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University of Groningen

The scalar glueball operator, the a-theorem, and the onset of conformality

da Silva, T. Nunes; Pallante, E.; Robroek, L.

Published in:

Physics Letters B

DOI:

10.1016/j.physletb.2018.01.047

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from

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Publication date:

2018

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):

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 ofthescalarglueballoperatorcontainsinformationonthe

mechanismthatleadstotheonsetofconformalityattheloweredgeoftheconformalwindowina

non-Abeliangaugetheory.Inparticular,itdistinguisheswhetherthemergingofanUVandanIRfixedpoint –

thesimplestmechanismassociatedtoaconformalphasetransitionandpreconformalscaling–doesor

doesnotoccur.Atthesametime,weshedlightonnewanalogiesbetweenQCDanditssupersymmetric

version.InSQCD,wederiveanexactrelationbetween

γ

G andthemassanomalousdimension

γ

m,and

we 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 and

γ

m,fully analogousto

SQCD, 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 leads

tothepossibilityofrealising“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 been

pro-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,andthe

associated 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

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.

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Itisthusrelevanttoidentifyobservablesthatcarrytheimprint ofthemechanismfortheonsetofconformalityatNcf,andatthe sametimearestringentlyconstrainedbyuniversalprinciples,such asexactsymmetriesandtheultraviolettoinfraredrenormalisation group(RG)flowgovernedbythea-theorem.

In this letter we show that the anomalous dimension

γ

G of

the 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 dimension

γ

m, thus relating

thetwo 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.Insection3we

analyse

γ

Gintwo-loopperturbationtheoryandclosetotheupper

edgeNf



NA Ff ,partlyreviewingknownresults,andwecomment

onthelimitsofapplicabilityofperturbationtheoryinthiscontext. 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

)

Ga

μν 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

)/∂

log

μ

for given N colours and Nf flavours; an analogous relation is valid in SQCD. We

shall 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

/

dlog

μ

=

0. Using inside the latter equation a Callan–SymanzikequationfortherenormalisedoperatorTr

(

G2

)

d d log

μ



ZG−1Tr

(G

2

)



=

0

,

γ

G

(g)

= −

log ZG

log

μ

,

(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 Nf

fixed,oneobtains d d log

μ



β(g)

g ZG



=

0 (3) and

γ

G

(g)

=

g

g



β(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

= −

2

β

0g2

+ . . .

,

β

0 from (7),and

γ

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

)

.Atafixedpoint

of the renormalization group flow, the solution of

β(

g

,

Nf

)

=

0

thus definesthe function g

(

Nf

)

offixed-point couplings on the

plane

(

g

,

Nf

)

,andequation(4)provides

γ

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 is

a physical property of the system, renormalisation scheme inde-pendent;thescalingdimensionofTr

(

G2

)

,dG

=

4

+

γ

G∗,thusenters

theexactconformalscalingofthecorrespondingcorrelatorsatthe fixedpoint.

3. PerturbativeresultsinQCD

Itisinstructivetofirstrecallsomefeaturesofperturbation the-ory.TheQCDbetafunctioncanbeexpressedasaseries

β(g)

= −

g3 ∞



l=0

β

lg2l

,

(6)

where

(

l

+

1

)

denotes the number of loops involved in the cal-culation of

β

l. The coefficients

β

0,1 are universal [1,28–30], i.e., renormalisationschemeindependent,givenby

β

0

=

1 3

(

4

π

)

2

(

11CA

4TfNf

)

β

1

=

1 3

(

4

π

)

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.The

quantity 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 g2

=

−β

0

1 isone rootofthe equation

β(

g

)

=

0 for some givenNf,

andfrom(5)

γ

G

= −

2

β

02

1.

We are interested inthe way

γ

G∗ variesalong thecurve of IR fixed points g

(

Nf

)

asNf decreasesintheconformal windowof

QCD,i.e.,forNf Diracfermionsinthefundamentalrepresentation;

in this case, NA F

f

= (

11

/

2

)

N, CA

=

N, Cf

= (

N2

1

)/(

2N

)

, and

Tf

=

1

/

2 in(7).IntheVenezianolimit, N

,

Nf

→ ∞

,holdingx

=

Nf

/

N andN g2constant,and



=

11

/

2

Nf

/

N



1,thatiscloseto

the upperedge,one obtains N g2

(4)

and

γ

G

 (

16



2

/

225

)(

1

+

O

(



))

positive.5 Its derivative with re-spectto Nf

=

xN withfixedN andx continuousintheVeneziano

limit d

γ

G

dNf

= −

32



225N

(

1

+

O

(



))

(8)

isnegativeandoforder



/

N,thus implyingthat

γ

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

=

2

β

20

1 remains indeed positive and monotonically increasing as Nf decreases on the entireinterval Ncf



Nf



NA Ff , andthe

IR zero disappears at Nc

f due to the change of sign of

β

1; for

N

=

3 andNf fundamentalfermions,thisoccursatNf

8

.

05.The

changeofsignof

β

1 wouldimplythat thefixed pointdisappears atinfinitecoupling g2

= −β

0

1

→ ∞

,andthe samesingularity occursfor

γ

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)withNf

fun-damental flavours Qi in the N representation and Q

˜

˜i in the N

¯

representation(i

,

˜

i

=

1

,

. . .

Nf),wheremanyresultscanbederived

exactly.Our goalinthissection isto determineexactconstraints on the UV to IR flow of

γ

G

(

g

,

Nf

)

andthe mass anomalous

di-mension

γ

m

(

g

,

Nf

)

.Lateron,insection5,wewillfindthatsome

properties of

γ

G can be proved to be equally true in the

mass-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 16

π

2 3N

Nf

+

Nf

γ

m

(

g) 1

N g2

/(

8

π

2

)

,

(9) with

γ

m

(g)

= −

g2 8

π

2 N2

1 N

+

O(g 4

)

(10)

themassanomalous dimensioncomputedinperturbation theory. ApowerfulpropertyofSQCDisthatitsexactbetafunctionandthe globalanomalyfreeRsymmetryatafixedpointdetermineexactly themassanomalousdimension

γ

m

(

Nf

)

alongthecurveofIRfixed

points, g

(

Nf

)

,intheconformalwindowofSQCD, whichextends

ontheinterval3N

/

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˜

=

3

2RQ Q˜

=

3R

=

3

Nf

N

Nf

,

(11)

withR theR-chargeofQ (Q ),

˜

andthelastequalitydictatedbythe R-chargeassignmentsofQ (Q )

˜

underU

(

1

)

R.UsingDQ Q˜

=

2

+

γ

m∗,

oneobtains

γ

m∗ exactly

γ

m

(N

f

)

=

1

3N

Nf

,

(12)

whichisindeedazeroofthebetafunction(9),providedthepole isnothit,i.e., N g2

/(

8

π

2

)

<

1.6 Equation(12)isthentakento de-termine

γ

m along the curve of IR fixed points in the conformal

window 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 implies

γ

m

= −

1, which in turn implies that the numerator of the beta

functionhasazeroforNf

=

3N

/

2.Thisidentifiesthelower edge

oftheconformalwindowforSQCD.

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

/

N

constant, with



=

3

Nf

/

N



1,i.e., closeto theupperedgeof

theconformalwindow.Thisgives[3]

N g2

=

8

3

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

)

= −

g3 16

π

2 Nf

γ

m ∗

(N

f

)

1

N g2 ∗

/(

8

π

2

)

,

(14)

where analogously to (5)

γ

m ∗

(

Nf

)

γ

m

(

g

,

Nf

)

|

g=g(Nf) and the

prime denotesthederivative withrespectto g.Equation(14) es-tablishes a usefulrelation betweenthe anomalous dimension

γ

G

and the derivative of

γ

m. This is a key resultthat we are going

to useinthe restofthissection. In particular,ourtaskis to de-rive constraints on the flow of

γ

m, its derivatives, and

γ

G, using

equation(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

/

2



Nf



3N,andcanbe summarised

asfollows:

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

(5)

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 the

fixed 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

/

2



Nf

<

3N,theexistenceofthe

IRfixedpointatnonzerocouplingexcludesan 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

/

2



Nf

<

3N.

3) We know exactly

γ

m

(

Nf

)

along the IR fixed point curve. If

Nf is assumed to be continuous, then d

γ

m

/

dNf

>

0, and

thus

γ

m

(

Nf

)

is strictly monotonic in Nf and decreases as

Nf decreasesalong g

(

Nf

)

.Also,d2

γ

m

/

dN2f

<

0 impliesthat

d

γ

m

/

dNf itself strictly increasesas Nf decreases. This result

comesstraightforwardlyfromtheexactsolutionfor

γ

m

(

Nf

)

in

SQCDandthea-theorem.

4) IntheVenezianolimit,toleadingorderinperturbationtheory andclosetotheupperedge,the IRfixedpoint couplingN g2

∗ isstrictlymonotonicinx

=

Nf

/

N,and,withabuseofnotation,

in Nf

=

xN withfixed N and x continuousinthe Veneziano

limit. 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

)

.Weaddthataresultfullyanalogous

tothatofQCDtwo-loopperturbationtheoryin(8),isobtained inSQCDifthetwo-loopSQCDbetafunctionisused.

Result 3) is straightforwardly implied by taking the derivatives of(12),specifically,d

γ

m

/

dNf

=

3N

/

N2f andd2

γ

m

/

dN2f

= −

6N

/

N

3 f.

The derivative of (13) with respect to Nf

=

xN for N fixed,

∂(

N g2

)/∂

Nf

= −

8

π

2

/

3N

(

1

+

O

(



))

is negative to leading order,

so is

g

/∂

Nf, thus providing result 4); it agrees with the

ob-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

mdNf

=

γ

m

(g,

Nf

)

Nf

g=g(Nf)

+

γ

m ∗

(N

f

)



g

Nf



(15) isknownexactly,d

γ

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.Thisresultisconsistentwith

γ

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.Thisfunctionalsoprovidesaneffectivemeasureof

thenumberofmasslessdegreesoffreedom,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 provides

aU 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-theoremdirectlyconstrainstheUVtoIRflowof

γ

m

(

g

,

Nf

)

inSQCD,and,via(14),itconstrainsthatof

γ

G

(

g

,

Nf

)

.

Withoutknowledgeofthea-theorem,equation(12)already im-pliesthat,alongtheIRfixedpointcurve, DQ Q˜ ,R and

γ

m∗ decrease

fromtheirvalueattheupperedge(DQ Q˜

=

2,R

=

2

/

3,

γ

m

=

0)to

theirvalueattheloweredge(DQ Q˜

=

1,R

=

1

/

3,

γ

m

= −

1),where

theunitarityboundissaturated.

The a-theorem allows us to further establish the monotonic variation of

γ

m

(

g

,

Nf

)

along anyvalidRGtrajectoryfromthe

ul-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.Weshallusethe

universal-ityofthea-functiontoalsoderiveconstraintsontheRGflowfrom ahypotheticalstronglycoupledUV fixedpointtotheweakly cou-pledIRfixedpoint;werefertothisflowasUVSC.

II.One candevisea flowinthespaceoftheoriesalongthe IR fixedpointcurve fromatheory withNf massless flavourstoone

withNf

n masslessflavours,andNf

n



Ncf,sothatboth

theo-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,oneobtainsfor

theflowUVA F [43] aU V

aI R

=

N Nf 48

γ

∗2 m

3

γ

m

=

N Nf 48



1

3N Nf

2



2

+

3N Nf



,

(18)

(6)

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 the

IR 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

/

2



Nf

<

3N.8

To establish result 1) of section 4.2 we use the interpolating function a

(

g

(

μ

))

for Nf fixed, given by aI R in (17) for

γ

m

γ

m

(

g

(

μ

))

.Accordingtothea-theorem,a

(

g

(

μ

))

isastrictly

mono-tonic function of the scale

μ

and decreases from the UV to the IR,anditisstationaryatafixedpoint.Thus,away fromthefixed point

β(

g

)

=

0 andalongtheflowsoftypeIwithfixedNf

da d log

μ

=

∂a

g

β(g) >

0 (19)

impliesthat

a

/∂

g hasthesamesignas

β(

g

)

with

a

g

= −

N Nf 16

γ

m

(

2

γ

m

)

γ

m

g

=

0

.

(20)

For

γ

m

<

0 and

γ

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 of

g awayfromfixedpoints. Importantly,thisresultapplies toboth flows,UVA F (

β(

g

)

<

0) andUVSC (

β(

g

)

>

0),giventhe

universal-ityoftheinterpolatinga-functionand(20).Thisisresult1),andit willimplytheincompatibilityofSQCDwithmergingandresult2) insection4.4.

At the nontrivial fixed point, UV or IR, the flow of the a-functionisstationary,da

/

dlog

μ

=

0,because

β(

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 of

γ

m

(

Nf

)

impliedby(12),d

γ

m

/

dNf

>

0,guaranteesthatda

/

dNf

=

9N4

/

8N3

f 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 to

bottom: 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 edge

oftheconformalwindowisincompatiblewithmerging.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

+

]

, with

α

±

=

α

c

±



the zeroes of

β(

α

,



)

;

α

± are

dis-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 occursat

α

cfor



=

0,to

α

>

α

c

(

α

<

α

c

)

and

α

<

α

<

α

+for



>

0.

Attheloweredge,



=

0,thebetafunction(22)developsalocal maximumat

α

c,thus

β



(

α

c

,



=

0

)

vanishes.SQCD, likeQCD,has

onecoupling,thegaugecoupling,andthelatterresult,via(5)and

α

g2,impliesthat

γ

G∗ vanishesforNf

=

Ncf,though thetheory

isinteracting.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 edge

γ

G

(

α

±

,



>

0

)

=

β



(

α

±

,



>

0

)

= ∓

f

(

α

±

)



is positiveat

α

− (IR)andnegativeat

9 f

(7)

α

+ (UV),then

γ

G

(

Nf

)

isnon monotonicalong theIRfixed point

curveifmergingoccurs.

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 theSQCDbetafunction

β(

α

)

=

2N g

β(

g

)

,with

β(

g

)

in(9).10

Mergingisrealised for f

(

α

)

=

α

2

/(

8

π

2

(

1

α

/(

8

π

2

)))

strictly increasingon

[

α

,

α

+

]

,whereasalways1

α

/

8

π

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 some





0,





1:

γ

m

(

α

,



)

=

−1

+ (2

/

3

)(

α

α

c

)

2 1

+

2



/

3

.

(24) Thus

γ

m

(

α

c

,



)

= −

1

/(

1

+

2



/

3

)

is a minimum of

γ

m

(

α

,



)

=

γ

m

(

α

c

,



)

+ (

2

/

3

)(

α

α

c

)

2

/(

1

+

2



/

3

)

on the interval

[

α

,

α

+

]

, with

α

±

=

α

c

±



.Atthezeroes,

γ

m

(

α

±

,



)

= −(

1

2



/

3

)/(

1

+

2



/

3

)

.Crucially,forany



>

0,

α

c doesnot correspondtoa fixed

point,nevertheless wehave foundthat theRGflow of

γ

m is

sta-tionary at

α

c, if merging is realised. Equation (20) then implies

that the a-function itself is stationary at

α

c, away from a fixed

point,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 point

curve 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 along

theIRcurve;inSQCD,through(14),thelatterpropertiesholdfor

γ

G

(

Nf

)

aswellasthederivativeofthemassanomalousdimension

γ

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 8

π

2



1

α 8π2



h(

α

)

= −

3N

+

Nf

Nf

γ

m

(

α

) .

(26)

Since both fixed points are at nonzero coupling, f

(

α

)

does not vanishontheclosedinterval

[

α

I R

,

α

U V

]

anditiscontinuous,with

1

α

/(

8

π

2

)

>

0.Hence,thebetafunctionvanishesonlyifits nu-merator vanishes,i.e.,

β(

α

I R,U V

)

=

0 onlyifh

(

α

I R,U V

)

=

0, andthe

continuityof

β(

α

)

impliesthecontinuityofh

(

α

)

.Then

β(

α

)

, con-tinuousandvanishing onlyatthe boundariesof

[

α

I R

,

α

U V

]

,hasa

maximumatsome

α

I R

<

α

<

α

U V,andh

(

α

)

alsohasa maximum

at some

α

I R

<

α

<

α

U V, and, by (26),

γ

m

(

α

)

has an extremum

at

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 is

fullyanalogous, 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

/

2



Nf

<

3N, the

existenceoftheIRfixedpointatnonzerocouplingexcludesanUV 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 and

α

U V

>

0. This

case 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

/(

4

π

2

)(

1

α

/(

16

π

2

))/(

1

α

/(

8

π

2

))

2

>

0, so that

β(

α

)

hasa maximum at some 0

<

α

<

α

U V while h

(

α

)

, and thus

γ

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 the

a-theorem requires that it varies (decreases) strictly monotoni-cally, i.e., h

(

α

)

= (β(

α

)/

f

(

α

))



<

0 on

(

0

,

α

U V

)

, or equivalently

β



(

α

)/β(

α

)

<

f

(

α

)/

f

(

α

)

on

(

0

,

α

U V

)

where f

,

f

>

0 and

β >

0.

For h

(

α

)

, and thus

γ

m

(

α

)

, strictly monotonic, and

γ

m

=

0

,

2,

equation (20) then implies a strictly monotonic a-function on

(

0

,

α

U V

)

. However, the a-theorem through (19) further requires

that

a

/∂

α

has the samesign as

β(

α

)

away froma fixed point, hence

a

/∂

α

>

0 for

β(

α

)

>

0.Equation(20)thenimpliesthe con-straints:

γ

m

/∂

α

>

0 for

γ

m

<

0 and

γ

m

>

2,and

γ

m

/∂

α

<

0 for

0

<

γ

m

<

2.Consider

α

inaneighbourhood oftheorigin

α

I R

=

0,

with

γ

m

(

0

)

=

0.Then,

γ

m

/∂

α

>

0 if

γ

m

(

α

)

>

0 and

γ

m

/∂

α

<

0

if

γ

m

(

α

)

<

0.None ofthelatter solutionssatisfies thea-theorem

constraintsabove.ThisimpliestheabsenceofanontrivialUVfixed pointabovetheconformalwindowofSQCD, Nf



3N.11

The second case in (25), where now

α

U V

=

0 and

α

I R

>

0 is

realised 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].

(8)

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 betafunction

hasbeen 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

log

μ

=

(27)

g3 16

π

2

(

4

π

)

2

β

0

N

(∂

log Z

/∂

log

μ

)

+

Nf

γ

m

(g)

1

N

g2

/

4

π

2

,

with

β

0in(7),theanomalousdimensionfactor

log Z

log

μ

=

2

γ

0



N g2

+ . . .



γ

0

=

5 3

(

4

π

)

2



1

2Nf 5N



(28) andthefermionmassanomalousdimension

γ

m

(g)

= −

9 3

(

4

π

)

2 N2

1 N g 2

+ . . . ,

(29)

bothstartingatorderN g2,andcomparingwith(9),oneconcludes thattheabsenceofsupersymmetrygeneratesthenewanomalous dimensioncontribution

log Z

/∂

log

μ

inthebetafunctionofQCD; itsstructureisotherwiseidenticalto(9).Equation(27)isexactin thelarge-Nlimit,i.e., toleading orderinthe1

/

N expansion,and itisexact toall ordersinthe O

(

1

)

ratio Nf

/

N intheVeneziano

limit.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 forwhich

γ0

in (28)changessign.Infact,

γ0

alsoenterstheglueballkineticterm, anditschangeofsignsignalsaphasetransitionfromconfinement toaphasewith

Tr

(

G2

)



=

0,theconformalCoulombphase.

Then, for Nf

/

N

=

5

/

2, barring the occurrence of a cusp

sin-gularity and noting that

log Z

/∂

log

μ

=

0,13 the beta function (27)vanishes for

γ

m

= −

4

/

5,a renormalisation-scheme

indepen-dentresult.Asanticipatedinsection3,thisresultsuggeststhatthe singularityof theQCD two-loop betafunction atthe loweredge, i.e., g

→ ∞

for

β

1

=

0,isindeedanartefactofthetruncated per-turbativeexpansion. Wedetermine

γ

G

(

Nf

)

using(27):

γ

G

(N

f

)

= −

g3 16

π

2

N(∂log Z

/∂

log

μ

)

 ∗

(N

f

)

+

Nf

γ

m ∗

(N

f

)

1

N g2 ∗

/(

4

π

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

/∂

log

μ

)



=

2

γ

0

(

2N g

+ . . .)

γ

m

= −

6

(

4

π

)

2

N2

1

N g

+ . . .

(31)

At the lower edge the derivative

(∂

log Z

/∂

log

μ

)

 ∗

(

Nc

f

)

vanishes

exactlysince

γ0

=

0.Therefore,attheloweredge(30)reducesto

γ

G

(N

cf

)

= −

g3 ∗ 16

π

2 Nf

γ

m ∗

(N

cf

)

1

N g2 ∗

/(

4

π

2

)

,

(32) which is the main result of this section, a relation between

γ

G∗ and

γ

m ∗ entirely analogous to SQCD. Like (27), (32) is exact in

the large-N limit and to all orders in the O

(

1

)

ratio Nf

/

N in

the Veneziano limit.Barringthe occurrenceofa cusp singularity, it suggeststhat the singular behaviour of two-loop perturbation theory,

γ

G

→ ∞

for

β

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 isatleast

verifiedin(31)toleadingorderinperturbationtheory,14andthere are no physicalconstraints that force

γ

m ∗ to vanish atthe lower

edge,analogouslytoSQCD.

Anonvanishing

γ

G∗attheloweredgewouldthenexclude

merg-ing, according to section 4.4, andit would lead to the following description. A phase transitionoccursat thelower edge,and

γ

G

developsafinitediscontinuity:

γ

G

(

Nf

)

isgivenby(32)andis

pos-itive forNf

/

N

=

5

/

2,while intheabsenceofa fixedpoint

γ

G is

givenby(4)andisnegativefor Nf

/

N

<

5

/

2,theconfiningphase,

without vanishing–note that

γ0

in(28)and

(∂

log Z

/∂

log

μ

)

 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

/

N



1, two-loop perturbation theory

veri-fies the a-theorem, i.e., aU V

aI R

>

0 and of order N2



2 [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

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