Higher order RG flow on the Wilson line in $\mathcal{N}=4$ SYM

Extending earlier work, we find the two-loop term in the beta-function for the scalar coupling $\zeta$ in a generalized Wilson loop operator of the $\mathcal{N}=4$ SYM theory, working in the planar weak-coupling expansion. The beta-function for $\zeta$ has fixed points at $\zeta=\pm 1$ and $\zeta=0$, corresponding respectively to the supersymmetric Wilson-Maldacena loop and to the standard Wilson loop without scalar coupling. As a consequence of our result for the beta-function, we obtain a prediction for the two-loop term in the anomalous dimension of the scalar field inserted on the standard Wilson loop. We also find a subset of higher-loop contributions (with highest powers of $\zeta$ at each order in `t Hooft coupling) coming from the scalar ladder graphs determining the corresponding terms in the five-loop beta-function. We discuss the related structure of the circular Wilson loop expectation value commenting, in particular, on consistency with a 1d defect version of the F-theorem. We also compute (to two loops in the planar ladder approximation) the two-point correlators of scalars inserted on the Wilson line.


Introduction and summary
The expectation value of the Wilson loop (WL) operator xTr Pe i ű A y is a key observable in gauge theories. In supersymmetric theories, it has a "locally supersymmetric" analog obtained by adding a particular coupling to extra fields in the gauge field multiplet. The prototypical example is the Wilson-Maldacena loop (WML) [1,2] in the N " 4 Super Yang-Mills (SYM) theory which contains an extra scalar coupling, and as a result has a particularly simple structure controlled by the underlying supersymmetry. When the Wilson loop contour is a circle or a straight line, both the WL and WML preserve a one-dimensional conformal symmetry SLp2, Rq and may be regarded as 1d conformal defects of the N " 4 SYM theory. A circular or straight WML also globally preserves half of the superconformal symmetry.
For a generic smooth contour, the standard WL in YM theory is known to be renormalizable: power divergences exponentiate and factorize (or simply absent in dimensional regularization) while the logarithmic divergences disappear after the renormalization of gauge coupling [3,4,5,6,7,8]. The latter are absent in N " 4 SYM so both WL and WML have finite expectation values which in the planar limit are given by the functions of the 't Hooft coupling λ " g 2 YM N . It is of interest to study a family of more general Wilson loop operators that interpolate between the WML and the standard WL [9]. This one-parameter family was introduced in [10] and further studied in [11,12]. It is of interest, in particular, in the context of the 1d defect QFT interpretation (see, in particular, [13,14,15,16,17,18,19]). Explicitly, this generalized Wilson loop operator depends on an arbitrary coefficient ζ in front of the coupling to the 6 scalars φ m and so it interpolates between the standard the WL (ζ " 0) and the WML (ζ " 1) cases [10] W pζq pCq " 1 N Tr P exp ¿ C dτ " i A µ pxq 9 x µ`ζ φ m pxq θ m | 9 x| ı , θ 2 m " 1 . (1.1) We may choose the unit vector θ m to be along 6-th direction, i.e. φ m θ m " φ 6 " φ. The expectation value xW pζq y for a smooth contour C will have logarithmic divergences that can be absorbed into a renormalization of the coupling ζ. Then the renormalized value of xW pζq y will be given by (in the planar limit) xW pζq y " W`λ; ζpµq, µ˘, where µ is a renormalization scale. 2 The running of ζ then defines a 1d RG flow between the WL and WML operators. Since φ m Ñ´φ m is a symmetry of the SYM path integral, W should be invariant under ζ Ñ´ζ. In what follows we shall assume that ζ ě 0. In the planar weak coupling expansion the leading order term in the beta-function was found in [10] to be β ζ "´λ 8π 2 ζp1´ζ 2 q`Opλ 2 q . (1. 3) The WL (ζ " 0) and WML (ζ " 1) cases in (1.1) are expected to be the only two fixed points also at higher orders in λ. 3 One may view the running of ζ as an RG flow in the effective 1d defect theory coupled to the bulk SYM theory. This interpretation may be made more explicit by representing the path ordering in (1.1) using the auxiliary 1d fermion path integral as in [5,6,20,21] and thus getting an interacting 1d defect action. Considering a circular contour, F "´log W may be interpreted as a 1d defect theory free energy on S 1 (normalized by the partition function of the bulk theory). It is then natural to expect that this quantity provides a defect analog of the F-theorem. Specifically, the d " 1 version of the generalized F-theorem [22,23,24] adapted to defects [25] requires that r F " sin πd 2 log ZpS d qˇˇd "1 " log ZpS 1 q "´F " log W decreases under RG flow:F UV ąF IR . This is analogous to the g-theorem [26,27] that applies to a 1d boundary of a 2d theory. The betafunction (1.3) implies that ζ " 0 is the UV fixed point and ζ " 1 the IR one, and so one shall find that logxW pζ"0q y ą logxW pζ"1q y . (1.4) This was indeed verified in [11] to hold both in perturbation theory and in the strong coupling expansion. A proof of the d " 1 defect version of the F-theorem was recently proposed in [19], where a quantity that monotonically decreases along the flow (and coincides withF at fixed points) was also given. On general grounds, consistent with the interpretation of log W as a defect free enerenergygy, we should have where the function C " Cpλ, ζq has the weak coupling expansion C " λ 4`O pλ 2 q [11]. In the case of a circle or straight line the flow of ζpµq is driven by the scalar operator φ 6 " φpxpτ qq in (1.1) restricted to the line. Then B Bζ xW pζq yˇˇζ "0,1 " 0 implies that its one-point function vanishes at the fixed points, as required by the 1d conformal invariance on the defect line. From (1.5) we also have According to (1.1) this second derivative is given by the integrated 2-point function of φ restricted to the line [11]. The latter is determined by the corresponding anomalous dimensions at ζ " 0 and ζ " 1. Indeed, from (1.3) one finds that Bβ ζ Bζ reproduces [10] the leading weak-coupling terms in the anomalous dimensions [9] of φ at the conformal points ζ " 1 and ζ " 0 Our main aim here will be to find the λ 2 term in the beta-function (1.3) which should also exhibit the factor ζp1´ζ 2 q. Let us start with writing down the general structure of β ζ .. are to be determined. The dependence on powers of ζ at each order in λ follows from the structure of the relevant perturbation-theory diagrams. An important observation is that the coefficients of the highest ζ 2n`1 powers at each λ n order in (1.9), i.e. b 1 , b 3 , b 6 , . . . , are determined by diagrams with maximal number of scalar propagators attached to the line. Thus they should not have internal vertices, i.e. should be given just by the scalar ladders. We shall compute them using the vertex renormalization method of [4] generalizing the one-loop computation in [10]. The resulting terms in β ζ may be written as where ζ n " ζpnq are the Riemann zeta-function values. In particular, at the two-loop order, we get from q 2 (1.11) To find the five-loop expression for the β ladder ζ in (1.10) we used planar loop equation (2.5) and dimensional regularization. As the higher coefficients q n are transcendental, the exact expression for this planar ladder-theory beta-function should be complicated. Note that as this is essentially a one-coupling (ξ " λζ 2 ) model, the three and higher loop coefficients are scheme-dependent (see below).
An indirect way to fix some combinations of other coefficients in the full beta-function (1.9) is to use the relation (1.7) between β ζ and the anomalous dimension of the scalar operator and the value of ∆ p1q that was already found earlier from diagrammatic computation for a cusp line in [28] (at the two-loop level) and from the quantum spectral curve in [29] (at several higher loop orders). Explicitly, according to [29] Comparing this with (1.7),(1.9) gives Using (1.11) then (1.14) implies that (1.16) Thus the explicit form of the 2-loop β ζ is given by This in turn implies that the two-loop terms in the anomalous dimensions (1.8) are given by where ∆ p1q is of course the same as in (1.13) while the two-loop term in ∆ p0q is a new non-trivial result. 4 Note that the sign-alternating structure of λ-expansion in (1.17) and (1.18) is consistent with expectation that the planar weak coupling expansion should have a finite radius of convergence | λ 4π 2 | " 1 with the strong-coupling λ " 1 asymptotics [9, 14, 11] ∆ p1q " 2´5 ?
It is important to stress that the two-loop coefficients b 2 , b 3 in (1.9),(1.17), are scheme independent: they are invariant under redefinitions of ζ that do not change the positions of the fixed points Since the β ζ transforms as a vector, we find that the coefficients in (1.9) change as This means that the beta-function is scheme independent at two loops, while the invariant combinations of the three-loop coefficients are b 4 and b 5`b6 . This is consistent with the fact that the dimensions ∆ p0q " 1`b 1 λ`b 2 λ 2`b 4 λ 3`. .. and ∆ p1q in (1.13) (and thus d n in (1.14),(1.15)) should be scheme independent.
Let us now comment on the implications of the above discussion for the structure of higher order terms the expectation value of xW pζq y in (1.2) on a circle (see [11,12]). Since the first derivative of W " xW pζq y at the conformal points ζ " 0, 1 should vanish (cf. (1.5)), we should have where [31,32,33] xW p1q y " The coefficients w 1 and w 2 are scheme-independent, while w 3 is finite after renormalization of ζ and in general contains log µ dependence on the renormalization scale [12]. Using the general relation (1.5) to the beta-function (1.9) where on general grounds we should have we find As b 1 , b 2 , b 3 , w 2 are scheme independent while w 3 is not this implies that c 3 in (1.23) is scheme dependent but 3c 2`c3 is scheme independent. 6 Here [11] (1.25) w 3 is determined by the contribution of scalar ladder diagrams (which is not UV finite at order λ 3 ). Its renormalized value was found (using a particular regularization scheme) in [12] to be w 3 "´1 96 p4π 2 q 2 " 5`6 logpµRq ‰ . Here R is the radius of the circle and the coefficient 6 of log µ in the bracket is related to the coefficient in the one-loop beta-function (1.3) (cf. (1.2)) while 5 is scheme-dependent. In view of (1.24) this then also fixes the value of c 3 in the same scheme. The value of w 2 remains currently unknown. If we choose a scheme in which the value of w 3 is equal to´w 2 then the λ 3 term in (1.21) will be proportional to p1´ζ 2 q 3 . One can then conjecture that there exists a scheme in which similar simplification happens to all orders in λ [11], i.e. one gets xW pζq y " xW p1q y where the function Fpxq has a regular power series expansion Fpxq " w 1 x`w 2 x 2`¨¨¨.
The rest of this paper is organized as follows. In section 2 we shall explain how to compute the coefficients in the beta-function (1.10) in ladder approximation using dimensional regularization. 6 This of course follows also directly from the transformation law C 1 pζ 1 q " p dζ 1 dz q´2 Cpζq with ζ 1 in (1.19): c 1 2 " c2´2c1z1 and c 1 3 " c3`6c1z1 so that 3c 1 2`c 1 3 " 3c2`c3. In general, for a set of couplings ζ i we have β i " µ dζ i {dµ transforming as a vector under their redefinitions and the relation B Bζ i log W " Cij β j means that Cij transforms as a tensor. At a conformal point β i pζ˚q " 0, one has B 2 Bζ i Bζ j log W| ζ"ζ˚" Cij Bβ j Bζ i | ζ"ζ˚. Here Bβ j Bζ i | ζ"ζ˚d oes not transform under ζ 1 " ζ`Xpζq that does not change the position of the conformal point, i.e. if Xpζ˚q " 0.
We shall mention that defining the effective ladder theory in d " 4´ǫ dimensions the corresponding beta-function has a Wilson-Fisher-like zero and the value of the Wilson loop at the corresponding IR fixed point is consistent with the 1d defect version of F-theorem [11,19].
In section 3 we shall consider the computation of two-point functions of scalars inserted on the Wilson loop at two loops in ladder approximation. We shall consider separately the cases of a "transverse" scalar not coupled to the loop and the scalar coupled to the loop. Using the Callan-Symanzik equation we shall find the corresponding anomalous dimensions of the scalar operators relating them to the beta-function β ζ and its derivative.
There are also a few technical Appendices. In particular, in Appendix D we shall present some details of the computation of the linear in ζ term in β ζ in (1.3) which were not spelled out in [10].

Scalar ladder contributions to the beta-function
In this section we shall present the derivation of the coefficients of the λ n ζ 2n`1 terms in the betafunction (1.10).

Vertex renormalization method
To compute the highest in ζ terms in each order of the small λ expansion in the beta-function we need to consider ladder graphs with only scalar propagators attached to Wilson loop. The perturbative expansion of the corresponding xW pζq y ladder is then given in the planar limit by a power series in a single effective coupling ξ where (for a closed contour parameterized by τ P p0, 2πq) Compared to (1.1) we have redefined the scalar φ Ñ ζ´1φ, and set φpτ q " φpxpτ qq. The averaging is done in the free adjoint scalar theory with the action In what follows we shall consider the case of a circular or straight line contour when the (unregularized) propagator Dpτ´τ 1 q " xφpτ q φpτ 1 qy has the following form 7 The problem of computing (2.2) with (2.4) is well defined and we expect W to admit a renormalizable perturbative expansion in ξ such that all UV divergences can be absorbed into a redefinition of 7 The original N " 4 SYM action is schematically S " 1 . . q, and λ " g 2 YM N . Eq. (2.4) takes into account a factor 1{2 from T a T a " N 2 1, valid for the generators T a of SU pN q in the fundamental representation.
ξ (any possible multiplicative renormalization of the loop operator should be absent in dimensional regularization).
In the planar limit the function Wpτ q in (2.2) obeys the following integral equation 8

BWpτ q
Bτ " Note that by definition in (2.2) we have Wp0q " 1 and in (2.5) one has also W 1 p0q " 0. Eq. (2.5) follows upon differentiation of the integral equation that is implied by the structure of the planar expansion of (2.2) in the free scalar theory (here thick line denotes the contour parameterised by τ ) Eq. (2.5) is valid for any propagator D. In particular, it can be applied to the case of the ladder contributions to the expectation value of WML (i.e. ζ " 1 case of (1.1)) for a circle when the effective propagator is constant [31] A riA a pτ q`φ a pτ qs where we used that T a T a " N 2 1. Taking the Laplace transform of (2.5) we have´1`s r Wpsq " D 0 r W 2 psq and thus finally reproducing (1.22) after setting τ " 2π as in (2.2). Similar Dyson integral equations also appear for the correlation function of two Wilson loops [34]. For a non-constant propagator Dpτ q, the solution of the loop equation (2.5) appears to be highly non-trivial, in particular, due to divergences starting at three loops. In Appendix B we demonstrate how to use (2.5) to reproduce the two-loop result for (1.21), i.e. find the value of w 1 in (1.25).
Following [4] and adapting their discussion to the present case, to compute the renormalization of ζ it is useful to consider the "one-point" scalar correlator on a straight Wilson line segment. Before performing the rescaling of ζ into φ introduced above, one starts with the quantity Note that this is not exactly the standard one-point function of the operator φ on the Wilson line, because we are integrating only over a segment τ 1 ă τ ă τ 2 (with far-separated points τ 1 , τ 2 ) and τ 0 is also assumed to be far from that interval (hence φpτ 0 q does not participate in the pathordering). Following [4], all UV divergences should come from "internal" coinciding points not involving τ 0 , τ 1 , τ 2 , so it is sufficient to look at this object in order to find the renormalization of ζ (or ξ). The normalization in (2.9) by the expectation value of the Wilson line segment without insertion is needed in order to remove some spurious divergences associated with the finite endpoints [4]. After performing the rescaling φ Ñ ζ´1φ, eq. (2.9) becomes (2.10) In the planar limit, the numerator in this expression satisfies a relation analogous to (2.6) and can be written as On the other hand, the denominator factor in (2.10) is simply equal to ζWpτ 2´τ1 q. When performing the renormalization of (2.10), we may effectively factor out the τ integration on the right-hand-side of (2.11), since the integral over τ cannot bring in new UV divergences as the point τ 0 is supposed to be far away from τ 1 , τ 2 . Therefore, one can see that in the planar limit the calculation reduces to study the renormalization of the "vertex function" where τ 1 , τ 2 are arbitrary (far separated) fixed points on the line. Note that at planar level the normalization by the denominator factor in (2.10) does not play an important role since, being equal to ζWpτ 2´τ1 q, it is essentially the same as the "square root" of the V function defined above (recall that ξ " λζ 2 ), and hence it is finite once V is made finite by renormalization (or vice-versa). At the non-planar level, however, the denominator factor is expected to play a non-trivial role.
To summarize, in the planar limit it will be sufficient to consider the renormalization of V as given in (2.12). The relevant collection of diagrams may be represented symbolically as Here we have chosen the "middle" point τ in (2.11) as 0 (using translational invariance) and replaced the argument τ 0 of φ by a generic point x 0 that may or may not lie on the line. The vertical line represents the propagator Dpτ 0´τ q or Dpx 0´x p0qq which will play only a spectator role.
The strategy is to find the divergent part of V in (2.12) and then absorb the divergences into the renormalization of ξ. This requires computing W from the corresponding sum of planar ladder diagrams (or using the loop equation (2.5)).
Let us note that the reason why this "scalar ladder" model is effectively an interacting one (despite the bulk theory being free) is due to the path ordering in (2.2). As was mentioned in the Introduction, one can cast the problem of renormalization of ζ or ξ in a more standard form by representing the path ordering using a functional integral over 1d fermions χ i pτ q in the fundamental representation. Integrating first over the free adjoint bulk scalar field we then get an effective 1d action of the following schematic form (2.14) Introducing a cutoff into the propagator and expanding in ξ one should be able then to compute the corresponding β ξ in the usual way.

Five-loop beta-function in ladder approximation
Let us consider the loop equation (2.5) on a line with a simple analytic regularization of the propagator This is essentially the standard dimensional regularization with d " 4´ǫ, ǫ ą 0, where we did not include the usual ǫ-dependent normalization factor (this is equivalent to a redefinition of the renormalization scale). Solving the loop equation perturbatively, i.e. expanding in powers of ξ we find at the leading order Integrating this with the boundary condition W 1 p0q " 0 gives The corresponding one-loop correction to the vertex (2.13) is represented by 1ˆW 1 pτ 1 q`W 1 pτ 2 qˆ1, i.e. by the following diagrams Next, W 2 pτ q is given by with the explicit expression being All higher order functions W n pτ q may be expressed as W n pτ q " K n pǫq Γp1`nǫq τ nǫ , (2.22) where the coefficients K n pǫq are determined by the recurrence relation This follows from the Laplace transform L of the loop equation on the line that reads 9 In the following we shall present the explicit results to five loop order, i.e. including W 5 pτ q in (2.16). We can then find the divergences in (2.12), i.e. coefficients of poles in ǫ Ñ 0 in and cancel them by replacing the bare coupling ξ in terms of the renormalized one using the familiar general relation 10 ξ " µ ǫ Requiring that the vertex (2.25) expressed in terms of ξpµq is finite gives and thus The perturbative solution takes the form r Wnpsq " Knpǫq s 1`nǫ , cf. (2.22). Replacing this in the loop equation and using p´Bsq α 1 s β " Γpα`βq Γpβq 1 s α`β gives (2.23). 10 Note that the factor µ ǫ is required to match the dimensions: defining the theory in d " 4´ǫ dimensions we keep the dimension of the scalar field to be 1 (as it is coupled to the line which still has dimension 1) so that the bulk action N ξ where we added also the five-loop contribution and ζ n are the zeta-function values. Recalling that according to (2.1) ξ " λζ 2 where λ is not running we may then read off the ladder contribution to the beta-function of ζ

Comment on Wilson-Fisher fixed point
Let us note that considering the theory in d " 4´ǫ dimensions and thus keeping the order ǫ term in the beta-function (2.29), we get from (2.26) (2.32) This implies that the effective 1d theory corresponding to the scalar ladder approximation has, in addition to the trivial UV fixed point ξ " 0, a non-trivial IR fixed point ξ˚" 4π 2 ǫ`2π 2 ǫ 2`¨¨¨. (2.33) The expectation value of xW pζq y for a circle found in d " 4´ǫ in ladder approximation may be written as [ Thus log W pξ˚q ă log W p0q " 0 in agreement with the 1d defect version of F-theorem. On the other hand, note that directly in d " 4, the beta-function in the ladder approximation is positive in perturbation theory, which means that the coupling ξ grows in the UV. From the point of view of the F-theorem, this is consistent with the fact that log W pξq " 1 128π 2 ξ 2`¨¨¨i s positive in perturbation theory.

Two-point function for "transverse" scalar
Using the scalar ladder approximation let us compute the defect two-point function for one of the "transverse" scalars φ 1 , ..., φ 5 (to be denoted by r φ) which is not coupled to the scalar Wilson line operator defined in (2.2) Here we shall consider an infinite straight line with L as an IR cutoff. We shall treat r φ and φ on equal footing rescaling both by ζ, i.e. the averaging is done with the free scalar action (2.3) where now S " N ξ ş d 4 x TrpBφBφ`B r φB r φq. Then ξ will appear in the propagators of both φ and r φ as in (2.4). Alternative equivalent option is to rescale both fields in (3.1) by ? ξ getting the factor ? ξ in the exponents in (3.1), ξ-independent propagator and the overall factor of ξ in (3.1). It is then natural to remove it by rescaling r G Ñ ξ´1 r G . (3. 2) The renormalization of r G and ξ´1 r G will differ just by a Z-factor corresponding to ξ, i.e. they will satisfy closely related Callan-Symanzik equations (see below).
At the tree and one loop level we will then have from (3.1) Using the propagator (2.15) and explicit expressions in (C.1)-(C.6), we get Here ξ is the bare coupling. Applying the renormalization, i.e. the redefinition of ξ as in (2.26)-(2.28), we find a finite result where ξ is now the renormalized coupling ξpµq and the limit L Ñ 8 is straightforward. Similarly, at two loops, from the results in (C.7)-(C.23) and using again the redefinition (2.26)-(2.28), we find a finite expression which in the limit L Ñ 8 gives Note that since r G ren pτ q requires only the renormalization of ξ for its finiteness (i.e. no extra Since for generic ξ its beta-function is non-zero, i.e. the conformal invariance is broken in the ladder theory in d " 4, this correlator cannot be put into the standard conformal form. A way to achieve conformal invariance is to consider the scalar ladder theory in d " 4´ǫ and specify to the Wilson-Fisher fixed point (2.33). Considering one loop order we have to keep the full ǫ dependence of the tree and one-loop terms. Then instead of (3.7) we get Evaluating this at ξ " ξ˚in (2.33), expanding in ǫ, and taking the infinite line limit L Ñ 8 gives simply The fact that all log τ terms cancel implies that the scaling with τ is non-anomalous at the conformal fixed point (i.e. the corresponding dimension is r ∆˚" 1), at least to two loops. This is analogous with what happens in full N " 4 SYM theory case, where the insertion of the "transverse" scalars into the WML is BPS-protected. Here in the ladder model we do not have an argument based on supersymmetry, but we can explain why r ∆˚" 1 as follows. If one starts with the WL coupled to two scalars as Tr P exp ż dτ pζ 1 φ`ζ 2 r φq , (3.11) then setting ζ 1 " ζ cos α, ζ 2 " ζ sin α and redefining the scalars (using SOp6q symmetry of the bulk scalar action) we get back to the WL coupled to a single scalar with the coupling ζ. Thus the beta-function cannot be a function of α, but only of ζ, i.e. the angle α should be a parameter of an exactly marginal deformation. For infinitesimal α the integrand in the exponent in (3.11) is ζφ`ζα r φ so that the insertion of r φ into the WL coupled to φ should be exactly marginal with ∆ " 1 at a fixed point.
(3.27) Like (3.7) this expression cannot be put into the conformal form (with all τ dependence appearing only as a power pµτ q 2∆ ) since the conformal invariance is broken for generic ξ.
The renormalized correlator satisfies the Callan-Symanzik equation We have written (3.28) for the rescaled (3.2) correlator corresponding to the picture where ? ξ coupling appears in the exponent of the WL operator and the two-point function is defined for the canonically normalized fields (i.e. without extra coupling factor in the bulk action compared to (2.3)). Thus ∆´1 in (3.28) is the anomalous dimension of the canonically normalized scalar φ.
Using (2.29), i.e. β ξ " 1 4π 2 ξ 2´1 32π 4 ξ 3`¨¨¨, we obtain This expression is in agreement with the general relation for the anomalous dimension ∆´1 " d dg β g where the coupling here is g " ? ξ appearing in front of the scalar φ in the exponent, i.e.
The dimension (3.29) in the effective ladder model corresponds to the terms with highest power of ζ in the dimension of the scalar φ " φ 6 in the full N " 4 SYM theory (cf. (1.17)) Similarly, the Callan-Symanzik equation (3.8) for the two-point function r G ren for the "transverse" scalar r φ in (3.7) rewritten for the rescaled (3.2) correlator, i.e. in the form (3.28), is iś It implies that the anomalous dimension r ∆´1 of the canonically normalized scalar field r φ found in the ladder approximation is directly proportional to the beta-function β ξ in (2.29) To reconstruct the expression for the corresponding anomalous dimension r ∆´1 in the full SYM theory (i.e. the analog of (3.31)) which is a function of λ and ζ we may use that (i) it should reduce to (3.33) if one keeps only highest powers of ζ at each order in λ; (ii) it should vanish at ζ "˘1; (iii) it should be equal to (3.31) at ζ " 0 as then all 6 scalars are on equal footing (not coupled to the loop). This gives This is simply related to the beta-function β ζ in (1.17) 14 Thus while the anomalous dimension of the coupled scalar (3.31) is given by the derivative of the beta-function of ζ, the anomalous dimension of the "transverse" scalar (3.35) is proportional to the beta-function itself. A way to understand why this relation of r ∆ pζq to the beta-function should hold in general let us start again with the loop (3.11) coupled to the two scalars (φ " φ 6 and a "transverse" r φ) with different coefficients, i.e.
In this case we may formally define two beta-functions β ζ i " µ d dµ ζ i but since α is an exactly marginal parameter not running with µ (see discussion below (3.11)) we should have At the same time, the general relations for the anomalous dimensions of φ and r φ at the point ζ 2 " 0 or α " 0 (when r φ is not coupled to the loop) are given by where in the last equality we used that B Bζ 2 β ζ 2ˇζ 2 "0 " pζ´1 cos α B Bα`s in α B Bζ qpsin α β ζ qˇˇα "0 " ζ´1β ζ . The above argument implies, in particular, that the fact that r ∆ " 1 at the fixed points when β ζ " 0 and ζ ‰ 0 follows essentially from the rotational symmetry in the scalar space. For the fixed point at ζ " 0, where all scalars are not coupled to the loop and the full SOp6q rotational symmetry is restored, this does not apply as here all scalars have then the same dimension ∆ p0q " 1`β 1 ζ | ζ"0 .
In computing higher-order corrections one also needs Useful formulae for diagram computations are " τ c 12`c23`c13`3 Γpc 12`1 qΓpc 23`1 q pc 12`c23`c13`2 qpc 12`c23`c13`3 q Γpc 12`c23`2 q , (A.8) Let us also recall some general relations between pole coefficients and beta-function in dimensional regularization. If ξpǫq is a bare coupling and ξpµq is a renormalized one we have where T n pξq have perturbative expansions Differentiating (A.10) over µ, using µ d dµ ξpǫq " 0 and setting to zero the resulting coefficients of poles in ǫ gives differential constraints on T ń Plugging here the expansions (A.11) gives coefficients of all higher poles in terms of the coefficients of the simple one

B Computation of Wilson loop on a circle in ladder approximation
Let us illustrate how to use the loop equation (2.5) to automatically generate relevant ladder diagrams in the planar expansion on the example of the two-loop calculation of the WL in (1.21). Following the mode regularization approach in [12], we replace the propagator on the circle in (2.4) by its regularized version where ε " a R Ñ 0 and a is a cutoff of dimension of length. Using this in the loop equation (2.5), the first-order term in the expansion (2.16) is simply Setting τ Ñ 2π before summing over n, one gets Next, using (B.2) in the loop equation, we obtain W 2 pτ q "  .

D Computation of order ζ term in the one-loop beta-function
Here we shall present some details of the computation of the one-loop term in β ζ in (1.3) that were not spelled out in [10]. Like the ζ 3 term in (1.3) that follows from the scalar ladder graphs (cf. (2.19)) the´ζ term comes from similar graphs with gluon propagator instead of the scalar one. For example, in the case of the Wilson line along Euclidean time t " τ direction the exponent in (1.1) is given by ş dτ piA t`φ q and thus the gluon contribution is minus that of the scalar due to extra i factor. This already reproduces the expression (1.3) for the one-loop term in β ζ .
(D. 16) Then one can check that the contributions of the triangular graph and self-energy correction indeed cancel (even exactly in ω), i.e. V`S " 0. (D.17)