cm . . __ e

__ ll!B ELSIWIER

24 April 1997

Physics Letters B 399 (1997) 163-171

PHYSICS LETTERS B

Unitarity bounds on the B ---$ T& form factors Damir Becirevic ’

Laboratoire de Physique Thcforique et Hautes Energies, Universite’ de Paris XI, B&iment 211, 91405 Orsay Cedex, France

Received 6 January 1997

Editor: R. Gatto

Abstract

Unitarity bounds for the B -+ n-f?& form factors are generated. The interpolation formula for the bounds constrained by lattice results is obtained. We note that the bounds are rather restrictive although do not allow for a definite conclusion on the needed f+( q*) functional behavior in the entire physical region. @ 1997 Elsevier Science B.V.

Keywords: Form factors; Lattice QCD; Dispersion relations; Unitarity bounds; H* spaces

1. Introduction

The study of the B + s-&t form-factors in the entire physical region in a model independent way is very important for the forthcoming experiments which will provide us with a way to extract the precise value of I&J which plays the crucial role in our understanding of the mechanism of CP violation. 1 l&l is mainly determined from the end-point of the lepton spectrum in inclusive B 4 X&e decays where the theory is very complicated and suffers from large uncertainties. Hence, we explore exclusive decay modes. The natural candidate is B --f n-( p)lFl. The determination of /I&,] depends on the knowledge of the physics at large distances, i.e. non-perturbative QCD. This problem is notoriously difficult. We cannot rely on the heavy quark symmetry to reduce number of the form factors or to fix their normalization at s:~,,, as in heavy-heavy transition. The knowledge of the form-factor at several points does not mean the knowledge of its functional behavior. Existing experimental values for the branching ratio are model dependent because of the dominant b -+ c transition background. The recent CLEO results are obtained by using several quark models in the efficiency calculation and they reported [ 11:

( (2.0 f 0.5 f 0.3) 1O-4 ISGWII [2]

Br(B’ -+ T-t+iq) = (1.8 f 0.5 f 0.3) 1O-4 WSB [3]

(1.9&0.5rt0.3)10-4 KS [4]

(1.8f0.4~t0.3)10-~ Melikhov [5]

(1)

The problems with quark models are related to the fact that they are not relativistic in some aspects and they do not provide real predictions for the form factors. Small recoil behavior of the form factor was also

’ E-mail: damir@qcd.th.u-psnd.fr

0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00284-O

164 D. Becireuic/Physics Letters B 399 (1997) 163-171

calculated by combined heavy quark effective theory and the chiral perturbation theory [6] which is valid only in the soft-pion limit. All those phenomenological approaches provide us with very useful guidelines, but for the model independent extraction of IVubl, we need the true QCD calculation of the corresponding form factors. There are two methods that are rooted in the QCD from the first principles: the lattice QCD [7] and the QCD sum rules [ 8,9]. While the lattice can be employed to explore the region of small recoils, sum rules give us the value of the form factors for small 4”. So far, in the lattice approach the values of form-factors were calculated at several points (several q2) by different groups [ 10-131, and then extrapolated according to an ansatz of functional dependence on 2. The general hope is that with faster computers and improved actions (for instance [ 141)) the extrapolations will be far more constrained and results less biased by anstitze. We concentrate on the bounds on the form factors with a consistent treatment of the existing lattice results in order to constrain these bounds. In a series of papers [ 15-171, the authors applied an old method [ 181 to the heavy quark systems. In fact, this approach was used for a study of heavy-heavy transition before [ 19,201, but the heavy-light decays were tackled for the first time in [ 161. The idea is to use crossing symmetry and dispersion relations in order to relate form factors with QCD perturbative calculations in an unphysical kinematic region. This procedure leads to the bounds on the interesting form factors which can be constrained by well lattice results. In order not to repeat the same analysis as in [ 211, we examined the approach in an alternative way.

2. Background

First, let us recall some basic formulae. Hadronic matrix element for the B -+ r&l semileptonic decay can

be parametrized as

(n-(p’)lV(O)IB(p)) = (P +d>@“f+(q2) + (P -P’)‘f-(q2) 2 p mi-mm,

q2 > f+(q2) + qp

m”B ;mQ4(q2)

where VP = iiy@b; f * (q”) are the form factors which contain the dynamical information on the decay, p and p’ are the momenta of B and v respectively and q = p - p’. In the equivalent parametrization (the second line) the scalar form factor f” ( q2) is introduced. The artificial singularity at q2 = 0 is removed by imposing:

fo(0) = f+(O). Th’ is gives an important constraint for the fit in lattice analyses. In the limit of massless leptons, the physical region of q’ accessible from this decay is 0 < q” 5 (mB - m.,,)2, and only f+(q2) contributes the decay rate

---f n+e-ve) = G2tV,d2 312 2 192T3m3 A (4 )lff(q2)12

B

(h(t) = (t+ rni - mi>2 - 4m$rni is the usual triangle function). It is obvious that for the extraction of VUb

from an experiment we must have the precise information on the functional dependence of ff (t) obtained from QCD. As mentioned in the introduction, such an information is not available, and we try to constrain the form factors. Detailed discussion in the framework of the two-dimensional QCD [ 231, suggested the pole dominance behavior. The extension to four dimensions is however not obvious at all. The method we use is well known. Namely, one starts from the two-point function:

W”“(q) G i I

d4xe’~“(olT(V~(x)V”+(0))10) = (qfiqV -q2gP”)&(q2) +gTI,(q2)

where IIrL ( q2) satisfy the once subtracted dispersion relations:

(3)

D. Becireuic/Ph.ysics Letters B 399 (1997) 163-171 165

(4)

The absorptive parts of the spectral functions ImII ~,r ( q2) can be obtained from the unitarity relation:

(4~~~4~-4~g~~)IrnII~~(t+i~) +gTml&(t+iE>

nf =- Cl 2 l- dpr(2~)4S(q-Pr)(01V~(0)Ir>(rlv~(o>lo) (5)

where I are all possible hadron states with appropriate quantum numbers, and the integration goes over the phase space allowed to each intermediate state. Crossing symmetry states that the matrix element (BTT VP (0) IO) is described by the same form factors as those for (~1 Vp( 0) IB), but real in t+ 5 t 5 cm region * . So, if we take I&Z-) as the lowest state contributing to the absorptive amplitude, we obtain the following inequalities

ImINt> 2 --/v(t) f_(t) + 16rrt [I mi;msf+(t) 2 + $f+(t)12

1 4(t - t+)

= &P(t) [ ~lf+(t)12+ (mi ;2mqp(t)/2]lY(t - t+)

ImIIL(t) L $rA’i2(t) f-(t) + rni-rni +

t f (t) 2a(t- t+)

= &A’/*(t)(mi - m~)2~.f(t)~28(t - t+)

In what follows, we use the second parametrization since we can profit from the condition f” (0) = f+ (0). By replacing the absorptive parts in dispersion relations (4)) we come to the inequalities

(6)

nf(m2, - mi)2

16?r2xr. (7)

t+

In these inequalities we used the fact that Q* = 0 is the point which is sufficiently far from the region where the current V, can create resonances. In fact, when (mb + m,) AQCD < (mb + m, ) 2 + Q2, the functions XT,L (Q*) can be calculated by means of perturbative QCD. At Q2 = 0, to leading order, their form is particularly simple:

(8)

(9)

166 D. Becireuic / Physics Letters B 399 (1997) 163-I 71

For a massless u( d)-quark, it gives: ~r( 0) = (877%~~) -’ and XL (0) = (8~~) -‘. The function in (6) is the sum of positive terms, and the inequality can be taken separately or if we chose [ 171

x = xdQ’> + ~x;(Q2)1~2=o

Nc +-~)j& X2(1 -x)2[(1+ ~>(W&-m,)(mbx-m,(l -x)) -t-4WJn,l

= 2(2n-)D/2 9

0 [m;x+m;(l -X)14-+

we obtain two separate inequalities for our form factors:

nf

48~~~ fr

nf (rni - mt)2

167r2xL t+

(10)

(11)

They provide us the bounds on the form factors on the cut.

3. Constrained bounds

To get some information about the form factors in the physical region for B ---f ?r.l?Ve, we map the complex t-plane onto the unit disk Iz j 5 1:

where N is a positive real constant. By this transformation, the regions t- < t < t+ and 0 5 t < t- are mapped

into the segments of the real axis ztin > z > -1 and zmas 2 z > ztin respectively, while two branches of the

root (two sides of the cut) are mapped to upper and lower semicircles of Iz 1 = 1. ztin and zmaX are given by

O-1 z -

m~+m,--2@%GG

max- m~+m,+2~~’ &in = -

C > X/R+1

Note that for N = 1 we have zk,, = 0. Physically, the relevant kinematic region for the process vacuum + Br now lies on the unit circle, while the region for the semileptonic B -+ dVe decay lies inside the unit circle, on the real axis. In the z-plane, the above inequalities ( lo), ( 11) can be rewritten in the form:

2lT 1

G .I Wi(8)Ifi(-9-)12 5 1 or &

s $l&(Z)4(Z)12 5 l (i=O~+) (13)

0 lzl=l

The reason to write the form factors as Fi( z ) will be clear below. The functions Wi( 8) are just transforms of the integrands in (lo), (11). &(z) are the analytic functions, which are the solutions of Dirichlet’s problem of finding an analytic function of the unit disk with I +i (3) I2 = wi( 8) on the circle:

2?r

+i(Z) = exp -& { J di?~lOgVV~(7Y)}.

0

(14)

D. Becirevic/Physics Letters B 399 (1997) 163-171 167

Their explicit forms read

cbo(z> = nf(mi - m$>2 4 -4 1-Z 1 16&mBm,NXL >

(l+z)(l-z)S l+z+2y&(l-z) [ 1 [ 1+z+-

JN 1 (W -5 4+(z) = n.f 1-Z 5

48n”mBm,Nx > 1 [ 1+z+- JN 1 (16)

The space of analytic functions on the unit disk with

(17)

on the boundary is the well known Hi space. To make use of the rich theory of HP spaces [ 241, we have to ensure our form factors to be analytic on the disk. While it is true for fo (z ) , it is not for f+ (z ) which has the pole at -1 < z,( t = M&) < zkn. The value of the residue of f’(z) at that pole (- f~*~s*s~) is rather badly known. To get rid of this singularity, we multiply the form factor by the Blaschke factor (analytic on the

disk) which render f+ (z > B, (z ) = F+ (z ) analytic,

B,(z) = z *

(18)

is unimodular, and therefore does not spoil ( 13). In fact, the condition ( 13) gives us the constraint 11 Fi 11 ,q, 5 1,

i.e. Fi(z) belong to the unit ball of H,. 2 Our goal is to constrain the form factors in the physical region for B + ?r!Ve decay when their values are known at several points. For this case, the interpolation problem was treated in [ 15,211, where the inner products (gj, gk) (j, k = 0,. . . , n 3 ) were constructed. The specific choice: go (z ) = & (z, ) F; (z ) and gi( z ) = ( 1 - z Zi) -’ allows to build the matrix whose determinant is positive which subsequently gives the bounds on Fi (z ) . Here we note that the positivity of the determinant is provided if the sequence { zj}i satisfies the condition [ 241:

1 - IZ.j+1l 5 CC1 - IZjl> (19)

where c < 1 is a real positive constant and Fi (zj) fixed. Here, we solve the interpolation problem by closely following [ 20,25,24]. If F( z ) is given at n-points then the general Hi function taking the specified values F( zi) = qi can be written in the form

F(z) = &P Bi(z)

im +B(z)g(z) i=l 1Z

where

(20)

(21)

and g( z ) is an arbitrary analytic function. F( zi) compatible with ( 13) form a convex and closed domain in the euclidean R” space. As argued in [ 25,201 the domain of admissible values F( zi) in this domain is described

by

(22)

3 n is the number of points at which the form factor values are known.

168 D. Becirevic/Physics Letters B 399 (1997) 163-171

This means that we minimize the norm

where from (20)

(23)

(24)

Now, we can apply duality theorem which converts the problem of minimization into the problem of maxi- mization in the normed dual space i.e.:

The supremum is taken upon all analytic functions in the unit sphere of H2,

G(z) =FG nzn9 n=O (gG3 1)

After the integration in (25) and taking the supremum, from (22) we obtain

c n F(Zi>F(Zj>~(Zi)~(Zji) Cl- 2) (1 - zj”) I l

i, j=O &(Zi)Bj(Z.j> 1 - ZiZj

(25)

(26)

(27)

Note that we used the fact that F (2) = F( 2) as well as the fact that the form factor values are known at real zi. Now we may give some comments. In [ 201, this analysis was used when the form factor is known at z = 0 and at zo fixed. Then the inequality similar to (27) is simplified in the sense that interpolation formula can be reliably applied for all ZO. Also, the physical region for the B --+ DC*) semileptonic decay is much smaller, so that the knowledge of the form factor at one point is “sufficient” to obtain well constrained bounds. The subtle point there is to constrain the values of the slope and the curvature, parameters that are needed in the fit for the model independent extraction of II&l. Then, some refinements like three-loop expressions for x( Q2) are in order [ 261. Also, the presence of subthreshold singularities must be treated carefully. In our case such refinements are not necessary at this stage and there are no branch points below the threshold. We rather insist on the point to get some reasonable constraints from the unitarity. O( LY,) corrections can be found in [ 271.

4. Results and comments

As we already mentioned, we do not have the normalization of the form factor at q&, but we can always choose N in (12) in such a way that the known value of the form factor is given at z = 0. This is quite natural since we take the lattice results which are for the time being obtained in the region of the high q2. Then from the values of the form factors for the highest to the lowest q” (keeping in mind the condition ( 19) ) the inequality (27) applies well. For the rest of the physical region we suppose that this formula applies equally well. In a certain sense, it is rather the recipe for extrapolation, which however has nothing to do with a simple ansatz (“guess”) like pole/double pole one. However the situation is improved when the constraint f+ = f” at q” = 0 is imposed. In Fig. 1, the bounds for the form factors are displayed, when the UKQCD results [ 12,211

D. Becirevic/Physics Letters B 399 (1997) 163-171 169

1.5

Pi- g 1.0

Q-u

0.5

5.0 10.0 * 15.0 20.0

9

10.02 15.0 20.0

4

o.ot”” t ” ” I ” ” ” ” 1’1 5.0 10.0, 15.0 20.0

9

Fig. 1. Bounds on the form factors p(q’) (a) and f+ (q2) (b) with UKQCD lattice results. (c) Bounds on f+(q2) additionally

constrained by the condition ff (0) = p (0).

are taken (Table 1). In this case N = 3.171 which is chosen in such way that q2 = 20 GeV2 ++ z, = 0. To have some feeling about the number, q2 = 0 is mapped on z = 0.278, while the position of the pole is at z* = -0.494. As it can be seen in Fig. la, the form factor FO = fo(q2) is rather well constrained, and we obtain 0.18 < p(O) < 0.71. This is to be used as the input to constrain F+ = f+(q2>B*(q2>. f+(q2) without and with this constraint is displayed on Fig. b and c respectively. This can be also considered as the verification of equivalence of the approach used in this letter and those used in [ 211.

As a short remark, we note that 0.30 < fo(0) < 0.73 if the Callan-Treiman relation (as suggested in Ref. [ 311) is taken as an additional constraint with fs = 150 f 25 MeV, the average obtained by different lattice groups [7]. In Fig. 2, the form factors are constrained with preliminary results of the FNAL lattice group [29]. For the bounds on fo, we choose N = 2.495 for the same reason as before. Here we have to be a bit more careful with data. Actually, we take only three points since they are almost equidistant as it is required in (19) (marked by stars in Table 2). We obtain -0.15 < p(O) < 0.48 which was taken as the input in the bounds for f+(q2). For f+(q2) N = 3.611, and all four points are taken into account since they are almost uniformly separated. In all bounds we take mb = 4.8tt.i GeV in a way that the bounds are more conservative. What we learn from the bounds is that the pole dominance behavior of f+ form factor receives rather strong corrections and have tendency of being flat in the region of small q2. This is in fact expected. It

is nothing new to say that the pole dominance is dubious for small and intermediate values of q”. Corrections can be taken by the fitted mass Mpoie which is in all analyses bigger then MB* (which means that the sign of residues coming from all radial excitation is positive). We also note that the double pole behavior (whatever it means) can be fitted for the behavior of f+( q2) for small q2. However, for q2 E { 1 M& - q&J, qiax} as

170 D. Becirevic/Physics Letters 3 399 (1997) 163-171

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Fig. 2. Bounds on the form factors j”( 17’) (a) and ft (q*) with ft (0) = jc (0) (b) with preliminary FNAL lattice results

F....s.. .,....“““‘l 5.0 10.0 2 15.0 20.0

9

Fig. 3. f+ ( q2) : Light thick lines show the bounds from UKQCD

lattice results (squares) ; Dark thick lines show the bounds from

preliminary FNAL lattice results (circles) ; Dotted line is the light

cone QCD sum rule prediction; Dashed line show the quark model

prediction.

Table 1

UKQCD lattice results [ 12,2 I].

q2 [ GeV2 ] f+(2)

14.9 0.85 + 0.20 17.2 l.lOztO.27

20 1.72 III 0.50

.P (s2)

0.46 i 0.10 0.49 It 0.10

0.56 xt 0.12

Table 2

F’NAL preliminary lattice results [ 291.

q2 [GeV2] f+($)

12.3 0.50 i 0.23

14.1 0.66 f 0.21

16.2 0.86 f 0.21

18.7 1.25 f 0.25

22 _

P(q2)

0.31 i 0.13

0.39 zt 0.12*

0.44zt0.11

0.54*0.11*

0.67 f 0.09*

argued in [ 301 should be pole dominated. There, we could estimate the value of gB*B, but our bounds are uncontrollably wide. The physical region is large and q” = 20 GeV2 is rather close to qk, and we may assume that from this point to qhx, the form factor f+(q2) is pole dominated. Note that we risk to have our upper

bound underestimated, but still is the only thing we can do since the information on the strength of even first radial excitation fB,gB,Br is not known. A rough estimate can be done by a quark model consideration and this question will addressed elsewhere. However, if our assumption is valid, then for the conservative I&j = 0.005 [ 341, we obtain: B(I?’ + r%ge) 5 2.49 x 10e4. At the end, the both sets of bounds are displayed on Fig. 3, against the light cone sum rules [ 321, and quark model [ 331 predictions.

Acknowledgements

We thank A. Le Yaouanc and JR Leroy for useful and motivating discussions. The important help of OS1 and R. Schuman Foundations is acknowledged.

D. Becireuic/Physics Letters B 399 (1997) 163-171

References

17

[ l] J.P. Alexander et al. (CLEO Collaboration), Phys. Rev. Lett. 77 (1996) 5000.

[2] N. Isgur, D. Scorn, B. Grinstein and M.B. Wise, Phys. Rev. D 39 (1989) 799; D 52 (1995) 2783.

[3] M. Wirbel, S. Stech and M. Batter, Z. Phys. C 29 (1985) 637; C 34 (1987) 103.

[4] J.G. Khmer and G.A. Schuler, Z. Phys. C 38 (1988) 511.

[5] D. Melikhov, Phys. Rev. D 53 (1996) 2160.

[6] R. Casalbuoni, A. Deandrea, N. Bartolomeo, R. Gatto, E Femglio and G. Nardulli, Phys. Lett. B 299 (1993) 139.

[7] G. Martinelli, talk given at 28th ICHEP, Warsaw, July 1996, hep-ph/9610455.

[8] P. Ball, Phys. Rev. D 48 (1993) 3190.

191 V.M. Belyaev, V.M. Braun, A. Khodjamirian and R. Rttckl, Phys. Rev. D 51 (1995) 6177.

[lo] A. Abada et al. (ELC Collaboration), Nucl. Phys. B 412 (1994) 675.

[ 1 l] J.P. Alton et al. (APE Collaboration), Phys. Lett. B 345 (1995) 513.

[ 121 D.R. Burford et al. (UKQCD Collaboration), Nucl. Phys. B 447 (1995) 425.

[ 131 S. Gtisken et al. (WUP Collaboration), Nucl. Phys. B (Proc. Supl.) 42 (1995) 412.

[ 141 M. Ltlscher, S. Sint, R. Sommer, P Weisz and U. Wolff, hep-lat/9609035.

[ 151 C.G. Boyd, B. Grinstein and RF Lebed, Phys. Rev. Lett. 74 (1995) 4603.

[ 161 C.G. Boyd, B. Grinstein and RF Lebed, Phys. Lett. B 353 (1995) 306.

[ 171 C.G. Boyd, B. Grinstein and RF Lebed, Nucl. Phys. B 461 (1996) 493.

[ 181 S. Okubo and IF Shih, Phys. Rev. D 4 (1971) 2020.

[ 191 E. de Rafael and J. Taron, Phys. Rev. D 50 (1994) 373.

[20] I. Caprini, Z. Phys. C 61 (1994) 651. [21] L. Lellouch, Nucl. Phys. B 479 (1996) 353.

[22] A. Khodjamirian and R. Rtlckl, talk given at 28th ICHEP, Warsaw, July 1996, hep-ph/9610367. 1231 B. Grinstein and P.E Mende, Nucl. Phys. B 425 (1994) 451-470.

[24] P Duren, Theory of HP Spaces (Academic, New York, 1970).

[25] I. Cap&, I. Guiagu and E.E. Radescu, Phys. Rev. D 25 ( 1982) 1808.

[26] I. Caprini and L. Micu, Phys. Rev. D 54 (1996) 5686.

[27] S. Narison, World Scientific Lecture Notes in Physics, Vol. 26 (1989) (World Scientific, Singapore).

[28] M.B. Voloshin, Sov. J. Nucl. Phys. 50 (1995) 105.

[29] J.N. Simone, private communication, see also J. Flynn, talk given at 28th ICHEP Warsaw, July 1996, hep-lat/9611016.

[ 301 G. Burdman and J.F. Donoghue, Phys. Lctt. B 280 (1992) 287.

[31] M.B. Voloshin, Sov. J. Nucl. Phys. 50 (1995) 105.

[32] A. Khodjamirian and R. Rtickl, talk given at 28th ICHEP, Warsaw, July 1996, hep-ph/9610367. [ 331 D. Meliihov and N. Nikitin, hep-ph/9609503.

[ 341 Particle Data Group, Phys. Rev. D 54 (1996).