Connecting Lattice QCD with Chiral Perturbation Theory at Strong Coupling

0.17 MB·English

by  Shailesh Chandrasekharan

Connecting Lattice QCD with Chiral Perturbation Theory at Strong Coupling
Shailesh Chandrasekharana and Costas G. Strouthosa,b
aDepartment of Physics, Box 90305, Duke University, Durham, North Carolina 27708.
bDivision of Science and Engineering, Frederick Institute of Technology, Nicosia 1303, Cyprus.
Westudy thedifficulties associated with detecting chiral singularities predicted bychiral pertur-
bationtheory(ChPT)inlatticeQCD.WefocusonthephysicsoftheremnantO(2)chiralsymmetry
of staggered fermions in thestrong coupling limit using the recently discovered directed path algo-
rithm. Since it is easier to look for power-like singularities as compared to logarithmic ones, our
calculationsareperformedat afixedfinitetemperatureinthechirallybrokenphase. Weshowthat
thebehaviorofthechiralcondensate,thepionmassandthepiondecayconstant,forsmall masses,
are all consistent with the predictions of ChPT. However, the values of the quark masses that we
need to demonstrate this are much smaller than those being used in dynamical QCD simulations.
Wealso need to use higher orderterms in thechiral expansion to fit our data.
4
0
INTRODUCTION T . ThistransitionbelongstotheO(2)universalityclass
0 c
2 [13]. Thus, at a fixed temperature below Tc, within the
n With the advent of fast computers it is now possible theO(2)scalingwindow,thelongdistancephysicsofour
a to calculate many hadronic quantities from first princi- modelisdescribedbyathree-dimensionalO(2)fieldthe-
J ples using lattice QCD. However, today these calcula- ory in its broken phase. At this temperature there is a

1 tions contain systematic errors due to finite lattice spac- range of quark masses where the light pions are describ-
ings,finitevolumesandquenching. Althoughinprinciple ableby the conventionalcontinuumChPT.The effective
1 all these errors can be controlled, the only clean way to action can be written in terms of S~, an O(2) vector field
v
2 reduce quenchingerrorsis to performunquenchedcalcu- with the constraint S~ S~ =1. At the lowest order this is
·
0 lations. Due to algorithmic difficulties today most un- given by [14]
0 quenched calculations are performed using large quark
1
40 mvaalusseessuasnindgth(qeureensuchltesdaarendthpeanrteixatlrlyapqoulaetnecdhetdo)reCahliPstTic. Seff = d3x F22 ∂µS~ ·∂µS~+Σ~h·S~ (1)
0 Unfortunately, recent attempts to connect lattice QCD Z h i
/ with the usual one-loop ChPT predictions have failed to
t
a giveclearanswers[1, 2, 3, 4]. Itis nowbelievedthatlat- where~histhemagneticfieldand ~h isidentifiedwiththe
-l ticeartifactsshouldbetakenintoconsiderationinChPT quarkmass. Thetwolowenergyc|on|stantsF2andΣthat
p
[5,6,7,8]. Ithasbeensuggestedin[9,10]thelatticedata appearat this order arethe piondecay constantandthe
e
is described better by the resulting more elaborate fit- chiralcondensaterespectively. Wenotethatsinceweare
h
: ting functions. There are alsoother interesting attempts discussing a three-dimensional effective theory, F2 has
v
to extract useful information using finite size effects in the dimensions of inverse length. Thus, we can define
i
X ChPT [11]. a correlation length ξ 1/F2. For massless quarks ξ
≡
r Given the difficulties associated with understanding diverges as (Tc T)−ν at Tc. In order to connect our
a −
chiral singularities in a realistic calculation of QCD, in model with ChPT and avoid lattice artifacts we need
this paper we explore the subject in strong coupling lat- to choose the temperature and quark masses such that
tice QCD with staggered fermions. We use a very effi- 1<<ξ <<1/M (lengths are measuredin lattice units)
π
cientalgorithmdiscoveredrecentlytosolvethismodelin where M is the pion mass. In this region we expect
π
the chiral limit [12]. Although the strong coupling limit observables such as the chiral condensate, the pion mass
suffers from severe lattice artifacts, when the quarks are and the pion decay constant satisfy the expansion
masslesslattice QCDwith staggeredfermionshasanex-
actO(2)chiralsymmetrywhichisbrokenspontaneously.
Thus,ourmodelcontainssomeoftheremnantphysicsof hOi=z0+z1√m+z2m+z3m√m+..., (2)
chiralsingularitiesexpectedinQCD.Inparticular,there
are light pions and it would be useful to understand the wherethe√mbehavioristhepower-likesingularityaris-
range of quark masses where the singularities predicted ing due to the infrared pion physics [15]. The main ob-
by conventional ChPT can be seen. jective of this paper is to detect these singularities in
Insteadoffocusingonchiralsingularitiesthatareloga- the context of strong coupling lattice QCD with stag-
rithmic, in this work we focus on power-likesingularities gered fermions. Such power-like singularities have been
that arise at finite temperatures. Recently, it was shown observedin spin models [16], but as far as we know have
with high precision that our model undergoes a second not been studied with precision in lattice QCD calcula-
order chiral phase transition at a critical temperature tions.
2
THE MODEL AND OBSERVABLES (iii) the helicity modulus
1
artTichlee ipsagritviteinonbyfunction of the model we study in this Y = L3* [ Jx,1]2+[ Jx,2]2+[ Jx,3]2 +, (7)
nXx Xx Xx o
where J =σ (b N/8), with σ =1 on even
Z(T,m)= [dU][dψdψ¯] exp S[U,ψ,ψ¯] , (3) x,µ x x,µ− x
− sites and σx = 1 on odd sites and
Z −
(cid:0) (cid:1)
(iv) the pion mass, obtained using the exponential de-
where [dU] is the Haar measure over U(3) matrices and
[dψdψ¯] specify Grassmann integration. At strong cou- cayofthecorrelationfunctionalongoneofthespa-
pling, the Euclidean space action S[U,ψ,ψ¯] is given by tial directions µ=1,2,3:
− ηx2,µ ψ¯xUx,µψx+µˆ−ψ¯x+µˆUx†,µψx −m ψ¯xψx, |xµli|m→∞Xx⊥hσxnxn0i=Ce−Mπ|xµ|, (8)
Xx,µ h i Xx
(4) where x refers to components of the coordinate x per-
⊥
where x refers to the lattice site on a periodic four-
pendicular to the µ direction. When m = 0 the cur-
dimensional hyper-cubic lattice of size L along the three
rent J is the conserved current associated with the
x,µ
spatial directions and size Lt along the euclidean time O(2) chiral symmetry. As discussed in [14], one can
direction. The index µ = 1,2,3,4 refers to the four
define the pion decay constant at a quark mass m to
mspaatcrei-xtirmeperedsiernecttiniogntsh,eUgxa,µug∈e fiUel(d3s), aisndthψe ,uψ¯suaalrelinthkes be Fm2 ≡ limL→∞Y. For m = 0 we then obtain
x x F = limm→0Fm, the pion decay constant introduced in
three-componentstaggeredquarkfields. Thegaugefields
Eq.(1). We canalsodefine Σ =lim φ the infinite
m L→∞
satisfy periodic boundary conditions while the quark volume chiral condensate. Again Σ=limmh→i0Σm.
fields satisfy either periodic or anti-periodic boundary
conditions. The factors η are the well-known stag-
x,µ
gered fermion phase factors. We choose them to be RESULTS
η2 = 1,µ = 1,2,3 (spatial directions) and η2 = T
x,µ x,4
(temporal direction), where the real parameter T acts We have done extensive computations at L = 4. It
t
like a temperature. By working on asymmetric lattices has been shown with high precision in [13] that this
withLt <<LatfixedLtandvaryingT continuouslyone model undergoes a chiralphase transition which belongs
can study finite temperature phase transitions in strong tothethree-dimensionalO(2)universalityclassatacriti-
coupling QCD [17]. We use U(3) gauge fields instead of caltemperatureT =7.47739(3). Figure1showstheplot
c
SU(3)inordertoavoidinefficienciesinthealgorithmdue ofF2 asafunctionofT. Basedonuniversalityweexpect
totheexistenceofbaryonicloops. Thisdistinctionisnot ξ−1 F2 =C(T T)ν forT <T withν =0.67155(27)
c c
important for our study since the baryons are expected [19].≡ If we fit o−ur results to this form in the range
to have a mass close to the cutoff. 7.05 T 7.42, with T and ν fixed to the expected
c
ThepartitionfunctiongiveninEq.(3)canberewritten value≤s, we≤find C = 0.2217(1) with χ2/DOF= 1.03. In-
inamonomer-dimerrepresentationasdiscussedindetail cluding T = 7.0 in the fit makes the χ2/DOF jump to
in[13,18]. Everyconfigurationinthenewrepresentation 2.0. At a temperature very close to T , we expect ξ to
c
is described by monomer variables nx = 0,1,2,3 associ- be extremely large and it would be difficult to satisfy
ated to the sites, and dimer variables bx,µ = 0,1,2,3 1/Mπ >> ξ with our limited computing resources. On
associated to the bonds connecting neighboring sites x the other handin orderto avoidlattice artifactsit is im-
and x+µˆ, along with the constraint that at each site, portant not to have ξ 1. Using the above analysis we
∼
nx+ µ[bx,µ+bx−µˆ,µ]=3. estimate that T = 7.0 is at the edge of the O(2) scaling
Inordertostudychiralphysicswefocusonthefollow- window and hence we choose to fix T at this value and
P
ing observables: study the long distancephysics nearthe chirallimit. We
vary the spatial lattice size from L = 8 to L = 96 for a
(i) The chiral condensate variety of masses 0 m 0.025. At m=0 we also have
≤ ≤
dataatL=120,144. Belowwediscussourmainresults.
1 1 ∂
φ = Z(T,m), (5)
h i L3Z ∂m
The Pion Decay Constant
(ii) the chiral susceptibility
Our results show that for L 32 the finite size effects
1 1 ∂2 on Y are smaller than the stat≥istical errors. Hence it is
χ= Z(T,m), (6)
L3Z ∂m2 relatively easy to compute F2 by extrapolating results
m
3
0.2
0.43
0.42
0.15
0.41
0.385
F2 0.1 Fm 0.4 0.38
0.39 Fm0.375
0.38
0.05 0.37
0.37 0.365
0 0.0005 0.001 0.0015
m
0.36
0 0 0.002 0.004 0.006 0.008 0.01 0.012
6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 m
T
fFuInGc.tio1n: 0.P2l2o1t7(o7f.4F7723v9s. TT).0.6T71h55e.solid line represents the FfuInGc.ti2o:n FPmlo=t 0o.f36F7m47v[1s.+m34..2mTh−e 1s8o5limd√linme].represents the
−
The Chiral Condensate
for Y at differentvolumes. InTable I we give the results
for some values of the quark mass. The first four terms
Chiral perturbation theory predicts that the first four
m Fm m Fm terms in the chiral expansion of Σm are given by
0.00000 0.3675(1) 0.0050 0.4063(1)
0.00025 0.3703(1) 0.0075 0.4188(1) Σm =Σ[1+b1√m+b2m+b3m√m]. (10)
0.00050 0.37300(15) 0.0100 0.42890(15)
As shown in [14], b1 (N 1), implying that b1 = 0 in
0.00100 0.37804(15) 0.0200 0.4583(2) our case. In order to∝test−how well Eq. (10) des6cribes
0.0025 0.3903(2) 0.0250 0.4693(2) our model, we compute Σ at small masses.
m
Atm=0wecomputeΣbyusingthefinitesizescaling
TABLE I: Fm as a function of m.
formula for χ given by [14]
CinhtPhTe cahrieragliveexnpabnysi[o1n4]ofFm inthree-dimensionalO(N) χ= N1 Σ2L3 1+0.226(N −1)F12L + Lα2 +... , (11)
h i
Fm =F[1+a1√m+a2m+a3m√m], (9) where α depends on the higher order low energy con-
where a1 (N 2). Since in our case N =2 we expect stantsofthechiralLagrangian. WhenF isfixedto0.3675
a1 = 0. I∝n ord−er to check if our results fit the predic- obtainedearlier,ourresultsfor16≤L≤144fitverywell
tions of ChPT we fit our data to Eq.(9) in the range tothisformula. WefindΣ=2.2648(10),α=4.6(3)with
0 m 0.00625. We find F = 0.3674(1), a1 = 0.01(2), χ2/DOF=0.87.
a2≤=34≤.0(7)and a3 = 182(6)with χ2/DOF=1.2. Our
−
data and the fit are shown in Fig. 2. The prediction m Σm m Σm
of ChPT that a1 = 0 is in excellent agreement with our 0.00000 2.2648(10) 0.0050 2.6340(08)
results. Fixing a1 = 0 in the fit yields F = 0.36747(6), 0.00025 2.2340(10) 0.0075 2.7278(15)
a2 = 34.3(3) and a3 = 185(3), while the χ2/DOF re- 0.00050 2.3560(10) 0.0100 2.8025(15)
−
mains essentially unchanged. Interestingly if we fit the 0.00100 2.4040(05) 0.0200 3.0199(11)
data in the range 0 m 0.00135 with a1 = a3 = 0 0.0025 2.5103(08) 0.0250 3.0986(08)
≤ ≤
fixed,wefindthatF =0.36764(7)anda2 =28.3(3)with
χ2/DOF=0.9. This shows that there are systematic er- TABLE II: Values of Σm at selected values of m.
rorsin evaluating the fitting parametersdue to contami-
nationfromhigherorderterms. Inthe insetofFig. 2we For m = 0 one can measure Σ by extrapolating the
m
6
focus on the extremely small mass region and show that resultsfor φ atdifferentvolumes. However,duetocrit-
h i
a1 = 0 is even clear to the eye. This is one of the main icalslowingdownitisdifficulttomeasurethecondensate
results of our paper. We suggest that a1 = 0 is a useful accurately at a fixed small mass on large volumes. With
signature of the O(2) universality and could be used in ourcomputingresourceswefindthatthisproceduregives
future studies of lattice QCD with staggeredfermions. reliable answers for Σ only up to m 0.001. On the
m
∼
4
0.4
2.8
2.7
0.3
2.45
2.6 0.15
Σm 2.4 Mπ
0.2
2.5 0.1
Σ M
m2.35 π
2.4 0.05
2.3 0.1
2.3
2.250 0.0005 0.001 0.0015 00 0.0005 0.001 0.0015
m m
2.2 0
0 0.002 0.004 0.006 0.008 0.01 0.012 0 0.002 0.004 0.006 0.008 0.01 0.012
m m
FIG. 3: Plot of Σm vs. m. The solid line represents the FIG. 4: Plot of Mπ vs. m. The solid line represents the
function Σm =2.2642[1+1.36√m+23m−135m√m]. function Mπ =4.095√m[1+0.68√m−25.8m+149m√m].
The Pion Mass
The first four terms in the chiral behavior of the pion
other hand when m = 0, in the large volume limit we
expect χ Σ2 L3 up6 to an additive constant plus ex- mass are predicted to be of the form
∼ m
ponentially small corrections. We find that the signal
Σ
fmourcΣhmcleeaxntrearcatetdsmfraolml mfiattsisnegs tthhaendtahtea stiogntahlisobfotramineids Mπ =rF2√m 1+c1√m+c2m+c3m√m . (12)
h i
from the direct measurement. Further, the values of
Σm obtainedusingthisprocedureagreewellwiththedi- Further,thechiralWardidentitiesimplythatc1 =b1/2−
rect measurement at larger quark masses. In table II we a1. Ourestimatesforthepionmassesforaselectedrange
of m are shown in Table III. Fitting our data to Eq.(12)
give the values of Σ obtained for selected values of the
m
after fixing Σ = 2.2648 and F = 0.3675 obtained above,
quarkmasses. FittingourresultstoEq.(10)intheregion
b02≤=m23(≤1)0,.b0306=25 g1i3v5es(1Σ0)=wi2th.26a4χ2(21/0D),ObF1==1.11..36O(4u)r, wweithfinχd2/cD1O=F=0.505.5(1.6W),ec2se=e t−ha2t1(o6u)radnadtac3is=con1s1i0s(t5en0t)
results are shown i−n Fig 3. We note that we cannot find withtherelationc1 =b1/2−a1althoughtheerrorinc1is
a mass range within our results in which we can find large. Fixing c1 =b1/2 = 0.68 obtained from fitting the
a good fit when we fix b2 = b3 = 0. However, in the chiralcondensate,yieldsc2 =−25.8(8)andc3 =149(14)
without changing the quality of the fit. This latter fit
range 0 m 0.00175 we can set b3 = 0 to obtain
Σ = 2.26≤43(10≤), b1 = 1.44(3) and b2 = 15.2(7) with a along with our results for Mπ are shown in Fig. 4.
χ2/DOF= 1. Note that b2 changes by about 30% when
this different fitting procedure is used, while b1 is more
DISCUSSION
stable.
Itisexpectedthatthenaturalexpansionparameterfor
ChPTinthreedimensionsisx M /(4πF2). Letusfind
π
≡
thevaluesofxwherethechiralexpansionuptoacertain
m Σm m Σm
power of x is sufficient to describe the data reasonably.
0.00000 0.000 0.0050 0.2812(8)
Of course this question is model-dependent, since some
0.00025 0.0650(3) 0.0075 0.3399(5)
models may have larger contributions at higher orders
0.00050 0.0920(4) 0.0100 0.3871(4) compared to others. Here we ask this question in the
0.00100 0.1295(4) 0.0200 0.5285(10) context of the model studied in this paper.
0.0025 0.2022(3) 0.0250 0.5837(8) In order to understand the dependence of F2 and Σ
m m
on x we plot in Fig. 5 these two observables as func-
TABLEIII: Values of Mπ at selected valuesof m. tions of x. The solid lines are the best fits to the form
z0 +z1x+z2x2 +z3x3, while the dot-dashed lines are
the best fits to a smaller range in x with z3 set to 0. As
5
Acknowledgments
0.18 2.7
Σ We thank M. Golterman, S. Hands, T. Mehen, S.
m
0.17 F 2 2.6 Sharpe, R. Springer and U.-J. Wiese for helpful com-
m ments. This work was supported in part by the Depart-
2.5 ment of Energy (DOE) grant DE-FG-96ER40945. SC is
0.16
2 Σ also supported by an OJI grant DE-FG02-03ER41241.
F 2.4
m m The computations were performed on the CHAMP, a
0.15
computerclusterfundedinpartbytheDOEandlocated
2.3
in the Physics Department at Duke University.
0.14
2.2
0.13 2.1
00 0.025 0.05 0.075 00..11 0.125 0.15 [1] S. Hashimoto et. al., Nucl. Phys. (Proc.Suppl.) 119, 332
x
(2003).
[2] S. Du¨rr, arXiv: hep-lat/0209319.
FIG. 5: Plot of Fm2 and Σm vs. x Mπ/(4πF2). The solid [3] I. Montvay,arXiv: hep-lat/0301013.
lines and dot-dashed lines are fits t≡o two different orders of [4] C.Bernardet.al.,Nucl.Phys.B.(Proc.Suppl.)119,170
chiral expansion discussed in thetext. (2003).
[5] W. Lee and S.Sharpe, Phys.Rev.D60, 114503 (1999).
[6] C. Aubinand C. Bernard, arXiv:hep-lat/0308036.
[7] G. Rupak and N. Shoresh, Phys. Rev. D66, 054503
the graph indicates the error in Σ due to the absence
m (2002).
ofthez3 termisdifferentfromtheerrorinFm2 atagiven [8] S. Aoki, Phys.Rev. D68, 054508 (2003).
value of x. In order to determine F2 within say 5% we
m [9] C. Aubinet. al., arXiv:hep-lat/0309088.
need the z3 term even at x 0.15. This shows that our [10] F.Farchioni,I.Montvay,E.ScholzandL.Scorzato,Eur.
∼
model contains important higher order terms. An inter- Phys. J. C31, 227 (2003).
esting question which we cannot answer at this point is [11] W.Bietenholz,T.Chiarappa,K.Jansen,K.I.Nagaiand
whether this property is generic or not. In any case we S. Shcheredin,arXiv:hep-lat/0311012.
[12] D.H.AdamsandS.Chandrasekharan,Nucl.Phys.B662,
have shown that the connection of lattice QCD data to
220 (2003).
ChPT is indeed possible as long as one does not assume
[13] S. Chandrasekharan and F.J.Jiang, Phys. Rev. D68,
thathigherordertermsarenegligible. For the lowestor- 091501(R) (2003).
dertermstodominateitmaybenecessarytogotomuch [14] P. Hasenfratz and H. Leutwyler, Nucl. Phys. B343, 241
smallerquarkmasesthanhasbeenpossibleuntilnow. In (1990).
factitismucheasiertoconnectourresultswithChPTat [15] D.J. Wallace and R.K.P.Zia, Phys. Rev. B12, 5340
m=0, which is called the ǫ-regime of ChPT [11]. Thus, (1975).
[16] J.Engels,S.Holtmann,T.MendezandT.Schulze,Phys.
findinganalgorithmtoworkdirectlyatm=0isauseful
Lett. B514, 299 (2001).
goalto strive for in the future. Finally we hope that our
[17] G. Boyd, J. Fingberg, F. Karsch, L. Karkkainen and B.
results will motivate further work to uncover the power-
Petersson Nucl. Phys.B376, 199 (1992).
law chiral singularities at finite temperatures in Lattice [18] P. Rossi and U. Wolff,Nucl. Phys. B248, 105 (1984).
QCD at weaker couplings. This should be easier than [19] M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi
lookingforlogarithmicsingularitiesatzerotemperature. and E. Vicari, Phys.Rev.B63, 214503 (2001).