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Comment on QCD sum rules and weak bottom decays
Nuclear Physics B219 (1983) 1-11 © North-Holland Publishing Company
COMMENT ON QCD SUM RULES AND WEAK BOTTOM DECAYS Branko GUBERINA* and Bruno MACHET**
Centre de Physique Th~orique, CNRS, Marseille, France Received 30 August 1982 (Revised 12 January 1983)
QCD sum rules derived by Bourrely et al. are applied to B-decays to get a lower and an upper bound for the decay rate. The sum rules are shown to be essentially controlled by the large mass scales involved in the process. These bounds combined with the experimental value of BR (B --, evX) provide an upper bound for the lifetime of the B + meson. A comparison is made with D-meson decays.
1. Introduction QCD sum rules provide us with a very powerful tool to study hadron physics. In a number of influential papers [1-3], they have been applied to the study of perturbative and non-perturbative effects, current quark masses, hadronic properties, etc. QCD sum rules have also been shown [2] to constrain the semileptonic decay rates. Under certain conditions one is able to get bounds on the partial rates a n d / o r lifetimes of the particles involved. The method could be very useful to study the heavy quark decays for the following reasons. (i) Our knowledge of the heavy quark decays is rather poor. One is forced to rely on the spectator decay model which was questioned in the case of D-decays where there are indications that additional mechanisms might be very important. (ii) Even if the additional mechanisms invoked in D-decays are expected to be less important or even negligible in B-decays, the spectator model in itself contains uncertainties about what kind of quark mass to be used, how to put phase space corrections (heavy particles in the final states), etc. Therefore, we find it important to try to study B-decays from the point of view of QCD sum rules. In a previous paper [2b] the decay D+--* K°e÷pe was studied along with 7re~and K e3 decays. It was shown that the form factors are constrained by the knowledge of the two-point function r Z . ' ( q ) = i f d 4 x e iq'x(OlTV ~'(x)v~*(0)10)
* Present address: Rudjer Bo~kovi6 Institute, Zagreb, Croatia, Yugoslavia. ** Present address: Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
in the deep euclidean region. The advantage of the method is that one needs not assume ad hoc functional dependences of the form factors. Once the value of the f+ form factor at q2 = 0 is fixed, the QCD sum rules give model-independent upper and lower bounds for the decay rate. The interesting property of the bounds is that, once combined with, say, inclusive semileptonic branching ratios, they provide an upper limit for the D ÷ lifetime. In the following we apply the techniques of ref. [2] to B-decays and show that in that case the QCD sum rules strongly constrain the form factors and lifetime. The bounds we obtain are given up to the value of the Kobayashi-Maskawa (KM) angles [4] and phases, which are not very well known. In that way, there is only one parameter left in addition, the value of the weak meson form factor at q2 = 0. In sect. 2 we briefly review for the sake of completeness the methods of ref. [2]. The results and discussions are given in sect. 3.
2. Review of the method
The hadronic part of the decay B +---, D°e+ve (see fig. 1 for notation) is governed by the matrix element
(D-°(p')lV~(O)lB+(p)) = ( p + p ' ) j + ( t ) + ( p - p ' ) j _ ( t ) ,
(2.1)
where V~(x) is the hadronic vectorial current of the b and c quarks V
(2.2)
(x) =
and t the momentum transfer squared t = (p - p,)2 -_ qZ.
(2.3)
Eq. (2.1) defines the two form factors f+ and f_. The decay rate can be expressed as
e÷
Fig. 1. The decay B + ~ D°e+ve at first order in the weak interaction.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
an integral over the physical domain of
3
t
m ~ < t ~< t, - (Ms
-
MD)
2,
(2.4)
r=G~l~bl =
1 ft, dt( m 2'2 (2,n.) 3 16M3"m~ t 1 - ~ d) X'/2(M2, M2, t)
2 I 2 X{m2eld(t)12+(2t+m,)~X(MB, MZ, t)lf(t)12),
(2.5)
where G F is the Fermi constant and Ucb is the appropriate combination of the KM mixing parameters (2.6)
f c b "~- ClC2S 3 "b $2C3eia .
Further, m e is the lepton (electron) mass, and
h(a, b, c) = a 2 + b 2 + c 2 - 2 a b - 2 a c - 2bc.
(2.7)
The form factor f ( t ) is directly proportional to f÷(t):
f(t)=2f+(t),
(2.8)
and d(t) is the so-called scalar form factor d(t) = ( M 2 - M~)f+(t)
+ tf_(t).
(2.9)
In the case where the outgoing lepton is an electron, one can see that the Id(t)[ 2 dependence in (2.5) may be neglected. The 2-point function relative to the hadronic current V~(x)
Ht,~(q) = ifd4xeiqx(OlTV,(x)V,*(O)[O),
(2.10)
may be decomposed as follows: / 7 ~ ( q ) = - (&,~q2_ q~G)H(q2)+ g~D(q2)
(2.11)
and computed perturbatively for spacelike q2. Using the positivity of its spectral function and saturating it by the intermediate state [B+D °) we obtain, via a dispersion relation, the inequality 2
x
O
2
2
~
Q2v/(t - q)(t - to)
212a,
4
B. Guberina,B. Machet / QCD sum rulesand weak bottom decays
which entails
2
Q2 [(t_t,)(t_to)]3/2lf(t)12.
-16or - 2 ,0 dt 1- -t3
(t + Q2)2
(2.12b)
Here Q2= _q2, q2< 0 where t I is defined in (2.4) and t o = ( M B + M D ) 2 is the beginning of the cut in the complex, t plane for the form factors f ( t ) and d(t). Once x(Q 2) is known, it is a standard technique based on analyticity and first suggested by Okubo [2] to derive from (2.12b) upper and lower bounds for If(t)[ in the physical domain defined by (2.4), which depend on X(Q 2), Q 2 and f+(0). The complete formulas are given in the appendix. From these, eq. (2.5) provides the corresponding bounds for the decay rate. COMPUTATIONOF x(Q 2) The exact expression for x(Q2), at the one-loop level is [2b] ~/'3 1 Q22x2(1 - x ) 2 x(')(O2)= 4orZ jo dx ~2x + ~2(1 _ x) + O2x( 1 - x)
(2.13)
All the dependence on the external renormalization has disappeared when we have differentiated/-/(Q 2) with respect to Q 2 (see (2.12a)). In eq. (2.13) ~ and ~ j denote the running masses of the c and b quarks. The 2-loop corrections are (we use the MS renormalization scheme): x(2)(a2)__- 1 a~(__Q2) 1 + O 4or 2 or
In m-~
'
(2.14)
and, as far as we can reliably take the heavy quarks vacuum condensates equal to zero, the only leading non-perturbative term is
1 2or as(O[Fa~Fa~l O) x"p(Q2)= 4or 2 3 Q,
(2.15)
The running mass is related to the invariant mass rh as ~ ( Q 2 ) __
rh (½ ln( Q2/A2 ) )-v,/a~ '
(2.16)
and the running coupling constant is given by 2or as (Q2) = _ fl,ln(QE/A2 )"
(2.17)
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
5
Here fit and "it are the first coefficients of the expansion in powers of a s of the functions fl(as) and V(as) respectively associated to the coupling constant and mass renormalization: 3N2-1 "Y'=2 2 N c ' fl'={(-llNc+2n')' (2.18) with Nc the number of colors and n f the number of flavors. We have taken the invariant masses as follows [5]: rh b = 7.9 GeV,
rh c = 1.8 GeV,
A = 0.150 GeV.
(2.19)
The above invariant masses correspond to the usual values of the running masses of current quarks ~c(rnc) = 1.25 GeV and r o b ( r o b ) ----4.25 GeV. 3. Results and discussion Naively one would expect the bounds for B-decays to be worse than for D-decays. This follows simply since one is forced in the B-meson case to go to a much higher Q2 value and the bounds generally diverge with Q2. However we found that the methods described in ref. [2b] work much better for B-decays due to the non-trivial Q2 dependance. From the main formula (A.7) one can see that the bounds are rather complicated functions of Q2 and the thresholds t o and t 1. We study that next in more detail. The function x ( Q 2 ) that enters the bound (A.7) receives contributions from both the perturbative and the non-perturbative part. The perturbative part (without QCD corrections) is plotted in fig. 2 for the B- and D-decays. One sees immediately that
)~ (o') .O3
.02
.01
0
10
20
Gev
Fig. 2. Plot of the functionx(Q 2) at the one-looplevel.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
Ct)
/"
s
/"
,,,'"
2
/" s"
r.,.(o)_-I
,-"
~ I=1 Gev I ---
DL'K-~e'v"~/t= . 2 G e v
~....""
....
1.5 f
0.5
i: 5Gev ~ --
....... ,,,
B'-,~',,~. ~:IG,~ ~
N ....
0
,
10
.
(~) ,
,,
,
,
20
.
.
.
.
11-
GeM
Fig. 3. Q2 dependence of the bounds for f+(t), taking f+(0) = 1, for the cases B+---,D°e+p e and D + ~ ~ 0 e + ~e.
x ( Q 2) approaches the asymptotic value (4¢r2) - 1 faster for D-decays. For large Q2 (we estimate = 30 GeV 2 for D-decays and - 100 GeV 2 for B), Xd)(Q 2) should be a very good approximation for the exact value of x ( Q 2 ) *. However, contrary to the case of D-decays, where one is forced to go to lower Q2 in order to get better bounds, the bounds for B-decay are very stable for Q2 high enough such that one could use only the perturbative part. The situation is quite clear if one plots the bounds on the form factors f+(t) as a function of Q2, for different values of t. Fig. 3 shows such a plot for the choice f+(0) = 1. It is obvious that the bounds for f÷(t) are much better for B-decays than for D-decays, indicating the non-trivial role played by thresholds t o and t r That becomes even more evident after integration over t: the bounds for F(B + ~ DOe +ve) and F ( D + ~ K.°e+ve) are plotted in fig. 4 as functions of 0 2 for fixed f÷(0). Both rates are divided by KM factors in order to avoid uncertainties in their values. The bounds are of course valid for Q2 large enough, since the non-perturbative and a s corrections are not taken into account. From fig. 4 one sees that the bounds are rather stable in the region Q2 _ (10-20 GeV) 2 i.e. in the range where non-perturbative terms mq(OlqqlO)/Q 2 and * For values smaller than -- 30 and = 100 GeV 2, this calculation cannot be trusted.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
/uo4
7
£+(0)=1
luo,i
10""Gev 10~Gev
/,'"
3
/
// //'" /
2 I"" /
.... °f
............... .
.
.
.
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,
.
.
.
.
fl 0
,
,
Q
,
20
,
,
)
Gev
Fig. 4. Q2 dependence of the bounds for IUcbl-2r(B+-, D°e+ve) and ]U~I-2F(D+--* K,°e+Ve) taking f + ( O ) = 1.
as(OJF2lO)/Q 4 are completely negligible. The a s corrections are given by (2.14), where the first term gives - 5% correction in x(Q 2) for Q2 = 100 GeV 2 and the unknown coefficient in the second term is multiplied by a number of the order 0.3. Fig. 5 shows a final plot for the bounds on IU~b1-2F(B+--* D°e+Ve) as a function of f+BO(0). Once the decay B + ~ D°e+p e is measured, the bounds obtained may be used to get information on Ucb and/or f+aO(0). The lower bound on F ( B + ~ D°e+ve) could be used to give an upper limit on the B-meson lifetime:
~-B~< B R ( B + ~ D°e+Pe) Inf £(B +-~ pOe + re)"
(3.1)
Since the BR in (3.1) isnot measured, one could use instead the measured value for the BR (B --* evX) [6], i.e.
zB
BR(B+- evX) r(B+-
(3.2)
This bound is plotted in fig. 6, as a function off+(0), for Q2 _ 100 OeV 2. In order to get ~'a, one has to multiply the plotted bound by [Ucb [ -2. Using Sakurai's value
8
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
10"' Gev
/"~'~.'~,/lu,,l' / Q = 10 Gev
Z
t
,, I
t
/" .,
b
,,,.'
i -" r s/ I /
.
.
.
.
I
.
.
.
.
1
.
.
.
•5
.
:11
r+(o>
Fig. 5. Bounds for F(B +--* D°e+ve) versus f+(0) for Q2
=
100 GeV 2.
IU cbl 2 MAX r~.
B.
I 0"14S
7
I
•
6
=
5 4
3 2
1
0
.5
1
F.,.(0)
Fig. 6. Upper bound for the product IUcbI2TB*for Q2 = 100 GeV 2.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
9
[7], I U~bl 2 = 0.2, and BR(B ---}evX) = (13.6 + 2.1 _ 1.7)% one ends up with a bound* ~B ~ 6.6 × 10-14 sec
(3.3)
for the choice of f ÷ ( 0 ) = 1. If we allow + 15% variation in f+(0), the bound stays in the range - ( 5 - 1 0 ) x 10 -t4 sec. Our bounds are sensitive to choice of f+(0). In principle one would expect f+(0) to be different from the SU(nf) symmetry value, since the flavor symmetry is badly broken. In fact, Bonvin and Schmid recently proposed [8] extracting f÷(0) for D ~ Key from experiment. Surprisingly enough, if one uses the recent Moriond - r7 average [9] "rD+t .8 +2"3~ _ t.5/ × 10-t3 sec one ends up with f+°r(0)= 1.07. Although the experimental errors are sizeable, i.e. of the order - 25%, this seems to indicate that f÷(0) = 1 as far as the masses involved in the transition are comparable. It would be useful to compare our bound on ~'B with predictions from the spectator model in order to get a feeling of how restrictive the bounds are. We do this using the experimental value for BR(B ~ evX) and the spectator estimate for F(B --* evX) in order to avoid an estimate of the non-leptonic contribution to the total rate. This procedure worked rather well for the D-decays [10]. In the spirit of the spectator model we use the dressed quark mass in calculations, m b = 4.8 GeV, which is very well defined and known [5, 11]. We estimate the rate F(B --, Dev) following a method described by Fakirov and Stech [12], assuming a constant f÷ = 1. Both approaches are normalized to Sakurai's values of K M parameters. One obtains
r ( B - , De ) r = F(B ---, evX) - 0.3, F(B --* evX) = 9.3 x 1012 sec- 1
(3.4) (3.5)
which gives ~'B = 1.5 × 10- ~4 sec.
(3.6)
One sees that our bound (3.3) is roughly 4 times larger than the estimate (3.6). In fact if one uses r = 0.3, i.e. BR(B --* D e v ) - 4%, the bound (3.1) comes very close to the spectator model calculation. In other words, the bound is very restrictive, provided the spectator model is the correct one. In conclusion, we believe that the bounds presented here provide us with a sort of independent method for the study of B-decays. These bounds can be useful to test the mutual consistency of the estimates obtained under different assumptions. * Implicit assumption here is that the SL branching ratio is the same for B° and B+ !
10
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
We would like to thank Eduardo de Rafael for many stimulating discussions and reading the manuscript. We acknowledge interesting discussions with Chris Quigg and the members of CPT section 2 in Marseille.
APPENDIX
EXACT EXPRESSIONSFOR THE BOUNDS ON f÷(t) For t ~
,0gUTz(t)
~+~oo
'
(A.1)
which corresponds to a conformal mapping in the complex t plane
( t - - t o ) 1/2
.l+z t -1- i - ~ =
\
to
(A.2)
"
Notice that z(0) = 0. Then we define by standard methods the function ~o(z)
- -
) 3/2
~ f t 0 -- t I + 1 + z to ~
q~(:) = 8 6 ¢ ~
f
2
l+,/2
(1 + z)2.
(a.3)
The constraint is given by the positivity of the determinant x ( Q 2)
f+ (0) ¢p(0) 1
1
1 - zZ(0)
1 - z(O)z(t)
1
1
1 - z(O)z(t)
1 - z2(t)
f+(0)~(0)
f+(t)~(z(t))
f+(t)ep(z(t)) >i 0,
(A.4)
i.e.
-B + vt~-AC A
- B - 7r-~-AC <~f+(t)<~ A '
(A.5)
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
11
with A = - ¢ p ( Z ( t ) ) 2,
(A.6a)
s
(A.6b)
z2(t) C = x ( Q 2) 1 - z2(t)
f2(0) ¢p2(0) 1 -z2(t)
(A.6c)
The final expression for the bounds can be written as
f+(O)
~p(z(t))
1-
1 - z2(t)
ep(0)
<<.f+(t)~
[J 1+
x(Q
f+2(O)ep2(O)
z2(t)
1~ t )
J
x(Q2) f2(0)cp:(o)
1
(A.7)
References [1] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385, 448, 519; L.J. Reinders, H.R. Rubinstein and S. Yazald, Nucl. Phys. B186 (1981) 109; Phys. Lett. 94B (1980) 203; 95B (1980) 103 [2] S. Okubo, Phys. Rev. D3 (1971) 2807; D4 (1971) 725; I. Fu. Shih and S. Okubo, Phys. Rev. D4 (1971) 3519 [2b] C. Bourrely, B. Machet and E. de Rafael, Nucl. Phys. B189 (1981) 157 [3] S. Narison and E. de Rafael, Phys. Lett. 103B (1981) 57; S. Narison, N. Paver, E. de Rafael and D. Treleani, Annecy Preprint, LAPP-TH 56 (1982) [4] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 0973) 652 [5] J. Gasser and H. Leutwyler, Phys. Reports 87 (1982) 79 [6] A. Silverman, in Proc. 1981 Int. Sym. on Lepton and photon interactions at high energies, p. 138, edit. W. Pfeil, Bonn 1981 [7] S.G. Wojcicki, talk at the annual meeting of the Division of Particle and Fields of the A.P.S., Santa Cruz, September 1981, SLAC Publ. 2837 (1981) [8] M. Bonvin and C. Schmid, Nucl. Phys. B194 (1982) 319 [9] F.C. Porter, SLAC-PUB-2895, March 1982 [10] N. Cabibbo and L. Maiani, Phys. Lett. 79B (1978) 109 [11] M.B. Voloshin, Sov. J. Nucl. Phys. 29 (1979) 703 [12] D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315 [13] H. Fritzsch, in Proc. 1981 Int. Sym. on Lepton and photon interactions at high energies, p. 786, edit. W. Pfeil, Bonn 1981; J.P. Leveille, University of Michigan Preprint UM HE 81-18 (1981); B. Guberina, Orsay Preprint LPTHE 82/5 (1982)
COMMENT ON QCD SUM RULES AND WEAK BOTTOM DECAYS Branko GUBERINA* and Bruno MACHET**
Centre de Physique Th~orique, CNRS, Marseille, France Received 30 August 1982 (Revised 12 January 1983)
QCD sum rules derived by Bourrely et al. are applied to B-decays to get a lower and an upper bound for the decay rate. The sum rules are shown to be essentially controlled by the large mass scales involved in the process. These bounds combined with the experimental value of BR (B --, evX) provide an upper bound for the lifetime of the B + meson. A comparison is made with D-meson decays.
1. Introduction QCD sum rules provide us with a very powerful tool to study hadron physics. In a number of influential papers [1-3], they have been applied to the study of perturbative and non-perturbative effects, current quark masses, hadronic properties, etc. QCD sum rules have also been shown [2] to constrain the semileptonic decay rates. Under certain conditions one is able to get bounds on the partial rates a n d / o r lifetimes of the particles involved. The method could be very useful to study the heavy quark decays for the following reasons. (i) Our knowledge of the heavy quark decays is rather poor. One is forced to rely on the spectator decay model which was questioned in the case of D-decays where there are indications that additional mechanisms might be very important. (ii) Even if the additional mechanisms invoked in D-decays are expected to be less important or even negligible in B-decays, the spectator model in itself contains uncertainties about what kind of quark mass to be used, how to put phase space corrections (heavy particles in the final states), etc. Therefore, we find it important to try to study B-decays from the point of view of QCD sum rules. In a previous paper [2b] the decay D+--* K°e÷pe was studied along with 7re~and K e3 decays. It was shown that the form factors are constrained by the knowledge of the two-point function r Z . ' ( q ) = i f d 4 x e iq'x(OlTV ~'(x)v~*(0)10)
* Present address: Rudjer Bo~kovi6 Institute, Zagreb, Croatia, Yugoslavia. ** Present address: Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
in the deep euclidean region. The advantage of the method is that one needs not assume ad hoc functional dependences of the form factors. Once the value of the f+ form factor at q2 = 0 is fixed, the QCD sum rules give model-independent upper and lower bounds for the decay rate. The interesting property of the bounds is that, once combined with, say, inclusive semileptonic branching ratios, they provide an upper limit for the D ÷ lifetime. In the following we apply the techniques of ref. [2] to B-decays and show that in that case the QCD sum rules strongly constrain the form factors and lifetime. The bounds we obtain are given up to the value of the Kobayashi-Maskawa (KM) angles [4] and phases, which are not very well known. In that way, there is only one parameter left in addition, the value of the weak meson form factor at q2 = 0. In sect. 2 we briefly review for the sake of completeness the methods of ref. [2]. The results and discussions are given in sect. 3.
2. Review of the method
The hadronic part of the decay B +---, D°e+ve (see fig. 1 for notation) is governed by the matrix element
(D-°(p')lV~(O)lB+(p)) = ( p + p ' ) j + ( t ) + ( p - p ' ) j _ ( t ) ,
(2.1)
where V~(x) is the hadronic vectorial current of the b and c quarks V
(2.2)
(x) =
and t the momentum transfer squared t = (p - p,)2 -_ qZ.
(2.3)
Eq. (2.1) defines the two form factors f+ and f_. The decay rate can be expressed as
e÷
Fig. 1. The decay B + ~ D°e+ve at first order in the weak interaction.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
an integral over the physical domain of
3
t
m ~ < t ~< t, - (Ms
-
MD)
2,
(2.4)
r=G~l~bl =
1 ft, dt( m 2'2 (2,n.) 3 16M3"m~ t 1 - ~ d) X'/2(M2, M2, t)
2 I 2 X{m2eld(t)12+(2t+m,)~X(MB, MZ, t)lf(t)12),
(2.5)
where G F is the Fermi constant and Ucb is the appropriate combination of the KM mixing parameters (2.6)
f c b "~- ClC2S 3 "b $2C3eia .
Further, m e is the lepton (electron) mass, and
h(a, b, c) = a 2 + b 2 + c 2 - 2 a b - 2 a c - 2bc.
(2.7)
The form factor f ( t ) is directly proportional to f÷(t):
f(t)=2f+(t),
(2.8)
and d(t) is the so-called scalar form factor d(t) = ( M 2 - M~)f+(t)
+ tf_(t).
(2.9)
In the case where the outgoing lepton is an electron, one can see that the Id(t)[ 2 dependence in (2.5) may be neglected. The 2-point function relative to the hadronic current V~(x)
Ht,~(q) = ifd4xeiqx(OlTV,(x)V,*(O)[O),
(2.10)
may be decomposed as follows: / 7 ~ ( q ) = - (&,~q2_ q~G)H(q2)+ g~D(q2)
(2.11)
and computed perturbatively for spacelike q2. Using the positivity of its spectral function and saturating it by the intermediate state [B+D °) we obtain, via a dispersion relation, the inequality 2
x
O
2
2
~
Q2v/(t - q)(t - to)
212a,
4
B. Guberina,B. Machet / QCD sum rulesand weak bottom decays
which entails
2
Q2 [(t_t,)(t_to)]3/2lf(t)12.
-16or - 2 ,0 dt 1- -t3
(t + Q2)2
(2.12b)
Here Q2= _q2, q2< 0 where t I is defined in (2.4) and t o = ( M B + M D ) 2 is the beginning of the cut in the complex, t plane for the form factors f ( t ) and d(t). Once x(Q 2) is known, it is a standard technique based on analyticity and first suggested by Okubo [2] to derive from (2.12b) upper and lower bounds for If(t)[ in the physical domain defined by (2.4), which depend on X(Q 2), Q 2 and f+(0). The complete formulas are given in the appendix. From these, eq. (2.5) provides the corresponding bounds for the decay rate. COMPUTATIONOF x(Q 2) The exact expression for x(Q2), at the one-loop level is [2b] ~/'3 1 Q22x2(1 - x ) 2 x(')(O2)= 4orZ jo dx ~2x + ~2(1 _ x) + O2x( 1 - x)
(2.13)
All the dependence on the external renormalization has disappeared when we have differentiated/-/(Q 2) with respect to Q 2 (see (2.12a)). In eq. (2.13) ~ and ~ j denote the running masses of the c and b quarks. The 2-loop corrections are (we use the MS renormalization scheme): x(2)(a2)__- 1 a~(__Q2) 1 + O 4or 2 or
In m-~
'
(2.14)
and, as far as we can reliably take the heavy quarks vacuum condensates equal to zero, the only leading non-perturbative term is
1 2or as(O[Fa~Fa~l O) x"p(Q2)= 4or 2 3 Q,
(2.15)
The running mass is related to the invariant mass rh as ~ ( Q 2 ) __
rh (½ ln( Q2/A2 ) )-v,/a~ '
(2.16)
and the running coupling constant is given by 2or as (Q2) = _ fl,ln(QE/A2 )"
(2.17)
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
5
Here fit and "it are the first coefficients of the expansion in powers of a s of the functions fl(as) and V(as) respectively associated to the coupling constant and mass renormalization: 3N2-1 "Y'=2 2 N c ' fl'={(-llNc+2n')' (2.18) with Nc the number of colors and n f the number of flavors. We have taken the invariant masses as follows [5]: rh b = 7.9 GeV,
rh c = 1.8 GeV,
A = 0.150 GeV.
(2.19)
The above invariant masses correspond to the usual values of the running masses of current quarks ~c(rnc) = 1.25 GeV and r o b ( r o b ) ----4.25 GeV. 3. Results and discussion Naively one would expect the bounds for B-decays to be worse than for D-decays. This follows simply since one is forced in the B-meson case to go to a much higher Q2 value and the bounds generally diverge with Q2. However we found that the methods described in ref. [2b] work much better for B-decays due to the non-trivial Q2 dependance. From the main formula (A.7) one can see that the bounds are rather complicated functions of Q2 and the thresholds t o and t 1. We study that next in more detail. The function x ( Q 2 ) that enters the bound (A.7) receives contributions from both the perturbative and the non-perturbative part. The perturbative part (without QCD corrections) is plotted in fig. 2 for the B- and D-decays. One sees immediately that
)~ (o') .O3
.02
.01
0
10
20
Gev
Fig. 2. Plot of the functionx(Q 2) at the one-looplevel.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
Ct)
/"
s
/"
,,,'"
2
/" s"
r.,.(o)_-I
,-"
~ I=1 Gev I ---
DL'K-~e'v"~/t= . 2 G e v
~....""
....
1.5 f
0.5
i: 5Gev ~ --
....... ,,,
B'-,~',,~. ~:IG,~ ~
N ....
0
,
10
.
(~) ,
,,
,
,
20
.
.
.
.
11-
GeM
Fig. 3. Q2 dependence of the bounds for f+(t), taking f+(0) = 1, for the cases B+---,D°e+p e and D + ~ ~ 0 e + ~e.
x ( Q 2) approaches the asymptotic value (4¢r2) - 1 faster for D-decays. For large Q2 (we estimate = 30 GeV 2 for D-decays and - 100 GeV 2 for B), Xd)(Q 2) should be a very good approximation for the exact value of x ( Q 2 ) *. However, contrary to the case of D-decays, where one is forced to go to lower Q2 in order to get better bounds, the bounds for B-decay are very stable for Q2 high enough such that one could use only the perturbative part. The situation is quite clear if one plots the bounds on the form factors f+(t) as a function of Q2, for different values of t. Fig. 3 shows such a plot for the choice f+(0) = 1. It is obvious that the bounds for f÷(t) are much better for B-decays than for D-decays, indicating the non-trivial role played by thresholds t o and t r That becomes even more evident after integration over t: the bounds for F(B + ~ DOe +ve) and F ( D + ~ K.°e+ve) are plotted in fig. 4 as functions of 0 2 for fixed f÷(0). Both rates are divided by KM factors in order to avoid uncertainties in their values. The bounds are of course valid for Q2 large enough, since the non-perturbative and a s corrections are not taken into account. From fig. 4 one sees that the bounds are rather stable in the region Q2 _ (10-20 GeV) 2 i.e. in the range where non-perturbative terms mq(OlqqlO)/Q 2 and * For values smaller than -- 30 and = 100 GeV 2, this calculation cannot be trusted.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
/uo4
7
£+(0)=1
luo,i
10""Gev 10~Gev
/,'"
3
/
// //'" /
2 I"" /
.... °f
............... .
.
.
.
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,
.
.
.
.
fl 0
,
,
Q
,
20
,
,
)
Gev
Fig. 4. Q2 dependence of the bounds for IUcbl-2r(B+-, D°e+ve) and ]U~I-2F(D+--* K,°e+Ve) taking f + ( O ) = 1.
as(OJF2lO)/Q 4 are completely negligible. The a s corrections are given by (2.14), where the first term gives - 5% correction in x(Q 2) for Q2 = 100 GeV 2 and the unknown coefficient in the second term is multiplied by a number of the order 0.3. Fig. 5 shows a final plot for the bounds on IU~b1-2F(B+--* D°e+Ve) as a function of f+BO(0). Once the decay B + ~ D°e+p e is measured, the bounds obtained may be used to get information on Ucb and/or f+aO(0). The lower bound on F ( B + ~ D°e+ve) could be used to give an upper limit on the B-meson lifetime:
~-B~< B R ( B + ~ D°e+Pe) Inf £(B +-~ pOe + re)"
(3.1)
Since the BR in (3.1) isnot measured, one could use instead the measured value for the BR (B --* evX) [6], i.e.
zB
BR(B+- evX) r(B+-
(3.2)
This bound is plotted in fig. 6, as a function off+(0), for Q2 _ 100 OeV 2. In order to get ~'a, one has to multiply the plotted bound by [Ucb [ -2. Using Sakurai's value
8
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
10"' Gev
/"~'~.'~,/lu,,l' / Q = 10 Gev
Z
t
,, I
t
/" .,
b
,,,.'
i -" r s/ I /
.
.
.
.
I
.
.
.
.
1
.
.
.
•5
.
:11
r+(o>
Fig. 5. Bounds for F(B +--* D°e+ve) versus f+(0) for Q2
=
100 GeV 2.
IU cbl 2 MAX r~.
B.
I 0"14S
7
I
•
6
=
5 4
3 2
1
0
.5
1
F.,.(0)
Fig. 6. Upper bound for the product IUcbI2TB*for Q2 = 100 GeV 2.
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
9
[7], I U~bl 2 = 0.2, and BR(B ---}evX) = (13.6 + 2.1 _ 1.7)% one ends up with a bound* ~B ~ 6.6 × 10-14 sec
(3.3)
for the choice of f ÷ ( 0 ) = 1. If we allow + 15% variation in f+(0), the bound stays in the range - ( 5 - 1 0 ) x 10 -t4 sec. Our bounds are sensitive to choice of f+(0). In principle one would expect f+(0) to be different from the SU(nf) symmetry value, since the flavor symmetry is badly broken. In fact, Bonvin and Schmid recently proposed [8] extracting f÷(0) for D ~ Key from experiment. Surprisingly enough, if one uses the recent Moriond - r7 average [9] "rD+t .8 +2"3~ _ t.5/ × 10-t3 sec one ends up with f+°r(0)= 1.07. Although the experimental errors are sizeable, i.e. of the order - 25%, this seems to indicate that f÷(0) = 1 as far as the masses involved in the transition are comparable. It would be useful to compare our bound on ~'B with predictions from the spectator model in order to get a feeling of how restrictive the bounds are. We do this using the experimental value for BR(B ~ evX) and the spectator estimate for F(B --* evX) in order to avoid an estimate of the non-leptonic contribution to the total rate. This procedure worked rather well for the D-decays [10]. In the spirit of the spectator model we use the dressed quark mass in calculations, m b = 4.8 GeV, which is very well defined and known [5, 11]. We estimate the rate F(B --, Dev) following a method described by Fakirov and Stech [12], assuming a constant f÷ = 1. Both approaches are normalized to Sakurai's values of K M parameters. One obtains
r ( B - , De ) r = F(B ---, evX) - 0.3, F(B --* evX) = 9.3 x 1012 sec- 1
(3.4) (3.5)
which gives ~'B = 1.5 × 10- ~4 sec.
(3.6)
One sees that our bound (3.3) is roughly 4 times larger than the estimate (3.6). In fact if one uses r = 0.3, i.e. BR(B --* D e v ) - 4%, the bound (3.1) comes very close to the spectator model calculation. In other words, the bound is very restrictive, provided the spectator model is the correct one. In conclusion, we believe that the bounds presented here provide us with a sort of independent method for the study of B-decays. These bounds can be useful to test the mutual consistency of the estimates obtained under different assumptions. * Implicit assumption here is that the SL branching ratio is the same for B° and B+ !
10
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
We would like to thank Eduardo de Rafael for many stimulating discussions and reading the manuscript. We acknowledge interesting discussions with Chris Quigg and the members of CPT section 2 in Marseille.
APPENDIX
EXACT EXPRESSIONSFOR THE BOUNDS ON f÷(t) For t ~
,0gUTz(t)
~+~oo
'
(A.1)
which corresponds to a conformal mapping in the complex t plane
( t - - t o ) 1/2
.l+z t -1- i - ~ =
\
to
(A.2)
"
Notice that z(0) = 0. Then we define by standard methods the function ~o(z)
- -
) 3/2
~ f t 0 -- t I + 1 + z to ~
q~(:) = 8 6 ¢ ~
f
2
l+,/2
(1 + z)2.
(a.3)
The constraint is given by the positivity of the determinant x ( Q 2)
f+ (0) ¢p(0) 1
1
1 - zZ(0)
1 - z(O)z(t)
1
1
1 - z(O)z(t)
1 - z2(t)
f+(0)~(0)
f+(t)~(z(t))
f+(t)ep(z(t)) >i 0,
(A.4)
i.e.
-B + vt~-AC A
- B - 7r-~-AC <~f+(t)<~ A '
(A.5)
B. Guberina, B. Machet / QCD sum rules and weak bottom decays
11
with A = - ¢ p ( Z ( t ) ) 2,
(A.6a)
s
(A.6b)
z2(t) C = x ( Q 2) 1 - z2(t)
f2(0) ¢p2(0) 1 -z2(t)
(A.6c)
The final expression for the bounds can be written as
f+(O)
~p(z(t))
1-
1 - z2(t)
ep(0)
<<.f+(t)~
[J 1+
x(Q
f+2(O)ep2(O)
z2(t)
1~ t )
J
x(Q2) f2(0)cp:(o)
1
(A.7)
References [1] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385, 448, 519; L.J. Reinders, H.R. Rubinstein and S. Yazald, Nucl. Phys. B186 (1981) 109; Phys. Lett. 94B (1980) 203; 95B (1980) 103 [2] S. Okubo, Phys. Rev. D3 (1971) 2807; D4 (1971) 725; I. Fu. Shih and S. Okubo, Phys. Rev. D4 (1971) 3519 [2b] C. Bourrely, B. Machet and E. de Rafael, Nucl. Phys. B189 (1981) 157 [3] S. Narison and E. de Rafael, Phys. Lett. 103B (1981) 57; S. Narison, N. Paver, E. de Rafael and D. Treleani, Annecy Preprint, LAPP-TH 56 (1982) [4] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 0973) 652 [5] J. Gasser and H. Leutwyler, Phys. Reports 87 (1982) 79 [6] A. Silverman, in Proc. 1981 Int. Sym. on Lepton and photon interactions at high energies, p. 138, edit. W. Pfeil, Bonn 1981 [7] S.G. Wojcicki, talk at the annual meeting of the Division of Particle and Fields of the A.P.S., Santa Cruz, September 1981, SLAC Publ. 2837 (1981) [8] M. Bonvin and C. Schmid, Nucl. Phys. B194 (1982) 319 [9] F.C. Porter, SLAC-PUB-2895, March 1982 [10] N. Cabibbo and L. Maiani, Phys. Lett. 79B (1978) 109 [11] M.B. Voloshin, Sov. J. Nucl. Phys. 29 (1979) 703 [12] D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315 [13] H. Fritzsch, in Proc. 1981 Int. Sym. on Lepton and photon interactions at high energies, p. 786, edit. W. Pfeil, Bonn 1981; J.P. Leveille, University of Michigan Preprint UM HE 81-18 (1981); B. Guberina, Orsay Preprint LPTHE 82/5 (1982)