Loop lessons from Wilson loops in N=4 supersymmetric Yang-Mills theory
aa r X i v : . [ h e p - t h ] F e b Preprint typeset in JHEP style - PAPER VERSION
Loop lessons from Wilson loops in N = 4 supersymmetric Yang-Mills theory Charalampos Anastasiou, Andrea Banfi
ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Abstract: N = 4 supersymmetric Yang-Mills theory exhibits a rather surprising dual-ity of Wilson-loop vacuum expectation values and scattering amplitudes. In this paper,we investigate this correspondence at the diagram level. We find that one-loop trian-gles, one-loop boxes, and two-loop diagonal boxes can be cast as simple one- and two-parametric integrals over a single propagator in configuration space. We observe that thetwo-loop Wilson-loop “hard-diagram” corresponds to a four-loop hexagon Feynman dia-gram. Guided by the diagrammatic correspondence of the configuration-space propagatorand loop Feynman diagrams, we derive Feynman parameterizations of complicated planarand non-planar Feynman diagrams which simplify their evaluation. For illustration, wecompute numerically a four-loop hexagon scalar Feynman diagram. Keywords:
NLO Computations, Supersymmetric gauge theory, ExtendedSupersymmetry, Supersymmetry and Duality. . Introduction N = 4 supersymmetric Yang-Mills theory is rich of symmetries and surprising dualities. Inthe late 90s, Maldacena proposed that this conformal field theory in flat space-time is dualin its planar limit to a string theory in anti-de Sitter space [1]. Later investigations showedthat anomalous dimensions of operators can be mapped to integrable quantum mechanicalsystems of spin-chains [2, 3, 4].A surprising discovery was that the two-loop four-point planar amplitude could beexpressed entirely in terms of the corresponding one-loop amplitude [5]. From the collinearlimit of gauge theory amplitudes it was conjectured that this factorization may hold forall two-loop planar MHV amplitudes, while the AdS/CFT correspondence was invitinga general factorization at all orders in perturbation theory [5]. A factorization of thethree-loop amplitude was proven in [6] where an explicit factorization ansatz valid at allorders was also formulated. Two-loop factorization was shown to hold for the five-pointplanar MHV amplitude [7, 8], however it was shown to break down for six-point scatteringamplitudes [9].Alday and Maldacena exploited the AdS/CFT correspondence to evaluate scatteringamplitudes of N = 4 supersymmetric Yang-Mills theory in the strong coupling limit, as aminimal surface on AdS bounded by a polygon with sides the momenta of the externalstates [10]. This has lead to the conjecture that scattering amplitudes are dual to thevacuum expectation value of Wilson-loops order by order in perturbation theory [11, 12].This duality has been tested with explicit comparisons at two loops for up to six-pointamplitudes and hexagon Wilson loops [9, 13, 14].The evaluation of Wilson-loop vacuum expectation values turns out to be much simplerthan the corresponding two-loop amplitudes. Two-loop Wilson loops are known analyti-cally up to hexagons [15, 16], and numerically for an arbitrary number of sides [17]. Therelative simplicity of two-loop Wilson loops is due to that their Feynman representationsrequire at most five integration variables irrespective of the number of the polygon sides.On the contrary, the number of Feynman parameters for an amplitude increases with thenumber of external legs. N = 4 supersymmetric Yang-Mills theory is rich of symmetries which constrain highlythe structure of scattering amplitudes. The amplitude/Wilson-loop duality may be at-tributed to these symmetries. Nevertheless, these amplitudes require the computation ofhighly complicated scalar integrals which are typically the most complicated “master” in-tegrals entering the evaluation of less symmetric amplitudes in theories such as QCD. Forexample, all four-point planar amplitudes in N = 4 supersymmetric Yang-Mills throughtwo loops are simple expressions in terms of the one-loop and two-loop box scalar (master)integrals [18, 19]. These very amplitudes being dual to a Wilson loop is not only a “magic”property of the theory but also a remarkable property of the one-loop box master integral.In this paper we show that there is a correspondence between individual Wilson-loopdiagrams and usual Feynman diagrams. We note that a scalar one-loop triangle is dual toa propagator of the Wilson-loop configuration space which joins a fixed point and a linesegment. We also find that the one-loop box in six dimensions and the two-loop diagonal-– 1 –ox in four dimensions with two light-like non-adjacent legs are dual to a configuration-space propagator joining two line segments.An intriguing Wilson-loop two-loop diagram is the so called “hard diagram”. It consistsof a triple-gluon vertex connected via gluon propagators to three sides of a polygon Wilson-loop. An analytic solution for this diagram is fully known for square and pentagon Wilsonloops [11, 20] while it is known in Regge kinematics for a hexagon Wilson-loop [15, 16]and for an octagon in special kinematic configurations [21]. The “hard diagram” has onlybeen computed numerically with standard methods for polygon Wilson loops with morethan six sides [17]. In this paper, we demonstrate that the Wilson-loop “hard diagram”is dual to a four-loop hexagon diagram with four one-loop triangle subgraphs. We alsoderive a representation of the propagator in configuration space connecting a fixed pointand a line segment as the product of two massive propagators. With this representation,we find that the “hard diagram” is dual to a one-loop hexagon massive Feynman integralintegrated over its own mass parameters.We hope that these representations of the “hard diagram” with an arbitrary numberof polygon sides will become a convenient starting point for an analytic evaluation of itin the future. However, this and other two-loop Wilson-loop diagrams are relatively easyto evaluate numerically [17]. We can then exploit the diagrammatic dualities of Wilsonloops and amplitudes to facilitate the computation of complicated Feynman integrals inamplitudes. For illustration, we present here a numerical evaluation of a scalar four-loophexagon integral with light-like legs which is mapped to a scalar hexagon Wilson-loop “harddiagram” times a 1 /ǫ prefactor.Inspired by the diagrammatic dualities we have found, we can derive simple represen-tations for multi-loop integrals with “easy-box” subgraphs (boxes with two non-adjacentlight-like legs). These are non-planar diagrams which are rather cumbersome to evaluatenaively. We replace “easy-box” subgraphs by a single propagator reducing the numberof loops by one. All two-loop non-planar integrals with such a subgraph are reduced toone-loop integrals where two of the external momenta are variables constructed as linearcombinations of the only two external momenta entering the “easy-box” subgraph. A two-fold integration is also required over a range of such linear combinations for the externalmomenta. These representations require a smaller number of integration variables thancanonical Feynman parameterizations. They also lead to simple Mellin-Barnes representa-tions.Our article is organized as follows. In Section 2 we derive the correspondence between aone-loop triangle and a propagator in configuration space. In Section 3 this correspondenceis exploited to rewrite planar easy boxes as a configuration-space propagator joining two linesegments. In Section 4 we consider the “hard diagram” contribution to a two-loop Wilsonloop and show that it can be mapped into a four-loop hexagon. This correspondence is thenused for a numerical evaluation of the four-loop hexagon, which we perform in Section 5.In Section 6 we derive an alternative representation of the configuration-space propagator,and use it to make the four-loop hexagon of Section 4 correspond to a one-loop hexagon This limit is sufficient for the exact determination of the two-loop hexagon Wilson-loop for arbitrarykinematics. – 2 –ith massive internal propagators, integrated over the internal masses. We conclude inSection 7 by applying the correspondence we have found for easy boxes to obtain betterFeynman parameterizations for non-planar diagrams.
2. A space-time propagator as a one-loop triangle
The basic object entering the evaluation of the vacuum expectation values of a Wilson loopis a scalar propagator in configuration space. For a massless scalar theory in D = 4 − ǫ UV dimensions, this is∆( x ) ≡ i Z d D k (2 π ) D e − ikx k + iε = 14 π D Γ (cid:0) D − (cid:1) ( − x + iε ) D − = 14 π − ǫ UV Γ(1 − ǫ UV )( − x + iε ) − ǫ UV . (2.1)There is a correspondence between this propagator and a scalar Feynman diagram. Con-sider a one-loop triangle with massless internal propagators (Fig. 1) p k+pkll+k+pl+k Figure 1:
A one-loop triangle with a massless leg p . Tria( k, p ; 1 , ,
1) = Z d D ℓiπ D ℓ + iε ) [( ℓ + k ) + iε ] [( ℓ + k + p ) + iε ] (2.2)with p = 0 , k , ( k + p ) = 0 . (2.3)We combine with a Feynman parameter the last two propagators of the above expression,Tria( k, p ; 1 , ,
1) = Z dτ Z d D ℓiπ D ℓ + iε ] [( ℓ + k + τ p ) + iε ] , (2.4)and integrate the loop momentum ℓ in D = 4 − ǫ dimensions. We obtain:Tria( k, p ; 1 , ,
1) = − Γ( − ǫ ) Γ(1 − ǫ )Γ(1 + ǫ )Γ(1 − ǫ ) Z dτ [ − ( k + τ p ) − iε ] ǫ . (2.5)The propagator in the above equation resembles already the configuration space propagatorin Eq. (2.1). To make the correspondence explicit we introduce a “space-time point” x = k and a “trajectory” in configuration space z ( τ ) = − τ p . By comparing Eq. (2.5) with thepropagator in Eq. (2.1), and identifying ǫ UV = − ǫ , we findTria( k, p ; 1 , ,
1) = − π ǫ Γ( − ǫ )Γ(1 − ǫ )Γ(1 − ǫ ) Z dτ ∆ ∗ ( x − z ( τ )) . (2.6)– 3 – ∗ ( x ) is the complex conjugate of the propagator ∆( x ) of Eq. (2.1), i.e. is an anti-causalpropagator, appearing when evaluating conjugate amplitudes. This correspondence is il-lustrated in Fig. 2. We notice that the result in Eq. (2.6) is invariant under translations, x x_ x k+pkp z Γ(−ε) ∼ Figure 2:
Pictorial representation of the mapping of the triangle into a Wilson line. so that we can equivalently define x = x + k and z ( τ ) = x − τ p , where x is an arbitraryspace-time point. We now consider a one-loop triangle (Fig. 3) with general powers of propagatorsTria( k, p ; ν , ν , ν ) = Z d D ℓiπ D ℓ ] ν [( ℓ + p ) ] ν [( ℓ − k ) ] ν . (2.7)We implicitly assume that all denominators are regularized by giving each of them apositive infinitesimal imaginary part. Following the same procedure as above, we derive:Tria( k, p, ν , ν , ν ) = ( − D Γ (cid:0) ν − D (cid:1) Γ( ν )Γ( ν )Γ( ν ) Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) Γ( D − ν ) × Z dτ (1 − τ ) ν − τ ν − [( k + τ p ) ] ν − D . (2.8)We notice that, unless ν = ν = 1, the interpretation of the one-loop triangle as a Wilson-line propagator is lost due to the non-trivial numerator of the integrand. For ν = ν = 1, ν ν ν k+pkp Figure 3:
A scalar triangle with arbitrary powers in the propagators. – 4 –he one-loop triangle can be interpreted as a propagator attached to a Wilson line, sinceEq. (2.8) can be rewritten as follows:Tria( k, p, , , ν ) = ( − ν Γ ( ν + ǫ )Γ( ν ) ×× Γ ( − ǫ ) Γ (2 − ǫ − ν )Γ(2 − ǫ − ν ) Z dτ − ( k + τ p ) ] ν + ǫ , (2.9)where we have set D = 4 − ǫ . Comparing with the scalar propagator in Eq. (2.1), we findthe correspondenceTria( k, p, , , ν ) = ( − ν π ǫ + ν Γ ( − ǫ )Γ( ν ) Γ (2 − ǫ − ν )Γ(2 − ǫ − ν ) ×× Z dτ ∆ ∗ ( x − z ( τ )) | ǫ UV = − ǫ − ν +1 , (2.10)where, as before, we have identified k with the space-time point x and have introduced thetrajectory z ( τ ) = − τ p . Therefore a one-loop triangle Tria( k, p, , , ν ) in D = 4 − ǫ isdual to the propagator in configuration space in D = 4 + 2 ν + 2 ǫ dimensions. We remarkthat the duality of the propagator in configuration space and the one-loop triangle hasbeen also discussed in Refs. [22, 23].
3. The two-loop diagonal box as a one-loop Wilson loop diagram
A question which arises from the duality of the previous section is whether similar dualitiesexist for more complicated diagrams. We shall show that the two-loop diagonal box diagramin Fig. 4 is dual to a one-loop Wilson-loop diagram. The two-loop diagonal box is ν ν ν ν ν p p Q Q Figure 4:
The diagonal box with massless internal propagators raised to arbitrary powers ν i . DBox( p , p , Q ; ν , ν , ν , ν , ν ) = Z d D kd D ℓi π D k ] ν [( k + p ) ] ν [ k ] ν [( k + p ) ] ν [ ℓ ] ν , (3.1) We thank Gregory Korchemsky for bringing these articles to our attention – 5 –here k = k + ℓ and k = k + p + Q . We can cast this integral asDBox( p , p , Q ; ν , ν , ν , ν , ν ) = Z d D kiπ D Tria( k, p, ν , ν , ν )[ k ] ν [( k + p ) ] ν . (3.2)and then use the triangle rule of Eq. (2.8) on Tria( k, p, ν , ν , ν ). We obtainDBox( p , p , Q ; ν , ν , ν , ν ) = ( − D Γ (cid:0) ν − D (cid:1) Γ( ν )Γ( ν )Γ( ν ) Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) Γ( D − ν ) ×× Z dτ Z d D kiπ D (1 − τ ) ν − τ ν − [( k + τ p ) ] ν − D [ k ] ν [( k + p ) ] ν . (3.3)We recognize that the k integral is another triangle Z d D kiπ D k + τ p ) ] ν − D [ k ] ν [( k + p ) ] ν = Tria (cid:18) Q + (1 − τ ) p , p , ν , ν , ν − D (cid:19) , (3.4)and apply again the triangle rule. Our representation of the two-loop diagonal box readsDBox( p , p , Q ; ν , ν , ν , ν , ν ) = ( − D Γ ( ν − D )Γ( ν )Γ( ν )Γ( ν )Γ( ν )Γ( ν ) ×× Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) ×× Z dτ Z dτ (1 − τ ) ν − τ ν − (1 − τ ) ν − τ ν − [((1 − τ ) p + τ p + Q ) ] ν − D . (3.5)In the special case ν = ν = ν = ν = 1, p Q Q Q τ p Q p p τ + + [Γ(−ε)] p ~ Figure 5:
Schematic illustration of the correspondence between the diagonal box and a one-loopscalar Wilson loop.
DBox( p , p , Q , , , , , ν ) = ( − ν Γ ( ν + 2 ǫ ) Γ(2 − ν − ǫ )Γ ( − ǫ )Γ(2 − ν − ǫ )Γ( ν ) ×× Z dτ Z dτ − (¯ τ p + τ p + Q ) ] ν +2 ǫ . (3.6)– 6 –e can interpret the two-loop diagonal box as one of the one-loop contributions to thevacuum expectation value of a Wilson loop with an arbitrary number of edges (see Fig. 5).This is accomplished by introducing the trajectories z ( τ i ) = x i − τ i p i , ending in the points¯ x i such that p i = x i − ¯ x i . In terms of the x i and ¯ x i the arbitrary momenta Q and Q aregiven by: Q = ¯ x − x , Q = ¯ x − x . (3.7)The exact correspondence isDBox( p , p , Q ; 1 , , , , ν ) = ( − ν π ν +2 ǫ Γ(2 − ν − ǫ )Γ ( − ǫ )Γ(2 − ν − ǫ )Γ( ν ) ×× Z dτ Z dτ ∆ ∗ ( z ( τ ) − z ( τ )) | ǫ UV =1 − ν − ǫ . (3.8)We further notice that for Q = 0 one recovers the cusp contribution to the one-loop Wilsonloop. It is known that the two-loop diagonal box of the previous section and the one-loop easy boxare dual. This can be seen easily by comparing their Mellin-Barnes representations [24].This duality is easy to prove by showing that the one-loop box (in D = 6 − ǫ dimensions)and the two-loop diagonal box (in D = 4 − ǫ dimensions) correspond to the same Wilson-loop diagram of the right-hand side of Eq. (3.8). ν ν ν ν p p Q Q Figure 6:
The easy box.
The one-loop “easy box” of Fig. 6 is defined as:Box( p , p , Q , ν , ν , ν , ν ) = Z d D kiπ D k ] ν [( k + p ) ] ν [ k ] ν [( k + p ) ] ν , (3.9)with k = k , k = k + p + Q and p = p = 0. We perform the Feynman parameterizationfound in Ref. [25]. We first join the two lines separated by p by introducing the Feynman We thank Bas Tausk for bringing this duality to our attention in private discussions immediately afterhis publication of Ref. [24], and Lance Dixon for presenting recently to us an independent derivation. – 7 –arameter τ Box( p , p , Q , ν , ν , ν , ν ) = Γ( ν )Γ( ν )Γ( ν ) Z dτ Z d D kiπ D ¯ τ ν − τ ν − [( k + τ p ) ] ν [ k ] ν [( k + p ) ] ν = Γ( ν )Γ( ν )Γ( ν ) Z dτ ¯ τ ν − τ ν − Tria( Q + ¯ τ p , p ; ν , ν , ν ) . (3.10)In the above, we used the shorthand notation ¯ τ i ≡ − τ i , and we recognized that the k integral corresponds to a triangle. Applying the triangle rule Eq. (2.8), we getBox( p , p , Q , ν , ν , ν , ν ) = ( − D Γ (cid:0) ν − D (cid:1) Γ( ν )Γ( ν )Γ( ν )Γ( ν ) ×× Γ (cid:0) D − ν (cid:1) Γ (cid:0) D − ν (cid:1) Γ ( D − ν ) Z dτ Z dτ ¯ τ ν − τ ν − ¯ τ ν − τ ν − [(¯ τ p + τ p + Q ) ] ν − D . (3.11)Comparing Eq. (3.11) with Eq. (3.5) we immediately obtain that the diagonal box isproportional to the one-loop easy box in 2( D − ν ) dimensions. The precise relation isBox D − ν ) ( p , p , Q , ν , ν ν , ν ) = ( − − ν Γ (cid:0) D − ν (cid:1) Γ(2 D − ν − ν ) ×× Γ( D − ν )Γ (cid:0) D − ν (cid:1) Γ( D − ν )Γ (cid:0) D − ν (cid:1) Γ( ν )Γ (cid:0) D − ν (cid:1) ×× DBox D ( p , p , Q , ν , ν , ν , ν , ν ) . (3.12)The connection to a Wilson-loop diagram is exact if we specialize Eq. (3.11) to thecase ν i = 1. Replacing D = 4 − ǫ we obtainBox( p , p , Q , , , ,
1) = 2 Γ( − ǫ ) Γ (2 + ǫ ) Γ(1 − ǫ )Γ(1 − ǫ ) ×× Z dτ Z dτ − (¯ τ p + τ p + Q ) ] ǫ . (3.13)and, equivalently,Box( p , p , Q , , , ,
1) = 8 π ǫ Γ( − ǫ ) Γ(1 − ǫ )Γ(1 − ǫ ) ×× Z dτ Z dτ ∆ ∗ ( z ( τ ) − z ( τ )) | ǫ UV = − − ǫ , (3.14)with the trajectories z ( τ i ) being the same as in Fig. 5. Therefore, the one-loop easy boxcan be interpreted as a one-loop diagram contributing to a Wilson loop with an arbitrarynumber of edges in D = 6 + 2 ǫ dimensions. Equivalently, the same Wilson-loop diagramin D = 4 + 2 ǫ dimensions corresponds to the one-loop easy box diagram in D = 6 − ǫ dimensions (which is finite). – 8 – z z z x x x x x x_ __ Figure 7:
The hard diagram.
4. The hard diagram as a four-loop hexagon
In this section we seek further dualities among Wilson loop diagrams and Feynman dia-grams. It has been shown by explicit calculations that the Wilson-loop and loop-amplitudeduality holds for two-loop six-leg amplitudes. In the previous section we saw that one-loopWilson-loop diagrams are mapped naturally to two-loop Feynman diagrams and also, witha handy duality of the two-loop diagonal box and the one-loop “easy box”, to one-loopFeynman diagrams.We now examine the so-called “hard diagram” which contributes to the two-loop Wil-son loop. It owes its name to the fact that this has been very difficult to compute ana-lytically for polygon Wilson loops with more than five sides. We shall show that the harddiagram is in fact a four-loop hexagon in disguise. We will first show that the hard diagramcan be obtained by acting with a suitable differential operator on a “scalar hard diagram”,a special two-loop diagram in configuration space. Then, we will use the triangle rule tomap the scalar hard diagram into a four-loop hexagon.The hard diagram is represented in Fig. 7. There we recognize three trajectories z ( τ i ) for i = 1 , ,
3, which depend on the three light-like momenta { p , p , p } via theparameterization z ( τ i ) = x i − τ i p i ≡ z i . (4.1)The other three positions { ¯ x , ¯ x , ¯ x } are related to three momenta { Q , Q , Q } , whichcan be off-shell or on-shell, as follows:¯ x − x = Q , ¯ x − x = Q , ¯ x − x = Q . (4.2)The above distances represent the insertion of an arbitrary number of sides which are notconnected with gluon propagators. The minimum number of sides for which all Q i arenon-zero is six, in which case { Q , Q , Q } are light-like.– 9 –eglecting colour and symmetry factors, the explicit expression for the hard diagramreads Hard( p , p , p , Q , Q , Q ) = p µ p µ p µ Z Y i =1 dτ i ! × V µ µ µ Z i d d zπ d/ ∆ ( z − z ( τ )) ∆ ( z − z ( τ )) ∆ ( z − z ( τ )) . (4.3)Here we have indicated with d = 4 − ǫ UV , the dimension of the configuration space inwhich the hard diagram lives, and V µ µ µ is a differential operator that represents thethree-gluon vertex: V µ µ µ = η µ µ ( ∂ µ − ∂ µ ) + η µ µ ( ∂ µ − ∂ µ ) + η µ µ ( ∂ µ − ∂ µ ) , (4.4)where we have used the notation ∂ µi ≡ ∂/∂z µi . Exploiting now the relations x − x = p + Q z − z = ¯ τ p + τ p + Q ,x − x = p + Q z − z = ¯ τ p + τ p + Q ,x − x = p + Q z − z = ¯ τ p + τ p + Q , (4.5)we can rewrite the tree-gluon vertex in terms of derivatives with respect to the externalmomenta { Q , Q , Q } only. Introducing the new differential operator˜ V µ µ µ = η µ µ (cid:18) ∂∂Q µ − ∂∂Q µ − ∂∂Q µ (cid:19) + η µ µ (cid:18) ∂∂Q µ − ∂∂Q µ − ∂∂Q µ (cid:19) + η µ µ (cid:18) ∂∂Q µ − ∂∂Q µ − ∂∂Q µ (cid:19) , (4.6)and exploiting the fact that the integral in Eq. (4.3) is finite, we can extract ˜ V µ µ µ fromthe z integral and writeHard( p , p , p , Q , Q , Q ) = p µ p µ p µ ˜ V µ µ µ SHard( p , p , p , Q , Q , Q ) . (4.7)Here we have introduced a “scalar hard diagram”, defined by:SHard( p , p , p , Q , Q , Q ) = Z Y i =1 dτ i ! Z i d d zπ d/ Y i =1 ∆ ( z − z ( τ i )) . (4.8)We can turn the scalar hard diagram into a Feynman diagram in momentum space.First, we introduce three momenta k i = z − x i and identify the position z we are integratingover with the loop momentum k = z − x = k . Substituting the representation in Eq. (2.6)for each propagator ∆( z − z i ) we obtainSHard( p , p , p , Q , Q , Q ) = (cid:18) − Γ(1 − ǫ )4 π ǫ Γ( − ǫ )Γ(1 − ǫ ) (cid:19) ×× Z i d d kπ d/ Y i =1 [Tria( k i , p i , , , ∗ , (4.9)– 10 –here, as in the previous section, we have taken the complex conjugate of the triangle toaccount for the fact that the iε prescription of the scalar propagators in Tria( k i , p i ; 1 , , k i , p i , , ,
1) with its explicit expression of Eq. (2.2), and rewritethe scalar hard diagram asSHard( p , p , p , Q , Q , Q ) = (cid:18) − Γ(1 − ǫ )4 π ǫ Γ( − ǫ )Γ(1 − ǫ ) (cid:19) ×× Z i d d kπ d/ Y i =1 d D ℓ i iπ D ℓ i ( ℓ i + k i ) ( ℓ i + k i + p i ) ! . (4.10)Notice that the dimension of the ℓ integral is D = 4 − ǫ , as in Eq. (2.2). The k integrationis, however, in d = 4 − ǫ UV = 4 + 2 ǫ dimensions. We remark that the diagram is finite for ǫ = − ǫ UV = 0, and insensitive to the regularization prescription of the various integrations.We observe now that, using the definitions of the momenta k i = z − x i in terms of thepositions x i , and the relations between the “distances” x i +1 − x i in Eq. (4.5), we canexpress k i in terms of the light-like momenta { p , p , p } and of three further momenta { Q , Q , Q } as follows k = k , k = k + p + Q , k = k − p − Q . (4.11)We then recognize that the integral in Eq. (4.10) resembles the four-loop hexagon shownin Fig. 8:Hexa (4) ( p , p , p , Q , Q , Q ) = Z d D kiπ D Y i =1 d D ℓ i iπ D ℓ i ( ℓ i + k i ) ( ℓ i + k i + p i ) ! . (4.12)We notice that the correspondence is exact only in strictly D = 4 dimensions because the Q p Q p Q p l l l k l + l k + l k + k l + p + l k + p + pl k + + Figure 8:
A four-loop scalar hexagon. – 11 – integrals in Eq. (4.10) and Eq. (4.12) have to be performed in two different dimensions.This correspondence, represented in Fig. 9, gives promise to derive the finite value of thehard diagram from the leading 1 /ǫ pole of the four-loop hexagon,SHard( p , p , p , Q , Q , Q ) | d =4 = (cid:18) π (cid:19) ×× lim ǫ → ǫ [Hexa (4) ( p , p , p , Q , Q , Q )] ∗ (cid:12)(cid:12)(cid:12) D =4 − ǫ , (4.13)using factorization properties of collinear singularities. The determination of this coefficient Q p Q p Q p x_ z x x_ z x x_ z x ∼ [Γ(−ε)] Figure 9:
Pictorial representation of the correspondence between a four-loop hexagon and a dia-gram in configure space. would give the analytic answer for the two-loop vacuum expectation value of a polygonWilson loop with an arbitrary number of sides. We leave this calculation for future work.
5. Numerical evaluation of a four-loop hexagon Feynman diagram
The dualities that we are exploring in this paper could facilitate Feynman diagram cal-culations. We have not yet explored their full potential. However, benefits in computingnumerically the multi-loop Feynman diagrams which enter these dualities are striking.From the Wilson-loop side of the correspondence, the hard diagram is only difficult tocompute analytically and it can be evaluated numerically rather easily. As we have seenin the previous section, this diagram is dual to a very complicated four-loop hexagon. Wecan use the duality to compute such a complicated Feynman integral. To the best of ourknowledge, there has been no four-loop Feynman integral with six legs evaluated in theliterature.We start from the explicit expression for the hexagon in Eq. (4.12) and replace eachtriangle subgraph with its representation in terms of a scalar propagator ∆( x ) in Eq. (2.6).Using the correspondence between momenta and space-time points introduced in Eq. (4.5),– 12 –e obtain the following representation:Hexa (4) ( p , p , p , Q , Q , Q ) = (cid:18) − π ǫ Γ( − ǫ ) Γ(1 − ǫ )Γ(1 − ǫ ) (cid:19) ×× Z
10 3 Y i =1 dτ i Z d D ziπ D ∆ ∗ ( z ) ∆ ∗ ( z + z ) ∆ ∗ ( z − z ) , (5.1)where the scalar propagators ∆( x ) are in 4 + 2 ǫ dimensions, while the z integral has tobe performed in 4 − ǫ dimensions. We now substitute the explicit expression for ∆( x ),introduce a further Feynman parameterization of the z integral so as to write the productof propagators as a single denominator, and finally perform the z integration to get:Hexa (4) ( p , p , p , Q , Q , Q ) = (cid:18) Γ( − ǫ ) Γ(1 − ǫ )Γ(1 − ǫ ) (cid:19) Γ(1 + 4 ǫ ) ×× Z Y i =1 dτ i ! Z Y i =1 dα i ! δ − X i =1 α i ! ( α α α ) ǫ [ − ( α α z + α α z + α α z + iε )] ǫ . (5.2)We obtain an expression for the four-loop hexagon that is a function of only five Feynmanparameters, and that depends on the coordinates z i only through the distances z i,i +1 in-troduced in Eq. (4.5). This representation offers the further advantage that all collinearsingularities of the hexagon graph have been captured as a divergent 1 /ǫ prefactor. Theintegral is finite in all limits and can be expanded directly around ǫ = 0.This scalar integral depends on twelve invariants: { s , . . . , s } = { ( p p ) , ( p p ) , ( p p ) , ( p Q ) , ( p Q ) , ( p Q ) , ( p Q ) , ( p Q ) , ( p Q ) , Q , Q , Q } . (5.3)For illustration of numerical results we consider the case in which all momenta, Q i , p i arelight-like, so that the final result depends only on { s , . . . , s } . We cast the result in theform: Hexa (4) ( p , p , p , Q , Q , Q ) = Γ(1 + 4 ǫ ) (cid:18) A − ǫ + A − ǫ + A − ǫ + A (cid:19) . (5.4)The results of the numerical integration for two specific sets of invariants are displayed intable 1. { s , . . . , s } A − A − A − A {− , − , − , − , − , − , − , − , − } − . . . . {− . , − . , − . , − . , − . , − . , − , − . , − . } − . − . − . . Table 1:
Numerical evaluation of the four-loop hexagon for different values of the kinematicalinvariants. – 13 –t is intriguing that this four-loop hexagon can be mapped to a one-loop triangle withnon-integers powers of propagators. From the above Feynman parameterization we readthat Hexa (4) ( p , p , p , Q , Q , Q ) = (cid:18) Γ( − ǫ )Γ(1 − ǫ )Γ(1 + ǫ )Γ(1 − ǫ ) (cid:19) Z dτ dτ dτ × Tria(¯ τ p + τ p + Q , ¯ τ p + τ p + Q , ǫ, ǫ, ǫ ) . (5.5)Equivalently, the scalar Wilson-loop “hard diagram” through order O ( ǫ ) is also propor-tional to the same integral over the one-loop triangle in the right-hand side of the aboveequation (with a non-divergent prefactor) [17],SHard( p , p , p , Q , Q , Q ) = 1(4 π ) Z dτ dτ dτ × Tria(¯ τ p + τ p + Q , ¯ τ p + τ p + Q , , ,
1) + O ( ǫ ) . (5.6)In this paper and in Ref. [17] equations like the above (5.5-5.6) have been used for numericalevaluations of the integrals on their left-hand side. However, we believe that they are likelyto be a good starting point for an analytic evaluation. We remark that the Tria functionis expressed rather compactly in terms of Appell F functions for arbitrary powers ofpropagators [26, 27, 28].
6. The “hard” Wilson-loop diagram as a massive one-loop hexagon
Wilson-loop integrals can be cast as massive loop Feynman integrals, integrated over theirmass parameters. For an arbitrary momentum q Z ∞ dm ( m ) − ǫ [( q − m + iε ) ] n = ( − n Γ(1 − ǫ )Γ( n − ǫ ) − n ) [ − q − iε ] n − ǫ . (6.1)For n = 2, we obtain a representation of the propagator in configuration space:∆ ∗ ( q ) = 14 π ǫ Γ(1 + ǫ )( − q − iε ) ǫ = 14 π ǫ Γ(1 − ǫ ) Z ∞ dm ( m ) − ǫ [ q − m + iε ] . (6.2)For a propagator connecting a fixed point x = − k and a light-like line segment x ( τ ) = τ p ,we have q = k + τ p . Performing the τ integral we have Z dτ ∆ ∗ ( k + τ p ) = 14 π ǫ Γ(1 − ǫ ) Z ∞ dm ( m ) − ǫ [ k − m + iε ] [( k + p ) − m + iε ] . (6.3)The same relation (up to prefactors) holds for the dual one-loop triangle. We findTria( k, p ; 1 , ,
1) = Γ( − ǫ )Γ(1 − ǫ ) Z ∞ dm ( m ) − ǫ ( k − m + iε ) [( k + τ p ) − m + iε ] . (6.4)This correspondence, pictorially represented in Fig. 10, leads to a different representation– 14 – mm k+pkp k+pk x x_ x z ~~ Figure 10:
Pictorial representation of the one-loop triangle as an integral over masses. for the four-loop hexagon considered in Section 4 and displayed in Fig. 8 and, equivalently,for the “hard” Wilson-loop diagram. We find:Hexa (4) ( p , p , p , Q , Q , Q ) = (cid:18) Γ( − ǫ )Γ(1 − ǫ ) (cid:19) ×× Z d kiπ Z ∞ Y i =1 dm i ( m i ) − ǫ ( k i − m i + iε )[( k i + p i ) − m i + iε ] , (6.5)where k = k , k = k + p + Q and k = k − Q − p . Similarly, the Wilson-loop harddiagram can be written as (see Fig. 11)SHard ∗ ( p , p , p , Q , Q , Q ) = (cid:0) − π ǫ Γ(1 − ǫ ) (cid:1) ×× Z d kiπ Z ∞ Y i =1 dm i ( m i ) − ǫ ( k i − m i + iε )[( k i + p i ) − m i + iε ] . (6.6)The four-dimensional massive one-loop hexagon can be evaluated analytically after reduc- Q p p Q p l l l k l + l k + l k + k l + p + l k + p + pl k + + Q x_ z x x_ z x x_ z x Q p p Q p m m m p k + k km k m m p k + p k + Q ~ ~ Figure 11:
Representation of the mapping of a four-loop hexagon into a one-loop hexagon inte-grated over internal masses. ing to box master integrals. The analytic integration over the masses m i is a formidabletask. However, it holds promise for achieving an analytic evaluation of the hard diagramfor a Wilson loop with an arbitrary number of sides.– 15 – . Non-planar Feynman integrals An intriguing feature of Eq. (3.11) is that an “easy box” is cast as a single scalar prop-agator raised to a dimension-dependent power. Such boxes can be subgraphs of morecomplicated higher loop non-planar diagrams. Eq. (3.11) can be then utilized to cast thesenon-planar diagrams as planar diagrams with one loop less, integrated over a range oflinear combinations for their external momenta. Q Q p p Q Q p y p y p _y p ~ ++y _ Figure 12:
Cross box diagrams contributing to 2 → The two-loop non-planar box integral XBox at the left of Fig. 12 has a one-loop “easy-box” subgraph. Applying Eq. (3.11), we obtain a representation as an integralXBox = 2 ( − − ǫ Γ(2 + ǫ ) Γ( − ǫ )Γ(1 − ǫ )Γ(1 − ǫ ) Z dy Z dy ×× F (cid:0) ( p + p ) , ( y p + y p + Q ) , ( y p + y p ) , (¯ y p + ¯ y p + Q ) , Q (cid:1) , (7.1)over a box function F (cid:0) ( p + p ) , ( y p + y p + Q ) , ( y p + y p ) , (¯ y p + ¯ y p + Q ) , Q (cid:1) = Z d D kiπ D k [( k + y p + y p ) ] ǫ ( k + p + p ) ( k + p + p + Q ) , (7.2)with one of the propagators raised to a non-integer power.We observe that this representation requires at most five Feynman parameters (threefor the box-function in the integrand and y , y ). A naive Feynman representation wouldyield a six-dimensional integral. One-loop box integrals have been studied extensively inthe literature. In Eq. (7.1) it is required a box integral with a non-integer power for oneof the propagators. For Q = Q this is known to be a sum of four F hypergeometricfunctions [28, 29] with well studied analytic continuation properties and asymptotic limitsin the mathematical literature.For a direct numerical evaluation with the method of sector decomposition [30] thisFeynman representation is better suited requiring less than half the number of sectors of anaive parameterization. Applying the non-linear transformations of Ref. [31] to factorizeinfrared singularities is also a much simpler task with our parameterization. Non-planarbox diagrams with light-like external legs pose an additional difficulty for their evaluationdue to not having a Euclidean region when the Mandelstam variables s = ( p + p ) , t = ( p + Q ) , u = ( Q + p ) are consistent with momentum conservation: Q = Q = s + t + u = 0 . (7.3)– 16 –he on-shell limit of an external leg does not commute with the ǫ = 0 of the dimensionalregulator due to the emergence of new collinear divergences. It is easy to obtain a Feynmanrepresentation with a Euclidean region for generic s, t, u from Eq. (7.1) which possesses thesame infrared singularities as for s, t, u = − s − t . This is,XBox = 2 ( − − ǫ Γ(2 + ǫ ) Γ( − ǫ )Γ(1 − ǫ )Γ(1 − ǫ ) Z dy Z dy ×× F ( s, y ¯ y t + y ¯ y u, y y s, ¯ y ¯ y s, , (7.4)where we have used momentum conservation only for the second argument of the F boxfunction.Using the Mellin-Barnes representation of the one-loop box with two off-shell legs, andintroducing one additional Mellin-Barnes integration, we can integrate out all Feynmanparameters. We obtainXBox( p , p , Q ; 0 , ,
0) = − − ǫ )Γ(1 − ǫ )Γ( − − ǫ )Γ(1 − ǫ ) Z i ∞− i ∞ Y i =1 dξ i πi Γ( − ξ i ) ! ×× Γ(3 + 2 ǫ + ξ )Γ(2 + ǫ + ξ )Γ( − − ǫ − ξ )Γ( − − ǫ − ξ ) ×× Γ(1 + ξ )Γ(1 + ξ )Γ(1 + ξ )Γ(1 + ξ )Γ(1 + ξ )Γ (2 + ξ ) ( − s ) − − ǫ − ξ ( − t ) ξ ( − u ) ξ , (7.5)which is the representation obtain in ref. [24].Similarly simplified Feynman parameterizations of non-planar diagrams with “easy-box” insertions can be also obtained for integrals with higher number of loops or legs.
8. Conclusions
We have observed dualities among diagrams which enter the calculation of the vacuumexpectation value of Wilson loops and scalar Feynman integrals. These can be pictured as: (a) (b) (c) where the meaning of each diagram is:(a) a one-loop triangle is dual to a scalar propagator in configuration space attached toa Wilson line;(b) a two-loop diagonal box (and a one-loop easy box) are dual to a scalar propagatorconnecting two Wilson lines, in fact a one-loop Wilson-loop with an arbitrary numberof edges; – 17 –c) a four-loop hexagon is dual to a three-boson vertex connected via scalar propagatorsto three Wilson lines, which is the so-called two-loop “hard diagram”.A remarkable feature of the representations we have obtained is that the Wilson-line di-agram appears always as a factor multiplying a singularity of the corresponding scalarmulti-loop diagram. This factorization property has two main advantages. On one hand,if one wishes to exploit a duality to compute an unknown Wilson-loop diagram, it is ingeneral enough to extract the coefficient of the leading singularity of the correspondingscalar integral, which is far easier than fully computing the integral itself. On the otherhand, given a scalar multi-loop integral, the corresponding Wilson-loop diagram providesautomatically a nice Feynman parameterization for it. In particular, given the fact thatpart of the singularities are already extracted and that Wilson loops are generally simpleto compute numerically, one can use the duality to tackle calculations that would appearimpossible at first sight. For illustration of the potential our observations, we compute forthe first time in the literature a four-loop hexagon integral, which is dual to the Wilson-loop“hard diagram”.The relations we have found concern planar diagrams only. However, they can be alsoused when a planar diagram is part of a larger non-planar diagram. For instance, one canuse the duality between the one-loop easy box and a scalar propagator to derive a simpleFeynman parameterization for the two-loop cross-box, yielding a better starting point forits numerical evaluation [31]. The very same parameterization simplifies the calculation ofthe cross-box using Mellin-Barnes techniques.To conclude, the dualities we have found look very promising, and can lead to signif-icant simplifications in the calculation of high-loop integrals appearing both in QCD andin other theories like N = 4 Super Yang-Mills or N = 8 Supergravity. We are lookingforward to further investigations on these dualities in the future. Acknowledgments.
We thank Andreas Brandhuber, Lance Dixon, Vittorio del Duca,Claude Duhr, Thomas Gehrmann, Gregory Korchemsky, Eric Laenen, Lorenzo Magnea,Volodya Smirnov, Bas Tausk and Gang Yang for useful discussions. We thank especiallyBas Tausk for communicating his insight on the diagonal-box/one-loop box duality in thepast. This work is supported by the ERC Starting Grant project IterQCD.
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