Progress on Perturbative Matching Calculations for the Charm Quark Mass using the HISQ Action
ABSTRACT The highly-improved staggered quark (HISQ) action is the most accurate discretization scheme to date for the charm quark. Here we report on the progress of perturbative matching for the quark mass using the HISQ action. The matching is done through O(α 2 s) using a combination of Monte Carlo simulations at weak coupling and diagrammatic perturbation theory. When combined with on-going simulation efforts using the HISQ action, a determination of the charm quark mass to a few percent accuracy can be achieved. Of particular interest will be a comparison with the recent sum rule determination of the charm mass due to Kühn et al. .
Progress on Perturbative Matching Calculations for
the Charm Quark Mass using the HISQ Action
Emel Dalgic∗ ab, Kit Wongc, Christine Daviesc, Eduardo Follanad,Alistair Harte,Ron
Horganf, Peter Lepageg, Quentin Masonf, Junko Shigemitsud, Howard Trottieraand
aSimon Fraser University, Burnaby BC, Canada V5A 1S6
bTRIUMF, Vancouver BC, Canada V6T 2A3
cUniversity of Glasgow, Glasgow, UK G12 8QQ
dThe Ohio State University, Columbus OH, USA 43210
eUniversity of Edinburgh, Edinburgh, UK EH9 3JZ
fUniversity of Cambridge, Cambridge, UK CB3 0HE
gCornell University, Ithaca NY, USA 14853
E-mail: email@example.com, firstname.lastname@example.org,
email@example.com, R.R.Horgan@damtp.cam.ac.uk ,
The highly-improvedstaggeredquark(HISQ)action is the most accurate discretizationscheme to
date for the charm quark. Here we report on the progress of perturbative matching for the quark
mass using the HISQ action. The matching is done through O(α2
Carlo simulations at weak coupling and diagrammatic perturbation theory. When combined with
on-going simulation efforts using the HISQ action, a determination of the charm quark mass to a
few percent accuracy can be achieved. Of particular interest will be a comparison with the recent
sum rule determination of the charm mass due to Kühn et al. .
s) using a combination of Monte
The XXV International Symposium on Lattice Field Theory
July 30-4 August 2007
c ? Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence.
Charm Mass with HISQ
Quark masses are fundamental parameters that go in the standard model and it is important to
determine them precisely to constrain models beyond the standard model and to use them as inputs
for phenomenological calculations. As we do not find free quarks in nature, their masses cannot
be directly measured. One needs to instead do a comparison between lattice and experiment to
extract quark masses. Quarks interact via the strong force, therefore such a calculation would be
nonperturbative. Lattice QCD methods are well suited for this problem. This work builds on recent
developments in staggered quarks .
Previously, the light quark masses mMS
rations. We list some mMS
determinations. The HPQCD collaboration used the AsqTad action
with 3 dynamical quarks to determine mMS
= 87±4±4 MeV . The CP-PACS and JLQCD col-
laborations used the Wilson action with 3 dynamical quarks to calculate mMS
The QCDSF-UKQCD collaborations used clover fermions with 2 flavors of sea quarks to get
= 110−130 MeV . The ALPHA collaboration used Wilson quarks with 2 dynamical
quarks and obtained mMS
= 97(22) MeV . The SPQcdR collaboration used Wilson Quarks with
2 dynamical flavors to get mMS
= 95±25 MeV .
Our goal is to also determine mcwith the HISQ (Highly Improved Staggered) quark action to
a few percent accuracy, which will be determined by taste-changing effects and discretization. The
taste changing effects for the HISQ action are up to 3-4 times smaller than for the Asqtad action
as detailed in , and a comparison of the two actions is shown in Fig. 1. It will be interesting to
compare our results for mcwith the sum rules calculation by Kühn et al. , where an error of 1%
Our aim is to do a perturbative matching calculation and obtain the renormalization factors for
mcin the MS scheme. Below we list the perturbative expansions of the pole mass in terms of the
bare mass, its relation to the MS mass, and the relevant mass renormalization factors:
have been determined by several collabo-
0) MeV . The Particle Data Group reports the value
Zm(µa,m0a) = 1+Zm,1(µa)αV(q∗)+Zm,2(µa)α2
mMS(µ) = mPole(1+Z1(+Z2(
Note that the correct expansion parameter to use is the renormalized coupling αV, the perturbative
series in αlatis not well behaved.
2. Diagrammatic Method
One way to do the matching is to use perturbation theory and compute all the diagrams up to
the order at which we work. Since matching corrects for the short distance effects brought about by
Charm Mass with HISQ
Figure 1: Comparison of results for the HISQ vs Asqtad actions . The HISQ results correspond (from
left to right) to 0-link, 1-link, 2-link and 3-link mesons. For Asqtad ηcwe show results for the 0- and 1-link
the finite lattice spacing, asymptotic freedom allows us to use perturbation theory. This approach
is very involved, as it requires calculating many diagrams, and the Feynman rules for the HISQ
action are extremely complicated. The relevant diagrams are shown in Fig. 2 .
3. Another Method: Weak Coupling Monte Carlo
An alternative to diagrammatic perturbation theory is to use Monte Carlo simulations at weak
couplings, where the theory enters the perturbative phase. Simulations involving a particular op-
erator, in this calculation the pole mass, are done at several values of the strong coupling, and
the resulting data are then fitted to an expansion in αVto yield the perturbative coefficients. The
expectation value of an observable can be calculated on the lattice using
< M > =
D3 D4D5 D6
D14-18 D19 D20
D25D26 D27 D28
CT1 CT2CT3 CT4 CT5CT6
Figure 2: Relevant diagrams for the diagrammatic method.
Charm Mass with HISQ
Figure 3: The renormalized mass M measured at different couplings and lattice volumes for the ASQTAD
action. The data points are fitted to an expansion in αV(q∗) (solid lines). The “slopes” and “curvatures” are
the first and second order coefficients respectively.
where β =10/g2. Itisworth reiterating that in practice itiscrucial touse the renormalized coupling
instead of the bare lattice coupling αlat≡ g2/4π, for which perturbation theory is very poorly
convergent. A good choice is αV(q∗) defined by the static potential, along with an estimate of the
optimal scale q∗for the quantity of interest. The coupling is then converted back to αlatusing the
known third order relation between αVand αlat.
As an example, Fig. 3 shows the renormalized mass M measured at different couplings and
lattice volumes for the ASQTAD action in the quenched approximation. The data points are fitted
to an expansion of the form
M = mtree+c1αV+cV
where mtreeis the tree-level mass. The “slope” of the curve gives the first order coefficient c1
(independent of the scheme) and the “curvature” is equal to cV
from one loop perturbation theory to determine c2more accurately. Fig. 4a shows the infinite
volume extrapolation of c1. For comparison, results from diagrammatic perturbation theory ,
both at finite volume and in the infinite volume limit, are also plotted. Numerical values of c2can
be calculated from Fig. 3 also by calculating the curvatures. To improve the accuracy, however, we
re-fit the data with c1fixed to the analytic values at finite volume. Our results are shown in Fig. 4b.
The overall agreement with our diagrammatic perturbation theory calculations is remarkable.
Fig. 5 shows the first order coefficients for the HISQ action, extrapolated to the infinite vol-
ume limit. For comparison, we also plot the same coefficients obtained using the diagrammatical
method, and observe that the agreement is again outstanding. The 1-loop perturbative calculations
at finite volume, which could be used to extract the second order coefficients from the data, are in
Table 1 shows the perturbative coefficient A20calculated with the AsqTad action, neglecting
sea quarks, compared to diagrammatic perturbation theory results. We find good agreement be-
tween the two sets of results.
We have demonstrated that perturbative coefficients for mass renormalization can be obtained
with high accuracy from Monte Carlo simulations at weak couplings. This numerical method
2. One can use the c1value calculated
Charm Mass with HISQ
00.04 0.080.12 0.16
Figure 4: a) Infinite volume extrapolation of the first order coefficients. The error bars are invisible at this
scale for the analytic results. b) Infinite volume extrapolation of the second order coefficients. Results are
obtained by fixing c1to the analytic values at finite volume.
Figure 5: Infinite volume extrapolation of the first order coefficients for the HISQ action. The error bars are
invisible at this scale for the analytic results.
Weak Coupling MC
Table 1: Asqtad quenchedA20, comparisonbetween diagrammaticperturbationtheory and high-betaMonte