cm __ __ li!B 2 November 1995

PHYSICS LETTERS 6

EISEVIER Physics Letters B 361(1995) 146-154

Photon energy spectrum in B -+ Xs + y and comparison with data

A. Alia, C. Greub b,1 a Hamburg, Germany

SLAC Theory Stanford University,

Received 24 1995; revised manuscript 25 August Editor: P.V.

Abstract

A comparison of the inclusive photon energy spectrum in the radiative decay B + X, -t y, measured recently by the CLEO CoIlaboration, with the standard model is presented, using a B-meson wave function mode1 and improving earlier perturbative QCD-based computations of the same. The dependence of the photon energy spectrum on the non-perturbative model parameters, pi, the b-quark Fermi momentum in the B hadron, and m4, the spectator quark mass, is explicitly shown, allowing a comparison of these parameters with the ones obtained from the analysis of the lepton energy spectrum in semileptonic B decays. Taking into account present uncertainties, we estimate BR( B -+ X, + y) = (2.55 f 1.28) x 10m4 in the standard model, assuming 1 V,, I/[&,[ = 1.0. Comparing this with the CLEO measurement BR(B + X, + y) = (2.32 f 0.67) x 10m4 implies IxSl/ll&l = 1.1 f 0.43, in agreement with the CKM unitarity.

1. Introduction

Recently, the CLEO Collaboration has reported the first measurement of the photon energy spectrum in the decay B + X,+y [ 11, following the measurement of the exclusive decay mode B --$ K* + y reported in 1993 by the same collaboration [2] . The inclusive branching ratio and the photon energy spectrum al- low a less model-dependent comparison with the un- derlying theory, more specifically the standard model (SM), as compared to the exclusive decay modes which require additionally decay form factors. The CLEO data have been compared in [ 1 ] with the SM- based theoretical computations presented in [ 3,4], al- lowing to draw the conclusion that agreement between

* Supported by Schweizerischer Nationalfonds.

Elsevier Science B.V. SSDI 0370-2693(95)01118-8

theoretical predictions and experiment is good, given that large uncertainties exist in both. In particular, the measured branching ratio BR(B + X, + y) = (2.32 f 0.67) x 10m4 [I] is in agreement with the SM-based estimates in [3], as well as with the ones in [5,6].

In this letter, we would like to report on an improved calculation of the photon energy spectrum compared to what we have presented earlier and which has been used in the CLEO analysis [ 11. The main theoretical difference lies in the inclusion of the complete op- erator basis 01, . . . . 0s for the effective Hamiltonian, defined below, in the computation of the partonic pro- cesses b -+ s + y and b + s + g + y, and in us- ing the complete leading-logarithmic computations of the anomalous dimension matrix presented in [ 71. In contrast, our previous calculations were done in the

A. Ali, C. Grab/ Physics Letters B 361(1995) 146-154 147

truncated approximation, where we had dropped the effects of the four-Fermi operators 03, . . . , 06 and the chromomagnetic operator 0s in the computation of the contributions from b + s + g + y. In addition, use was made of the anomalous dimension matrix de- rived in [ 81. The other ingredient of our calculation, namely a specific B-meson wave function model [9] to incorporate the non-perturbative effects on spectra, remains unaltered. However, since data are now avail- able, we fit the normalized photon energy spectrum with the improved theoretical framework to determine from the shape the non-perturbative parameters of the wave function model being used, namely the Fermi motion parameter PF and the spectator quark mass PI*. These in turn determine within the model the b- quark mass. There is considerable theoretical interest in these parameters, in particular pi, which is a good measure of the kinetic energy of the b quark in the B meson, and the b-quark mass. While, admittedly, present errors are large preventing us from drawing sharp conclusions, some valuable insight on the shape parameters and normalization can already be obtained and we quantify this information.

This letter is organized as follows: In Section 2, we introduce the effective Hamiltonian for the decay B -+ X, +y and present the Wilson coefficients numerically. The bulk of this section contains an anatomy of the partonic processes b + s + y and b + s + g + y, where the essential steps in the derivation of the matrix elements are given. The photon energy spectrum at the partonic level is derived in Section 3, where we also summarize the B-meson wave function model [9] to get the spectrum in the decays B + X, +y. Numerical results for the branching ratio BR( B --) X, + y) in the SM, the ratio of the

An anatomy of the decays b + sy and b + syg

The framework we use here is that of an effective theory with five quarks, obtained by integrating out the heavier degrees of freedom, which in the standard model are the top quark and the W&son. A complete set of dimension-6 operators relevant for the processes

Table 1 Wilson coefficients C&L) at the scale p = mw = 80.33 GeV (“matching conditions”) and at thme other scales, Jo = 10.0 GeV, CL = 5.0 GeV and ,U = 2.5 GeV, evaluated with two-loop /3- function and the leading-order anomalous-dimension matrix. The entries correspond to the top quark mass Eii( m, @C) = 170 Gev

(equivalently, m, Ple = 180 GeV) and the QCD parameter with 5 flavours As = 195 MeV (equivalently, as(mi) = 0.117), both in the MS scheme

Cl 0.0 -0.158 -0.235 -0.338 c2 1.0 1.063 1.100 1.156 c3 0.0 0.007 0.011 0.016 c4 0.0 -0.017 -0.024 -0.034 CS 0.0 0.005 0.007 0.009 Gi 0.0 -0.019 -0.029 -0.044 Cl -0.193 -0.290 -0.333 -0.388 C8 -0.096 -0.138 -0.153 -0.171

c7 eff -0.193 -0.273 -0.306 -0.347

Citf -0.096 -0.132 -0.146 -0.162

b-+s+yand b+s+y+giscontainedinthe effective Hamiltonian

Hdb --t v)

where GF is the Fermi constant coupling constant, Cj (CL) are the Wilson coefficients evaluated at the scale CL, and At = V&y: with qj being the CKM matrix elements. The operators Oj are listed in [ 31, We note here that the explicit mass factors which appear in 07 and Og (see [ 31) are the running quark masses. Using the renormalization group equation the Wilson coeffi- cient can be calculated at the scale p w mb which is the relevant scale for B decays. At this scale the large logarithms are contained in the Wilson coefficients. The leading logarithmic expressions for these coeffi- cients, which we use in this paper, are given in the literature [ 7,5,10]. Their numerical values * are given in Table 1. For subsequent discussion we define two effective Wilson coefficients CTff (p) and C’i* ( CL) be-

low and give their numerical values in Table 1: Cyff z

c7 - c5 /3 - G, and Ciff = Cs + c5.

2The results given here for the entries concerning 07 and 0s correspond to the naive dimensional regular&&ion scheme (NDR) , which we use in the calculation of all the matrix elements.

148 A. Ali, C. Greub/ Physics Letters B 36111995) 146-154

We summarize the principal points in the deriva- tion of the photon energy spectrum in the decay b -+

+g) in this

Eg -t 0 or E, + 0, respectively. The singular configurations in b --f syg are cancelled in a distribu- tion sense in the photon energy spectrum if one also takes into account virtual corrections to the two-body process b -+ sg and b + sy, order by order in per- turbation theory. We will take into account only those virtual correction diagrams which are needed to can- cel the infrared singularities from b + syg.

We first discuss the result for b -+ sy. It turns out that the effects of the four-Fermi operators to b -+ sy can be absorbed into a redefinition of the coefficient

Cl + C, eff, with CT* defined above. This not only holds for the b + sy amplitude without virtual correc- tions but also for those infrared-sensitive virtual cor- rections which we have mentioned above. One there- fore has to calculate the matrix element of CT607 for b + sy including virtual corrections. The result of this calculation, which was derived in d = 4 - 2~ di- mensions, can be expressed as 131

where we have split this quantity into an infrared-finite and an infrared-singular part:

r;:;,, mzl = -s (1 - tq3 (1 + r) ICTff GF Atl*cr,xn a,

2(1+r) logr

I l-r E ’ (3)

and

r virt mz 7,fin=~(1-r)3(1+r)/C7effG~Al12aem

x (1+zr),

l+r 7-z - K ) l-r

(log2r -410gr

+4logr log(1 -r))

-8+3logr+8log(l-r) -2logk . m; >

(5)

The quantity r is defined as r = ( m,/mb)2. The in- frared singularity (3) will be cancelled when taking into account the gluon bremsstrahlung diagrams in- volving the operator 07.

As we already noted, the process b + syg has also an infrared singularity as the photon becomes soft. These singularities are cancelled analogously by virtual photon corrections to the matrix element for b -+ sg, i.e. in CiffOs. The singular part is obtained

from Eq. (3) by replacing CTff by Q&‘iff and the finite part reads

r vin =*(1-r)3(1+r)JQdCiffG,A,/2 &fin g(j,$

where r is given in Eq. (5) and Qd = - l/3. A remark concerning the quark masses is in order here. When calculating the matrix elements we have used the on- shell subtraction prescription for the quark masses. Due to the explicit factors of the running quark masses in the operators 07 and Os, the rnz factor contained in I’Tifl and Iii’ given above should be replaced by the following product:

mi - m&otemb(rU)2f (7)

where mb,p& and mb ( &6) denote the pole mass and the running mass of the b quark, respectively. In actual practice, we identify all the masses mb in the various intermediate expressions with mb,p& and multiply at the end the so-derived decay width I( B + X, + r> with a correction factor R:

R = (mb(P)/mb,pole j2. (8)

We now take up the matrix elements for the pro- cess b -+ syg. As the explicit expressions are too long to be presented here, we only point out the ba- sic structure and give the complete formulae else- where [ 111. We first concentrate on the contributions of the four-Fermi operators 01, . . . . 06. It turns out that

A. Ali, C. Greub/ Physics Letters B 361 (1995) 146-154 149

the diagrams which do not involve both gauge par- ticle radiation from an internal quark are either zero or can be absorbed into a redefinition of the coeffi- cients (c7,c8) + (c7 eff, Ciff ) . The remaining case, in which both gauge particles are emitted from the in- ternal fermion line, is discussed now. There are two such diagrams associated with each four-Fermi opera- tor, whose sum is ultraviolet (and infrared) finite. We denote these matrix elements by Mi

Mi = ~(r~glCiOi]b) (i = 1,2, . ...6) . (9)

Analogously, the matrix elements of Cyff07 and CiffOs are denoted by M7 and Ms, respectively. Adding all these contributions, the complete matrix element for b -+ syg is denoted by M, with M = cf Mi. When squaring M and summing over the polarizations and spins of the particles, it turns out that only IM$ and ]Ms]2, are infrared-singular. The other squared ampli- tudes and all the interference terms are infrared-safe. We therefore make the decomposition

I@ = F + lM71; + lM31: 9 (10)

where F denotes the infrared-safe contributions which will be taken into account numerically.

3. The photon energy spectrum in B + X, + y

In the following, it is useful to define the dimension- less photon energy x through the relation Ey = (mi - m~)x/(2m~);xthenvariesintheinterval[O,l].Tbe dimensionally regularized contribution dI’~ms/dx as- sociated with 1 MT\: reads (using 5 = ( 1 - r)x)

dI$= x-2E

dx =C,(l -‘)-4E(l_X),+2EIE(r)

-21 +x)1%(1 -5) --Lo(x)

with

(11)

x (1+r)(l-r)3,

I,(x) = la(x) + Ib(X

43 4r

=-4-1-t #$ --4(1+r)log(l-0,

rb = 8(1-t) log( 5

1 - 5) - +-$log(l-5)-+$

Li(0 +8(l+r)---

2(1 + r) 5 5 log2(1 -5). (12)

Here the symbol Li stands for the Spence function. As dl?~mms/dx has a non-integrable singularity at

x = 0 (i.e., E, = 0)) we should work in d-dimensions as well. However, as the photon energy spectrum is of no experimental interest at very small energies, i.e. E, + 0, we remove the infrared regularization imme- diately. As for the branching ratio, we note that the dimensionally regularized total integral l?iErns can be obtained from the analogous expression for I$“” by doing obvious replacements. The result for driXms/dx reads

drp”” dx=m

5 I Gd,~~~;72 b 961~5 aernffs

x (l+r>(l-r>3(1-x)

X 1 2[(1-r)x(x-2) +2(1+r)1 l-5 X(1 -x)(1--r)

log - r

+ (1 -r>(l -x)(1 -2x)

(1 -5)2

(1+(2x2-x+1) 8 - l-5

-- . >

(13) X

Returning to the discussion of the infrared singularity of the quantity dI’!“ms/dx in Eq. ( 11) , which occurs for x -+ 1, i.e., in the experimentally interesting re- gion, we recall that this singularity is cancelled in a distribution sense if one takes into account the virtual QCD corrections to the tree level matrix element of 07 for the two-body process b -+ sy, given in Eqs. ( 3) and (4). Technically, it is useful to define the in- tegrated quantity

1

r7ho) = I[

dry= -+ r;Q(i -xl dn, (14) dx 1 so

150 A. Ali, C. Greub/Physics Letters B 361(1995) 146-154

in which the singularities cancel manifestly. Eq. ( 14) is identical with the one we have derived and presented earlier [4], with the only difference that the r.h.s. in Eq. ( 14) is proportional to CTff instead of CT as in Ref. [ 41. This is a consequence of taking into account all four-Fermi operators. As the end-point spectrum shows sensitivity to the leftover effects of the (can- celled) infrared singularity, we resum the leading (Su- dakov type [ 12 ] ) infrared logarithms to all orders. As all these steps are described in detail in Ref. [ 41, we do not repeat this discussion here.

Before leaving this section, we point out that there is a mismatch between the maximum energy of the photon at the parton level, which for the decay of a b- quark at rest corresponds to y = (mz - rn:> / (2mb) and in the physical threshold in B -+ X, + y, which foilows from Mp = mK + m,. In the model being considered this gap is filled by non-perturbative ef- fects. We also stress that the coefficients of the non- leading logarithmic and non-logarithmic terms in the end-point spectra in the inclusive decay B -+ X, + y and the semileptonic decay B --f X&Q are different. This difference can be manifestly seen if one calcu- lates thex-moments M,(B + X,-t-y) and M,(B + X,lVe ) :

Mslmb 1

M,(B+X,+y) E r s

dl- dxx”-’ dx , (1%

0

The behaviour near the end-point region in n corre- sponds to the large-n limit of the moments M,. They have been worked out in the leading non-trivial order in perturbation theory and the results can be expressed as

M, N 1 + $~(Alog*n+Blogn+const.),

(16)

where CF = 4/3, the leading coefficient is universal with A = - 1 [ 121, and the non-leading coefficients are process dependent; B = 7/2 [ 31 and B = 31/6 [ 131, for B -+ X, + y and B -+ X&e, respectively. The main contribution of this paper is to calculate the

constant and non-leading terms which are important for the bulk of the photon energy spectrum.

In order to implement the B-meson bound state ef- fects on the photon energy spectrum, we continue to use the wave-function model [9] that we have used in our earlier work [3,4]. This model has received renewed interest in the context of heavy quark effec- tive theory [ 141. In this model, which we refer to as the Fermi motion model, the B-meson consists of a b-quark and a spectator q and the four-momenta of the constituents are required to add up to the four- momentum of the B-meson. In the rest frame of the B- meson the b-quark and the spectator fly back-to-back with three momenta pg = -p4 z p. Energy conserva- tion then implies the equation

which can only hold for all values of (pi, if at least one of the masses becomes momentum dependent. We treat the spectator quark m, as a momentum- independent parameter; the b-quark mass is then mo- mentum dependent and we denote it by W(p):

W2(p> = Me* + mq2 - 2M~dp* + m9*. (17)

The b-quark, whose decays determine the dynamics, is given a non-zero momentum having a Gaussian dis- tribution, with the width determined by the parameter

PF:

P = IPI 5 (18)

with the normalization &O” dp p* 4(p) = 1. The pho- ton energy spectrum from the decay of the B-meson at rest is then obtained by convoluting the appropri- ately boosted partonic spectrum with the wave func- tion. The spectrum is described essentially by two pa- rameters, PF, determining the non-perturbative width, and Mb, which determines the peak position.

4. Estimates of BR(B + X, + y) in the SM and the parameters @F, t?zb)

It has become customary to calculate the branching ratio for the inclusive decay B -+ X, + y in terms of the semileptonic decay branching ratio

A. Ali, C. Grab/Physics Lcrtcrs B 361(1995) 146-154 151

JWB -+ XY)

= r(B4y+Xs) *BR(B+Xhq), rS1 1

(19)

where, in the approximation of including the leading- order QCD correction, rsr is given by the expression

rs, = G M2 5 1g2r3 mbdmc/mb)

x (1 -2/3CY"f(mc/mb)). 7T

(20)

The phase space function g( z, ) and the function f( z ) , obtained by one-loop QCD corrections to lYst, can be seen in 191. To get the branching ratio in Eq. (19) one can either take the partonic (purely perturbative) expressions for both r( B + X,y) and rst, or else one can first implement the wave function effects and then integrate the spectra. In the latter case, the dominant wave function effects in the quantity rsl are included if one identifies mb in E!q. (20) with the effective value Wee of the b quark mass, which is the value of the floating b-quark mass averaged over the Gaussian distribution:

where W(p) is given in Rq. (17). We remark that these procedures yield an almost identical branching ratio (within l%), which shows that the Gaussian- distributed Fermi motion model [9] is in agreement with the result that power corrections to the inclusive decay widths r( B --+ X, + y) and I’(B + X&), calculated in the context of the heavy quark effective theory [ 151, cancel out in their ratio.

We now estimate BR( B + X, + y) in the standard model and theoretical uncertainties on this quantity. The parameters that we have used in estimating the inclusive rates for BR(B -+ X, + 7) are summarized in Table 2. The largest theoretical uncertainty stems from the scale dependence of the Wilson coefficients. As given explicitly in the preceding section, the decay rate for B --t X, + y depends on seven of the eight Wilson coefficients given earlier, once one takes into account the bremsstrahlung corrections and is not fac- tored in terms of a single (effective) coefficient, that one encounters for the two-body decays b -+ s + y [5]. To get some insight in the errors we enumerate

Table 2 Values of the parameters used in estimating the branching ra- tio BR(B -+ X, + y) in the standard model. The range of As is taken from the present world average (corresponding to a,(mi) = 0.117 f 0.005, using the two-loop p-function [ 161) and the semileptonic branching ratio from [ 171

Parameter

nZr (GeV)

CL (GeV)

A5 (GeV)

a( B -t X&q)

m&b

mw (GeV) $A

Range

170f 11 5.o+5.O -2.5

o.195+o.WS -0.05 ( 10.4 f 0.4)% 0.29 f 0.02 80.33 130.0

here the values of the two dominant effective coeffi- cients, CT* and Ciff, as one varies ,u, As and m, in

the range given in Table 2: CJff = -0.306 f 0.050

and Cs”’ = -0.146 f 0.020. The present theoretical uncertainties on these coefficients represent the domi- nant contribution to the theoretical error on BR( B + X, + y), contributing about f35%.

The second source of scale-dependence is due to the appearance of the running quark masses in the opera- tors 07 and 0s. As discussed in Section 2, this brings into fore the extra (scale-dependent) multiplicative factor R = (mb(P)/mb,pole )* for the branching ra- tio BR(B -+ X, + y). Intrinsic uncertainties in the concept of the pole mass due to infrared renormalons suggest that one should express all physical results in terms of the running masses [ 181. This requires recal- culating the decay rate BR( B -+ X, + y) with the run- ning masses, incorporating resummations of the kind recently undertaken for the semileptonic B decay rates [ 191. This remains to be done.

The next largest error arises from the parameters which are extrinsic to the decay B --+ X, + y and have crept in due to normalizing the branching ra- tio BR(B -+ X, + y) in terms of the semileptonic branching ratio. To estimate this extrinsic error, we note that the mass difference mb-mc is known through the lfmp expansion [ 141 or from the semileptonic b -+ c spectrum; to a rather high accuracy its value

is mb - m, = 3.40 GeV [ 201. The b-quark mass mb is, however, not known so precisely. Using for the b quark pole mass mb,p& = 4.8 ItO. GeV [21], one gets mc/mb = 0.29 f 0.02, which is consistent with

152 A. Ali, C. Greub/ Physics Letters B 361 (1995) 146-154

, C_____________________________________________J

c 1

01 150 160 170 180 190

mtDp t9evl

Fig. 1. BR( B + X,y) as a function of the m-top quark mass. The three solid lines correspond to the variation of the parameters ( /.L, As) as described in the text. The experimental (5 1 (T) -bounds from CLEO [ I] are shown by the dashed lines.

the determination of the same from the lepton energy spectrum mc/mb = 0.316 f 0.013 [22] but less pre- cise. This leads to f8.1% error on the branching ratio BR( B -+ X, + y). Taking into account the experi- mental error of &4.1% on BR( B + X.hq ) [ 171, and adding these errors linearly, one gets an error off 12% on BR( B + X, + y) from the semileptonic decays.

The procedure that we have adopted in estimating the theoretical uncertainties on BR(B --f X, -t- 7) is as follows: We propagate the errors due to the scale p, the QCD scale parameter, As, and the top quark mass in our calculations. As remarked already, this constitutes the largest error. The extrinsic errors (from rnc/mb and the semileptonic branching ratio), being obviously uncorrelated, are then added in quadrature in quoting the branching ratio.

We now proceed to discuss our results. Assuming

IVrslllvCbl = 1 1161, we plot in Fig. I the branching ratio BR(B -+ X,y) as a function of the top quark mass m,. For all three solid curves, representing the SM-branching ratio, we have used m,/mb = 0.29. The top solid curve is drawn for ,u = 2.5 GeV and A5 = 0.260 GeV. The bottom solid curve is for ,u = 10 GeV and A5 = 0.145 GeV, and the middle solid curve corresponds to the central values of the input param- eters in Table 2. Using the m-top quark mass Zit = ( 170f 11) GeV, and adding the extrinsic error, we get

BR(B 4 X, + y) = (2.55 f 1.28) x 1O-4, (22)

0 I,, , , I I,

0 1 2 3 (IV,l/lV,,l)”

Fig. 2. BR(B -+ XSy) as a function of ([V,,l/lVcbl)*. The solid lines correspond to the variation of the parameters (p, A5, at,) in the limits specified in Table 2. The experimental (f la)-bounds from CLEO [ 11 are shown by the dashed lines.

to be compared with the CLEO measurement BR(B --+ X, + y) = (2.32 f 0.67) x 10m4. The ( * 1 a) -upper and -lower bound from the CLEO mea- surement are shown in Fig. 1 by dashed lines. We see that the agreement between SM and experiment is good. The theoretical errors estimated by us are, however, larger, than for example in [ 51, for reasons that we have explained above.

In Fig. 2 we show the branching ratio BR(B + X, + y) as a function of the CKM matrix element ratio squared ~V,s~/~Vc~~2, varying m,, p and AS in the range specified in Table 2. Using the ( f 1~) -experimental bounds on the branching ratio (dashed lines) we infer

IKJ/]&?l= l.10f0.43, (23)

which is consistent with the indirect constraints from the CKM unitarity [ 161, but imprecise.

Now we discuss the photon energy spectrum and fit of the parameters pF and mq by using the CLEO data which has been corrected for the detector effects. The photon yield from the decay B -+ X, + y is given in Table 3 in photon energy bins having a width of 250 MeV starting with EY = 1.95 GeV. We note that these entries are based on the weighted average of the two different methods (event shape and B reconstruc- tion) used by the CLEO Collaboration in the analysis of their data [ 231. The photon energy has been mea- sured in the laboratory frame (i.e. in the rest frame of Y (4.9 ) and the numbers in Table 3 are presented in this frame. This implies that the B mesons from

A. Ali, C. Greub/ Physics Letters B 361(1995) 146-154 153

Table 3

Photon yield in the laboratory frame from the decay B -+ X, + y.

obtained from the measurement of the photon energy spectrum by the CLEO Collaboration [l] based on a sample of 2.152 million BB events. The data shown have been corrected due to the detector acceptance [ 231

./+interval Number of events

1.95-2.20 GeV 2.20-2.45 GeV 2.45-2.70 GeV 2.70-2.95 GeV

229 f 256 484f 163 381 f 105

12f 59

the Y (45) decay have a momentum of x 350 MeV, and in doing the analysis we have boosted the theo- retical rest-frame spectra accordingly. As the theoret- ical uncertainties are mainly in the normalization of the spectrum, we normalized both the theoretical pre- dictions (parametrized in terms of PF and rnq) and the experimental data to unit area in the interval be- tween 1.95 GeV and 2.95 GeV. We then performed a x2 analysis. The experimental errors are still large and the fits result in relatively small x2 values. We find the following ranges: 0 I m4 I 340 MeV and 260 MeV 5 PF 5 900 MeV, with the two ranges some- what correlated [ 111. The minimum, ~2, = 0.038, is obtained for PF = 450 MeV and mb(pole) = 4.77 GeV, in good agreement with theoretical estimates of the same, namely mb(pole) = 4.8 f 0.15 GeV [2l] and pi = ,&2 = 0.25 ho.05 GeV2 obtained from the QCD sum rules [24]. In Fig. 3 we have plotted the photon energy spectrum normalized to unit area in the interval between 1.95 GeV and 2.95 GeV for the pa- rameters which correspond to the minimum x2 (solid curve) and for another set of parameters that lies near the x2-boundary obtained from the requirement x2 = ~2, + 1. Data from CLEO [ 1 ] are also shown. The photon energy spectrum in B + X, + y has also been investigated recently with a somewhat different non- perturbative (model) and perturbative( QCD) input in [25]. Due to detailed differences in the underlying theoretical frameworks, and in particular in the analy- sis of data, no direct quantitative comparison with this work is attempted here.

In summary, we have presented an improved the- oretical calculation of the branching ratio BR(B -+ X, + y) and the photon energy spectrum in B 4 X, + y, using perturbative QCD and a Gaussian Fermi

_ pF = 450 MeV, mg = 0

3 1 ----- PP = 310 MeV, mq = 300 MeV

E,[GeVl

Fig. 3. Comparison of the normalized photon energy distribution using the corrected CLBO data [ l] and our theoretical distribu- tions, both normalized to unit area in the photon energy interval between 1.95 GeV and 2.95 GeV. The solid curve corresponds to the values with the minimum x2, (m,,p~)=(0,450 MeV), and the dashed curve to the values (m,,p~)=(300 MeV, 310 MeV).

motion model. We estimate BR( B -+ X, + y) = (2.55 f 1.28) x 10e4 in the SM, in agreement with the corresponding branching ratio BR( B -+ X, +y) = (2.32 f 0.67) x 10e4 reported by the CLEO Collabo- ration. Our model calculations provide a good account of the measured photon energy distribution, and the best-fit parameters correspond to PF = 450 MeV and

mbpk = 4.77 GeV. The errors on these parameters are still large but within errors both pi and mb,n& de- termined from the radiative B decays are compatible with the corresponding theoretical estimates using the QCD sum rules [ 2 1,241. Precise comparison requires improved measurements and theory, which we hope are forthcoming.

Acknowledgements

We are very grateful to the members of the CLEO Collaboration, in particular Tomasz Skwarnicki and Ed Thorndike, for providing Table 3 and for numerous helpful discussions. We acknowledge helpful corre- spondence and discussions with Matthias Neubert and Vladimir Braun on quark masses. Discussions with Guido Martinelli, Thomas Mannel, Giulia Ricciardi, Arkady Vainshtein and Daniel Wyler are also thank- fully acknowledged. We also thank Frank Cuypers for providing us with a program to draw the contour plot and for useful general discussions on statistics. One

I54 A. Ali, C. Greub/Physics Letters B 361fI995) 146-154

of us (C.G.) would like to thank the DESY theory group for its hospitality.

Note added in proof

A part of the Wara,) corrections to the photon energy spectrum, called drpms/dx and given in Eq. ( 13), has also been derived recently in [ 261. Our emphasis here is on the large-E, region, where the difference between the summed form of Eq. ( 13), as advocated in [ 261, and our Eq. ( 13) is numerically unimportant. As we treat the s-quark mass non-zero in our calculations, the collinear singularity in the photon energy spectrum in B + X, + y is cut off.

References

[ I ] M.S. Alam et al., (CLEG Collaboration) Phys. Rev. Len. 74 (1995) 2885.

121 R. Ammar et al., (CLEO Collaboration) Phys. Rev. L&t. 71 (I 993) 674.

[3] A. Ali and C. Grub, Z. Phys. C 49 (1991) 431; C 60 (1993) 433; Phys. Lett. B 259 (1991) 182.

[4] A. Ali and C. Greub, Phys. Lett. B 287 ( 1992) 191. [5] A.J. Bums, M. Misiak, M. Miinz and S. Pokorski, Nucl.

Phys. B 424 (1994) 374. 161 M. Ciuchini et al., Phys. Lett. B 334 (1994) 137. 171 M. Ciuchini et al., Phys. Lett. B 316 (1993) 127; Nucl.

Phys. B 41.5 (1994) 403; G. Cella et al., Phys. I&t. B 325 (1994) 227; M. Misiak, Nucl. Phys. B 393 (1993) 23: B 439 (1995) 461 (E).

[ 81 B. Grinstein, R. Springer and M.B. Wise, Phys. Lett. B 202 (1988) 138; Nucl Phys. B 339 (1990) 269.

[91 A. Ali and E. Pietarinen, Nucl. Phys. B 154 (1979) 519; G. Altarelli et al., Nucl. Phys. B 208 ( 1982) 365.

I101 A. Ali, G. Giudice and T. Mannel, Z. Phys. C 67 ( 1995) 417.

I 11 I A. Ali and C. Greub, DESY Report (in preparation), 1121 V. Sudakov, Zh. Eksp. Teor. Fiz. 30 (1956) 87 [Sov. Phys.

JETP 3 (1956) 651. [ 131 M. Jezabek and J.H. Ktlhn, Nucl. Phys. B 320 (1989) 20. 1141 I. Bigi et al., Phys. Rev. Lett. 71 (1993) 496; Int. Joum.

Mod. Phys. A 9 (1994) 2467. 1151 J. Chai, H. Georgi and 8. Grinstein, Phys. Lett. B 247

( 1990) 399; 1. Bigi, N. Uraltsev and A. Vainshtein, Phys. Len. B 293 (1992) 430; B 297 (1993) 477 (E); A. Falk, M. Luke and M. Savage, Phys. Rev. D 49 (1994) 3367.

[16] L. Montanet et al. (Particle Data Group), Phys. Rev. D 50 (1994) 1173.

[ 171 L. Gibbons (CLEO Collaboration), in: Proceedings of the XXX Rencontres de Moriond. Les Arcs, March 1994.

[ 181 I. Bigi et al., Phys. Rev. D 50 (1994) 2234; M. Beneke and V.M. Braun, Nucl. Phys. B 426 ( 1994) 301.

[ 191 I? Ball, M. Beneke and V.M. Braun, Preprint CERN-TH/95- 65 [ hep_ph/9503492].

[ZO] M. Shifman, N.G. Uraltsev and A. Vainshtein, Phys. Rev. D 51 (1995) 2217; M.B. Voloshin, TPI-MINN-94138-T (hep-ph/9411296).

1211 E. Bagan. I? Ball, V.M. Braun and H.G. Dosch, Phys. L&t. B 278 (1992) 457; M. Neubert, Phys. Rep. 245 (1994) 259.

[22] R. Riickl, MPI-Ph/36/89. [23] E.H. Thomdike and T. Skwarnicki (private communication). [ 241 I? Ball and V. Braun, Phys. Rev. D 49 (1994) 2472;

V.M. Braun (private communication). [25] R.D. Dikeman, M. Shifman and R.G. Uraltsev, Preprint TPI-

MINN-95/9-T, UMN-TH-1339-95, UND-HEP-95.BIG05 (hep-ph/9505397).

[26] A. Kapustin, Z. Ligeti and H.D. Politzer, Preprint CALT- 68-2009 (hep-ph19507248).