Page 1
arXiv:1002.0227v2 [hep-ph] 6 Apr 2010
TTP10-11
Constraining the MSSM sfermion mass matrices with light fermion masses
Andreas Crivellin and Jennifer Girrbach
Institut f¨ ur Theoretische Teilchenphysik
Karlsruhe Institute of Technology, Universit¨ at Karlsruhe,
D-76128 Karlsruhe, Germany
(Dated: February 2010)
We study the finite supersymmetric loop corrections to fermion masses and mixing matrices in
the generic MSSM. In this context the effects of non-decoupling chirally-enhanced self-energies are
studied beyond leading order in perturbation theory. These NLO corrections are not only necessary
for the renormalization of the CKM matrix to be unitary, they are also numerically important
for the light fermion masses. Focusing on the tri-linear A-terms with generic flavor-structure we
derive very strong bounds on the chirality-changing mass insertions δf LR,RL
naturalness criterion. In particular, the NLO corrections to the up quark mass allow us to constrain
the unbounded element δu RL
13
if at the same time δu LR
for single-top production at the LHC.
IJ
by applying ’t Hooft’s
13
is unequal to zero. Our result is important
PACS numbers: 11.10.Gh,12.15.Ff,12.60.Jv,14.80.Ly
I.INTRODUCTION
A major challenge in particle physics is to understand
the pattern of fermion masses and mixing angles. With
the discovery of neutrino oscillations flavor has become
even more mysterious since the nearly tri-bimaximal mix-
ing strongly differ from the quark sector.
mal supersymmetric standard model (MSSM) does not
provide insight into the flavor problem by contrast the
generic MSSM contains even new sources of flavor and
chirality violation, stemming from the supersymmetry-
breaking sector which are the sources of the so-called su-
persymmetric flavor problem. The origin of these flavor-
violating terms is obvious: In the standard model (SM)
the quark and lepton Yukawa matrices are diagonalized
by unitary rotations in flavor space and the resulting ba-
sis defines the mass eigenstates. If the same rotations
are carried out on the squark fields of the MSSM, one
obtains the super–CKM/PMNS basis in which no tree–
level FCNC couplings are present. However, neither the
3 × 3 mass terms m2
handed and right–handed sfermions nor the tri-linear
Higgs–sfermion–sfermion couplings are necessarily diag-
onal in this basis. The tri-linear QHdAddR, QHuAuuR
and LHdAleRterms induce mixing between left–handed
and right–handed sfermions after the Higgs doublets Hd
and Hu acquire their vacuum expectation values (vevs)
vd and vu, respectively. In the current era of precision
flavor physics stringent bounds on these parameters have
been derived from FCNC processes in the quark and in
the lepton sector, by requiring that the gluino–squark
loops and chargino–sneutrinos/neutralino–slepton loops
do not exceed the measured values of the considered ob-
servables [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
However, in [14, 15] it is shown that all flavor viola-
tion in the quark sector can solely originate from trilinear
SUSY breaking terms because all FCNC bounds are sat-
isfied for MSUSY ≥ 500GeV. Dimensionless quantities
are commonly defined in the mass insertion parametriza-
The mini-
˜ Q, m2
˜ u, m2
˜d, m2
˜Land m2
˜ eof the left–
tion as:
δf XY
IJ
=
?∆m2
m2
F
?IJ
XY
?
˜fIXm2
˜fJY
. (1)
In Eq. (1) I and J are flavour indices running from 1 to
3, X,Y denote the chiralities L and R,?∆m2
mass matrix (see Appendix A2) and m˜f2
corresponding diagonal ones. In this article we are going
to complement the analysis of [14] with respect to three
important points:
F
?IJ
XYwith
F = U,D,L is the off-diagonal element of the sfermion
IX, m2
˜ fJYare the
• Electroweak correction are taken into account.
Therefore, we are able to constrain also the flavor-
violating and chirality-changing terms in the lepton
sector.
• The constraints on the flavor-diagonal mass inser-
tions δu,d,lLR
11,22
are obtained from the requirement
that the corrections should not exceed the mea-
sured masses. This has already been done in the
seminal paper of Gabbiani et al. [2]. We improve
this calculation by taking into account QCD cor-
rections and by using the up-to-date values of the
fermion masses.
• The leading chirally-enhanced two-loop corrections
are calculated. As we will see, this allows us to
constrain the elements δf RL
same time, also δf LR
13
(δdLR
23
13
(and δdRL
) is different from zero.
23
), if at the
Our paper is organized as follows: In Sec. II we study
the impact of chirally enhanced parts of the self-energies
for quarks and leptons on the fermion masses and mix-
ing matrices (CKM matrix and PMNS matrix). First,
we introduce the general formalism in Sec. IIA and then
specify to the MSSM with non-minimal sources of flavor
violation in Sec. IIB where we compute the chirally en-
hanced parts of the self-energies for quarks and leptons
Page 2
2
fJ
−iΣf
IJ
fI
FIG. 1: Flavor-valued wave-function renormalization.
taking into account also the leading two-loop corrections.
Sec. III is devoted to the numerical analysis. Finally we
conclude in Sec. IV.
II. FINITE RENORMALIZATION OF FERMION
MASSES AND MIXING MATRICES
We have computed the finite renormalization of the
CKM matrix by SQCD effects in Ref. [14, 16] and of
the PMNS matrix in Ref. [17]. In this section we com-
pute the finite renormalization of fermion masses and
mixing angles induced through one-particle irreducible
flavor-valued self-energies beyond leading-order. We first
consider the general case and then specify to the MSSM.
fJ
−iΣf (1)
KJ
fK
−iΣf (1)
IK
fI
FIG. 2:
structed out of two one-loop self energies with I ?= J ?= K.
One-particle irreducible two-loop self-energy con-
A. General formalism
In this section we consider the general effect of one-
particle irreducible self-energies.
compose any self-energy in its chirality-changing and its
chirality-flipping parts in the following way:
It is possible to de-
Σf
IJ(p) =
?
+
Σf LR
IJ
?
(p2) + p /Σf RR
IJ
(p2)
?
PR
?
Σf RL
IJ
(p2) + p /Σf LL
IJ
(p2)PL.
(2)
Note that chirality-changing parts Σf LR
mass dimension 1, while Σf LL
sionless. With this convention the renormalization of the
fermion masses is given by:
IJ
and Σf RL
are dimen-
IJ
have
IJ
and Σf RR
IJ
m(0)
fI→ m(0)
fI+ Σf LR
II
(m2
fI) +1
2mfI
?
Σf LL
II
(m2
fI) + Σf RR
II
(m2
fI)
?
+ δmfI= mphys
fI
. (3)
If the self-energies are finite, the counter-term δmfIin
Eq. (3) is zero in a minimal renormalization scheme like
MS. In the following we choose this minimal scheme
for two reasons: First, A-terms are theoretical quantities
which are not directly related to physical observables.
For such quantities it is always easier to use a minimal
scheme which allows for a direct relation between theo-
retical quantities and observables. Second, we consider
the limit in which the light fermion masses and CKM el-
ements are generated radiatively. In this limit it would
be unnatural to have tree-level Yukawa couplings and
CKM elements in the Lagrangian which are canceled by
counter-terms as in the on-shell scheme.
The self-energies in Eq. (2) do not only renormalize the
fermion masses. Also a rotation 1+∆Uf L
which has to be applied to all external fields is induced
through the diagram in Fig. 1:
IJin flavor-space
∆Uf L
IJ=
1
m2
fJ− m2
2Re
fI
?
?m2
m2
fJΣf LL
IJ
?m2
fI
?+ mfJmfIΣf RR
?m2
IJ
?m2
?
fI
?+ mfJΣf LR
Σf LL′
II
IJ
?m2
fI
?+ mfIΣf RL
?m2
IJ
?m2
fI
??
for I ?= J,
∆Uf L
II
=1
?
Σf LL
II
fI
?+ 2mfIΣf LR′
II
fI
?+ m2
fI
?m2
fI
?+ Σf RR′
II
fI
???
.(4)
The prime denotes differentiation with respect to the ar-
gument. The flavor-diagonal part arises from the trun-
cation of flavor-conserving self-energies. Eq. (3) and
Page 3
3
Eq. (4) are valid for arbitrary one-particle irreducible
self-energies.
B. Self-energies in the MSSM
Self-energies with supersymmetric virtual particles are
of special importance because of a possible chiral en-
hancement which can lead to order-one corrections. In
this section we calculate the chirally enhanced (by a fac-
AIJ
f
MSUSYYIJ
f
or tanβ) parts of the fermion self-energies
in the MSSM. Therefore it is only necessary to evaluate
the diagrams at vanishing external momentum.
We choose the sign of the self-energies Σ to be equal
to the sign of the mass, e.g. calculating a self-energy
diagram yields −iΣ. Then, with the conventions given
in the Appendix A, the gluino contribution to the quark
self-energies is given by:
tor
Σ˜ g
qIL−qJR= −
6
?
i=1
m˜ g
16π2
?Γ˜ g˜ qi
m˜ gW(J+3)i∗
Q
qJR
?∗Γ˜ g˜ qi
qILB0(m2
˜ g,m2
˜ qi) (5)
=αs
2πCF
6
?
i=1
WIi
QB0(m2
˜ g,m2
˜ qi) (6)
and for the neutralino and chargino contribution to the
quark self-energy we receive:
Σ˜ χ0
dIL−dJR= −
6
?
i=1
4
?
j=1
m˜ χ0
16π2Γ
j
˜ χ0
dJRΓ
j˜di∗
˜ χ0
dILB0(m2
j˜di
˜ χ0
j,m2
˜di)
(7)
Σ˜ χ±
dIL−dJR= −
6
?
i=1
2
?
j=1
m˜ χ±
16π2Γ
j
˜ χ±
dJR
j˜ ui∗
Γ
˜ χ±
dILB0(m2
j˜ ui
˜ χ±
j,m2
˜ ui)
(8)
The self-energies in the up-sector are easily obtained by
interchanging u and d. We denote the sum of all contri-
bution as:
Σq LR
IJ
= Σ˜ g
qIL−qJR+ Σ˜ χ0
qIL−qJR+ Σ˜ χ±
qIL−qJR
(9)
Note that the gluino contribution are dominant in the
case of non-vanishing A-terms, since they involve the
strong coupling constant. In the lepton case, neutralino–
slepton and chargino–sneutrino loops contribute the non-
decoupling self-energy ΣℓLR
IJ. With the convention in the
Appendix A the self-energies are given by:
Σ˜ χ±
ℓIL−ℓJR= −
2
?
j=1
3
?
k=1
m˜ χ±
16π2Γ
j
˜ χ±
ℓJR
j˜ νk∗
Γ
˜ χ±
ℓIL
j˜ νk
B0(m2
˜ χ±
j,m2
˜ νk),
(10)
Σ˜ χ0
ℓIL−ℓJR= −
6
?
i=1
4
?
j=1
m˜ χ0
16π2Γ
j
˜ χ0
ℓJRΓ
j˜ℓi∗
˜ χ0
ℓILB0(m2
j˜ℓi
˜ χ0
j,m2
˜ℓi).
(11)
Again, we denote the sum of all contribution as:
ΣℓLR
IJ
= Σ˜ χ0
ℓIL−ℓJR+ Σ˜ χ±
ℓIL−ℓJR.(12)
With I = J we arrive at the flavor-conserving case. This
can lead to significant quantum corrections to fermion
masses, but except for the gluino, the pure bino (∝ g2
and the negligible small bino-wino mixing (∝ g1g2) con-
tribution, they are proportional to tree-level Yukawa cou-
plings. However, if the light fermion masses are gener-
ated radiatively from chiral flavor-violation in the soft
SUSY-breaking terms, then the Yukawa couplings of the
first and second generation even vanish and the latter ef-
fect is absent at all. Radiatively generated fermion mass
terms via soft tri-linear A-terms corresponds to the up-
per bound found from the fine-tuning argument where
the correction to the mass is as large as the physical mass
itself. This fine-tuning argument is based on ’t Hooft’s
naturalness principle: A theory with small parameters is
natural if the symmetry is enlarged when these param-
eters vanish. The smallness of the parameters is then
protected against large radiative corrections by the con-
cerned symmetry.If such a small parameter, e.g.
fermion mass, is composed of several different terms there
should be no accidental large cancellation between them.
We will derive our upper bounds from the condition that
the SUSY corrections should not exceed the measured
value.
If we restrict ourself to the case with vanishing first
and second generation tree-level Yukawa couplings, the
off-diagonal entries in the sfermion mass matrices stem
from the soft tri-linear terms. Thus we are left with δf LR
only. In the mass insertion approximation with only LR
insertion the flavor violating self-energies simplifies. For
the gluino (neutralino) self-energies which are relevant
for our following discussion for the quark (lepton) case
we get:
1)
a
IJ
Σ˜ g
qIX−qJY=2αs
3πM˜ gm˜ qJYm˜ qIXδq XY
IJ
C0
?M2
?
1,m2
˜ qJY,m2
˜ qIX
(13)
?,
Σ˜ B
ℓIX−ℓJY=α1
4πM1m˜ℓJYm˜ℓIXδℓXY
IJ
C0
M2
1,m2
˜ℓJY,m2
˜ℓIX
(14)
?
.
Since the sneutrino mass matrix consists only of a LL
block, there are no chargino diagrams in the lepton case
with LR insertions at all.
Since the SUSY particles are known to be much heav-
ier than the five lightest quarks it is possible to evaluate
the one-loop self-energies at vanishing external momen-
tum and to neglect higher terms which are suppressed by
powers of m2
SUSY. The only possibly sizable decou-
pling effect concerning the W vertex renormalization is a
loop-induced right-handed W coupling (see [16]). There-
fore Eq. (2) simplifies to
fI/M2
Σf (1)
IJ
= Σf LR (1)
IJ
PR+ Σf RL (1)
IJ
PL
(15)
Page 4
4
at the one-loop level (indicated by the superscript (1)).
In this approximation the self-energies are always chiral-
ity changing and contribute to the finite renormalization
of the quark masses in Eq. (3) and to the flavor-valued
wave-function renormalization in Eq. (4). At the one-
loop level we receive the well known result
m(0)
fI→ m(1)
fI= m(0)
fI+ Σf LR (1)
II
(16)
for the mass renormalization in the MS scheme. Accord-
ing to Eq. (4) the flavor-valued rotation which has to be
applied to all external fermion fields is given by:
∆Uf L (1)=
0
mf2Σf LR (1)
12
+ mf1Σf RL (1)
f2− m2
12
m2
f1
mf3Σf LR (1)
13
+ mf1Σf RL (1)
f3− m2
+ mf2Σf RL (1)
m2
f2
13
m2
f1
mf1Σf LR (1)
21
+ mf2Σf RL (1)
f1− m2
+ mf3Σf RL (1)
m2
f3
21
m2
f2
0
mf3Σf LR (1)
23
23
f3− m2
mf1Σf LR (1)
31
31
f1− m2
mf2Σf LR (1)
32
+ mf3Σf RL (1)
f2− m2
32
m2
f3
0
. (17)
The corresponding corrections to the right-handed wave-
functions are obtained by simply exchanging L with R
and vice versa in Eq. (17). Note that the contributions
of the self-energies Σf RL (1)
IJ
by small mass ratios. Therefore, the corresponding off-
diagonal elements of the sfermion mass matrices cannot
with J > I are suppressed
be constrained from the CKM and PMNS renormaliza-
tion. However, since we treat, in the spirit of Ref. [18],
all diagrams in which no flavor appears twice on quark
lines as one-particle irreducible, chirally-enhanced self-
energies can also be constructed at the two-loop level
(see Fig. (2)):
Σf RR (2)
IJ
?p2?=
?p2?=
?
?
K?=I,J
Σf RL (1)
IK
p2− m2
Σf LR (1)
IK
Σf LR (1)
KJ
fK
,Σf LL (2)
IJ
?p2?=
?p2?=
?
?
K?=I,J
Σf LR (1)
IK
p2− m2
Σf RL (1)
IK
Σf RL (1)
KJ
fK
,
Σf LR (2)
IJ
K?=I,J
mfK
Σf LR (1)
KJ
p2− m2
fK
,Σf RL (2)
IJ
K?=I,J
mfK
Σf RL (1)
KJ
p2− m2
fK
.
(18)
Therefore, the chiral-enhanced two-loop corrections to the masses and the wave-function renormalization are given
by:
m(0)
f1
m(0)
f2
m(0)
f3
→
???Σf LR (1)
Σf RL (1)
31
mf2mf3
m(0)
f1+ Σf LR (1)
11
−Σf LR (1)
12
Σf LR (1)
21
mf2
−Σf LR (1)
−Σf LR (1)
Σf LR (1)
32
mf3
13
Σf LR (1)
31
mf3
m(0)
f2+ Σf LR (1)
22
23
m(0)
f3+ Σf LR (1)
33
,(19)
∆Uf (2)
L
=
−
???Σf LR (1)
Σf RL (1)
23
12
2m2
???
2
f2
−
13
2m2
???
2
f3
−Σf LR (1)
???Σf LR (1)
−Σf RL (1)
13
Σf LR (1)
32
mf2mf3
???
Σf LR (1)
12
mf2mf3
Σf LR (1)
12
Σf RL (1)
23
m2
f3
−
23
2m2
2
f3
−
???Σf LR (1)
12
2m2
???
2
f2
Σf LR (1)
21
Σf RL (1)
13
m2
f3
???
Σf RL (1)
32
Σf RL (1)
21
mf2mf3
31
−
???Σf LR (1)
13
2m2
2
f3
−
???Σf LR (1)
23
2m2
???
2
f3
,(20)
Page 5
5
where we have neglected small mass ratios. In the quark
case, we already know about the hierarchy of the self-
energies from our fine-tuning argument.
Eq. (20) is just necessary to account for the unitarity of
the CKM matrix [14]. However, the corrections to m(0)
in Eq. (19) can be large. For this reason we can also
constrain Σf LR (1)
31
with ’t Hooft’s naturalness criterion
if at the same time Σf LR (1)
13
In this case
f1
is different from zero.
III. NUMERICAL ANALYSIS
In this section we are going to give a complete nu-
merical evaluation of the all possible constraints on the
SUSY breaking sector from ’t Hooft’s naturalness argu-
ment. This criterion is applicable since we gain a flavor
symmetry [14] if the light fermion masses are generated
radiatively. Therefore the situation is different from e.g.
the little hierarchy problem, where no additional symme-
try is involved. First of all, it is important to note that all
off-diagonal elements of the fermion mass matrices have
to be smaller than the average of their assigned diagonal
elements
?∆m2
F
?IJ
XY<
?
m2
˜fIXm2
˜fJY,(21)
since otherwise one sfermion mass eigenvalue is negative.
We note that in Ref. [2] this constraint is not imposed.
All constraints in this section are non-decoupling since
we compute corrections to the Higgs-quark-quark cou-
pling which is of dimension 4.
straints on the soft-supersymmetry-breaking parameters
do not vanish in the limit of infinitely heavy SUSY
masses but rather converge to a constant [14].
ever, even though δf LR
IJ
is a dimensionless parame-
ter it does not only involve SUSY parameter.
also proportional to a vacuum expectation and therefore
scales like v/MSUSY. Thus, our constraints on δf LR
not approach a constant for MSUSY → ∞ but rather
get stronger. Similar effects occur in Higgs-mediated
FCNC processes which decouple like 1/M2
than 1/M2
SUSY[19, 20, 21].
effects can only be induced within supersymmetry in
the presence of non-holomorphic terms which are not
required for our constraints.
decoupling Higgs-mediated FCNC process is the observ-
able RK = Γ(K → eν)/Γ(K → µν) that is currently
analyzed by the NA62-experiment. In this case Higgs
contributions can induce deviations from lepton flavor
universality [10, 22, 23].
Therefore, our con-
How-
It is
IJ
do
Higgsrather
However, Higgs-mediated
An example of a non-
A. Constraints on flavor-diagonal mass insertions
at one loop
The diagonal elements of the A-terms can be con-
strained from the fermion masses by demanding that
1
1.5
2
2.5
3
3.5
5001000 15002000
500
1000
1500
2000
mg
uLR?10?1
? in GeV
mq
? in GeV
∆11
u LR?10?4
0.4
0.6
0.8
1
1.2
1.4
50010001500 2000
500
1000
1500
2000
mg
d LR?10?4
?in GeV
mq
?in GeV
∆22
1.5
2
3
4
5
6
6
500100015002000
500
1000
1500
2000
mg
d LR?10?2
? in GeV
mq
? in GeV
∆11
0.4
0.5
0.6
0.8
1
1.4
500 10001500 2000
500
1000
1500
2000
mg
? in GeV
mq
? in GeV
∆22
FIG. 3: Constraints on the diagonal mass insertions δu,d LR
obtained by applying ’t Hooft’s naturalness criterion.
11,22
Page 6
6
1
1.5
2
3
4
200400600800 1000 1200 1400
M1
∆22
200
400
600
800
1000
1200
1400
me?
∆11
l LR? 10?4
0.2
0.3
0.4
0.6
1
200400600800 1000 1200 1400
M1
200
400
600
800
1000
1200
1400
mΜ?
l LR
FIG. 4: Contraints on the diagonal mass insertion δℓ LR
function of M1 and m˜ e, m˜ µ.
11,22as a
Σf LR (1)
II
conserving A-term for the up, charm, down and strange
quarks are shown in Fig. (3) and the constraints from the
electron and muon mass are depicted in Fig. (4). The up-
per bound derived from the fermion mass is roughly given
by
≤ mfI[see Eq. (16)]. The bounds on the flavor-
???δq LR
II
??? ?
3π mqI(MSUSY)
αs(MSUSY)MSUSY
(22)
for quarks and
??δℓLR
II
???
8πmℓI
α1MSUSY
(23)
for leptons in the case of equal SUSY masses. In the
lepton case Eq. (23) can be further simplified, since we
can neglect the running of the masses:
|δℓLR
??δℓLR
11
| ? 0.0025
??? 0.5
?
500 GeV
MSUSY
500GeV
MSUSY
?
.
,
22
?
?
(24)
However, as already pointed out in Ref. [24] a muon
mass that is solely generated radiatively potentially leads
to measurable contributions to the muon anomalous mag-
netic moment. This arises from the same one-loop dia-
gram as Σℓ LR
22
with an external photon attached. There-
fore, the SUSY contribution is not suppressed by a loop
factor compared to the case with tree-level Yukawa cou-
plings.
B.Constraints on flavor-off-diagonal mass
insertions from CKM and PMNS renormalization
1.CKM matrix
A complete analysis of the constraints for the CKM
renormalization was already carried out in Ref. [14]. The
numerical effect of the chargino contributions is negligi-
ble at low tanβ and the neutralino contributions amount
only to corrections of about 5% of the gluino contribu-
tions. Therefore, we refer to the constraints on the off-
diagonal elements δq LR
IJ
given in Ref. [14].
2.Threshold corrections to PMNS matrix
Up to now, we have only an upper bound for the matrix
element Ue3= sinθ13e−iδand thus for the mixing angle
θ13; the best-fit value is at or close to zero: θ13= 0.0+7.9
[25]. 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 predic-
tion θ13= 0◦at the weak scale. 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,
−0.0
|∆Ue3| ≤
???Uphys
e3
???.(25)
The renormalization of the PMNS matrix is described in
detail in [17], where the on-shell scheme was used. As
discussed in Sec. (II) we also use the MS scheme in this
section. Then the physical PMNS matrix is given by:
Uphys= U(0)+ ∆U ,(26)
where ∆U should not be confused with the wave function
renormalization ∆Uf L. Then ∆U is given by
∆U =?∆UℓL?TU(0). (27)
Note that in contrast to the corrections to the CKM ma-
trix, there is a transpose in ∆UℓL, because the first index
of the PMNS matrix corresponds to down-type fermions
and not to up-type fermion as in the CKM matrix. Only
the corrections to the small element Ue3can be sizeable,
since all other elements are of order one. If we set all
Page 7
7
off-diagonal element to zero except for δℓLR
13
?= 0, we get
∆Ue3=∆UℓL
31Uphys
τ3
1 +??∆UℓL
mτ
− Uphys
e3
??∆Uℓ L
31
??2
11
??2
≈ −Uphys
τ3
ΣℓRL
31
.
(28)
Note that here, in contrast to the renormalization of the
CKM matrix, the physical PMNS element appears. This
is due to the fact that one has to solve the linear system
in Eq. (27) as described in [17]. By means of the fine-
tuning argument we can in principle derive upper bounds
for δℓLR
13
. The results depend on the SUSY mass scale
MSUSYand the assumed value for θ13.
Here, we consider the corrections stemming from
flavor-violating A-terms to the small matrix element Ue3.
The δℓLL
13 -contribution was already studied in [17] with
the result that they are negligible small. We also made a
comment about the δℓLR
13
-contribution which is outlined
in more detail. Our results depend on the overall SUSY
mass scale, the value of θ13 and of δℓLR
you can see the percentage deviation of Ue3through this
SUSY loop corrections in dependence of δℓLR
θ13 (bottom) for MSUSY = 1000 GeV. The constraints
on δℓLR
13
get stronger with smaller θ13 and with larger
MSUSY. In Fig. (6) the excluded
below the curves for different MSUSYscales. The derived
bound can be simplified to
13
. In Fig. (5)
13
(top) and
?θ13,δℓ LR
13
?-region is
??δℓLR
13
??? 0.2
?500GeV
MSUSY
?
|θ13in degrees|.(29)
Exemplarily, we get for reasonable SUSY masses of
MSUSY = 1000 GeV and θ13 = 3◦an upper bound of
??δℓLR
derived bounds if θ13is non-zero. As an important con-
sequence, we note that τ → eγ impedes any measurable
correction from supersymmetric loops to Ue3 : E.g. for
sparticle masses of 500 GeV we find |∆Ue3| ≤ 10−3cor-
responding to a correction to the mixing angle θ13of at
most 0.06◦. That is, if the DOUBLE CHOOZ experiment
measures Ue3 ?= 0, one will not be able to ascribe this
result to the SUSY breaking sector. Stated positively,
Ue3 ? 10−3will imply that at low energies the flavor
symmetries imposed on the Yukawa sector to motivate
tri-bimaximal mixing are violated. This finding confirms
the pattern found in [17] where the product δℓ LL
has been studied instead of δℓ LR
13
??≤ 0.3. The constraints on δℓLR
13
from τ → eγ are
of the order of 0.02 [17] and in general better than our
13
δℓ LR
33
13
.
C.Constraints from two-loop corrections to
fermion masses
Combining two flavor-violating self-energies can have
sizable impacts on the light fermion masses according to
Eq. (19). Requiring that no large numerical cancella-
tions should occur between the tree-level mass (which is
0.000.050.100.150.200.25
0
50
100
150
200
∆LR
l 13
?? UL13
l
??Ue3in ?
012345
0
100
200
300
400
500
Θ13ino
?? UL13
l
??Ue3in ?
FIG. 5: |∆Ue3|/Ue3 in %. Top: as a function of δℓ LR
MSUSY = 1000 GeV and different values of θ13(green 1◦; blue:
3◦; red: 5◦). Bottom: as a function of θ13 for MSUSY = 1000
GeV and different values of δℓ 13
δℓ LR
13
= 0.3; green: δℓ LR
13
= 0.1) (both from top to bottom)
13
for
LR (red: δℓ LR
13
= 0.5; blue:
0.00.10.20.30.40.5
0
2
4
6
8
10
12
14
∆13
l LR
Θ13ino
FIG. 6: The excluded?θ13,δℓ LR
for (from bottom to top) MSUSY = 500 GeV (red), 1000
GeV (blue), 2000 GeV (green) and 5000 GeV (yellow). The
black dashed line denotes the future experimental sensitivity
to θ13 = 3◦.
13
?-region is below the curves
absent in the case of a radiative fermion mass) and the
supersymmetric loop corrections we can derive bounds
on the products δf LR
KI
which contain the so far less
constrained elements δf LR
KI, K > I.
IKδf LR
We apply the fine-tuning argument to the two-loop
contribution originating from flavor-violating A-terms,
e.g.
11
sponds to a 100% change in the fermion mass through su-
persymmetric loop corrections which is equivalent to the
case that the fermion Yukawa coupling vanishes. The up-
per bound depends on the overall SUSY mass scale and
???Σf LR(2)
??? ≤ mf1. The bound Σf LR(2)
11
= mf1corre-
Page 8
8
is roughly given as
???δq LR
I3
δq LR
3I
??? ?9π2mqImq3(MSUSY)
???64π2mℓ1mℓ3
(αs(MSUSY)MSUSY)2, I ?= 3(30)
for quarks and
??δℓ LR
13
δℓLR
31
(α1MSUSY)2
(31)
for leptons. Again, Eq. (31) can be further simplified
??δℓLR
13
δℓLR
31
??≤ 0.021
?500GeV
MSUSY
?2
.(32)
The contributions proportional to δf LR
important, since these elements are already severely con-
strained by FCNC processes [26]. As studied in Ref. [27],
single-top production involves the same mass insertion
δu LR
31
which can also induce a right-handed W coupling
if at the same time δdLR
33
?= 0 [16]. Therefore our bound
can be used to place a constraint on this cross section.
Also the product δu,ℓLR
23
δu,ℓLR
32
since the muon and the charm are too heavy. However,
δdLR
23
δdLR
32
can be constrained as shown in Fig. (10). Our
results for the up, down, and electron mass are depicted
in Fig. (8),(9) and (7). In the quark case also the bounds
from the CKM renormalization on δq LR
account.
13
δf LR
31
cannot be
cannot be constrained,
13,23are taken into
IV.CONCLUSIONS
According to ’t Hooft’s naturalness principle, the
smallness of a quantity is linked to a symmetry that is
restored if the quantity is zero. The smallness of the
Yukawa couplings of the first two generations (as well as
the small CKM elements involving the third generation)
suggest the idea that Yukawa couplings (except for the
third generation) are generated through radiative correc-
tions [14, 15, 24, 28, 29, 30]. It might well be that the
chiral flavor symmetry is broken by soft SUSY-breaking
terms rather than by the trilinear tree-level Yukawa cou-
plings.
We use ’t Hooft’s naturalness criterion to constrain
the chirality-changing mass insertion δu,d,ℓLR
mass and CKM renormalization. Therefore, we compute
the finite renormalization of fermion masses and mixing
angles in the MSSM, taking into account the leading two-
loop effects. These corrections are not only important, in
order to obtain a unitary CKM matrix, they are also nu-
merically important for light fermion masses. This allows
us to constrain the product δf LR
which is important, especially with respect to the before
unconstrained element δu RL
13
. All constraints given in this
paper are non-decoupling. This means they do not van-
ish in the limit of infinitely heavy SUSY masses unlike
the bounds from FCNC processes. Therefore our con-
straints are always stronger than the FCNC constraints
for sufficiently heavy SUSY (and Higgs) masses.
IJ
from the
13
δf LR
31
(and δdLR
23
δdLR
32
)
0.00.10.20.3 0.40.5
0.0
0.1
0.2
0.3
0.4
0.5
∆31
l LR
∆13
l LR
Constraints on ∆13
l LR∆31
l LRfrom me
200 400600
MSUSY
8001000
?0.10
?0.05
0.00
0.05
0.10
∆LR
l 13∆LR
l 31
Constraints on ∆13
l LR∆31
l LRfrom me
FIG. 7: Results of the two-loop contribution to the electron
mass. Above: Region compatible with the naturalness prin-
ciple for (from top to bottom) MSUSY = 200 GeV (yellow),
500 GeV (green), 800 GeV (blue), 1000 GeV (red). Bottom:
Allowed range for δℓ LR
13
δℓ LR
31
as a function of MSUSY.
The PMNS renormalization is a bit more involved
since the matrix is not hierarchical. The radiative de-
cay τ → eγ severely limits the size of the loop correction
∆Ue3to the PMNS element Ue3. In a previous paper we
have studied this topic for effects triggered by the prod-
uct δℓ LL
13
δℓLR
33
[17]. In this paper we have complemented
that analysis by investigating δℓLR
reasonable slepton masses and noting that the Daya Bay
neutrino experiment is only sensitive to values of θ13
above 3◦, we conclude that the threshold corrections to
Ue3are far below the measurable limit. Consequently, if
a symmetry at a high scale imposes tri-bimaximal mix-
ing, SUSY loop corrections cannot spoil this prediction
θ13= 0 at the weak scale. This is an important result for
the proper interpretation of a measurement of θ13. Thus
if DOUBLE CHOOZ or Daya Bay neutrino experiment
will measure a non-zero θ13 then this is also true at a
high energy scale.
13
instead. Assuming
Page 9
9
0.0 0.10.20.3 0.40.5
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
∆31
u LR
∆13u LR
Constraints on ∆13
u LR∆31
u LRfrom mu
400600800 100012001400
?0.02
?0.01
0.00
0.01
0.02
MSUSY
∆13u LR∆13u RL
Constraints on ∆13
u LR∆13
u RLfrom mu
FIG. 8: Results of the two-loop contribution to the up quark
mass. Above: Region compatible with the naturalness prin-
ciple (100% bound) for (from top to bottom) MSUSY = 500
GeV (yellow), 1000GeV (green), 1500 GeV (blue), 2000 GeV
(red). Bottom: Allowed range for δu LR
MSUSY.
13
δu LR
31
as a function of
Acknowledgments
We like to thank Ulrich Nierste for helpful discus-
sions and proofreading the article.
ported by BMBF grants 05HT6VKB and 05H09VKF
and by the EU Contract No. MRTN-CT-2006-035482,
“FLAVIAnet”. Andreas Crivellin and Jennifer Gir-
rbach acknowledge the financial support by the State of
Baden-W¨ urttemberg through Strukturiertes Promotion-
skolleg Elementarteilchenphysik und Astroteilchenphysik
and the Studienstiftung des deutschen Volkes, respec-
tively.
This work is sup-
0.00.1 0.20.3 0.40.5
0.0000
0.0005
0.0010
0.0015
∆13
0.0020
0.0025
0.0030
∆31
d LR
d LR
Constraints on ∆13
d LR∆31
d LRfrom md
4006008001000 1200 1400
?0.0006
?0.0004
?0.0002
0.0000
0.0002
0.0004
0.0006
MSUSY
∆13d LR∆31d LR
Constraints on ∆13
d LR∆31
d LRfrom md
FIG. 9: Results of the two-loop contribution to the down
quark mass. Above: Region compatible with the naturalness
principle for (from top to bottom) MSUSY = 500 GeV (yel-
low), 1000GeV (green), 1500 GeV (blue), 2000 GeV (red).
Bottom: Allowed range for δd LR
13
δd LR
31
as a function of MSUSY.
Appendix A: Conventions
1.Loop integrals
For the self-energies, we need the following loop inte-
grals:
B0(x,y) = −∆ −
x
x − ylnx
∆ =1
ǫ− γE+ ln4π.
xy lnx
µ2−
y
y − xlny
µ2,(A1)
with
C0(x,y,z) =
y+ yz lny
(x − y)(y − z)(z − x)
z+ xz lnz
x
.(A2)
2. Diagonalization of mass matrices and Feynman
rules
For the vacuum expectation value we choose the nor-
malization without the factor√2 and define the Yukawa
Page 10
10
0.00.10.20.30.40.5
0.000
0.005
0.010
0.015
0.020
0.025
0.030
∆32
d LR
∆23d LR
Constraints on ∆23
d LR∆32
d LRfrom ms
400600800100012001400
?0.015
?0.010
?0.005
0.000
0.005
0.010
0.015
MSUSY
∆23d LR∆32d LR
Constraints on ∆23
d LR∆32
d LRfrom ms
FIG. 10: Results of the two-loop contribution to the strange
quark mass. Above: Region compatible with the naturalness
principle (100% bound) for (from top to bottom) MSUSY =
500 GeV (yellow), 1000GeV (green), 1500 GeV (blue), 2000
GeV (red). Bottom: Allowed range for δd LR
tion of MSUSY.
23
δd LR
32
as a func-
couplings in the following way:
v =
?
v2
u+ v2
d= 174 GeV,tanβ =vu
vd,(A3)
ml= −vdYl,md= −vdYd,mu= vuYu.(A4)
Neutralinos ˜ χ0
i
In the following we mainly use the convention of [31].
Ψ0=
?˜B,˜ W,˜H0
d,˜H0
u
?
,
L˜ χ0
mass= −1
2(Ψ0)⊤MNΨ0+ h.c.
M1
0
0M2
−g1vd
g1vu
√2
MN=
−g1vd
g2vd
√2
0
−µ
√2
g1vu
√2
−g2vu
−µ
0
√2
√2
g2vd
√2
−g2vu
√2
. (A5)
MN can be diagonalised with an unitary transformation
such that the eigenvalues are real and positive.
Z⊤
NMNZN= MD
N=
m˜ χ0
1
0
...
0m˜ χ0
4
.
(A6)
For that purpose, Z†
ZN consists of the eigenvectors of the Hermitian ma-
trix M†
NMN. Then the columns can be multiplied with
phases eiφ, such that ZT
real diagonal elements.
NM†
NMNZN= (MD
N)2can be used.
NMNZN= MD
Nhas positive and
Charginos ˜ χ±
i
Ψ±=
?
˜ W+,˜H+
u,˜ W−,˜H−
d
?
,
L˜ χ±
mass= −1
?0 X⊤
2
?Ψ±?⊤MCΨ±+ h.c.
0
MC=
X
?
,X =
?M2 g2vu
g2vd
µ
?
.(A7)
The rotation matrices for the positive and negative
charged fermions differ, such that
ZT
−XZ+=
?m˜ χ1
0
0m˜ χ2
?
. (A8)
Sleptons
The sleptons˜LI
charged mass eigenstates˜Li, i = 1...6:
2= ˜ eIL and˜RI= ˜ e+
IRmix to six
˜LI
2= WIi∗
?(m2
L
˜ℓ−
i,
˜RI= W(I+3)i
?
L
˜ℓ+
i,
W†
L
L)LL (m2
(m2
L)LR
L)RR
L)†
RL(m2
WL= diag
?
m2
˜ℓ1,...,m2
˜ℓ6
?
,
and the slepton mass matrix is composed of
(m2
L)IJ
LL=
e2?v2
+(m2
d− v2
4s2
u
Wc2
??1 − 2c2
W
W
?
δIJ+ v2
dY2
ℓIδIJ
˜L)T
IJ,
(m2
L)IJ
RR= −e2?v2
L)IJ
d− v2
2c2
W
+ vdAIJ∗
u
?
δIJ+ v2
dY2
ℓIδIJ+ m2
˜eIJ,
(m2
LR= vuµYIJ∗
ℓℓ
.
Page 11
11
Lepton-slepton-neutralino coupling
Feynman rule for incoming lepton ℓI, outgoing neu-
tralino and slepton˜ℓi:
iΓ
˜ χ0
ℓI
j˜ℓi
=i
?WIi
?
+ i
L
√2
?
g1Z1j
N+ g2Z2j
N
?
j˜ℓi
ℓIL
+ YℓIW(I+3)i
L
Z3j
N
?
???
=Γ
˜ χ0
PL
?
?
−g1
√2W(I+3)i
L
Z1j∗
N
??
+ YℓIWIi
LZ3j∗
N
?
?
=Γ
˜ χ0
j˜ℓi
ℓIR
PR.
(A9)
Lepton-sneutrino-chargino coupling
Feynman rule for incoming lepton ℓI,
chargino and sneutrino ˜ νJ:
outgoing
iΓ˜ νJ˜ χ±
ℓI
i
= −i?g2Z1i
+PL+ YℓIZ2i∗
−PR
?WIJ∗
ν
.
Down-squarks
The down-squarks˜QI
mass eigenstates˜di, i = 1...6:
2=˜dILand˜DI=˜d∗
IRmix to six
˜QI
2= WIi∗
?(m2
and the downs-squark mass matrix is composed of
D
˜d−
i,
˜DI= W(I+3)i
?
D
˜d+
i,
W†
D
D)LL (m2
(m2
D)LR
D)RR
D)†
RL(m2
,WD= diag
?
m2
˜d1,...,m2
˜d6
?
,
(m2
D)IJ
LL= −e2?v2
+v2
d− v2
12s2
u
??1 + 2c2
W
˜ Q)T
?
+ vdAIJ∗
d
W
?
Wc2
δIJ
dY2
dIδIJ+ (m2
IJ,
(m2
D)IJ
RR= −e2?v2
D)IJ
d− v2
6c2
W
u
δIJ+ v2
dY2
dIδIJ+ m2
˜dIJ,
(m2
LR= vuµYIJ∗
d
.
Up-squarks
Finally, one has six up-squarks ˜ uicomposed from fields
˜QI
IR
1= ˜ uILand˜UI= ˜ uI
˜QI
1= WIi
?(m2
U˜ u+
U)LL (m2
(m2
i,
˜DI= W(I+3)i∗
U
˜ u−
i,
WT
U
U)LR
U)RR
U)†
RL(m2
?
,W∗
U= diag?m2
˜ u1,...,m2
˜ u6
?.
(m2
U)IJ
LL= −e2?v2
+v2
d− v2
12s2
u
??1 − 4c2
W
W
?
IJ,
Wc2
δIJ
uY2
uiδIJ+ (V m2
e2?v2
W
LR= −vdµYIJ∗
˜ QV†)T
(m2
U)IJ
RR=
d− v2
3c2
u
?
− vuAIJ∗
δIJ+ v2
uY2
uIδIJ+ m2
˜uIJ,
(m2
U)IJ
uu
.
Quark-squark-gluino coupling
Feynman rule for incoming quark dI, uI, outgoing
gaugino and squark˜di, ˜ ui:
iΓ˜ g˜di
dI= igs
√2Ta?
√2Ta?
−WIi
−WIi∗
DPL+ W(I+3)i
D
PR
?
,(A10)
iΓ˜ g˜ ui
uI= igs
UPL+ W(I+3)i∗
U
PR
?
.(A11)
Quark-squark-neutralino coupling
Feynman rule for incoming quark dI, uI, outgoing neu-
tralino and squark˜di, ˜ ui:
iΓ
˜ χ0
dI
j˜di
=i
?WIi
?
?WIi∗
?
D
√2
?
√2g1
3
−g1
3Z1j
N+ g2Z2j
N
?
+ YdIWIi
+ YdIW(I+3)i
D
Z3j
N
?
PL
+ i−
W(I+3)i
D
Z1j∗
N
DZ3j∗
N
?
PR,
iΓ
˜ χ0
uI
j˜ ui
=i
U
√2
?
−g1
3Z1j
N− g2Z2j
N
?
− YuIWIi∗
− YuIW(I+3)i∗
U
Z4j
N
?
PL
+ i
2√2g1
3
W(I+3)i∗
U
Z1j∗
N
U Z4j∗
N
?
PR.
Quark-squark-chargino coupling
Feynman rule for incoming quark dI, uI, outgoing
chargino and squark ˜ ui,˜di:
iΓ
˜ χ±
dI
j˜ ui
=i
?
−g2WJi∗
+ i−YdIWJi∗
?
+ iYuIWJi
U Z1j
++ YuJW(J+3)i∗
?
DZ1j
?
U
Z2j
+
?
VJIPL
?
U Z2j∗
−
VJIPR.,
iΓ
˜ χ±
uI
j˜di
=i−g2WJi
?
−− YdJW(J+3)i
DZ2j∗
+
VJI∗PR.
D
Z2j
−
?
VJI∗PL
Page 12
12
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