ELSEVIER

18 August 1994

Physics Letters B 334 (1994) 399-404

PHYSICS LETTERS B

On the 1/m 2 corrections to the form-factors in A b > A: e decays

J.G. Ktrner l, D. Pirjol 2 Instttut fiir Phystk (THEP), Johannes Gutenberg-Umversttat, Staudmgerweg 7, D-55099 Mamz, Germany

Received 26 May 1994 Editor P.V. Landshoff

Abstract

Model-independent bounds are presented for the form-factors describing the semileptonic decay Ab--* A,n~, at the end-point of the lepton energy spectrum. For the axial-vector form-factorf A ( 1 ) we obtainf A ( 1 ) ~< 0.960 -- 0.935, where the uncertainty arises from the value o f /z 2, the average kinetic energy of the b quark inside the Ab baryon. A bound is given on the forward matrix element of the axial current in a Ab baryon which is relevant for the inclusive semileptonic decays of the polarized Ab decays.

1. Recently it has been shown [ 1 ] how to obtain model-independent bounds on form-factors describing semileptonic transitions of heavy hadrons. It had been known for some time that some of the form-factors describing the decay B--*D*e~ do not receive 1/m corrections at the so-called "no- reco i l " kinematical point [2]. The only corrections to the heavy-quark symmetry limit appear at order 1/m 2 and they have been evaluated in the past by making use of model- dependent results [4 -7] . In [ 1 ] a model-independent bound on the magnitude of these corrections has been given. The aim of this paper is to present a similar bound on the 1/m E corrections to the heavy quark sym- metry predictions for the semileptonic decay Ab---> Ace~ e.

2. We begin by describing the kinematics of the decay. The Ab baryon is polarized with the spin vector s and is taken to be at rest in the laboratory frame. It decays into a final hadronic state conta in ing one

1 Supported m part by the B ~ , Germany under contract 06MZ730 2 Supported by the Gradulertenkolleg Tellchenphyslk, Umversltat Mainz

0370-2693/94/$07 00 © 1994 Elsevmr Scmnce B V All rights reserved SSDI0370-2693(94)00805-H

charmed quark of invariant mass mx and a lepton pair effe of 4-momentum q. The inclusive decay width summed over all final hadronic states with the same invariant mass can be expressed in terms of the had- ronic tensor

W~,~= (2~') 3 ~ (Ab(V, s)[J~(0)IX> x

x <XlJ~(0) lab(v, s) >

× 8 ( p x - p + q ) , (1)

with p~¢ = m z and p = mA~V the momentum of the Ab and J is the weak current. We normalize the [ Ab) state as usual to Vo. The most general decomposition into covariants for the hadronic tensor W~,, has the form

Wjzv = - g jzvWl + v ~v vW2 -- i E u ~ Z v ~ q Z W 3

+quq~W4 + (quv~+q~v~,)W5

- q" s [ - g g ~ G l + v~,v ~G2 - ieu ~ z v ~q aG 3

+q~,q~G4 + (q~,v~+q~v~,)Gs]

+ (s~,v~+s~v~,)G6 + (suq~+s~q~,)G7

+ ie~ , ,~v "s ~G8 + i ~ , , ~ q " s / ~ G 9 . (2)

400 J G Korner, D Pirjol/Phystcs Letters B 334 (1994) 399--404

Here the same parametrization has been used as in [ 8]. The hadronic tensor W, v is given by the disconti-

nuity of the forward scattering amplitude

Tu~(q. v, q2) __ _ i J dax exp( - iq.x)

)< (Ab(V, s) I TJ*~(x)J~(O) lAb(v, s ) ) , (3)

across the cut in the complex plane of the variable v. q which corresponds to the process under consideration. In our case the cut extends from ~q2 to 1 / (2mAb) (m 2 b _ mA + q 2). The invariant mass of the final hadronic state varies from mac on the right edge of the cut up to mab - ~qZ on the left edge.

We will parametrize the forward scattering ampli- tude T~, ~ in terms of invarlant form-factors T1,5 and $1.9 (defined in analogy to Wl,~ and G1, 9 in (2)) .

For our purposes it will prove more convenient to study the analyticity properties of T~,, as a function of complex v. q at fixed velocity v ' of the final hadronic state (instead of qZ). In these variables, for a given to = v. v', the cut corresponding to the physical decay process ranges between

1 mat , to W to 2]ti--~l_ 1 ~ v " q ~ m ab -- mac to . (4)

The invariant mass of the final hadronic state corre- spondingly takes values between

1 mAb t o + ~ >lm X > l m A c , ( 5 )

and q2 is a function of v .q along the cut, given by

q2 = _ mAb(rnnb _ 2v.q)

1 + - ~ ( m / t o - - V ' q ) 2 • (6)

The discontinuity of T ~ across this cut is related to the hadronic tensor (1) by

1 W~,~ = - 2"--~ disc T ~ . (7)

In the following we will construct two sum rules for the form-factors describing the semileptonic decays of the Ab baryon in analogy to the ones presented in [ 1 ]. The method is based on the positivity of certain com- binations of invariant form-factors in (2). These can be obtained from inequalities of the form

n Un * 'qV u~ >~ O (8)

and follow from the fact that this quantity is directly related to the decay rate Ab ~ W(q, n~)Xc of the A b

baryon into a virtual W boson with the polarization vector n. Let us work in the rest frame of the Ab baryon.

One positive-definite combination of invariant form- factors in (2) is obtained by using in (8) a longitudinal polarization vector. This gives

qo 2 _q2 WE = Wl + q2 WE

qo 2 _q2 -q.,(o,+ 7o ) qo

+ 2 ~'5 (s 'q)G6. (9)

Another possible choice is the total decay rate into a circularly polarized W boson, which is proportional to

WTL,, = W1 -- q" sG1

+ ( ~ - q 2 ( W 3 - q ' s G 3 )

q ' s . ) + ~ (Gs+qoG9) • (10)

The total decay rate into a transversally polarized W is

WT =2W1 - -2q 'sGl . (11)

Finally, the case of a scalar (or " temporal") polar- ization gives

qo Wo = - W1 + ~-7 W2 +q2W4 +2qoW5

qg - q ' s ( - G l + --qTGz +q2G4 + 2qoGs)

qo +2~-~ (s'q)G6

+2(s 'q )G7. (12)

These quantities must be positive for any possible orientation of the spin vector s with respect to q and for any current J in (1).

J G Kdrner, D Ptrjol/Physics Letters B 334 (1994) 399-404 401

3. The forward scattering amplitude T . ~ can be given a representauon in terms of physical intermediate states as

(Ab I J*~(0)IX)(XlJ~(O)lAb) E

X(px=mAb.--q) mat, -- Ex -- qo + ie

(Ab IJ~(0)IX)(XlJ~(O)l&) + y" r n A t - - E x + q o + i ~

X<pX =mAb. +q) (13)

Only the first term above gives a contribution to the discontinuity across the cut corresponding to the decay process of interest. We will be mainly interested in the contribution of the intermediate state A~. We will con- sider first the case of an axial current J~, = gy,ysb. The corresponding matrix element of the axial current in (13) can be parametrized as usual in terms of 3 form- factors, defined through

(ac (v ' , s ' ) ](yu y sb lab (v , s) )

=if(v,, s,) [fA A ~ , 1 Y ~ + f 2 v ~ W f 3 v ~ ]

× ysu(v, s) . (14)

We have used a parametrization appropriate to the heavy-mass limit, in terms of the velocities of the par- ticles involved. The spinors are normalized so that a u = I.

By taking the imaginary part of (13), the hadronic tensor W.~ can be also written as a sum over interme- diate states. The contribution of the lowest-lying state A~ to the positive-definite combinations of structure functions defined in (9), (10), (12) is

o9+1 WE = [ (mA~--mA~)f A 2q 2

--mA~(W-- 1)f A

3] , (15) _mat,(o9_ l ) f A 2

1+O9 WwL~ = If ~ I 2 " - 7 (1 ___if-s), (16)

o9--1 W°= ~ [(rnAt,+mA~)fA

-- (mAt, -- mA~og)f A

-- (mAt , tO -- mA~) f A] 2 (17)

A factor of 1 / o)6(qo -- mAt, "1- mac to) has been removed from these expressions. Here r~ is a unit vector collinear

with the vector q. If I q I = 0, it defines the quantization axis along which the spin of the virtual W boson is aligned. The above formulas can be simply related to helicity amplitudes for the process Ab(AAt,)~ Ac(AAc) + W(Aw) listed for different other cases of physical interest in [9].

For the case of a vector current J . = gy.b, we define in an analogous way to (14)

(Ac(v', s') I(y~,blAb(v, s) )

=ff(v ' , s ' ) [ f v y ~ + f v v ~ + f Vv, lu(v , s) . (18)

In terms of these form-factors, the contribution of the intermediate state A~ reads

o9--1 WE = 2q2 [ ( m a b + m A c ) f v

+mAc(o9+ l ) f v

+ mat,( o9+ l ) f V] 2 , (19)

WTL -----I/Vl z o 9 - 1 • - - 7 (1 -t-a-s), (20)

o9+1 Wo= ~ [ (mAt,--mAc)f v

+ (mAb - - m a c o 9 ) f v

-t- (mat, o9--rnac) f 7]2 . (21)

As before, a factor of 1/o96(qo--mAt, "+'mAyo9) has to be added on the r.h.s.

4. On the other hand, it has been recently shown [8,10-13] that it is possible to reliably calculate the forward scattering matrix element (3) in a region which is far away from the physical cuts. The result takes the form of an expansion in powers of 1/mc,b and depends on a few nonperturbative matrix elements of dimension-6 operators taken between heavy hadron states. For the case of an axial (vector) current J~, = ?y~ysb, the results for T1,5 can be taken from [ 11 ]. As for the spin-dependent structure functions one can see from (9), (10), (12) that at the equal-velocity point to = 1, only G8, 9 contribute. The corresponding amplitudes $8,9 can be straightforwardly calculated fol- lowing the method given in [ 8,12 ] with the result (for J~. = gy~ysb; the result for the vector current case can be obtained by making the replacement mc ~ - mc)

402 J.G. KOrner, D Pirjol / Physws Letters B 334 (1994) 399-404

1[_7/ /2__~2 /22] (mb +m~) $8= i t \ l+ rn~ + 6m~]

5/2 + 3~bA 2 v'q(mb+m¢)

4/2 2 + - ~ T [(v'q)Z--qZ](rnb+m~) ,

S9=

(22)

/22 (v.q+](mb+mc)) mb A2

4/22 [ (v.q)Z_q2 ] (23) 3A3

where

A=m 2 --2mbqo +qE--m2 +ie, (24)

and

/2zrr= --(Ab(V, S)Ihb(iD)Ehb lAb(V, S))sp . . . . . . . ged" (25)

/2 2 represents the average kinetic energy of the b quark inside a Ab baryon. At present very little is known about its numerical value. A previous estimate [4] of the corrections of order I /m: to the Ab ~ Ac form-factors used/22 = 1 GeV 2. The corresponding parameter in the B-B* system has the approximate value 0.5 GeV 2. Using the mass formula

mab =mb + A + /2~ (26) 2rob

and a similar one for the Ac baryon, together with the mass values [ 14] mAb " = - - 5.641 GeV, mAc = 2.285 GeV, mb= 4.8 GeV and m~ = 1.39 GeV, yields

12 2 = 0.211 GeV 2 . (27)

For the purpose of illustration we will vary the value of /2~ between 0.211 and 0.5 GeV 2

The other parameter in (22), (23) is /2~, which is defined as the 1/m~-correction to the forward matrix element of the axial current in a Ab state:

(Ab(V, s)Iby,,ysblAb(v, s) )

= 1+ m~ + ' " u ~ y s u . (28)

This matrix element does not receive any corrections at order 1/mb [3]. /22 is related to the parameter eb defined in [ 8] by/22 = m2eb" Very little is known about its precise numerical value. We will show below that it must be negative and will give an upper bound for its value.

Inserting q2 from (6) in the theoretical expression for T~, ~ and taking the imaginary part yields the QCD prediction for the hadronic tensor, which is a singular function of qo consisting of 6 functions and its deriva- tives. Integrating the QCD prediction for W1 over the physical cut in the qo complex plane at fixed to = 1 one obtains

f dqoWl(qo, to= 1)

/22 /22 - 1 6mcmb 4m 2 4m 2 . (29)

As one can see from (11), this quantity is positive definite, which means that it represents an upper bound for the contribution of any physical state to the hadronic tensor. In particular, from (15), (16) one obtains a bound on the invariant form-factor f ~ (1) which is relevant for the decay rate of the process Ab ~ Ace~e near the endpoint of the electron spectrum:

/22 /22 /22 If ~'(1) 12~< 1 . (30)

6mcmb 4m 2 4m2c

This bound is similar to the bound recently derived in [ 1 ]. Numerically, this bound is

1 - If A(1)12>---0.165 /22 (GeV 2)

= ( 3 . 5 - 8 . 3 ) % . (3~)

This result is comparable with the estimate of the same quantity in [4] which gave about 6.2%.

This bound can be improved if the contribution of the spin-dependent structure functions is taken into account. Let us consider for example the combination WrL defined in (10), at to = 1. The Ac contribution to its integral along the cut is

f dqoWTL(qo, oJ= 1 )= If ~(1)12(1 +r~'s) (32)

whereas the QCD prediction for the same quantity is

J.G. K6rner, D Ptrjol / Phystcs Letters B 334 (1994) 399-404 403

f dqo WTL(qO, w = ) 1

6m~mb 4m~

1 + ( . . s ) •

/ ~ (1 +t~-s) 4m 2 ]

(33)

Requiring the positlvity of this quantity for any value of t~-s gives the inequality

1 2 /x~ + ~/~,~<0, (34)

from which one obtains an upper bound on/z~z:

1 2 /z 2 ~< - ~/z,~- ( - 0 . 0 7 0 + - 0.167)GeV 2. (35)

In [4] ]z~ 2 has been esumated to be /z 2 = - / z 2 / 3 , assuming that the contribution of terms arising from double insertions of chromomagnetic operator can be neglected. The present method gives also the sign of the corrections to this estimate.

The inequality (34) implies that the bound (30) on If ~ ( 1 ) I can be improved, provided that a determi- nation (calculation) of /z 2 becomes available. For the moment, we will restrict ourselves to the simple bound (30).

In a completely analogous way one can obtain a different bound on the vector-current form-factors

V f 1.2.3 defined in (18). This time the bound results from requiring the positivity of the scalar combination of form-factors (12). From (21), the contribution of the Ac state reads

f dqoWo(qo, w= 1) I f v(1) + f v ( 1 ) + f v ( 1 ) ] 2 ,

(36)

whereas the QCD analysis yields for the same quantity

f dqoWo(qo, o J = l ) = l - tX2 (m lb')2 4 ~ . (37)

It has been shown [3] that the combination of form- factors appearing in (36) receives no 1/m corrections at to= 1. Putting together (36) and (37) gives a lower bound on the magnitude of the 1/m 2 corrections to the leading-order result:

1 - [f v(1) +f v(1) +f v(1)]2 1) 2

>I 4 \me mb =0"065/x2

= (1 .4+3 .2 )%. (38)

which is in agreement with the 19% estimate of this quantity in [ 4 ]. On the other hand, in ] 7 ] Y-f v ( 1 ) was found to be larger than unity. As will be shown below, the radiative corrections could account for such an increase.

These bounds get changed when radiative correc- tions are incorporated. Their effect is to modify the heavy quark prediction for the form-factors at leading order in 1/m by a multiphcative factor. To one-loop order, the factor multiplyingf ~ (1) is [ 15]

rl~rt = l + a.__~ (rnb + mC log mb 8) 7r m b - - m c m c

= 0.958, (39)

and the factor multiplying the form-factor combination f v ( 1 ) + f v ( 1 ) + f v ( 1 )

r/~ ~t= 1+ cJL( mb~+m~ log mb - 2 ) kmb - mc mc

= 1.025, (40)

where AQCD = 0.250 GeV has been used. As a result we finally obtain

f ~(1) ~< 0.960 + 0.935, (41 )

fv(1)+fv(1)+fv(1)<~l .O05+0.996. (42)

In principle these bounds could be improved if the contributions of any other states to the integrals of the positive-definite quantities (11), (12) are known or if they can be estimated in some way. In particular, the contribution of two-body intermediate states like Act/ and ~cTr can be estimated by making use of the heavy hadron chiral perturbation theory [ 16,17]. Unfortu- nately, the coupling of the light Goldstone bosons to the antitriplet of heavy baryons containing one heavy quark is at present unknown and any estimate of these contributions will necessarily entail model-dependent assumptions.

404 J.G. Kdrner, D. Ptrjol / Phystcs Letters B 334 (1994) 399-404

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