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PoS(LATTICE 2007)239

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

Jackson Wub

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: emel@triumf.ca, k.wong@physics.gla.ac.uk,

c.davies@physics.gla.ac.uk,e.follana@physics.gla.ac.uk,

a.hart@ed.ac.uk, R.R.Horgan@damtp.cam.ac.uk ,

gpl@mail.lns.cornell.edu, quentin-mason@cornell.edu,

shige@pacific.mps.ohio-state.edu,trottier@sfu.ca, jwu@triumf.ca

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. [1].

s) using a combination of Monte

The XXV International Symposium on Lattice Field Theory

July 30-4 August 2007

Regensburg, Germany

∗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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PoS(LATTICE 2007)239

Charm Mass with HISQ

Emel Dalgic

1. Introduction

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 [2].

Previously, the light quark masses mMS

rations. We list some mMS

s

determinations. The HPQCD collaboration used the AsqTad action

with 3 dynamical quarks to determine mMS

s

= 87±4±4 MeV [3]. 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

mMS

s

= 110−130 MeV [5]. The ALPHA collaboration used Wilson quarks with 2 dynamical

quarks and obtained mMS

s

= 97(22) MeV [6]. The SPQcdR collaboration used Wilson Quarks with

2 dynamical flavors to get mMS

s

= 101(8)(25

mMS

s

= 95±25 MeV [8].

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 [2], 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. [1], where an error of 1%

is quoted.

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:

u,dand mMS

s

have been determined by several collabo-

s

= 91.114.6

6.2MeV[4].

0) MeV [7]. The Particle Data Group reports the value

mPole= m0[1+αlat(A11log(m0a)+A10),

+ α2

latlog2(m0a)+A21log(m0a)+A20+...],

µ

mPole)αMS

π

mMS(µ) =am0

a

Zm(µa,m0a) = 1+Zm,1(µa)αV(q∗)+Zm,2(µa)α2

mMS(µ) = mPole(1+Z1(+Z2(

µ

mPole)α2

MS

π2),

Zm(µa,m0a),

V+...

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

2

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Charm Mass with HISQ

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Figure 1: Comparison of results for the HISQ vs Asqtad actions [9]. 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

pseudoscalars.

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 > =

?

[DU][dψDψ]M(U,ψ)e−βS[U,ψ],

D3 D4 D5D6

D7-11D12 D13

D14-18 D19D20

D21 D22 D24

D25 D26 D27 D28

CT1 CT2CT3 CT4 CT5 CT6

Figure 2: Relevant diagrams for the diagrammatic method.

3

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Charm Mass with HISQ

Emel Dalgic

0.3

0.31

0.32

0.33

0.34

0

0.02 0.04

? (q??)

V

0.06 0.08

6???x?12

3

8???x?16

3

10???x?20

3

12???x?24

3

20???x?24

3

*

renormalized?mass?M

ASQTAD

m???=?0.3

0

“slope”?=?c

“curvature”?=?c

1

2

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[10].

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

2α2

V+...,

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 [11],

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

progress.

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

4

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Charm Mass with HISQ

Emel Dalgic

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.04 0.08 0.120.16

ASQTAD,?c

m??=?0.7

0

m??=?0.3

0

m??=?0.4

0

1

1/L

c1

MC?simulations

diagrammatic?PT

(a)

1

1.5

2

2.5

3

3.5

0 0.04 0.08 0.120.16

1/L

ASQTAD,?c2

V

m??=?0.7

0

m??=?0.3

0

m??=?0.4

0

c2

V

MC?simulations

diagrammatic?PT

(b)

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.

0.2

0.4

0.6

0.8

0

0.04 0.080.12 0.16

1/L

m??=?0.85

0

m??=?0.43

0

m??=?0.66

0

HISQ,?c

MC?simulations

diagrammatic?PT

1

c1

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.

am

0.3

0.4

0.5

0.6

0.7

Diagrammatic PT

5.78(5)

5.61(7)

5.47(6)

5.23(6)

5.15(6)

Weak Coupling MC

5.26(33)

5.30(28)

5.25(24)

5.18(22)

5.05(21)

Table 1: Asqtad quenchedA20, comparisonbetween diagrammaticperturbationtheory and high-betaMonte

Carlo simulations.

5

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Charm Mass with HISQ

Emel Dalgic

Figure 6: There are only 4 diagrams to compute in the fermionic part.

Figure 7: Fattening the link. This needs to be done twice, once with the thin link, and once more with the

already fattened link.

provides a valuable alternative to diagrammatic perturbation theory.

4. Strategy

Our short term goal is to obtain the gluonic part of our calculation by the weak coupling Monte

Carlo method, and get the fermionic part using diagrammatical perturbation theory. The diagrams

needed are shown in Fig. 6.

In the long term, it would be desirable to do the entire calculation diagrammatically and also

to compute the gluonic part using the Monte Carlo weak coupling approach in order to compare

the two methods.

5. Handling HISQ Perturbation Theory

The HISQ action is obtained by fattening a simple link variable, reunitarizing the result, and

fattening the resulting link again. Fattening isnecessary for suppressing taste changing interactions,

and fattening twice suppresses taste changing further. Reunitarization is necessary to suppress the

unphysical tadpole diagrams which only exist in the lattice discretization and not the continuum

limit. Fattening of a link can be visualised as in Fig. 7. Pieces of the action (in this case the links

shown above) can be combined by doing Fourier convolutions, which means one assigns the gluons

6

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Charm Mass with HISQ

Emel Dalgic

Figure 8: Convolution of two operators for two gluons.

to the pieces in all possible ways. Fig. 8 is a simple illustration with two operators and two gluons.

Once we perform the convolutions and reunitarize, we obtain an improved link variable. We need

to fatten this yet again. We therefore repeat the same process of convoluting, this time with the

already fattened and reunitarized link, to obtain the full HISQ vertices.

6. Conclusions

Our project involves performing perturbative matching calculations to find mcthrough O(α2

using the HISQ quark action. To achieve this, work is underway to do the fermionic part of the

calculation diagrammatically, while the gluonic part is computed using the weak coupling Monte

Carlo method. When both calculations are complete, the results will be combined to obtain per-

turbative coefficients for the mCmass renormalization. Our initial tests show that perturbative

coefficients can be obtained accurately using the weak coupling Monte Carlo method. For the

longer term goal of doing the entire calculation diagrammatically, the vertex functions necessary

to achieve this aim have been prepared.

s)

References

[1] J.H. Kühn et al., arXiv:hep-ph/0702103v1.

[2] E. Follana et al., Phys.Rev.D75, 054502 (2007).

[3] Q. Mason et al., Phys.Rev.D73,114501(2006).

[4] T. Ishikawa et al., Nucl.Phys.Proc.Suppl.140,225(2005).

[5] M. Göckeler et al., PoS LAT2005, 078 (2006).

[6] M. Della Morte et al., Nucl.Phys.B729,117(2005).

[7] D. Becirevic et al., Nucl.Phys.B729,117(2005).

[8] W. M. Yao et al., J. Phys. G33, 1 (2006).

[9] C. Davies et al., PoS LAT2006, 082 (2006).

[10] Q. Mason et al., HPQCD Collaboration, Phys. Rev. Lett. 95, 052002 (2005).

[11] Q. Mason et al., HPQCD Collaboration, Phys. Rev. D73, 114501 (2006).

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