Wilson loops in SYM theory: From weak to strong coupling

SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 108 (2002) 106-I 12

ELSEVIER

Wilson loops in SYM theory: Gordon

W. SemenoF

We review

and Astronomy,

of Theoretical calculations

from weak to strong coupling

and K. Zarembob*

a Department of Physics Canada V6T 1Zl b Department

www.elsevier.com/locale/npe

Physics,

of Wilson

University

Uppsala

loops in

of British

University,

Columbia,

Vancouver,

Box 803, S-751 08 Uppsala,

N = 4 supersymmetric

Yang-Mills

theory

correspondence

; Fiv - (D,@“)2 + ; [Qi, m’12}(1)

&I4 The metric in Ad& x ,635is such that the line element is dimensionless, which reflects scale invariance of the dual field theory: ds2 =

5 (dz; + dz2) + dR2,s.

(2)

The coordinate z runs from the boundary at z = 0 to the horizon at z = 0;). The string *Also at ITEP, Moscow, Russia

Sweden

of perturbative series test of the AdS/CFT

and Yang-Mills coupling constants are related by g9 = 47rg&,, and the string tension, which is di-

The gravity and gauge field degrees of freedom which emerge from string theory are related in a way which has some surprising and far reaching consequences. An important example is the holographic duality which appears to relate certain gravity and gauge field theories [1,2]. The most precise statement of holographic duality is contained in the Maldacena conjecture [3]. The conjecture in its strongest form asserts an exact duality between four dimensional N = 4 super-YangMills theory (SYM) with gauge group SU(N) and string theory on the spacetime Ad& x Sg which has the 4-dimensional space as its boundary. N = 4 SYM is a conformal field theory with vanishing beta function. Its field content consists of gauge fields, six scalars and four Majorana fermions in the adjoint representation. The bosonic part of the Lagrangian is C=Itr

Columbia,

using both the AdS/CFT

correspondence and perturbation theory. We emphasize the cases in which resummation give exact results. Agreement of these result with string theory predictions is a stringent correspondence.

1. The AdS/CFT

British

mensionless in AdS space, is T = dm/21r. Weaker and computationally more useful versions of this duality are obtained by taking limits (table 1). The ‘t Hooft limit of the gauge theory [4] takes gyM -+ 0 and N + oo with the ‘t Hooft coupling X q gtM N held fixed. In the string theory, this corresponds to vanishing string coupling, g8 + 0. It has long been anticipated that the large-l\’ limit of a gauge theory is equivalent to a non-interacting string theory [4]. The AdS/CFT correspondence is the first quantitative example of such an equivalence. The g8 + 0 limit of string theory in AdS space is still a complicated dynamical theory. It simplifies when the string tension is also taken to be large. Then only massless states of the string are important. Massive states become infinitely heavy and decouple. The string theory is then approximated by classical IIB supergravity on the background Ad& x SS. In the gauge theory, the limit of large tension corresponds to the limit of large ‘t Hooft coupling X + co. Thus, the strongly coupled large N limit of Yang-Mills theory should be equivalent to IIB supergravity in Anti-de-Sitter space. Even the last, weakest version of the AdS/CFT duality has profound consequences. Previous to it, the main quantitative tool which could be used for super-Yang-Mills theory was perturbation theory in gcM, which is limited to the regime

0920-5632/02/$ - see front matter 0 2002 Elsevier Science B.V All rights reserved PI1 SO920-5632(02)01312-9

G. F! Semenoff; K. Zarembo/Nuclear

Table

Physics B (Proc. Suppl.) 108 (2002) 106112

107

1

Different limits of the AdS/CFT

correspondence

N=4SYti Yang-Mills Number

String theory in Ad&

coupling:

gYM

of colors:

N

String String

coupling:

x S5

gs

T

tension:

Level 1: Exact equivalence Level 2: Equivalence in the ‘t Hooft limit N -+ co,

X = g$MN-fixed

(planar

limit)

gs -+ 0,

T-fixed

(non-interacting

strings)

Level 3: Equivalence at strong coupling N+co,

A>>1

gs + 0, (classical

where gcM and X are both small. The conjectured duality allows one to do concrete computations in the large ‘t Hooft coupling limit [5,6]. The supersymmetry and conformal invariance of N=4 super-Yang-Mills theory significantly restrict the form of correlation function and in some cases protect them from radiative corrections so that they have only a trivial dependence on the coupling constant. A number of these have been computed using the AdS/CFT correspondence and have been found to agree with their free field limit. This can be viewed as a simultaneous confirmation of the non-renormalization theorem and the prediction of AdS/CFT. Examples are the two-point functions and the large N limit of three point functions of chiral primary operators [7]. Because AdS/CFT and perturbation theory compute different limits, it is difficult to obtain an explicit check of the AdS/CFT correspondence for any quantity which has non-trivial dependence on the coupling constant. There are now a few examples of such quantities and whose large N limit is computable and is thought to be known to all orders in perturbation theory in planar diagrams. The first example is the circular Wilson loop. Its expectation value was computed in ref. [8,9]. The second example is the correlation functions of a circular Wilson loop with chiral primary operators [lo]. A review of these and

T >>1

supergravity)

related results is the central theme of this Paper. The notation and conventions which we follow are described in detail in [8] and [lo]. 2. Wilson loops at strong coupling The Wilson loop operator which we will discuss is associated with the holonomy of the heavy WBoson which arises when the SU(N + 1) gauge symmetry is broken to SU(N) x U(1): W(C)

= $trPefc

dr(iA,(z)i,+~i(z)ei151).

(3)

Here, C is a closed curve parameterized by z”(r) and Bi is a unit 6-vector, e2 = 1, in the direction of the symmetry breaking condensate. The holographic dual of the Wilson loop is a fundamental string that propagates in the bulk of Ad& x Sg. The expectation value of the Wilson loop is given by the string partition function with the Dirichlet boundary condition that the worldsheet boundary is the contour C, which is also located at the boundary of Ad&, x Ss. When X is large, the integral over string world sheets is semiclassical and is saturated by the saddle point of the string action - the surface of a minimal area: (W(C))

= const Xe314 exp (-$-4(C)).

(4)

G. W Semenon

108

K. ZaremboINuclear

Physics B (Proc. Suppl.) 108 (2002) 106-112

The overall factor of X-3/4 is due to zero modes that arise in the process of gauge fixing the integral over metrics on the world sheet [9,11]. Consequently, Wilson loops obey an area law at strong coupling. This statement requires certain refinement, since the area of a surface which extends to the boundary in AdS space is infinite. The infinite part is proportional to the length of the loop and should be subtracted. It turns out that the subtracted term is always greater than the original area, so the renormalized area is always negative. Thus, there are three universal predictions of the AdS/CFT correspondence, in the strong ‘t Hooft coupling limit: Wilson loop expectation values exponentiate, the exponent is proportional to the square root of X and the coefficient is positive: In (W(C))

= 6

x (positive

number).

(5)

It is curious that the area law, which is usually associated with confinement, arises in a conformally invariant field theory. In fact, the geometry of AdS space is such that the area law leads to the Coulomb potential which is compatible with conformal invariance. The static potential is computed from an expectation value of the rectangular Wilson loop:

$

V(L) = - JiI+ym In (W(CLxT)). Solving for the minimal area one finds [12,13]:

V(L) = -

surface and evaluating

its

4nsJr; r4(1/4)

L’

Another example, in which the minimal area can be easily found is a circular loop. Calculation of the minimal are gives a very simple result [14,15]: A(circle) = -2 x and for the expectation value of the circular loop we get: W(circle)

= const A-314 e JT;.

The minimal surface for the straight of Ad&. line is really simple: it is a half-plane of Ad& geometry whose area, after the divergence is removed, is zero. Consequently, the expectation value for the Wilson line is equal to one. This is consistent with the fact that the Wilson line is a BPS operator, it commutes with half of the supercharges and cancellation of quantum corrections might be anticipated. The expectation values for the circle and for the straight line are not the same in apparent contradiction with the conformal symmetry. Hence, conformal invariance must be violated. Not surprisingly, it is violated by regularization. The subtraction of the divergence from the minimal area leaves a finite piece that does not vanish when the regularization is removed. The expectation value for the circular loop is entirely due to this anomaly.

(8)

This result does not look suspicious, unless one wonders how it was derived. The easiest way to solve for the minimal surface is to map a circle onto a line by a conformal transformation which extends to a diffeomorphism zp + zJz2,

2.1. Operator product expansion (OPE) When probed from a distance much larger than the size of the loop, the Wilson loop operator should behave effectively as a point-like object. For this purpose, it can be expanded in a series of local operators [16,14]: W(C)

= (W(C))

c

C/@@(O)

(9)

where oA(0) is an operator evaluated at the center of the loop, AA is the conformal dimension of oA(z) and R is the radius of the loop. The coefficients of the OPE can be read off from the correlation functions of the Wilson loop with local operators. We can choose the basis of unit normalized primary operators (those which have the lowest dimension in a given representation of the conformal algebra):

(~A(x)oB(!d) =

,x

_

iTA,,,

(10)

Their OPE coefficients can be extracted from the large distance behavior of the correlator: (W(C)QA(L)>,

(W(C))

_ RAA - cA ~ + ...

where L >> R. The omitted terms correspond descendants and are of higher’order in R/L.

(11) to

G. W Semenofl

K. Zarembo/Nuclear

Physics B (Proc. Suppl.) 108 (2002) 106-112

109

The strong-coupling evaluation of the OPE coefficients involves a hybrid of string and supergravity calculations: The classical string world sheet created by the Wilson loop absorbs the supergravity mode emitted at the point of operator insertion. Explicit calculations for a number of chiral operators can be found in Ref. [14]. The chiral primary operators (CPOs) are: Figure

1. String breaking.

(12) where C;, ,,,ik are totally symmetric traceless tensors normalized as C[,,.i,Ci.,,il. = brJ (We use the conventions of Refs. [7,14]). The overall coefficient in the definition of CPOs is chosen to unit The twonormalize their two-point functions. point correlators of chiral operators are protected by supersymmetry and do not receive radiative corrections, which insures the correct normalization to all orders of perturbation theory. This will be important when we will compare perturbative calculations to supergravity predictions. The AdS duals of CPOs are linear combinations of spin-zero Kaluza-Klein modes of the supergravity fields on 5”. For this reason, each CPO is associated with a spherical function: Yr(e)

= c;l,,,il.eil

. . .eif-.

(13)

The OPE coefficients of a Wilson loop must be proportional to Y’(8). Explicit calculations for the circular contour at strong coupling give [14]:

2.2. Wilson loop correlator The minimal area law has an interesting and somewhat unexpected consequence. It predicts that, in the large X limit, the two-point correlator of Wilson loops will undergo a phase transition as the distance between the loops changes. This phase transition is a consequence of the string breaking [17]. At short distances, the correlator of two Wilson loops is saturated by the string stretched between the contours. The area of the string world sheet grows with the separation between the loops. Since the string has tension,

eventually the world sheet breaks into two minimal surfaces that span each of the contours separately (Fig. 1). In between the surfaces, the string world sheet degenerates into an infinitely thin tube which describes propagation of individual supergravity modes. The two regimes are separated by the Gross-Ooguri phase transition, and the correlator is not analytic in the distance between t,he loops [18-201. As an example, we plot the logarithm of the correlator of two circular loops as a function of the distance L between them in Fig. 2. The first derivative of the correlator is discontinuous at the critical separation, so the Gross-Ooguri transition is first order in this case. The Gross-Ooguri transition is an inherently stringy phenomenon and is rather counterintuitive from the field theory perspective. Indeed, any Feynman diagram that contributes to the Wilson loop correlator is an analytic function of the distance between the loops. Of course, one has to sum an infinite series of all planar diagrams to reach the strong-coupling limit on the field theory side. Surprisingly, even partial resummation that takes into account only planar graphs without internal vertices reveals the Gross-Ooguri transition at strong coupling [21]. It is also possible to see how the Gross-Ooguri transition disappears as one gradually decrease the coupling on the string side of the correspondence [X3]. The string fluctuations, that should be taken into account at finite X, smooth the transition, which becomes a crossover and is completely washed out at weak coupling.

G. W Semenofl K. Zarembo/Nuclear

110

Physics B (Proc. Suppl.) 108 (2002) 106-112

toleading order corrections go in the right direction for simplest contours. For instance, the first perturbative correction to the static potential is negative [23]:

6. 4.

V(L) = -

2.

Another

IL

(

0.5

1

1.5

2

2.5

3

R

Figure 2. X- ‘I2 In (W(Ci)W(C&)) vs. the distance between the loops Ci and C2 for concentric circles of radius R [19]. The Gross-Ooguri phase transition is of the first order and takes place at L, = 0.91R [18].

& - $1,;

example

In (IV(circle))

+...

is the circular

= g - &

i. >

(17)

loop, for which

+ ... .

Perturbative calculations that lead to this formula can be generalized to include a particular class of diagrams to all orders of perturbation theory, namely diagrams without internal vertices (rainbow diagrams). The sum of these diagrams is believed to give an exact result for the circular Wilson loop. 3. Exact results for circular Wilson loop

2.3. Wilson loops in perturbation To the leading order in perturbation

(W(C)) = 1 +A

theory theory, +.*.

(15)

The integrand is the sum of scalar and vector exchanges. For a loop without cusps or selfintersections, the integral is finite. An expectation value for a smooth contour is known to be finite at two [8] and three [22] loops. The cancellations are likely to persist to higher orders of perturbation theory, though no rigorous proof has been given. The integrand in (15) is non-negative. The extremal case is the straight line, for which the correction is strictly zero, as it should be for a BPS operator. For any other contour, In (W(C))

= X x (positive

number).

(16)

This is a general prediction of perturbation theory. Comparing to the string theory prediction (5), we see that a Wilson loop expectation value interpolates between linear and square-root scaling with X as we go from weak to strong coupling. One would expect that this interpolation is smooth. In particular, higher-order perturbative corrections should decrease In (W(C)). Explicit calculations indeed demonstrate that next-

The circular Wilson loop is conformally equivalent to the straight line, which does not receive radiative corrections. This equivalence is spoiled by an anomaly in the conformal symmetry and the circular loop gains an expectation value, which is a non-trivial function of the ‘t Hooft coupling. Still, the supersymmetry should lead to many cancellations among quantum correction for the circular loop. It was shown to leading order in ref. [8] and argued generally by Drukker and Gross [9] that the rainbow diagrams exhaust all corrections that survive supersymmetry cancellations. 3.1.

Expectation value to all orders in perturbation theory In this section, we consider a circular Wilson loop, whose radius we can assume to be unity. A convenient parameterization of this loop is z(r) = (cos 7, sin 7,0,0). We will sum all planar diagrams which have no internal vertices. It is instructive to consider first the lowest order diagram (15). For a circular loop, that expression greatly simplifies:

(19) independently of ri and rz. The contour integrals in (15) are trivial and just give an overall

G. W Semenoff; K. Zarembo/Nuclear

Physics B (Proc. Suppl.)

of the first term factor of (27r)2. Computation in perturbative series (18) turns out very simple. The only complication we encounter at higher orders is path ordering and necessity to keep only planar diagrams. In virtue of (19), the gluon and the scalar propagators, whose ends lie on the same circle, always combine to a constant. This observation makes the problem of resummation of rainbow diagrams essentially zero-dimensional. In fact, we can express the circular loop in terms of a correlator in a zero-dimensional field theory: (lV(circle))

= ($

where the “path tion function Z=

J

tr e”) integral”

, is defined

dNZMexp

by the parti-

(21)

Averaging over M correctly accounts for the combinatorics of rainbow diagrams and the measure is chosen to reproduce the field-theory propagator. It is now straightforward to compute the expectation value of the circular loop using classic results of random matrix theory [24]. The eigenvalues of the Gaussian random matrix M have a continuous distribution with finite support in the large-N limit and obey the semi-circle law: (k

trf(M))

Substituting (w(circle))

= ~/~dm~~f(m)~(22) f(m)

= em, we find:

= -& Ii (Li)

,

where II is modified Bessel function. We can compare this result with the prediction of AdS/CFT correspondence (8) by taking the large-X limit: (lV(circle))

= fiAm3j4

e fi.

(24

We find the complete agreement with string theory prediction! The exact expression (23) smoothly interpolates between the small and large X limits.

111

108 (2002) 106-I 12

OPE coefficients for chiral primary operators At weak coupling, the OPE coefficient of the circular Wilson loop for dimension-k CPO (12) is proportional to Xki2:

3.2.

(25) Comparing this with the AdS/CFT prediction (14) which is of order Xl/2 for all Ic, we see that OPE coefficients are non-trivial functions of X. Again, appealing to supersymmetry and conformal invariance, we can argue that correlators of the circular Wilson loop with chiral operators are saturated by free fields. Therefore, calculation of these correlators amounts to resummation of all planar rainbow diagrams. This is a rather lengthy exercise for arbitrary Ic which involves the use of loop equations [25,26] for the matrix model (21). The details may be found in the original reference [lo). Here, we only quote the result:

(w(c)% _ 2+la1k W(C)) -

I1

(fi) Rk Y’(e)(26) (A)

3

This expression, which of course reduces to (25) at weak coupling, is expected to be exact in the large-N limit. At strong coupling we expect to reproduce the AdS/CFT prediction (14), and this is indeed the case, as can be checked easily by taking the large fi limit of the modified Bessel functions in (26). The universality of the strong coupling limit (that for all k the r.h.s. of (14) is N 6) appears because all modified Bessel functions have the same asymptotics at infinity. The coefficients also turn out to be identical. This provides an infinite series of correlation functions, for which resummed perturbative series allow to trace an interpolation between weak coupling regime and the strongcoupling prediction of string theory in Anti-deSitter space. 4. Remarks In conclusion, study of Wilson loops in N = 4 SYM theory allows to test the AdS/CFT correspondence in a very stringent way. Moreover,

112

G.cI! Semenon

K. Zarembo/Nuclear

Wilson loops probe string theory in Ads, even at strong coupling, so these tests verify Maldacena conjecture in its strongest form2 suggesting that AdS/CFT correspondence indeed is a string/gauge theory duality. The current status of this subject leaves many questions unanswered. Some of the immediate questions are l It has recently been demonstrated that radiative corrections to the connected correlator of two circular loops from diagrams with internal vertices cancel up to order g6 [22,27]. There should be a more rigorous proof that radiative corrections actually cancel to all orders. An approach suggested in [9] is to show that the result for the circle comes from a conformal anomaly. Establishing this at a rigorous level would be an important step in the right direction. l It should be possible to study other kinds of Wilson loops [23,28,22]. l Most desirable would be to obtain some results for non-conformally invariant gauge theories. At this point this appears to be very difficult as most of the analytic computations that have been done so far depend heavily on conformal invariance. Acknowledgments

The work of K.Z. Swedish Academy of grant IG 2001-062. the Natural Sciences Council of Canada.

was supported by Royal Sciences and by STINT G.W.S. is supported by and Engineering Research

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Physics B (Proc. Suppb) 108 (2002) 106-112

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