University of Groningen
The scalar glueball operator, the a-theorem, and the onset of conformality
da Silva, T. Nunes; Pallante, E.; Robroek, L.
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Physics Letters B
DOI:
10.1016/j.physletb.2018.01.047
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da Silva, T. N., Pallante, E., & Robroek, L. (2018). The scalar glueball operator, the a-theorem, and the
onset of conformality. Physics Letters B, 778, 316-324. https://doi.org/10.1016/j.physletb.2018.01.047
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Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
The
scalar
glueball
operator,
the
a-theorem,
and
the
onset
of
conformality
T. Nunes da Silva
1,
E. Pallante
∗
,
L. Robroek
VanSwinderenInstituteforParticlePhysicsandGravity,Nijenborgh4,9747AG,Groningen,TheNetherlands
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory:
Received 24 September 2016
Received in revised form 26 December 2017 Accepted 17 January 2018
Available online 3 February 2018 Editor: G.F. Giudice
Keywords:
Non-Abelian gauge theories QCD
Conformal symmetry Conformal window
Weshowthattheanomalousdimension
γ
G ofthescalarglueballoperatorcontainsinformationonthemechanismthatleadstotheonsetofconformalityattheloweredgeoftheconformalwindowina
non-Abeliangaugetheory.Inparticular,itdistinguisheswhetherthemergingofanUVandanIRfixedpoint –
thesimplestmechanismassociatedtoaconformalphasetransitionandpreconformalscaling–doesor
doesnotoccur.Atthesametime,weshedlightonnewanalogiesbetweenQCDanditssupersymmetric
version.InSQCD,wederiveanexactrelationbetween
γ
G andthemassanomalousdimensionγ
m,andwe prove that the SQCD exact beta functionis incompatible with merging as a consequenceof the
a-theorem;wealsoderivethegeneralconditionsthatthelatterimposesontheexistenceoffixedpoints,
andprovetheabsenceofanUVfixedpointatnonzerocouplingabovetheconformalwindowofSQCD.
Perhaps notsurprisingly, wethenshow thatanexact relationbetween
γ
G andγ
m,fully analogoustoSQCD, holdsforthe masslessVenezianolimitoflarge-NQCD. Weargue, basedonthelatterrelation,
thea-theorem,perturbationtheoryandphysicalarguments,thattheincompatibilitywithmergingmay
extendtoQCD.
©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense
(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Fora sufficiently large number Nf of massless fermions, itis
believed that a new phase of QCD arises [1,2]. It is called the conformal window, ranging from a value Ncf, where the zero-temperature theory deconfines and chiral symmetry is restored, toavalue NA F
f ,abovewhichasymptoticfreedomislost.Theories
withNc
f
<
Nf<
NA Ff haveanontrivial,i.e.,interactinginfrared(IR)fixed point where they are conformal. A conformal window also arises in supersymmetricversions of non-Abelian gauge theories
[3] and generalisations of QCD with fermions in higher dimen-sional representations and/or other gauge groups. Theories with Nf
>
Ncf maythusleadtonewpossibilitiesforparticledynamics.Above theconformal window, Nf
>
NA Ff , infrared freedom leadstothepossibilityofrealising“asymptoticallysafe”theories,witha nontrivialultraviolet(UV)fixedpoint,seee.g.[4].
*
Corresponding author.E-mailaddresses:t.j.nunes@posgrad.ufsc.br(T. Nunes da Silva), e.pallante@rug.nl (E. Pallante), l.l.robroek@student.rug.nl(L. Robroek).
1 Present address: Departamento de Física, CFM, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil.
Just belowthe conformal window, Nf
Ncf,it has beenpro-posedthe phenomenologicallyinteresting possibilityofa precon-formal behaviour characterised by a walking, i.e., slow-running2 gaugecoupling[5,6].Sincetheorieswithapreconformalbehaviour wouldnot differfromQCDasfarastheirfixed pointstructure is concerned,theymustbeconfiningandasymptotically free.3 How-ever, the preconformal behaviour is entangled to the nature of the mechanism that opens the conformal window at Ncf, and it shouldbeexpectedtomodifytheevolutionfromtheUVtotheIR of observables.Ithas beenshownthat a phasetransition named conformal in [7–9] – the equivalent of a Berezinskii–Kosterlitz– Thouless (BKT) phasetransition intwo-dimensional spin systems
[10–12]–leadstothewalkingphenomenonforNf
Ncf,andtheassociated preconformalbehaviour ofphysicalobservablesknown as Miransky orBKT scaling [7–12]. Interestingly,it was then ob-served[13]thatthemergingofapairofUVandIRfixedpointsat Ncf isasimplewayofrealisingpreconformalscaling.Alternatively, andamongother possibilities,afirstorderphasetransitionatNcf wouldnotleadtoprecursoreffects,see[14]inthiscontext.
2 At least on a finite energy range [μ
I R,μU V].
3 In other words, no phase transition is expected to occur between QCD and pre-conformal theories with NfNcf at zero temperature.
https://doi.org/10.1016/j.physletb.2018.01.047
0370-2693/©2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.
Itisthusrelevanttoidentifyobservablesthatcarrytheimprint ofthemechanismfortheonsetofconformalityatNcf,andatthe sametimearestringentlyconstrainedbyuniversalprinciples,such asexactsymmetriesandtheultraviolettoinfraredrenormalisation group(RG)flowgovernedbythea-theorem.
In this letter we show that the anomalous dimension
γ
G ofthe scalar glueball operator at a fixed point is such an observ-able;itsultraviolettoinfraredflowdetermines whetheran UV-IR fixedpointmergingoccurs.Wealsoshowthat forbothSQCD, for which an exact beta function is known [15–17], and the mass-less Veneziano limit of large-N QCD, whose exact beta function hasbeenrecentlyproposed[18–20],thereisanexactrelation be-tween
γ
G andthe mass anomalous dimensionγ
m, thus relatingthetwo RGflows. These are inturn governedby the a-theorem, whichallows ustoprovetheincompatibility ofSQCDwith merg-ingtoallordersinperturbationtheory,anddirectlyconstrainthe existenceofanUV fixedpointatnonzerocoupling.Theanalogies withSQCDandtheuniversalityofthea-theoremsuggestthatthe sameincompatibilitymayextendtoQCD.Indeed,thoughtheexact betafunctionproposedin[19]forVenezianolarge-NQCDhasbeen obtainedby meansofhomologymethods [18–20]that arenot as muchconsolidated in quantum field theory astheir cohomologi-cal counterparts involving supersymmetry, itpasses a numberof perturbative and nonperturbative consistency checks, as we will discussinsection5.
The letter is organised as follows. In section 2 we review a knownformulafor
γ
G basedonthetraceanomaly.Insection3weanalyse
γ
Gintwo-loopperturbationtheoryandclosetotheupperedgeNf
NA Ff ,partlyreviewingknownresults,andwecommentonthelimitsofapplicabilityofperturbationtheoryinthiscontext. Insection 4we derive results inSQCD, andprove the incompat-ibility withmergingin 4.4.In section 5 we discussthe massless Venezianolimit oflarge-NQCD,andinvestigateto whatextentit reproducesthe resultsofSQCD. Asa sidenote, insection 5.4we discusswhytheadditionofeffectivefour-fermion operatorsdoes notlead to alternativeviable realisations ofmerging inQCD. We concludeinsection6.
2. Thescalarglueballoperatoranditsanomalousdimension It is well known that the anomalous dimension of the scalar glueball operator Tr
(
G2)
≡
Gaμν Gaμν is constrained by the trace anomaly,i.e.,thenonzerocontributiontothetraceoftheenergy– momentum tensor, see,e.g., [21] andmore recently [22,23]. The trace anomaly of QCD that enters the matrix elements of renor-malisedgaugeinvariantoperatorsis4
Tμμ
=
β(g)
2g Tr
(G
2
)
+
fermion mass contribution,
(1)withthebeta function
β(
g)
= ∂
g(
μ
)/∂
logμ
for given N colours and Nf flavours; an analogous relation is valid in SQCD. Weshall restrict ourselves here to the massless theory. The non-renormalisation of Tμμ implies that the renormalised operator ORGI
≡ (β(
g)/
g)
Tr(
G2)
is also renormalisation-group (RG) invari-ant, i.e., dORGI/
dlogμ
=
0. Using inside the latter equation a Callan–SymanzikequationfortherenormalisedoperatorTr(
G2)
d d log
μ
ZG−1Tr(G
2)
=
0,
γ
G(g)
= −
∂
log ZG∂
logμ
,
(2)4 We are thus not interested in the most general expression, which also involves gauge-fixing and EoM operators, see [21,24,25].
with
γ
G(
g)
the anomalous dimension of Tr(
G2)
for N and Nffixed,oneobtains d d log
μ
β(g)
g ZG=
0 (3) andγ
G(g)
=
g∂
∂
gβ(g)
g= β
(
g)−
β(g)
g (4)fora theory withgiven N and Nf.This equation reproduces the
known resultinperturbative QCD [22,26,27],
γ
G= −
2β
0g2+ . . .
,β
0 from (7),andγ
G isnegative, so that the operator Tr(
G2)
be-comesincreasinglyrelevanttowardstheinfrared.
We shall be interested in the g and Nf dependence of the
anomalousdimensions,thusingeneral
γ
G(
g,
Nf)
.Atafixedpointof the renormalization group flow, the solution of
β(
g,
Nf)
=
0thus definesthe function g∗
(
Nf)
offixed-point couplings on theplane
(
g,
Nf)
,andequation(4)providesγ
G atg∗(
Nf)
:γ
G∗(N
f)
≡
γ
G(g,
Nf)
|
g=g∗(Nf)= β
(
g,Nf)
|
g=g∗(Nf),
(5)where the prime will always denote the derivative withrespect to g. The fixed-point anomalous dimension
γ
G∗ fora given Nf isa physical property of the system, renormalisation scheme inde-pendent;thescalingdimensionofTr
(
G2)
,dG=
4+
γ
G∗,thusenterstheexactconformalscalingofthecorrespondingcorrelatorsatthe fixedpoint.
3. PerturbativeresultsinQCD
Itisinstructivetofirstrecallsomefeaturesofperturbation the-ory.TheQCDbetafunctioncanbeexpressedasaseries
β(g)
= −
g3 ∞ l=0β
lg2l,
(6)where
(
l+
1)
denotes the number of loops involved in the cal-culation ofβ
l. The coefficientsβ
0,1 are universal [1,28–30], i.e., renormalisationschemeindependent,givenbyβ
0=
1 3(
4π
)
2(
11CA−
4TfNf)
β
1=
1 3(
4π
)
4 34C2A−
4(
5CA+
3Cf)T
fNf,
(7)here written in terms of the quadratic Casimir invariants Cf
≡
C2
(
R)
and CA≡
C2(
G)
, for, respectively, the representation R to whichthe Nf fermionsbelongandtheadjointrepresentation.Thequantity Tf
≡
T(
R)
isthetraceinvariantfortherepresentation R.Coefficientsofhigherorderarerenormalisationschemedependent
[31,32]andhavebeencalculatedup tofive-looporderintheM S scheme[33–36].
To two loops, a nontrivial IR fixed point with coupling g∗2
=
−β
0/β
1 isone rootofthe equationβ(
g)
=
0 for some givenNf,andfrom(5)
γ
G∗= −
2β
02/β
1.We are interested inthe way
γ
G∗ variesalong thecurve of IR fixed points g∗(
Nf)
asNf decreasesintheconformal windowofQCD,i.e.,forNf Diracfermionsinthefundamentalrepresentation;
in this case, NA F
f
= (
11/
2)
N, CA=
N, Cf= (
N2−
1)/(
2N)
, andTf
=
1/
2 in(7).IntheVenezianolimit, N,
Nf→ ∞
,holdingx=
Nf
/
N andN g2constant,and=
11/
2−
Nf/
N1,thatisclosetothe upperedge,one obtains N g2
and
γ
G∗(
162
/
225)(
1+
O(
))
positive.5 Its derivative with re-spectto Nf=
xN withfixedN andx continuousintheVenezianolimit d
γ
G∗dNf
= −
32225N
(
1+
O(
))
(8)isnegativeandoforder
/
N,thus implyingthatγ
G∗,as N g2∗,isa strictlymonotonicfunctionofNf alongtheIRfixedpointcurve,at
leastintheneighbourhoodoftheupperedge,anditincreasesas Nf decreases.Inother words,theuniversaltwo-loopcontribution
inperturbationtheoryisconsistentwithanincreasinglyirrelevant operatorTr
(
G2)
asapproachingtheloweredge.Wefinally observethat,beyondtheVenezianolimit and mov-ing away from the upper edge, the two-loop expression
γ
G∗=
−
2β
20/β
1 remains indeed positive and monotonically increasing as Nf decreases on the entireinterval Ncf Nf NA Ff , andtheIR zero disappears at Nc
f due to the change of sign of
β
1; forN
=
3 andNf fundamentalfermions,thisoccursatNf∼
8.
05.Thechangeofsignof
β
1 wouldimplythat thefixed pointdisappears atinfinitecoupling g2∗
= −β
0/β
1→ ∞
,andthe samesingularity occursforγ
G∗.Thisbehaviour, however,is likelytobe anartefact ofthe truncatedperturbative expansion,aswe furtherdiscussin section5.Also,since theperturbativeseries(6)isatbest asymp-totic,weshouldtakethetwo-loop,orhigherorder,resultsatmost asqualitativeindications.4. ResultsinSQCD
Weconsider SU
(
N)
supersymmetricQCD(SQCD)withNffun-damental flavours Qi in the N representation and Q
˜
˜i in the N
¯
representation(i
,
˜
i=
1,
. . .
Nf),wheremanyresultscanbederivedexactly.Our goalinthissection isto determineexactconstraints on the UV to IR flow of
γ
G(
g,
Nf)
andthe mass anomalousdi-mension
γ
m(
g,
Nf)
.Lateron,insection5,wewillfindthatsomeproperties of
γ
G can be proved to be equally true in themass-less Veneziano limit of large-N QCD. For our purpose, we make useofSeiberg’ssolutionforthephasesofSQCD[3],theNSVZ ex-actbetafunction[15–17],andthea-theoremontheirreversibility ofrenormalisationgroup(RG)flowsinfour-dimensionalfield theo-ries[39],ageneralisationtohigherdimensionsofZamolodchikov’s c-theorem[40]intwodimensions.
4.1. KnownresultsinSQCD
TheNSVZexactbetafunctionforgivenN andNf reads[15–17]
β(g)
= −
g 3 16π
2 3N−
Nf+
Nfγ
m(
g) 1−
N g2/(
8π
2)
,
(9) withγ
m(g)
= −
g2 8π
2 N2−
1 N+
O(g 4)
(10)themassanomalous dimensioncomputedinperturbation theory. ApowerfulpropertyofSQCDisthatitsexactbetafunctionandthe globalanomalyfreeRsymmetryatafixedpointdetermineexactly themassanomalousdimension
γ
m∗(
Nf)
alongthecurveofIRfixedpoints, g∗
(
Nf)
,intheconformalwindowofSQCD, whichextendsontheinterval3N
/
2<
Nf<
3N [3].5 For a study of higher orders in perturbation theory see [37,38].
In more detail, the exact R symmetry at a fixed point allows us to determine the anomalous dimension of spinlesschiral pri-maryoperators fromtheir R-charge.Forthegaugeinvariant com-posite mesonoperator M
= ˜
Q Q ,withscaledimension DQ Q˜ and R-charge RQ Q˜ ,onehas[3]DQ Q˜
=
32RQ Q˜
=
3R=
3Nf
−
NNf
,
(11)withR theR-chargeofQ (Q ),
˜
andthelastequalitydictatedbythe R-chargeassignmentsofQ (Q )˜
underU(
1)
R.UsingDQ Q˜=
2+
γ
m∗,oneobtains
γ
m∗ exactlyγ
m∗(N
f)
=
1−
3NNf
,
(12)whichisindeedazeroofthebetafunction(9),providedthepole isnothit,i.e., N g2
∗
/(
8π
2)
<
1.6 Equation(12)isthentakento de-termineγ
m along the curve of IR fixed points in the conformalwindow with varying Nf; indeedit vanishes atthe upper edge,
Nf
=
3N,wherethetheoryisIRfree,anditisnegativebelowit.The lower edge is signaled by a physical condition, i.e., a re-normalisation-scheme independent condition. This is the satura-tionoftheunitarity boundinSeiberg’ssolutionforthephasesof SQCD, andsuch a condition is independent ofthe beta function. Specifically,thesaturationoftheunitarityboundDQ Q˜
=
1 impliesγ
m∗= −
1, which in turn implies that the numerator of the betafunctionhasazeroforNf
=
3N/
2.Thisidentifiesthelower edgeoftheconformalwindowforSQCD.
Equation (12) is implicitly a function of the coupling g∗
(
Nf)
along theIRfixed pointcurve.Onecandetermine g∗
(
Nf)
pertur-batively, by taking N
,
Nf→ ∞
and holding N g2 and x=
Nf/
Nconstant, with
=
3−
Nf/
N1,i.e., closeto theupperedgeoftheconformalwindow.Thisgives[3]
N g∗2
=
83
π
2
+
O(2
) .
(13)4.2. NewresultsinSQCD
Equation (5) implies that the derivative of (9) with respect to the coupling, evaluated at a nontrivial fixed point, gives the anomalous dimension ofthe scalar glueballoperator atthefixed pointasafunctionofNf
γ
G∗(N
f)
= −
g∗3 16π
2 Nfγ
m ∗(N
f)
1−
N g2 ∗/(
8π
2)
,
(14)where analogously to (5)
γ
m ∗(
Nf)
≡
γ
m(
g,
Nf)
|
g=g∗(Nf) and theprime denotesthederivative withrespectto g.Equation(14) es-tablishes a usefulrelation betweenthe anomalous dimension
γ
Gand the derivative of
γ
m. This is a key resultthat we are goingto useinthe restofthissection. In particular,ourtaskis to de-rive constraints on the flow of
γ
m, its derivatives, andγ
G, usingequation(14),Rsymmetryandthea-theorem.
It is convenient to immediatelysummarise the main new re-sults of section 4 forSQCD. They are all valid in the conformal window andits edges, 3N
/
2Nf3N,andcanbe summarisedasfollows:
6 Note, however, that the cusp singularity in (9) for N g2/(8π2)=1 is a renormalisation-scheme dependent condition; it cannot occur if a physical zero of the numerator of (9)occurs. The role of the cusp singularity in SUSY Yang–Mills
1)
γ
m(
g,
Nf)
isa strictly monotonic functionof g for Nf fixed,for anyvalid RGflow froman UV fixed point to an IR fixed point, andit may be stationaryat the fixed points. This re-sult is implied by the a-theorem andproved in section 4.3, equation (20). The strict monotonicity of
γ
m away from thefixed point will be sufficient to prove the incompatibility of theSQCDexactbetafunction(9)withmerginginsection4.4. 2) Aresultstrongerthantheincompatibilitywithmergingisalso
provedinsection4.4:InSQCD,thea-theoremimpliesthrough equation (20) that the beta function, if continuous and thus free from cusp singularities, does not admit more than one fixedpointatnonzerocoupling.Hence,intheconformal win-dowanditsloweredge,3N
/
2Nf<
3N,theexistenceoftheIRfixedpointatnonzerocouplingexcludesan UVfixedpoint atnonzerocoupling.Ifinsteadoneofthetwofixedpoints oc-curs atzerocoupling,thea-theoremcanbe satisfied,butnot always.Wefindunderwhichconditionsthea-theoremis sat-isfied.
Results1)and2)areneverthelessnotabletodetermineif
γ
G∗(
Nf)
isstrictlypositivealongthenontrivialIRfixedpointcurve g∗
(
Nf)
ofSQCD, oritvanishes. Results3)to5) belowprovidearguments infavourofastrictlypositive
γ
G∗(
Nf)
for3N/
2Nf<
3N.3) We know exactly
γ
m∗(
Nf)
along the IR fixed point curve. IfNf is assumed to be continuous, then d
γ
m∗/
dNf>
0, andthus
γ
m∗(
Nf)
is strictly monotonic in Nf and decreases asNf decreasesalong g∗
(
Nf)
.Also,d2γ
m∗/
dN2f<
0 impliesthatd
γ
m∗/
dNf itself strictly increasesas Nf decreases. This resultcomesstraightforwardlyfromtheexactsolutionfor
γ
m∗(
Nf)
inSQCDandthea-theorem.
4) IntheVenezianolimit,toleadingorderinperturbationtheory andclosetotheupperedge,the IRfixedpoint couplingN g2
∗ isstrictlymonotonicinx
=
Nf/
N,and,withabuseofnotation,in Nf
=
xN withfixed N and x continuousinthe Venezianolimit. This result is fully analogous to the perturbative QCD resultinsection3.
5) IntheVenezianolimit,toleadingorderinperturbationtheory andclosetotheupperedge,thesolutionfortheIRfixedpoint of SQCD is consistent with
γ
m ∗(
Nf)
<
0, and, through (14),a strictlypositive
γ
G∗(
Nf)
.WeaddthataresultfullyanalogoustothatofQCDtwo-loopperturbationtheoryin(8),isobtained inSQCDifthetwo-loopSQCDbetafunctionisused.
Result 3) is straightforwardly implied by taking the derivatives of(12),specifically,d
γ
m∗/
dNf=
3N/
N2f andd2γ
m∗/
dN2f= −
6N/
N3 f.
The derivative of (13) with respect to Nf
=
xN for N fixed,∂(
N g2∗)/∂
Nf= −
8π
2/
3N(
1+
O(
))
is negative to leading order,so is
∂
g∗/∂
Nf, thus providing result 4); it agrees with theob-servation that the theory is increasingly strongly coupled as Nf
decreases.
Result5) followsfrom (14) andthe properties of
γ
m. In fact,thederivativewithrespectto Nf
d
γ
m∗ dNf=
∂
γ
m(g,
Nf)
∂
Nfg=g∗(Nf)
+
γ
m ∗(N
f)
∂
g∗∂
Nf (15) isknownexactly,dγ
m∗/
dNf=
3N/
N2f.Therhsof(15)canbedeterminedintheVenezianolimit with
1,andtakingderivativeswithrespectto Nf
=
xN forN fixed.Equation(10)gives
∂
γ
m/∂
Nf=
0 toleadingorderand,using(13),theexpansion
γ
m ∗(N
f)
∂
g∗∂
Nf=
1 3N 1−
1 N2(
1+
O())
(16)reproduces theexpansiond
γ
m∗/
dNf=
3N/
N2f=
1/(
3N)(
1+
O(
))
totheleading 1
/
N order.Thisresultisconsistentwithγ
m ∗(
Nf)
<
0 and,through(14),
γ
G∗(
Nf)
strictlypositive.4.3. Implicationsofthea-theorem
The a-theorem for four-dimensional RG flows establishes the existence of a monotonically decreasing function that interpo-lates between the Euler anomalies of an UV and an IR CFT, i.e., aU V
−
aI R>
0.Thisfunctionalsoprovidesaneffectivemeasureofthenumberofmasslessdegreesoffreedom,consistentlywiththe intuitionthatthisnumberdecreasesasweintegrateouthigh mo-menta. Cardy’s conjectured a-function[42], givenby the integral of the trace ofthe energy–momentumtensor on the sphere S4, haspassedall testsinthecontext oftheoriesthat arefree inthe UVandwhoseIRdynamicscanbe computed.Therecentproofof thea-theorem[39,42]requirestherathergeneralprerequisiteofa unitarySmatrix.
Weuseheretheinterpolatinga-functionforSQCDinthe con-formalwindowobtainedin[43],whoseIRvaluecanbecomputed from the U
(
1)
RF F , U(
1)
R and U(
1)
3R anomalies, and providesaU V
−
aI R intermsoftheanomalyfreeR-chargeofthefield Q (Q ).˜
The latter is a function of
γ
m via (11) and DQ Q˜=
2+
γ
m. This meansthatthea-theoremdirectlyconstrainstheUVtoIRflowofγ
m(
g,
Nf)
inSQCD,and,via(14),itconstrainsthatofγ
G(
g,
Nf)
.Withoutknowledgeofthea-theorem,equation(12)already im-pliesthat,alongtheIRfixedpointcurve, DQ Q˜ ,R and
γ
m∗ decreasefromtheirvalueattheupperedge(DQ Q˜
=
2,R=
2/
3,γ
m∗=
0)totheirvalueattheloweredge(DQ Q˜
=
1,R=
1/
3,γ
m∗= −
1),wheretheunitarityboundissaturated.
The a-theorem allows us to further establish the monotonic variation of
γ
m(
g,
Nf)
along anyvalidRGtrajectoryfromtheul-traviolettotheinfrared.Inparticular,itallowsustoderiveresults 1)and2)ofsection4.2.
TwotypesofUVtoIRflowsareofinterestinthisanalysis,both werediscussedin[43]andtheyareillustratedinFig. 1:
I.For Nf fixed, the theory flows fromthe asymptotically free
fixed point (UV) to the nontrivial IR fixed point, the horizontal line in Fig. 1; we refer to this flow as UVA F. The interpolating
a-functiona
(
g(
μ
))
,withrenormalisation-scaledependentcoupling g(
μ
)
,variesfromitsvalueaU V toaI R.Weshallusetheuniversal-ityofthea-functiontoalsoderiveconstraintsontheRGflowfrom ahypotheticalstronglycoupledUV fixedpointtotheweakly cou-pledIRfixedpoint;werefertothisflowasUVSC.
II.One candevisea flowinthespaceoftheoriesalongthe IR fixedpointcurve fromatheory withNf massless flavourstoone
withNf
−
n masslessflavours,andNf−
nNcf,sothatboththeo-riesareinthesamephase.Thiscanbeachievedbyaddingamass deformationforn flavours.Theinterpolatinga-functionvariesfrom aU V
=
a(
Nf)
toaI R=
a(
Nf−
n)
.IncaseI,fromtheIREuleranomalycoefficient[43]7 aI R
=
3 32 2(N
2−
1)
+
2NfN(1−
R) 1−
3(
1−
R)
2,
(17)withR
= (
2+
γ
m∗)/
3,andaI R→
aU V forR→
2/
3,oneobtainsfortheflowUVA F [43] aU V
−
aI R=
N Nf 48γ
∗2 m 3−
γ
m∗=
N Nf 48 1−
3N Nf2
2+
3N Nf,
(18)Fig. 1. Flow
I (horizontal line) from the UV (g
=0) to the nontrivial IR fixed point for a theory with Nf massless flavours in the conformal window. Flow II along theIR fixed point curve from a theory with Nf massless flavours (UV) to one with
Nf<Nf (IR).
where (12) is used in the second line. It vanishes at the upper edge,Nf
=
3N,anditsatisfiesaU V−
aI R>
0,aI R>
0,for3N/
2Nf
<
3N.8To establish result 1) of section 4.2 we use the interpolating function a
(
g(
μ
))
for Nf fixed, given by aI R in (17) forγ
m∗→
γ
m(
g(
μ
))
.Accordingtothea-theorem,a(
g(
μ
))
isastrictlymono-tonic function of the scale
μ
and decreases from the UV to the IR,anditisstationaryatafixedpoint.Thus,away fromthefixed pointβ(
g)
=
0 andalongtheflowsoftypeIwithfixedNfda d log
μ
=
∂a
∂
gβ(g) >
0 (19)impliesthat
∂
a/∂
g hasthesamesignasβ(
g)
with∂
a∂
g= −
N Nf 16γ
m(
2−
γ
m)
∂
γ
m∂
g=
0.
(20)For
γ
m<
0 andγ
m>
2 (0<
γ
m<
2), it follows from (20) that∂
γ
m/∂
g=
0 andofthesamesign(oppositesign)of∂
a/∂
g.There-fore, for Nf fixed
γ
m must be a strictly monotonic function ofg awayfromfixedpoints. Importantly,thisresultapplies toboth flows,UVA F (
β(
g)
<
0) andUVSC (β(
g)
>
0),giventheuniversal-ityoftheinterpolatinga-functionand(20).Thisisresult1),andit willimplytheincompatibilityofSQCDwithmergingandresult2) insection4.4.
At the nontrivial fixed point, UV or IR, the flow of the a-functionisstationary,da
/
dlogμ
=
0,becauseβ(
g)
=
0,and(20)does not constrain
∂
γ
m/∂
g – unless one is able to prove that∂
a/∂
g=
0 foranyg=
0.Incase II, using(17) and(12) along theIR fixed point curve, onehas a(Nf
)
−
a(Nf−
n)=
9N4 16 1(N
f−
n)2−
1 N2f>
0,
(21)witha
(
Nf)
= (
3N2/
16)(
1−
3N2/
N2f)
.Inother words,theflow ofγ
m∗(
Nf)
impliedby(12),dγ
m∗/
dNf>
0,guaranteesthatda/
dNf=
9N4
/
8N3f is also positive, and results 3) and 4) can be
re-interpretedasconsequencesofthea-theorem.
Consistency of the a-theoremwith result 5), through (20), is obviousatthispoint,because(20)doesnotconstrain
∂
a/∂
g along theIRcurve.8 Corrections to a
I Rfrom a possible accidental symmetry due to the violation of
the unitarity bound at Nf=3N/2 vanish [43].
Fig. 2. The
beta function
β(α,Nf)with f(α)=1 in (22)for decreasing Nf, top tobottom: for Nf>Ncf there is a pair of fixed points at α−(IR) and α+(UV). They
merge at αcfor Nf=Ncf and disappear for Nf<Ncf.
4.4. ProofoftheabsenceofmerginginSQCD
Inthissectionwe exploreaspecificmechanismthatmaylead totheoccurenceoftheloweredgeofaconformalwindow,guided by the idea that such a mechanismis itself a powerfulprobe of theunderlyingtheory.We considerthepossibilitythata nontriv-ial,i.e.,interactingUVfixedpointexistsintheconformalwindow andmergeswiththeIR fixedpoint atthe loweredge.The possi-bility of an additional, more strongly coupled UV fixed point in the QCD conformal window was put forward in [2]. The merg-ing oftheUV-IR pairof fixedpoints atthelower edge[13,44] is phenomenologicallyinteresting,sinceitnaturallyleadsto BKT/Mi-ransky scaling[7–12] anda “walking” gauge couplingjustbelow theconformalwindow.
Firstly, we establish a generalresultvalid forSQCDandQCD: A strictly monotonic
γ
G∗(
Nf)
andnonvanishingatthelower edgeoftheconformalwindowisincompatiblewithmerging.Secondly, as an instructiveexercise, we analyse merging in the context of SQCDandprovetheincompatibilityoftheSQCDexactbeta func-tion withmerging,by useofthea-theoremandresult1)of sec-tion4.2.
ClosetoNcf,theansatzforthebetafunctionthatrealises merg-inghastheform[13]sketchedinFig. 2:
β(
α
,
)
=
f(
α
)
− (
α
−
α
c)
2,
(22)where
= (
Nf−
Ncf)/
N,α
is (apowerof) acoupling,and f(
α
)
is a strictly monotonic function of
α
,9 nonzero on the interval[
α
−,
α
+]
, withα
±=
α
c±
√
the zeroes of
β(
α
,
)
;α
± aredis-tinct andrealfor
>
0,α
+=
α
−=
α
c for=
0,andcomplexfor<
0,thusleadingtothedisappearanceoftheconformalwindow. We note that the only effect ofa strictly increasing (decreasing) f(
α
)
in(22)istoshiftthemaximumofthebetafunction,which occursatα
cfor=
0,toα
∗>
α
c(
α
∗<
α
c)
andα
−<
α
∗<
α
+for>
0.Attheloweredge,
=
0,thebetafunction(22)developsalocal maximumatα
c,thusβ
(
α
c,
=
0)
vanishes.SQCD, likeQCD,hasonecoupling,thegaugecoupling,andthelatterresult,via(5)and
α
∼
g2,impliesthatγ
G∗ vanishesforNf=
Ncf,though thetheoryisinteracting.Inotherwords,anonvanishing
γ
G∗attheloweredge ofSQCD,andQCD,isincompatiblewithmerging.Besides, since
γ
G∗ also vanishes atthe upper edge,where the theory is IR free, and below the upper edgeγ
G(
α
±,
>
0)
=
β
(
α
±,
>
0)
= ∓
f(
α
±)
√
is positiveat
α
− (IR)andnegativeat9 f
α
+ (UV),thenγ
G∗(
Nf)
isnon monotonicalong theIRfixed pointcurveifmergingoccurs.
We now specialise to SQCD. The incompatibility ofthe SQCD exact beta function (9) with merging is a direct consequence of thea-theorem, throughresult1) insection4.2,applied totheRG flowofthetheory froma hypotheticalstrongly coupledUV fixed pointtotheweaklycoupledIRfixedpoint.Toproveit,weimpose that(9)realisesthemergingform(22)inthesurroundingsofthe lower edge,
0 with
1. We equate (22)with
α
=
N g2 to theSQCDbetafunctionβ(
α
)
=
2N gβ(
g)
,withβ(
g)
in(9).10Mergingisrealised for f
(
α
)
=
α
2/(
8π
2(
1−
α
/(
8π
2)))
strictly increasingon[
α
−,
α
+]
,whereasalways1−
α
/
8π
2>
0,and−3
+ (3
/
2+
)(
1−
γ
m(
α
,
))
=
− (
α
−
α
c)
2,
(23)wherewe used Nc
f
/
N=
3/
2 and Nf/
N=
3/
2+
.The condition (23)determines the RG flow of
γ
m onthe interval[
α
−,
α
+]
, for some0,
1:
γ
m(
α
,
)
=
−1
+ (2
/
3)(
α
−
α
c)
2 1+
2/
3.
(24) Thusγ
m(
α
c,
)
= −
1/(
1+
2/
3)
is a minimum ofγ
m(
α
,
)
=
γ
m(
α
c,
)
+ (
2/
3)(
α
−
α
c)
2/(
1+
2/
3)
on the interval[
α
−,
α
+]
, withα
±=
α
c±
√
.Atthezeroes,
γ
m(
α
±,
)
= −(
1−
2/
3)/(
1+
2/
3)
.Crucially,forany>
0,α
c doesnot correspondtoa fixedpoint,nevertheless wehave foundthat theRGflow of
γ
m issta-tionary at
α
c, if merging is realised. Equation (20) then impliesthat the a-function itself is stationary at
α
c, away from a fixedpoint,thusviolatingthea-theorem. Thisestablishes animportant result,toallordersinperturbationtheory:IftheSQCDexactbeta functionsatisfiesthea-theorem,thenitcannotrealisemerging.
Even without the aid of an exact solution for the underly-ing theory, the non monotonicity of the scalar glueball anoma-lous dimension
γ
G∗(
Nf)
with merging along the IR fixed pointcurve seems at odds with the simple fact that interactions be-come stronger as Nf becomes smaller along the IR fixed point
curve, a feature implicitin thea-theorem. Infact, two-loop per-turbation theory for QCD in section 3, as well as results 3), 4) and5)forSQCDinsection4.2areconsistentwithanonvanishing
γ
G∗(
Nf)
everywhere belowthe upperedge, andmonotonic alongtheIRcurve;inSQCD,through(14),thelatterpropertiesholdfor
γ
G∗(
Nf)
aswellasthederivativeofthemassanomalousdimensionγ
m ∗(
Nf)
.Finally,notethatinthepresenceofmergingtheoperatorTrG2 wouldbe irrelevantalongtheIRcurve andrelevantalong theUV curve, marginal at the lower edge. Thus, plausibly, the UV fixed pointcurvewouldbealineofcriticalpointsintheconformal win-dow,whereaphasetransitionoccursinthecontinuumtheory;this isadistinctivesignatureofmerging.
4.5.Proofofresult2)
Result2)ofsection4.2,aresultstrongerthanthe incompatibil-itywithmerging,followsstraightforwardly fromasimilar lineof reasoning.ConsidertheRGflowfromahypotheticalnontrivialUV fixedpoint, withcoupling
α
U V=
0,to a nontrivialIRfixed point,with coupling
α
I R=
0, for N,
Nf fixed. If the beta function (9)iscontinuousandonlyvanishes atthe fixedpoints
β(
α
I R,U V)
=
0,thentwocasesarepossible:
10 This guarantees the correct N
,Nf counting for SQCD in the presence of
merg-ing.
a)If 0
<
α
I R<
α
U V, β(
α
) >
0 on(
α
I R,
α
U V)
b)If 0
<
α
U V<
α
I R, β(
α
) <
0 on(
α
U V,
α
I R) .
(25)Weconsiderthefirst case,thetopcurve inFig. 2,andfor conve-niencewewrite(9)asfollows:
β(
α
)
=
f(
α
)
h(α
)
f(
α
)
=
α
2 8π
21−
α 8π2 h(α
)
= −
3N+
Nf−
Nfγ
m(
α
) .
(26)Since both fixed points are at nonzero coupling, f
(
α
)
does not vanishontheclosedinterval[
α
I R,
α
U V]
anditiscontinuous,with1
−
α
/(
8π
2)
>
0.Hence,thebetafunctionvanishesonlyifits nu-merator vanishes,i.e.,β(
α
I R,U V)
=
0 onlyifh(
α
I R,U V)
=
0, andthecontinuityof
β(
α
)
impliesthecontinuityofh(
α
)
.Thenβ(
α
)
, con-tinuousandvanishing onlyatthe boundariesof[
α
I R,
α
U V]
,hasamaximumatsome
α
I R<
α
<
α
U V,andh(
α
)
alsohasa maximumat some
α
I R<
α
<
α
U V, and, by (26),γ
m(
α
)
has an extremumat
α
,away from a fixed point. Equation(20) then impliesa sta-tionarya-functionawayfromafixedpoint,hencetheviolationof thea-theorem.For thesecond casein (25),with 0
<
α
U V<
α
I R,the proof isfullyanalogous, withtheobviousexchangesofmaximaand min-ima,IRandUV.
This proves that the a-theorem implies that the SQCD beta function does not admit more than one fixed point at nonzero coupling.Hence, intheconformal window, 3N
/
2Nf<
3N, theexistenceoftheIRfixedpointatnonzerocouplingexcludesanUV fixedpointatnonzerocoupling.
If one of the two fixed points occurs instead at zero cou-pling, the a-theorem can be satisfied, but not always. Consider the first case in (25), where now
α
I R=
0 andα
U V>
0. Thiscase could be realised above the conformal window, Nf
3N,once asymptotic freedom is lost. This time f
(
α
I R)
=
0 and f(
α
)
is strictly positive andstrictly increaseson
(
0,
α
U V]
, i.e., f(
α
)
=
α
/(
4π
2)(
1−
α
/(
16π
2))/(
1−
α
/(
8π
2))
2>
0, so thatβ(
α
)
hasa maximum at some 0<
α
<
α
U V while h(
α
)
, and thusγ
m(
α
)
,are allowed to vary strictly monotonically on
(
0,
α
U V)
.Specifi-cally, h
(
α
)
should vary from h(
α
I R)
>
0 to h(
α
U V)
=
0 and thea-theorem requires that it varies (decreases) strictly monotoni-cally, i.e., h
(
α
)
= (β(
α
)/
f(
α
))
<
0 on(
0,
α
U V)
, or equivalentlyβ
(
α
)/β(
α
)
<
f(
α
)/
f(
α
)
on(
0,
α
U V)
where f,
f>
0 andβ >
0.For h
(
α
)
, and thusγ
m(
α
)
, strictly monotonic, andγ
m=
0,
2,equation (20) then implies a strictly monotonic a-function on
(
0,
α
U V)
. However, the a-theorem through (19) further requiresthat
∂
a/∂
α
has the samesign asβ(
α
)
away froma fixed point, hence∂
a/∂
α
>
0 forβ(
α
)
>
0.Equation(20)thenimpliesthe con-straints:∂
γ
m/∂
α
>
0 forγ
m<
0 andγ
m>
2,and∂
γ
m/∂
α
<
0 for0
<
γ
m<
2.Considerα
inaneighbourhood oftheoriginα
I R=
0,with
γ
m(
0)
=
0.Then,∂
γ
m/∂
α
>
0 ifγ
m(
α
)
>
0 and∂
γ
m/∂
α
<
0if
γ
m(
α
)
<
0.None ofthelatter solutionssatisfies thea-theoremconstraintsabove.ThisimpliestheabsenceofanontrivialUVfixed pointabovetheconformalwindowofSQCD, Nf
3N.11The second case in (25), where now
α
U V=
0 andα
I R>
0 isrealised by the conformal window and can indeed be shownto be allowedby thea-theoremfollowingafullyanalogousproof.It is worth to note that noneof these proofsmake use ofspecific assignmentsofR-chargesatfixedpoints,noroftheiruniqueness.
11 This result was argued with different methods, using specific values of R-charges at the fixed points, in [45].
5. Large-NQCDintheVenezianolimit
We now investigate to what extent the results obtained in SQCDremain validinthemassless Veneziano limit (Nf
,
N→ ∞
,Nf
/
N=
const)of large-N QCD,for whichan exact betafunctionhasbeen proposed [19],asa generalisation ofthelarge-N Yang– Millsexactbetafunctionderivedonthebasisoftheloopequations forcertainquasi-BPSWilsonloops[18].Thisbetafunction remark-ablymanifestssalientanalogieswiththeexactNSVZbetafunction in (9), with one crucial difference. From inspection of the beta functionforgivenN, Nf12[18,19]
β(g)
=
∂
g∂
logμ
=
(27)−
g3 16π
2(
4π
)
2β
0−
N(∂
log Z/∂
logμ
)
+
Nfγ
m(g)
1−
Ng2/
4π
2,
withβ
0in(7),theanomalousdimensionfactor∂
log Z∂
logμ
=
2γ
0 N g2+ . . .
γ
0=
5 3(
4π
)
2 1−
2Nf 5N (28) andthefermionmassanomalousdimensionγ
m(g)
= −
9 3(
4π
)
2 N2−
1 N g 2+ . . . ,
(29)bothstartingatorderN g2,andcomparingwith(9),oneconcludes thattheabsenceofsupersymmetrygeneratesthenewanomalous dimensioncontribution
∂
log Z/∂
logμ
inthebetafunctionofQCD; itsstructureisotherwiseidenticalto(9).Equation(27)isexactin thelarge-Nlimit,i.e., toleading orderinthe1/
N expansion,and itisexact toall ordersinthe O(
1)
ratio Nf/
N intheVenezianolimit.Indeed,onecanverifythat itsweakcouplingexpansion re-producesthe universalpartoftheperturbative betafunction,i.e., thetwo-looporder,uptothelastcontributiontothetwo-loop co-efficient
β
1 in (7), whichis1/
N2 suppressed withrespect tothe leadingcontribution[18,19].Anotherveryinterestingresult[19]isthedeterminationofthe loweredgeoftheconformalwindow,withinthelocal approxima-tion ofthe glueball effective action validin the confiningphase. Thelower edgeoccursat Nf
/
N=
5/
2, thevalue forwhichγ0
in (28)changessign.Infact,γ0
alsoenterstheglueballkineticterm, anditschangeofsignsignalsaphasetransitionfromconfinement toaphasewithTr
(
G2)
=
0,theconformalCoulombphase.Then, for Nf
/
N=
5/
2, barring the occurrence of a cuspsin-gularity and noting that
∂
log Z/∂
logμ
=
0,13 the beta function (27)vanishes forγ
m= −
4/
5,a renormalisation-schemeindepen-dentresult.Asanticipatedinsection3,thisresultsuggeststhatthe singularityof theQCD two-loop betafunction atthe loweredge, i.e., g∗
→ ∞
forβ
1=
0,isindeedanartefactofthetruncated per-turbativeexpansion. Wedetermineγ
G∗(
Nf)
using(27):γ
G∗(N
f)
= −
g3∗ 16π
2−
N(∂log Z/∂
logμ
)
∗(N
f)
+
Nfγ
m ∗(N
f)
1−
N g2 ∗/(
4π
2)
,
(30)12 Ref.[19]writes (27)in terms of the ’t Hooft coupling g
c=
√
N g.
13 The anomalous dimension term has an exact expression with overall coefficient γ0in terms of the Wilsonian coupling [18,19].
wherefrom(28)thederivativeswithrespectto g are
(∂
log Z/∂
logμ
)
=
2γ
0(
2N g+ . . .)
γ
m= −
6(
4π
)
2N2
−
1N g
+ . . .
(31)At the lower edge the derivative
(∂
log Z/∂
logμ
)
∗(
Ncf
)
vanishesexactlysince
γ0
=
0.Therefore,attheloweredge(30)reducestoγ
G∗(N
cf)
= −
g3 ∗ 16π
2 Nfγ
m ∗(N
cf)
1−
N g2 ∗/(
4π
2)
,
(32) which is the main result of this section, a relation betweenγ
G∗ andγ
m ∗ entirely analogous to SQCD. Like (27), (32) is exact inthe large-N limit and to all orders in the O
(
1)
ratio Nf/
N inthe Veneziano limit.Barringthe occurrenceofa cusp singularity, it suggeststhat the singular behaviour of two-loop perturbation theory,
γ
G∗→ ∞
forβ
1=
0,isan artefactofthe perturbative ex-pansion.In full analogywith SQCD, equation (32)impliesthat
γ
G∗(
Nc f)
is strictlypositive, if
γ
m ∗(
Ncf)
<
0.The lattercondition isatleastverifiedin(31)toleadingorderinperturbationtheory,14andthere are no physicalconstraints that force
γ
m ∗ to vanish atthe loweredge,analogouslytoSQCD.
Anonvanishing
γ
G∗attheloweredgewouldthenexcludemerg-ing, according to section 4.4, andit would lead to the following description. A phase transitionoccursat thelower edge,and
γ
Gdevelopsafinitediscontinuity:
γ
G∗(
Nf)
isgivenby(32)andispos-itive forNf
/
N=
5/
2,while intheabsenceofa fixedpointγ
G isgivenby(4)andisnegativefor Nf
/
N<
5/
2,theconfiningphase,without vanishing–note that
γ0
in(28)and(∂
log Z/∂
logμ
)
in(31)nolongervanishbelowtheloweredge. 5.1. QCDandthea-theorem
What aboutthe a-theoremandits constraintson QCD orany of its limits? Thea-theorem, as proved in[39],wouldimply the existence ofa properUV to IRinterpolatinga-functionfora vast class of four-dimensionalfield theories wherea unitary S matrix exists, thus including SQCD, aswell as QCD. On the other hand, one can construct an a-function and studyaU V and aI R only in
a limitedsetofexamples.InSQCD,supersymmetryandtheexact anomaly-freeRsymmetryatthefixedpointarethekeyproperties thatallowtheexplicitconstructionoftheinterpolatinga-function discussedinsection 4.3.Mostimportantly,they allowusto show howthea-functionevolutiondirectlyconstrains theultravioletto infraredflowofthemassanomalousdimensionanditsderivatives. In QCD, some results are also available. Cardy’s conjectured a-function,whichcoincidesbyconstructionwiththeEuleranomaly coefficientaU V (aI R)attheUV(IR)CFTs,hasbeenshowntosatisfy
aU V
−
aI R>
0 in theconfined andchirallybrokenphaseof QCD,when its infrared realisation is assumed to have N2f
−
1 mass-lessGoldstonebosonsthatarefreeinthelongdistancelimit[42]. This resultis not basedon perturbation theory.Close to the up-per edge oftheQCD conformalwindow, inthelarge-N limitand with=
11/
2−
Nf/
N1, two-loop perturbation theoryveri-fies the a-theorem, i.e., aU V
−
aI R>
0 and of order N22 [46].
The a-theoremin thecontext ofthe massless Veneziano limit of large-NQCDdeservesfurtherinvestigation.
14 Note, however, that g
∗comes from the cancellation of a priori infinitely many