Rare B-meson decays in supersymmetry

IL NUOVO CIMENTO VOL. 106 A, N. 8 Agosto 1993

Rare B-Meson Decays in Supersymmetry(*).

T. M. ALIEV(**), G. TURAN and O. YILMAZ

Physics Department, Middle East Technical University - Ankara, Turkey

(ricevuto il 18 Dicembre 1992; approvato il 12 Febbraio 1993)

Summary. -- The weak processes B --~ Ket ~, B ---) H* et~(e = e or ?), B --. K~ and B --. K* ~ are studied in the framework of the supersymmetric theory (SUSY). It is shown that the branching ratio (BR) for B ---) Ket ~ decay is in the order of 10 .6 and it is - 10 .5 + 10 .6 for B ---) K* Ct ~. It is found that there are mass regions where the contribution of the SUSY particles to the decay rate exceeds that of the SM. Using the existing experimental upper bound to BR restrictions on the masses of the SUSY particles ~g and m3/2 are obtained.

PACS 12.90 - Miscellaneous theoretical ideas and models.

1. - I n t r o d u c t i o n .

The main goal of the future B-factories is to preform more careful and comprehensive investigations on the B-meson decays and the poorly studied aspects of the standard model. For example, one may recall the reliable determination of the Kobayashi-Maskawa matrix elements Vbu, Vcb (which will be in turn an indirect test of the existence of the t-quark and fourth generation), CP violation, etc.

B-meson decays are rather suitable tools to probe new physics beyond SM, in particular for supersymmetry. Usually searching for new physics is performed in two different ways. One of them is direct observation of new particles which is one of the essential parts in the experimental programmes of high-energy colliders SpaS, LEPI I , UNH, HERA. The second one is to make high-precision experiments at low energies and look for deviations from the SM predictions. SM suppressed rare decays have been very suitable candidates to this aim. The decay rate can be considerably enhanced because of the new particles which are predicted by models beyond SM.

From the theoretical point of view, rare decays take place via flavour-changing

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) Permanent address: Institute of Physics, Azerbaijanian Academy of Sciences, Baku, Azerbaijan.

1059

1060 T.M. ALIEV, G. TURAN and O. YILMAZ

neutral current (FCNC). According to the Glashow-Weinberg-Paschos theorem [1] these decays occur in higher levels of perturbation theory. So, rare decays are very interesting objects for testing SM prediction at loop level. The rare decays B --, K(e, B ---) K* ((, B --~ K~ and B ~ K* ~ seem to be the most promising candidates to this aim. They have been investigated in the framework of the SM in ref. [2] and [3].

While in the SM the b--. s transitions are dominated by one-loop contributions with the exchange of a virtual W and the top quark, in SUSY new sources of FCNC are present due to the interactions between gluino, quark and scalar quark. Since these interactions are non-diagonal with respect to fiavours, gluino can mediate FCNC[4]. In this case the strong-coupling constant ~ appears in the decay amplitude instead of the Fermi coupling constant GF in SM. Besides, scalar quarks and gluino which are predicted by SUSY theories may be lighter than W- and Z-bosons. All these factors can significantly enhance the decay amplitude. Therefore it is necessary to investigate rare B-meson decays more comprehensively.

This work is an extended version of our previous study on the above-mentioned rare B-processes in the context of the minimal N = 1 supersymmetric theories [5].

2 . - F o r m a l i s m .

Calculation of the weak-decay amplitude for B --* xee (x = K, K* and e = e, ~, ,) is performed in two steps. The first step is to derive the effective Hamiltonian which describes weak decays at quark level. The second one is to obtain the weak-decay matrix elements starting from this Hamiltonian.

The effective Hamiltonian for the weak decays of the type B --+ Kee is derived by integrating out the heavy degrees of freedom. In the framework of the SM, the effective Hamiltonian (HsM) at quark level for the transition b --* sfe has the following form [3]:

(1) HSM--~ ~2 Ytb Y~sj=l,2~, 7,8,9 Cj(lz)Oj([z).

The explicit form of the coefficients Cj(/z) and operators Oj(~) can be found in [3]. Note that without strong-interaction corrections the amplitude of the transition b --. see has been calculated in [2].

Let us now calculate the weak B ---) Xee decays in the context of SUSY. Diagrams contributing to these processes can be divided into three classes (see fig. 1): the box diagrams, Z and 7 exchange diagrams generated by the induced b~Z and bg7 vertices, correspondingly. Calculations show that the contributions of the SUSY box diagrams are very small in comparison with the other diagrams, therefore we do not present their explicit forms here.

Before going into the details of the calculations we would like to make the following remarks. In the gluino exchange diagrams we consider only left-handed quarks in the external legs because the dominant contribution to flavour-changing gluino-quark-scalar-quark vertices comes from the left-handed sector [6]. The second remark is about the scalar b and ~ quark mass difference. In the framework of the model which we are using the scalar quark masses are given by ~ 2 = m212 + cmt2, ~ - - ~ - - m 2 / 2 , where c is the model-dependent parameter between 0.1 and 1 [4, 6, 7] and m3/2 is the gravitino mass. The third remark concerns the charged

RARE B - M E S O N DECAYS IN SUPERSYMMETRY 1061

e - ~ / e § ~?,Z

u, c, t

t~',~

W

t'-f )~ t '+ a)

g

b w -_ , , v c c v v ~ . = e +

u,r I b,c,t ~ v v w c ~ - - ~ - - - - - e - s W

g

- 7i. ~T E 7

b)

, z z z

g g g

tz ~, ~ ~z -~i~ fi fi c) fi fi ~z

Fig. 1. - Feynman diagrams describing the B ~ XeC decay in a) SM and b), c) SUSY theories, respectively. Here if, ~,/-] and ti, j denote the gluino, scalar quark, higgsino and scalar top quark fields.

higgsino diagrams. The advantage of the changed higgsino consideration is that in this case the super GIM suppression is absent and Yukawa coupling of the top quark is included into the game. Finally, we note that the sum of the Z-boson exchange diagrams vanishes for zero external momenta. In the higgsino exchange case, it is necessary to take into account the tL-tr mixing. The related mass matrix is given by

(2) ( m~/2 + m2t + cm~ Am3~2 mt 1.

Am3~2 net m~/2 + m~ ] After diagonalization of this matrix, we have two states with masses n~tl and ~ a :

mr1, t2 + Am3/2 mt

1062

and also

T. M. ALIEV, G. TURAN and o. YILMAZ

2Am3 /2 mt (4) tg2/3 = ,

cm 2

where fl is the mixing angle between {-L and t-R states. Since cm 2 << 2Am3/2mt, the mixing angle/3 = rr/4.

Now, the effective Hamiltonian for the transition b ~ see in s u s Y can be written in the following form:

(5) HsvsY= 1-~2 VtbVr sy'Lb(~[~y,f + B['y~ysf) +iG~%~-~(m~L +mbR)b~y ~ ,

where L = (1 - y5)/2 and R = (1 + y5)/2. We first consider the photon exchange diagrams. After some calculations we

get:

(6)

z ~ m H / L l

~=0,

cos2/3(F2 (~h)

3F +sin2/3 F l (~ t 2) - 3F3(Str ) - ~ 4(~t2)

+

/

-2 -2 = m q / m g , ati where ~q -2 -2 =mt~/mHforq=b , s a n d i = l, 2. H e r e ~ g a n d ~ H a r e t h e

masses of gluino and higgsino, Vtb and l?t~ are the super Kobayashi-Maskawa matrix elements. Following ref. [4, 7] in our calculations we t a k e Yij -- V/j , where Vii are the ordinary Kobayashi-Maskawa matrix elements in the standard model. In eq. (6),

RARE B-MESON DECAYS IN SUPERSYMMETRY

F(5)'S are some functions defined as

1 0 6 3

(7)

1

F 1 ( 5 ) = f dx(1 - x) 3 x + 5(1 - x) '

0

1 I dxx(1 - x) 2

F 2 ( 5 ) = x + 5(1 - x)

o

1

f dxx3 F 8 ( 5 ) =

x + 5(1 - x) o

1 f dxx3

F4(5) = Ix + 5(1 - X)] 2 ' o

1 I d x x 2 ( 1 - x )

F5 (5) = x + 5(1 - x)

o

and h t = 2rot ~ / - G - - ~ v 2 Iv1 is the Yukawa coupling of the top quark and vl and v2 are the vacuum expectation values of the two Higgs doublets which give the masses of up and down quarks in SUSY.

Now we consider the Z exchange diagrams. Explicit calculations show that in zero external momentum the sum of the first three diagrams in fig. lc) gives zero. From the remaining diagrams we obtain

(8)

~[z = - 1---I h t 12sin22fl(l _

/

[ ~ z = - ( I - 4 sin 20w)

~ = 0 ,

where

(9)

[ ] 1 A(s t i ) = 1 1 In + -

( ~ t i 1 ) 2 5ti 5t~ 1 ,

2 52 ] 3 5t2 In 5t2 t-----L-' In / - -

5 t~ - 1 s t , - 1 st, 2 A ( % , s t , ) - - -

5t2 -- 5tl

So, we have completed the first step in calculating the weak-decay amplitude for B --~ xee . For the second step, we first consider the hadronic matrix elements for the

70 - II Nuovo Cimento A

1064

above-mentioned decays. They are defined as

(10)

and

(11)

T. M. ALIEV, G. TURAN and o. YILMAZ

{ ( K ( P K ) I ~ b l B ( P s ) ) = (PB + Pg),f+ (q~) + q J - ( q e ) ,

(K(PK) I~%~ b lB(P~) ) = i[(PB + PK), q~ -- (Ps + PK)~ q,] h(q ~)

(K* (PK.)I-~T~ b lB(PB)) = ie~,~: E~(PK .) (PB + PK*) ~ q: T1 (q~) +

+[e,(PK.)(m~ - m~.) - (e .q)(P~ + PK*),] T2(q2),

(12) (K*(PK,)I~7~,LblB(PB)) = iC~,~:e~(PK,)(PB + PK,)~q:V(q 2) +

+ e~ (PK*) (m~ - m~.)A1 (q2) _ (e .q) (PB + PK,)~]A2 (q2),

where T, = - i%~q~(1 + ys)/q 2 . Since the coefficients off_ and (PB § PK)~ in eq. (10) are proportional to the lepton masses we can safely omit them.

Here, we have also considered the hadronic matrix elements evaluated in terms of the universal Isgur-Wise function which appear in the effective heavy-quark theory [8]. They are given by

(13) { (K(v ' ) l~r~(1 -~ rs)blB(v)) = V ~ K m B ~ ( v ' v ' ) ( v , + v~),

(g(v')l~(1 + y~)blB(v) ) = mV~-~Km~(v'v')(1 + v . v ' )

and

(14) t (K*(v ' , e) l~r , (1 �9 rPb lB(v ) ) = m~K.m~dv'v ' )"

�9 ((1 + v . v ' ) e~ - v~ (e.v) �9 is ,~rv 'ZeY),

( (g*(v ' ) l~ (1 + ys)blB(v) ) = m~K.mB~(V'V ) (e 'V) .

In our numerical calculations, we have used the following form of the Isgur-Wise function:

(15) ~(v.v*) = ~o02- 2 + 2(v-v')

with co0 ~" 1.80. After performing the standard calculations, we have for the differential decay rates

dF(B --. Kee)/ds and dF(B -o K* ee)/ds (s = q2/m~)

dr G~ a2 m~ (16) ~-s (B -~ Kee) - 293r:a IVtbV~12[ImbCTh + Csf+ I ~ + IC~f+ I~] �9

RARE B-MESON DECAYS IN SUPERSYMMETRY 1065

and

(17)

where

(B--eK*eC)- CfF~-------m-B2937~5 IVtbV~s128 s 2 - 2 s 1 + m--~B + 1 -- m~ ] j

{[( ( )] �9 2 1 - ms 2 - s 2 + 2 m~ - s [ImbCTTl+CsVI2+ ICgVI2]+

+ 1 m--~ - 2 + 2 m ~ - s \ 4 - ~ , [ m----~ s - s +

m~ ( m 2 , ) ( inK2, ) ( m2, )] +2m~----~ 1 m~ 1 + 3 m----~--s 1 m~ s �9

From eqs. (6) and (8) we have

C7 = 2C sM + C sUsY ,

c8 -- c2 M + c s'sY ,

c~ = c~ M + c $ 'SY .

(18)

�9 [ ImbC7T2 + CsAl l 2 + IC9A, 12]],

c#usy= V~ e~ , 4GF

c~,~sy = X/2 (. ~# +., , ' ) , 4G~

cSUS Y= V ~ . 4GF

(Here, we use the notation of ref. [2] and the explicit forms of C sM , C sM and C sM can also be found there.) We note that the strong-interaction corrections are small in SUSY models[4]. Therefore we have only considered contributions without a~ corrections.

It is also possible to show that differential decay rates evaluated in terms of the

1066 T.M. ALIEV, G. TURAN and o. YILMAZ

Isgur-Wise function (15) have the following form (see also [9]):

(19) ~s-s (B --) Kff ) =

and

G 2 2 .5 F ~ l fbB 12 2n3~ ~ IVtb V~' l (Q+ + r)2 $Z(v'v')[IC7 + Cs[2 + IC912]

"•8 f* 2 2 . 5 O" F O~ "r B i 2

(20) ( B - * K * e e ) = 2113rr 5 [VtbV~ 1Q§

�9 [([Csi2+ ICg i2 ) (4s~ ( l+r2 - s2 )+Q+(1- r )2 )+4 iCT]2(4 ( (1 - rZ )2 - s ) / s+Q+)+

+ 16 Re (C7 Cs) (1 - r 2 - s2)].

Here, various abbreviations are defined as

r = mK(*)/mB , Q+_ = (1 _ r) 2 - s ,

(v .v ' ) = (1 + r 2 - s2)/2r.

We have also calculated the contributions of the SUSY particles to the decay rate of the B --) K ~ and B --) K* ~; and found that it is negligible in comparison with the SM. Therefore, we shall not present the related expressions for these decays here.

For the hadronic form factors f+, h, A1, As, T1, T2 and V, we use the one-pole dominance model. It this model, we have [10]

f+ (s) = f+ (0)/(1 - s),

h(s) = h(0)/(1 - s),

V(s) -- V(O)/(mB + mE*) (1 -- S),

T l ( 8 ) -~ Tl(O)/m~s(1 - s),

A 1 (s) - A2 (s) -~ (,41 (0)/V(0)) V(s),

Te(S) = (AI(O)/V(O)) TI(S),

where f+ (0) = 0.34, h(0) =f+ (0)/2mb, V(0) ~ TI(0) ~ 0.37 and AI(0) = 0.33. Note that the operator ~%~ b has non-renormalizable quantities, therefore h also depends on the subtraction point/~ [11]. But for simplicity we do not take into account this dependence.

We conclude this section by writing out the input parameters which we have used in our numerical calculations: ms=5.277GeV, mb=5.12GeV, V2/Vl----1, IVtbVt~ [ -- 5 . 1 0 - 2 , v B --- 1.3.10-12s.


3. - R e s u l t s a n d d i s c u s s i o n .

The resul ts of our calculations for the B --) Ke f and B --~ K* f f for f = e including SUSY contributions for different masses of the SUSY particles ~ n , mg and m3/2 as a function of the top quark mass are presented in fig. 2 and 3, respectively. Here, the hadronic form factors f § h, ..., T2 are computed according to the one-pole dominance model (model I), whereas fig. 4 and 5 show the same results with the hadronic matrix elements replaced by the ones evaluated in t e rms of the Isgur-Wise functions (model II). For comparison, we have also presented predictions of the SM in both figures.

F rom fig. 2-5 we see tha t when m t - 200 GeV, BR(B - . K f f ) - 1 .10 -6 and B --~ --* K* ~'f - 3" 10-6 for f = e. We note tha t we have also calculated the BR(B --~ Xef ) - m t dependence for f = ~ and the resul ts we obtained are ve ry similar to the resul ts of the f = e case. According to the es t imates in future B-factories we are expecting (3 + 5). l0 s B-mesons per yea r [12]. So the ra re B -~ K f f and B -~ K* f f decays seem to be quite detectable in the future.

Now we t ry to answer the following question: using the experimental upper limit for B R ( B - . K f f )~< 3.2.10-4112] what kind of restrict ions on the masses of SUSY particles ~ g and m3/2 do we obtain for fixed ~ H and mt? To answer this question we consider the plot of ~ g vs. m3/2 for different sets of ~ H and mr. Referr ing to fig. 5 we find tha t ms/2 >i 30GeV and n~g~>46GeV at i t = 100GeV, ~ H = 150GeV in model I.

We have made a similar calculation for B --) K* f f by using the experimental upper limit for BR which is ~< 2.3.10 -5 , and obtained tha t m3/2 >t 74 GeV and ~ g I> 100 GeV at again i t -- 100 GeV and ~ H = 150 GeV. We note tha t these bounds are s t ronger than the restrict ions obtained f rom collider exper iments [13].

2.4

2.0

1.6

1.2

0.8

0.4

0.( 100

5.6

4.8

4.0

~ 3.2

~2.4

1.6

. - . ; ' / / /

/ / . - " / / 7 " /

0.8

0.0 100

J l * l l l , l l l r 1 6 2 I I c I l l I I l I , l l l l I L I I I I L , I I , , , , I , , ~ I [ I

140 180 220 140 180 220 mt(GeV) mt(GeV)

Fig. 2. Fig. 3.

Fig. 2. - Branching ratio for the decay B --) KCf in units of 10 -6 as a function of the top quark mass mt in model I. SM; - - - - SUSY (mH = 80 GeV, ~ g = 60 GeV, ma/2 = 50 GeV); - - - SUSY (mH = 200GeV, n~g= 60GeV, m~/2 = 50GeV); . . . . . SUSY (mH = 200GeV, ' ~ g = = 60 GeV, ms~2 = 60 GeV).

Fig. 3. - Branching ratio for the decay B --) K* ef in units of 10 .6 as a function of the top quark mass m t in model I. The meaning of the curves is the same as in fig. 2.

1068 T.M. ALIEV, G. TURAN and O. YILMAZ

1.4 4.0

1"2 1 / /

1.0 . . "

~o0.8 ~

0.6

0.4

0.2

0 , 0 I I n ' l ' ' a n l l h h n l n i L ~ l n n n ~ l n t ~ l l n ' n n l n

100 140 180 220 m t (GeV)

Fig. 4.

~.6

3.2 . " "

2.8 --

2.0L4 , ~ ' ' , - ' ' " ~

1.6 " / / 1.2 ~

0.8 0 ,4 i t t J I L ~ t t I L I p , l , , , , I , ~ , l l n n n , l l , , , l l

100 140 180 220 rn t (GeV)

Fig. 5.

Fig. 4. - Branching ratio for the decay B ~ Ket ~ in units of 10 -~ as a function of the top quark mass m t in model II. SM; - - - - SUSY (mn = 100 GeV, ~g = 60 GeV, m8/2 = 50 GeV).

Fig. 5. - Branching ratio for the decay B ~ K* ~'C in units of 10 -6 as a function of the top quark mass m t in model II. The meaning of the curves is the same as in fig. 4, except for m3/z = 40 GeV.

40

35

~3o

25

20

Fig. 6.

"'~%~ ",,x

"~'~-,~,~,,~ -..\

",,,x,,~, ~

30 40 50 60 70 ~ (GeV)

00,

9 0 1 -

801406070!50 ~ "':":::":':":::"::':"2"::::'::::':"~~2 ~':?:,

x", ,

30 "':",

20 40 60 80 100 120 ~g (GeV)

Fig. 7.

Fig. 6. - Dependence of the masses of ~t~g on m3/2 at BR(B --* Ket ~ = 3.2.10 -4 in model I. - . . . . r~ H = 80 GeV, m t = 150 GeV; mH = 100 GeV, m t = 150 GeV; ma = 150 GeV, m t = 100 GeV.

Fig. 7. - Same as fig. 6 but at BR(B ~ K* ee ) = 2.3.10 -5.

In conclusion, in this work we have studied the role of the SUSY particles in the rare decays B --, Kee, B --. K* CC, B --. K ~ and B --. K* ~ . We have found that there are some mass values of new particles where the effect of SUSY exceeds that of SM for B--~ Kee and B--* K'e/7, whereas the contributions to the decays B--* K ~ and B--> K*,~ are negligible in comparison with the SM.


After completing this work we learned about the article of Bertolini et aL [14] studying the SUSY contributions to the effective Hamiltonian. Our calculations agree with theirs.

This work was supported partially by the M E T U Research Grant Project under Grant No. AFP-92-01-05-08.

R E F E R E N C E S

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A. SONI and G. L. KANE: Phys. Rev. Lett., 57, 106 (1986); N. G. DESHPANDE and J. TRAM- PETIC: Phys. Rev. Lett., 60, 2583 (1988); WEI-SHU Hou and R. S. WILLEY: Nucl. Phys. B, 326, 54 (1989).

[3] B. GRINSTEIN, M. J. SAVAGE and M. B. WISE: Nucl. Phys. B, 319, 271 (1989); B. GRINSTEIN, R. SPRINGER and M. B. WISE: NucL Phys. B, 339, 269 (1990); A. ALI: DESY Rep. 91-080 (1991); A. ALI, T. MANNEL and T. MOROZUMI: Phys. Lett. B, 273, 505 (1991).

[4] S. BERTOLINI and A. MASIERO: Phys. Left. B, 174, 343 (1986). [5] T. M. ALIEV, O. YILMAZ and G. TURAN: preprint ICTP-92-209, Trieste (1992). [6] R. LANGACKER and R. SATHIAPALAN: Phys. Lett. B, 144, 401 (1984). [7] J. F. DONOGHUE, H. P. NILLES and D. WAYLER: Phys. Left. B, 128, 55 (1983). [8] N. ISGUR and M. WISE: Phys. Lett. B, 232, 113 (1989); 237, 527 (1990). [9] A. ALI: DESY Report 92-058 (1992).

[10] M. BAUER, B. STECH and M. WIRBEL: Z. Phys. C, 29, 637 (1985); 34, 103 (1987). [11] N. ISGUR and M. B. WISE: Phys. Rev. D, 42, 2388 (1990). [12] WEI-SHU HOU, R. S. WILLEY and A. SONI: Phys. Rev. Lett., 58, 1608 (1987). [13] PARTICLE DATA GROUP: Phys. Lett. B, 239, 1 (1990). [14] S. BERTOLINI, F. BORZUMATI, A. MASIERO and G. RIDOLFI: Nucl. Phys. B, 353, 591

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Rare B-meson decays in supersymmetry