Pionic coupling constants of heavy mesons in the quark model

7 Pages·0.15 MB·English

by  D. Melikhov

Pionic coupling constants of heavy mesons in the quark model
Dmitri Melikhova,∗ and Michael Beyerb
aLPTHE, Universit´e de Paris XI, Bˆatiment 211, 91405 Orsay Cedex, France
bPhysics Department, Rostock University, D-18051 Rostock, Germany
WeanalysepioniccouplingsofheavymesonscombiningPCACwiththedispersionquarkmodel
tocalculatetherelevanttransitionformfactors. Groundstatesandradialexcitationsareconsidered.
For the ground state coupling constants the values gˆ = 0.5±0.02 in the heavy quark limit, and
gB∗Bπ = 40±3, gD∗Dπ = 16±2 are obtained. A sizeable suppression of the coupling constants
describing thepionic decays of theradial excitations is observed.
Pionic coupling constants of heavy mesons are basic quantities to understand the heavy meson lifetimes: they
govern strong decays of heavy mesons by the emission of a pion which in practice turn out to be most important
9 strong decay modes. The knowledge of the pionic coupling constants of the ground-stateas well as the excited heavy
9
mesons is timely in particular in view of the new experimental data on the excited heavy mesons [1].
9
The heavyquark(HQ)symmetryprovidesimportantrelationsbetweenthe couplingconstantsofpionictransitions
1
of various mesons containing a heavy quark. In the leading 1/m order (LO) the heavy quark spin decouples from
Q
n
other degrees of freedom [2]. Thus in the limit of an infinitely heavy spectator quark, the respective pionic decay
a
rates for mesons with different spins but same quantum numbers of the light degrees of freedom (total momentum of
J
the light degrees of freedom j, angular momentum L, etc) [3] are equal.
0
Note, however that the heavy quark expansion which provides many rigorous results for semileptonic transitions
1
betweenheavymesons,turnsouttobemuchlessefficientinthecaseofthestrongtransitions. Infact,thesetwotypes

1 of processes have quite different dynamics. Semileptonic transitions of heavy mesons are induced by the heavy quark
v weak transition and in this case HQET predicts not only the structure of the 1/m expansion of the long-distance
Q
1 (LD) contributions to the form factors, but also provides important absolute normalization of the LO form factors.
6
On the other hand, in the case of strong pionic decay the heavy quark remains spectator and the dynamics of the
2
processisdetermined bythe lightdegreesoffreedom. As a consequenceHQ symmetry doesnotprovideanyabsolute
1
normalization,thusmakingthepionicdecaysofheavymesonsamorecomplicatedproblemtoanalysecomparedwith
0
9 semileptonic decays.
9 The amplitudes of the strong pionic vector-to-vectorand pseudoscalar-to-vectortransitions have the structure
/
h
V(p )π(q)P(p ) =q ε (p )g
p h 2 | 1 i ν ν 2 VPπ
- V(p )π(q)V(p ) =ε (p )ε (p )p p ǫνµαβg , (1)
2 1 ν 1 µ 2 1α 2β VVπ
p h | i
e
where ε denotes the polarization vector of the vector meson. The coupling constants can be expanded in a 1/m
h Q
series as follows
:
v
Xi fπ g gˆ =gˆ+ gˆP(1) +...,
VPπ PVπ
2√M M ≡ m
r P V Q
a
f gˆ(1)
πg gˆ =gˆ+ V +..., (2)
VVπ VVπ
2 ≡ m
Q
andduetotheHQsymmetryintheleading1/m orderbothconstantsaregovernedbythesamequantitygˆ,whereas
Q
the higher order terms are different.
The pionic coupling constants for heavy mesons gDD∗π and gBB∗π have been analysed within various versions of
the constituent quark model [4], sum rules [5] and using the lattice simulations [6].
The results of the quark model strongly depend on the particular version of the QM used: the values of gˆ range
from 1 (nonrelativistic quark model) to 1/3 (the Salpeter equation with massless quarks) [4]. LCSR [5] obtained a
small value 0.32 0.02. Recent lattice simulations reported 0.42 with however large statistical errors 0.04 0.08.
± ± ±
Thus at the moment no reliable predictions are available even for the coupling constants of the groundstate mesons.
An analysis of the pionic decays of radially excited states is almost lacking. This calls for further investigation and a
better understanding of the problem.
∗On leave of absense from NuclearPhysics Institute, Moscow State University,Moscow, 119899, Russia
1
AlthoughthepresentQMresultsarestronglyscattered,onefeatureofthisapproachstillseemstobeveryattractive:
namely,variousprocessescanbeconnectedtoeachotherviathewavefunctionsofthemesonsinvolved,onceaproper
formulation of the QM is used. We believe that such a proper formulation should (i) be based on a relativistic
considerationand (ii) reproduce rigorousQCD results in the known limits, e.g. in the limit when the meson decay is
inducedbytheHQtransition. Thedispersionformulationofref[7]wasshowntosatisfytheserequirements. Recently,
we have applied this dispersion approach to analyse the B π transition form factors and determined the relevant
→
quark model parameters and the wave function of the B meson [8]. Once the whole framework and the numerical
parameters are fixed we expect to perform a reliable analysis of the pionic coupling constants of the heavy mesons.
An additional attractive feature of this approach is that it is straight forward to incorporate the excited states, in
particular radial excitations.
The dispersion approachis based on the spectral representationsof the form factors throughthe wave functions of
the initial and final mesons. The double spectral densities are obtained from the relevant Feynman graphs, whereas
the subtractionterms remainaprioriambiguous and should be determined fromsome other arguments. In ref [7] the
subtraction terms have been determined such that the correct structure of the 1/m expansion in the leading and
Q
subleadingordersisreproduced. Suchaprocedureofdeterminingthesubtractiontermsprovidesareliabledescription
oftheformfactorsifthedecayingquarkisheavyforbothheavyandlightquarksproducedintheweakdecay. However
such a strategymight be not efficient for the case when the interacting quark is light but the spectator is heavy, that
is exactly the case of the pionic decay of a heavy meson, and requires proper modification.
In this letter we analyse the pionic transitions of the ground state and radially excited heavy mesons, studying
in parallel the V Vπ and V Pπ cases. We make use the PCAC which allows one to reduce the calculation
→ →
of the complicated amplitude involving three hadrons to a considerably more simple amplitudes of the P V and
→
V V transitions induced by the light-quarkaxial-vectorcurrent. The correspondingamplitudes have the following
→
structure
V(p )A V(p ) =iε (p )ε (p )ǫ P h(q2)/2+...,
2 µ 1 α 1 β 2 µναβ ν
h | | i
V(p2)Aµ P(p1) =iεν(p2)[gµνf(q2)+a+(q2)Pµqν +a−(q2)qµqν], (3)
h | | i
where q = p p , P = p +p , and the dots denote terms transverse with respect to q the particular form of
1 2 1 2 µ
−
which is not important for us. The form factors a− and h contain a pole at q2 = m2π due to the contribution of the
intermediate pion state in the q2-channel, and the residues in these poles are expressed through the pionic coupling
constants g and g as follows
VPπ VVπ
f q2g
h(q2)= π VVπ +h¯(q2)
m2 q2
π−
f g
a−(q2)= π VVπ +a¯−(q2), (4)
m2 q2
π−
where h¯ and a¯− are regular at q2 =m2π.
In the V V case, we find that the function Φ (q2) h(q2)(m2 q2) is smooth between the point q2 = 0 and
→ V ≡ π −
q2 =m2 and due to the smallness of the pion mass we assume that Φ (0) Φ (m2). This yields the relation
π V ≃ V π
g =h(0)/f . (5)
VVπ π
For the P V case an additional step is necessary: taking first the divergence of the axial-vector current we find
→
Φ (q2)
P
<V(p )q A P(p )>=q ǫ (p )
2 | µ µ| 1 ν ν 2 m2 q2
π−
withΦP(q2)= f(q2)+(p21−p22)a+(q2)+q2a¯−(q2) (m2π−q2)+q2fπgVPπ. AstandardPCACassumptionΦP(m2π)≃
Φ (0) yields
P
(cid:2) (cid:3)
1
g = [f(0)+(p2 p2)a (0)]. (6)
PVπ f 1− 2 +
π
Notice that the pion should not be necessarily soft and the relations (5) and (6) can be readily applied to the case
when the masses M and M are substantially different as e.g. for the transition between the ground-state and
1 2
radially-excited heavy mesons.
The relations(5)and(6)representthe couplingconstantsof interestthroughthe transitionformfactors,forwhich
we apply the dispersion approach.
2
I. SPECTRAL REPRESENTATIONS OF THE COUPLING CONSTANTS
We start with considering the spectral representation of the form factors along the lines of ref [7]. We apply this
method to the case of the heavy spectator quark and propose proper modifications to calculate the pionic coupling
constants.
The dispersion approach gives the transition form factors of the meson M to the meson M as double relativistic
1 2
spectralrepresentationsintermsofthesoftwavefunctionsψ (s )=G (s )/(s M2)andψ (s )=G (s )/(s M2),
1 1 1 1 1− 1 2 2 2 2 2− 2
of the initial and final mesons, respectively. To be specific, in the case of the weak decay PQ Vq′ induced by the
′ ′ →
weak quark transition Q q where q might be both heavy and light the form factor can be represented as follows
→
[7]
ds G (s ) ds G (s )
f(q2)= 1 1 1 2 2 2 f˜ (s ,s ,q2)+ (s M2)+(s M2) g˜ (s ,s ,q2)
s M2 s M2 D 1 2 1− 1 2− 2 D 1 2
Z 1− 1 2− 2 h+(s M2)ξ (s(cid:0),s )+(s M2)ξ (s ,s(cid:1)) , (7)
1− 1 1 1 2 2− 2 2 1 2
where f˜ = disc disc f (s ,s ,q2) and g˜ = disc disc g (s ,s ,q2) are directly cal(cid:3)culated from the triangle
D s1 s2 D 1 2 D s1 s2 D 1 2
Feynman graph with pointlike vertices as double spectral densities of the relevant form factors. The functions ξ
1
and ξ are not known precisely but are known to behave as 1/m in the limit m , where m is the mass of
2 Q Q Q
→ ∞
the initial heavy quark. It is important to point out at the hierarchy of different terms in eq (7) in case the initial
active quark is heavy: namely, the term proportional to disc disc f contributes in the LO (and all other orders
s1 s2 D
as well), the subtraction term proportionalto disc disc g contributes starting from the subleading order, and the
s1 s2 D
subtraction terms proportional to ξ contribute only in the higher orders. So this expansion works well if the initial
active quark is heavy but the spectator quark is light, both for the cases of the light and heavy active quark in the
final state.
However in case the spectator is heavy and the active quark is light, the subtraction term proportional to g˜ as
D
wellasthefunctionsξ containpowersofthespectatormass,andalltermshavethesameorderofmagnitudeandthus
shouldbe consideredonanequalfooting. Hence, the expansion(7)becomes ineffective in practiceand the procedure
requires a modification.
To find such proper modification it is convenient to start with considering the V V transition induced by
1 2
→
the axial-vector current. In the quark model the process is represented by the triangle graph with relevant vertices
describing the quark structure of the vector meson states. The procedure described in detail in [7] yields for the
g =h(q2 =0)/f the following spectral representation
V1V2π π
∞
m
1
g = dsΨ (s)Ψ (s)
V1V2π 4π2f V1 V2
π(m1+Zm3)2
λ1/2 1 s+m2 m2 m2
m log(r)+(m m ) + 1− 3λ1/2 1 log(r) , (8)
1 3 1
× − s √s+m +m s − 2
(cid:20) 1 3 (cid:18) (cid:19)(cid:21)
with r =(s+m2 m2+λ1/2)/(s+m2 m2 λ1/2), λ=(s m2 m2)2 4m2m2, and m and m the masses of
1− 3 1− 3− − 1− 3 − 1 3 1 3
the active and the spectator constituent quarks, respectively (we follow the notations of [7]).
Thisspectralrepresentationcanberewritteninamoreconventionalformasanintegraloverthelight-conevariables
as follows
m dxdk2 2k2
1 ⊥ ⊥
g = Ψ (s)Ψ (s) m x+m (1 x)+ . (9)
V1V2π 2π2f x(1 x)2 V1 V2 1 3 − √s+m +m
π Z − (cid:20) 1 3(cid:21)
Here s= m21 + m23 + k⊥2 , and x and 1 x are the fraction of the light-cone momentum carried by the spectator
1−x x x(1−x) −
and the active quark, respectively.
The expression (8) is obtained as a double dispersion representationin the invariant masses of the initial and final
qq¯ pairs, the spectral density of which is calculated from the triangle Feynman graph. At q2 = 0 it is reduced to
a simple form (8). In principle subtraction terms can be added to this double spectral representation. However, in
the case of the V V transitions there are no reasons dictating a necessity of subtractions, and we assume that the
→
subtraction terms are absent.
The representation (8) describes the π 2γ decay if we set m = m and take the quark-photon vertex in the
0 1 3
→
form q¯γ q. In this case only the first term in (8) is present, and the photon wave function corresponding to the
µ
3
pointlike interaction reads Ψ (s) = 1/s. So, g becomes independent of the quark mass and we simply reproduce
γ γγπ
the value of the axial anomaly [10] from the imaginary part of the triangle graph (cf [11] and refs therein).
The axial-vector current satisfies the equation of motion ∂ A = 2mj , so the coupling constant g can be
µ µ 5 V1V2π
equivalently calculated directly from the amplitude V (p )2mj (0)V (p ) =ε (p )ε (p )ǫ q P h(q2)/2.
2 2 5 1 1 α 1 β 2 µναβ ν ν
h | | i
We proceed the same way to analyse the P V transition: instead of calculating the full representations for f
→
and a and then taking their linear combination (6), we focus on the amplitude V(p )2mj P(p ) . Similar to the
+ 2 5 1
h | | i
V V case,thisamplitudeprovidesthecorrectdoublespectraldensityofthespectralrepresentationforg which
VPπ
→
takes the form
∞
m
1
g = dsΨ (s)Ψ (s)
VPπ 4π2f P V
π(m1+Zm3)2
2m
s (m m )2 log(r) 1+ 1 (s+m2 m2)log(r) 2λ1/2
× − 1− 3 − √s+m +m 1− 3 −
m(cid:20)(cid:0) dxdk2 (cid:1) (cid:18) 1 3(cid:19)(cid:16)k2 2m (cid:17)(cid:21)
= 1 ⊥ Ψ (s)Ψ (s) s (m m )2 ⊥ 1+ 1 . (10)
4π2f x(1 x)2 P V − 1− 3 − 1 x √s+m +m
π Z − (cid:20) − (cid:18) 1 3(cid:19)(cid:21)
The form of the relations (6) and (7) however prompts that in distinction to the V V transition, in the P V
→ →
transitionthesubtractiontermisnonzero. The lattercannotbe determineduniquelywithinthedispersionapproach.
ItisveryimportanthoweverthattheHQsymmetryensuresthesubtractiontermtocontributeonlyinthesubleading
1/m order. Namely, the HQ symmetry predicts the LO relation between the coupling constants g = m g .
Q VPπ Q VVπ
The double spectral densities of the representations for g (9) and g (10) satisfy this relation. Hence the
VVπ VPπ
subtraction terms in g and g should also have the same LO 1/m behavior. Since the subtraction term in
VVπ VPπ Q
g is absent, the subtraction term in g does not contribute in the LO. Although we cannot determine the
VVπ VPπ
subtraction term uniquely, several reasonable ways of fixing this subtraction term yield a numerical uncertainty in
g to be not more than 10%.
PVπ
The normalization condition for the radial wave functions Ψ of the ground state and the radial excitation of the
vector and the pseudoscalar mesons has the form
1 λ1/2 1 dxdk2
dsΨ (s)Ψ (s) [s (m m )2]= ⊥ Ψ (s)Ψ (s)[s (m m )2]=δ . (11)
8π2 i j s − 1− 3 8π2 x(1 x) i j − 1− 3 ij
Z Z −
It is easy to see that in the nonrelativistic (NR) limit ~k = λ1/2/2√s m ,m the coupling constants take the
1 3
| | ≪
values gNR =2M/f and gNR =2/f . Moreover,in the NR limit the coupling constants of the transition between
PVπ π VVπ π
the ground state and the radial excitation vanish, i.e. gNR =gNR =0.
PV′π VV′π
For the analysis of the HQ expansion it is convenient to introduce a new variable z such that s=(z+m +m )2
1 3
and use the fact that the soft wave functions φ are localized in the region z Λ . Performingthe HQ expansion
0 QCD
≃
of all quantities including Ψ in the inverse powers of m =m and keeping m =O(Λ ) (see [7]) we find
3 Q 1 QCD
→∞
1 z+m + z(z+2m )
gˆ = dzφi(z)φj(z)m m log 1 1 +2 z(z+2m ) . (12)
ij 2Z 0 0 1" 1 z+m1−pz(z+2m1)! p 1 #
p
Inthis expressionφi,j arethe LOsoftwavefunctionofthegroundstateandtheradiallyexcitedmesonswhichsatisfy
0
the normalization condition
dzφi(z)φj(z)√z(z+2m )3/2 =δ (13)
0 0 1 ij
Z
Notice that the Isgur-Wisefunction is directly expressedthroughφ (see [7]). These formulasprovidea possibility to
0
evaluate the coupling constants of interest if the numerical parameters of the model are known.
II. NUMERICAL ANALYSIS
WenowprovidethenumericalestimatesoftheLOquantitygˆfortheground-stateandradiallyexcitedmesons. The
spectral representation for this quantity is completely fixed and we need to choose the proper numerical parameters
ofthe model. As foundinmany applicationsofthe dispersionapproach(see[7,8]andrefstherein), anapproximation
4
ofthesoftwavefunctionbyanexponentprovidesreasonableestimates. Assumingthewavefunctiontohavetheform
Ψ(s) exp( ~k2/2β2), the LO radial wave function of the ground state reads [7]
≃ −
z+m z(z+2m )
1 1
φ(z) exp . (14)
≃rz+2m1 (cid:18)− 2β∞2 (cid:19)
Similarly, we assume for the wave function of the radial excitation the form
z(z+2m ) z(z+2m )
1 1
φ (z) 1 C exp , (15)
r ≃ − r 2β∞2 − 2β∞2
(cid:18) (cid:19) (cid:18) (cid:19)
where the normalizationfactorandthe coefficientC are fixedby the normalizationconditions andthe orthogonality
r
between the wave functions of the ground state and the radial excitation.
The light-quark mass was determined from the numerical matching of the transition form factors in the B π,ρ
→
decay to the available lattice data [9] and was found to be tightly restricted to the range m = 0.23 0.01 GeV.
1
±
Therefore the only unknown parameter is β∞. The analysis of the leptonic decay constants of heavy mesons shows
that simple approximate relations
βP(mQ)=β∞(1 CP/mQ), βV(mQ)=β∞(1 CV/mQ) (16)
− −
with CP 0.1, CV 0.2 and β∞ = 0.5 yield the values of β which provide reasonable leptonic constants of the
≃ ≃
charm and beauty mesons (see Table 1) calculated within the quark model through formulas given in [7,12]. The
slope parameter of the IW function for such β∞ is ρ2 =1.2 0.03 [7]. The eq (12) then yields
±
gˆ=0.5, gˆ =0.11. (17)
r
Thus for the LO quantity the orthogonality of the radial wave functions provides a strong suppression of gˆ 1.
r
The results of calculating the pionic coupling constants of the B and D mesons are given in Table 2. Whereas the
g isdeterminedquitereliablywithinthedispersionapprach,theg suffersfromtheintrinsicuncertaintyofthe
VVπ PVπ
approachbased on the impossibility to fix the subtraction terms. Assuming severalreasonablechoices of subtraction
terms in the form factors f, a and g which provide the correct behaviour in the limit of the heavy active quark,
+
but differ for the case of the heavy spectator we have found that in all cases, the difference in the coupling constants
of the ground state is not more than 10%. This minor difference is easily understood taking into account that the
subtraction terms contribute only in the subleading order.
Notice thatthere is a possibility of analternativedeterminationof the gB∗Bπ fromthe formfactorsof the semilep-
tonic B π transition near zero recoil. Namely,
→
f (M2) f (M∗2)= gB∗BπfB∗(1+r′(1)), (18)
0 B ≃ 0 B 2MB∗
where r(qˆ2 =q2/M2 ) is defined as follows
B∗
r(qˆ2)= f−(q2) MB2∗ , (19)
−f (q2)M2 M2
+ B− π
Thefunctionr(qˆ2)satisfiestheconditionr(1)=1andisknownnumericallyfromtheresultsofthelatticesimulations
in the region qˆ2 0.7 [9], in particular r(0.7) 0.9. The function r(qˆ2) is regular near qˆ2 =1 and thus there are no
≤ ≃
reasons for fast variations of this function near qˆ2 = 1. Then, taking into account the lattice data we can estimate
r′(1) 0.8. Using the current algebra relation f (M2) = f /f , neglecting the 1/m corrections which are small,
≤ 0 B B π b
and taking the ratio of the leptonic constants fB∗/fB 1.2 we find
≃
1
gˆ= 0.45 (20)
fB∗/fB(1+r′(1)) ≥
1Notice that in contrast to the PCAC based approaches, the NR quark-model calculations based on taking into account the
quarkstructureoftheemittedpionthroughthepionwavefunctionyieldbiggervaluesofgˆanddonotyieldasuppressionofgˆr
since theproduct of the orthogonal wave function is smeared bytheintegration with thepion wave function and as theresult
theorthogonality is not efficient [13].
5
in agreement with (17) and with the recent result of the lattice simulation [6]. We would like to point out that the
∗
value gˆ 0.27 proposed in [14] as the preferrable solution found from the description of the D decays falls far out
≃
of this estimate.
Thecalculationofthecouplingconstantsofthepionictransitionbetweenthegroundstateandtheradialexcitation
is more involved. In this case the LO term is itself strongly suppressed because of the orthogonality of the wave
functions. So the dependence on the parameters of the wave function of the radial excitation in the case of the
gVV′π and, in addition to this, also the dependence on the particular choice of the subtraction procedure in gPV′π is
more sizeable. Table 2 presents numerical results. The slope parameter of the Gaussian wave function of the radial
excitation at finite masses is not known, so for estimating gVV′π we assume that this parameter lies within a 10%
interval around the corresponding ground-state slope. In the case of gPV′π the relation (6) prompts that the 1/mQ
corrections numerically might be sizeable if the M2 M2 is big as it is for the radial excitations of heavy mesons.
V′ − P
In this case we cannot obtain a reliable estimate and Table 2 provides the lower bounds found from the spectral
representations without subtractions.
III. CONCLUSION
We have analysed the pionic coupling constants of heavy mesons within the framework based on the combination
of PCAC with the dispersion approachand obtained the following results:
1. Spectral representations of the coupling constants in terms of the wave functions of the initial and final heavy
mesons have been obtained. The double spectral densities of these spectral representations for g and g are
PVπ VVπ
equal in the HQ limit in agreement with the HQ symmetry.
In the case of the V V transition the calculation of the g is equivalent to the calculation of the amplitudes
VVπ
→
V 2mj V . There is no reasondictating a necessity of any subtractionterm in the spectralrepresentationfor g
5 VVπ
h | | i
and thus the latter is determined unambiguously within our approach.
In the P V case the double spectral density of g can be also found from the amplitude V 2mj P . The
VPπ 5
→ h | | i
spectral representation for g , however, contains a subtraction term which cannot be fixed within our approach.
VPπ
Importantisthatdue tothe HQsymmetrythis subtractiontermdoesnotcontributeinthe leading1/m order2 and
Q
thus numerically is not essential at least for the gB∗Bπ.
2. A spectral representationfor the LO 1/m coupling constant gˆ which describes the pionic transition in the HQ
Q
limit has been obtained. The gˆ is represented through the LO wave functions of the heavy mesons which determine
alsotheIsgur-Wisefunction. The gˆisdeterminedquite reliablywithinthedispersionapproachsinceitisnotaffected
by the uncertainties in the subtraction procedure.
Forthetransitionbetweentheground-stateheavymesonswehavefoundgˆ=0.5 0.02. Wearguethatthebehavior
±
of the B π transition form factors near zero recoil provide independent arguments in favour of such considerably
→
large value of gˆ.
For the pionic transition between the ground state and the radial excitation in the HQ limit a strong suppression
gˆ =0.11 0.02 is observed as a consequence of the orthogonality of the corresponding radial wave functions.
r
±
3. Usingthe QMparametersobtainedfromtheanalysisoftheB π decaywehavecalculatedthe pioniccoupling
→
constants of the ground state charm and bottom mesons. Our final numerical estimates are listed in Table 2.
For the V V transition higher-order 1/m effects in g are found to be small and not to exceed 5-6%.
Q VVπ
→
In the P V case, the coupling constant g depends on the choice of the subtraction procedure which cannot
PVπ
→
be fixed unambiguously. However, numerically the uncertainty estimated by using several reasonable subtraction
procedures is not more than 10%.
∗′ ∗′
4. We estimated the pionic couplings of the radially excited states B and D . The errors turn out to be much
biggercomparedwiththecorrespondinggroundstates: theLOcontributionisstronglysuppressedandthusthe1/m
Q
effects are found to be numerically more important. As a result the coupling constants are sensitive to the specific
form of the radial-excitationwave function and gPV′π, in addition to this, strongly depends on the the details of the
subtraction procedure. Therefore only lower bounds on gPV′π are given.
2Strictly speaking, we cannot completely exclude a possibility of subtraction terms both in gˆVVπ and gˆPVπ which give the
same contributions to both quantities in the LO. However, unless no reasons for such subtraction terms are found we do not
include them into consideration. In a recent analysis of the B → π form factors [8] we have found that the behavior of f0 at
zerorecoiliscompatiblewithgˆB∗Bπ ≃0.5−0.7. ThevaluesofgˆB∗Bπ asbigas0.6−0.7canbeobtainedonlybyassumingthe
subtraction terms both in gB∗Bπ and gB∗B∗π which give a nonvanishingcontribution already in theleading 1/mQ order.
6
IV. ACKNOWLEDGEMENTS
We are grateful to D. Becirevic, A. Le Yaouanc and O. P´ene for stimulating discussions of this subject. The work
was supported in part by DFG under grants 436 RUS 18/7/98 and 436 RUS 17/26/98,and by the NATO Research
Fellowships Program.
[1] DELPHI Collaboration, P. Abreu et al.,Phys. Lett. 426 (1998) 231.
[2] N.Isgur and M. B. Wise, Phys. Lett. B 232 (1989) 113; ibid, 237 (1990) 527; M. Luke,Phys. Lett. B 252 (1990) 447.
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TABLE I. Quark masses and the slope parameters of the soft meson wave functions and the calculated leptonic decay
constants (in GeV)
mb mc mu β∞ βB βB∗ βD βD∗ fB fB∗ fD fD∗
4.85 1.4 0.23 0.5 0.49 0.49 0.46 0.43 0.16 0.175 0.2 0.22
TABLEII. Pioniccouplingconstantsofheavymesons. Theerrorbarscorrespond tothevariationsoftheslopeparameters
β around theaverage values of Table 1 yielding a 10% variations of theleptonic decay constants.
gˆVVπ gˆVPπ gˆV′Vπ gˆV′Pπ
D,D∗ 0.53±0.03 0.53±0.05 0.15±0.03 >0.14
B,B∗ 0.5±0.02 0.5±0.04 0.12±0.03 >0.12
HQ-limit 0.5±0.02 − 0.11±0.02 −
7