Table Of ContentA QCD sum rules calculation of the J/ψD D strong coupling
s∗ s
constant
B. Oso´rio Rodrigues,1,2 M. E. Bracco,3 M. Chiapparini,1 and A. Cerqueira Jr.3
1Instituto de F´ısica, Universidade do Estado do Rio de Janeiro,
Rua S˜ao Francisco Xavier 524, 20550-900, Rio de Janeiro, RJ, Brazil.
2Centro Brasileiro de Pesquisas F´ısicas - CBPF,
5
1 Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil.
0
2 3Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro,
n
Rod. Presidente Dutra Km 298, P´olo Industrial, 27537-000 , Resende, RJ, Brazil.
a
J
3 Abstract
1
In this work, we calculate the form factors and the coupling constant of the strange-charmed
]
h
vertex J/ψD∗D in the framework of the QCD sum rules by studying their three-point correlation
p s s
-
p functions. Allthepossibleoff-shellcasesareconsidered,D ,D∗ andJ/ψ,resultinginthreedifferent
s s
e
h
form factors. These form factors are extrapolated to the pole of their respective off-shell mesons,
[
1 giving the same coupling constant for the process. Our final result for the J/ψD∗D coupling
s s
v
8 constant is g = 4.30+0.42GeV−1.
8 J/ψDs∗Ds −0.37
0
3
0
.
1
0
5
1
:
v
i
X
r
a
1
I. INTRODUCTION
Inthepastfewdecades, theQuantumChromodynamicsSumRules(QCDSR)community
has been actively working in the calculation of strong coupling constants of a great variety
of vertices. Particularly, our group has calculated a large number of charmed vertices as the
D∗Dπ [1], D∗Dρ [2], J/ψD∗D [3] among many others that were revisited in the review of
ref. [4]. Some recent works in QCDSR target vertices with the heavy beauty mesons (B,
B∗, B , B∗, ...) as the B∗BK [5], B∗B η((cid:48)) [6], B(∗)B Υ [7, 8],B(∗)B J/ψ [8] and/or strange-
s s s s s c c c c
charmed mesons (D , D∗, D , D∗ ,...) as the J/ψD D [9, 10], J/ψD∗D∗ [10], J/ψD∗ D ,
s s s1 s0 s s s s s0 s1
φD∗ D [11], D∗D η((cid:48)) [6] and many others.
s0 s1 s s
Regarding the strange-charmed vertices, which are a subject of major interest nowadays,
due to the detection of various new exotic states, it is noticeable that some of them have
(∗) (∗)
masses that lie just above the D D mass threshold. By exotic, we mean that the usual
s s
quark-model description as qq¯ pairs does not hold for these new states. An example of
such exotic state is the Y(4140), first observed by the CDF collaboration [12], which has an
observed decay into the pair J/ψφ. Some interpretations for its internal structure are the
tetraquark state [13], D D¯∗ molecular state [14], D∗D¯∗ molecular state [15–17] and hybrid
s s s s
state [15]. Among them, we highlight the molecular D∗D¯∗ hypothesis, so that the decay
s s
Y(4140) J/ψφ can be understood with the help of an intermediate decay as Y(4140)
→ →
D∗D¯∗ J/ψφ. Inthisdecay, thevertexJ/ψD∗D ispresent, thusamorepreciseknowledge
s s → s s
of its coupling constant and form factors may help in the understanding of the fundamental
constitution of the Y(4140) meson. Also, it may be useful in the studies of other exotic
states as the X(4350) [18–21], Y(4274) [22] and the Y(4660) [23, 24], for example.
In this work, we investigate the J/ψD∗D vertex within the QCDSR formalism, following
s s
thedevelopmentofourpreviousworkfortheJ/ψD D vertex[9]. Thisdevelopmentconsists
s s
in calculating the form factors from the three point correlation function in two ways: one
with quarks and gluons degrees of freedom, the OPE side, and the other with hadronic
degrees of freedom, the phenomenological side. After that, a double Borel transform is
performed on each side and using the quark-hadron duality principle, we equate both OPE
andphenomenologicalsides. Ifawindowofstabilityisobtained,theformfactorandcoupling
constant can be extracted.
In the following sects., we present the QCDSR formalism and our results for this calcu-
2
lation.
II. FORMALISM
Thestartingpointtoperformthesumrule,isthecalculationofthethreepointcorrelation
function of the vertex. In this case, the J/ψD∗D is a vector-vector-pseudoscalar (VVP)
s s
vertex. Each meson will be represented by an interpolating current, which contains the
quantum numbers of the meson.
We can calculate three correlation functions [4]: one with the vector meson J/ψ off-shell,
another with the vector meson D∗ off-shell and a third one with the pseudo-scalar meson
s
D off-shell:
s
(cid:90)
Γ(M)(p,p(cid:48)) = d4xd4yeip(cid:48)xe−iqyΓ(M)(x,y) (1)
where q = p(cid:48) p is the transferred momentum, M is the off-shell meson (M = D ,J/ψ,D∗)
− s s
and Γ(M)(x,y) is presented in eqs. (2)-(4) below,
Γ(J/ψ)(x,y) = 0(cid:48) T jDs(x)jJ/ψ†(y)jDs∗†(0) 0(cid:48) , (2)
µν (cid:104) | { 5 µ ν }| (cid:105)
Γ(Ds∗)(x,y) = 0(cid:48) T jDs(x)jDs∗†(y)jJ/ψ†(0) 0(cid:48) , (3)
µν (cid:104) | { 5 ν µ }| (cid:105)
Γ(Ds)(x,y) = 0(cid:48) T jDs∗(x)jDs†(y)jJ/ψ†(0) 0(cid:48) , (4)
µν (cid:104) | { ν 5 µ }| (cid:105)
where T is the time ordered product and jJ/ψ, jDs∗ and jDs are the interpolating currents of
µ ν 5
the mesons J/ψ, D∗ and D respectively.
s s
According to the QCDSR, these correlation functions can be calculated in two different
ways: using hadron degrees of freedom, called the phenomenological side, or using quark
degrees of freedom, called the OPE side. In the following subsects., we calculate the corre-
lations functions for the phenomenological and OPE sides, and the sum rule.
A. The phenomenological side
The effective Lagrangian, which represents the process in the J/ψD∗D vertex, is [25, 26]:
s s
(cid:0) (cid:1)
= g εαβγδ∂ ψ ∂ D∗−D+ +∂ D∗+D− , (5)
LJ/ψDs∗Ds − J/ψDs∗Ds α β γ sδ s γ sδ s
where ε0123 = +1 is the Levi-Civita totally antisymmetric tensor.
3
From this Lagrangian, we can obtain the vertices of the hadronic process necessary to the
calculation of the phenomenological side of the QCDSR. For the J/ψ, D∗ and D off-shell
s s
cases they are, respectively:
D∗(p)J/ψ(q) D (p(cid:48)) = ig(J/ψ) (q2)(cid:15) (q,λ)(cid:15) (p,λ)q p εαβγδ, (6)
(cid:104) s | s (cid:105) J/ψDs∗Ds β δ α γ
J/ψ(p)D∗(q) D (p(cid:48)) = ig(Ds∗) (q2)(cid:15) (p,λ)(cid:15) (q,λ)p q εαβγδ, (7)
(cid:104) s | s (cid:105) J/ψDs∗Ds β δ α γ
J/ψ(p)D (q) D∗(p(cid:48)) = ig(Ds) (q2)(cid:15) (p,λ)(cid:15)∗(p(cid:48),λ)p p(cid:48) εαβγδ, (8)
(cid:104) s | s (cid:105) − J/ψDs∗Ds β δ α γ
where g(M) (q2) is the form factor of the J/ψD∗D vertex with meson M off-shell (M =
J/ψDs∗Ds s s
J/ψ,D ,D∗).
s s
Inthecalculationofthephenomenologicalside,wealsomakeuseofthefollowinghadronic
matrix elements:
m2
0 jP P(q) = P(q) jP 0 = f P , (9)
(cid:104) | 5 | (cid:105) (cid:104) | 5 | (cid:105) Pm +m
q1 q2
V(q) jV 0 = f m (cid:15)∗(q), (10)
(cid:104) | x | (cid:105) V V x
0 jV V(q) = f m (cid:15) (q), (11)
(cid:104) | x | (cid:105) V V x
where P is the pseudo-scalar meson (D ), V is a vector meson (J/ψ or D∗), q is the four-
s s
momentum of the respective meson, m2 is the squared mass of the meson, f are the
P,V P,V
decayconstants,m andm aretheconstituentquarksofmesonP and(cid:15) isthepolarization
q1 q2 x
vector, with x = µ(ν) for J/ψ(D∗).
s
Finally, the correlation functions (2)-(4) of the phenomenological side for each off-shell
case read:
Cg(J/ψ) (q2)p(cid:48)λpσε
Γphen(J/ψ) = J/ψDs∗Ds µνλσ +h.r., (12)
µν (p2 +m2 )(q2 +m2 )(p(cid:48)2 +m2 )
Ds∗ J/ψ Ds
Cg(Ds∗) (q2)p(cid:48)λpσε
Γphen(Ds∗) = − J/ψDs∗Ds µνλσ +h.r., (13)
µν (p2 +m2 )(q2 +m2 )(p(cid:48)2 +m2 )
J/ψ Ds∗ Ds
Cg(Ds) (q2)p(cid:48)λpσε
Γphen(Ds) = J/ψDs∗Ds µνλσ +h.r., (14)
µν (p2 +m2 )(q2 +m2 )(p(cid:48)2 +m2 )
J/ψ Ds Ds∗
where h.r. stands for the contributions of higher resonances and continuum states of each
meson and C is defined as:
f f f m2 m m
C = Ds Ds∗ J/ψ Ds Ds∗ J/ψ . (15)
(m +m )
c s
From eqs. (12)-(14), it is clear that this vertex has only one tensor structure to work
within the formalism of the QCDSR.
4
B. The OPE side
The OPE side is obtained from eqs. (2)-(4), with the interpolating currents written in
terms of the quark fields: jµJ/ψ = c¯γµc, jνDs∗− = c¯γνs and j5Ds− = ic¯γ5s. By construction,
the OPE side is given by an expansion known as Wilson’s Operator Product Expansion,
dominated by the perturbative term and followed by non-perturbative contributions:
ΓOPE(M) = Γpert(M) +Γnon-pert(M), (16)
µν µν µν
pert(M) non-pert(M)
where Γ is the perturbative term and Γ are the non-perturbative contribu-
µν µν
tions to the correlation function. When calculating the form factor, this expansion exhibits
a rapid convergence and can be truncated after a few terms [4]. Considering the similarities
between the J/ψD∗D and both J/ψD D and J/ψD∗D vertices, we expect, from the for-
s s s s
mer, a similar behavior regarding the OPE series as seen in the two latter [3, 9]. Therefore, it
should be adequate to consider non-perturbative contributions up to the mixed quark-gluon
condensate:
Γnon-pert = Γ(cid:104)q¯q(cid:105) +Γmq(cid:104)q¯q(cid:105) +Γ(cid:104)g2G2(cid:105) +Γ(cid:104)q¯gσGq(cid:105) +Γmq(cid:104)q¯gσGq(cid:105). (17)
µν µν µν µν µν µν
Calculating eq. (16) using the non-perturbative contributions of eq. (17) corresponds to
calculating the diagrams of fig. 1. As a consequence of the use of the double Borel transform,
only the J/ψ off-shell case has contributions from all the non-perturbative terms of eq. (17).
All the non-perturbative terms, except the gluon condensates (fig. 1d-1i), are suppressed
for both D and D∗ off-shell cases. This has been taken into account in fig. 1, wherefore
s s
these contributions were omitted from the beginning. The D off-shell case is also omitted
s
in fig. 1, as it is just an interchange between the mesons D∗ and D of the D∗ off-shell case.
s s s
Using dispersion relations, the perturbative term (fig. 1a) for a given meson M off-shell
can be written in the following form:
1 (cid:90) ∞(cid:90) ∞ ρpert(M)(s,u,t)
Γpert(M)(p,p(cid:48)) = µν dsdu, (18)
µν −4π2 (s p2)(u p(cid:48)2)
0 0
− −
pert(M)
where the spectral density ρ (s,u,t) can be obtained from the Cutkosky’s rules. The
µν
quantities s = p2, u = p(cid:48)2 and t = q2 are the Mandelstam variables.
In general, eq. (18) is the main contributing term of the OPE series in a QCDSR calcu-
lation. For a VVP vertex, the spectral density can be parametrized as:
3
ρpert(M)(s,u,t) = F(M)(s,u,t)p(cid:48)λpσε , (19)
µν √λ µνλσ
5
where λ = (u+s t)2 4us and F(M) is an invariant amplitude. For the J/ψ and D off
s
− −
shell, the invariant amplitude can be written as:
F(J/ψ) = (m m )(A+B) m , (20)
s c s
− −
F(Ds) = F(Ds∗) = (m m )B m , (21)
c s c
− − −
where
(cid:34) (cid:35)
k¯ p(cid:48) (cid:126)k cosθ (cid:126)k cosθ
A = 0 0| | , B = | | ,
√s − p(cid:126)(cid:48) √s p(cid:126)(cid:48)
| | | |
2p(cid:48)k¯ u+( 1)((cid:15))(m2 m2)
cosθ = 0 0 − − s − c ,
2 p(cid:126)(cid:48) (cid:126)k
| || |
(cid:113) s+(cid:15)(m2 m2)
(cid:126)k = k¯2 (cid:15)m2 (1 (cid:15))m2, k¯ = s − c ,
| | 0 − s − − c 0 2√s
s+u t √λ
p(cid:48) = − , p(cid:126)(cid:48) = ,
0 2√s | | 2√s
(∗)
and (cid:15) = 0(1) for D (J/ψ) off-shell.
s
The first non-perturbative contribution to the correlation function (16) is the quark con-
densate showed in fig. 1b:
s¯s p(cid:48)λpσε
Γ(cid:104)s¯s(cid:105)(J/ψ) = −(cid:104) (cid:105) µνλσ . (22)
µν (p2 m2)(p(cid:48)2 m2)
− c − c
Figure 1c is the first order mass term for the strange quark condensate contribution to
eq. (16), numerically less important and usually negligible in the three point QCDSR:
m m s¯s (cid:0)p(cid:48)2 +p2 2m2(cid:1)
Γms(cid:104)s¯s(cid:105)(J/ψ) = c s(cid:104) (cid:105) − c p(cid:48)λpσε . (23)
µν 2(p2 m2)2(cid:0)p(cid:48)2 m2(cid:1)2 µνλσ
− c − c
Full expressions for the gluon condensates ( g2G2 ) of fig. 1d-1i and mixed quark-gluon
(cid:104) (cid:105)
condensates ( s¯gσ Gs ) of fig. 1j-1o for the J/ψ off-shell case can be found in Appendix A.
(cid:104) · (cid:105)
C. The QCD Sum Rule
TwomorestepsarenecessaryinordertocalculatetheQCDSRtoobtaintheformfactors.
First, we make the change of variables p2 P2, p(cid:48)2 P(cid:48)2 and q2 Q2 followed by
→ − → − → −
a double Borel transform [27, 28] to both sides of the QCDSR in eqs. (12)-(14) and (16),
which involves the transformation: P2 M2 and P(cid:48)2 M(cid:48)2, where M and M(cid:48) are the
→ →
6
Borel masses. After that, we equate the phenomenological and OPE sides, invoking the
quark-hadron duality from which the sum rule is obtained.
Thesecondstepistoeliminatetheh.r.termsfromthephenomenologicalsideineqs.(12)-
(14). This is achieved by the introduction of the continuum threshold parameters s and u
0 0
in the OPE side and by taking advantage of the quark-hadron duality. These parameters
fulfill the following relations: m2 < s < m(cid:48)2 and m2 < u < m(cid:48)2, where m and m are the
i 0 i o 0 o i o
masses of the incoming and outcoming mesons respectively and m(cid:48) is the mass of the first
excited state of these mesons.
After performing these steps, we obtain the QCDSR expressions for the form factors:
3 (cid:82)s0 (cid:82)u0 √1 F(J/ψ)e−Ms2e−Mu(cid:48)2dsdu+ [Γnon-pert]
g(J/ψ) (Q2) = −4π2 sinf uinf λ BB , (24)
J/ψDs∗Ds (Q2+Cm2 )e−m2Ds∗/M2e−m2Ds/M(cid:48)2
J/ψ
(cid:104) (cid:105)
3 (cid:82)s0 (cid:82)u0 √1 F(Ds∗)e−Ms2e−Mu(cid:48)2dsdu+ Γ(cid:104)g2G2(cid:105)
g(Ds∗) (Q2) = 4π2 sinf uinf λ BB , (25)
J/ψDs∗Ds (Q2+Cm2 )e−m2J/ψ/M2e−m2Ds/M(cid:48)2
Ds∗
(cid:104) (cid:105)
3 (cid:82)s0 (cid:82)u0 √1 F(Ds)e−Ms2e−Mu(cid:48)2dsdu+ Γ(cid:104)g2G2(cid:105)
g(Ds) (Q2) = −4π2 sinf uinf λ BB , (26)
J/ψDs∗Ds (Q2+Cm2 )e−m2J/ψ/M2e−m2Ds∗/M(cid:48)2
Ds
(cid:104) (cid:105)
where [ ] stands for the double Borel transform and [Γnon-pert] = Γ(cid:104)g2G2(cid:105) for
BB BB BB
(∗)
the D off-shell cases, as already mentioned. The constant C is defined in eq. (15).
s
As in previous works, the definition for the coupling constant g is given by [29]:
J/ψDs∗Ds
lim g(M) (Q2) . (27)
Q2→−m2 J/ψDs∗Ds
M
In order to calculate this, it is necessary to extrapolate the results for the form factor to the
regionofQ2 < 0. Fromeqs.(24)-(26)itisclearthatwecanobtainthecouplingconstantfrom
three different form factors, one for each meson off-shell. However, the coupling constant
must be the same regardless the form factor used for the extrapolation. This condition is
used to minimize the uncertainties in the calculation of the coupling constant, which will
become quite clear in the following section.
7
III. RESULTS AND DISCUSSION
Equations (24)-(26) show the three different form factors of the J/ψD∗D vertex. The
s s
numerical calculation of these form factors give results that must be fitted to an analytical
function of Q2. In order to minimize the uncertainties regarding the fitting procedure, it is
required that these three form factors lead to the same coupling constant [29]. This condi-
tion also helps to reduce the errors of finding the Borel masses and continuum thresholds
necessary to the calculation.
Regarding the Borel masses, which satisfy the relation M(cid:48)2 = m2oM2, they can assume
m2
i
any value within the Borel window of stability, in which the phenomenologial and OPE
side should be compatible. The Borel window can be reduced by requiring that the pole
contribution must be bigger than the continuum contribution by at least 50% and that the
perturbative term contributes with more than 50% to the total correlation function. We
have worked with the average values of the form factors within the Borel window. Hence, it
is not necessary to use any specific Borel mass value, minimizing the uncertainties associated
to the choice of this quantity [9]. The procedure is briefly sketched as follows: the average
value of the form factor is calculated in the Borel window for each value of Q2 used. The
standard deviation is then used to automatize the analysis of the stability of the form factor
regarding the Borel masses and continuum threshold parameters. This stability criterion
then indicates the optimal values of the continuum threshold parameters and the Borel
windows we have to work with. Thus, not only is a good stability ensured in the Borel
window but also in the whole Q2 interval, without the need to chose a specific value for the
Borel mass.
The continuum threshold parameters, s and u , are defined as s = (m + ∆ )2 and
0 0 0 i i
u = (m +∆ )2, where the quantities ∆ and ∆ have been determined imposing the most
0 o o i o
stable Borel window, as mentioned before. In order to include the pole and to exclude the
contributions from higher resonances and continuum states, the values for ∆ , ∆ and
J/ψ D∗
s
∆ cannot be far from the experimental value (when available) of the distance between
Ds
the pole and the first excited state [30, 31]. Our analysis has found that the best values
are ∆ = 0.6 GeV and ∆ = ∆ = 0.5 GeV, which leads to remarkably stable Borel
Ds Ds∗ J/ψ
windows for the three off-shell cases, as it can be seen in fig. 2 for J/ψ and D off-shell.
s
Figures for D∗ off-shell are omitted as they are very similar to the D off-shell case.
s s
8
Alsoinfig.2, wecancheckthattheperturbativetermistheleadingoneintheOPEseries,
followedbythequarkcondensate(fortheJ/ψ off-shellcase)andthegluoncondensates. The
summed contribution of the non-perturbative terms, m s¯s , s¯gσGs and m s¯gσGs , is
s s
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
small ( 2%), which means we could easily neglect these terms without changing our results
≈
significantly. Likewise, the gluon condensates have a very small contribution to the total
correlation function in the D and D∗ off-shell cases and they may also be neglected.
s s
Besides the values of the Borel masses and the continuum thresholds, we also need to
know the values of decay constants, condensates and hadrons and quark masses. The hadron
masses used are m = 1.968 GeV, m = 2.112 GeV and m = 3.097 GeV [30]. The
Ds Ds∗ J/ψ
other parameters are presented in table II.
In table I, we present the Q2 and Borel windows found for each form factor g(M) (Q2)
J/ψDs∗Ds
(where M is the meson off-shell), together with its parametrization and the corresponding
coupling constant g . Its associated error σ is calculated using the method that will
J/ψDs∗Ds
be explained later in this work.
TABLE I: Parametrization of the form factors and numerical results for the coupling constant of
this work.
Quantity J/ψ off-shell D∗ off-shell D off-shell
s s
Q2 (GeV2) [0.5, 2.0] [1.0, 3.0] [1.0, 4.0]
M2 (GeV2) [7.0, 7.9] [5.3, 7.3] [4.9, 5.9]
g(M) (Q2) A Ae−Q2/B Ae−Q2/B
J/ψDs∗Ds B+Q2
A 193.4 GeV 2.003 GeV−1 2.330 GeV−1
B (GeV2) 54.46 6.027 6.122
g(M) σ (GeV−1) 4.31+0.41 4.20+0.22 4.39+0.27
J/ψDs∗Ds ± −0.38 −0.18 −0.23
TheBorelwindowspresentedintableI(M2 row)satisfythealreadymentionedconditions
regarding the pole and continuum contributions, as shown in fig. 3.
The form factor obtained for the J/ψ off-shell case was well adjusted by a monopo-
lar curve, while for the D∗ and D off-shell cases, the form factors were well adjusted by
s s
exponential curves (fig. 4).
The coupling constants of each off-shell case are shown in table I. They present different
values among them, however, when the uncertainties are taken into account, they are all
9
compatible, as suggested by fig. 4, where the error bars stand for the uncertainties (σ) of
the coupling constants also presented in table I.
The uncertainties were calculated following the same procedures used in our previous
works [2, 5, 9]. They are estimated by studying the behavior of the coupling constants when
each parameter is varied individually between its upper and lower limits. Experimental
parameters have their own errors (shown in table II), while for the QCDSR parameters, we
varied the thresholds ∆ and ∆ in 0.1 GeV and the Q2 window in 20%. For the error
s u
± ±
due to the Borel mass (M2), we used the standard deviation of the form factor within the
Borel window. After that, we calculated the average of all these contributions and their
respective standard deviations to obtain the coupling constant and its uncertainty for a
given off-shell case. In table II, we can see how the variance of each parameter affects the
final value of the coupling constant.
TABLE II: Percentage deviation of the coupling constants related to each parameter.
Parameter ∆g(J/ψ) (%) ∆g(Ds∗) (%) ∆g(Ds) (%)
J/ψDs∗Ds J/ψDs∗Ds J/ψDs∗Ds
f = 257.5 6.1(MeV)[30] 1.94 1.93 1.94
Ds ±
fD∗ = 301 13(MeV)a 3.53 3.53 3.53
s ±
f = 416 6(MeV)[30] 1.17 1.18 1.17
J/ψ
±
m = 1.27+0.07 (GeV)[30] 19.5 12.9 16.6
c −0.09
m = 101+29 (MeV)[30] 2.56 0.96 2.25
s −21
M2 (GeV2)b 1.42 4.46 0.12
∆ 0.1(GeV),∆ 0.1(GeV) 8.93 2.49 2.08
s u
± ±
Q2 20% (GeV2)b 4.94 1.10 0.98
±
s¯s = (290 15)3 (MeV3)[32] 1.83 - -
(cid:104) (cid:105) − ±
g2G2 = 0.88 0.16(GeV4)[33] 1.45 0.18 0.73
(cid:104) (cid:105) ±
s¯gσ Gs = (0.8 0.2) s¯s (GeV5)[34] 0.49 - -
(cid:104) · (cid:105) ± (cid:104) (cid:105)
Fitting parameters (A and B) 14.9 0.12 0.06
aValue obtained using fDs∗/fDs = 1.17 from [35, 36].
bThe intervals for these quantities are those of table I.
According to table II, the variation of most of the parameters (the decay constants,
condensates and the Borel mass for example) has a small impact on the value of the coupling
10
