Rare B decays in the Standard Model

EISEVIER

Nuclear Insuuments and Methods in Physics Research A 384 ( 1996) 8-16 NUCLEAR

INSTRUMENTS & METHODS IN PHYSICS RESEARCH

Section A

Rare B decays in the Standard Model

A. Ali * Deursches Elekrronen-Svnchrotron DESK Notkestrasse 85, D-22603 Hamburg, German>

Abstract We discuss the electromagnetic-penguin-dominated radiative B decays B + X, + y. B*(O) -+ K**‘“’ + y, and B, --+

(b + 7 in the context of the Standard Model (SM) and their Cabibbo-Kobayashi-Maskawa (CKM)-suppressed counterparts, B --+ xd + y, B* -+ p* + y, B” --t (p’,o) + y, and B, --t K” + y, using QCD sum rules for the exclusive decays. The importance of these decays in determining the parameters of the CKM matrix [l] is emphasized. The semileptonic decays B + XJJ+e- are also discussed in the context of the SM and their role in determining the Wilson coefficients of the effective theory is stressed. Comparison with the existing measurements are made and SM-based predictions for a large number of rare B decays are presented.

1. Estimates of B(B ---$ X, + y) and IV,1 in the Standard Model

The Standard Model (SM) of particle physics does not admit flavour-changing-neutral-current (FCNC) transitions in the Born approximation. However, they are induced through the exchange of W* bosons in loop diagrams. The short-distance contribution in rare decays is dominated by the (virtual) top quark contribution. Hence the decay characteristics provide quantitative information on the top quark mass and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements v;; i = d, s, b [ 11. We shall discuss repre- sentative examples from several such transitions involving B decays, starting with the decay B + X, + y, which has been measured by CLEO [2]. This was preceded by the measurement of the exclusive decay B -+ K’ + y by the same collaboration [ 31. The present measurements give [4] :

B( B + X, + y) = (2.32 * 0.57 * 0.35) x 10-4, (1)

L3(B + K’ + y) = (4.2 f 0.8 & 0.6) x 10-5, (7-j

yielding an exclusive-to-inclusive ratio:

These decay rates determine the ratio of the CKM matrix elements 1 V, I/ 1 Vcb 1 and the quantity Rx’ provides information on the decay form factor in B + K’ + y. In what follows we take up these points briefly.

The leading contribution to b -+ s + y arises at one-loop from the so-called penguin diagrams. With the help of the

l E-mail [email protected]

unitarity of the CKM matrix, the decay matrix element in the lowest order can be written as:

x (F2(xO - F2(Xc))qPEYSapv(mhR+ msL)b. (4)

where Xi = mj/mf, and qp and l ,, are, respectively, the photon four-momentum and polarization vector. The GIM mechanism [ 5 ] is manifest in this amplitude and the CKM- matrix element dependence is factorized in At = t&V,,. The (modified) Inami-Lim function F2( Xi) derived from the (one-loop) penguin diagrams is given by [ 61:

X

F2(x) = 24(x _ 114 [6x( 3x - 2) log x

-(x-1)(8x2+5x-7)]. (5)

The measurement of the branching ratio for B -+ X, + y can be readily interpreted in terms of the CKM-matrix element product A,/IV,b( or equivalently IVBI/IVcbl. For a quantitative determination of 1 V, I/ 1 &b 1, however, QCD radiative corrections have to be computed and the contribution of the so-called long-distance effects estimated.

The appropriate framework to incorporate QCD corrections is that of an effective theory obtained by integrating out the heavy degrees of freedom, which in the present context are the top quark and W’ bosons. The operator basis depends on the underlying theory and for the SM one has (keeping operators up to dimension 6)

R

‘,‘&(b -+ S-k y) = -4GFV,‘vb ~ci(I*)~i(~) 3

fi (6)

i=l

0168-9002/96/$15.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved

PIISO168-9002(96)00908-4

A. Ali/Nucl. Instr. and Meth. in Php. Rex A 3U4 f 1996) X-16 9

where the operator basis, the “matching conditions” C,(mw ), and the solutions of the renormalization group equations C, (1~) can be seen in Ref. [ 71. The perturbative QCD corrections to the decay rate r ( B -+ X, + y) have

two distinct contributions: - Corrections to the Wilson coefficients Ci(,U), calculated

with the help of the renormalization group equation, whose solution requires the knowledge of the anomalous dimension matrix in a given order in as.

- Corrections to the matrix elements of the operators 0, entering through the effective Hamiltonian at the scale P = O(mh).

The anomalous dimension matrix is needed in order to sum up large logarithms, i.e., terms like (Y:( mw) log”*( mh/bf).

whereM=m,ormwandm<n(withn=O,l,Z ,... ).At present only the leading logarithmic corrections (m = n)

have been calculated systematically and checked by several independent groups in the complete basis given in Eq. (6) [ 81. First calculations of the NLO corrections to the anomalous dimension matrix have been recently reported by Misiak [ 91 and are found to be small. Next-to-leading order corrections to the matrix elements are now available completely. They are of two kinds: - QCD bremsstrahlung corrections b + sy + g, which are

needed both to cancel the infrared divergences in the decay rate for B -+ X, + y and in obtaining a non-trivial QCD contribution to the photon energy spectrum in the inclusive decay B -+ X, + y:

- next-to-leading order virtual corrections to the matrix elements in the decay b ---) s + y.

The bremsstrahlung corrections were calculated in Refs. [ 10. I I ] in the truncated basis and last year also in the com-

plete operator basis [ 12,131. The higher order matching conditions, i.e.. C;( mw ) , are known up to the desired accuracy, i.e., up to O( CI~( Mw ) ) terms [ 141. The next-to-leading or-

der virtual corrections have also been calculated [ 151. They reduce the scale-dependence of the inclusive decay width.

The branching ratio f3(B + X, + y) can be expressed in terms of the semileptonic decay branching ratio

B(B --+ X&Y) = r(B+y+X,) Ih

rSL 1 f3(B - Xev,, . (7)

where the leading-order QCD corrected expression for rs~ can be seen in Ref. [ 71. The leading order ( 1 /mh) power corrections in the heavy quark expansion are identical in the inclusive decay rates for B -+ X, + y and B + XI&, entering in the numerator and denominator in the square

bracket, respectively, and hence drop out.

In Ref. [7], the present theoretical errors on the branching ratio B( B -P X,y) are discussed, yielding:

B( B + X, + y) = (3.20 f 0.30 f 0.38 + 0.32) x IO-’

(8)

where the first error comes from the combined effect of Am, and Ap ( the scale dependence), the second error arises from

the extrinsic sources (such as A ( mh) , A ( BRSL 1). and the third error is an estimate (f 10%) of the NLO anomalous

dimension piece in Cy’, the coefficient of the magnetic moment operator. Combining the theoretical errors in quadra-

ture gives [7]

a(B + X, + y) = (3.20f 0.58) x lo-’ 1 (9)

which is compatible with the present measurement t3( B --)

X, + y) = (2.32f 0.67) x 10-j [2]. Expressed in terms

of the CKM matrix element ratio, one gets

IVSl/l&, = 0.85 f O.l2(expt) rfr: O.lO(th) . ( IO)

which is within errors consistent with unity, as expected from the unitarity of the CKM matrix.

2. Inclusive radiative decays B -+ X, + y

The theoretical interest in studying the (CKM-

suppressed) inclusive radiative decays B --t X,1 + y lies

in the first place in the possibility of determining the parameters of the CKM matrix. We shall use the Wolfenstein

parametrization Il6], in which case the matrix is deter-

mined in terms of the four parameters A. A = sin Bc. /J and 71. The quantity of interest in the decays B + Xd + y

is the end-point photon energy spectrum, which has to be measured requiring that the hadronic system Xd recoiling against the photon does not contain strange hadrons to sup-

press the large-E, photons from the decay B + X,, + y. Assuming that this is feasible, one can determine from the ratio of the decay rates B(B -+ Xd + y) /B( B - X, + y) the CKM-Wolfenstein parameters p and 7~. This measure-

ment was first proposed in Ref. [ I1 1. where the photon energy spectra were also worked out.

In close analogy with the B -+ X, + y case discussed

earlier, the complete set of dimension-6 operators relevant for the processes b 4 dy and b -+ dyg can be written as:

x

‘&r(b + d) = -%& c

fi ,=I C,(P)B,(P). (11)

where 5, = V,hVjz for j = t,c, u. The operators Gj. j = 1,2. have implicit in them CKM factors. In the Wolfenstein

parametrization [ 161, one can express these factors as:

5” = AA+p - iq), & = -A,?, 51 = -& - &. ( 12)

We note that all three CKM-angle-dependent quantities li are of the same order of magnitude, 0( A’). This aspect can be taken into account by defining the operators 6, and & entering in ‘H,fy( b -+ d) as follows [ I I ] :

1. THEOKY

IO A. Ali/Nucl. Insrr. and Meth. in Phys. Res. A 384 (19%) 8-16

with the rest of the operators (&I; j = 3.. .8) defined like their counterparts Oj in 7&(b -+ s), with the obvious replacement s -+ d. With this choice, the matching conditions Cj (mw ) and the solutions of the RG equations yielding Cj (p) become identical for the two operator bases Oj and dj. The essential difference between T(B + X, + y) and I(B --) Xd + y) lies in the matrix elements of the first two operators 01 and 02 (in ‘&s(b --) s)) and 81 and 82 (in &r( b -+ d) ). The branching ratio B( B --) Xd + y) in the SM can be written as:

-Cl-PP2-q&+04), (14)

where the functions Di depend on the parameters ml, mh, mc, ,u, as well as the others we discussed in the context of a( B --) X, + y). These functions were first calculated in Ref. [ 111 in the leading logarithmic approximation. Recently, these estimates have been improved in Ref. [ 191, making use of the NLO calculations in Ref. [ 151. To get an estimate of the inclusive branching ratio, the CKM parameters p and v have to be constrained from the unitarity fits. Present data and theory restrict them to lie in the following range (at 95% CL) [ 171:

0.20 5 q < 0.52,

-0.35 I p < 0.35, (15)

which, on using the current lower bound from LEP on the By-3 mass difference AM, > 9.2 ps-’ [20], is reduced to -0.25 5 p 5 0.35 using & = 1 .l, where & is the SU( 3) -breaking parameter & = fa,&,/&,&, . The pre- ferred CKM-fit values are (p. 77) = (0.05,0.36), for which one gets [ 191

S(B 4 Xd + y) = 1.63 X 1o-5, (16)

whereas a( B --+ Xd + y) = 8.0 x lo-” and 2.8 X lo-’ for the other two extremes p = 0.35. 17 = 0.40 and p = -3 = -0.25, respectively. In conclusion, we note that the func- tional dependence of B(B -+ Xd + y) on the Wolfenstein parameters (p, 77) is mathematically different than that of AM,. However, qualitatively they are very similar. From the experimental point of view, the situation p < 0 is favourable for both the measurements as in this case one expects (rel- atively) smaller values for AM, and larger values for the branching ratio f?( B -+ Xd + y), as compared to the p > 0 case which would yield larger AM, and smaller B(B ---) xd + r>.

2.1. 8( B -+ V + y) and constraints on the CKM parameters

Exclusive radiative B decays B + V + y, with V = K’, p, w, are also potentially very interesting from the point of view of determining the CKM parameters [ 211. The extraction of these parameters would, however, involve a trustworthy estimate of the SD- and LD-contributions in the decay amplitudes.

The SD-contribution in the exclusive decays (B*, B”) -+ (K**,K*‘) +y. (B*:,B’) --) (p*.~‘) +y,B’-+ w-ty and the corresponding B, decays, B, -+ $I + y and B, -+ K” + y, involve the magnetic moment operator 07 and the related one obtained by the obvious change s --+ d, &. The transition form factors governing the radiative B decays B -+ V + y can be generically defined as:

Here V is a vector meson with the polarization vector e’“) , V = p, w, K’ or 4; B is a generic B-meson B*, B” or B,, and + stands for the field of a light u, d or s quark. The vec- tOrS pB, pv and q = pB - pv correspond to the 4-momenta of the initial B-meson and the outgoing vector meson and photon, respectively. In Eq. (17) the QCD renormalization of the &rPvq’b operator is implied. Keeping only the SD-contribution leads to obvious relations among the exclusive decay rates, exemplified here by the decay rates for (B,,Bd) -+p+yand(B,,Bd) -+K*+y:

I((B$,B:) --) (P*,P’) +Y) r((B,f,Bj) ---) (K**,K*O) + y)

tv,l 2 “vd jiq ’ [ 1

(18)

where @u.d is a phase-space factor which in all cases is close to 1 and pi E [Fs(Bi -+ m)/Fs(Bi + K’y)]‘. The transition form factors FS are model dependent. Estimates of Fs in the QCD sum rule approach in the normalization of Eq. ( 17) range between Fs (B -+ K’y) = 0.3 1 (Narison in Ref. [22]) to Fs(B -+ K’y) = 0.37 (Ball in Ref. [22]), with a typical error of *15%, and hence are all consistent with each other. This, for example, gives RK* = 0.16*0.05, using the result from Ref. [ 211, which is in good agreement with data. The ratios of the form factors, i.e. Ki, should therefore be reliably calculable as they depend essentially only on the SU( 3)-breaking effects which have been estimated [ 21,221.

The LD-amplitudes in radiative B decays from the light quark intermediate states necessarily involve other CKM matrix elements. Hence, the simple factorization of the decay rates in terms of the CKM factors involving It&] and IV,/ no longer holds, thereby invalidating relation ( 18) given above. In the decays B 4 V + y they are induced by the matrix elements of the four-Fennion operators 81 and 82

(likewise 01 and 02). Estimates of these contributions re-

A. Ali/Nucl. Instr. and Meth. in Phys. Rex A 384 (1996) 8-16 11

quire non-perturbative methods. This problem has been in- vestigated in Refs. [23,24] using a technique [25] which treats the photon emission from the light quarks in a the- oretically consistent and model-independent way. This has been combined with the light-cone QCD sum rule approach to calculate both the SD and LD - parity conserving and parity violating - amplitudes in the decays (B’, B”) --) (p*, p/w) + y. To illustrate this, we concentrate on the B* decays B* -+ p* + y and take up the neutral B decays B” -+ p(o) + y at the end. The LD-amplitude of the four-Fermion operators 61, 82 is dominated by the contribution of the weak annihilation of valence quarks in the B meson and it is color-allowed for the decays of charged B* mesons. Using factorization, the LD-amplitude in the decay B, -+ p* + y can be written in terms of the form factors Ff and Fk,

dronr = - * V”r, V;L, (c2 + +I> m,@ El’)

v5 c

Thus, the CKM factors suppress the LD-contributions. The analogous LD-contributions to the neutral B de-

cays B” -+ py and B” -+ oy are expected to be much smaller. The corresponding form factors for the decays

;: + p”(w) y are obtained from the ones for the decay + p*y discussed above by the replacement of the light

quark charges e, -+ ed. which gives the factor - I/2; in addition, and more importantly, the LD-contribution to the neutral B decays is colour-suppressed, which reflects itself through the replacement of the factor al by a~. This yields for the ratio

~a’-PY r-/s eda

Bf-.p*‘y =- N -0.13 f 0.05) (24)

R L/s eual

where the numbers are based on using az/ar = 0.27 k 0.10

[26]. This would then yield R$Tfl N R$yy = 0.05, which in turn gives

1 a2Fk(q2) dB”PY Ions - < 0.02.

fe ““uPp,qp. 2&q*)) .

dr5O-m -

(19) shoti

(25)

J This, as well as the estimate in Eq. (23), should be taken

Again, one has to invoke a model to calculate the form only as indicative in view of the approximations made

factors. Estimates from the light-cone QCD sum rules give in Refs. [ 23,241. That the LD-effects remain small in

[24]: B” + p-y has been supported in a recent analysis based

3 =00125*0.0010 2 =00155*00010

on the soft-scattering of on-shell hadronic decay products

* (20) B0 -+ pop0 -+ py [ 271, though this paper estimates them

Fs ’ ‘Fs’ * somewhat higher (between 4% and 8%).

where the errors correspond to the variation of the Bore1 parameter in the QCD sum rules. Including other possible uncertainties, one expects an accuracy of the ratios in (20)

Restricting to the colour-allowed LD-contributions, the relations, which are obtained ignoring such contributions (and isospin invariance),

of order 20%. The parity-conserving and parity-violating amplitudes turn out to be numerically close to each other in

r(B* + p*y) =2r(B” + $y) =2r(B” -+ WY),

the QCD sum rule approach, Fk N Fk G FL, hence the ratio (26) of the LD- and the SD- contributions reduces to a number

r241 get modified to

A long cn=R

Bf +p*Y &bV$

L/S qf. (21)

W* -+ P*Y) w* + P'Y) 2r(B0 --+ py) = 2r(BO -+ oy)

Using C2 = I .lO, CI = -0.235, CtR = -0.306 from Ref. 171 (corresponding to the scale p = 5 GeV) gives:

2

R B* L/S

--‘PiY _ 4dmp(C2 + Cl/NC) F:i*p*r

mbC7eff F,B* +P*Y

= -0.30 f 0.07 ( (22)

Al --PI-T2 = 1 + 2Rrlst&i

(l-P)*+11*

P’ +v2 + (RL,s)~K: c1 _ pj2 +72 . (27)

which is not small. To get a ball-park estimate of the ratio where RLIS z R$*p*y. The ratio r(B* + P*Y)12w”

drong/dsbon. we take the central value from the CKM fits, yielding ]&b]/]xd:dl N 0.33 [ 171:

+ py) (= T’(B* -+ p*y)/2T(B” -+ WY)) is shown in Fig. 1 as a function of the parameter p, with q= 0.2, 0.3 and

A 0.4. This suggests that a measurement of this ratio would

long

I-1

B f -4 Y

A shon = (R,B,~-p*y,~ N 10%. (23)

constrain the Wolfenstein parameters (p, q), with the dependence on p mo& marked than on v. In particular, a negative value of p leads to a constructive interference in B, + py

I. THEORY

12 A. Ali/Nucl. Instr. and Meth. in Phy. Res. A 384 (1996) X-16

I \ 5 I

\

Fig. I. Ratio of the neutral and charged B-decay rates

r(B* --t p*y)/2r( B” - py) as a function of the Wolfenstein param-

eter p. with 7 = 0.2 (short-dashed curve). 7 = 0.3 (solid curve), and

7 = 0.4 (long-dashed curve). (Figure taken from Ref. [ 241.)

decays, while large positive values of p give a destructive

interference. The ratio of the CKM-suppressed and CKM-allowed de-

cay rates for charged B mesons gets modified due to the LD

contributions. Following earlier discussion, we ignore the LD-contributions in T(B -+ K*y). The ratio of the decay

rates in question can therefore be written as:

r(B* + P*Y) T(B* + K**y)

=K”A2[(1 -p?+$l

X P( 1 - PI - 77*

1 + 2RLlsKd (, _ Pj2 + ,+

P2 + v2 +(RL,s)~~+, _Pj2+772

Using the central value from the estimates of the ratio of the

form factors squared K” = 0.59 * 0.08 [2l], we show the ratio (28) in Fig. 2 as a function of p for 7 = 0.2.0.3, and 0.4. It is seen that the dependenceof this ratio is rather weak

on v but it depends on p rather sensitively. The effect of the

Fig. 2. Ratio of the CKM-suppressed and CKM-allowed radiative B-decay

rates: r(Bu - py)/lY(B - K’y) (with B = B, or Bd) as a function of

the Wolfenstein parameter p, with 7 = 0.2 (short-dashed curve), 7 = 0.3

(solid curve), and TJ = 0.4 (long-dashed curve). (Figure taken from Ref.

1241 )

LD-contributions is modest but not negligible, introducing an uncertainty comparable to the -15% uncertainty in the

overall normalization due to the SU( 3) -breaking effects in the quantity K”.

Neutral B-meson radiative decays are less prone to the

LD-effects, as argued above, and hence one expects that to a good approximation (say, better than 10%) the ratio of the decay rates for neutral B meson obtained in the approx-

imation of SD-dominance remains valid [ 2 I ] :

r&++~;;jy) = KdA*[ (1 - /I)’ + 7’1 , (29)

where this relation holds for each of the two decay modes

separately. Finally, combining the estimates for the LD- and SD-form

factors in Ref. [ 241 and Ref. [ 211, respectively, and re-

stricting the Wolfenstein parameters in the range -0.25 < p _< 0.35 and 0.2 5 77 5 0.4, as suggested by the CKM-fits

[ 171, we give the following ranges for the absolute branch-

ing ratios:

a( B” -+ py) N a( B” + wy) = (0.65 f 0.35) x 10-h,

(30)

where we have used the experimental value for the branching ratio a( B + K* + y) [ 31, adding the errors in quadrature. The large error reflects the poor knowledge of the CKM

matrix elements and hence experimental determination of

these branching ratios will put rather stringent constraints on the Wolfenstein parameter p.

In addition to studying the radiative penguin decays of the

B,f and Bi mesons discussed above, hadron machines such as HERA-B will be in a position to study the corresponding

decays of the Bf meson and Ah baryon, such as By + 4 + y and Ah + A + y, which have not been measured so far. We list below the branching ratios in a number of interesting

decay modes calculated in the QCD sum rule approach in

Ref. [2l].

f?(B, -+ @y) = a(Bd --+ K’y) = (4.2 & 2.0) x 1W5,

U(Bs -+ K*Y)

@Bd --+ K’Y)

=+ &B, --) K*y) = (0.75 f 0.5) x 1O-6.

The estimated branching ratios in. a number of inclusive and exclusive radiative B decay modes are given in Table 1, where we have also listed the branching ratios for B, + yy

and Bd -+ yy.

2.2. Inclusive rare decays B -+ X,$?‘e- in the SM

The decays B + Xse+e-, with ! = e, ,u, 7, provide a

more sensitive search strategy for finding new physics in

A. Ali/Nucl. Instr. and Meth. in Phy. Res. A 384 (1996) 8-16 13

Table I Estimates of the branching fractions for FCNC B-decays in the standard

model taking into account the uncertainties in the input parameters as

discussed in Ref. [ 71. The entries in the second column correspond to the

short-distance contributions only, except for the radiative decays B* -+

p* + Y and B” -+ (#, o) + Y. where long-distance effects have also

been included. For the two-body branching ratios, we have used fB, = 200

MeV and fB,/fBd = I .l6. Experimental measurements and upper limits

are also listed. In the second row, the statistical and systematic uncertainties

have been combined to give the quoted experimental uncertainty.

Decay modes B(SM) Measurements and

90% CL upper limits

(B*.B”) +Xsy (3.2f0.6) x 1O-4

(B*,B”)-K*Y (4.Ozt2.0) x lO-5

(B*,B”)+X‘,y (1.6+1.2)x lo--’

B*+$ +y (l.5fl.l)x10-6

B” + p” + Y (0.651kO.35) x IO-”

B” -+6JfY (0.65f0.35) x IO-’

Bs-4+Y (4.2zt2.0) x lO-5

Bs-+K* +y (0.8+0.5) x~O-~ f-

(B,j.Bu)dXse e (8.4f2.2) x lO-6

(Bd,B,)+Xde+e- (4.9f2.9) x IO-’ + -

(Bd.Bu)+XsP P (5.7*1.2)x10-”

(Bd, Bu) - x,,P+/.‘- (x3*1.9) x 10-7 +- (Bd,Bu)-+Xsr T (2.6ztO.5) x IO-’

(Bd.Bu, -+X,,T+,- (I.~~O.~)XIO-~

(Bd.B”) -Ke+e- (5.9*2.3) x IO-’

(B,.B,)-KP+II- (4.0fl.5) x 10-7

(Brj,B,)--‘KI.L+fi- (4.Ozk1.5) x IO--’

( Bd, Bu ) - K*e+e- (2.3f0.9) x lO-h

(Bd,Bu)-+K*P+P- (l.5f0.6)x10-h

( Bd, Bu) -t XSYV (4.0fl.O) x 10-5

(&j,Bu)+@‘fi (2.3fl.5) x lO-6

(Bd,Bu)+bD (3.2*1.6)x lO-6

(Bd, B,) -+K*vD (l.lf0.55)xl0-5

BS+YY (3.0&1.0)X 10-7

Bd-+YY (1.2f0.8) x IO-’

Bs -T+T- (7.4ztl.9) x 10-7

t- B,,+r 7 (3.1fl.9)x10-8 $ -

BS - .u LC (3.5fl.O) x 10-y

Bd ‘P+P- (1.5f0.9) x 10-l” +-

Bs+e e (8.013.5) x IO-l4

Bd - e+e- (3.4f2.3) x IO-t5

(2.3f0.7) x 1O-4 I21

(4.2Srl.0) x IO-’ [41

<1.1x10-~

< 3.9 x 10-5

< I.3 x 10-5

141 141 141

< 3.6 x lO-5 I491

< 1.2x 10-5

< 0.9 x 10-5

< 0.9 x lo-’

< 1.6x lO-5

< 2.5 x lo-’

< 7.7 x 10-4

< I.1 x 10-4

< 3.8 x lO-5

< 8.4 x lo-6

< 1.6x lO-h

1401 I401 1401 I401 I471 1461

1501 I501

I471 1471

rare B decays than for example the decay B -+ X,y , which constrains the magnitude of C;“. The sign of C7eff, which depends on the underlying physics, is not determined by the measurement of B(B --+ X, + y). This sign, which in our convention is negative in the SM, is in general model dependent. It is known (see for example Ref. [ 281) that in super- symmetric (SUSY) models, both the negative and positive signs are allowed as one scans over the allowed SUSY parameter space. We recall that for low dilepton masses, the differential decay rate for B + X,e+f!- is dominated by the contribution of the virtual photon to the charged lepton pair, which in turn depends on the effective Wilson coefficient Cqff. However, as is well known, the B -+ X,!+e- amplitude in the standard model has two additional terms,

arising from the two FCNC four-Fermi operators ’ , which are not constrained by the B + X, + y data. Calling their coefficients Ca and Cia, it has been argued in Ref. [ 281 that the signs and magnitudes of all three coefficients Ctff, Cg and Cia can, in principle, be determined from the decays B 4 X, + y and B -+ X&!-.

The SM-based rates for the decay b + &E-, calculated in the free quark decay approximation, have been known in the LO approximation for some time [ 291. The LO calculations have the unpleasant feature that the decay distributions and rates are scheme-dependent. The required NLO calculation is in the meanwhile available, which reduces the scheme-dependence of the LO effects in these decays [ 301. In addition, long-distance (LD) effects, which are expected to be very important in the decay B ---) X$0-. have also been estimated from data on the assumption that they arise dominantly due to the charmonium resonances .I/$ and $’ through the decay chains B --+ X,J/$($‘, . .) -+ X,l?e-. Likewise, the leading ( 1 /rnt,‘) power corrections to the par- tonic decay rate and the dilepton invariant mass distribution have been calculated with the help of the operator product expansion in the effective heavy quark theory [ 3 1 ] The results of Ref. [ 3 1 ] have, however, not been confirmed in a recent independent calculation [ 321, which finds that the power corrections in the branching ratio B( B -+ X,P’e- ) are small (typically - 1.5%). The corrections in the dilepton mass spectrum and the FB asymmetry are also small over a good part of this spectrum. However, the end-point dilepton invariant mass spectrum is not calculable in the heavy quark expansion and will have to be modeled. Non- perturbative effects in B + X,e+!- have been estimated using the Fermi motion model in Ref. [33]. These effects are found to be small except for the end-point dilepton mass spectrum where they change the underlying parton model distributions significantly and have to be taken into account in the analysis of data [ 321.

The amplitude for B -+ X,e’!- is calculated in the effective theory approach, which we have discussed earlier, by extending the operator basis of the effective Hamiltonian introduced in Eq. (6) :

Fh( b -+ s + y; b + s + t+t-)

= 3iedb ---t s + y) - %$4i, [CYOY + CI&I,] , di

(32)

where the two additional operators are:

(33)

’ This also holds for a large class of models such as MSSM and the

two-Higgs doublet .models but not for all SM-extensions. In LR symmet-

ric models, for example, there are additional FCNC four-Fermi operators

involved (341.

I. THEORY

14 A. Ali/Nucl. Instr. and Meth. in Phys. Res. A 384 (19%) R-16

The analytic expressions for Cg( mw ) and Citi (mw) can be seen in Ref. [30] and will not be given here. We recall that the coefficient Cp in LO is scheme-dependent. However, this is compensated by an additional scheme-dependent part in the (one loop) matrix element of (39. We call the sum CG”, which is scheme-independentand enters in the physical decay amplitude given below:

C,;‘“(s^) Z Csq(s^) + Y(s^). (35)

The function Y(g) is the one-loop matrix element of 0s and can be seen in literature [ 30,7]. A useful quantity is the differential FB asymmetry in the c.m.s. of the dilepton defined in Ref. [ 35 ] :

e+ and the b-quark. This can be expressed as:

I -I

dd(s^) dB dB

dd(i)

-= --

=-Bs,~~u2W

dS

d5

s s dz

c

dz’

xc10 [S!J?(C,r”(s^)) +2c;“‘(1 +&]

(36)

I,

(37)

0

where z = cos 8, with 19 being the angle between the lepton

however, not an independent measure, as it is directly proportional to the FB asymmetry discussed above. The relation is [32]:

I A(;) =Bx A. (39)

This is easy to notice if one writes the Mandelstam variable u(b) in the dilepton cm. and the B-hadron rest systems.

Next, we discuss the effects of LD contributions in the processes B + X,e+e-. Note that the LD contributions due to the vector mesons such as J/I,+ and I++‘, as well as the continuum CC contribution already discussed, appear as an effective (SLY~~L)(&V) interaction term only, i.e. in the operator 09. This implies that the LD-contributions should change C9 effectively, C7 as discussed earlier is dominated by the SD-contribution, and Crc has no LD-contribution. In accordance with this, the function Y(s^) is replaced by

Y(i) -+ Y’(5) E Y(S) + Y,,(Z),

where Y,,( s^) is given as [ 351

(40)

Y,,(i) = -$ (3CI + cz + 3c3 + c4 + 3c5 + G)

X c VI-(Vi --t 1+1-)Mv,

Mt, - j;in~ - ikfv,T~, ’ (41)

v,=l/*.$‘.“’

The Wilson coefficients CJ’r, Ci’r and Ci(i appearing in the above equation and the dilepton spectrum (see, for example, Ref. [ 321) can be determined from data by solving the partial branching ratio B(As^) and partial FB asymmetry A( Ai), where A3 defines an interval in the dilepton invariant mass [28].

3 and 4, respectively. This can be used to test the SM, as the signs of the Wilson coefficients in general are model depen-

where K is a fudge factor, which appears due to the inade-

dent. For further discussions we refer to Ref. [32] where also theoretical dispersion on the decay distributions due to

quacy of the factorization framework in describing data on

various input parameters is worked out.

B + J/$X,. The long-distance effects lead to significant interference effects in the dilepton invariant mass distribution and the FB asymmetry in B -+ Xs!+!- shown in Figs.

There are other quantities which one can measure in the decays B --+ X,e+t- to disentangle the underlying dynam- ics. We mention here the longitudinal polarization of the lepton in B + Xsl+C-, in particular in B --+ X,r’r-, proposed by Hewett [ 361. In a recent paper, Ktiiger and Sehgal [ 371 have stressed that complementary information is contained in the two orthogonal components of polarization (Pr, the component in the decay plane, and PN, the component nor- mal to the decay plane), both of which are proportional to ml/mh. and therefore significant for the ~+r- channel. A third quantity, called energy asymmetry, proposed by Cho, Misiak and Wyler [ 381, defined as

A= N(E,- > Eat) - N(Ep+ > Et--)

N(E,- > El+ ) + N(Er+ > Et- ) (38)

where N( Ep- > Et+ ) denotes the number of lepton pairs where e+ is more energetic than C- in the B-rest frame, is,

Taking into account the spread in the values of the input parameters, /_L, A, m,, and &I. discussed in the previous section in the context of a(B -+ X, + y), we estimate the following branching ratios for the SD-piece only (i.e., from the intermediate top quark contribution only) [ 321:

f?(B + X,e+e-) = (8.4 f 2.3) x 10e6,

B(B --+ X,,u+,u-) = (5.7f 1.2) x 10-h,

f3(B -+ X,r+r-) = (2.6 !c 0.5) x 10-7, (42)

where theoretical errors and the error on &L have been added in quadrature. The present experimental limit for the inclusive branching ratio in B 3 X&e- is actually still the one set by the UA 1 collaboration some time ago [ 391, namely L3(B + X,p+p-) > 5.0 x 10e5. As far as we know, there are no interesting limits on the other two modes, involving X,e’e- and Xsr+r-.

A. Ali/Nucl. Insrr. and Merh. in Phm Res. A 384 (1996) R-16 IS

Fig. 3. Dilepton invariant mass distribution in B -+ X&t!- in the SM

with the next-to-leading order QCD corrections and non-perturbative effects calculated in the Fermi motion model (solid curve), and including the LD-contributions (dashed curve). The model parameters (pi, mq) are

indicated in the figure. Note that the height of the J/e peak is suppressed due to the linear scale. (Figure taken from Ref. I32 I .)

It is obvious from Fig. 3 that only in the dilepton mass region far away from the resonances there is a hope of ex- tracting the Wilson coefficients governing the short-distance physics. The region below the J/r+0 resonance is well suited for that purpose as the dilepton invariant mass distribution there is dominated by the SD-piece. Including the LD- contributions, the following branching ratio has been estimated for the dilepton mass range 0.2 5 ? < 0.36 in Ref. [32]:

B(B -+ X,/A+,~A-) = (1.3 & 0.3) x l0-h, (43)

with B(B -+ X,e+e-) N a(B -+ X,p+p”-). The FB- asymmetry is estimated to be in the range IO-27%, as can be seen in Fig. 4.

The experimental limits on the decay rates of the exclusive decays B + (K, K*)!?- [ 26,401, while arguably closer

04, I

Fig. 4. Normalized FB asymmetry in B + X&e- in the SM as a function of the dilepton invariant mass calculated using the next-to-leading order QCD correction and the Fermi motion effects (solid curve), and including the LD-contributions (dashed curve). The Fermi motion parameters are indicated in the figure. (Figure taken from Ref. [ 321,)

to the SM-based estimates, can only be interpreted in spe- cific models of form factors, which hinders somewhat their transcription in terms of the information on the underlying Wilson coefficients. Using the exclusive-to-inclusive ratios R~rr s P(B -+ K!+!-)/P(B --* X,!+!-) = 0.07 f 0.02 and R~*pp 3 T(B -+ K*e+!-)/F(B + X,l’e-) = 0.27 f 0.0.07, which were estimated in Ref. [42]. the results are presented in Table I.

In conclusion, the semileptonic FCNC decays B + X,e’!- (and also the exclusive decays) will provide very precise tests of the SM, as they will determine the signs and magnitudes of the three Wilson coefficients, CT. Ct” and CIII. This, perhaps, may also reveal physics beyond-the-SM if it is associated with not too high a scale. The MSSM model is a good case study where measurable deviations from the SM are anticipated and worked out [ 28,381.

2.3. Summary and overview of rare B decays in the SM

The rare B decay mode B -+ X,r@, and some of the exclusive channels associated with it, have comparatively larger branching ratios. The estimated inclusive branching ratio in the SM is [42-441:

D(B -+ X,V@) = (4.0f 1.0) x 10-j. (4)

where the main uncertainty in the rates is due to the top quark mass. The scale-dependence, which enters indirectly through the top quark mass, has been brought under control through the NLL corrections, calculated in Ref. [45]. The corresponding CKM-suppressed decay B -+ X@ is related by the ratio of the CKM matrix element squared [ 421:

(45)

Similar relations hold for the ratios of the exclusive decay rates which depend additionally on the ratios of the form factors squared, which deviate from unity through SU( 3) - breaking terms, in close analogy with the exclusive radiative decays discussed earlier. These decays are particularly attractive probes of the short-distance physics, as the long- distance contributions are practically absent in such decays. Hence, relations such as the one in (45) provide, in principle, one of the best methods for the determination of the CKM matrix element ratio ]&dj/l&,I [42]. From the prac- tical point of view, however, these decay modes are rather difficult to measure, in particular at the hadron colliders and probably also at the B factories. The best chances are in the Z’-decays at LEP, from where the present best upper limit stems [46] :

L3(B + XVE) < 7.7 x 10-4. (46)

The estimated branching ratios in a number of inclusive and exclusive decay modes are given in Table 1, updating the estimates in Ref. [ 71.

1. THEORY

16 A. Ali/Mucl. Instr. and Meth. in Phvs. Res. A 384 (1996) R-16

Further down the entries in Table 1 are listed some two- body rare decays, such as (Bf, Bi) -+ 77, studied in Ref. [48], where only the lowest order contributions are calculated, i.e., without any QCD corrections, and the LD-effects, which could contribute significantly, are neglected. The decays (Bf, By) -+ efe- have been studied in the next-to- leading order QCD in Ref. [45]. Some of them, in particular, the decays Bf + pFL+pL- and perhaps also the radiative decay By -+ yy, have a fighting chance to be measured at LHC. The estimated decay rates, which depend on the pseu- doscalar coupling constant fa, (for B,-decays) and fa, (for Bd-decays) , together with the present experimental bounds are listed in Table 1. Since no QCD corrections have been included in the rate estimates of ( B,, Bd) -+ yy, the branching ratios are rather uncertain. The constraints on beyond- the-SM physics that will eventually follow from these decays are qualitatively similar to the ones that (would) follow from the decays B + X, + y and B + X,el^P-, which we have discussed at length earlier.

Acknowledgements

I would like to thank Hrachia Asatrian, Vladimir Braun. Christoph Greub and Tak Morozumi for helpful discussions. The warm hospitality extended by Fernando Ferroni and his collaborators in Rome during the Beauty ‘96 workshop is thankfully acknowledged.

References

ill

PI 131 141

151 161 171

181

L9l

1101

1111

1121 I131

I141 I151

N. Cabibbo, Phys. Rev. Len. 10 (1963) 531:

M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

MS. Alam et al. (CLEG Collab.), Phys. Rev. Lett. 74 (1995) 2885.

R. Ammar et al. (CLEO Collab.), Phys. Rev. Lett. 71 ( 1993) 674.

R. Ammar et al. (CLEO Collab.), contributed paper to the hit. Conf.

on High Energy Physics, Warsaw, 1996, CLEO CONF 96-05.

S.L. Glashow. J. lliopoulos and L. Maiani, Phys. Rev. D 2 ( 1970) 1285.

T. lnami and C.S. Lim. Prog. Theor. Phys. 65 ( 1981) 297.

A. Ali. preprint DESY 96- 106 [ hep_ph/9606324], to appear in Proc.

20th Int. Nathiagali Conf. on Physics and Contemporary Needs.

Bhurban. Pakistan, 1995 (Nova Science Publishers, New York, 1996).

M. Ciuchini et al.. Phys. Lett. B 316 (1993) 127; Nucl. Phys. B 415

(1994) 403;

G. Cella et al., Phys. Len. B 325 (1994) 227;

M. Misiak, Nucl. Phys. B 393 ( 1993) 23; [E. B 439 ( 1995) 461 I. M. Misiak, contribution to the Int. Conf. on High Energy Physics.

Warsaw, 1996.

A. Ah and C. Greub, Z. Phys. C 49 (1991) 431: Phys. Len. B 259

(1991) 182: Z. Phys. C 60 (1993) 433.

A. Ali and C. Greub, Phys. Len. B 287 (1992) 191.

A. Ali and C. Greub. Phys. Lett. B 361 ( 1995) 146.

N. Pott. Phys. Rev. D 54 (1996) 938.

K. Adel and Y.-p. Yao. Phys. Rev. D 49 (1994) 4945.

C. Greub. T. Hurth and D. Wyler, Phys. Lett. B 380 (1996) 385;

Phys. Rev. D 54 ( 1996) 3350.

] 161 L. Wolfenstein, Phys. Rev. Lett. 51 (1983)

[ 171 A. Ali and D. London, preprint DESY 96-140, UdeM-GPP-TH-96-

45, [ hep-ph/9607392], to appear in Proc. QCD Euroconference 96.

Montpellier. France, 1996.

IIf31

1191 1201

(211

1221

1231 u41

1251

[26l I271

1281

1291

1301

1311

[321

1331

r341

1351 1361

I371

I381 1391

[401

I411

1421

1431

1441

I451 [461

I471

1481

[491

[501

R.M. Barnett et al. (Particle Data Group), Phys. Rev. D 54 ( 1996) I. A. Ali. H.M. Asatrian and C. Greub, to be published.

L. Gibbons. plenary talk at the hit. Conf. on High Energy Physics

(ICHEP96). Warsaw. 1996.

A. Ah. VM. Braun and H. Simma, Z. Phys. C 63 ( 1994) 437.

P. Ball. TU-Mllnchen Report TUM-T3l-43/93 (1993);

P. Colangelo et al., Phys. Len B 317 (1993) 183:

S. Narison, Phys. Len. B 327 ( 1994) 354;

J. M. Soares. Phys. Rev. D 49 ( 1994) 283.

A. KhodzhamirianG. StollandD. Wyler, Phys.L.ett. B 358 ( 1995) 129.

A. Ali and V.M. Braun, Phys. Lett. B 359 ( 1995) 223.

1.1. Balitsky, V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. 312

( 1989) 509.

T.E. Browder and K. Honscheid, Prog. Part, Nucl. Phys. 35 ( 1995) 8 I,

1.F. Donoghue, E. Golowich and A.A. Petrov, preprint UMHEP-433

( 1996) [hep-ph/9609530].

A. Ali, G. F. Giudice and T. Mannel, Z. Phys. C 67 ( 1995) 417.

S. Bertolini. F. Borzumati and A. Masiero, Phys. Rev. Lett. 59 ( 1987)

180;

R. Grigjanis et al., Phys. Lett. B 213 (1988) 355;

B. Grinstein, R. Springer, and M.B. Wise, Phys. Lett. 202 (1988)

138; Nucl Phys. B 339 ( 1990) 269;

G. Cella et al., Phys. Lett. B 248 ( 1990) 181.

M. Misiak, Nucl. Phyn. B 393 (1993) 23; [E. B 439 (1995) 4611:

A.J. Buras and M. Mtlnz. Phys. Rev. D 52 ( 1995) 186.

A. F. Falk. M. Luke and M. J. Savage, Phys. Rev. D 49 ( 1994) 3367.

A. Ali. G. Hiller, L.T. Handoko and T. Morozumi. preprint DESY

96-206; Hiroshima univ. report HUPD-9615 1 hep-ph/9609449].

A. Ali and E. Pietarinen, Nucl. Phys. B I54 ( 1979) 519;

G. Altarelli et al., Nucl. Phys. B 208 ( 1982) 365.

K. Fujikawa and A. Yamada, Phys. Rev. D 49 (1994) 5890;

P. Cho and M. Misiak, Phys. Rev. D 49 ( 1994) 5894.

A. Ali, T. Mannel and T. Morozumi, Pbys. Lett. B 273 ( 1991) 505.

J. Hewett, Phys. Rev. D 53 ( 1996) 4964.

F. Kruger and L.M. Sehgal. Phys. Lett. B 380 ( 1996) 199.

P. Cho. M. Misiak and D. Wyler, Phys. Rev. D 54 ( 1996) 3329.

C. Albajar et al. (UAI), Phys. Lett. B 262 (1991) 163.

T. Skwarnicki, Proc. 17th Int. Symp. on Lepton Photon Interactions,

Beijing. P.R. China, 1995, eds. Zheng Zhi-Peng and Chen He-Sheng.

CDF Collaboration, Fermilab Conf. 95 1201 -E ( 1995 )

A. Ali. C. Greub and T. Mannel. DESY Report 93-016 ( 1993). and

in B-Physics Working Group Report, ECFA Workshop on a European

B-Meson Factory, ECFA 93/l 51, DESY 93-053 ( 1993). eds. R.

Aleksan and A. Ali.

G. Buchalla. A.J. Buras and M.E. Lautenbacher, MPI-Ph/95-104:

TUM-T3 I - IOO/95; FERMILAB-PUB-95/305-T; SLAC-PUB 7009;

[ hep-phi95 123801.

Y. Grossman, Z. Ligeti and E. Nardi, Nucl. Phys. B 465 ( 1996) 369.

G. Buchalla and A.J. Buras, Nucl. Phys. B 400 ( 1993) 225.

Contributed paper by the ALEPH collaboration to the Int. Conf. on

High Energy Physics ( ICHEP96), Warsaw, 1996, PA I O-01 9.

CDF Collaboration. Fermilab Conf. 951201-E (1995);

P Sphicas (CDF), private communication.

G.-L. Lin. J. Liu. and Y.-P. Yao, Phys. Rev. D 42 (1990) 2314;

H. Simma and D. Wyler, Nucl. Phys. B 344 ( 1990) 283;

S. Herrlich and 1. Kalinowski. Nucl. Phys. 381 (1992) 501;

T.M. Aliev and G. Turan, Phys. Rev. D 48 (1993) 1176;

G.G. Devidze, G.R. Jibuti and A.G. Liparteliani, preprint HEP

Institute, Univ. of Tbilisi (revised version) (1995).

S. Abachi et al. (W Collab.), contributed paper to the Int. Conf. on

High Energy Physics (ICHEP96). Warsaw, 1996.

L3 Collaboration, contributed paper to the EPS Conf. (EPS-0093-2).

Brussels. 1995.

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Rare B decays in the Standard Model