An HHL 3-point correlation function in the η-deformed AdS5×S5

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

An HHL

3-point

correlation

function

in

the

η

-deformed

AdS

₅

×

S

5

Changrim Ahn

a

,

b

,

∗

,

Plamen Bozhilov

b

,

c

a_{InstitutfürMathematik,InstitutfürPhysikand IRISAdlershof,Humboldt-UniversitätzuBerlin,ZumGroßenWindkanal6,12489Berlin,Germany} b_{DepartmentofPhysics,EwhaWomansUniversity,DaeHyun11-1,Seoul120-750,RepublicofKorea}

c_{InstituteforNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,Bulgaria}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received11January2015 Accepted13February2015 Availableonline18February2015 Editor:L.Alvarez-Gaumé

We derivethe 3-pointcorrelation functionbetween two giant magnons heavy stringstates and the light dilatonoperator with zero momentuminthe

η

-deformed AdS5×S5 validfor any J1 and

η

in

the semiclassical limit. We show that thisresult satisfies aconsistencyrelation betweenthe 3-point correlationfunctionandtheconformaldimensionofthegiantmagnon.Wealsoprovidealeadingfinite

J1correctionexplicitly.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

TheAdS/CFTduality

[1]

betweenstringtheoriesoncurvedspace–times withanti-deSittersubspacesandconformalfieldtheoriesin differentdimensionshasbeenactively investigatedinthelastyears.Alotofimpressiveprogresshas beenmadeinthisfield ofresearch basedmainlyon theintegrability structuresdiscoveredon bothsidesofthe correspondence(forrecentreview onthe AdS/CFTduality, see [2]). Forthe most studied case of the

N =

4 super Yang–Mills theory, the anomalous dimensions of gauge-invariant single-trace operatorsmatchnon-perturbativelywiththestringenergiesinthecurvedAdS5

×

S5 background.Integrabilityprovidestoolstosolvethe finite-volumespectralproblemexactly.

Afterthesesuccesses,onedirectionofinterestingdevelopment istogeneralizethedualitytolargertheorieswhichincludetheoriginal AdS/CFTasaspecialcaseandtheotheristogobeyondthespectralproblembycomputinggeneralcorrelationfunctions,inparticular,the three-pointfunctions,orthestructureconstants.

Aninteresting developmentfortheformerdirectionistostudythestringtheory onthe

η

-deformedAdS5

×

S5 background

[3]

.The bosonic part of the superstring sigma model Lagrangian on this

η

-deformed background and perturbative worldsheet S-matrix were obtainedin

[4]

.The TBAforspectrum andexplicitdispersion relationforgiant magnon

[5]

have beenderived in

[6]

.Finite-size effect onthegiantmagnonspectrumhasbeencomputedin

[7]

.Forthree-pointcorrelationfunctions,quitealotofinterestingresultsonboth strongandweakcouplingregionswereaccumulatedalthoughnon-perturbativeresultsaremuchmoredifficultthanthespectralproblem. Inthisletter,we computethethree-point correlationfunctionoftwo giantmagnonheavy operatorswithfinite-size J1 andasingle dilatonlightoperator ofthestringtheory withthe

η

-deformedAdS5

×

S5 background

[3]

.Then,we show thatthisresultisconsistent withthedispersionrelationofthefinite-sizegiantmagnonsolutionobtainedin

[7]

usingMathematicacode.

Thepaperisorganizedasfollows.InSection2,wederivetheexactsemiclassicalstructureconstantvalidforany J1and

η

andproveits consistency.InSection3,weexpanditforthecaseof J1



T (T isthetensionofstring)andobtainexplicitexpression.Abriefconclusion isinSection4andashortMathematicacodefortheconsistencyconditionisprovidedin

Appendix A

.

2. Exactsemiclassicalstructureconstant

Accordingto [8],the three-point functionsof two “heavy” operators anda “light” operatorcan be approximatedby a supergravity vertexoperatorevaluatedatthe“heavy”classicalstringconfiguration:



VH

(x

1

)V

H

(x

2

)V

L

(x

3

)

 =

VL

(x

3

)

classical

.

*

Correspondingauthor.

E-mailaddresses:[email protected](C. Ahn),[email protected](P. Bozhilov). http://dx.doi.org/10.1016/j.physletb.2015.02.032

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

For

|

x1

|

= |

x2

|

=

1,x3

=

0,thecorrelationfunctionreducesto



VH

(x

1

)V

H

(x

2

)V

L

(

0

)

 =

C123

|

x1

−

x2

|

2H

.

Then,thenormalizedstructureconstant

C

3

=

C123

C12 canbefoundfrom

C

3

=

cVL

(

0

)

classical

,

(2.1)

wherec isthenormalizedconstantofthe“light”vertexoperator.Actually,wearegoingtocomputethenormalizedstructureconstant

(2.1).Forthecaseunderconsideration,the“light”stateisrepresentedbythedilatonwithzeromomentum.

Accordingto

[9]

,C3fortheinfinite-sizegiantmagnonsanddilatonwithzeromomentumintheundeformedAdS5

×

S5 isgivenby

C3

=

cd +∞



−∞ d

τ

e cosh4

(

κτ

e

)

+∞



−∞ d

σ



κ

2

+ ∂

XK

¯∂

XK



=

4cd 3

κ

+∞



−∞ d

σ



κ

2

+ ∂

XK

¯∂

XK



,

(2.2)

wheret

=

κτ

e istheEuclidean AdS timeandtheterm

∂

XK

¯∂

XK isproportional tothe stringLagrangian on S2 computedonthe giant

magnonsolutionlivinginthe Rt

×

S2 subspace.

Sincehereweareinterestedinfinite-size giantmagnons,wehavetoreplace

+∞



−∞ d

σ

→

+L



−L d

σ

=

2 θmax



θmin dθ

θ



,

whereL givesthesizeofthegiantmagnonand

θ

isthenon-isometricangleonthetwo-sphere

[11]

.

Goingtothe

η

-deformed AdS5

×

S5 case, we haveto compute theterm

∂

XK

¯∂

XK forthisbackground,which isproportional to the

stringLagrangianon S2_{η for}finite-size giantmagnons:

L_S2 η

= −

T

2

∂

XK

¯∂

XK

,

where XK

= (φ

1

,

θ )

aretheisometricandnon-isometricstringcoordinatesonS2η correspondingly.

Workinginconformalgaugeandapplyingtheansatz

φ

1

(

τ

,

σ

)

=

τ

+

F1

(ξ ),

θ (

τ

,

σ

)

= θ(ξ), ξ =

ασ

+ β

τ

,

α

, β

−

constants

,

onefinds L_S2 η

= −

T 2



(

α

2

− β

2

₎

θ

2 1

+ ˜

η

2

₍

₁

₋

_χ

₎

+ (

1

−

χ

)



(

α

2

− β

2

₎₍

_F 1

)

2

−

2

β

F1

−

1



,

(2.3)

where

η

˜

isrelatedtothedeformationparameter

η

accordingto

[4]

˜

η

=

2

η

1

−

η

2

,

(2.4)

andanew variable

χ

isdefinedby

χ

=

cos2

θ.

Theprimehereandbelowisaderivatived

/

d

ξ

.ThestringtensionT forthe

η

deformedcaseisrelatedtothecouplingconstant g by

T

=

g

1

+ ˜

η

2

_.

_(2.5)

Thefirstintegralsoftheequationsofmotion F₁ and

θ

canbewrittenas

F₁

=

β

α

2

_{− β}

2

−

κ

2 1

−

χ

+

1



,

(2.6)

θ

2

=

1

+ ˜

η

2

₍

₁

₋

_χ

₎

(

α

2

_{− β}

2

₎

2



(

α

2

_{+ β}

2

₎

_κ

2

₋

β

2

κ

4 1

−

χ

−

α

2

₍

₁

₋

_χ

₎

.

(2.7)

Inserting

(2.6)

,

(2.7)

into(2.3),weobtain:

L_S2 η

= −

T 2

β

2

κ

2

₊

_α

2

₍

_κ

2

₋

₂

₍

₁

₋

_χ

₎₎

α

2

_{− β}

2

.

(2.8)

Nowweintroducethenewparameters v

= −

β

α

,

W

=

κ

2

_,

whichleadsto L_S2 η

= −

T 2

(

1

+

v2

)W

−

2

(

1

−

χ

)

1

−

v2

.

(2.9)

Therefore,forthecaseathand,thenormalizedstructureconstanttakestheform

Cη₃˜

=

8c d  3

√

W χp



χm d

χ

χ





W

+

(

1

+

v 2

_)W

₋

₂

₍

₁

₋

_χ

₎

1

−

v2

,

(2.10) where

χ

m

=

χ

min

,

χ

p

=

χ

max

.

OnecanrewriteEq.(2.7)as

χ



=

2

η

˜

1

−

v2

(

χ

η

−

χ

)(

χ

p

−

χ

)(

χ

−

χ

m

)

χ

,

(2.11) where

[7]

χ

m

=

1

−

W

,

χ

p

=

1

−

v2W

,

χ

η

=

1

+

1

˜

η

2

.

(2.12)

Usingthis,wecanexpressalltheresultsintermsof

χ

p

,

χ

m byeliminatingv,W .

Replacing

(2.11)

in(2.10)andusing

(2.12)

,wecanexpressC₃η˜ by

Cη₃˜

=

8c d  3

η

˜

√

1

−

χ

m χp



χm



χ

−

χ

m

(

χ

η

−

χ

)(

χ

p

−

χ

)

χ

d

χ

.

(2.13)

TheintegralcanbeeasilyexpressedbyK and



,thecompleteellipticintegralsofthefirstandthethirdkind,respectively,areas follows:

Cη₃˜

=

16c d  3

η

˜

χ

m



χ

p

(

1

−

χ

m

)(

χ

η

−

χ

m

)





1

−

χ

m

χ

p

,

1

−





−

K

(

1

−



)



,

(2.14)

whereweintroducedashortnotation



by



=

χ

m

(

χ

η

−

χ

p

)

χ

p

(

χ

η

−

χ

m

)

.

(2.15)

Eq.(2.14)isthemainresultofthispaper,whichisanexact semiclassicalresultforthenormalizedstructureconstantCη₃˜ validforany value of

η

˜

and J1. Here,

χ

p and

χ

m are determined by theangularmomentum J1 andworld-sheet momentum p from the following equations1: J1

=

2T

˜

η

1



χ

p

(

χ

η

−

χ

m

)



χ

pK

(

1

−



)

−

χ

m



1

−

χ

m

χ

p

,

1

−





,

(2.16) p

=

2

χ

m

˜

η



1

−

χ

p

χ

p

(

1

−

χ

m

)(

χ

η

−

χ

m

)



K

(

1

−



)

− 

χ

p

−

χ

m

χ

p

(

1

−

χ

m

)

,

1

−





.

(2.17)

Theworld-sheetenergyofthegiantmagnonisgivenby

E

=

2T

˜

η

χ

p

−

χ

m



χ

p

(

1

−

χ

m

)(

χ

η

−

χ

m

)

K

(

1

−



) .

(2.18)

Oneofnontrivialchecks isthat the g derivativeof



=

E

−

J1 shouldbeproportionaltothenormalizedstructureconstantCη3˜ since the g derivativeof thetwo-point function insertsthe dilaton(Lagrangian) operatorintothe two-point functionof theheavy operators

[10].Thiscanbeexpressedby

Cη₃˜

=

8c d  3



1

+ ˜

η

2

∂

∂

g

.

(2.19) 1 _{Weexpress J}

TocheckthatEqs.(2.14),

(2.16)

,and

(2.18)

satisfyEq.(2.19),weusethefactthat

∂

J1

∂

g

=

∂

p

∂

g

=

0 (2.20)

asnoticedin

[11]

forthecaseofundeformedgiantmagnon.Fromthese,we canobtaintheexpressionsfor

∂

χ

p

/∂

g and

∂

χ

m

/∂

g which

canbe insertedinto

∂/∂

g.The

η

-deformed caseinvolvesmuch morecomplicatedexpressionswhich canbe dealtwith Mathematica. In

Appendix A

,we provideourMathematicacodewhichconfirmsthatthestructureconstant Cη₃˜ inEq.(2.14) dosatisfy theconsistency condition

(2.19)

exactly.

Inthelimit

η

˜

→

0 with

η

˜

2

_χη

_→

_{1,Eq.(2.14)becomes}

C₃0

=

16c d  3



χ

p 1

−

χ

m [E

(

1

−



)

−



K

(

1

−



)

]

,



=

χ

m

χ

p (2.21)

whereweusedthe identity

(

1

−

a

)(

a

,

a

)

=

E

(

a

)

.Thisisthestructureconstantoftheundeformedtheoryderivedin

[11]

.

3. Leadingfinite-sizeeffectonCη₃˜

Itisstraightforward tocomputealeadingfinite-sizeeffectonC₃η˜ for J1



g bythelimit



→

0 in

(2.14)

. Firstweexpandtheparameters

χ

p,W andv forsmall



asfollows:

χ

p

=

χ

p0

+ (

χ

p1

+

χ

p2log



)



,

W

=

1

+

W1



,

v

=

v0

+ (

v1

+

v2log



)



.

(3.1)

InsertingintoEq.(2.14),weobtain

C₃η˜

≈

16c d  3

η

˜

2



₁

₊

1 ˜ η2



χ

p0



(

1

+ ˜

η

2

₎

_χ

p0arctanh

˜

η

√

χ

p0



1

+ ˜

η

2

−



W1 2

(

1

+ ˜

η

2

₎

_χ

p0arctanh

˜

η

√

χ

p0



1

+ ˜

η

2

+

˜

η

4



1

+ ˜

η

2

₍

₁

₋

_χ

p0

)



×



(

1

+ ˜

η

2

)(

χ

p0

−

2

χ

p1

)

−

4



(

1

+ ˜

η

2

)

χ

p0

+

2W1



1

+ ˜

η

2

(

1

−

χ

p0

)



log 2





−

η

˜

4



1

+ ˜

η

2

₍

₁

₋

_χ

p0

)





(

1

+ ˜

η

2

)(

χ

p0

−

2

χ

p2

)

+

2W1



1

+ ˜

η

2

(

1

−

χ

p0

)





log





.

(3.2)

InviewofEqs.(2.12)and

(2.15)

,wecanexpressalltheauxiliaryparametersintermsofv (oritscoefficientsv0,v1,andv2):

χ

p0

=

1

−

v20

,

χ

p1

=

1

−

v20

−

2v0v1

−

(

1

−

v2₀

)

2 1

+ ˜

η

2_v2 0

,

χ

p2

= −

2v0v2

,

W1

= −

(

1

+ ˜

η

2

₎₍

₁

₋

_v2 0

)

1

+ ˜

η

2_v2 0

.

(3.3) Thisleadsto C₃η˜

≈

16c d  3

η

˜



arctanh

˜

η

1

−

v2₀



1

+ ˜

η

2

+

1 4

(

1

+ ˜

η

2

₎₍

₁

₋

_v2 0

)



1

+ ˜

η

2_v2 0



2

×



(

1

+ ˜

η

2

)

⎛

⎜

⎝(

1

−

v2₀

)



1

+ ˜

η

2v2₀



⎛

_⎜

⎝

2



1

+ ˜

η

2



₍₍

₁

₋

_v2 0

)

arctanh

˜

η

1

−

v2₀



1

+ ˜

η

2

− ˜

η

log 16

⎞

⎟

⎠

− ˜

η



1

−

v0

(

3v0

−

2v30

−

4v1

+

v0

(

1

−

v20

−

4v0v1

)

η

˜

2

)





+

η

˜

(

1

+ ˜

η

2

)(

1

−

v20

−

4v0v2

)

4

(

1

+ ˜

η

2

₎₍

₁

₋

_v2 0

)(

1

+ ˜

η

2v20

)



log





.

(3.4)

Tofixv0,v1,andv2,onecanusethesmall



expansionoftheangulardifference

whereweidentifiedtheangulardifference

φ

1 withthemagnonmomentum p ontheworldsheet.Theresultis

[7]

v0

=

cotp₂

˜

η

2

₊

_csc2 p 2

,

(3.5) and v1

=

v0

(

1

−

v20

)



1

−

log 16

+ ˜

η

2



2

−

v2₀

(

1

+

log 16

)



4

(

1

+ ˜

η

2_v2 0

)

,

v2

=

1 4v0

(

1

−

v 2 0

).

(3.6) Byusing

(3.5)

,

(3.6)

in

(3.4)

,onefinds Cη₃˜

≈

16c d  3

η

˜



arcsinh



η

˜

sinp 2



+

(

1

+ ˜

η

2

)

sin 2 p 2 4

˜

η

2

₊

_csc2 p 2

×



2



˜

η

2

₊

_csc2p 2arcsinh



˜

η

sinp 2



− ˜

η

(

1

+

log 16

)





+ ˜

η

log





.

(3.7)

Theexpansionparameter



intheleadingorderisgivenby

[7]



=

16 exp

⎡

⎣−



J1 g

+

2



1

+ ˜

η

2

˜

η

arcsinh



˜

η

sinp 2



!

"

_"

#

_

1

+ ˜

η

2sin2 p2 1

+ ˜

η

2



_sin2 p 2

⎤

⎦ .

(3.8)

HereweusedEq.

(2.5)

forthestringtensionT .

Thefinalexpressionforthenormalizedstructurecostantisgivenby

Cη₃˜

≈

16c d  3

η

˜



arcsinh



˜

η

sinp 2



−

4

η

˜

(

1

+ ˜

η

2

₎

_sin3 p 2

1

+ ˜

η

2_sin2 p 2



1

+

J1 g



˜

η

2

₊

_csc2 p 2 1

+ ˜

η

2

×

exp

⎡

⎣−



J1 g

+

2



1

+ ˜

η

2

˜

η

arcsinh



˜

η

sinp 2



!

"

"

_#

1

+ ˜

η

2_sin2 p 2



1

+ ˜

η

2



_sin2 p 2

⎤

⎦



.

(3.9)

Letuspointoutthatinthelimit

η

˜

→

0,

(3.9)

reducesto

C3

≈

16 3 c d sin p 2



1

−

4 sinp 2

sinp 2

+

J1 g



exp



−

J1 g sinp₂

−

2



,

whichreproducestheresultfortheundeformedcasefoundin

[11]

.AnothercheckisthatthissatisfiesEq.(2.19)with



computedin

[7]



≡

E

−

J1

≈

2g

1

+ ˜

η

2



1

˜

η

arcsinh



˜

η

sinp 2



−

4

(

1

+ ˜

η

2

₎

_sin3 p 2

1

+ ˜

η

2_sin2 p 2

×

exp

⎡

⎣−



J1 g

+

2



1

+ ˜

η

2

˜

η

arcsinh



˜

η

sinp 2



!

"

"

_#

1

+ ˜

η

2_sin2 p 2



1

+ ˜

η

2



_sin2 p 2

⎤

⎦



.

(3.10) 4. Concludingremarks

Hereweobtainedtheexact semiclassical3-point correlationfunctionbetweentwofinite-sizegiantmagnons“heavy”stringstatesand the“light”dilatonoperatorwithzeromomentuminthe

η

-deformed AdS5

×

S5.Itisgivenintermsofthecompleteellipticintegralsof thefirst and thirdkind. We proved theconsistency ofourresultby takinga derivative ofthe conformaldimension w.r.t.the coupling constant.Wealsoprovidedtheleadingfinite-sizeeffectexpansionofthestructureconstant.

Itwill be interesting to compute other three-point correlation functionsof the

η

-deformed backgroundsuch as HHHto whichour resultmaybeuseful.

Acknowledgements

Thiswork was partially supported by the Korean–Eastern European cooperation in research anddevelopment through the National ResearchFoundation ofKorea (NRF) (NRF-2013K1A3A1A39073412)(CA) andtheBrain PoolProgram (131S-1-3-0534)(PB) bothfunded bythe Ministry of Science,ICTandFuturePlanning. CA isalsosupported in partby theSFB647 “Raum-Zeit-Materie. Analytischeund GeometrischeStrukturen” andthe MarieCurie NetworkGATIS oftheEuropean Union’s Seventh FrameworkProgramme FP7-2007-2013 undergrantagreementNo.317089.

Appendix A. TheMathematicacodefortheconsistencycheck(Eq.(2.19)) J

[

g₋

] :=

2T

η

√

(

xn

−

xm

[

g

])

xp

[

g

]

xp

[

g

]

EllipticK



(

xp

[

g

] −

xm

[

g

])

xn

(

xn

−

xm

[

g

])

xp

[

g

]



−

xm

[

g

]

EllipticPi



1

−

xm

[

g

]

xp

[

g

]

,

(

xp

[

g

] −

xm

[

g

])

xn

(

xn

−

xm

[

g

])

xp

[

g

]



;

p

[

g₋

] :=

2xm

[

g

]

η



1

−

xp

[

g

]

(

1

−

xm

[

g

])(

xn

−

xm

[

g

])

xp

[

g

]

EllipticK



(

xp

[

g

] −

xm

[

g

])

xn

(

xn

−

xm

[

g

])

xp

[

g

]



−

EllipticPi



xp

[

g

] −

xm

[

g

]

(

1

−

xm

[

g

])

xp

[

g

]

,

(

xp

[

g

] −

xm

[

g

])

xn

(

xn

−

xm

[

g

])

xp

[

g

]



;

En

[

g₋

] :=

2T

(

xp

[

g

] −

xm

[

g

])

η

√

(

1

−

xm

[

g

])(

xn

−

xm

[

g

])

xp

[

g

]

EllipticK



(

xp

[

g

] −

xm

[

g

])

xn

(

xn

−

xm

[

g

])

xp

[

g

]



;

T

=

1

+

η

2_g

_;

Eq1

=

D

[

J

[

g

],

g

] ==

0

;

Eq2

=

D

[

p

[

g

],

g

] ==

0

;

sol

=

Solve

[{

Eq1

,

Eq2

}, {

D

[

xm

[

g

],

g

],

D

[

xp

[

g

],

g

]}];

xpd

=

D

[

xp

[

g

],

g

]/.

sol

[[

1

]];

xmd

=

D

[

xm

[

g

],

g

]/.

sol

[[

1

]];

threept

=

8c 3



1

+

η

2FullSimplify

[

D

[

En

[

g

] −

J

[

g

],

g

]/.{

D

[

xp

[

g

],

g

] →

xpd

,

D

[

xm

[

g

],

g

] →

xmd

}]

16c



−

EllipticK



xn(−xm[g]+xp[g]) (xn−xm[g])xp[g]



+

EllipticPi



1

−

xm_xp[[_gg_]]

,

xn(_(xn−₋xm_xm[_[g_g]+_])xpxp_[[g_g])_]



xm

[

g

]

3

η

√

(

−

1

+

xm

[

g

])(−

xn

+

xm

[

g

])

xp

[

g

]

References

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