Heavytolight scalar form factors from Muskhelishvili–Omnès dispersion relations
Eur. Phys. J. C
Heavytolight scalar form factors from MuskhelishviliOmnès dispersion relations
D.L. Yao 2
P. FernandezSoler 2
M. Albaladejo 1
F.K. Guo 0 3
J. Nieves 2
0 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences , Zhong Guan Cun East Street 55, Beijing 100190 , China
1 Departamento de Física, Universidad de Murcia , 30071 Murcia , Spain
2 Instituto de Física Corpuscular (centro mixto CSICUV), Institutos de Investigación de Paterna , Apartado 22085, 46071 Valencia , Spain
3 School of Physical Sciences, University of Chinese Academy of Sciences , Beijing 100049 , China
By solving the MuskhelishviliOmnès integral equations, the scalar form factors of the semileptonic heavy meson decays D → π ¯ν , D → K¯ ¯ν , B¯ → π ν¯ and B¯s → K ν¯ are simultaneously studied. As input, we employ unitarized heavy mesonGoldstone boson chiral coupledchannel amplitudes for the energy regions not far from thresholds, while, at high energies, adequate asymptotic conditions are imposed. The scalar form factors are expressed in terms of Omnès matrices multiplied by vector polynomials, which contain some undetermined dispersive subtraction constants. We make use of heavy quark and chiral symmetries to constrain these constants, which are fitted to lattice QCD results both in the charm and the bottom sectors, and in this latter sector to the lightcone sum rule predictions close to q2 = 0 as well. We find a good simultaneous description of the scalar form factors for the four semileptonic decay reactions. From this combined fit, and taking advantage that scalar and vector form factors are equal at q2 = 0, we obtain Vcd  = 0.244 ± 0.022, Vcs  = 0.945 ± 0.041 and Vub = (4.3 ± 0.7) × 10−3 for the involved CabibboKobayashiMaskawa (CKM) matrix elements. In addition, we predict the following vector form factors at q2 = 0:  f+D→η(0) = 0.01 ± 0.05,  f+Ds →K (0) = 0.50 ± 0.08, f Ds →η(0) = 0.73 ± 0.03 and  f+B¯ →η(0) = 0.82 ± 0.08,  + which might serve as alternatives to determine the CKM elements when experimental measurements of the corresponding differential decay rates become available. Finally, we predict the different form factors above the q2−regions accessible in the semileptonic decays, up to moderate energies amenable to be described using the unitarized coupledchannel chiral approach. 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 2 Theoretical framework . . . . . . . . . . . . . . . . 3 2.1 Form factors in H 3 decays . . . . . . . . . . . 3 2.2 MuskhelishviliOmnès representation . . . . . 4 2.3 Inputs and MO solutions . . . . . . . . . . . . 5 2.3.1 The (S, I ) = (1, 0) sector . . . . . . . . 6 2.3.2 The (S, I ) = (0, 1/2) sector . . . . . . . 8 2.4 Chiral expansion of the form factors and the MO polynomial . . . . . . . . . . . . . . . . . . . 10 2.4.1 Form factors in heavy meson chiral perturbation theory . . . . . . . . . . . . . . 10 2.4.2 Matching . . . . . . . . . . . . . . . . . 12 3 Numerical results and discussion . . . . . . . . . . 13 3.1 Fit to the LQCD+LCSR results in the bottom sector . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Extension to the charm sector and combined fit 15 3.2.1 Further considerations . . . . . . . . . . 18 3.3 Extraction of CKM elements and predictions . 19 3.4 Scalar form factors above the qm2ax−region . . . 21 4 Summary and outlook . . . . . . . . . . . . . . . . 21 Appendix A: Heavyquark mass scaling of the LECs in the Dφ interactions . . . . . . . . . . . . . . . . . . 23 Appendix B: Bottom form factors and quadratic MO polynomials . . . . . . . . . . . . . . . . . . . . . 24 References . . . . . . . . . . . . . . . . . . . . . . . . 25

Contents
1 Introduction
Exclusive semileptonic decays play a prominent role in the
precise determination of the Cabibbo–Kobayashi–Maskawa
(CKM) matrix elements, which are particularly important to
test the standard model (SM) – any violation of the unitarity
of the CKM matrix would reveal new physics beyond the SM
(see for instance the review on the CKM mixing parameters
by the Particle Data Group (PDG) [1]). Experimental and
theoretical efforts have been devoted to multitude of
inclusive and exclusive semileptonic decays driven by electroweak
charge currents. For instance, the K 3 decays and those of the
type H → φ ¯ ν and H → φ ν¯ (hereafter denoted by H 3
or H → φ), where H ∈ {D, B¯ } is an open heavyflavor
pseudoscalar meson and φ ∈ {π, K , K¯ , η} denotes one of
the Goldstone bosons due to the spontaneous breaking of the
approximate chiral symmetry of Quantum Chromodynamics
(QCD), are important in the extraction of some of the CKM
matrix elements. Experimentally, significant progresses have
been achieved and absolute decay branching fractions and
differential decay rates have been accurately measured [2–
11]. On the theoretical side, determinations of the form
factors in the vicinity of q2 = 0 (with q2 the invariant mass of
the outgoing lepton pair) using lightcone sum rules (LCSR)
have significantly improved their precision [12,13], and have
reached the level of twoloop accuracy [14]. Meanwhile,
improvements have been made by using better actions in
lattice QCD (LQCD), which have allowed to extract CKM
matrix elements with significantly reduced statistical and
systematical uncertainties [15–21]. As a result of this activity in
the past decade, lattice calculations on the scalar form
factors in heavytolight semileptonic transitions have been also
reported by the different groups (see the informative review
by the Flavour Lattice Averaging Group (FLAG) [22]).
The extraction of the CKM mixing parameters from K 3
and/or H 3 decays relies on the knowledge of the vector
[ f+(q2)] and scalar [ f0(q2)] hadronic form factors that
determine the matrix elements of the charged current between the
initial and final hadron states.1 Various parameterizations,
such as the Isgur–Scora–Grinstein–Wise updated model [23]
or the series expansion proposed in Ref. [24], are extensively
used in LQCD and experimental studies. In this work, we
will study the scalar form factors in H 3 decays by using the
Muskhelishvili–Omnès (MO) formalism, which is a model
independent approach to account for H φ coupledchannel
rescattering effects. The coupledchannel MO formalism
has been extensively applied to the scalar π π , π K and π η
1 The contribution of the scalar form factor to the decay width is
suppressed since it vanishes in the limit of massless leptons. However, both
scalar and vector form factors take the same value at q2 = 0, and thus an
accurate determination of the q2−dependence of the scalar form factor
can be used to constrain the vector one in this region.
form factors, see, e.g., Refs [25–29]. It builds up an elegant
bridge to connect the form factors with the corresponding
Swave scattering amplitudes via dispersion relations. The
construction of those equations is rigorous in the sense that
the fundamental principles, such as unitarity and analyticity,
and the proper QCD asymptotic behaviour are implemented.
The first attempts to extend this method to the investigation of
the scalar H → φ form factors were made in Refs. [30–38],
but just for the singlechannel case. A similar dispersive MO
approach has been also employed to study the semileptonic
B¯ → ρlν¯l [37,39–41] and B¯s → K¯ ∗lν¯l [37] decays and the
possible extraction of the CKM element Vub from data on
the fourbody B¯ → π πlν¯l and B¯s → K¯ πlν¯l decaymodes.
The study of heavylight form factors using the MO
representation incorporating coupledchannel effects has not been
undertaken yet. This is mainly because of the poor
knowledge on the H φ interactions up to very recent years.
However, a few intriguing positiveparity charmed mesons, like
the Ds∗0(2317), have been recently discovered [1], giving
support to a new paradigm for heavylight meson spectroscopy
[42] that questions their traditional qq¯ constituent quark
model interpretation. Hence, the study of the H φ
interactions aiming at understanding the dynamics of these newly
observed states has become an interesting subject by itself,
see, e.g., Refs. [43–54] for phenomenological studies and
[55–61] for LQCD calculations. For the D 3 decays,
several LQCD results on the relevant form factors have been
recently reported, see, e.g., Refs. [16,17,21]. This situation
makes timely the study of the scalar D → φ form factors by
means of the MO representation incorporating our current
knowledge of Dφ interactions. The extension to the H = B¯
case is straightforward with the help of heavy quark flavour
symmetry (HQFS). Based on HQFS, the low energy
constants (LECs) involved in the Dφ interactions or D → φ
semileptonic form factors are related to their analogues in
the bottom sector by specific scaling rules. It is then feasible
either to predict quantities in the bottom (charm) sector by
making use of the known information in the charm (bottom)
case or to check how well HQFS works by testing the scaling
rules.
In the present study, we construct the MO representations
of the scalar form factors, denoted by f0(q2), for the
semileptonic D → π and D → K¯ transitions, which are related to
the unitarized Swave scattering amplitudes in the Dφ
channels with strangeness (S) and isospin (I ) quantum numbers
(S, I ) = (
0, 21
) and (S, I ) = (
1, 0
), respectively. These
amplitudes are obtained by unitarizing the O( p2)
heavymeson chiral perturbative ones [62], with LECs determined
from the lattice calculation [56] of the Swave scattering
lengths in several (S, I ) sectors. The scheme provides an
accurate description of the Dφ interactions in coupled
channels. For instance, as it is shown in Ref. [52], the finite volume
energy levels in the (S, I ) = (0, 1/2) channel calculated with
the unitarized amplitudes, without adjusting any parameter,
are in an excellent agreement with those recently reported
by the Hadron Spectrum Collaboration [60]. In addition, it is
demonstrated in Ref. [42] that these well constrained
amplitudes for Goldstone bosons scattering off charm mesons
are fully consistent with recent high quality data on the
B− → D+π −π − final states provided by the LHCb
experiment [63].
The unitarized chiral scattering amplitudes are used in
this work as inputs to the dispersive integrals. However, these
amplitudes are valid only in the low Goldstoneboson energy
region. Hence, asymptotic behaviors at high energies for the
phase shifts and inelasticities are imposed in the solution of
the MO integral equations. The Omnès matrices obtained in
this way incorporate the strong final state interactions, and the
scalar form factors are calculated by multiplying the former
by polynomials. The (a priori unknown) coefficients of the
polynomials are expressed in terms of the LECs appearing
at nexttoleading order (NLO) in the chiral expansion of the
form factors [64,65].
The scheme employed in the charm sector is readily
extended to the bottom one. Afterwards, the LECs could
be either determined by fitting to the results obtained in the
LQCD analyses of the D → π(K¯ ) decays carried out in Refs.
[16,17,21] or to the LQCD and LCSR combined B¯ → π and
B¯s → K scalar form factors reported in Refs. [15,18–20] and
[12–14], respectively. In both scenarios LQCD and LCSR
results are well described using the MO dispersive
representations of the scalar form factors constructed in this work.
However, our best results are obtained by a simultaneous
fit to all available results, both in the charm and bottom
sectors. As mentioned above, all of the LECs involved in the
B¯(s)φ interactions or B¯(s) → φ¯ semileptonic transitions are
related to those in the charm sector by making use of the
heavy quark scaling rules [65], which introduce some
constraints between the polynomials that appear in the different
channels. Thus, assuming a reasonable effect of the HQFS
breaking terms, a combined fit is performed to the D → π/K¯
and the B¯ → π and B¯s → K scalar form factors, finding
a fair description of all the fitted data, and providing
reliable predictions of the different scalar form factors in the
whole semileptonic decay phase space, which turn out to
be compatible with other theoretical determinations by, e.g.,
perturbative QCD [66,67]. The results of the fit allow also to
predict the scalar form factors for the D → η, Ds → K and
Ds → η transitions in the charm sector, and for the very first
time for the B¯ → η decay. In some of these transitions, the
form factors are difficult for LQCD due to the existence of
disconnected diagrams of quark loops.
Based on the results of the combined fit, and taking
advantage of the fact that scalar and vector form factors are equal
at q2 = 0, we extract all the heavylight CKM elements and
test the secondrow unitarity by using the Vcb value given
in the PDG [1]. We also predict the different form factors
above the q2−regions accessible in the semileptonic decays,
up to energies in the vicinity of the involved thresholds, which
should be correctly described within the employed unitarized
chiral approach.
This work is organized as follows. In Sect. 2.1, we
introduce the definitions of the form factors for H 3 decays. A
general overview of the MO representation of the scalar form
factors is given in Sect. 2.2, while the inputs for the MO
problem and the solutions are discussed in Sect. 2.3. In Sect. 2.4,
we derive the scalar as well as vector form factors at NLO
in heavymeson chiral perturbation theory and then perform
the aforementioned matching between the MO and the chiral
representations close to the corresponding thresholds.
Section 3 comprises our numerical results and discussions, with
details of the fits given in Sects. 3.1 and 3.2. With the results
of the combined charmbottom fit, in Sect. 3.3, we extract
the related CKM elements and make predictions for the
values of f+(0) in various transitions. Predictions for
flavourchanging b → u and c → d, s scalar form factors above the
q2−regions accessible in the semileptonic decays are given
and discussed in Sect. 3.4. We summarize the results of this
work in Sect. 4. Finally, the heavyquark scaling rules of
the LECs involved in the H φ interactions are discussed in
Appendix A, while some further results for b → u form
factors, obtained with quadratic MO polynomials, are shown in
Appendix B.
2 Theoretical framework
2.1 Form factors in H 3 decays
For a semileptonic decay of the type H ( p) → φ¯ ( p )
( p ) ν¯ ( pν ), the Lorentz invariant Feynman amplitude is
proportional to
G F
M ∝ √2
u¯( p )γ μ(1 − γ5)v( pν )
× VQq φ¯ ( p )q¯ γμ(1 − γ5)QH ( p)
where G F is the Fermi constant, and VQq is the CKM matrix
element corresponding to the flavour changing Q → q
transition. The terms in the first and second curly brackets stand
for the weak and hadronic matrix elements, respectively. In
the hadronic matrix element, the axialvector part vanishes
due to the parity conservation, while the remaining vector
part is parametrized in a conventional form as
φ¯ ( p )q¯ γ μ QH ( p) = f+(q2)
μ
−
m2H − M 2
φ qμ
q2
+ f0(q2) m2H − M 2
φ qμ ,
q2
(
2
)
where f+(q2) and f0(q2) are the vector and scalar form
factors, respectively, with qμ = pμ − p μ and μ = pμ + p μ.
Note that both form factors should be equal at q2 = 0. As
discussed in Ref. [68], they specify the Pwave ( J P = 1−) and
Swave ( J P = 0+) of the crossedchannel matrix elements,
0q¯ γ μ QH ( p)φ (− p ) = φ¯ ( p )q¯ γ μ QH ( p) .
(
3
)
Both the scalar and vector form factors contribute to the
differential decay rate, see e.g., Ref. [22]. Nevertheless, when
the lepton mass is neglected, the differential decay rate in the
H −meson rest frame can be simply expressed in terms of
the vector form factor via
d (H → φ¯ ν¯ )
dq2
G2F
= 24π 3  p 3VQq 2 f+(q2)2 .
(
4
)
It is then possible to extract the CKM element VQq  even
for a single value of the fourmomentum transfer, provided
one simultaneously knows the vector form factor and the
experimental differential decay width. A possible choice is
q2 = 0, where the scalar and vector form factors coincide,
f+(0) = f0(0).
In the next two subsections, Sects. 2.2–2.3, we give
specific details on the MO representation of the form factors. For
brevity, in some occasions we will focus on the formalism
for the case of the charm sector (H = D). The extension to
the bottom sector (H = B¯ ) is straightforward using HQFS,
though some aspects are explicitly discussed in Sect. 2.3.2b.
2.2 Muskhelishvili–Omnès representation
We now discuss the dispersive representation of the scalar
form factors within the MO formalism. Throughout this
work, isospin breaking terms are not considered, and
therefore it is convenient to work with the isospin basis. Before
proceeding, we first discuss the relation of the form factors
expressed in the particle and isospin bases. We start defining
the phase convention for isospin states:
D+
K 0
 ¯
1 1
= − 2 , + 2 ,
1 1
= − 2 , + 2 ,
π +
B¯ 0
= −1, +1 ,
1 1
= − 2 , + 2 ,
while the other states are defined with a positive sign in front
of the I, I3 state. The form factors involving the c → d
transition are those appearing in the D0 → π −, D+ → π 0,
D+ → η, and Ds+ → K 0 semileptonic decays. (Note that
D0 → π − and D+ → π 0 transitions are related by an
isospin rotation). The details of these form factors close to
the zero recoil point, where the outgoing Goldstone boson
is at rest, are greatly influenced by the π D, Dη, and Ds K¯
scattering amplitudes in the (S, I ) = (0, 1/2) sector. Note
that to be consistent with the convention of Refs. [56,62], we
with λ(x , y, z) = x 2 + y2 +z2 −2x y −2yz −2x z, the Kählén
function. Invariance under timereversal together with the
opticaltheorem leads to
Im T −1(s + i ) = − (s).
2 Here also D0 → K − and D+ → K¯ 0 form factors are related by an
isospin rotation.
use π D instead of Dπ to construct the isospin 1/2 state. We
then define the vectorcolumn F (0,1/2) as
F (0,1/2)(s) ≡ ⎜⎜
⎝
⎛
3 f D0→π− (s) ⎞
2 0
ffD0Ds++→→Kη0((ss)) ⎠⎟⎟ .
0
We shall use the shorthands f0Dπ (s) = f0D0→π− (s),
f0Dη(s) = f0D+→η(s), and f0Ds K¯ (s) = f0Ds+→K 0 (s).
Likewise for the c → s transitions, we have the D0 → K −,
D+ → K¯ 0, and Ds+ → η semileptonic decays, related to
the D K and Ds η scattering amplitudes in the (S, I ) = (
1, 0
)
sector. We thus define:2
F (
1,0
)(s) ≡
−
√2 f0D0→K − (s)
f0Ds+→η(s)
,
(
6
)
(
7
)
(
9
)
(
10
)
for which we will also use the notation f0D0→K − (s) =
f0DK (s) and f0Ds+→η(s) = f0Ds η(s). Here, and again to be
consistent with the convention of Refs. [56,62], we use D K
instead of K D to construct the isoscalar state, which is just
the opposite convention to that used for the isospin π D state.
With these definitions, the unitarity relation for any of the
F (s) can be compactly written as:
(
5
)
σab(s) =
λ1/2(s, ma2, m2b)
s
s − (ma + mb)2 ,
F (s + i ) − F (s − i )
2i
= Im F (s + i )
= T ∗(s + i ) (s)F (s + i ) , (
8
)
where T (s) stands for the coupledchannel Swave
scattering amplitude in the corresponding (S, I ) sector, which
will be discussed further in Sect. 2.3. The diagonal matrix
(s) contains the phase space factors. For (0, 1/2), one
has
(s) = diag σDπ (s), σDη(s), σDs K¯ (s) , whereas in
the (
1, 0
) sector (s) = diag σDK (s), σDs η . The function
σab(s) is defined as:
(
11
)
(
12
)
(
13
)
(
15
)
(
18
)
(
19
)
(
16
)
(
17
)
with sth the lowest threshold, which is referred to as the MO
integral equation [72]. Taking a polynomial P (s) of rank
(n − 1) would require the knowledge of F (s) for n values
of s. Unlike the single channel case, there is no analytical
solution [in the sense of Eq. (
14
)] for the coupledchannel
MO problem and it has to be solved numerically. The MO
equation can be written in an alternative form:
Re
Im
with P
1
(s) = π P
sth
(s) = X (s) Re
∞
ds
s − s
(s) ,
Im
(s ),
denoting the principal value and
X (s) = Im T (s) [Re T (s)]−1 ,
which is expressed in terms of the T matrix and encodes the
information on the Dφ rescattering. The linear MO system,
Eq. (
20
), can be solved by following the appropriate
numerical method described in Ref. [71].
(s) = exp
s
π
∞
(m D+Mπ )2
ds
δ(s )
s (s − s)
,
(
14
)
2.3 Inputs and MO solutions
with H (s) = (I + 2i T (s) (s)). Furthermore since H (s)H ∗(s) = I,
though H (s) is not the Smatrix in the coupledchannel case, it follows
det [ (s + i )] = e2iφ(s)det [ (s − i )] ,
with exp 2i φ (s) = det [H (s)]. This is to say that the determinant of the
matrix (s) satisfies a singlechannel Omnèstype relation [71], which
is extensively used in this work to check the accuracy of the numerical
calculations. Note that above all thresholds, det [H (s)n] = det [S(s)]
and therefore in the elastic case (ηi → 1 ∀i ), φ (s) = i=1, δi (s), with
n the number of channels.
To solve the MO equation and obtain the (s) matrix, the T
matrix is needed as an input. We will use here the amplitudes
based on unitarized chiral effective theory as computed in
Refs. [56, 62]. Because of the normalizations used in Eqs. (
8
)
and (
10
), the unitarized amplitude in Ref. [56], denoted here
by TU (s), is related to the T matrix introduced in the above
subsection by:
1
T (s) = − 16π TU (s) .
This unitary T matrix is written as:
TU−1(s) = V −1(s) − G(s) ,
where the elements of the diagonal matrix G(s) are the
twomeson loop functions [56], and the matrix V (s) contains the
interaction potentials which, as pointed out above, are taken
from the O( p2) chiral Lagrangians deduced in Ref. [62] (see
Ref. [52] for more details).4 In Ref. [56] all the parameters
involved in the T matrix, which are a few LECs and one
subtraction constant, have been pinned down by fitting to the
lattice calculation of the Swave scattering lengths in several
(S, I ) sectors. The obtained interaction potentials
successfully describe [52] the charm (S, I ) = (0, 1/2) finite volume
energy levels, without adjusting any parameter, calculated by
the Hadron Spectrum Collaboration [60]. Moreover, as
mentioned in the Introduction, these well constrained amplitudes
4 The most recent unitarized amplitudes based on oneloop potentials
are given in Refs. [50,53]. However, we will not use them here since
the LECs involved in the oneloop analyses can not be well determined
due to the lack of precise data, as pointed out in Ref. [73].
In this convention, the T  and Smatrices are related by
S(s) = I + 2i 21 (s)T (s) 21 (s).
The unitarity relation of Eq. (
8
) can be used to obtain
dispersive representations for the form factors. We start considering
the D → π transition in the single channel case (elastic
unitarity) where the form factor satisfies
Im f0Dπ (s) = [t π D (s)]∗σDπ (s) f0Dπ (s) , s ≡ q2 ,
with t π D (s) the π D Swave elastic scattering amplitude.
Equation (
12
) admits an algebraic solution [69],
f0D→π (s) =
(s) P (s) ,
where P (s) is an undetermined polynomial, and the Omnès
function (s) is given by
with δ(s) the elastic 0+ π D phase shift, in accordance with
the Watson final state interaction theorem [70]. This was the
scheme adopted in the previous studies carried out in Refs.
[30–36, 38].
For coupled channels the solution for F (s) takes the form:
F (s) =
(s) · P (s)
being P (s) a vector of polynomials with real coefficients and
(s) the Omnès matrix that satisfies3
Im
(s + i ) = T ∗(s + i ) (s) (s + i )
which leads to the following unsubtracted dispersion relation:
1
(s + i ) = π
sth
∞ T ∗(s )
(s )
s − s − i
(s )
ds ,
3 Taking into account that the (s) matrix should have only a
righthand cut and it should be real below all thresholds, Eq. (
18
) is equivalent
to
(s + i ) = H (s + i ) (s − i ),
(
20
)
(
21
)
(
22
)
(
23
)
turned out to be fully consistent [42] with the LHCb data on
the B− → D+π −π − reaction [63].
Since the amplitudes are based on chiral potentials, the
obtained T matrices are only valid in the energy region not
far from the corresponding thresholds. We thus adopt such
T matrices only up to a certain value of s, denoted by sm .
Above that energy, the T matrix elements are computed as an
interpolation between their values at s = sm and the
asymptotic values at s = ∞. The interpolation still gives a
unitary T matrix since, as we will specify below, it is actually
performed on the phase shifts and the inelasticities.
Moreover, the approximation of quasitwobody channels cannot
hold for arbitrarily large energies and Eq. (
8
) is a
reasonable approximation to the exact discontinuity only in a finite
energy range. However, as we are interested in constructing
the form factors in a finite energy region also, the detailed
behaviour of the spectral function at much higher energies
should be, in principle, unimportant. As we will see below,
this is not entirely correct, in particular for the B¯ → π
semileptonic transition, because of the large q2−phase space
accessible in this decay. Nevertheless, we will assume that
Eq. (
8
) holds up to infinite energies, only requiring that
the T matrix behaves in a way that ensures an appropriate
asymptotic behaviour of the form factors, and we will discuss
the dependence of our results on the contributions from the
high energy region. In general, except for the B¯ decays5, the
asymptotic conditions on the T matrix are chosen such that:
n
lim
s→∞ i=1
lim Ti j (s) = 0 for i = j ,
s→∞
δi (s) = nπ ,
where n is the number of channels involved in the T matrix
and δi (s) are the phase shifts. These conditions ensure (in
general) that the unsubtracted dispersion relation for the
Omnès matrix in Eq. (
19
) has a unique solution, albeit a
global normalization [72] (see also in particular Sect. 4.3 of
Ref. [71]). The condition of Eq. (
25
) guaranties that
lim det [ (s)] → 1/sn
s→∞
as can be deduced from the discussion of Eqs. (
16
) and
(
17
). Note that the normalization of the Omnès matrix is
completely arbitrary, and the computed form factors do not
depend on it.6
5 We will specifically discuss the situation for these transitions below.
6 For example, let us consider Omnès matrices and ¯ normalized to
(0) = I or ¯ (sn) = A (sn the normalization point, sn sth and A
a real matrix), respectively. The matrix ¯ (s) is readily obtained from
(s) as ¯ (s) = (s) −1(sn)A. The form factors can then also be
written as F(s) = (s)P(s) = ¯ (s)A−1 (sn)P(s) ≡ ¯ (s)P¯ (s),
where the definition of P¯ (s) reabsorbs the constant matrix A−1 (sn).
(
24
)
(
25
)
(
26
)
In what follows, we detail the T matrices and the
specific shape of the asymptotic conditions for the two
coupledchannel cases [(S, I ) = (
1, 0
) and (0, 1/2)] that will be
analyzed below.
2.3.1 The (S, I ) = (
1, 0
) sector
In this sector, we will consider two coupled channels, D K
(
1
) and Ds η (
2
), and above all thresholds, the T matrix is
parametrized in terms of two phase shifts and one inelasticity
parameter,
with the phase φ12 = δ1 + δ2 + mod(π ) and 0 ≤ η ≤ 1.
To solve the MO integral equation [cf. Eq. (
20
)], we use a
T matrix of the form
T (s) =
− 161π TU(
1,0
)(s) sth
s
sm ,
TH (s)
s
sm ,
with sth = (m D + MK )2 the lowest threshold, TU defined in
Eq. (
23
) and TH the asymptotic matrix that will be discussed
below. Phase shifts, inelasticities and amplitude moduli from
TU(
1,0
) are displayed in Fig. 1 up to √s = 2.8 GeV, slightly
above sm = (2.7 GeV)2. Above this scale, the T matrix
elements are computed as an interpolation between their values
at s = sm and the asymptotic values at s = ∞, given in
Eqs. (
24
) and (
25
). Thus, TH is constructed from Eq. (
27
)
using the following parameterizations for phase shifts and
inelasticities:
η(s) = η(∞) + [η(sm ) − η(∞)]
2
δi (s) = δi (∞) + [δi (sm ) − δi (∞)] 1 + (s/sm )3/2 ,
2
1 + (s/sm )3/2
as suggested in Ref. [71]. As discussed above, the Omnès
matrix is uniquely determined by choosing η(∞) = 1 and
δ1(∞) + δ2(∞) = 2π . The only remaining freedom is the
distribution of 2π over the two phase shifts. Note that, δi (sm )
is defined modulo π and this ambiguity is fixed by
continuitycriteria. Here for the D K − Ds η coupled channels, we choose
δ1(∞) = 2π , δ2(∞) = 0. Different choices of the
asymptotic values or of the interpolating functions in Eq. (
29
) will
modify the shape of the Omnès solution far from the chiral
region. The numerical effect of such freedom on the derived
scalar form factors should be safely compensated by the
undetermined polynomial in front of the Omnès matrix.
In Fig. 2, we show the solution of the MO integral
equation (
20
), with the input specified above, and the contour
condition (qm2ax) = I, with qm2ax = (m D − MK )2. We
display results only up to s = sm that would be later used to
(
27
)
(
28
)
(
29
)
20
Fig. 1 Phase shifts, inelasticities [see Eq. (
27
)] and amplitude moduli from T (
1,0
). The vertical line indicates the Ds η threshold. Error bands have
U
been obtained by Monte Carlo propagating the uncertainties of the LECs quoted in Ref. [56]
Re{Ω}
Im{Ω}
0
0.5
2
2.5
0
0.5
2
2.5
1
1.5
√s [GeV]
1
√s [GeV]
1.5
2
1
0
1
2
)
s
(
2
1
Ω
2.5
2
1.5
evaluate the scalar form factors entering in the D → K¯ and
Ds → η semileptonic transitions. Note that the imaginary
parts are zero below the lowest threshold sth = (m D + MK )2,
and how the opening of the Ds η threshold produces clearly
visible effects in the Omnès matrix. At very high energies,
not shown in the figure, both real and imaginary parts of all
matrix elements go to zero, as expected from Eq. (
26
).
T12
1.0
0.9
Dπ
Dη
DsK¯
160
120
a. Charm sector: Here we consider three channels, Dπ
(
1
), Dη (
2
), and Ds K¯ (
3
), and above all thresholds, the
Smatrix can be still specified7 by the elastic parameters, i.e.,
three phase shifts and three inelasticities [74, 75],
⎛
S(s) = ⎝
η1e2iδ1 γ12eiφ12 γ13eiφ13 ⎞
γ12eiφ12 η2e2iδ2 γ23eiφ23
γ13eiφ13 γ23eiφ23 η3e2iδ3 ⎠
.
(
30
)
2 1
γi j = 2
sin αi j =
Furthermore, the parameters in the offdiagonal elements are
related to the diagonal ones δi and ηi by
1 + ηk2 − ηi2 − η2j , i = j = k = i ,
φi j = δi + δ j + αi j + mod(π ), i, j, k = 1, 2, 3 ,
and αi j is determined as
− (ηi − η j )2
≡ Xi j .
(
31
)
Note that the solutions for αi j can be either arcsin( Xi j ) or
π − arcsin( Xi j ). The inelasticity parameters should satisfy
the following boundary conditions:
1 − η j − ηk  ≤ ηi ≤ 1 − η j − ηk , i = j = k .
(
32
)
To solve the MO integral Eq. (
20
), we use a T matrix similar
to that in Eq. (
28
), with the obvious substitution of T (
1,0
)(s)
U
7 The T matrix is obtained from Eq. (
30
).
1
4ηi η j
γi2k γ j2k
γi2j
0 ≤ ηi ≤ 1 ,
by TU(
0, 21
)(s). In addition, sth = (m D + Mπ )2 and we now take
sm = (2.6 GeV)2. Phase shifts, inelasticities and amplitude
moduli from T (0,1/2) are displayed in Fig. 3 for sth ≤ s ≤ sm .
U
Above sm = (2.6 GeV)2, TH is constructed from Eq. (
30
)
using interpolating parameterizations for phase shifts and
inelasticities similar to those given in Eq. (
29
), imposing
continuity of phase shifts and of the T matrix, and taking
δ1(∞) = 3π ,
η j (∞) = 1 ,
δi (∞) = 0, i = 2, 3,
j = 1, 2, 3 .
(
33
)
With the inputs specified above, the threedimensional
(S, I ) = (0, 1/2) Omnès matrix can be numerically
computed and its complex elements are shown in Fig. 4 up to
√s ≤ 2.6 GeV.
b. Bottom sector: In Figs. 5 and 6, we show phase shifts,
inelasticities and the solution of the MO integral equation for
the (S, I ) = (0, 1/2) channel in the bottom sector. The chiral
amplitudes8 are used in Eq. (
19
) up to sm = (6.25 GeV)2,
and from there on, the asymptotic forms of the amplitudes
are employed. As we will show in the next section, in the case
of B¯ decays, the accessible phase space is quite large, and q2
varies from around m2B at zero recoil [qm2ax = (m B − Mφ )2]
down to zero, when the energy of the outgoing light meson
is about m B /2 far from the chiral domain. The B¯ π scalar
form factor decreases by a factor of five, and the LQCD
8 The values of the involved LECs in the B¯ φ interactions are determined
from their analogues in the charm sector by imposing the heavyquark
mass scaling rules [52] discussed in the Appendix A.
2
1
1
)
s
(
11 0
Ω
Re{Ω}
Im{Ω}
Fig. 4 Charm (S, I ) = (0, 1/2) Omnès matrix solution of the MO
integral equation (
20
) with the contour condition (m D − Mπ )2 = I,
and asymptotic phase shifts δDπ (∞) = 3π , δDη(∞) = 0 and
results around qm2ax and the LCSR predictions in the vicinity
of q2 = 0 are not linearly connected. In the present approach,
as we will discuss, we multiply the MO matrix by a
rankone polynomial, and thus the extra curvature provided by
the MO matrix becomes essential. While (s) around qm2ax
is rather insensitive to the adopted asymptotic behaviour of
the T matrix, since it is dominated by the integration region
close to threshold (s < sm ) where the chiral amplitudes are
being used9, this is not the case for low values of q2 close
to 0, quite far from the twobody scattering thresholds. This
unwanted dependence, due to the large extrapolation, could
be compensated in the form factors by using higher rank
polynomials, but that would introduce additional undetermined
parameters. Conversely, this dependence of (0), relative
to the results at qm2ax, on the details of the amplitudes at
high energies will be diminished by solving a MO integral
equation involving several subtractions, instead of the
unsubtracted one of Eq. (
19
). This, however, will also introduce
some more free parameters [34]. The situation is better in the
9 In general, the MO matrix in the chiral domain, between the qm2ax and
scattering (below sm ) regions is rather insensitive to the high energy
behaviour of the amplitudes.
δDs K¯ (∞) = 0. Error bands have been obtained by Monte Carlo
propagating the uncertainties of the LECs quoted in Ref. [56]
charm sector, where the needed q2−range is much reduced,
and thus most of the contributions to the MO matrix come
from integration region within the chiral regime. Indeed in
the bottom sector we need to use δ1(∞) = 2π, δ2(∞) = 2π
and δ3(∞) = 0, instead of the choice of Eq. (
33
) used for the
charm decays, to find acceptable fits to the LQCD and LCSR
predictions of the B¯ π and B¯s K scalar form factors. With this
choice, we find theoretically sound fits where the LECs10
that determine the rankone Omnès polynomials describe
the LQCD data close to qm2ax, within the range of expected
validity of the chiral expansion, while the LCSR results are
reproduced thanks to the nonlinear behaviour encoded in
the MO matrix (s). This picture will be reinforced by the
consistent results that will be obtained, assuming a
reasonable effect of the HQFS breaking terms, from combined fits
to the D → π/K¯ and the B¯ → π and B¯s → K scalar form
factors.
We do not really have an explanation of why the above
choice of phase shifts at infinity works better in the
bottom sector than the usual one in Eq. (
33
) and adopted in the
10 These are β1P and β2P , to be introduced in Sect. 2.4, that appear in
the chiral expansion of the form factors at NLO.
B¯π
B¯η
B¯sK¯
Fig. 5 Phase shifts, inelasticities and amplitude moduli from T (0,1/2)
U
in the bottom sector. The vertical lines indicate the B¯ η and B¯s K¯
thresholds. The values of the involved LECs in the B¯ φ interactions are
determined from their analogues in the charm sector by imposing the
heavycharm meson decays. We would like, however, to mention
the different behaviour of the unitarized chiral phase shifts
in the charm and bottom sectors. In both cases, the chiral
amplitudes give rise to two resonances [52]: the first one, the
nonstrangeness flavor partner of the Ds∗0(2317), quite broad,
and located around 100 MeV above the Dπ or B¯ π thresholds
and the second one placed below the heaviest of the
thresholds, Ds K¯ and B¯s K¯ , respectively. In the charm sector, the
second resonance does not produce clear signatures in the
phase shifts of the two open channels Dπ and Dη, while it
is clearly visible in the phase shifts of the bottom B¯ π and
B¯ η channels. Moreover, the second resonance is significantly
narrower for the latter heavyquark sector than for the former
one (70 versus 270 MeV).
Note also that now lims→∞ i3=1 δi (s) = 4π > 3π ,
which implies a slightly faster decreasing of the MO matrix
elements at high energies.
2.4 Chiral expansion of the form factors and the MO
polynomial
Once the Omnès matrix is obtained, the form factor F (s) is
determined, according to Eq. (
15
), up to a polynomial P(s)
that contains unknown coefficients. We will match the
dispersive and the NLO chiral representations of the form factors
in a region of values of s where the latter are supposed to be
still valid. Besides the theoretical benefit of this constraint,
it has also the practical advantage of expressing the
coefficients of the polynomials in terms of the few LECs used in
quark mass scaling rules discussed in the Appendix A. Error bands
have been obtained by Monte Carlo propagating the uncertainties of
the LECs quoted in Ref. [56]
the chiral expansion of the form factors. Since, as will be
discussed below (cf. Eqs. (
40
) and (
41
) and the discussion that
follows), the NLO chiral expansion of the form factors used
here is appropriate only up to terms linear in s, we should
also take linear forms for the MO polynomials,
P(s) = α0 + α1 s .
Since the Omnès matrix elements asymptotically behave as
1/s [see Eq. (
26
)], due to the chosen asymptotic conditions,
this implies that the form factors will tend to a constant11
for s → ∞. Note that one would rather expect the form
factors to vanish in this limit [76]. To achieve such asymptotic
behaviour one should employ order zero polynomials.
However, since we are interested in the region 0 s smax, with
smax in the vicinity of (m H − Mφ )2, we prefer to keep the
linear behaviour of the polynomials, since this allows for a
better matching of the coefficients α0,1 with the LECs that
appear in the NLO chiral calculation of the form factors.
(
34
)
2.4.1 Form factors in heavy meson chiral perturbation
theory
The leadingorder (LO) coupling of the charm (D and Ds ) or
bottom (B¯ and B¯s ) mesons to the Nambu–Goldstone bosons
11 This not strictly true in the case of B¯(s)−decays since, as discussed
above, different asymptotic conditions have been assumed in the bottom
sector and the Omnès matrix elements are expected to decrease slightly
faster than 1/s.
Re{Ω}
Im{Ω}
Fig. 6 Bottom (S, I ) = (0, 1/2) Omnès matrix solution of the MO
integral equation (
20
) with the contour condition (m B − Mπ )2 = I,
and asymptotic phase shifts δB¯ π (∞) = 2π, δB¯ η(∞) = 2π and
L0 =
2 fP i m˚ Pμ∗ + ∂μP u† J μ,
of the spontaneous breaking of the approximate chiral
symmetry of QCD, through the chargedcurrent lefthanded
currγeLμnt=J μγ μ=(1 −Q¯γγ5L)μ,uis, dQ¯eγscLμrdib,eQd¯ γbLyμstheT f,owlliotwhiQng
=chicra,lbe,ffaencdtive Lagrangian [64,65,77]
√
(
35
)
where P and P∗ are the pseudoscalar and vector heavylight
mesons with content (Qu¯, Qd¯, Qs¯), respectively, which
behave as SU (
3
) light flavor triplets. Here m˚ denotes the
degenerate mass of the P(s) and P(∗s) mesons in the chiral and
heavyquark limits, and fP is the pseudoscalar heavylight
meson decay constant defined as
0 J μP( p1) = i
√
μ
2 fP p1 .
The chiral block is defined by u2 = U = exp[i √2 /F0],
where is the octet of the Nambu–Goldstone bosons
= ⎜⎝
⎛ √12 π 0 + √16 η
π −
K −
π +
− √12 π 0 + √16 η
K 0
¯
K + ⎞
K 0 ⎟ ,
− √26 η⎠
(
36
)
(
37
)
δB¯s K¯ (∞) = 0. Error bands have been obtained by Monte Carlo
propagating the uncertainties of the LECs quoted in Ref. [56]
with F0 the pion decay constant in the chiral limit (we will
take the physical value for the decay constant F0 92 MeV).
The relevant NLO chiral effective Lagrangian reads [65]
L1 = −β1P P u (∂μU †) J μ −β2P (∂μ∂ν P) u (∂νU †) J μ . (
38
)
We need the LO PP∗φ interaction as well, which is given
by [64,65,77] ,
LPP∗φ = g˜ Pμ∗ uμP† + P uμPμ∗† ,
(
39
)
where uμ = i (u†∂μu − u ∂μu†) and g˜ ∼ gm˚ , with g ∼ 0.6 a
dimensionless and heavy quark mass independent constant.
The topologies of relevant Feynman diagrams are shown in
Fig. 7. The vector and scalar form factors, in the (strangeness,
isospin) basis, at O(Eφ ) (i.e., NLO) in the chiral expansion
read [65]
f [Pφ](S,I)
+
(S,I )
C[Pφ] f P
(s) = √2F0 √2 +
√2 g˜ m˚ f P
m2R − s
+ β1P − β22P ( Pφ − s) ,
(
40
)
P
φ
(a)
(
1, 0
)
D K
−√2
Ds η
Fig. 7 Topologies of the relevant Feynman diagrams contributing to
the hadronic matrix elements. The solid circle denotes the LO PP∗φ
interaction and the solid square represents the lefthanded current
Table 1 Strangenessisospin coefficients appearing in the chiral
expansion of the form factors
P
{Qs¯, Qd¯, Qu¯} and φ ∈ {π, K , K¯ η}. The coefficients C[(PS,φI])
are collected in Table 1. Moreover, we have fixed m R to
m D∗ (m B∗ ) and to m Ds∗ (m Bs∗ ) for the (S, I ) = (0, 1/2) and
(S, I ) = (
1, 0
) charm (bottom) channels, respectively. In
principle, at LO in the heavy quark expansion, m R should
be set to m˚ , however the use of the physical vector mass
is quite relevant for the vector form factor, because of the
propagator structure, though it has much less relevance for the
scalar form factor that we study in this work. We should also
note that all kinematical factors are always calculated using
physical masses of the involved mesons. It is worthwhile
to notice that s is of the order m2P ∼ O(Eφ0 ), Pφ − s ∼
O(Eφ ) and Pφ − s ∼ O(Eφ ), and Pφ + s ∼ O(Eφ0 ),
so that a small change of O(Eφ ) in Pφ + s only leads to
a higher order effect. Thus, Pφ + s should be regarded
as basically a constant with s ∼ m2P , and the expression in
Eq. (
41
) should be matched to a rank1 MO polynomial as
mentioned before (see below for details of the matching).
Finally, we should mention that the LECs β1P and β2P scale
with the heavy quark mass as [65]
β1P ∼ √m P ,
β2P ∼ 1/ m3P
neglecting logarithmic corrections.
Ds K¯
B¯s K¯
1
(
41
)
(
42
)
2.4.2 Matching
At energies close to the thresholds, the scalar form factors in
Eq. (
15
) should have the same structure as the ones obtained
from chiral perturbation theory, given in Eq. (
41
). We match
the two representations at a point s = s0 located in the valid
region of the chiral expansion. Namely, we take s0 = qm2ax =
(m P − Mπ )2 for the (0, 1/2) form factors, and s0 = (m D −
MK )2 for the charm (
1, 0
) case, since this is the point in
which the momentum of the lightest meson is zero. Imposing
that the dispersive form factors and their first derivative to be
equal to the chiral ones at s = s0, the coefficients in the
polynomials, Eq. (
34
), can be expressed as:
α0 =
α1 =
−1(s0) · Fχ (s0) − α1 s0,
−1(s0) · Fχ (s0) −
(s0) ·
−1(s0) · Fχ (s0) ,
where the stands for a derivative with respect to s. The
vectors Fχ contain the chiral form factors,
Fχ(
0, 21
)(s) ≡ ! f0Dπ(
0, 21
) , f0Dη(
0, 21
) , f0Ds K¯ (
0, 21
) "T ,
Fχ(
1,0
)(s) ≡
f0DK (
1,0
) , f0Ds η(
1,0
) T ,
(
43
)
(
44
)
(
45
)
with all the elements given in Eq. (
41
). Similar expressions
are used for the (
0, 21
) channel in the bottom sector. In other
words, the vectors Fχ (s) contain the form factors defined
in Eqs. (
6
) and (
7
), but computed according to the chiral
expansion.
It is worth noting that the NLO LECs β1P and β2P
determined from a fit to data using the MO scheme would have
some residual dependence on the matching point. To
minimize such dependence, we have chosen s0 = qm2ax, where
the momentum of the Goldstone bosons is close to zero and
higher order chiral corrections are expected to be small.
Different choices of the matching point, within the chiral regime,
will amount to changes in the fitted (effective) β1P and β P
2
LECs driven by higher order effects.
In the charm (
1, 0
) sector, due to the presence of the
Ds∗0(2317) state as a bound state in the T matrix, the solution
of Eq. (
15
) gets modified. The contribution from Ds∗0(2317)
is easily incorporated as follows:
(
1,0
)
· P(
1,0
)(s) →
(
1,0
) · # sβ−0 s p + P(
1,0
)(s)$,
(
46
)
where β0 is an unknown parameter which characterizes the
coupling of Ds∗0(2317) to the lefthand current. contains
m B
m Bs
f B
f Bs
m B∗
gDs η [GeV]
Table 2 Properties of the Ds∗0(2317) pole from the unitarized chiral
amplitudes derived in Ref. [56]
the couplings of the Ds∗0(2317) to the D K  Ds η system12,
namely, = (gD K , gDs η)T . This bound state is dynamically
generated in the unitarized amplitudes given in Ref. [56],
that we employ here. The couplings are computed from the
residue of the amplitude at the pole,
gi g j
Ti j (s) = s − s p + . . .
The Ds∗0(2317) pole position s p , together with gD K and gDs η,
are collected in Table 2.
3 Numerical results and discussion
So far, the theoretical MO representations of the scalar form
factors have been constructed. In this section, we want to
confront the soobtained form factors to the LQCD and LCSR
results. In what follows, we first fit to the B¯ → π and
B¯s → K scalar form factors, where we expect the 1/m P
corrections to the chiral expansion in Eq. (
41
) to be
substantially suppressed. Next, we will carry out a combined fit to all
the data in both charm and bottom sectors, by adopting some
(approximate) heavyquark flavor scaling rule [65] for the
12 Note that the first term in the bracket of Eq. (
46
) should have a more
general form, s−β0sp , with β0 a vector with two independent components,
β0 = (β0a , β0b)T . The specific form in Eq. (
46
) reduces the number of
free parameters, by forcing β0a /β0b = gDK /gDs η. On the other hand,
this has the effect that the form factors f0DK and f0Ds η are not exactly
independent of the choice of the point sn where one normalizes the
Omnès matrix, (sn ) = I. Nonetheless, we have checked that this
choice, varying sn from zero to qm2ax, has no practical effect in the
determination of β0, which indicates that our assumption is reasonable.
We also remark that this discussion has no effect at all in the (0, 1/2)
sector.
β1P and β2P LECs in Eq. (
41
). Using the results of our
combined fit, we will: (i) determine the CKM elements, Vcd ,
Vcs  and Vub, (ii) predict form factors, not computed in
LQCD yet, and that can be used to overconstrain the CKM
matrix elements from analyses involving more semileptonic
decays, and (iii) predict the different form factors above the
q2−regions accessible in the semileptonic decays, up to
moderate energies amenable to be described using the unitarized
coupledchannel chiral approach.
Masses and decay constants used in this work are compiled
in Table 3. In addition, the mass of the heavylight mesons
in the chiral limit, see Eq. (
35
), is set to m˚ = (m P + m Ps )/2,
for simplicity the same average is used to define m¯ P in the
Appendix A and in the relations given in Eq. (
53
). The P P ∗φ
axial coupling constant g˜ in Eq. (
39
) can be fixed by
calculating the decay width of D∗+ → D0π + [1], which leads13 to
g ∼ 0.58 and hence g˜ D∗ Dπ ∼ 1.113 GeV. In the bottom
sector we use a different value for g, around 15% smaller,
consistent with the lattice calculation of Ref. [78], where g ∼ 0.51
(or g˜ B¯ ∗ B¯ π ∼ 2.720 GeV) was found. Note that the difference
is consistent with the expected size of heavyquarkflavor
symmetry violations. In addition, there exist sizable SU (
3
)
corrections to the overall size of the P ∗ pole contribution
to both f+ and f0 form factors. Thus, such contribution is
around ∼ 20% smaller for B¯ ∗ B¯s K than for B¯ ∗ B¯ π [32, 38].
According to [79] this suppression is mainly due to a
factor Fπ /FK ∼ 0.83 [22]. We will implement this correction
in the pole contribution to f0 in Eq. (
41
) when the
Goldstone boson is either a kaon or an eta meson (for simplicity,
we also take Fη ≈ FK ), and both in the bottom and charm
sectors.
3.1 Fit to the LQCD+LCSR results in the bottom sector
We are first interested in the B¯ → φ transitions induced by
the b → u flavourchanging current, which include B¯ → π ,
B¯ → η and B¯s → K . The scalar form factors involved in
those transitions can be related to the Omnès matrix through
⎛
⎜⎜
⎝
3 f B¯ 0→π + (s) ⎞
2 0
ff0B0¯Bs0−→→Kη+((ss)) ⎟⎟⎠
=
(
0, 21
)(s) · PB¯
(
0, 21
)(s) .
B
¯
(
48
)
Within the present approach, and considering just rankone
MO polynomials, there are only two undetermined
parameters: the NLO LECs β1P and β2P that appear in the chiral
expansion of the form factors in Eq. (
41
). We fit these
parameters, in the bottom sector, to LQCD (UKQCD [20], HPQCD
[15, 18] and Fermilab Lattice & MILC (to be referred to
as FLMILC for brevity) [19]) and LCSR [12, 13] results
13 Errors on g determined from the decay D∗+ → D0π + are very
small of the order of 1%.
for the scalar form factors in B¯ 0 → π + and B¯s0 → K +
semileptonic decays. Lattice results are not available for
the whole kinematic region accessible in the decays, and
they are restricted to large values of q2 ≥ 17 GeV2, where
momentumdependent discretization and statistical errors are
under control. To constrain the behaviour of the scalar form
factors at small values of q2, we take four LCSR points
(equallyspaced) in the interval q2 = 0 − 6 GeV2 for each
decay.
The UKQCD Collaboration [20] provides data for both
B¯ → π and B¯s → K form factors together with statistical
and systematic correlation matrices for a set of three form
factors computed at different q2 (Tables 8 and 9 of this
reference). In the case of HPQCD [18] B¯s → K and FLMILC
[19] B → π form factors, we have read off four points from
the final extrapolated results (bands) given in these
references, since in both cases, originally only four momentum
configurations (0 → 0, 0 → 1, 0 → √2 and 0 → √3) were
simulated. Finally, we also include in the fit the five B → π
points provided by the HPQCD Collaboration in the erratum
of Ref. [15].
Thus, the χ 2 function reads
χ 2 = (χc2ov)UB¯K→QπCD + (χc2ov)UB¯sK→QCKD + (χ 2)FB¯L→−πMILC
+(χ 2)HB¯P→QπCD + (χ 2)HB¯sP→QCKD
+(χ 2)LB¯C→SRπ + (χ 2)LB¯Cs→SRK ,
where χ 2 is the usual uncorrelated Gaussian merit function,
and χc2ov is defined as
2
χcov =
% f0(qi2) − f0i &(C−1)i j % f0(q 2j) − f0j & .
i, j=1
with the covariance matrix C constructed out the statistical
and systematic correlation matrices and uncertainties given
in Ref. [20]. Here f0(q2) stands for the theoretical scalar
form factor obtained from the MO representation.
The chisquared fit results are
β1B = (0.27 ± 0.12 ± 0.07) GeV
β2B = (0.037 ± 0.004 ± 0.003) GeV−1
with χ 2/do f = 4.2 for 25 degrees of freedom, and a
correlation coefficient 0.999 between the two fitted parameters. The
first set of errors in the parameters is obtained from the
minimization procedure, assuming Gaussian statistics, while the
second one accounts for the uncertainties of the LECs quoted
in Ref. [56] that enter in the definition of the chiral
amplitudes. Such a correlation coefficient so close to 1 indicates
that the considered data can not properly disentangle both
LECs,14 and that different (β1B , β2B ) pairs belonging to the
straight line
(
52
)
in the vicinity of the best fit values (β1B , β2 ), with σ1B,2
B
the corresponding errors, quoted in Eq. (
51
) lead to
similar descriptions of the data (see the dashedblue line in the
right panel of Fig. 10). The scalar form factors obtained
are displayed in Fig. 8. We find a fair description of the
LQCD and LCSR results for the B¯s0 → K + scalar form
factor, while we face some problems for the B¯ 0 → π +
decay. The large value of χ 2/do f reported in Eq. (
51
)
is mainly due to the existing tension between the LQCD
results from different collaborations in this latter decay. The
disagreement between UKQCD and HPQCD B¯ 0 → π +
scalar form factors was already highlighted in the topright
panel of Fig. 23 of the UKQCD work [20], where it is
noted that the HPQCD calculation used only a single lattice
spacing.
In addition, as we discussed before, the B¯ π −scalar form
factor decreases by a factor of five in the q2−range
accessible in the decay, and the LQCD results around qm2ax and the
LCSR predictions in the vicinity of q2 = 0 are not linearly
connected at all. In the current scheme, where only rankone
MO polynomials are being used, this extra needed
curvature should be provided by the q2−dependence of the MO
matrix, , whose behaviour near q2 = 0, far from qm2ax, is not
determined by the behaviour of the amplitudes in the chiral
regime. Indeed, it significantly depends on the highenergy
input.15 This is an unwanted feature, source of systematic
uncertainties. To minimize this problem, in the next
subsection we will perform a combined fit to transitions induced by
the b → u and c → d, s flavourchanging currents. The latter
ones describe D → π and D → K¯ semileptonic decays for
which there exist recent and accurate LQCD determinations
of the scalar form factors. Moreover, in these latter transitions
the q2−ranges accessible in the decays and the form factor
14 This can be easily understood since these LECs enter in the definition
of α0 and α1 in the combinations β1−m2P β2 and β1−m P (m P −2Mφ)β2,
which are identical up to some small SU (
3
) corrections.
15 The results displayed in Fig. 8 might suggest that the present
approach hardly provides enough freedom to simultaneously
accommodate the near q2 = 0 (LCSR) and qm2ax (LQCD) determinations of
the B¯ π scalar form factor. The situation greatly improves when only
the HPQCD, among all LQCD calculations, B¯ π results are considered
in the qm2ax region, being then possible to find an excellent combined
description of the LCSR and HPQCD results with χ2 = 9.65 for a
total of 18 degrees of freedom (see dashedred curve in the right plot of
Fig. 10), which leads to χ2/do f = 0.5. The parameters β1B,2 come out
still to be almost totally correlated as in Eq. (
51
), and moreover they
lie, within great precision, in the straight line of Eq. (
52
), but in the
β B
1 ∼ 0.7 GeV region.
1.6
1.4
1.2
1
¯πB0 0.8
f 0.6
0.4
0.2
0
Fig. 8 Fitted B¯ 0 → π +, B¯s0 → K + (top) and predicted B¯ − → η
(bottom) scalar form factors. Besides the fitted data (UKQCD [20],
HPQCD [15,18], FLMILC [19] and LCSR [12,13]), and for
comparison, predictions from the NLO perturbative QCD approach of Ref. [66]
for the B¯ 0 → π + decay are also shown. Statistical (stat) and statistical
plus systematic (stat & sys) 68% – confident level (CL) bands are also
variations are much limited, becoming thus more relevant the
input provided in the chiral regime.
To finish this subsection, we would like to stress that given
the large value found for χ 2/do f , statistical errors should be
taken with some care. Indeed, one can rather assume some
systematic uncertainties affecting our results, that could be
estimated by considering in the best fit alternatively only the
HPQCD or the UKQCD and the FLMILC sets of
predictions. We will follow this strategy to obtain our final results
for the CKM matrix elements and form factors at q2 = 0
from the combinedfit to charm and bottom decays detailed
in the next subsection.
3.2 Extension to the charm sector and combined fit
Besides the parameter β0 introduced in Eq. (
46
) to account
for the effects on the Ds∗0(2317) state in the c → s decays,
one should also take into account that the LECs β1P and β P
2
depend on the heavy quark mass. The scaling rules given in
Eq. (
42
) can be used to relate the values taken for these LECs
in the bottom (βiB ) and charm sectors (βiD). We will assume
some heavy quark flavor symmetry violations and we will
use
β1D
β B =
1
' m¯ D (1 + δ),
m¯ B
ββ2DB = ' mm¯¯ 33DB (1 − 3δ)
2
(
53
)
shown. The systematic uncertainties are inherited from the errors on
the LECs quoted in Ref. [56], that enter in the definition of the chiral
amplitudes, and are added in quadratures to the statistical uncertainties
to obtain the outer bands. To estimate the systematic uncertainties for
each set of LECs we redo the best fit
where, one should expect the new parameter, δ, to be of the
order QCD/m¯ D. Note that we are correlating the heavy
quark flavor symmetry violations in the LECs β1 and β2.
There is not a good reason for this other than avoiding to
include new free parameters. On the other hand, at the charm
scale, one might also expect sizable corrections to the LO
prediction f0(s) ∼ C × f P /F0 of Eq. (
41
), even more bearing
in mind the large (40 − 50%) heavyquark symmetry
violations inferred from the ratio f B / f D quoted in Table 3. (Note
that at LO in the inverse of the heavy quark mass, this ratio
should scale as (m¯ D/m¯ B )1/2). Thus, we have also introduced
an additional parameter, δ , defined through the replacement
f P → f P × (1 + δ )
(
54
)
when Eq. (
41
) is applied to the c → d, s decays. Thus,
we have three new parameters β0, δ, δ , which in addition to
β1B,2, will be fitted to the LQCD & LCSR results for the scalar
form factors in the B¯ → π , B¯s → K , D → π and D → K¯
semileptonic decays.
First we need to incorporate the c → d, s input into the
merit function χ 2, which was defined in Eq. (
49
) using only
bottom decay results. In the last 10 years, LQCD
computations of the relevant D → π and D → K¯ semileptonic
decay matrix elements have been carried out by the HPQCD
[16,17] and very recently by the ETM [21] Collaborations.
Compared with the former, the latter corrects for some
hypercubic effects, coming from discretization of a quantum field
theory on a lattice with hypercubic symmetry [80], and uses a
large sample of kinematics, not restricted in particular to the
parent D meson at rest, as in the case of the HPQCD
simulation. Moreover, it is argued in Ref. [21] that the restricted
kinematics employed in the simulations of Refs. [16, 17]
may obscure the presence of hypercubic effects in the
lattice data, and these corrections can affect the extrapolation
to the continuum limit in a way that depends on the specific
lattice formulation. This might be one of the sources of the
important discrepancies found between the D → π form
factors reported by the HPQCD and ETM Collaborations in
the region close to qm2ax = (m D − Mπ )2, as can be seen in
the left top panel of Fig. 9.
Here, we prefer to fit to the most recent data together with
the covariance matrices provided by the ETM Collaboration.
This analysis is based on gauge configurations produced with
N f = 2 + 1 + 1 flavors of dynamical quarks at three
different values of lattice spacing, and with pion masses as small
as 210 MeV. Lorentz symmetry breaking due to hypercubic
effects is clearly observed in the ETM data and included in
the decomposition of the current matrix elements in terms
of additional form factors. Those discretization errors have
not been considered in the HPQCD analyses, and for this
reason we have decided to exclude the results of these latter
collaboration in our fits.
The scalar form factors involved in the D → π and D →
K¯ transitions are related to the Omnès matrices displayed in
Figs. 2 and 4 through16 Eqs. (
15
) and (
6
) and (
7
). Hence, the
bottomcharm combined χ 2 now reads
χ 2 = (χc2ov)UB¯ K→QπCD + (χc2ov)UB¯sK→QCKD + (χ 2)FB¯L→−πMILC
+ (χ 2)HB¯ P→QπCD + (χ 2)B¯s →K
HPQCD
+ (χ 2)LB¯C→SRπ + (χ 2)B¯s →K
LCSR
+ (χc2ov)D→π + (χc2ov)D→K¯ ,
(
55
)
where we have added sixteen ETM points, eight for each
of the two D → π and D → K¯ decay modes. Each of
the new eightpoint sets is correlated and the corresponding
covariance matrices17 have been obtained from the authors of
Ref. [21]. Thus, we are fitting five parameters to a total of 43
points. The bestfit results for the five unknown parameters
16 Notice that the particle charges are not specified in the notation used
in Lattice QCD, for instance, D0 → π − in Eq. (
6
) is simplified to
D → π , to be used below and denoted by Dπ in the lattice paper.
17 The D → π and D → K¯ scalar form factor covariancematrices
have troublesome small eigenvalues, as small as 10−6 or even 10−9. Due
to this, the fitting procedure could be easily spoiled since a tiny error
in the fitting function yields a huge χ 2 value (specific examples can be
found in Ref. [81]). We have used the singular value decomposition
(SVD) method to tackle this issue, which is widely used by a number
of lattice groups [82–84].
and their Gaussian correlation matrix are collected in Table 4
and the resulting scalar form factors are shown in Fig. 9.
The results for the bottom scalar form factors are almost
the same as the ones shown in Fig. 8, while the ETM c → d, s
transition form factors are remarkably well described within
the present scheme. As in the former bestfit to only the B¯(s)
results, the large value obtained for χ 2/do f is mainly due to
the existing tension between the LQCD results from different
collaborations in the B¯ → π decay.
Due to the hypercubic effects, there might be
inconsistencies between the ETM and HPQCD analyses for the Dπ
scalar form factor in the region close to qm2ax = (m D − Mπ )2
[21]. As one can see, our result disagrees with the HPQCD
data in that region too. We have checked that if we fit to the
HPQCD instead of the ETM data in the charm sector, the best
fit still tends to coincide with the ETM data. This observation
is important and it seems to indicate that the Lorentz
symmetry breaking effects in a finite volume, due to the hypercubic
artifacts, could be important in the LQCD determination of
the form factors in semileptonic heavytolight decays, as
pointed out in Ref. [21].
The HQFS breaking parameters δ and δ turn out to be
quite correlated and their size is of the order QCD/mc.
As expected, δ presents also a high degree of correlation
with β1B and β B , and on the other hand, the combined fit
2
does not reduce the large correlation between these two
latter LECs, while the central values (errors) quoted for them
in Table 4 are compatible within errors with (significantly
smaller than) those given in Eq. (
51
), and obtained from the
fit only to b → u transitions. In addition, the values quoted
for (β1B , β2B ) in Table 4 perfectly lie in the straight line of
Eq. (
52
), deduced from the fit to only bottom form factors
carried out in the previous Sect. 3.1. Indeed, the straight line
that one can construct with the results of Table 4 in the
(β1B , β2B )−plane is practically indistinguishable from that of
Eq. (
52
). All this can be seen in the left plot of Fig. 10, where
both straight lines are depicted, together with the statistical
68% CL ellipses and the onesigmarectangle bands obtained
by minimizing the merit function given in Eq. (
49
) or
alternatively in Eq. (
55
), and considering only bottom or bottom
and charm scalar form factors, respectively.
In the right plot of Fig. 10, we show the dependence of
χ 2 on β1B for different situations. We display the combined
charmbottom and the bottomonly fits, and in both cases, we
have considered results obtained when all B¯ → π LQCD
form factors or only the HPQCD or the UKQCD and the
FLMILC subsets of results are considered in the fits. The
circles stand for the different bestfit results, accounting for
variations of χ 2 up to one unit from the minimum value18,
while the dashed and solid curves have been obtained by
18 Thus, the ranges marked by the circles show the statistical errors of
β1B in each fit.
0
stat
Fig. 9 Scalar form factors from the D 3, B¯ 3 and (B¯s ) 3 combined fit
(see Table 4 for details). The three bottom panels are similar to those
depicted in Fig. 8, but computed from the results of the combined
bestfit. The four panels in the first two rows show form factors for c → d, s
semileptonic transitions. Only ETM results, corrected for some
hypercubic (discretization) effects [21], have been considered in the fit of
Table 4. For comparison, predictions from the HPQCD [16,17]
Collabrelating β1B and β2B through Eq. (
52
) and minimizing χ 2 with
respect to the other parameters, β0, δ and δ , respectively.
Several conclusions can be extracted from the results shown
in the figure:
• The combined charmbottom analyses (solid lines)
provide large curvatures of χ 2 as a function of β B , hence
1
leading to better determinations of this latter LEC, always
in the 0.2 GeV region, as we also found in Eq. (
49
) from
oration are also displayed. Differences between ETM and HPQCD sets
of D → π and D → K form factors are clearly visible in the vicinity of
qm2ax, in particular for the D → π case. Statistical (stat) and statistical
plus systematic (stat & sys) 68%confident level bands are also given
and are calculated as explained in Fig. 8. Finally, predictions for the
D → η, Ds → K and Ds → η scalar form factors are also shown
the best fit to only the bottom results. A value for β B
1
close to this region, taking into account errors, is also
found from a fit where only the bottom form factors are
considered, but without including the HPQCD B¯ → π
results (dashedgreen curve). Only the dashedred line
(fit only to the bottom results, but without including in
this case the UKQCD and FLMILC B¯ → π scalar form
factors) turns out to be incompatible with the combined
fit presented in Table 4. Thus, we find some arguments to
Table 4 Results from the bottomcharm combined fit, with χ 2 defined
in Eq. (
55
) and a total of 38 degrees of freedom. The first set of errors
in the bestfit parameters is obtained from the minimization procedure,
assuming Gaussian statistics, while the second one accounts for the
uncertainties of the LECs quoted in Ref. [56] that enter in the definition
of the chiral amplitudes. The LECs β0 and β1B (β2B ) are given in units
of GeV (GeV−1)
dχo2f = 2.77
β0
β1B
β2B
δ
δ
Fig. 10 Left plot: statistical 68% CL ellipses and onesigmarectangle
bands in the (β1B , β2B )−plane obtained by minimizing the merit
functions given in Eqs. (
49
) and (
55
). In addition the correlation relation of
Eq. (
52
) is also displayed. A similar relation deduced from the bestfit
results of Table 4 is also shown, though it is hardly distinguishable from
the former one. Right plot: Dependence of χ 2 on β1B for different
situsupport the range of values quoted in Table 4 for the
parameters (β1B , β2B ) that appear in the heavy meson
chiral perturbation theory (HMChPT) expansion of the
scalar form factors at the bottom scale.
• The existing tension between HPQCD, and UKQCD and
FLMILC sets of B¯ → π form factors leads to large
values of χ 2. Thus, as mentioned above, statistical errors
should be taken with some care, and some systematic
uncertainties would need to be considered in derived
quantities, as for instance in the values of the form factors
at q2 = 0 or in the CKM mixing parameters. We note
that this source of systematics also induces variations on
the fitted parameters in Table 4 which range between 50
(β0 and β B ) and 100% (β1B , δ and δ ) of the statistical
2
errors quoted in the table.
bottom & charm [all data]
bottom [all data]
bottom & charm [HPQCD+LCSR in B¯ → π]
bottom [HPQCD+LCSR in B¯ → π]
bottom & charm [UKQCD+FLMILC+LCSR in B¯ → π]
bottom [UKQCD+FLMILC+LCSR in B¯ → π]
2χ
140
120
100
80
60
40
20
ations. The circles stand for the different bestfit results, accounting for
variations of χ 2 up to 1 unit from the minimum value, while the dashed
and solid curves have been obtained by relating β1B and β2B through
Eq. (
52
) and minimizing χ 2 with respect to the other parameters, β0, δ
and δ , respectively
Our predictions for the scalar form factors for the Ds → η,
D → η and Ds → K transitions, for which there are no
lattice results as yet, are also shown in Fig. 9. Note that
transitions involving the η meson in the final state are more
difficult to be evaluated in LQCD simulations. Interestingly, the
D → η scalar form factor in the threechannel (0, 1/2)−case
is largely suppressed, similar to the component regarding the
K → η transition in the strangenesschanging scalar form
factors as shown in Ref. [26].
3.2.1 Further considerations
We have also obtained results using constant and quadratic
MO polynomials. In the first case, the dispersive
representations of the form factors should be matched to the LO chiral
ones, where the terms driven by the β1P and β2P LECs are
dropped out. The first consequence is that bottom and charm
sectors are no longer connected since, in addition, we are
not enforcing the heavyquark scaling law for the decay
constants. To better describe the data, one might perform separate
fits to bottom and charm form factors with free α0 parameters
in Eq. (
34
). Fits obviously are poorer, and moreover, they do
not necessarily provide reliable estimates of the form factors
at qm2ax, since the fitted parameters are obtained after
minimizing a merit function constructed out of data in the whole
q2−range accessible in the decays. For charm decays, the
description of the D → π form factor is acceptable, while
that of the D → K¯ is in comparison worse, mostly because
the s−dependence induced by the Ds∗0(2317) can not be now
modulated by the MO polynomial. In the bottom sector, as
one should expect, the simultaneous description of LQCD
and LCSR form factors in the vicinity of qm2ax and q2 = 0,
respectively, becomes poorer. Indeed, since the LQCD input
has a larger weight in the χ 2 than the LCSR one, the latter
form factors are totally missed by the new predictions, which
now lie below the lower error bands of the LCSR results.
The consideration of quadratic MO polynomials solves
this problem, as shown in Fig. 12 of Appendix B. Indeed, it
is now possible to improve the description of the B¯ → π
LCSR form factors, providing still similar results in the
qm2ax−region, where the LQCD data are available. Thus, for
instance, we get f B¯ →π (0) = 0.248(
10
) using the new fit to
+
be compared with 0.211(
10
) obtained using the parameters
of the fit of Eq. (
51
) (form factors displayed in Fig. 8).
Nevertheless, as we will see in the next subsection, there exist
some other systematic errors, which practically account for
the latter difference, and thus this source of uncertainty will
be considered in the determination of the CKM matrix
element Vub. In addition, though the χ 2/do f obtained with
quadratic MO polynomials is better, it is still large (around
3.7) due to the tension between the B¯ → π LQCD results
from different collaborations. Moreover β1B and β2B are still
fully correlated, and the quadratic terms of the MO
polynomials that multiply the elements i j , (i = 1, 2, 3, j = 2, 3) of
the matrix displayed in Fig. 6 are almost undetermined (see
the large errors in the parameters α2,3 and especially α2,2
given in Table 6). Finally, the centralvalue predictions, that
will be shown in Sect. 3.4, for the form factors above qm2ax
and to moderate energies amenable to be described using the
unitarized coupledchannel chiral approach, are not affected
by the inclusion of quadratic terms in the MO
polynomials, though errors are enhanced. For all of this, we consider
our best estimates for the form factors those obtained using
rankone polynomials.
We do not discuss quadratic terms in the charmsector
because rankone MO polynomials led already to excellent
reproductions of the form factors (see Fig. 9), in part due
(
56
)
(
57
)
(
58
)
(
59
)
(60)
to the smaller q2−range involved in these decays. Moreover
a correct charmbottom combined treatment will require the
matching at nexttonexttoleading order (NNLO) in the
chiral expansion, which is beyond the scope of this work.
Taking advantage that scalar and vector form factors are equal
at q2 = 0, the results of the combined charmbottom fit
presented in the previous subsection can be used to extract the
vector form factor, f+, at q2 = 0 for various semileptonic
decays studied in this work. Moreover, given some
experimental input for the quantity VQq  f+(0), with Qq = bu, cd
or cs, we can extract the corresponding CKM matrix element
using the present MO scheme. Measurements of the
differential distribution d (H → φ¯ ν¯ )/dq2 at q2 = 0 will directly
provide model independent determinations of VQq  f+(0),19
while measurements of the total decay width could be used to
estimate this latter quantity only after relying on some model
for the q2−dependence of f+.
In the charm sector from the fit presented in Table 4 and
Fig. 9, we find
f D→π (0) = 0.585(
35
)stat(
19
)sys1 (
32
)sys2 ,
+
f D→K¯ (0) = 0.765(
30
)stat(
4
)sys1 (
14
)sys2 ,
+
where the first and second sets of errors are similar to those
quoted in Table 4 and account for statistical (propagated from
the 1σ fluctuations of the fitted parameters) and chiral
systematical (propagated from the errors of the LECs that enter
in the computation of the MO matrix) uncertainties,
respectively. The third set of errors (sys2) takes into account the
variations that are produced when in the best fit one
considers alternatively only HPQCD or UKQCD and FLMILC
¯ → π form factors. The results of Eqs. (
56
) and (
57
) are
B
in good agreement with our preliminary estimates reported
in [85], where we fitted only to the charm ETM LQCD form
factors.
In combination with the experimental values
f D→π (0)Vcd  = 0.1426(
19
) ,
+
f D→K¯ (0)Vcs  = 0.7226(
34
) ,
+
Vcd  = 0.244(
22
)
Vcs  = 0.945(
41
)
19 Neglecting the lepton masses.
taken from the report by the Heavy Flavor Averaging Group
(HFLAV) [86], we obtain
for the corresponding CKM matrix elements. The dominant
error is the theoretical one affecting the determination of
the form factors at q2 = 0, within the scheme presented in
this work. As expected, these results nicely agree with those
reported by the ETM Collaboration [21] since we describe
rather well the LQCD scalar form factors calculated in this
latter work. The values of Eq. (
59
) agree within around 1σ
with the average ones20 given in Ref. [1],
Vcd  = 0.220(
5
) ,
Vcs  = 0.995(
16
) .
On the other hand, the test of the secondrow unitarity of the
CKM matrix is satisfied within errors
Vcd 2 + Vc2s  + Vcb2 = 0.95(
9
) ,
where Vcb = 0.0405(
15
) from PDG [1] has been used.
Likewise, in the bottom sector, we obtain from the
combined bottomcharm fit
where we see that the error budget is now dominated by
the inconsistency between HPQCD and UKQCD and
FLMILC sets of results for f0B¯ →π for high q2, above 17 GeV2.
Dropping out the UKQCD and FLMILC sets of results for
B¯ → π , the LCSR and HPQCD results for this transition
can be significantly better described simultaneously, leading
to values of the form factor at q2 = 0 around 0.24 for the
combined charmbottom fit, in the highest edge of the
interval quoted in Eq. (64), and compatible within errors with
the result of 0.26+−00..0043 predicted in Ref. [12] using LCSR.21
However, the description of the B¯s → K and D → π scalar
form factors gets somewhat worse, being thus the situation
unclear.
In principle, based on the above values, the CKM element
Vub could be determined as in the charm sector. However,
the full kinematic region in the bottom case is very broad,
and the experimental determination of f+(0)Vub might
suffer from large systematical uncertainties. A customary way
to extract Vub has been to perform a joint fit to the LQCD
and LCSR theoretical results for f+(q2) and to
measurements of the differential decay width, with Vub being a free
parameter, see, e.g., Refs. [33, 87, 88]. This is not feasible
to us, since we only know the value of the vector form
factor at zero momentum by using the relation f+(0) = f0(0).
20 Determinations from leptonic and semileptonic decays, as well as
from neutrino scattering data in the case of Vcd , are used to obtain the
PDG averages.
21 Fitting only to the b → u data and not considering UKQCD and
FLMILC sets of B¯ → π results, we find f0B¯ →π (0) ∼ 0.27, even in
better agreement with the LCSR determination.
However the latest Belle [9] and BaBar [10] works reported
accurate measurements of the B¯ → π partial branching
fractions in several bins of q2 that are used to extract the f+
form factor shape and the overall normalization determined
by Vub. As a result, Belle and BaBar obtained values of
(9.2 ± 0.3) × 10−4 and (8.7 ± 0.3) × 10−4 for f+(0)Vub,
respectively. Though the latter values were extracted from
direct fits to data, they might be subject to some
systematic uncertainties, since they were obtained using some
specific q2 parameterizations (Becirevic and Kaidalov [89] and
Boyd–Grinstein–Lebed [90] in the Belle and BaBar works,
respectively). Nevertheless, we average both determinations
and we take
f B¯ →π (0)Vub = (8.9 ± 0.3) × 10−4.
+
Using this latter value and our estimate for the form factor at
q2 = 0 given in Eq. (64), we get
103Vub = 4.3(
7
).
103Vub = 4.49(
16
)(
17
)
103Vub = 3.72(
19
)
There exist tensions between the inclusive and exclusive
determinations of Vub [1]:
(inclusive),
(exclusive).
(67)
(68)
(65)
(66)
(69)
and combining both values, Kowalewski and Mannel quote
an average value of
103Vub = 4.09(
39
)
in the PDG review [1], which is in good agreement with our
central Vub result of Eq. (66). We should mention that it
is higher than the typical values obtained from LQCD and
LCSR determinations of the B¯ → π f+(q2) form factor,
combined with measurements of the q2 distribution of the
differential width. Thus, the FLAG review [22] gives an
average value (in 103 units) of 3.67 ± 0.09 ± 0.12. Nevertheless,
this latter value is still compatible, taking into account the
uncertainties, with our result.
These extractions of the CKM elements rely strongly on
the results either from LQCD in the high q2 region or from
LCSR in the vicinity of q2 = 0 (the latter only in the
bottom sector), which are used in the combined fit, and hence
are not ab initio predictions. However, our extractions
incorporate the influence of general Smatrix properties, in the
sense that unitarity and analyticity are implemented in the
MO representation of the scalar form factors. Moreover, one
of the advantages of our approach is that we can make
predictions for the channels related by chiral SU (
3
) symmetry
of light quarks. In some of these channels, the form factors
are difficult for LQCD due to the existence of disconnected
diagrams of quark loops. The D → η, Ds → K , Ds → η
and B¯ → η scalar form factors were already shown in Fig. 9
for the whole kinematical regions accessible in the decays.
On the other hand, their values at q2 = 0 are particularly
important, since they might serve as alternatives to
determine the CKM elements when experimental measurements
of the corresponding differential decay rates become
available. Our predictions for the absolute values of the vector
form factors at q2 = 0 are (we remind once more here that
vector and scalar form factors coincide at q2 = 0)
f D→η(0) = 0.01(
3
)stat(
2
)sys1 (
4
)sys2 ,
 +
f Ds →K (0) = 0.50(
6
)stat(
3
)sys1 (
5
)sys2 ,
 +
For the decay B¯s → K , we find, adding errors in
quadrature, f B¯s →K (0) = 0.30 ± 0.03 in perfect agreement with the
+
results obtained from the LCSR (0.30−+00..0034 [13]) and HPQCD
(0.32 ± 0.06 [18]) analyses, but about 1 sigma above the
LQCD result of the UKQCD Collaboration [20]. The
singlechannel Omnèsimproved constituent quark model study of
Ref. [38] led to 0.297 ± 0.027, which is also in good
agreement with our result.
3.4 Scalar form factors above the qm2ax−region
It is worth recalling here the relation between the results
obtained for the form factors and the scattering amplitudes
used as input of the MO representation. If we focus, for
instance, on the charm form factors, the lightest opencharm
scalar resonance, called D0∗(2400) by the PDG [1], lies in
the (S, I ) = (0, 1/2) sector. In Refs. [43,45,47], two
different states, instead of only one, were claimed to exist in
the energy region around the nominal mass of the D0∗(2400).
These studies were based on chiral symmetry and unitarity.
This complex structure should be reflected in the scattering
regime of the form factors. Indeed, this can be seen in the
first row of panels of Fig. 11, where form factors for different
semileptonic transitions are shown above the qm2ax−region.
As discussed in Sect. 2.3, here we use the O( p2) HMChPT
amplitudes obtained in Refs. [56,62], which also successfully
describe the (0, 1/2) finitevolume energy levels reported in
the recent LQCD simulation of Ref. [60] (see Ref. [52] for
details) and are consistent with the precise LHCb data [63]
for the angular moments of the B− → D+π −π − [42]. These
chiral amplitudes predict the existence of two scalar broad
resonances, instead of only one, with masses around 2.1 and
2.45 GeV, respectively [42,52], which produce some
signatures in the D → π , D → η and Ds → K form factors
at around q2 = 4.4 and 6 GeV2, as can be appreciated in
Fig. 11. The effect of this twostate structure is particularly
visible in the Ds → K form factor. Note that this twostate
structure should have also some influence in the region below
qm2ax, where we have fitted the LQCD data. Below qm2ax, the
sensitivity of the form factors to the details of the two
resonances is however smaller than that of the energy levels
calculated in the scattering region, since the former ones are
given below the lowest threshold, while the latter ones are
available at energies around and above it. Nonetheless, the
success in describing the LQCD results for the D → π scalar
form factor clearly supports the chiral input, and the
predictions deduced from it, used in the current scheme. If better
determined form factors were available in all of the
channels, perhaps the two state structure for the D0∗(2400) could
be further and more accurately tested.
A similar pattern is found in the bottom sector [42,52],
as expected from the approximate heavyflavor symmetry of
QCD. The twostate structure is clearly visible, more than
that in the charm sector, in the corresponding form factors
(three bottom plots of Fig. 11), and it has a certain impact
in the form factors close to qm2ax, where LQCD results are
available.
In the charm (S, I ) = (
1, 0
) sector the effect of the
narrow Ds∗0(2317) resonance, which is the SU (
3
) flavor partner
of the lighter one of the two D0∗ states, predicted by the
unitarized NLO chiral amplitudes [42,52], is clearly visible in
the scalar D → K¯ and Ds → η form factors, and it fully
dominates these form factors in the vicinity of the pole, as
can be seen in the second row of panels of Fig. 11. Indeed,
this state also influences the D → K¯ form factor below
(near) qm2ax, where the LQCD results are available22, and the
excellent description of the ETM results gives clear support
to the coupledchannel MO representation of the D → K¯
and Ds → η scalar form factors derived in this work.
4 Summary and outlook
We have studied the scalar form factors that appear in
semileptonic heavy meson decays induced by the
flavourchanging b → u and c → d, s transitions using the MO
formalism. The coupledchannel effects, due to rescattering
of the H φ (H = D, B¯ ) system, with definite strangeness
and isospin, are taken into account by solving coupled
integral MO equations. We constrain the subtraction constants
in the MO polynomials, which encodes the zeros of the form
22 Indeed, the existence of the Ds∗0(2317) was suggested in [34] by
fitting the single channel MO representation of the D → K¯ scalar form
factor, constructed out the unitarized LO chiral elastic D K amplitude,
to LQCD results of the scalar form factor below qm2ax.
stat
2
0
0
HPQCD
ETM
HPQCD
ETM
1
2
FLMILC
pQCD
UKQCD
HPQCD
LCSR
Fig. 11 Scalar form factors for different b → u and c → d, s
transitions. They have been computed in this work using the MO
matrices derived in Sect. 2.3 from the NLO HMChPT amplitudes of Refs.
[56,62], and the LECs, compiled in Table 4, obtained from a fit to LQCD
factors, thanks to lightquark chiral SU (
3
) and heavyflavor
symmetries.
The H φ interactions used as input of the MO equations
are well determined in the chiral regime and are taken from
previous work. In addition, some reasonably behaviors of the
amplitudes at high energies are imposed, while appropriate
heavyflavor scaling rules are used to relate bottom and charm
form factors. We fit our MO representation of the scalar form
factors to the latest c → d, s and b → u LQCD and b → u
LCSR results and determine all the involved parameters, in
particular the two LECs (β1P and β2P ) that appear at NLO
and LCSR results below qm2ax. Statistical (stat) and statistical plus
systematic (stat & sys) 68%confident level bands are also given and are
calculated as explained in Fig. 8
in the chiral expansion of the scalar and vector form
factors near qm2ax, which are determined in this work for first
time. We describe the LQCD and LCSR results rather well,
and in combination with experimental results and using that
f0(0) = f+(0), we have also extracted the Vub, Vcd  and
Vcs  CKM elements, which turn out to be in good agreement
with previous determinations from exclusive decays.
We would like to stress that we describe extremely well the
recent ETM D → π scalar form factor, which largely
deviates from the previous determination by the HPQCD
Collaboration, providing an indication that the Lorentz symmetry
breaking effects in a finite volume, due to the hypercubic
artifacts, could be important in the LQCD determination of
the form factors in semileptonic heavytolight decays, as
claimed in Ref. [21]. As it is also pointed out in the
previous reference, this is a very important issue, which requires
further investigations, since it might become particularly
relevant in the case of the determination of the form factors
governing semileptonic B¯ −meson decays into lighter mesons.
We have also predicted the scalar form factors, which
are in the same strangenessisospin multiplets as the fitted
D → π , D → K¯ , B¯ → π and B¯s → K ones. Our
prediction of the form factors in such channels (D → η, Ds → K ,
Ds → η, and B → η) are difficult for LQCD simulations
due to the existence of disconnected diagrams. These form
factors are related to the differential decay rates of different
semileptonic heavy meson decays and hence provide
alternatives to determine the CKM elements with the help of future
experimental measurements.
Moreover, we also find that the D → η scalar form factor
is largely suppressed compared to the other two components
(D → π , D → K¯ ) in the threechannel (0, 1/2)−multiplet,
which is similar to what occurs for the K → η
strangenesschanging scalar form factor in Ref. [26].
Our determination of the form factors has the advantage
that the constraints from unitarity and analyticity of the
Smatrix have been taken into account, as well as the
stateoftheart H φ chiral amplitudes. Thus, our predictions for
the flavourchanging b → u and c → d, s scalar form
factors above the q2−region accessible in the semileptonic
decays, depicted in Fig. 11, should be quite accurate23 and
constitute one of the most important findings of the current
research. Indeed, we have shown how the form factors in
this region reflect details of the chiral dynamics that
govern the H φ amplitudes, and that give rise to a new paradigm
for heavylight meson spectroscopy [42] which questions the
traditional qq¯ constituent quark model interpretation, at least
in the scalar sector.
As an outlook, the scheme presented here will also be
useful to explore the H φ interactions by using the lattice data
for the scalar form factors in semileptonic decays of B¯ or D
mesons. As pointed out in Ref. [73], more data are needed
to fix the LECs in the NNLO potentials. Since the dispersive
calculation of Dφ and B¯ φ scalar form factors depend on the
scattering amplitudes of these systems, the LQCD results for
the form factors can be used to mitigate the lack of data and
help in the determination of the new unknown LECs.
One might also try to extend the MO representation to a
formalism in a finite volume with unphysical quark masses,
23 Note that for the moderate q2−values shown in Fig. 11, the form
factors are largely insensitive to the highenergy input in the MO
dispersion relation, and they are almost entirely dominated by the lowenergy
(chiral) amplitudes.
such that comparisons to the discretized lattice data could be
directly undertaken. On the other hand, the chiral matching of
the form factors can be carried out at higher order to take into
account the expected sizable corrections in SU (
3
) HMChPT.
Moreover, this improved matching will in practice suppose
to perform additional subtractions in the dispersive
representations of the form factors, and it should reduce the
importance of the highenergy input used for the H φ amplitudes.
The high energy input turns out to be essential to describe
the scalar B¯ → π form factor near q2 = 0, and it represents
one of the major limitations of the current approach.
Both improvements would lead to a more precise and
modelindependent determination of the CKM matrix
elements related to the heavytolight transitions.
Acknowledgements DLY would like to thank YunHua Chen and
Johanna Daub for helpful discussions on solving the MO problem. We
would like to thank the authors of Ref. [21] for providing us the
covariance matrices and J. Gegelia for comments on the manuscript. P. F.S.
acknowledges financial support from the Ayudas para contratos
predoctorales para la formació de doctores program (BES2015072049)
from the Spanish MINECO and ESF. This research is supported by the
Spanish Ministerio de Economía y Competitividad and the European
Regional Development Fund, under contracts
FIS201451948C21P, FIS201784038C21P and SEV20140398, by Generalitat
Valenciana under contract PROMETEOII/2014/0068, by the National Natural
Science Foundation of China (NSFC) under Grant No. 11747601, by
NSFC and DFG though funds provided to the SinoGerman CRC 110
“Symmetries and the Emergence of Structure in QCD” (NSFC Grant
No. 11621131001), by the Thousand Talents Plan for Young
Professionals, by the CAS Key Research Program of Frontier Sciences under
Grant No. QYZDBSSWSYS013, and by the CAS Center for
Excellence in Particle Physics (CCEPP).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Heavyquark mass scaling of the LECs in
the Dφ interactions
Thanks to heavy quark symmetry, the Dφ and B¯ φ
interactions share the same effective Lagrangian with the
correspondence D ↔ B¯ . The NLO LECs hi ’s scale as [56,62,91]
h0,1,2,3 ∼ m Q ,
equivalently,
1
h4,5 ∼ m Q ,
B m¯ B h0D,1,2,3 ,
h0,1,2,3 = m¯ D
B m¯ D h4D,5 .
h4,5 = m¯ B
Here m¯ D (m¯ B ) is the average of the physical masses of
the D (B¯ ) and Ds (B¯s ) mesons. In addition, in the
unita(A1)
(A2)
Table 5 LECs and subtraction
constants used in this work to
compute the UChPT Dφ and
B¯ φ amplitudes
Dφ
B¯ φ
a(μ = 1 GeV)
rized ChPT (UChPT) amplitudes there appears one
subtraction constant, a(μ), with μ = 1 GeV the scale introduced in
dimensional regularization. In the (S, I ) = (0, 1/2) channel,
the subtraction constants in the charm, denoted by a D (μ),
and in the bottom, denoted by a B (μ), sectors are related as
follows [45, 52]:
1. First, given the phenomenological value of a D (μ), a
sharpcutoff, qmax, is determined by requiring the
dimensionally and the sharpcutoff regularized Ds K loop
functions to be equal at threshold (see Eq. (
52
) of Ref. [92]).
This cutoff turns out to be qmax = 0.72−+00..0067 GeV.
2. Next, qmax is used to determine a B (μ) by requiring that
the dimensionally and the sharpcutoff regularized B¯s K
loop functions to be also equal at threshold.
LECs and subtraction constants for Dφ and B¯ φ interactions
used in this work are collected in Table 5. Those in the charm
sector are taken from Ref. [56].
Appendix B: Bottom form factors and quadratic MO
polynomials
We show in Fig. 12, the scalar B¯ → π, η and B¯s → K form
factors obtained using ranktwo MO polynomials.
Specifically, we replace Eq. (
34
) by
P (s) = α0 + α1 s + α2 s2 ,
where α2, together with β1B,2, are fitted to the merit function
defined in Eq. (
49
). (Note that α0,1 are still expressed in terms
of β1B,2.) Best fit results are compiled in Table 6.
(B1)
10
q2 [GeV2]
Fig. 12 Same as Fig. 8, but using a quadratic MO polynomial (see Eq. (B1)), and parameters given in Table 6
Table 6 Results from the fit to the scalar B¯ → π and B¯s → K LQCD
& LCSR form factors using ranktwo MO polynomials (see Eq. (B1)),
and with χ 2 defined in Eq. (
49
). There is a total of 22 degrees of
freedom and the bestfit gives χ 2/do f = 3.7. The LECs β1B , β2B and α2,i
are given in units of GeV, GeV−1 and GeV−4, respectively
(β2B × 10)
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