Efficient Simulation of Heavy Quark Vacuum Polarization

0.13 MB·English

by  Matthew Nobes

Efficient Simulation of Heavy Quark Vacuum
Polarization
5 Matthew Nobes∗
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2 February 1, 2008

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a
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Abstract
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1 We outline a simple way to include heavy quark vacuum polarization
in lattice QCD simulations. The method, based on effective field theory,

2 requiresonlyatrivialmodificationofthegluonactionandhasnoimpact
v on simulation times. We assess the range of validity for this procedure,
9 and theimpact that it may have.
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1 1 Introduction
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The inclusion of light quark vacuum polarization in lattice QCD Monte Carlo
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/ simulationshaslongbeenamajorproblem. Withthe developmentofimproved
t
a staggeredfermions [1] it has become possible to do high-precisionlattice QCD,
l with good control over all systematic errors [2]. One such systematic error is
-
p the continued exclusion of heavy quark vacuum polarization. State of the art
e
lattice simulations only include the effects of light (s,u and d) quark vacuum
h
polarization.
:
v The exclusion of heavy quark vacuum polarization is a good approximation
i
X because, as we shall discuss, their effect on simulation results is suppressed by
a factor of q¯2/m2, where q¯ is a typical lattice momentum. For many lattice
r
a calculations (hadron spectra, decay constants) q¯ ≈ Λ ≈ 250MeV. For a
QCD
charm quark m ≈ 1GeV this could be as large as a 6% effect. Todays lattice
c
calculationsaspireto few percent(1−3%)errors[2], so fortruly high-precision
work, this error should be removed. It would be good to have a method for
removingitwithouthavingtodofulldynamicalsimulationsoftheheavyquarks.
Fortunately these effects can be handled using effective field theory.
2 The Effective Lagrangian
The earliest example of an effective field theory is the Euler–Heisenberg non-
linear photon theory [3]. This is an expansion of QED for low-energy photons
∗Institute for High Energy Phenomenology, Cornell University, Ithaca, New York, USA,
14853-5001. Email: [email protected]
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withaverageenergiesω ≪m . EulerandHeisenbergwereconcernedwith non-
e
linear interactions between four photons, these corrections begin at O(ω4/m4),
e
however there is a correction to the photon propagator which is a O(ω2/m2)
e
effect [4]. Similar expansions apply to QCD.
Itisstraightforwardtowritedownthemostgeneraleffectiveactionthatwill
account for these effects
S = d4x −1FµνFa + COn O (1)
EH Z  4 a µν XQ XOn mnQ+1 n!
 
whereO areoperatorsofdimension2(n+1)builtoutofF ,F˜ andD that
n µν µν µ
respectthe symmetries ofQCD, CO arethe coefficientsof these operatorsand
n
the sum runs over all heavy quark flavors Q = c,b,t. The operators of higher
dimensions are suppressed by powers of the heavy quark masses. In this note,
we will be concerned with only the operators of dimension six.
At dimension six there are two operators we need to consider [5]
G = g f Fµ Fν Fσ (2)
1 s abc aν bσ cµ
1
G = D FµσD Fa (3)
2 2 µ a ν νσ
With these operators the effective action is
S = d4x −1FµνFa + CG1G + CG2G (4)
EH Z  4 a µν Q m2Q 1 m2Q 2!
X
 
This actioncan be matched to full QCD order by order in perturbation theory,
since a heavy quark loop is highly virtual. Matching at leading order gives the
coefficients [6]
α α
s s
CG1 =−CF360π, CG2 =−CF15π, (5)
where C =4/3 for SU(3).
F
For matching to the lattice theory it is convenient to rewrite the action in
terms of a different basis of operators. We introduce the operator
G =DλFµνD Fa , (6)
3 a λ µν
and make use of the relation [5]
1
d4xG = d4x G −2G (7)
1 3 2
2
Z Z (cid:20) (cid:21)
to rewrite the action as
S = d4x −1FµνFa + CG1 G + (CG2 −2CG1)G . (8)
EH Z  4 a µν Q 2m2Q 3 m2Q 2!
X
 
2
In this brief note we are primarily concerned with the effect of heavy quark
vacuum polarization on the gluon action. With this in mind we make a field
transformation
C
Aa →Aa + α DνFa (9)
µ µ m2 s νµ
Q
which gives
8C 1
Fa Fµν →Fa Fµν − α G +O . (10)
µν a µν a m2 s 2 m4
Q Q!
We can pick the constant C to eliminate G from the gluon action1. This
2
transformation will introduce additional four-quark operators which should be
incorporated into the fermion actions used in the simulations.
Finally, we rescale the fields A→A/g , and express the action as
s
1
S = − d4x Tr[F F ]
EH 2g2 µν µν
s Z (µν
X
1 α
s
+ C Tr[D F D F ] . (11)
m2  F180π λ µν λ µν 
XQ Q Xµνλ 
 
Ourgoalistoconstructalatticegluonactionthatreproducesthisaction. Before
we do this, it is useful to consider how large the mass m should be in order
Q
that the expansion in m−2 is sensible.
Q
3 Limits on m
Q
In order to set limits on the expansion in m−2 we look at one part of the
Q
matching calculation for (4). To match the coefficient CG2 one can consider
(light) quark-quark scattering at one loop order in continuum QCD. The only
effect of virtual heavy quarks at this order will be their contribution to the
gluonvacuumpolarization. Therefore,we cangetacheckonthe validity ofthe
expansion by investigating the one-loop gluon propagator in continuum QCD.
InfullQCDtherenormalizedgluonpropagatorisgiven(inFeynmangauge)
by
−ig
D (q2)= µν . (12)
µν
q2 1−Πˆ(q2)
h i
We separate the contribution of heavy quark loops out of the self energy
Πˆ(q2)= Πˆ (q2)+Πˆ (q2). (13)
Q QCD
Q
X
1Thistechnique isusedtoeliminatethesameoperator inthelatticetheory,withm−2→
a2.
3
Thelightquarkandgluonicpartsoftheselfenergywillbedescribedbythestan-
dard QCD action, so we’ll just retain the heavy quark contribution. Therefore,
we will consider the propagator
−ig
D (q2)= µν . (14)
µν
q2 1− Πˆ (q2)
Q Q
h i
The contribution to the self energy can bePfound in any textbook (for example
[7]) it’s just the QED contribution times the color factor C .
F
2α 1 1
Πˆ (q2)=−C s dxx(1−x)log . (15)
Q F π Z0 1− mq22 x(1−x)
Q
 
Expanding in q2/m2 we find
Q
α q2
Πˆ (q2)=−C s . (16)
Q F15πm2
Q
The range of validity for the expansion in 1/m2 is set by the radius of
Q
convergence of the logarithim in (15),
|q|<2m . (17)
Q
Sinceourgoalistousethisexpansioninlatticesimulationsitisusefultoexpress
this in lattice units, multiplying both sides by a, and taking the momentum
|q|=Λ
QCD
m a>0.5Λ a (18)
Q QCD
For a 0.1 fm lattice Λ a ≈ 0.1−0.3, so we get a very loose limit, m a >
QCD Q
0.05−0.15. Atthislatticespacing,m a=0.5,sothisboundissatisfied. Notice
c
thatforlowmomentumprocessesitisnotnecessaryform atobegreaterthan
Q
one, fundamentally this is an expansion in |q|/m not 1/(m a). For processes
Q Q
that involve larger internal momenta this limit is more strict. For example, the
semileptonic form factor for B → π +ℓ+ν can involve internal momenta as
high as
Λ m
QCD b
≈1GeV. (19)
2
r
With |q|=1GeV (17)is still satisfiedfor acharmquark,but forlighterquarks
itwouldn’tbe. Whenwematchtothelatticetheorywewillfindanotherbound
on m a.
Q
4 Matching to the Lattice Theory
The leading correction in (11) is trivially included in the widely used improved
gluon action
2 1 1 1
S = β ReTr[1−U ]+β ReTr[1−U ]+β ReTr[1−U ]
L g2 P3 P R3 R 63 6
0 x (cid:26) (cid:27)
X
(20)
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where U is the plaquette, U is the rectangle, and U is the parallelogram
P R 6
term2. We define the operators
F = Tr(F F )
0 µν µν
µν
X
F = Tr(D F D F ) (21)
1 µ µν µ µν
µν
X
F = Tr(D F D F )
2 σ µν σ µν
µνσ
X
in terms of which we have
1 a4 a6
ReTr[1−U ] = − F + F +O a8
P 0 1
3 4 24
1 5a6 (cid:0) (cid:1)
ReTr[1−U ] = −2a4F + F +O a8 (22)
R 0 1
3 6
1 a6 a6 (cid:0) (cid:1)
ReTr[1−U ] = −2a4F + F + F +O a8 .
6 0 1 2
3 3 6
(cid:0) (cid:1)
The choice3
β = 0
6
1
β = − β (23)
R 20u2 P
0
5
β =
P
3
gives [10]
1
S =− d4xF +O α a2,a4,α /(m a)2 . (24)
L 0 s s c
2g
0 Z
(cid:0) (cid:1)
In order to reproduce (11) we need to have a non-zero value for β . When
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we do this, we are be forced to included anadditional correctionto β in order
R
to cancel any potential contribution from F . We take
1
β α
β = P s C[1]
6 u2 (am )2
0 Q
1 α
β = − β 1+ s R[1] (25)
R 20u2 P (am )2
0 (cid:18) Q (cid:19)
5
β =
P
3
2Corrections to the standard Wilson plaquette action due to heavy quark vacuum polar-
izationhavebeeninvestigated in[8,9].
3Thefactorofu0 isatadpoleimprovementfactor,itplaysnoroleinourdiscussion.
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which gives
1 α 40 2
S = − a4 1− s C[1]− R[1] F
L 2g (am )2 3 3 0
0 x (cid:26)(cid:18) Q (cid:20) (cid:21)(cid:19)
X
α 5 10
+ a2 s R[1]− C[1] F
(am )2 18 3 1
Q (cid:20) (cid:21)
10α
− a2 s C[1]F , (26)
9(am )2 2
Q (cid:27)
where for simplicity we just include only one flavor of heavy quark. Setting
1 1
C[1] =−C =− (27)
F
200π 150π
producesthe correctcoefficientforthe F term. The additionalcontributionto
2
thecoefficientoftheF termproducesanunobservableshiftinthewavefunction
0
renormalization for the gluons, so we are free to drop it. This leaves only the
F term. We tune R[1] to insure that its coefficient remains zero. This is easily
1
accomplished with
2
R[1] =12C[1] =− . (28)
25π
With these values the lattice action correctly reproduces (11) with
O a4,α a2,α /(am )4,α2/(am )2 (29)
s s c s c
(cid:0) (cid:1)
errors.
5 Impact on One-Loop Corrections
We combine our results with the known one loop corrections to β and β ,
R 6
β 2 1
P
β = − 1+α (π/a) 0.4805+X N − (30)
R 20u2  s  R f 25π (am )2
0 Q Q
X
  
β 1 1
P
β = − α (π/a) 0.03325+X N + . (31)
6 u2 s  6 f 150π (am )2
0 Q Q
X
 
The one loop terms due to the gluons are well known [11, 12], those due to dy-
namicallightfermionsarebeingcomputed[13]. Witham ≈1thesecorrections
c
to the one-loop terms are 0.025 for β and 0.0021 for β .
R 6
Theseexpressionsalsolimithowsmallonecantakea. Thecouplingconstant
α (π/a) goes to zerologarithmically as a→0,whereas the correctionfrom the
s
heavy quark vacuum polarization is growing as 1/a2. Clearly, if we take a
too small the correction term will end up larger than the tree level term. For
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example, on an ultra-fine lattice, with a=0.01 fm, we have m a=0.05, which
c
gives (setting N =0)
f
β β
P P
β =− [1−9.7α (π/a)], β =− 0.88α (π/a). (32)
R 20u2 s 6 u2 s
0 0
In this case, the “correction”to β is larger than the leading order term. With
R
m a = 0.13 the coefficient of the correction term becomes one, so we take this
c
as a rough lower limit, a=0.025 fm.
A more realistic lattice spacing is a=0.09 fm, which gives m a=0.45 and
c
β β
P P
β = − [1+(0.481−0.126)α (π/a)]=− [1+0.355α (π/a)],
R 20u2 s 20u2 s
0 0
β β
P P
β = − (0.0333+0.0105)α (π/a)=− 0.0437α (π/a). (33)
6 u2 s u2 s
0 0
In the this case, the corrections due to heavy quark vacuum polarization are
significant but do not cause convergence problems.
6 Conclusion
Inthisnotewehavedemonstratedthatheavyquarkvacuumpolarizationcanbe
included in Monte Carlo lattice simulations with no additional computational
cost. The action is the standard Symanzik improved gluon action
2 1 1 1
S = β ReTr[1−U ]+β ReTr[1−U ]+β ReTr[1−U ]
L g2 P3 P R3 R 63 6
0 x (cid:26) (cid:27)
X
(34)
with modified coefficients
β 2 1
P
β = − 1+α (π/a) 0.4805+X N −
R 20u2  s  R f 25π (am )2
0 Q Q
X
  
β 1 1
P
β = − α (π/a) 0.03325+X N + (35)
6 u2 s  6 f 150π (am )2
0 Q Q
X
5  
β = .
P
3
This action reproduces continuum QCD up to corrections of order a4,(α a)2,
s
α /(am )4 and α2/(am )2. In order to insure reasonable behaviour of the per-
s c s c
turbationseriesm a&0.13isrequired. Withthisconstraint,thisactionshould
c
simulate processes with average internal momentum |q| ≪ 2m . Ideally, the
c
charmquark would be treated dynamically, then these constraints wouldapply
m a instead.
b
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Acknowledgments
We thank Peter Lepage for many useful discussions and Quentin Mason for a
discussion regarding the improved action. This work was supported in part by
the National Science Foundation under grant number PHY0098631.
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