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Rare B decays in the Standard Model
Nuclear Insuuments
and Methods in Physics Research A 384 ( 1996) 8-16
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
EISEVIER
Rare B decays in the Standard Model A. Ali * Deursches Elekrronen-Svnchrotron DESK Notkestrasse 85, D-22603 Hamburg, German>
Abstract We discuss the electromagnetic-penguin-dominated radiative B decays B + X, + y. B*(O) -+ K**‘“’ + y, and B, --+ (b + 7 in the context of the Standard Model (SM) and their Cabibbo-Kobayashi-Maskawa (CKM)-suppressed counterparts, B --+ xd + y, B* -+ p* + y, B” --t (p’,o) + y, and B, --t K” + y, using QCD sum rules for the exclusive decays. The importance of these decays in determining the parameters of the CKM matrix [l] is emphasized. The semileptonic decays B + XJJ+e- are also discussed in the context of the SM and their role in determining the Wilson coefficients of the effective theory is stressed. Comparison with the existing measurements are made and SM-based predictions for a large number of rare B decays are presented.
unitarity of the CKM matrix, the decay matrix element in the lowest order can be written as:
1. Estimates of B(B ---$ X, + y) and IV,1 in the Standard Model The Standard Model (SM) of particle physics does not admit flavour-changing-neutral-current (FCNC) transitions in the Born approximation. However, they are induced through the exchange of W* bosons in loop diagrams. The short-distance contribution in rare decays is dominated by the (virtual) top quark contribution. Hence the decay characteristics provide quantitative information on the top quark mass and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements v;; i = d, s, b [ 11. We shall discuss representative examples from several such transitions involving B decays, starting with the decay B + X, + y, which has been measured by CLEO [2]. This was preceded by the measurement of the exclusive decay B -+ K’ + y by the same collaboration [ 31. The present measurements give [4] : B( B + X, + y) = (2.32 * 0.57 * 0.35) x 10-4,
(1)
L3(B + K’ + y) = (4.2 f 0.8 & 0.6) x 10-5,
(7-j
yielding an exclusive-to-inclusive ratio:
These decay rates determine the ratio of the CKM matrix elements 1V, I/ 1Vcb1and the quantity Rx’ provides information on the decay form factor in B + K’ + y. In what follows we take up these points briefly. The leading contribution to b -+ s + y arises at one-loop from the so-called penguin diagrams. With the help of the
x
E-mail
F2(Xc))qPEYSapv(mhR+
X
-(x-1)(8x2+5x-7)].
(5)
The measurement of the branching ratio for B -+ X, + y can be readily interpreted in terms of the CKM-matrix element product A,/IV,b( or equivalently IVBI/IVcbl. For a quantitative determination of 1V, I/ 1&b 1, however, QCD radiative corrections have to be computed and the contribution of the so-called long-distance effects estimated. The appropriate framework to incorporate QCD corrections is that of an effective theory obtained by integrating out the heavy degrees of freedom, which in the present context are the top quark and W’ bosons. The operator basis depends on the underlying theory and for the SM one has (keeping operators up to dimension 6) R
-+
S-k y) =
-4GFV,‘vb fi
@
(4)
msL)b.
F2(x) = 24(x _ 114 [6x( 3x - 2) log x
[email protected]
0168-9002/96/$15.00 Copyright PIISO168-9002(96)00908-4
-
where Xi = mj/mf, and qp and l,, are, respectively, the photon four-momentum and polarization vector. The GIM mechanism [ 5 ] is manifest in this amplitude and the CKMmatrix element dependence is factorized in At = t&V,,. The (modified) Inami-Lim function F2( Xi) derived from the (one-loop) penguin diagrams is given by [ 61:
‘,‘&(b l
(F2(xO
1996 Elsevier Science B.V. All rights reserved
~ci(I*)~i(~) i=l
3
(6)
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where the operator basis, the “matching conditions” C,(mw ), and the solutions of the renormalization group
equations C, (1~) can be seen in Ref. [ 71. The perturbative QCD corrections to the decay rate r ( B -+ X, + y) have two distinct contributions: - Corrections to the Wilson coefficients Ci(,U), calculated with the help of the renormalization group equation, whose solution requires the knowledge of the anomalous dimension matrix in a given order in as. - Corrections to the matrix elements of the operators 0, entering through the effective Hamiltonian at the scale P = O(mh). The anomalous dimension matrix is needed in order to sum up large logarithms, i.e., terms like (Y:(mw) log”*( mh/bf). whereM=m,ormwandm
[ 91 and are found to be small. Next-to-leading order corrections to the matrix elements are now available completely. They are of two kinds: - QCD bremsstrahlung corrections b + sy + g, which are needed both to cancel the infrared divergences in the decay rate for B -+ X, + y and in obtaining a non-trivial QCD contribution to the photon energy spectrum in the inclusive decay B -+ X, + y: - next-to-leading order virtual corrections to the matrix elements in the decay b ---) s + y. The bremsstrahlung corrections were calculated in Refs. [ 10. I I ] in the truncated basis and last year also in the complete operator basis [ 12,131. The higher order matching conditions, i.e.. C;( mw ) , are known up to the desired accuracy, i.e., up to O( CI~(Mw ) ) terms [ 141. The next-to-leading order virtual corrections have also been calculated [ 151. They reduce the scale-dependence of the inclusive decay width. The branching ratio f3(B + X, + y) can be expressed in terms of the semileptonic decay branching ratio
B(B --+ X&Y)=
r(B+y+X,) rSL
1
Ih
f3(B -
Xev,, . (7)
where the leading-order QCD corrected expression for rs~ can be seen in Ref. [ 71. The leading order ( 1/mh) power corrections in the heavy quark expansion are identical in the inclusive decay rates for B -+ X, + y and B + XI&, entering in the numerator and denominator in the square bracket, respectively, and hence drop out. In Ref. [7], the present theoretical errors on the branching ratio B( B -P X,y) are discussed, yielding: B( B + X, + y) = (3.20 f 0.30 f 0.38 + 0.32) x IO-’
Rex A 3U4
9
f1996)X-16
the extrinsic sources (such as A ( mh), A ( BRSL1). and the third error is an estimate (f 10%) of the NLO anomalous dimension piece in Cy’, the coefficient of the magnetic moment operator. Combining the theoretical errors in quadrature gives [7] a(B
+ X, + y) = (3.20f
0.58) x lo-’
1
(9)
which is compatible with the present measurement X, + y) = (2.32f 0.67) x 10-j [2]. Expressed of the CKM matrix element ratio, one gets IVSl/l&,
= 0.85 f O.l2(expt)
t3( B --) in terms
rfr:O.lO(th) .
( IO)
which is within errors consistent with unity, as expected from the unitarity of the CKM matrix.
2. Inclusive
radiative decays B -+ X, + y
The theoretical interest in studying the (CKMsuppressed) inclusive radiative decays B --t X,1 + y lies in the first place in the possibility of determining the parameters of the CKM matrix. We shall use the Wolfenstein parametrization Il6], in which case the matrix is determined in terms of the four parameters A. A = sin Bc. /J and 71. The quantity of interest in the decays B + Xd + y is the end-point photon energy spectrum, which has to be measured requiring that the hadronic system Xd recoiling against the photon does not contain strange hadrons to suppress the large-E, photons from the decay B + X,, + y. Assuming that this is feasible, one can determine from the ratio of the decay rates B(B -+ Xd + y) /B( B - X, + y) the CKM-Wolfenstein parameters p and 7~. This measurement was first proposed in Ref. [ I1 1. where the photon energy spectra were also worked out. In close analogy with the B -+ X, + y case discussed earlier, the complete set of dimension-6 operators relevant for the processes b 4 dy and b -+ dyg can be written as: x
‘&r(b + d) = -%& fi
c C,(P)B,(P). ,=I
(11)
where 5, = V,hVjz for j = t,c, u. The operators Gj. j = 1,2. have implicit in them CKM factors. In the Wolfenstein parametrization [ 161, one can express these factors as: 5” = AA+p - iq),
& = -A,?,
51 = -&
- &.
( 12)
We note that all three CKM-angle-dependent quantities li are of the same order of magnitude, 0( A’). This aspect can be taken into account by defining the operators 6, and & entering in ‘H,fy(b -+ d) as follows [ I I ] :
(8) where the first error comes from the combined effect of Am, and Ap ( the scale dependence), the second error arises from
1.
THEOKY
IO
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2.1. 8( B -+ V + y) and constraints on the CKM parameters
with the rest of the operators (&I; j = 3.. .8) defined like their counterparts Oj in 7&(b -+ s), with the obvious replacement s -+ d. With this choice, the matching conditions Cj (mw ) and the solutions of the RG equations yielding Cj (p) become identical for the two operator bases Oj and dj. The essential difference between T(B + X, + y) and I(B --) Xd + y) lies in the matrix elements of the first two operators 01 and 02 (in ‘&s(b --) s)) and 81 and 82 (in &r( b -+ d) ). The branching ratio B( B --) Xd + y) in the SM can be written as:
-Cl-PP2-q&+04),
(14)
where the functions Di depend on the parameters ml, mh, mc, ,u, as well as the others we discussed in the context of a( B --) X, + y). These functions were first calculated in Ref. [ 111 in the leading logarithmic approximation. Recently, these estimates have been improved in Ref. [ 191, making use of the NLO calculations in Ref. [ 151. To get an estimate of the inclusive branching ratio, the CKM parameters p and v have to be constrained from the unitarity fits. Present data and theory restrict them to lie in the following range (at 95% CL) [ 171:
-0.35 I p < 0.35,
(15)
which, on using the current lower bound from LEP on the By-3 mass difference AM, > 9.2 ps-’ [20], is reduced to -0.25 5 p 5 0.35 using & = 1.l, where & is the SU( 3) -breaking parameter & = fa,&,/&,&, . The preferred CKM-fit values are (p. 77) = (0.05,0.36), for which one gets [ 191 X
1o-5,
(16)
whereas a( B --+ Xd + y) = 8.0 x lo-” and 2.8 X lo-’ for the other two extremes p = 0.35. 17= 0.40 and p = -3 = -0.25, respectively. In conclusion, we note that the functional dependence of B(B -+ Xd + y) on the Wolfenstein parameters (p, 77) is mathematically different than that of AM,. However, qualitatively they are very similar. From the experimental point of view, the situation p < 0 is favourable for both the measurements as in this case one expects (relatively) smaller values for AM, and larger values for the branching ratio f?( B -+ Xd + y), as compared to the p > 0 case which would yield larger AM, and smaller B(B ---) xd
+
r>.
Here V is a vector meson with the polarization vector e’“) , V = p, w, K’ or 4; B is a generic B-meson B*, B” or B,, and + stands for the field of a light u, d or s quark. The vectOrS pB, pv and q = pB - pv correspond to the 4-momenta of the initial B-meson and the outgoing vector meson and photon, respectively. In Eq. (17) the QCD renormalization of the &rPvq’b operator is implied. Keeping only the SD-contribution leads to obvious relations among the exclusive decay rates, exemplified here by the decay rates for (B,,Bd) -+p+yand(B,,Bd) -+K*+y: I((B$,B:) r((B,f,Bj)
0.20 5 q < 0.52,
S(B 4 Xd + y) = 1.63
Exclusive radiative B decays B + V + y, with V = K’, p, w, are also potentially very interesting from the point of view of determining the CKM parameters [ 211. The extraction of these parameters would, however, involve a trustworthy estimate of the SD- and LD-contributions in the decay amplitudes. The SD-contribution in the exclusive decays (B*, B”) -+ (K**,K*‘) +y. (B*:,B’) --) (p*.~‘) +y,B’-+ w-ty and the corresponding B, decays, B, -+ $I + y and B, -+ K” + y, involve the magnetic moment operator 07 and the related one obtained by the obvious change s --+ d, &. The transition form factors governing the radiative B decays B -+ V + y can be generically defined as:
--) (P*,P’) +Y) ---) (K**,K*O) + y)
tv,l
“vd[ jiq
2 1 ’
(18)
where @u.d is a phase-space factor which in all cases is close to 1 and pi E [Fs(Bi -+ m)/Fs(Bi + K’y)]‘. The transition form factors FS are model dependent. Estimates of Fs in the QCD sum rule approach in the normalization of Eq. ( 17) range between Fs (B -+ K’y) = 0.3 1 (Narison in Ref. [22]) to Fs(B -+ K’y) = 0.37 (Ball in Ref. [22]), with a typical error of *15%, and hence are all consistent with each other. This, for example, gives RK* = 0.16*0.05, using the result from Ref. [ 211, which is in good agreement with data. The ratios of the form factors, i.e. Ki, should therefore be reliably calculable as they depend essentially only on the SU( 3)-breaking effects which have been estimated [ 21,221. The LD-amplitudes in radiative B decays from the light quark intermediate states necessarily involve other CKM matrix elements. Hence, the simple factorization of the decay rates in terms of the CKM factors involving It&] and IV,/ no longer holds, thereby invalidating relation ( 18) given above. In the decays B 4 V + y they are induced by the matrix elements of the four-Fennion operators 81 and 82 (likewise 01 and 02). Estimates of these contributions re-
A. Ali/Nucl.
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Thus, the CKM factors suppress the LD-contributions. The analogous LD-contributions to the neutral B decays B” -+ py and B” -+ oy are expected to be much smaller. The corresponding form factors for the decays + p”(w) y are obtained from the ones for the decay ;: + p*y discussed above by the replacement of the light quark charges e, -+ ed. which gives the factor - I/2; in addition, and more importantly, the LD-contribution to the neutral B decays is colour-suppressed, which reflects itself through the replacement of the factor al by a~. This yields for the ratio
quire non-perturbative methods. This problem has been investigated in Refs. [23,24] using a technique [25] which treats the photon emission from the light quarks in a theoretically consistent and model-independent way. This has been combined with the light-cone QCD sum rule approach to calculate both the SD and LD - parity conserving and parity violating - amplitudes in the decays (B’, B”) --) (p*, p/w) + y. To illustrate this, we concentrate on the B* decays B* -+ p* + y and take up the neutral B decays B” -+ p(o) + y at the end. The LD-amplitude of the four-Fermion operators 61, 82 is dominated by the contribution of the weak annihilation of valence quarks in the B meson and it is color-allowed for the decays of charged B* mesons. Using factorization, the LD-amplitude in the decay B, -+ p* + y can be written in terms of the form factors Ff and Fk,
where the numbers are based on using az/ar = 0.27 k 0.10
dronr = - *
[26]. This would then yield R$Tfl which in turn gives
V”r,V;L,(c2 + +I>
m,@
El’)
c
v5
1a2Fk(q2) fe ““uPp,qp.
cn=R
Bf
+p*Y
L/S
=-
eda
N
eual
L/s
-0.13 f 0.05)
(24)
N R$yy
= 0.05,
dB”PY
(25)
shoti 2&q*))
J
.
(19)
‘Fs’
2
=00155*00010 *
*
(20)
where the errors correspond to the variation of the Bore1 parameter in the QCD sum rules. Including other possible uncertainties, one expects an accuracy of the ratios in (20) of order 20%. The parity-conserving and parity-violating amplitudes turn out to be numerically close to each other in the QCD sum rule approach, Fk N Fk G FL, hence the ratio of the LD- and the SD- contributions reduces to a number r241 A long
R
Bf-.p*‘y
Ions < 0.02. dr5O-m -
Again, one has to invoke a model to calculate the form factors. Estimates from the light-cone QCD sum rules give [24]: 3 =00125*0.0010 Fs ’
~a’-PY r-/s
&bV$
(21)
qf.
This, as well as the estimate in Eq. (23), should be taken only as indicative in view of the approximations made in Refs. [ 23,241. That the LD-effects remain small in B” + p-y has been supported in a recent analysis based on the soft-scattering of on-shell hadronic decay products B0 -+ pop0 -+ py [ 271, though this paper estimates them somewhat higher (between 4% and 8%). Restricting to the colour-allowed LD-contributions, the relations, which are obtained ignoring such contributions (and isospin invariance), r(B*
+ p*y) =2r(B”
+ $y)
=2r(B”
-+ WY), (26)
get modified to
W* -+P*Y) 2r(B0 --+ py)
w* +
P'Y) = 2r(BO -+ oy) 2
Using C2 = I .lO, CI = -0.235, CtR = -0.306 from Ref. 171 (corresponding to the scale p = 5 GeV) gives: R
B* --‘PiY
_
4dmp(C2
L/S
+ Cl/NC)
mbC7eff =
= 1 + 2Rrlst&i
F:i*p*r F,B*
(l-P)*+11*
+P*Y
-0.30 f 0.07 (
(22)
which is not small. To get a ball-park estimate of the ratio drong/dsbon. we take the central value from the CKM fits, yielding ]&b]/]xd:dlN 0.33 [ 171: A long A shon I-1
B f -4
Y
= (R,B,~-p*y,~
N 10%.
--PI-T2
Al
(23)
+
(RL,s)~K:
P’
+v2
c1 _ pj2
where RLIS z R$*p*y.
+72
.
(27)
The ratio r(B*
+ P*Y)12w” + py) (= T’(B* -+ p*y)/2T(B” -+ WY)) is shown in Fig. 1as a function of the parameter p, with q= 0.2, 0.3 and 0.4. This suggests that a measurement of this ratio would constrain the Wolfenstein parameters (p, q), with the dependence on p mo& marked than on v. In particular, a negative value of p leads to a constructive interference in B, + py
I. THEORY
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A 384 (1996)
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LD-contributions is modest but not negligible, introducing an uncertainty comparable to the -15% uncertainty in the overall normalization due to the SU( 3) -breaking effects in the quantity K”. Neutral B-meson radiative decays are less prone to the LD-effects, as argued above, and hence one expects that to a good approximation (say, better than 10%) the ratio of the decay rates for neutral B meson obtained in the approximation of SD-dominance remains valid [ 2 I ] :
5
Fig.
I.
Ratio
r(B*
--t
p*y)/2r(
eter p. with 7 = 0.4
of
7
the B” -
= 0.2
(long-dashed
\
I
I
neutral py)
curve).
and
charged
as a function
(short-dashed
r&++~;;jy)=KdA*[ (1 -
curve).
(Figure
7
B-decay
of the Wolfenstein = 0.3
taken from
Ref.
(solid
rates param-
curve),
and
[ 241.)
decays, while large positive values of p give a destructive interference. The ratio of the CKM-suppressed and CKM-allowed decay rates for charged B mesons gets modified due to the LD contributions. Following earlier discussion, we ignore the LD-contributions in T(B -+ K*y). The ratio of the decay rates in question can therefore be written as:
P*Y)
+ K**y)
X
(29)
where this relation holds for each of the two decay modes separately. Finally, combining the estimates for the LD- and SD-form factors in Ref. [ 241 and Ref. [ 211, respectively, and restricting the Wolfenstein parameters in the range -0.25 < p _< 0.35 and 0.2 5 77 5 0.4, as suggested by the CKM-fits [ 171, we give the following ranges for the absolute branching ratios:
a( B” -+ py) N a( B” + wy) = (0.65 f 0.35) x 10-h, (30)
r(B* + T(B*
/I)’ + 7’1 ,
\
1 +
=K”A2[(1
-p?+$l
P( (,
+
2RLlsKd
1 - PI _ Pj2
77* ,+
P2 + v2 _Pj2+772
+(RL,s)~~+,
Using the central value from the estimates of the ratio of the form factors squared K” = 0.59 * 0.08 [2l], we show the ratio (28) in Fig. 2 as a function of p for 7 = 0.2.0.3, and 0.4. It is seen that the dependenceof this ratio is rather weak on v but it depends on p rather sensitively. The effect of the
where we have used the experimental value for the branching ratio a( B + K* + y) [ 31, adding the errors in quadrature. The large error reflects the poor knowledge of the CKM matrix elements and hence experimental determination of these branching ratios will put rather stringent constraints on the Wolfenstein parameter p. In addition to studying the radiative penguin decays of the B,f and Bi mesons discussed above, hadron machines such as HERA-B will be in a position to study the corresponding decays of the Bf meson and Ah baryon, such as By + 4 + y and Ah + A + y, which have not been measured so far. We list below the branching ratios in a number of interesting decay modes calculated in the QCD sum rule approach in Ref. [2l]. f?(B, -+ @y) = a(Bd --+ K’y) = (4.2 & 2.0) U(Bs
-+
x
1W5,
K*Y)
@Bd --+ K’Y) =+ &B,
--) K*y) = (0.75 f 0.5) x 1O-6.
The estimated branching ratios in. a number of inclusive and exclusive radiative B decay modes are given in Table 1, where we have also listed the branching ratios for B, + yy and Bd -+ yy. Fig. 2. Ratio rates: r(Bu
of the CKM-suppressed -
the Wolfenstein (solid 1241
curve),
)
py)/lY(B
-
parameter and TJ = 0.4
K’y) p, with
and CKM-allowed (with
B = B,
7 = 0.2
(long-dashed
or Bd)
(short-dashed
curve).
(Figure
radiative
B-decay
as a function curve),
7 = 0.3
taken from
2.2. Inclusive rare decays B -+ X,$?‘e- in the SM
of
Ref.
The decays B + Xse+e-, with ! = e, ,u, 7, provide a more sensitive search strategy for finding new physics in
A. Ali/Nucl.
I
Table
Estimates model
of the branching
taking
into
short-distance
account
contributions -+
+ Y and B”
been included. MeV
o)
except
for
branching
Experimental
Decay
in the input
parameters
correspond
the radiative
decays B* effects
as
to the -+
have also
ratios, we have used fB,
= 200
measurements
limits
and upper
and systematic
to give the quoted experimental
modes
in the standard
long-distance
are also listed. In the second row, the statistical have been combined
B-decays
in the second column
+ Y. where
For the two-body I .l6.
for FCNC
the uncertainties
only,
(#,
fB,/fBd =
and
fractions
[ 71. Theentries
discussed in Ref.
p*
Instr. and Meth. in Phy.
uncertainties
uncertainty.
Measurements
B(SM)
90%
CL
and
upper limits
(3.2f0.6)
x 1O-4
(2.3f0.7)
x 1O-4
I21
(B*,B”)-K*Y
(4.Ozt2.0)
x lO-5
(4.2Srl.0)
x IO-’
[41
(B*,B”)+X‘,y
(1.6+1.2)x
lo--’
(B*.B”)
+Xsy
B*+$
+y
(l.5fl.l)x10-6
B” +
p” + Y
(0.651kO.35)
B” -+6JfY
Bs-+K*
+y f-
(B,j.Bu)dXse
e
(Bd,B,)+Xde+e+
(Bd.Bu)+XsP
P
-
< 3.9 x 10-5
x IO-’
< I.3 x 10-5
141 141 141
< 3.6 x lO-5
I491
1401 I401 1401 I401 I471 1461
<1.1x10-~
(0.65f0.35)
Bs-4+Y
x IO-”
(4.2zt2.0)
x lO-5
(0.8+0.5)
x~O-~
(8.4f2.2)
x lO-6
(4.9f2.9)
x IO-’
(5.7*1.2)x10-”
(Bd,Bu)- x,,P+/.‘(Bd,Bu)-+Xsr+-T
(x3*1.9)
x 10-7
(2.6ztO.5)
x IO-’
(Bd.Bu,
-+X,,T+,-
(I.~~O.~)XIO-~
(Bd.B”)
-Ke+e-
(5.9*2.3)
x IO-’
< 1.2x
(4.0fl.5)
x 10-7
< 0.9 x 10-5
(Brj,B,)--‘KI.L+fi-
(4.Ozk1.5)
x IO--’
< 0.9 x lo-’
(Bd,
(2.3f0.9)
x lO-h
(B,.B,)-KP+II-
Bu
)-
K*e+e-
(l.5f0.6)x10-h
( Bd,Bu) -t XSYV (&j,Bu)+@‘fi (Bd,Bu)+bD
(4.0fl.O)
x 10-5
(2.3fl.5)
x lO-6
-+K*vD
(3.2*1.6)x
lO-6
(3.0&1.0)X
Bd-+YY
(1.2f0.8)
x IO-’
(7.4ztl.9)
x 10-7
t-
B,,+r
$
7 LC
< 7.7 x 10-4
(l.lf0.55)xl0-5
BS+YY
Bs -T+T-
lO-5
< 2.5 x lo-’
(Bd,Bu)-+K*P+P-
(Bd, B,)
< 1.6x
10-5
10-7
<
I.1 x 10-4
< 3.8 x lO-5
(3.1fl.9)x10-8 (3.5fl.O)
x 10-y
< 8.4 x lo-6
Bd ‘P+P+Bs+e e
(1.5f0.9)
x 10-l”
< 1.6x
(8.013.5)
x IO-l4
Bd -
(3.4f2.3)
x IO-t5
BS -
.u
e+e-
1501 I501
lO-h
I471 1471
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13
8-16
arising from the two FCNC four-Fermi operators ’ , which are not constrained by the B + X, + y data. Calling their coefficients Ca and Cia, it has been argued in Ref. [ 281 that the signs and magnitudes of all three coefficients Ctff, Cg and Cia can, in principle, be determined from the decays B 4 X, + y and B -+ X&!-. The SM-based rates for the decay b + &E-, calculated in the free quark decay approximation, have been known in the LO approximation for some time [ 291. The LO calculations have the unpleasant feature that the decay distributions and rates are scheme-dependent. The required NLO calculation is in the meanwhile available, which reduces the scheme-dependence of the LO effects in these decays [ 301. In addition, long-distance (LD) effects, which are expected to be very important in the decay B ---) X$0-. have also been estimated from data on the assumption that they arise dominantly due to the charmonium resonances .I/$ and $’ through the decay chains B --+ X,J/$($‘, . .) -+ X,l?e-. Likewise, the leading ( 1/rnt,‘) power corrections to the partonic decay rate and the dilepton invariant mass distribution have been calculated with the help of the operator product expansion in the effective heavy quark theory [ 3 1] The results of Ref. [ 3 1] have, however, not been confirmed in a recent independent calculation [ 321, which finds that the power corrections in the branching ratio B( B -+ X,P’e- ) are small (typically - 1.5%). The corrections in the dilepton mass spectrum and the FB asymmetry are also small over a good part of this spectrum. However, the end-point dilepton invariant mass spectrum is not calculable in the heavy quark expansion and will have to be modeled. Nonperturbative effects in B + X,e+!- have been estimated using the Fermi motion model in Ref. [33]. These effects are found to be small except for the end-point dilepton mass spectrum where they change the underlying parton model distributions significantly and have to be taken into account in the analysis of data [ 321. The amplitude for B -+ X,e’!- is calculated in the effective theory approach, which we have discussed earlier, by extending the operator basis of the effective Hamiltonian introduced in Eq. (6) : Fh( b -+ s + y; b + s + t+t-)
rare B decays than for example the decay B -+ X,y ,which constrains the magnitude of C;“. The sign of C7eff,which depends on the underlying physics, is not determined by the measurement of B(B --+ X, + y). This sign, which in our convention is negative in the SM, is in general model dependent. It is known (see for example Ref. [ 281) that in supersymmetric (SUSY) models, both the negative and positive signs are allowed as one scans over the allowed SUSY parameter space. We recall that for low dilepton masses, the differential decay rate for B + X,e+f!- is dominated by the contribution of the virtual photon to the charged lepton pair, which in turn depends on the effective Wilson coefficient Cqff. However, as is well known, the B -+ X,!+eamplitude in the standard model has two additional terms,
= 3iedb ---t s + y) -
%$4i, [CYOY+CI&I,] , di (32)
where the two additional operators are:
(33) ’ This
also holds for
two-Higgs ric models, involved
a large class of models
such as MSSM
doublet .models but not for all SM-extensions. for example,
there are additional
FCNC
In LR
four-Fermi
and the symmetoperators
(341.
I. THEORY
A. Ali/Nucl.
14
Instr. and Meth. in Phys. Res. A 384 (19%)
The analytic expressions for Cg( mw ) and Citi (mw) can be seen in Ref. [30] and will not be given here. We recall that the coefficient Cp in LO is scheme-dependent. However, this is compensated by an additional scheme-dependent part in the (one loop) matrix element of (39. We call the sum CG”, which is scheme-independentand enters in the physical decay amplitude given below:
C,;‘“(s^)Z
Csq(s^)
+
Y(s^).
(35)
The function Y(g) is the one-loop matrix element of 0s and can be seen in literature [ 30,7]. A useful quantity is the differential FB asymmetry in the c.m.s. of the dilepton defined in Ref. [ 35 ] : I
dd(s^) -= dS
s I,
however, not an independent measure, as it is directly proportional to the FB asymmetry discussed above. The relation is [32]:
I
A(;)
=Bx
A.
(39)
This is easy to notice if one writes the Mandelstam variable u(b) in the dilepton cm. and the B-hadron rest systems. Next, we discuss the effects of LD contributions in the processes B + X,e+e-. Note that the LD contributions due to the vector mesons such as J/I,+ and I++‘,as well as the continuum CCcontribution already discussed, appear as an effective (SLY~~L)(&V) interaction term only, i.e. in the operator 09. This implies that the LD-contributions should change C9 effectively, C7 as discussed earlier is dominated by the SD-contribution, and Crc has no LD-contribution. In accordance with this, the function Y(s^) is replaced by Y(i) -+ Y’(5) E Y(S) + Y,,(Z),
(40)
where Y,,( s^) is given as [ 351
-I
dB -dz
Y,,(i) = -$
dB dz’ s 0
(36) X
where z = cos 8, with 19being the angle between the lepton e+ and the b-quark. This can be expressed as:
dd(i) =-Bs,~~u2W d5
c
xc10 [S!J?(C,r”(s^)) +2c;“‘(1
+&]
(37)
The Wilson coefficients CJ’r, Ci’r and Ci(i appearing in the above equation and the dilepton spectrum (see, for example, Ref. [ 321) can be determined from data by solving the partial branching ratio B(As^) and partial FB asymmetry A( Ai), where A3 defines an interval in the dilepton invariant mass [28]. There are other quantities which one can measure in the decays B --+ X,e+t- to disentangle the underlying dynamics. We mention here the longitudinal polarization of the lepton in B + Xsl+C-, in particular in B --+ X,r’r-, proposed by Hewett [ 361. In a recent paper, Ktiiger and Sehgal [ 371 have stressed that complementary information is contained in the two orthogonal components of polarization (Pr, the component in the decay plane, and PN, the component normal to the decay plane), both of which are proportional to ml/mh. and therefore significant for the ~+r- channel. A third quantity, called energy asymmetry, proposed by Cho, Misiak and Wyler [ 381, defined as A=
R-16
N(E,-
> Eat)
N(E,-
> El+ ) + N(Er+
- N(Ep+
> Et--) > Et- )
(38)
where N( Ep- > Et+ ) denotes the number of lepton pairs where e+ is more energetic than C- in the B-rest frame, is,
(3CI + cz + 3c3 +
c
c4
+
3c5 + G)
VI-(Vi --t 1+1-)Mv,
v,=l/*.$‘.“’
Mt,
-
j;in~
-
ikfv,T~,
’
(41)
where K is a fudge factor, which appears due to the inadequacy of the factorization framework in describing data on B + J/$X,. The long-distance effects lead to significant interference effects in the dilepton invariant mass distribution and the FB asymmetry in B -+ Xs!+!- shown in Figs. 3 and 4, respectively. This can be used to test the SM, as the signs of the Wilson coefficients in general are model dependent. For further discussions we refer to Ref. [32] where also theoretical dispersion on the decay distributions due to various input parameters is worked out. Taking into account the spread in the values of the input parameters, /_L,A, m,, and &I. discussed in the previous section in the context of a(B -+ X, + y), we estimate the following branching ratios for the SD-piece only (i.e., from the intermediate top quark contribution only) [ 321: f?(B + X,e+e-) B(B --+ X,,u+,u-) f3(B -+ X,r+r-)
= (8.4 f 2.3) x 10e6, = (5.7f
1.2) x 10-h,
= (2.6 !c 0.5) x 10-7,
(42)
where theoretical errors and the error on &L have been added in quadrature. The present experimental limit for the inclusive branching ratio in B 3 X&e- is actually still the one set by the UA 1 collaboration some time ago [ 391, namely L3(B + X,p+p-) > 5.0 x 10e5. As far as we know, there are no interesting limits on the other two modes, involving X,e’e- and Xsr+r-.
A. Ali/Nucl.
Insrr. and Merh. in Phm
Fig. 3. Dilepton invariant mass distribution in B -+ X&t!in the SM with the next-to-leadingorder QCD correctionsand non-perturbativeeffects calculated in the Fermi motion model (solid curve), and including the LD-contributions (dashed curve). The model parameters (pi, mq) are indicated in the figure. Note that the height of the J/e peak is suppressed due to the linear scale. (Figure taken from Ref. I32 I .)
It is obvious from Fig. 3 that only in the dilepton mass region far away from the resonances there is a hope of extracting the Wilson coefficients governing the short-distance physics. The region below the J/r+0resonance is well suited for that purpose as the dilepton invariant mass distribution there is dominated by the SD-piece. Including the LDcontributions, the following branching ratio has been estimated for the dilepton mass range 0.2 5 ? < 0.36 in Ref. [32]: B(B -+ X,/A+,~A-) = (1.3 & 0.3) x l0-h,
(43)
with B(B -+ X,e+e-) N a(B -+ X,p+p”-). The FBasymmetry is estimated to be in the range IO-27%, as can be seen in Fig. 4. The experimental limits on the decay rates of the exclusive decays B + (K, K*)!?[ 26,401, while arguably closer 04,
I
Fig. 4. Normalized FB asymmetryin B + X&ein the SM as a function of the dilepton invariant mass calculated using the next-to-leading order QCD correction and the Fermi motion effects (solid curve), and including the LD-contributions (dashed curve). The Fermi motion parameters are indicated in the figure. (Figure taken from Ref. [ 321,)
Res. A 384 (1996)
IS
R-16
to the SM-based estimates, can only be interpreted in specific models of form factors, which hinders somewhat their transcription in terms of the information on the underlying Wilson coefficients. Using the exclusive-to-inclusive ratios R~rr s P(B -+ K!+!-)/P(B --* X,!+!-) = 0.07 f 0.02 and R~*pp 3 T(B -+ K*e+!-)/F(B + X,l’e-) = 0.27 f 0.0.07, which were estimated in Ref. [42]. the results are presented in Table I. In conclusion, the semileptonic FCNC decays B + X,e’!- (and also the exclusive decays) will provide very precise tests of the SM, as they will determine the signs and magnitudes of the three Wilson coefficients, CT. Ct” and CIII. This, perhaps, may also reveal physics beyond-the-SM if it is associated with not too high a scale. The MSSM model is a good case study where measurable deviations from the SM are anticipated and worked out [ 28,381. 2.3. Summary and overview of rare B decays in the SM The rare B decay mode B -+ X,r@, and some of the exclusive channels associated with it, have comparatively larger branching ratios. The estimated inclusive branching ratio in the SM is [42-441: D(B -+ X,V@) = (4.0f
1.0) x 10-j.
(4)
where the main uncertainty in the rates is due to the top quark mass. The scale-dependence, which enters indirectly through the top quark mass, has been brought under control through the NLL corrections, calculated in Ref. [45]. The corresponding CKM-suppressed decay B -+ X@ is related by the ratio of the CKM matrix element squared [ 421: (45) Similar relations hold for the ratios of the exclusive decay rates which depend additionally on the ratios of the form factors squared, which deviate from unity through SU( 3) breaking terms, in close analogy with the exclusive radiative decays discussed earlier. These decays are particularly attractive probes of the short-distance physics, as the longdistance contributions are practically absent in such decays. Hence, relations such as the one in (45) provide, in principle, one of the best methods for the determination of the CKM matrix element ratio ]&dj/l&,I [42]. From the practical point of view, however, these decay modes are rather difficult to measure, in particular at the hadron colliders and probably also at the B factories. The best chances are in the Z’-decays at LEP, from where the present best upper limit stems [46] : L3(B + XVE) < 7.7 x 10-4.
(46)
The estimated branching ratios in a number of inclusive and exclusive decay modes are given in Table 1, updating the estimates in Ref. [ 71.
1. THEORY
A. Ali/Mucl.
16
Instr. and Meth. in Phvs. Res. A 384 (1996)
Further down the entries in Table 1 are listed some twobody rare decays, such as (Bf, Bi) -+ 77, studied in Ref. [48], where only the lowest order contributions are calculated, i.e., without any QCD corrections, and the LD-effects, which could contribute significantly, are neglected. The decays (Bf, By) -+ efe- have been studied in the next-toleading order QCD in Ref. [45]. Some of them, in particular, the decays Bf + pFL+pL-and perhaps also the radiative decay By -+ yy, have a fighting chance to be measured at LHC. The estimated decay rates, which depend on the pseudoscalar coupling constant fa, (for B,-decays) and fa, (for Bd-decays) , together with the present experimental bounds are listed in Table 1. Since no QCD corrections have been included in the rate estimates of ( B,, Bd) -+ yy, the branching ratios are rather uncertain. The constraints on beyondthe-SM physics that will eventually follow from these decays are qualitatively similar to the ones that (would) follow from the decays B + X, + y and B + X,el^P-, which we have discussed at length earlier.
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NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
EISEVIER
Rare B decays in the Standard Model A. Ali * Deursches Elekrronen-Svnchrotron DESK Notkestrasse 85, D-22603 Hamburg, German>
Abstract We discuss the electromagnetic-penguin-dominated radiative B decays B + X, + y. B*(O) -+ K**‘“’ + y, and B, --+ (b + 7 in the context of the Standard Model (SM) and their Cabibbo-Kobayashi-Maskawa (CKM)-suppressed counterparts, B --+ xd + y, B* -+ p* + y, B” --t (p’,o) + y, and B, --t K” + y, using QCD sum rules for the exclusive decays. The importance of these decays in determining the parameters of the CKM matrix [l] is emphasized. The semileptonic decays B + XJJ+e- are also discussed in the context of the SM and their role in determining the Wilson coefficients of the effective theory is stressed. Comparison with the existing measurements are made and SM-based predictions for a large number of rare B decays are presented.
unitarity of the CKM matrix, the decay matrix element in the lowest order can be written as:
1. Estimates of B(B ---$ X, + y) and IV,1 in the Standard Model The Standard Model (SM) of particle physics does not admit flavour-changing-neutral-current (FCNC) transitions in the Born approximation. However, they are induced through the exchange of W* bosons in loop diagrams. The short-distance contribution in rare decays is dominated by the (virtual) top quark contribution. Hence the decay characteristics provide quantitative information on the top quark mass and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements v;; i = d, s, b [ 11. We shall discuss representative examples from several such transitions involving B decays, starting with the decay B + X, + y, which has been measured by CLEO [2]. This was preceded by the measurement of the exclusive decay B -+ K’ + y by the same collaboration [ 31. The present measurements give [4] : B( B + X, + y) = (2.32 * 0.57 * 0.35) x 10-4,
(1)
L3(B + K’ + y) = (4.2 f 0.8 & 0.6) x 10-5,
(7-j
yielding an exclusive-to-inclusive ratio:
These decay rates determine the ratio of the CKM matrix elements 1V, I/ 1Vcb1and the quantity Rx’ provides information on the decay form factor in B + K’ + y. In what follows we take up these points briefly. The leading contribution to b -+ s + y arises at one-loop from the so-called penguin diagrams. With the help of the
x
F2(Xc))qPEYSapv(mhR+
X
-(x-1)(8x2+5x-7)].
(5)
The measurement of the branching ratio for B -+ X, + y can be readily interpreted in terms of the CKM-matrix element product A,/IV,b( or equivalently IVBI/IVcbl. For a quantitative determination of 1V, I/ 1&b 1, however, QCD radiative corrections have to be computed and the contribution of the so-called long-distance effects estimated. The appropriate framework to incorporate QCD corrections is that of an effective theory obtained by integrating out the heavy degrees of freedom, which in the present context are the top quark and W’ bosons. The operator basis depends on the underlying theory and for the SM one has (keeping operators up to dimension 6) R
-+
S-k y) =
-4GFV,‘vb fi
@
(4)
msL)b.
F2(x) = 24(x _ 114 [6x( 3x - 2) log x
[email protected]
0168-9002/96/$15.00 Copyright PIISO168-9002(96)00908-4
-
where Xi = mj/mf, and qp and l,, are, respectively, the photon four-momentum and polarization vector. The GIM mechanism [ 5 ] is manifest in this amplitude and the CKMmatrix element dependence is factorized in At = t&V,,. The (modified) Inami-Lim function F2( Xi) derived from the (one-loop) penguin diagrams is given by [ 61:
‘,‘&(b l
(F2(xO
1996 Elsevier Science B.V. All rights reserved
~ci(I*)~i(~) i=l
3
(6)
A. Ali/Nucl.
Instr. and Meth. in Php.
where the operator basis, the “matching conditions” C,(mw ), and the solutions of the renormalization group
equations C, (1~) can be seen in Ref. [ 71. The perturbative QCD corrections to the decay rate r ( B -+ X, + y) have two distinct contributions: - Corrections to the Wilson coefficients Ci(,U), calculated with the help of the renormalization group equation, whose solution requires the knowledge of the anomalous dimension matrix in a given order in as. - Corrections to the matrix elements of the operators 0, entering through the effective Hamiltonian at the scale P = O(mh). The anomalous dimension matrix is needed in order to sum up large logarithms, i.e., terms like (Y:(mw) log”*( mh/bf). whereM=m,ormwandm
[ 91 and are found to be small. Next-to-leading order corrections to the matrix elements are now available completely. They are of two kinds: - QCD bremsstrahlung corrections b + sy + g, which are needed both to cancel the infrared divergences in the decay rate for B -+ X, + y and in obtaining a non-trivial QCD contribution to the photon energy spectrum in the inclusive decay B -+ X, + y: - next-to-leading order virtual corrections to the matrix elements in the decay b ---) s + y. The bremsstrahlung corrections were calculated in Refs. [ 10. I I ] in the truncated basis and last year also in the complete operator basis [ 12,131. The higher order matching conditions, i.e.. C;( mw ) , are known up to the desired accuracy, i.e., up to O( CI~(Mw ) ) terms [ 141. The next-to-leading order virtual corrections have also been calculated [ 151. They reduce the scale-dependence of the inclusive decay width. The branching ratio f3(B + X, + y) can be expressed in terms of the semileptonic decay branching ratio
B(B --+ X&Y)=
r(B+y+X,) rSL
1
Ih
f3(B -
Xev,, . (7)
where the leading-order QCD corrected expression for rs~ can be seen in Ref. [ 71. The leading order ( 1/mh) power corrections in the heavy quark expansion are identical in the inclusive decay rates for B -+ X, + y and B + XI&, entering in the numerator and denominator in the square bracket, respectively, and hence drop out. In Ref. [7], the present theoretical errors on the branching ratio B( B -P X,y) are discussed, yielding: B( B + X, + y) = (3.20 f 0.30 f 0.38 + 0.32) x IO-’
Rex A 3U4
9
f1996)X-16
the extrinsic sources (such as A ( mh), A ( BRSL1). and the third error is an estimate (f 10%) of the NLO anomalous dimension piece in Cy’, the coefficient of the magnetic moment operator. Combining the theoretical errors in quadrature gives [7] a(B
+ X, + y) = (3.20f
0.58) x lo-’
1
(9)
which is compatible with the present measurement X, + y) = (2.32f 0.67) x 10-j [2]. Expressed of the CKM matrix element ratio, one gets IVSl/l&,
= 0.85 f O.l2(expt)
t3( B --) in terms
rfr:O.lO(th) .
( IO)
which is within errors consistent with unity, as expected from the unitarity of the CKM matrix.
2. Inclusive
radiative decays B -+ X, + y
The theoretical interest in studying the (CKMsuppressed) inclusive radiative decays B --t X,1 + y lies in the first place in the possibility of determining the parameters of the CKM matrix. We shall use the Wolfenstein parametrization Il6], in which case the matrix is determined in terms of the four parameters A. A = sin Bc. /J and 71. The quantity of interest in the decays B + Xd + y is the end-point photon energy spectrum, which has to be measured requiring that the hadronic system Xd recoiling against the photon does not contain strange hadrons to suppress the large-E, photons from the decay B + X,, + y. Assuming that this is feasible, one can determine from the ratio of the decay rates B(B -+ Xd + y) /B( B - X, + y) the CKM-Wolfenstein parameters p and 7~. This measurement was first proposed in Ref. [ I1 1. where the photon energy spectra were also worked out. In close analogy with the B -+ X, + y case discussed earlier, the complete set of dimension-6 operators relevant for the processes b 4 dy and b -+ dyg can be written as: x
‘&r(b + d) = -%& fi
c C,(P)B,(P). ,=I
(11)
where 5, = V,hVjz for j = t,c, u. The operators Gj. j = 1,2. have implicit in them CKM factors. In the Wolfenstein parametrization [ 161, one can express these factors as: 5” = AA+p - iq),
& = -A,?,
51 = -&
- &.
( 12)
We note that all three CKM-angle-dependent quantities li are of the same order of magnitude, 0( A’). This aspect can be taken into account by defining the operators 6, and & entering in ‘H,fy(b -+ d) as follows [ I I ] :
(8) where the first error comes from the combined effect of Am, and Ap ( the scale dependence), the second error arises from
1.
THEOKY
IO
A. Ali/Nucl.
Insrr.
and
Meth.
in Phys. Res. A 384 (19%)
8-16
2.1. 8( B -+ V + y) and constraints on the CKM parameters
with the rest of the operators (&I; j = 3.. .8) defined like their counterparts Oj in 7&(b -+ s), with the obvious replacement s -+ d. With this choice, the matching conditions Cj (mw ) and the solutions of the RG equations yielding Cj (p) become identical for the two operator bases Oj and dj. The essential difference between T(B + X, + y) and I(B --) Xd + y) lies in the matrix elements of the first two operators 01 and 02 (in ‘&s(b --) s)) and 81 and 82 (in &r( b -+ d) ). The branching ratio B( B --) Xd + y) in the SM can be written as:
-Cl-PP2-q&+04),
(14)
where the functions Di depend on the parameters ml, mh, mc, ,u, as well as the others we discussed in the context of a( B --) X, + y). These functions were first calculated in Ref. [ 111 in the leading logarithmic approximation. Recently, these estimates have been improved in Ref. [ 191, making use of the NLO calculations in Ref. [ 151. To get an estimate of the inclusive branching ratio, the CKM parameters p and v have to be constrained from the unitarity fits. Present data and theory restrict them to lie in the following range (at 95% CL) [ 171:
-0.35 I p < 0.35,
(15)
which, on using the current lower bound from LEP on the By-3 mass difference AM, > 9.2 ps-’ [20], is reduced to -0.25 5 p 5 0.35 using & = 1.l, where & is the SU( 3) -breaking parameter & = fa,&,/&,&, . The preferred CKM-fit values are (p. 77) = (0.05,0.36), for which one gets [ 191 X
1o-5,
(16)
whereas a( B --+ Xd + y) = 8.0 x lo-” and 2.8 X lo-’ for the other two extremes p = 0.35. 17= 0.40 and p = -3 = -0.25, respectively. In conclusion, we note that the functional dependence of B(B -+ Xd + y) on the Wolfenstein parameters (p, 77) is mathematically different than that of AM,. However, qualitatively they are very similar. From the experimental point of view, the situation p < 0 is favourable for both the measurements as in this case one expects (relatively) smaller values for AM, and larger values for the branching ratio f?( B -+ Xd + y), as compared to the p > 0 case which would yield larger AM, and smaller B(B ---) xd
+
r>.
Here V is a vector meson with the polarization vector e’“) , V = p, w, K’ or 4; B is a generic B-meson B*, B” or B,, and + stands for the field of a light u, d or s quark. The vectOrS pB, pv and q = pB - pv correspond to the 4-momenta of the initial B-meson and the outgoing vector meson and photon, respectively. In Eq. (17) the QCD renormalization of the &rPvq’b operator is implied. Keeping only the SD-contribution leads to obvious relations among the exclusive decay rates, exemplified here by the decay rates for (B,,Bd) -+p+yand(B,,Bd) -+K*+y: I((B$,B:) r((B,f,Bj)
0.20 5 q < 0.52,
S(B 4 Xd + y) = 1.63
Exclusive radiative B decays B + V + y, with V = K’, p, w, are also potentially very interesting from the point of view of determining the CKM parameters [ 211. The extraction of these parameters would, however, involve a trustworthy estimate of the SD- and LD-contributions in the decay amplitudes. The SD-contribution in the exclusive decays (B*, B”) -+ (K**,K*‘) +y. (B*:,B’) --) (p*.~‘) +y,B’-+ w-ty and the corresponding B, decays, B, -+ $I + y and B, -+ K” + y, involve the magnetic moment operator 07 and the related one obtained by the obvious change s --+ d, &. The transition form factors governing the radiative B decays B -+ V + y can be generically defined as:
--) (P*,P’) +Y) ---) (K**,K*O) + y)
tv,l
“vd[ jiq
2 1 ’
(18)
where @u.d is a phase-space factor which in all cases is close to 1 and pi E [Fs(Bi -+ m)/Fs(Bi + K’y)]‘. The transition form factors FS are model dependent. Estimates of Fs in the QCD sum rule approach in the normalization of Eq. ( 17) range between Fs (B -+ K’y) = 0.3 1 (Narison in Ref. [22]) to Fs(B -+ K’y) = 0.37 (Ball in Ref. [22]), with a typical error of *15%, and hence are all consistent with each other. This, for example, gives RK* = 0.16*0.05, using the result from Ref. [ 211, which is in good agreement with data. The ratios of the form factors, i.e. Ki, should therefore be reliably calculable as they depend essentially only on the SU( 3)-breaking effects which have been estimated [ 21,221. The LD-amplitudes in radiative B decays from the light quark intermediate states necessarily involve other CKM matrix elements. Hence, the simple factorization of the decay rates in terms of the CKM factors involving It&] and IV,/ no longer holds, thereby invalidating relation ( 18) given above. In the decays B 4 V + y they are induced by the matrix elements of the four-Fennion operators 81 and 82 (likewise 01 and 02). Estimates of these contributions re-
A. Ali/Nucl.
Instr. and Meth. in Phys. Rex A 384 (1996)
8-16
11
Thus, the CKM factors suppress the LD-contributions. The analogous LD-contributions to the neutral B decays B” -+ py and B” -+ oy are expected to be much smaller. The corresponding form factors for the decays + p”(w) y are obtained from the ones for the decay ;: + p*y discussed above by the replacement of the light quark charges e, -+ ed. which gives the factor - I/2; in addition, and more importantly, the LD-contribution to the neutral B decays is colour-suppressed, which reflects itself through the replacement of the factor al by a~. This yields for the ratio
quire non-perturbative methods. This problem has been investigated in Refs. [23,24] using a technique [25] which treats the photon emission from the light quarks in a theoretically consistent and model-independent way. This has been combined with the light-cone QCD sum rule approach to calculate both the SD and LD - parity conserving and parity violating - amplitudes in the decays (B’, B”) --) (p*, p/w) + y. To illustrate this, we concentrate on the B* decays B* -+ p* + y and take up the neutral B decays B” -+ p(o) + y at the end. The LD-amplitude of the four-Fermion operators 61, 82 is dominated by the contribution of the weak annihilation of valence quarks in the B meson and it is color-allowed for the decays of charged B* mesons. Using factorization, the LD-amplitude in the decay B, -+ p* + y can be written in terms of the form factors Ff and Fk,
where the numbers are based on using az/ar = 0.27 k 0.10
dronr = - *
[26]. This would then yield R$Tfl which in turn gives
V”r,V;L,(c2 + +I>
m,@
El’)
c
v5
1a2Fk(q2) fe ““uPp,qp.
cn=R
Bf
+p*Y
L/S
=-
eda
N
eual
L/s
-0.13 f 0.05)
(24)
N R$yy
= 0.05,
dB”PY
(25)
shoti 2&q*))
J
.
(19)
‘Fs’
2
=00155*00010 *
*
(20)
where the errors correspond to the variation of the Bore1 parameter in the QCD sum rules. Including other possible uncertainties, one expects an accuracy of the ratios in (20) of order 20%. The parity-conserving and parity-violating amplitudes turn out to be numerically close to each other in the QCD sum rule approach, Fk N Fk G FL, hence the ratio of the LD- and the SD- contributions reduces to a number r241 A long
R
Bf-.p*‘y
Ions < 0.02. dr5O-m -
Again, one has to invoke a model to calculate the form factors. Estimates from the light-cone QCD sum rules give [24]: 3 =00125*0.0010 Fs ’
~a’-PY r-/s
&bV$
(21)
qf.
This, as well as the estimate in Eq. (23), should be taken only as indicative in view of the approximations made in Refs. [ 23,241. That the LD-effects remain small in B” + p-y has been supported in a recent analysis based on the soft-scattering of on-shell hadronic decay products B0 -+ pop0 -+ py [ 271, though this paper estimates them somewhat higher (between 4% and 8%). Restricting to the colour-allowed LD-contributions, the relations, which are obtained ignoring such contributions (and isospin invariance), r(B*
+ p*y) =2r(B”
+ $y)
=2r(B”
-+ WY), (26)
get modified to
W* -+P*Y) 2r(B0 --+ py)
w* +
P'Y) = 2r(BO -+ oy) 2
Using C2 = I .lO, CI = -0.235, CtR = -0.306 from Ref. 171 (corresponding to the scale p = 5 GeV) gives: R
B* --‘PiY
_
4dmp(C2
L/S
+ Cl/NC)
mbC7eff =
= 1 + 2Rrlst&i
F:i*p*r F,B*
(l-P)*+11*
+P*Y
-0.30 f 0.07 (
(22)
which is not small. To get a ball-park estimate of the ratio drong/dsbon. we take the central value from the CKM fits, yielding ]&b]/]xd:dlN 0.33 [ 171: A long A shon I-1
B f -4
Y
= (R,B,~-p*y,~
N 10%.
--PI-T2
Al
(23)
+
(RL,s)~K:
P’
+v2
c1 _ pj2
where RLIS z R$*p*y.
+72
.
(27)
The ratio r(B*
+ P*Y)12w” + py) (= T’(B* -+ p*y)/2T(B” -+ WY)) is shown in Fig. 1as a function of the parameter p, with q= 0.2, 0.3 and 0.4. This suggests that a measurement of this ratio would constrain the Wolfenstein parameters (p, q), with the dependence on p mo& marked than on v. In particular, a negative value of p leads to a constructive interference in B, + py
I. THEORY
A. Ali/Nucl.
12
Instr.
Meth.
and
in Phy.
Res.
A 384 (1996)
X-16
LD-contributions is modest but not negligible, introducing an uncertainty comparable to the -15% uncertainty in the overall normalization due to the SU( 3) -breaking effects in the quantity K”. Neutral B-meson radiative decays are less prone to the LD-effects, as argued above, and hence one expects that to a good approximation (say, better than 10%) the ratio of the decay rates for neutral B meson obtained in the approximation of SD-dominance remains valid [ 2 I ] :
5
Fig.
I.
Ratio
r(B*
--t
p*y)/2r(
eter p. with 7 = 0.4
of
7
the B” -
= 0.2
(long-dashed
\
I
I
neutral py)
curve).
and
charged
as a function
(short-dashed
r&++~;;jy)=KdA*[ (1 -
curve).
(Figure
7
B-decay
of the Wolfenstein = 0.3
taken from
Ref.
(solid
rates param-
curve),
and
[ 241.)
decays, while large positive values of p give a destructive interference. The ratio of the CKM-suppressed and CKM-allowed decay rates for charged B mesons gets modified due to the LD contributions. Following earlier discussion, we ignore the LD-contributions in T(B -+ K*y). The ratio of the decay rates in question can therefore be written as:
P*Y)
+ K**y)
X
(29)
where this relation holds for each of the two decay modes separately. Finally, combining the estimates for the LD- and SD-form factors in Ref. [ 241 and Ref. [ 211, respectively, and restricting the Wolfenstein parameters in the range -0.25 < p _< 0.35 and 0.2 5 77 5 0.4, as suggested by the CKM-fits [ 171, we give the following ranges for the absolute branching ratios:
a( B” -+ py) N a( B” + wy) = (0.65 f 0.35) x 10-h, (30)
r(B* + T(B*
/I)’ + 7’1 ,
\
1 +
=K”A2[(1
-p?+$l
P( (,
+
2RLlsKd
1 - PI _ Pj2
77* ,+
P2 + v2 _Pj2+772
+(RL,s)~~+,
Using the central value from the estimates of the ratio of the form factors squared K” = 0.59 * 0.08 [2l], we show the ratio (28) in Fig. 2 as a function of p for 7 = 0.2.0.3, and 0.4. It is seen that the dependenceof this ratio is rather weak on v but it depends on p rather sensitively. The effect of the
where we have used the experimental value for the branching ratio a( B + K* + y) [ 31, adding the errors in quadrature. The large error reflects the poor knowledge of the CKM matrix elements and hence experimental determination of these branching ratios will put rather stringent constraints on the Wolfenstein parameter p. In addition to studying the radiative penguin decays of the B,f and Bi mesons discussed above, hadron machines such as HERA-B will be in a position to study the corresponding decays of the Bf meson and Ah baryon, such as By + 4 + y and Ah + A + y, which have not been measured so far. We list below the branching ratios in a number of interesting decay modes calculated in the QCD sum rule approach in Ref. [2l]. f?(B, -+ @y) = a(Bd --+ K’y) = (4.2 & 2.0) U(Bs
-+
x
1W5,
K*Y)
@Bd --+ K’Y) =+ &B,
--) K*y) = (0.75 f 0.5) x 1O-6.
The estimated branching ratios in. a number of inclusive and exclusive radiative B decay modes are given in Table 1, where we have also listed the branching ratios for B, + yy and Bd -+ yy. Fig. 2. Ratio rates: r(Bu
of the CKM-suppressed -
the Wolfenstein (solid 1241
curve),
)
py)/lY(B
-
parameter and TJ = 0.4
K’y) p, with
and CKM-allowed (with
B = B,
7 = 0.2
(long-dashed
or Bd)
(short-dashed
curve).
(Figure
radiative
B-decay
as a function curve),
7 = 0.3
taken from
2.2. Inclusive rare decays B -+ X,$?‘e- in the SM
of
Ref.
The decays B + Xse+e-, with ! = e, ,u, 7, provide a more sensitive search strategy for finding new physics in
A. Ali/Nucl.
I
Table
Estimates model
of the branching
taking
into
short-distance
account
contributions -+
+ Y and B”
been included. MeV
o)
except
for
branching
Experimental
Decay
in the input
parameters
correspond
the radiative
decays B* effects
as
to the -+
have also
ratios, we have used fB,
= 200
measurements
limits
and upper
and systematic
to give the quoted experimental
modes
in the standard
long-distance
are also listed. In the second row, the statistical have been combined
B-decays
in the second column
+ Y. where
For the two-body I .l6.
for FCNC
the uncertainties
only,
(#,
fB,/fBd =
and
fractions
[ 71. Theentries
discussed in Ref.
p*
Instr. and Meth. in Phy.
uncertainties
uncertainty.
Measurements
B(SM)
90%
CL
and
upper limits
(3.2f0.6)
x 1O-4
(2.3f0.7)
x 1O-4
I21
(B*,B”)-K*Y
(4.Ozt2.0)
x lO-5
(4.2Srl.0)
x IO-’
[41
(B*,B”)+X‘,y
(1.6+1.2)x
lo--’
(B*.B”)
+Xsy
B*+$
+y
(l.5fl.l)x10-6
B” +
p” + Y
(0.651kO.35)
B” -+6JfY
Bs-+K*
+y f-
(B,j.Bu)dXse
e
(Bd,B,)+Xde+e+
(Bd.Bu)+XsP
P
-
< 3.9 x 10-5
x IO-’
< I.3 x 10-5
141 141 141
< 3.6 x lO-5
I491
1401 I401 1401 I401 I471 1461
<1.1x10-~
(0.65f0.35)
Bs-4+Y
x IO-”
(4.2zt2.0)
x lO-5
(0.8+0.5)
x~O-~
(8.4f2.2)
x lO-6
(4.9f2.9)
x IO-’
(5.7*1.2)x10-”
(Bd,Bu)- x,,P+/.‘(Bd,Bu)-+Xsr+-T
(x3*1.9)
x 10-7
(2.6ztO.5)
x IO-’
(Bd.Bu,
-+X,,T+,-
(I.~~O.~)XIO-~
(Bd.B”)
-Ke+e-
(5.9*2.3)
x IO-’
< 1.2x
(4.0fl.5)
x 10-7
< 0.9 x 10-5
(Brj,B,)--‘KI.L+fi-
(4.Ozk1.5)
x IO--’
< 0.9 x lo-’
(Bd,
(2.3f0.9)
x lO-h
(B,.B,)-KP+II-
Bu
)-
K*e+e-
(l.5f0.6)x10-h
( Bd,Bu) -t XSYV (&j,Bu)+@‘fi (Bd,Bu)+bD
(4.0fl.O)
x 10-5
(2.3fl.5)
x lO-6
-+K*vD
(3.2*1.6)x
lO-6
(3.0&1.0)X
Bd-+YY
(1.2f0.8)
x IO-’
(7.4ztl.9)
x 10-7
t-
B,,+r
$
7 LC
< 7.7 x 10-4
(l.lf0.55)xl0-5
BS+YY
Bs -T+T-
lO-5
< 2.5 x lo-’
(Bd,Bu)-+K*P+P-
(Bd, B,)
< 1.6x
10-5
10-7
<
I.1 x 10-4
< 3.8 x lO-5
(3.1fl.9)x10-8 (3.5fl.O)
x 10-y
< 8.4 x lo-6
Bd ‘P+P+Bs+e e
(1.5f0.9)
x 10-l”
< 1.6x
(8.013.5)
x IO-l4
Bd -
(3.4f2.3)
x IO-t5
BS -
.u
e+e-
1501 I501
lO-h
I471 1471
Res. A 384 (1996)
13
8-16
arising from the two FCNC four-Fermi operators ’ , which are not constrained by the B + X, + y data. Calling their coefficients Ca and Cia, it has been argued in Ref. [ 281 that the signs and magnitudes of all three coefficients Ctff, Cg and Cia can, in principle, be determined from the decays B 4 X, + y and B -+ X&!-. The SM-based rates for the decay b + &E-, calculated in the free quark decay approximation, have been known in the LO approximation for some time [ 291. The LO calculations have the unpleasant feature that the decay distributions and rates are scheme-dependent. The required NLO calculation is in the meanwhile available, which reduces the scheme-dependence of the LO effects in these decays [ 301. In addition, long-distance (LD) effects, which are expected to be very important in the decay B ---) X$0-. have also been estimated from data on the assumption that they arise dominantly due to the charmonium resonances .I/$ and $’ through the decay chains B --+ X,J/$($‘, . .) -+ X,l?e-. Likewise, the leading ( 1/rnt,‘) power corrections to the partonic decay rate and the dilepton invariant mass distribution have been calculated with the help of the operator product expansion in the effective heavy quark theory [ 3 1] The results of Ref. [ 3 1] have, however, not been confirmed in a recent independent calculation [ 321, which finds that the power corrections in the branching ratio B( B -+ X,P’e- ) are small (typically - 1.5%). The corrections in the dilepton mass spectrum and the FB asymmetry are also small over a good part of this spectrum. However, the end-point dilepton invariant mass spectrum is not calculable in the heavy quark expansion and will have to be modeled. Nonperturbative effects in B + X,e+!- have been estimated using the Fermi motion model in Ref. [33]. These effects are found to be small except for the end-point dilepton mass spectrum where they change the underlying parton model distributions significantly and have to be taken into account in the analysis of data [ 321. The amplitude for B -+ X,e’!- is calculated in the effective theory approach, which we have discussed earlier, by extending the operator basis of the effective Hamiltonian introduced in Eq. (6) : Fh( b -+ s + y; b + s + t+t-)
rare B decays than for example the decay B -+ X,y ,which constrains the magnitude of C;“. The sign of C7eff,which depends on the underlying physics, is not determined by the measurement of B(B --+ X, + y). This sign, which in our convention is negative in the SM, is in general model dependent. It is known (see for example Ref. [ 281) that in supersymmetric (SUSY) models, both the negative and positive signs are allowed as one scans over the allowed SUSY parameter space. We recall that for low dilepton masses, the differential decay rate for B + X,e+f!- is dominated by the contribution of the virtual photon to the charged lepton pair, which in turn depends on the effective Wilson coefficient Cqff. However, as is well known, the B -+ X,!+eamplitude in the standard model has two additional terms,
= 3iedb ---t s + y) -
%$4i, [CYOY+CI&I,] , di (32)
where the two additional operators are:
(33) ’ This
also holds for
two-Higgs ric models, involved
a large class of models
such as MSSM
doublet .models but not for all SM-extensions. for example,
there are additional
FCNC
In LR
four-Fermi
and the symmetoperators
(341.
I. THEORY
A. Ali/Nucl.
14
Instr. and Meth. in Phys. Res. A 384 (19%)
The analytic expressions for Cg( mw ) and Citi (mw) can be seen in Ref. [30] and will not be given here. We recall that the coefficient Cp in LO is scheme-dependent. However, this is compensated by an additional scheme-dependent part in the (one loop) matrix element of (39. We call the sum CG”, which is scheme-independentand enters in the physical decay amplitude given below:
C,;‘“(s^)Z
Csq(s^)
+
Y(s^).
(35)
The function Y(g) is the one-loop matrix element of 0s and can be seen in literature [ 30,7]. A useful quantity is the differential FB asymmetry in the c.m.s. of the dilepton defined in Ref. [ 35 ] : I
dd(s^) -= dS
s I,
however, not an independent measure, as it is directly proportional to the FB asymmetry discussed above. The relation is [32]:
I
A(;)
=Bx
A.
(39)
This is easy to notice if one writes the Mandelstam variable u(b) in the dilepton cm. and the B-hadron rest systems. Next, we discuss the effects of LD contributions in the processes B + X,e+e-. Note that the LD contributions due to the vector mesons such as J/I,+ and I++‘,as well as the continuum CCcontribution already discussed, appear as an effective (SLY~~L)(&V) interaction term only, i.e. in the operator 09. This implies that the LD-contributions should change C9 effectively, C7 as discussed earlier is dominated by the SD-contribution, and Crc has no LD-contribution. In accordance with this, the function Y(s^) is replaced by Y(i) -+ Y’(5) E Y(S) + Y,,(Z),
(40)
where Y,,( s^) is given as [ 351
-I
dB -dz
Y,,(i) = -$
dB dz’ s 0
(36) X
where z = cos 8, with 19being the angle between the lepton e+ and the b-quark. This can be expressed as:
dd(i) =-Bs,~~u2W d5
c
xc10 [S!J?(C,r”(s^)) +2c;“‘(1
+&]
(37)
The Wilson coefficients CJ’r, Ci’r and Ci(i appearing in the above equation and the dilepton spectrum (see, for example, Ref. [ 321) can be determined from data by solving the partial branching ratio B(As^) and partial FB asymmetry A( Ai), where A3 defines an interval in the dilepton invariant mass [28]. There are other quantities which one can measure in the decays B --+ X,e+t- to disentangle the underlying dynamics. We mention here the longitudinal polarization of the lepton in B + Xsl+C-, in particular in B --+ X,r’r-, proposed by Hewett [ 361. In a recent paper, Ktiiger and Sehgal [ 371 have stressed that complementary information is contained in the two orthogonal components of polarization (Pr, the component in the decay plane, and PN, the component normal to the decay plane), both of which are proportional to ml/mh. and therefore significant for the ~+r- channel. A third quantity, called energy asymmetry, proposed by Cho, Misiak and Wyler [ 381, defined as A=
R-16
N(E,-
> Eat)
N(E,-
> El+ ) + N(Er+
- N(Ep+
> Et--) > Et- )
(38)
where N( Ep- > Et+ ) denotes the number of lepton pairs where e+ is more energetic than C- in the B-rest frame, is,
(3CI + cz + 3c3 +
c
c4
+
3c5 + G)
VI-(Vi --t 1+1-)Mv,
v,=l/*.$‘.“’
Mt,
-
j;in~
-
ikfv,T~,
’
(41)
where K is a fudge factor, which appears due to the inadequacy of the factorization framework in describing data on B + J/$X,. The long-distance effects lead to significant interference effects in the dilepton invariant mass distribution and the FB asymmetry in B -+ Xs!+!- shown in Figs. 3 and 4, respectively. This can be used to test the SM, as the signs of the Wilson coefficients in general are model dependent. For further discussions we refer to Ref. [32] where also theoretical dispersion on the decay distributions due to various input parameters is worked out. Taking into account the spread in the values of the input parameters, /_L,A, m,, and &I. discussed in the previous section in the context of a(B -+ X, + y), we estimate the following branching ratios for the SD-piece only (i.e., from the intermediate top quark contribution only) [ 321: f?(B + X,e+e-) B(B --+ X,,u+,u-) f3(B -+ X,r+r-)
= (8.4 f 2.3) x 10e6, = (5.7f
1.2) x 10-h,
= (2.6 !c 0.5) x 10-7,
(42)
where theoretical errors and the error on &L have been added in quadrature. The present experimental limit for the inclusive branching ratio in B 3 X&e- is actually still the one set by the UA 1 collaboration some time ago [ 391, namely L3(B + X,p+p-) > 5.0 x 10e5. As far as we know, there are no interesting limits on the other two modes, involving X,e’e- and Xsr+r-.
A. Ali/Nucl.
Insrr. and Merh. in Phm
Fig. 3. Dilepton invariant mass distribution in B -+ X&t!in the SM with the next-to-leadingorder QCD correctionsand non-perturbativeeffects calculated in the Fermi motion model (solid curve), and including the LD-contributions (dashed curve). The model parameters (pi, mq) are indicated in the figure. Note that the height of the J/e peak is suppressed due to the linear scale. (Figure taken from Ref. I32 I .)
It is obvious from Fig. 3 that only in the dilepton mass region far away from the resonances there is a hope of extracting the Wilson coefficients governing the short-distance physics. The region below the J/r+0resonance is well suited for that purpose as the dilepton invariant mass distribution there is dominated by the SD-piece. Including the LDcontributions, the following branching ratio has been estimated for the dilepton mass range 0.2 5 ? < 0.36 in Ref. [32]: B(B -+ X,/A+,~A-) = (1.3 & 0.3) x l0-h,
(43)
with B(B -+ X,e+e-) N a(B -+ X,p+p”-). The FBasymmetry is estimated to be in the range IO-27%, as can be seen in Fig. 4. The experimental limits on the decay rates of the exclusive decays B + (K, K*)!?[ 26,401, while arguably closer 04,
I
Fig. 4. Normalized FB asymmetryin B + X&ein the SM as a function of the dilepton invariant mass calculated using the next-to-leading order QCD correction and the Fermi motion effects (solid curve), and including the LD-contributions (dashed curve). The Fermi motion parameters are indicated in the figure. (Figure taken from Ref. [ 321,)
Res. A 384 (1996)
IS
R-16
to the SM-based estimates, can only be interpreted in specific models of form factors, which hinders somewhat their transcription in terms of the information on the underlying Wilson coefficients. Using the exclusive-to-inclusive ratios R~rr s P(B -+ K!+!-)/P(B --* X,!+!-) = 0.07 f 0.02 and R~*pp 3 T(B -+ K*e+!-)/F(B + X,l’e-) = 0.27 f 0.0.07, which were estimated in Ref. [42]. the results are presented in Table I. In conclusion, the semileptonic FCNC decays B + X,e’!- (and also the exclusive decays) will provide very precise tests of the SM, as they will determine the signs and magnitudes of the three Wilson coefficients, CT. Ct” and CIII. This, perhaps, may also reveal physics beyond-the-SM if it is associated with not too high a scale. The MSSM model is a good case study where measurable deviations from the SM are anticipated and worked out [ 28,381. 2.3. Summary and overview of rare B decays in the SM The rare B decay mode B -+ X,r@, and some of the exclusive channels associated with it, have comparatively larger branching ratios. The estimated inclusive branching ratio in the SM is [42-441: D(B -+ X,V@) = (4.0f
1.0) x 10-j.
(4)
where the main uncertainty in the rates is due to the top quark mass. The scale-dependence, which enters indirectly through the top quark mass, has been brought under control through the NLL corrections, calculated in Ref. [45]. The corresponding CKM-suppressed decay B -+ X@ is related by the ratio of the CKM matrix element squared [ 421: (45) Similar relations hold for the ratios of the exclusive decay rates which depend additionally on the ratios of the form factors squared, which deviate from unity through SU( 3) breaking terms, in close analogy with the exclusive radiative decays discussed earlier. These decays are particularly attractive probes of the short-distance physics, as the longdistance contributions are practically absent in such decays. Hence, relations such as the one in (45) provide, in principle, one of the best methods for the determination of the CKM matrix element ratio ]&dj/l&,I [42]. From the practical point of view, however, these decay modes are rather difficult to measure, in particular at the hadron colliders and probably also at the B factories. The best chances are in the Z’-decays at LEP, from where the present best upper limit stems [46] : L3(B + XVE) < 7.7 x 10-4.
(46)
The estimated branching ratios in a number of inclusive and exclusive decay modes are given in Table 1, updating the estimates in Ref. [ 71.
1. THEORY
A. Ali/Mucl.
16
Instr. and Meth. in Phvs. Res. A 384 (1996)
Further down the entries in Table 1 are listed some twobody rare decays, such as (Bf, Bi) -+ 77, studied in Ref. [48], where only the lowest order contributions are calculated, i.e., without any QCD corrections, and the LD-effects, which could contribute significantly, are neglected. The decays (Bf, By) -+ efe- have been studied in the next-toleading order QCD in Ref. [45]. Some of them, in particular, the decays Bf + pFL+pL-and perhaps also the radiative decay By -+ yy, have a fighting chance to be measured at LHC. The estimated decay rates, which depend on the pseudoscalar coupling constant fa, (for B,-decays) and fa, (for Bd-decays) , together with the present experimental bounds are listed in Table 1. Since no QCD corrections have been included in the rate estimates of ( B,, Bd) -+ yy, the branching ratios are rather uncertain. The constraints on beyondthe-SM physics that will eventually follow from these decays are qualitatively similar to the ones that (would) follow from the decays B + X, + y and B + X,el^P-, which we have discussed at length earlier.
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