# Charmed-meson decay constants in three-flavor lattice QCD.

**ABSTRACT** We present the first lattice QCD calculation with realistic sea quark content of the D+-meson decay constant f(D+). 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 f(D+)=201+/-3+/-17 MeV, where the errors are statistical and a combination of systematic errors. We also obtain f(Ds)=249+/-3+/-16 MeV for the Ds meson.

**0**Bookmarks

**·**

**84**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We modify the mesonic wave function by using a short distance scale r 0 in analogy with hydrogen atom and estimate the values of masses and decay constants of the open flavour charm mesons D, D s and B c within the framework of a QCD potential model. We also calculate leptonic decay widths of these mesons to study branching ratios and lifetime. The results are in good agreement with experimental and other theoretical values.Pramana 12/2012; 79(6). · 0.72 Impact Factor - SourceAvailable from: Heinrich LeutwylerSinya Aoki, Yasumichi Aoki, Claude Bernard, Tom Blum, Gilberto Colangelo, Michele Della Morte, Stephan Dürr, Aida X. El Khadra, Hidenori Fukaya, Roger Horsley, [......], Carlos Pena, Christopher T. Sachrajda, Stephen R. Sharpe, Junko Shigemitsu, Silvano Simula, Rainer Sommer, Ruth S. Van de Water, Anastassios Vladikas, Urs Wenger, Hartmut Wittig[Show abstract] [Hide abstract]

**ABSTRACT:**We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi of decay constants and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory and review the determination of the BK parameter of neutral kaon mixing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, for this review, we focus on D- and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit.European Physical Journal C 10/2013; 74(9). · 5.44 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Many new results on charm physics were obtained recently. We review of the experimental status in charm and charmonium decay, production and spectroscopy, together with the theoretical interpretations and the open questions.International Journal of Modern Physics A 01/2012; 22(30). · 1.09 Impact Factor

Page 1

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 [1]. 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[2]. 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 [3]. 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-

Experi-

One can, thus, determine |Vcd|fD+,

sured fD+ with a 20% error [4], 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 [5]. 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,

(1)

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 [6]. These results are more reliable than

older calculations [7] 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. [3] 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

(2)

Page 2

2

TABLE I: Notation for quark masses used in this Letter.

m

mc

ms

mu Physical up quark

md

Physical down quark

mh Simulation’s heavier sea quark

ml

Simulation’s lighter sea quark

mq

Simulation’s light valence quark 0.1ms ≤ mq ? ms

DescriptionRemark

Charmed quark

Physical strange quark

From mDs[10, 11]

From m2

mu = ms/45.5 [9]

md= ms/19.6 [9]

mh≈ 1.1ms

0.1ms ≤ ml? 0.8ms

K[8]

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 [20]. 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 [21]. 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 [22]. Discretization effects are entangled

with the heavy-quark expansion, so we use heavy-quark

effective theory (HQET) as a theory of cutoff effects [23].

This provides good control, as discussed in Ref. [24], and

the framework has been tested with the (successful) pre-

diction of the Bcmeson mass [14]. 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 [25]

?0|Aµ|Dq? = ifDqpµ, (3)

where Aµ = ¯ qγµγ5c is an electroweak axial vector cur-

rent.

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. [26]. 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-

relations.

The lattice axial vector current must be multiplied by

a renormalization factor ZAcq

ρAcq

malization factors ZVcc

4

and ZVqq

nonperturbatively. The remaining factor ρAcq

close to unity because the radiative corrections mostly

cancel [28].A one-loop calculation gives [29] 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-

As

Dq(t)ODq(0)?, CA(t) =

4. We write [27] ZAcq

4

=

4(ZVcc

4ZVqq

4)1/2, because the flavor-conserving renor-

4

are easy to compute

4should be

4

=

4− 1) ≈ 1.3%; αsis the strong coupling.

M2

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 [30].

A symmetry as ma,mb→ 0 ensures that ∆P = 0. The

“pion” cloud in the simulation includes all these pseu-

doscalars.

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 [30] and the inequality (in general) of the

valence and sea quark masses [21]. One finds [31]

∆fq= −1 + 3g2

2(4πfπ)2

?¯hq+ hI

q+ a2?δ′

AhA

q+ δ′

VhV

q

??, (6)

Page 3

3

where fπ≈ 131 MeV is the pion decay constant, g is the

D-D∗-π coupling [32], and δ′

arise only at non-zero lattice spacing [30]. The terms¯hq,

hI

qare functions of the pseudoscalar meson

masses. The last two, hA

to write out here. It is instructive to show the other two,

¯hqand hI

A, δ′

Vparametrizeeffects that

q, hA

q, and hV

qand hV

q, are too cumbersome

q, when mq= mlor mh:

¯hq =

hI

hI

1

16

?

ξnξ

?2I(M2

ql,ξ) + I(M2

qh,ξ)?, (7)

l = −1

h= −I(M2

2I(M2

ll,I) +1

6I(M2

η,I), (8)

hh,I) +2

3I(M2

η,I), (9)

where I(M2) = M2lnM2/Λ2

and M2

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

η,I= (M2

ll,I+ 2M2

qreceives

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 [33] 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′

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 [9]. The rest—Φ, g2, f1,

and f2—appear only for charmed mesons. We constrain

g2to its experimentally measured value, within its mea-

sured uncertainty [34]. 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

A, and

A,V= 0.

0.0 0.2 0.4

0.6

0.81.0 1.2

mq/ms

0.80

0.90

1.00

Rq/s

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.

0.00 0.010.020.030.04

a2 (fm2)

0.30

0.35

0.40

0.45

0.50

φs (GeV3/2)

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 [5] 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. [5].

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

Page 4

4

TABLE II: Error budget (in per cent) for Rd/s, φs, φd.

source

statistics

input parameters a and mc

higher-order ρAcq

heavy-quark discretization

light-quark discretization and χPT fits

static χPT

finite volume

total systematic

Rd/s

0.5

0.6

φs

1.4

2.8

1.3

φd

1.5

2.9

1.3

4

0

0.5

5.0

1.4

1.4

5.4

4.2

3.9

0.5

0.5

6.5

4.2

6.3

1.5

1.5

8.5

error to be 1.5% or less. Finite-volume effects also mod-

ify I(M2): based on our experience with fπ and fK [9]

and on continuum χPT [35], 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,

(11)

(12)

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 [4], fDs= 267±33 MeV [25], 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

−9MeV [36].

[1] See, for example, the CKM Unitarity Triangle Workshop,

http://ckm2005.ucsd.edu/.

[2] V. Lubicz, Nucl. Phys. Proc. Suppl. 140, 48 (2005);

M. Wingate, ibid. 140, 68 (2005).

[3] C. T. H. Davies et al., Phys. Rev. Lett. 92, 022001 (2004).

[4] G. Bonvicini et al., Phys. Rev. D 70, 112004 (2004).

[5] For a preliminary report of this work, see J. N. Simone

et al., Nucl. Phys. Proc. Suppl. 140, 443 (2005).

[6] M. Wingate et al., Phys. Rev. Lett. 92, 162001 (2004).

[7] For example, A. X. El-Khadra et al., Phys. Rev. D 58,

014506 (1998); C. Bernard et al., ibid. 66, 094501 (2002).

[8] C. Aubin et al., Phys. Rev. D 70, 031504 (2004).

[9] C. Aubin et al., Phys. Rev. D 70, 114501 (2004).

[10] M. Di Pierro et al., Nucl. Phys. Proc. Suppl. 129, 328

(2004).

[11] M. Di Pierro et al., Nucl. Phys. Proc. Suppl. 129, 340

(2004).

[12] C. Aubin et al., Phys. Rev. Lett. 94, 011601 (2005).

[13] M. Okamoto et al., Nucl. Phys. Proc. Suppl. 140, 461

(2005); C. Aubin et al., in preparation.

[14] I. F. Allison et al., Phys. Rev. Lett. 94, 172001 (2005).

[15] M. Ablikim et al., Phys. Lett. B 597, 39 (2004).

[16] G. S. Huang et al., Phys. Rev. Lett. 94, 011802 (2005).

[17] J. M. Link et al., Phys. Lett. B 607, 233 (2005).

[18] D. Acosta et al., hep-ex/0505076.

[19] C. Bernard et al., Phys. Rev. D 64, 054506 (2001);

C. Aubin et al., ibid. 70, 094505 (2004).

[20] T. Blum et al., Phys. Rev. D 55, 1133 (1997); K. Orginos

and D. Toussaint, ibid. 59, 014501 (1999); J. Lag¨ ae

and D. Sinclair, ibid. 59, 014511 (1999); G. P. Lepage,

ibid. 59, 074502 (1999); K. Orginos, D. Toussaint and

R. L. Sugar, ibid. 60, 054503 (1999); C. Bernard et al.,

ibid. 61, 111502 (2000).

[21] C. Bernard and M. Golterman, Phys. Rev. D 49, 486

(1994); S. R. Sharpe and N. Shoresh, ibid. 62, 094503

(2000).

[22] A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie,

Phys. Rev. D 55, 3933 (1997).

[23] A. S. Kronfeld, Phys. Rev. D 62, 014505 (2000).

[24] A. S. Kronfeld, Nucl. Phys. Proc. Suppl. 129, 46 (2004).

[25] S. Eidelman et al., Phys. Lett. B 592, 1 (2004).

[26] M. Wingate et al., Phys. Rev. D 67, 054505 (2003)

[27] A. X. El-Khadra et al., Phys. Rev. D 64, 014502 (2001).

[28] J. Harada et al., Phys. Rev. D 65, 094513 (2002).

[29] M. Nobes et al., private communication.

[30] W. Lee and S. Sharpe, Phys. Rev. D 60, 114503 (1999);

C. Bernard, ibid. 65, 054031 (2002); C. Aubin and

C. Bernard, ibid. 68, 034014 (2003); 68, 074011 (2003).

[31] C. Aubin and C. Bernard, Nucl. Phys. Proc. Suppl. 140,

491 (2005).

[32] B. Grinstein et al., Nucl. Phys. B 380, 369 (1992);

J. L. Goity, Phys. Rev. D 46, 3929 (1992).

[33] A. Kronfeld and S. Ryan, Phys. Lett. B 543, 59 (2002).

Page 5

5

[34] A. Anastassov et al., Phys. Rev. D 65, 032003 (2002).

[35] D. Arndt and C.-J. Lin, Phys. Rev. D 70, 014503 (2004).

[36] M. Artuso et al., hep-ex/0508057.

#### View other sources

#### Hide other sources

- Available from Dru Renner · May 27, 2014
- Available from arxiv.org
- Available from arxiv.org