R(D ∗), V cb , and the Heavy Quark Symmetry relations between form factors
HJE
), jVcbj, and the Heavy Quark Symmetry
Dante Bigi 0 1
Paolo Gambino 0 1
Stefan Schacht 0 1
0 I10125 Torino , Italy
1 Dipartimento di Fisica, Universita di Torino & INFN , Sezione di Torino
Stringent relations between the B( ) ! D( ) form factors exist in the heavy quark limit and the leading symmetry breaking corrections are known. We reconsider their uncertainty and role in the analysis of recent Belle data for B modelindependent parametrizations and in the related prediction of R(D( )). We nd jVcbj = 41:5(1:3) 10 3 and jVcbj = 40:6(+11::23) 10 3 using input from Light Cone Sum Rules,
Heavy Quark Physics; Quark Masses and SM Parameters

R(D
! D( )`
with
and R(D ) = 0:260(8).
1 Introduction
2
3
4
5
Uncertainty of the relations between form factors
Strong unitarity bounds for B ! D
form factors
Fits to B ! D `
data
Calculation of R(D )
5.1
5.2
5.3
The standard way: normalizing P1 to A1
Normalizing P1 to V1
Enforcing a constraint at q2 = 0
6
Conclusions
1
Introduction
Among the various avour observables showing a signi cant deviation from their Standard
Model (SM) predictions ( avour anomalies ), those related to treelevel semileptonic B
decays have received remarkable attention, as they potentially represent clean signals of
New Physics. In addition to the longstanding discrepancy between the determination of
the CKM element jVcbj from inclusive and exclusive semileptonic B decays, there is a
anomaly [1] in the ratios of exclusive semileptonic decays to tau and to light leptons,
R(D( )) = B(B ! D( )
)
;
B(B ! D( )` )
which could indicate a violation of lepton universality, and hence a clear departure from
the SM. A full understanding of these relatively simple B decays is a necessary condition
to pro t from the potential of the BelleII and LHCb experiments in the search for New
Physics, independently of these anomalies.
For what concerns the B ! D channel, recent progress in the determination of the
relevant form factors in lattice QCD [2, 3] and a new analysis of the q2 spectrum in
B ! D`
by the Belle Collaboration [4] have resulted [5, 6] in a more precise value of
jVcbj in reasonable agreement with the inclusive determination [7, 8] and in the precise
prediction R(D) = 0:299(3).
The situation is not yet so favourable in the B ! D channel, which has so far provided
the most accurate exclusive determination of jVcbj. First, unquenched lattice calculations
of the relevant form factor [9, 10] are still limited to the zerorecoil point, where the
jVcbj = 39:05(75) 10 3 [1], the
parametrization [13] prefer a much higher value [14{16],
ts performed with the BoydGrinsteinLebed (BGL)
jVcbj = (38:2
1:5) 10 3;
jVcbj = (41:7
parametrization [11], and have published results in terms of the few parameters of this
parametrization. Only recently, Belle has published a preliminary analysis [12] which, for
the rst time, includes deconvoluted kinematic and angular distributions, without relying
on a particular parametrization of the form factors.
The new Belle results have allowed for ts of the experimental spectra with di erent
CLN t leads to
parameterizations, with surprising consequences on the resulting value of jVcbj: while a
HJEP1(207)6
well compatible with the most recent inclusive result, jVcbj = 42:00(63) 10 3 [8].
As we have emphasized in ref. [14], this strong dependence of jVcbj on the
parameterization should be interpreted with great care because i) it refers to a speci c set of data
and the large discrepancy between eqs. (1.2) and (1.3) may not carry on to other sets of
data; ii) the physical information encoded in the CLN and BGL parametrizations are not
equivalent. Although they are grounded in the same foundations (analyticity, crossing
symmetry, operator product expansion), the CLN parametrization makes use of Heavy Quark
E ective Theory (HQET) relations between the B( )
! D( ) form factors in various ways
in order to reduce the number of independent parameters. Indeed, Heavy Quark Symmetry
requires all of these form factors to be proportional to the IsgurWise function, and the
leading symmetry breaking corrections of O( s; =mc;b) are known [17{19]. However, the
residual uncertainty is not negligible and should be taken into account in the analysis of
experimental data. There are also a few precise lattice QCD calculations which test and
complement these relations and should be taken into account.
The main purpose of the present paper is to investigate to which extent the Heavy
Quark Symmetry relations between the form factors a ect the results of our previous
analysis, once their uncertainty is properly accounted for. The methodology developed
to this end will then be applied to the calculation of R(D ), where the only available
information on the scalar form factor comes from the form factor relations.
In determining the uncertainty of the HQET relations between form factors we will
use the form factor ratios and their uncertainties in deriving strong unitarity bounds on
the coe cients of the BGL parametrization. We will then use these bounds directly in
the t to experimental data, without deriving a simpli ed parametrization like in ref. [11].
Finally, we will apply the results of our ts to the calculation of R(D ).
Our paper is organized as follows: in the next section we discuss the uncertainties
due to higher order e ects in the HQET relations between form factors. In section 3 we
compute the strong unitarity bounds on the form factors that enter the B ! D ` decay
rate taking into account their uncertainties. In section 4 we discuss our new
ts to the
preliminary Belle data which incorporate the strong unitarity conditions. In section 5 we
Uncertainty of the relations between form factors
As explained in refs. [11, 13] the unitarity constraints on the parameters of the zexpansion
can be made stronger by adding other hadronic channels which couple to c b currents.
While in general this would require nonperturbative information on each form factor, in
the case of the B( ) ! D( ) transitions the form factors are all related by Heavy Quark
Symmetry, which can be used to simplify the task. These transitions are described by a
total of 20 helicity amplitudes, which provide an appropriate basis of form factors. In the
following we will adopt the notation of ref. [11]  see in particular eqs. (A.3){(A.6), in
which all form factors reduce to the IsgurWise function (w) in the heavy quark limit. In
this notation S1 3 couple with a scalar charmbottom current, P1 3 with a pseudoscalar
current, V1 7(A1 7) with a vector (axialvector) current. For later convenience, we also
provide in table 1 the relation with the notation of ref. [13].
The HQET ensures that the form factor Fi(w) (here w = v v0, with v and v0 the
fourvelocities of the incoming and outgoing mesons) can be expanded in inverse powers of
the heavy quark masses and in s, which at the NexttoLeading order (NLO) results in
Fi(w) = (w) h1 + ci s
s + cib b + cic c + : : : ;
i
where b;c =
s(pmcmb)
=2mb;c, the strong coupling is typically evaluated at
0:26, and the dots represent higher order corrections. We recall that the
pmcmb with
IsgurWise function (w) is normalized to 1 at zero recoil, (1) = 1. Some of the ratios are
known at O( 0 s2) but the extra corrections are very small [24].
We will follow here the calculation of ref. [20] which updates those employed in the
CLN paper. In particular, we will adopt the values of quark masses and of the subleading
parameters given there,
f+ =
1 + r
2pr V1


B
! D
g^ =
1
mB pr V5
f^ =
mB (1 + w)pr A2
F^1 = m2B (1 + w) (1
r) prA6

F^2 =
1 + r
p
r
P2
g =
B ! D
1
mBpr V4
f = mBpr(1 + w)A1 ;
the more common notation f0 = f0BGL=(m2B
m2D), used, e.g., in ref. [5].
As the IsgurWise function cancels out in the ratios of form factors, the latter can be
computed more accurately in the heavy quark expansion. The central values of the ratios
of form factors Fj to V1 computed in this way and expanded in w1 = w
1,
Fj (w)
V1(w)
=Aj 1 + Bj w1 + Cj w12 + Dj w13 + : : : ;
are given in table 2, which updates table A.1 of ref. [11] and has very similar results.
There is no obvious way to estimate the size of the higher order corrections to the NLO
HQET expressions. Parametrically they are O( s2), O( s c), and O( c2), where roughly
(2.3)
(2.4)
2
s
s c
2
c
6%;
{ 4 {
HJEP1(207)6
which introduce a link between form factors protected by Luke's theorem and others which
are not protected. As an e ect, the 1=m corrections tend to be smaller in V1;2;3 and P1;2;3 as
well, as a sort of indirect Luke's protection. Finally, there are the following exact relations
between form factors at zerorecoil (w = 1):
S1;2;3(wmax) = V1;2;3(wmax);
P1;2;3(wmax) = A5;6;7(wmax)
S2(1) = S3(1);
A1(1) = A5(1);
A2(1) = A6(1);
A3(1) = A4(1) = A7(1):
but the choice of mc or of the scale s can easily change this estimate. Most importantly,
the coe cients in front of these parameters can enhance or suppress signi cantly their
contribution. For instance, the perturbative expansion is actually an expansion in
s=4
and the twoloop is generally enhanced by 0
9. In the following we will mostly worry
about power corrections.
It is useful to recall that several of the form factors do not receive NLO power
corrections at zero recoil because of Luke's theorem [18]. In particular, all of the scalar and
axialvector form factors, S1 3 and A1 7, do not receive 1=m corrections at zero recoil.
There are also exact kinematic relations between the (pseudo)scalar and (axial)vector form
factors at maximal recoil w = wmax (corresponding to q2 = 0)
In some cases, such as V1, the leading power corrections at zerorecoil are known to be
suppressed [25].
The actual pattern of the NLO HQET corrections re ects these two qualitative
suppressions: the form factors protected directly or indirectly by Luke's theorem receive small
or moderate power corrections over the whole w range (1
while the others (V4;5;6;7) are a ected by leading power corrections as large as 50%. The
magnitude of the coe
cients of b;c reaches 2.1. The total NLO correction is almost 60%
in V6=V1, see table 2. As Luke's theorem does not protect the form factors from 1=m2
corrections, it is therefore natural to expect 1=m2 corrections of order 10{20%, and one
cannot exclude that occasionally they can be even larger.
The comparison with recent lattice QCD results is instructive, even though it is limited
to the few cases for which the form factors have been computed at zero or small recoil.
Considering only unquenched lattice results, we average those by the Fermilab/MILC and
HPQCD collaborations [2, 3, 9, 10], neglecting correlations between their results. We also
mention that there is some tension between the preliminary value of A1(1) = 0:857(41) by
HPQCD and the result of Fermilab/MILC, A1(1) = 0:906(13). Incidentally we note that
the rst value agrees well with the heavy quark sum rule estimate of ref. [26]. The results
w < 1:59(1:51) for B ! D( )),
{ 5 {
(2.5)
(2.6)
S1
S2
S3
P1
P2
P3
1.0208
1.0208
0:0436
0:0749
0:0710
0:2164
0:0949
0:2490
0
0:2251
0:2651
0:1492
0:0440
0:1835
0:1821
0:0704
0:0280
0:0629
0:0009
0:3488
0:2548
0:0528
0:0201
0:0846
0:1903
0:0026
0:0034
0:0030
0
0:0000
0:0000
0:0012
0:0014
0:0009
0:0011
0:0580
0:0074
0:0969
0:1475
0:2944
0:0978
0:0942
0:0105
0:0418
0:0947
0:0007
0:0009
0:0008
0
at or near zero recoil are
from which it follows that
S1(w) = 1:027(8)
V1(w) = 1:053(8)
A1(1) = 0:902(12) ;
1:154(32)(w
1:236(33)(w
Notice that in the case of S1=V1 both numerator and denominator have been computed
at small recoil by the Fermilab/MILC and HPQCD collaborations, and we therefore have
also a lattice determination of the slope of the ratio.
{ 6 {
On the other hand, the HQET calculation at NLO of ref. [20] gives
S1(w)
V1(w) HQET
A1(1)
V1(1) HQET
S1(1)
where the errors represent only the parametric uncertainty on mb, s and the QCD sum
HJEP1(207)6
rules parameters.
Comparing the zerorecoil values of the ratios in eqs. (2.8) to those in eqs. (2.9) one
observes deviations between 5% and 13%, which are obviously due to higher order corrections
unaccounted for in eq. (2.9). In all cases the deviation is larger than the NLO correction.
While it is quite possible that lattice uncertainties are somewhat underestimated, here we
are not interested in a precision determination. What matters here is that the size of these
deviations is consistent with our discussion above. The slope of the ratio S1=V1 computed
on the lattice has a di erent sign from the one in (2.9) and their di erence induces a 6%
shift at maximal w. However, since S1=V1 = 1 at maximal recoil, it is not surprising that
higher order corrections are moderate in this case.
In conclusion, higher order corrections to the form factor ratios computed in HQET
at NLO are generally sizeable and can naturally be of the order of 10{20%.1
3
Strong unitarity bounds for B
D
form factors
!
In the following we refer to the setup based on [13] which we have employed in [14] to
perform a t to the recent Belle B ! D ` di erential distributions. In this framework
the generic form factor Fi (already in CLN notation) can be expressed as
Fi(w) =
pi(w)
Bi(z) i(z)
N
X a(ni)zn
n=0
1The CLN form factors Fi we consider are helicity amplitudes which are linear combinations of the
form factors in terms of which the matrix elements are decomposed. This sometimes leads to a correlation
between numerator and denominator in the ratios of helicity amplitudes, which could a ect our error
estimates. There are only a few such cases among the ratios we employ in this paper: S1=V1, P1=A1,
A5=A1, P1=A5. The correlation is maximal at zero recoil where A1 = A5 and one has
V1
S1 = 1 +
0:48. This suggests that higher order corrections are somewhat suppressed, as
in fact we found by comparing with LQCD above. In the second line the NLO corrections are sizeable despite
the suppression factor, and also here one can naturally expect NNLO corrections between 10% and 20%.
(2.9)
(3.1)
B D
B D
Type
Mass (GeV)
Method
Decay const.(GeV)
resonance as it is very close to threshold and its value is very
uncertain. Predictions for the decay constant of the second 0
resonance are also very uncertain
and we do not include them here.
where z = ( w + 1
p
p2)=(pw + 1 + p2) and the prefactors pi(w) are the ratios between
helicity amplitudes in the CLN and BGL notations which can be read o
series in z in (3.1) is truncated at power N and we will set N = 2 from the outset, which
is su cient at the present of level accuracy as 0 < z < 0:056 in the physical region for
semileptonic B ! D decays to massless leptons.
The Blaschke factors, Bi(z), take into account the subthreshold Bc resonances with
the same quantum numbers as the current involved in the de nition of Fi. As the exact
location of the threshold (mB( ) + mD( ) )2 depends on the particular B( )
! D( ) channel,
Bi(z) may di er even between form factors with the same quantum numbers. We will
employ the resonances given in table 3. Finally, the outer functions i(z) can be read from
eq. (4.23) and tables I and IV of ref. [13] and are given explicitly in a few cases in [5, 14].
We will use nI = 2:6 for the number of spectator quarks (three), decreased by a large
and conservative SU(3) breaking factor. The other inputs we use are given in table 4 (all
uncertainties are small and can be neglected), where the ~ are the
constants after taking
into account the oneparticle exchanges, see [5, 11].
Analyticity ensures that the coe cients of the zexpansion (3.1) for the form factor Fi
satisfy the weak unitarity condition
but there are a number of two body channels (BD; BD ; B D; B D ; b c; : : : ) with the
right quantum numbers, as well as higher multiplicity channels, that give positive
contriN
n=0
X(a(ni))2 < 1;
{ 8 {
[27{29]
L
1+ are needed for the scalar and pseudoscalar formfactors [13]; they are related to 0+ and 0 ,
respectively.
numbers 1+,
butions to the absorptive part to the twopoint function and can strengthen the unitarity
bound on the coe cients of each form factor. For instance, the form factors A1;5 which
appear in the B ! D ` decays both contribute to the same unitarity sum with quantum
aA1 2
n
+ aA5 2
n
< 1:
However, the (strong ) unitarity sums including all the B( )
numbers 0+, 0 , 1 , 1+ are
! D( ) channels with quantum
Input
Now we can use the relations between the 20 form factors we have presented in the previous
section to obtain constraints on the coe cients of any speci c form factor Fi [5, 11, 13]. It
is su cient to replace Fj by (Fj =Fi) Fi and expand the product in powers of z to reexpress
each coe cient anFj in terms of a linear combination of the anFi . In the case N = 2 which is
relevant here, each unitarity sum can then be reduced to a quadratic form in a0Fi , a1Fi , a2Fi ,
and each unitarity condition in eqs. (3.4) represents an ellipsoid in the (a0Fi , a1Fi , a2Fi ) space.
To take into account the uncertainties in the relations between form factors we generate
replicas of the set of ratios Fj =Fi which satisfy the kinematic relations of eqs. (2.5), (2.6)
and incorporate the lattice QCD results of eqs. (2.8) within their uncertainties. Each
replica must also have all of the ratios contained within a band around their central values
computed at NLO in HQET as presented in the previous section, improved whenever
{ 9 {
3
X
N
X
possible with existing lattice data. At zerorecoil the band has a width corresponding
to the maximum between 25% and 15% + 2 HQET. At the endpoint, corresponding to
q
2 = 0, the width is slightly larger and corresponds to the maximum between 30% and
20% + 2 HQET. Here HQET is the total parametric relative uncertainty of the NLO HQET
calculation, obtained combining in quadrature the uncertainty from the QCD sum rule
parameters, mb, and s
.
Another condition that the replicas (Fj =Fi) must comply with is that the coe cients
a(nFj) of the form factor Fj , computed by expanding in powers of z the expression
Fj (w) = (Fj =Fi) (Fi=V1) V exp(w)
1
(3.5)
satisfy weak unitarity, i.e. eq. (3.2). Here V exp(w) is the result of the t to B ! D`
1
experimental data and lattice results performed in [5]. The replicas (Fj =Fi) are acceptable
if unitarity is satis ed for values of coe cients of V1exp(w) within 3 s from their central
values. Each replica therefore represents a viable model of the form factors. An example
of a few replicas of the ratios S1;2=A1 passing all tests is shown in gure 1.
We will be primarily interested in the four form factors which enter the B ! D `
decays, namely A1; A5, V4, and P1. The latter contributes only for massive nal leptons.
Each set of replicas of Fi=A1 gives rise to four ellipsoids in the (a0A1 ; a1A1 ; a2A1 ) space,
corresponding to the four possible conditions in eq. (3.4). As A1(1) is known relatively well
from lattice QCD calculations, we can x a0A1 and obtain 4 ellipses in the (a1A1 ; a2A1 ) plane.
Samples of such ellipses from the S and V sectors are shown in
gure 2: there is very little
sign of correlation between a1A1 , a2A1 , but the regions identi ed in the two cases are
similar. We have repeated the same procedure with a large number of replicas. The envelope
formed by all the ellipses represents the allowed region in the (a1A1 ; a2A1 ) plane and is shown
in gure 3. It is quite remarkable that the allowed regions are very similar for the S, P, V
channels, while the A channel is less constraining. The intersection of the S, P, V and A
channels is the allowed region we will consider in the following.
In the same way we have derived bounds in the (a1A5 ; a2A5 ) plane. Indeed, A5(1) is
xed by the same lattice calculations which x A1(1). The nal results are also shown in
0.6
0.4
0.2
1
A2
gure 3. The case of V4 is slightly di erent because there is no lattice calculation that xes
a0V4 . In principle one should keep the threedimensional envelope of all ellipsoids. However,
in line with the previous discussion, we can assume that V4(1) is within about 30% from
the values it takes when one uses A1(1) or V1(1) together with HQET form factor ratios.
This leads to 0:0209 < aV4 < 0:0440. The bounds in the (a1V4 ; a2V4 ) plane depend little on
0
the exact value of a0V4 in that range, besides being anyway much weaker than those on
the coe cients of A1;5. Therefore, also in this case we obtain a twodimensional allowed
region, shown in
gure 3. The case of P1 is very similar to that of V4 and one similarly
nds 0:041 < aP1 < 0:089 and then a twodimensional allowed region in the (a1P1 ; a2P1 ).2
0
Two comments are in order at this point. First, the weak or absent correlation between
a1 and a2 that we observe in most cases does not imply the absence of a strong correlation
between slope and curvature of the form factors when they are expressed in terms of the
variable w. The latter was observed long ago in refs. [11, 13] and is a simple consequence
of the change of variable from z to w and of the outer functions structure, combined with
the weak unitarity bounds on a1;2. Indeed, if we proceed as in ref. [11], we con rm their
bounds on slope and curvature of V1. The only exception are the constraints from the
vector channel, which we nd more constraining than in [11].3
Second, we see no point in modifying the parametrization to include these stronger
unitarity bounds. The bounds we have found should be used directly in ts to experimental
and lattice data based on the BGL parametrization. In the future, when new lattice
information on the slopes of these form factors will become available, the bounds can be
simpli ed; they will become onedimensional bounds on a2Fi only.
2The twodimensional numerical regions are available from the authors upon request.
3We traced the origin of the discrepancy to the exponent of ( j2
(w + 1)=2) in the denominator of the
third row in their eq. (5) (sum over j = 4
7) which should be 4 instead of 5. The main results of [11] are
una ected.
1.0
0.5
(S), pseudoscalar (P), vector (V) and axialvector (A) channels.
4
Fits to B
D `
data
!
We will now employ the results of the previous section in a t to the available experimental
data for B ! D ` , in order to illustrate the relevance of strong unitarity bounds in the
present situation.
To this end, we repeat here the analysis of ref. [14], based on the
preliminary Belle data of [12], and refer to [14] for all the details. The only additional
piece of data we will include in the t is the HFLAV average for the branching ratio of
B0
! D +
where we added the errors in quadrature.
Combining it with the total lifetime
B0 = (152:0
0:4) 10 14s [
27
], we get a rather precise value for the total width of this
decay. The above branching ratio can be compared with
B(B0
! D +
l l) = 0:0495
0:0025
reported in [12]. One would expect the lower value in (4.1) to drive the t towards slightly
lower values of jVcbj but we will see that the precision of the new input changes the t
in an unexpected way.
We stress that the branching ratio is, to good approximation,
independent of the parametrization of the form factors used in the experimental analyses
and it is therefore the only piece of data that we can use from older experimental results. We
will also neglect all correlations of the total width with the binned angular and kinematic
distributions included in the t. For what concerns the lattice determination of the form
BGL Fit: Data + lattice
Data + lattice + LCSR
Data + lattice
Data + lattice + LCSR
unitarity
2=dof
jVcbj
aA1
0
aA1
1
aA1
2
aA5
1
aA5
2
aV4
0
aV4
1
aV4
2
weak
28:2=33
unitarity constraints. In the BGL ts a0A5 is related to the value of a0A1 , a0A5 = 0:1675 a0A1 .
factor at zero recoil, A1(1), we will use the average given in eq. (2.7), which di ers slightly
from the value employed in ref. [14].
The results of the constrained t are shown in table 5, where we consider ts in the BGL
parametrization with weak and strong unitarity bounds, with and without the inclusion of
the constraints computed with Light Cone Sum Rules at w = wmax in [36]:
A1(wmax) = 0:65(18);
R1(wmax) = 1:32(4);
R2(wmax) = 0:91(17);
(4.2)
where
V4(w)
A1(w)
;
R1(w) =
R2(w) =
w
w
r
1
1
1
w
r A5(w)
r A1(w)
:
We have also performed
ts with the CLN parametrization (with free parameters
A1(1); 2; R1(1); R2(1)) in the same way as in [14]. We obtain jVcbj = 0:0393(12) ( 2=dof =
35:4=37) without the LCSR and jVcbj = 0:0392(12) ( 2=dof = 35:9=40) with the LCSR.
As expected, the di erence between the values of jVcbj obtained with the BGL and CLN
parametrization is reduced by the use of strong unitarity bounds, but it remains as large
as 3.5{5%, depending on whether LCSR results are included or not.
Comparing the
ts in table 5 with those in ref. [14] we note that the inclusion of
the world average for the branching ratio has a signi cant impact on jVcbj: the central
value increases by 1.2 to 1.7% and the error is reduced by 10{20%. Using the average of
eq. (2.7) instead of the Fermilab/MILC result alone also leads to a minor increase of the
jVcbj central value.
Comparing the ts in table 5 with weak and strong unitarity bounds we observe that
the strong constraints decrease jVcbj by 1.5{2.2% and tighten its uncertainty quite a bit,
especially in the less constrained t without LCSR input.
HJEP1(207)6
,121.5
R
combined in quadrature with a 15% theoretical uncertainty.
It is also interesting to compare the e ects of the strong unitarity bounds we have
derived with the help of heavy quark symmetry relations with a naive rescaling of the weak
unitarity conditions of eq. (3.3). This gives an idea of how strong the strong unitarity
bounds really are and helps us understanding their usefulness. The e ects of the strong
unitarity bounds is roughly similar to that of using
2
n=0
X(aVn4 )2 <
2
n=0
X[(anA1 )2 + (anA5 )2] <
V ;
A
with
A;V < 0:2, depending on the inputs. In e ect, the strong unitarity bounds
introduce little correlations among the aiFj coe cients: they mostly bound their individual
size. This is unsurprising, as the unitarity sum rules cannot be saturated by one or two
amplitudes only.
We now want to verify a posteriori that the results of our ts are compatible with
heavy quark symmetry within reasonable uncertainties. Indeed, the form factor ratios
R1;2(w) de ned above after eq. (4.2) can be determined from the results of our ts. A
deviation from the NLO HQET predictions signi cantly larger than
20% would signal
an unexpected and unnatural breakdown of the heavy mass expansion. This point has been
emphasised in refs. [14] and [37]. The two plots in gure 4 show that the ts without/with
LCSR lead to R2 in good agreement with HQET (with input from QCD sum rules) and the
same holds for R1 when LCSR are included. On the other hand, without LCSR R1 is well
compatible with HQET only at small or moderate recoil: at large w there is a clear tension
with both HQET and LCSR predictions. Lattice calculations will compute A1 and R1;2 at
small recoil in the near future4 and are likely to settle the whole jVcbj determination. In
the meantime, the t without LCSR appears somewhat disfavoured.
Finally, we comment on the di erences of our ts with those performed in [20]. The
main di erences are that the authors of ref. [20] employ the CLN parametrization for
4Preliminary and incomplete results have been presented recently [38]. They seem to exclude large
deviations from HQET at small recoil.
2q2
(q2)3
d
dw
;
3
m2(m2 q2)2r3(1+r)2(w2 1) 2 P1(w)2
:
Here d =dw represents the di erential width for the decay to massless leptons, see e.g. [14],
and depends on the form factors A1;5 and V4. In the second term k = E2WjVcbj2G2F m5B=32 3,
r = mD =mB, with EW ' 1:0066 the leading QED correction. The second term depends
on a new form factor, P1(w), whose zexpansion is
the reference form factor A1 and assume the NLO HQET calculation for the form factor
ratios R1;2 without accounting for a theoretical uncertainty due to unknown higher order
corrections. They also perform a combined t to B ! D and B ! D
Belle data. In our ts
we do not employ directly the HQET relations because we believe their present uncertainty
does not make them useful and therefore a combined t to B ! D and B ! D data would
give the same results of the two separate ts presented here and in [5]. Indeed, the coupling
between the two sets of data through the unitarity bounds would be extremely small.
5
In the case of a massive ( ) lepton the di erential width for B ! D
can be written as
the sum of two terms
HJEP1(207)6
where
where
where the outer function is given by
P1(w) =
p
r
(1 + r)B0 (z) P1 (z) n=0
1
X anP1 zn ;
P1 =
s
nI
~1L+ (0) p1
p
8 2 r2(1 + z)2
z ((1 + r)(1
z) + 2pr(1 + z))4
:
The Blaschke factor B0 (z) takes into account the rst three 0 resonances, see table 3.
The ratio R(D ), de ned in eq. (1), can be split into two parts
R(D ) = R ;1(D ) + R ;2(D ) ;
R ;1(D ) =
R ;2(D ) =
1
1
R w ;max dw d ;1=dw
1R wmax dw d =dw
R w ;max dw d ;2=dw
1R wmax dw d =dw
;
;
w ;max = (m2B + m2D
m2)=(2mBmD )
1:355:
(5.1)
(5.2)
(5.3)
(5.4)
Unfortunately, the experimental q2spectrum of B ! D
cannot be reliably used to
constrain the form of P1 and there is no lattice calculation of this form factor. We therefore
consider three options:
to use P1 = (P1=A1)A1 where A1(w) is taken from the t and the ratio from HQET;
to use P1 = (P1=V1)V1 where V1(w) is taken from the t of [5] and the ratio from
HQET;
to use the HQET expression for P1(1) and the constraint P1(wmax) = A5(wmax)
HJEP1(207)6
together with unitarity.
Having three alternative derivations will give us an additional handle to estimate the overall
uncertainty. For a reference, we recall that the experimental world average for R(D ) is [1]
(see update online)
R(D )exp = 0:304
0:013
The standard way: normalizing P1 to A1
The rst option corresponds to the usual way of computing R(D ), see e.g. [20, 39]. The
relevant ratio is traditionally denoted by R0:
R0(w) =
P1(w)
A1(w)
;
and of course R0(w) ! 1 in the heavy quark limit. We use the updated NLO HQET
calculation for R0 [20], include gaussian uncertainties from the QCD sum rules parameters
and mb in the same way as [20], and in addition we assign to the NLO HQET result a 15%
uncertainty from higher order corrections. This corresponds to over 30% uncertainty on
R ;2(D ). Here and in the remainder of this section all the errors are meant to be gaussian
errors and are combined in quadrature whenever appropriate: this di ers from what we
did in section 3 where we were looking for absolute bounds. In addition, we also impose
strong unitarity constraints on the parameters of the zexpansion for P1, see gure 3: this
moves the central value slightly o the one computed using the central values in table V
and reduces somewhat the uncertainty.
Using our t with LCSR and strong unitarity bounds, we nd
R ;1(D ) = 0:232
R ;2(D ) = 0:026 ;
R(D ) = 0:258(5)(+87) ;
(5.6)
where the rst error refers to the B ! D `
t parameters and the full parametric
uncertainty of R0, while the second one is related to the 15% uncertainty due to higher order
corrections to R0. The contribution of R ;2(D ) to the
nal result is about 10%. The
uncertainty on P1, which a ects only R ;2(D ), has therefore a comparably small impact
on the total uncertainty of the SM prediction of R(D ). It turns out, however, that this
is the largest single source of uncertainty. The results obtained with the t without LCSR
and with strong unitarity bounds are very similar,
where the two errors have the same meaning as in (5.6). Combining all errors in quadrature
we end up with R(D ) = 0:258(+190) and 0:257(+180), respectively. These results agree
well with those obtained in ref. [20] using the same normalization to A1, except for the
uncertainty due to higher order corrections which is not considered there. They are also
0:003 [39] which has been used so far as reference SM
prediction in most papers on the subject.
Let us now proceed to compute R(D ) in the second way. Only the calculation of R ;2(D )
is di erent from the above derivation. Here we use the precise determination of V1(w) from
experimental B ! D` data and lattice QCD calculations of [5]. In particular, with the
BGL N = 2 parametrization of V1 and our t with LCSR and strong unitarity bounds
we get
where the rst error comes from parametric and t uncertainties, and the second one from
the 15% higher orders error. Using instead our t without LCSR input one gets
R ;1(D ) = 0:232 ;
R ;2(D ) = 0:036 ;
R(D ) = 0:268(+98)(+1120) ;
R ;1(D ) = 0:232 ;
R ;2(D ) = 0:038 ;
R(D ) = 0:270(+98)(+1120) :
The values of R(D ) in eqs. (5.8), (5.9) are substantially higher and have a larger
uncertainty than those obtained with the rst method, although they are compatible within
errors. The higher value of R(D ) is mostly due to the large di erence, already noticed
in section 2, between the NLO HQET and the lattice QCD predictions for A1(1)=V1(1),
see eqs. (2.8), (2.9). A lattice QCD determination of the form factor P1, even only at
zerorecoil, would drastically decrease the uncertainty in R(D ).
5.3
Enforcing a constraint at q2 = 0
The ts presented in section 3 allow for a 5% determination of A5 at the endpoint w = wmax.
This is outside the physical range for the semileptonic decay to taus, see (5.4), but the
relation P1(wmax) = A5(wmax) still constrains P1(w) signi cantly. We will now use only
the t with strong unitarity bounds and LCSR, which gives
A5(wmax) = 0:545
0:025:
(5.10)
(5.7)
(5.8)
(5.9)
This is signi cantly lower than P1(wmax) ' 0:69 obtained using the normalization to
V1 considered in the previous subsection, and also lower than the P1(wmax) ' 0:62
obtained normalizing P1 to A1. For what concerns the value at zero recoil, w = 1, we can
again use P1(1) = (P1=V1)HQETV1(1)lat or P1(1) = (P1=A1)HQETA1(1)lat, where the
lattice values V1(1)lat and A1(1)lat are taken from eq. (2.7), leading to P1(1) = 1:27(21) and
P1(1) = 1:12(18). Here we have combined in quadrature the parametric uncertainty with
a 15% theoretical uncertainty. An intermediate choice consists in using the HQET relation
between P1 and the IsgurWise function, which is 1 at zero recoil. At the NLO we nd
where the rst error is parametric, and the second corresponds to the 15% theoretical
uncertainty considered above. Using eq. (5.1) this amounts to a determination of a0P1 ,
P1(1) = 1:21
0:06
0:18
which can be combined with (5.10) to derive
a0P1 = 0:0595
0:0093;
a1P1 =
0:318
0:170
0:056a2P1
where the last term must satisfy ja2P1 j < 1 and a1P1 is consistent with strong unitarity for
almost any a2P1 . Using the last two relations, scanning in the relevant range of a2P1 , and
combining the errors in quadrature we get
R ;2(D ) = 0:028 ;
R(D ) = 0:260(5)(6) ;
where the rst error refers to the parametric uncertainty in R ;1(D ) and the second one
is related to P1 and parametric uncertainty in R ;2(D ) only. The correlation between the
two errors is small. We observe that the uncertainty is slightly smaller than those of the
other methods.
The three methods we have employed to compute R(D ) lead to results which are
consistent within uncertainties. The third method has a slightly smaller error and bene ts
from an important constraint at q2 = 0 which is not taken into account with the rst two
methods. In particular, eqs. (5.8), (5.9) are likely to somewhat overestimate R(D ). We
therefore adopt as our nal result the one obtained with the third method,
R(D ) = 0:260
0:008;
which still di ers 2:6 from the experimental world average, but it is higher and has an
uncertainty almost three times larger than existing estimates.5
We apply the same methodology also to the prediction of the longitudinal
lepton
polarization [39{42]
5Only ref. [16], which appeared together with the rst version of this paper, has a larger uncertainty,
nding R(D ) = 0:257(5).
(5.11)
(5.12)
where in our notation
and H00; H
P =
0:38
in the ts, is
H0t = mB
p
pr(1 + r)pw2
1 + r2
2wr
1
P1 ;
P =
0:47
are the integrated decay rates for de nite
lepton helicity. One has [39]
d
dq2 =
d +
dq2 =
G2F jVcbj2jp~jq2
Recently, Belle reported the measurement
0:51+00::2116 [43, 44]. Our SM prediction, independently of the use of LCSR
6
Unitarity bounds are an essential part of the model independent form factor
parametrization in semileptonic B decays. They can be made stronger using Heavy Quark Symmetry
relations between the B( )
! D( ) form factors, and are solid and reliable constraints,
provided one takes into account conservative uncertainties and recent input from lattice
calculations and experiment.
In this paper we have obtained bounds on the zexpansion parameters of the form
factors relevant in the calculation of B ! D `
decays. Since we keep only terms up to
z2 in the expansion, and we generally have lattice QCD information on the rst coe cient
a0, the bounds are expressed as allowed regions in the (a1; a2) planes for each of the form
factors, see gure 3. As lattice QCD calculations extend beyond the zero recoil point, they
will soon provide a relatively precise determination of the slopes of some of the form factors.
Our bounds will then become rather strict onedimensional bounds on the curvature, or
on the a2 parameters.
In practice, we have revisited the CLN methodology 20 years later, and used
experimental and lattice data to estimate the uncertainties in the HQET relations and to reduce
the errors. Unlike CLN, however, we do not provide a simpli ed parametrization. On the
contrary, our results on unitarity bounds applied to the BGL parametrization should form
the basis of a new generation of model independent analyses of B ! D ` data at both
BelleII and LHCb.
For what concerns the determination of jVcbj, we con rm and reinforce the conclusions
of our recent analysis [14]. The present world average of the exclusive determination of
jVcbj [1] relies on the CLN parametrization, but does not include a reliable estimate of the
related theoretical uncertainties and is likely to be biased. Although the strong unitarity
bounds have important consequences on the determination of jVcbj and reduce its value by
about 2%, our ts to recent Belle's and lattice data (complemented by the world average
for the B0 ! D +` branching ratio) show a large persisting di erence (3.5{5%) in the
value of jVcbj extracted using the BGL and CLN parametrizations. As already observed
in [14], it is possible that such a large di erence is accidentally related to the only Belle
data we could analyse for the B ! D ` di erential distributions, and that future global
averages of Babar and Belle data will lead to a smaller di erence between the CLN and BGL
ts. However, our approach now includes HQET constraints with realistic uncertainties
and improves on the CLN parametrization in several ways. Our nal results for jVcbj are
consistent with the inclusive determination but the error is signi cantly larger, about 3%
instead of 1.5%.
We have also reconsidered the SM prediction of R(D ) in the light of the above results.
Our analysis points to a higher central value and a signi cantly larger theoretical error
than found in previous analyses [20, 39]. Our nal result is reported in eq. (5.11) and its
uncertainty is dominated by the uncertainty in the normalization of the P1 form factor,
which will be certainly reduced by future lattice QCD calculations. Although we nd that
its signi cance is slightly reduced, this intriguing avour anomaly remains a challenge for
model builders.
Acknowledgments
We are grateful to Marcello Rotondo, Soumitra Nandi, Matthias Neubert, and Christoph
Schwanda for useful discussions.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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