B^sub S^ [right arrow] D^sub S^lv^sub l^ near zero recoil in and beyond the Standard Model

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We compute the normalization of the form factor entering the ...... decay amplitude by using numerical simulations of QCD on the lattice. From our study with ...... dynamical light quarks, and by employing the maximally twisted Wilson quark action, we obtain in the continuum limit ....... We also compute the scalar and tensor form factors in the region near zero recoil and find ......, ......, for ....... The latter results are useful for searching the effects of physics beyond the Standard Model in ...... decays. Our results for the similar form factors relevant to the non-strange case indicate that the method employed here can be used to achieve the precision determination of the ...... decay amplitude as well.

Eur. Phys. J. C (2014) 74:2861DOI 10.1140/epjc/s10052-014-2861-z

Regular Article - Theoretical Physics

Bs Ds near zero recoil in and beyond the Standard Model

Mariam Atoui1,a, Vincent Mornas1,b, Damir Beirevi2,c, Francesco Sanlippo2,d

1 Laboratoire de Physique Corpusculaire, Universit Blaise Pascal, CNRS/IN2P3, 63177 Aubire Cedex, France

2 Laboratoire de Physique Thorique (Bt 210), Universit Paris Sud and CNRS, Centre dOrsay, 91405 Orsay-Cedex, France

Received: 10 January 2014 / Accepted: 12 April 2014 / Published online: 8 May 2014 The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract We compute the normalization of the form factor entering the Bs Ds decay amplitude by using numeri

cal simulations of QCD on the lattice. From our study with Nf = 2 dynamical light quarks, and by employing the max

imally twisted Wilson quark action, we obtain in the continuum limit G(1) = 1.052(46). We also compute the scalar

and tensor form factors in the region near zero recoil and nd f0(q20)/f+(q20) = 0.77(2), fT (q20, mb)/f+(q20) = 1.08(7),

for q20 = 11.5 GeV2. The latter results are useful for search

ing the effects of physics beyond the Standard Model in Bs Ds decays. Our results for the similar form fac

tors relevant to the non-strange case indicate that the method employed here can be used to achieve the precision determination of the B D decay amplitude as well.

1 Introduction

Inclusive and exclusive semileptonic b c decays, with

{e, }, have been subjects of intensive research over

the past two decades. Within the Standard Model (SM) the main target of that research was, and still is, the accurate determination of the CabibboKobayashiMaskawa matrix element |Vcb|, which is extracted from the comparison of

theoretical expressions with experimental measurements of the partial or total decay widths. It turns out, however, that the value for |Vcb| obtained from the exclusive decays agrees

only at the 2.1 level with the one extracted from inclusive decays. More specically, the two independent analyses of inclusive decays [1,2] (updated in Refs. [3]) give completely

a e-mail: [email protected]

b e-mail: [email protected]

c e-mail: [email protected]

d e-mail: [email protected]

consistent results, of which the average reads1

|Vcb|incl. = 41.90(70) 103. (1)

The analyses of exclusive decays, instead, are performed by tting the experimental data to the shapes of the form factors parameterized according to the expressions proposed and derived in Ref. [5], so that the nal results are then reported in the following form:

|Vcb|F(1) = 35.90(45) 103,

|Vcb|G(1) = 42.6(1.5) 103, (2) as obtained from B(B D) and B(B D),

respectively. F(1) and G(1) are the relevant hadronic form

factors at the zero-recoil point. Thanks to the heavy quark symmetry, and up to perturbative QCD corrections, both these form factors are equal to 1 in the limit of mc,b

[6,7]. To compute the deviation of these form factors

from unity, it is necessary to include all the non-perturbative order 1/mnc,b QCD corrections. The only model independent method allowing one to compute F(1) and G(1) from the

rst theory principles is lattice QCD. Using the most recent estimates of the above form factors, F(1) = 0.902(17) [8,9]

and G(1) = 1.074(24) [10], one arrives at

|Vcb|excl. = 38.56(89) 103, (3)

a number that obviously differs from |Vcb|incl. given in

Eq. (1). In principle the exclusive decay modes are better suited for the precision determination of |Vcb| because fewer

theoretical assumptions are needed to compute the corresponding decay rates. The main obstacle is, however, the necessity for a reliable, high precision, lattice QCD estimate of F(1) and G(1). Furthermore, for the required per

cent precision of |Vcb| it is important to have a good control

over the structure dependent soft photon B D() soft,

which could otherwise be misidentied as pure semileptonic

1 The most recent update of the analysis of inclusive decays using the kinetic scheme can be found in Ref. [4].

123

2861 Page 2 of 13 Eur. Phys. J. C (2014) 74:2861

decays. This problem is less acute for the down-type spectator, i.e. B0 D()+

, than for the up-type spectator quark, B D()0

[11]. For that reason it is desirable to consider the charged and neutral B-meson decay modes separately. In this paper we will mainly discuss Bs Ds

decay for which the soft photon pollution is smaller. Moreover, this mode is much more affordable numerically because the valence s-quark is easily accessible in numerical simulations of QCD on the lattice which is not the case with the physical u/d-quark. In this paper we will also comment on the non-strange case when appropriate. Finally, we prefer to focus on B(s) D(s) , rather than B(s) D(s) , because

the hadronic matrix element involves much fewer form factors and the decay rate is therefore likely to be less prone to systematic uncertainties.

Despite the importance of G(1), only a few lattice QCD

studies have been performed so far. The methodology described in Ref. [12] has been implemented in unquenched simulations with N f = 2 + 1 dynamical staggered light

quarks in Refs. [10,13] where the propagating heavy quarks on the lattice have been interpreted by means of an effective theory approach. In Refs. [14,15], an alternative method to compute the B D semileptonic form factors has been

proposed and implemented in quenched approximation. The latter method, based on the use of the step scaling function, allows one to compute the same form factors without recourse to heavy quark effective theory. The price to pay, however, is that the method of Refs. [14,15] is computationally very costly and to this date it has not been extended to unquenched QCD. In this paper we use a modication of the proposal of Refs. [14,15], presented in Ref. [16] and also implemented in the computation of the decay constant fB and the b-quark mass, cf. Ref. [17]. The remainder of this paper is organized as follows: In Sect. 2 we dene the form factors and express them in a way that is suitable for the strategy used for their computation which is described in Sect. 3. Details of our lattice computations and the results for G(1) are given in

Sect. 4, while the results concerning the scalar and tensor form factors in the region close to zero recoil are discussed and presented in Sect. 5. We nally conclude in Sect. 6.

2 Denitions

The hadronic matrix element describing the Bs Ds

decay in the SM, Ds| b(1 5)c|Bs Ds| bc|Bs , is

parameterized in terms of the hadronic form factors f+,0(q2)

as

Ds(k)|V|Bs(p) = (p + k) f+(q2)

+ q

m2Bs m2Ds q2

where V = bc, q = p k, and q2 (0, q2max], with

q2max = (mBs mDs )2. The extraction of the form factors

becomes particularly simple if we use the projectors

P0 =

q q2max

, P+ =

q2 mBs EDs

,

q

, (5)

so that

P0 Ds(k)|V|Bs(p) =

mBs + mDs

mBs mDs

f0(q2),

P+ Ds(k)|V|Bs(p) =

f+(q2). (6)

In our computations we will consider the situations with

| p| = 0 and q = (mBs EDs ,

q2 2mBs

mBs EDs

k). Another frequently used parameterization of this matrix element, motivated by the heavy quark expansion, reads

1 mBs mDs Ds(k)|V|Bs(p) = v + v h+(w)

+ v v h(w), (7)

where v = p/mBs , v = k/mDs , and the relative velocity

w = v v = (m2Bs + m2Ds q2)/(2mBs mDs ). From the

comparison of Eqs. (4) and (7) one gets

f+(q2) =

mBs + mDs

4mBs mDsh+(w) 1

mBs mD

mBs + mDs

h(w)

h+(w)

,

f0(q2) =

mBs mDs

mBs + mDs

(w + 1) h+(w)

1

mBs + mDs

mBs mDs

w 1

w + 1

h(w)

h+(w)

. (8)

The form factor G(w) used in experimental analyses of the

B(s) D(s) decay is proportional to f+(q2), and reads

G(w) = h+(w) 1

mBs mDs

mBs + mDs

h(w)

h+(w)

, (9)

h+(w)

1

mBs mDs

mBs + mDs

2 H(w)

where, for convenience, we introduced

H(w) =

mBs + mDs

mBs mDs

h(w)

h+(w)

. (10)

Our main target is the determination of G(1), and therefore

we are particularly interested in the dominant h+(1) term

that can easily be obtained from f0(q2max),

h+(1) =

mBs + mDs

4mBs mDsf0(q2max). (11)

Unfortunately, however, H(1) is not directly accessible from the lattice. Instead, we need to compute the form factors

f0(q2) f+(q2) , (4)

123

Eur. Phys. J. C (2014) 74:2861 Page 3 of 13 2861

f0,+(q2) at several small values of the D-meson three-

momentum and then extrapolate H(w) to H(1). As we shall see in the following, we manage to work with w [greaterorsimilar] 1 but by staying very close to zero recoil and the uncertainty associated with this extrapolation is completely negligible. To be more specic, at |

k| = 0, we compute

R0(q2) =

f0(q2)

f+(q2)

, (12)

so that

H(w) =

S q(x, 0; U) = ei x/L Sq(x, 0; U), (16) can then be combined with the ordinary (untwisted) propagator, Sq(x, 0; U), into a two-point correlation function.

The resulting lowest lying state extracted from the exponential fall-off has a three-momentum different from zero,

|

k| = |

|/L [1821]. Then, by choosing

= (1, 1, 1)0,

which also minimizes the discretization errors, one can tune 0 to an arbitrary small value and therefore explore the kinematical region of Bs Ds decay very close to zero

recoil, w [greaterorsimilar] 1 (q2 [lessorsimilar] q2max).

For a given 0, the continuum energy-momentum relation would give

w =

1 +

R0(mBs + mDs )2 2mBs mDs (w + 1) R0(mBs mDs )2 2mBs mDs (w 1)

, (13)

where, for shortness, we write R0 R0(q2(w)).

3 Strategy

To extract the form factors f0,+(q2) from numerical simula

tions on the lattice, one rst computes the correlation functions

C(

q; t) = x, y

3202

m2Ds L2

, (17)

which is a good approximation for the small values of 0 chosen in this study. Otherwise one can use w = E(0)/mDs ,

and the free boson dispersion relation on the lattice:

4 sinh2 ED2 = 4 3 sin2

0

2L

+ 4 sinh2

mD

2 .

Pbs( 0, 0)V( x, t)Pcs( y, tS)ei q( x y) ,

In terms of quark propagators the correlation function (14) reads

C(

q; t)

=

x, y

(14)

where the interpolating source operators, the pseudoscalar densities Pcs and Pbs, are sufciently separated in the time direction so that for 0 t tS one can isolate the low

est lying states with J P = 0 that couple to two source

operators, and then extract the matrix element of the vector current between the two. The simplest choice would be the local operator, Phs = h5s, but for practical convenience

one often resorts to the smearing technique that helps to signicantly reduce the couplings to radially excited states. In other words, the lowest lying states are better isolated when smeared source operators are used and when the corresponding time interval, within which the matrix elements are extracted, becomes larger. As mentioned in the previous section, we also need to give the Ds meson a few momenta

|

k| = 0, in order to study the behavior of H(w) as a function

of w and extrapolate to H(1). To make those momenta small and remain close to the zero-recoil point, it is convenient to compute the quark propagators, Sq(x, 0; U) q(x) q(0) ,

by imposing the twisted boundary conditions [18,19]. Those are easily implemented by rephasing the gauge eld congurations according to

U(x) U(x) = ei/LU(x), (15)

where U(x) stands for the gauge links, = (0,

), and L

is the size of the spatial side of the cubic lattice. The quark propagator computed on such a rephased conguration,

.

(18)

In addition to the above three-point correlation functions, we also computed the two-point correlators that are necessary to remove the source operators and gain access to the vector current matrix element. From the large time behavior of the two-point correlation functions

x

Phs(

Tr

5Ss(0, y)5S c(y, x; U)Sb(x, 0; U)

x; t)Phs(0; 0)

t 0

cosh[mH (T/2 t)]mH emHT/2, (19)

we can extract mH and ZH = 0| h5s|H , where h (H)

stands for either c (Ds) or b (Bs), and T is the size of the temporal extension of the lattice. With these ingredients we are able to extract the desired hadronic matrix element from the decomposition

C(

q; t)

0 t tS

ZBs ZDs

4mBs EDs exp m

|ZH|2

Bs t

exp EDs (tS t) Ds(

k)|V|Bs(

0) , (20)

123

2861 Page 4 of 13 Eur. Phys. J. C (2014) 74:2861

and then by using the projectors (5) we get the form factors f0,+(q2), as indicated in Eq. (6). As already mentioned

above, to make sure that the lowest lying states are well isolated, we employed the smearing procedure discussed in our previous works [2224] where the full information concerning the smearing parameters can be found as well.

While working with the fully propagating b-quark on the lattice, i.e. without resorting to an effective theory approach, the most difcult problem is to deal with large discretization errors. The reason is mostly practical since the lattice spacings (a) accessible in current lattice QCD simulations are not small enough to satisfy mba < 1. The charm quark, on the other hand, can be simulated directly and the discretization effects associated with its mass can be monitored by working at several small lattice spacings. Hence, the strategy is to perform the computations starting from the charm quark mass and then successively increase the heavy quark mass by a factor of so that after n + 1 steps one arrives at

mb. For each value of the heavy quark mass mh = k+1mc

we compute the form factor G(1, mh, mc), and evaluate the

ratio of form factors computed at two successive heavy quark masses, while keeping other valence quarks and the lattice spacing xed. In practice we compute

k(1) = G

(1, k+1mc, mc, a2)

G(1, kmc, mc, a2) , (21)

where the rst argument in the form factor is w = 1, the

second is the heavy quark mass that we want to send to the physical b-quark, while mc and a are the charm quark mass and the lattice spacing, both of which are kept xed. Each of these ratios can then be extrapolated to the continuum limit, lima0 k(1) = k(1).

The advantage of considering k(1) instead of G(1, k+1

mc, mc) becomes apparent when considering the heavy quark mass dependence. In the continuum limit, thanks to the heavy quark symmetry, the form factor scales with the inverse heavy quark mass as

G(1, mh, mc) = g0 +

g1 mh +

one actually interpolates to (mb). In the continuum limit, we then t the lattice data to

(1, mh, mc) (1, mh) = 1 +

s1 mh +

s2m2h +

, (23)

determine s1,2, and interpolate to (mb). Another interesting feature is that the expansion in inverse heavy quark mass is strictly valid in the heavy quark effective theory (HQET) and had we used Eq. (22) we would have had to include the perturbative matching between our results (obtained in full QCD) to HQET, and then the result of extrapolation to the b quark mass should have been converted back to QCD. In the ratios of form factors, k (k), the matching to HQET and back becomes completely immaterial as the matching factors cancel to a large extent. We attempted including these corrections to our interpolation to (1, mb), and the results remained would change by a few per-mil level only, thus completely immaterial for our purpose.

To get the physically relevant G(1), one starts from the

elastic form factor, the value of which is by denition

G(1, mc, mc) = 1. The physically interesting B(s) D(s)

form factor is then obtained as a product of k(1) factors discussed above, namely

G(1) G(1, mb, mc)

= nn1 . . . 10 G(1, mc, mc)

=1

. (24)

In this study we choose n = 8, which gives

=

mb mc

1 n+1

g2m2h +

, (22)

where the non-perturbative coefcients g0,1,2,... should be determined from the t to the lattice data. Keeping in mind the practical limitations that prevent us from ensuring that the heavy quark masses are smaller than the inverse lattice spacing, it is clear that it is very challenging to disentangle the physical effects from lattice artifacts in the gi, and in the dominant term g0 in particular. Consequently the resulting

G(1) G(1, mb, mc) suffers from systematic uncertainty,

the size of which is very difcult (if not impossible) to assess and therefore cannot be used for a precision determination of |Vcb| from B(B(s) D(s) ). In contrast, the successive

ratios of the form factors satisfy limmh(mh) = 1, and

therefore instead of extrapolating to the inverse b-quark mass,

= 1.176, (25)

where we used mMSc(2 GeV) = 1.14(4)GeV [25], and

mMSb(2 GeV) = 4.91(15) GeV [17].

4 Lattice details

In this work we use the publicly available gauge eld congurations that include Nf = 2 dynamical light quarks, gen

erated according to the twisted mass QCD action with maximal twist [28] by the European Twisted Mass Collaboration (ETMC) [26,27]. Main details as regards 13 ensembles of gauge eld congurations are collected in Table 1. We computed all quark propagators by using stochastic sources, and then applied the so-called one-end trick to compute the correlation functions [26,27].

Since we use the smeared source operators, the factor

|ZDs | needed in Eq. (20) depends on the momentum

k given

to the Ds meson. We computed |ZDs (

k)| for each of our 0-

values and after dividing out the correlation function C(t,

k)

in Eq. (18), by the exponentials and couplings |ZDs (

k)| and

|ZBs |, we looked for the plateau region to extract the desired

123

Eur. Phys. J. C (2014) 74:2861 Page 5 of 13 2861

Table 1 Ensembles of gauge eld condigurations used in this work. Lattice spacing is xed by using the parameterr0/a [29], withr0 = 0.440(12) fm

xed by matching f obtained on the same lattices with its physical value (cf. Ref. [25]). Bare quark masses i are given in lattice units. Quoted values of the renormalization constant ZT refer to the MS renormalization scheme and = 2 GeV

Ensemble = 6/g20 sea L3 T # meas. s c a [fm] ZV (g20) ZT (g20)

I 3.80 0.0110 243 48 240 0.0194 0.2331 0.098(3) 0.5816(2) 0.73(2)

II 0.0080 240

III 3.90 0.0100 243 48 240 0.0177 0.2150 0.085(3) 0.6103(3) 0.750(9)

IV 0.0085 240

V 0.0064 240

VI 0.0040 288

VII 0.0040 323 64 240 VIII 0.0030 240

IX 4.05 0.0080 323 64 686 0.0154 0.1849 0.067(2) 0.6451(3) 0.798(7)

X 0.0060 400

XI 0.0030 750

XII 4.20 0.0065 323 64 480 0.0129 0.1566 0.054(1) 0.686(1) 0.822(4) XIII 0.0020 483 96 100

0.6

to the values of 0 in (18)]. In Table 1 we gave the value of the charm quark mass in lattice units. Other heavy quark masses are simply obtained after successive multiplication by , with the physical b = 9c. We note that the errors

on the form factors become large for very heavy quarks.

As mentioned in Sect. 3 the computation of H(w) can be made only at w = 1. We tuned the values of the twisting

angle 0 for each of our lattices in such a way as to make the corresponding w xed [cf. Eq. (17)]. More specically, apart from the zero-recoil point w = 1, we computed the

form factors with four different non-zero momenta given for Ds, corresponding to

w {1.004, 1.016, 1.036, 1.062}. (27) Clearly, only a tiny extrapolation H(w) to H(1) is needed. A linear and a quadratic t in w to reach H(1) lead to indistinguishable results, both results being small and further suppressed by the mass factor in Eq. (9) so that G(1) is largely

dominated by the form factor h+(1) evaluated according to

Eq. (11). For very heavy quarks (mh close to mb) the effect of that extrapolation becomes visible, but since the form factor computed with such a heavy quark is dominated by discretization and large statistical errors they do not have any signicant effect on our nal results.

We then computed the ratios of the form factors G(1)

obtained at each two successive heavy valence quarks as indicated in Eq. (21). Importantly, a strong cancelation of statistical errors leads to very accurate ks. The values of all ks are presented in Table 2. In Fig. 2 we illustrate the situation for two values of k. From these plots we can see that our lattice data exhibit very little or no dependence on the light sea quark mass, nor on the lattice spacing. Note

0.5

Ds (k) Vi Bs (0)

0.4

0.3

0.2

0.1

0.0

5 10 15 20

t a

Fig. 1 Plateaus on which the matrix element is extracted according to Eq. (20) for 5 different values of momenta corresponding to w = 1

(when this matrix element is zero) and four other momenta corresponding to the w given in Eq. (27). Plotted are the data from the ensemble

IV (cf. Table 1) and for mb = mh = 4mc (N.B. mphys.b = 9mc)

matrix element [cf. Eq. (20)]. After inspection, we xed the plateaus to the intervals

t [10, 13]3.8, [10, 13]3.9, [12, 17]4.05, [17, 19]4.2, (26)

in an obvious notation. Those plateaus are chosen to be common to all the sea quark masses considered at a given lattice spacing, and to all the heavy valence quark masses. Note that the three-point correlation functions (14) are computed with tS = T/2 for all of our lattices. As an example, we illus

trate in Fig. 1 the quality of the plateaus corresponding to the matrix element Ds(

k)|Vi|Bs(

0) , and their sensitivity to the

values of the three-momentum

k used in this paper [or better,

123

2861 Page 6 of 13 Eur. Phys. J. C (2014) 74:2861

Table 2 Results of the ratios of the form factor G(1) computed at successive heavy quark masses according to Eq. (21), as computed on all of our

ensembles of gauge eld congurations

Ensemble 0(1) 1(1) 2(1) 3(1) 4(1) 5(1) 6(1) 7(1) 8(1)

I 1.001(3) 1.001(4) 1.001(4) 0.997(5) 0.993(9) 0.981(17) 0.934(35) 0.762(81) 0.020(334)

II 1.005(4) 1.010(7) 1.011(8) 1.012(12) 1.007(22) 1.024(39) 1.011(86) 0.859(242) 0.121(717)

III 0.997(2) 1.004(4) 1.002(3) 0.997(6) 1.000(6) 0.998(10) 1.003(19) 1.028(35) 1.121(90)

IV 0.995(2) 0.994(4) 0.994(3) 0.991(4) 0.989(4) 0.986(8) 0.970(17) 0.960(40) 0.892(126)

V 0.999(2) 1.000(3) 1.000(5) 0.999(6) 0.999(6) 1.002(8) 1.002(16) 1.031(47) 1.115(140)

VI 0.996(2) 1.003(4) 1.002(3) 1.003(3) 1.000(6) 0.996(11) 0.988(25) 1.004(64) 1.102(194)

VII 1.000(3) 1.001(3) 1.002(4) 1.002(5) 1.002(7) 1.006(9) 1.021(14) 1.039(36) 1.037(133)

VIII 1.002(2) 1.002(4) 1.003(5) 1.000(5) 1.020(20) 0.974(23) 0.976(16) 0.932(33) 0.851(85)

IX 0.996(2) 1.004(4) 0.998(4) 0.998(5) 0.995(5) 0.989(8) 0.979(13) 0.970(24) 0.926(51)

X 0.997(2) 0.996(3) 1.005(4) 0.999(4) 1.005(6) 1.007(10) 1.013(16) 1.027(30) 1.069(76)

XI 0.998(1) 1.009(2) 1.004(2) 1.007(2) 1.008(3) 1.011(4) 1.016(7) 1.025(14) 1.025(27)

XII 0.996(1) 1.010(3) 1.000(2) 1.007(2) 1.005(1) 1.005(3) 1.008(3) 1.008(3) 1.018(7)

XIII 1.001(6) 1.000(9) 0.997(8) 0.994(11) 0.988(12) 0.981(15) 0.970(21) 0.954(34) 0.925(93)

1.10

1.10

1.05

1.05

0(1)

1.00

3(1)

1.00

0.95

0.95

0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.90 0.0 0.1 0.2 0.3 0.4 0.5 0.6

sea s

sea s

Fig. 2 Values of 0 and 3 as obtained on all of the lattices used in this work is shown as a function of the light sea quark mass (divided by the physical strange quark mass). Different symbols are used to distinguish the lattice data obtained at different lattice spacings: open circles

= 3.80, open squares = 3.90 (243), lled squares for = 3.90

(323), lled circles = 4.05, and right-pointed triangle for = 4.20.

The result of continuum extrapolation is also indicated at the point corresponding to the physical ud/s mud/ms = 0.037(1) [25]

also that for larger heavy quark masses the errors on k are larger, and therefore the corresponding continuum value k will have larger error as well.

The extrapolation of k(1) to the continuum limit is performed by using the following form

k(1) = k + k

msea

ms + k

a a=3.9

2, (28)

and then identify

k(1) = lim

a0

mseamud

k(1), (29)

where mud stands for the average of the physical up and down quark masses computed on the same lattices [25]. As anticipated from Fig. 2 the values of k and k, as obtained from

the t of our data to Eq. (28), are consistent with zero. The resulting k(1) = (1, k+1mc) (1, mh) are given in

Table 3. Since our data do not exhibit a dependence on the sea quark mass we also attempted extrapolating k(1) k(1)

by imposing k = 0 in Eq. (28). The results for the rst

few k(1) remain practically indistinguishable from those obtained by letting k as a free t parameter. For higher masses, namely for 48(1), the results of two continuum

extrapolations remain compatible but the error bars in the case of a free k are considerably larger. The problem of larger errors for large quark masses is circumvented by the interpolation formula (23). Clearly the data with larger error bars become practically irrelevant in the t because the intercept of the t is xed to unity by the heavy quark symmetry. In other words, the result of interpolation to (1, mb) remains

123

Eur. Phys. J. C (2014) 74:2861 Page 7 of 13 2861

Table 3 Results of the continuum extrapolation of k(1) to k(1) = (1, k+1mc) (1, mh) using Eq. (28). Results of extrapolation with k

as a free parameter are shown separately from those in which the observed independence on the sea quark mass is imposed in the t (28) by setting k = 0

k 0 1 2 3 4 5 6 7 8

1/mh [GeV] 0.739 0.629 0.534 0.454 0.386 0.328 0.279 0.237 0.202

(1, mh)k=0 0.991(1)

1.007(2) 1.008(2) 1.006(2) 1.013(2) 1.013(5) 1.019(6) 1.022(11) 1.060(31)

(1, mh)k free 0.996(2)

1.004(4) 1.005(5) 1.005(6) 1.012(8) 1.005(10) 1.006(23) 0.975(48) 0.853(111)

1.10

1.05

(1,m )

h

1.00

0.95

0.0 0.2

0.4 0.6 0.8

1 m h [GeV ]

1

Fig. 3 We show our data

for k(1) = (1, mh) with mh = k+1mc,

and show the result of the

t in 1/mh to the form given in Eq. (30) as a function of the inverse

heavy quark mass with mh = k+1mc. Filled

symbols correspond to (

1, mh) extrapolated to the continuum limit by using Eq. (28) with all

parameters free, whereas the empty symbols refer to the results obtained by

imposing k = 0. Fitting curves of the central

values, together with the

bounds (dashed lines), are also displayed. The gray vertical line indicates

to point corresponding to the inverse of the physical b-quark mass

unchanged regardless

of whether we include our results for 48(1) in the t or

not. We stress again that instead of extrapolating G(1, mh

, mc) in inverse heavy quark mass to the physically interesting

point [G(1, mb, mc)] one interpo

lates (1, mh) to (1

, mb) since limmh(mh) = 1. In

practice we identify

k(1) = (1, k+1mc), and then like

suggested in Eq. (23),

t our results to

(1, mh) = 1 +

s1 mh

+

s2 m2h

, (30)

which is illustrated in

Fig. 3. We then proceed as in Eq. (24)

and obtain

G(1) = 1.073(17)(k = 0),

G(1) = 1.052(46)(k = 0). (31)

The rst (more

accurate) result agrees with the only existing unquenched lattice

QCD result, obtained for the light non-strange spectator

quark [10]. To calculate our results in Eq. (31) no renormalization

constant was actually needed.

This is convenient but

not particularly beneciary for our computation since the

vector current renormalization con-

stants have been computed non-perturbatively in Ref. [30] to a very good accuracy (cf. values listed in Table 1). Therefore, we were able to perform several checks and instead of starting from the elastic form factor G(1, mc, mc) = 1, we could

have started from a k < n to compute G(1, k+1mc, mc) in

the continuum limit, and then applied k+1 . . . n to reach

the b-quark mass. For example, by using k = 3,

G(1, mb, mc) = 87654 G(1, 4mc, mc)

= 1.059(47), (32)

in the case with k = 0. This results is obviously completely

consistent with the number given in Eq. (31). To get the above result we also needed to perform a continuum extrapolation of G(1, 4mc, mc) by using the expression analogous to the

one shown in Eq. (28). We checked and observed that our lattice data for the form factor are also independent on the light sea quark when the valence quark masses are xed, a behavior very similar to what is shown in Fig. 2. Furthermore we checked that, after adding the cubic term in 1/mh to Eq. (30), the resulting G(1) = 1.047(61), remains fully

consistent with our main result given in Eq. (31). Although the nite volume effects are not expected to affect the quantities computed in this paper, they could appear when the dynamical (sea) quark mass is lowered. In order to check for that effect we can compare our results obtained on the ensembles VI and VII which differ by the volume. The situation shown in Fig. 2 is a generic illustration of the situation we see with all the other quantities: the form factors are completely insensitive to a change of the lattice volume.

All these checks suggest that our result (31) obtained by using k as a free parameter, remains stable and we take it for our nal result, namely

G(1) = 1.052(46). (33)

Finally, we repeated the whole computation for the non-strange case, i.e. by keeping the sea and valence light quarks degenerate in mass. We obtained G(1) = 1.079(29) and

G(1) = 1.033(95), corresponding to k = 0 and k = 0,

respectively. The latter number is not helpful in reducing the error bar of |Vcb| extracted from B D decays. It shows,

however, that the method employed in this work can be used to get a percent precision of G(1) even in the non-strange

123

2861 Page 8 of 13 Eur. Phys. J. C (2014) 74:2861

Table 4 All physically relevant results of this study: G(w) is the dominant form factor governing the hadronic matrix element relevant to Bs Ds

( {e, }) computed in the zero-recoil region, f0(q2)/f+(q2) is needed for Bs Ds in the Standard Model and for all the leptons in the

case of helicity enhanced contributions present in the models beyond Standard Model. The tensor form factor fT (q2), also needed in some NP scenarios, is computed at = mb in the MS renormalization scheme

w

q2BsDs [GeV2] G(w)

fT (q2)

f+(q2)

f0(q2)

f+(q2)

k = 0 k = 0 k = 0 k = 0 k = 0 k = 01.(11.54) 1.052(47) 1.073(17)

1.004(11.46) 1.052(47) 1.075(16) 0.766(19) 0.752(7) 1.076(68) 1.078(43)

1.016(11.20) 1.029(49) 1.063(15) 0.781(24) 0.757(9) 1.062(76) 1.064(49)

1.036(10.79) 1.044(51) 1.034(17) 0.787(34) 0.760(16) 0.975(94) 0.997(64)

1.062(10.23) 0.986(57) 1.004(20) 0.825(59) 0.761(34) 0.920(111) 1.004(76)

case provided the statistical quality of the data is substantially improved. Note also that our G(1) in Eq. (33) agrees

with the result obtained by the expansion around the BPS limit in Ref. [31].

We end this discussion with a comment concerning the non-zero-recoil situation (w = 1). The analysis is essentially

the same as in the zero-recoil case described above. From the correlation functions (18) and by using the projector P+ (5)

we get the form factor f+(q2), which is proportional to the

desired G(w, kmc, mc), cf. Eqs. (8, 9). The observations

made in the analysis of G(1) concerning the independence on

the light sea quark mass and on the lattice spacing remain true after switching from w = 1 to w = 1. The values are given in

Table 4, where we again report our results both in the case in which the parameter k in the continuum extrapolation (28)

is left free and in the case in which k = 0 is imposed.

The net effect in the latter case is that the resulting error is considerably smaller. Using the parameterization of Ref. [5], which takes into account the relation between the curvature and the slope of G(w), namely

G(w)

G(1) = 1 82z + (512 10)z2

(2522 84)z3, (34)

with z = (w + 1 2)/(w + 1 + 2), one could

attempt to extract the slope 2 from our data. Knowing that the window of the w we consider here is very short, see (27), a clean determination of 2 would require very accurate values of G(w). In our case we only obtain 2 = 1.2(8), or

in the case where we dismiss the dependence on the sea quark mass (when the errors on G(w) are smaller) we get

2 = 1.1(3), both being consistent with the experimentally

established 2 = 1.19(4)(4) [32]. The same quality of the

result for 2 is obtained if the data are t to [3335]

G(w)

G(1) =

21 + w

, (36)

where the renormalization scale dependence reects the fact that the tensor density in QCD is a logarithmically divergent operator. In what follows the -dependence will be tacitly assumed. In this paper we report the result of the rst lattice QCD computation of the tensor form factor in the region close to q2max.

More specically, in this section we compute

R0(q2) =

f0(q2)

f+(q2)

, (37)

which directly enter the expression for differential decay rate that can be found in eg. Ref. [48]. Since the form factors f+,T (q2) are not accessible at q2max (zero recoil, w = 1), we

computed R0,T (w(q2)) at w [greaterorsimilar] 1.

, RT (q2) =

fT (q2)

f+(q2)

22. (35)

5 Scalar and tensor form factors

Recently measured B(B D ) by the BaBar collab

oration indicated about 2-discrepancy with respect to the SM estimate, obtained by combining the measured B(B

D) and the known information about the scalar form factor [36]. Since then, a number of studies appeared trying to explain that discrepancy by interpreting it as a potential signal of New Physics (NP) [3748]. In the models with two Higgs doublets (2HDM), the charged Higgs boson can mediate the tree-level processes, including B D , and considerably

enhance the coefcient multiplying the scalar form factor in the decay amplitude. For that reason it becomes important to get a lattice QCD estimate of f0(q2). Furthermore, the model independent considerations of NP also allow for a possibility of having a non-zero tensor coupling, in which case one more form factor appears in the decay amplitude. The tensor form factor fT (q2) is dened via,

Ds(k)| bc|Bs(p)

= i pk k p

2 fT (q2, )

mBs + mDs

123

Eur. Phys. J. C (2014) 74:2861 Page 9 of 13 2861

Table 5 Results of the ratios of R0(w3) computed at successive heavy quark masses according to Eq. (39), on all of our ensembles of gauge eld congurations. w3 = 1.016

Ensemble 01(w3) 2(w3) 03(w3) 04(w3) 05(w3) 06(w3) 07(w3) 08(w3)

I 0.986(3) 0.986(2) 0.984(3) 0.982(6) 0.983(11) 0.992(27) 1.054(97) 0.9(1.9)

II 0.993(6) 0.991(4) 0.988(5) 0.987(8) 0.990(18) 1.015(48) 1.181(23) 0.5(3.2)

III 0.988(4) 0.986(3) 0.981(4) 0.975(5) 0.969(7) 0.965(13) 0.964(32) 0.974(90)

IV 0.996(4) 0.992(2) 0.987(2) 0.982(4) 0.977(7) 0.974(16) 0.991(35) 1.064(94)

V 0.991(4) 0.988(3) 0.983(3) 0.976(4) 0.966(7) 0.947(13) 0.904(27) 0.810(64)

VI 0.991(5) 0.989(3) 0.986(3) 0.983(3) 0.981(5) 0.978(11) 0.974(27) 0.957(70)

VII 1.000(4) 0.994(2) 0.989(2) 0.984(3) 0.980(4) 0.980(9) 0.994(19) 1.048(48)

VIII 0.995(4) 0.991(2) 0.988(2) 0.984(3) 0.983(5) 0.988(11) 1.020(30) 1.169(128)

IX 0.992(3) 0.989(2) 0.985(3) 0.981(4) 0.978(5) 0.975(9) 0.967(20) 0.946(55)

X 0.999(7) 0.9892(5) 0.987(5) 0.983(7) 0.980(11) 0.980(20) 0.984(43) 0.99(12)

XI 0.991(2) 0.988(1) 0.983(1) 0.978(2) 0.972(2) 0.967(4) 0.966(8) 0.981(22)

XII 0.992(2) 0.988(3) 0.982(1) 0.977(1) 0.971(1) 0.965(1) 0.959(2) 0.951(3)

XIII 0.999(7) 0.992(5) 0.987(5) 0.983(5) 0.980(11) 0.980(20) 0.984(43) 0.99(12)

5.1 R0(q2)

The extraction of R0(q2) is practically straightforward. After applying the projectors P+,0 (5) to the matrix element extracted from the correlation functions (18), we combine them in the ratios R0(w, mh, mc) = R0(w, kmc, mc). Our

goal is again to use the ratios of R0(w) computed at successive heavy quark masses, and then reach the point corresponding to the physically relevant R0(w, mb, mc) through interpolation in inverse heavy quark mass. To this end, we rst form

0k(w) =

R0(w, k+1mc, mc, a2)

R0(w, kmc, mc, a2) , (38) where we indicate the momentum transfer w, the masses of quarks entering the weak vertex (mc, mh), and the fact that the 0k(w) are obtained at xed lattice spacing, a. In

Table 5 we present the results for 0k(w) for a specic value of w = 1.016. Before discussing the heavy quark mass

dependence we need to extrapolate to the continuum limit, lima0 0k(w) = 0k(w) by using a form similar to (28):

0k(w, msea, a2) = k(w) + k(w)

msea

ms

+ k(w)

a a=3.9

2, (39)

R0(w, mb) = 0n . . .

0k+1 0k

thus assuming the linear dependence on the dynamical (sea) quark mass and on the square of the lattice spacing. Since we work with maximally twisted QCD on the lattice, the leading discretization errors are proportional to a2 [28].

After inspection, we found again that the dependence of the form factors on the sea quark mass is indiscernible from our data and that our results depend very mildly on the lattice

spacing. Since the dependence on the sea quark mass is negligible, we again consider the continuum extrapolation by setting k(w) = 0, separately from the case in which k(w)

are left as free parameters. The net effect is that the error on 0k(w) = k(w, 0, 0) is considerably smaller in the case

with k(w) = 0 and the data better respect the heavy quark

mass dependence.

With several 0k(w) in hands, we need to discuss the heavy quark interpolation. We rst discuss its value in the innitely heavy quark mass limit. Using the heavy quark effective theory mass formula mBs,Ds = mb,c+ +(1+32)/mb,c [49

51], we can consider the ratio of form factors given in Eq. (8), knowing that h+(w) scales as a constant with inverse heavy

quark mass. One then deduces that, for the charm quark xed to its physical value,

R0(w, mh, mc) 1/mh. (40)

Equivalently,

R0(w, mh, mc) = mh R0(w, mh, mc), scales as a constant in the heavy quark mass limit, and the corresponding

0(w) can then be described by a form similar to Eq. (23) and the physically relevant

R0(w, mb) could be

obtained from

R0(w, kmc). (41)

We can also rewrite the above formula in terms of R0(w), as

n+1mc R0(w, mb) =

0n . . .

0k+1 0kkmc R0(w, kmc)

R0(w, mb) = n+k1

0k+1 0k

0n . . .

0

n ...0k+10k

R0(w, kmc), (42)

123

2861 Page 10 of 13

Eur. Phys. J. C (2014) 74:2861

1.10

1.00

1.05

0.95

fq2

1.00

0 (w 3 ,m )

h

0.90

(

0.95

2 +

f 0(q))

0.85

0.90

0.80

0.85

0.75

0.80

0.0 0.2

0.4 0.6 0.8

0 2 4 6 8 10

1 mh [GeV ]

1

q2Bs Ds [GeV2

]

Fig. 4 Fit of our data

to Eq. (43). Empty symbols denote the results computed in the continuum

by setting k(w) = 0 in Eq. (39). Filled

symbols correspond to the

results obtained after allowing k(w) = 0 in

Eq. (39). Plotted are the

data with w = w3 = 1.016

and therefore 0k =

0n/. In other words the interpolation formula to be used in

this case is

0(w, mh) =

Fig. 5 Results for R0(q2) = f0(q2)/f+(q2) presented in this paper in

the case of Bs Ds are linearly t to the form R0(q2) = 1 q2. As

in the previous plots, the empty/lled symbols correspond to the results obtained with k(w) = 0/ k(w = 0 in Eq. (39)

strange case (Bud Dud ) in which a constituent quark

model has been employed. That obviously did not allow one to keep track of the QCD anomalous dimension. However, as we shall see, their result [ fT (q2)/f+(q2) = 1.03(1)]

is rather close to what we obtain from our lattice simulations. Furthermore, in Ref. [56] it was found that this ratio RT (q2) = fT (q2)/f+(q2) is a at function of q2.

On the lattice, the extraction of the form factor fT (q2) is completely analogous to what we explained in the previous sections for f+(q2) and f0(q2). Heavy quark behavior of

fT (q2) is similar to that of f+(q2), which is simple to see

after applying the heavy quark equation of motion to the b-quark,2 1+/v2 b = b, which in the b rest frame reads 0b =

b, and therefore c0ib = i cib, so that the heavy quark

behavior of the form factor fT (q2) resembles that of f+(q2).

We again dene the ratios computed at two successive quark masses that differ by a factor of ,

Tk (w) =

1 +

s

1(w) mh +

s2(w) m2h

, (43)

where, again, our

= 1.176. An illustration of that interpo

lation is provided in

Fig. 4 for one specic case of w. We see that our results

obtained by assuming the independence of R0(w) on the sea

quark scale better with the heavy quark mass than those

obtained by letting the parameter k(w) in Eq. (39) free, although

the two are compatible within the

error bars.

Our nal results

for f0(q2)/f+(q2) with q2 = q2max

2mBs mDs (w 1),

are given in Table 4. Knowing that the form factors satisfy

the constraint f0(0) = f+(0), one can

then attempt to t

linearly in q2, as R0(q2) = 1 q2.

From the results

obtained with k = 0 we obtain =

0.021(1) GeV2, while from the data with k free, we get = 0.020(1) GeV2. This is illustrated in Fig. 5.

It is interesting to note that these results are consistent with the values that can be obtained from the results quoted in recent literature (in the non-strange case). More specifically, from the lattice results of Refs. [14,15] one nds = 0.020(1) GeV2, while from those reported in Ref. [13]

one nds = 0.022(1) GeV2. Recent QCD sum rule anal

yses give = 0.021(2) GeV2 [5254].

Note also that near zero recoil, the central value of our result R0(q2) = 0.77(2), coincides with the quark model

results of Refs. [55,56].

5.2 RT (q2)

To our knowledge, there is no QCD-based determination of the B(s) D(s) transition tensor form factor. The only

existing result is the one presented in Ref. [56] for the non-

RT (w, k+1mc, mc, a2)

RT (w, kmc, mc, a2) , (44) which we then extrapolate to the continuum limit by using

Tk (w, msea, a2) = k(w) + k(w)

msea

ms

+ k(w)

a a=3.9

2. (45)

Like in the previous cases, we observe that Tk (w) does not depend on the sea quark mass and its dependence on lattice spacing is insignicant within our error bars. For that reason we made the continuum extrapolation by imposing k(w) = 0 and by leaving k(w) as a free parameter.

2 We stress again that the c-quark mass in our simulations is always kept xed to its physical value.

123

Eur. Phys. J. C (2014)

74:2861 Page 11 of 13 2861

1.15

0 =

13, 1 =

439216, 2 = 4.002,

0 =

25 12, 1 =

77 24, 2 =

21943

3456 ,

where we also gave the rst few -function coefcients. Finally, in the computation we used MSNf=4 = 296(10) MeV

[58].

As can be seen from Table 4 the error on fT (q2)/f+(q2) is

getting larger for larger values of w. We are therefore unable to check on the atness of RT (q2) being valid in the innitely heavy quark mass limit (Table 6).

6 Summary and perspectives

In this paper we presented the results of our lattice QCD study of the exclusive semileptonic Bs Ds decay form fac

tors in the region near zero recoil (close to q2max). The method employed here is the one proposed in Ref. [16] that allows for circumventing the problem of extrapolation in the inverse heavy quark mass and to reach the physical answer through interpolation. This is achieved by studying the successive ratios of form factors computed with heavy b-quark mass differing by a xed factor of . In that way, in the continuum limit, these ratios have a xed value for mb and

instead of extrapolating, one interpolates to the (inverse) b-quark mass.

We rst computed the normalization to the vector form factor relevant to the extraction of the CKM matrix element

|Vcb| from B(Bs Ds ) with the light lepton in the nal

state {e, }. We obtained

G(1) = 1.052(46), (50) and we found that the method used here can also be employed to compute G(1) for the non-strange decay modes B(B

D ), provided the statistical sample of gauge eld congurations were larger. We also observe that the above error bar can be signicantly reduced if one imposed the condition that the form factor ratios and the form factors themselves do not depend on the mass of the dynamical (sea) quark, which is essentially what we see with all of our lattice data (at all values of the lattice spacing).

Thanks to the use of twisted boundary conditions imposed on the valence charm quark, we were able to explore the region of very small momenta given to Ds, and therefore to compute the form factors for small recoil momenta w [greaterorsimilar] 1.

Since we restrained our analysis to very small ws, we could not estimate the accurate value of the slope of the form factor

G(w).

Instead, we computed the ratio of the scalar to vector form factors, R0(q2) = f0(q2)/f+(q2), which is needed to inter

pret the recent discrepancy between the experimentally measured B(B D )/B(B D) and its theoretical

1.10

1.05

T w 3 ,m)

( h

1.00

0.95

0.90

0.85

0.0 0.2

0.4 0.6 0.8

1 m ]

h [GeV1

Fig. 6 Plot analogous to

Fig. 4 but for the case of T (w, mh) as given in Eq. (43). The vertical

gray line corresponds to the inverse b-quark

mass

Results of that

extrapolation, Tk (w) T (w, mh, mc) with

then interpolated in heavy quark mass to the b-quark, according

to

T (w, mh) = 1 + s1

mh = k+1mc, are

(w)mh + s2

(w) m2h

, (46)

which is shown in Fig.

6. As in the case of R0(q2), we also here need to extrapolate

one of the ratios RT (q2) to the continuum limit. We

checked that for either k = 2, or 3 or 4 we

end up with completely

consistent results for

RT (w, mb) = Tk

(w) . . . T8 (w) RT (w, k+1mc). (47)

The results given

in Table 4 are obtained by choosing k = 3. Furthermore,

we included the evolution of the tensor density from =

2 GeV, at which the renormalization constants have been

computed, to = mb by using

fT (q2, ) =

cT () cT (0)

fT (q2, 0) (48)

where

cloT() = as()0/0,

cnloT() = cloT()

1 +

10 01 20

as()

,

cnnloT() = cnloT() +

1

2cloT()

10 0120

2

+

2 0

as()2, (49)

and where we used as() s()/, for shortness. The

anomalous dimension coefcients are known to three loops in perturbation theory and for Nf = 4 in the MS scheme their

values are [57]:

210

+ 30

11 + 20 20

123

2861 Page 12 of 13 Eur. Phys. J. C (2014) 74:2861

Table 6 Same as in Table 5 but for the ratios of RT (w3) and computed following Eq. (45)

Ensemble T1 (w3) T2 (w3) T3 (w3) T4 (w3) T5 (w3) T6 (w3) T7 (w3) T8 (w3)

I 0.992(6) 0.988(8) 0.985(11) 0.983(18) 0.988 (31) 1.024 (61) 1.21 (18) 0.3 (4.6)

II 0.986(11) 0.979(14) 0.966(20) 0.942(31) 0.899(57) 0.82(13) 0.5(5) 1.8(9.7)

III 1.006(4) 1.006(5) 1.005(7) 1.003(11) 1.000(18) 0.994(32) 0.984(53) 0.942(95)

IV 1.010(5) 1.010(6) 1.010(7) 1.011(8) 1.012(11) 1.025(22) 1.049(52) 1.12(20)

V 0.998(4) 0.997(5) 0.996(6) 0.995(8) 0.994(12) 0.995(20) 0.978(43) 0.938(94)

VI 0.996(4) 0.993(5) 0.989(7) 0.985(10) 0.980(17) 0.972(28) 0.944(50) 0.86(11)

VII 1.002(3) 1.001(4) 1.001(5) 1.001(7) 1.000(12) 0.996(24) 0.998(50) 1.02(13)

VIII 1.002(3) 0.999(5) 1.003(6) 0.992(10) 1.019(17) 1.030(21) 1.101(50) 1.37(21)

IX 0.998(5) 0.997(6) 0.996(7) 0.996(10) 0.999(13) 1.007(18) 1.023(32) 1.080(69)

X 0.988(12) 0.982(14) 0.972(19) 0.958(25) 0.936(36) 0.896(58) 0.817(159) 0.62(50)

XI 1.000(1) 0.999(2) 0.998(2) 0.996(3) 0.994(5) 0.990(7) 0.986(12) 0.993(24)

XII 1.003(1) 1.002(1) 1.001(2) 1.001(2) 1.001(3) 0.998(5) 0.998(5) 0.993(8)

XIII 0.988(11) 0.982(15) 0.972(19) 0.958(26) 0.936(36) 0.896(58) 0.87(13) 0.62(50)

prediction within the Standard Model. Since the scalar form factor contribution to the decay rate is helicity suppressed in the Standard Model, it is much more signicant for the case of the -lepton in the nal state than in the case of .This contribution is very important in various NP scenarios.In this paper we computed f0(q2)/f+(q2) by using the same

method of ratios and by restraining our attention to the small recoil region. For several small ws, we quote

f0(q2)

f+(q2)

""""

q2=11.5 GeV2=

0.77(2). (51)

Finally, in the models of physics beyond the Standard Model in which the tensor coupling to a vector boson is allowed, a third form factor might become important. Here we provide the rst lattice QCD estimate of this (tensor) form factor fT (q2) with respect to the vector one, f+(q2). By employing

the same methodology as above, in the MS renormalization scheme and at = mb we obtainfT (q2)

f+(q2)

""""

q2=11.5 GeV2=

1.08(7). (52)

We attempted repeating the same analysis for the case of the non-strange decay mode and found f0(q20)/f+(q20) =

0.73(4) for q20 = 11.6 GeV2 (and f0(q20)/f+(q20) = 0.75(2)

if neglecting the dependence on the sea quark mass), in agreement with the results obtained in Refs. [1315]. Finally, in the non-strange case we get fT (q20)/f+(q20) = 1.06(12), which

becomes 1.10(7) if neglecting the dependence on the sea quark mass. These values show that the prospects of using this method for computing the form factors for the non-strange decay modes, B D , are promising provided the sta

tistical quality of the data is improved.

Acknowledgments We thank the ETMC for making their gauge eld congurations publicly available. We are also grateful to B. Blossier,

V. Lubicz, and O. Pne for discussions related to the subject of this paper. Computations of the relevant correlation functions are made on the HPC resources of IDRIS Orsay, thanks to the computing time given to us by GENCI (2013-056808). M.A. is grateful to the CNRS Liban for funding her research.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP3 / License Version CC BY 4.0.

References

1. O. Buchmuller, H. Flacher, Phys. Rev. D 73, 073008 (2006). [http://arxiv.org/abs/hep-ph/0507253

Web End =hep-ph/0507253 ]

2. C.W. Bauer, Z. Ligeti, M. Luke, A.V. Manohar, M. Trott, Phys. Rev. D 70, 094017 (2004). [http://arxiv.org/abs/hep-ph/0408002

Web End =hep-ph/0408002 ]

3. J. Beringer et al., Particle Data Group Collaboration. Phys. Rev. D 86, 010001 (2012)

4. P. Gambino, C. Schwanda, http://arxiv.org/abs/1307.4551

Web End =arXiv:1307.4551 [hep-ph]5. I. Caprini, L. Lellouch, M. Neubert, Nucl. Phys. B 530, 153 (1998). [http://arxiv.org/abs/hep-ph/9712417

Web End =hep-ph/9712417 ]

6. N. Isgur, M.B. Wise, Phys. Lett. B 232, 113 (1989)7. N. Isgur, M.B. Wise, Phys. Lett. B 237, 527 (1990)8. J.A. Bailey et al., Fermilab Lattice and MILC Collaborations. PoS LATTICE 2010, 311 (2010). http://arxiv.org/abs/1011.2166

Web End =arXiv:1011.2166 [hep-lat]

9. C. Bernard et al., Phys. Rev. D 79, 014506 (2009). http://arxiv.org/abs/0808.2519

Web End =arXiv:0808.2519 [hep-lat]

10. M. Okamoto, C. Aubin, C. Bernard, C.E. DeTar, M. Di Pierro, A.X. El-Khadra, S. Gottlieb, E.B. Gregory et al., Nucl. Phys. Proc. Suppl. 140, 461 (2005). [http://arxiv.org/abs/hep-lat/0409116

Web End =hep-lat/0409116 ]

11. D. Becirevic, N. Kosnik, Acta Phys. Polon. Suppl. 3, 207 (2010). http://arxiv.org/abs/0910.5031

Web End =arXiv:0910.5031 [hep-ph]

12. S. Hashimoto, A.X. El-Khadra, A.S. Kronfeld, P.B. Mackenzie, S.M. Ryan, J.N. Simone, Phys. Rev. D 61, 014502 (1999). [http://arxiv.org/abs/hep-ph/9906376

Web End =hep-ph/9906376 ]

13. J.A. Bailey, A. Bazavov, C. Bernard, C.M. Bouchard, C. DeTar, D. Du, A.X. El-Khadra, J. Foley et al., Phys. Rev. D 85, 114502 (2012) [Erratum-ibid. D 86, 039904 (2012)] http://arxiv.org/abs/1202.6346

Web End =arXiv:1202.6346 [hep-lat]

123

Eur. Phys. J. C (2014) 74:2861 Page 13 of 13 2861

14. G.M. de Divitiis, E. Molinaro, R. Petronzio, N. Tantalo, Phys. Lett.

B 655, 45 (2007). http://arxiv.org/abs/0707.0582

Web End =arXiv:0707.0582 [hep-lat]

15. G.M. de Divitiis, R. Petronzio, N. Tantalo, JHEP 0710, 062 (2007). http://arxiv.org/abs/0707.0587

Web End =arXiv:0707.0587 [hep-lat]

16. B. Blossier et al., ETM Collaboration. JHEP 1004, 049 (2010). http://arxiv.org/abs/0909.3187

Web End =0909.3187 [hep-lat]

17. P. Dimopoulos et al., ETM Collaboration. JHEP 1201, 046 (2012). http://arxiv.org/abs/1107.1441

Web End =1107.1441 [hep-lat]

18. P.F. Bedaque, Phys. Lett. B 593, 82 (2004). [http://arxiv.org/abs/nucl-th/0402051

Web End =arXiv:nucl-th/0402051 ]

19. G.M. de Divitiis, R. Petronzio, N. Tantalo, Phys. Lett. B 595, 408 (2004). http://arxiv.org/abs/hep-lat/0405002

Web End =hep-lat/0405002

20. C.T. Sachrajda, G. Villadoro, Phys. Lett. B 609, 73 (2005). http://arxiv.org/abs/hep-lat/0411033

Web End =hep-lat/0411033

21. D. Guadagnoli, F. Mescia, S. Simula, Phys. Rev. D 73, 114504 (2006). http://arxiv.org/abs/hep-lat/0512020

Web End =hep-lat/0512020

22. D. Becirevic, F. Sanlippo, JHEP 1301, 028 (2013). http://arxiv.org/abs/1206.1445

Web End =arXiv:1206.1445 [hep-lat]

23. D. Becirevic, F. Sanlippo, Phys. Lett. B 721, 94 (2013). http://arxiv.org/abs/1210.5410

Web End =arXiv:1210.5410 [hep-lat]

24. D. Becirevic, B. Blossier, A. Gerardin, A. Le Yaouanc, F. Sanlippo, Nucl. Phys. B 872, 313 (2013). http://arxiv.org/abs/1301.7336

Web End =arXiv:1301.7336 [hep-ph]

25. B. Blossier et al., ETM Collaboration. Phys. Rev. D 82, 114513 (2010). http://arxiv.org/abs/1010.3659

Web End =arXiv:1010.3659 [hep-lat]

26. P..Boucaud et al. [ETM Collaboration], Phys. Lett. B 650, 304 (2007). [http://arxiv.org/abs/hep-lat/0701012

Web End =hep-lat/0701012 ]

27. P. Boucaud et al. [ETM Collaboration], Comput. Phys. Commun. 179, 695 (2008). http://arxiv.org/abs/0803.0224

Web End =arXiv:0803.0224 [hep-lat]

28. R. Frezzotti, G.C. Rossi, JHEP 0408, 007 (2004). http://arxiv.org/abs/hep-lat/0306014

Web End =hep-lat/0306014 29. R. Sommer, Nucl. Phys. B 411, 839 (1994). http://arxiv.org/abs/hep-lat/9310022

Web End =hep-lat/9310022 30. M. Constantinou et al., ETM Collaboration. JHEP 1008, 068 (2010). http://arxiv.org/abs/arXiv:1004.1115

Web End =arXiv:1004.1115 [hep-lat]; http://arxiv.org/abs/1201.5025

Web End =arXiv:1201.5025 [hep-lat]

31. N. Uraltsev, Phys. Lett. B 585, 253 (2004). [http://arxiv.org/abs/hep-ph/0312001

Web End =hep-ph/0312001 ]32. Y. Amhis et al. [Heavy Flavor Averaging Group Collaboration], http://arxiv.org/abs/1207.1158

Web End =arXiv:1207.1158 [hep-ex]

33. A. Le Yaouanc, L. Oliver, J.C. Raynal, Phys. Rev. D 69, 094022 (2004). [http://arxiv.org/abs/hep-ph/0307197

Web End =hep-ph/0307197 ]

34. A. Le Yaouanc, L. Oliver, J.C. Raynal, Phys. Rev. D 67, 114009 (2003). [http://arxiv.org/abs/hep-ph/0210233

Web End =hep-ph/0210233 ]

35. A. Le Yaouanc, L. Oliver, J.C. Raynal, Phys. Lett. B 557, 207 (2003). [http://arxiv.org/abs/hep-ph/0210231

Web End =hep-ph/0210231 ]

36. J.P. Lees et al., BaBar Collaboration. Phys. Rev. Lett. 109, 101802 (2012). http://arxiv.org/abs/arXiv:1205.5442

Web End =arXiv:1205.5442 [hep-ex]; http://arxiv.org/abs/1303.0571

Web End =arXiv:1303.0571 [hep-ex]

37. S. Fajfer, J.F. Kamenik, I. Nisandzic, Phys. Rev. D 85, 094025 (2012). http://arxiv.org/abs/1203.2654

Web End =arXiv:1203.2654 [hep-ph]

38. S. Fajfer, J.F. Kamenik, I. Nisandzic, J. Zupan, Phys. Rev. Lett. 109, 161801 (2012). http://arxiv.org/abs/1206.1872

Web End =arXiv:1206.1872 [hep-ph]

39. A. Celis, M. Jung, X.-Q. Li, A. Pich, JHEP 1301, 054 (2013). http://arxiv.org/abs/1210.8443

Web End =arXiv:1210.8443 [hep-ph]

40. A. Crivellin, C. Greub, A. Kokulu, Phys. Rev. D 86, 054014 (2012). http://arxiv.org/abs/1206.2634

Web End =arXiv:1206.2634 [hep-ph]

41. P. Biancoore, P. Colangelo, F. De Fazio, Phys. Rev. D 87, 074010 (2013). http://arxiv.org/abs/1302.1042

Web End =arXiv:1302.1042 [hep-ph]

42. O. Eberhardt, U. Nierste, M. Wiebusch. http://arxiv.org/abs/1305.1649

Web End =arXiv:1305.1649 [hep-ph]43. J.A. Bailey et al., Phys. Rev. Lett. 109, 071802 (2012). http://arxiv.org/abs/1206.4992

Web End =arXiv:1206.4992 [hep-ph]

44. M. Tanaka, R. Watanabe, Phys. Rev. D 87, 034028 (2013). http://arxiv.org/abs/1212.1878

Web End =arXiv:1212.1878 [hep-ph]

45. I. Dorsner, S. Fajfer, N. Kosnik, I. Nisandzic. http://arxiv.org/abs/1306.6493

Web End =arXiv:1306.6493 [hep-ph]

46. Y. Sakaki, M. Tanaka, A. Tayduganov, R. Watanabe. http://arxiv.org/abs/1309.0301

Web End =arXiv:1309.0301 [hep-ph]

47. X.-G. He, G. Valencia, Phys. Rev. D 87, 014014 (2013). http://arxiv.org/abs/1211.0348

Web End =arXiv:1211.0348 [hep-ph]

48. D. Becirevic, N. Kosnik, A. Tayduganov, Phys. Lett. B 716, 208 (2012). http://arxiv.org/abs/1206.4977

Web End =arXiv:1206.4977 [hep-ph]

49. A.V. Manohar, M.B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000)

50. M. Neubert, Phys. Rep. 245, 259 (1994). [http://arxiv.org/abs/hep-ph/9306320

Web End =hep-ph/9306320 ]51. N. Uraltsev, in Heavy Flavor Physics: A Probe of Natures Grand Design. Proceedings of 137th Course of the International School of Physics Enrico Fermi, Varenna, Italy, 818 July 1997 (IOS Press, Amsterdam, Netherlands, 1998), pp. 329409. [http://arxiv.org/abs/hep-ph/9804275

Web End =hep-ph/9804275 ]

52. S. Faller, A. Khodjamirian, C. Klein, T. Mannel, Eur. Phys. J. C

60, 603 (2009). http://arxiv.org/abs/0809.0222

Web End =arXiv:0809.0222 [hep-ph]

53. K. Azizi, Nucl. Phys. B 801, 70 (2008). http://arxiv.org/abs/0805.2802

Web End =arXiv:0805.2802 [hep-ph]54. P. Blasi, P. Colangelo, G. Nardulli, N. Paver, Phys. Rev. D 49, 238 (1994). [http://arxiv.org/abs/hep-ph/9307290

Web End =hep-ph/9307290 ]

55. R.N. Faustov, V.O. Galkin, Phys. Rev. D 87, 034033 (2013). [http://arxiv.org/abs/1212.3167

Web End =arXiv:1212.3167 ]

56. D. Melikhov, B. Stech, Phys. Rev. D 62, 014006 (2000). [http://arxiv.org/abs/hep-ph/0001113

Web End =hep-ph/0001113 ]

57. J.A. Gracey, Phys. Lett. B 488, 175 (2000). [http://arxiv.org/abs/hep-ph/0007171

Web End =hep-ph/0007171 ]58. S. Bethke, Nucl. Phys. Proc. Suppl. 234, 229 (2013). http://arxiv.org/abs/1210.0325

Web End =arXiv:1210.0325 [hep-ex]

123