On the cusp anomalous dimension in the ladder limit of N=4 SYM
OOn the cusp anomalous dimension in theladder limit of N = SYM
Matteo Beccaria a , b , Alberto Fachechi a , b , Guido Macorini a , b a Dipartimento di Matematica e Fisica Ennio De Giorgi,Università del Salento, Via Arnesano, 73100 Lecce, Italy b INFN, Via Arnesano, 73100 Lecce, Italy
E-mail: [email protected] , [email protected] , [email protected] A BSTRACT : We analyze the cusp anomalous dimension in the (leading) ladder limit of N = φ . The first is the light-like regimewhere x = e i φ →
0. This limit is characterised by a non-trivial expansion of the cuspanomaly as a sum of powers of log x , where the maximum exponent increases with theloop order. The coefficients of this expansion have remarkable transcendentality featuresand can be expressed by products of single zeta values. We show that the whole logarith-mic expansion is fully captured by a solvable Woods-Saxon like one-dimensional poten-tial. From the exact solution, we extract generating functions for the cusp anomaly as wellas for the various specific transcendental structures appearing therein. The second limitthat we discuss is the regime of small cusp angle. In this somewhat simpler case, we showhow to organise the quantum mechanical perturbation theory in a novel efficient way bymeans of a suitable all-order Ansatz for the ground state of the associated Schrödingerproblem. Our perturbative setup allows to systematically derive higher-order perturba-tive corrections in powers of the cusp angle as explicit non-perturbative functions of theeffective coupling. This series approximation is compared with the numerical solution ofthe Schrödinger equation to show that we can achieve very good accuracy over the wholerange of coupling and cusp angle. Our results have been obtained by relatively simpletechniques. Nevertheless, they provide several non-trivial tests useful to check the appli-cation of Quantum Spectral Curve methods to the ladder approximation at non zero φ , inthe two limits we studied. a r X i v : . [ h e p - t h ] A p r ontents x → φ κ Ω
14B Convergence of the expansion (2.13) 15C Small x limit as perturbation around a finite depth well 16D List of higher order corrections for small φ up to O ( φ ) The study of cusped Wilson loops W was initiated by Polyakov in [1] while attemptingto view gauge fields as chiral fields on a loop space. Gauge interactions are interpretedin terms of the propagation of infinitely thin rings formed by the lines of color-electricflux. The analysis of the quantum properties of W led to the introduction of the cuspanomalous dimension Γ cusp ( φ ) depending on the Euclidean cusp angle φ and appearingin the relation (cid:104) W (cid:105) ∼ e − Γ cusp ( φ ) log Λ UV Λ IR , (1.1)where Λ UV, IR are ultraviolet and infrared energy cutoffs. In QCD, the cusp anomalousdimension was first computed at two loops in [2, 3] and has been recently extended tothree loops in [4, 5], see also [6]. In supersymmetric theories, it is possible to introducea locally supersymmetric Wilson loop that couples to scalars in addition to gluons. In N = φ can be tuned to discuss several interesting physical regimes.At small cusp angle, we are doing perturbation around the straight line configuration, φ =
0, that is half BPS and has no quantum corrections. The first non-trivial term in thesmall angle expansion is Γ cusp ( φ ) = − B ( λ ) φ + O ( φ ) where B ( λ ) is the so-called Brem-strahlung function (depending on the planar ’t Hooft coupling λ ) which is fully known– 1 –11, 12]. From Γ cusp ( φ ) , we can extract the potential for a quark anti-quark pair living on a3-sphere and separated by the angle δ = π − φ . In the limit δ →
0, the flat space potentialis recovered. Finally, one can analytically continue in the cusp angle and consider thelimit ϕ = i φ → ∞ where it turns out that Γ cusp ∼ ϕ Γ ∞ cusp . The coefficients Γ ∞ cusp is theanomalous dimension of a null Wilson loop and is related to the high-spin behaviour ofanomalous dimensions of composite operators [23–25], governed by the celebrated BESexact integral equation derived in [26] by integrability methods in N = N = (cid:82) dt Φ · n | ˙ x ( t ) | , where Φ is the vector of the 6 scalars of N = x ( t ) is the piece-wise straight quark (anti-quark) trajectory [18]. The unitvector n ∈ S is constant apart from a discontinuous turn at the cusp by the angle θ . Acrucial remark made in [9] is that the extra parameter θ may be used to study the scalinglimit i θ → ∞ , (cid:98) λ = λ e i θ fixed. (1.2)The limit (1.2) is interesting because it selects ladder diagrams and, remarkably, the cuspanomalous dimension may be identified with the ground state energy of a 1d Schrödingerequation. In standard notation, it is a function Γ lad ( κ , φ ) = − Ω ( κ , φ ) where κ = (cid:98) λ / π and Ω > (cid:18) − d dw + V ( w , φ ) (cid:19) ψ ( w ) = − Ω ( κ , φ ) ψ ( w ) , w ∈ R , V ( w , φ ) = − κ w + cos φ . (1.3)Despite its simplicity, the ladder approximation is quite interesting and various remark-able feature of the function Γ lad ( κ , φ ) have been investigated at generic φ in [9, 28]. Inthis paper, we reconsider its perturbative expansion in two special limits where we areable to provide new exact results.
Light-like limit x → , where x = e i φ . As remarked in [28], the limit x → Γ lad = ∞ ∑ n = (cid:20) b ( n ) ( x ) x n + O ( x n + ) (cid:21) ( κ /4 ) n , b ( n ) ( x ) = n ∑ k = b ( n ) k log n − k x , (1.4) The quark anti-quark potential is known at 3 loops at weak coupling [13, 14, 9, 15–17, 8] and at one loopat strong coupling [18–21]. It may be treated at all orders by means of the Quantum algebraic curve [22]. The φ → π limit is particularly interesting because it allows to extract the flat space quark-antiquarkpotential. However, it is difficult because the Schrödinger ground state energy is not analytic in the coupling,see [22, 29]. The expansion (1.4) is what is found in the ladder approximation. The true cusp anomaly is linear ∼ log x for x → x , see for instance [10]. – 2 –here b ( n ) is rational, b ( n ) = b ( n ) is a rational multiple of ζ ( ) , b ( n ) is a rational multipleof ζ ( ) , b ( n ) is a rational multiple of ζ ( ) , and all next coefficients b ( n ) k are linear combina-tions of products of simple ζ values with transcendentality degree d = k −
1. The degreed gets a contribution equal to n from ζ n (for even n the involved transcendental constantis π n ) and is additive with respect to multiplication d ( AB ) = d ( A ) + d ( B ) . The expan-sion in (1.4) has been determined at six loops by the algorithm discussed in [28] involvingharmonic polylogarithms and their small x expansions. In that approach, it is non trivialto explore the above properties of the coefficients in (1.4). Here, we study the x → b ( n ) k . They are exact in κ and can be used to systematically obtain longexpansions at higher-loop order. Small angle φ For φ =
0, the ladder approximation reduces to a Schrödinger equation with solvablePöschl-Teller potential, see (1.3). Perturbation theory in φ is analytic and takes the form Γ lad = ∞ ∑ n = c n ( κ ) φ n , κ = (cid:98) λπ . (1.5)In [9], first order Rayleigh-Schrödinger perturbation theory has been applied to providethe results c ( κ ) = − √ + κ c ( κ ) = − κ
16 1 + √ + κ + κ + √ + κ . (1.6)The coefficients in the expansion (1.6) are interesting because they are non-perturbativein the effective ’t Hooft coupling (cid:98) λ = π κ . We show that it is possible to systematicallyimprove (1.6) by implementing a perturbation method originally proposed in [30] thattypically works in the case of polynomial perturbations. This method has the advantage ofbypassing the machinery of the Rayleigh-Schrödinger approach. The resulting algorithmis applied to obtain the coefficient functions c n ( κ ) in closed form for very high n . Theassociated long series expansion is successfully compared with the numerical solution ofthe Schrödinger problem in the whole range of physical parameters κ , φ .The plan of the paper is the following. In Sec. (2) we study the light-like limit and pro-vide a master equation that permits to easily extract the whole logarithmic expansion in(1.4) at any loop order. In Sec. (2.1), we further manipulate the master equation showinghow to determine a compact generating function for the various transcendentality struc-tures appearing in (1.4). In Sec. (3), we treat the small φ perturbative expansion of thecusp anomalous dimension. The higher order results are checked at large κ in Sec. (3.1).In Sec. (3.2), we show that our expansions can be used to provide the correct cusp anomalyat all κ and φ with great accuracy. Various appendices collect long results and related dis-cussions. – 3 – The light-like limit x → As we discussed in the introduction, the first limit we want to treat is x = e i φ →
0. To ex-plain what we are going to compute, it is convenient to recall the results of [28] providingthe weak-coupling expansion (1.4) at 6-loops. We give it in terms of the coefficients Ω n ( x ) appearing in, see their Eq. (3.40), Γ lad = − ∞ ∑ n = (cid:18) κ x − x (cid:19) n Ω n ( x ) . (2.1)When x →
0, the coefficient functions Ω n ( x ) have the structure outlined in (1.4). Writingonly the first four non vanishing leading terms, the six loop results obtained in [28] are Ω ( x ) = − x , Ω ( x ) =
43 log x + ζ log x + ζ + O ( x ) , Ω ( x ) = −
85 log x − ζ log x − ζ log x − ζ log x + O ( ) , Ω ( x ) = x + ζ log x + ζ log x + ζ log x + O ( log x ) , (2.2) Ω ( x ) = − x − ζ log x − ζ log x − ζ log x + O ( log x ) , Ω ( x ) = x + ζ log x + ζ log x + ζ log x + O ( log x ) .The omitted terms have a uniform transcendentality as discussed in the introduction. They can be found in [28] up to 6-loops. Just to give an example, the complete expressionof Ω is Ω ( x ) = x + ζ log x + ζ log x + π
63 log x + (cid:18) π ζ + ζ (cid:19) log x + (cid:18) ζ + π (cid:19) log x + (cid:18) π ζ + π ζ + ζ (cid:19) log x + (cid:18) π ζ (2.3) + ζ ζ + π (cid:19) log x + (cid:18) ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x + (cid:18) π ζ The exact expression at two loops is quite simple and reads Ω ( x ) = − ( x ) + ( x ) log x +
43 log x + π log x + ζ . The degree 4 terms in (2.2) are proportional to π . This is written as a rational multiple of ζ , but ofcourse one may also use ζ . – 4 – ζ + π ζ ζ + ζ ζ + π (cid:19) log x + π ζ + π ζ + ζ ζ + π ζ + π ζ + π ζ + ζ . (2.4)An algorithm to compute the full x -dependence of Ω n has been proposed in [28] and isbased on a recursive representation in terms of harmonic polylogarithms. The light-likelimit can be treated as a special case or by a simplification of the algorithm. The extensionto higher loops is certainly possible, but quite involved. In [28], it has been remarkedthat only powers of single zeta values appear in the asymptotic expansion, at least upto six loops. Here, we show how to generate expansions like (2.3) in a simple way. Wewant to select the logarithmic terms in (2.2) and neglect O ( x ) corrections. To this aim,it is convenient to scale the independent variable in the Schrödinger equation (1.3) andintroduce τ by setting w = Λ τ , where Λ = − log x is a parameter that will be sent to + ∞ .The potential becomes, for τ ≥ V = − κ ( τ log x ) + ( x + x − ) = − κ x + e Λ ( τ − ) + e − Λ ( τ + ) + e − Λ = − κ x + e Λ ( τ − ) + O ( x ) . (2.5)For τ <
0, the potential in (2.5) is extended by τ → − τ symmetry. In the following, weshall restrict to the region τ ≥
0. The potential in (2.5) is a one-dimensional version of theWoods-Saxon confining model. When Λ → + ∞ , we have a negative constant − κ x for0 ≤ τ < τ >
1, see Fig. (1). The Schrödinger equation with the potential
Figure 1 . Plot of the approximate potential + e Λ ( τ − ) . As Λ = − log x is increased, the potentialapproaches a finite depth square well. (2.5) can be solved exactly with boundary conditions ψ (cid:48) ( ) = ψ (+ ∞ ) =
0. (2.6)– 5 –he solution vanishing at infinity is (up to a complex normalization constant) ψ ( τ ) = y i β ( Ω + i β ) Γ ( Ω + i β ) Γ ( + i β ) F (cid:18) − Ω + i β Ω + i β + i β ; − y (cid:19) − c.c. (2.7)where y = x e Λ τ , β = (cid:112) κ x − Ω . (2.8)Imposing the second boundary condition ψ (cid:48) ( ) = x → x , we arrive at the master equation2 cos ( Λ β ) = β + i Ω β − i Ω Γ ( + i β ) Γ ( − i β ) (cid:20) Γ ( Ω − i β ) Γ ( Ω + i β ) (cid:21) + β − i Ω β + i Ω Γ ( − i β ) Γ ( + i β ) (cid:20) Γ ( Ω + i β ) Γ ( Ω − i β ) (cid:21) . (2.9)Expansion of (2.9) is quite simple. One simply writes Ω as a power series in κ (actually κ x ) starting at order O ( κ ) , and uses the definition of β in (2.8). This procedure fullyreproduces the six loop results in (2.2), and can be extended at higher orders. For instance,at seven loops, we find the new expression Ω ( x ) = − x − π x − ζ log x − π x − (cid:18) π ζ + ζ (cid:19) log x − (cid:18) ζ + π (cid:19) log x − (cid:18) π ζ + π ζ + ζ (cid:19) log x − (cid:18) π ζ + ζ ζ + π (cid:19) log x − (cid:18) ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x − (cid:18) π ζ + ζ + π ζ ζ + ζ ζ + π (cid:19) log x − (cid:18) π ζ + π ζ + ζ ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x (2.10) − (cid:18) ζ + π ζ + π ζ + π ζ ζ + π ζ ζ + ζ ζ + ζ ζ + π (cid:19) log x − π ζ − ζ ζ − π ζ − π ζ ζ − π ζ − ζ ζ − π ζ − π ζ − π ζ − ζ + O ( x ) .The similar 8-loop result is collected in App. (A).– 6 – .1 Generating functions and transcendentality expansion Given the plain structure in (2.2) and (2.10), it is tempting to understand what is the gen-erating function for the various coefficients b ( n ) k , see (1.4). This may be achieved by meansof the following trick. The Γ functions in the r.h.s. of (2.9) are the only source of transcen-dental contributions. We can consider the r.h.s. of (2.9) with fixed ratio β / Ω and expandat small β the resulting expression. This expansion readsr.h.s. of (2.9) = β − Ω β + Ω − β Ω ( β + Ω ) π + β Ω ζ (2.11) + β Ω ( Ω − β ) ( β + Ω ) π + β Ω (cid:20) π ζ ( β − Ω ) + ζ ( Ω − β ) (cid:21) ++ β ( β + Ω ) (cid:20) π Ω ( β + β Ω + Ω ) − ζ ( β − Ω ) ( β + Ω ) (cid:21) − β Ω (cid:20) − ζ ( Ω − β ) + π ζ ( β − β Ω + Ω )+ π ζ ( β + β Ω + Ω ) (cid:21) + · · · ,and additional contributions may be obtained with no problems. In (2.11), we have writ-ten in bold face the transcendental constants. The terms in (2.11) are naturally ordered byincreasing transcendentality degree (we remind that π n and ζ n contribute n units). Thismeans that we can decompose Ω as a sum of contributions of increasing degree and solve(2.9) for each of them. The first term in (2.11) has degree zero and gives the leading ordercondition (cid:113) κ x − ω tan (cid:18) log x (cid:113) κ x − ω (cid:19) + ω =
0. (2.12)Expanding (2.12) at small x gives ω = − x log x κ + x log x (cid:18) κ (cid:19) − x log x (cid:18) κ (cid:19) + x log x (cid:18) κ (cid:19) − x log x (cid:18) κ (cid:19) + x log x (cid:18) κ (cid:19) + · · · . (2.13)Comparing (2.13) with (2.2), we see that the compact relation (2.12) captures all the lead-ing logarithms for x → ω = f ( κ x ) with2 k (cid:20) log x f ( k ) − (cid:21) f (cid:48) ( k ) + f ( k ) − log x k =
0. (2.14)Starting with f ( k ) = − x k + · · · , one gets from (2.14) a simple recursion for thecoefficients in (2.13). The convergence properties of the expansion (2.13) are discussed inApp. (B). To illustrate what happens beyond the leading (rational) logarithms, we present– 7 –he contributions with transcendentality up to 5, i.e. proportional to π , ζ , π , ζ , and π ζ as an illustrative example. After some some straightforward manipulations, we getfrom (2.9) using (2.11) Ω = ω + ζ ω ( κ x − ω ) ω log x − − ζ κ x ( κ x − ω ) ( ω log x − ) − ζ ω ( κ x − ω ) ( ω log x − ) × (2.15) (cid:20) log x ω ( κ x − ω ) − log x ω ( κ x − ω ) + ( κ x − ω ) (cid:21) + ζ κ x ( κ x − ω ) ( κ x − ω ) ( ω log x − ) + π ζ κ x ( κ x − ω )( κ x + x ω − ω ) ( ω log x − ) + · · · ,where again we have written in bold face the transcendental constants. Dots in (2.15)stand for terms with transcendentality d >
5. This is a compact generating function forthe considered terms. If we plug (2.13) into (2.15), we recover the associated contributionsin (2.2). For instance, one obtains immediately the following expansions valid up to 14loops (recall that Ω and Ω are given in (2.10) and (A.1) respectively) Ω = − x − ζ log x − ζ log − ζ log − (cid:18) π ζ + ζ (cid:19) log x + . . . Ω = x + ζ log x + ζ log x + ζ log x + (cid:18) π ζ + ζ (cid:19) log x + . . . , Ω = − x − ζ log x − ζ log x − ζ log x − (cid:18) π ζ + ζ (cid:19) log x + . . . , Ω = + ζ log x (2.16) + ζ log x + ζ log x + (cid:18) π ζ + ζ (cid:19) log x . . . , Ω = − x − ζ log x − ζ log x − ζ log x − (cid:18) π ζ + ζ (cid:19) log x + . . . , Ω = x + ζ log x + ζ log x + ζ log – 8 – (cid:18) π ζ + ζ (cid:19) log x + . . . .and so on. Having long series of this kind allows to recognize some interesting pattern.For instance, the ratio of the NLO logarithm coefficient (proportional to ζ ) to the coeffi-cient of the LO logarithm is simply, see (1.4) b ( n ) b ( n ) = ( n − )( n − ) n ζ , (2.17)where n is the loop order. This relation can be proved rigorously starting from (2.14)and converting the factors in (2.17) into differential operators n → κ∂ κ . Another simplerelation concerns the ratio of the NNLO logarithm coefficient (proportional to ζ ), b ( n ) b ( n − ) = ( − n ) ζ , (2.18)where the shift in the index of b is a non trivial fact. The vanishing b ( n ) = φ To study the problem (1.3) at small φ , we begin by setting y = w + y w + cos φ = y + y φ + (cid:18) y − y (cid:19) φ + (cid:18) y − y + y (cid:19) φ + · · · . (3.2)The kinetic term is also simple ψ (cid:48)(cid:48) ( w ) = y ( − y ) ψ (cid:48)(cid:48) ( y ) + y ( − y ) ψ (cid:48) ( y ) . (3.3)The exact ground state wavefunction at φ = ψ ( y ) = y Ω /2 , with κ = Ω ( Ω + ) . (3.4) The invertibility of (3.1) is not an issue here. Our aim is to write the wave function in terms of the variable y that will capture in a simple way the dependence on w . Boundary conditions are obvious from (1.3). – 9 –e now make the educated perturbative Ansatz ψ ( y , φ ) = y Ω /2 (cid:18) + ∞ ∑ n = f n ( y ) φ n (cid:19) , κ = Ω ( Ω + ) + ∞ ∑ n = δκ n ( Ω ) φ n , (3.5)where δκ n ( Ω ) are functions of the ground state energy Ω .We emphasize that it is crucial to set up the perturbative expansion according to (3.5), i.e. expanding κ , instead of writing a more natural perturbative expansion of Ω as a func-tion of κ . This has the advantage that the unperturbed ground state in (3.4) – appearing asa factor in (3.5) – is not changed during the procedure. Replacing (3.5) into the Schrödingerequation, we get y − Ω (cid:112) − y ddy (cid:20) y Ω + (cid:112) − y f (cid:48) ( y ) (cid:21) + y Ω ( Ω + ) + δκ =
0, (3.6) y − Ω (cid:112) − y ddy (cid:20) y Ω + (cid:112) − y f (cid:48) ( y ) (cid:21) + y Ω ( Ω + ) + f ( y ) (cid:18) δκ + y Ω ( Ω + ) (cid:19) + y ( y − ) Ω ( Ω + ) + δκ + y δκ = · · · and so on. Clearly, the equations in the chain (3.6) can be integrated one after the other.Imposing boundary conditions, we fix the coefficients δκ n . Actually, inspection of theresults shows that the general form of the corrected wave-function is ψ ( y , φ ) = y Ω /2 (cid:18) + ∞ ∑ n = φ n n ∑ k = a n , k y k (cid:19) . (3.7)In other words, the corrections f n ( y ) are simple polynomials in y ! This remarkable fea-ture is recurrent in quantum mechanical problems with a perturbation in the form of apolynomial, see the original proposal in [30] or, for instance, [31]. The explicit solution forthe first three non trivial corrections is f ( y ) = y Ω ( Ω + ) Ω +
12 , f ( y ) = y Ω ( Ω + )( Ω + )( Ω + ) ( Ω + )( Ω + ) − y Ω ( Ω + ) (cid:0) Ω + Ω + Ω + (cid:1) ( Ω + ) ( Ω + ) , (3.8) f ( y ) = y Ω ( Ω + )( Ω + )( Ω + )( Ω + )( Ω + ) ( Ω + )( Ω + )( Ω + ) − y Ω ( Ω + )( Ω + )( Ω + ) (cid:0) Ω + Ω + Ω + (cid:1) ( Ω + ) ( Ω + )( Ω + ) + y Ω ( Ω + ) (cid:0) Ω + Ω + Ω + Ω + Ω + Ω + (cid:1) ( Ω + ) ( Ω + )( Ω + ) .(3.9)and δκ = − Ω ( Ω + ) Ω + κ = Ω ( Ω + ) (cid:0) Ω + Ω + Ω + (cid:1) ( Ω + ) ( Ω + ) , (3.10) δκ = − Ω ( Ω + ) (cid:0) Ω + Ω + Ω + Ω + Ω + Ω + (cid:1) ( Ω + ) ( Ω + )( Ω + ) .The expansion of κ in (3.5), may be turned into an expansion of Ω according to Ω = t − + ( t − )( t + ) φ t ( t + ) +( t − )( t + ) (cid:0) t + t + t + t + t + (cid:1) φ t ( t + ) ( t + ) + (3.11) ( t − )( t + ) t ( t + ) ( t + ) ( t + ) (cid:16) t + t + t + t + t + t + t + t + t + t + t + (cid:17) φ + . . .where t = √ κ +
1. The general structure of the higher order corrections is Ω = ∞ ∑ n = P n ( t ) D n ( t ) φ n , (3.12)where D n ( t ) is the rational function D n ( t ) = ( t + ) ( t − )( t + ) (cid:2) ( n + ) − t n − Γ ( ( n + ) − ) (cid:3) ∏ n + j = t ( t + )( j + t ) − j + n + , (3.13)and P n ( t ) are polynomials. The first cases are collected in App. (D). They have an increas-ing complexity, but may be generated quite easily by the above procedure. κ The expansion (3.11) may be checked at weak coupling, i.e. in the limit κ →
0, using thesix-loop expressions derived in [28]. At strong coupling, we can evaluate systematicallythe perturbative expansion of the (ladder) cusp anomaly for any φ . This can be achievedby the same strategy described in the previous section, i.e. by an Ansatz similar to (3.5).We begin by multiplying the Schrödinger equation by µ / κ with µ = √ κ ( cos φ + ) .After a rescaling x = µ X / √ κ , we obtain (cid:18) − d dX + V ( X , φ ) (cid:19) ψ ( X ) = E ψ ( X ) , V ( X , φ ) = − √ κ + X + cos φ − √ κ X + −
56 cos φ + cos ( φ ) κ X + . . . , E = − cos φ + √ κ Ω . (3.14)The perturbative correction to the ground state energy E of the (formal) problem − ψ (cid:48)(cid:48) ( X ) + (cid:18) X + ∞ ∑ n = ε n − c n X n (cid:19) ψ ( X ) = E ψ ( X ) , (3.15)– 11 –an be found efficiently by the Ansatz ψ ( X ) = e − X (cid:18) + ∞ ∑ n = p n ( X ) ε n (cid:19) , (3.16)where p n ( X ) is an even polynomial with degree 4 n and without constant term. The firstthree cases are explicitly p ( X ) = − c X − c X p ( X ) = (cid:0) c − c (cid:1) X + (cid:0) c − c (cid:1) X + (cid:0) c − c (cid:1) X + c X , p ( X ) = − (cid:0) c − c c + c (cid:1) X + (cid:0) − c + c c − c (cid:1) X + (cid:0) − c + c c − c (cid:1) X + (cid:0) − c + c c − c (cid:1) X − c (cid:0) c − c (cid:1) X − c X ,and lead to the perturbed energy E = + c ε − ( c − c ) ε + ( c − c c + c ) ε + · · · . (3.18)The associated expansion of the cusp anomalous dimension is therefore Γ lad = √ κ φ (cid:20) − √ κ + − cos φ κ + κ − +
20 cos φ + cos ( φ ) + κ −
317 cos φ − ( φ ) − ( φ ) + · · · (cid:21) , (3.19)in full agreement with what one finds by expanding (3.11) at large t . In this section, we compare the exact numerical solution of the Schrödinger problem (1.3)with the small φ expansion of its ground state energy, see (3.12). We shall consider threereference values κ =
1, 2, 5 and explore convergence with respect to φ in the physicalinterval 0 ≤ φ < π . The first comparison is with the naive partial sums of (3.12), ( recallthat Γ lad = − Ω ) Ω N = N ∑ n = P n D n φ n . (3.20)This is shown in the left panel of Fig. (2). One sees that including terms up to N = φ not too close to π . As φ → π , convergence slows down and the series expansion cannot provide an accurateestimate of the correct (ladder) cusp anomaly. This is clearly illustrated by the points at φ = π . However, the strong coupling expansion (3.19) suggests that the singularity at φ = π is simply due to an overall factor 1/ cos ( φ /2 ) . From the physical point of view,– 12 – igure 2 . The plots show the convergence of the perturbative expansion of Ω at small φ for κ =
1, 2, 5. The many thin dashed lines show results with lower values of N in (3.20) (left panel) and(3.21) (right panel). The four red dots are the (numerical) exact values of Ω obtained solvingnumerically the Schrödinger equation at φ = π , π , π , π , and π , see also Tab. (1). the φ → π limit is a flat space limit where that singularity is nothing but the overall scale1/ r in the quark-antiquark at distance r ∼ ( φ ) [9]. Hence, we also compare thenumerical values of the cusp anomaly with the improved summation (cid:101) Ω N = φ (cid:34) cos φ N ∑ n = P n D n φ n (cid:35) truncated at φ N . (3.21)The right panel of Fig. (2) shows that this is a major improvement. The convergence in N is now greatly increased and accurate results are obtained even quite near the singularpoint φ = π . This is illustrated in a more quantitative way in Tab. (1) where we collectsome reference numerical values shown in Fig. (2). In all cases, the relative accuracy of(3.21) is well below the 10 − level. In this paper we have considered various properties of the perturbative expansion ofthe cusp anomalous dimension in N = x = e i φ and (ii) small φ corrections. In the former case, we showed that all the log-arithmic corrections are captured by a Wood-Saxon type solvable problem. Besides, wehave shown how to generate all such corrections at higher-orders by compact generatingfunctions that encode them and are naturally organised in increasing transcendentalitydegree. Our approach explains the remarkable regularities observed in the past. In the– 13 – φ Ω exact Ω N = (cid:101) Ω N = π π π π π π π π π π π π π π π Table 1 . This table compares the exact values of Ω with the approximations in (3.20) and (3.21)for N =
20. The comparison is done at κ =
1, 2, 5 and various φ . The first column contains the(numerical) exact value corresponding to the red dots in Fig. (2). Digits highlighted in blue are inagreement with the exact solution. small φ regime, we showed that is possible to work out the quantum mechanical pertur-bation expansion bypassing the Rayleigh-Schrödinger scheme. This is due to the simplestructure of the perturbed wave-function associated with the ground state. Our remarkleads to a quite efficient algorithm. The associated long expansion in powers of φ hasbeen shown to provide, after some educated manipulations, an accurate representationof the ladder cusp anomaly in the whole range of couplings and angles. We believe thatit would be very interesting to look at our results from the perspective of the QuantumSpectral Curve by extending the analysis of [22] to the case of a generic cusp angle φ . Ournew results could certainly be useful as a non-trivial check of that method. A Complete expression of the eight-loops term Ω Ω ( x ) = x + π log x + ζ log x – 14 – π log x + (cid:18) π ζ + ζ (cid:19) log x + (cid:18) ζ + π (cid:19) log x + (cid:18) π ζ + π ζ + ζ (cid:19) log x + (cid:18) π ζ + ζ ζ + π (cid:19) log x + (cid:18) ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x + (cid:18) π ζ + π ζ ζ + ζ ζ + ζ + π (cid:19) log x + (cid:18) π ζ + ζ ζ + π ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x + (cid:18) ζ + π ζ + π ζ ζ + π ζ ζ + ζ ζ + π ζ + ζ ζ + π (cid:19) log x + (cid:18) π ζ + π ζ ζ + ζ ζ + ζ ζ + π ζ + π ζ + π ζ + π ζ + π ζ + ζ (cid:19) log x + (cid:18) π ζ + ζ ζ + π ζ + π ζ ζ + π ζ ζ + π ζ ζ + ζ ζ + π ζ + ζ + π ζ ζ + ζ ζ + π (cid:19) log x + ζ + π ζ + ζ + π ζ ζ + π ζ + π ζ ζ + π ζ + π ζ ζ + ζ ζ ζ + π ζ + ζ ζ + π ζ + π ζ + π ζ + ζ + O ( x ) . (A.1) B Convergence of the expansion (2.13)
The expansion (2.13) solves (2.12). The dependence on powers of x and log x is clearlytrivial. Indeed, introducing the variables κ = κ x log x , ω = ω log x , (B.1)we can rewrite (2.12) as (cid:113) κ − ω tan (cid:18) (cid:113) κ − ω (cid:19) + ω =
0. (B.2)– 15 –his equation admits a solution ω ( κ ) that is analytic in a disc of radius R . This conver-gence radius is determined by a branch point at κ < κ = − R >
0, we determine R byeliminating w in the two equations F ( R , w ) = w − (cid:112) R + w tanh (cid:18) (cid:112) R + w (cid:19) = ∂∂ w F ( R , w ) =
0. (B.3)This gives w =
2. Then, R is found as the unique positive root of √ R + (cid:18) √ R + (cid:19) =
2, (B.4)that is R = | κ | < R / ( x log x ) . C Small x limit as perturbation around a finite depth well The potential in (2.5) can also be split in the following way, where we have again rescaled w by log x introducing w = τ log x for τ > − x cosh ( τ log x ) + ( x + x − ) = − Θ ( − τ ) + ( − τ ) + e − log x | τ − | + . . . . (C.1)where Θ ( τ ) is the Heaviside step function. In other words, the potential looks like aFermi-Dirac distribution. The unperturbed shape is a finite depth well, while the correc-tion – the second term in (C.1) – captures its deviation in the small strip τ − ∼
1/ log x .The solution of the Schrödinger equation for the finite depth well is elementary andreads, for τ > ψ ( τ ) = A cos (cid:18) log x √ κ x − Ω τ (cid:19) , 0 < τ < B exp (cid:18) Ω log x τ (cid:19) , τ >
1. (C.2)At τ =
1, continuity fixes the ratio A / B , while continuity of ψ (cid:48) determines the relationbetween Ω and κ to be precisely (2.12).This first approximation may be improved by taking into account the correction in(C.1). Treating it at first order in perturbation theory amounts to compute its integraltimes | ψ ( τ ) | . But this is accomplished as follows (we omit a trivial factor 2) (cid:90) ∞ d τ sign ( − τ ) + e − log x | τ − | F ( τ ) = (cid:90) d τ F ( τ ) + e − log x ( − τ ) − (cid:90) ∞ d τ F ( τ ) + e − log x ( τ − ) = − x (cid:90) − log x d µ F ( + µ log x ) + e µ + x (cid:90) ∞ d µ F ( − µ log x ) + e µ (C.3)– 16 –p to corrections O ( x ) – from the upper integration limit in the first integral – we canexpand at small x and obtain (cid:90) ∞ d τ sign ( − τ ) + e − log x | τ − | F ( τ ) = (cid:90) ∞ d µ + e µ (cid:20) F ( − ) − F ( + ) log x − µ F (cid:48) ( − ) + F (cid:48) ( + ) log x + µ F (cid:48)(cid:48) ( − ) − F (cid:48)(cid:48) ( + ) x + · · · (cid:21) . (C.4)In our application F ( τ ) = | ψ ( τ ) | , and F (cid:48) ( τ ) are continuous at τ =
1, so (cid:90) ∞ d τ sign ( − τ ) + e − log x | τ − | F ( τ ) = − ζ log x F (cid:48) ( ) + ζ log x [ F (cid:48)(cid:48) ( − ) − F (cid:48)(cid:48) ( + )] + · · · . (C.5)Collecting all pieces, we obtain the simple formula Ω = Ω + κ x ( δ E + δ E + + δ E − ) , (C.6)where Ω is the leading order, i.e. the solution of (2.12), and δ E = ζ N ψ ( ) ψ (cid:48) ( ) log x , δ E ± = ∓ ζ N ψ ( ) ψ (cid:48)(cid:48) ( ± ) + ψ (cid:48) ( ) log x , N = (cid:90) ∞ d τ | ψ ( τ ) | . (C.7)After some simplification, it is possible to show that (C.6) is indeed equivalent to the firstthree terms of (2.15). In this approach, the appearance of the simple transcendental zetavalues if simply due to the elementary integral (cid:90) ∞ d µ µ n + e µ = ( − − n ) n ! ζ n + . (C.8)The leading perturbative calculation in (C.6), already captures the exact expansion atNNLO. The second order perturbation with respect to the second term in (C.1) will givea ζ contribution that mixes with a genuine ζ term from dots in (C.5). Despite its sem-plicity, this method cannot be extended in a simple way to higher orders because of thecomplexity of the expressions for the higher order correction to the energy. These involvethe full spectrum as well as infinite sums over the unperturbed wave-functions. However,as we discussed in the main text, this complexity is only apparent due to the solvabilityof the potential in (2.5). D List of higher order corrections for small φ up to O ( φ ) P ( t ) = P ( t ) = t + t + t + t + t + P ( t ) = t + t + t + t + t + t + t + t + t + t + t + P ( t ) = t + t + t + t + t + t + t + – 17 – t + t + t + t + t + t + t + t + t + t + t + P ( t ) = t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + P ( t ) = t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + P ( t ) = t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + – 18 – eferences [1] A. M. Polyakov, Gauge Fields as Rings of Glue , Nucl. Phys.
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