Large spin expansion of the long-range Baxter equation in the sl(2) sector of N=4 SYM

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

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arXiv:0710.1991v3 [hep-th] 24 Jan 2008MatteoBeccariaandFrancescaCatinoDipartimentodiFisica,Universita’delSalento,ViaArnesano,73100LecceINFN,SezionediLecceE-mail:matteo.beccaria@le.infn.it,ABSTRACT:Recently,severalmulti-loopconjectureshavebeenproposedforthespinde-pendenceofanomalousdimensionsoftwist-2and3operatorsintheslsectorofSYM.Currently,theseconjecturesarenotproven,althoughseveralconsistencycheckshavebeenperformedontheirlargespinexpansion.Inthispaper,weshowhowtheseexpansionscanbeef cientlycomputedwithoutresortingtoanyconjecture.Tothisaimwepresentinfulldetailsamethodtoexpandatlargespinthesolutionofthelong-rangeBaxterequation.Wetreatthetwist-2and3casesattwoloopsandthetwist-3caseatthreeloops.Severalsubtletiesarisewhoseresolutionleadstoasimplealgorithmcomputingtheexpansion.

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

1.Introduction

IntegrabilityoffourdimensionalplanarYang-Millstheoriesisaquiterelevantfeaturethatpermitsinprincipletoobtainnon-perturbativeresults

[1].It rstappearedasasurprisingfactintheone-loopanalysisofspecialQCDsubsectors[2].Later,itsuniversalitybecamemoreandmoreclearduetothemanyextensionsathigherorders[3,4,5].

Currently,adeepunderstandingofitsoriginisachievedbymeansofMaldacena

Yang-Millstheory,AdS/CFTcorrespondence[6].Inthemaximalsupersymmetric

integrabilityofthesuperconformalsideisrelatedtointegrabilityofthedualsuperstring

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

theoryon[7].Asamajoroutcome,adeeplymotivatedhigherloopproposalforthe-matrixofthepairSYM/typeIIonisnowavailable[8,9,10,11,12,13,14,15,16,17].Inthispaper,weshallworkwithinthisframeworkassumingthevalidityoftheaboveAnsatz,althoughthesubjectisstillunderdevelopment[18].

Werestrictourconsiderationstotheso-calledslsectorofSYMwhichisaninvariantsubsectorclosedunderperturbativerenormalizationmixing.Itisspannedbysingletraceoperatorsobtainedaslinearcombinationsofthebasicobjects

Tr(1.1)

whereisoneofthethreecomplexscalar eldsofSYMandisalight-conecovariantderivative.Thenumbersarenon-negativeintegersandtheirsumisthetotalspin.Asusual,thenumberof eldsiscalledthetwistoftheoperator.Itequalstheclassicaldimensionminusthespin.Thesubsectorofstateswith xedspinandtwistisalsoperturbativelyclosedunderrenormalizationmixing.

Theslsectorisveryrichandinteresting.Evenwhenthetwistiskeptlow,itisanon-trivialin nitedimensionalsectorwithcertainsimilaritieswithanalogousQCDcom-positeoperatorsappearingindeepinelasticscattering.

Atoneloop,thedilatationoperatorcanbeinterpretedastheintegrableHamiltonianofaspinchainwithsites.Ateachsite,thedegreesoffreedom,associatedwith,transforminthein nitedimensionalslrepresentation[19].

Goingbeyondoneloop,asymptoticall-orderBetheAnsatzequationshavebeenpro-posed[9].Importantsuccessfulchecksaredescribedin[20,21,13].Thankstosupersym-metry,wrappingproblems(see[22,23]forrecentdevelopments)onlyoccuratlooporderintwist-operators[10,11].Hence,ifweareinterestedinathreeloopanalysis,thetwistsand3aresafe.Weshallfocusonthesecases.

ScalingcompositeoperatorsareelementsinthespanofEq.(1.1)whichareeigenvec-torsofthedilatationoperator.Theeigenvaluesaretheanomalousdimensionswhereisthe’tHooftplanarcoupling(isthenumberofcolorssenttoin nitywithconstant)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Recently,workontwist-2and3operatorshasledtohigherorderconjecturesforthefunctionsinvarioustwistsandingeneralizedsubsectors[22,25,26,27,28].How-ever,proofsaremissing,atleastbeyondone-loop.Thegeneralwaytodeducesuchconjecturesissomewhatdeceiving.OnestartsbysolvingtheBetheAnsatzequationsatvarious.Generally,theBetherootsarenon-trivial

turnsouttobearationalnumbertoanypreci-algebraicnumbers.Nevertheless,

sion.Givenasuf cientlylongsequenceofsuchrationalnumbers,onemakesaninspiredguessabouttheclosedformulafor.GeneralargumentsfromFeynmandiagramcalculationsanddeeppropertieslikereciprocity[29,30,26]areinvokedtoconstrainthegeneralexpression.Inseveralcases,asimpleexpressionisfound.

However,ifitwerenotfortheamountofinspiration,theseconjectureswouldbenoth-ingbutinterpolationformulae,althoughsurprisinglysimple.Itwouldbeverynicetoprovetheseformulaewithoutmakingconjectures.Threenaturallinesofinvestigationarethefollowing:

1.BetheAnsatzequations.Aswementioned,theydealwiththeBetherootsasthebasicobject.Thisisabitunnaturalifoneisultimatelyinterestedintheanomalousdi-mensionswhichareamuchsimplerquantity.TheBetheequationscanbeusedto

limit[13,16,31].How-obtaintheasymptoticdensityofBetheroots,i.e.the

ever,systematiccorrectionsatlargeappeartoberatherinvolvedandpracticallyrestrictedtoafeworders[32].

2.Baxterequation.AnalternativeapproachtothecalculationofisbasedontheBaxterapproachoriginallyformulatedin[33].ThecrucialingredientistheBax-

obeyarelativelysimplefunctionalequation.teroperatorwhoseeigenvalues

isassumedtobeapolynomial,thentheBaxterequationisequivalenttoIf

thealgebraicBetheAnsatzequationsforitsrootstobeidenti edwiththeBetheroots.Amoregeneraldiscussionwithreferencestothenon-polynomialcasescanbefoundin[34,35].Remarkably,thepolynomialturnsouttohaverationalcoef cientsinallcaseswhereaclosedAnsatzforhasbeenproposed.Accord-

.ingly,theanomalousdimensionisarationalcombinationofderivativesof

Thisapproachleadstotheknownexactone-loopformulae.Thekeyfeatureisthattheone-loopBaxterpolynomialisknownasananalyticfunctionofinthiscase.Unfortunately,suchaniceresultisnotavailablebeyondone-loop.

3.Conformalmethods.Thesearereviewedintherecentpaper[36].Leadingtwistcon-formalprimaryoperatorslieontheunitarityboundandhenceareconserved(irre-ducible)inthefreetheory.Itispossibletocleverlyexploitthepatternofbreakingoftheirreducibilityconditionsintheinteractingtheory.Intheend,itispossibletogainanorderofperturbationtheoryandinfertwo-loopresultsfromthelowestordercalculations.

Inthispaper,weshallreconsidertheBaxterapproach.Wegiveupthetaskofcom-putingtheexactBaxterpolynomialatmorethanoneloopparametricallyinthespin.Instead,wedevelopatechniquetosystematicallyderivethelargespinexpansionof

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

athighordersinthenaturallogarithmicexpansion.Thiswillpermittoderivefrom rstprinciplestheexpansionofalthoughwithoutknowingitsresum-mation.Thiscanberegardedasastrongcheckoftheproposedconjecturesatlargespin.Itisalsoausefulresultbyitselfif,forinstance,oneisinterestedinprovinglargespinpropertieslikereciprocity.

Thesketchofthepaperisthefollowing.First,weillustratethemethodintheeasyone-loopcaseintwist-2and3.Atthislevel,wedonothingmorethanrephraseaverytrickytechniqueoriginallydevisedbyG.Korchemskyin[35].Theresultingprocedurewillbecalled-methodforbrevity.Itworkswellandthedesiredexpansionissystem-aticallyobtainedinafewlinesofcalculations,easilyimplementedonsymbolicalgebrapackages.

Then,wemovetothetwo-loopcase.Here,we ndsomesurprise.The-method

startingfromtheterms.Curiously,failstoreproducetherationalpartof

thefailureisnotsignaledbyanyapparentpathology.Thelessonisthatthe-methodcanbemisleading,althoughitworkswellforthenonrationalpartandforallthelogarithmi-callyenhancedcontributions.Theproblemisnotpresentintwist-3asweillustrateandpartiallyexplain.

Asafurtherstep,weanalyzewhatishappeningintwist-2andproposeasafeim-provedexpansionoftheBaxterequationwhichcorrectlyreproducesthefullanomalousdimension.Finally,weanalyzethetwist-3caseatthreelooptoshowthatagainthereisafailureintherationalpartof.Thesamesolutionadoptedforthetwist-2,2-loopcase,solvescompletelytheproblemandappeartobethenecessaryuniversalrecipeforthiskindofexpansions.

2.Theone-loopBaxterequationinthesl

2.1GeneralstructureoftheBaxterequationsector

Inthenotationof[37],theone-loopBaxterequationinsl-likesectorsis

(2.1)

whereisthetwist,theconformalspin,andapolynomial

(2.2)

Forbrevity,inthefollowing,weshallreferto

abuseoflanguage.

Thesecondchargeasthetransfermatrixwithasmallisexplicitlyknownandreads

(2.3)

whereisthespinquantumnumberofthesolution.Weshallbeinterestedinthepoly-nomialsolution

(2.4)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

ReplacingintotheBaxterequation,weobtaintheBetheAnsatzequationsfortheslspinchain

(2.5)

Theone-loopanomalousdimension/energyandquasi-momentumaregivenintermsoftheBetherootsby

(2.6)

TheycanbecalculatedfromtheBaxterpolynomial,completelybypassingtheknowledgeof

.Alsoweshallalwaystakeeven.Inthiscase,

thegroundstatewithminimalanomalousdimensionisnondegenerateandhas

withautomaticallyvanishingquasi-momentum.Thisiscorrectforsingle-tracecompositeoperatorsinthegaugetheory.Exploitingparity,theanomalousdimensionissimply

(2.8)

2.2Groundstatesolutionintwist2and3

Thestructureofthegroundstate,i.e.withminimalanomalousdimension,israthersimpleforevenandtwist2or3.Inthetwist-2case,doesnotdependonanyunknownquantumnumber

(2.9)

(2.11)

Fromthisexplicitexpressionwecancomputetheone-loopanomalousdimension

(2.12)

where(nested)harmonicsumsarede nedasusualby

signb(2.13)

Inparticularistheharmonicsum

(2.14)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Thetwist-3caseisalsosimplebecauseagain

quantumnumber.donotdependonanyunknown

(2.15)

(2.18)

2.3Largeexpansionoftheexact

Thelargeexpansionof

de nedasisthatofthefunctionforsmallvaluesofthevariable

(2.19)twist-2

Theexpansioniswell-knowninanalyticformatallordersandreads

(2.20)

whereareBernoullinumbersde nedby

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

2.4.1Twist-2

WewritetheBaxterequationas

(2.26)

Thelinearoperatoronther.h.s.canbeevaluatedinclosedformwhenactingonmanyspe-

.Inparticular,thisistrueforallthatwillappearinthisandlatercalculations.cial

Thesimplestcaseisthatofpolynomial

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Also,for,wehavetheveryusefulidentity

Takingderivatives,wealsoobtain

Goingbacktotheproblemofdetermining,weimmediately nd(2.29)(2.30)

(2.32)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

forImbyassumingthatthetermisdominantandtreatingthesecondpieceasaperturbation.Thisseemstobequitereasonablesinceintheendwewanttocomputeinwherethesecondpiecevanishes.Ourimplementationstartingdirectlyfromalogarithmicexpansionoffollowsthissimpleidea.Noticehoweverthattheexpansionisguaranteedtoworkforlargeenough,dependingontheorderoftheexpansion.Intheoriginalwork[35],onehastoanalyticallycontinuedownto

whichappearstobedangerous.Nevertheless,themachineryworkswellatone-loop.Weshall ndsurprisesattwoloops.

Asasecondcomment,wenoticethattheBaxterequationisinvariantunderanytrans-formation

with(2.33)

Inotherword,isdeterminedmoduloanadditivetermwithperiodicity1inthevariable.Intheaboveone-loopexamples,thistermisabsent.Heuristically,thiscanbe

,itseemsreasonabletoconcludethatatexplainedasfollows.Givenapolynomial

eachorderin,mustgrowatmostpolynomiallywith.Aperiodiccontribu-tioninthevariablewouldinsteadleadtoexponentiallygrowingquantities.Certainly,itwouldbenicetounderstandthispointbetter.

Wenowmovetothemoreinterestingtwo-loopcasewhereweshallshowthepartialfailureofthe-method.

3.Thetwo-loopBaxterequationforthegroundstate

3.1Generalstructure

SYMisdiscussedatthreeloopThelongrangeBaxterequationfortheslsectorof

accuracyin[40].Here,weshallbeinterestedinthetwoloopreductionthatwediscussinfulldetailsinthecaseofthegroundstate.

Weshallrequirethefollowingde nitions

(3.2)

Also,for,weintroduce

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

wheregetsradiativecorrectionstotheconservedcharges.Forthetwist-2andtwist-3groundstate,thissimplymeans

(3.5)

(3.6)

whereistheone-loopchargeandatwoloopcorrectiontobecomputed.

Inthegroundstate,thisequationmustbesolvedintermsof

(3.7)

,areevenandwithdegrees,respectively.Ofwherethepolynomials

istheone-loopBaxterpolynomial.Given,thetwo-loopanomalousdi-course

mensionis

(3.10)

Togoon,weneedthetwo-loopcharge

wehave.Thisiseasytocompute.Forboth

(3.13)

WereplacetheseexpansionsintheBaxterequationandmatchtheleadingtermspowersof.Afterashortcalculation,weobtainthefollowingcompactresults

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

twist-2

(3.16)

twist-3

(3.17)

Theseequationsarerathercomplicatedbutindeedadmitthedesiredpolynomialso-lutions.Asaconcreteexample,welistheretheirsolutionsat

twist-2

(3.20)

N(3.23)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Thenestedsumisabittricky.Aconvenientheuristicprocedureisasfollows.Westartfromtherecurrencerelation

even(3.24)

andinsertinplaceofagenericlogarithmicexpansionforlarge

canbeeasilymatchedandthe nalresultisterms.Thevarious

where

.Intheaboveexpressions,

wehavede ned

(3.28)

Thelargeexpansioncanbeevaluatedwithoutparticulardif cultiesandit nallyreads

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

we ndthatthenewcontributioncanbeplainlymatchedtotheequationifittakesthegeneralform(werecallagainthatinthefollowingweset

(3.33)

Theexpansioncanapparentlybecontinuedtoanyorderandnospecialproblemsdoappear.However,ifwecomputethepredicted,weencounterasubtleproblemalreadyatorder.Theanomalousdimensionisgivenby

(3.34)

Evaluatingbythesamemethodsweusedatone-loopandsolvingthedifferenceequationsassociatedwith,weeasily nd

(3.35)

ComparingwiththeexactresultEq.(3.26),oneseesthatamismatchappearsatorderExpandingthecalculationtohigherorders,one nds.

(3.36)

Alltermsoftheabovemismatcharenottranscendentalneitherhavelogarithmicenhance-ment.So,weconcludethatthe-methodworksatleadinglogarithmicaccuracyinclud-ingalsothenon-enhancedtranscendentalterms.However,itdoesnotworkfortheratio-

!nalcontributionsstartingfrom

Thisfailureseemstoberelatedtothefactthatthetwo-looptermsinthel.h.s.

.Later,weshallclarifythispoint.oftheBaxterequationsdonotvanishat

3.4LargeexpansionfromtheBaxterequation:-methodintwist-3

Thetwist-3caseismuchmoreeasybecausethespecialstructureofthetwo-loopBaxter

inthel.h.s.oftheBaxterequationsvanishat.Aequation.Theterms

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

straightforwardcalculationprovidesthefollowingexpressionofthetwo-loopdifferencefunction

Thecontributionsfromtherationalfunctionscanbeevaluatedin

particularthespecialvaluesbyusingin

(3.37)

AshortcalculationshowsthatthecorrectcompletetwolooplargespinexpansionoftheanomalousdimensionEq.(3.29)isperfectlyreproducedbystartingfromtheaboveex-pression.

4.TheimprovedexpansionoftheBaxterequationintwist-2

Letusgobacktotheweakpointsofthe-method.Weareassuminganexpansionfor

validintheBaxterequationforbothandinaneighborhoodof.Theassumedexpansionisclearlywrongasitstands.Forinstance,byparityinvariance

whichisnotobviousintheexpansionderivedatlargewehaverigorously

.

expansionof.Toseewhatishappening,welookattheleadingterminthe

Itiseasilyderivedas

(4.1)

If,weobtain

(4.2)

(4.3)

ThedifferentsignsareresponsibleforthesuppressionofthesecondpieceoftheBaxterequation

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

However,ifisaround,thevalueweareinterestedin,we nd,forsmallenough

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Afterthesereplacements,weinsertintheBaxterequationthegeneralizedasymptoticexpansionoffordescribedinEq.(4.9).

Wegivealltheexpressionsthatareneededtoreproducetheanomalousdimensionat

included.Thisrequirestoconsiderthefollowingfunctionsorder

Firstthenon-anomalousone-loopcontributions(4.12)

(4.15)

(4.17)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Eachofthesefunctionshasvanishingderivativein.Thismeansthatnoanoma-louscontributionsappearatone-loop,recoveringthecorrectnessofthe-methodatoneloop.However,two-loopnontrivialcontributionsareassociatedwiththethirdderiva-tive.Forinstance

(4.23)

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Theanomalouscontributionscanalsobecomputed,howeverwehavecheckedthatinallcasestheydonotgivecontributionstotheanomalousdimension.Thisisduetothefactthattheoneloopcontributionsarewhilethetwoloopcontributionsare.Someexamplesatone-loopare

(4.32)

(4.38)

.whichispreciselytherequiredpiecetocorrectthemismatchinEq.(3.36)toorder

Wehaveextendedthecalculationtohigherorders.Thewholeprocedureiseasilyautom-atizedandinallcases,themismatchiscorrected.

5.ThethreeloopBaxterequation:Twist-3

Toconclude,wenowanalyzethethreeloopBaxterequationintwist-3toshowthateveninthiscase,itisnecessarytoincludeanomaloustermstomatchtherationalpartofthelargespinanomalousdimension.

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

Wehavetoextendthenotationweintroducedforthetwoloopcase.Tothisaim,wede nefor

(5.1)

(5.2)

TheBaxterequationreadsagain

(5.3)

Inthegroundstatethetransfermatrixgetsradiativecorrectionstotheuniquenon-trivialcharge

(5.4)

Thisequationmustbesolvedintermsof

(5.5)

,,areevenandwithdegrees,,wherethepolynomials

tively.The3-loopanomalousdimensionisconvenientlyexpressedintermsofrespec-

(5.6)

as

(5.8)

aswellasthethreeloopcorrectiontothesecondcharge

Recently, several multi-loop conjectures have been proposed for the spin dependence of anomalous dimensions of twist-2 and 3 operators in the sl(2) sector of N=4 SYM. Currently, these conjectures are not proven, although several consistency checks have bee

(5.10)

Aswementioned,thethreelooptermsinthesecondtermofthel.h.s.donotvanishas

.Thissuggeststhatanomalouscontributionswillbepresentatthisorder.

Again,anexamplecanbeusefultochecktheaboveequation.Foritssolutionis

Noticethatgenericdoublelogarithmsappearintheexpansion.Formalpropertiesofthisexpressionaswellasitsfourloopextensionarediscussedin[26].

5.2LargeexpansionfromtheBaxterequation:-method

Followingthe-methodweobtainthefollowingthreeloopcontributiontothefunction

.Itisconvenienttowriteitintermsof

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