Table Of ContentConformal or confining – results from lattice gauge
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BenjaminSvetitsky∗
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p RaymondandBeverlySacklerSchoolofPhysicsandAstronomy,TelAvivUniversity,69978Tel
e Aviv,Israel
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E-mail:[email protected]
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We have calculated the running coupling in SU(2), SU(3), and SU(4) gauge theories to see
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7 whethertheyhaveinfraredfixedpoints. Aninfraredfixedpointmeansnoconfinement:Itmeans
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that the long-distancephysicsis conformal, withouta mass scale and indeed withouta particle
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. spectrum.Allthesetheoriesrunslowlyinperturbationtheory,soanonperturbativedetermination
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XthQuarkConfinementandtheHadronSpectrum
8–12October2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Conformalorconfining BenjaminSvetitsky
1. Introduction
ForseveralyearsmycollaboratorsandIhavebeenstudyingSU(N)gaugetheories,N=2,3,4,
with fermions in the two-index symmetric representation of color [1]–[6]. In each case, we have
examined the theory with2flavors. Asafunction of the number offlavors, these theories lie near
the borderline between confinement and screening [7]; screened gauge theories are believed to be
governed by an infrared fixed point, and hence are called conformal. The main question we have
been asking is whether each theory is above or below the borderline, that is, whether it is inside
theconformalwindowinN . Withresultsforthesethreetheories inhand,itisagoodtimetosum
f
thingsup.
QCD, familiar and much-studied, has a rapidly running coupling that becomes strong in the
infrared, where the consequences are quark confinement and spontaneous breaking of chiral sym-
metry. The monotonic running of the coupling is expressed by the strictly negative b function of
QCD. AsweincreasethenumberofmasslessflavorsinQCD,weexpect(basedontheformofthe
two-loop b function [8, 9]) that the b function will develop a shape that includes a zero-crossing
at some coupling g∗; this is an infrared fixed point, where the theory is trapped as we go to large
distance scales. Ofspecial interest istheregimein N wherethetheoryisjustshort offormingan
f
IRFP;thisisatheorywithab function thatliesneartheaxis,awalkingtheory.
Ourapproachtothetheoriesathandisviatheirb functions. WeusetheSchrödingerfunctional
method [10, 11] to calculate the running coupling and to see how it changes with scale; the scale
chosen is the volume of the lattice, which affects the coupling through a background field. No
lattice calculation, however, can proceed without understanding the phase diagram of the theory,
and indeed the phase diagram offers its own hints about the physics of the continuum theory.
Finally, once the Schrödinger functional is implemented, it is straightforward to go beyond the b
function to calculate the anomalous dimensions of various operators. The anomalous dimension
g of the fermion mass parameter [12] is in fact easier to calculate than the b function, and it is
m
notwithoutphenomenological interest. Iwilldescribe the methodandresultsinthistalk. Butfirst
I’d like to make some general comments about the difficulty of life near the sill of the conformal
window.
The cases of walking and of an IRFP are both hard cases for study, and they are hard to tell
apart. They are characterized by slow running (at best), so that strong coupling in the IR—where
we look for the fixed point—is also strong coupling in the UV—that is, at the scale of the lattice
spacinga. AQCD-styleweak-coupling continuum limitwouldrequireanenormousratiobetween
thesizeLofthelatticeandthelatticespacinga. Ontheotherhand,anylatticeofordinary,practical
dimensionsallowsyoutolookonlyatanarrowrangeofscales. Infact,eveninaclearlyconformal
theorythelargestscaleLmaystillleaveyoufarfromtheIRFP,sothatyoumissthescaleinvariance
entirely.
Another way of saying this [13] is to note that if there is an IRFP then the gauge coupling is
an irrelevant coupling under the RG, while the quark mass m is a relevant parameter, as is 1/L.
q
Thismeansthatthemasslesslimitwouldbringintoplayvery badfinite-size effects.
AllthissuppliesmotivationtousetheSchrödingerfunctional, whichturnsfinitevolumefrom
ahindrance intoamethod.
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Conformalorconfining BenjaminSvetitsky
2. The b function inthe masslesstheory
TheSchrödinger functional method defines therunning coupling asfollows. Takeahypercu-
bical Euclidean box of volume L4 and a massless gauge theory. We fix the gauge field Ai(x,t) on
the two time boundaries of the box, t =0,L. This sets up a background electric field in the box;
this field is the unique minimum of the classical action Scl = d4xFmn2 . Wescale the field withL
so that L is the only scale in the problem. Now we calculate R(in principle) the quantum partition
function Z and, from it, the effective action G =−logZ of the background field and its quantum
fluctuations. Inperturbationtheoryithastobeproportionaltotheclassicalaction,andtheconstant
ofproportionality isjusttherunning coupling atthescale L,
1
G = S . (2.1)
cl
g2(L)
The SF method takes this to be the nonperturbative definition of the running coupling: It is the
ratioofG totheclassical actionofthebackground field.
Letmeskipmostofthedetailsofhowwecarryoutsuchacalculationonthelattice. Themost
important fact is that, since G cannot be calculated directly, we work with its derivative dG /dh
with respect to some parameter h in the boundary gauge field. This can be calculated as a lattice
correlation function, andwethencompareitsvaluetothesamederivativeofS toobtaing2(L).
cl
WeworkwithWilson–cloverfermions,usingfatlinksinthefermionactiontosuppresslattice
artifacts. For each value of the (bare) lattice coupling g , we find the critical hopping parameter
0
k (g ) (related to the bare mass) by demanding that the current mass m of the fermion be zero.
c 0 q
ThecurrentmassisdefinedbyanaxialWardidentity thatrelates correlation functions oftheaxial
current tothoseofthepseudoscalar density,
¶ m hAam (x)Oai=2mqhPa(x)Oai, (2.2)
where Oa is a convenient source on a wall of the lattice. Then, given g and k , we calculate the
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running coupling forarangeoflatticesizes L=6a,...,16a.
Let’s focus now on the SU(2) theory [4]. (The fermions are color triplets.) If we plot the
running coupling against logL (Fig. 1) wefind that the data for each value of g line up. Starting
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with L=6a, let’s define a variable for the inverse coupling at scale factor s according to u(s)≡
1/g2(sL). Thebetafunction foruis
b˜(u)≡du/dlogs=2b (g2)/g4. (2.3)
TheslopesinFig.1wouldgiveb˜(u)directly, exceptthattherunningcoupling g2(L)changesasL
is increased for any g . Here, however, we can take advantage of the fact that the theory is either
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conformal or close to it: The running of the coupling is so slow that the coupling changes little
along the lines drawn in the figure, and hence b˜(u)—the slope of each line—also changes little.
Extractingtheslopesandplottingthemagainstthecouplingg(L=8a)givestheapproximate b˜(u)
plotted in the right-hand figure. Evidently the b function crosses zero, an infrared fixed point.
Interestingly, thelattice result tracksthetwo-loop b function fairlywell. Comparison oftheblack
squares (the full fits) to the red circles (obtained by dropping the L=6a lattices) gives some idea
ofthesensitivity oftheresulttothelatticespacing.
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Conformalorconfining BenjaminSvetitsky
0.5
0.4
1/g20.3
0.2
0.1
0
16 12 8 6
L/a
Figure1: SU(2)/triplettheory[4]. Left: SFcouplingvs.sizeLofthelattice,forbarecouplingsg ranging
0
fromweak(top)tostrong(bottom). Thefittedslopesgivetheb˜ function. Thedottedlinehastheslopeof
b˜ calculatedintheweakcouplinglimit(oneloop). Right: Theb˜ functionextractedfromtheseslopes. The
dash-dottedlineistheone-loopvalueandthedottedcurveisthetwo-loopresult.
mq< 0
mq= 0
1st
k
confining k
c
non−conf.
1st
No critical point
b
Figure2:SketchedphasediagramfortheSU(3)/sextettheory.
3. PhaseDiagram
Let’s pause for a look at a phase diagram. Fig. 2 is a sketch of the phase diagram we’ve
explored [3] for the SU(3) theory (with sextet fermions) on a finite lattice. The lower left-hand
cornerofthediagram isastrong-coupling, confining phase, walledoffby(1)thefirst-orderfinite-
temperature transition to the right, which connects to (2) a first-order phase boundary reached by
increasingk inthesmallb (≡6/g2)regime. Therearetwoimportantconsequencesofthisdiagram
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forourwork.
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Conformalorconfining BenjaminSvetitsky
1. How is one to contemplate a continuum limit for this theory? In QCD, the horizontal part
of the phase boundary is a second-order transition to the Aoki phase. One reaches a theory
with massless pions on this curve; then, as L is increased, the corner of the confining phase
slidestotheright,towardsacontinuum limitwithamassless(oratleastalight)pion. Inthe
sextet theory, on the other hand, the confining phase is strictly massive and so is the phase
boundary; it is hard to imagine taking such a limit, unless there is a dramatic change in the
phase boundary when L is sufficiently large. The alternative is to take a continuum limit in
the non-confining phase on the right, on the k (b ) curve where m =0. Is this a sign of
c q
a conformal infrared theory? (This conundrum was posed long ago [14, 15] for QCD with
N &8. Seealso[16].)
f
2. For the SF method we require m = 0. This can only be achieved in the right side of the
q
diagram. Aswegotowardsstrongercoupling insearchofanIRFP,wewillbeforcedtohalt
where the k curve hits the phase boundary, since there is nolocus where m =0tothe left
c q
ofthatpoint. (m changessigndiscontinuously onthehorizontal phaseboundary.)
q
The second point is the reason that we cannot continue measuring the b function plotted in
Fig. 1 at yet stronger couplings; we are lucky to be able to reach the IRFP in the SU(2)/triplet
theory. In the SU(3)/sextet and SU(4)/decuplet theories the consequences are yet more serious,
since the phase boundary is reached at relatively weak couplings, wellshort ofthe two-loop fixed
point in each case. Our solution to this problem is to generalize the lattice action to include a
pure-gauge plaquette termbuiltoutofthefatlinksusedinthefermionterm,
b b
(cid:229) f (cid:229)
S = TrU + TrV . (3.1)
g p p
2N 2d
c f
Judicious choice of b moves the phase boundary and thus expands the accessible range of the
f
running coupling.
Ishowtheb˜ functionsfortheSU(3)[5]andSU(4)[6]theoriesinFig.3. ItlooksliketheSU(3)
theory possesses an IRFP, but the error bars on the leftmost points don’t allow this conclusion to
bestatedwithconfidence. Itisquitepossible thattheb function turnsawayfromtheaxiswithout
crossing it,asenvisioned ina“walking” scenario. ThisistrueallthemorefortheSU(4)theory. It
is important to note that wecannot yet provide asystematic continuum limitforthe b function in
anyofthesetheories becauselargefluctuations preventasmoothextrapolation.
4. Massanomalous dimension
From the axial Ward identity (2.2), one derives that the anomalous dimension g of the mass
m
parametercanbeextractedfromthenormalization Z ofthepseudoscalardensityPa(x). Weobtain
P
Z fromacorrelation function,
P
Pb(t)Ob(t′=0) =ZPZOe−mp L/2, (4.1)
D E(cid:12)t=L/2
(cid:12)
whichpropagates thelightestparticleinthetheo(cid:12)ryfromasourceO onthewalltothecenterofthe
lattice;weremovethenormalizationZO ofO aswellasmp bycalculatingawall-to-wallcorrelator,
Ob(t =L)Ob(t′=0) =Z2 e−mp L . (4.2)
O
D E
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Conformalorconfining BenjaminSvetitsky
b = 0 b = 0
6 10
b = 0.5 b = 1
6 10
one loop one loop
two loops two loops
0
0
u) u)
~( ~(
b b
-0.05
-0.05
-0.1
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
2 2
u = 1/g u = 1/g
Figure3: b˜ functionsfortheSU(3)/sextet(left)andSU(4)/decuplet(right)theories[5,6]. Blacksquares
vs. redcirclesdenotethetwodifferentactionsusedforeachtheory.
1
N = 2
N = 3
0.8 NN == 4¥
0.6
g
m
0.4
0.2
0
0 10 20 30 40
2
g N
Figure4: Left: SU(2)/triplettheory[4]. PseudoscalarrenormalizationconstantZ plottedagainstthesize
P
Lofthelattice,forbarecouplingsg rangingfromweak(top)tostrong(bottom). Thefittedlog–logslopes
0
give−g . Right:g (g2)fortheSU(2,3,4)theories,plottedtogether[6].
m m
Uponextracting Z ,weuseitsL-dependence tofindg via
P m
Z (L) L −gm
P
= , (4.3)
Z (L ) (cid:18)L (cid:19)
P 0 0
which again incorporates the observation that the running is slow and hence g (g2) is approxi-
m
matelyconstantasLchanges atfixedg .
0
The extraction of g (g2) from a log–log fit is illustrated in Fig. 4. I’ve plotted there the re-
m
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Conformalorconfining BenjaminSvetitsky
0.8
b = 0.0, a/L
6
b = 0.0, (a/L)2
6
0.6 b 6 = 0.5, a/L
b = 0.5, (a/L)2
6
one loop
g
m0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10 11 12
2
g
Figure 5: SU(3)/sextettheory[5]: Continuumextrapolationsof g (g2). the blacksquaresandredcircles
m
originatefromtwodifferentlatticeactions,whileempty(filled)symbolsrepresentlinear(quadratic)extrap-
olations.
sults for all three theories, using the large-N variable g2N as a common abscissa [6]. This plot is
remarkable forthreereasons:
1. Thecurvesforallthreetheories departfromtheone-loop lineandleveloff.
2. Theyleveloffataboutthesamevalue.
3. Thisvalueismuchlessthanone.
The last point is significant because various approximations and models [17, 18, 19] predict that
theoriesnearthesilloftheconformalwindow,andparticularly walkingtheories, shouldgiveg ≃
m
1. Oncemore, however, I haveplotted results thatdo nottake account offinitelattice error. Since
manypeopleareupsetbythefindingthatg neverapproaches 1,wehaveinvestedextraeffortand
m
produced a continuum extrapolation forthe SU(3)/sextet theory [5]. Ascan be seen in Fig. 5, the
error bars have grown and there is added uncertainty in how to extrapolate, but on the whole the
extrapolations drive g furtherdown.
m
Acknowledgments
Mycollaborators inall this work have been TomDeGrand and Yigal Shamir, but some inter-
pretationpresentedhereismyown. ThisworkwassupportedbytheUSDepartmentofEnergyand
by the Israel Science Foundation. Computer time was granted by the US National Science Foun-
dation through the TeraGrid program and by the US Department of Energy through the USQCD
collaboration. Oursoftwaregrewoutofthepublicly available codeoftheMILCcollaboration.
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Conformalorconfining BenjaminSvetitsky
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