3 October 1996
PHYSICS LElTERS B
ELSEWIER Physics Letters B 386 (1996) 328-334
On the decay mode B- ---) ,u-Y,y
I? Colangelo a,, F. De Fazio a,b, G. Nardulli a,b,2 Istituto Nuzionale di Fisicu Nucleure. Sezione di Bari, Italy
h Dipartimento di Fisica. Universitd di Bun. Italy
Received IO June 1996
Editor: R. Gatto
A QCD relativistic potential model is employed to compute the decay rate and the photon spectrum of the process B- + p-V,,y. The result f3( B- -+ /_-Fry) z 1 x 1 O- confirms the enhancement of this decay channel with respect to the purely leptonic mode, and supports the proposal of using this process to access relevant hadronic quantities such as the B-meson leptonic decay constant and the CKM matrix element I/;rb.
Noticeable theoretical attention has been recently given to the weak radiative decay
B- + p-F,y. (1)
The reason is in the peculiar role of this decay mode for the understanding of the dynamics of the annihilation processes occurring in heavy mesons [ I-51. Moreover, it has been observed that ( 1) can be studied to obtain indications on the value of the B-meson leptonic constant fa using a decay channel which differs from the purely leptonic modes B- -+ L-V!, and is not hampered by the limitations affecting those latter processes. Such difficulties mainly consist in low decay rates 3 (using VUb = 3 x 10e3, fs = 200 MeV and rB- = 1.646 * 0.063 ps  one predicts LJ( B- + e-Fe) E 6.6 x lO_ and a( B- 4 p -tip) cv 2.8 x 10e7 ) or in reconstruction problems for B- + T-P~.
In Ref.  heavy quark symmetry and experimental data on D* -+ Day have been exploited to study the dependence of t3(B- + p-V,y) on the heavy meson decay constant fi/&, which is the common value of fa and fs. (modulo logarithmic factors) in the limit mb + 03. The analysis is based on the dominance of polar diagrams contributing to the process B- --f ,x -l,y, the pole being either the vector meson B* or the positive parity Jp = l+ state Bi (see Ref. [ 51 for further details). According to the analysis in , in
correspondence to the expected range of values of P: E N 0.35 GeV3j2, the branching ratio f3( B- ----f p-V,y) should be O( 10W6), which represents an enhancement with respect to the purely leptonic mode.
E-mail address: COLANGELO@BARI.INFN.lT. ? E-mail address: NARDULLI@BARI.INFN.IT. ? Present bounds are: B(B- + e-0,) < 1.5 x 10e5, LT(B- + p-i;,) < 2.1 x 10e5 [71
0370-2693/%/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved, PII SO370-2693(96)00955-O
P. Colangelo et al./Physics Letters B 386 (1996) 328-334 329
In order to give further arguments in support of that analysis, we want to consider the process ( 1) in a different context. More precisely, whereas in  we have studied the feasibility of extracting fs = p;lJma from future experimental data, in this letter we study the decay (1) within a well defined theoretical model in order to have an independent estimate of the decay rate.
We employ a relativistic constituent quark model already used to study several aspects of the B-meson phenomenology [g-lo]. Within this model the mesons are represented as bound states of valence quarks and antiquarks interacting via a QCD inspired instantaneous potential with a linear dependence at large distances, to account for confinement, and a modified coulombic behaviour at short distances to include the asymptotic freedom property of QCD. We adopt the interpolating form between such asymptotic dependences provided by the Richardson potential [ 1 l] 4. In the rest frame, the state describing a B, meson is represented as
where LY and p are colour indices, r and s are spin indices, bt and dA are creation operators of the quark b and the antiquark L&, carrying momenta kt and -kt respectively. The B-meson wave function $s(kt ) satisfies a wave equation with relativistic kinematics (Salpeter equation) [ 121 taking the form (in the meson rest frame) :
where V( kr , k, ) is the interaction potential in the momentum space and @B is covariantly normalized:
/ (25-)3, d~,I+~(h~l*=2 MB,. (4)
Solving Eq. (3) by numerical methods one obtains the wave function; we shall present this result later on. It can be mentioned that by this model a number of predictions have been derived; for example, the heavy meson spectrum, leptonic constants [ 8,9], semileptonic form factors and strong decay constants [ lo] _
In the framework of the relativistic QCD potential model the wave function $s is the main dynamical quantity governing the decay ( 1) . As a matter of fact, the amplitude of the process B- (p) -+ CL- (pt ) ~~ ( p2) y( k, F) can be written as
where GF is the Fermi constant, VUb is the CKM matrix element involved in the decay, L@ = ,G(pt )rp( 1 - ys)v(p2) is the weak leptonic current, and IIP is defined by II, = nPve* (E is the photon polarization vector) ; nFv represents the correlator
In Eq. (6) q is q = p1 + ~2, J,(x) = ii(x)yP(l - y,)b(x) is the weak hadronic current and V,(O) =
ie ii(O)y,u(O) - ie 6(0)y,b(O) is the electromagnetic (e.m.) current. The two pieces in the e.m. current cor- respond to the coupling to the light quark and to the heavy quark, respectively. The corresponding contributions
to D,UY will be referred to as II&, and II:,, depicted in Figs. la,b. II:, contains the light quark propagator:
4 A smearing of the Richardson potential at short distances has also been introduced to take into account the effects of the relativistic kinematics; see Ref. 191 for the explicit form of the potential.
330 P. Colungelo et al./Physics Letters B 386 (1996) 328-334
Fig. I. Diagrams describing the decay B- - p-Pti~.
() = s -e*(, + VI,), d4t (zT)4 e2 _ ,$ while Iby contains the analogous b quark propagator.
The calculation of the time-ordered product appearing in (6) gives
The analogous expression for II: can be obtained by the replacements: i H -i, m, H rnt,, $,, ++ .ii,,.
The operators ji,, and jiV, which depend on the integration variable X, are written as $, = ti(O)l$,b(x)
and .&, = .G(x)$,b(O), with the I matrices given by Ii, = yy(!+ m,>y,( 1 - ys) and I$, = yP( 1 - ~5) (/ + mb) yu. In the constituent quark model such operators can be expressed in terms of quark operators; for example one has
where Eq (q) = ,/q2 + rn$ and uy (I:,) are quark (antiquark) spinors.
By exploiting anticommutation relations among annihilation and creation operators, we obtain, in the B meson rest frame
Eb(kl )E,(kl) I
where 41 = (E,, -kl ) and q2 = (Eb, kl ). We recognize in the two factors in the curly brackets the contributions
of I$ and fli,,, respectively.
P. Colangelo et al./Physics Letters B 386 (1996) 328-334 331
Since II,, only depends on two vectors, the B meson momentum pP and the photon momentum k,, it can be written in terms of six independent Lorentz structures:
QLv = a PEPS + P k,kv + f k,p, + 8 pllk, + 5 g,, + i 17 ECLvpvpPkg. (11)
By gauge invariance one has (Y = 0 and 5 = -p . k 5; moreover, after saturation by E* one gets
np = n+* = [5 (kg, -P . kg,,) + i rl l pvpdkal E*, (12) i.e. only the terms proportional to 7 and JJ survive: they are the vector and the axial vector contribution, respectively. It is convenient to compute Eq. (12) in the B rest frame p = (MB,O), with k = (kO,O, 0, k); the result reads
At this point, it is straightforward to calculate the rate of the decay process ( 1) ; one obtains (m, z 0)
T(B- -+ ,u-cKy) = ~ 3(2~)~ s
dkko(& - 2k0)[lII,,/2+ )n,2)21,
II,, = - ,~~JdCOS~ ~~,XI,~,Xl,us(~XI,)[EbE~(Eb+:b)(E~+m~,)]l-2 -I 0
n12 = &-dcos~ J' l~lldl~~l~n(lk~l)[E~E,~Eb+~h)iE,+m,~]2 -I 0
kO[(Eb+mb)(E,+m,) -(k1/21 +Ik I I cos8(mb-mu)(ELt+mu+Eb+mb)
In Eqs. ( 15), ( 16) the quantities f and g are defined as
f = ntfj + Mi - 2MBk0 - 2MBEb + 2Ebko - 2kJkl) cost? - rn:, (17)
g=mt+Mi- 2MBk0 - ~MBE, + 2Euko + 2k0/kl 1 cos 8 - rni, (18)
332 P. Colangelo et al./Physics Letters B 386 (1996) 328-334
whereas the S-wave reduced function u is related to es:
lkll UB(lkil) = - A7i#B(k); (19)
we plot in Fig. 2 the function UB obtained as a solution of the wave Eq. (3). Let us now point out that in computing the diagrams in Figs. la,b and, therefore, in Eqs. ( IS), ( 16), we
have so far imposed 4-momentum conservation for the physical particles B, p, v and y in the process (1). On the other hand, energy conservation has to be imposed also at quark level since, otherwise, ( 15) and ( 16) would present spurious kinematical singularities. In order to deal with this problem we follow the approach originally proposed within the ACCMM model [ 131 for the decay b --) u ! Ft. One assumes that the spectator quark has a definite mass, while the active quark has a running mass, defined consistently with the energy
Therefore, as in the case of the ACCMM model, the running mass of the active b quark can be defined by:
mi(ki) = IW~ + ln: - ~MB J ky + mf. (21)
Moreover, by requiring that the right hand side of Eq. (21) is positive, an upper bound on the quark momentum lk, I: can be obtained
Notice that the masses of the light constituent quarks, as obtained by fits to the meson spectrum, are m, = md = 38 MeV.
The contribution of the two