The width of the decay B0 → π+e−ve in QCD

Volume 229, number 1,2

PHYSICS LETTERS B

5 October 1989

THE WIDTH OF THE DECAY B °--,n+e-ge I N QCD A.A. O V C H I N N I K O V Institute for Nuclear Research of the Academy of Sciences of the USSR, 117312 Moscow, USSR Received 8 March 1989; revised manuscript received 10 July 1989

E

The formfactor of the decay B°--,n +e- 9~ is found in the physical region of the transverse momentum squared by means of the QCD sum rules method. The full width of the decay is estimated. The results can serve as a basis for the determination of the Kobayashi-Maskawa matrix element for the b~u transition.

1. Introduction A complete determination o f the K o b a y a s h i Maskawa matrix elements is a very important problem o f electroweak and strong interaction theory at present. For instance the determination o f the matrix element Vbu corresponding to the transition b--.u would allow to connect the recently measured B°-B ° oscillation parameter with such an important parameter o f the theory as the t-quark mass (see ref. [ 1 ] for example). The hope for the determination o f Vbu is mainly connected now with the determination of any exclusive decay of the B-meson into hadrons which do not contain c-quarks, because the b--*c transition is dominant. For example, recently the events B + ~ p l ) n + and B°~pf~n+n - were reconstructed by the A R G U S Collaboration [ 2 ]. However, at present there are no methods to get reliable predictions for these decays. So the experimental determination o f simple B-meson semileptonic decays into the lowest states such as n, p, etc. is o f interest. These semileptonic decays allow at present a quite reliable theoretical description. The decays B~nev~, B--,pev~ determine the electron spectrum near its endpoint, where the contribution o f the transition b ~ c is absent [electron energy Ee> ½roB( 1 - m 2 / m 2) ]. That also can serve as a basis for measuring Vbu. Here we will consider the simplest decay B ° ~ n + e - 9~ with the help o f the Q C D sum rules method [ 3 ]. The other semileptonic decays of the B-meson can be described in a similar way.

2. The method The hadronic formfactors o f the B ° --,n+e-ge are defined in a standard way:

decay

( n + ( p ) laVublB°(p ' ) ) = ( P + P ' ) u f + (t) + ( p - p ' ) ~ , f _ ( t ) , t= ( p - p ' ) 2 .

(1)

Only the formfactorf+ (t) is of interest to us because the f o r m f a c t o r f _ (t) does not contribute to the observables of the decay due to the smallness of the electron mass. It is necessary to obtain the value of f+ (t) at the transverse m o m e n t u m squared t varying from zero to the maximal value tmax= (mB--m~) 2 (mB and m~ are the masses of the B- and n-mesons). In the case of the decay B ° ~ + e - % , in contrast with the decays K--,nev~, D--,Keve and B~Dev~ (the description of the last two decays in the framework o f Q C D sum rules can be found in refs. [4,5 ] ), it is not possible to assume f+ (t) to be constant in the physical region o f t, because it has a pole at t = m 2. =/max (mB. is the mass o f the B*-meson, which is connected with the vector current a~,ub). Thus besides the value at zero transverse m o m e n t u m squared f + ( 0 ) it is necessary to calculate also the effective coupling constant o f the transition B*--. Bn determining the behaviour off+ (t) at large t ~ m 2. The value f+ (0) can be obtained with the help of the standard Q C D sum rules method for the threepoint Green functions utilizing the double dispersion 127

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relation. This can be done in the same way as in ref. [4] where the decay D+-oK°e+v~ was considered. Substituting in the formulae of ref. [ 4 ] the standard values of the parameters (ms=5.3 GeV, rob=4.8 GeV, (#q ( 1 GeV) ) _~ ( - 2 4 0 MeV)3,fB ~ 110 MeV [6], ot~(1 G e V ) / n = 0 . 1 and mo2=0.8_+0.2 GeV 2 [ 7 ] ), we find after the standard numerical analysis of the resulting sum rules (see for example refs. [4,5]) f+ (0) = 0 . 2 7 .

(2)

The accuracy of this estimate is ~ 20%. Note that the operator product expansion [ 4 ] does not contain the so-called bilocal operators [ 8 ] due both to the fact that there are no diagrams which are disconnected with the lines of the heavy quark and to the fact that the heavy quark propagator does not have a singularity at zero momentum. Contrary to the case o f t = 0 this method is inapplicable to the calculation off+ (t) at sufficiently large t. Really at large euclidean transverse momentum t~ rn~ the large distance contribution in the vector current channel becomes important and the standard operator product expansion [ 4 ] becomes wrong: it is singular at t= m~. For the evaluation o f f + (t) at large t we use the following method. Let us consider the two-point correlator

Hu(p, q) = f dx eipx(0 IT ~7sd(x) ayub(O)ln(q ) ) =-IrI(p2, pq)(P+q)~,+IT(p2, pq)(p--q) u .

(3)

Due to the large mass of the b-quark we can use the non-relativistic approximation in this mass, i.e. the limit mb---*co analogously to ref. [9 ], where this approximation has been used to calculate the constants fD and fa. Let us p u t p = q = 0 (q°=m~), Po=mb--E in the correlator (3). Under these conditions p2_~ mZ--2mbE, and since pq=O (in the chiral limit m~ = 0 ) the correlator (3) can be calculated at sufficiently large E with the help of the operator product expansion for the currents in (3) as a series in 1/E. The large distance contribution is now contained in the pion matrix elements of different operators. The amplitude II(p 2, p q = 0 ) =-H(E) at mb---,ov satisfies the dispersion relation of the form 128

5 October 1989 O9

II(E)= f du p(u) u+E" 0

(4)

The spectral density p (u) can be obtained by saturation of (3) by a complete set of intermediate states I n ( p ) ) connected with the pseudoscalar current ( 016ysdl n) ~ 0. The lowest energy states which give poles in H(E)are the states [B) and [B'n) carrying the momentum p = 0 . For the latter state, only the non-connected contribution in the vector current matrix element ~ × ( n ' l n ) results in the pole in H ( E ) . Obviously, the contribution of the IB)-state is proportional to the desired quantity f+ (tmax) and that of the [B*n)-state to the unknown formfactor ( 0 [675d1 B ' n ) . Nevertheless, due to the condition p---q=0 both formfactors are taken at the points of large transverse momentum squared ~rn~ and thus can be reliably parametrized by a single effective B*-B-n coupling constant. At large t, f+ (t) can be expressed through this constant g by the equation g

f+(t)=famnm~._ t , t.~m2a,

(5)

since in the limit mb~ COthe residue (01 a~,ublB* (p, 2) ) =eu(p, A). Fb is connected with the B-meson leptonic decay constant by the equation Fb=mRfa [9] (this value differs from the one obtained with the help of relativistic QCD sum rules [ 10] by less than 30%). Thus we find the lowest states contribution to the amplitude H i n the form 12[

1

uRg~E+Am

1 '~ E + Am*+m~ J Am* + rn,~ 1 - Am '

where Am = m s - mb ~ 500 MeV and Am* = mR. - mb. Taking into account that mn. - rnn~0 at rnw-. co and assuming m~=0 we find the phenomenological expression for the amplitude: g HexP(E)=-If~ ( E + A m ) 2 + A H ,

(6)

where the higher states contribution is denoted by M-/. As we shall see from theoretical calculations, the spectral density p(u) approaches zero as u-,co so it seems in analogy with refs. [ 11,12 ] that the higher states contribution in (6) can be parametrized in the form l-_d'~R/(E+ml) , where ml=mw--mb~_l.5

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GeV and R is some effective residue. However, in the derivation of eq. (6) we have not taken into account that besides the pole parts (infinite in the limit m ~ 0 , mb~) of the formfactors (nl~7~blB) and (015)'5d[ B ' n ) , which cancel each other, giving the finite expression (6), there are some finite parts of these formfactors. These contributions do not cancel each other in general, which results in the term ~ 1/ ( E + A m ) in eq. (6). In other words, if the lowest states contribution (6) can be viewed as coming from the effective BB*n coupling, the last term is due to the coupling of the type B'B*n, BB*'n, and so on. This term is not suppressed exponentially after the Borel transformation (see section 3), but there is no way to avoid taking it into account. Thus we choose AHin eq. (6) to be 1 2 // H R A / / = -~f a ~E ~---~m + E---~ml) and the phenomenological part of the sum rule (6) contains three unknown constants g, H and R. Thus our problem is reduced to the calculation of the constant g, which then must be substituted into eq. ( 5 ), from the sum rules. Now in order to get the sum rules for the evaluation o f g we must carry our a theoretical calculation of the amplitude//(E).

3. T h e s u m rules

Let us calculate the correlator (3) by means of the Wilson operator expansion of the current product in (3) with the subsequent evaluation of the pionic matrix elements ( 0 [ . . . [ n ) . It is convenient to use the operator version of the external field calculations technique [ 13 ]. We obtain 1

1-l/,(p, q) = - i ( 0 1 ay, ~+m b - i 9 ysdl n(q) ) , where Vu is the covariant derivative Vu=Ou 1"~g~l ~ a t a , Sp (t at b) = 2~ ab. We decompose this expression into a series of 9/E, then calculate the matrix elements and put pq= O.We restrict ourselves to terms up to ~ 1/E 3. Thus we get

l ( 1 /t 5~2 ) //th .... (E)=-~U~ ~ S + ~ - 5 + 1 - - ~ + . . . ,

(7)

5 October 1989

1 m2 I
(7cont'd)

where fi2 is defined by the equation _

ta

(OlgsayaG~u-~ dlo(q) ) = -if~O2qu, 62 = 0.2 + 0.02 GeV [ 11 ]. The quark condensate in (7) is normalized in the low normalization point about the characteristic value of E (or M, see below) ~ 1 GeV. Note that atpq/(m 2 --/3 2) ~ 1 it is not sufficient to know the matrix elements of the local operators, but the so-called n-meson "wave functions" corresponding to various operators are necessary. Comparing the "theoretical" (7) and the "experimental" (6) parts of the sum rules and applying the Borel transformation [ 3 ] in the variable E (E-,M), we obtain g as a function of the Borel parameter M:

g+H.M --MI Am/M

-

e

~ -f n- ~ a ( l + ~]A

"~

5~ 2 3--~j--Re-ml/M]. (8)

The region of investigation for M is determined by the condition for the power corrections series to converge: M>~ 0.9 GeV. On the other hand the region for M must be bounded from above by the condition for the term which is proportional to R to be rather small (say, less than 30% of the first term) in order to be sure that the higher states contribution is small. Let us assume the constant R to be not very large in order for the interval for M to exist. One readily finds the value of R that results in an excellent stability of the function (g+H.M) (M) up to a very large M: R ~ 7 GeV-~. In this case the interval exists and we get g ~ 16 and H ~ 0 (the slope of the curve). If we take R = 0 we see that the value of g is almost the same g ~ 13 although the value of H is different. It is easy to verify now that for an arbitrary value of R almost the same value o f g is obtained: g = 16 +_3. The other way to extract the information on g from the sum rule obtained is to apply the operator ( 1 - M d / d M ) to both sides of the equation for g+H.M. In this way the unknown residue H can be excluded and the value of R can be found from the condition of good stability of the corresponding 129

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function in the allowed region for M. This gives R ~ 10 GeV- ~and the same value ofg. Thus we obtain g=16+3. This result is obtained at f R~- 110 MeV, which is the average of the two values obtained with the help of the QCD sum rules method [ 6 ]. In the final answer the uncertainties in the value offR at the level of ~ 20% are taken into account among the others (they are not large). So the formfactor f÷ (t) at large t is found. If we extrapolate the tbrmula (5) to the low t region, we find f ÷ ( 0 ) = 0 . 3 3 , which is rather close to the obtained above value f + ( 0 ) = 0 . 2 7 (the accuracy ~20%). One can construct a formula interpolating between this value and (5) at t~m~. For that purpose we add one more pole term in (5), corresponding to the B*'-meson with mass mR.,- mR. + 1 GeV and find the corresponding residue from the condition f+ (0) - 0 . 2 7 . It should be noted that the derivat i v e f ~ ( t ) at t = 0 can also be found from the sum rules [4]. It is in agreement with the result of the above mentioned procedure although its accuracy is rather low. Thus jr+ (t) at all physical t is found. Taking into account all the uncertainties involved we find for the full width of the decay

F(B°--.n+ e-~,e) /_F(b--,uev ) =0.03-0.06, F(b~uev) =

G2m~

1927t3 [ Vbu [

2

1.5X 10141Vbu 12 l/s.

(9)

The electron spectrum near its endpoint depends essentially on the f÷ (t) behaviour at large t, i.e. on the constant g. The beginning of the pion spectrum (IP~[ <
F= mSG2l Vbu 12 16ZC3

( 1 -- x//~) 2

f 0

× [2x( 130

dy

Xmax

f axf +(y)

Xrnin

1+ y - a ) - 4 x 2 - y ] ,

5 October 1989

where the following notations are used: t

Y=m~-

(Pa - P n ) 2

m~

Xmax ( y ) ----y( 1 - I - y - - a T rain

[Pe [

m2

, x=--,mR a=--,m~ ~ / ( 1 - - y q - a ) 2 - 4 a ) -~ .

Now let us compare the obtained results with the results of other authors considering the decay B--, ne v. Most of them use the constituent quark models, which are not well grounded in the framework of QCD and of which the accuracy is not known. In ref. [ 14 ] in the framework of the non-relativistic constituent quark model the value ~ 0.02 was obtained for the ratio (9), which is somewhat lower than our estimate. In ref. [ 15 ] within some region of the parameter characterizing the relativistic wave functions of quarks in mesons in the infinite momentum frame, a close value 0.04-0.07 was found. The authors of ref. [ 16 ], using the bound-state picture for B and B* and some considerations connected with the PCAC technique, got for the constant g, see (5), a value about 3 times larger than our result (8) and a value of the width that is too large. An analogous method was applied in ref. [17]. And, finally, in ref. [10] the current algebra sum rule similar to the AdlerWeisberger relation was used. The width is overestimated 2-3 times here which may be due to the roughness of the estimate of the higher states contribution in the sum rule obtained. To summarize, in the framework of the QCD sum rules method both the value of the formfactor f÷ (0) of the decay B-*nev~ and the effective coupling constant of the B*-,Bn transition are estimated, which determines the behaviour of the formfactor f+ (t) in the whole physical region 0 < t < ( m R - m~) 2. The full width of the decay is obtained at the level of ~ 5% from the total inclusive rate of B-meson semileptonic decay (for the b-~u transition). In comparison with the estimates of other authors [ 10,14-17 ], the obtained results are quite reliable since they are based on a method utilizing mainly the first principles of QCD. Our results can serve as a basis for measuring the b--, u Kobayashi-Maskawa matrix element.

Volume 229, number 1,2

PHYSICS LETTERS B

Acknowledgement T h e a u t h o r is g r a t e f u l t o A.A. P i v o v a r o v f o r u s e f u l d i s c u s s i o n s a n d to t h e referee for s o m e useful r e m a r k s .

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[7] V.M. Belyaev and B.L. Ioffe, Zh. Eksp. Teor. Fiz. 83 (1982 ) 876; A.A/Ovchinnikov and A.A. Pivovarov, Yad. Fiz. 48 ( 1988 ) 1135. [8] I.I. Balitsky and A.V. Yung, Phys. Lett. B 114 (1982) 53; B.L. Ioffe and A.V. Smilga, Pisma Zh. Eksp. Teor. Fiz. 37 (1983) 250; K.G. Chetyrkin et al., Moscow preprint INR P-0337 (1984). [9] E.V. Shuryak, Nucl. Phys. B 198 (1982) 83. [ 10 ] C.A. Dominguez and N. Paver, DESY report DESY-88-063 (1988). [ 11 ] V.A. Novikov et al., Nucl. Phys. B 237 (1984) 525. [ 12 ] A.B. Krasulin, A.A. Ovchinnikov and A.A. Pivovarov, Yad. Fiz. 48 (1988) 1029. [ 13] A.I. Vainshtein et al., Yad. Fiz. 39 (1984) 124. [ 14] B. Grinstein, M. Wise and N. lsgur, Phys. Rev. Lett. 56 (1986) 298. [15 ] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637. [ 16] M. Suzuki, Phys. Rev. D 37 (1988) 239. [ 17 ] S. Nussinov and W. Wetzel, Phys. Rev. D 36 ( 1987 ) 130.

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