EL.SEVIER

5 October 1995

PHYSICS LETTERS 6

Physics Letters B 359 (1995) 223-235

Estimates of the weak annihilation contributions to the decays B-+p+yandB--,w+y

A. Ali, V.M. Braun ’ Deutsches Elektronen Synchrotron DESK D-22603 Hamburg, Federal Republic of Germany

Received 15 June 1995; revised manuscript received 4 August 1995

Editor: P.V. Landshoff

Abstract

The dominant long-distance contributions to the exclusive radiative decays B -+ p(w) + y involve photon emission from light quarks at large distances, which cannot be treated perturbatively. We point out that this emission can be described in a theoretically consistent way as the magnetic excitation of quarks in the QCD vacuum and estimate the corresponding parity-conserving and parity-violating amplitudes using the light-cone QCD sum-rule approach. These are then combined with the corresponding short-distance contribution from the magnetic moment operator in the same approach, derived earlier, to estimate the decay rates r( B -+ p(w) + y). The implications of this result for the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements in radiative B decays are worked out.

1. Exclusive radiative B decays provide a valuable source of information about the CKM matrix. Assuming

that the short-distance (SD) contribution of the magnetic moment operator is derives [ 1 ]

where <u,d takes into account SU(3) breaking in the decay form factors differences. With this assumption and isospin invariance one also expects:

dominating these transitions, one

and the (nominal) phase space

(2)

These relations have been used to convert the experimental upper bound on the ratio of the exclusive radiative B decays [2]:

WB -+ P(W)Y) l3(B --* K*y)

< 0.34 (90% C.L.), (3)

’ On leave of absence from St. Petersburg Nuclear Physics Institute, 188350 Gatchina, Russia.

0370-2693/95/.$X)9.50 @ 1995 Elsevier Science B.V. All rights reserved

SSDIO370-2693(95)01087-4

224 A. Ali, V.M. Braun / Physics Letters B 359 (1995) 223-235

into a bound (V&J/IV,,/ < 0.64-0.76, depending on the estimate of the SU(3)-breaking parameters in the SD piece [ 1,3,4]. While this bound is at present not competitive with the corresponding bound from the CKM unitarity [5] and from the fits of the CKM matrix elements [6], which yield

(4)

one anticipates that the increased sensitivity in the radiative B decay modes projected for the upgraded CLEO

detector and the high-luminosity B factories will allow one to test the relationships (l), (2) quantitatively. The troublesome point, which has been raised in a number of papers [ 7-131, is the possibility of significant

long-distance (LD) contributions to radiative B decays from the light quark intermediate states. Their amplitudes necessarily involve other CKM matrix elements and hence the simple factorization of the decay rates in terms of the CKM factors involving 1 i&l and IV,,1 no longer holds, thereby invalidating the relationships ( 1) and (2), A redeeming feature, however, is that the experiments by testing (2) can determine in a model-independent way the extent to which the radiative transitions in question are dominated by the SD contributions. In general, light quark (u, c) intermediate states enter in the magnetic moment transitions through the corresponding Wilson coefficient CTff ( ,u) (see below), and through the transitions induced by the matrix elements of the four-fermion operators. In fact, the effect of the light quarks to CTff (,u) is completely negligible as demonstrated through the explicit calculations [ 14,151, being suppressed by powers O(mi/mw)* and 0(mi/m,)2 with mi = m, or m, * . Estimates of the contribution of the four-fermion operators require a certain nonperturbative technique and have been worked out using quark models and the vector meson dominance (VDM) approximation. The purpose of this Letter is to suggest an alternative technique, which treats the photon emission from the light quarks in a theoretically consistent and model-independent way. We then combine this treatment with the light-cone QCD sum-rule approach to calculate both the SD and LD - parity conserving and parity violating - amplitudes in the decays Bu,d --+ p(o) + y. We find that the LD contributions are negligible in the neutral B-meson decays Bd -+ p(w) + y but they may contribute up to f20% corrections in the decay rate of the charged B meson B; --f p* + y. We work out the modified form of the relations ( 1) and (2)) and work out the consequences of this result for the extraction of the CKM parameters from exclusive radiative B decays.

2. For subsequent use, we collect the definitions used in this work. Radiative weak transitions at the B-meson scale are governed by the effective Hamiltonian

(5)

where we have shown only the contributions which will be important for what follows. The 01, 02 are the standard four-fermion operators

01 = (ZP$?) (&r&T), 02 = a~p~a)&?~,~,), (6)

P = yl*( 1 + ys) 3 and a, B are color indices, the Ct , C2 are the corresponding coefficients (depending on the scale ,x> ; 07 is the magnetic moment operator

07 = ~&,,( 1 - ys)Ffi”“b (7)

and F&,, is the electromagnetic field strength tensor, which we take, for the photon emission with momentum q and polarization ep), to be

F,, (x) = i (eiy)qF - ec)qv) e@ . (8)

* See also [ 131 for a recent discussion of this point.

3 Our conventions for the ys-matrix and •&,,~P tensor conform to those in [ 161.

A. Ali, V.M. Braun/ Physics Letters B 359 (1995) 223-235 225

a b

C

Fig. I. Weak annihilation contributions in BU -+ m involving the operators 01 and 02 denoted by @ with the photon emission from (a) the loop containing the bquark, (b) the loop containing the light quark, and (c) the tadpole which contributes only with additional gluonic corrections.

The coefficient CTff includes also the effect of the four-fermion operators 0s and 06. For details and numerical

values of these coefficients, see [ 17,181. We concentrate on the B: decays, B: -+ p* + y and take up the neutral B decays Bd -+ p(w) + y at the end. The SD contribution to the decay rate involves the matrix element

(~yl07IB) = 34)~ (y)~(~I&wd’(l - ys)blB(p)), (9)

which is parametrized in terms of two invariant form factors

Here p and k = p - q are, respectively, the B-meson and p-meson momentum and e(P) is the polarization vector of the p meson. For the real photon emission the two form factors coincide, F[ = F. G FS. This form factor was calculated in [ 1 ] using the light-cone QCD sum rules, and we shall follow this approach in this paper as

well. Combining the above expressions we get the SD decay amplitude

A shon = -GF~b&~c7$E(y)‘~(P)y {Epv,&fqP - i [g,,(q .p) - p,&,]} . 2Fs(q2 = 0). Jz

(11)

The LD contributions of the four-fermion operators 01, 02 involve two possibilities for the flavor flow, shown schematically in Fig. 1. In this paper we consider the contribution of the weak annihilation of valence quarks

226 A. Ali, VM. Braun/ Physics Letters B 359 (1995) 223-235

in the B meson, Figs. la, lb. It is color-allowed for the decays of charged B mesons, and is expected to give the dominant LD contribution in this case. In the factorization approximation, we write

(mI02IB) = (F~~Q~NTW“W~ + (~rl~~~~lo)(Ol~~~~lB),

and make use of the definitions of the decay constants

(12)

(Old&+) = &.fB, (PI~~~~O)=E~)~,~~, (13)

to reduce the problem at hand to the calculation of simpler form factors induced by vector and axial-vector currents.

The factorization approximation assumed in this paper has been tested (to some extent) in two-body and quasi-two-body nonleptonic B decays involving the transitions b + cEs and b + ciid. It has not been tested experimentally in radiative B decays. In our framework, nonfactorizable contributions belong to either the 0( cr,) (and higher order) radiative corrections or to contributions of higher-twist operators to the sum rules. Since a11 such corrections are neglected in our paper, the assumption of factorization is consistent with our theoretical accuracy. In the present analysis we use the factorization approximation for the color-allowed radiative decays for quantitative estimates, in which case it is expected to hold better. For color-suppressed contributions, such as B” --+ ( p”, w) + y, we invoke the factorization approximation to argue that they are sufficiently smaller than the color-allowed ones and can be neglected.

The two terms in (12) correspond in an obvious way to the contributions of photon emission from the loop containing the b-quark, Fig. la, and from the loop of the light quarks, Fig. lb, respectively. The latter subprocess involves an (axial) vector transition with the momentum transfer mf. According to our estimates, this contribution is much smaller than the one coming from the first term, which is dominated by the photon emission from a soft u-quark and which we shall consider in detail. To that end, we write down

fp(yIXvblB) =-cE(~)~{-~ [g,,(q.p) -pclq,,] .2Ff(q2) +~~~~ap~q~~2Fk(q~) +contact terms},

(14)

where the two form factors describe parity-violating and parity-conserving amplitudes, respectively. Note that we have included the factor fp on the 1.h.s. of (14) to make the form factors dimensionless. The contact terms indicated on the r.h.s. of ( 14) depend on the gauge chosen for the electromagnetic field and should be omitted as they are identically canceled by similar gauge-dependent contact terms contributing to the second term in ( 12). Physically, they correspond to contributions of photon emission from the W* boson and arise in intermediate steps because the factorization approximation in (14) introduces charged weak currents. Adding the contribution of the operator 01, we obtain the LD amplitude to the decay B, -+ p + y in terms of the form factors Ff and Ft,

Along = - 5 d vubvu*d (CT2 + &) m,Ep;P)

x {-i [g,v(q.p) -p&] .2F?q2) ++aPp”qP.2F;(q2)}.

We now proceed to estimate the form factors defined in ( 14).

(15)

3. The major difficulty is that the photon emission from the u-quark involves contributions of large distances and cannot be treated perturbatively. Although the approach which we outline below is quite general, it is most easily implemented in the framework of QCD sum rules, in the particular modification suggested in [ 19-21,1 I. The essence of this technique is to avoid introduction of model-dependent wave functions of the B meson, replacing it by a suitable interpolation operator, and using dispersion relations and duality to pick up the contribution of the B meson. Following this method, we consider the correlation function

A. Ali, V. M. Braun / Physics Letters B 359 f 1995) 223-235 227

Tpv (P3 4) = i2 Jdxe-ipx J dy eiqy(O~T{j~m(y>ii(0)~,b(O)b(x)iy~u(x)}~O)

=i[(p.q)g,v - PpqvlTl (P2> + ~fiv,pp,q~T2tp2) + Otp,) + contact terms, t 16)

with fixed q2 = 0 and (p - q>2 = rnfj. As we are going to demonstrate, the invariant functions Ti and T2 can be

calculated in QCD at large negative p 2 4 . On the other hand, the discontinuity of the functions T, ( p2), T2 ( p2)

at positive p2 is saturated by contributions of meson states:

fipm2, 2Ft2tO) 7i,2(p2> = -A +. . . .

mb m2,-p2 (17)

Equating the two representations and making the usual assumption that the B-meson contribution corresponds to the spectral density integrated over the “region of duality”, one arrives at the sum rule for the form factors

of interest.

To calculate the “theoretical part” of the sum rule, we observe that at large negative p2 the b-quark propagates only at short distances, and we can use the perturbative expansion of its propagator in external soft fields. To

the leading order, one gets

T”‘(p q) =i2 dx dk 1

P ’ JJ __ e-O-kb (27r)4i rng - k2 J dy eiqy(~l~{j~mm(y)~(~)~~(mb+ I>ir54~>)lo).

(18)

The integration over y involves contributions of small (of order l/mb) and large (of order 1 /&co) distances. A consistent tool to separate them is provided by the Operator Product Expansion and is most easily implemented using the external field technique [ 221.

The trick is to consider quark propagation in the background of the external electromagnetic field VP, adding to the QCD Lagrangian density an extra term SL = -ejzm( X) VP (x), where yPrn is the electromagnetic current. The photon emission is obtained as the linear term in the expansion of the relevant Green functions in the external field. The corresponding formalism has been worked out in detail in [22,23,19,20] and in this paper we only give a summary of the results.

Notice that in ( 18) x2 N l/p2 --+ 0, that is, the light quark propagates close to the light cone. In this case,

it is possible to show (see [ 1,191) that the charge photon emission does not involve contributions of large distances. On the other hand, all the necessary information about the magnetic photon emission from large

distances is contained in a single (nonperturbative) matrix element of the gauge-invariant nonlocal operator I,& (0) e(x) at light-like separations x2 = 0 [ 19,201:

The (. . .)F denotes the vacuum expectation value in the external field F. Note that the path-ordered exponential factor includes both the gluon field A, and the photon field VP. It can be shown (see, e.g. [ 191) that this factor indeed arises in the expansion in (18) from higher-order terms in the expansion of the b-quark propagator in the external strong and electromagnetic fields. In what follows we drop the gauge factors, assuming the Fock-Schwinger gauge xpA, = xpV’ = 0. The normalization is chosen in such a way that &t du &,(a) = 1,

($$) is the quark condensate and the dimensionful constant x has the physical meaning of being the magnetic

4 By contact terms we indicate contributions which do not vanish after multiplication by qr. They should be calculated and subtracted to

separate the gauge-invariant Lorentz structures. Another possibility is to use the Fock-Schwinger gauge for the photon field xpV, (x) = 0, V,(O) = 0. see below, in which case the contact terms vanish.

228 A. Ali, V.M. Braun / Physics Letters B 359 (1995) 223-23.5

susceptibility of the quark condensate. In the limit of a constant external magnetic field F,p(x) = const., the spins of quarks in the vacuum tend to get oriented along the field direction, with the average spin being proportional to the quark density (I,&+), the applied field F,P, the quark charge ee and the magnetic susceptibility of the medium [ 221:

The

(‘k&b = etix($dF,p.

value of the magnetic susceptibility is large [ 251

x(,u = 1 GeV) = -4.4GeV2,

(20)

(21)

and because of this the nonperturbative photon emission is numerically very important 5 . The physical meaning of the response function &(u) becomes transparent for the particular choice of the

external field as a plain wave (8). In this case (19) can be rewritten as

1

I dy eigY(Ol~~~~m(~)~(0)~~~~~~~~lO) = e~x(++)[qpg,, -qagpp] J

du eiU4X&(a). (22)

0

Comparing this expression with the usual definition of the light-cone hadron wave functions [24] it is easy to see that &(a) defines the distribution amplitude (photon wave function) describing the photon dissociation into a quark-antiquark pair with the variable u being just the momentum fraction carried by the quark. The shape of this distribution in general depends on the normalization scale for the quark fields. However, an accurate analysis shows [ 191 that unlike meson wave functions [ 241, this distribution is close to its asymptotic form already at low virtualities p N 1 GeV:

d+(u) =6u(l -u>. (23)

Corrections to ( 19) come from higher-twist contributions which are formally suppressed by powers of the deviation from the light cone x2. For hadron wave functions these can be attributed to contributions of higher Fock components in the wave function with a larger number of constituents. The case of the photon is still specific, since additional contributions arise that related to operators involving the photon field instead of the gluon field6. These specific contributions are more important than those involving quark-antiquark-gluon operators (whose photon-to-vacuum matrix elements are numerically small, see [ 19,201 for the list of all the contributions to twist-4 accuracy and numerical estimates) and can be calculated exactly in terms of the quark condensate.

The most general decomposition of the relevant matrix element to the twist-4 accuracy involves two new invariant functions (distribution amplitudes) g$‘) (u) and gi2) (u) :

($(O)~M?#(x))F =e+(@b) / duFap(ux) [,yq$(u) + x2g$‘)(u)]

0

+e* bw J dug~)W [xp~+& - x,x9Fpq - x2Fap] (ux).

0

For one particular projection of Lorentz indices the answer is given in [ 19,20 ] :

(24)

5 A larger value x = -5.7 GeV-‘, given in the first of Refs. [25], corresponds to a lower normalization point p = 500 MeV. 6 These additional contributions can also be thought of as contact terms, produced by operators which vanish on using the equations of

motion. The external field technique avoids all contact terms at the cost of introducing additional operators.

A. Ali, KM. Braun/ Physics Letters B 359 (1995) 223-235 229

f~up$($(o>flap$/(x))~ =etiOW$~apjj duF,p(nx) [XC&(U) -$x*(1-u)+gluons+O(~~)],

0

(25)

which implies

g;“(U) - #‘<U> = -i( 1 - u).

To determine the functions gr (t ) (u) and g$*)( U) separately, we use the operator identity

(26)

(27)

To prove (27)) consider a simpler identity

which can easily be checked integrating by parts. Then substitute u( d/ du) = xt (d/&g) and subtract from

(28) the first term on the r.h.s. of (27). By simple algebra the difference can be written as

where &#(x) E (d/ax,)+(x) . Since for arbitrary Lorentz vectors X~ , yp, one has

(x ’ ybap - Yp~a?lxq + Ya~pqxq = &3~p~y,qpy5, (30)

this expression can be written in the form of the second term on the r.h.s. of (27) using equations of motion and neglecting gluon corrections.

Note that the vacuum-to-photon transition matrix element of this term vanishes. Substituting the general

expression (24) for the matrix elements in the first term, one obtains the (exact) relation

2u3 J dL’ (1) -+ (u> =#‘(u). (31)

U

Solving the system of Eqs. (26)) (3 1) we get:

g”‘(U) = -i(l - U)(3 -U), Y gC2)(u) Y =-a<1 -U)2, (32)

which is our final result for the photon emission to twist-4 accuracy. The contributions of the quark-antiquark-

gluon operators can be added using (27) and the expressions given in [20]. They are numerically small and are omitted in the numerical analysis given below. Taking the external field in the form of the plain wave (8) we get an equivalent representation:

230 A. Ali, V.M. Braun/ Physics Letters B 359 (1995) 223-235

1

s dy eiqY(~l~{~~m(y)~(f.N fl~3ti(x>NV = e~M~>[4p~a~ - ampI

J dueiuqx [x&(u) +X*&‘)(U)]

0

+e++b&) {(w)[q3ga, - xagppl ++[qpxa -4kq31 -x2[qpgaN - 4ampl) ) du e’UqXgy)(U).

0

(33)

4. The expressions given above are sufficient for the description of real photon emission in the case that

the kinematics of the particular process ensures that the quark propagates near the light cone and allow one to derive the sum rules for the form factors of interest expressed in terms of integrals of photon wave functions.

In this Letter we cannot give a detailed derivation and refer the reader to [ 1 ] for technical discussion.

Repeating the by now standard steps, we arrive at the sum rules:

$$*Ft(O) exp (-(mg - mi)/t) = i $ exp (-(~?/u>mQf) B(s0 - “t/u>

0

x euG4CI) m%(u) { 1

4(m'+ut)g$1)(u) u*t* 1 + $(2u- 1) [(e, - eb)ii -eblnu] 1 ,

1

$2Fk(O) exp (-(& - &/t) = J $ exp (-(E/u>n$j/t) Nso - n$/U)

0

x eU(iti> X&(u) 1 1

4(m2, + ut)

ll*t* [g”‘(U) -g’*‘(u)] +

Y Y 1 $[(e,-eb)E-eblnu] , 1

(34)

where we have introduced a short-hand notation U = 1 - u and neglected rns compared to rnz, t, and so.

Here I is the Bore1 parameter and SO is the continuum threshold which stands for the cutoff in the dispersion

relation restricting the region of duality for the B meson. In both the sum rules, the first term in the curly bracket corresponds to nonperturbative contributions of the photon wave function, and the two last terms are perturbative contributions of the photon emission from U- and b-quark, respectively. Numerically, the

nonperturbative contribution of leading twist dominates and is the same in both the form factors. Thus, in our approach the parity-violating and parity-conserving LD amplitudes are close to each other. For completeness,

we quote the corresponding sum rule from [ 1 ] for the SD form factor:

1

.fBm2B a2Fs(0) exp (-(m2, - mt>/t) =

J $ exp (-(ii/u)mE/t) B(s0 - mQu>

0

{

m2,+ut x mbqb,l(u> + um,g~)(u)~m,&‘(U) -

1

where ~$f (u), gy'( u) and gy’ (u) are the leading-twist p-meson wave functions, specified in Section 4 of [ 11. It is seen that the structure of sum rules for the SD and LD form factors is very similar, and many of the uncertainties (such as the dependence on fs and f,) cancel in their ratio.

Before proceeding to present our numerical results, we would like to remark that the calculations presented here suggest that the VDM approach underestimates the LD amplitudes for the photon emission. The difference between the VMD approach and our method is two-fold. First, the value of the quark magnetic susceptibility

A. Ali, V. M. Braun / Physics Letters B 359 f 1995) 223-235 231

in the VDM approximation Xvo~ = -2/m; [ 231 appears to be significantly below the result in (21) . Second, the p-meson wave function is more “narrow” than the photon wave function, see [ 24,19,1] and since the decay kinematics picks up contributions with almost the entire momentum carried by one of the constituents,

the p-meson contribution in VMD is additionally suppressed. Thus, we expect that the VDM-type relations between B -+ py and B -+ pp amplitudes receive significant (- 50%) corrections from contributions of the excited states.

5. In performing the numerical analysis we conform to the values of the parameters given in [ 11. We note

that all the three sum rules are dominated by the first term in the curly brackets, and since also the wave

functions & and 4,’ are similar, the ratio of the SD and LD form factors takes a simple form:

(37)

The only important correction to this estimate corresponds to the perturbative emission while nonperturbative twist-4 contributions are in fact negligible. Assuming the interval t N 5 - 10 GeV* for the Bore1 parameter, we obtain:

F,L/FS = 0.0125 l 0.0010, F,L/FS = 0.0155 zt 0.0010, (38)

where the errors correspond to the variation of the Bore1 parameter. Including other possible uncertainties, we expect an accuracy of the ratios in (38) to be of order 20%. This can be combined with the result of [ 1 ]

F& = fiF&,v = @,,,, = 0.24 f 0.04, (39)

to extract the absolute values of the LD form factors. Since the parity-conserving and parity-violating amplitudes turn out to be close to each other, the ratio of the LD and the SD contributions reduces to a number

Using C2 = 1.10, Cr = -0.235, Cyff = -0.306 (at the scale ,!L = 5 GeV) [ 17,181, we get

R B,-~~ _ 4r2mp(C2 + Cl/N,) L/S =

FL = _. 3. f o o7 .- eff

mbC7 FS . *’

(40)

(41)

Since the Wilson coefficients are scale-dependent, this ratio is in general also scale-dependent (unless canceled by a compensating dependence in the form factors). Varying the scale ,u of the coefficients in the range

mh/2 < p < 2mb, an additional dependence of 510% is introduced in RfiT’, which, however, is smaller than

the error given above. To get a ball-park estimate of the ratio d&dshort, we take the central values of the CKM matrix elements,

Vud = 0.9744 f 0.0010 [ 51, l&d1 = (1.0 f 0.045) . lo-*,

(&.,,I = 0.041 f 0.004, j&&b] = 0.08 f 0.02 [ 61,

yielding,

l-Atong/dst,ort)BU-py = lR$+n,w N 10%. (42)

A quantitative analysis of the relative decay rates in terms of the CKM parameters is presented below.

232 A. Ali, V.M. Braun /Physics Letters B 359 (1995) 223-235

Fig. 2. Ratio of the neutral and charged B-decay rates T(Bu + py)/?!T( Bd - py) as a function of the Wolfenstein parameter

r] = 0.2 (short-dashed curve), v= 0.3 (solid curve), and 7 = 0.4 (long-dashed curve). P. with

The analogous LD contributions to the neutral B decays Bd --) py and Bd -t oy are much smaller, a point that has also been noted in the context of the VMD and quark-model based estimates. In our approach, the corresponding form factors for the decays Bd -+ p(w) y are obtained from the ones for the decay B, -+ py reported above by the replacement e, -+ ed, which gives the factor -5; in addition, and more importantly, the LD contribution to the neutral B decays is color-suppressed, which reflects itself through the replacement of the factor ai s C2 + Cl/NC in (2) by a2 E Ci + G/NC. This yields for the ratio

R Bd’PY L/S eda -=-

R B,,+pY N -0.13 f 0.05,

LIS wl

(43)

where the numbers are based on using az/al = 0.27 f 0.10 [ 287. This would then yield at most

which in turn gives d~n~py/d$,~py < 0.02. We note that the LD contribution just discussed is of the same order as the one expected from the diagrams in Figs. lb, lc which we do not consider in this paper. In what follows we shall therefore neglect the LD contribution to the neutral B decays. Restricting ourselves to the color-allowed LD contributions only, the relation (2) gets modified to:

2r( Bd --+ py) = 2i-( &j --) WY) = 1 + ,+‘PY ub ud

LIS v v* tb td

= 1 + 2 . &&d PC1 -P) -q2 (1 -p)2+q2

(44)

A. Ali. V.M. Bran/ Physics Letters B 359 (1995) 223-235 233

where RLIS E Rf;F and p, ~7 are the Wolfenstein parameters [26]. The ratio T(B, --t py)/2r( Bd --f Py)

(= r(B, 4 Pr)/=Y&i + oy)) is shown in Fig. 2 as a function of the parameter p, with 77 = 0.2, 0.3 and 0.4. This suggests that a measurement of this ratio would constrain the Wolfenstein parameters (p, v), with the dependence on p more marked. In particular, a negative value of p leads to a constructive interference in B, -+ Py decays, while large positive values of p give a destructive interference. This behavior is in qualitative

agreement with what has been also pointed out in [ 91. The ratio of the CKM-suppressed and CKM-allowed decay rates in (1) for charged B mesons likewise gets

modified due to the LD contributions. Following [ 271, we ignore the LD contributions in r( B + K*y) The ratio of the decay rates in question can be written as:

r(Bu+py) =‘%*[(l-p)*+v*] r(B + ICKY) ]+2.RL,Sv~d~/1_-p~~I~~ + (R~/s)*v$

(45)

where A = 0.2205. Using the central value from the estimate 5” = 0.59 f 0.08 [ l] we show in Fig. 3a the ratio

(45) as a function of p for 77 = 0.2, 0.3 and 0.4. It is seen that the dependence of this ratio is rather weak on

the parameter q but it depends on p rather sensitively. To show the effect of the LD contribution, we compare

in Fig. 3b the predictions for the ratio (45) as a function of p with and without the LD contribution, fixing 77 = 0.3, the central value for this parameter obtained from the CKM fits [ 61. It is seen that the effect of the

LD contributions on this ratio is small, and is comparable to N 15% uncertainty in the normalization due to the SU( 3) -breaking effects in the form factors. Neglecting the color-suppressed LD contributions, as argued above,

the ratio of the decay rates for neutral B meson is not effected and to a good approximation the SD-dominated result

r(Bd + PY, WY) T(B + K’y)

=5dA*[(l -p)*+q*l

still holds. Finally, combining the estimates presented here and in [I] for the form factors and restricting the

Wolfenstein parameters in the range -0.4 6 p 6 0.4 and 0.2 < 77 < 0.4, as suggested by the CKh4 fits [6], we give the following range for the absolute branching ratios:

B(B,+py)=(1.9f1.6)~10-~, B(Bd -+ py) ? B(Bd + oy) = (0.85 *0.65) x 10-6. (47)

where we have used the experimental value for the branching ratio B( B + K*+y) = (4.5f1.5f0.9) x 1o-5

[ 291, 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 parameters p and 7. Similar constraints would also follow from the measurement of the inclusive radiative decays B -+ Xd + y [ 301, for which a branching ratio D( B --+ Xd + y) = ( 1.0 f0.8) x lop5 is estimated using the measured inclusive rate f?( B ---f X, + y) = (2.32 f 0.57 f 0.35) x 1O-4 [ 311.

To summarize, we have presented a QCD sum-rule-based calculation of the contribution of the weak annihi- lation to exclusive radiative rare B decays, which is expected to dominate the LD contribution for the decays of charged B mesons, B: + p* + y. The numerical effect of the weak annihilation amplitude depends on the value of the CKM-Wolfenstein parameter p, and is typically estimated to be in the 10% range for the presently allowed values of the parameters p and r]. The corresponding LD contributions in the neutral B-meson decays

Bd + p( w)y are found to be insignificant. The interference of LD and SD contributions provides a possibility to measure the sign of the Wolfenstein parameter p from the ratio (44). The branching ratios f3( B, + py) and B( Bd -+ p( w)y) likewise are sensitive to the sign and magnitude of p, increasing with large negative values of p. Our investigations strengthen the expectations that exclusive radiative rare B decays, very much like their inclusive counterparts, are dominated by SD contributions, which in the context of the standard model implies that these decays are valuable in quantifying the CKM parameters.

234 A. Ali, V.M. Braun/ Physics Letters B 359 (1995) 223-235

0.08

0.06

0.08

0.06

0.04

0.02

0

-0.4 -0.2 J

0.2 0.4

P Fig. 3. Ratio of the CKM-suppressed and CKM-allowed radiative B-decay rates f (4 + py)/T( B - K*y) (with B = BU or Ed) as a

function of the Wolfenstein parameter p, (a) with 7 = 0.2 (short-dashed curve), 7) = 0.3 (solid curve), and 71 = 0.4 (long-dashed curve);

(b) comparison of the result obtained by neglecting the LD contribution (dashed curve) with the one including the LD contribution (solid curve), both evaluated with Q = 0.3.

A. Ali, V.M. Braun /Physics Letters B 359 (1995) 223-235 235

One of us ( A.A.) would like to thank Amarjit Soni and Giulia Ricciardi for vigorous discussions on radiative B decays. The other (V.M.B.) would like to thank Alexander JChodzhamirian for a correspondence and helpful

discussions. After this paper was completed, we became aware of the work [ 321, where exclusive radiative B and D

decays are worked out in an approach very similar to ours, reaching very similar conclusions as the ones

presented here.

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