arXiv:hep-lat/0506030v2 8 Sep 2005
Charmed-Meson Decay Constants in Three-Flavor Lattice QCD
C. Aubin,1C. Bernard,2C. DeTar,3M. Di Pierro,4E. D. Freeland,5Steven Gottlieb,6U. M. Heller,7
J. E. Hetrick,8A. X. El-Khadra,9A. S. Kronfeld,10L. Levkova,6P. B. Mackenzie,10D. Menscher,9F. Maresca,3
M. Nobes,11M. Okamoto,10D. Renner,12J. Simone,10R. Sugar,13D. Toussaint,12and H. D. Trottier14
(Fermilab Lattice, MILC, and HPQCD Collaborations)
1Physics Department, Columbia University, New York, New York, USA
2Department of Physics, Washington University, St. Louis, Missouri, USA
3Physics Department, University of Utah, Salt Lake City, Utah, USA
4School of Computer Science, Telecommunications and Information Systems, DePaul University, Chicago, Illinois, USA
5Liberal Arts Department, The School of the Art Institute of Chicago, Chicago, Illinois, USA
6Department of Physics, Indiana University, Bloomington, Indiana, USA
7American Physical Society, Ridge, New York, USA
8Physics Department, University of the Pacific, Stockton, California, USA
9Physics Department, University of Illinois, Urbana, Illinois, USA
10Fermi National Accelerator Laboratory, Batavia, Illinois, USA
11Laboratory of Elementary-Particle Physics, Cornell University, Ithaca, New York, USA
12Department of Physics, University of Arizona, Tucson, Arizona, USA
13Department of Physics, University of California, Santa Barbara, California, USA
14Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada
(Dated: September 8, 2005)
We present the first lattice QCD calculation with realistic sea quark content of the D+-meson
decay constant fD+. We use the MILC Collaboration’s publicly available ensembles of lattice gauge
fields, which have a quark sea with two flavors (up and down) much lighter than a third (strange).
We obtain fD+ = 201±3±17 MeV, where the errors are statistical and a combination of systematic
errors. We also obtain fDs= 249 ± 3 ± 16 MeV for the Ds meson.
PACS numbers: 13.20.Fc,12.38.Gc
Flavor physics currently plays a central role in elemen-
tary particle physics . To aid the experimental search
for physics beyond the standard model, several hadronic
matrix elements must be calculated nonperturbatively
from quantum chromodynamics (QCD). One of the most
important of these is the decay constant of the B meson
fB. Any framework for calculating fBshould, there-
fore, be subjected to stringent tests, and such a test is a
key aim of this Letter.
The most promising method for these nonperturbative
calculations is numerical lattice QCD. For many years
the results suffered from an unrealistic treatment of the
effects of sea quarks. In the last few years, however, this
obstacle seems to have been removed: with three flavors
of sea quarks lattice QCD now agrees with experiment for
a wide variety of hadronic quantities . This validation
of lattice QCD has been realized, so far, only for so-called
“gold-plated” quantities: masses and matrix elements of
the simplest hadronic states. Note, however, that many
of the hadronic matrix elements relevant to flavor physics
are in this class, including fB.
The challenges in computing fB are essentially the
same for the D+-meson decay constant fD+.
ments have observed the leptonic decay D+→ l+νl, but
not B+→ l+νl.
where Vcd is an element of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix. Taking |Vcd| from elsewhere,
one gets fD+. In 2004 the CLEO-c Collaboration mea-
One can, thus, determine |Vcd|fD+,
sured fD+ with a 20% error , and a more precise mea-
surement is expected soon.
This Letter reports the first lattice-QCD calculation of
fD+ with three flavors of sea quarks . We find
fD+ = 201 ± 3 ± 6 ± 9 ± 13 MeV,
where the uncertainties are statistical, and a sequence of
systematic effects, discussed below. We also obtain the
decay constant of the Dsmeson,
fDs= 249 ± 3 ± 7 ± 11 ± 10 MeV.
The second result is more precise than a recent lattice-
QCD calculation with the same sea quark content but
non-relativistic heavy quarks, which found fDs= 290 ±
20 ± 41 MeV . These results are more reliable than
older calculations  because we now incorporate (three)
sea quarks and, for fD+, also because the light valence
quark masses are smaller than before.
These results test the methods of Ref.  because they
are predictions. The input parameters have been fixed
previously [3, 8, 9, 10, 11], and, once comparably precise
experimental measurements become available, one can
see how Eqs. (1) and (2) fare. Indeed, this work is part of
a program to calculate matrix elements for leptonic and
semileptonic decays [10, 12, 13], neutral-meson mixing,
and quarkonium [11, 14]. So far, these lattice QCD cal-
culations agree with experiment for the normalization of
TABLE I: Notation for quark masses used in this Letter.
mu Physical up quark
Physical down quark
mh Simulation’s heavier sea quark
Simulation’s lighter sea quark
Simulation’s light valence quark 0.1ms ≤ mq ? ms
Physical strange quark
From mDs[10, 11]
mu = ms/45.5 
md= ms/19.6 
0.1ms ≤ ml? 0.8ms
D-meson semileptonic form factors [12, 15, 16]. They also
have predicted correctly the form-factor shape [12, 17],
as well as the mass of the Bcmeson [14, 18].
In this set of calculations we use ensembles of un-
quenched lattice gauge fields generated by the MILC Col-
laboration [9, 19], with lattice spacing a = 0.175, 0.121,
and 0.086 fm. The key feature of these ensembles is that
they incorporate three flavors of sea quarks, one whose
mass is close to that of the strange quark, and two with
a common mass taken as light as possible.
For the sea quark and light valence quark we use the
“Asqtad” staggered-fermion action . Several differ-
ent quark masses appear in this calculation; for conve-
nience, they are defined in Table I. At a = 0.175, 0.121,
and 0.086 fm there are, respectively, 4, 5, and 2 ensem-
bles with various sea quark masses (ml,mh) [9, 19]. The
larger simulation mass, mhis close to the physical strange
quark mass ms. The light pair’s mass mlis not as small
as those of the up and down quark in Nature, but the
range 0.1ms ≤ ml ? 0.8ms suffices to control the ex-
trapolation in quark mass with chiral perturbation the-
ory (χPT). For carrying out the chiral extrapolation, it
is useful to allow the valence mass mqto vary separately
from the sea mass . At a = 0.175, 0.121, and 0.086 fm
we have, respectively, 6, 12, and 8 or 5 values of the va-
lence mass, in the range 0.1ms≤ mq? ms.
A drawback of staggered fermions is that they come
in four species, called tastes. The steps taken to elim-
inate three extra tastes per flavor are not (yet) proven,
although there are several signs that they are valid. Cal-
culations of fD+ and fDsare sensitive to these steps:
if Eqs. (1) and (2) agree with precise measurements, it
should be more plausible that the techniques used to re-
duce four tastes to one are correct.
For the charmed quark we use the Fermilab action for
heavy quarks . Discretization effects are entangled
with the heavy-quark expansion, so we use heavy-quark
effective theory (HQET) as a theory of cutoff effects .
This provides good control, as discussed in Ref. , and
the framework has been tested with the (successful) pre-
diction of the Bcmeson mass . Nevertheless, heavy-
quark discretization effects are the largest source of sys-
tematic error in fDs, and the second-largest in fD+.
The decay constant fDq, for a Dq meson with light
valence quark q and momentum pµ, is defined by 
?0|Aµ|Dq? = ifDqpµ, (3)
where Aµ = ¯ qγµγ5c is an electroweak axial vector cur-
rectly from the lattice Monte Carlo calculations.
usual in lattice gauge theory, we compute two-point cor-
relation functions C2(t) = ?O†
?A4(t)ODq(0)?, where ODqis an operator with the quan-
tum numbers of the charmed pseudoscalar meson, and
A4is the (lattice) axial vector current. The operators are
built from the heavy-quark and staggered-quark fields as
in Ref. . We extract the Dq mass and the ampli-
tudes ?D|ODq|0? and ?0|A4|D? from fits to the known t
dependence. Statistical errors are determined with the
bootstrap method, which allows us to keep track of cor-
The lattice axial vector current must be multiplied by
a renormalization factor ZAcq
malization factors ZVcc
nonperturbatively. The remaining factor ρAcq
close to unity because the radiative corrections mostly
cancel .A one-loop calculation gives  ρAcq
1.052, 1.044, and 1.032 at a = 0.175, 0.121, and 0.086 fm.
We estimate the uncertainty of higher-order corrections
to be 2αs(ρAcq
The heart of our analysis is the chiral extrapolation,
from the simulated to the physical quark masses. It is
necessary, and non-trivial, because the cloud of “pions”
surrounding the simulated Dq mesons is not the same
as for real pions. With staggered quarks the (squared)
pseudoscalar meson masses are
The combination φq = fDq√mDqemerges di-
Dq(t)ODq(0)?, CA(t) =
4.We write  ZAcq
4)1/2, because the flavor-conserving renor-
are easy to compute
4− 1) ≈ 1.3%; αsis the strong coupling.
ab,ξ= (ma+ mb)µ + a2∆ξ,(4)
where ma and mb are quark masses, µ is a parameter
of χPT, and the representation of the meson under the
taste symmetry group is labeled by ξ = P,A,T,V,I .
A symmetry as ma,mb→ 0 ensures that ∆P = 0. The
“pion” cloud in the simulation includes all these pseu-
According to next-to-leading order χPT the decay con-
stant takes the form
φq= Φ[1 + ∆fq(mq,ml,mh) + pq(mq,ml,mh)],(5)
where Φ is a quark-mass-independent parameter. ∆fq
arises from loop processes involving light pseudoscalar
mesons, and pqis an analytic function. To obtain them
one must take into account the flavor-taste symmetry of
the simulation  and the inequality (in general) of the
valence and sea quark masses . One finds 
∆fq= −1 + 3g2
where fπ≈ 131 MeV is the pion decay constant, g is the
D-D∗-π coupling , and δ′
arise only at non-zero lattice spacing . The terms¯hq,
qare functions of the pseudoscalar meson
masses. The last two, hA
to write out here. It is instructive to show the other two,
q, and hV
q, are too cumbersome
q, when mq= mlor mh:
ql,ξ) + I(M2
l = −1
where I(M2) = M2lnM2/Λ2
contributions only from taste-singlet mesons (represen-
tation I). The term¯hq receives contributions from all
representations, with multiplicity nξ = 1,4,6,4,1 for
ξ = P,A,T,V,I, respectively. The analytic function is
χ(with Λχthe chiral scale),
hh,I)/3. The term hI
pq= (2ml+ mh)f1(Λχ) + mqf2(Λχ) + O(a2),(10)
where f1and f2are quark-mass-independent parameters.
They are essentially couplings of the chiral Lagrangian,
and their Λχdependence must cancel that of ∆fq. This
specifies O(a2) terms proportional to f1 and f2, which
can be removed after our fit. We estimate the remaining
O(a2) effects of light quarks to be small: around 4% at
a = 0.121 fm and 1.4% at a = 0.086 fm.
The salient feature  of the chiral extrapolation of φq
is that ∆fqcontains a “chiral log” I(2mqµ) ∼ mqlnmq,
which has a characteristic curvature as mq→ 0. Equa-
tions (4)–(8) show that the chiral log is diluted by dis-
cretization effects, because a2∆ξ?= 0 for ξ ?= P.
We can now discuss how we carry out the chiral ex-
trapolation. Recall that we compute φq for many com-
binations of the valence and light sea quark masses. At
each lattice spacing, we fit all results for φqto the mass
dependence prescribed by Eqs. (4)–(10). Of the twelve
parameters, eight—µ, the four non-zero ∆ξ, fπ, δ′
V—appear in the χPT for light pseudoscalar mesons.
We constrain them with prior distributions whose central
value and width are taken from the χPT analysis of pseu-
doscalar meson masses and decay constants on the same
ensembles of lattice gauge fields . The rest—Φ, g2, f1,
and f2—appear only for charmed mesons. We constrain
g2to its experimentally measured value, within its mea-
sured uncertainty . Thus, only three parameters—Φ,
f1, and f2—are determined solely by the φqfit. To obtain
physical results we reconstitute the fit setting the light
sea quark mass ml→ (mu+md)/2, and ∆ξ= δ′
For φd(φs) we set the light valence mass mq→ md(ms).
To isolate the uncertainties of the chiral extrapolation
from other sources of uncertainty, we consider the ratio
Rq/s= φq/φs. Figure 1 shows Rq/sat a = 0.121 fm as a
function of mq/ms, projected onto mq = ml. The gray
FIG. 1: Chiral extrapolation of Rq/sat a = 0.121 fm. Data
points show only statistical errors, but the systematic error
of fitting is shown at left.
FIG. 2: Dependence of φson a2. Circles result from removing
the O(a2) pieces in Eq. (10); squares omit this step.
(red) curve is the result of the full fit of φqto the separate
sea- and valence-mass dependence. The black curve, and
the extrapolated value at mq/ms = 0.05, results from
setting ∆ξ = δ′
the other lattice spacings we obtain similar results.
The precision after the chiral extrapolation is, however,
a bit illusory. We tried several variations in the fit proce-
dure: fitting the ratio directly; adding terms quadratic in
the quark masses to Eq. (10); variations in the widths of
the prior constraints of the parameters. When these pos-
sibilities are taken into account, the extrapolated value
of Rd/svaries by 5%, which we take as a systematic un-
certainty. This variation could be reduced with higher
statistics at the lightest sea quark masses.
The lattice spacing dependence of φs = fDs√mDsis
shown in Fig. 2. The (blue) circles are the main results.
In a preliminary report of this work  the O(a2) terms
in φswere not removed. The (red) squares illustrate the
effect of omitting this step. As one can see, the effect is
small at a = 0.086 fm, but it is the main reason why the
results in Eqs. (1) and (2) are smaller than in Ref. .
The χPT expressions for φq assume that the Dq me-
son is static. Since its mass is around 1900 MeV and
the pseudoscalars are a few hundred MeV, this is a good
starting point. Some corrections to this approximation
can be absorbed into the fit parameters, with no real
change in the analysis. A more interesting change arises
in the one-loop self-energy diagrams, for which the func-
tion I(M2) is modified, and depends on mD∗−mDas well
as M. By replacing our standard extrapolation by one
using the modified function, we estimate the associated
A,V= 0 when reconstituting the fit. At
TABLE II: Error budget (in per cent) for Rd/s, φs, φd.
input parameters a and mc
light-quark discretization and χPT fits
error to be 1.5% or less. Finite-volume effects also mod-
ify I(M2): based on our experience with fπ and fK 
and on continuum χPT , we estimate a further error
of 1.5% or less.
Although χPT is able to remove (most of) the light-
quark discretization errors, heavy-quarkdiscretization ef-
fects remain. We estimate this uncertainty using HQET
as a theory of cutoff effects [23, 24]. To arrive at a nu-
merical estimate, one must choose a typical scale¯Λ for
the soft interactions; we choose¯Λ ≈ 500–700 MeV. We
then estimate a discretization uncertainty of 2.7–4.2% at
a = 0.086 fm. Similarly, the results at a = 0.121 fm are
expected to lie within 1–2% of those at a = 0.086 fm.
Because we cannot disentangle heavy- and light-quark
discretization effects, to quote final results we average the
results at a = 0.086 and 0.121 fm. We then find
Rd/s = 0.786(04)(05)(04)(42)
φs = 0.349(05)(10)(15)(14) GeV3/2,
which are the principal results of this work. The uncer-
tainties (in parentheses) are, respectively, from statistics,
input parameters a and mc, heavy-quark discretization
effects, and chiral extrapolation. A full error budget is
in Table II; all uncertainties are reducible in future work.
The results for fD+ and fDsin Eqs. (1) and (2) are ob-
tained via fDs= φs/√mDs, fD+ = Rd/sφs/√mD+, by
inserting the physical meson masses.
Present experimental measurements, fD+ = 202±41±
17 MeV , fDs= 267±33 MeV , are not yet precise
enough to put our results in Eqs. (1) and (2) to a strin-
gent test. The anticipated measurements of fD+ and,
later, fDsfrom CLEO-c are therefore of great interest.
If validated, our calculation of fD+ has important impli-
cations for flavor physics. For B physics it is crucial to
compute the decay constant fB. To do so, we must sim-
ply change the heavy quark mass. In fact, heavy-quark
discretization effects, with the Fermilab method, are ex-
pected to be smaller, about half as big.
We thank the U.S. National Science Foundation, the
Office of Science of the U.S. Department of Energy, Fer-
milab, and Indiana University for support, particularly
for the computing needed for the project. Fermilab is
operated by Universities Research Association Inc., un-
der contract with the U.S. Department of Energy.
Note added: After this Letter was submitted, the
CLEO-c Collaboration announced a new measurement,
fD+ = 223 ± 16+7
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