Charm physics with highly improved staggered quarks

Abstract

We use a relativistic highly improved staggered quark action to discretize charm quarks on the lattice. We calculate the masses and the dispersion relation for heavy-heavy and heavy-light meson states, and show that for lattice spacings below .1 fm, the discretization errors are at the few percent level. We also discuss the prospects for accurate calculations at the few percent level of f_D_s, f_D, and the leptonic width of the psi and phi.

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arXiv:hep-lat/0610110v1 20 Oct 2006
Charm physics with highly improved staggered
quarks
HPQCD and UKQCD: C.T.H. Davies,aE. Follana∗,aK. Hornbostel,bG.P. Lepage,cQ.
Mason,dH. Trottier,eJ. Shigemitsufand K. Wonga
aUniversity of Glasgow, Glasgow, UK
bDallas Southern Methodist University, Dallas, Texas, USA
cCornell University, Ithaca, New York, USA
dCambridge University, Cambridge, UK
eSimon Fraser University, Vancouver, British Columbia, Canada
fOhio State University, Columbus, Ohio, USA
We use a relativistic highly improved staggered quark action to discretize charm quarks on the
lattice. We calculate the masses and the dispersion relation for heavy-heavy and heavy-light
meson states, and show that for lattice spacings below .1 fm, the discretization errors are at the
few percent level. We also discuss the prospects for accurate calculations at the few percent level
of fDs, fD, and the leptonic width of the ψ and φ.
XXIVth International Symposium on Lattice Field Theory
July 23-28, 2006
Tucson, Arizona, USA
∗Speaker.
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
http://pos.sissa.it/

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Charm physics with highly improved staggered quarks
E. Follana
0.91 1.1
LQCD/Exp’t (nf= 0)LQCD/Exp’t (nf= 0)
0.911.1
LQCD/Exp’t (nf= 3) LQCD/Exp’t (nf= 3)
Υ(1P − 1S)
Υ(3S − 1S)
Υ(2P − 1S)
Υ(1D − 1S)
ψ(1P − 1S)
2MBs− MΥ
2MDs− Mηc
MΞ− MN
MΩ
fK
fπ
Figure 1: Comparison between quenched and unquenched simulation results.
1. Introduction
Improved staggered quarks have proved very effective in obtaining precise results of phe-
nomenological interest in the unquenched light valence sector [1] (see figure 1 for a comparison of
quenched and unquenched results.) On the other hand, non-relativistic effective field formulations
are very successful in the bottom sector [2, 3]. Although a non-relativistic formulation can also be
applied, in principle, to the charm sector, the errors are much larger.
Highly improved staggered quarks have very small discretization errors. Combining this with
fine enough lattices may provide a good method of handling charm quarks.
CLEO-c is making precise measurements of several quantities (for example fDsfD) in the
charm system with small errors (≈ 4%). For comparison, Fermilab results have errors of ≈ 8% [4].
This provides a good opportunity to test our methods.
2. Improved Staggered Quarks
The massless one-link (Kogut-Susskind) staggered Dirac operator is defined as:
D(x,y) =
1
2au0
d
∑
µ=1
ηµ(x)?Uµ(x)δx+ ˆ µ,y−H.c.?,
ην(x) = (−1)∑µ<νxµ
(2.1)
with u0an optional tadpole-improvement factor.
Thisoperator suffers from doubling: there are four “tastes” (non-physical flavours) of fermions
in the spectrum, which couple through taste-changing interactions. These are lattice artifacts of or-
der a2, involving at leading order the exchange of a gluon of momentum q≈π/a. Such interactions
are perturbative for typical values of the lattice spacing, and can be corrected systematically a la
Symanzik. By judiciously smearing the gauge field we can remove the coupling between quarks
and high momentum gluons.
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Charm physics with highly improved staggered quarks
E. Follana
++
++=
=
(Naik)
c5’
c1
(Fat link)
c3’
c5 c7c3
Figure 2: Paths used for smearing in the ASQTAD staggered action.
The most widely used improved staggered action is called ASQTAD, and removes all tree-
level a2discretization errors [5, 6, 7]. The paths used to smear the gauge-fields are shown in figure
2.
The HISQ (highly improved staggered quarks) staggered Dirac operator involves two levels of
smearing with an intermediate projection onto SU(3). It is designed so that, as well as eliminating
all a2discretization errors, it further reduces the one-loop taste-changing errors (for a more detailed
discussion see [8].)
Thisaction has been showntosubstantially reduce the errors associated withthetaste-changing
interactions [8, 9, 10].
3. Charm Sector
When we put massive quarks on the lattice, the discretization errors grow with the quark mass
as powers of am. Therefore to obtain small errors we would need am ≪ 1. For heavy quarks this
would require very small lattice spacings. On the other hand, to keep our lattice big enough to
accommodate the light degrees of freedom, we need La ≫ m−1
different scales in the problem makes difficult a direct solution. What we can do instead is to take
advantage of the fact that m is large, by using an effective field theory (NRQCD, HQET). This
program has been very successful for b quarks [2, 3, 4].
The charm quark is in between the light and heavy mass regime. It is quite light for an easy
application of NRQCD,but quite large for the usual relativistic quark actions, amc<
we use a very accurate action (HISQ) and fine enough lattices (fine MILC ensembles), it is possible
to get results accurate at the few percent level.
π. The fact that we have two very
∼1. However, if
4. Results
We use 2+1-flavours unquenched configurations generated by the MILC collaboration [11,
12, 13]. We present here results obtained from an ensemble with ml= ms/5, where msis the
light and msthe strange quark mass, and with a lattice spacings of 1/11fm. The extent of the
corresponding lattices is 283x 96.
The bare mass used for the charm valence quarks is fixed by adjusting the “Goldstone” pseu-
doscalar mass to the experimental value.
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Charm physics with highly improved staggered quarks
E. Follana
2.8
3
3.2
3.4
3.6
3.8
4
mass /GeV
charmonium masses, hisq ma=0.43 on fine MILC
ηc(1S)
ψ(1S)
ηc(2S)
ψ(2S)
χc0
χc1
Figure 3: Charmonium spectrum on fine MILC configurations.
4.1 c¯ c pseudoscalar and vector
We show in figure 3 some of our results for the charmonium spectrum, as a check of the
formalism. We haven’t optimized in any way our operators for the calculation of the excited states,
which therefore have large errors.
In figure 4 we show the results for the pseudoscalar and vector mesons. In this sectors we
have very small statistical errors. In the staggered formalism we have 16 different mesons of each
type, with, in general, different masses, due to the taste-changing interactions. We can see that
such mass splitting is mostly noticeable for the pseudoscalar states, but is almost negligible for the
vector mesons, where it’s below our statistical error. This is consistent with previous results [14],
although the actual size of the splitting is very much reduced with respect to the one using one-link
staggered fermions. The total splitting for ηcis only about 10 Mev for HISQ, and is also much
smaller than the one for ASQTAD. We obtain an hyperfine splitting of ≈ 110MeV, to be compared
with the experimental value of 117MeV.
4.2 Speed of Light
It is important to check that the discretization artifacts at the masses we use in our simulation
do not spoil the relativistic invariance of the action (as is bound to happen at large enough mass.)
One way to do this is by calculating the dispersion relation of a meson, or equivalently, that the
squared speed of light is still 1 at small non-zero momenta.
In figure 5 we show the results for c2at a small non-zero momentum for pseudoscalar heavy-
heavy and light-light mesons, where the heavy mass is set to the charm mass and the light one to
the strange mass. We show the results for both HISQ and ASQTADactions. c2is one at the strange
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Charm physics with highly improved staggered quarks
E. Follana
2.95
3
3.05
3.1
3.15
3.2
mass /GeV
charmonium, fine MILC, asqtad vs hisq
ηc
ψ
hisq
asqtad
hisq
expt
Figure 4: Charm pseudoscalar and vector states for HISQ and ASQTAD on fine MILC configurations.
0.9
0.95
1
1.05
1.1
c2
c2, fine MILC, hisq vs asqtad
hh llhh ll
hisq
asqtad
Figure 5: Speed of light squared for HISQ and ASQTAD on fine MILC configurations.
5