PPreprint typeset in JHEP style - PAPER VERSION
November 2014QMUL-PH-14-26
Dilogarithm ladders from Wilson loops
Marco S. Bianchi a and Matias Leoni b , ca Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of London, Mile End Road, London E1 4NS, UK b Physics Department, FCEyN-UBA & IFIBA-CONICET Ciudad Universitaria,Pabell´on I, 1428, Buenos Aires, Argentina c Instituto de F´ısica de La Plata, CONICET, UNLP C.C. 67, 1900 La Plata,ArgentinaE-mail: [email protected], [email protected]
Abstract:
We consider a light-like Wilson loop in N = 4 SYM evaluated on a regular n -polygon contour. Sending the number of edges to infinity the polygon approximatesa circle and the expectation value of the light-like WL is expected to tend to thelocalization result for the circular one. We show this explicitly at one loop, providinga prescription to deal with the divergences of the light-like WL and the large n limit.Taking this limit entails evaluating certain sums of dilogarithms which, for a regularpolygon, evaluate to the same constant independently of n . We show that this occursthanks to underpinning dilogarithm identities, related to the so-called “polylogarithmladders”, which appear in rather different contexts of physics and mathematics andenable us to perform the large n limit analytically. Keywords:
Wilson loops, Dilogarithm identities. a r X i v : . [ h e p - t h ] N ov ontents
1. Introduction 12. Contour parametrization 33. One-loop integrals 44. Dilogarithm identities and circular limit 65. Higher order in dimensional regularization parameter 9
1. Introduction
In this note we consider a limit on light-like Wilson loops on a regular polygonalcontour in N = 4 SYM. By regular we mean that all the light-like sides of the polygonhave equal euclidean norm. Then, sending the number of edges to infinity, we expectthe contour to approximate a smooth circle. Consequently, the expectation value ofthe light-like Wilson loop is supposed to reproduce that of the -BPS circular one,whose exact expression is known from a matrix model computation [1] as a result oflocalization [2].Light-like Wilson loops in N = 4 SYM are dual to MHV scattering amplitudes [3–6]. The fact that the limit we are taking on the light-like Wilson loop is known, impliesin turn that the same limit should also hold for MHV scattering amplitudes, with specialkinematics. This could be in principle used as a check on potential expressions for MHVscattering amplitudes of N = 4 SYM for any number of external particles, though thelimit could be hard to perform analytically in practice. Nevertheless, a similar argumentwas used in [7] (approximating a rectangular Wilson loop by a sequence of light-likesegments) to find an inconsistency of the BDS proposal [8] for multileg amplitudes atstrong coupling [9].Despite the simplicity of this idea we are not aware of any direct computation inthe literature where such kind of limit has been explicitly checked perturbatively. Herewe provide an explicit computation of the expectation value of a N = 4 SYM light-likeWilson loop approximating the circular one, at one loop in perturbation theory.– 1 –ight-like Wilson loops suffer from ultraviolet divergences, which are dual to in-frared singularities of scattering amplitudes. At one loop, these arise from gluon ex-changes between adjacent edges. Therefore one has to introduce a regularization todeal with them. In particular we use dimensional regularization. On the other handthe -BPS Wilson loop evaluated on a smooth contour, such as a circle, is finite [1]. Ininterpolating between the two the limit of infinite number of edges is taken. We give aprescription to deal with this limit in such a way that the result is finite, as expected.This effectively removes any contribution from divergent diagrams, leaving a sum overfinite ones, where we can set the regularization parameter to zero. Remarkably, oncethis is done, and for the particular regular contour we have chosen, the result of the sumover these one-loop contributions turns out to be constant, i.e. independent of n . Sincethe integrals involved in the computation are expressible in terms of dilogarithms, thisimplies in turn identities for the sum of dilogarithms evaluated at particular values oftheir arguments. When these are powers of the same algebraic number, such identitiesare called dilogarithm ladders and there exists a vast literature on these (see [10, 11] fora review, or [12] and [13] for more recent developments). This is not precisely the kindof identities we encounter, though they should be related by use of the Abel identity.Nevertheless, other identities relating combinations of dilogarithms were studied in thepast, emerging from various problems of mathematical physics, such as in the contextof Heisenberg spin chains [14], two-dimensional integrable and lattice models [15–17]and CFT’s [18, 19] whose central charge can be expressed as a sum of dilogarithmsevaluated at particular algebraic numbers ∗ . We find that these dilogarithm identities,which have been proved [21, 22], include those we face in our computation as particu-lar subcases. Thanks to these results we can analytically prove that the limit on theregular polygon Wilson loops converges to the one-loop expansion of the localizationexpression for the circular one.At one loop one can straightforwardly compute the expectation value of the circu-lar Wilson loop by a Feynman diagram computation. Insisting on using dimensionalregularization we can extend the comparison of the light-like and circular Wilson loopsto finite values of the regularization parameter (cid:15) . In this case we still verify that thefinite part of the light-like Wilson loop converges to the circular one in the large n limit,for any value of (cid:15) , although the two results do not coincide at finite n , in contrast withthe (cid:15) → ∗ For a more comprehensive list of references and applications of dilogarithms in physics and math-ematics we refer the reader to [20] and the references therein. – 2 – a) n = 16 (b) n = 32(c) n = 64 Figure 1: Examples of the contour for n = 16, 32 and 64. The polygon is formedby connecting with light-like lines two sets of points lying on a circle at t = 0 and at t = 2 sin πn respectively ( t is the vertical axis in the figures). In the limit n → ∞ thecontour becomes a space-like circle of radius 1.
2. Contour parametrization
In this section we derive a parametrization of the polygonal contour with n edges, onwhich we want to evaluate the light-like Wilson loop. We restrict our analysis to regularlight-like polygons P = (cid:83) ni =1 x i,i +1 with an even number of edges n . We parametrizetheir vertices as follows x k +1 = (cid:18) πn , sin 2 π (2 k + 1) n , cos 2 π (2 k + 1) n (cid:19) x k = (cid:18) , sin 4 πkn , cos 4 πkn (cid:19) , k = 0 , , . . . n/ − t = 2 sin πn in the time direction.The overall radii of the circles do not play any role thanks to conformal invariance andare set to unity (see Figure 1).Given this parametrization, the relevant invariants on which the Wilson loop candepend on can be separated into two categories: odd-to-odd and odd-to-even distances(even-to-even are equal to odd-to-odd by symmetry). They are evaluated from (2.1) x k, l = 4 sin π ( k − l ) nx k, l +1 = 4 sin π (2 k − l − n − πn (2.2)For a polygon with n edges there are n/ n/ −
1) odd-to-even indepen-dent distances if n = 0 mod n − / n = 2 mod
4, as reviewed in Table 1.– 3 – x k, l x k, l +1 ↑↑ + ↓↓ ↑↓ + ↓↑ m m m − m (2 m −
1) 4 m m + 2 m m m (2 m + 1) (2 m + 1) Table 1: For different number of edges we report the number of independent invariants(central columns) and the number of diagrams with gluon exchanges between edgesseparated by an odd (segments pointing both to the future or to the past) or even(segments pointing in opposite directions in time) number of sides.
3. One-loop integrals
In this section we evaluate the expectation value of the light-like Wilson loop on thepolygon P at one loop in perturbation theory. We use the position space gluon propa-gator in dimensional regularization (cid:104) A µ ( x ) A ν ( y ) (cid:105) = − Γ(1 − (cid:15) )4 π − (cid:15) η µν [ − ( x − y ) ] − (cid:15) (3.1)where η µν is the Minkowski metric. An overall factor ( ig ) N = λ will be understood inthe following. There is only one kind of integral to be considered at one loop [6] I (1) i,j ≡ Γ(1 − (cid:15) )8 π − (cid:15) (cid:90) dτ i dτ j P + Q − s − t [ − ( P + τ i ( s − P ) + τ j ( t − P ) + ( P + Q − s − t ) τ i τ j )] − (cid:15) (3.2)which is conveniently expressed in terms of the distances s ≡ x i,j , t ≡ x i +1 ,j +1 , P ≡ x i +1 ,j and Q ≡ x i,j +1 , as in Figure 2. Its limit where s, t, P →
0, namely whenever thegluon is exchanged between two adjacent edges, yields divergent contributions whichcan be evaluated in dimensional regularization I (1) i,i +1 = −
12 Γ(1 − (cid:15) )4 π − (cid:15) ( − Q ) (cid:15) (cid:15) (3.3)Whenever a gluon is exchanged between two edges separated by only one light-like side,we have, e.g., P = 0. The corresponding contribution can be smoothly obtained as alimit of the integral (3.4). For nonvanishing invariants this integral is finite and can beevaluated at (cid:15) = 0 I (1) i,j = 18 π (cid:2) − Li (1 − as ) − Li (1 − at ) + Li (1 − aP ) + Li (1 − aQ ) (cid:3) (3.4)– 4 – s P ti i + 1 jj + 1 Figure 2: One-loop contribution to the Wilson loop.where a ≡ P + Q − s − tP Q − st (3.5)Using regular polygons yields further simplifications. First, thanks their symmetrieswe have that s = t in each exchange. Second, the integral only depends on the integernumber ∆ i , which is the difference between the labels of the two edges, between whichthe gluon is exchanged. We will call it ∆ i , hereafter. At one loop, there is a distinc-tion between gluon exchanges connecting edges pointing in the same time direction orpointing in the opposite. Let us analyze the latter case first: we have that P Q = st ,meaning that a diverges. The limit a → ∞ cannot be taken from the result (3.4).Rather, it is more convenient to implement this condition on (3.2) directly. When thisis done the integrand factorizes and can be integrated straightforwardly I (1)2 k, l +1 = I (1)2 k +1 , l = −
12 Γ(1 − (cid:15) )4 π − (cid:15) ( − P ) − (cid:15) (cid:104)(cid:16) P s (cid:17) (cid:15) − (cid:105) (cid:15) (3.6)Even though the result is finite, we evaluate it in dimensional regularization for futureconvenience.Finally, the case where the gluon is exchanged between two edges pointing in thesame direction is completely regular and we can use (3.4) directly.– 5 –s a whole we have I (1) (∆ i ) = −
12 Γ(1 − (cid:15) )4 π − (cid:15) (cid:15) (cid:0) πn (cid:1) (cid:15) ∆ i = 1 −
12 Γ(1 − (cid:15) )4 π − (cid:15) (cid:15) (cid:16) π (∆ i − n (cid:17) − (cid:15) (cid:20)(cid:18) sin π (∆ i − n sin π ∆ in − sin πn (cid:19) (cid:15) − (cid:21) ∆ i odd π (cid:20) Li (cid:18) cos π (∆ i +1) n cos πn (cid:19) + Li (cid:18) cos π (∆ i − n cos πn (cid:19) − (cid:16) − sin π ∆ in cos πn (cid:17)(cid:21) ∆ i even (3.7)
4. Dilogarithm identities and circular limit
With a little combinatorics on the polygons, we can simplify the final expression forthe one-loop correction of the Wilson loop expectation value. Gluons can be exchangedbetween edges separated by ∆ i = 1 , , . . . n/
2. Even and odd separations correspondto different cases and are treated separately. For each separation there are n differentgluon exchanges, apart from the extremal case ∆ i = n/
2, where there are only n/ i are equal to each other. This can be used to reduce the double sumsover the indices of the edges of gluon exchanges, implicit in the total contribution, tosingle sums over the separations. Then the combinatorics vary according to whetherthe total number of edges n = 0 mod n = 2 mod
4, as reviewed in the Table 1.The expression for the one-loop Wilson loop expectation value reads in all cases (cid:104) W (cid:105) (1) = − n
12 Γ(1 − (cid:15) )4 π − (cid:15) (cid:15) (cid:18) πn (cid:19) (cid:15) + − n n/ − (cid:88) k =1
12 Γ(1 − (cid:15) )4 π − (cid:15) (cid:15) (cid:18) πkn (cid:19) − (cid:15) (cid:34)(cid:32) sin πkn sin π (2 k +1) n − sin πn (cid:33) (cid:15) − (cid:35) ++ n n/ − (cid:88) k =1 π (cid:34) Li (cid:32) cos π (2 k +1) n cos πn (cid:33) + Li (cid:32) cos π (2 k − n cos πn (cid:33) − (cid:32) − sin πkn cos πn (cid:33)(cid:35) (4.1)In the first line the n divergent contributions from exchanges between adjacent edgesappear. The second line represents the remaining n/ n/ −
2) terms from even-to-oddexchanges. They sum up for a total of ( n/ terms, as in the last column of Table 1.Finally the third line is given by the n/ n/ −
1) contributions from odd-to-odd andeven-to-even contributions, as in the fourth column of Table 1.– 6 –ext we take the n → ∞ limit of such an expression. In this regime, we expectthe light-like polygonal contour to approximate a circle. The supersymmetric Wilsonloop on a circle has been given an exact result through localization on a four-sphere,yielding a Gaussian matrix model. Such a Wilson loop operator features a coupling tothe scalar fields of N = 4 SYM. On the other hand, for a light-like contour such termsare dropped and one recovers the standard Wilson loop. Even with this difference inthe couplings, we expect the expectation value of the light-like polygonal Wilson loopto tend to that of the circular one in the large n limit.Since the circular Wilson loop is finite, we expect the ultraviolet divergences fromthe cusps of the light-like Wilson loop to drop out in such a regime. In order toensure this, we take this limit on the dimensionally regularized result (3.3). In the n → ∞ limit the invariants vanish and the relevant integral becomes scaleless andis thus discarded in dimensional regularization. This result is still to be summed onorder n terms with n tending to infinity, meaning that choosing a correct prescriptionamounts to an order of limits problem. We take the n → ∞ first, then in the spiritof dimensional regularization we analytically continue the result from the region in (cid:15) space where the sum converges. Eventually we take the (cid:15) → (cid:15) →
0, which ensures that theseterms also go to 0 in the large n limit.Finally the most interesting contribution comes from the even-to-even terms. In-deed this piece evaluates to a constant for any value of n , namely n n/ − (cid:88) k =1 (cid:34) Li (cid:32) cos π (2 k +1) n cos πn (cid:33) + Li (cid:32) cos π (2 k − n cos πn (cid:33) − (cid:32) − sin πkn cos πn (cid:33)(cid:35) = π (4.2)Surprisingly, even if one would have expected an n -dependent expression which shouldevaluate to π in the limit of large n , a numerical inspection shows that the left handside of (4.2) evaluates to π for any value of n . For the moment this is just a case bycase empirical observation that we extrapolate to be valid for n → ∞ (we shall laterprove this is actually correct by recasting the result in terms of know identities). Inconclusion, performing the large n limit is trivial and, taking into account the 1 / (8 π )factor in (4.1), the expectation value of the Wilson loop gives 1 /
8. This coincides withthe first order contribution to the circular Wilson loop at weak coupling (cid:104) W (cid:13) (cid:105) = 1 + λ . . . (4.3)Besides triggering the limit of the light-like Wilson loop to the circular one, theidentity (4.2) is very interesting by itself, since it relates combinations of dilogarithms– 7 –t particular values of their arguments. Such relations are of mathematical interestand there exists a vast literature on them. When they relate polylogarithms of thesame weight evaluated at powers of some algebraic number φ , they are known as poly-logarithm ladders. Such identities are usually written more compactly in terms of theRogers function L which is defined to beL( x ) ≡ Li ( x ) + 12 log x log(1 − x ) (4.4)for 0 ≤ x ≤ x through the reflectionand inversion identities L( x ) = π − L (cid:18) x (cid:19) x > x ) = L (cid:18) − x (cid:19) − π x < x ) + L( y ) = L( xy ) + L (cid:18) x (1 − y )1 − xy (cid:19) + L (cid:18) y (1 − x )1 − xy (cid:19) (4.6)for 0 < x, y <
1. As an example of an early dilogarithm ladder we quote one of Watson’sidentities [23] 2L( β ) + L( β ) = 107 L (1) (4.7)where β = sec π is one of the roots of the equation x − x − x + 1 = 0 (4.8)We can now inspect formula (4.2) in order to see if it can be related to known iden-tities in such a way to analytically prove the agreement with the circular Wilson loop.It is convenient to analyze the n = 2 ( mod
4) and n = 0 ( mod
4) cases separately † . Fromthe former case we can massage (4.2) (rearranging the sum) to obtain the equivalentidentity n/ − / (cid:88) k =1 L (cid:32) sin kn π cos π n (cid:33) − π n (cid:0) n + 3 (cid:1) = 0 n = 1 ( mod
2) (4.9)In this form we can compare it to the result by Kirillov [22] (eq. (1.16) of [20], properlyrewritten) n/ − / (cid:88) k =1 L (cid:32) sin j +1) πn sin k ( j +1) πn (cid:33) − π n (cid:0) (3 j + 1)( n − − j − n (cid:1) = 0 (4.10) † In the following steps we also find convenient to send n → n . – 8 – = 1 ( mod
2) 0 ≤ j ≤ n − g.c.d ( n, j + 1) = 1We specialize to the case j = n − and, using the inversion functional equation of theRogers function, and after some further rearrangements of the sum, it can be broughtto the form (4.9), thus proving their equivalence.In the n = 0 ( mod
4) case, using again functional identities of the Rogers functionand manipulations of the sum, we can derive from (4.2) the equivalent identity n − (cid:88) k =1 L (cid:32) cos kn π cos πn (cid:33) −
12 L (cid:16) cos πn (cid:17) + π n (cid:0) n − n + 12 (cid:1) = 0 n = 0 ( mod
2) (4.11)This can be shown to coincide with a special case of Kirillov’s result [22] (eq. (1.28)of [20]) n − (cid:88) k =1 L (cid:32) sin j +1) πn sin k +1)( j +1) n π (cid:33) = π (cid:18) n − n − j ( j + 2) n + 6 Z + (cid:19) (4.12)setting j = n/ − j .From these formulae it is possible to derive some classic ladder identities, by us-ing the Abel identity and the algebraic equations satisfied by the algebraic numbersappearing in them. For instance Watson’s identity (4.7) can be obtained from (4.9),taking n = 7 with some manipulations involving the Abel identity.In other words, using dilogarithm identities allows us to prove that the limit of thelight-like Wilson loop expectation value coincides with that of the circular one at oneloop. It is amusing that in our setting this limit is driven by nontrivial dilogarithmidentities which appeared in rather disparate contexts in theoretical physics.One could also think the phenomena we found the other way around. We havechosen a contour which is parametrized by trigonometric functions evaluated in rationalportions of π which are algebraic numbers. Since the finite pieces of the result for theWilson loop are formed by a uniform transcendentality combination of polylogarithmicfunctions, we have somehow “generated” a non trivial set of polylogarithm identitiesof algebraic numbers. It would be interesting to understand if there is an underlyingexplanation of such a surprising result and if it would be possible to generalize it toother contours and higher loops leading to identities about which much less is known.
5. Higher order in dimensional regularization parameter
The one-loop expectation value of both the circular and light-like Wilson loops can bederived at finite (cid:15) . – 9 –n the former case we have (cid:104) W (cid:13) (cid:105) (1) = 2 (cid:15) − π (cid:15) − Γ(1 − (cid:15) )Γ (cid:0) (cid:15) + (cid:1) Γ( (cid:15) + 1) (5.1)this is the result of a straightforward Feynman diagram computation which consistentlycoincides with the localization result at (cid:15) = 0.For the light-like Wilson loop we can take the n = 2 ( mod
4) case for simplicityand compute the relevant integral (3.2) to all order in (cid:15) . This yields −
12 Γ(1 − (cid:15) )4 π − (cid:15) a(cid:15) ( (cid:15) + 1) (cid:0) ( P ) (cid:15) +1 2 F (cid:0) , (cid:15) + 1; (cid:15) + 2; a P (cid:1) ++( Q ) (cid:15) +1 2 F (cid:0) , (cid:15) + 1; (cid:15) + 2; a Q (cid:1) − s (cid:15) +1 2 F (1 , (cid:15) + 1; (cid:15) + 2; a s ) (cid:1) (5.2)except for the case where s = P (or s = Q ), which appears on the regular polygonfor k = n − and k = n +24 respectively, where the integral becomes12 Γ(1 − (cid:15) )4 π − (cid:15) ( Q − s ) s (cid:15) − F (cid:18) , , − (cid:15) ; 2 ,
2; 1 − Q s (cid:19) (5.3)Performing the relevant sum (4.1) we can compute numerically the one-loop expecta-tion value as a function of n and (cid:15) . At fixed finite (cid:15) the sums are not constant anylonger, nevertheless for large values of n one can verify that the light-like Wilson loopapproximates the circular Wilson loop expectation value at any value of (cid:15) .As an example Figure 3 shows a plot of the relevant part of the light-like Wilsonloop expectation value (blue) against the circular one (yellow) as a function of n and (cid:15) . For large n the two surfaces coincide. Acknowledgements
We thank Lorenzo Bianchi, Gast´on Giribet and Gabriele Travaglini for very usefuldiscussions. Part of this work was performed at the 2014 Spring School on SuperstringTheory and Related Topics held at ICTP. We want to thank the organizers for providingthe stimulating environment. This work was supported in part by the Science andTechnology Facilities Council Consolidated Grant ST/L000415/1
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