- Email: [email protected]
Form factors of exclusive b→u transitions
24 September 1998
Physics Letters B 436 Ž1998. 344–350
Form factors of exclusive b ™ u transitions Michael Beyer a , Dmitri Melikhov
b,1
a
b
Physics Department, Rostock UniÕersity, D-18051 Rostock, Germany LPTHE, UniÕersite´ de Paris XI, Batiment 211, 91405 Orsay Cedex, France ˆ Received 12 June 1998; revised 29 June 1998 Editor: P.V. Landshoff
Abstract We present the form factors of the B ™ p , r transitions induced by the b ™ u quark currents at all kinematically accessible q 2. Our analysis is based on the spectral representations of the form factors within the constituent quark picture: we fix the soft meson wave functions and the constituent quark masses by fitting A1Ž q 2 . and T2 Ž q 2 . to the lattice results at small recoils Ž17 Q q 2 Q 20 GeV 2 .. We then calculate the B ™ p , r transition form factors down to q 2 s 0. For the B ™ p case the region q 2 Q 20 GeV 2 however does not cover the whole kinematically accessible range. Due to the smallness of the pion mass the region of small recoils is close to the nearby B ) Ž5234. resonance. We develop a parametrization which includes the B ) dominance of the form factors fq and fy at small recoils and numerically reproduces the results of < < 2 y1 and G Ž B ™ r l n . s 15.8 " 2.3 < Vu b < 2 psy1. calculations at q 2 Q 20 GeV 2. We find G Ž B ™ p l n . s 8.0q0.8 y0.2 Vu b ps q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction First measurements of the semileptonic ŽSL. B ™ Žp , r . l n branching fractions by CLEO w1,2x opened a possibility to determine < Vu b <. Precise knowledge of this element of the Cabbibo-Kobayashi-Maskawa matrix which describes the quark mixing in the Standard Model ŽSM. is necessary both for understanding the dynamics of the SM and the origin of CP violation. However, for a proper extraction of < Vu b < from the SL decays one needs a reliable knowledge of the meson transition form factors which
1 On leave of absence from Nuclear Physics Institute, Moscow State University, Moscow, 119899, Russia
encode the long-distance ŽLD. contributions to the exclusive b ™ u transitions. Various nonperturbative theoretical frameworks have been applied to the description of the meson transition form factors induced by the b ™ u weak transition: among them are the constituent quark models w3–12x, QCD sum rules w13–15x, lattice QCD w16x, and analytical constraints w17,18x. Lattice QCD simulations provide the most fundamental nonperturbative approach and thus should lead to the most reliable results. Still, some restrictions remain to be solved in the context of heavy-tolight transitions. One of them is the necessity to extrapolate the transition form factors in the heavy quark mass from the values of order m c utilized in the lattice approach to m b . Another problem is that lattice calculations provide the form factors only in a
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 4 7 - 8
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
region excluding large recoils. Therefore to obtain form factors in the whole kinematical decay region one has to rely on some extrapolation procedures. QCD sum rules give a complementary information on the form factors as they allow one to determine the form factors at not very large momentum transfers and therefore also do not cover the whole kinematically accessible q 2-range w13x. In practice, however, various versions of QCD sum rules give rather uncertain predictions dependent on the technical subtleties of the particular version w13,15x. Various models based on the constituent quark picture have been used for considering meson decays Žsee, e.g. a talk of A. Le Yaouanc for a detailed review w12x.. An attractive feature of the approaches based on the concept of constituent quarks is that these approaches provide a physical picture of the process. However, a long-standing problem of the constituent quark model ŽQM. applications to meson decays is a strong dependence of the predictions on the QM parameters. Although none of these approaches is able at the moment to provide the form factors in the whole accessible kinematical region of the B decay, a combination of different approaches might be fruitful. For instance, in Ref. w16x a simple lattice-constrained parametrization based on approximate relations obtained within the constituent quark picture w7x and pole dominance have been proposed. However, within this approach the B meson decays induced by the different quark transitions, e.g. b ™ u and b ™ s, remain largely disconnected. In Ref. w20x it was noticed that determining the soft meson wave functions by matching the quark model calculations of the transition form factors to the lattice results at small recoils allows one to connect many decay processes to each other. In this letter we apply such an approach to a study of the B ™ p , r transition form factors. Namely, we fix the meson soft wave functions and the constituent quark masses by fitting the lattice results to the form factors A1Ž q 2 . and T2 Ž q 2 . at small recoils w16x, and then calculate the form factors in the region 0 - q 2 Q 20 GeV 2 through the spectral representations of the quark model w10x. These spectral representations respect rigorous QCD constraints in the limit of heavy meson decays both to heavy and light mesons and thus we expect them to supply
345
a reliable continuation of the lattice results to the lower q 2 region. Thus, for the B ™ r transition we calculate the form factors at all kinematically accessible q 2 . For the B ™ p case this range given above does not cover the whole kinematically accessible region. Also, no lattic points are provided for fq Ž q 2 . above q 2 ) 20 GeV 2 . To extrapolate the form factors fq and fy to larger q 2 , note that the momentum transfers become rather close to the B ) Ž5234. resonance. We therefore propose a parametrization which takes into account the B ) dominance in the region of small recoils and reproduces the results of calculations at q 2 Q 20 GeV 2 . The form factors of interest are connected with the meson transition amplitudes induced by the vector Vm s q2 gm q1 , axial-vector Am s q2 gm g 5q1 , and tensor Tmn s q2 smn q1 , q1 ™ q2 quark transition currents as follows Žsee notations in Ref. w20x. ² P Ž M2 , p 2 . < Vm Ž 0 . < P Ž M1 , p 1 . : s fq Ž q 2 . Pm q fy Ž q 2 . qm , ² V Ž M2 , p 2 , e . < Vm Ž 0 . < P Ž M1 , p 1 . : s 2 g Ž q 2 . emn a b e ) n p 1a p 2b , ² V Ž M2 , p 2 , e . < Am Ž 0 . < P Ž M1 , p 1 . : sie ) a
f Ž q 2 . gma
qaq Ž q 2 . p 1 a Pm q ay Ž q 2 . p 1 a qm , ² P Ž M2 , p 2 . < Tmn Ž 0 . < P Ž M1 , p 1 . : s y2 i s Ž q 2 . Ž p 1 m p 2 n y p 1 n p 2 m . , ² V Ž M2 , p 2 , e . < Tmn Ž 0 . < P Ž M1 , p 1 . : sie ) a
gq Ž q 2 . emn a b P b q gy Ž q 2 . emn a b q b
qg 0 Ž q 2 . p 1 a emnbg p 1b p g2 ,
Ž 1.
where q s p 1 y p 2 , P s p 1 q p 2 . The dispersion approach of Refs. w10,19x gives the transition form factors of the meson M1 to the meson M2 as double relativistic spectral representations through the soft wave functions of the initial and final mesons, c 1Ž s1 . and c 2 Ž s2 ., respectively f i Ž q 2 . s ds1 c 1 Ž s1 . ds2 c 2 Ž s2 . f˜i Ž s1 , s2 ,q 2 . , Ž 2 .
H
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
346
where s1 Ž s2 . is the invariant mass of the initial Žfinal. qq pair. The double spectral densities f˜i of the representation Ž2. for the 0y™ 0y,1q meson decays induced by the vector, axial-vector and tensor quark currents have been calculated in w10,19x. The representation Ž2. is valid for q 2 F Ž m 2 y m1 . 2 . It is important to notice that the form factors Ž2. develop the correct structure of the heavy-quark expansion in accordance with QCD in the leading and next-to-leading 1rm Q orders if the soft wave functions c i are localized in the momentum space in a region of the order of the confinement scale. The spectral densities for all the form factors Ž1. have been calculated in w19x. The spectral representations Ž2. take into account LD contributions connected with the meson formation in the initial and final channels. At large q 2 the LD effects in the q 2-channel become more essential and thus one should properly replace f M 1 ™ M 2Ž q
2
. ™ f q1 ™ q 2Ž q
2
. f M 1 ™ M 2Ž q
2
.,
Ž 3.
where the quark transition form factor f q1 ™ q 2Ž q 2 . is introduced that accounts for the LD effects at large q 2 given by the relevant hadronic resonances and continuum states. The form factor f q1 ™ q 2Ž q 2 . equals unity at q 2 far below the resonance region and 2 contains poles at q 2 s Mres . Notice that the particular form of the quark transition form factor does not depend on the initial and final mesons involved but rather depends on the set of the relevant hadronic resonances and is different for the vector, axial-vector etc. channels.
2. B ™ r transitions We consider the meson wave functions and the constituent quark masses as variational parameters 2
Table 1 Quark masses and the slope parameters of the soft meson wave functions Žin GeV. mb
bB
mu
bp
br
4.85"0.03 0.23"0.01 0.54"0.04 0.36"0.02 0.31"0.03
and determine them from fitting the lattice results to reproduce T2 Ž q 2 . and A1Ž q 2 . at q 2 s 19.6 and 17.6 GeV 2 w16x by the double spectral representations Ž2. and assuming f b ™ u s 1 in the region q 2 Q 20 GeV 2 . The soft w ave function of a m eson M w q Ž m q . q Ž m q .x can be written as
(
s 2 y m 2q y m2q
p c Ž s. s
'2
ž
2
(s y Ž m y m . q
wŽ k2 .
/
s 3r4
2
q
,
Ž 4.
where k 2 s lŽ s,m2q ,m2q .r4 s with lŽ s,m2q ,m 2q . s Ž s y m 2q y m2q . 2 y 4 m 2q m2q , and the ground-state radial S-wave function w Ž k 2 . is normalized as Hw 2 Ž k 2 . k 2 dk s 1. For the functions w Ž k 2 . we assume a simple gaussian form w Ž k 2 . A exp Ž yk 2r2 b 2 . ,
Ž 5.
where b to be obtained by a fit. The ranges of the B and r are shown in Table 1. The values of the constituent quark masses and the slope parameter br are fixed rather tightly by the x 2 fit to the lattice data, whereas bB cannot be fixed with a good accuracy. We determine the ranges of bB such that the leptonic decay constant f B calculated through the relation w10x
2
One comment on the previous application of the dispersion quark model to meson decays is in order. In w10x it was shown that the form factors calculated with the QM parameters of the ISGW2 model w4x Žwhich differs considerably from the ISGW2 model for the transition form factors. provide a good description of all experimental data on semileptonic B and D decays. However, the form factors of w10x have a much flatter q 2-dependence and do not match the lattice results at large q 2 .
(
f P s Nc Ž m q q m q . ds c Ž s .
=
H
l1r2 s,m2q ,m2q s y Ž m q y m q . 2
ž
8p 2 s
/
s
.
Ž 6.
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
347
Fig. 1. The form factors of the B ™ r and B ™ p transitions through the b ™ u quark currents vs. lattice data w16x and calculations within different approaches. A1 s frŽ MB q Mr ., A 2 s yŽ MB q Mr . aq, A 0 s w q 2 ayq f q Ž MB2 y Mr2 . aq xr2 Mr , V s Ž MB q Mr . g, T1Ž q 2 . s ygqr2, T2 s y 12 Ž gqq q 2 gyrŽ MB2 y Mr2 ... Solid lines - our QM results, dotted lines - lattice-constrained parametrizations of w16x, dashed lines - LCSR w15x.
lies in the interval f B s 170 " 30 MeV in accordance with the lattice estimates w16x. Once the wave functions and the quark masses are determined, we use the spectral representations Ž2. for calculating all the form factors for the B ™ r transition in the whole kinematically accessible region. Fig. 1 illustrates the calculated form factors versus the lattice
data. Table 2 gives parameters of a convenient interpolation of the results of the calculation in the form f Ž 0. f Ž q2 . s , Ž 7. 1 y s 1 qˆ 2 q s 2 qˆ 4 where we have introduced qˆ 2 s q 2rM B2) with MB) s 5.324 GeV. Since we have calculated the form
Table 2 Parameters of the fits to the calculated B ™ p , r transition form factors in the form Ž10., Ž15. for f " and Ž7. for all other form factors. The numbers correspond to the central values of the QM parameters given in Table 1
f Ž0. s1 s2
fq
fy
s
g
f
aq
ay
gq
gy
g0
0.284 0.184 y0.52
y0.247 0.16 y0.577
0.05 1.5 0.5
0.051 1.60 0.60
1.55 0.69 0.041
y0.04 1.40 0.50
0.044 1.49 0.54
y0.27 1.60 0.60
0.25 1.61 0.60
0.00374 2.36 1.64
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
348 Table 3 Decay rates in units < Vu b < 2 psy1 Ref.
G ŽB™p l n .
G ŽB™ r l n .
GL rGT
This work ISGW2 QM w4x Lat w16x LCSR w13x
8.0q0.8 y0 .2
15.8"2.3 14.2 16.5q3.5 y2.3 13.5" 4.0
0.88"0.08 0.3 0.80q0.04 y0.03 0.52"0.08
9.6 .4 8.5q3 y0 .9 y
states. It should be noted, that both the vector 1y and scalar 0q resonances contribute to fy whereas only vector 1y states contribute to fq Žsee e.g. w21x.. Regular terms might be taken into account by assuming a single-pole form for the form factors with a modified q 2-dependent ’residue’ as follows f"Ž q2 . s
factors at all kinematically accessible q 2 the particular form of the fit function is not important. The interpolations Ž7. deviate from the results of calculation by less than 1%. The calculated decay rates are given in Table 3.
fˆ" Ž qˆ 2 . 1 y qˆ 2
,
Ž 10 .
where fˆq Ž 1 . s
g B ) Bp f B ) 2 MB )
,
fˆy Ž 1 . s yfˆq Ž 1 .
MB2 y Mp2 MB2)
.
Ž 11 . 3. B ™ p transitions For the transition B ™ p a new wave function parameter bp appears. It is not independent and strongly correlates with m u through fp given by eq Ž6.. Requiring f P s 132 MeV this implicitly determines bp once m u is fixed. With the wave functions gived, we calculate the B ™ p transition form factors at 0 - q 2 Q 20 GeV 2 . The form factors versus the lattice results shown in Fig. 1 are found to be in perfect agreement. This confirms our assumption f b ™ u s 1 at q 2 Q 20 GeV 2 . This region however does not cover the whole kinematically accessible range. To find the form factors at larger q 2 we must use some extrapolation procedure. In the region of small recoils the form factors are dominated by the neighbouring B ) poles and one finds fq Ž q 2 . s
g B ) Bp f B ) 2 MB ) Ž 1 y q 2rMB2) . q regular terms at q 2s MB2) ,
fy Ž q 2 . s
g B ) Bp f B )
MB2 y Mp2
2 MB ) Ž 1 y q 2rMB2) .
MB2)
q regular terms at q 2s MB2) ,
Ž 8.
Ž 9.
where the B ) Bp coupling constant g B ) Bp is defined through ²p Ž p 2 . B ) Ž q .< B Ž p 1 .: s g B ) Bp ea) Ž q . p 2a. The regular terms here stand for the contribution of other resonances and continuum hadronic
Using the PCAC prescription for the pion field, the B ) Bp coupling constant can be estimated at the unphysical point g B ) Bp Ž p 12 s MB2 ,q 2 s MB2 , p 22 s 0.. At this point the coupling constant is represented through the meson transition form factor f P Ž M B .™ V Ž M B . which can be calculated within the same dispersion approach. Namely, we find ²p Ž p 2 . V Ž q . < P Ž p 1 . : lim
s
p 22™0, q 2™p 12
1 fp
ea) Ž q . p 2a f Ž p 22 , p 12 ,q 2 .
qaq Ž p 22 , p 12 ,q 2 .Ž p 12 y q 2 . qay Ž p 22 , p 12 ,q 2 . p 22 1 s fp
ea) Ž q . p 2a f Ž 0, MP2 , MP2 . ,
Ž 12 .
and the form factor f Ž0, MB2 , MB2 . of the B ™ B ) transition is calculated through the spectral representation Ž2. assuming identical radial wave functions of B ) and B mesons. In the heavy quark limit this is a rigorous property, and we expect this approximation to work well for real B and B ) mesons. To get to the physical point g B ) Bp Ž mp2 , MB2 , MB2) . one needs to perform a continuation which is not unique. However due to the small difference of the B and B ) meson masses we expect g B ) Bp Ž mp2 , MB2 , MB2) . , g B ) Bp Ž0, MB2 , MB2 .. The result of the calculation of f Ž0, MB2 , MB2 . is weakly sensitive to the values of the quark masses but mostly depends on the B wave function. The
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
value f Ž0, MB2 , MB2 . strongly correlates with f B such that the relation g B ) Bp s
9 " 0.4 GeV
Ž 13 .
fB
is fulfilled for the range of the QM parameters which reproduce f B s 170 " 30 MeV. The Sum Rule analysis of the g B ) Bp and references to other results can be found in w22x. Finally, the residue of the form factor fq at the B ) pole takes the value fˆq Ž 1 . s Ž 0.8 " 0.04 . f B ) rf B ,
Ž 14 .
and for further numerical estimates we use f B )rf B s 1.2 " 0.1. For the quantities fˆ" we assume a smooth parametrization fˆ" Ž qˆ 2 . s
f " Ž 0.
Ž 1 y s1"qˆ 2 q s 2"qˆ 4 .
,
Ž 15 .
where the coefficients s 1,2 are not independent: Eq. Ž14. gives fq Ž 0 . q 1 y sq 1 q s2
s Ž 0.8 " 0.04 .
fB )
Ž 16 .
fB
and the relation Ž11. leads to fq Ž 0 .
Ž 1 y sq1 q sq2 .
q
fy Ž 0 .
MB) 2
Ž 1 y sy1 q sy2 . MB2 y Mp2
s 0.
Ž 17 . The parameters s 1,2 are determined from the x 2-fit to the results of the calculation at q 2 Q 20 GeV 2 . Table 2 presents the relevant numbers. At q 2 G 20 GeV 2 the parametrizations are used for extrapolation of the form factors f " to all kinematically accessible q 2 Žsee Fig. 1.. For the form factor f 0 Ž q 2 . s fq Ž q 2 . q 2 q fy Ž q 2 .rPq a combination of PCAC and current algebra yields the relation w23x f 0 MB2
Ž
. s f Brfp .
Ž 18 .
Using the value f B s 170 " 30 MeV we obtain f 0 Ž MB2 . s 1.35 " 0.3, which is found to be in a reasonable agreement with the results of our extrapolating formulas.
349
The calculated B ™ p l n decay rate is gived in Table 3. Notice that the details of the high-q 2 behavior of the form factors which depend on the extrapolation procedure do not affect considerably the decay rate. The latter is mostly determined by the region q 2 Q 20 GeV 2 where the form factors are calculated directly. Fig. 1 compares our results with recent light-cone sum rule calculations available at q 2 Q 16 GeV 2 w15x and lattice-constrained parametrizations of Ref. w16x. One can see that the results of different approaches to the form factors do not differ significantly. However, it should be taken into account that in the case of the B ™ r transition this minor difference in the form factors provides rather sizeable spread of predictions for the decay rates. Summing up, we have analyzed the form factors of the exclusive b ™ u transition using the spectral representations based on constituent quark picture and obtain form factors in the whole kinematically accessible region. The meson wave functions and the constituent quark masses have been determined by describing the results of lattice simulations of the form factors A1Ž q 2 . and T2 Ž q 2 . at small recoils. This allowed us to calculate the form factors at q 2 Q 20 GeV 2 which cover all kinematically accessible q 2 in the B ™ r transition. In the B ™ p case the interval q 2 Q 20 GeV 2 does not cover the kinematically accessible region and an extrapolation to higher q 2 is necessary. To this end we take into account the dominance of the form factors at small recoils by the B ) pole. The calculated B ™ p l n decay rate is found to be only slightly sensitive to the particular details of the extrapolation procedure.
Acknowledgements We take the pleasure in thanking D. Becirevic and H. Schroder ¨ for helpful discussions which considerably influenced the final form of the paper. We are grateful to A. Le Yaouanc and O. Pene for the interest in this work and to DFG for financial support under grant 436 RUS 18r7r98. D.M. is grateful to the Physics Department of the Rostock University for hospitality.
350
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
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w13x P. Ball, V.M. Braun, Phys. Rev. D 55 Ž1997. 5561; V.M. Braun, in: Progress in Heavy Quark Physics, Proceedings of the IV International Workshop, Rostock, 1997, edited by M. Beyer, T. Mannel, H. Schroder, p. 105 ŽUniversity of Ros¨ tock, 1998., hep-phr9801222. w14x A. Khodjamirian, R. Ruckl, S. Weinzierl, O. Yakovlev, Phys. ¨ Lett. B 410 Ž1997. 275; E. Bagan, P. Ball, V.M. Braun, Phys. Lett. B 417 Ž1998. 154. w15x P. Ball, hep-phr9803501, P. Ball, V.M. Braun, hepphr9805422. w16x J.M. Flynn, C.T. Sachrajda, e-print hep-latr9710057, to appear in Heavy Flavours Ž2nd ed.., ed. by A.J. Buras, M. Linder ŽWorld Scientific, Singapore.; UKQCD Collaboration, L. Del Debbio et al., Phys. Lett. B 416 Ž1998. 392. w17x L. Lellouch, Nucl. Phys. B 479 Ž1996. 353. w18x D. Becirevic, Phys. Rev. D 54 Ž1996. 6842; Phys. Lett. B 399 Ž1997. 163. w19x D. Melikhov, Phys. Rev. D 56 Ž1997. 7089. w20x D. Melikhov, N. Nikitin, S. Simula, Phys. Rev. D 57 Ž1998. 6814. w21x B. Grinstein, P. Mende, Nucl. Phys. B 451 Ž1994. 451. w22x V.M. Belyaev, V.M. Braun, A. Khodjamirian, R. Ruckl, ¨ Phys. Rev. D 51 Ž1995. 6177. w23x M.B. Voloshin, Sov. J. Nucl. Phys. 50 Ž1989. 105; C.A. Dominguez, J.G. Korner, K. Schilcher, Phys. Lett. B 248 ¨ Ž1990. 399.
Physics Letters B 436 Ž1998. 344–350
Form factors of exclusive b ™ u transitions Michael Beyer a , Dmitri Melikhov
b,1
a
b
Physics Department, Rostock UniÕersity, D-18051 Rostock, Germany LPTHE, UniÕersite´ de Paris XI, Batiment 211, 91405 Orsay Cedex, France ˆ Received 12 June 1998; revised 29 June 1998 Editor: P.V. Landshoff
Abstract We present the form factors of the B ™ p , r transitions induced by the b ™ u quark currents at all kinematically accessible q 2. Our analysis is based on the spectral representations of the form factors within the constituent quark picture: we fix the soft meson wave functions and the constituent quark masses by fitting A1Ž q 2 . and T2 Ž q 2 . to the lattice results at small recoils Ž17 Q q 2 Q 20 GeV 2 .. We then calculate the B ™ p , r transition form factors down to q 2 s 0. For the B ™ p case the region q 2 Q 20 GeV 2 however does not cover the whole kinematically accessible range. Due to the smallness of the pion mass the region of small recoils is close to the nearby B ) Ž5234. resonance. We develop a parametrization which includes the B ) dominance of the form factors fq and fy at small recoils and numerically reproduces the results of < < 2 y1 and G Ž B ™ r l n . s 15.8 " 2.3 < Vu b < 2 psy1. calculations at q 2 Q 20 GeV 2. We find G Ž B ™ p l n . s 8.0q0.8 y0.2 Vu b ps q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction First measurements of the semileptonic ŽSL. B ™ Žp , r . l n branching fractions by CLEO w1,2x opened a possibility to determine < Vu b <. Precise knowledge of this element of the Cabbibo-Kobayashi-Maskawa matrix which describes the quark mixing in the Standard Model ŽSM. is necessary both for understanding the dynamics of the SM and the origin of CP violation. However, for a proper extraction of < Vu b < from the SL decays one needs a reliable knowledge of the meson transition form factors which
1 On leave of absence from Nuclear Physics Institute, Moscow State University, Moscow, 119899, Russia
encode the long-distance ŽLD. contributions to the exclusive b ™ u transitions. Various nonperturbative theoretical frameworks have been applied to the description of the meson transition form factors induced by the b ™ u weak transition: among them are the constituent quark models w3–12x, QCD sum rules w13–15x, lattice QCD w16x, and analytical constraints w17,18x. Lattice QCD simulations provide the most fundamental nonperturbative approach and thus should lead to the most reliable results. Still, some restrictions remain to be solved in the context of heavy-tolight transitions. One of them is the necessity to extrapolate the transition form factors in the heavy quark mass from the values of order m c utilized in the lattice approach to m b . Another problem is that lattice calculations provide the form factors only in a
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 4 7 - 8
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
region excluding large recoils. Therefore to obtain form factors in the whole kinematical decay region one has to rely on some extrapolation procedures. QCD sum rules give a complementary information on the form factors as they allow one to determine the form factors at not very large momentum transfers and therefore also do not cover the whole kinematically accessible q 2-range w13x. In practice, however, various versions of QCD sum rules give rather uncertain predictions dependent on the technical subtleties of the particular version w13,15x. Various models based on the constituent quark picture have been used for considering meson decays Žsee, e.g. a talk of A. Le Yaouanc for a detailed review w12x.. An attractive feature of the approaches based on the concept of constituent quarks is that these approaches provide a physical picture of the process. However, a long-standing problem of the constituent quark model ŽQM. applications to meson decays is a strong dependence of the predictions on the QM parameters. Although none of these approaches is able at the moment to provide the form factors in the whole accessible kinematical region of the B decay, a combination of different approaches might be fruitful. For instance, in Ref. w16x a simple lattice-constrained parametrization based on approximate relations obtained within the constituent quark picture w7x and pole dominance have been proposed. However, within this approach the B meson decays induced by the different quark transitions, e.g. b ™ u and b ™ s, remain largely disconnected. In Ref. w20x it was noticed that determining the soft meson wave functions by matching the quark model calculations of the transition form factors to the lattice results at small recoils allows one to connect many decay processes to each other. In this letter we apply such an approach to a study of the B ™ p , r transition form factors. Namely, we fix the meson soft wave functions and the constituent quark masses by fitting the lattice results to the form factors A1Ž q 2 . and T2 Ž q 2 . at small recoils w16x, and then calculate the form factors in the region 0 - q 2 Q 20 GeV 2 through the spectral representations of the quark model w10x. These spectral representations respect rigorous QCD constraints in the limit of heavy meson decays both to heavy and light mesons and thus we expect them to supply
345
a reliable continuation of the lattice results to the lower q 2 region. Thus, for the B ™ r transition we calculate the form factors at all kinematically accessible q 2 . For the B ™ p case this range given above does not cover the whole kinematically accessible region. Also, no lattic points are provided for fq Ž q 2 . above q 2 ) 20 GeV 2 . To extrapolate the form factors fq and fy to larger q 2 , note that the momentum transfers become rather close to the B ) Ž5234. resonance. We therefore propose a parametrization which takes into account the B ) dominance in the region of small recoils and reproduces the results of calculations at q 2 Q 20 GeV 2 . The form factors of interest are connected with the meson transition amplitudes induced by the vector Vm s q2 gm q1 , axial-vector Am s q2 gm g 5q1 , and tensor Tmn s q2 smn q1 , q1 ™ q2 quark transition currents as follows Žsee notations in Ref. w20x. ² P Ž M2 , p 2 . < Vm Ž 0 . < P Ž M1 , p 1 . : s fq Ž q 2 . Pm q fy Ž q 2 . qm , ² V Ž M2 , p 2 , e . < Vm Ž 0 . < P Ž M1 , p 1 . : s 2 g Ž q 2 . emn a b e ) n p 1a p 2b , ² V Ž M2 , p 2 , e . < Am Ž 0 . < P Ž M1 , p 1 . : sie ) a
f Ž q 2 . gma
qaq Ž q 2 . p 1 a Pm q ay Ž q 2 . p 1 a qm , ² P Ž M2 , p 2 . < Tmn Ž 0 . < P Ž M1 , p 1 . : s y2 i s Ž q 2 . Ž p 1 m p 2 n y p 1 n p 2 m . , ² V Ž M2 , p 2 , e . < Tmn Ž 0 . < P Ž M1 , p 1 . : sie ) a
gq Ž q 2 . emn a b P b q gy Ž q 2 . emn a b q b
qg 0 Ž q 2 . p 1 a emnbg p 1b p g2 ,
Ž 1.
where q s p 1 y p 2 , P s p 1 q p 2 . The dispersion approach of Refs. w10,19x gives the transition form factors of the meson M1 to the meson M2 as double relativistic spectral representations through the soft wave functions of the initial and final mesons, c 1Ž s1 . and c 2 Ž s2 ., respectively f i Ž q 2 . s ds1 c 1 Ž s1 . ds2 c 2 Ž s2 . f˜i Ž s1 , s2 ,q 2 . , Ž 2 .
H
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
346
where s1 Ž s2 . is the invariant mass of the initial Žfinal. qq pair. The double spectral densities f˜i of the representation Ž2. for the 0y™ 0y,1q meson decays induced by the vector, axial-vector and tensor quark currents have been calculated in w10,19x. The representation Ž2. is valid for q 2 F Ž m 2 y m1 . 2 . It is important to notice that the form factors Ž2. develop the correct structure of the heavy-quark expansion in accordance with QCD in the leading and next-to-leading 1rm Q orders if the soft wave functions c i are localized in the momentum space in a region of the order of the confinement scale. The spectral densities for all the form factors Ž1. have been calculated in w19x. The spectral representations Ž2. take into account LD contributions connected with the meson formation in the initial and final channels. At large q 2 the LD effects in the q 2-channel become more essential and thus one should properly replace f M 1 ™ M 2Ž q
2
. ™ f q1 ™ q 2Ž q
2
. f M 1 ™ M 2Ž q
2
.,
Ž 3.
where the quark transition form factor f q1 ™ q 2Ž q 2 . is introduced that accounts for the LD effects at large q 2 given by the relevant hadronic resonances and continuum states. The form factor f q1 ™ q 2Ž q 2 . equals unity at q 2 far below the resonance region and 2 contains poles at q 2 s Mres . Notice that the particular form of the quark transition form factor does not depend on the initial and final mesons involved but rather depends on the set of the relevant hadronic resonances and is different for the vector, axial-vector etc. channels.
2. B ™ r transitions We consider the meson wave functions and the constituent quark masses as variational parameters 2
Table 1 Quark masses and the slope parameters of the soft meson wave functions Žin GeV. mb
bB
mu
bp
br
4.85"0.03 0.23"0.01 0.54"0.04 0.36"0.02 0.31"0.03
and determine them from fitting the lattice results to reproduce T2 Ž q 2 . and A1Ž q 2 . at q 2 s 19.6 and 17.6 GeV 2 w16x by the double spectral representations Ž2. and assuming f b ™ u s 1 in the region q 2 Q 20 GeV 2 . The soft w ave function of a m eson M w q Ž m q . q Ž m q .x can be written as
(
s 2 y m 2q y m2q
p c Ž s. s
'2
ž
2
(s y Ž m y m . q
wŽ k2 .
/
s 3r4
2
q
,
Ž 4.
where k 2 s lŽ s,m2q ,m2q .r4 s with lŽ s,m2q ,m 2q . s Ž s y m 2q y m2q . 2 y 4 m 2q m2q , and the ground-state radial S-wave function w Ž k 2 . is normalized as Hw 2 Ž k 2 . k 2 dk s 1. For the functions w Ž k 2 . we assume a simple gaussian form w Ž k 2 . A exp Ž yk 2r2 b 2 . ,
Ž 5.
where b to be obtained by a fit. The ranges of the B and r are shown in Table 1. The values of the constituent quark masses and the slope parameter br are fixed rather tightly by the x 2 fit to the lattice data, whereas bB cannot be fixed with a good accuracy. We determine the ranges of bB such that the leptonic decay constant f B calculated through the relation w10x
2
One comment on the previous application of the dispersion quark model to meson decays is in order. In w10x it was shown that the form factors calculated with the QM parameters of the ISGW2 model w4x Žwhich differs considerably from the ISGW2 model for the transition form factors. provide a good description of all experimental data on semileptonic B and D decays. However, the form factors of w10x have a much flatter q 2-dependence and do not match the lattice results at large q 2 .
(
f P s Nc Ž m q q m q . ds c Ž s .
=
H
l1r2 s,m2q ,m2q s y Ž m q y m q . 2
ž
8p 2 s
/
s
.
Ž 6.
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
347
Fig. 1. The form factors of the B ™ r and B ™ p transitions through the b ™ u quark currents vs. lattice data w16x and calculations within different approaches. A1 s frŽ MB q Mr ., A 2 s yŽ MB q Mr . aq, A 0 s w q 2 ayq f q Ž MB2 y Mr2 . aq xr2 Mr , V s Ž MB q Mr . g, T1Ž q 2 . s ygqr2, T2 s y 12 Ž gqq q 2 gyrŽ MB2 y Mr2 ... Solid lines - our QM results, dotted lines - lattice-constrained parametrizations of w16x, dashed lines - LCSR w15x.
lies in the interval f B s 170 " 30 MeV in accordance with the lattice estimates w16x. Once the wave functions and the quark masses are determined, we use the spectral representations Ž2. for calculating all the form factors for the B ™ r transition in the whole kinematically accessible region. Fig. 1 illustrates the calculated form factors versus the lattice
data. Table 2 gives parameters of a convenient interpolation of the results of the calculation in the form f Ž 0. f Ž q2 . s , Ž 7. 1 y s 1 qˆ 2 q s 2 qˆ 4 where we have introduced qˆ 2 s q 2rM B2) with MB) s 5.324 GeV. Since we have calculated the form
Table 2 Parameters of the fits to the calculated B ™ p , r transition form factors in the form Ž10., Ž15. for f " and Ž7. for all other form factors. The numbers correspond to the central values of the QM parameters given in Table 1
f Ž0. s1 s2
fq
fy
s
g
f
aq
ay
gq
gy
g0
0.284 0.184 y0.52
y0.247 0.16 y0.577
0.05 1.5 0.5
0.051 1.60 0.60
1.55 0.69 0.041
y0.04 1.40 0.50
0.044 1.49 0.54
y0.27 1.60 0.60
0.25 1.61 0.60
0.00374 2.36 1.64
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
348 Table 3 Decay rates in units < Vu b < 2 psy1 Ref.
G ŽB™p l n .
G ŽB™ r l n .
GL rGT
This work ISGW2 QM w4x Lat w16x LCSR w13x
8.0q0.8 y0 .2
15.8"2.3 14.2 16.5q3.5 y2.3 13.5" 4.0
0.88"0.08 0.3 0.80q0.04 y0.03 0.52"0.08
9.6 .4 8.5q3 y0 .9 y
states. It should be noted, that both the vector 1y and scalar 0q resonances contribute to fy whereas only vector 1y states contribute to fq Žsee e.g. w21x.. Regular terms might be taken into account by assuming a single-pole form for the form factors with a modified q 2-dependent ’residue’ as follows f"Ž q2 . s
factors at all kinematically accessible q 2 the particular form of the fit function is not important. The interpolations Ž7. deviate from the results of calculation by less than 1%. The calculated decay rates are given in Table 3.
fˆ" Ž qˆ 2 . 1 y qˆ 2
,
Ž 10 .
where fˆq Ž 1 . s
g B ) Bp f B ) 2 MB )
,
fˆy Ž 1 . s yfˆq Ž 1 .
MB2 y Mp2 MB2)
.
Ž 11 . 3. B ™ p transitions For the transition B ™ p a new wave function parameter bp appears. It is not independent and strongly correlates with m u through fp given by eq Ž6.. Requiring f P s 132 MeV this implicitly determines bp once m u is fixed. With the wave functions gived, we calculate the B ™ p transition form factors at 0 - q 2 Q 20 GeV 2 . The form factors versus the lattice results shown in Fig. 1 are found to be in perfect agreement. This confirms our assumption f b ™ u s 1 at q 2 Q 20 GeV 2 . This region however does not cover the whole kinematically accessible range. To find the form factors at larger q 2 we must use some extrapolation procedure. In the region of small recoils the form factors are dominated by the neighbouring B ) poles and one finds fq Ž q 2 . s
g B ) Bp f B ) 2 MB ) Ž 1 y q 2rMB2) . q regular terms at q 2s MB2) ,
fy Ž q 2 . s
g B ) Bp f B )
MB2 y Mp2
2 MB ) Ž 1 y q 2rMB2) .
MB2)
q regular terms at q 2s MB2) ,
Ž 8.
Ž 9.
where the B ) Bp coupling constant g B ) Bp is defined through ²p Ž p 2 . B ) Ž q .< B Ž p 1 .: s g B ) Bp ea) Ž q . p 2a. The regular terms here stand for the contribution of other resonances and continuum hadronic
Using the PCAC prescription for the pion field, the B ) Bp coupling constant can be estimated at the unphysical point g B ) Bp Ž p 12 s MB2 ,q 2 s MB2 , p 22 s 0.. At this point the coupling constant is represented through the meson transition form factor f P Ž M B .™ V Ž M B . which can be calculated within the same dispersion approach. Namely, we find ²p Ž p 2 . V Ž q . < P Ž p 1 . : lim
s
p 22™0, q 2™p 12
1 fp
ea) Ž q . p 2a f Ž p 22 , p 12 ,q 2 .
qaq Ž p 22 , p 12 ,q 2 .Ž p 12 y q 2 . qay Ž p 22 , p 12 ,q 2 . p 22 1 s fp
ea) Ž q . p 2a f Ž 0, MP2 , MP2 . ,
Ž 12 .
and the form factor f Ž0, MB2 , MB2 . of the B ™ B ) transition is calculated through the spectral representation Ž2. assuming identical radial wave functions of B ) and B mesons. In the heavy quark limit this is a rigorous property, and we expect this approximation to work well for real B and B ) mesons. To get to the physical point g B ) Bp Ž mp2 , MB2 , MB2) . one needs to perform a continuation which is not unique. However due to the small difference of the B and B ) meson masses we expect g B ) Bp Ž mp2 , MB2 , MB2) . , g B ) Bp Ž0, MB2 , MB2 .. The result of the calculation of f Ž0, MB2 , MB2 . is weakly sensitive to the values of the quark masses but mostly depends on the B wave function. The
M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
value f Ž0, MB2 , MB2 . strongly correlates with f B such that the relation g B ) Bp s
9 " 0.4 GeV
Ž 13 .
fB
is fulfilled for the range of the QM parameters which reproduce f B s 170 " 30 MeV. The Sum Rule analysis of the g B ) Bp and references to other results can be found in w22x. Finally, the residue of the form factor fq at the B ) pole takes the value fˆq Ž 1 . s Ž 0.8 " 0.04 . f B ) rf B ,
Ž 14 .
and for further numerical estimates we use f B )rf B s 1.2 " 0.1. For the quantities fˆ" we assume a smooth parametrization fˆ" Ž qˆ 2 . s
f " Ž 0.
Ž 1 y s1"qˆ 2 q s 2"qˆ 4 .
,
Ž 15 .
where the coefficients s 1,2 are not independent: Eq. Ž14. gives fq Ž 0 . q 1 y sq 1 q s2
s Ž 0.8 " 0.04 .
fB )
Ž 16 .
fB
and the relation Ž11. leads to fq Ž 0 .
Ž 1 y sq1 q sq2 .
q
fy Ž 0 .
MB) 2
Ž 1 y sy1 q sy2 . MB2 y Mp2
s 0.
Ž 17 . The parameters s 1,2 are determined from the x 2-fit to the results of the calculation at q 2 Q 20 GeV 2 . Table 2 presents the relevant numbers. At q 2 G 20 GeV 2 the parametrizations are used for extrapolation of the form factors f " to all kinematically accessible q 2 Žsee Fig. 1.. For the form factor f 0 Ž q 2 . s fq Ž q 2 . q 2 q fy Ž q 2 .rPq a combination of PCAC and current algebra yields the relation w23x f 0 MB2
Ž
. s f Brfp .
Ž 18 .
Using the value f B s 170 " 30 MeV we obtain f 0 Ž MB2 . s 1.35 " 0.3, which is found to be in a reasonable agreement with the results of our extrapolating formulas.
349
The calculated B ™ p l n decay rate is gived in Table 3. Notice that the details of the high-q 2 behavior of the form factors which depend on the extrapolation procedure do not affect considerably the decay rate. The latter is mostly determined by the region q 2 Q 20 GeV 2 where the form factors are calculated directly. Fig. 1 compares our results with recent light-cone sum rule calculations available at q 2 Q 16 GeV 2 w15x and lattice-constrained parametrizations of Ref. w16x. One can see that the results of different approaches to the form factors do not differ significantly. However, it should be taken into account that in the case of the B ™ r transition this minor difference in the form factors provides rather sizeable spread of predictions for the decay rates. Summing up, we have analyzed the form factors of the exclusive b ™ u transition using the spectral representations based on constituent quark picture and obtain form factors in the whole kinematically accessible region. The meson wave functions and the constituent quark masses have been determined by describing the results of lattice simulations of the form factors A1Ž q 2 . and T2 Ž q 2 . at small recoils. This allowed us to calculate the form factors at q 2 Q 20 GeV 2 which cover all kinematically accessible q 2 in the B ™ r transition. In the B ™ p case the interval q 2 Q 20 GeV 2 does not cover the kinematically accessible region and an extrapolation to higher q 2 is necessary. To this end we take into account the dominance of the form factors at small recoils by the B ) pole. The calculated B ™ p l n decay rate is found to be only slightly sensitive to the particular details of the extrapolation procedure.
Acknowledgements We take the pleasure in thanking D. Becirevic and H. Schroder ¨ for helpful discussions which considerably influenced the final form of the paper. We are grateful to A. Le Yaouanc and O. Pene for the interest in this work and to DFG for financial support under grant 436 RUS 18r7r98. D.M. is grateful to the Physics Department of the Rostock University for hospitality.
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M. Beyer, D. MelikhoÕr Physics Letters B 436 (1998) 344–350
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