Table Of ContentLattice Study of the Extent of the Conformal Window
in Two-Color Yang-Mills Theory
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a Gennady Voronov∗(for the LSD Collaboration)
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DepartmentofPhysics,YaleUniversity,NewHaven,CT06511,USA
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E-mail: [email protected]
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a WeperformalatticecalculationoftheSchrödingerfunctionalrunningcouplinginSU(2)Yang-
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- Mills theory with six massless Wilson fermions in the fundamental representation. The aim of
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thisworkistodeterminewhethertheabovetheoryhasaninfraredfixedpoint. Duetosensitivity
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h oftheSFrenormalizedcouplingtothetuningofthefermionbaremasswewereunabletoreliably
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extracttherunningcouplingforstrongerbarecouplings.
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XXIXInternationalSymposiumonLatticeFieldTheory
July10-162011
SquawValley,LakeTahoe,California
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
1. Introduction
A SM Higgs adequately accounts for all electroweak measurements (for now). However, it
hasanumberoftheoreticalshortcomings,chieflyamongthemthehierarchyproblem. Technicolor
is a promising alternative for electroweak symmetry breaking that avoids the introduction of a
fundamental light scalar particle. In particular, a Technicolor model based on a walking gauge
theory can properly account for the Standard Model fermion masses[1]. Since such a theory is
expected to reside just below the conformal window, it is essential to narrow down the extent of
this window. The goal of this work is to do that for the SU(2) gauge theories with N fermion
f
flavors in the fundamental representation. These gauge theories are special in that there is an
enhanced SU(2N ) global symmetry and it is currently unknown what implications this will have
f
fortechnicolormodelbuilding.
The asymptotic properties of a field theory are encoded in the RG flow of the couplings. We
areparticularlyinterestedintheinfrareddynamicsoftwo-colorasymptoticallyfreetheories. Such
theorieshavetwopossibledistinctIRbehaviors. Onepossibilityisthattheβ functionhasnozero
aside from the one that leads to the Gaussian UV fixed point. In this case, the running coupling
will increase until it is sufficiently strong to break chiral symmetry. All fermions will screen out
and the coupling will run as in the pure gauge theory. Alternatively the β function may have an
additionalzeroleadingtoanIRfixedpoint.
ThereexistsalargeparameterspaceofYang-Millstheories. Onecanconsiderdifferentgauge
symmetries and different numbers of fermion flavors transforming under various representations
of the gauge group. Restricting ourselves to N flavors of fermions transforming under the fun-
f
damentalrepresentationofSU(N )andfixingN , perturbationtheorypreciselytellsushowmany
c c
fermionsflavorsarerequiredtomaintainasymptoticfreedom. Fortwo-colors,asymptoticfreedom
sets in for N < 11. At a fixed color, as we lower the number of fermion flavors to just below
f
where asymptotic freedom sets in, we have a weakly-coupled IR fixed point and therefore an IR
conformaltheory[2]. However,forsmallN weknowthatchiralsymmetryisbrokenintheIRand
f
thetheoryisconfining. Therefore,weexpectthatthereissomecriticalnumberoffermionflavors,
Nc atwhichatheoryatfixedcolortransitionsfromconfiningtoconformalIRbehaviorandwecan
f
talk of a conformal window, i.e. the range of N at which the theory is both conformal in IR and
f
asymptoticallyfreeinUV.
Theβ functioncanbeexpandedinperturbationtheory. Thefirsttwoperturbativecoefficients
are universal and are given in [2]. The two loop expansion suggests that 5<Nc <6 and the six
f
flavor theory has an IRFP around g¯2 ≈ 144. In perturbation theory, the IRFP at the upper end
∗
of the conformal window is quite weak. As we proceed in decreasing N , the strength of the
f
perturbativeIRFPgrows. Therefore,atthelowerendoftheconformalwindowperturbationtheory
is likely unreliable and we will need to narrow down the extent of the conformal window using
non-perturbativemethods.
TheSU(2)gaugetheorieswithN flavorsoffermionsinthefundamentalrepresentationhave
f
been previously studied. The N =6 theory is studied by Bursa et. al. They show evidence that
f
theN =6theoryisconsistentwithanIRFP5<g¯2<6(intheSFscheme)[3]. Ohkiet. al. show
f ∗
evidence that the N =8 theory has a fixed point g¯2 >7.5 using a twisted Wilson/Polyakov loop
f ∗
methodin[4]. ItispeculiarthatthesixflavortheoryasstudiedbyBursaet. al. wouldhaveaweaker
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LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
fixedpointthantheeightflavortheory,especiallygivenhowstrongthesixflavorperturbativefixed
pointis. OurgoalistothoroughlystudytheSU(2)gaugetheorieswithfermionsinthefundamental
representation. We would like to narrow down the extent of the conformal window for this class
of theories. The unexpected results for the six flavor theory prompted us to begin with it, to see
if we could either confirm or refute the result. Since this work was undertaken, Karavirta et. al.
presentedanSU(2)SF studyoftheN =4,6,and10theories. TheyfindnoevidenceofanIRFP,
f
inconclusive evidence of an IRFP with g¯2 >11, and evidence of an IRFP g¯2 ≈1 in the four, six,
∗ ∗
andtenflavortheoriesrespectively[5].
2. SchrödingerFunctionalSchemeandStepScaling
TheSchrödingerfunctional(SF)Z allowsustodefineanon-perturbativerenormalizedcou-
pling [6]; it is given by a path integral over gauge and fermion fields that reside within a four-
dimensional Euclidean box of spatial extent L with periodic boundary conditions in spatial direc-
tions and Dirichlet boundary conditions in the time direction. We choose gauge boundary con-
ditions [7], U(x,k)| = exp(cid:2)−iηaτ (cid:3) and U(x,k)| = exp(cid:2)−i(π−η)aτ (cid:3), and fermion
x0=0 L 3 x0=L L 3
boundary conditions [8], P ψ| = ψ¯P | = P ψ| = ψ¯P | =0. The gauge bound-
+ x0=0 − x0=0 − x0=L + x0=L
ary conditions classically induce a constant chromoelectric background field whose strength is
characterized by the dimensionless parameter η. With these boundary conditions we see that the
SFZ(η,L)=(cid:82) D[U,ψ,ψ¯]e−S[U,ψ,ψ¯;η].
Therunningcoupling,intheSFscheme,isdefinedby,
(cid:12) (cid:28) (cid:29)
k = ∂ logZ(cid:12)(cid:12) = ∂S , (2.1)
g¯2(L) ∂η (cid:12) ∂η
η=π/4
(cid:104) (cid:105)
where k = −24(L/a)2sin (a/L)2(π−2η) is chosen so that the renormalized coupling agrees
with the bare coupling at tree-level. Note that the first two perturbative coefficients of the SF
runningcouplingareexactlytheuniversalcoefficientsgivenin[2]. Wenowhaveanon-perturbative
definitionforarenormalizedcouplinginaformthatisamenabletoalatticecalculation.
For this work we used the standard Wilson plaquette gauge action and the Wilson fermion
action. An advantage of working in the SF scheme is that we can evaluate the running coupling
alongthem (cid:0)g2(cid:1)curve. m (g2)isdefinedasthebaremassvaluethatresultsinazeroPCACquark
c 0 c 0
mass [9]. We determined m for a range of bare coupling on 83×16 and 163×32 lattices. We
c
foundm tobeconsistentwithinstatisticsbetweenthetwovolumesandconsequentlyinterpolated
c
an m curve in the m −g2 plane using the smaller volume data. For g2 ≤0.5, we determine m
c 0 0 0 c
usingtwo-loopperturbationtheory[10]. Otherwise,weusetheprocedureoutlinedabove.
We are interested in investigating the running of the coupling over a large range of scales.
A step scaling analysis enables us to do this in a manner that is computationally feasible [11].
We begin by calculating the SF renormalized coupling over a range of bare couplings and lattice
volumes. Latticeperturbationtheorygivesg2/g¯2 asanexpansioninpowersofg2. Thismotivates
0 0
aninterpolatingfit[12],
1 1 n m (cid:16)a(cid:17)j
− = ∑ ∑a g2i . (2.2)
g¯2(cid:0)g2,L(cid:1) g2 i,j 0 L
0 a 0 i=0j=0
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LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
(a) (b)
Figure1: Criticalbaremasstoreproducemasslessfermions. In(a)continuumextrapolationofm ,
c
2-loopperturbativem , andvaluesusedincalculationshown. In(b)continuumm shownagainst
c c
locationofbulkphasetransition.
Thisprocedureproducesasmoothfunctionoftherenormalizedcouplingversusthebarecoupling
and inverse lattice volume. We note that it is advantageous to perform one global fit to all data
ratherthanfittinganinterpolatingpolynomialtoeachlatticevolume. Thisallowsustointerpolate
to additional lattice volumes and smooths out the approach to the continuum limit. In Figure 2a
below we show a bootstrap replication of our renormalized coupling data plotted against a global
fit.
Nowwedefinethediscretestepscalingfunction,
(cid:18) (cid:19)(cid:12)
Σ(cid:16)u,a(cid:17)≡ g¯2 g2 ,sL (cid:12)(cid:12) , (2.3)
L 0∗ a (cid:12)
g¯2(g2 ,L)=u
0∗ a
itisthevalueoftherenormalizedcouplingonavolumeof(sL)4 andbarecouplingtunedsuchthat
we have a renormalized coupling of u on a lattice of volume L4. We arrive at a continuum step
scalingfunction,
(cid:16) a(cid:17)
σ(u)= limΣ u, , (2.4)
a→0 L
L
by taking the continuum limit of the discrete step scaling function. In practice, these functions
are evaluated using the interpolating polynomial specified in Eq. (2.2) and with the choice s=2.
We take the continuum limit by evaluating Σ for various values of a/L, taking care to only to use
interpolatedvaluesofa/L,andextrapolatingtozero.
3. Results
We calculated the renormalized coupling for a range of values of the bare coupling g2 and
0
latticevolumesL/a. ConfigurationsweregeneratedwiththeChromaimplementationoftheHMC
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LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
(a) (b)
Figure 2: In (a), all renormalized coupling data points in a typical bootstrap ensemble plotted in
form 1 − 1 against a global interpolating fit to data restricted to g2 ≤0.5. In (b), a plot of the
g2 g¯2 0
0
discretebetafunctionderivedfromfitshownin(a).
(a) (b)
Figure 3: g¯2 evaluated at g2 =0.6 and 0.8, in (a) and (b) respectively, for various lattice volumes
0
andfermionbaremasses.
algorithm and ∂S is evaluated on every configuration [13]. We aimed to acquire on the order of
∂η
100Kconfigurationssince ∂S isanoisyobservablewithlongautocorrelationtimes.
∂η
Upon studying the mass tuning on additional lattice volumes and obtaining additional statis-
tics, we see that our initial observation of no statistically significant dependence on the lattice
volume was incorrect. We calculated an improved m curve by fitting our non-perturbatively de-
c
termined critical bare mass values to an interpolating polynomial m (cid:0)g2,a(cid:1) and extrapolating to
c 0 L
a/L→0. In Figure 1a, we show the the values of m used in our simulations, the two-loop per-
c
turbative m curve, and our improved non-perturbative m curve. It is clear that the values of m
c c c
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LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
used in our evaluations of the renormalized coupling deviate significantly from the appropriate
values. Moreover, Figure 2a, which shows our renormalized coupling measurements versus the
barecoupling,indicatesajumpdiscontinuityatpreciselythevalueofthebarecouplingwherewe
switchfromusingtheperturbativelydeterminedm toourmiss-tunedvaluesofm . Weperformed
c c
a global fit to all of our renormalized coupling data that was calculated with a perturbative value
of m , i.e. for g2 ≤0.5. The discontinuity in g¯2, is highlighted by extrapolating our fit to the first
c 0
data points which are mistuned in the bare mass. To quantify this we evaluated a limited sample
ofrenormalizedcouplingsatavarietyofbaremassesnearm . Figure3shows1/g¯2 versusm for
c 0
various lattice volumes and g2 =0.6 and 0.8. This figure demonstrates that g¯2 is trending in the
0
correctdirectiontoremovethediscontinuityinFigure2aandthatamistuningofthebaremassby
as little as 4% results in an error in the renormalized coupling that dominates all other sources of
error. Moreover, such an error can easily introduce or obscure an IRFP, especially if the the RG
flowoftherunningcouplingisparticularlyslow.
Inordertoguaranteethatwecantakeacontinuumlimit,weneedtoensurethatweobtaindata
fromtheweak-couplingsideofanyspuriouslatticephasetransition. Withthisinmind,wescanned
throughthebareparameterspaceandlocatedpeaksintheplaquettesusceptibilityonaL/a=83×9
lattice. Thissearchindicatesalineinthem −g2 planeoffirstorderphasetransitionthatendsata
0 0
criticalpoint ataroundg2 ≈2.2. Forg2 (cid:46)2.2, we seecrossoverbehavior. InFigure 1b, weshow
0 0
theabovetransitionlineplottedalongwithm (g2). Figure1bindicatesthatthesixflavormassless
c 0
fundamentalWilsonfermionactionhasasensiblecontinuumlimitonlyforg2(cid:46)2.2. Therefore,we
0
expectthatwewillnotbeabletoexaminetherunningcouplingatsufficientlystrongbarecoupling
withourcurrentaction.
Finally, using only data generated with a properly tuned bare mass (g2 ≤0.5), we study the
0
continuumrunningofthistheory. Towardsthisend,wegenerated2000bootstrapensemblesfrom
ourrestricteddatasetandapplythefittingproceduredescribedintheprevioussectiontoeachindi-
vidualensemble. Astepscalinganalysis,usingfitsliketheoneshowninFig2a,withs=2,using
a linear extrapolation to the continuum using linearly spaced values between L/a=7 and 10, was
then used to produce curves, for each individual ensemble, of the discrete beta function σ(u)/u.
Figure2bshowsσ(u)/uplottedagainstone-loopperturbationtheory. Eachpoint,withtwo-sided
errorbars, was obtained via the BC method [14]. This plot demonstrates, that our methods can
a
properlyreproduceperturbationtheorywhenthebaremassistunedappropriately.
4. ConclusionsandOutlook
We were unable to extract a renormalized coupling flow outside the perturbative region due
to mistuning of the bare fermion mass. We have since properly tuned the bare mass. Figure 2b
demonstrates that our production and analysis applications can reproduce perturbation theory in
a robust manner. We found that these figures do not qualitatively change as we change the fit
parametersanddetailsofthecontinuumextrapolation.
The RG flow of the running coupling in the six-flavor SU(2) theory is particularly slow and
consequentlytherenormalizedcouplingsensitivelydependsonthebarefermionmass. Weempha-
size that in such theories the bare mass must be tuned very carefully. We have since done this for
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LatticeStudyoftheExtentoftheConformalWindowinTwo-ColorYang-MillsTheory GennadyVoronov
theSU(2)six-flavortheory. Inordertoreliablystudytherunningcouplingofthistheorywewould
simplyhavetoreproduceourdatausingthecorrectvalueofthefermionbaremass.
Studies of the plaquette and plaquette susceptibility indicate the presence of bulk phase tran-
sitionalongthem lineatg2 ≈2.2. Thiseffectivelylimitshowstrongofarenormalizedcoupling
c 0
thatwecaninvestigatewiththestandardWilsonfermionaction. Otherinvestigationsintothesix-
flavor theory suggest that if there exists an IRFP then it likely resides at a stronger coupling that
can be probed with our current action before encountering a bulk phase transition [5]. Therefore,
whilewecanregenerateanewdatasetwithanimprovedestimateofm , webelievethatitwould
c
beamisallocationofresourcestopursuethistheorywithoutfirstswitchingtoanimprovedaction
wherethebulkphasetransitionoccursatastrongercoupling.
References
[1] T.Appelquist,D.Karabali,L.Wijewardhana,Chrialhierarchiesandtheflavorchangingneutral
currentproblemintechnicolor,Phys.Rev.Lett.57(1986)957.
[2] W.Caswell,Asymptoticbehaviorofnon-abeliangaugetheoriestotwo-looporder,Phys.Rev.Lett.33
(1974)244.
[3] F.Bursa,L.DelDebbio,L.Keegan,C.PicaandT.Pickup,Massanomalousdimensionandrunningof
thecouplinginSU(2)withsixfundamentalfermions,PoSLattice2010(2010)070[arXiv:1010.0901].
[4] H.Ohki,StudyofthescalingpropertiesinSU(2)gaugetheorywitheightflavors,PoSLattice2010
(2010)066[arXiv:1011.0373].
[5] T.Karavirta,J.Rantaharju,K.RummukainenandK.Tuominen,Determiningtheconformalwindow:
SU(2)gaugetheorywithN =4,6and10fermionflavors,arXiv:1111.4104.
f
[6] M.Lüscher,R.Narayanan,P.WeiszandU.Wolff,TheSchrödingerfunctional-arenormalizable
probefornon-abeliangaugetheories,Nucl.Phys.B384(1992)168-228[arXiv:hep-lat/9207009].
[7] M.Lüscher,R.Sommer,U.WolffandP.Weisz,ComputationoftherunningcouplingintheSU(2)
Yang-Millstheory,Nucl.Phys.B389(1993)247-264[arXiv:hep-lat/9207010].
[8] S.Sint,OntheSchrödingerfunctionalinQCD,Nucl.Phys.B421(1994)135-158
[arXiv:hep-lat/9312079].
[9] M.Lüscher,S.Sint,R.Sommer,P.WeiszandU.Wolff,Non-perturbativeO(a)improvementof
latticeQCD,Nucl.Phys.B491(1997)323-343[arXiv:hep-lat/9609035].
[10] H.PanagopoulosandY.Proestos,ThecriticalhoppingparameterinO(a)improvedLatticeQCD,
Phys.Rev.D65(2002)014511[arXiv:hep-lat/0108021].
[11] M.Lüscher,P.WeiszandU.Wolff,Anumericalmethodtocomputetherunningcouplingin
asymptoticallyfreetheories,Nucl.Phys.B359(1991)221-243.
[12] T.Appelquist,G.Fleming,andE.Neil,LatticestudyofconformalbehaviorinSU(3)Yang-Mills
theories,Phys.Rev.D79(2009)076010[arXiv:0901.3766].
[13] R.EdwardsandB.Joó,TheChromasoftwaresystemforlatticeQCD,Nucl.Phys.Proc.Suppl.140
(2005)832[arXiv:hep-lat/0409003].
[14] B.Efron,R.Tibshirani,Anintroductiontothebootstrap,Chapman&Hall,BocaRaton1993.
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