Wrapping corrections beyond the sl(2) sector in N=4 SYM

0.12 MB·English

by  CarloAlberto Ratti

FortschrittederPhysik,27January2012
Wrapping corrections beyond the sl(2) sector in N = 4 SYM
MatteoBeccaria1,∗,GuidoMacorini2,∗∗,andCarloAlbertoRatti1,∗∗∗
1 DipartimentodiFisica,Universita’delSalento,ViaArnesano,73100Lecce&
INFN,SezionediLecce.
2 NielsBohrInternationalAcademyandDiscoveryCenter,
2 Blegdamsvej17DK-2100Copenhagen,Denmark.
1
0
ReceivedDayMonth2012,revisedDayMonth2012,acceptedDayMonth2012
2
PublishedonlineDayMonth2012

n
a
Keywords Integrability,Y-system.
J

6 Thesl(2)sector of N = 4SYMtheory hasbeen muchstudied andtheanomalous dimensions of those
2 operators are well known. Nevertheless, many interesting operators are not included in this sector. We
consider a class of twist operators beyond the sl(2) subsector introduced by Freyhult, Rej and Zieme.

] They are spin n, length-3 operators. At one-loop they can be identified with three gluon operators. At
h strongcoupling,theyareassociatedwithspinningstringswithtwospinsinAdS spaceandchargeinS5.
t
- We exploit the Y-system to compute the leading weak-coupling four loop wrapping correction to their
p anomalous dimension. Theresult iswritteninclosedformasafunction ofthespinn. Wecombine the
e wrappingcorrectionwiththeknown four loopasymptoticBetheAnsatzcontributionandanalyzespecial
h limitsinthespinn. Inparticular,atlargen,weprovethatageneralizedGribov-Lipatovreciprocityholds.
[
Atnegativeunphysicalspin,wepresentasimpleBFKL-likeequationpredictingtherightmostleadingpoles.

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2
2
5 1 Introduction
.
1
The mirror thermodynamic Bethe Ansatz (TBA) for the AdS ×S5 superstring [1] provides a general
0 5
2 approachto the study of finite size correctionsto states/operatorsin AdS/CFT. The associated Y-system
1 has been proposed in [2] based on symmetry argumentsand educated guesses about the analyticity and
: asymptoticpropertiesoftheY-functions.
v
i Verypowerfulexplicittestsofthesemethodshavebeenmostlydoneinthesl(2)sectorofN =4SYM
X
[3]. Beyondsl(2),ourknowledgeofthelargerpartofthefullpsu(2,2|4)structureofthetheoryislimited.
r
Inthisbriefnote, following[4], [5], we showsomerecentprogressin thecomputationofthe leading
a
orderwrappingcorrectionsoftheFreyhult-Rej-Zieme(FRZ)operators,aspecialclassofoperatorsbeyond
thesl(2)sectorfirstdiscussedatone-loopinthegeneralanalysis[6]andfurtheranalyzedatallordersin
[7]. TheresultsfollowfromadirectapplicationoftheY-systemtechniques.
FRZoperatorsaretwist3operators. Theirgeneralformcanbewrittenbyinsertingcovariantandanti-
covariant derivatives D, D¯ into the half-BPS state TrZ3 (Z being one of the three complex scalars of
N =4SYMtheory):
OFRZ =Tr(Dn+mD¯mZ3)+··· . (1)
n,m
Atstrongcoupling,theseoperatorsaredualtospinningstringconfigurationswithtwospinsS =n+m−1
1 2
andS = m− 1 inAdS andchargeJ = L = 3inS5. Thedistributionoftherootsonthenodesofthe
2 2 5
∗ [email protected].
∗∗ [email protected].
∗∗∗ [email protected].
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2 M.Beccaria,G.Macorini,andC.Ratti:Wrappingcorrectionsbeyondthesl(2)sectorinN =4SYM
psu(2,2|4)algebraissummarizedinFigure2:
+1
(cid:0)❅♥ ♥ (cid:0)❅♥ ♥ (cid:0)❅♥ ♥ (cid:0)❅♥ (2)
n+2m m m−1 m−2
Nontrivialrootsoutofthesl(2)sectoroccurform ≥ 2. Intheparticularm = 2case,theoperatorsare
called3-gluonoperators.
In general, the wrapping effects are expected to appear at weak coupling for the OFRZ operators at
n,m
fourloop. Inpaper[8],theasymptoticpartoftheanomalousdimensionsoftheOFRZhasbeencomputed
n,2
exactly up to four loops. The result is given as a closed formula in the spin parameter n. Up to this
level, reciprocityholdsfor the asymptotic contributions. By following[4], fromthe Y-system equations
wederivepreciselyasimilarclosedformulafortheleadingwrappingcorrections. Thisformula,together
withtheresultsof[8],completesthestudyoftheanomalousdimensionsforthe3-gluonoperatorsupfour
loop. Asabyproduct,weshallbeabletotestpositivelyreciprocityaswellasdiscusstheBFKLpolesof
the full fourloopresult. We also show that a very simple and naturalmodificationof the twist-2 BFKL
equationpredictsthecorrectpolestructure.
For the general OFRZ FRZ operators, no asymptotic closed formula for the anomalous dimension is
n,m
known beyond one-loop. Thus, the research of such a closed formula for the wrapping corrections is
hopelessanduseless. Neverthelessthelargenexpansionoftheasymptoticminimalanomalousdimension
ofOFRZisknownforfixedration/morfixedm[7].Theexpansionisobtainedatallordersinthecoupling
n,m
andincludingtheleadingterm∼lognaswellasthesubleadingasymptoticallyconstantcorrection∼n0.
Thesetwocontributionsareexpectedtobefreeofwrappingcorrections.
Following[5],weconsiderpreciselythelargenexpansionofleadingorderwrappingcorrectionwhich
appearsatfourloop. WeprovideanalgorithmtocomputethroughtheY-systemthelargenexpansionfor
fixedmandpresentexplicitnewresultsform = 2,3,4. Form = 2wematchthelargen expansionof
theclosedformulaforthe3-gluonoperators[4]. Theexpansionsfortheothertwovaluesarenew. Infull
generality,weprovethe logn scalingbehaviouratlargenthusconfirmingtheassumptionin[7].
n2
Theplanofthepaperisthefollowing:InthenextSectionwegivetheminimalsetofY-systemequations
requiredtocomputetheleadingorderwrappingcorrections.InSection3wegivetheresultfortheleading
wrappingcorrectionsofthe3-gluonoperators.InSection4weconsiderthelargenlimitofFRZoperators
OFRZ. Thegeneralstrategyisdescribedandresultsaregivenform=(3,4).
n,m
2 Leading wrapping corrections from Y-system
TheY-systemisasetoffunctionalequationsforthefunctionsY (u)definedonthefat-hookofpsu(2,2|4)
a,s
[2]. Theseequationsare
Y+ Y− (1+Y )(1+Y )
a,s a,s = a,s+1 a,s−1 . (3)
Y Y (1+Y )(1+Y )
a+1,s a−1,s a+1,s a−1,s
Theirboundaryconditionsare discussed in [9]. Theanomalousdimensionofa genericstate isgivenby
theTBAformula
∞ du ∂ǫ⋆
E = g2ℓγ = ǫ (u ) + a log(1+Y⋆ (u)), (4)
ℓ-loop 1 4,i R 2πi ∂u a,0
ℓ=0 i a≥1Z
X X X
asymptoticγasy wrappingW
where the asymptotic and the w|rapp{izng co}ntrib|utions are well{zdistinct. In for}mula (4), the dispersion
relationis
2ig 2ig
ǫ (u)=a+ − , (5)
a x[a] x[−a]
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andthestarmeansevaluationinthemirrorkinematics1.
ThecrucialassumptionintheidentificationofrelevantsolutionstotheY-systemis
x[−a] L
Y⋆ ∼ , (6)
a≥1,0 x[+a]
(cid:18) (cid:19)
for large L, or large u (or small g). In this limit, it can be shown that the Hirota equation splits in two
su(2|2) wings. Onecanhavea simultaneousfinitelargeLlimitonbothwingsafterasuitablegauge
L,R
transformation.Thesolutionisthen
x[−a] L φ[−a]
Y (u)≃ TL TR , (7)
a,0 x[+a] φ[+a] a,1 a,1
(cid:18) (cid:19)
where φ[−a] isthefusionformfactorandTL,R arethetransfermatricesoftheantisymmetricrectangular
φ[+a] a,1
representationsof su(2|2) . They can be explicitly computedby an appropriategeneratingfunctional
L,R
[10].
Atweakcoupling,theformula(4)fortheleadingwrappingcorrectionstakesthesimplerform
∞ ∞
1
W =−π RduYa⋆,0 =−2i Resu=i2aYa⋆,0. (8)
a=1Z a=1
X X
Theformula(7)fortherelevantY functionsgetssimplifiedtoo. Inparticular,theY’scanbeexpressedas
functionsofthe1-loopBaxterpolynomialsQ ’s. Weget[5]:
i
x[−a] L ⋆ 4g2 L
= ,
(cid:18)x[+a](cid:19) ! (cid:18)a2+4u2(cid:19)
φ[−a] ⋆ = Q+(0) 2 Q[41−a] Q[5−a] Q7(0),
(cid:18)φ[+a](cid:19) 4 Q[4−1−a]Q[4a−1]Q[4a+1] Q5(0) Q[7−a]
(cid:2) (cid:3)
T∗,L =(log(Q′))|u=+2i g2(−1)a+1 a Q[4−1−k]−Q[41−k] +O(g4),
a,1 4 u=−2i Q[41−a] kX=−a u−ik2 (cid:12)(cid:12)Q[4−1−a],Q[4−1−a]→0
∆k=2 (cid:12)
Q[a]Q[−a] a−1 Q[k] Q[k+2]−Q[k] Q[k(cid:12)−2]−Q[k]
T∗ =(−1)a+1 5 7 4 6 6 + 6 6 +O(g2). (9)
a,1 Q[1−a] Q[k] Q[k+1]Q[k+1] Q[k−1]Q[k−1]!
4 k=1−a 6 5 7 5 7
X
∆k=2
TheexpressionsfortheQ ’sfortheFRZoperatorsareknownandcanbefoundin[7],[8]. Theseexpres-
i
sions togetherwith equations(7), (8) and(9) are in principleall whatone needsto computethe leading
wrappingcorrectionsfortheFRZoperators.
3 The 3-gluonoperators OFRZ case
n,2
FRZ operatorsOFRZ with m = 2 are amongthe simplest nontrivial operatorsthatcan be built beyond
n,2
the sl(2) sector of N = 4 SYM theory. In this Section we give a general result for their leading order
wrappingcorrectionsW . Moredetailsabouttheresultsofthissectioncanbefoundin[4].
n
ThestartingpointistheobservationthatbyspecializingtheformulasofSection2tothesimplem=2
caseandfixingn,thecomputationofthewrappingcorrectionsisstraightforward.In[5]weproducedalist
±···±
1 ShiftedquantitiesaredefinedasF| {az }(u)=F[±a](u)=F(cid:0)u±ia(cid:1).
2
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4 M.Beccaria,G.Macorini,andC.Ratti:Wrappingcorrectionsbeyondthesl(2)sectorinN =4SYM
ofresultsforthewrappingupton =70. Thenwewereabletocondensatethesedatainaclosedformula
thatreplicatesandgeneralizestoanyvalueofnthenumericalresultsofthelist. Theformulais
W = (r +r ζ +r ζ ) g8,
n 0,n 3,n 3 5,n 5
2 1
r = 80 4S + +4 −4(N +1)+ ,
5,n 1
N +1 N +1
(cid:18) (cid:19) (cid:18) (cid:19)
2
r = 16 4S + +4 ×
3,n 1
N +1
(cid:18) (cid:19)
2 2 1
8(N +1)S +8+ (2−S )− − ,
2 N +1 2 (N +1)2 (N +1)3
(cid:20) (cid:21)
2
r = 2 4S + +4 ×
0,n 1
N +1
(cid:18) (cid:19)
4
16(N +1)(2S −S )+32S + (S −2S +4S )+
2,3 5 3 5 2,3 3
N +1
(cid:20)
8 4 4 1
+ (−S +2)+ (−S +4)− − . (10)
(N +1)2 3 (N +1)3 3 (N +1)5 (N +1)6
(cid:21)
HereS ≡ S (N)andN = n/2+12. Notethateachoftherationalcoefficientr canbewritten
a,b,... a,b,... i
asr =γ r ,whereγ =S + 2 +4istheoneloopanomalousdimension.
i,n 1-loop i,n 1-loop 1 N+1
TheAnsatz(10)completesthefourloopexpressionoftheenergyspectrumfortheOFRZoperators,the
n,2
otherrelevantcoentributionsuptothisorderbeingthefirstfourasymptoticorders. Theirexpressionscan
befoundin[8].
Uptofourloop,theasymptoticpartofthespectrumshowsthegeneralizedGribov-Lipatovreciprocity
property[8]. Formula(10)allowstocheckwhetherthispropertyextendstothefullfourloopresult. This
isindeedthecase:ThelargenexpansionofW /γ reads
n 1-loop
ζ r +ζ r +r =
5 5,n 3 3,n 0,n
32 −32ζ 232ζ3 − 352 4834 − 2344ζ3 3544ζ3 − 83956 271768 − 9512ζ3
3 3 + 5 15 + 105 35 + 35 945 + 1485 55 +
e J2e e J4 J6 J8 J10
1872392ζ3 − 20053258 87933002 − 524872ζ3 4917304ζ3 − 5747755528
5005 45045 + 61425 455 + 935 883575 +..., (11)
J12 J14 J16
where we introduced the charge J2 = N(N +2). All the odd powers of 1/J cancel proving that the
reciprocitypropertydoeshold.Itisremarkablethatthispropertyisaconsequenceofnon-trivialcancella-
tionsofodd1/J termswhicharepresentintheexpansionofeachsinglecoefficientr . Thepresenceof
i,n
reciprocityisreallyappreciable,sinceitallowstopredictahalfofthelargenexpansionterms(expressed
asfunctionsofthesamen)ascombinationsoftheotherhalf.
e
TheAnsatz(10)allowsalsotostudytheBFKLpolesofthefullfourloopresult. Ingeneral,atℓ-loop
theanalyticcontinuationofγ inthevariableN aroundN =−1isexpectedtobehaveatworstasω−ℓ,
ℓ-loop
whereωisasmallexpansionparameterdefinedbyN =−1+ω. Atfourloopstheasymptoticanomalous
dimensionγasy presentsinsteadalso polesin ω−k with k = (7,6,5). Thesepolesgetindeedcompen-
4-loop
satedinsidethefullγ =γasy +W wherepreciselythewrappingcontribution(10)isincluded. The
4-loop 4-loop
finalexpressionsfortheexpansionsofthefirstfourγ are
ℓ-loop
4 8 4π2
γ =− +..., γ = + +..., (12)
1-loop ω 2-loop ω2 3ω
0 16 −3ζ +π2+12 32(1+2ζ ) 160ζ
3 3 3
γ = − +..., γ =− + +....
3-loop ω3 3ω2 4-loop ω4 ω3
(cid:0) (cid:1)
2 Weremindthatnisanevenpositiveinteger.ThismeansthatNisaninteger,N ≥2.
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Strikingly,theleadingpolescanbereproducedbyaBFKL-likeequationthatlinksωtothefullanomalous
dimensionγ
ω γ
− = χ , χ (z)=S (z)+S (z+1). (13)
g2 1 2 1 1 1
(cid:16) (cid:17)
Infact,theweakcouplingexpansionofthisequationreadsprecisely
4 8 0 32(1+2ζ )
γ = − +... g2+ +... g4+ +... g6+ − 3 +... g8+....
ω ω2 ω3 ω4
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(14)
4 General FRZoperators: Largen expansion
ForthegeneralOFRZ onlythelargenlimitoftheasymptoticpartoftheanomalousdimensionisknown
n,m>2
forfixedration/morfixedm[7]. Thus,asfarasweconsiderthewrappingcorrections,weareprimarily
interestedinthecomputationoftheirlargenexpansions.
TheY-systemdescribedinSec. 2canbeoptimizedtogetonlythelargencontributions.Thecomputa-
tionalstrategyisthefollowing:
• Startingfromeq. (8)oneevaluatestheresidueatfixeda=1,2,... withoutassigningn;
• Analyzingthedependenceonnoftheresidue,onerealizesthatncomesfromtheBaxterpolynomials
Q ,itsderivatives(whicharewrittenintermsofthebasichypergeometricfunctionF )andfrom
4 n,m
theexplicitn-dependentcoefficientsoftheotherBaxterpolynomials. Atthispointthelimitovern
canbetakenintwodistinctsteps;
• Using the Baxter equation, it is possible to shift the argumentof F to some minimal value and
n,m
takethelargenlimitonthecoefficients.Thisgivesafirstexpansioncontainingvariousderivativesof
thelogarithmofF ;
n,m
• Thederivativesofthelogarithmcanbesystematicallycomputedbymeansofthemethodexplained
in[11];
• Theoutcomeofthisprocedurearesequencesofrationalnumbersbeingthea-dependentcoefficients
of the large n expansion of Resu=ia2 Ya⋆,0. These sequences turn out to be rather simple rational
functionswhichareeasilyidentified. Thesumoveraoftheserationalfunctionsgivesthewrapping
W asdefinedbyeq. (8).
Followingthiscomputationalmethod,in[5]wecomputedthelargenexpansionofthewrappingcor-
rectionsfortheoperatorsOFRZ withm = 2,3,4. Them = 2caseagreeswiththelargenexpansionof
n,m
eq. (10). Form=3,4wegetthesenewresults
4 2 logn¯+1 88 logn¯
g−8W = − (36ζ +5π2−3) + (36ζ +5π2−3) + (15)
n,m=3 3 3 n2 3 3 n3
2
− 4(9108ζ +1289π2−615) logn¯−14868ζ −2017π2+1527 +...,
9n4 3 3
1024(cid:2) 3 logn¯ +4 7168 6 logn¯ +5 (cid:3)
g−8W = − (81ζ −32) 2 + (81ζ −32) 2 + (16)
n,m=4 1215 3 n2 1215 3 n3
1024 n¯
− 138240(1971ζ −760) log +143289ζ −53248 +....
8505n4 3 2 3
h i
Theextensionoftheseresultstolargervaluesofmisaplaintask.
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6 M.Beccaria,G.Macorini,andC.Ratti:Wrappingcorrectionsbeyondthesl(2)sectorinN =4SYM
Thenumericalcheckin Table(17) showsthatthere isgoodagreementbetweenourexpansionsanda
numericalestimateofthewrappingcorrections.
m=3 n estimate (LO NLO NNLO) fullexpansion diff%
10 −1.857411 (−4.038730 3.785374 −2.548066) −2.801422 51%
30 −0.438781 (−0.594614 0.193683 −0.045355) −0.446286 1.7%
50 −0.197270 (−0.238478 0.047207 −0.006716) −0.197986 0.36%
m=4 n estimate (LO NLO NNLO) fullexpansion diff%
10 −1.201460 (−2.908787 3.493848 −3.576129) −2.991068 149%
30 −0.293765 (−0.424071 0.176476 −0.061888) −0.309484 5.4%
50 −0.134246 (−0.169551 0.042847 −0.009090) −0.135794 1.2%
(17)
Followingthegeneralideaofreciprocity,weareledtorewritetheabovelargenexpansionsintermsof
thequantity
J2 =n(n+a ). (18)
m m
It turns out that the coefficients of the two odd terms 1/J3 and logJ/J3 indeed vanish for the choice
a = 8, 11, 14. Thisis notcompletelytrivialsince we haveoneparameterandtwo structures. It is
2,3,4
temptingtoconjecturethesimplerelationa = 3m+2andtoclaimthatreciprocityintheabovesense
m
holdsforthefullanomalousdimensionaswell.
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