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Weak radiative B-meson decay beyond leading logarithms
8 May 1997
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 400 (1997) 206-2 19
Weak radiative B-meson decay beyond leading logarithms Konstantin
Chetyrkin a,1,2,Mikolaj Misiak bv3y4,Manfred Miinz cs
a Max-Planck-lnstitutf
Physik, Werner-Heisenberg-ltitut, D-80805 Miinchen, Germany b Ins&t fiir Theoretische Physik der Universitiit Ziirich, CH-SO57 Zfirich, Switzerland ’ Physik Department, Technische Universitiit Miinchen, D-85748 Garching, Germany
Received 7 February 1997 Editor: PV. Landshoff
Abstract We present our results for order in QCD. We combine leading order prediction for smaller than in the previously with the CLEO measurement
three-loop anomalous dimensions necessary in analyzing B--+X,y decay at the next-to-leading them with other recently calculated contributions, obtaining a practically complete next-tothe branching ratio B( B-+X,y)= (3.28 -+ 0.33) x 10e4. The uncertainty is more than twice available leading order theoretical result. The Standard Model prediction remains in agreement at the 2a level. @ 1997 Elsevier Science B.V.
1. Weak radiative B-meson decay is known to be a very sensitive probe of new physics [ 1,2]. Heavy Quark Effective Theory tells us that inclusive B-meson decay rate into charmless hadrons and the photon is well approximated by the corresponding partonic decay rate I( B-+X,y) -T(b+X,y). The accuracy of this approximation is expected to be better than 10% [ 31. The inclusive branching ratio B(b-+X,y) was extracted two years ago from CLEO measurements radiative B-meson decay. It amounts to [4] a(b+X,y)
(1) of weak
= (2.32 f 0.57 f 0.35) x 10-4.
(2)
In the forthcoming few years, much more precise measurements of B( B+X,y) areexpected from the upgraded CLEO detector, as well as from the B-factories at SLAC and KEK. Acquiring experimental accuracy of below 10% is conceivable. Thus, the goal on the theoretical side is to reach the same accuracy in perturbative ’ This work was partially supported by INTAS under Contract INTAS-93-0744. 2 Supported in part by Schweizerischer Nationalfonds and Polish Commitee for Scientific Research
(grant 2 PO3B 180 09, 199597). 3 Supported in part by Deutsches Bundesministerium ftir Bildung und Forschung under contract 06 TM 743. 4 Permanent address: Institute of Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia. ’ Permanent address: Institute of Theoretical Physics, Warsaw University, Hoia 69, 00-681 Warsaw, Poland. 0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI1 SO370-2693(97>00324-9
K. Chetyrkin et al./Physics
Letters B 400 (1997) 206-219
207
calculations of b--+X,y decay rate. This requires performing a complete calculation at the next-to-leading order (NLO) in QCD [ 1,5]. All the next-to-leading contributions to b-+X,y are collected for the first time in the present letter. 6 One of the most complex ingredients of the NLO calculation consists in finding three-loop anomalous dimensions in the effective theory used for resummation of large logarithms ln( M&/m;). This is the main new result presented in this letter. However, we have recalculated all the previously found anomalous dimensions, too. The remaining ingredients of the NLO calculation are already present in the literature [ 6,8-171. We use the published results for the two-loop matrix elements calculated by Greub, Hurth and Wyler [ 91, Bremsstrahlung corrections obtained by Ali, Greub [ 15,161 and Pott [ 171, and the matching conditions found by Adel and Yao [131. The following three sections of the present paper are devoted to presenting the complete NLO formulae for b+X,y. Next, we analyze them numerically. Finally, we discuss predictions for the branching ratio B( B-+X,y) including a discussion of the relevant uncertainties. 2. The analysis
of b+X,y
decay begins with introducing
an effective hamiltonian
(3) where x,j are elements of the CKM matrix, Pi (,u) are the relevant operators and Ci( ,u) are the corresponding Wilson coefficients. The complete set of physical operators necessary for b--+X,y decay is the following:
(4) where T” stand for SU(3) cotor generators. The small CKM matrix element l&, as well as the s-quark mass are neglected in the present paper. Our basis of four-quark operators is somewhat different than the standard one used in Refs. [ 1,6,18], although the two bases are physically equivalent. We would like to stress that, in our basis, Dirac traces containing ys do not arise in effective theory calculations performed at the leading order in the Fermi coupling GF (but to all orders in QCD). This allows us to consistently use a fully anticommuting ys in dimensional regularization, which greatly simplifies multiloop calculations. Our choice of color structures is dictated only by convenience in computer algebra applications. Throughout the paper, we use the MS scheme with fully anticommuting ys. Such a scheme is not uniquely defined until one chooses a specific form for the so-called evanescent operators which algebraically vanish in four dimensions [ 19,201. In our three-loop calculation, eight evanescent operators were necessary. We list them explicitly in Appendix A. Resummation of large logarithms ln( M$/m$ is achieved by evolving the coefficients Ci ( p) from ,u = Mw to ,U = ,Ub N mb according to the Renormalization Group Equations (RGE) . Instead of the original coefficients Ci( ,u) , it is convenient to use certain linear combinations of them, called the “effective coefficients” [ 1,9] 6 Except for the unknown
but negligible
two-loop
matrix elements
of the penguin operators
which we denote further by 4,...,P6
K. Chetyrkinetal./Physics
208
LenersB400(1997)206-219
for i= 1,...,6
G(P) 6 C7(p)
qqpu>
for i = 7
+ CyiCi(p)
=
(5)
i=l
6
for i= 8.
Cg(LL)+C&i(p) i=l
The numbers yi and zi are defined so that the leading-order b + sy and b --t s gluon matrix elements of the effective hamiltonian are proportional to the leading-order terms in CTff and Ciff, respectively [ 11. In the MS scheme with fully anticommuting ys, we have y = (O,O,-i,-$,-?,--y) and z = (O,O, 1,-i,20,-y).7 The leading-order contributions to the effective coefficients Cp” ( ,w) are regularizationand renormalizationscheme independent, which would not be true for the original coefficients C7 ( ,u) and Cs (/A). The effective coefficients evolve according to their RGE
(6) driven by the anomalous
a,(P) v%-d = FY
matrix yff (p) . One expands this matrix perturbatively
dimension
n(O)&
as follows:
a264 1)eff+ ... LfJ
+
(7)
(471.)*
independent, while y(‘jeff is not. In the MS scheme The matrix F(Ojeff is renormalization-scheme anticommuting ys (and with our choice of evanescent operators) we obtain --4; 12 0
Y *(Oferr=
0
-$
0
0
-E
0
$
0
0
416 s, -+$5 _E
0 00 0 0 _$
-q _y
0 $
2 2
0
-y
0
20 -F
0
0
00+
$
00
0
000
00
0
0
$L$
g ZT! 27 % _B$
(8)
5Z$j
E 22 s
00-y
with fully
F 0 28 s
and
7 These numbers
are different from those in Section 4 of Ref. [l]
because we use a different basis of four-quark
operators
here.
Letters B 400 (1997) 206 -21 9
K. Chetyrkin et al./Physics -355
9
-$?$
-u&
_g?
134 Tm
!280 81
c(l)eff
Y
_
-
0
0
_y
_a&%9
0
0
-a&!
59399 243
g.J
-g
-%
56 81
35 21
m 81
1841 108
glJ 8,
3313 -RR
22348 243
10178 81
%-z
12899 648
1;;;
172471 648
1183696 129
0 0
0
58640 243
26348 243
1;:32”
0
0
0
0
0
0
10
0
0
0
0
0
209
2&55 2:;;;
(9)
2901296 243 3296251 129
0
4688 21
4063
-a$
21
The matrix T(‘jeff is the main new result of this paper. Its evaluation required performing three-loop renormalization of the effective theory (3). Details of this calculation will be given in a forthcoming publication
r211. We expand the coefficients
in powers of ay, as follows:
as(P) c,:ff(,> = Ci’O’“QL)+ TCi
(1)eff
(ILL) +
(10)
...
At p = Mw, the values of the coefficients are found by matching the effective theory amplitudes Standard Model ones. The well-known leading-order results read [ 16,221
c!“‘“ff(Mw) 1
= C$O’(Mw)
0 1 3x3 - 2x2 4(x _ 1>4
=
-3x2 4(x - 1)4
with the full
for i= 1,3,4,5,6 for i = 2 lnx+
lnx+
--8x3 - 5x2 + 7x 24(x - 1)3 -x3 + 5x2 + 2x 8(x - 1)s
for i = 7
(11) for i=8,
where
(12) The next-to-leading contributions to the four-quark operator coefficients at ,u = Mw can be found either by transforming the results of Ref. [lo] to our operator basis, or by a direct computation. We find for i = 1 for i=2,3,5,6 fori=4,
15 0 E(x)--:
C!l)eff( Mw) = Ci”’ ( Mw) = 1
(13)
where E(x)
=
x(18-11x-x2) 12( 1 - x)3
+
The next-to-leading contributions the following equations: C(l)eff(Mw) 7
= -$&Ch;;
x2(15-
16x+4x2)
6(1 -x)~ to CTff(Mw)
+ A7 - $ (E(x)
lnx - $lnx.
and Ciff (Mw)
- $1,
can be extracted
(14) from Ref. [ 131 according
to
(15)
210
K. Chetyrkin et d/Physics
C(l)eff(Mw) 8
= 8-3:;
+A,
- i (E(x)
Letters B 400 (1997) 206-219
(16)
- 5))
s
where the quantities CA:,) and Chtj are given in Eq. (5) of Ref. [ 131 with p = Mw. The terms denoted by A7 and As stand for shifts one needs to make when passing from the so-called R* renormalization scheme used in Ref. [ 131 to the MS scheme used here Ai =
2
[c,'O'(M~)
(x -+ x - ?xfnx)
- c/o)(M~)
(x)]
for i=7,8.
(17)
Only the leading (zeroth-order) term in LY,needs to be retained in the above equation. The resulting explicit expressions for CFff(Mw) and Ciff (Mw) in the MS scheme read C(l)eff(Mw) I
=
-102x5
-16x4
-9’:‘” x
- 8xLi2 cl _ ;)
+ 6x4 + 46x3 - 28x2 ln2 x 3(X - 1)s
- 588x4 - 2262x3 + 3244x2 - 1364x + 208 lnx
+
81(x - 1)5 1646x4 + 12205x3 - 10740x2 + 2509x - 436
+
(18)
486(x - 1)4
‘8 (ly
Mw)
-17x3
=
- 31x2
2(x - 1)s -210x5
+ +
:,0x’
ln2 x
+ 1086x4 + 4893x3 + 2857x2 - 1994x + 280 lnx 216(x - 1)5
737x4 - 14102x3 -28209x2
+ 610x - 508 (19)
1296(x - 1)4
Having specified the initial conditions Group equations (6) for CTff( ,LQ)
at ,U = Mw, we are ready to write the solution
to the Renormalization
(Mw)
+(p
297664 14283 77 fi *’ -
~357075 q 14 7164416 a + 8
(21) i=l
where ~7= a,( Mw)/cu,(pb), ai = ( ei = (
and the numbers
14 z> 4661194 816831 ’
fi = ( -17.3023, gi = (
14.8088,
hi = (
626126’ 212277
16 TiT3 -g,
6 z?T
ai-hi -g
are as follows: ?
0.4086,
-0.4230,
-0.8994,
0.1456)
-1.9043,
-0.1008,
0.1216,
0.0183)
0.7476,
-0.5385,
0.0914)
0,
0,
8.5027,
4.5508,
0.7519,
2.0040,
-10.8090,
-0.8740,
0.4218,
-2.9347,
0.3971,
0.1600,
0.0225)
-iz 1 >
-0.6494,
-0.0380,
-0.0186,
-0.0057).
5628 1 51730’
-k
K. Chetyrkin et al./Physics
AS far as the other Wilson coefficients are concerned, only the leading-order necessary in the complete NLO analysis of b---+X, y. We obtain 8
C(O)&) 1
=
+ii
@(pb)
=
$3
C,(O’(/&)
=
_Ly-u 21
C’O’(/Ub) = 4
+
23
-# 477
-0.0659~“~4086 + 0.059517-“.4230 - 0.0218~~-~.~~~~ + 0.0335.170.‘456,
&+
+0.023777a.4as6 _ 0.017371-a.42so _ 0.133671-o.s994 _ 0.0316$.‘456,
$$
_ 1126r)g
+0.009477a~40s6 - o.0100~-a~4230 + o.oo10~-a~*994 - 0.0017770.‘456,
- &$
+0.010877°~4086 + 0.0163r]-“.4230 + 0.010377-“.8994 + 0.0023r10.‘4s6,
C,(O)(,ub) = -$7-g
-
= (CS’O’ ( Mw)
+ g$!)
$
0.913577°.4086 + 0.0873~7-‘.~~~’ - 0.057177-“.8994 + 0.0209q”.‘456.
In our numerical analysis, the values of aS ( ,u) in all the above formulae expression for the strong coupling constant a,(p)
to them are
g$i,
+
+
108
(O)eff( /I&)
contributions
+$3,
C~o’(~ub) = r,-g
‘8
211
Letters B 400 (1997) 206-219
=-
a,(Mz)
1_
V(P)
fic+(M~) -___
PO
4%-
InO)
V(P)
(22) are calculated
with use of the NLG
1
(23)
’
where (24)
As easily can be seen from Eqs. ( lo)-(24), the Wilson coefficients at J.L = pb depend on only five parameters: Mz, Mw, n,( Mz), ?r~,,~,,i~and pb. In our numerical analysis, the first two of them are fixed to Mz = 9 1.187 GeV and Mw = 80.33 GeV [ 231. For the remaining three, we take LYE
=0.118&0.003
mr,rote = 175 i 6 GeV
[24],
(25)
[251
(26)
and ,UI, E [2.5 GeV, 10 GeV].
(27)
For ,Ub = 5 GeV and central values of the other parameters, c’“‘eff(&
= (-0.480,
C(l)eff(,Ub) I
= -0.995
1.023, -0.0045,
-0.0640,
we find
0.0004,
LY,(,U(LD) = 0.212,
0.0009,
-0.312,
-0.148),
+ 1.443 = 0.448.
In the intermediate step of the latter equation, we have shown the C’l)eff(pb). The first of them originates from terms proportional to i.d. it is due to the NLO matching. The second of them comes from next-to-leading anomalous dimensions. Quite accidentally, the two
(28) relative importance of two contributions to C$lfeff( Mw) and Cs(l’eff( Mw) in Eq. (21)) the remaining terms, i.e. it is driven by the terms tend to cancel each other. In effect,
8 Analogous expressions in Ref. [ 11 are somewhat different, because we use a different basis of four-quark operators here.
212
K. Chetyrkin et d/Physics
Letters B 400 (1997) 204-219
the total influence of C:‘)eff(pb) on the b+X,y decay rate does not exceed 6%. (Explicit expressions for the decay rate are given below.) However, if the relative sign between the two contributions was positive, then the decay rate would be affected by almost 30%. This observation illustrates the importance of explicit calculation of both the NLO matching conditions and the NLO anomalous dimensions. 3. At this point, we are ready to use the calculated coefficients at fl = ,Ub 21 mb as an input in the NLO expressions for b+X,y decay rate given by Greub, Hurth and Wyler 191. However, we make an explicit lower cut on the photon energy in the Bremsstrahlung correction E, > (1 -,)E~
= (1 -6)F.
(29)
We write the decay rate as follows: I[b+&y]E+(l-S)E~-
= T[b + sy]
+ T[b + sy gluon]E~>(‘-“)E~~ (30)
where %tpb)
D = C,(OJeff(pb) + 7
C:‘)eff(pul,) + ~~~“‘eff(pb)
[ri +yj,O’eff~n~
(31)
2
(32)
i=l
and A = (e- cys(/.u,)lnS(7+21nS)/3~
_
1) ]C$“)eff(,&,)12
+
+ i,
j=l ,
Ci’“‘eff(~b)Clo’eff(~b)fii(S). il.1
D with respect The contribution denoted by 1D 12 is independent of the cutoff parameter 6. By differentiating to ,Ub and using the RGE (6)) one can easily find out that the leading fib-dependence cancels out in D. When calculating 1D12 in our numerical analysis, we consistently set the 0( (Y:) term to zero. The contribution denoted by A vanishes in the limit cu,-+O. The first term in A contains the exponentiated (infrared) logarithms of 6 which remain after the IR divergences cancel between the virtual and Bremsstrahlung corrections to b-+X, y. The terms denoted by ri in D depend on what convention is chosen for additive constants in the functions Jln (S) and f7s (8). We fix this convention by requiring that all fij (6) vanish in the formal limit 6 -+ 0. In order to complete the full presentation of b-+X,y results, it remains to give explicitly the constants ri and the functions fi,i (6). The constants rs and r-2 are exactly as given in Ref. [9] rs=-&(-33+2??-6ir), i-2 = & (-833
-I- 14422
(33) 3’2 + [1728 - ISOd
+ [648 + 72~~ + (432 - 216r2)L I%{-5+[45-3m2+9L+9L2]
- 1296&(3) + (1296 - 324r2)L
+ 36L3] z2 + [-54 z+[-37i)+9L2]
- 842
r1 = -Igrz.
+ 1092L - 756L2] z3}
z2+[28-12LJ
where z = mz/rng and L = In z . In our operator basis (4), the constant to r-2
+ 108L2 + 36L3] z
z3}+U(z4L4),
(34)
t-1 does not vanish, but is proportional
(35)
K. Chetyrkin et d/Physics
The constant
213
Letters B 400 (1997) 206-219
~7 turns out to be9 8 2
y7=+-gr,
(36)
The quantities ~3, . . . , t-6 originate from two-loop matrix elements of the penguin operators Ps , . . . , Pe. They remain the only still unknown elements of the formally complete NLO analysis of b-+X,y. However, the Wilson coefficients Ci are very small for i = 3,. . . ,6. In consequence, setting ~3,. . ., t-6 to zero (which we do in our numerical analysis) has a negligible effect, i.e. it can potentially affect the decay rate by only about 1%. Let us now turn to the functions fi.i (8). One can easily obtain f77( 8) from Eqs. (23) and (24) of Ref. [ 161. (These equations were also the source for our ~-7and the exponent in Eq. (32) above.) =~6+~62-~~3+~~(~-4)ln6.
f77(6)
The function fss(6)
(37)
fss( S) can be found by integrating
Eq. ( 18) of the same paper [ 161
-21nmom, [62 +26+2ln(l-S)]
=-& C
.
+2Li2(1-S)-~-~(1+2S)ln~+81n(l-S)-$3+362+7S
(38)
I
The logarithm of m, in the above equation is the only point in our analysis at which the s-quark mass cannot be neglected. Collinear divergences which arise in the massless s-quark limit have been discussed in Ref. [ 261. The properly resumed photon spectrum was there found to be finite in the m, + 0 limit, and suppressed with respect to the case when ln(mb/m,) in fss(b) is used. Even with m, as small as 0.1 GeV, the term proportional to In(m,,/m,) is negligible, i.e. it affects the decay rate by less than 1%. Therefore, for simplicity, we will just use the naive expression (38) with mb/m, = 50 in our numerical analysis. The functions f22( a), f27( 8)) f28 (8) and f7s (6) can be found from appendix B of Ref. [ 91. After performing some of the phase-space integrations, one finds (l-@/z
162 = 27
f22(@
J’
6
dr(l-Zt)lF
+z I’
6
/
l/z dt
Re(G(t)+i)+
0 f28(&
= -f
j-78(8)
=i
=
[Li2(1--6)
-~712--15ln6+!+S-
,
$2+1+3],
(39)
and
C
-2 arctan dm, -n2/2+21n2[(fi+Jt-Zi)/2]-2i7rln[(fi+Jt-;E)/2],
= Q-22(&,
f12(@
for t < 4, fort24.
ftj (S) do not vanish, but are proportional
= -$22@),
g It differs from the one given in Ref. [9] where a formal limit 8 r7.
dt(l-tt)Re(G(t)+k)
(l-@/z
In our operator basis (4), the functions fll(@
/
f27(&,
where, as before, z = rnz/mE G(t)
dr(1-zt)‘~~+;2
(1---6)/z
(l-W/z = -$
/
+
0
f27(6)
, 11 1
l/z
f17c8)
to the functions
(40) fgj (S)
= -$27(&
1 was taken and the contribution
due to
f77( 1) was absorbed into
214
K. Chetyrkin et d/Physics
Letters B 400 (1997) 206-219
The only functions fi,i( 6) which have not yet been given explicitly are the ones with at least one index corresponding to the penguin operators Ps, .... Ps. As checked in Ref. [ 171, these functions cannot affect the decay rate by more than 2%. Therefore, we neglect them in our numerical analysis. However, for completeness, a formula from which they can be obtained is given in our Appendix B. 4. The decay rate given in Eq. (30) suffers from large uncertainties due to F&,~,~ and the CKM angles, One can get rid of them by normalizing b+X,y decay rate to the semileptonic decay rate of the b-quark
r[b -+ &e~,l =
G2F m5 b,poleK(z)
Iv,b12,
(41)
192T3gtz)
where g(z)
= 1 - 82 + 8z3 - z4 - 12z2 In z
is the phase-space
h(z) &T(Z)
is a sizable next-to-leading analytically in Ref. [28] ,qz)
= -(I
(42)
factor, and 2%(mb) 3T
tc(z)=l-
) (z = 2;:;:)
-
(43)
QCD correction
z*) (F -
to the semileptonic
yz + Fz2) + z lnz
decay [ 271. The function
(20 + 9Oz - $z2 + yz3)
+(l-z2)(~-~z+~z2)ln(l-z)-4(l+30z2+z4)lnz +z4)[6Li2(z)
-Q?]
-32~~/~(1+z) quantity
+ z21n2 z (36 + z2>
ln(l-z)-(1+16z2 d-4I&(&j-l-4&(-~‘?) [
Thus, the final perturbative
h( z ) has been given
we consider
-21nz
ln
(&$>I.
is the ratio (44)
where D and A are given in Eqs. (3 1) and (32)) respectively, 1 F= _ K(Z)
““~,=~))‘=_J_(l-~~).
and (45)
(
5. Let us now turn to the numerical results. Besides the five parameters listed above Eq. (25)) the quantity I&k(8) in Eq. (44) depends on a few more SM parameters. They are the following: (i) The ratio mc,pole/fnb.pole~ Similarly to Ref. [ 91, we will use !!&Z!Z = 0.29 i 0.02, ~~/J,pOle (ii)
(46)
which is obtained from m&p& 7 4.8 f 0.15 GeV and m&p& - mc,pole = 3.40 GeV. The b-quark mass itself. Apart ‘from the ratio m,/mb, the b-quark mass enters our formulae only in two places: in the explicit logarithm in D (31) and in the factor F (45)) as the argument of as. In both cases, the mb-dependent terms are next-to-leading, and their mb-dependence is logarithmic. Changing mb from 4.6 GeV to 5 GeV in these places affects the b+X,y decay rate by less than 2%. Thus, it does not really matter whether one uses the pole mass or the MS mass there. We choose to use mb,pole = 4.8 f 0.15 GeV.
K. Chetyrkin et al./Physics
Letters B 400 (1997) 206-219
215
3.1 2.9
2.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 1. Rqua,k( 8) as a function
6
(iii)
(iv)
of 6.
The electromagnetic coupling constant CZ~~.It is not a priori known whether this constant should be renormalized at /.L - mb or /_LN Mw, because the QED corrections have not been included. We allow CX,, to vary between (Y,, (mb) and my,, (Mw), i.e. we take LY;~ = 130.3 f 2.3. The ratio of CKh4 factors. We will use
I I GKb
-
Vcb
= 0.976 f 0.010
(47)
given in Ref. 1291. It corresponds to gaussian error analysis, which is in accordance with interpreting the quoted error as a la uncertainty. The dependence of Rquark(8) on S is shown in Fig. 1. The middle curve is the central value. The uncertainty of the prediction is described by the shaded region. It has been found by adding in squares all the parametric uncertainties mentioned above in points (i)-(iv) and in Eqs. (25)-( 27). The ratio $&,.k(@ is divergent in the limit 6 -+l, which is due to the function fss (8). In this limit, the physical quantity to consider would be only the sum of b -+X,y and b-+X, decay rates, in which this divergence would cancel out. The divergence at 6+1 is very slow. It manifests itself only by a small kink in Fig. 1 where values of 6 up to 0.999 were included. For 6 = 0.99, the contribution from fss(8) to &u&(S) is we11 below 1%. In order to allow easy comparison with previous experimental and theoretical publications, we choose P” = 0.99 as the first particular value of 6 to consider. This value of 6 will be assumed in discussing the “total” B-XJ decay rate below. A value of 6 which is closer to what is actually measured is 6 = z = (m,/I?‘@)2. It corresponds to counting only the photons with energies above the charm production threshold in the b-quark rest frame. This is the second value of 6 we will consider below. We find R quark(8max) = (3.12i0.29)
&,,,k(6
x 10-3,
(48)
= Z) = (2.86 f 0.24) x 10-3.
In both cases, the uncertainty is below 10%. The dominant importance of various uncertainties is shown in Table 1.
(49) sources
of it are m,/i’?‘& and ,Ub. The relative
216
K. Chetyrkin et al./Physic.s
Table 1 Unceminties
Letters B 400 (1997) 206-219
in Rquark (6) due to various sources Source &(MZ)
+.po1e
/-Q
‘QJOk
2.5% 2.2%
1.7% 1.7%
6.6% 3.3%
5.2% 6.6%
/i”‘b.pole
‘nb,pole
%m
CKM angles
0.5% 0.6%
1.9% 1.9%
2.1% 2.1%
Let us note that setting 6 = Z, introduces an additional spurious inaccuracy due to m,/mb in &,,.k. However, this additional uncertainty is significantly smaller than the original inaccuracy due to m,/mb for S uncorrelated with m,/mb. It is quite surprising that the pb-dependence is weaker for 6 = z than for 8 = P”. Naively, one would expect larger pb-dependence for small values of 6, when the first term in Eq. (32) becomes more and more important. Clearly, 6 = z is not small enough for this effect to show up. The weaker pb-dependence for 6 = z seems to be quite accidental.
6. In the end, we need to pass from the calculated b-quark decay rates to the B-meson on the Heavy Quark Effective Theory (HQET) calculations we write
decay rates. Relying
(50) where 8:’ and 8:: p arametrize nonperturbative corrections to the semileptonic and radiative B-meson rates, respectively. Following Refs. 13,301, we express 8:’ and 8:: in terms of the HQET parameters Al and AZ:
decay
(51) (52) where g(z) is the same as in Eq. (42). The poorly known parameter A1 cancels out in the r.h.s. of Eq. (50) which depends only on the difference 8:’ - 8::. The value of A2 is known from B*-B mass splitting A2 = i(rni,
- rni)
N 0.12 GeV2.
(53)
The two nonperturbative corrections in Eq. (50) are both around 4% in magnitude, and they tend to cancel each other. In effect, they sum up to only around 1%. Such a small number has to be taken with caution, because the four-quark operators 2’1, . . . , P6 have not been included in the calculation of 8::. Contributions from these operators could potentially give one- or two-percent effects. Nevertheless, it seems reasonable to conclude that the total nonperturbative correction to Eq. (50) is well below lo%, i.e. it is smaller than the inaccuracy of the perturbative calculation of &u&( am”). When the above caution is ignored and 13(B-+XceGc) = (10.4 f 0.4)% [23] is used, one obtains the following numerical prediction for B-tX,y branching ratio B(B--+X,y)
= (3.28 f 0.33)
x 10-4.
(54)
K. Chetyrkin etal./Physics LettersB 400 (1997) 206-219
217
The central value of this prediction is outside the la experimental error bar in Eq. (2). lo However, the experimental and theoretical error bars practically touch each other. Therefore, we conclude that the present B~x,y measurement remains in agreement with the Standard Model. This conclusion holds in spite of that the theoretical uncertainty is now more than twice smaller than in the previously available leading order prediction [ 1 I. In the measurement of B+X,y branching fraction, one needs to choose a certain lower bound on photon energies. Instead of talking about the “total” decay rate, it is convenient to count only the photons with energies above the charm production threshold in the B-meson rest frame
mi--m2, N 2.31 GeV.
Ey>
(55)
2m B
Most of the photons in B-X,y survive this energy cut, and a huge background from charm production is removed. Let us denote the corresponding branching fraction by i?( B-+Xsy)above. If the b-quark was infinitely heavy, it would not move inside the B-meson. The restriction (55) would then be equivalent to setting 6 = z at the quark level. Since the Fermi motion of the b-quark inside the B-meson is a O(&/mb) effect, we can write similarly to Eq. (50)
a(B~X,y)"boYe=B(B~X,ev) Rquark(8=z) (l+O(&/mb)).
(56)
The 0 (l?/mb) part of the latter equation can be calculated using specific models for the Fermi motion of the b-quark inside the B-meson, as it has been done, e.g.. in Ref. [ 161 (see also Ref. [31] ). Performing a similar calculation with use of the NLO values of the Wilson coefficients is beyond the scope of the present paper. Such an analysis will be necessary when more statistics allows to reduce experimental errors in measurements of a( B+Xsy)abo"e. 7.To conclude, we have presented practically complete NLO formulae for b-+X,y decay in the Standard Model. They include previously published contributions as well as our new results for three-loop anomalous dimensions. Our prediction for the B --+X,y branching fraction in the Standard Model is (3.28 f 0.33) x lop4 which remains in agreement with the CLEO measurement at the 20 level. Clearly, an interesting test of the SM will be provided when more precise experimental data are available. More importantly, the B+X,y mode will have more exclusion power for extensions of the SM. Appendix A. Here, we give the eight evanescent operators we have used in our anomalous dimension computation. Giving them explicitly is necessary in order to fully specify our renormalization scheme, and thus give meaning to the anomalous dimension matrix presented in Eq. (9),
PII = ~~~Y~,Y~~Y~~~*~L)(~LY~'~~*~~~~=~L) - I@'1 P12 = (~LY,,Y,,Y,,~L)(~L~~~Y~~~~~~~L.) - 16f’2 P15 = (SLY~,Y~LZY~~Y~~Y~CL.F~L) ~GY~'Y'~Y~~Y~~Y'W
- 208
+ 64P3
4 p16 = (SLy~,y~~y~3y~~y~LgTflbL)
C(yy'L'y"y'3y~4y'ST"q)
- 2OP6
+64P4
rl P21 =
(~LY,,Y,,Y,,Y,Y,,T~~L)(~LY~~Y~~Y~~Y~~Y~~T~~L) -2091-2569
loWe identify the b-quark identification acceptable.
decay rate given by CLEO with the B-meson
decay rate. Large statistical
errors in the CLEO result make this
218
K. Chetyrkin et al./ Physics Letters B 400 (19971206-219 P22 = ~~~~,,y,,y,,y,,y,,T~~)(~~y~‘y~~y~~y~~y~~b~)
- 2092 - 256P2
P25 = (SLY I4 Y /J2 Y IQ.Y w y P5 y I4 y P7 bL) ~(4~“‘y”~y~~y~~y~~y~y~~~)
- 336P5 + l28OP3
4
P26 = (SLY /4y p2y P3 y P4 y P _y M y I.0 T”bL) C(q~~‘y~‘yll~y~y~~y~y~~T’q)
- 336P6 + 1280P4.
(A.11
4
Appendix B. Here, we express the functions f~ (S) present in Eq. 32) in terms of the quantities given in Eq. (27) of Ref. [ 171. For each (ij) # (77), fij( 6) is given by
fij(S)
=:
& k,l=l,
1
Wik
dt Mkl (tv U>
k
Wjl,
Mkl( t, u)
tB.1)
where
-2-i 1 0 0 0
0 0 0 1 0 0 0 0 1 0 1 0 _11_1100
0 0 0 0 00 -00 In four space-time
Pi
000 000 000
03.2)
1666 4Yoo -$S-$200 000010 000001 dimensions,
‘2 f: wikdk.
the matrix L? transforms
our operator basis (4) into the one used in Ref. [ 171
03.3)
k=l References [I] A.J. Bums, M. Misiak, M. Mtinz and S. Pokorski, Nucl. Phys. B 424 (1994) 374. [2] J. Hewett, Top ten models constrained by h -+ sy, published in Proceedings Spin structure in high energy processes, p. 463, Stanford, 1993. [3] A.F. Falk, M. Luke and M. Savage, Phys. Rev. D 49 (1994) 3367. [4] M.S. Alam et al., Phys. Rev. Lett. 74 (1995) 2885. [ 51 A. Ah and C. Greub Phys. Lett. B 293 (1992) 226. [6] B. Grinstein, R. Springer and M.B. Wise, Nucl. Phys. B 339 (1990) 269. [7] R. Grigjanis, P.J. O’Donnell, M. Sutherland and H. Navelet, Phys. Lett. B 213 (1999) 355; B 286 (1992) 413 (E). [8] M. Ciuchini, E. France, G. Martinelli, L. Reina and L. Silvestrini, Phys. L&t. B 316 (1993) 127; M. Ciuchini, E. France, L. Reina and L. Silvestrini, Nucl. Phys. B 421 (1994) 41. [9] C. Greub, T. Hurth and D. Wyler, Phys. Lett. B 380 (1996) 385; Phys. Rev. D 54 (1996) 3350. [lo] A.J. Buras, M. Jamin, M.E. Lautenbacher and PH. Weisz, Nucl. Phys. B 370 (1992) 69; B 375 (1992) 501 (addendum). [ 1 l] M. Ciuchini, E. France, G. Martinelli and L. Reina, Nucl. Phys. B 415 (1994) 403. 1121 M. Misiak and M. Mhnz, Phys. Lett. B 344 (1995) 308. [ 131 K. Adel and Y.P Yao, Phys. Rev. D 49 ( 1994) 4945. [ 141 P Cho, B. Grinstein, Nucl. Phys. B 365 (1991) 279. 1151 A. Ali and C. Greub, 2. Phys. C 49 (1991) 431; Phys. Lett. B 259 (1991) 182. [ 161 A. Ah and C. Greub, Phys. Len. B 361 (1995) 146. [ 171 N. Pott, Phys. Rev. D 54 (1996) 938.
K. Chetyrkin et al./Physics [ 18 ] 1191 [ 20 ] [21] [22] [23] [24] [25] [26] [27 ] [28] [ 291 [30] 13 I ]
Letters B 400 (1997) 206-219
F.J. Gilman and M.B. Wise Phys. Rev. D 20 (1979) 2392. M.J. Dugan and B. Grinstein, Phys. Lett. B 256 (1991) 239. S. Herrlich and U. Nierste, Nucl. Phys. B 455 (1995) 39. K. Chetyrkin, M. Misiak and M. M&z, in preparation. T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297, 1772 (E). Particle Data Group, Phys. Rev. D 54 ( 1996) 1. M. Schmelling, plenary talk given at the ICHEP96 conference, Warsaw, July 1996. P. Tipton, plenary talk given at the ICHEP96 conference, Warsaw, July 1996. A. Kapustin, Z. Ligeti and H.D. Politzer, Phys. Lett. B 357 (1995) 653. N. Cabibbo and L. Maiani, Phys. Len. B 79 (1978) 109. Y. Nir, Phys. L&t. B 221 (1989) 184. A.J. Buras, plenary talk given at the ICHEP96 conference, Warsaw, July 1996 (hep-ph/9610461). M. Neubert, Proc. of the 17th Symposium on Lepton-Photon Interactions, Beijing, 1995, eds. Z. Zhi-Peng Scientific 1996 (and references therein). M. Neubert, Phys. Rev. D 49 ( 1994) 4623
219
and C. He-Sheng,
World
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 400 (1997) 206-2 19
Weak radiative B-meson decay beyond leading logarithms Konstantin
Chetyrkin a,1,2,Mikolaj Misiak bv3y4,Manfred Miinz cs
a Max-Planck-lnstitutf
Physik, Werner-Heisenberg-ltitut, D-80805 Miinchen, Germany b Ins&t fiir Theoretische Physik der Universitiit Ziirich, CH-SO57 Zfirich, Switzerland ’ Physik Department, Technische Universitiit Miinchen, D-85748 Garching, Germany
Received 7 February 1997 Editor: PV. Landshoff
Abstract We present our results for order in QCD. We combine leading order prediction for smaller than in the previously with the CLEO measurement
three-loop anomalous dimensions necessary in analyzing B--+X,y decay at the next-to-leading them with other recently calculated contributions, obtaining a practically complete next-tothe branching ratio B( B-+X,y)= (3.28 -+ 0.33) x 10e4. The uncertainty is more than twice available leading order theoretical result. The Standard Model prediction remains in agreement at the 2a level. @ 1997 Elsevier Science B.V.
1. Weak radiative B-meson decay is known to be a very sensitive probe of new physics [ 1,2]. Heavy Quark Effective Theory tells us that inclusive B-meson decay rate into charmless hadrons and the photon is well approximated by the corresponding partonic decay rate I( B-+X,y) -T(b+X,y). The accuracy of this approximation is expected to be better than 10% [ 31. The inclusive branching ratio B(b-+X,y) was extracted two years ago from CLEO measurements radiative B-meson decay. It amounts to [4] a(b+X,y)
(1) of weak
= (2.32 f 0.57 f 0.35) x 10-4.
(2)
In the forthcoming few years, much more precise measurements of B( B+X,y) areexpected from the upgraded CLEO detector, as well as from the B-factories at SLAC and KEK. Acquiring experimental accuracy of below 10% is conceivable. Thus, the goal on the theoretical side is to reach the same accuracy in perturbative ’ This work was partially supported by INTAS under Contract INTAS-93-0744. 2 Supported in part by Schweizerischer Nationalfonds and Polish Commitee for Scientific Research
(grant 2 PO3B 180 09, 199597). 3 Supported in part by Deutsches Bundesministerium ftir Bildung und Forschung under contract 06 TM 743. 4 Permanent address: Institute of Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia. ’ Permanent address: Institute of Theoretical Physics, Warsaw University, Hoia 69, 00-681 Warsaw, Poland. 0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PI1 SO370-2693(97>00324-9
K. Chetyrkin et al./Physics
Letters B 400 (1997) 206-219
207
calculations of b--+X,y decay rate. This requires performing a complete calculation at the next-to-leading order (NLO) in QCD [ 1,5]. All the next-to-leading contributions to b-+X,y are collected for the first time in the present letter. 6 One of the most complex ingredients of the NLO calculation consists in finding three-loop anomalous dimensions in the effective theory used for resummation of large logarithms ln( M&/m;). This is the main new result presented in this letter. However, we have recalculated all the previously found anomalous dimensions, too. The remaining ingredients of the NLO calculation are already present in the literature [ 6,8-171. We use the published results for the two-loop matrix elements calculated by Greub, Hurth and Wyler [ 91, Bremsstrahlung corrections obtained by Ali, Greub [ 15,161 and Pott [ 171, and the matching conditions found by Adel and Yao [131. The following three sections of the present paper are devoted to presenting the complete NLO formulae for b+X,y. Next, we analyze them numerically. Finally, we discuss predictions for the branching ratio B( B-+X,y) including a discussion of the relevant uncertainties. 2. The analysis
of b+X,y
decay begins with introducing
an effective hamiltonian
(3) where x,j are elements of the CKM matrix, Pi (,u) are the relevant operators and Ci( ,u) are the corresponding Wilson coefficients. The complete set of physical operators necessary for b--+X,y decay is the following:
(4) where T” stand for SU(3) cotor generators. The small CKM matrix element l&, as well as the s-quark mass are neglected in the present paper. Our basis of four-quark operators is somewhat different than the standard one used in Refs. [ 1,6,18], although the two bases are physically equivalent. We would like to stress that, in our basis, Dirac traces containing ys do not arise in effective theory calculations performed at the leading order in the Fermi coupling GF (but to all orders in QCD). This allows us to consistently use a fully anticommuting ys in dimensional regularization, which greatly simplifies multiloop calculations. Our choice of color structures is dictated only by convenience in computer algebra applications. Throughout the paper, we use the MS scheme with fully anticommuting ys. Such a scheme is not uniquely defined until one chooses a specific form for the so-called evanescent operators which algebraically vanish in four dimensions [ 19,201. In our three-loop calculation, eight evanescent operators were necessary. We list them explicitly in Appendix A. Resummation of large logarithms ln( M$/m$ is achieved by evolving the coefficients Ci ( p) from ,u = Mw to ,U = ,Ub N mb according to the Renormalization Group Equations (RGE) . Instead of the original coefficients Ci( ,u) , it is convenient to use certain linear combinations of them, called the “effective coefficients” [ 1,9] 6 Except for the unknown
but negligible
two-loop
matrix elements
of the penguin operators
which we denote further by 4,...,P6
K. Chetyrkinetal./Physics
208
LenersB400(1997)206-219
for i= 1,...,6
G(P) 6 C7(p)
qqpu>
for i = 7
+ CyiCi(p)
=
(5)
i=l
6
for i= 8.
Cg(LL)+C&i(p) i=l
The numbers yi and zi are defined so that the leading-order b + sy and b --t s gluon matrix elements of the effective hamiltonian are proportional to the leading-order terms in CTff and Ciff, respectively [ 11. In the MS scheme with fully anticommuting ys, we have y = (O,O,-i,-$,-?,--y) and z = (O,O, 1,-i,20,-y).7 The leading-order contributions to the effective coefficients Cp” ( ,w) are regularizationand renormalizationscheme independent, which would not be true for the original coefficients C7 ( ,u) and Cs (/A). The effective coefficients evolve according to their RGE
(6) driven by the anomalous
a,(P) v%-d = FY
matrix yff (p) . One expands this matrix perturbatively
dimension
n(O)&
as follows:
a264 1)eff+ ... LfJ
+
(7)
(471.)*
independent, while y(‘jeff is not. In the MS scheme The matrix F(Ojeff is renormalization-scheme anticommuting ys (and with our choice of evanescent operators) we obtain --4; 12 0
Y *(Oferr=
0
-$
0
0
-E
0
$
0
0
416 s, -+$5 _E
0 00 0 0 _$
-q _y
0 $
2 2
0
-y
0
20 -F
0
0
00+
$
00
0
000
00
0
0
$L$
g ZT! 27 % _B$
(8)
5Z$j
E 22 s
00-y
with fully
F 0 28 s
and
7 These numbers
are different from those in Section 4 of Ref. [l]
because we use a different basis of four-quark
operators
here.
Letters B 400 (1997) 206 -21 9
K. Chetyrkin et al./Physics -355
9
-$?$
-u&
_g?
134 Tm
!280 81
c(l)eff
Y
_
-
0
0
_y
_a&%9
0
0
-a&!
59399 243
g.J
-g
-%
56 81
35 21
m 81
1841 108
glJ 8,
3313 -RR
22348 243
10178 81
%-z
12899 648
1;;;
172471 648
1183696 129
0 0
0
58640 243
26348 243
1;:32”
0
0
0
0
0
0
10
0
0
0
0
0
209
2&55 2:;;;
(9)
2901296 243 3296251 129
0
4688 21
4063
-a$
21
The matrix T(‘jeff is the main new result of this paper. Its evaluation required performing three-loop renormalization of the effective theory (3). Details of this calculation will be given in a forthcoming publication
r211. We expand the coefficients
in powers of ay, as follows:
as(P) c,:ff(,> = Ci’O’“QL)+ TCi
(1)eff
(ILL) +
(10)
...
At p = Mw, the values of the coefficients are found by matching the effective theory amplitudes Standard Model ones. The well-known leading-order results read [ 16,221
c!“‘“ff(Mw) 1
= C$O’(Mw)
0 1 3x3 - 2x2 4(x _ 1>4
=
-3x2 4(x - 1)4
with the full
for i= 1,3,4,5,6 for i = 2 lnx+
lnx+
--8x3 - 5x2 + 7x 24(x - 1)3 -x3 + 5x2 + 2x 8(x - 1)s
for i = 7
(11) for i=8,
where
(12) The next-to-leading contributions to the four-quark operator coefficients at ,u = Mw can be found either by transforming the results of Ref. [lo] to our operator basis, or by a direct computation. We find for i = 1 for i=2,3,5,6 fori=4,
15 0 E(x)--:
C!l)eff( Mw) = Ci”’ ( Mw) = 1
(13)
where E(x)
=
x(18-11x-x2) 12( 1 - x)3
+
The next-to-leading contributions the following equations: C(l)eff(Mw) 7
= -$&Ch;;
x2(15-
16x+4x2)
6(1 -x)~ to CTff(Mw)
+ A7 - $ (E(x)
lnx - $lnx.
and Ciff (Mw)
- $1,
can be extracted
(14) from Ref. [ 131 according
to
(15)
210
K. Chetyrkin et d/Physics
C(l)eff(Mw) 8
= 8-3:;
+A,
- i (E(x)
Letters B 400 (1997) 206-219
(16)
- 5))
s
where the quantities CA:,) and Chtj are given in Eq. (5) of Ref. [ 131 with p = Mw. The terms denoted by A7 and As stand for shifts one needs to make when passing from the so-called R* renormalization scheme used in Ref. [ 131 to the MS scheme used here Ai =
2
[c,'O'(M~)
(x -+ x - ?xfnx)
- c/o)(M~)
(x)]
for i=7,8.
(17)
Only the leading (zeroth-order) term in LY,needs to be retained in the above equation. The resulting explicit expressions for CFff(Mw) and Ciff (Mw) in the MS scheme read C(l)eff(Mw) I
=
-102x5
-16x4
-9’:‘” x
- 8xLi2 cl _ ;)
+ 6x4 + 46x3 - 28x2 ln2 x 3(X - 1)s
- 588x4 - 2262x3 + 3244x2 - 1364x + 208 lnx
+
81(x - 1)5 1646x4 + 12205x3 - 10740x2 + 2509x - 436
+
(18)
486(x - 1)4
‘8 (ly
Mw)
-17x3
=
- 31x2
2(x - 1)s -210x5
+ +
:,0x’
ln2 x
+ 1086x4 + 4893x3 + 2857x2 - 1994x + 280 lnx 216(x - 1)5
737x4 - 14102x3 -28209x2
+ 610x - 508 (19)
1296(x - 1)4
Having specified the initial conditions Group equations (6) for CTff( ,LQ)
at ,U = Mw, we are ready to write the solution
to the Renormalization
(Mw)
+(p
297664 14283 77 fi *’ -
~357075 q 14 7164416 a + 8
(21) i=l
where ~7= a,( Mw)/cu,(pb), ai = ( ei = (
and the numbers
14 z> 4661194 816831 ’
fi = ( -17.3023, gi = (
14.8088,
hi = (
626126’ 212277
16 TiT3 -g,
6 z?T
ai-hi -g
are as follows: ?
0.4086,
-0.4230,
-0.8994,
0.1456)
-1.9043,
-0.1008,
0.1216,
0.0183)
0.7476,
-0.5385,
0.0914)
0,
0,
8.5027,
4.5508,
0.7519,
2.0040,
-10.8090,
-0.8740,
0.4218,
-2.9347,
0.3971,
0.1600,
0.0225)
-iz 1 >
-0.6494,
-0.0380,
-0.0186,
-0.0057).
5628 1 51730’
-k
K. Chetyrkin et al./Physics
AS far as the other Wilson coefficients are concerned, only the leading-order necessary in the complete NLO analysis of b---+X, y. We obtain 8
C(O)&) 1
=
+ii
@(pb)
=
$3
C,(O’(/&)
=
_Ly-u 21
C’O’(/Ub) = 4
+
23
-# 477
-0.0659~“~4086 + 0.059517-“.4230 - 0.0218~~-~.~~~~ + 0.0335.170.‘456,
&+
+0.023777a.4as6 _ 0.017371-a.42so _ 0.133671-o.s994 _ 0.0316$.‘456,
$$
_ 1126r)g
+0.009477a~40s6 - o.0100~-a~4230 + o.oo10~-a~*994 - 0.0017770.‘456,
- &$
+0.010877°~4086 + 0.0163r]-“.4230 + 0.010377-“.8994 + 0.0023r10.‘4s6,
C,(O)(,ub) = -$7-g
-
= (CS’O’ ( Mw)
+ g$!)
$
0.913577°.4086 + 0.0873~7-‘.~~~’ - 0.057177-“.8994 + 0.0209q”.‘456.
In our numerical analysis, the values of aS ( ,u) in all the above formulae expression for the strong coupling constant a,(p)
to them are
g$i,
+
+
108
(O)eff( /I&)
contributions
+$3,
C~o’(~ub) = r,-g
‘8
211
Letters B 400 (1997) 206-219
=-
a,(Mz)
1_
V(P)
fic+(M~) -___
PO
4%-
InO)
V(P)
(22) are calculated
with use of the NLG
1
(23)
’
where (24)
As easily can be seen from Eqs. ( lo)-(24), the Wilson coefficients at J.L = pb depend on only five parameters: Mz, Mw, n,( Mz), ?r~,,~,,i~and pb. In our numerical analysis, the first two of them are fixed to Mz = 9 1.187 GeV and Mw = 80.33 GeV [ 231. For the remaining three, we take LYE
=0.118&0.003
mr,rote = 175 i 6 GeV
[24],
(25)
[251
(26)
and ,UI, E [2.5 GeV, 10 GeV].
(27)
For ,Ub = 5 GeV and central values of the other parameters, c’“‘eff(&
= (-0.480,
C(l)eff(,Ub) I
= -0.995
1.023, -0.0045,
-0.0640,
we find
0.0004,
LY,(,U(LD) = 0.212,
0.0009,
-0.312,
-0.148),
+ 1.443 = 0.448.
In the intermediate step of the latter equation, we have shown the C’l)eff(pb). The first of them originates from terms proportional to i.d. it is due to the NLO matching. The second of them comes from next-to-leading anomalous dimensions. Quite accidentally, the two
(28) relative importance of two contributions to C$lfeff( Mw) and Cs(l’eff( Mw) in Eq. (21)) the remaining terms, i.e. it is driven by the terms tend to cancel each other. In effect,
8 Analogous expressions in Ref. [ 11 are somewhat different, because we use a different basis of four-quark operators here.
212
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Letters B 400 (1997) 204-219
the total influence of C:‘)eff(pb) on the b+X,y decay rate does not exceed 6%. (Explicit expressions for the decay rate are given below.) However, if the relative sign between the two contributions was positive, then the decay rate would be affected by almost 30%. This observation illustrates the importance of explicit calculation of both the NLO matching conditions and the NLO anomalous dimensions. 3. At this point, we are ready to use the calculated coefficients at fl = ,Ub 21 mb as an input in the NLO expressions for b+X,y decay rate given by Greub, Hurth and Wyler 191. However, we make an explicit lower cut on the photon energy in the Bremsstrahlung correction E, > (1 -,)E~
= (1 -6)F.
(29)
We write the decay rate as follows: I[b+&y]E+(l-S)E~-
= T[b + sy]
+ T[b + sy gluon]E~>(‘-“)E~~ (30)
where %tpb)
D = C,(OJeff(pb) + 7
C:‘)eff(pul,) + ~~~“‘eff(pb)
[ri +yj,O’eff~n~
(31)
2
(32)
i=l
and A = (e- cys(/.u,)lnS(7+21nS)/3~
_
1) ]C$“)eff(,&,)12
+
+ i,
j=l ,
Ci’“‘eff(~b)Clo’eff(~b)fii(S). il.1
D with respect The contribution denoted by 1D 12 is independent of the cutoff parameter 6. By differentiating to ,Ub and using the RGE (6)) one can easily find out that the leading fib-dependence cancels out in D. When calculating 1D12 in our numerical analysis, we consistently set the 0( (Y:) term to zero. The contribution denoted by A vanishes in the limit cu,-+O. The first term in A contains the exponentiated (infrared) logarithms of 6 which remain after the IR divergences cancel between the virtual and Bremsstrahlung corrections to b-+X, y. The terms denoted by ri in D depend on what convention is chosen for additive constants in the functions Jln (S) and f7s (8). We fix this convention by requiring that all fij (6) vanish in the formal limit 6 -+ 0. In order to complete the full presentation of b-+X,y results, it remains to give explicitly the constants ri and the functions fi,i (6). The constants rs and r-2 are exactly as given in Ref. [9] rs=-&(-33+2??-6ir), i-2 = & (-833
-I- 14422
(33) 3’2 + [1728 - ISOd
+ [648 + 72~~ + (432 - 216r2)L I%{-5+[45-3m2+9L+9L2]
- 1296&(3) + (1296 - 324r2)L
+ 36L3] z2 + [-54 z+[-37i)+9L2]
- 842
r1 = -Igrz.
+ 1092L - 756L2] z3}
z2+[28-12LJ
where z = mz/rng and L = In z . In our operator basis (4), the constant to r-2
+ 108L2 + 36L3] z
z3}+U(z4L4),
(34)
t-1 does not vanish, but is proportional
(35)
K. Chetyrkin et d/Physics
The constant
213
Letters B 400 (1997) 206-219
~7 turns out to be9 8 2
y7=+-gr,
(36)
The quantities ~3, . . . , t-6 originate from two-loop matrix elements of the penguin operators Ps , . . . , Pe. They remain the only still unknown elements of the formally complete NLO analysis of b-+X,y. However, the Wilson coefficients Ci are very small for i = 3,. . . ,6. In consequence, setting ~3,. . ., t-6 to zero (which we do in our numerical analysis) has a negligible effect, i.e. it can potentially affect the decay rate by only about 1%. Let us now turn to the functions fi.i (8). One can easily obtain f77( 8) from Eqs. (23) and (24) of Ref. [ 161. (These equations were also the source for our ~-7and the exponent in Eq. (32) above.) =~6+~62-~~3+~~(~-4)ln6.
f77(6)
The function fss(6)
(37)
fss( S) can be found by integrating
Eq. ( 18) of the same paper [ 161
-21nmom, [62 +26+2ln(l-S)]
=-& C
.
+2Li2(1-S)-~-~(1+2S)ln~+81n(l-S)-$3+362+7S
(38)
I
The logarithm of m, in the above equation is the only point in our analysis at which the s-quark mass cannot be neglected. Collinear divergences which arise in the massless s-quark limit have been discussed in Ref. [ 261. The properly resumed photon spectrum was there found to be finite in the m, + 0 limit, and suppressed with respect to the case when ln(mb/m,) in fss(b) is used. Even with m, as small as 0.1 GeV, the term proportional to In(m,,/m,) is negligible, i.e. it affects the decay rate by less than 1%. Therefore, for simplicity, we will just use the naive expression (38) with mb/m, = 50 in our numerical analysis. The functions f22( a), f27( 8)) f28 (8) and f7s (6) can be found from appendix B of Ref. [ 91. After performing some of the phase-space integrations, one finds (l-@/z
162 = 27
f22(@
J’
6
dr(l-Zt)lF
+z I’
6
/
l/z dt
Re(G(t)+i)+
0 f28(&
= -f
j-78(8)
=i
=
[Li2(1--6)
-~712--15ln6+!+S-
,
$2+1+3],
(39)
and
C
-2 arctan dm, -n2/2+21n2[(fi+Jt-Zi)/2]-2i7rln[(fi+Jt-;E)/2],
= Q-22(&,
f12(@
for t < 4, fort24.
ftj (S) do not vanish, but are proportional
= -$22@),
g It differs from the one given in Ref. [9] where a formal limit 8 r7.
dt(l-tt)Re(G(t)+k)
(l-@/z
In our operator basis (4), the functions fll(@
/
f27(&,
where, as before, z = rnz/mE G(t)
dr(1-zt)‘~~+;2
(1---6)/z
(l-W/z = -$
/
+
0
f27(6)
, 11 1
l/z
f17c8)
to the functions
(40) fgj (S)
= -$27(&
1 was taken and the contribution
due to
f77( 1) was absorbed into
214
K. Chetyrkin et d/Physics
Letters B 400 (1997) 206-219
The only functions fi,i( 6) which have not yet been given explicitly are the ones with at least one index corresponding to the penguin operators Ps, .... Ps. As checked in Ref. [ 171, these functions cannot affect the decay rate by more than 2%. Therefore, we neglect them in our numerical analysis. However, for completeness, a formula from which they can be obtained is given in our Appendix B. 4. The decay rate given in Eq. (30) suffers from large uncertainties due to F&,~,~ and the CKM angles, One can get rid of them by normalizing b+X,y decay rate to the semileptonic decay rate of the b-quark
r[b -+ &e~,l =
G2F m5 b,poleK(z)
Iv,b12,
(41)
192T3gtz)
where g(z)
= 1 - 82 + 8z3 - z4 - 12z2 In z
is the phase-space
h(z) &T(Z)
is a sizable next-to-leading analytically in Ref. [28] ,qz)
= -(I
(42)
factor, and 2%(mb) 3T
tc(z)=l-
) (z = 2;:;:)
-
(43)
QCD correction
z*) (F -
to the semileptonic
yz + Fz2) + z lnz
decay [ 271. The function
(20 + 9Oz - $z2 + yz3)
+(l-z2)(~-~z+~z2)ln(l-z)-4(l+30z2+z4)lnz +z4)[6Li2(z)
-Q?]
-32~~/~(1+z) quantity
+ z21n2 z (36 + z2>
ln(l-z)-(1+16z2 d-4I&(&j-l-4&(-~‘?) [
Thus, the final perturbative
h( z ) has been given
we consider
-21nz
ln
(&$>I.
is the ratio (44)
where D and A are given in Eqs. (3 1) and (32)) respectively, 1 F= _ K(Z)
““~,=~))‘=_J_(l-~~).
and (45)
(
5. Let us now turn to the numerical results. Besides the five parameters listed above Eq. (25)) the quantity I&k(8) in Eq. (44) depends on a few more SM parameters. They are the following: (i) The ratio mc,pole/fnb.pole~ Similarly to Ref. [ 91, we will use !!&Z!Z = 0.29 i 0.02, ~~/J,pOle (ii)
(46)
which is obtained from m&p& 7 4.8 f 0.15 GeV and m&p& - mc,pole = 3.40 GeV. The b-quark mass itself. Apart ‘from the ratio m,/mb, the b-quark mass enters our formulae only in two places: in the explicit logarithm in D (31) and in the factor F (45)) as the argument of as. In both cases, the mb-dependent terms are next-to-leading, and their mb-dependence is logarithmic. Changing mb from 4.6 GeV to 5 GeV in these places affects the b+X,y decay rate by less than 2%. Thus, it does not really matter whether one uses the pole mass or the MS mass there. We choose to use mb,pole = 4.8 f 0.15 GeV.
K. Chetyrkin et al./Physics
Letters B 400 (1997) 206-219
215
3.1 2.9
2.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 1. Rqua,k( 8) as a function
6
(iii)
(iv)
of 6.
The electromagnetic coupling constant CZ~~.It is not a priori known whether this constant should be renormalized at /.L - mb or /_LN Mw, because the QED corrections have not been included. We allow CX,, to vary between (Y,, (mb) and my,, (Mw), i.e. we take LY;~ = 130.3 f 2.3. The ratio of CKh4 factors. We will use
I I GKb
-
Vcb
= 0.976 f 0.010
(47)
given in Ref. 1291. It corresponds to gaussian error analysis, which is in accordance with interpreting the quoted error as a la uncertainty. The dependence of Rquark(8) on S is shown in Fig. 1. The middle curve is the central value. The uncertainty of the prediction is described by the shaded region. It has been found by adding in squares all the parametric uncertainties mentioned above in points (i)-(iv) and in Eqs. (25)-( 27). The ratio $&,.k(@ is divergent in the limit 6 -+l, which is due to the function fss (8). In this limit, the physical quantity to consider would be only the sum of b -+X,y and b-+X, decay rates, in which this divergence would cancel out. The divergence at 6+1 is very slow. It manifests itself only by a small kink in Fig. 1 where values of 6 up to 0.999 were included. For 6 = 0.99, the contribution from fss(8) to &u&(S) is we11 below 1%. In order to allow easy comparison with previous experimental and theoretical publications, we choose P” = 0.99 as the first particular value of 6 to consider. This value of 6 will be assumed in discussing the “total” B-XJ decay rate below. A value of 6 which is closer to what is actually measured is 6 = z = (m,/I?‘@)2. It corresponds to counting only the photons with energies above the charm production threshold in the b-quark rest frame. This is the second value of 6 we will consider below. We find R quark(8max) = (3.12i0.29)
&,,,k(6
x 10-3,
(48)
= Z) = (2.86 f 0.24) x 10-3.
In both cases, the uncertainty is below 10%. The dominant importance of various uncertainties is shown in Table 1.
(49) sources
of it are m,/i’?‘& and ,Ub. The relative
216
K. Chetyrkin et al./Physic.s
Table 1 Unceminties
Letters B 400 (1997) 206-219
in Rquark (6) due to various sources Source &(MZ)
+.po1e
/-Q
‘QJOk
2.5% 2.2%
1.7% 1.7%
6.6% 3.3%
5.2% 6.6%
/i”‘b.pole
‘nb,pole
%m
CKM angles
0.5% 0.6%
1.9% 1.9%
2.1% 2.1%
Let us note that setting 6 = Z, introduces an additional spurious inaccuracy due to m,/mb in &,,.k. However, this additional uncertainty is significantly smaller than the original inaccuracy due to m,/mb for S uncorrelated with m,/mb. It is quite surprising that the pb-dependence is weaker for 6 = z than for 8 = P”. Naively, one would expect larger pb-dependence for small values of 6, when the first term in Eq. (32) becomes more and more important. Clearly, 6 = z is not small enough for this effect to show up. The weaker pb-dependence for 6 = z seems to be quite accidental.
6. In the end, we need to pass from the calculated b-quark decay rates to the B-meson on the Heavy Quark Effective Theory (HQET) calculations we write
decay rates. Relying
(50) where 8:’ and 8:: p arametrize nonperturbative corrections to the semileptonic and radiative B-meson rates, respectively. Following Refs. 13,301, we express 8:’ and 8:: in terms of the HQET parameters Al and AZ:
decay
(51) (52) where g(z) is the same as in Eq. (42). The poorly known parameter A1 cancels out in the r.h.s. of Eq. (50) which depends only on the difference 8:’ - 8::. The value of A2 is known from B*-B mass splitting A2 = i(rni,
- rni)
N 0.12 GeV2.
(53)
The two nonperturbative corrections in Eq. (50) are both around 4% in magnitude, and they tend to cancel each other. In effect, they sum up to only around 1%. Such a small number has to be taken with caution, because the four-quark operators 2’1, . . . , P6 have not been included in the calculation of 8::. Contributions from these operators could potentially give one- or two-percent effects. Nevertheless, it seems reasonable to conclude that the total nonperturbative correction to Eq. (50) is well below lo%, i.e. it is smaller than the inaccuracy of the perturbative calculation of &u&( am”). When the above caution is ignored and 13(B-+XceGc) = (10.4 f 0.4)% [23] is used, one obtains the following numerical prediction for B-tX,y branching ratio B(B--+X,y)
= (3.28 f 0.33)
x 10-4.
(54)
K. Chetyrkin etal./Physics LettersB 400 (1997) 206-219
217
The central value of this prediction is outside the la experimental error bar in Eq. (2). lo However, the experimental and theoretical error bars practically touch each other. Therefore, we conclude that the present B~x,y measurement remains in agreement with the Standard Model. This conclusion holds in spite of that the theoretical uncertainty is now more than twice smaller than in the previously available leading order prediction [ 1 I. In the measurement of B+X,y branching fraction, one needs to choose a certain lower bound on photon energies. Instead of talking about the “total” decay rate, it is convenient to count only the photons with energies above the charm production threshold in the B-meson rest frame
mi--m2, N 2.31 GeV.
Ey>
(55)
2m B
Most of the photons in B-X,y survive this energy cut, and a huge background from charm production is removed. Let us denote the corresponding branching fraction by i?( B-+Xsy)above. If the b-quark was infinitely heavy, it would not move inside the B-meson. The restriction (55) would then be equivalent to setting 6 = z at the quark level. Since the Fermi motion of the b-quark inside the B-meson is a O(&/mb) effect, we can write similarly to Eq. (50)
a(B~X,y)"boYe=B(B~X,ev) Rquark(8=z) (l+O(&/mb)).
(56)
The 0 (l?/mb) part of the latter equation can be calculated using specific models for the Fermi motion of the b-quark inside the B-meson, as it has been done, e.g.. in Ref. [ 161 (see also Ref. [31] ). Performing a similar calculation with use of the NLO values of the Wilson coefficients is beyond the scope of the present paper. Such an analysis will be necessary when more statistics allows to reduce experimental errors in measurements of a( B+Xsy)abo"e. 7.To conclude, we have presented practically complete NLO formulae for b-+X,y decay in the Standard Model. They include previously published contributions as well as our new results for three-loop anomalous dimensions. Our prediction for the B --+X,y branching fraction in the Standard Model is (3.28 f 0.33) x lop4 which remains in agreement with the CLEO measurement at the 20 level. Clearly, an interesting test of the SM will be provided when more precise experimental data are available. More importantly, the B+X,y mode will have more exclusion power for extensions of the SM. Appendix A. Here, we give the eight evanescent operators we have used in our anomalous dimension computation. Giving them explicitly is necessary in order to fully specify our renormalization scheme, and thus give meaning to the anomalous dimension matrix presented in Eq. (9),
PII = ~~~Y~,Y~~Y~~~*~L)(~LY~'~~*~~~~=~L) - I@'1 P12 = (~LY,,Y,,Y,,~L)(~L~~~Y~~~~~~~L.) - 16f’2 P15 = (SLY~,Y~LZY~~Y~~Y~CL.F~L) ~GY~'Y'~Y~~Y~~Y'W
- 208
+ 64P3
4 p16 = (SLy~,y~~y~3y~~y~LgTflbL)
C(yy'L'y"y'3y~4y'ST"q)
- 2OP6
+64P4
rl P21 =
(~LY,,Y,,Y,,Y,Y,,T~~L)(~LY~~Y~~Y~~Y~~Y~~T~~L) -2091-2569
loWe identify the b-quark identification acceptable.
decay rate given by CLEO with the B-meson
decay rate. Large statistical
errors in the CLEO result make this
218
K. Chetyrkin et al./ Physics Letters B 400 (19971206-219 P22 = ~~~~,,y,,y,,y,,y,,T~~)(~~y~‘y~~y~~y~~y~~b~)
- 2092 - 256P2
P25 = (SLY I4 Y /J2 Y IQ.Y w y P5 y I4 y P7 bL) ~(4~“‘y”~y~~y~~y~~y~y~~~)
- 336P5 + l28OP3
4
P26 = (SLY /4y p2y P3 y P4 y P _y M y I.0 T”bL) C(q~~‘y~‘yll~y~y~~y~y~~T’q)
- 336P6 + 1280P4.
(A.11
4
Appendix B. Here, we express the functions f~ (S) present in Eq. 32) in terms of the quantities given in Eq. (27) of Ref. [ 171. For each (ij) # (77), fij( 6) is given by
fij(S)
=:
& k,l=l,
1
Wik
dt Mkl (tv U>
k
Wjl,
Mkl( t, u)
tB.1)
where
-2-i 1 0 0 0
0 0 0 1 0 0 0 0 1 0 1 0 _11_1100
0 0 0 0 00 -00 In four space-time
Pi
000 000 000
03.2)
1666 4Yoo -$S-$200 000010 000001 dimensions,
‘2 f: wikdk.
the matrix L? transforms
our operator basis (4) into the one used in Ref. [ 171
03.3)
k=l References [I] A.J. Bums, M. Misiak, M. Mtinz and S. Pokorski, Nucl. Phys. B 424 (1994) 374. [2] J. Hewett, Top ten models constrained by h -+ sy, published in Proceedings Spin structure in high energy processes, p. 463, Stanford, 1993. [3] A.F. Falk, M. Luke and M. Savage, Phys. Rev. D 49 (1994) 3367. [4] M.S. Alam et al., Phys. Rev. Lett. 74 (1995) 2885. [ 51 A. Ah and C. Greub Phys. Lett. B 293 (1992) 226. [6] B. Grinstein, R. Springer and M.B. Wise, Nucl. Phys. B 339 (1990) 269. [7] R. Grigjanis, P.J. O’Donnell, M. Sutherland and H. Navelet, Phys. Lett. B 213 (1999) 355; B 286 (1992) 413 (E). [8] M. Ciuchini, E. France, G. Martinelli, L. Reina and L. Silvestrini, Phys. L&t. B 316 (1993) 127; M. Ciuchini, E. France, L. Reina and L. Silvestrini, Nucl. Phys. B 421 (1994) 41. [9] C. Greub, T. Hurth and D. Wyler, Phys. Lett. B 380 (1996) 385; Phys. Rev. D 54 (1996) 3350. [lo] A.J. Buras, M. Jamin, M.E. Lautenbacher and PH. Weisz, Nucl. Phys. B 370 (1992) 69; B 375 (1992) 501 (addendum). [ 1 l] M. Ciuchini, E. France, G. Martinelli and L. Reina, Nucl. Phys. B 415 (1994) 403. 1121 M. Misiak and M. Mhnz, Phys. Lett. B 344 (1995) 308. [ 131 K. Adel and Y.P Yao, Phys. Rev. D 49 ( 1994) 4945. [ 141 P Cho, B. Grinstein, Nucl. Phys. B 365 (1991) 279. 1151 A. Ali and C. Greub, 2. Phys. C 49 (1991) 431; Phys. Lett. B 259 (1991) 182. [ 161 A. Ah and C. Greub, Phys. Len. B 361 (1995) 146. [ 171 N. Pott, Phys. Rev. D 54 (1996) 938.
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219
and C. He-Sheng,
World