Frequency and wave number dependence of the shear correlator in stronglycoupled hot Yang-Mills theory

K. Kajantie,1,2,* Martin Krssak,3,† M. Vepsalainen,1,‡ and Aleksi Vuorinen3,§

1Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland2Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland

3Faculty of Physics, University of Bielefeld, D-33501 Bielefeld, Germany(Received 6 June 2011; published 13 October 2011)

We use AdS/QCD duality to compute the finite temperature Green’s function Gð!; k;TÞ of the shear

operator T12 for all !, k in hot Yang-Mills theory in the strongly coupled domain. The goal is to assess

how the existence of scales like the transition temperature and glueball masses affect the correlator

computed in the scalefree conformal N ¼ 4 supersymmetric Yang-Mills theory. We observe sizeable

effects for T close to Tc which rapidly disappear with increasing T. Quantitative agreement of these

predictions with future lattice Monte Carlo data would suggest that QCD matter in this temperature range

is strongly interacting.

DOI: 10.1103/PhysRevD.84.086004 PACS numbers: 11.25.Tq, 05.70.Ce, 11.10.Wx

I. INTRODUCTION

There is a standard framework for computing Green’sfunctions in supersymmetric N ¼ 4 Yang-Mills theoryusing AdS/CFT duality [1–7]. In [8], this was carried outexplicitly over the entire !, k plane for an operator cou-pling to a scalar field. The specific goal was to compute thereal static (! ¼ 0) Green’s function both directly and byintegrating over !. The result was then Fourier trans-formed to a spatial Green’s function and compared witha computation in next-to-leading order QCD perturbationtheory [9,10], see also [11,12].

The previous calculation was carried out in a scalefreetheory. The purpose of this article is to see what effectsarise from �QCD, i.e., scale invariance breaking in SU(Nc)

Yang-Mills theory. Physically, this manifests itself in theexistence of glueballs and a first-order phase transition.1

The predictions are quantitative and it is suggested thattheir comparison with lattice Monte Carlo data may give abetter handle on verifying the strongly interacting nature ofQCD matter in the temperature range Tc . . . 10Tc. There isdefinite phenomenological evidence for this (magnitude ofpressure, small shear viscosity, jet quenching), but nodefinite nonperturbative test [14]. In fact, how can thesystem be strongly interacting while so many latticeresults [15] agree with ideal gas? One concrete test[5,16] would be to verify if the shear spectral functionhas a transport peak at ! ¼ 0 or not; our calculationaddresses this question.

There are many candidate gravity duals of SU(Nc)Yang-Mills theory. The dual we want to apply is

improved holographic QCD (IHQCD) as developed in[17–20]. This is a bottom-up Anti de Sitter (AdS)gravityþ dilaton model which combines elegantlyasymptotic freedom in the ultraviolet (UV) with a givenbeta function and confinement in the infrared (IR).IHQCD has to be solved numerically and we can so faronly compute the imaginary part of the Green’s functionthe case k ¼ 0 as a function of !. Since we want to treatthe entire !, k plane, and for increased transparency, weshall in this paper also use a simplified version ofIHQCD for which the gravity background is analyticallyknown [18]. It contains qualitatively the relevant featuresof IHQCD and computations in the overlapping regionconfirm that it gives good idea of what one can expect ofscale invariance breaking effects on Green’s functions inSUðNcÞ Yang-Mills theory.It should be emphasized that IHQCD is not a full

gravity dual of Y-M theory in the weak coupling domain,it is only constructed to describe a limited range ofproperties: a coupling running logarithmically to zeroin the UV, its beta function of asymptotically free formand correct bulk thermodynamics at large T. However, it,for example, gives �=s ¼ 1=ð4�Þ at all T > Tc. Relatedto this, it leads to a monotonically increasing shearspectral function �ð!Þ and thus will not contain thetransport peak at small !, characteristic of the weaklycoupled perturbative domain. The full dual of Y-M theoryis rather a string theory, or in the supergravity language,would contain an infinite set of higher derivativecorrections.IHQCD contains the background metric and the scalar

dynamically coupled, and fluctuation analysis should becarried out taking this into account [21–23]. The transversetensor mode [T12 with k ¼ ð0; 0; kÞ, also called shear modeor scalar mode] does not couple to other ones and we thusconsider only this case and study the retarded Green’sfunction

*keijo.kajantie@helsinki.fi†krssak@physik.uni-bielefeld.de‡mikko.vepsalainen@helsinki.fi§vuorinen@physik.uni-bielefeld.de1A computation in a nonconformal theory but without a phase

transition has been carried out in [13].

PHYSICAL REVIEW D 84, 086004 (2011)

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Gð!;k;TÞ¼Z 1

�1dtd3xei!t�ikx3hi�ðtÞ½T12ðt;xÞ;T12ð0;0Þ�iT

¼Z 1

�1d!0

�

�ð!0;kÞ!0 �!� i�

: (1)

On scales and notation: in IHQCD, the QCD-like scale� arises as an integration constant of the dynamical dila-ton field equation, giving the z-dependence of the dilatonfield, z being the fifth dimension. In the model studiedhere, it is introduced as a constant of dimension 1 by the zdependence of the ansatz (15). The spin 2 glueball spec-trum (squared) then will be 4�2ð2þ nÞ, n ¼ 0; 1; 2; . . .(analytical), and the transition temperature Tc ¼ 0:400�(numerical result). Energy (and wave number) can bemade dimensionless by scaling with either T or � andwe shall define

! ¼ �T ~! ¼ 2�! � 5:00Tc!; (2)

similarly for k. As the dimensionless fifth coordinate, weshall use

y ¼ �2z2: (3)

Unless specifically noted, results for the retarded Green’sfunction G are given without writing explicitly the dimen-sionless factorL3=ð4�G5Þ, whereL is the AdS radius andG5 the 5 dim gravitational constant.

II. GRAVITY DUAL

A. IHQCD

IHQCD is based on the gravityþ scalar action, in stan-dard notation,

S ¼ 1

16�G5

�Zd5x

ffiffiffiffiffiffiffi�gp �

R� 43ð@��Þ2 þ Vð�Þ

�

�Z

d4xffiffiffiffiffiffiffiffi��

p �2K þ 6

LþL

2Rð�Þ

��: (4)

The background, with boundary at z ¼ 0, is chosen to be

ds2 ¼ b2ðzÞ��fðzÞdt2 þ dx2 þ dz2

fðzÞ�; (5)

where bðzÞ, fðzÞ, �ðzÞ are solutions of the first-ordersystem

_W ¼ 4bW2 � 1

f

�W _fþ 1

3bV

�; W ¼ � _b=b2; (6)

_b ¼ �b2W; _ ¼ 3

2

ffiffiffiffiffiffiffiffib _W

p; (7)

€f ¼ 3 _fbW; (8)

see Appendix A of [19]. The coupling of the boundary

theory is associated with ðzÞ ¼ e�ðzÞ, and a specific com-bination of the fields, arising in connection of solving

Einstein’s equations ( _b ¼ db=dz),

ðÞ ¼_

_b=b; (9)

is associated with the beta function of the boundary theory.An important general property of the solutions is that b andf are monotonically decreasing and is monotonicallyincreasing as functions of z.The solutions have to satisfy the following properties. In

the UV, z ! 0,

bðzÞ ! Lz: (10)

Imposing asymptotic freedom to leading order, one thenhas,

ðÞ ¼_

_b=b¼ �b0

2 ¼ �zd

dz; (11)

implying that the leading term of ðzÞ near the boundary is

b0ðzÞ ¼ 1

L; L � log

1

�z; (12)

� ¼ constant of integration. This is how the dimensionaltransmutation of SUðNcÞ YM theory appears in the gravitydual. Note that on the boundary ! 0 but � ! �1.Including more powers of in the beta function, one can

derive expansions in for all the fields. For the record, wehave collected some of them in Appendix A.In the IR, for z ! 1, the crucial confinement criterion is

that out of the many different solutions of Einstein’s Eq. (8),one should choose the one satisfying

ðÞ ¼ � 3

2

�1þ 3ð�� 1Þ

4�

1

logþO

�1

log2

��: (13)

The parameter �> 1 here is connected with the IRbehavior of all the other background functions so that, forexample,

bðzÞ!e�ð�zÞ�ð�zÞp; �¼number��; p¼number:

(14)

The appropriate IR relations are summarized inAppendix B. We shall next choose p ¼ �1 to extend theform to all z and � ¼ 2 to have a glueball spectrum withm2 � integer [24].

B. Model: Approximate version of IHQCD

Our model for simulating the essential properties ofIHQCD is defined by a simple ansatz [18] for bðzÞ:

bðzÞ ¼ Lzexp

�� 1

3�2z2

�¼ L�ffiffiffi

yp exp

�� 1

3y

�; (15)

where we often set L ¼ 1 and where we use the dimen-sionless distance variable y in (3).Starting from (15), one can derive all the other bulk

fields. First, integrating (7) leads to

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

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�0ðyÞ ¼ 0ðyÞðyÞ ¼

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 9

2y

s; (16)

which integrates to

ðyÞ=0 ¼ exp

0@12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyðyþ 9

2

s 1A0@ ffiffiffi

yp þ

ffiffiffiffiffiffiffiffiffiffiffiffiyþ 9

2

s 1A9=4

: (17)

Now that ðyÞ is known, the function as specified byIHQCD is obtained:

ðÞ ¼ �0ðyÞb0ðyÞ=bðyÞ ¼ �3

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 9=ð2yÞp1þ 3=ð2yÞ : (18)

In the IR, this goes like

ðÞ¼�32

�1þ 3

4yþ . . .

�¼�3

2

�1þ 3

8logþ . . .

�; (19)

and thus satisfies explicitly the IHQCD confinement

criterion. In the UV, ðyÞ goes to ð0Þð1þ 3ffiffiffiffiffiffiffiffiy=2

p þ9y=4þ . . .Þ, i.e., this ansatz does not produce a ðyÞ van-ishing logarithmically at the boundary. The far UV regionis inessential for the treatment of thermodynamics inIHQCD [25] and the absence of the ‘‘spurious logarithms’’[26] simplifies the AdS analysis significantly.

To get fðzÞ, we note that (8) integrates to (C is a constant)

_fðzÞ ¼ C=b3ðzÞ: (20)

The greatest utility of the ansatz (15) lies in that this isintegrable in closed form. Using fð0Þ ¼ 1, fðyhÞ ¼ 0, onehas

fðyÞ ¼ 1� ðy� 1Þey þ 1

ðyh � 1Þeyh þ 1¼ yh � 1� ðy� 1Þey�yh

yh � 1þ e�yh

¼ 1

yh � 1þ e�yh

X1n¼1

yh þ n� 1

n!ð�1Þnþ1ðyh � yÞn:

(21)

Finally, the scalar potential VðÞ can be solved from (6)with the answer (keepingL here to show where the dimen-sions of V come from)

V½ðyÞ� ¼ 12

L2� eð2=3Þy

��13y

2 þ 56yþ 1

�fðyÞ

��12 þ 1

3y

�yf0ðyÞ

�: (22)

In thevacuum f ¼ 1,f0 ¼ 0 and, using (17), the potential atlarge is seen to behave as

V½ðyÞ� ¼ 12

L24=3

ffiffiffiffiffiffiffiffiffiffilog

p ½1þOð1= logÞ�; (23)

again in agreement with the confining potential criteria inIHQCD. A conceptual problem with VðÞ in (22) is itsdependence on yh. However, what matters for solving thebackground is the range 0< y< yh, and in this range the

potentials computed from (22) for f ¼ 1 and f ¼ fðyÞ arevery close to each other. They differ maximally at y ¼ yhand this difference is maximally 4.7% if yh ¼ 0:79.

1. Thermodynamics

To assess the relevance of the model bðzÞ in (15), we firstwork out its thermodynamics. From 4�T ¼ � _fðzhÞ and(21), one obtains the entire thermodynamics parametrizedby yh (including now the factor L in bðzÞ):

TðyhÞ ¼ �

2�

y3=2h

yh � 1þ e�yh� 2�T; (24)

sðyhÞ ¼ L3

4G5

�3y�3=2h e�yh ; (25)

pðyhÞ ¼Z 1

yh

dx½�T0ðxÞ�sðxÞ: (26)

Using (17), we may replace yh by h, which is used in [18].The overall structure of T is such that it diverges both forsmall yh (�T ! �=

ffiffiffiffiffiyh

p, the big black hole limit) and for

large yh (�T ! 12�

ffiffiffiffiffiyh

p, the small BH limit). Thus, TðyhÞ

has in between a minimum which lies at

yh ¼ 2:149; Tmin ¼ 0:3962�: (27)

For T > Tmin, the system is in a deconfined plasma phasewith pressure �N2

c , but actually in a part of this range,Tc > T > Tmin, the plasma phase is metastable. The stablephase for 0< T < Tc is a thermal glueball gas phase withpressure �N0

c , i.e., in this approach p ¼ 0. The transitiontemperature Tc is where the pressure vanishes, whichhappens at

yh¼yc¼1:6863; Tc¼0:4000�¼1:0096Tmin: (28)

In the UV, where yh ! 0 and �T ! �=ffiffiffiffiffiyh

p,

p

T4 ! L3

4G5

�3

4; (29)

independent of the IR scale �. This can be used to fixthe dimensionless combination L3=ð4G5Þ, which thenbecomes �N2

c . For example, in the present AdS/QCDcontext, one may match at some large T � Tc to thepressure of weakly interacting gluon gas, p=T4 ¼�2N2

c=45. Weakly interacting N ¼ 4 supersymmetricYM matter has p=T4 ¼ �2N2

c=6 while the strongly inter-acting one has p=T4 ¼ �2N2

c=8 with

L3

4�G5

¼ N2c

2�2: (30)

The resulting thermodynamics is plotted in Fig. 1. Theinteraction measure is broader than that of pure YM atlarge Nc and the latent heat smaller. The latent heat isrelated to the large T asymptotics by

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L¼ �cT4c

¼ scT3c

¼32

3

ðyc�1þe�ycÞ3y6c

e�yc�

T4

��������T!1: (31)

The smallness of L is due to the fact that the yc-dependentterm at yc ¼ 1:6863 is already considerably reduced fromits maximum value 1=8. There is no parameter with whichone could tune L to a larger value.

Finally, the sound velocity is given by

c2s ¼ 1

3þ 2yh

3� yh � ð3þ 2yhÞe�yh

yh � 1þ e�yh

¼ 1

3� 1

ð3�TÞ2 þ . . . (32)

At Tc, c2s ¼ 0:024. Note that the Chamblin-Reallbackground used in [13] has a constant sound velocityand does not have a phase transition, since for it always� ¼ constant� p.

2. Glueball masses

Glueball masses are obtained as poles of Green’s func-tion and we shall later see how this goes through.Effectively, this leads to a Schrodinger-type equation,

� c 00ðzÞ þ VSchrðzÞc ðzÞ ¼ m2c ðzÞ; (33)

where, for the background (5) and for scalar glueballs,

VSchr¼3

2

� €bbþ

_b2

2b2

�þ €X

Xþ3

_b

b

_X

X; X�ðÞ

3; (34)

for tensor glueballs the X-terms are missing. With theabove explicit expressions, it is trivial to evaluate thepotentials. Scaling out � by writing V=�2 ! V, �z ! z,the tensor potential is

VtðzÞ ¼ 15

4z2þ 2þ z2; (35)

and the scalar one

VsðzÞ¼VtðzÞþ 1

ð3þ2z2Þ2ð9þ2z2Þ2

���2187

z2�5346�1224z2þ216z4þ48z6

�: (36)

The tensor equation is analytically solvable [24] and leadsto the mass spectrum

m2T=�

2 ¼ 4ðnþ 2Þ; n ¼ 0; 1; 2; . . . (37)

Solving numerically the Schrodinger equation, one findsthe scalar glueball mass spectrum

m2S=�

2 ¼ 6:27; 10:27; 14:25; . . . (38)

The lowest tensor and scalar states are in the ratiomT1=mS1¼1:13 (in the full IHQCD this ratio is 1.36[20]), but the m2 spacing is about 4�2 also for scalars.In the full IHQCD, the ratiomS2=mS1 ¼ 1:61, while here itis 1.28.

III. FLUCTUATION EQUATION

To calculate correlators [1–4] of the operator T12 in thebackground (5), one has to solve the fluctuation equation

€hþ d

dzlogðb3fÞ _hþ

�!2

f2�k2

f

�h� €hþP _hþQh¼0: (39)

Here, h � hKðzÞ, K ¼ ð!;kÞ, and

hðx;zÞ¼Z d4K

ð2�Þ4e�i!tþik�xh0ðKÞhKðzÞ; hKð0Þ¼1; (40)

T4

3 p

T4

3 p

T4

T Tc

b z1

ze

2 z2 3

1 2 3 4 5

0.2

0.4

0.6

0.8 T4

103 p

T4

T Tc

0.995 1.005

0.02

0.04

0.06

FIG. 1 (color online). Thermodynamics from the model bðzÞ. The curves are normalized to their T ! 1 asymptotic values, whichare �N2

c . At Tc, there is a weak first-order transition with �c=T4c ¼ 0:06. The right panel shows the region around Tc with the

supercooled (dashed) and unstable (dotted) phases. The curve for pressure is multiplied by 10 for clarity. The lattice data for theinteraction measure ð�� 3pÞ=T4 at large Nc is very similar to the curve ð�� 3pÞ=T4 here with the difference that the maximum (hereat T ¼ 1:5Tc) is pulled to T ¼ Tc so that the latent heat is �c=T

4c ¼ 0:3.

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

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is the fluctuation of the 12 component of the backgroundmetric: g12 ¼ b2ðzÞhðt; x3; zÞ. This tensor fluctuation doesnot mix with other ones. In IHQCD, the scalar one doesand the fluctuation equation permitting computations of theGreen’s function of the scalar operator F2

�� will contain

more terms [21–23].The characteristic exponents of (39) at z ¼ zh are

i!= _fh. Equation (39) should be solved so that at thehorizon (infalling boundary condition, di are calculablecoefficients)

hKðz ! zhÞ ¼ ðz� zhÞi!= _fh½1þ d1ðz� zhÞþ d2ðz� zhÞ2 þ . . .� (41)

and that at the boundary, hKð0Þ ¼ 1 (correct normaliza-tion). These solutions will satisfy

h�KðzÞ ¼ hKðzÞ: (42)

Given a solution, a computation along the lines of [27]shows that the retarded Green’s function is obtained from

GðKÞ ¼ 1

16�G5

½fb3 _hKh�K þ ðcounter termsÞ � hKh�K�;(43)

where the hKh�K ¼ hKhK terms with unspecified coeffi-

cient are real.For the imaginary part of G, the spectral function, the

situation is particularly simple:

ImGðkÞ � �ð!; kÞ ¼ 1

16�G5

fb3Im _hKh�K

¼ 1

16�G5

fb3WðhK; hKÞ=ð2iÞ; (44)

where one has noted that hK and hK are two independentsolutions of (39) and that their Wronskian arises when ImGis evaluated. However, the Wronskian of any two linearlyindependent solutions of (39) is, integrating _W=W ¼ �P,

Wð�1; �2Þ � �1_�2 ��2

_�1 ¼ W0ðKÞb3f

; (45)

where W0 is a z-independent constant (but will depend onK ¼ !, k). Thus, the result in (44) is a ratio of twoWronskians and independent of z.

To find the constant W0 for the Wronskian WðhK; hKÞ,we simply evaluate it in the limit z ! zh using the expan-sion (41) and find

WðhK; hKÞ ¼2i!

fb3b3ðzhÞ: (46)

The result thus automatically contains the entropy density,which in IHQCD is

sðTÞ ¼ A

4G5

¼ 1

4G5

b3ðzhÞ: (47)

However, so far this is for a solution normalized as in (41),which leads to some value hKð0Þ at the boundary.Correcting for the normalization, the result is

�ð!; kÞ ¼ 1

4�sðTÞ !

jhKð0Þj2: (48)

A practical note on the application of this formula: in ithKðzÞ arises from numerical integration which cannot beextended to z ¼ 0 since this is a singular point of theequation. However, in (48) hKðzÞ has to be evaluatedprecisely at z ¼ 0 and this value has to be found byextrapolation. Numerical effect is significant at large !,

where jhKð0Þj � 1=!3=2.We shall also need the real part, for example, the static

Green’s function Gð0; kÞ is purely real. For the real part,one has to make further assumptions, corresponding tosubtractions in the dispersion relation (1). The standardprocedure here [3] is to expand the solution hKðzÞ com-puted with the boundary condition (41) in terms of the twoindependent solutions at z ¼ 0:

hKðzÞ ¼ AðKÞ�uðz; KÞ þ BðKÞ�nðz; KÞ; (49)

where

�n ¼ z4ð1þ c1z2 þ c4z

4 þ . . . :Þ;�u ¼ 1þ f1z

2 þ f3z6 þ . . .þ c logz�nðz; KÞ: (50)

In IHQCD, the expansion coefficients will contain loga-rithmic UV coefficients of the type in Appendix A [17].Noting that the normalization factor hKð0Þ in (48) is AðKÞ,the result (43) can be evaluated using

hK _hKz3jhKð0Þj2

¼ 1

z3jAj2 ðA�u þ B�nÞðA _�u þ B _�nÞ

¼ 2f1z2

þ 4c logzþ cþ 2f21 þ 4BðKÞAðKÞ þOðz2Þ

¼ !2 � k2

2z2� ð!2 � k2Þ2

16ð4 logz� 1Þ

þ 4BðKÞAðKÞ þOðz2Þ; (51)

where the last form applies for conformal expansions (50).The scheme is to neglect the real divergent termsþconstant in (51) and to evaluate the entire Green’s functionfrom

GðKÞ ¼ L3

4�G5

BðKÞAðKÞ ;

BðKÞAðKÞ ¼ �WðhK;�uÞ

WðhK;�nÞ ¼ � hK _�u ��u_hK

hK _�n ��n_hK:

(52)

Equivalently, one may say that this is a way to derive thecounterterms in (43) [28]. In [8], this scheme was tested byevaluating the purely real static Green’s function Gð0; kÞ

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also by an !-integral over the complex Gð!; kÞ. It alsogave correctly the real and imaginary parts on the lightcone, for Gðk; kÞ.

SinceGðKÞ in (52) is given as a ratio of twoWronskians,it is independent at what z it is evaluated, to the extent hK,�u, �n are correct solutions. Since hK is integrated nu-merically, it certainly is a solution. In practice, the accu-racy and constancy then depends on how many terms areincluded in small-z expansions of �u, �n. If the numericalintegration of hK from (39) is extended down to somesmall �, the ratio of Wronskians can be evaluated atthis �. This is to be contrasted with (48) in which hKð0Þhas to be extrapolated to z ¼ 0 for accuracy. Of course,(48) only gives the imaginary part while both the real andimaginary parts are obtained from (52).

A. Equations for the model

Next, we specialize to the model defined in Sec. II B. Interms of y ¼ �2z2, the fluctuation Eq. (39) becomes

�00KðyÞ þ PðyÞ�0

KðyÞ þQðyÞ�KðyÞ ¼ 0; (53)

where

PðyÞ ¼ d

dylogðb3f ffiffiffi

yp Þ

¼ � 1

y� 1þ y

y� 1þ ð1� yhÞeyh�y ; (54)

QðyÞ ¼ !2

yf2ðyÞ �k2

yfðyÞ ; (55)

where the dimensionless !, k were defined in (2).Equation (53) is to be solved so that at the horizon (the

y� yh term is for k ¼ 0)

�KðyÞ ¼ ðy� yhÞp�1þ pð1þ yhÞ � 2pð2þ yhÞ

2yhð1þ 2pÞ ðy� yhÞ

þOððy� yhÞ2Þ�; (56)

where

p ¼ �i!

4�T¼ i

!ffiffiffiffiffiyh

pf0ðyhÞ ¼ �i

!

y3=2h

ðyh � 1þ e�yhÞ: (57)

For correct normalization, the solution is divided by�Kð0Þ.The Green’s function now is

GðKÞ¼ 2�

16�G5

½fb3 ffiffiffiy

p�0

K��Kþcounter terms ��K��K�:(58)

For the spectral function, one again needs theWronskian ofthe two independent solutions at the horizon,

Wð�K;�KÞ ¼

2i!

fb3ffiffiffiy

p b3ðyhÞ; (59)

and the result (48) is unchanged. For the real part andevaluation of the counterterms in (58), the expansionsaround y ¼ 0 have to be worked out. One writes

�KðyÞ ¼ AðKÞ�uðy; KÞ þ BðKÞ�nðy; KÞ; (60)

where

�nðy; KÞ ¼ y2�1þ 1

3ð2þ k2 � !2Þyþ . . .

�; (61)

�uðy;KÞ¼1þð!2�k2Þy�1

2½ð!2� k2Þ2�ð!2� k2Þ�

� logy ��nðyÞþOðy3Þþ . . . ; (62)

(note no y2 term; it is only in �n). The real counter termsneglected now are

2�4

�!2 � k2

y� 1

2½ð!2 � k2Þ2 � ð!2 � k2Þ�ð2 logyþ 1Þ

�;

(63)

and, solving A and B with Wronskians,

GðKÞ ¼ L3

4�G5

��4 BðKÞAðKÞ : (64)

Now, the correct dimension 4 comes from �4. In units of�T, using (24):

GðKÞð�TÞ4 ¼ L3

4�G5

� 16ðyh � 1þ e�yhÞ4y6h

� BðKÞAðKÞ : (65)

We are interested in the high T phase and, as was derivedearlier, this means T�Tc¼0:4000� or yh < yc ¼ 1:6863.

B. Vacuum spectral function and the limitat large ! or k

To obtain the vacuum Green’s function of the model, weput f ¼ 1 in (53) and obtain the equation

y�00ðyÞþð�1�yÞ�0ðyÞþ K2�ðyÞ¼0; K2� !2� k2:

(66)

This also gives the leading terms of the !; k � �; T limitof the solutions of (53). Understanding this limit analyti-cally is important for numerical accuracy. This equation isthe equation for the confluent hypergeometric functionwith a solution �1 growing like a power at large y:

�1ðyÞ ¼ y2Uð2� K2; 3; yÞ! yK

2½1þ K2ð2� K2Þy�1 þOðy�2Þ�: (67)

Its small y expansion is

�1ðyÞ¼ 1

�ð2�K2Þ�1þK2yþ1

2K2ð1�K2Þ

��logyþc ð2�K2Þþ2�E�3

2

�y2þOðy3Þ

�: (68)

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The second solution �2ðyÞ¼y2Lð2ÞK2�2

ðyÞ!eyy�K2þ1þ...,

grows exponentially and is unacceptable. If we nowdecompose the solution �1 as in (60), B is immediately

read as the coefficient of the y2 term and A¼1=�ð2�K2Þ.The vacuum (this is vacuum since no T is involved)Green’s function then is

GvacðKÞ¼�4 �12K2ð1� K2Þ½c ð2� K2Þþ2�E� 3

2�: (69)

Since

c ðzÞ ¼ ��E þ X1m¼0

�1

mþ 1� 1

mþ z

�; (70)

the result in (69) has poles at

!2 � k2 ¼ 4�2ðmþ 2Þ: (71)

This reproduces the tensor glueball mass spectrum in (37),now derived as poles of Green’s function. Taking ! !!þ i� and plotting ImGvac from (69) for finite but small �,one finds Gaussian peaks at glueball masses. In the limit� ! 0 [29],

�vac� ImGvac

¼ �

32ð!2�k2Þð!2�k2�4�2ÞX1

m¼0

�mþ2�!2�k2

4�2

�:

(72)

We shall later use this as a baseline to assess the magnitudeof finite T effects [see Eq. (79)].

The above was for vacuum, f ¼ 1. The asymptotic large!, k limit for f � 1 can be argued as follows. Taking first

!> k> 0, the large K2 > 0-limit of c in (69) is

c ð2� K2Þ ¼ � cotð�K2Þ þ logK2 � 3

2K2� 13

12K4þ . . . :

(73)

Note now that � cotðxþ iyÞ ! �i� if y ! 1. If now

the analytic continuation is performed so that also ImK2 !1, then (69) and (73) give

ImGðKÞ�4

¼ �

2½ð!2 � k2Þ2 � ð!2 � k2Þ þOð1Þ�: (74)

Both terms are very clearly seen in numerics, but we havebeen unable to fit a constant (or possibly a logarithmic)term nor compute it analytically.

In contrast, in the static ! ¼ 0 case, K2 ¼ �k2,the constant term (and higher corrections Oð1=k2Þ)can be evaluated. Now Eq. (69) is purely real. Expandingc ð2þk2Þ!logk2þ3=ð2k2Þ�13=ð12k4ÞþOð1=k6Þ in (69)gives a constant term �5=24. However, this is not theentire constant term, which is crucial for numerical analy-sis. A careful analysis (see Appendix C) permits one tocompute it with the result (C9), which is the same aswriting (69) in the form

B

A! � 1

2k2ð1þ k2Þ

�c ð2þ k2Þ þ 2�E � 3

2

�

þ e�yh

20ðyh � 1þ e�yhÞ : (75)

Thus, there is an overall constant term �5=24þe�yh=ð20ðyh�1þe�yhÞÞ��5=24þT4

c=ð5TÞ4. This is clearlyseen in numerics.In the thermal theory with f � 1, the glueballs form a

glueball gas with pressure OðN0cÞ and AdS/QCD duality

says little about its properties [30], it is the phase at p ¼ 0in Fig. 1. In the high T phase at T � Tmin, the deltafunction peaks are not seen but still some structure remains(see Figs. 7 and 8 below). The broadening of hadronicstates in a thermal ensemble has been studied extensivelyin connection with quarkonium physics [31].

IV. NUMERICAL RESULTS

We are interested in computing the finite temperatureretarded Green’s function of T12 in a QCD-like stronglyinteracting theory, for various frequencies and wave num-bers k ¼ ð0; 0; kÞ. At large T, say, T > 10Tc, one expectsthe theory to approach a conformal theory with Green’sfunctions explicitly plotted in [4,5,8]. In the range Tc to10Tc, there are important nonconformal effects, reflectedin the large value of the interaction measure ð�� 3pÞ=T4.This is the range of T to be studied here.We remind that the results for G or � ¼ ImG are given

without an overall factor of L3=ð4�G5Þ.

A. Spectral function for k ¼ 0

This is clearly a very interesting range due to its relationto viscosity and lattice determinations thereof [32–38].Consider first the results for the model (15). As discussedin Sec. III B, the large-! limit is, measuring both G and !in units of �T,

�asð!;kÞð�TÞ4 ¼ ImGasð!;k;TÞ

ð�TÞ4

¼ �

32

�ð ~!2� ~k2Þ2�25:0T2

c

ð�TÞ2 ð ~!2� ~k2Þ

��ð!�kÞ:

(76)

This is an asymptotic formula and both terms are clearlyseen in numerics. However, we do not know the terms oflower order in !2 � k2, but (76) should be applicable onlyfor !2 > k2 þ 25:0T2

c , it is negative at smaller !.Results for the model (15) are shown in Fig. 2 together

with the leading asymptotic behavior �!4=32. The spec-tral function is shown both as such or divided by ! so thatthe value at ! ¼ 0 is essentially the shear viscosity. Oneobserves a sizeable effect near Tc but already at 2Tc thecurves are very close to the conformal situation, approxi-mated by the 10Tc curve. The main T-dependent effect is

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that contained analytically in the asymptotic behavior (76).Even for T � Tc, the nonconformal effects disappear at! * 3Tc; remember that the lightest scalar glueball massis � 6:2Tc.

Consider then the results, shown in Fig. 3, computedfrom full numerics2 of IHQCD, using the formula (48).This gives only the imaginary part; due to the complicatedlogarithmic structure of IHQCD (see Appendix A) it willbe more complicated to evaluate the Wronskians in (52)and compute the real part also. The overall pattern at k ¼ 0is very similar. An interesting detail is that in IHQCD theapproach to the asymptotic limit is rather slow, possiblyrelated to the UV logarithms.

The imaginary time T12 correlator at k ¼ 0 can now beintegrated from

Gð�;0;TÞ¼Z 1

0

d!

��ð!;0;TÞcosh½ð1�2T�Þ�2!�

sinhð�2!Þ ; (77)

with everything in units of �T. Note first that in theN ¼ 4 conformal case the answer is [8]

Gð�;0;TÞ¼ 3

4�5½�ð5;T�Þþ�ð5;1�T�Þ�þGðT�Þ; (78)

where the first terms come from the integration of �!4=32and G from �ð!; 0Þ � �!4=32. The thermal part G

varies very slowly from Gð12Þ ¼ 0:0710 at the symmetry

point to the value 0.13 at � ¼ 0. At T� ¼ 12 the vacuum

part, i.e., the � function terms, contribute 93�ð5Þ=ð2�5Þ ¼0:1576 and dominate totally at smaller �.The outcome in the nonconformal cases is shown

in Fig. 4. The T ¼ 10Tc curve is very close to the con-formal curve (78). A particularly interesting point isthe middle one, � ¼ 1=ð2TÞ. Figure 5 shows the corre-sponding integrand of (77), in which large ! values arestrongly suppressed, Fig. 6 shows the resulting value ofGð� ¼ 1=ð2TÞÞ.To assess the significance of the result, it is very useful to

generate a comparison curve by inserting the vacuumspectral function (72) with delta function peaks at theposition of glueball masses to the representation (77) at� ¼ 1=ð2TÞ. The answer is (� ¼ 2:5Tc)

Gvacð�¼ 1=ð2TÞÞð�TÞ5 ¼ 1

2

�2:5Tc

�T

�5 X1m¼0

ffiffiffiffiffiffiffiffiffiffiffiffimþ2

p ðmþ1Þsinhð2:5 ffiffiffiffiffiffiffiffiffiffiffiffi

mþ2p

Tc=TÞ:

(79)

The resulting curve is also plotted in Fig. 6, the limit atlarge T is 93�ð5Þ=ð2�5Þ, the same as for the vacuum curvein (78). One sees that the imaginary time correlator gen-erated by the vacuum spectral function behaves similarly to

ImG , 0; T

T 4

T

T 10 Tc

2 Tc 1.5 Tc

32

4

T 4

Tc

1 2 3 4

0.01

0.1

1

10

ImG , 0; T

T 3

T

T 10 Tc2 Tc

1.5 Tc

32

3

T 3

Tc

1 2 3 4

0.050.10

0.501.00

5.00

FIG. 2 (color online). The spectral function ImGð!; 0;TÞ=ð�TÞ4 in units ofL3=ð4�G5Þ plotted vs !=ð�TÞ (left panel) or divided by!=ð�TÞ (right panel), for the model (15). The dashed line is the leading large-! limit.

Tc

2Tc

T

T 10 TcImG , 0; T

T 4

32

4

T 4

0 1 2 3 4

0.01

0.1

1

10

T 10 Tc

2 Tc

Tc

ImG , 0; T

T 3

T

0 1 2 3 4

0.050.10

0.501.00

5.00

FIG. 3 (color online). The spectral function ImGð!; 0;TÞ=ð�TÞ4 in units ofL3=ð4�G5Þ plotted vs !=ð�TÞ (left panel) or divided by!=ð�TÞ (right panel) in exact numerics of IHQCD.

2The parameters of the scalar potential were otherwise thesame as those in [20] but V0 ¼ 1 instead of V0 ¼ 0:04128; thismakes numerics simpler and does not affect the results.

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

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the genuine finite T-correlator, finite T just enhances themagnitude. The vacuum correlator has nothing like a trans-port peak at ! ¼ 0, it is identically zero up to the lowest

glueball mass at 2ffiffiffi2

p�. Still the outcomes are very similar.

We do not attempt a detailed comparison with latticeMonte Carlo data. However, the results are in qualitativeagreement with the plots in [34]. An agreement confirmedby careful analysis would support the present picture ofstrongly interacting QCD matter, but Fig. 6 illustrates thedifficulty of making definite conclusions.

Since one knows that the viscosity prediction holdsindependent of T, it is useful to present the data sothat this is explicitly seen. Data for the dimensionlessquantity

4�

sðTÞ�ð!; 0;TÞ

!; (80)

computed for the model (15) in the region of small !, arepresented in Fig. 7, left panel. The normalization is suchthat the standard viscosity prediction �=s ¼ 1=ð4�Þcorresponds to the value 1 at ! ¼ 0. The right panel ofFig. 7 shows the spectral function with the asymptoticbehavior (76) subtracted. The subtraction is reliable onlyfor ! * 5:0Tc=ð�TÞ.Finally, Fig. 8 shows the subtracted Green’s function

plotted in nonthermal units, vs !=ð2�Þ, computed forthe model (15). This is what remains of the infinite set ofdelta function peaks in (72). One may try to define aglueball mass at finite temperature by the positionof the first peak in this curve. This grows very accuratelylinearly with T:

m GðTÞ ¼ mGðTÞmG=

ffiffiffi2

p ¼ 1:016T

Tc

� 0:241; (81)

where the lowest tensor glueball mass is mG ¼ 2ffiffiffi2

p�.

G ; T

T 5

2 TT 10 Tc

T Tc

0.2 0.4 0.6 0.8 1.0

1

10

100

1000G ; T

T 5

2 T

T 10 Tc

T Tc

0.5 0.6 0.7 0.8 0.9 1.0

0.5

1.0

1.5

2.0

FIG. 4 (color online). The imaginary time Green’s function Gð�; k ¼ 0Þ computed from (77). The right panel shows the region near� ¼ 1=ð2TÞ on a linear scale, the dashed curves come from full numerical evaluation of IHQCD.

, 0; T

sinh2

T

3.8 Tc

1.4 Tc

Tc

0 2 4 6 80.00

0.05

0.10

0.15

T

, 0; T

sinh2

Tc

1.4 Tc

3.8 Tc

0 2 4 6 80.000.020.040.060.080.100.120.14

FIG. 5 (color online). The weighted spectral distribution, integrand of (77), at the middle point � ¼ 1=ð2TÞ for both the model (15)(left) and full numerics of IHQCD (right), at various temperatures.

T Tc

G 12 T

; T

T 5

Model

IHQCD

Conf

vac,model

0 1 2 3 4 50.00

0.05

0.10

0.15

0.20

0.25

FIG. 6 (color online). The temperature dependence ofGð1=ð2TÞÞ for both the model (15) and full numerics ofIHQCD. The arrow shows the value for the conformal N ¼ 4theory [8] The dashed curve shows the plot of (79), generatedfrom the T-independent vacuum spectral function (72) via thespectral representation (77).

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B. Real static correlator, ! ¼ 0

The static correlator is defined by

GðrÞ ¼Z

0d�hT12ð�;xÞT12ð0; 0ÞiT ¼

Z

0d�Gð�;xÞ

¼ 1

2�2r

Z 1

0dkk sinðrkÞGðkÞ (82)

and can also be related to the !-dependent spectralfunction by

GðkÞ ¼Z 1

0

d!

�

2�ð!; kÞ!

: (83)

It is determined numerically by solving (39) with ! ¼ 0so that the boundary condition at the horizon is (41) with

! ¼ 0 and solving Gð0; kÞ from (52) as a ratio of twoWronskians. It is also important to eliminate the large-kbehavior given in (75), including the constant term. Upontransforming to coordinate space, these give rise to ðxÞor its derivatives. To get Gð0; k;TÞ=T4, one further has touse (65).Subtracting the expansion (75) from the numerically

computed B=A gives the result in Fig. 9 for the staticcorrelator. The curves are of the same shape as those inthe conformal case [8], where one plots B/A vs k=ð�TÞ.When T ! 1, the value at k ¼ 0 of the curves approachthe value ��4=10 of the conformal case. The dominantscale breaking effect is the rapid decrease of the correlatorwhen T decreases towards Tc. The next subsection explainshow this is via Eq. (83) related to the behavior of �ð!; kÞnear the light cone, see Fig. 12.In the conformal case, one can define a second-order

dissipative coefficient by the expansion

Gð0; k;TÞ ¼ G0 þ 1

2�ðTÞk2 þOðk4Þ (84)

and its value is [38]

� ¼ L3

4�G5

1

4�2T2 ¼ 1

8N2

cT2 ¼ s

4�2T: (85)

Inserting the normalization (30), our computation of � at10Tc is 1% from the conformal value in (85). At smaller T,

T

T Tc

T 2 Tc

4

s T

, 0; T

T 1.5 Tc

0.0 0.1 0.2 0.3 0.4 0.51.00

1.05

1.10

1.15

1.20

1.25

4

s T

T

1.5 Tc2 Tc

5 Tc 10 Tc

, 0; T as

1 2 3 4 5 60.5

0.0

0.5

1.0

1.5

FIG. 7 (color online). The imaginary part of the Green’s function Gð!; 0;TÞ normalized as in (80) plotted vs !=ð�TÞ for small !(left panel) or larger ! with the asymptotic behavior (76) subtracted (right panel). The subtraction is reliable only for !=ð�TÞ *5:0Tc=ð�TÞ. The model (15) is used here.

1.5 Tc

2 Tc5 Tc

10 Tc

4

s T

, 0; T as

2

2 4 6 8 10 120.5

0.0

0.5

1.0

1.5

FIG. 8 (color online). The imaginary part of the Green’sfunction Gð!; 0;TÞ with the asymptotic behavior subtracted,normalized as in (80) plotted vs ! ¼ !=ð2�Þ. The model (15)is used here.

G 0, k; T

T 4

kTc

2 Tc

3 Tc

T

5 Tc

5 10 15

10

8

6

4

2

0

2

G 0, k T ; T

T4

k

T

T Tc

2 Tc

5 Tc

1 2 3 4 5 6

10

8

6

4

2

0

2

FIG. 9 (color online). The (real) static Green’s function Gð0; k;TÞ=T4 in units of L3=ð4�G5Þ with the asymptotic behavior (75)subtracted plotted vs k ¼ k=ð2�Þ (left panel) or vs k=ð�TÞ (right panel).

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

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there is additional suppression on top of the T2 depen-dence. This is quantitatively shown in Fig. 10.

Given the momentum-dependent static Green’s functionGðkÞ, the coordinate-dependent correlator GðrÞ can becomputed by carrying out the integral (82). The shape ofthe curves is very similar to that plotted explicitly for theconformal case in [8].

C. Green’s function for any !, k

The dominant feature on the !; k plane is the structurearound the light cone. In [8], this was solved analyticallyand, for example, one could prove that exactly on the lightcone

Gðk;kÞ¼ ð1þ iffiffiffi3

p Þ �ð1=3Þ8 �61=3�ð2=3Þk

4=3þOð1Þ

�ð0:136092þ i0:235718Þk4=3; k�k=ð�TÞ: (86)

Results for the spectral function as a function of ! atfixed values of k are shown in Fig. 11. The left panel showswhat effects changing T at fixed value of k=ð�TÞ has: thecorrelation function monotonically decreases. The rightpanel shows that essentially the conformal set of curvesis obtained for T ¼ 10Tc, for the model (15).

Another useful way of analyzing the light cone regionis to see how the spectral function varies whencrossing the light cone perpendicular to it, i.e., at fixed

!þ ¼ ð!þ kÞ= ffiffiffi2

p. The outcome is shown in Fig. 12 at a

fixed value of !þ. At larger !þ, the peak gets narrower

�1=k1=3 but higher �k1=3 so that the integral over !,which via Eq. (83) leads to the static correlator, is essen-tially constant. This is as shown in [8] for the conformalcase. The new element is that when T decreases towardsTc, the correlator also decreases. After integration over !,according to (83), this correlates with the rapid decrease ofthe static correlator, already shown in Fig. 9.

V. CONCLUSIONS

In this paper, we have computed the retarded Green’sfunctionGð!; k;TÞ, k ¼ ð0; 0; kÞ, of the 1, 2 component ofthe energy momentum tensor in the high-temperaturephase of a QCD-like strongly coupled theory with a first-order phase transition and glueballs, using gauge/gravityduality. Temperatures between Tc and 10Tc were numeri-cally studied. Results were given for the !-dependence ofthe spectral function at k ¼ 0, from which one couldcompute the T-dependence of the imaginary time correla-tor Gð�;TÞ, measured on the lattice. The real staticcorrelator at ! ¼ 0 was also computed and related tostructure observed near the light cone, ! � k. Overall, at10Tc, the correlators were very close to the conformalsituation [8], but large effects were observed when Tdecreased towards Tc.The gravity background used here is IHQCD as

developed in [17–20] for k ¼ 0 and a simplified and

T T2

10 Tc 10 Tc2

T Tc

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

FIG. 10 (color online). The expansion coefficient �ðTÞ definedby (84), normalized to the value of �ðTÞ=T2 at 10Tc.

, 2; T

T

k

T2 10 Tc5 Tc

2 Tc

1.5 Tc

Tc

1 2 3 4 510

0

10

20

30

40

k 0

k 2

k 5k 10

0.236 1 3

, k; 10 Tc as

2 4 6 8 10 120.10.00.10.20.30.40.50.6

FIG. 11 (color online). The imaginary part of the Green’s function Gð!; k ¼ 2�T;TÞ=T4 in units ofL3=ð4�G5Þ plotted vs !=ð�TÞ,for the model (15). The right panel shows the behavior near the peaks at k ¼ 2, 5, 10�T. The curve 0:236!1=3 is the conformalbehavior of the correlator on the light cone, ! ¼ k.

T 10 Tc

1.5 Tc

Tc

as

10 2

2 1 1 20.1

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 12 (color online). The quantity ð�ð!; k;TÞ ��asð!; kÞÞ=! when crossing the light cone in the perpendiculardirection, i.e., at fixed !þ ¼ ð!þ kÞ= ffiffiffi

2p

(here !þ ¼ 10ffiffiffi2

p)

with varying !�, for various temperatures.

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quasianalytically tractable version thereof for all !, k.IHQCD describes very successfully the properties oflarge Nc Yang-Mills theory both at T ¼ 0 and at finite T,but its gravity background, in general, has to be evaluatednumerically. The model used here retains the attractivefeatures of IHQCD in the infrared, so that it has confine-ment, glueballs, and a first-order phase transition, whilesimplifying the complicated logarithmic structure ofIHQCD in the ultraviolet. The chief virtue of the modelis that both bðzÞ and fðzÞ, appearing in the fluctuationequation, are analytically known. Another analyticapproximation to IHQCD [25,39], would be quantitativelymore accurate but does not permit analytic evaluation offðzÞ. In any case, we expect that the results of thispaper should be qualitatively valid for IHQCD, too, forall !, k. This is confirmed by numerical computationsat k ¼ 0.

There is ample phenomenological evidence for thestrongly interacting nature of QCD matter at T * Tc

(p ¼ 34pideal, small viscosity, jet quenching), but one has

not been able to develop a first-principle numerically veri-fiable criterion for this. We have suggested that a quanti-tative verification of our predictions could serve thispurpose.

There are several obvious directions of further study.For the first, a full numerical computation in IHQCDshould be carried out, also for nonzero k. This is techni-cally complicated due to the fact that the expansions nearthe boundary are not power series in z but contain powersof logz, studied here in Appendix A. Secondly, to extendthe calculation to correlators of the scalar operator TrF2

��,

one should use the more complicated fluctuation equationin [23], coupling the dilaton and the scalar component ofthe tensor fluctuation. A third open problem is the compu-tation of more terms in the large! expansion (74). Finally,it will be interesting to compare the results with those fromnext-to-leading order perturbative QCD, when they areavailable.

ACKNOWLEDGMENTS

We thank J. Alanen, Jorge Casalderrey-Solana, MikkoLaine and Francesco Nitti for discussions and advice. F.Nitti has given us the Mathematica code used in [20]. Thework of M.V. has been supported by Academy of Finland,Contract No. 128792 and that of M.K. and A.V. by theSofja Kovalevskaja program of the Alexander vonHumboldt foundation. K. K. and A.V. thank the 2010ESI workshop ‘‘AdS Holography and Quark-GluonPlasma’’ in Vienna for hospitality.

APPENDIX A: UV EXPANSION IN IHQCD

In this appendix, we shall give small-z logarithmic ex-

pansions of bðzÞ, ðzÞ and WðzÞ � � _b=b2, adding a fewterms to those given in Appendix A of [17]. Also, fðzÞ will

obtain similar logarithmic corrections to the z4 term byintegration of (20).The expansions are given as powers of ðzÞ or as inverse

powers of

L � log1

�z; (A1)

where � is the integration constant of

d

dz¼ �WbðzÞ: (A2)

The correction terms arise from the UV beta functionexpansion written in the form

b0ðÞ¼�ðb0Þ2�b1b20

ðb0Þ3�b2b30

ðb0Þ4�b3b40

ðb0Þ5þ . . . ;

(A3)

where only b0, b1 are scheme independent. It will appearthat all quantities are functions of

b0;bibiþ10

; b � b1b20

; (A4)

[b � b1=b20 is a standard notation, not to be confused with

bðzÞ].First, by integrating db=b¼d=ðÞ¼db0=½b0ðÞ�,

(note the different constants b0 and b0)

b

b0¼ exp

�1

b0

�ðb0Þb1=b20

�1þ

�b2b30

�b2�b0

þ�bð1þbÞ

�b2�2

b2b30

�þb22b60

þb3b40

�1

2ðb0Þ2þ . . .

�:

(A5)

The constant of integration b0 is fixed so that b0 appearsin the power-like term.

For W � � _b=b2, one can derive

WðÞ¼Wð0Þexp��4

9

Z

0d �

ð �Þ�2

�; Wð0Þ¼ 1

L: (A6)

Expanding this, one has

LW ¼ 1þ 4

9b0þ

�8

81þ 2

9

b1b20

�ðb0Þ2

þ 4

�8

37þ 2

81

b1b20

þ 1

27

b2b30

�ðb0Þ3 þ . . . (A7)

The z dependence is obtained by combining the equa-

tions W ¼ � _b=b2 and ¼ b _= _b into (A2) so that, inte-grating,

z ¼Z z

0dz ¼

Z ðzÞ

0

d

�Wb: (A8)

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

086004-12

Noting thatW and b contain the constants 1=L and b0, thisequation naturally defines the scale

� ¼ b0L

(A9)

of z. Equation (A8) now has to be inverted for : oneexpands the RHS around ¼ 0 using the expansionsabove, integrates term by term and takes the limit ! 0.Finally, the result is inverted recursively to obtain ¼ ðzÞ. The expansion one obtains is

b0¼ 1

L�blogL

L2þ�b2log2L�b2 logLþ4

9�b2þb2

b30

�1

L3þ��b3log3Lþ5

2b3log2L�

�3b

b2b30

�2b3þ4

3b

�logL

�4

9þ2

3b�1

2b3þ b3

2b40

�1

L4þ�b4log4L�13

3b4log3Lþ

�3

2b4þ8

3b2þ6b2

b2b30

�log2L

þ�4b4�4b2þ16

9bþ3b2

b2b30

�2bb3b40

�logLþ104

81�4

3b�10

9b2þ7

6b4þ52

27

b2b30

�3b2b2b30

þ5

3

b22b60

�1

6bb3b40

þ1

3

b4b50

�

� 1

L5þO

�1

L6

�: (A10)

To get corrections to bðzÞ, one writes (A2) in the form

bðzÞ ¼ 0ðzÞ�W

; 0ðzÞ ¼ � 1

z

d

dL; (A11)

uses (A10) and expands in . This leads to

bðzÞ ¼ Lz

�1� 4

9b0þ

�44

81� 2

9b

�ðb0Þ2 �

�2408

2187� 80

81bþ 4

27

b2b30

�ðb0Þ3 þOð4Þ

�

¼ Lz

�1� 4

9Lþ 2ð18b logLþ 22� 9bÞ

81L2þ�4

�710

2187� 20

81b� 1

9b2 þ 4

27

b2b30

þ�22

81b� 2

9b2�logLþ 1

9b2log2L

�

� 1

L3þO

�1

L4

��: (A12)

The three first terms in the second form were given in (A.28) of [17].Finally, the UVexpansion of the potential V is very simply directly obtained by inserting the expansions ofW and to

VðÞ ¼ 12W2ðÞ½1� ð=ð3ÞÞ2�:

V ¼ 12

L2

�1þ 8

9b0þ�2381 þ 4

9b

�ðb0Þ2 þ

�402187 þ 14

81bþ 827

b2b30

�ðb0Þ3 þOð4Þ

�: (A13)

APPENDIX B: IR EXPANSION

One cannot uniquely solve Eq. (8) at large z, large , butassume that bðzÞ behaves as

bðzÞ ¼ b0e�ð�zÞ�ð�zÞp; � � 1; p real: (B1)

� here in the IR is a number times � in the UVand, to emphasize this difference, we use a differentletter for it. Inserting this to the equations of motion

above and evaluating successively _b=b, bW, W, _W=W,

one can write _� in a form which is easily integrable.If 0 is the constant of integration of this equation,one finds that, up to corrections 1þOð1=z�Þ � 1þOð1= logÞ,

bðzÞ ¼ b0e�ð�zÞ�ð�zÞp ¼ b0

2=3

�2

3log

�ð��1Þ=ð2�Þ; (B2)

WðzÞ ¼ �

Leð�zÞ�ð�zÞ��1�p ¼ 1

L2=3�

�2

3log

�ð��1Þ=ð2�Þ;

L ¼ b0�

;(B3)

ðzÞ ¼ e3ð�zÞ�=2ð�zÞ3ð��1�2pÞ=4;

eð�zÞ� ¼ 2=3ð�zÞp��=2þ1=2;

�z ¼�2

3log

�1=�

;

(B4)

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ðÞ ¼ � 3

2

�1þ �� 1

2�ð�zÞ� þO�1

z2�

��

¼ � 3

2

�1þ 3ð�� 1Þ

4� logþO

�1

log2

��; (B5)

VðzÞ ¼ 9W2ðzÞ ¼ 9�2

L2e2ð�zÞ�ð�zÞ2ð��1�pÞ

¼ 9�2

L24=3

�2

3log

�ð��1Þ=ð�Þ; (B6)

where always � =0, apart from the leading termin ðÞ, where the leading term is �3=2 for any 0.By choosing p suitably, one can make any of the quantitiesa pure exponential in z�. However, when expressed interms of , the leading behavior is always independent

of p. Note that b2=3 � bs � ðlogÞð��1Þ=ð2�Þ is the stringframe b, which is used to express the confinementcriterion. Note also that the confining beta function is notassumed here, it follows from the assumed form ofbðzÞ. The scalar potential V satisfies V ¼ 12W2ð1� X2Þ,X � =ð3Þ.

Finally, Eq. (20) now is

_fðzÞ ¼ Ce3ð�zÞ�ð�zÞ�3p: (B7)

In our model with � ¼ 2, p ¼ �1 applies at all z and issimply integrable in closed form. If one wants just a large zapproximation for fðzÞ, one can integrate it up to correc-tions of order 1=z� by extending the integration to z ¼ 0and imposing the condition fð0Þ ¼ 1. This leads to

fðzÞ ¼ 1��z

zh

�1���3p

e3��ðz��z�

hÞ;

� _fðzhÞ ¼ 4�T ¼ 3��ð�zhÞ��1:(B8)

APPENDIX C: ULTRAVIOLET LIMIT OF THESTATIC CORRELATOR

In this Appendix, we derive the high-k limit the staticcorrelator Gð! ¼ 0;k;TÞ. Eliminating the first derivativein (53) by writing

c ðyÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffib3f

ffiffiffiy

pq��ðyÞ (C1)

and setting ! ¼ 0, we have the equation

c 00ðyÞ ¼�

k2

yfðyÞ ��f0ðyÞ2fðyÞ

�2 þ 1

4þ 1

2yþ 3

4y2

�c ðyÞ

� ½k2FðyÞ þGðyÞ�c ðyÞ ¼ 0:(C2)

We wish to solve this around y ¼ 0 for k ! 1. Thesystematic approach given in [40] starts by defining

� �Z y

0

dxffiffiffiffiffiffiffiffiffiffiffiffixfðxÞp ; c ðyÞ ¼

�d�2

dy

��1=2Wð�Þ: (C3)

The idea with replacing y by the new variable � is that theresulting equation is in a Schrodinger-like form in whichonly k2 multiplies the function to be solved. IntroducingWeliminates the first derivative generated. The solution thenis given in the form3

Wð�Þ ¼ �K2ðk�ÞX1s¼0

Asð�Þk2s

� �2

kK3ðk�Þ

X1s¼0

Bsð�Þk2s

; (C4)

where the coefficients As and Bs are computed from thecorresponding expansion coefficients As describing thesolution away from the singular point � ¼ 0:

Asð�Þ ¼X2sj¼0

ajð3Þ��jA2s�jð�Þ;

Bsð�Þ ¼X2sþ1

j¼0

ajð2Þ��j�1A2s�j�1ð�Þ;(C5)

where ajð�Þ ¼ Qjk¼1ð4�2 � ð2k� 1Þ2Þ=8k. Functions As

in turn are given by

ASþ1ð�Þ¼�1

2A0sð�Þþ1

2

Zd��ð�ÞAsð�Þ

¼�1

2

ffiffiffiffiffiffiffiffiffiffiffiffiyfðyÞ

qA0sðyÞþ1

2

Z�ðyÞAsðyÞ dyffiffiffiffiffiffiffiffiffiffiffiffi

yfðyÞp : (C6)

The integration constants in each step are chosen so thatwhen we retrace our steps back to the original modefunction �ðyÞ, we have �ð0Þ finite and equal to 1. Thefunction �ðyÞ in the above recursion relation is given bythe coefficient functions FðyÞ, GðyÞ of Eq. (C2), and reads

�ðyÞ ¼ GðyÞFðyÞ � FðyÞ�3=4 d2

dy2FðyÞ�1=4

� 15

16yþ 1

2þ 8yh � 8þ 5e�yh

32ðyh � 1þ e�yhÞ yþ . . . (C7)

Working out the series expansions of As, s ¼ 0 . . . 4(with As ¼ constant), we can read out the coefficients ofthe y0 and y2 terms in

�ðyÞ ¼ ðb3f ffiffiffiy

p Þ�1=2c ðyÞ ¼ ð2�b3 ffiffiffif

p Þ�1=2wð�Þ (C8)

to finally arrive at

B

A¼ �k4

�lnkþ �E � 3

4

�� k2ðlnkþ �EÞ � 5

24

þ e�yh

20ðyh � 1þ e�yhÞ : (C9)

3For the coefficients As and Bs, we follow here Exercise 5.2 ofChapter 12 in [40].

K. KAJANTIE et al. PHYSICAL REVIEW D 84, 086004 (2011)

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