Lepton flavour violation in the MSSM

Abstract

We derive new constraints on the quantities δ

XY

ij
, X, Y = L,R, which parametrise the flavour-off-diagonal terms of the charged slepton mass matrix in the MSSM. Considering mass and anomalous
magnetic moment of the electron we obtain the bound | dLL13dRR13 | <~0.1 \left| {\delta_{LL}^{13}\delta_{RR}^{{13}}} \right| \lesssim 0.{1} for tan β = 50, which involves the poorly constrained element δ

RR

13. We improve the predictions for the decays τ → μγ, τ → eγ and μ → eγ by including two-loop corrections which are enhanced if tan β is large. The finite renormalisation of the PMNS matrix from soft SUSY-breaking terms is derived and applied to the charged-Higgs-lepton
vertex. We find that the experimental bound on BR(τ → eγ) severely limits the size of the MSSM loop correction to the PMNS element U

e3, which is important for the proper interpretation of a future U

e3 measurement. Subsequently we confront our new values for δ

LL

ij
with a GUT analysis. Further, we include the effects of dimension-5 Yukawa terms, which are needed to fix the Yukawa unification
of the first two generations. If universal supersymmetry breaking occurs above the GUT scale, we find the flavour structure
of the dimension-5 Yukawa couplings tightly constrained by μ → eγ.

KeywordsSupersymmetry Phenomenology

Full text

Page 1

arXiv:0910.2663v4 [hep-ph] 22 Jun 2010
TTP09-38
SFB/CPP-09-94
arXiv:0910.2663
October 2009
Lepton flavour violation in the MSSM
Jennifer Girrbach1, Susanne Mertens1,2, Ulrich Nierste1and Sören Wiesenfeldt1,3
1Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology,
Universität Karlsruhe, 76128 Karlsruhe, Germany
2Institut für Experimentelle Kernphysik, Karlsruhe Institute of Technology,
Universität Karlsruhe, 76128 Karlsruhe, Germany
3Helmholtz Association, Anna-Louisa-Karsch-Straße 2, 10178 Berlin, Germany
Abstract
We derive new constraints on the quantities δij
diagonal terms of the charged slepton mass matrix in the MSSM. Considering mass and anomalous
magnetic moment of the electron we obtain the bound |δ13
the poorly constrained element δ13
RR.We improve the predictions for the decays τ → µγ, τ →
eγ and µ → eγ by including two-loop corrections which are enhanced if tanβ is large. The finite
renormalisation of the PMNS matrix from soft SUSY-breaking terms is derived and applied to the
charged-Higgs-lepton vertex. We find that the experimental bound on BR(τ → eγ) severely limits
the size of the MSSM loop correction to the PMNS element Ue3, which is important for the proper
interpretation of a future Ue3measurement. Subsequently we confront our new values for δij
GUT analysis. Further, we include the effects of dimension-5 Yukawa terms, which are needed to fix
the Yukawa unification of the first two generations. If universal supersymmetry breaking occurs above
the GUT scale, we find the flavour structure of the dimension-5 Yukawa couplings tightly constrained
by µ → eγ.
XY, X,Y = L,R, which parametrise the flavour-off-
LLδ13
RR|<
∼0.1 for tanβ = 50, which involves
LLwith a
1. Introduction
Weak-scale supersymmetry (SUSY) is an attractive framework for physics beyond the standard model (SM) of
particle physics. The SM fields are promoted to superfields, with additional constituents of opposite spin. Due
to their identical couplings, they cancel the quadratic divergent corrections to the Higgs mass. Since none of the
SUSY partners have been observed in experiments, supersymmetry must be broken and the masses of the SUSY
partners are expected to be in the multi-GeV region.
A supersymmetric version of the standard model requires a second Higgs doublet in order to cancel the Higgsino-
related anomalies and to achieve electroweak symmetry breaking. At tree level, one of the Higgs doublets, Hu,
couples to the up-type particles, whereas the other doublet, Hd, couples to the down type particles. The Yukawa
couplings of the minimal supersymmetric standard model (MSSM) read
WMSSM= Yij
uuc
iQjHu+ Yij
ddc
iQjHd+ Yij
lec
iLjHd+ µHdHu.
(1a)
1

Page 2

1Introduction2
Neutrinos are massless in the MSSM; however, experiments and cosmological observations consistently point at
small but non-vanishing masses in the sub-eV region. We will therefore consider an extended MSSM with three
right handed neutrinos, where the Yukawa couplings are given by
W = WMSSM+ Yij
ννc
iLjHu+1
2Mij
Rνc
iνc
j.
(1b)
Here, Q and L denote the chiral superfields of the quark and lepton doublets and uc, dc, ecand νcthe up and
down-quark, electron and neutrino singlets, respectively. Each chiral superfield consists of a fermion and its scalar
partner, the sfermion. The Yukawa coupling matrices Yu,d,l,νare defined with the right-left (RL) convention. The
field νcis sterile under the SM group, so we allow for a Majorana mass term in addition to the Dirac coupling.
The respective mass matrix is denoted by MRand the scale of MRis undetermined but expected to be above the
electroweak scale, Mew(see Sec. 4).
The Higgs fields acquire the vacuum expectation values (vevs)
?Hu? = vu,?Hd? = vd.
(2)
where |vu|2+ |vd|2= v2= (174 GeV)2. The ratio of the two vevs is undetermined and defines the parameter
tanβ,
vu
vd
=: tanβ .
(3)
While tanβ is a free parameter of the theory, there exist lower and upper bounds on its value. Experimentally,
Higgs searches at LEP rule out the low-tanβ region in simple SUSY models [1]. This result fits nicely with the
theoretical expectation that the top Yukawa coupling should not be larger than one. The region of the MSSM
parameter space with large values of tanβ is of special importance for the flavour physics of quarks and leptons.
We therefore have a brief critical look at the upper bounds on this parameter: Demanding a perturbative bottom
Yukawa coupling ybnaïvely leads to an upper limit on tanβ of about 50 inferred from the tree-level relation
yb= −
mb
v · cosβ≈ −mb
v
tanβ .
(4)
Similarly, the MSSM provides a natural radiative breaking mechanism of the electroweak symmetry as long as
yb< ytat a low scale [2]. At tree level, the ratio of the Yukawa couplings is given by
1 >
????
yb
yt
????=mb
mt
tanβ .
(5)
Since mt(µ)/mb(µ) ≈ 60 at the electroweak scale, tanβ should not exceed this value.
Both arguments, however, do only hold at tree level. In particular, down quarks as well as charged leptons
couple to Hu via loops. As a result, if we take tanβ-enhanced contributions into account, an explicit mass
renormalisation changes the relation of Yukawa coupling and mass [3–5]. The tanβ enhancement of the bottom
coupling in Eq. (4) can be compensated; similarly, the ratio of the Yukawa couplings is changed due to an explicit
bottom quark mass renormalisation. We will find that values of tanβ up to 100 both provide small enough
Yukawa couplings and do not destroy natural electroweak symmetry breaking.
Large values for tanβ are interesting for two reasons. One, in various grand-unified theories (GUTs), top and
bottom Yukawa couplings are unified at a high scale. In this case, it is natural to expect tanβ = mt/mb, as shown
above. Two, many supersymmetric loop processes are tanβ-enhanced due to chirality-flipping loop processes with
supersymmetric particles in the loop. This enhancement can compensate the loop suppression and therefore large
values of tanβ lead to significant SUSY corrections.
In this paper, we will study the lepton sector in the (extended) MSSM. Since the neutrinos are massive, the
leptonic mixing matrix, UPMNS, is no longer trivial and leads to lepton flavour violation (LFV). In its standard
parametrisation, it reads
UPMNS=


1
0
0
00
c23
−s23
s23
c23




c13
0
0
1
0
s13eiδ
0
c13
−s13eiδ




c12
−s12
0
s12
c12
0
0
0
1




eiα1
0
0
2
00
0
1
eiα2
0
2

,
(6)

Page 3

1Introduction3
with sij = sinθij and cij = cosθij. The two phases α1,2appear if neutrinos are Majorana particles. They are
only measurable in processes which uncover the Majorana nature of neutrinos, such as neutrinoless double beta
decay.
The PMNS matrix allows for flavour transitions in the lepton sector, in particular neutrino oscillations, through
which its parameters are well constrained. Compared with the mixing angles of the quark mixing matrix, VCKM,
two mixing angles, namely the atmospheric and solar mixing angles, θ23= θatmand θ12= θsol, are surprisingly
large, whereas the third mixing angle is small. The current experimental status at 1σ level is as follows [6]:1
θ12= 34.5 ± 1.4 ,∆m2
21= 7.67+0.22
?
+2.46± 0.15 · 10−3eV2
−0.21· 10−5eV2,
−2.37± 0.15 · 10−3eV2
θ23= 42.3+5.1
−3.3,∆m2
31=
inverted hierarchy,
normal hierarchy,
θ13= 0.0+7.9
−0.0.
(7)
These values are determined by the atmospheric and solar mass splitting ∆m2
the absolute mass scale open. The pattern of mixing angles is close to tri-bimaximal, corresponding to θ23= 45◦,
θ12≃ 35◦, and θ13= 0◦[8]. Due to the smallness of θ13, the CP phase δ is unconstrained. Tri-bimaximal mixing
can be motivated by symmetries (see Ref. [9] and references therein), which constrain fundamental quantities
like Yukawa couplings or soft SUSY-breaking terms. Measurable quantities like Ue3usually do not point directly
to fundamental parameters, but are sensitive to corrections from all sectors of the theory. The analysis of such
corrections is therefore worthwhile. A large portion of this paper is devoted to the influence of supersymmetric
loops and higher-dimensional Yukawa terms on observables in the lepton sector of the MSSM.
In a supersymmetric framework, additional lepton flavour violation can be induced by off-diagonal entries in the
slepton mass matrix, which parametrise the lepton-slepton misalignment in a model independent way. However, a
generic structure of the soft masses is already excluded because too large decay rates for lj→ liγ would arise. To
avoid this flavour problem, the SUSY breaking mechanism is often assumed to be flavour blind, yielding universal
soft masses at a high scale. Then the PMNS matrix is the only source of flavour violation in the lepton sector, as
is the CKM matrix for the quarks; this ansatz is called minimal flavour violation. The soft terms do not cause
additional flavour violation and the various mass and coupling matrices are flavour-diagonal at some scale in the
basis of fermion mass eigenstates, e.g.,
atm= ∆m2
13, ∆m2
sol= ∆m2
21, leaving
m2
˜L= m2
˜ e= m2
01,m2
Hu= m2
Hd= m2
0,Al= A0Yl.
(8)
Here, m2
Higgs doublets, and Alis the trilinear coupling matrix of the leptons.
Even if the soft terms are universal at the high scale, renormalisation group equations (RGE) can induce non
vanishing off-diagonal entries in the slepton mass matrix at the electroweak scale. Lepton flavour violation can be
parametrised by non-vanishing δij
as the ratio of the flavor-violating elements of the slepton mass matrix (87) and an average slepton mass (see
Eq. (16)),
˜L,˜ edenote the soft mass matrices of the sleptons (see Eq. (87)), m2
Hthe analogous soft masses of the
XYat the electroweak scale in a model-independent way, where δij
XYis defined
δij
XY=
∆mij
?
XY
m2
iXm2
jY
,X,Y = L,R,i,j = 1,2,3(i ?= j) .
(9)
The flavour-off-diagonal elements ∆mij
Eq. (1a) is diagonal. According to the chiralities of the sfermion involved, there are four different types, δLL, δRR,
δLR, and δRL. The tolerated deviation from alignment can be quantified by upper bounds on δij
above and are already extensively studied in the literature (see for example [10–12] and references therein).
Being generically small, the sfermion propagator can be expanded in terms of these off-diagonal elements,
corresponding to the mass insertion approximation (MIA) [13,14]. Instead of diagonalising the full slepton mass
matrix and dealing with mass eigenstates and rotation matrices at the vertices, in MIA one faces flavour-diagonal
couplings and LFV appears as a mass insertion in the slepton propagator. This approach is valid as long as
???δij
XYare defined in a weak basis in which the lepton Yukawa matrix Yl in
XY, as discussed
XY
??? ≪ 1 and makes it possible to identify certain contributions easily. For a numerical analysis an exact
1Recently, a hint for non-zero θ13, sin2θ13= 0.016 ± 0.010 (1σ), was claimed in Ref. [7].

Page 4

2Upper bound for tanβ
4
diagonalisation of all mass matrices is, of course, possible. In Ref. [12] a systematic comparison between the full
computation and the MIA both in the slepton and chargino/neutralino sector clarifies the applicability of these
approximations.
This paper provides a comprehensive analysis of the lepton sector in the MSSM, focusing on the phenomeno-
logical constraints on the parameters δij
XYin Eq. (9). In Sec. 2 we briefly review the supersymmetric threshold
corrections to ybin Eq. (4) and relax the usually quoted upper bounds on tanβ. In Sec. 3 we derive new con-
straints on the δij
XY’s by studying loop corrections to the electron mass, finite renormalisations of the PMNS
matrix and the magnetic moment of the electron at large tanβ. As a byproduct we identify all tanβ-enhanced
corrections to the charged-Higgs coupling to leptons. We then improve the MSSM predictions for the decay rates
of lj→ liγ by including tanβ-enhanced two-loop corrections. In Sec. 4 we embed the MSSM into SO(10) GUT
scenarios and study RGE effects. Even for flavour-universal soft breaking terms at the GUT scale MGUTsizable
flavour violation can be generated between MGUTand the mass scales of the right-handed neutrinos [15]. Com-
paring the results to the upper bounds on |δij
scenarios. We include corrections to the Yukawa couplings from dimension-5 terms and constrain their possible
flavour misalignment with the dimension-4 Yukawa matrices. In Sec. 5 we summarise our results. Our notations
and conventions are listed in three appendices.
XY| found in Sec. 3 enables us to draw general conclusions on GUT
2. Upper bound for tanβ
The tree level mass of a particle receives corrections due to virtual processes. Supersymmetric loop corrections
are small unless a large value for tanβ compensates for the loop suppression. We will therefore start with a
discussion of tanβ-enhanced loops and derive a relation between Yukawa coupling and mass, coming from the
resummation of tanβ-enhanced corrections to the mass to all orders.
Corrections to the mass with more than one loop and more than one coupling to Higgs fields do not produce
further factors of tanβ. Nevertheless, tanβ-enhanced loops can become important in an explicit mass renor-
malisation, where tanβ-enhanced contributions to all orders are taken into account, because counterterms are
themselves tanβ-enhanced.
Down-quarks and charged leptons receive tanβ-enhanced corrections to their masses, due to loops with Hu.
As a coupling to Hudoes not exist at tree level, tanβ-enhanced loops are finite; there is no counterterm to this
loop induced coupling. Moreover, tanβ-enhanced contributions to self-energies do not decouple. The coupling of
Huto the charged slepton is proportional to µ with µ = O(MSUSY). On the other hand, the integration over the
loop momentum gives a factor 1/MSUSY. Thus the dependence on the SUSY mass scale cancels out. In the large
tanβ regime, neutralino-slepton and chargino-sneutrino loops can significantly change the relation between the
Yukawa couplings and masses [16–18],
m(0)
l
= mphys
l
+
∞
?
n=1
(−∆)nmphys
l
=mphys
1 + ∆l
l
,
(10)
where the corrections ∆l are related to the self-energy Σl as ∆l = −Σl/ml. This relation includes all tanβ-
enhanced contributions and can be determined by only calculating two diagrams, according to chargino-sneutrino
and neutralino-slepton loops (Figure 1). As a result, the relation between Yukawa coupling and physical mass is
given by
yl= −m(0)
l
vd
= −
mphys
l
vd(1 + ∆l).
(11)
The individual contributions from the two diagrams to the self-energy Σ = Σ˜ χ±+ Σ˜ χ0are
Σ˜ χ±
lL−lR=
1
16π2
?
?
j=1,2
M˜ χ±
jPLΓ
˜ χ±
l
j˜ νl∗
Γ
˜ χ±
l
j˜ νl
PLB0
?
M2
˜ χ±
j,m2
˜ νl
?
,
Σ˜ χ0
lL−lR=
1
16π2
i=1,2
4
?
j=1
M˜ χ0
jPLΓ
˜ χ0
l
j˜li∗
Γ
˜ χ0
l
j˜li
PLB0
?
M2
˜ χ0
j,m2
˜li
?
.
(12)

Page 5

3Constraints on the flavour-violating parameters5
lL
lR
˜ νl
˜ χ±
j
lL
lR
˜li
˜ χ0
j
Figure 1: Contribution to the self-energy Σ arising from chargino-sneutrino and neutralino-slepton loops
0 20 406080100
0.0
0.5
1.0
1.5
tanΒ
yb
Figure 2: Bottom quark coupling as a function of tanβ for ǫb= 0 (black-dashed) and ǫb= 0.008 (red-solid).
The loop integrals, couplings and rotation matrices are defined in the appendices A and B; M˜ χ± and M˜ χ0 denote
the chargino and neutralino masses, respectively. The tanβ-enhanced transitions require a chirality flip and are
given by
∆l= −Σl
ml
= ǫltanβ =α1
4πM1µ tanβ
−α2
?1
2f1
?1
?
M2
1,µ2,m2
˜lL
?
− f1
?
?
M2
1,µ2,m2
˜lR
?
+ f1
?
M2
1,m2
˜lL,m2
˜lR
??
4πM2µ tanβ
2f1
?
M2
2,µ2,m2
˜lL
+ f1
?M2
2,µ2,m2
˜ νl
??
,
(13)
with the loop-function f1(x,y,z) defined in Eq. (74).
The improved relation between the Yukawa coupling and the physical mass (11) relaxes the upper bound for
tanβ, as indicated in the Introduction. Equation (4) changes to [16]
yb= −mb
v
tanβ
1 + ǫbtanβ.
(14)
For down-quarks, the SUSY contributions are dominated by gluino loops such that
ǫb≃2αs
3πm˜ gµf1
?m˜b1,m˜b2,m˜ g
?.
(15)
A typical value is ǫb≈ 0.008, leading to yb= O(1) for tanβ ≈ 100 (see Fig. 2). Similarly, the bound for natural
electroweak symmetry breaking shifts to tanβ ? 100.
In the following, we will use this relation (11) to constrain lepton flavour violating parameters.
3. Constraints on the flavour-violating parameters
Various processes can be used to constrain lepton flavour violating (LFV) parameters. Remarkably, we can also
use lepton flavor conserving (LFC) observables, due to double lepton flavour violation (LFV). Two LFV vertices
lead to lepton flavor conservation (LFC) and so contribute to the LFC self-energies. In a similar manner, we will
consider multiple flavour changes contributing to the magnetic moment of the electron. In addition, we consider
LFV processes, in particular radiative decays.

Page 6

3Constraints on the flavour-violating parameters6
As mentioned in the Introduction, we introduce the dimensionless parameters δij
XYvia
∆mij
XY= δij
XY
?
mij
XY
?2
= δij
XY
?
iXand ∆mij
m2
iXm2
jY,X,Y = R,L ,i ?= j .
(16)
Note that ∆mij
the slepton mass matrix (87), so mij
The effects discussed in this section stem from chirally enhanced self-energies, which involve an extra factor of
tanβ compared to the tree-level result, analogous to ǫbin Eq. (15). Such effects have been widely studied before
in the quark sector, yet most authors have performed their studies in the decoupling limit MSUSY≫ v, where
MSUSY denotes the mass scale of the soft SUSY breaking parameters. (For a guide through the literature see
Ref. [18].) If one relaxes the condition MSUSY≫ v, novel effects (namely those which vanish like some power of
v/MSUSY) can be analysed. Analytical results covering the case MSUSY∼ v have been derived in Refs. [16,18–20],
numerical approaches were pursued in Refs. [21,22]. Superparticle contributions to physical processes vanish for
MSUSY→ ∞, typically as v2/M2
by combining the decoupling supersymmetric loop with resummation formulae valid for MSUSY ≫ v one can
correctly reproduce the resummed all-order result to leading non-vanishing order in v/MSUSY. This approach has
been used in an analysis of “flavoured” electric dipole moments in Ref. [23].
The possibility to constrain δij
XYthrough LFC processes has been pointed out in Ref. [24], which addresses
leptonic Kaon decays. Here we analyse the constraints from two LFC observables which have not been considered
before: In Sec. 3.1 we apply a naturalness argument to the electron mass, demanding that the supersymmetric
loop corrections are smaller than the measured value. In Sec. 3.5 we study the anomalous magnetic moment of
the electron.
XYhas mass-dimension two. Both m2
XYare the diagonal and off-diagonal entries of
XYis an average slepton mass.
SUSY, and can only be addressed with the methods of the latter papers. However,
3.1. Flavour-conserving self-energies
The masses of the SM fermions are protected from radiative corrections by the chiral symmetry Ψ → eiαγ5Ψ.
According to Weinberg, Susskind and ’t Hooft, a theory with small parameters is natural if the symmetry is
enhanced when these parameters vanish. The smallness of the parameters is then protected against large radiative
corrections by the concerned symmetry. This naturalness principle makes the smallness of the electron mass
natural. Radiative corrections are proportional to the electron mass itself δme∝ meln(Λ/me) and vanish for
me= 0. If such a small parameter is composed of some different terms and one does not want any form of fine-
tuning, one should require that all contributions should be roughly of the same order of magnitude; no accidental
cancellation between different terms should occur. Hence, the counterterm of the electron mass should be less
than the measured electron mass,
????
δme
mphys
e
????=
?????
m(0)
e
− mphys
mphys
e
e
?????< 1 .
(17)
This naturalness argument for the light fermion masses was already discussed in the pioneering study of Ref. [10].
Since the authors of this analysis want to provide a model-independent analysis on classes of SUSY theories they
consider only photinos and do not include flavour violation in the corrections to light fermion masses. Their
derived upper bound for δ11
LRdepends on the overall SUSY mass scale and actually becomes stronger for larger
SUSY masses. The case of radiatively generated fermion masses via soft trilinear terms is studied in [25] and
an updated version including two flavour-violating self-energies can be found in [26]. Here we concentrate on
the chirality-conserving flavour-violating mass insertions and use this argument to restrict the product δ13
which is then independent of the SUSY scale. Considering double lepton flavour violation, we demand that the
radiative corrections should not exceed the tree-level contribution. For the electron mass, the dominant diagrams
involve couplings to the third generation. As a result, we can constrain the product
has so far been unconstrained from radiative decay lj→ liγ as we will discuss shortly in the following Section 3.2.
This cancellation of the RR sensitivity is analysed in [11,12] with the conclusion that even a better experimental
sensitivity on BR(lj→ liγ) can not help to set strong constraints in the RR sector. However, with double mass
insertions and the bound from µ → eγ it is possible to derive bounds on products like δ23
The diagram in Figure 3(a) achieves an mτtanβ enhancement only if there is a helicity flip in the stau
propagator. Since α1/(4π) ≫ Y2
LLδ13
RR
??δ13
LLδ13
RR
??. Note that
??δ13
RR
??
LLδ31
RR.
e/?16π2?, the higgsino contribution is negligible. A chargino loop can also

Page 7

3Constraints on the flavour-violating parameters7
eL
eR
˜ τj
˜ eX
˜ eY
˜ χ0
(a)
eL
eR
H0∗
u
˜ τL
˜ τR
˜ eL
˜ eR
˜B
(b)
Figure 3: (a) LFC self-energy through double LFV and (b) dominant double LFV contribution to the electron
mass renormalisation.
SPS
m0
A0
m1/2
tanβ
sgn(µ)
µ(MZ)
1a
100
1b
200
234AB
145090400 500500
−100
250
000000
400
30
+1
507
300
10
+1
422
400
10
+1
516
300
50
+1
388
500
40
+1
614
500
10
+1
629
10
+1
352
Table 1: Snowmass Points and Slopes [27] and two additional scenarios (masses in GeV)
be neglected, because only left-left (LL) insertions for the sneutrinos can be performed and the helicity flip is
associated with an electron Yukawa coupling. The left-right (LR) insertions are either not associated with an
tanβ-enhanced contribution or suppressed by v/MSUSY, compared to right-right (RR) and LL-insertions. Thus
the dominant diagram involves a Bino and the scalar tau-Higgs coupling, as shown in Figure 3(b). For simplicity,
we choose all parameters real and obtain
ΣFV
e
≃α1
4πM1µmphys
τ
1 + ∆τ
tanβ
∆m13
LL∆m13
RRF0
?M2
?M2
RR. This term is proportional to mτ, in contrast
1,m2
˜ eL,m2
˜ eR,m2
˜ τL,m2
˜ τR
?
≃ −α1
4πM1µmphys
τ
1 + ∆τ
tanβ
∆m13
LL∆m13
RRf′′
1
1,m2
˜L,m2
˜ R
?.
(18)
For equal SUSY masses this simplifies to ΣFV
to the LFC self energy, which is proportional to me. Thus the counterterm receives an additional constant term
ΣFV
e ,
e
= −α1
48π
mτtanβ
1+∆τδ13
LLδ13
−imphys
→ mphys
l
+
iΣ(1)= −mphys
+δm(1)
l, one gets the second order contributions since the only real diagrams of order
n are one-loop diagrams in which a counterterm of order n − 1 is inserted.
We will use the on-shell renormalisation scheme, where the mass and the wave-function counterterms cancel
all loop contributions to the self-energy such that the pole of the lepton propagator is equal to the physical mass
of the lepton. Then we obtain the relation
l
∆l
+
iΣFV
l
+
−iδm(1)
l
!=
−imphys
l
Substituting mphys
ll
m(0)
l
= mphys
l
+
∞
?
n=1
δm(n)
l
=
mphys
l
1 + ∆l
+
ΣFV
l
1 + ∆l
.
(19)
For the numerical analysis, let us consider the mSUGRA scenario SPS4; its parameter values are given in
Table 1. For this model, the constraint reads (Figure 4)
??δ13
RRδ13
LL
??<
?
0.097,
if δ13
RRδ13
LL> 0
0.083,
if δ13
RRδ13
LL< 0
(20)

Page 8

3Constraints on the flavour-violating parameters8
30
30
30
30
100
100
100
100
250
250
250
250
500
500
500
500
750
750
750
750
1000
1000
1000
1000
?1.0
?0.50.0
δ13
RR
0.5 1.0
?1.0
?0.5
0.0
0.5
1.0
δ13
LL
Figure 4: Percentage deviation of the electron mass through SUSY loop correction in dependence of real δ13
δ13
LLfor SPS4.
RRand
scenario
x = 0.3
0.261
0.234
0.301
0.269
0.292
0.235
0.734
0.702
0.731
0.693
x = 1
0.073
0.059
0.083
0.067
0.082
0.067
0.210
0.190
0.205
0.179
x = 1.5
0.050
0.040
0.057
0.045
0.057
0.042
0.142
0.127
0.137
0.116
x = 3.0
0.026
0.023
0.029
0.024
0.031
0.027
0.071
0.064
0.067
0.054
for
LL> 0
RRδ13
δ13
LL> 0
δ13
LL< 0
δ13
LL> 0
δ13
LL< 0
δ13
LL> 0
δ13
LL< 0
δ13
LL> 0
δ13
LL< 0
1
M1= M2= mL= mR
δ13
δ13
RRδ13
RRδ13
RRδ13
RRδ13
RRδ13
RRδ13
RRδ13
RRδ13
RRδ13
LL< 0
2
3M1= M2= mL= mR
3
M1= M2= 3mL= mR
4
M1= M2=mL
3= mR
5
3M1= M2= mL= 3mR
Table 2: Different mass scenarios and the corresponding upper bounds for
right and left-handed slepton mass, respectively, M1and M2the bino and wino masses. In all scenarios, tanβ = 50
and sgn(µ) = +1. The upper line is valid for δ13
??δ13
LL< 0.
RRδ13
LL
??. mR,Ldenotes the average
RRδ13
LL> 0, the lower for δ13
RRδ13
As discussed above, the constraint is independent of the overall SUSY mass scale. It does, however, depend on
the ratio of the various masses, in particular x = µ/mR. mR,Ldenotes the average right and left-handed slepton
mass matrix, respectively. To be sure that this result is no special feature of SPS4, we consider other scenarios in
Table 2, which differ by ratios of the SUSY breaking masses. The sparticle mass spectrum at the electroweak scale
for SPS4 corresponds most likely to scenario 2 with x ≈ 0.9− 1. The bounds are very strong for larger values of
x and weaken for a small µ-parameter. The upper bounds do not change considerably for values of x larger than
one and therefore are relatively stable and independent of the parameter space. Electroweak symmetry breaking
yields a relation between the µ-Parameter, the mass of the Z boson and the two Higgs fields such that in absence
of any fine tuning µ2should be within an order of magnitude of M2
We will see in the following section that the dominating RR terms in the flavor violating self-energy τ → e
cancel in part of the parameter space. That is why so far no upper bound on δRRcould be derived [11,12]. In
these regions, we can use the constraint on
??δ13
3.2. Lepton-flavour violating self-energies
Z(also known as the µ problem).
RRδ13
LL
??as a restriction on δ13
RR.
Lepton flavour violating self-energies can be tanβ-enhanced and, moreover, they also have a non-decoupling
behaviour. They occur in the renormalisation of the PMNS matrix and lead to a correction of the radiative
decays lj→ liγ.

Page 9

3Constraints on the flavour-violating parameters9
τR
eL
˜ τR
˜ τL
˜ eL
˜B
τR
eL
˜ τL
˜ eL
˜H
˜B
vu
τR
eL
˜ τL
˜ eL
˜H
˜ W
vu
τR
eL
˜ ντ
˜ νe
˜H
˜ W
vu
Figure 5: Dominant diagrams for the neutralino-slepton and chargino-sneutrino loop for the τR→ eL-transition.
We consider all diagrams with one LFV MI in the slepton propagator and start with τR→ eL; the dominant
diagrams are shown in Figure 5. In fact, we can do an approximate diagonalisation of the neutralino mass matrix
(83) and use the MI approximation in the slepton propagator. Furthermore, we will choose µ to be real.
The dominant diagrams are proportional to the mass of the tau and sensitive to the LL element (see diagram
1 to 3 in Fig. (5)), while the RR dependence is suppressed,
Σ˜ χ0
τR−eL≃
mphys
τ
1 + ∆τ
µ tanβ m˜ eLm˜ τLδ31
LL
?
−α1
4πM1f2
?M2
α2
4πM2
1,m2
˜ eL,m2
˜ τR,m2
˜ τL
?
+
?
−1
2
α1
4πM1 +1
2
?
f2
?M2
1,µ2,m2
˜ τL,m2
˜ eL
??
.
(21)
This self-energy will contribute to the renormalisation of the PMNS matrix (Section 3.3). As in the previous
section, this contribution potentially leads to an upper bound on δ13
the PMNS element Ue3. Note that in Eq. (21) the LFC mass renormalisation is already taken into account.
The dominant contributions for the opposite helicity transition, τL→ eR, are analogous to the first and second
diagrams in Fig. 5,
LLwhen a naturalness argument is applied to
Σ˜ χ0
τL−eR≃α1
4π
mphys
τ
1 + ∆τ
M1µ tanβ m˜ eRm˜ τRδ31
RR
?f2
?M2
1,µ2,m2
˜ τR,m2
˜ eR
?− f2
?M2
1,m2
˜ eR,m2
˜ τR,m2
˜ τL
??.
(22)
They are sensitive to the RR element; however, the relative minus sign due to the different hyperchargespotentially
leads to cancellations. In this approximation, the RR sensitivity vanishes completely for µ = m˜ τLand hence no
upper bounds on δij
RRhas been derived, as mentioned in the previous section. In Ref. [12] the processes µ → eγ and
µ → e conversion in nuclei are combined. The corresponding one-loop amplitudes suffer from similar cancellations,
albeit in different regions of the parameter space leading to the constraint δ12
RR≤ 0.2.
Let us now turn to the chargino-sneutrino loops. As the left-handed charged sleptons and sneutrinos form a
doublet, they have the same SUSY breaking soft mass and therefore the same off-diagonal elements. The neutrino
is always left-handed so that the chargino loop can only be sensitive to the LL element and a chirality flip of the
charged lepton is needed. The higgsino component of the chargino couples to the right-handed lepton and the
wino part to the left-handed lepton. Thus, the self-energy is proportional to the mass of the right-handed lepton,
Σ˜ χ±
liL−ljR=
g2
16π2Yj
3
?
n=1
?
k=1,2
Z2k
−Z1k
+Zjn
νZin∗
ν
m˜ χkB0
?
m2
˜ νn,m2
˜ χ±
k
?
2,µ2?
≃α2
4π
mphys
j
1 + ∆lj
M2µ tanβ m˜ νim˜ νjδij
LLf2
?
m2
˜ νi,m2
˜ νj,M2
.
(23)
In the second line, we used the MI approximation in the chargino propagator and for the LFV (see last diagram
in Fig. (5)).
3.3. PMNS matrix renormalisation
Up to now, we have only an upper bound for the matrix element Ue3 and thus for the mixing angle θ13; the
best-fit value is at or close to zero (cf. Eq. (7)). It might well be that it vanishes at tree level due to a particular
symmetry and obtains a non-zero value due to corrections. So we can ask the question if threshold corrections to
the PMNS matrix could spoil the prediction θ13= 0◦at the weak scale. What does it mean for the physics at the

Page 10

3Constraints on the flavour-violating parameters10
high scale if experiment will tell us that θ13does not vanish? As before, we demand the absence of fine-tuning
for these corrections and therefore require that the SUSY loop contributions do not exceed the value of Ue3,
|δUe3| ≤
???Uphys
e3
???.
(24)
Then we can in principle use the smallness of Ue3to constrain δ13
In an effective field theory approach, the renormalisation of the PMNS matrix is done via rotation matrices
that diagonalise the mass matrix, which receives contribution from both the tree-level coupling of the fermions to
Hdand the loop-induced coupling to Hu(see Refs. [13,21,28,29] for the quark sector). Lepton flavour violating
self-energies induce off-diagonal entries in the mass matrix. In order to deal with physical fields, one has to rotate
them in flavour space to achieve a diagonal mass matrix.
As a drawback, the effective field theory method is only valid if the masses
of the supersymmetric particles in the loop are much larger than v = 174 GeV.
For sleptons and neutralinos this assumption is doubtful, so that we resort to
the diagrammatic method of Refs. [16–18] which does not rely on any hierarchy
between MSUSY and v. In our diagrammatic approach, we consider chargino
and neutralino loops in the external lepton propagator and resum all tanβ-
enhanced corrections explicitly. Once again the on-shell scheme is used. The
loop corrections are finite and the counterterms are defined such that they exactly
cancel the loop diagrams:
?
The first-order correction is displayed in the adjoining figure. The counterterm reads
LL.
νk
ll
lj
Σ
Wµ
U(0)= Uphys+
n
δU(n)= Uphys+ δU .
(25)
δU(1)
jk=
?
??
l?=j
Uphys
lk
(/ pj+ ml)
p2
j− m2
l
?Σlj−ll
?ΣljR−llL
ml
?∗
?∗,
≃
l?=jUphys
−?
lk
1
mj
j > l , i.e., heavy particle as external leg;
l?=jUphys
lk
1
?ΣljL−llR
?∗,j < l , i.e., heavy particle as internal propagator.
(26)
As for the mass renormalisation there are no genuine tanβ-enhanced two-loop diagrams. The corrections in
second order come from one-loop diagrams in which a counterterm of first order is inserted, corresponding to the
substitution Uphys
lk
→ Uphys
are not directly proportional to the PMNS-element under consideration. The sum of the counterterms has to
cancel the corrections up to that order, so at the nthorder, one gets


Now we can take the limit n → ∞ and obtain a linear system of equations for the U(0)elements (k = 1,2,3):
U(0)
ek+
lk
+δU(1)
lk. In contrast to the resummation of the mass counterterms, these counterterms
n
?
m=1
δU(m)
jk
=

?
−?
l?=j
?
Uphys
lk
?
+?n−1
Uphys
lk
m
δU(m)
lk
δU(m)
?
1
mj
?
?ΣljR−llL
ml
?∗,j > l
l?=j
+?n−1
mlk
1
?ΣljL−llR
?∗, j < l
.
(27)
1
mµΣµR−eLU(0)
1
mµΣeL−µRU(0)
1
mτΣeL−τRU(0)
µk+
1
mτΣτR−eLU(0)
1
mτΣτR−µLU(0)
1
mτΣµL−τRU(0)
τk=Uphys
ek
,
(28a)
U(0)
µk−
ek+
τk=Uphys
µk,
(28b)
U(0)
τk−
ek−
µk=Uphys
τk
.
(28c)
In the MSSM, we have Σ = Σ˜ χ0+ Σ˜ χ±. As shown above, ΣτR−eLis sensitive to δ13
to avoid accidental cancellations and set all off-diagonal elements to zero except for δ13
explicitly solve the linear system of equations
LLand so is δUe3. We aim
2. In this case we can
LL
U(0)
e3=
Uphys
e3
−
1
mτΣτR−eLUphys
τ3
2
1 +
???
1
mτΣτR−eL
???
.
(29)
2In [26] we study the case with nonvanishing δ13
LR.

Page 11

3Constraints on the flavour-violating parameters11
0.10.20.30.4 0.50.60.7
0
50
100
150
∆13LL
tanΒ ? 90
tanΒ ? 70
tanΒ ? 50
tanΒ ? 30
bound
Corrections in %
0.10.20.30.40.50.60.7
0
50
100
150
∆13LL
tanΒ ? 90
tanΒ ? 70
tanΒ ? 50
tanΒ ? 30
bound
Corrections in %
Figure 6:
??δU13??/Uphys
13
in percent as a function of δ13
LLand tanβ for θ13= 3◦(left) and θ13= 1◦(right).
By means of Eq. (24), we can in principle derive upper bounds for δ13
on tanβ and the assumed value for Uphys
After three years of running, the DOUBLE CHOOZ experiment will be sensitive to θ13= 3◦, which corresponds
to Ue3= 0.05. A future neutrino factory may probe θ13down to θ13= 0.6◦[30]. In general, even with future
experimental facilities, we can conclude that the corrections from SUSY loops to the small element Ue3 stay
unobservably small. This means at the same time that if some experiment measures θ13?= 0, this will not be
compatible with tri-bimaximal mixing at the high scale and moderate sparticle masses, since SUSY threshold
corrections cannot account for such an effect: Even for large tanβ the already existing constraints on δ13
τ → eγ are stronger assuming reasonable SUSY masses. However, since τ → eγ decouples, our method leads to
a sharper bound for very large SUSY masses, especially with θ13= 1◦and large tanβ.
LL. As shown in Figs. 6, they strongly depend
e3
.
LLfrom
3.4. Counterterms in the flavour basis and charged Higgs couplings
Neutrinos are both produced and detected as flavour eigenstates. In order to have flavour diagonal W couplings,
however, it is necessary to introduce counterterms, δVij, which cancel the LFV loops. By doing this you perform
a renormalisation of the unit matrix. In an effective field theory approach this is achieved via a wave function
renormalisation by rotating the lepton fields leading to a diagonal mass matrix and physical fields. This rotation
of the fields induce LFV in the charged Higgs coupling to lepton and neutrino and the same is true for the
counterterms in the diagrammatic approach.
The first-order correction is displayed in the figure above; the flavour-diagonal vertices do not get any coun-
terterms, since the external loops are already included in the mass renormalisation. We obtain
δV =



0−
1
mphys
µ
ΣeL−µR
0
−
−
1
mphys
τ
1
mphys
τ
ΣeL−τR
ΣµL−τR
0
1
mphys
µ
1
mphys
τ
ΣµR−eL
ΣτR−eL
1
mphys
τ
ΣτR−µR

.

(30)
You can translate this to the mass eigenstate basis used in Eq. (28) via δUik= δV∗
induce LFV in the charged Higgs coupling to lepton and neutrino, due to the different helicity structure of the
Higgs and W coupling and the different lepton masses. The H+eντvertex can be of particular importance, since
it is possible to pick up terms with a tau Yukawa coupling. As discussed before, this coupling is enhanced in the
large tanβ regime and can partly compensate the loop suppression factor.
The chargino contributions from the counterterm and the LFV loop cancel in the charged Higgs coupling as
the chargino loop is exactly proportional to the mass of the right handed lepton. Therefore only the neutralino
contributions remain.
ijU(0)
jk. These counterterms

Page 12

3Constraints on the flavour-violating parameters12
The charged Higgs coupling to electrons reads
iΓH+
eντ=
ig2
√2MW
tanβ
?
?
m(0)
e δV13+ m(0)
τ
ΣeR−τL
mphys
τ
?
?
=
ig2
√2MW
tanβ
?
?
−mphys
mphys
−mphys
mphys
e
τ
ΣeL−τR
1 + ∆e
+ΣeR−τL
1 + ∆τ
?
?
,
(31a)
iΓH+
eνµ=
ig2
√2MW
tanβm(0)
eδV12+ m(0)
µ
ΣeR−µL
mphys
µ
=
ig2
√2MW
?
tanβ
e
µ
ΣeL−µR
1 + ∆e
+ΣeR−µL
1 + ∆µ
,
(31b)
iΓH+
eνe=
ig2
√2MW
mphys
e
1 + ∆etanβ
?
1 +mphys
mphys
τ
e
tanβ
1 + ∆τ∆e
LR
,
(31c)
where ΣFV
lepton mass cancels out in ΣeR−ℓiL/mℓi, the LFV loop contributions with ντ and νµdiffer by a factor mτ/mµ.
Similarly, we obtain for the coupling to muons,
e
=
mphys
τ
1+∆τtanβ ∆e
LR. We see that the counterterms are suppressed with the electron mass. As the
iΓH+
µντ=
ig2
√2MW
tanβ
?
m(0)
µδV23+ m(0)
τ
ΣµR−τL
mphys
τ
?
=
ig2
√2MW
?
tanβ
?
−mphys
mphys
µ
τ
ΣµL−τR
1 + ∆µ
+ΣµR−τL
1 + ∆τ
?
,
(32a)
iΓH+
µνµ=
ig2
√2MW
mphys
µ
1 + ∆µtanβ
?
1 +mphys
mphys
τ
µ
tanβ
1 + ∆τ∆µ
LR
.
(32b)
iΓH+
µνe=
ig2
√2MW
tanβ
?
m(0)
µδV21− m(0)
e
ΣµR−eL
mphys
e
?
=
ig2
√2MW
tanβ
?
ΣµL−eR
1 + ∆µ
−mphys
mphys
e
µ
ΣµR−eL
1 + ∆e
?
,
(32c)
While the first term is similar to the couplings to electrons, the counterterm dominates over the loop contribution
if there is an electron neutrino in the final state.
Finally, for the τ coupling one finds
iΓH+
τντ=
ig2
√2MW
mphys
τ
1 + ∆τ
tanβ ,
(33a)
iΓH+
τνµ=
ig2
√2MW
tanβ
?
?
m(0)
τδV32− m(0)
µ
ΣτL−µR
mphys
τ
?
?
=
ig2
√2MW
tanβ
?
?ΣτR−eL
ΣτR−µL
1 + ∆τ
−mphys
mphys
µ
τ
ΣτL−µR
1 + ∆µ
?
?
,
(33b)
iΓH+
τνe=
ig2
√2MW
tanβm(0)
τδV31− m(0)
e
ΣτL−eR
mphys
τ
=
ig2
√2MW
tanβ
1 + ∆τ
−mphys
mphys
e
τ
ΣτL−eR
1 + ∆e
.
(33c)
The results of Eqs. (31)-(33) are given in Eqs. (92-95) of Ref. [23] for the decoupling limit MSUSY ≫ v. In
Appendix C of Ref. [23] an iterative procedure (analogous to the one in Ref. [21]) has been outlined which achieves
the all-order resummation of the tanβ-enhanced higher-order corrections. Eqs. (31)-(33) comprise analytical
formulae for the limits to which this iterative procedure converges.
The tanβ-enhanced lepton flavour violating Higgs couplings can become important in the leptonic decay of
charged Kaons, K → lν, where they potentially induce lepton non-universality. Then the current experimental
data and our fine-tuning argument together constrain the various terms in Eqs. (31), as they contribute to the
electron self-energy as well. In particular, if the second term in Eq. (31c) had a significant effect in the ratio
RK= Γ(K → eν)/Γ(K → µν), as was assumed in Ref. [24], ∆e
mass [31]. (The value ∆11
RL
a more than 2000% correction to the electron mass.) While in the improved analysis [32] the contribution of
ΣFV
e
to the electron mass was not considered, their scanned values of δ13
tuning argument. The scan respects
??δ13
is used in Ref. [33] to derive large, phenomenologically interesting values for δ13
LRwould give a large contribution to the electron
RRδ13
∧= ∆e
LR= 10−4[24] corresponds to δ13
LL≈ 2 in the SPS4 scenario and thus gives
LL,RRare in agreement with the fine-
LLδ13
RR
??≤ 0.25, in marginal agreement with our results of Sects. 3.1 and
3.5. The NA62 experiment at CERN aims to reduce the error of RKfrom 1.3% to 0.3%. This prospective error
LLand δ13
RR.
3.5. Anomalous Magnetic Moment of the Electron
The anomalous magnetic moment of the electron plays a central role in quantum electrodynamics. The precise
measurements provide the best source of the fine structure constant αemif one assumes the validity of QED [34].

Page 13

3Constraints on the flavour-violating parameters13
Conversely, one can use a value of αemfrom a (less precise) measurement and insert it into the theory prediction
for aeto probe new physics in the latter quantity. The most recent calculation yields [35]
ae= 1 159 652 182.79(7.71) × 10−12,
(34)
where the largest uncertainty comes from the second-best measurement of αemwhich is α−1
from a Rubidium atom experiment [36].
Supersymmetric contributions to the magnetic moment are usually small, due to the smallness of the electron
Yukawa coupling and the SUSY mass suppression. However, multiple flavour changes, resulting in a LFC loop,
insert the τ Yukawa coupling, which strongly enhances the amplitude. As a result, supersymmetric contributions
can be as large as O(10−12), comparable to the weak or hadronic contributions [35]. The amplitude can exceed
a 3σ deviation of the theoretical mean value, which enables us to constrain the LFV parameters δ13
In Ref. [11] the magnetic and electric dipole moments aiand diof the charged lepton ℓiwere calculated in the
MSSM, considering flavour-conserving and flavour-violating contributions within the mass insertion approxima-
tion. The authors found that the naïve mass scaling can be overcome with double mass insertions. However, in
their phenomenological analysis to constrain the flavour-violating parameters δij
experimental bounds on dµ and de but did not consider ae. Our consideration of ae adds a novel aspect to
the phenomenological study of LFV parameters in the MSSM and complements the analysis of Ref. [11] in this
respect.
em= 137.03599884(91)
LLand δ13
RR.
XY, they only used aµ and the
The supersymmetric contributions to the anomalous magnetic moment aeare generated by chargino and neu-
tralino loops, where the photon couples to any charged particle in the loop. The full analytic result can be found
in Ref. [37]. Here, we will neglect the terms which are both proportional to the electron mass and not (potentially)
tanβ-enhanced and are therefore left with
aχ0
e = −me
16π2
4
?
?
A=1
6
?
3
?
X=1
mχ0
3m2
A
˜lX
Re?NL
Re?CL
1AXNR∗
1AX
?FN
?FC
2(xAX),xAX=
m2
˜ χ0
A
m2
˜lX
,
(35a)
aχ±
e
=
me
16π2
A=1,2X=1
2mχ±
3m2
A
˜ νX
1AXCR∗
1AX
2(xAX),xAX=
m2
˜ χ±
A
m2
˜ νX
.
(35b)
The loop functions are listed in Eq. (80) and the couplings read [38]
NL
iAX= −√2g1
iAX=(Zi,X
?
Zi+3,X
L
?∗
Z1A
N+ Yli
?
Zi,X
L
?∗
+ Yli
Z3A
N =
?
?∗?
Γ˜ χ0
liR
A˜lX
?∗
,
(36a)
NR
L)∗
√2
?
g1
?Z1A
=
N
?∗+ g2
?
=
?Z2A
?∗
A˜ νX
liL
N
?∗?
?Z3A
N
Zi+3,X
L
?∗
=
?
Γ˜ χ0
liL
A˜lX
?∗
,
(36b)
CL
iAX= −YliZ2A
CR
iAX= −g2
−Zi,X
?Z1A
ν
Γ˜ χ±
liR
A˜ νX
,
(36c)
+
?∗Zi,X
ν
?
Γ˜ χ±
?∗
.
(36d)
The mixing matrices are defined in Appendix B. Note that they are 6 × 6 matrices, in order to allow for flavour
changes in the loop.
The dependence on tanβ in Eqs. (35) is hidden in the mixing matrices. In principle, tanβ comes from a
chirality flip on the selectron line and in the chargino case from the combination of vacuum expectation value vu
and the Yukawa coupling, yevu= metanβ. We can, however, simplify the expressions significantly as follows:
We assume a universal SUSY mass, real parameters and the same signs for M1and M2[39], then expand aein
powers of MW/MSUSYor 1/tanβ. Then we obtain
aχ0
e = sgn(µM2)g2
1− g2
192π2
g2
2
32π2
2
m2
e
M2
m2
SUSY
tanβ
?
1 + O
?
1
tanβ,
MW
MSUSY
MW
MSUSY
??
,
aχ±
e
= sgn(µM2)
e
M2
SUSY
tanβ
?
1 + O
?
1
tanβ,
??
.
(37)
The result is again finite. The 1/M2
thermore, we note that a large value for tanβ can dilute the 1/M2
computed with the exact formula in Eqs. (35).
SUSYdependence reflects the decoupling behaviour of supersymmetry. Fur-
SUSYsuppression. The numerical results are

Page 14

3 Constraints on the flavour-violating parameters14
(a) (b)
Figure 7: Supersymmetric contributions to ae as a function of δ13
level: δ13
MSUSY = 500, tanβ = 50, and µ = MSUSY. The light, medium, and dark grey regions correspond to the
theoretical 1σ, 2σ, and 3σ regions, respectively. In (b), the dashed curve shows the result without the mass
correction.
LLfor δ13
RR= 0.6 (green); 0.2 (red)) of Tab. 2 with
RRfor (a) scenario 5 (from steep to
RR= 0.6 (green); 0.4 (blue); 0.2 (red)); (b) scenario 2 (δ13
So far, the Yukawa couplings are unrenormalised; the inclusion of the mass renormalisation amounts to a loop
contribution to aewhich approximately grows as tan2β [17]. Diagonalising the mixing matrices perturbatively,
one finds a linear dependence on the Yukawa coupling of the remaining second terms of Eqs. (35). In this way
we find an easy expression, which takes the corrections into account by a global factor,
aSUSY,1L
e
+ aSUSY,∆e
e
= aSUSY,1L
e
?
1
1 + ∆e
?
,
(38)
where aSUSY,1L
For the numerical analysis, we only allow δ13
flavour violation. The theoretical uncertainty in Eq. (34) is taken as 1σ deviation and we require that the SUSY
contribution to aeis less than 3σ.
We show the results for our scenarios 2 and 5 (see Table 2) in Fig. 7. As δ13
becomes stronger and vice versa. The bound strongly depends on the SUSY mass. Since aedecouples for large
SUSY masses, the bounds become very loose for MSUSY? 500 GeV. On the other hand, small SUSY masses lead
to complex slepton masses, resulting in a lower bound on the SUSY mass. For this reason, the upper bounds
on δ13
RRare limited by the SUSY mass constraints. We find
coinciding with our non-decoupling bound in Eq. (20).
e
= aχ0
e + aχ±
e, as discussed in Ref. [17].
LLand δ13
RRto be non-zero such that they are the only source of
RRincreases, the bound on δ13
LL
LLand δ13
??δ13
LL· δ13
RR
??< 0.1 for MSUSY
<
∼500 GeV,
3.6. The radiative decay lj→ liγ
Since their SM branching ratios are tiny, supersymmetric contributions to lepton flavour violating decays li→ ljγ
can be sizable and vastly dominate over the SM values. As indicated above, these decays currently give the
best constraints on the left-left (LL) and left-right (LR) lepton flavour violating parameters. At one-loop level
and within MIA, li→ ljγ has for example extensively been studied in Ref. [11], constraining e.g. the mSUGRA
parameters M1 and mR. In this section, we compute the supersymmetric contributions to li → ljγ, including
both the mass renormalisation and the two-loop contributions coming from flavour-violating loops. The current
upper bounds for the branching ratios are listed in Table 3.

Page 15

3Constraints on the flavour-violating parameters15
Let us briefly summarise the formalism. Three SUSY diagrams contribute to the amplitude of lj → liγ,
corresponding to the coupling of the photon to lj, li, and the charged particle in the loop. The off shell amplitude
can be written as [41]
iM = ieǫµ∗ui(p − q)?q2γµ(AL
1PL+ AR
1PR) + mljiσµνqν(AL
2PL+ AR
2PR)?uj(p) ,
(39)
where ǫ∗is the photon polarisation vector. If the photon is on shell, the first part of the off-shell amplitude
vanishes.
The coefficients A contain chargino and neutralino contributions,
AL,R= A(˜ χ0)L,R+ A(˜ χ±)L,R,i = 1,2 ,
(40)
so ALis given by the sum of [42]
A(˜ χ0)L
2
=
1
32π2
4
?
A=1
6
?
X=1
2
?
1
m2
˜lX
?
NL
iAXNL∗
jAX
1
12FN
1(xAX) + NL
iAXNR∗
jAX
m˜ χ0
3mlj
A
FN
2(xAX)
?
,
(41)
A(˜ χ±)L
2
= −
1
32π2
A=1
3
?
X=1
1
m2
˜ νX
?
CL
iAXCL∗
jAX
1
12FC
1(xAX) + CL
iAXCR∗
jAX
2m˜ χ±
3mlj
A
FC
2(xAX)
?
,
(42)
with the couplings given in Eqs. (36). We get ARby simply interchanging L ↔ R.
Finally, the decay rate is given by
Γ(lj→ liγ) =
e2
16πm5
lj
???AL
2
??2+??AR
2
??2?
.
(43)
Both the flavor-conservingtransition li→ liγ and the flavour-changingself-energies are tanβ-enhanced. For this
reason, we do not only consider the effect of the mass renormalisation but also include the two-loop contributions.
Because of the double tanβ enhancement they can compete with the first non-vanishing contribution. As for the
corresponding counterterms, mass counterterms have to be inserted. In addition, wave-function renormalisation
counterterms play a role as the above-quoted result for lj→ liγ presumes an expansion in the external momenta of
the lepton. Therefore, to be consistent, the counterterm has to be given in higher order of the external momentum.
However, only the mass counterterm will be tanβ-enhanced because of the chirality flip involved. Corresponding
diagrams are shown in Figs. 8.
The wave-function and mass counterterms are given by:
?
where the fields and masses with a superscript 0 are the unrenormalised fields. In order to identify the countert-
erms, one first considers the kinetic and the mass term of the Lagrangian. The one-loop self-energy of the lepton
can be divided into a scalar and a vector-type part, where the latter can further be divided in a left-left and a
right-right transition,
l0
L=1 +1
2δlL
?
lL,l0
R=
?
1 +1
2δlR
?
lR,m0
l= ml+ δml,
(44)
iΣl(p) = iΣS
lL−lR(p) + i/ pΣlL−lL(p)PL+ i/ pΣlR−lR(p)PR.
(45)
Now we demand that the additional terms in the mass Lagrangian cancel the scalar-part of the one-loop self-
energy whereas the additional terms in the wave-function Lagrangian cancel the vector-type part. Therefore the
counterterms have to fulfil the following conditions:
δlL= −ΣlL−lL(p2= m2
δml= ΣlL−lR(p2= m2
l),δlR= −ΣlR−lR(p2= m2
l) ,
(46)
l) −ml
2(δlL+ δlR) .
(47)
experimental upper bounds
BR(µ → eγ)
BR(τ → eγ)
BR(τ → µγ)
1.2 · 10−11
1.1 · 10−7
6.8 · 10−8
Table 3: Current upper bounds for BR(lj→ liγ), j > i [40].