Supersymmetric penguin contributions to the decay $b \to s \gamma$ with non-universal squarks masses

arXiv:hep-ph/0012113v3 13 Dec 2001

Supersymmetri penguin ontributions to the de ay

b → sγ

with non-universal squarks masses

M.B. Causse

∗

and J. Orlo

†

Laboratoirede Physique Corpus ulaire, Université Blaise Pas al,

F-63177Aubière Cedex

Abstra t

We give expli it expressions for the amplitudes asso iated with the supersymmetri (SUSY) ontributionstothepro ess:

b

→ sγ

inthe ontextofSUSYextensionsofStandard Model(SM)withnon-universalsoftSUSYbreakingterms. Fromexperimentaldata,we de-du e limits on the squark mass insertions obtained from dierent ontributions (gluinos, neutralinos and harginos).

1 Introdu tion

Therare

B

de aysrepresentagoodtestfornewphysi sbeyondtheSMsin etheyarenotae ted appre iably by un ertainties due to long distan e ee ts. Here, in the ontext of spontaneously broken minimal

N = 1

supergravity [1℄, we study penguin diagrams with gluinos, neutralinos and harginos, whi h are responsible

∆S = 1

radiative mesoni de ays. In parti ular, we study

the

b → sγ

de ay [2℄ that gives the most stringent lower bounds on the average squark mass.

We know that in generi msugra models [3℄, the soft universal breaking terms lead to a high degenera y in the sfermioni se tor. Flavor hanging neutral urrent (FCNC) tests play an important role to onstrain the SUSYmass spe trum.

Wethus onsider the SUSYextensions of the SMwith non-universalsoft breakingterms [5℄. We shall use the mass insertionmethodby whi h itis possible to obtain a set of upper bounds on the o-diagonalterms (

∆

) in the sfermion mass matri es (the mass terms relating sfermion of the same ele tri harge but dierentavor). Obviouslythe mass insertionmethodoers the major advantage that one doesnot need the full diagonalizationof the sfermion mass matri es. Then only a smallnumber of ee tive parameters (

δ

) summarize the ee ts. We have applied this method to the gluinos, neutralinos and harginos ontributions to the de ay

b → sγ

, the

∗

email: mb wanadoo.fr

†

bounds onthe o-diagonaltermsin the sfermion mass matri es(for squarks down and up). This paper isorganizedasfollows. Inse tion2 wepresent generalitiesonthe minimalSUSY models and the mass insertion method. In se tion 3, we give the expli it expressions of the amplitudes asso iated with the gluinos, neutralinos and harginos ontributions to the de ay:

b → sγ

. Finally in se tion 4, we nd expli it expressions for the bran hing ratio

BR(b → sγ)

andtheupperboundsobtainedfortheo-diagonalterms(

∆

)inthesquarkmassmatri es. Inthe annex,we re allthe analyti expressions forthe Feynmanintegralswhi h ariseinthe evaluation of these amplitudes.

2 Generalities on the minimal SUSY models and the mass insertion method.

The minimalsupersymmetri standard model(MSSM), obtained by supersymmetrizingthe SM eld ontents and allowing for all possible soft susy-breaking terms, ontains a huge number of freeparameters. Inthisnote,we on entrateonaspe i setofmodelswherethesesoft-breaking terms are lose tothe msugrauniversality.

By minimal supergravity models [3℄ (msugra) we belowmean, the low energy limitof spon-taneously broken

N = 1

supergravity theories whi h supersymmetrize the SM and present the following two features :

• R

-parity is implemented so that no baryon and/or lepton number violating terms appear inthe superpotential,

•

the Kälhermetri is at, i.e. allthe s alar kineti termsare anoni al.

These features bring about new sour es of FCNC (avor hanging neutral urrent) ee ts. The experimental limitson

B

meson physi s and in parti ular

b → sγ

, onstitute interesting FCNC tests for these MSS Models, typi ally requiring squark masses of the same ele tri harge to be relativelydegenerate,i.e. theirmassdieren emust besmallerthan theiraveragevalue. Briey, we review the majoringredientswhi h give rise tothis new sour e of FCNC.

The lowenergy Lagrangian onsists of:

•

The superpotentialof the

N = 1

globally supersymmetri SM:

W = h

u

QH

1

u

c

+ h

d

QH

2

d

c

+ h

L

LH

2

e

c

+ µH

2

H

1

(1)

•

The s alar part of SUSYsoft breaking terms for the minimal

N = 1

supergravity theories :

L

scal

sof t

= m

2

×

X

i=scalar

|ϕ

i

|

2

+

h

Am



h

u

QH

e

1

u

e

c

+ h

u

QH

e

2

d

e

c

+ h

L

LH

e

2

e

e

c



+ BmµH

2

H

1

+ h.c

i

(2)

where

A

and

B

are two dimensionless free parameters of the trilinear and bilinear s alar ontributions;

m

denotethe s aleof the lowenergy SUSYbreaking.

From equations (1)and (2), we obtain the squarkdown

6 × 6

mass matrix

(Q = −

1

3

)

M

_df

2

_e

_d

_∗

=

"

m

2

f

d

L

d

f

∗

L

m

2

f

d

L

d

f

c

L

m

2

f

d

∗

L

d

f

c∗

L

m

2

f

d

c

L

d

f

c∗

L

#

(3) Where

m

2

f

d

L

d

f

∗

_L

= m

2

f

d

c

L

d

f

c∗

L

= m

d

m

+

_d

+ m

2

× 1

(4) and

m

2

_d

_f

L

d

f

c

L

= Amm

d

+ µm

d

hH

1

i/hH

2

i

(5) with

m

d

= 3 × 3

quarks

D

mass matrix and

e

D

= −

1

3

is the ele tri harge. At this stage, it is lear that the

d

L

− f

d

+

L

− eg

oupling annot lead to avor hanges (FC). Indeed, if we diagonalize

m

d

m

+

d

, we diagonalize at the same time

m

2

f

d

L

d

f

∗

L

. However, this is no longertrue if we renormalize

m

2

f

d

L

d

f

∗

_L

: its value stays (4)atthe high s ale, but itsevolution down tothe

M

W

s ale annot be diagonaldue to the

h

u

QH

1

u

c

term in the superpotential(1). Hen e the

3 × 3

mass matrix of

m

2

e

df

d

∗

renormalizedto the

M

W

s ale, reads:

m

2

_d

_f

L

d

f

∗

L

(q

2

= M

_w

2

) = m

d

m

d

+

+ m

2

× 1 + cm

_u

m

+

u

(6)

Wherethe oe ient

c

anbe omputedbysolvingtheset ofrenormalizationgroupequationsfor theevolutionoftheSUSYquantities. Fromequ. (6)weseethatthesimultaneousdiagonalization of

m

2

f

d

L

d

f

∗

_L

and

m

d

m

d

+

is no longer possible due to the presen e of the

cm

u

m

+

u

term. The avor

hangeisproportionalto

c

andtotheusual(CKM)angles. Inabasiswhere

d

L

− f

d

+

L

−eg

ouplings are avor diagonal, the avor mixing o urs in the squark propagators. The above remark an be summarizedin the followings hemati way:

∆

ij

_LL

˜

d

iL

d

˜

jL

→

∆

ij

_LL

= c(V.[m

diag

u

]

2

.V

†

)

ij

(7)

Forthe

L → R

transitions, wehave:

∆

ij

_LR

˜

d

iL

d

˜

c

jL

→

∆

ij

_LR

= ∆

bs

LR

(8)

Thequantities

∆

ij

aremassinsertions onne tingavors

i

and

j

alongasfermionpropagatorand the indi es

L, R

referto the heli ity of the fermion partners. There are three typesof sfermions

mixing:

∆

LL

,

∆

RR

and

∆

LR

. In the MSSM ase with universal soft SUSY breaking (msugra), there exists a kind of hierar hy among mass insertions that is

(∆

LL

)

ij

≫ (∆

LR

)

ij

≫ (∆

RR

)

ij

. Thisisnolongertrueifavor hangingisprodu edbyanotherkindofinitial onditions. Then, generally, nothing an be said about the hierar hy of these 3 ontributions. In that ase, one needs a model-independent parameterization of the avor hanging (FC) and CP quantities in SUSYtotest variantsof theuniversalMSSM. The hosen parameterizationisthemass insertion approximation [6℄-[8℄. It on erns the most pe uliar sour e of FCNC SUSY ontributions that do not arise from the mere supersymmetrization of the FCNC in the SM. They originate from the FC oupling of gluinos, neutralinos and harginos to fermions and sfermions. One hooses a basis for fermions and sfermions states where all ouplings of these parti les to gauginos are avor diagonal,while the FC originatesfrom non-diagonalsfermion mass terms inpropagators. Denotingby

∆

the o-diagonalterms inthe sfermionsmassmatrix (i.e. the massterms relating sfermions of the same ele tri harge, but dierent avor), the sfermion propagators an be expanded as a series of

δ = ∆/ ˜

m

2

where

m

e

is an average sfermions mass and a typi al s ale of the SUSYbreaking. As longasthe ratio ofnon-diagonalentries (

∆

) tothe average squarkmass isasmallparameter[5℄,the rsttermintheexpansion, obtainedfromthenon-diagonalinsertion ofmassbetweentwodiagonalsquarkpropagators,representsanreasonableapproximation. This method has the advantage that one doesnot need the full diagonalizationof the sfermion mass matri es. So, from the FCNC experimental data we may derive upper bounds on the dierent

δ

's.

In the following se tion, we give the expli it expression of the amplitudes asso iated to the gluinos,neutralinos and harginos ontributions tothe de ay:

b → sγ

[2℄,[10℄,[11℄.

3 Amplitudes ontributing to the de ay

b → sγ

TheMSSMFeynmanrulesusedforthe al ulationoftheamplitudes anbefoundinthereferen e [9℄. The al ulationoftheseamplitudesisdonewiththeseFeynmanrulesandthemass insertion approximation. Supersymmetri penguindiagrams ontributing tothe de ay

b → sγ

are:

•

gluinos(pengluinos): gures 1(a) and 1(b)

•

neutralinos(penneutralinos): gures2(a) and 2(b)

•

harginos (pen harginos): gures 3(a) and 3(b); gures 4(a) and 4(b); gures 5(a) and 5(b).

These diagrams indu e the ee tive operator

O

LR

= m

b

ε

µ

(q)s(p − q)σ

µν

_q

ν

P

R

b(p)

,

q

is the out-going momentum of the photon.

Nowwe angivetheexpli itexpressionsoftheamplitudesasso iatedwiththesupersymmetri penguin diagrams.

3.1 The pengluinos

b

R

p

˜

d

iL

k − p

1(a)

˜

d

iL

s

L

p − q

˜

g

q

b

R

p

˜

d

iR

k − p

1(b)

˜

d

iL

s

L

p − q

˜

g

q

Figure1: Gluino ontribution(pengluinos).

T

g

′

_g

LL

= e

D

C

2

(R)α

s

1

√

π

√

α

δ

LL

M

2

D

ε

µ

(q)s(p

− q)σ

µν

q

ν

P

R

[pH(X

_eg

) + qH(X

_eg

)]b(p)

(9) where

D

i

with

i = 1...6

aresquarkdown masseigenstates,

e

D

istheele tri hargeofthesquark D;

X

eg

=

M

2

e

g

M

2

D

and the fun tion

H(X

eg

)

is given in annex;

δ

LL

is the mass insertion onne ting avors

b

and

s

with the heli ity

L

:

δ

LL

=

6

X

i=1

(M

2

Di

− M

D

2

)Z

D

si∗

Z

D

bi

M

2

D

=

∆

LL

M

2

D

(10)

inwhi h

Z

D

is amixingmatrix dened by:

diag(M

_D1

2

...M

_D6

2

) = Z

_D

+



M

2

LL

M

2+

LR

M

2

LR

M

RR

2



Z

D

;

M

2

D

is the average squark down mass and

C

2

(R) =

4

3

(fundamentalrepresentation ).So the

T

′

g

g

LL

expression be omes:

T

g

_g

′

LL

= e

D

C

2

(R)α

s

δ

LL

M

2

D

√

π

√

αε

µ

(q)s(p − q)σ

µν

q

ν

P

R

[pH(X

_eg

) + qH(X

_eg

)]b(p)

From the diagram drawn ingure 1(b) we have

T

_g

_g

′

_LR

= e

D

C

2

(R)α

s

M

_eg

δ

LR

M

2

D

√

π

√

αM

1

(X

eg

)ε

µ

(q)s(p − q)σ

µν

q

ν

P

R

b(p)

with

δ

LR

=

6

X

i=1

(M

Di

2

− M

D

2

)Z

D

si∗

Z

(b+3)i

D

M

2

D

=

∆

LR

M

2

D

Thanks to the experimental limits, it will be possible to put upper bounds on the dierent

δ

's, that ison the non-diagonal termsin the sfermion mass matrix.

b

R

p

˜

d

iL

k − p

2(a)

˜

d

iL

s

L

p − q

χ

0

j

q

b

R

p

˜

d

R

k − p

2(b)

˜

d

iL

s

L

p − q

χ

0

j

q

Figure2: Neutralino ontributions (penneutralinos).

b

R

p

˜

u

iL

k − p

3(a)

˜

u

iL

s

L

p − q

χ

−

j

q

b

R

p

˜

u

iL

k − p

3(b)

˜

u

iL

s

L

p − q

χ

−

j

q

Figure 3: Chargino ontributions with photon ouplingtoup squarkand mass insertion

LL

.

3.2 The penneutralinos

The penneutralinosare illustrated ingures 2(a) and 2(b).

•

for the diagram(a)

T

_χ

′

0

LLj

= e

D

α

w

δ

LL

2 cos

2

_(θ

w

)M

D

2

√

_π

√

α(z

Lχ

0

j

)s(p − q)ε

µ

(q)σ

µν

_q

ν

P

R

[pH(X

0j

) − qH(X

0j

)/3]b(p)

j = 1...4

the four neutralinosindi es,

X

0j

=

M

2

χ0

_j

M

2

D

with

M

χ

0

j

the neutralino mass. and

z

Lχ

0

j

=









1

3

Z

1j

N

sin θ

w

− Z

2j

N

cos θ

w









2

Clearly,

z

Lχ

0

j

is less or equal to

1

. In msugra, for example, we will have:

z

Lχ

0

1

≈

sin

2

(θ

w

)

9

be ause

Z

11

N

≈ 1

for the lightest neutralino ( bino-like), and

z

Lχ

0

2

∼

= 0.8

for

Z

22

N

≈ 1

•

for the diagram(b)

T

_χ

′

0

RLj

= −e

D

α

w

sin(θ

w

)δ

LR

M

χ

0

j

3 cos

2

_(θ

w

)M

D

2

√

π

√

α(z

Rx

0

j

)M

1

(X

0j

)s(p

− q)ε

µ

(q)σ

µν

_q

ν

P

R

b(p)

with

z

Rχ

0

j

= (

1

3

Z

1j∗

N

sin θ

w

− Z

N

2j∗

cos θ

w

)Z

N

1j∗

where,in msugra,we an have

z

Rχ

0

2

≈ 1

b

R

p

˜

u

iL

k − p

4(a)

˜

u

iL

s

L

p − q

χ

−

j

q

b

R

p

˜

u

iL

k − p

4(b)

˜

u

iL

s

L

p − q

χ

−

j

q

Figure4: Chargino ontributions with photon ouplingto the hargino and mass insertion

LL

.

3.3 The pen harginos

Duetothephoton- hargino oupling,thereare

6

diagrams. Forthemassinsertion

LL

,4diagrams illustratedingures3(a)and3(b)andgures4(a-b) ontribute. Whentheheli ityipisrealized inthequark

b

externalline,onlythewino omponentof harginois on ernedinthe al ulationof theamplitude(diagrams b).Butinthe ase wheretheheli ityipisrealizedonthe harginoline the higgsino omponents are taken into a ount(diagrams a). From the pen hargino diagrams: gures3(a) and 3(b), wherethe photonis oupledtothe squarks, we obtain:

•

for the diagram(a)

T

_χ

′

−

LLj

= e

u

α

w

m

b

δ

LLχ

j

M

χ

−

j

Z

+∗

1j

Z

2j

−∗

M

w

cos(β)M

U

2

√

2π

√

αM

1

(X

j

)s(p − q)ε

µ

(q)σ

µν

q

ν

P

R

b(p)

where

j = 1, 2

are the two hargino states,

X

j

=

M

2

χ−

_j

M

2

U

and

M

χ

−

j

is the

j

hargino mass,

e

u

= 2/3

isthe squarks up ele tri harge,

M

U

the squarkup average mass. And

δ

LLχ

j

=

6

X

i=1

3

X

J=1

3

X

K=1

(1 − δ

J K

)

(M

2

U i

− M

U

2

)

M

2

U

Z

U

Ki

V

sK

Z

U

J i∗

V

∗

bJ

(11)

in whi h

J

and

K

run over the 3 generations of squarks and

U

i

with

i = 1, . . . 6

are up squarks mass eigenstates. As in

δ

LL

,

δ

LLχj

ontains squark mixing fa tors

Z

, but in addition,there are some known Cabbiboquarks mixingfa tors (e.g.

V

bc

).

•

for the diagram(b)

T

_χ

′

−

LLj

= e

u

α

w

δ

LLχ

j

Z

+∗

1j

Z

1j

+

M

2

U

√

π

√

αs(p − q)ε

µ

(q)σ

µν

q

ν

P

R

[pH(X

j

) + qH(X

j

)/3]b(p)

Inmsugraforthelightest hargino,wehave

Z

+∗

1j

≈ 1

when

Z

−∗

2j

≈ 0

;insu h ase,weremark that onlythe diagram (b) ( wino omponentof hargino) ontributeto the amplitude.

Thepen harginosillustratedingures4(a)and4(b),wherethephotonis oupledtothe hargino, give:

•

for diagram(a):

T

_χ

′

−

LLj

= −α

w

m

b

δ

LLχ

j

M

χ

−

j

Z

+∗

1j

Z

−∗

2j

M

2

w

cos(β)M

U

2

√

2π

√

αF (X

j

)s(p − q)ε

µ

(q)σ

µν

q

ν

P

R

b(p)

b

R

p

˜

u

iL

k − p

5(a)

˜

u

iR

s

L

p − q

χ

−

j

q

b

R

p

˜

u

iL

k − p

5(b)

˜

u

iR

s

L

p − q

χ

−

j

q

Figure 5: Chargino ontributions with mass insertion

LR

.

•

for the diagram(b)

T

_χ

′

−

LLj

= α

w

δ

LLχ

j

Z

+∗

1j

Z

1j

+

M

2

U

√

π

√

αs(p − q)ε

µ

(q)σ

µν

q

ν

P

R

[pG(X

j

) + qG(X

j

)/2]b(p)

We dene the fra tion of gaugino in the hargino

j

by:

z

χLj

= Z

+∗

1j

Z

1j

+

. the

F (X

j

)

and

G(X

j

)

are given inannex 6. As above, inmsugra, onlythe diagram (b) will ontribute to the amplitude.

The

LR

mass insertionfor the hargino ontribution are illustrated ingures 5(a) and 5(b). Only the higgsino omponents of the hargino ontributes to the amplitude. Therefore, due to the

s

L

− χ

−

j

− U

iR

ouplinggivinga fa tor

U

J

_Z

(J+3)i

U

Z

+∗

2j

P

R

V

sJ

where

U

J

is a Yukawa oupling proportional to the asso iated quark mass, the top quark ontribution overwhelms the up and harm ones.

From diagram (a)ingure 5 wethus obtain:

T

_χ

′

−

LRj

= −e

u

α

w

m

t

m

b

δ

LRχ

j

z

Rj

M

χ

−

j

M

2

W

cos(β) sin(β)M

U

2

√

π

√

αM

1

(X

j

)s(p − q)ε

µ

(qσ

µν

q

ν

P

R

b(p)

where 1

δ

LRχ

j

=

6

X

i=1

(M

2

U i

− M

U

2

)

M

2

U

Z

_U

(t+3)i

V

st

Z

U

ti∗

V

∗

bt

(12)

and for diagram(b) :

T

_χ

′

−

LRj

= α

w

m

t

m

b

δ

LRχ

j

z

Rj

M

χ

−

j

2M

2

W

cos(β) sin(β)M

U

2

√

π

√

αF (X

j

)s(p − q)ε

µ

(q)σ

µν

q

ν

P

R

b(p)

with:

z

Rj

= Z

+∗

2j

Z

2j

−∗

, the fra tion of higgsino in the hargino

j

, its greatest value is

1

and the minimum value

1

2

for one of the

2

harginos. In the following se tion we give the expli it expression of the bran hing ratio for the de ay

b → sγ

and the upper bounds on the mass insertions.

1

It would be desirableto pullout CKMfa tors from

δ

's, sothat theyonly ree tsquarkproperties. While thisisarguablypossiblefor

δ

LRχ

,itrequiresnon-trivialassumptionsfor

δ

LLχ

,whi hiswhywekeptCKMfa tors inthedenitionofboth.

4

BR(b → sγ)

Expression and the upper bounds on the mass insertions

δ

.

The de ay

b → sγ

is very interested be ause the rare

B

de ay represent a good test for new physi sbeyond SMsin e they are not ae tedappre iably by un ertainties due tolong distan e ee ts.

The bran hing ratio [10℄ is

BR(b → sγ) =

BR(B → X

s

γ)

BR(B → X

c

eν

e

)

=

Γ(b → sγ)

Γ(b → ceν

e

)

where

b → ceν

e

is dominant then

BR(b → sγ) = Γ(b → sγ)τ

B

.The expli it expression of

BR(b → sγ)

obtained fromthe al ulationexposed inthe se tion3 is:

BR(b → sγ) =

m

3

b

α τ

B

16 π

2









m

b

α

s

e

_M

D

2

C

2

(R)

D

δ

LL

H(X

eg

)

(13)

+

α

s

e

D

C

2

(R)M

eg

M

2

D

δ

LR

M

1

(X

eg

) −

e

D

α

w

M

χ

0

j

sin

2

_(θ

w

) z

Rχ

0

j

9 M

2

D

δ

LR

M

1

(X

0j

)

+

e

D

α

w

m

b

sin

2

_(θ

w

) z

Lχ

0

j

18 M

2

D

cos

2

(θ

w

)

δ

LL

H(X

0j

) +

m

b

α

w

M

2

U

(G(X

j

) + e

U

H(X

j

)) δ

LLχ

j

+

α

w

m

b

m

t

M

χ

−

j

M

2

w

cos(β) sin(β)M

U

(

F (X

j

)

2

− e

U

M

1

(X

j

))δ

LRχ

−

j











2

Byimposingthatea hindividualterm, intheabove equation,doesnotex eed inabsolutevalue theexperimentaldataof

BR(b → sγ)

thatis

(1 − 4)10

−4

(whi hin ludestheQCDun ertainties following [4℄), we give upperbounds on the dierent

δ

.

Wehave hosenthevaluesforthesupersymmetri parti lesfromtheexperimentaldatagiven in[12℄. Moreover, we haveimposed the following onditions :

•

the average squarkmass:

M

U

= M

D

= M

e

q

.

•

Forthe neutralino masses: we hoose the LSPmass, with

M

χ

0

1

≈

M

_e

g

6

(GUT relation n)

•

The lightest stop mass:

M

et

≥ M

χ

0

1

+ 30Gev

•

The harginomass:

M

χ

−

1

≈ 2M

χ

0

1

(GUT)

Otherwise, we lay down

X

eg

=

M

2

e

g

M

2

e

q

,

X

0

=

M

2

χ0

M

2

˜

q

,

X =

M

2

χ−

M

2

˜

q

. The others experimental data hosen are dened in the followingtable :

α

s

α

α

w

sin

2

(θ

w

) BR(b → sγ)

M

W

m

b

m

t

τ

B

0.12 1/127.9

α/s

2

w

0.2315

1 → 4 10

−4

80.41 4.5 170 1.49

10

−12

s

The results are given, in tables 1 and 2, for dierent

tan(β)

values ( i.e.

tan(β) =

2, 5, 10, 20 et 40) and for 2 values of

M

q

˜

:

M

q

˜

= 300GeV

et

M

q

˜

= 500GeV

(the orresponding results

M

_eg

X

˜

g

δ

LL

δ

LR

M

χ

0

1

X

0

δ

LL

z

Lχ

0

1

δ

LR

z

Rχ

0

1

300

1

2.96

10

−2

50

3 × 10

−2

_7.1

_0.34

(0.36)

(8.2)

_{(2.7 × 10}

−2

₎

₍₁₀

−2

₎

_(19.7)

_(0.94)

600

4

9.5

_{1.8 × 10}

−2

100

10

−2

_8.4

_0.25

(1.44) (26.4) (4.9 × 10

−2

₎

_{(4 × 10}

−2

₎

_(23.25)

_(0.7)

800

7

17.6

_{2.6 × 10}

−2

130

0.19

9.8

0.23

(2.56) (48.8) (7.2 × 10

−2

₎

_{(7 × 10}

−2)

_(27.3)

_(0.64)

Table 1: Limitsontheo-diagonalterms

δ

LL

and

δ

LR

fordown squarks with

M

q

˜

= 300

GeV (or

M

q

˜

= 500

GeV), oming fromgluino and neutralino ontributions.

M

χ

−

X

δ

_LLχ

z

_χL1

δ

_LRχ

z

_R1

δ

_LRχ

z

_R1

δ

_LRχ

z

_R1

δ

_LRχ

z

_R1

δ

_LRχ

z

_R1

tan(β) = 2

M

U=M

q

tan(β) = 5

tan(β) = 10

tan(β) = 20 tan(β) = 40

100

0.1

0.57

0.08

0.04

0.02

0.01

0.0051

(4 × 10

−2

₎

_(1.58)

_(0.09)

_(0.045)

_(0.023)

_(0.011)

_(0.0058)

200

0.44

1.23

1.6

0.76

0.39

0.2

0.099

(0.16)

(3.41)

(0.2)

(0.095)

(0.049)

(0.025)

(0.012)

300

1

2.34

0.3

0.14

0.075

0.03

0.019

(0.36)

(6.5)

(0.83)

(0.4)

(0.2)

(0.1)

(0.052)

Table 2: Limits on the o-diagonal terms

δ

LL

and

δ

LR

for up squarks with

M

q

˜

= 300

GeV (or

M

q

˜

= 500

GeV), oming from harginos ontributions.

are notedbetween parenthesisin tables). Wehave obtained the

LL

insertionlimitsbyimposing

BR(b → sγ) = (4) × 10

−4

and for

LR

insertion

BR(b → sγ) = (2) × 10

−4

tobe onsistent with referen e[6℄ inthe gluino ase.For hargino ontribution wetake

M

U

= M

q

˜

.

From the results obtained we remark that:

•

be ause of our use of the mass insertions, we are limited to

δ < 1

for the bran hing ratio expression (13) to make sense. When larger than one, experimental bounds on

δ

like

δ

LL

< 2.96

thus really mean that the maximal ee t of su h terms (for

δ

LL

∼ 1

) only ontributes a fra tion of about

1/9

th of the experimental bound;

•

in the gluino ase,

δ

LL

is more sensitive to the gluino mass than

δ

LR

, be ause this last ontribution has anamplitude enhan ement fa tor of

M

eg

. Then the dependen e on

M

˜

g

of the

H(X

g

)

fun tion is partially ompensated (see the denitions in Annex 6 and the plot 6);

•

for the neutralino, the limit on

δ

LL

is less sensitive to the neutralino mass, thanks to the small values of

X

0

ontained in the same

H(X

0

)

fun tion. However, even for the upper value for

z

Lχ

0

j

= 1

(e.g. for

j = 2

), the limit annot be more ompetitive than the gluino limit,ex ept if

M

_χ0

M

g

_e

is smaller than inthe msugra models. The

δ

LR

limitsde rease weakly with the enhan ement of

M

χ

0

, due to the additional power of

M

χ

0

in the amplitude and thefa t thatthe fun tion

M

1

(X

0

)

isfairly onstantfor small

X

0

(see Annex6and plot 6);

•

the hargino ontributionistheonlyone onstrainingthedieren esbetweentheupsquark masses. However the expressions for

δ

LLχ

and

δ

LRχ

ontain Cabbibo mixingfa tors. For

δ

LRχ

, the dominating top ontribution allows to extra t a simple fa tor of

V

ts

V

tb

≈

1

30

. The rst remark above then applies for interpreting our mass insertions results, on e

δ

LLχ

> 0.03

. Nevertheless, we obtain a limit that qui kly be omes more onstraining with in reasing

tan(β)

. For

δ

LLχ

, the fa torization of CKM elements annot be so easily justied: if for instan e there is large mixing between the squarks 2 and 3, the leading CKM fa tors are of order one, and the expression (11) for

δ

LLχ

with up squarks be omes thesameas

δ

LL

fordown squarks in(10). Thesensitivityon

M

q

˜

ofthe boundsisabout the samefa tor of 3forthe hargino

δ

LLχ

asfor the neutralino

δ

LL

and

δ

LR

while the hargino

δ

LRχ

is less sensitive forsmall harginomasses;

•

the limitsintables (1) and (2) an easilybe generalized for dierent values of squark and gaugino masses: they are inversely proportional to the fun tions given in Annex 6 and plottedin gure6,

•

nally,if following [13℄, we use the latestCLEO data [14℄ and substra t the SM ontribu-tion [15℄, we obtain a tighter bound, in the absen e of an ellations between the various ontributions that was assumed throughout this paper. The various results in the tables are then redu ed by afa tor of about 20.

5 Con lusion

Wehavegivenexpli itexpressionsfortheamplitudesasso iatedtothesupersymmetri ontribu-tionstothe de ay

b → sγ

inthe ontextofsupersymmetri extensions ofSMwith non-universal soft SUSY breaking terms. The model independent parameterization whi h we have hosen is the mass insertion approximation. From the FCNC experimental data, we have derived upper bounds on the dierent

δ

's. The ontribution from the hargino and neutralino ex hanges are less sensitive tothe gaugino mass than the gluino ontribution.

6 Annex

H(X) =

−1 + 9X + 9X

2

_{− 17X}

3

_{+ 6X}

2

_{(X ln X + 3 ln X)}

12(X − 1)

5

M

1

(X) =

1 + 4X − 5X

2

_{+ 2X(X + 2) ln(X)}

8(X − 1)

4

= L(X)/2

F (X) =

5 − 4X − X

2

_{+ 2 ln(X) + 4X ln(X)}

2(X − 1)

4

G(X) =

1 + 9X − 9X

2

_{− X}

3

_{+ 6X(1 + X) ln(X)}

3(X − 1)

5

−G(X)

−F (X)

M

1

(X)

H(X)

X

10 1 0.1 0.1 0.01

Figure6: Thedierentfun tionsen ounteredintheevaluationoftheamplitudes,fortheavailable values of

X = M

2

gaugino

/M

squarks

2

ona logarithmi s ale.

Note added

This work was supported in part by the GdRSupersymetrie of the Fren h CNRS.

Whilethisworkwasbeingrefereed,ageneralstudyof

B → X

s

γ

appeared[16 ℄withinteresting resultsontheinterferen es betweenvarious ontributions,in ludingtheones presentedhere. For theparameters studiedthere(

µ = 300

GeV,

M

q

˜

= 500

GeV,

M

2

= 100 → 230

GeV),we agreethat there is no onstrain on up squarks from table 2: the light hargino is a gaugino with

z

R1

≪ 1

and the heavy hargino gives the last line in parenthesis, with

z

R2

= 1

. Taking the Cabbibo fa tors into a ount, the strongest bound on the o-diagonal element

δ

u,LR,23

is just about one for

tan β = 50

, whi h means no ontraintin the mass insertionlogi s.

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