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arXiv:0903.1664v2 [hep-lat] 9 Nov 2009

DESY 09-029

Semileptonic form factors D → π,K and

B → π,K from a fine lattice

A. Al-Haydari,a

G. N. Lacagnina,c

A. Ali Khan,a

M. Panero,b,d

V. M. Braun,b

A. Sch¨ aferb

S. Collins,b

and G. Schierholzb,e

M. G¨ ockeler,b

aDepartment of Physics, Faculty of Science, Taiz University, Taiz, Yemen

bInstitute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany

cINFN, Sezione di Milano, 20133 Milano, Italy

dInstitute for Theoretical Physics, ETH Z¨ urich, 8093 Z¨ urich, Switzerland

eDeutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany

QCDSF Collaboration

Abstract

We extract the form factors relevant for semileptonic decays of D and B mesons from

a relativistic computation on a fine lattice in the quenched approximation. The lattice

spacing is a = 0.04 fm (corresponding to a−1= 4.97 GeV), which allows us to run very

close to the physical B meson mass, and to reduce the systematic errors associated with

the extrapolation in terms of a heavy quark expansion. For decays of D and Dsmesons,

our results for the physical form factors at q2= 0 are as follows: fD→π

fD→K

+

(0) = 0.78(5)(4) and fDs→K

+

(0) = 0.68(4)(3). Similarly, for B and Bs we find:

fB→π

+

(0) = 0.27(7)(5), fB→K

+

(0) = 0.32(6)(6) and fBs→K

our results with other quenched and unquenched lattice calculations, as well as with light-

cone sum rule predictions, finding good agreement.

+

(0) = 0.74(6)(4),

+

(0) = 0.23(5)(4). We compare

PACS numbers: 11.15.Ha, 12.38.Gc, 13.20.Fc, 13.20.He

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1Introduction

Heavy meson decays are the main source of precision information on quark flavor mixing pa-

rameters in the Standard Model. The over-determination of the sides and the angles of the

CKM unitarity triangle is the aim of an extensive experimental study: It addresses the question

whether there is New Physics in flavor-changing processes and where it manifests itself. One of

the sides of the unitarity triangle is given by the ratio |Vub/Vcb|. Vcbis known to approximately

2% accuracy from b → cℓνℓtransitions [1, 2] whereas the present error on Vubis much larger

and there is also some tension between the determinations from inclusive and exclusive decay

channels. Reduction of this error requires more experimental statistics but—even more so—an

improvement of the theoretical prediction of the semileptonic spectra and decay widths.

This is the prime motivation for the study of semileptonic form factors of decays of a heavy

meson H = B,D into a light pseudoscalar meson P = π,K, which are usually defined as

?P(p)|Vµ|H(pH)? =m2

H− m2

q2

P

qµf0(q2) +

?

pµ

H+ pµ−m2

H− m2

q2

P

qµ

?

f+(q2).(1)

Here Vµ= q2γµq1is the vector current in which q1(q2) denotes a light (heavy) quark field; p

(pH) is the momentum of the light (heavy) meson with mass mP (mH), and q := pH− p is the

four-momentum transfer. The f0(q2) and f+(q2) form factors are dimensionless, real functions

of q2(in the physical region), which encode the strong interaction effects. They are subject to

the kinematic constraint f+(0) = f0(0).

In the approximation of massless leptons (which is highly accurate for ℓ = e or ℓ = µ), the

differential decay rate for the H → Pℓνℓprocess involves f+(q2) only:

dΓ

dq2=G2

192π3m3

H

F|Vq2q1|2

??m2

H+ m2

P− q2?2− 4m2

Hm2

P

?3/2 ??f+(q2)??2.(2)

Another motivation for our study is that f0(q2) and f+(q2) enter as ingredients in the analysis

of nonleptonic two-body decays like B → ππ and B → πK in the framework of QCD factor-

ization [3, 4], with the objective to extract CP-violating effects and in particular the angle α

of the CKM triangle. One issue that is especially important in this respect is the question of

flavor SU(3) violation in the form factors of the decay B → π vs. the rare decay B → K.

High-statistics unquenched lattice calculations of D-meson (and also B-meson) decay form

factors in the kinematic region where the outgoing light hadron carries little energy (small

recoil region) have been performed recently [5, 6, 7] and attracted a lot of attention. Direct

simulations at large recoil, q2≪ m2

2 GeV, prove to be difficult and require a very fine lattice which is so far not accessible in

calculations with dynamical fermions. This problem is aggravated by the challenge to consider

heavy quarks which either calls for using effective heavy quark theory methods or, again, a

very fine lattice. In practice, one is forced to rely on extrapolations from larger momentum

transfer q2and/or smaller heavy quark masses. Several extrapolation procedures have been

suggested [8, 9, 10, 11, 12] that incorporate constraints from unitarity and the scaling laws in

the heavy quark limit. Alternatively, B-meson form factors in the region of large recoil have

been estimated using light-cone sum rules [13, 14] (for recent updates see [15, 16, 17, 18]).

In this paper we report on a quenched calculation of semileptonic H → Pℓνℓform factors

with lattice spacing a ∼ 0.04 fm using nonperturbatively O(a) improved Wilson fermions and

B, with light hadrons carrying large momentum of order

1

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O(a) improved currents. On such a fine lattice a relativistic treatment of the c quark is justified

and also the extrapolation to the physical b quark mass becomes much more reliable compared

to similar calculations on coarser lattices. In addition, we can explore possible subtleties in

approaching the continuum limit in form factor calculations: In our previous work [19] we did

find indications for a substantial discretization error in the decay constants fDsetc.; similar

conclusions have also been reached in Ref. [20]. This is particularly relevant in view of the

claims of evidence for New Physics from comparison with recent dynamical simulations—see,

e.g. Ref. [21] for a discussion.

On physical grounds, one may expect a nontrivial continuum limit because form factors at

large momentum transfer are determined by the overlap of very specific kinematic regions in

hadron wave functions (either soft end-point, or small transverse separation). The common

wisdom that hadron structure is very “smooth”—and that numerical simulations on a coarse

lattice could thus be sufficient to capture the continuum physics—may not work in this par-

ticular case. This can be especially important for SU(3) flavor-violating effects, which are of

major interest for the phenomenology. Inclusion of dynamical fermions and the approach to the

chiral limit are certainly also relevant problems, but not all issues can be addressed presently

within one calculation.

This work should be viewed as a direct extension of the investigation of the APE collaboration

in Ref. [22], who performed a quenched calculation with the same nonperturbatively O(a)

improved action and currents. Also their data analysis is similar. However, they use coarser

lattices with a ∼ 0.07 fm (β = 6.2). On the other hand, the spatial volume of their lattices is

very close to ours (L ∼ 1.7 fm). So the main difference lies in the lattice spacing, and a direct

comparison of the results is possible yielding information on the size of lattice artefacts, while

there is no need for us to perform simulations on a coarser lattice ourselves.

The presentation is organized as follows. Our strategy is discussed in detail in Sec. 2. It

allows us to run fully relativistic simulations for values of mHup to the vicinity of mB: This is

achieved by using a lattice characterized by a very fine spacing a. The extraction of physical

quantities from our data and the final results with the associated error budget are presented

in Sec. 3. The final Sec. 4 contains a summary and some concluding remarks. Some technical

details and intermediate results of our calculation are shown in the Appendix. Preliminary

results of this study have been presented in Refs. [23, 24].

2Simulation details

The lattice study of heavy hadrons is an issue that involves some delicate technical aspects:

The origin of the problem stems from the fact that, typically, the lattice cutoff is (much) smaller

than the mass of the B meson.

Common strategies to solve this problem are based on heavy quark effective theory (HQET),

i.e. expanding the relativistic theory in terms of m−1

One can simulate in the static limit [25] or keep correction terms in the action to simulate

at finite mQ(non-relativistic QCD or NRQCD) [26]. These approaches have been employed

effectively for studying B physics (see, for example, Refs. [27, 28, 29, 30]).

However, for smaller quark masses like the c quark in D mesons a large number of terms in

the expansion must be included, making the simulations less attractive. The Fermilab group

Q, where mQis the mass of the heavy quark.

2

Page 4

developed an approach which interpolates between the heavy- and light-quark regimes [31].

The coefficients accompanying each term in the action are functions of the quark mass and

in practice, normally, the lowest-level action is used.

improved relativistic action with a re-interpretation of the results. Except for HQET [32, 33],

the associated renormalization constants for these approaches are only known perturbatively.

We reduce the uncertainties related to the extrapolation to the physical heavy meson mass

by using lattices with a small spacing in conjunction with a non-perturbatively improved O(a)

relativistic quark action. This theoretically clean approach enables one to get sufficiently close

to the mass of the physical B meson, so that the heavy-quark extrapolation is short-ranged.

In addition, in the region of the D meson mass, the discretization errors are reduced to around

1%, see Section 3.1.

This corresponds to using the O(a)

L3× T

β

403× 80

6.6

0.04 fm

1.63× 3.2 fm4

4.97 GeV

114

0.135472(11)

0.13, 0.129, 0.121, 0.115

0.13519, 0.13498, 0.13472

526 MeV, 690 MeV, 856 MeV

1.467

0.8118

1.356

−0.0874

lattice spacing a

physical hypervolume

a−1

# of configurations

κcritical

κheavy

κlight

mP

cSW

ZV

bV

cV

Table 1: Parameters of the lattice calculation (see the text for the definition of the various

quantities).

Table 1 summarizes basic technical information about our study. We use the standard Wilson

gauge action to generate quenched configurations with the coupling parameter β = 6.6. For this

parameter choice, the lattice spacing in physical units determined from Ref. [34] using Sommer’s

parameter r0= 0.5 fm is a = 0.04 fm. Our calculation is based on the O(a) improved clover

formulation for the quark fields [35], with the nonperturbative value of the clover coefficient cSW

taken from Ref. [36]. We use O(a) improved definitions of the vector currents in the form [37]

Vµ= ZV

?

1 + bVamq2+ amq1

2

?

(q2γµq1+ iacV∂νq2σµνq1)(3)

with σµν =

as cV are known nonperturbatively [38, 39, 40]. All statistical errors are evaluated through

a bootstrap procedure with 500 bootstrap samples. We consider three hopping parameters

corresponding to “light” quarks, κlight (the corresponding masses of the light pseudoscalar

meson states mPare also given in Table 1), and four hopping parameters, κheavy, corresponding

i

2[γµ,γν]. The renormalization factor ZV, the improvement coefficient bV as well

3

Page 5

p

H

decay product

quark

boson

W

lepton

neutrino

spectator quark

p

q

heavy quark

t t = T/2

yx

H

Vµ

0

PS

S

Figure 1: Diagram representing the semileptonic decay of a heavy-light pseudoscalar meson

to a light pseudoscalar meson (left panel). A schematic representation of the corresponding

three-point correlator calculated on the lattice is also shown (right panel).

to “heavy” quarks; in particular, κ = 0.13498 and κ = 0.129 are found to correspond to quark

masses close to the physical strange and charm quark mass, respectively.

The extraction of the matrix element appearing in Eq. (1) from the lattice can be done

by considering the large time behavior of three-point correlation functions C(3)

pseudoscalar light meson sink at time t = 0, a vector current at time tx, and a pseudoscalar

heavy-light meson source at time ty= T/2 (see Fig. 1):

µ (0,tx,ty) for a

C(3)

µ(0,tx,ty) =

?

? x,? y

e−i? pH·? yei? q·? x?HS(? y,ty)Vµ(? x,tx)PS(0)? .(4)

Here, HSand PSare Jacobi-smeared operators of the form qhγ5qsand qlγ5qs, respectively; qh

denotes the heavy quark, qlis the decay-product quark, while qsis the “spectator” quark.

For sufficiently large time separations (i.e. 0 ≪ tx≪ T/2 or T/2 ≪ tx≪ T), C(3)

behaves as:

with ZS

light (heavy) meson. To extract the matrix elements we divide the three-point functions by the

prefactors, which are extracted from fits to smeared-smeared two-point functions. The matrix

element is then obtained by fitting this result to a constant, in an appropriate time range where

a clear plateau forms (for example, for 12 ≤ tx≤ 28).

We consider three-point functions associated with different combinations of the momenta p

and pH, which are listed in Table 2. In particular, we focus our attention onto three-momenta

of modulus 0 and 1 [in units of 2π/(aL)], since they yield the most precise signal, restricting

ourselves to the cases where ? p and ? pH lie in the same direction. Thus we measure directly

5 different values for the form factors, for every κlightand κheavycombination. The full form

factors can then be constructed from the data points obtained this way, by making an ansatz

for the functional form of f0(q2) and f+(q2).

In the present work, we fit our data with the parametrization proposed by Be´ cirevi´ c and

Kaidalov [8]:

f0(q2) =cBK· (1 − α)

1 − ˜ q2/β

µ (0,tx,ty)

C(3)

µ(0,tx,ty) −→

ZS

4EHEe−Etxe−EH(ty−tx)?H(pH)|Vµ|P(p)?

HZS

for tx< T/2

±ZS

HZS

4EHEe−E(T−tx)e−EH(tx−ty)?H(pH)|Vµ|P(p)? for tx> T/2

H= |?0|HS|H(pH)?| and ZS= |?0|PS|P(p)?|, while E (EH) denotes the energy of the

,(5)

,f+(q2) =

cBK· (1 − α)

(1 − ˜ q2)(1 − α˜ q2),(6)

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Page 6

? pH

? p? q

(0,0,0)

(1,0,0)

(0,0,0)

(1,0,0)

(1,0,0)

(1,0,0)

(−1,0,0)

(0,0,0)

(0,0,0)

(1,0,0)

(−1,0,0)

(2,0,0)

(0,0,0)

(1,0,0)

(0,0,0)

Table 2: Momentum combinations considered in the analysis of the three-point functions, in

units of 2π/(aL).

where ˜ q := q/mH⋆, mH⋆ being the mass of the lightest heavy-light vector meson.

The parametrization for the form factors given in Eq. (6) accounts for the basic properties

that come from the heavy-quark scaling laws in the limits of large and small recoil and also

satisfies the proportionality relation derived in Ref. [41]. It is also consistent with the trivial

requirement that the l.h.s. of Eq. (1) be finite for vanishing momentum transfer, which implies

f0(0) = f+(0). The results that we obtained for the three parameters entering Eq. (6) from a

simultaneous fit to f0and f+are presented in the Appendix.

Some alternative ans¨ atze for the functional form of f+(q2) were proposed in Refs. [15, 9, 10]

and are discussed in Ref. [42]: They yield results essentially compatible with each other and

with the Be´ cirevi´ c-Kaidalov parametrization Eq. (6). More recently, Bourrely, Caprini and

Lellouch [12] discussed the representation of f+(q2) as a (truncated) power series in terms of

an auxiliary variable z. A similar parametrization has also been recently used by the Fermilab

Lattice and MILC collaborations, see Refs. [43, 44] for a discussion.

3 Extraction of physical results

In order to extract physical results from our simulations, we follow a method analogous to

Ref. [22]. We first perform a chiral extrapolation in the light quark masses. For a given

quantity Φ [one of the BK parameters appearing in Eq. (6)], the extrapolation relevant for

decays to a pion is performed as follows: We fit the results obtained at different values of the

mass of the pseudoscalar state linearly in m2

P,

Φ = c0+ c1· m2

P, (7)

and extrapolate to m2

extrapolations are shown in Figure 2 for the case of Φ = f+(0), α and β at κheavy= 0.115. On

the other hand, for decays to a kaon, we hold the hopping parameter of one of the two final

quarks fixed to κ = 0.13498, which, for our configurations, corresponds to the physical strange

quark at a high level of precision [19], and perform a short-ranged extrapolation of the curve

obtained from the linear fit in m2

Then we perform the interpolation to the physical c quark mass in terms of a heavy-quark

expansion for the D (or Ds) meson decays, or the extrapolation to the physical b quark mass

for the B (or Bs) meson. For our data, the extrapolation of the heavy quark mass to the

physical b mass is short-ranged: for the heaviest κheavy= 0.115, it turns out that the inverse

of the pseudoscalar meson mass (with the light quark mass already chirally extrapolated to its

P= m2

π, where mπ is the mass of the physical pion. Examples of the

Pto the square of the mass of the physical K meson.

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Page 7

00.010.02 0.030.04

0.05

(amP)2

0

0.2

0.4

0.6

0.8

1

f+(0)

00.010.020.030.04

0.05

(amP)2

0

0.5

1

1.5

α

00.010.020.030.04

0.05

(amP)2

0

0.5

1

1.5

2

2.5

3

β

Figure 2: Extrapolation of the Be´ cirevi´ c–Kaidalov parameters to the chiral limit, for decays to

a pion, at a fixed value κheavy= 0.115. The parameters obtained for κdecay product= κspectator

are extrapolated linearly in m2

P. The extrapolated values are shown as the full black dots.

Decay

B,D → π

fitl0

l1

l2

–

χ2/d.o.f.

0.1377/2

0.021/1

0.3247/2

0.03813/1

0.4025/2

0.001888/1

linear

quadratic

linear

quadratic

linear

quadratic

4.1+1.3

−1.0

5.1+2.9

−2.1

4.9+1.1

−0.9

6.3+2.4

−1.9

3.4+0.9

−0.8

4.9+1.8

−1.3

−4.1+1.6

−9.3+6.9

−5.4+1.5

−12.2+6.3

−2.9+1.3

−11.0+4.6

−2.3

−9.6

5.9+8.5

−5.9

–

7.7+7.3

−5.4

–

9.7+5.4

−4.6

B,D → K

−1.9

−8.2

Bs,Ds→ K

−1.7

−6.0

Table 3: The coefficients obtained from the fits to m3/2

Eq. (8) for different decays.

Hf+(0) in powers of m−1

Haccording to

physical value) is about m−1

physical B meson. The extrapolation can be performed by taking advantage of the fact that, in

the infinitely heavy quark limit, the Be´ cirevi´ c–Kaidalov parameters appearing in Eq. (6) enjoy

certain scaling relations: cBK√mH, (β−1)mHand (1−α)mHare expected to become constant

in the mH→ ∞ limit. For finite mH, one can parametrize the scaling deviations in powers of

m−1

ϕ = l0+ l1· m−1

where ϕ ∈ {cBK√mH, (β −1)·mH, (1−α)·mH}. Note that, since f+(0) = cBK·(1−α), one

can also use ϕ = f+(0) · m3/2

The extrapolation of m3/2

vantage of simulating on a fine lattice, which allows us to probe a mass range very close to

the physical B meson mass. We compare the results obtained from an extrapolation to the

inverse of the physical B meson mass using either a first- or a second-order polynomial in m−1

for the fit function, finding consistency (within error bars), for all decays. The corresponding

fit results are listed in Table 3. In the following we refer to this first method as the “coefficient

extrapolation” method.

An alternative method to extract the physical form factors from the lattice data was pro-

posed by the UKQCD collaboration [45]. It consists of performing the chiral and heavy quark

extrapolations at fixed v·p = (m2

meson and p is the four-momentum of the light meson. The following steps are performed:

H= 0.243 GeV−1, to be compared with m−1

B= 0.189 GeV−1for the

H:

H+ l2· m−2

H+ ...(8)

H—which was, in fact, our choice.

Hf+(0) is presented in Figure 3. The figure clearly shows the ad-

H

H+m2

P−q2)/(2mH), where v is the four-velocity of the heavy

1. fit of the form factors measured from the lattice simulations to the parametrization in

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00.1 0.20.3

-1 [GeV-1]

0.4

0.50.6

0.7

mH

0

1

2

3

4

5

mH

3/2 f+(0) [GeV3/2]

B, D to π decays

00.10.20.3

-1 [GeV-1]

0.4

0.50.6

0.7

mH

0

1

2

3

4

mH

3/2 f+(0) [GeV3/2]

Bs, Ds to K decays

Figure 3: Left panel: The green (red) squares denote the interpolated (extrapolated) form

factor m3/2

(solid line). A quadratic fit to the data is also shown (dashed line). Right panel: the results

for the case of decay of a Ds(Bs) meson into a kaon.

Hf+(0) to the physical D (B) meson, for a decay to a pion, using a linear fit in 1/mH

Eq. (6);

2. interpolation of the form factors at given values of v · p within the range of simulated

data;

3. chiral extrapolation of the points thus obtained, via a linear extrapolation in m2

m2

Pto either

πor m2

K(as described above);

4. linear or quadratic extrapolation in m−1

for the quantities:

Hto the inverse of the physical heavy meson mass

?αs(mB)

αs(mH)

?−

˜ γ0

2β0f0(v · p)√mH,

?αs(mB)

αs(mH)

?−

˜ γ0

2β0 f+(v · p)

√mH

(9)

which enjoy scaling relations at fixed v·p [46, 47]. Here, β0is the first β-function coefficient,

while ˜ γ0= −4 denotes the leading-order coefficient of the anomalous dimension for the

vector current in HQET. It yields a (subleading) logarithmic dependence on mH—see

also Refs. [22, 45] for further details;

5. final fit of the points thus obtained to the parameterization in Eq. (6).

For comparison, we also calculate the physical form factors using this alternative approach,

finding consistent results. This is illustrated in Table 4 which summarizes the results for f+(0)

from both methods.

For our final results we take those obtained from the coefficient extrapolation method. We

found this method to be superior in our case as the UKQCD method suffered from the fact that

there was only a small region of overlap in the ranges of v · p for the form factors at different

7

Page 9

κlightand κheavy. In addition, since the data can be fitted with both a linear and quadratic

function in m−1

differences in the results from the linear and quadratic fit to estimate the systematic errors, as

discussed in the next section. Our results for the form factors at finite q2are shown in Figs. 4

and 5.

H, we use the linear fits for the central values and statistical errors and use the

Coefficient extrapolation

linear in m−1

H

0.74+6

−6

0.78+5

−5

0.68+4

−4

0.27+8

−6

0.32+6

−5

0.23+5

−4

UKQCD method

linear in m−1

H

0.69+5

−5

0.75+4

−5

0.68+4

−4

0.29+13

−8

0.35+11

−8

0.23+6

−5

Decay

D → π

D → K

Ds→ K

B → π

B → K

Bs→ K

Final results for the physical values of the f+(0) form factor, for different decays,

with statistical errors only. We compare the results obtained from the coefficient extrapolation

and UKQCD methods as well as different truncations of the heavy quark expansion when

extrapolating or interpolating in m−1

quadratic in m−1

0.73+5

0.77+5

0.67+4

0.30+11

0.35+9

0.26+7

H

quadratic in m−1

0.69+5

0.75+4

0.67+4

0.31+15

0.34+12

0.27+8

H

−6

−6

−5

−5

−4

−4

−8

−10

−8

−9

−5

−6

Table 4:

H.

3.1Systematic uncertainties

Systematic uncertainties affecting our lattice calculation include: the quenched approximation,

the method to set the quark masses, the chiral extrapolation for the light quarks, discretization

effects, the extrapolation (interpolation) of the heavy quark to the physical b (c) mass, finite

volume effects, uncertainties in the renormalization coefficients, and effects related to the model

dependence for f0,+(q2). Let us now consider each source of error in turn.

Quenched approximation: the size of the error this approximation introduces is not

known. However, one can take as an estimate the variation in the results if different quantities

are used to set the scale. In the quenched approximation different determinations of the lattice

spacing vary by approximately 10% [48]. By repeating the full analysis, we find that varying

the lattice spacing by 10% induces an uncertainty of approximately 2% for the D → π decay,

and of approximately 12% for B → π.

Setting the quark masses: we use the κ values corresponding to the light (u/d) and

strange quarks determined in Ref. [19]: κl= 0.135456(10) and κs= 0.134981(9). The uncer-

tainty in these determinations leads to a very small uncertainty in the form factors. For the

c and b quarks we do not quote the corresponding κ values. We interpolate (or extrapolate)

our results directly to the physical masses of the pseudoscalar heavy-light states. The result-

ing uncertainty is determined by the statistical errors of the masses used for the interpolation

(or extrapolation). The latter are found to contribute only a negligible amount to the overall

systematic uncertainty.

Chiral extrapolation: the method we used to perform the chiral extrapolation of our

simulation results is discussed above. Note that the use of a large lattice practically constrains

us to use only a few and relatively large values for the light quark mass (so that the masses

of our lightest pseudoscalar mesons are far from the physical pion mass). However, as the

8

Page 10

-2 -1.5 -1 -0.50

0.5

1

1.5

2

2.5

q2 [GeV2]

0.5

1

1.5

2

f+

f0

D → π

QCDSF (this work)

Abada et al. [22]

Aubin et al. [5]

-2 -1.5 -1 -0.50

0.5

1

1.5

2

2.5

q2 [GeV2]

0.5

1

1.5

2

f+

f0

D → K

QCDSF (this work)

Abada et al. [22]

Aubin et al. [5]

-2 -1.5 -1 -0.50

0.5

1

1.5

2

2.5

q2 [GeV2]

0.5

1

1.5

2

f+

f0

Ds→ K

Figure 4: Physical form factors for D and Dsdecays as a function of q2from this work and

other quenched and dynamical studies. The solid black lines are the form factors obtained from

the coefficient extrapolation method where Eq. (8) has been truncated at O(m−1

dashed black lines indicate the error on the form factors. The range of v · p values achieved in

our simulations approximately corresponds to −1.5 GeV2? q2? 2 GeV2. The dashed red lines

are the results for the coefficient extrapolation method from Ref. [22]. The open red squares

and circles are their results obtained using the UKQCD method.

H), while the

examples in Figure 2 show, the dependence of our results on the light quark mass is rather

mild. So the size of the uncertainty arising from the chiral extrapolation though difficult to

estimate is unlikely to be large.

Discretization effects: as it was already remarked above, the leading discretization effects

in our calculation are reduced to O(a2); given that our lattice is very fine (a = 0.04 fm), the

associated systematic error can be estimated to be of the order of 1% (10%) for the decays of

charmed (beautiful) mesons [19].

Extrapolation/interpolation of the heavy quark: our data can be fitted to both a

linear and quadratic function in m−1

fit for our final results and the difference between the linear and quadratic fit as an indication

of the systematic uncertainty. This leads to approximately a 1% uncertainty for D decays and

Hwith a reasonable χ2. We use the results for the linear

9

Page 11

0

5

10

q2 [GeV2]

15

20

25

0

0.5

1

1.5

2

2.5

3

3.5

4

f+

f0

B → π

QCDSF (this work)

Abada et al. [22]

UKQCD [45]

Fermilab and MILC [44]

HPQCD [7]

0

5

10

q2 [GeV2]

15

20

25

0

0.5

1

1.5

2

2.5

3

3.5

4

f+

f0

B → K

0

5

10

q2 [GeV2]

15

20

25

0

0.5

1

1.5

2

2.5

3

3.5

4

f+

f0

Bs→ K

Figure 5: Same as in Fig. 4, but for B and Bs decays; in this case, the v · p values of our

simulations are in the range 14 GeV2? q2? 23 GeV2. For B → π, the dashed and solid

magenta lines in the range q2= 0 − 14 GeV2indicate the prediction from light-cone sum

rules [15, 16].

8% uncertainty for B decays.

Finite volume effects: for our calculation, finite-volume effects are not expected to be

severe; in particular, the correlation length associated with the lightest pseudoscalar state that

we simulated (for κlight= 0.13519) corresponds to approximately 9 lattice spacings, which is

more than four times shorter than the spatial extent of our lattice. Systematic infrared effects

can thus be quantified around 1–2%. This is comparable with the estimate of Ref. [44], in

which, using chiral perturbation theory [49, 50], the finite volume effects for their calculation

with 2 + 1 flavors of staggered quarks and values of mPL between 4 and 6 are estimated to be

less than 1%.

Renormalization coefficients: the uncertainty associated with the ZV coefficient, as de-

termined in Ref. [38] for the quenched case, is about 0.5%. The same article also quotes a

10

Page 12

1% uncertainty for bV, which induces an error about 1% for decays of D mesons and about

3% for B mesons. Concerning cV, a look at the results displayed in Fig. 2 of Ref. [39] would

suggest that the relative error in the region of interest (g2

30%; however, it should be noted that cV itself is a relatively small number, of the order of

9%, and the impact of the uncertainty on cV on our results is about 1% (2%) for decays of D

(respectively: B) mesons.

Model dependence: finally, the systematic effect related to the ansatz to parametrize the

form factors was estimated in Ref. [42], through a comparison of different functional forms that

satisfy analogous physical requirements. For the B → π decay, it turns out to be of the order

of 2%.

Combining the systematic errors in quadrature we arrive at an overall error of 5% for D

decays and about 18% for B decays.

0≃ 0.91) may be quite large, around

3.2Comparison with previous results

Our results can be compared to other lattice calculations of these quantities and also with

results of light-cone sum rules (LCSR) [13, 14]. Table 5 summarizes the comparison for f+(0),

while for finite q2the form factors from other studies are displayed in Figs. 4 and 5. In the

following we discuss in detail the comparison with these works, highlighting the advantages and

limitations of the different approaches, as well as the possible sources of discrepancies.

Our results can be closely compared with those obtained by the APE collaboration in

Ref. [22], reporting a calculation very similar to ours. They worked in the quenched approx-

imation, using the same non-pertubatively O(a) improved action and currents and a similar

analysis; on the other hand, their simulations were performed on a coarser lattice, with β = 6.2,

yielding a lattice spacing a = 0.07 fm, or a−1≃ 2.7 GeV. The table and figures show that their

values for the form factors lie around 3σ (D → π) and 2.5σ (D → K) below our results, in terms

of the statistical errors, in the region of q2= 0. If we adjust the APE results to be consistent

with setting the lattice spacing using r0instead of the mass of the K∗(used in Ref. [22]), the

discrepancy reduces slightly, down to roughly 2.5σ (D → π) and 2σ (D → K). Assuming that

O(a2) errors are the dominant source of the discrepancy, the difference in the results of the two

studies is consistent with an upper limit on the discretization errors of approximately 0.08, or

slightly above 1σ in our results for fD→π

+

(0) and 0.23 or 3 − 4σ in the APE results.

For B decays we are not able to make such a close comparison, because the study in Ref. [22]

extrapolates to the B meson from results in the region of 1.7 − 2.6 GeV for the heavy-light

pseudoscalar meson mass.Although one would expect larger discretization effects for the

B decay form factors, we find close agreement between our values and those from the APE

collaboration. However, we should point out that any potential discrepancy may be masked by

the long-ranged extrapolation in the heavy quark mass.

Several unquenched calculations have been performed recently, which are based on the MILC

Nf= 2+1 dynamical rooted staggered fermions configurations [51]. Results are available from

joint works from the Fermilab, MILC and HPQCD collaborations for D decays [5], and from

Fermilab and MILC [52, 44] and (separately) HPQCD [7] for B decays. These results were

obtained using the MILC “coarse” lattices with a = 0.12 fm for D decays and including a finer

lattice with a = 0.09 fm for the B decays. While these lattices are much coarser than those

used in both our and the APE study a detailed analysis of the chiral extrapolation was possible

11

Page 13

through the use of 5 light quark masses for the 0.12 fm lattice (only two values were used for

a = 0.09 fm).

The Fermilab, MILC and HPQCD joint work for D → π and D → K used an improved

staggered quark action (“Asqtad”) [51] for the light quarks and the Fermilab action for the

heavy quark. To the order implemented in the study, the Fermilab action corresponds to a re-

interpretation of the clover action. This approach can be used to simulate directly at the charm

and bottom quark mass at the expense of more complicated discretisation effects. Discretisation

errors arising from the final state energy (5%) and the heavy quark (7%) are estimated to lead

to the largest systematic uncertainties in the calculation (compared to the 3% error from the

chiral extrapolation). Given the coarseness of the lattice used, repeating the analysis on a

much finer lattice would enable the estimates of the the discretisation errors to be confirmed.

Overall, the results are consistent with ours, which suggests that the systematic effects due to

the quenched approximation are not the dominant source of error.

For the decay B → π, Fermilab and MILC used the same quark actions as for the study

of D decays. Using the 5 light quark masses at a = 0.12 fm and 2 light quark masses at

a = 0.09 fm they performed a joint continuum and chiral extrapolation which removed some

of the discretisation effects. They estimated that a 3% discretisation error arising from the

heavy quark remains after the extrapolation. The results at finite q2are compared with ours

in Figure 5, with statistical and chiral extrapolation errors only (which cannot be separated).

A value for f+(0) is not given in Ref. [44] which focuses on extracting |Vub| at finite q2using

the parameterisation of Bourrely, Caprini and Lellouch [12]. However, an earlier analysis on

the 0.12 fm lattices only was reported in Ref. [52]. Their result for f+(0) is given in Table 5.

HPQCD performed the calculations for the B → π decay on the MILC configurations using

Asqtad light quarks and NRQCD for the b quark. Use of the latter enables direct simulations

at the b quark mass. However, as NRQCD is an effective theory the continuum limit cannot

be taken and scaling in the lattice results must be demonstrated at finite a. Results from the

coarse lattice are shown in Figure 5, with statistical and chiral extrapolation errors only and for

f+(0) in Table 5. A limited comparison of results on the finer lattice for one light quark mass

did not indicate that the discretisation errors are large. The systematic errors are dominated

by the estimated 9% uncertainty in the renormalisation factors which are calculated to 2 loops

in perturbation theory.

The Fermilab-MILC and HPQCD results are consistent with each other to within 2σ and

are also consistent with our results and those of the APE collaboration. As for the studies of

D decays this suggests that quenching is not the dominant systematic error in the calculation

of B → π decay. Similarly, unquenched results on finer lattices are needed to investigate the

discretisation effects. Finally, note that in order not to overload Figures 4 and 5, we do not

show the (older) quenched results of the Fermilab group [53]. For B → π decays these results

are within the range of the other existing calculations, whereas for D-decays the form factors

come out 10 − 20% larger compared to most other calculations and also the new unquenched

results obtained with similar methods.

A different type of comparison can be made with the estimates obtained in the framework of

LCSR. This analytical approach is, to some extent, complementary to lattice calculations, since

it allows one to calculate the form factors directly at large recoil, albeit with some assumptions.

Figure 5 compares our extrapolation of the fB→π

+

with the direct LCSR calculation [15, 16]. Their predictions are compatible with our results.

(q2) form factor in the region q2< 12 GeV2

12

Page 14

Similar consistency is found between lattice and LCSR calculations of f+(0), as seen in Table 5,

for both B and D decays. Note that the uncertainty quoted for f+(0) for B decays is smaller

than that for D meson decays, and comparable with the precision of the lattice results. However,

while LCSR provides a systematic approach for calculating these quantities it is by definition

approximate and the errors cannot be reduced below 10 − 15%, unlike the lattice approach,

which is systematically improveable.

Decay

D → π

This work

0.74(6)(4)

Other results

0.64(3)(6)

0.57(6)(1)

0.65(11)

0.63(11)

0.73(3)(7)

0.66(4)(1)

0.78(11)

0.75(12)

Source Method

Fermilab-MILC-HPQCD [5]

APE [22]

Khodjamirian et al. [54]

Ball [55]

Fermilab-MILC-HPQCD [5]

APE [22]

Khodjamirian et al. [54]

Ball [55]

Nf= 2+1 LQCD

Nf= 0 LQCD

LCSR

LCSR

Nf= 2+1 LQCD

Nf= 0 LQCD

LCSR

LCSR

D → K0.78(5)(4)

Ds→ K

B → π

0.68(4)(3)

0.27(7)(5)0.23(2)(3)

0.31(5)(4)

0.26(5)(4)

0.258(31)

0.26(4)

0.26(5)

0.331(41)

0.36(5)

0.33(8)

0.30(4)

Fermilab-MILC [52]

HPQCD [7]

APE [22]

Ball and Zwicky [15]

Duplanˇ ci´ c et al. [16]

Wu and Huang [18]

Ball and Zwicky [15]

Duplanˇ ci´ c et al. [17]

Wu and Huang [18]

Duplanˇ ci´ c et al. [17]

Nf= 2+1 LQCD

Nf= 2+1 LQCD

Nf= 0 LQCD

LCSR

LCSR

LCSR

LCSR

LCSR

LCSR

LCSR

B → K0.32(6)(6)

Bs→ K

Table 5: Comparison of the results for f+(0) of the present work with other calculations,

obtained from lattice QCD (LQCD) simulations or from light-cone sum rules (LCSR) by various

groups. Where two errors are quoted the first is statistical and the second is the combined

systematic errors.

0.23(5)(4)

4Conclusions

In this article we have presented a lattice QCD calculation of the form factors associated with

semileptonic decays of heavy mesons.

We have performed a quenched calculation on a very fine lattice with β = 6.6 (a = 0.04

fm), which allows us to treat the D meson decays in a fully relativistic setup, and to get

close to the region corresponding to the physical B meson mass. The importance of small

lattice spacings for heavy-quark simulations has recently become clear in the context of the

determination of fDs, the decay constant of the Dsmeson. In spite of O(a) improvement, a

continuum extrapolation linear in a2seems to be reliable only for lattice spacings below about

13

Page 15

0.07 fm in the quenched approximation [19, 20]. Depending on the particular improvement

condition, even a non-monotonous a dependence can appear on coarser lattices.

In this work we have investigated to which extent the systematic effects caused by lattice

discretization and long-ranged extrapolations to the physical heavy meson masses may influence

the results of different lattice calculations in which all other sources of systematic errors are

treated in a similar way. For these reasons, the results of our study can be directly compared

with those by the APE collaboration in Ref. [22], which reports a very similar calculation on a

coarser lattice at β = 6.2 (a ≃ 0.07 fm) with the same lattice action and currents. Adjusting

the APE results so that they comply with our procedure for setting the physical value of the

lattice spacing, we find quite large discrepancies of roughly 2.5σ (D → π) and 2σ (D → K). If

we assume that O(a2) errors are the dominant source of this effect, the difference in the results

of the two studies suggests an upper limit on the discretization errors of approximately 0.08 or

slightly above 1σ in our numbers for fD→π

+

(0) and 0.23 or 3 − 4σ in the APE results.

It is, however, to be noted that the interpretation of this difference as a mere discretization

error is somewhat more ambiguous than in the case of the decay constants considered in [19, 20],

because the momentum transfer q2adds another parameter that has to be adjusted before the

comparison can be attempted. The corresponding comparison for B decays can, in addition,

be undermined by the long-ranged extrapolations in the heavy quark mass and/or q2. These

results suggest that, for high-precision phenomenological applications, completely reliable rel-

ativistic lattice calculations of these form factors could require even finer spacings, and that,

for dynamical simulations at realistic pion masses, this goal might be difficult to achieve in the

near future. While we believe that the progress in computational power will eventually allow

one to realize this formidable task, it is fair to say that, for the moment, the less demanding

approaches which interpolate between the D meson scale and non-relativistic results provide a

valid alternative.

Finally, a few words are in order about the general perspective for calculations of the semilep-

tonic form factors of heavy mesons. Form factors of B decays at small values of the relativistic

momentum transfer q2involve a light meson with momentum up to 2.5 GeV in the final state

and are very difficult to calculate on the lattice, mainly because no lattice effective field theory

formulation is known for this kinematics that would allow for the consistent separation of the

large scales of the order of the heavy quark mass, as implemented in the Soft-Collinear Effective

Theory.

Thus one is left with two choices. The first one is to calculate the form factors at moderate

recoil (m2

extrapolate to large recoil (m2

of this approach is that the calculations can be performed on relatively coarse and thus not

very large (in lattice units) lattices. Therefore dynamical fermions may be included, high

statistical accuracy can be achieved as well as a better control over the chiral extrapolation.

The disadvantage is that a reliable extrapolation from the q2> 12−15 GeV2regime accessible

in this method to q2= 0 may be subtle. However, this problem may be alleviated by a promising

new approach, “moving NRQCD” [56], which formulates NRQCD in a reference frame where the

heavy quark is moving with a velocity v. By giving the B meson significant spatial momentum,

relatively low q2can be achieved for lower values of the final state momentum thus avoiding

large discretisation effects.

For the particular case of the B → π semileptonic decay the problem of simulating at

B− q2∼ O(mBΛQCD)) using, e.g., the HQET or NRQCD expansion and then to

B−q2∼ O(m2

B)) guided by the dispersion relations. The advantage

14

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