal A terms:
(FAm ALR)IJK =
ie2MA2 sin
p2c2W
M1 C00(jM1j; me~RJ ; me~LI )
MLIJR
E.5
4lepton 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 (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model
Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1
[arXiv:1207.7214] [INSPIRE].
[2] CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS
experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
[3] F. Borzumati and A. Masiero, Large Muon and electron Number Violations in Supergravity
Theories, Phys. Rev. Lett. 57 (1986) 961 [INSPIRE].
[4] J.A. Casas and A. Ibarra, Oscillating neutrinos and
! e , Nucl. Phys. B 618 (2001) 171
[hepph/0103065] [INSPIRE].
[5] I. Masina and C.A. Savoy, Sleptonarium: Constraints on the CP and
avor pattern of
scalar lepton masses, Nucl. Phys. B 661 (2003) 365 [hepph/0211283] [INSPIRE].
[6] A. Brignole and A. Rossi, Anatomy and phenomenology of mutau lepton avor violation in
the MSSM, Nucl. Phys. B 701 (2004) 3 [hepph/0404211] [INSPIRE].
[7] P. Paradisi, Constraints on SUSY lepton avor violation by rare processes, JHEP 10 (2005)
006 [hepph/0505046] [INSPIRE].
[8] T. Fukuyama, A. Ilakovac and T. Kikuchi, Lepton
avor violating leptonic/semileptonic
decays of charged leptons in the minimal supersymmetric standard model, Eur. Phys. J. C
56 (2008) 125 [hepph/0506295] [INSPIRE].
[9] P. Paradisi, Higgsmediated
and
! e transitions in II Higgs doublet model and
supersymmetry, JHEP 02 (2006) 050 [hepph/0508054] [INSPIRE].
[10] A. Dedes, S. Rimmer and J. Rosiek, Neutrino masses in the lepton number violating MSSM,
JHEP 08 (2006) 005 [hepph/0603225] [INSPIRE].
[11] A. Dedes, H.E. Haber and J. Rosiek, Seesaw mechanism in the sneutrino sector and its
consequences, JHEP 11 (2007) 059 [arXiv:0707.3718] [INSPIRE].
[12] S. Antusch and S.F. King, Lepton Flavour Violation in the Constrained MSSM with
Constrained Sequential Dominance, Phys. Lett. B 659 (2008) 640 [arXiv:0709.0666]
in constrained MSSMseesaw scenarios, JHEP 06 (2008) 079 [arXiv:0803.2039] [INSPIRE].
[14] A. Ilakovac and A. Pilaftsis, Supersymmetric Lepton Flavour Violation in LowScale Seesaw
Models, Phys. Rev. D 80 (2009) 091902 [arXiv:0904.2381] [INSPIRE].
[15] L. Calibbi, J. JonesPerez, A. Masiero, J.h. Park, W. Porod and O. Vives, FCNC and CP
Violation Observables in a SU(3) avoured MSSM, Nucl. Phys. B 831 (2010) 26
[arXiv:0907.4069] [INSPIRE].
[16] W. Altmannshofer, A.J. Buras, S. Gori, P. Paradisi and D.M. Straub, Anatomy and
Phenomenology of FCNC and CPV E ects in SUSY Theories, Nucl. Phys. B 830 (2010) 17
[arXiv:0909.1333] [INSPIRE].
[17] L. Calibbi, A. Faccia, A. Masiero and S.K. Vempati, Lepton avour violation from
SUSYGUTs: Where do we stand for MEG, PRISM/PRIME and a super avour factory,
Phys. Rev. D 74 (2006) 116002 [hepph/0605139] [INSPIRE].
[18] L. Calibbi, M. Frigerio, S. Lavignac and A. Romanino, Flavour violation in supersymmetric
SO(10) uni cation with a type II seesaw mechanism, JHEP 12 (2009) 057
[arXiv:0910.0377] [INSPIRE].
[19] J. Hisano, M. Nagai, P. Paradisi and Y. Shimizu, Waiting for
! e from the MEG
experiment, JHEP 12 (2009) 030 [arXiv:0904.2080] [INSPIRE].
[20] J. Girrbach, S. Mertens, U. Nierste and S. Wiesenfeldt, Lepton avour violation in the
MSSM, JHEP 05 (2010) 026 [arXiv:0910.2663] [INSPIRE].
[21] C. Biggio and L. Calibbi, Phenomenology of SUSY SU(5) with type I+III seesaw, JHEP 10
(2010) 037 [arXiv:1007.3750] [INSPIRE].
[22] J.N. Esteves, J.C. Romao, M. Hirsch, F. Staub and W. Porod, Supersymmetric typeIII
seesaw: lepton
avour violating decays and dark matter, Phys. Rev. D 83 (2011) 013003
[arXiv:1010.6000] [INSPIRE].
[23] A. Ilakovac, A. Pilaftsis and L. Popov, Charged lepton
avor violation in supersymmetric
lowscale seesaw models, Phys. Rev. D 87 (2013) 053014 [arXiv:1212.5939] [INSPIRE].
[24] M. AranaCatania, S. Heinemeyer and M.J. Herrero, New Constraints on General Slepton
Flavor Mixing, Phys. Rev. D 88 (2013) 015026 [arXiv:1304.2783] [INSPIRE].
[25] T. Goto, Y. Okada, T. Shindou, M. Tanaka and R. Watanabe, Lepton avor violation in
the supersymmetric seesaw model after the LHC 8 TeV run, Phys. Rev. D 91 (2015) 033007
[arXiv:1412.2530] [INSPIRE].
avor
violation in lowscale seesaw models: SUSY and nonSUSY contributions, JHEP 11 (2014)
048 [arXiv:1408.0138] [INSPIRE].
[27] A. Vicente, Lepton
avor violation beyond the MSSM, Adv. High Energy Phys. 2015 (2015)
686572 [arXiv:1503.08622] [INSPIRE].
[28] C. Bonilla, M.E. Krauss, T. Opferkuch and W. Porod, Perspectives for Detecting Lepton
Flavour Violation in LeftRight Symmetric Models, JHEP 03 (2017) 027
[arXiv:1611.07025] [INSPIRE].
Magnetic Moment and Lepton Flavor Violation, Phys. Rept. 731 (2018) 1
[arXiv:1610.06587] [INSPIRE].
[29] M. Lindner, M. Platscher and F.S. Queiroz, A Call for New Physics: The Muon Anomalous
[30] L. Calibbi and G. Signorelli, Charged Lepton Flavour Violation: An Experimental and
Theoretical Introduction, Riv. Nuovo Cim. 41 (2018) 1 [arXiv:1709.00294] [INSPIRE].
[31] A. Dedes, M. Paraskevas, J. Rosiek, K. Suxho and K. Tamvakis, Mass Insertions vs. Mass
Eigenstates calculations in Flavour Physics, JHEP 06 (2015) 151 [arXiv:1504.00960]
[INSPIRE].
[32] J. Rosiek, MassToMI  A Mathematica package for an automatic Mass Insertion
expansion, Comput. Phys. Commun. 201 (2016) 144 [arXiv:1509.05030] [INSPIRE].
[33] F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, A complete analysis of FCNC and
CP constraints in general SUSY extensions of the standard model, Nucl. Phys. B 477
(1996) 321 [hepph/9604387] [INSPIRE].
[34] M. Misiak, S. Pokorski and J. Rosiek, Supersymmetry and FCNC e ects, Adv. Ser. Direct.
High Energy Phys. 15 (1998) 795 [hepph/9703442] [INSPIRE].
[35] B. de Wit, M.T. Grisaru and M. Rocek, Nonholomorphic corrections to the one loop N = 2
superYangMills action, Phys. Lett. B 374 (1996) 297 [hepth/9601115] [INSPIRE].
[36] M. Matone, Modular invariance and structure of the exact Wilsonian action of N = 2
supersymmetric YangMills theory, Phys. Rev. Lett. 78 (1997) 1412 [hepth/9610204]
[INSPIRE].
[37] D. Bellisai, F. Fucito, M. Matone and G. Travaglini, Nonholomorphic terms in N = 2
SUSY Wilsonian actions and the renormalization group equation, Phys. Rev. D 56 (1997)
5218 [hepth/9706099] [INSPIRE].
[38] M. Dine and N. Seiberg, Comments on higher derivative operators in some SUSY eld
theories, Phys. Lett. B 409 (1997) 239 [hepth/9705057] [INSPIRE].
[39] N. ArkaniHamed and R. Rattazzi, Exact results for nonholomorphic masses in softly broken
supersymmetric gauge theories, Phys. Lett. B 454 (1999) 290 [hepth/9804068] [INSPIRE].
[40] F. GonzalezRey and M. Rocek, Nonholomorphic N = 2 terms in N = 4 SYM: One loop
calculation in N = 2 superspace, Phys. Lett. B 434 (1998) 303 [hepth/9804010] [INSPIRE].
[41] E.I. Buchbinder, I.L. Buchbinder and S.M. Kuzenko, Nonholomorphic e ective potential in
N = 4 SU(N ) SYM, Phys. Lett. B 446 (1999) 216 [hepth/9810239] [INSPIRE].
[42] S.P. Martin, Dimensionless supersymmetry breaking couplings, at directions, and the origin
of intermediate mass scales, Phys. Rev. D 61 (2000) 035004 [hepph/9907550] [INSPIRE].
[43] J. Rosiek, Complete Set of Feynman Rules for the Minimal Supersymmetric Extension of
[44] J. Rosiek, Complete set of Feynman rules for the MSSM: Erratum, hepph/9511250
[46] S.M. Barr and A. Zee, Electric Dipole Moment of the Electron and of the Neutron, Phys.
Rev. Lett. 65 (1990) 21 [Erratum ibid. 65 (1990) 2920] [INSPIRE].
[47] D. Chang, W.S. Hou and W.Y. Keung, Two loop contributions of avor changing neutral
! e , Phys. Rev. D 48 (1993) 217 [hepph/9302267] [INSPIRE].
[48] J. Hisano, M. Nagai and P. Paradisi, New Twoloop Contributions to Hadronic EDMs in
the MSSM, Phys. Lett. B 642 (2006) 510 [hepph/0606322] [INSPIRE].
[49] M. Jung and A. Pich, Electric Dipole Moments in TwoHiggsDoublet Models, JHEP 04
(2014) 076 [arXiv:1308.6283] [INSPIRE].
[50] T. Abe, J. Hisano, T. Kitahara and K. Tobioka, Gauge invariant BarrZee type
contributions to fermionic EDMs in the twoHiggs doublet models, JHEP 01 (2014) 106
[Erratum ibid. 04 (2016) 161] [arXiv:1311.4704] [INSPIRE].
[51] V. Ilisie, New BarrZee contributions to (g
2) in twoHiggsdoublet models, JHEP 04
(2015) 077 [arXiv:1502.04199] [INSPIRE].
[52] A. Crivellin, J. Heeck and P. Sto er, A perturbed leptonspeci c twoHiggsdoublet model
facing experimental hints for physics beyond the Standard Model, Phys. Rev. Lett. 116
(2016) 081801 [arXiv:1507.07567] [INSPIRE].
[53] A.J. Buras, P.H. Chankowski, J. Rosiek and L. Slawianowska,
Md;s; B0d; s !
+
and
B ! Xs in supersymmetry at large tan , Nucl. Phys. B 659 (2003) 3 [hepph/0210145]
[55] A. Crivellin, L. Hofer and J. Rosiek, Complete resummation of chirallyenhanced
loope ects in the MSSM with nonminimal sources of avorviolation, JHEP 07 (2011) 017
[arXiv:1103.4272] [INSPIRE].
[56] A. Crivellin and C. Greub, Twoloop supersymmetric QCD corrections to Higgsquarkquark
couplings in the generic MSSM, Phys. Rev. D 87 (2013) 015013 [Erratum ibid. D 87 (2013)
079901] [arXiv:1210.7453] [INSPIRE].
[57] L.J. Hall, R. Rattazzi and U. Sarid, The top quark mass in supersymmetric SO(10)
uni cation, Phys. Rev. D 50 (1994) 7048 [hepph/9306309] [INSPIRE].
[58] M. Carena, M. Olechowski, S. Pokorski and C.E.M. Wagner, Electroweak symmetry
breaking and bottom  top Yukawa uni cation, Nucl. Phys. B 426 (1994) 269
[hepph/9402253] [INSPIRE].
[hepph/9912516] [INSPIRE].
[59] M. Carena, D. Garcia, U. Nierste and C.E.M. Wagner, E ective Lagrangian for the tbH+
interaction in the MSSM and charged Higgs phenomenology, Nucl. Phys. B 577 (2000) 88
[60] C. Bobeth, T. Ewerth, F. Kruger and J. Urban, Analysis of neutral Higgs boson
contributions to the decays B( s) ! `+` and B ! K`+` , Phys. Rev. D 64 (2001) 074014
[61] K.S. Babu and C.F. Kolda, Higgs mediated B0 !
Rev. Lett. 84 (2000) 228 [hepph/9909476] [INSPIRE].
in minimal supersymmetry, Phys.
[62] G. Isidori and A. Retico, Scalar avor changing neutral currents in the large tan beta limit,
JHEP 11 (2001) 001 [hepph/0110121] [INSPIRE].
[63] A. Dedes and A. Pilaftsis, Resummed e ective Lagrangian for Higgs mediated FCNC
interactions in the CP violating MSSM, Phys. Rev. D 67 (2003) 015012 [hepph/0209306]
[64] L. Hofer, U. Nierste and D. Scherer, Resummation of tanbetaenhanced supersymmetric
loop corrections beyond the decoupling limit, JHEP 10 (2009) 081 [arXiv:0907.5408]
[65] D. Noth and M. Spira, Supersymmetric Higgs Yukawa Couplings to Bottom Quarks at
nexttonexttoleading Order, JHEP 06 (2011) 084 [arXiv:1001.1935] [INSPIRE].
[66] J. Rosiek, P. Chankowski, A. Dedes, S. Jager and P. Tanedo, SUSY FLAVOR: A
Computational Tool for FCNC and CPviolating Processes in the MSSM, Comput. Phys.
Commun. 181 (2010) 2180 [arXiv:1003.4260] [INSPIRE].
[67] A. Crivellin, J. Rosiek, P.H. Chankowski, A. Dedes, S. Jaeger and P. Tanedo,
SUSY FLAVOR v2: A computational tool for FCNC and CPviolating processes in the
MSSM, Comput. Phys. Commun. 184 (2013) 1004 [arXiv:1203.5023] [INSPIRE].
[68] J. Rosiek, SUSY FLAVOR v2.5: a computational tool for FCNC and CPviolating processes
in the MSSM, Comput. Phys. Commun. 188 (2015) 208 [arXiv:1410.0606] [INSPIRE].
[69] Particle Data Group collaboration, K.A. Olive et al., Review of Particle Physics, Chin.
Phys. C 38 (2014) 090001 [INSPIRE].
[70] S.A.R. Ellis and A. Pierce, Impact of Future Lepton Flavor Violation Measurements in the
Minimal Supersymmetric Standard Model, Phys. Rev. D 94 (2016) 015014
[arXiv:1604.01419] [INSPIRE].
[71] R. Kitano, M. Koike and Y. Okada, Detailed calculation of lepton
avor violating muon
electron conversion rate for various nuclei, Phys. Rev. D 66 (2002) 096002 [Erratum ibid.
D 76 (2007) 059902] [hepph/0203110] [INSPIRE].
[72] A. Crivellin, S. Davidson, G.M. Pruna and A. Signer, Renormalisationgroup improved
analysis of
! e processes in a systematic e ective eldtheory approach, JHEP 05 (2017)
117 [arXiv:1702.03020] [INSPIRE].
[73] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Remarks on Higgs Boson Interactions
with Nucleons, Phys. Lett. B 78 (1978) 443 [INSPIRE].
[74] V. Cirigliano, R. Kitano, Y. Okada and P. Tuzon, On the model discriminating power of
! e conversion in nuclei, Phys. Rev. D 80 (2009) 013002 [arXiv:0904.0957] [INSPIRE].
[75] A. Crivellin, M. Hoferichter and M. Procura, Accurate evaluation of hadronic uncertainties
in spinindependent WIMPnucleon scattering: Disentangling two and three avor e ects,
Phys. Rev. D 89 (2014) 054021 [arXiv:1312.4951] [INSPIRE].
[76] A. Crivellin, M. Hoferichter and M. Procura, Improved predictions for
nuclei and Higgsinduced lepton
[arXiv:1404.7134] [INSPIRE].
Determination of the PionNucleon
Term from RoySteiner Equations, Phys. Rev. Lett.
115 (2015) 092301 [arXiv:1506.04142] [INSPIRE].
[78] P. Junnarkar and A. WalkerLoud, Scalar strange content of the nucleon from lattice QCD,
Phys. Rev. D 87 (2013) 114510 [arXiv:1301.1114] [INSPIRE].
[79] J.M. Alarcon, J. Martin Camalich and J.A. Oller, The chiral representation of the N
[80] J.M. Alarcon, L.S. Geng, J. Martin Camalich and J.A. Oller, The strangeness content of the
nucleon from e ective
eld theory and phenomenology, Phys. Lett. B 730 (2014) 342
[arXiv:1209.2870] [INSPIRE].
[81] A. Czarnecki, W.J. Marciano and K. Melnikov, Coherent muon electron conversion in
muonic atoms, AIP Conf. Proc. 435 (1998) 409 [hepph/9801218] [INSPIRE].
[82] T. Suzuki, D.F. Measday and J.P. Roalsvig, Total Nuclear Capture Rates for Negative
Muons, Phys. Rev. C 35 (1987) 2212 [INSPIRE].
[83] J. Rosiek, General Mass Insertion Expansion in Flavor Physics, in 5th Large Hadron
Collider Physics Conference (LHCP 2017) Shanghai, China, May 15{20, 2017,
arXiv:1708.06818 [INSPIRE].
[84] E. Arganda, M.J. Herrero, R. Morales and A. Szynkman, Analysis of the h; H; A !
decays induced from SUSY loops within the Mass Insertion Approximation, JHEP 03
(2016) 055 [arXiv:1510.04685] [INSPIRE].
and
[85] BaBar collaboration, B. Aubert et al., Searches for Lepton Flavor Violation in the Decays
, Phys. Rev. Lett. 104 (2010) 021802 [arXiv:0908.2381]
arXiv:1801.04688 [INSPIRE].
at p
[arXiv:1408.5774] [INSPIRE].
[86] Belle and BelleII collaborations, K. Hayasaka, Results and prospects on lepton avor
violation at Belle/Belle II, J. Phys. Conf. Ser. 408 (2013) 012069 [INSPIRE].
[87] G. Cavoto, A. Papa, F. Renga, E. Ripiccini and C. Voena, The quest for
! e and its
experimental limiting factors at future high intensity muon beams, Eur. Phys. J. C 78
(2018) 37 [arXiv:1707.01805] [INSPIRE].
[88] Belle collaboration, K. Hayasaka et al., New Search for
and
! e Decays at
Belle, Phys. Lett. B 666 (2008) 16 [arXiv:0705.0650] [INSPIRE].
[89] MEG collaboration, J. Adam et al., New constraint on the existence of the + ! e+
decay, Phys. Rev. Lett. 110 (2013) 201801 [arXiv:1303.0754] [INSPIRE].
[90] MEG II collaboration, A.M. Baldini et al., The design of the MEG II experiment,
[91] ATLAS collaboration, Search for the lepton
avor violating decay Z ! e in pp collisions
s = 8 TeV with the ATLAS detector, Phys. Rev. D 90 (2014) 072010
Phys. B 299 (1988) 1 [INSPIRE].
11 (1975) 2856 [INSPIRE].
[93] SINDRUM collaboration, U. Bellgardt et al., Search for the Decay + ! e+e+e , Nucl.
[94] A. Blondel et al., Research Proposal for an Experiment to Search for the Decay
[96] K. Hayasaka et al., Search for Lepton Flavor Violating Tau Decays into Three Leptons with
HJEP06(218)3
719 Million Produced Tau+Tau Pairs, Phys. Lett. B 687 (2010) 139 [arXiv:1001.3221]
and emu at p
s = 8 TeV, CMSPASHIG14040 [INSPIRE].
e in protonproton collisions at p
s = 13 TeV, [arXiv:1712.07173] [INSPIRE].
[97] CMS collaboration, Search for lepton avour violating decays of the Higgs boson to
and
[98] CMS collaboration, Search for lepton avourviolating decays of the Higgs boson to etau
[99] SINDRUM II collaboration, W.H. Bertl et al., A search for muon to electron conversion in
muonic gold, Eur. Phys. J. C 47 (2006) 337 [INSPIRE].
[100] Mu2e collaboration, R.J. Abrams et al., Mu2e Conceptual Design Report,
[101] T. Appelquist and J. Carazzone, Infrared Singularities and Massive Fields, Phys. Rev. D
[102] E. Arganda, A.M. Curiel, M.J. Herrero and D. Temes, Lepton
avor violating Higgs boson
decays from massive seesaw neutrinos, Phys. Rev. D 71 (2005) 035011 [hepph/0407302]
[103] A. Azatov, S. Chang, N. Craig and J. Galloway, Higgs ts preference for suppressed
downtype couplings: Implications for supersymmetry, Phys. Rev. D 86 (2012) 075033
[arXiv:1206.1058] [INSPIRE].
[104] C. Petersson, A. Romagnoni and R. Torre, Liberating Higgs couplings in supersymmetry,
Phys. Rev. D 87 (2013) 013008 [arXiv:1211.2114] [INSPIRE].
[105] A. Bartl, H. Eberl, E. Ginina, K. Hidaka and W. Majerotto, h0 ! cc as a test case for
quark
avor violation in the MSSM, Phys. Rev. D 91 (2015) 015007 [arXiv:1411.2840]
[106] M. AranaCatania, E. Arganda and M.J. Herrero, Nondecoupling SUSY in LFV Higgs
decays: a window to new physics at the LHC, JHEP 09 (2013) 160 [Erratum ibid. 10
(2015) 192] [arXiv:1304.3371] [INSPIRE].
[107] D. Aloni, Y. Nir and E. Stamou, Large BR(h !
) in the MSSM?, JHEP 04 (2016) 162
[arXiv:1511.00979] [INSPIRE].
[108] G. Barenboim, C. Bosch, J.S. Lee, M.L. LopezIban~ez and O. Vives, Flavorchanging Higgs
boson decays into bottom and strange quarks in supersymmetric models, Phys. Rev. D 92
(2015) 095017 [arXiv:1507.08304] [INSPIRE].
[109] M.E. Gomez, S. Heinemeyer and M. Rehman, Lepton
avor violating Higgs Boson Decays
in Supersymmetric High Scale Seesaw Models, arXiv:1703.02229 [INSPIRE].
Higgs boson in MSSM, JHEP 11 (2014) 137 [arXiv:1409.6546] [INSPIRE].
the Tevatron and LHC, Phys. Rev. D 79 (2009) 055006 [arXiv:0812.4320] [INSPIRE].
the Standard Model , Phys. Rev. D 41 ( 1990 ) 3464 [INSPIRE].
[45] B.C. Allanach et al., SUSY Les Houches Accord 2, Comput. Phys. Commun . 180 ( 2009 ) 8 avor violation , Phys. Rev. D 89 ( 2014 ) 093024 [77] M. Hoferichter , J. Ruiz de Elvira, B. Kubis and U.G. Mei ner, HighPrecision [ 95 ] Mu3e collaboration, N. Berger, The Mu3e Experiment, Nucl . Phys. Proc. Suppl . 248  250 [110] A. Dedes , M. Paraskevas , J. Rosiek , K. Suxho and K. Tamvakis , Rare Topquark Decays to [111] A. Crivellin , A. Kokulu and C. Greub , Flavorphenomenology of twoHiggsdoublet models with generic Yukawa structure , Phys. Rev. D 87 ( 2013 ) 094031 [arXiv: 1303 .5877] [112] A. Dedes , J. Rosiek and P. Tanedo , Complete OneLoop MSSM Predictions for B ! ``0 at