Virtual scaling function of AdS/CFT

Lisa Freyhult1 and Stefan Zieme2,3

1Department of Physics and Astronomy, Uppsala University, SE-751 08 Uppsala, Sweden2Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Am Muhlenberg 1, 14476 Golm, Germany

3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA(Received 25 January 2009; published 14 May 2009)

We write an integral equation that incorporates finite corrections to the large spin asymptotics of N ¼4 supersymmetric Yang-Mills theory twist operators from the nonlinear integral equation. We show that

these corrections are an all-loop result, not affected by wrapping effects, and agree, after determining the

strong coupling expansion, with string theory predictions.

DOI: 10.1103/PhysRevD.79.105009 PACS numbers: 11.15.Me, 11.25.Tq

I. INTRODUCTION

Twist operators have proved to be very important opera-tors for studying the planar limit of the anti–de Sitter/conformal field theory correspondence. Their special scal-ing behavior at large spin [1] was established and anintegral equation, the BES equation, describing the loga-rithmic growth of the anomalous dimension was derived[2]. In the weak coupling regime its expansion agrees withthe explicit four-loop N ¼ 4 result [3].

The strong coupling limit was studied in [4,5] and shownto match the available two-loop string data [6] whilepredictions for higher loops were also made in [4].

In particular, the anomalous dimension of twist L op-erators exhibits the large spin M scaling behavior

�Lðg;MÞ ¼ fðgÞðlogMþ �E � ðL� 2Þ log2Þ þ BLðgÞþ � � � : (1)

The scaling function fðgÞ is commonly denoted the cuspanomalous dimension fðgÞ ¼ 2�cuspðgÞ as it describes thelogarithmic growth of the anomalous dimension of light-like Wilson loops with a cusp [7]. In analogy with the QCDsplitting functions [8] we denote the finite order correctionBLðgÞ as the virtual part.

In a refined limit for the scaling function an all-loopstring calculation from the bosonic Oð6Þ sigma model hasbeen performed [9]. It agrees to two loops with the directstring computation [10]. The same limit in the weak cou-pling regime leads to a generalized scaling function [11]which agrees with the predictions from the Oð6Þ sigmamodel [9] upon continuation to infinite coupling, as wasdemonstrated in [12] and further studied in [13]. For analternative approach to the generalized scaling function,see [14].

Anomalous dimensions of twist operators can also becomputed for finite values of the spin in terms of harmonicsums [15]. For the leading twist-two operators the asymp-

totic Bethe ansatz and Baxter equation, however, breakdown at four-loop order [16] and wrapping effects haveto be taken into account.For the lowest M ¼ 2 state, the Konishi operator, the

complete anomalous dimension including wrapping ef-fects, has been successfully computed from different per-spectives [17]. Subsequently, using the Luscher formalismthe wrapping effects for twist-two operators, which curethe asymptotic Bethe ansatz result, were computed [18].The result satisfies all Balitskii-Fadin-Kuraev-Lipatov [19]constraints at negative spins and predicts that at largevalues of the spin M the first contributions from wrappingeffects will be of order Oðlog2M=M2Þ [20]. This indicatesthat finite order correctionsOðM0Þ to the large spin scalingcan also be computed from the asymptotic integrablestructure to all-loop orders in the Yang-Mills coupling.In what follows we will compute these finite order

effects to the logarithmic scaling behavior for operatorsof general twist L. In the strong coupling regime we canrewrite the finite order corrections in terms of the functionsthat determine �cusp and recover the string results [21],

which shows that the virtual corrections BLðgÞ are indeednot affected by wrapping effects. We also determine sub-leading corrections in 1=g. For twist-two operators we canalso compute further subleading corrections in M up towrapping order. They equally match string theorypredictions.The cusp anomalous dimension describes the leading

singularities in the logarithm of planar multiloop gluonscattering amplitudes. The virtual part enters the sublead-ing divergencies as a part of the collinear anomalousdimension [22]. The latter are known to fourth order [23].

II. FINITE ORDER INTEGRAL EQUATION

From the nonlinear integral equation (NLIE) (see [24]and references therein) of the slð2Þ sector [11] one canextract the following equation for the density for the dis-tribution of Bethe roots including corrections of OðM0Þ:

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�ðtÞ ¼ t

et � 1

�Kð2gt; 0ÞðlogMþ �E � ðL� 2Þ log2Þ

� L

8g2tðJ0ð2gtÞ � 1Þ

þ 1

2

Z 1

0dt0

�2

et0 � 1

� L� 2

et0=2 þ 1

�ðKð2gt; 2gt0Þ

� Kð2gt; 0ÞÞ � 4g2Z 1

0dt0Kð2gt; 2gt0Þ�ðt0Þ

�; (2)

with the anomalous dimension given by �Lðg;MÞ ¼16g2�ð0Þ. The integral kernel Kðt; t0Þ ¼ K0ðt; t0Þ þK1ðt; t0Þ þ Kdðt; t0Þ can be written in terms of Bessel func-tions [2]. The parity even/odd components are given by

K0ðt; t0Þ ¼ 2

tt0X1n¼1

ð2n� 1ÞJ2n�1ðtÞJ2n�1ðt0Þ;

K1ðt; t0Þ ¼ 2

tt0X1n¼1

ð2nÞJ2nðtÞJ2nðt0Þ;(3)

respectively. The dressing kernel Kdðt; t0Þ is given by theconvolution

Kdðt; tÞ ¼ 8g2Z 1

0dt00K1ðt; 2gt00Þ t00

et00 � 1

K0ð2gt00; t0Þ:(4)

We will drop the part �ðlogMþ �E � ðL� 2Þ log2Þ,readily identifiable as the contribution to the cusp anoma-lous dimension analyzed in [4]. Decomposing the densityinto parity even/odd parts

et � 1

t�ðtÞ ¼ �þð2gtÞ

2gtþ ��ð2gtÞ

2gt; (5)

one can express the functions ��ðtÞ in the form of aNeumann series over Bessel functions

�þðtÞ ¼ 2X1n¼1

ð2nÞJ2nðtÞ�2n;

��ðtÞ ¼ 2X1n¼1

ð2n� 1ÞJ2n�1ðtÞ�2n�1;

(6)

which satisfy an infinite system of equations. We want toobtain an equation for BLðgÞ in terms of the solution to theBES equation only, similar to the case of the generalizedscaling function analyzed in [12]. Therefore, we introducea label j such that the system of equations becomes

Z 1

0

dt

t

��þðt; jÞ

1� e�t=2g� ��ðt; jÞ

et=2g � 1

�J2nðtÞ ¼ jL

8ngþ jh2n;

Z 1

0

dt

t

���ðt; jÞ

1� e�t=2gþ �þðt; jÞ

et=2g � 1

�J2n�1ðtÞ

¼ 1� j

2�n;1 þ jh2n�1; (7)

with hn ¼ hnðgÞ given by

hnðgÞ ¼Z 1

0

dt

4

�2

et � 1� L� 2

et=2 þ 1

��Jnð2gtÞ

gt� �n;1

�:

(8)

For j ¼ 0 one recovers the solution of the BES equationanalyzed in [4], while j ¼ 1 leads to the system of equa-tions that determines BLðgÞ. To derive this set of equationswe made use of the summation formula of even Besselfunctions 2

P1n¼1 J2nðtÞ ¼ ð1� J0ðtÞÞ and the orthogonal-

ity relation of even/odd Bessel functions.As was shown in detail in [12], we choose some refer-

ence j0 and multiply both sides of the system (7) withð2nÞ�2nðg; j0Þ and ð2n� 1Þ�2n�1ðg; j0Þ, respectively.Summing over all n � 1 and making use of the expansionformulas of the even/odd parts (6) we obtain two equationsfor the even/odd parts ��ðt; j0Þ. Subtracting the even fromthe odd part one obtains an integral kernel which is invari-ant under j $ j0 as such should be its solution. Thisproperty, supplemented with the explicit form of hn (8),can be used to obtain �1ðg; jÞ in terms of the solution to the

BES equation, �ð0Þ� ðtÞ � ��ðt; 0Þ and �ð0Þ

1 ðtÞ � �1ðt; 0Þ ¼fðgÞ=ð16g2Þ, by putting j0 ¼ 0 and taking j ¼ 1. Hence weobtain

�1ðg; 1Þ ¼ 1

4

Z 1

0dt

�2

et � 1� L� 2

et=2 þ 1

�

���ð0Þ� ð2gtÞ � �ð0Þ

þ ð2gtÞgt

� 2�1ðg; 0Þ�

� L

2g

X1n¼1

�2nðg; 0Þ: (9)

With the orthogonality of Bessel functions we obtain from(6)

�2n ¼Z 1

0

dt

tJ2nðtÞ�þðtÞ; (10)

which can be used to derive for BLðgÞ ¼ 16g2�1ðg; 1Þ thefinal equation

BLðgÞ ¼ 4g2Z 1

0dt

�2

et � 1� L� 2

et=2 þ 1

�

���ð0Þ� ð2gtÞ � �ð0Þ

þ ð2gtÞgt

� 2�ð0Þ1 ðgÞ

�

� 4gLZ 1

0

dt

t�ð0Þþ ð2gtÞ: (11)

At weak coupling one finds with the solution to the BESequation

��ð2gtÞ ¼�1� g2

�2

3

�J1ð2gtÞ þOðg4Þ;

�þð2gtÞ ¼ 4g3�ð3ÞJ2ð2gtÞ þOðg5Þ;(12)

that BLðgÞ, in agreement with [11], has the expansion

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BLðgÞ ¼ �8g4ð7� 2LÞ�ð3Þ þOðg6Þ: (13)

III. STRONG COUPLING EXPANSION

In order to find the strong coupling expansion, according

to [4], we rewrite �1ðg; 1Þ � �ð1Þ1 ðgÞ as

�ð1Þ1 ðgÞ ¼ 1

2

Z 1

0dt

��ð0Þ� ðtÞ

gtðet=2g � 1Þ þ�ð0Þþ ðtÞ

gtðe�t=2g � 1Þ

� �ð0Þ1 ðgÞ

gðet=2g � 1Þ�� ðL� 2Þ

4

�Z 1

0dt

��ð0Þ� ðtÞ

gtðet=4g þ 1Þ þ�ð0Þþ ðtÞ

gtðe�t=4g þ 1Þ

� �ð0Þ1 ðgÞ

gðet=4g þ 1Þ�: (14)

Following [4] we introduce the change of variables

2��ðtÞ ¼ ð1� sechðt=2gÞÞ��ðtÞ � tanhðt=2gÞ��ðtÞ;(15)

and obtain �ð1Þ1 ðgÞ as a function of the first generalized

scaling function �1 and the solution to the BES equationonly

�ð1Þ1 ðgÞ ¼ 1

16g2ðL� 2Þ�1ðgÞ þ �ð0Þ

1 ðgÞðL� 2Þ log2

�Z 1

0dt

�1

4gtð�ð0Þ

þ ðtÞ þ �ð0Þ� ðtÞÞ

þ �ð0Þ1

2gðet=2g � 1Þ�; (16)

where �1 is defined as in [12]. With the expansion of ��

�þðtÞ ¼X1k¼0

ð�1Þðkþ1ÞJ2kðtÞ�2k;

��ðtÞ ¼X1k¼0

ð�1Þðkþ1ÞJ2k�1ðtÞ�2k�1;

(17)

according to [4] we have �0 ¼ 4g�ð0Þ1 and for (k � �1)

�k ¼ � 1

2�ð0Þk þ X1

p¼1

1

gpðc�p �ð2p�1Þ

k þ cþp �ð2pÞk Þ; (18)

with the coefficients c�p given by c�p ¼ Pr�0g

�rc�p;r and

�ðpÞ2m ¼ �ðmþ p� 1

2Þ�ðmþ 1Þ�ð12Þ

; �ðpÞ2m�1 ¼

ð�1Þp�ðm� 12Þ

�ðmþ 1� pÞ�ð12Þ:

(19)

The part proportional to �ð0Þ1 of (16) and the integral over

Bessel functions can be performed to obtain

�ð1Þ1 ðgÞ ¼ 1

16g2ðL� 2Þ�1ðgÞ þ �ð0Þ

1 ðgÞðL� 2Þ log2

þ �ð0Þ1 ð��E � loggÞ � 1

4g

X1k¼1

ð�1Þkþ1

2k�2k

� 1

4g

X1k¼0

ð�1Þkþ1

2k� 1�2k�1: (20)

At strong coupling the first generalized scaling functionhas been analyzed in [12,25] and is given by

�1ðgÞ ¼ �1þOðe��gÞ: (21)

It does not receive perturbative 1=g corrections, but non-trivial exponential corrections in the coupling which arerelated to the mass gap parameter of theOð6Þmodel [9,12].An all-order quantization condition that determines thecoefficients c�p has been computed in [4]. Knowing these

coefficients to a certain order readily gives BLðgÞ to thesame order. To the leading order in 1=g they are given by

cþ1 ¼ � 3 log2

�þ 1

2þOð1=gÞ;

c�1 ¼ 3 log2

4�� 1

4þOð1=gÞ;

(22)

and for twist-two operators L ¼ 2, we find

B2ðgÞ ¼ ð��E � loggÞfðgÞ � 4gð1� log2Þ

��1� 6 log2

�þ 3ðlog2Þ2

�

�þOð1=gÞ: (23)

The virtual correction thus cancels the �E dependence of(1) and with the explicit first two orders strong couplingscaling function fðgÞ ¼ 4g� 3 log2=� and our choice of

g ¼ ffiffiffiffi�

p=4� we predict the string energy up to one loop

E� S ¼ Lþ �L

� ffiffiffiffi�

p4�

; S

���������L¼2

¼� ffiffiffiffi

�p�

� 3 log2

�

�log

4�Sffiffiffiffi�

p þffiffiffiffi�

p�

ðlog2� 1Þ

þ 1þ 6 log2

�� 3ðlog2Þ2

�; (24)

and determine the constant c of [21] to be

c ¼ 6 log2þ � (25)

in agreement with [26]. With the result for B2 given in (23),BL follows straightforwardly from (20) for general, finitevalues of L.With the all-loop quantization condition of [4] we can

determine subleading strong coupling corrections to thevirtual correction B2ðgÞ by solving for the c�p and accord-

ingly performing the sums in (20). For the first terms wefind

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B2ðgþ c1Þ ¼�log

2

g� �E

�fðgþ c1Þ � 4g� 1þ 1

g

K

2�2

� 1

g29�ð3Þ28�3

þ 1

g3

�9�ð4Þ27�4

� K2

27�4

�

� 1

g4

�6831�ð5Þ218�5

� 423K�ð3Þ213�5

�þOð1=g5Þ;

(26)

where c1 ¼ 3 log24� . We made the same shift of the coupling

constant as was suggested in [4].

Subleading spin corrections

The NLIE is written in terms of the roots of the transfermatrix, the so-called holes; see [1,11] for details. For L ¼2, using the explicit expression for the holes [27],

uh ¼ � 2g2 þ q2ffiffiffiffiffiffiffiffiffiffiffiffi�2q2p ;

q2 ¼ � 1

4ðð2Mþ 2þ �2ðgÞÞð2Mþ �2ðgÞÞ þ 2Þ;

(27)

it is possible to further expand the NLIE of [11].Expanding to OððlogMÞ2=M2Þ changes the factor thatmultiplies Kð2gt; 0Þ in (2) to

logMþ �E þ fðgÞ2

logMþ �E

Mþ 1þ B2ðgÞ

2M; (28)

and apart from this the integral equation remains un-changed. In analogy with the above computation we find,for twist-two operators,

�2ðg;MÞ ¼ fðgÞ�logMþ �E þ fðgÞ

2

logMþ �E

M

þ 1þ B2ðgÞ2M

�þ B2ðgÞ þ � � � ; (29)

in perfect agreement with the result from string theory [21]and in agreement with reciprocity relations [28]. It ispossible to formally continue the expansion but the next

correction OððlogMÞ2=M2Þ receives contributions fromwrapping effects [18], and the NLIE therefore does notproduce the correct result.

IV. CONCLUSION

We have computed the finite order correction to thelogarithmic scaling of large spin operators of arbitrarytwist at weak and strong coupling. As we obtained theintegral equation from the infinite-volume NLIE we haveunequivocally shown that the asymptotic Bethe ansatz [29]is capable to determine the first subleading term in thelarge spin expansion of twist operators, unaffected bywrapping effects.At strong coupling and twist L ¼ 2 we can reproduce

the string theory results of [21] and determine the constantthat arises in the one-loop sigma-model calculation. Forthis special value of twist we also determined the sublead-ing correction in spinM up to wrapping order in agreementwith string theory.We have also established the first steps in order to fill the

gap between certain Wilson line expectation values andlogarithms of multiloop gluon scattering amplitudes as thevirtual part B is responsible for the difference between thesubleading singularities in these two quantities. It remainsa challenge to ascertain an operator interpretation of themissing link to the complete collinear anomalous dimen-sion, denoted as Geik; see [22].

ACKNOWLEDGMENTS

We have benefited from discussions with Lance Dixon,Adam Rej, Matthias Staudacher and Valentina Forini. Weare most grateful to Benjamin Basso and GregoryKorchemsky for valuable comments and sharing their in-sights into the all-loop quantization condition with us. Thisresearch was supported in part by the National ScienceFoundation, under Grant No. PHY05-51164, and theSwedish research council (VR).

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