# Constraining the MSSM sfermion mass matrices with light fermion masses

**ABSTRACT** 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 delta^{f\,LR,RL}_{IJ} by applying 't Hooft's naturalness criterion. In particular, the NLO corrections to the up quark mass allow us to constrain the unbounded element delta^{u\,RL}_{13} if at the same time $\delta^{u\,LR}_{13}$ is unequal to zero. Our result is important for single-top production at the LHC. Comment: 12 pages, 10 figures

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**ABSTRACT:**We study the decay modes {overline B_s} to φ {π^0} and {overline B_s} to φ {ρ^0} using Soft Collinear Effective Theory. Within Standard Model and including the error due to the SU(3) breaking effect in the SCET parameters we find that BR {overline B_s} to φ {π^0} = 7_{ - 1 - 2}^{ + 1 + 2} × {10^{ - 8}} and BR {overline B_s} to φ {π^0} = 9_{ - 1 - 4}^{ + 1 + 3} × {10^{ - 8}} corresponding to solution 1 and solution 2 of the SCET parameters respectively. For the decay mode {overline B_s} to φ {ρ^0}, we find that BR {overline B_s} to φ {ρ^0} = 20.2_{ - 1 - 12}^{ + 1 + 9} × {10^{ - 8}} and BR {overline B_s} to φ {ρ^0} = 34.0_{ - 1.5 - 22}^{ + 1.5 + 15} × {10^{ - 8}} corresponding to solution 1 and solution 2 of the SCET parameters respectively. We extend our study to include supersymmetric models with non-universal A-terms where the dominant contributions arise from diagrams mediated by gluino and chargino exchanges. We show that gluino contributions can not lead to an enhancement of the branching ratios of {overline B_s} to φ {π^0} and {overline B_s} to φ {ρ^0}. In addition, we show that SUSY contributions mediated by chargino exchange can enhance the branching ratio of {overline B_s} to φ {π^0} by about 14 % with respect to the SM prediction. For the branching ratio of {overline B_s} to φ {ρ^0}, we find that SUSY contributions can enhance its value by about 1 % with respect to the SM prediction.Journal of High Energy Physics 08/2012; · 6.22 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the decay modes ${\overline B_s} \to \phi {\pi^0}$ and ${\overline B_s} \to \phi {\rho^0}$ using Soft Collinear Effective Theory. Within Standard Model and including the error due to the SU(3) breaking effect in the SCET parameters we find that BR ${\overline B_s} \to \phi {\pi^0} = 7_{ - 1 - 2}^{ + 1 + 2} \times {10^{ - 8}}$ and BR ${\overline B_s} \to \phi {\pi^0} = 9_{ - 1 - 4}^{ + 1 + 3} \times {10^{ - 8}}$ corresponding to solution 1 and solution 2 of the SCET parameters respectively. For the decay mode ${\overline B_s} \to \phi {\rho^0}$ , we find that BR ${\overline B_s} \to \phi {\rho^0} = 20.2_{ - 1 - 12}^{ + 1 + 9} \times {10^{ - 8}}$ and BR ${\overline B_s} \to \phi {\rho^0} = 34.0_{ - 1.5 - 22}^{ + 1.5 + 15} \times {10^{ - 8}}$ corresponding to solution 1 and solution 2 of the SCET parameters respectively. We extend our study to include supersymmetric models with non-universal A-terms where the dominant contributions arise from diagrams mediated by gluino and chargino exchanges. We show that gluino contributions can not lead to an enhancement of the branching ratios of ${\overline B_s} \to \phi {\pi^0}$ and ${\overline B_s} \to \phi {\rho^0}$ . In addition, we show that SUSY contributions mediated by chargino exchange can enhance the branching ratio of ${\overline B_s} \to \phi {\pi^0}$ by about 14 % with respect to the SM prediction. For the branching ratio of ${\overline B_s} \to \phi {\rho^0}$ , we find that SUSY contributions can enhance its value by about 1 % with respect to the SM prediction.Journal of High Energy Physics 2012(8). · 6.22 Impact Factor - SourceAvailable from: Andreas Crivellin
##### Article: 1-loop SQCD corrections to the decay of top-squarks to charm and neutralino in the generic MSSM

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**ABSTRACT:**In this article we calculate the 1-loop supersymmetric QCD (SQCD) corrections to the decay $\tilde u_1\to c \tilde\chi^0_1$ in the MSSM with generic flavour structure. This decay mode is phenomenologically important if the mass difference between the lightest squark $\tilde u_1$ (which is assumed to be mainly stop-like) and the neutralino LSP $\tilde \chi^0_1$ is smaller than the top mass. In such a scenario $\tilde u_1\to t \tilde\chi^0_1$ is kinematically not allowed and searches for $\tilde u_1\to W b \tilde\chi^0_1$ and $\tilde u_1\to c \tilde\chi^0_1$ are performed. A large decay rate for $\tilde u_1\to c \tilde\chi^0_1$ can weaken the LHC bounds from $\tilde u_1\to W b \chi^0_1$ which are usually obtained under the assumption ${\rm Br}[\tilde u_1\to W b \chi^0_1]=100\%$. We find the SQCD corrections enhance $\Gamma[\tilde u_1\to c \tilde\chi^0_1]$ by approximately 10\% if the flavour-violation originates from bilinear terms. If flavour-violation originates from trilinear terms, the effect can be $\pm 50\%$ or more, depending on the sign of $A^t$. We note that connecting a theory of SUSY breaking to LHC observables, the shift from the $\overline{\rm DR}$ to the on-shell mass is numerically very important for light stop decays.10/2014;

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

50010001500 2000

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

500100015002000

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

5001000 15002000

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

200400 600800 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

)

0.00.10.2 0.3 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

200400 600

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

0.00.10.20.3 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

400600800100012001400

?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-

0.00.10.2 0.30.40.5

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

400 600800 10001200 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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