YYITP-SB-11-33
Gauge theory webs and surfaces
Ozan Erdo˘gan, George Sterman
C.N. Yang Institute for Theoretical Physics and Department of Physics and AstronomyStony Brook University, Stony Brook, New York 11794-3840, USA (Dated: October 29, 2018)
Abstract
We analyze the perturbative cusp and closed polygons of Wilson lines for massless gauge theoriesin coordinate space, and express them as exponentials of two-dimensional integrals. These integralshave geometric interpretations, which link renormalization scales with invariant distances. a r X i v : . [ h e p - t h ] J a n . INTRODUCTION Gauge field path-ordered exponentials [1–3] or Wilson lines, represent the interaction ofenergetic partons with relatively softer radiation in gauge theories. For constant velocities,ordered exponentials of semi-infinite length correspond to the eikonal approximation forenergetic partons. Classic phenomenological applications of ordered exponentials include softradiation limits in deeply inelastic scattering [4] and parton pair production and electroweakannihilation [5–7]. They appear as well in the treatment of parton distributions [8, 9]. In allthese cases, the electroweak current is represented by a color singlet vertex at which lines inthe same color representation but with different velocities are coupled. This vertex is oftenreferred to as a cusp.Cusps also appear as vertices in polygons formed from Wilson lines [10], which have beenstudied extensively in the context of their duality to scattering amplitudes in N = 4 SYMtheory [11–15]. In the strong-coupling limit of this theory, gauge-gravity duality relatesthe cusp and polygons to the exponentials of two-dimensional surface integrals. Surfacesbounded by open and closed paths of ordered exponentials are also a classic ingredient inlattice [3] and large- N c [16] paradigms for confinement in quantum chromodynamics.In this paper, we show that in any gauge theory with massless vector bosons the cuspmatrix element for lightlike Wilson lines can be expressed as the exponential of an integralover a two-dimensional surface, a result with applications as well to polygons formed fromordered exponentials. The corresponding integrand is an infrared finite function of the gaugetheory coupling, evaluated for each point on the surface at a scale given by the invariantdistance from that point to the cusp vertex. This result extends to all orders in perturbationtheory.The set of all virtual corrections for the cusp [17] is formally identical to a vacuumexpectation value, and can be written asΓ ( f ) ( β , β ) = (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) T (cid:18) Φ ( f ) β ( ∞ ,
0) Φ ( f ) β (0 , −∞ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:29) , (1)in terms of constant-velocity ordered exponentials,Φ ( f ) β i ( x + λβ i , x ) = P exp (cid:18) − ig (cid:90) λ dλ (cid:48) β i · A ( f ) ( x + λ (cid:48) β i ) (cid:19) . (2)Here f labels a representation of the gauge group and β i is a four-velocity, taken lightlikein the following. The combination of ordered exponentials in Eq. (1) corresponds to a2artonic process with spacelike momentum transfer. For correspondence to a timelike processlike pair creation, Φ ( f ) β (0 , −∞ ) can be replaced by Φ ( ¯ f ) β ( ∞ , II. EXPONENTIATION AND MOMENTUM-SPACE WEBS
The cusp has long been known [32] to be the exponential of a sum of special diagramscalled webs, which are irreducible by cutting two eikonal lines. We represent this result asΓ( β , β , ε ) = exp E ( β , β , ε ) , (3)3n D = 4 − ε dimensions. The exponent E equals a sum over web diagrams, d , each givenby a group factor multiplied by a diagrammatic integral, E ( β , β , ε ) = (cid:88) webs d C d F d ( β , β , ε ) , (4)where F d represents the momentum- or coordinate-space integral for diagram d . The coef-ficients of these integrals, C d are modified color factors. Two-loop examples are shown inFig. 1.In momentum space we can write the exponent E as the integral over a single, overallloop momentum that runs through the web and the cusp vertex, assuming that all loopintegrals internal to the web have already been carried out. The web is defined to includethe necessary counterterms of the gauge theory [8, 30, 33, 34]. Taking into account theboost invariance of the cusp for massless loop velocities, and the invariance of the orderedexponentials under rescalings of the velocities β i , we have for the exponent the form, E ( β , β , ε ) = (cid:90) d D k (2 π ) D β · β k · β k · β k ¯ w (cid:18) k µ , k · β k · β µ β · β , α s ( µ , ε ) , ε (cid:19) . (5)In addition, the webs themselves are renormalization-scale independent, µ ddµ ¯ w (cid:18) k µ , k · β k · β µ β · β , α s ( µ , ε ) , ε (cid:19) = 0 . (6)This renormalization-scale invariance allows us to choose µ equal to either of the kinematicarguments in the web. A further important property of webs is the absence of collinear andsoft subdivergences in the sum of all web diagrams. That is, in Eq. (5), collinear poles are (a) (b) (c) (d) FIG. 1. Two-loop web diagrams, referred to in the text as: (a) E cross , (b) E g , (c)–(d) E se . Webdiagram (a) has the modified color factor, C a C A /
2, where a refers to the representation of theWilson lines. For diagram (a), the web color factor differs from its original color factor, whileall other color factors are the same as in the normal expansion. Diagrams related by top-bottomreflection are not shown. k and either k · β or k · β vanish, infrared poles only when all threevanish and the overall ultraviolet poles only when all components of k diverge. Equation (5)thus organizes the same double poles found in the corresponding partonic form factors [34–36]. Arguments for these properties in momentum space are given in Ref. [34], based on thefactorization of soft gluons from fast-moving collinear partons. These considerations suggestthat when embedded in an on-shell amplitude or cross section, the web acts as a unit, almostlike a single gluon, dressed by arbitrary orders in the coupling. In the following, we observethat this analogy can be extended to coordinate space.The form given above, in terms of webs, is for the unrenormalized cusp. When renormal-ized by the minimal subtraction of ultraviolet poles, the exponent E can be written in theform [37], E ren ( α s ( µ ) , ε ) = − (cid:90) µ dξ ξ (cid:20) Γ cusp (cid:16) α s (cid:0) ξ (cid:1) (cid:17) log (cid:18) µ ξ (cid:19) − G eik ( α s (cid:0) ξ ) (cid:1)(cid:21) , (7)where µ is the renormalization scale, and where, here and below, we have set β · β = 1.At order α ns , the leading pole behavior of this exponent is proportional to Γ (1)cusp α ns (1 /ε ) n +1 ,with Γ (1)cusp ( α s /π ) the one-loop cusp anomalous dimension. Nonleading poles are generatedfrom higher orders in Γ cusp , from G eik , and from the ε -dependence of the running couplingin D dimensions [35]. After renormalization in this manner, the cusp is a sum of infraredpoles in one-to-one correspondence with the ultraviolet poles that are subtracted. The cuspanomalous dimension is given to two loops byΓ cusp , a = (cid:16) α s π (cid:17) C a (cid:104) (cid:16) α s π (cid:17) K (cid:105) ,K = (cid:18) − π (cid:19) C A − n f T f , (8)with C a = 4 / , a = q, g for QCD, n f the number of fermion flavors, and T f = 1 /
2. Atone loop, G eik is zero, and we will derive its two-loop form below. Equation (7) gives all thepoles of the cusp, when reexpanded in terms of the coupling at any fixed scale. We notethat for timelike kinematics, the renormalization scale µ should be chosen negative [37].5 II. WEBS AND SURFACES IN COORDINATE SPACEA. The unrenormalized exponent and its surface interpretation
The coordinate-space analog of Eq. (5) is a double integral over two parameters, σ and λ that measure distances along the Wilson lines β and β , respectively, with a new webfunction, w , which depends on these variables through the only available dimensionlesscombination, λσµ , E = (cid:90) ∞ dλλ (cid:90) ∞ dσσ w ( α s ( µ , ε ) , λσµ , ε ) . (9)Here and below, we choose timelike kinematics. We emphasize that we are interested pri-marily in the form and symmetries of the integrand, rather than its convergence properties.Nevertheless, to separate infrared and ultraviolet poles in the integration, it is necessarythat the integrand, w in Eq. (9) be free of both infrared and ultraviolet divergences at ε = 0in renormalized perturbation theory (aside from the renormalization of the cusp itself). Aswe shall see below, Eq. (9) with a finite web function leads to a renormalized cusp that isfully consistent with the momentum-space form, Eq. (7). In this construction, all ε poles ofthe exponent, and therefore the cusp, are then associated with the integrals over λ and σ in (9).A direct, coordinate-space demonstration of the finiteness of the web function is interest-ing in its own right, and is given in Ref. [38]. Formally, such a demonstration is necessary toextend the proof of renormalizability for cusps connecting massive lines [30] to the masslesscase [10]. Here, we simply mention the essential ingredients of such an argument.Diagram by diagram, one may use the analytic structure of the coordinate integra-tions [39] combined with a coordinate-space power-counting technique to identify the mostgeneral singular subregions in coordinate space [40]. In coordinate space, nonlocal ultra-violet subdivergences arise when a subset of vertices line up at finite distances from thecusp along either of the lightlike Wilson lines, while other, “soft” vertices remain at finitedistances. Such subdiagrams factorize, however, in much the same manner as in momentumspace [41, 42]. Once in factorized form, combinatoric arguments show that divergent inte-grals cancel when all web diagrams are combined at a given order [38] in coordinate space,in much the same way as in the momentum-space treatment of Ref. [34]. Finally, taking λ and σ as the positions of vertices in the web diagrams furthest from the cusp, there are no6 a)(b) FIG. 2. Representation of singular regions for a two-loop web diagram. (a) Single-scale regions,characteristic of webs. (b) Multiple-scale regions, associated with subdivergences that cancel inthe sum of web diagrams. soft (infinite wavelength) divergences from integrations over the internal vertices of webs incoordinate representation, as shown in Ref. [40].As we shall shall illustrate below, it is possible to implement the cancellation of subdiver-gences at fixed positions, λ and σ , along the ordered paths, specified by the vertices furthestfrom the cusp. Once this is done and the subdivergences thereby eliminated, the integralsover all vertices of the web diagrams converge on scales set by λ and σ in (9), and the webacts as a unit. Singular behavior of the cusp arises as λ and/or σ vanish, and in these limitsall web vertices approach the directions of β or β together, as in Fig. 2(a). This is theperturbative realization of the web as a geometrical object. Subdivergent configurationsthat cancel are illustrated in Fig. 2(b).The web function w constructed this way is again a renormalization group invariant, sothat in (9), we may shift the renormalization scale to the product ( λσ ) − , which results in anexpression with the coupling running as the leading vertices move up and down the Wilsonlines, E = (cid:90) ∞ dλλ (cid:90) ∞ dσσ w ( α s (1 /λσ ) , ε ) . (10)In this all-orders form, dependence on the product λσ is entirely through the running cou-pling, aside from the overall dimensional factor. For N = 4 SYM theory, Eq. (10) forthe cusp holds as well at strong coupling [12, 13, 43], where the coordinates λ and σ also7arametrize a surface. The generality of these results can be traced to the symmetries ofthe problem [43]. It is interesting to note, however, that in the strong-coupling analysis, theproduct of internal coordinates λσ , which serves as the renormalization scale in Eq. (10),relates the plane of the Wilson lines to a minimal surface in five dimensions. B. Web renormalization in coordinate space
To derive a renormalized exponent for the cusp in coordinate space, we will find it usefulto expand the unrenormalized web function in (10) in explicit powers of ε , E ( ε ) = ∞ (cid:88) n =0 ε n (cid:90) ∞ dλλ (cid:90) ∞ dσσ w n ( α s (1 /λσ )) , (11)where w n is the coefficient of ε n , noting that the coupling retains implicit ε -dependence.As noted above, the renormalized exponent is determined by the ultraviolet poles of thesescaleless integrals. With this in mind, consistency with momentum-space pole structure inEq. (7) then clearly requires w ( α s (1 /λσ )) = −
12 Γ cusp (cid:0) α s (1 /λσ ) (cid:1) . (12)For finite values of λ and σ , only w contributes to the unrenormalized integral in the ε → λ = 0 and σ = 0, which produce poles that cancompensate explicit powers of ε in Eq. (11). Such boundary contributions from terms ε n w n with n > G eik in the renormalized form,Eq. (7).To compute G eik , we recall that the running coupling α s (1 /λσ ) remains a function of ε when reexpanded in terms of the coupling at any fixed scale, µ , which we represent as α s (1 /λσ ) = α s ( µ ) ( µ λσ ) ε (cid:18) α s ( µ )4 π b ε (cid:2) ( µ λσ ) ε − (cid:3) + . . . (cid:19) ≡ ¯ α s (cid:0) α s ( µ ) , ( µ λσ ) ε , ε (cid:1) , (13)where we exhibit only the dependence to order α s , which is all we need here, and where b = (11 / C A − (4 / n f T f . The subleading anomalous dimension G eik is found from singlepoles in E ( ε ) after the λ and σ integrations. These can arise at any order by combinationsof an overall factor ε n in (11) with poles in the expansion of the coupling, (13). To identify8uch terms, we may conveniently take σ < λ and multiply by 2, and reexpand α s (1 /λσ ) interms of α s (1 /λ ), schematically, E ( ε ) = − (cid:90) ∞ dλλ (cid:90) ∞ dσσ Γ cusp (cid:0) α s (1 /λσ ) (cid:1) (14)+ 2 ∞ (cid:88) n =1 ε n (cid:90) ∞ dλλ (cid:90) λ dσσ w n (cid:0) ¯ α s (cid:0) α s (1 /λ ) , ( σ/λ ) ε , ε (cid:1)(cid:1) . The renormalized exponent is defined as the remainder when all ultraviolet poles are sub-tracted minimally at an arbitrary, fixed scale µ . Leading and nonleading poles are thengenerated by E ren ( ε, α s ( µ )) = − (cid:90) ∞ /µ dλλ (cid:90) ∞ /µ dσσ Γ cusp (cid:0) α s (1 /λσ ) (cid:1) + (cid:90) ∞ /µ dλλ G eik (cid:0) α s (1 /λ ) (cid:1) , (15)where the integrals are now defined by infrared regularization ( ε < C. Lowest orders
The lowest order expression for Eq. (9) already illustrates the nontrivial relationshipbetween the renormalization scale and the positions of the vertices. It is found directly fromthe coordinate-space gluon propagator in Feynman gauge, D µν ( x ) = (cid:90) d D k (2 π ) D e − ik · x − i g µν k + i(cid:15) = Γ(1 − ε )4 π − ε − g µν ( − x + i(cid:15) ) − ε . (16)The resulting expression for the unrenormalized exponent is E (LO) = − C F Γ(1 − ε )2 (cid:90) ∞ dλλ dσσ (cid:18) α s ( µ ) π (cid:19) (2 πλσµ ) ε , (17)= − C F (cid:18) ε π (cid:19) (cid:90) ∞ dλλ dσσ (cid:18) α s ( µ ) π (cid:19) (2 πe γ E λσµ ) ε , where in the second form we have expanded the integrand to order ε . The correspondingrenormalized exponent is E (LO)ren = − C F Γ(1 − ε )2 (cid:90) ∞ /µ dλλ (cid:90) ∞ /µ dσσ (cid:18) α s ( µ ) π (cid:19) (2 πλσµ ) ε , (18)9hich is precisely Eq. (15) to lowest order. Here and below, for definiteness we choose theWilson lines in fundamental representation.At two loops, the diagrams of Fig. 1 can be used to illustrate both the cancellationof subdivergences in the sum of web diagrams, and the manner in which we identify theparameters λ and σ , which together define the position of the web function. Our calculationsare carried out with ultraviolet regularization ( D < E cross = N cross ( ε ) (cid:90) ∞ dλ (cid:90) λ dλ (cid:90) ∞ dσ (cid:90) σ dσ λ σ + i(cid:15) ) − ε λ σ + i(cid:15) ) − ε , (19)where the prefactor is given by N cross ( ε ) ≡ − (cid:16) α s π (cid:17) C A C F Γ (1 − ε )2 ( πµ ) ε . (20)For the color factor in this web diagram, we keep only the C A C F / ε >
0, we choose to integrate over the inner eikonal parameters, andidentify λ ≡ λ and σ ≡ σ in the general form of Eq. (10), giving E cross = − (cid:16) α s π (cid:17) C A C F Γ (1 − ε )8 ε (2 πµ ) ε (cid:90) ∞ dλ dσ ( λσ ) − ε . (21)This expression has overall double ultraviolet poles in addition to two scaleless (surface)integrals along the Wilson lines. The singular behavior of the coefficient arises from λ (cid:28) λ and σ (cid:28) σ , a “subdivergent” configuration, in which the two gluons approach differentWilson lines. The contributions from these regions will be canceled by corresponding termsfrom the three-gluon diagrams.We now turn to the diagrams with a three-gluon coupling, one of which is shown inFig. 1(b), referred to below as E g . In the expression for E g , we introduce upper limits, Λand Σ on the two paths. For the simple cusp, we will take the limit Λ , Σ → ∞ . We returnto the finite case in the discussion of polygons.10fter evaluation of the three-gluon vertex, using β = 0, E g can be written as E g = N g ( ε ) (cid:90) d D x (cid:90) Σ0 dσ − x + 2 σx · β + i(cid:15) ) − ε × (cid:20)(cid:90) Λ0 dλ (cid:90) Λ λ dλ − x + 2 λ x · β + i(cid:15) ) − ε x · β (1 − ε )( − x + 2 λ x · β + i(cid:15) ) − ε − (cid:90) Λ0 dλ (cid:90) λ dλ x · β (1 − ε )( − x + 2 λ x · β + i(cid:15) ) − ε − x + 2 λ x · β + i(cid:15) ) − ε (cid:21) = N g ( ε ) (cid:90) d D x (cid:90) Σ0 dσ − x + 2 σx · β + i(cid:15) ) − ε (22) × (cid:20)(cid:90) Λ0 dλ − x + 2 λ x · β + i(cid:15) ) − ε (cid:90) λ dλ ∂∂λ (cid:18) − x + 2 λ x · β + i(cid:15) ) − ε (cid:19) − (cid:90) Λ0 dλ − x + 2 λ x · β + i(cid:15) ) − ε (cid:90) Λ λ dλ ∂∂λ (cid:18) − x + 2 λ x · β + i(cid:15) ) − ε (cid:19)(cid:21) , where in this case the numerical prefactor is N g ( ε ) = − i (cid:16) α s π (cid:17) C A C F Γ (1 − ε )8 π − ε ( πµ ) ε . (23)In the second equality of Eq. (22), we isolate two total derivatives, in the variables λ and λ . We shall carry out these two integrals first, at fixed values of the other path parametersand of x µ .There is a suggestive way of interpreting the total derivatives in Eq. (22), starting byrecognizing that the “propagator” for the Wilson line is a step function, for example, θ ( λ ),with “equation of motion” ∂ λ θ ( λ ) = δ ( λ ). In these terms, the λ or λ integrals over totalderivatives can also be thought of as the result of integration by parts and the use of theequation of motion. In the term with ∂/∂ λ , the equation of motion sets λ = λ and λ = Λ.As Λ → ∞ for fixed x µ , the term with λ = Λ vanishes as a power for any ε < /
2. Thevanishing of such contributions, through the cancellation of propagators, is an ingredientin the gauge invariance of the cusp, which generalizes to the gauge invariance of partonicamplitudes [44]. We shall take the limit Λ → ∞ first, at fixed values of the remainingintegration variables after using the eikonal equation of motion. We will confirm below thatthis prescription gives a gauge-invariant result for the cusp after summing over diagrams.We will evaluate the term from λ = Λ, which by itself is gauge dependent, in the Appendix.Returning to Eq. (22), we now integrate over the total-derivative integrals, λ in the first11erm and over λ in the second, and get E g = N g ( ε ) (cid:90) d D x (cid:90) Σ0 dσ − x + 2 σx · β + i(cid:15) ) − ε × (cid:90) Λ0 dλ (cid:20) − − x + i(cid:15) ) − ε − x + 2 λx · β + i(cid:15) ) − ε + 2( − x + 2 λx · β + i(cid:15) ) − ε − − x + 2Λ x · β + i(cid:15) ) − ε − x + 2 λx · β + i(cid:15) ) − ε (cid:21) ≡ E s + 2 E pse + E end . (24)Here we have relabeled the remaining parameters as σ and λ in both terms. The three termsidentified in the second relation correspond to the three terms in square brackets of the firstrelation. These terms involve scalar propagators only, and are represented by Fig. 3. Werefer to the first term in brackets as the 3-scalar integral, E s (Fig. 3(a)), in which the end ofone of the scalar propagators is fixed at the cusp by the eikonal equation of motion. We willcall the second term the “pseudo-self-energy”, E pse [Fig. 3(b)], since two scalar propagatorsform a loop and attach to the Wilson line at the same point. Finally, the third term, E end [Fig. 3(c)], in which λ = Λ for finite Λ will be referred to as the “end-point” diagram forthis case. As noted above, the cusp itself is defined without the end-point diagram, but wewill return to it in our discussion of Wilson line polygons below.We can identify the sources of subdivergences in the expressions of Eq. (24) by findingpoints where the x µ integral is pinched between coalescing singularities [40]. In the 3-scalarterm E , the integration contours of the light cone component β · x and two-dimensionaltransverse components x ⊥ are pinched when x µ = ζβ µ , with 0 < ζ < σ , and also when x µ = ηβ µ , with 0 < η < λ . For fixed λ and σ these are the singular subdivergences referredto above, in which the point x µ approaches the path in the β or β directions, respectively.In either case two lines are forced to the light cone on one of the Wilson lines, while the third (a) (b) (c) FIG. 3. (a) 3-scalar diagram (b) Pseudo-self-energy diagram (c) End-point diagram. λ and σ , which is removed by the standard renormalizationof the gauge theory.The integration of the 3-scalar term has been in the literature for a long time, but somedetails are given in the Appendix, to derive it as a coefficient times the scaleless integralsover parameters λ and σ . We find E s = (cid:16) α s π (cid:17) C A C F Γ(1 − ε )Γ(1 + ε )Γ(1 − ε )16 ε (2 πµ ) ε (cid:90) ∞ dλ dσ ( λσ ) − ε . (25)We have taken the upper limits to infinity at this point, because we are interested in the(unrenormalized) cusp integral.The pseudo-self-energy term in Eq. (24) inherits the entire ultraviolet divergence of thediagram E , Fig. 1(b) at fixed λ and σ , and requires a counterterm that is part of the web,rather than cusp, renormalization. The result is E pse = − (cid:16) α s π (cid:17) C A C F ε (cid:90) ∞ dλ dσλσ (cid:20) Γ (1 − ε )1 − ε (2 πµ λσ ) ε − Γ(1 − ε )(2 πµ λσ ) ε (cid:21) , (26)with the same scaleless integral times a single-scale constant. Finally, for the gluon self-energy diagrams, Figs. 1(c)–1(d), we use the renormalized one-loop gluon Green functionin coordinate space. The result for the self-energy contribution, E se of Fig. 1(c), where thegluon connects both Wilson lines, can be written as E se = − (cid:16) α s π (cid:17) C F ε (cid:90) ∞ dλ dσλσ (cid:20) Γ (1 − ε )1 − ε (cid:26) (5 − ε ) C A − T f n f (1 − ε )3 − ε (cid:27) (2 πµ λσ ) ε − Γ(1 − ε ) (cid:26) C A − T f n f (cid:27) (2 πµ λσ ) ε (cid:21) + E long , (27)where the (unrenormalized) longitudinal part of the Green function is given by E long = − (cid:16) α s π (cid:17) C F Γ (1 − ε )32 ε (1 + ε )(1 − ε ) (cid:26) (5 − ε ) C A − T f n f (1 − ε )3 − ε (cid:27) (28) × (cid:90) ∞ dλ dσ ∂∂λ ∂∂σ (cid:104) ( πµ ( β λ − β σ ) ) ε (cid:105) . The function E long comes from the coordinate-space transform of the q µ q ν term in the gluonself energy, and reduces to total derivatives in both σ and λ . In momentum space, the q µ q ν terms decouple from the gauge-invariant cusp algebraically in the sum over diagrams,assuming that the external Wilson lines carry no momentum. To define such derivative terms13n coordinate space for the cusp requires the introduction of small but nonzero β and β ,and with this infrared regularization, the longitudinal term above cancels the correspondingterm for the self-energy diagram of Fig. 1(d), up to end-point contributions analogous to E end in Eq. (24), which we have discarded in the calculation of the cusp contribution from E g above. We will once again neglect such terms for the purposes of this calculation, butwill return to this question in the next subsection.To check the finiteness and structure of the sum of these two-loop web diagrams, weexpand them in ε , keeping all terms that can contribute ultraviolet poles to the cusp. The(two) three-gluon diagrams plus the crossed ladder gives E cross + 2 E s = 18 (cid:16) α s π (cid:17) C F C A (cid:0) πe γ E µ (cid:1) ε (cid:18) π ε ζ + O ( ε ) (cid:19) (cid:90) ∞ dλ dσ ( λσ ) − ε . (29)Thus, as anticipated, the ultraviolet poles from the subdivergences of the web cancel, leavingonly the overall scaleless integrals, whose singular behavior can be associated with hard,soft, and collinear configurations for all of the lines of the web together. The π term willcontribute to Γ cusp and the εζ term to G eik . We next expand the integrands of E se and E pse at two loops, Eqs. (27) and (26) to order ε , E se + 4 E pse = − (cid:16) α s π (cid:17) C F (cid:90) ∞ dλ dσλσ (cid:20)(cid:26) ε π (cid:27) ε b (cid:2) (2 πµ e γ E λσ ) ε − (2 πµ e γ E λσ ) ε (cid:3) + (cid:26)(cid:18) C A − n f T f (cid:19) + ε (cid:18) C A − n f T f + π b (cid:19)(cid:27) (2 πµ e γ E λσ ) ε (cid:21) . (30)The terms proportional to b /ε serve to evolve the one-loop web, Eq. (18) to the scale 1 /λσ times constants.Combining Eqs. (29) and (30), we find the explicit terms in the web expansion, Eq. (11).In a scheme where logs of factors 2 πe γ E are absorbed into the definition of α s (1 /λσ ), wehave for the terms in Eq. (11), w ( α s ) = − α s π C F − (cid:16) α s π (cid:17) C F (cid:18)(cid:20) − π (cid:21) C A − n f T f (cid:19) + . . . ,w ( α s ) = − (cid:16) α s π (cid:17) C F (cid:18)(cid:20) − ζ (cid:21) C A − n f T f + ζ b (cid:19) + . . . ,w ( α s ) = − α s π C F π
12 + . . . , (31)where omitted terms are higher order in α s or do not contribute to the cusp ultraviolet poles.The term linear in ε begins at order α s , but the single pole also gets a contribution from the ε term at one loop, when combined with the running of the coupling. With these results14n hand, we can return to Eq. (11) and expand α s (1 /λσ ) in terms of the coupling at a fixedscale, α s ( µ ) using (13). This enables us to derive the single ultraviolet pole in E to order α s , and hence the anomalous dimension G eik at two loops, G eik = 12 C F C A (cid:16) α s π (cid:17) (cid:20)(cid:26) − π − ζ (cid:27) C A + (cid:26) − π (cid:27) n f T f (cid:21) . (32)In Sec. IV, we will see the close relation of this result to the “collinear anomalous dimension”derived long ago in Ref. [10] for a closed polygon of Wilson lines of finite size. D. Web integrals, end points and gauge invariance
A self-contained coordinate-space derivation of Eq. (9), generalizing the renormalizationanalysis of Ref. [30] for massive Wilson lines is given in [38]. Here, however, we will generalizeour prescription for the calculation of the gauge-invariant cusp anomalous dimension. As wehave seen, this requires us to find in coordinate space the analog of the action of momentum-space Ward identities that ensure the gauge invariance of the S-matrix [44].In the following brief but all-orders discussion we follow Ref. [45] and write the exponentas a sum over the numbers, e a , of gluons attached to the two Wilson lines, of velocity β a , a = 1 ,
2. We note, however, that the argument extends to any number of lines. Theweb diagrams are integrals over the positions λ j β and σ k β of these ordered vertices ofa function W e ,e ( { λ j } , { σ k } ), which includes the integrals over all the internal vertices ofthe corresponding web diagrams. In the notation of Ref. [45] we then have at n th order( n ≥ e + e ), E ( n ) = n − (cid:88) e =1 n − e (cid:88) e =1 e (cid:89) j =1 (cid:90) ∞ λ j − dλ j e (cid:89) k =1 (cid:90) ∞ σ k − dσ k W ( n ) e ,e ( { λ j } , { σ k } ) , (33)with λ , σ ≡
0. Here we expand functions as E = (cid:80) ( α s /π ) n E ( n ) . We can use the notationof Eq. (33) to generalize our treatment of the three-gluon diagram and self-energy diagramsabove. First, we isolate those contributions to W ( n ) e ,e ( { λ j } , { σ k } ) that are of the form oftotal derivatives in the largest path parameters, λ e , σ e , and whose upper limits vanishwhen the end points of ordered exponentials are taken to infinity for fixed values of the15nternal vertices of the web. We represent this separation as, W ( n ) e ,e ( { λ j } , { σ k } ) = ∂∂ λ e X ( λ )( n ) e ,e ( { λ j } , { σ k } ) + ∂∂ σ e X ( σ )( n ) e ,e ( { λ j } , { σ k } ) (34)+ ∂∂ λ e ∂∂ σ e X ( λσ )( n ) e ,e ( { λ j } , { σ k } ) + W ( n ) e ,e ( { λ j } , { σ k } ) , where the X ( I ) , I = λ, σ , λσ , are functions whose derivatives are taken by λ e , σ e orboth, and which vanish when λ e and/or σ e are taken to infinity with other integrationvariables held fixed. The function W is the remaining web integrand. To determine thecusp, we evaluate the total derivatives at the lower limits, λ e = λ e − , σ e = σ e − or both,discarding the upper limits, as E end in the two-loop case above. We then relabel the largestremaining λ j integral (either λ e or λ e − ) as λ , and integrate over the rest of the λ j , up to λ . The σ k parameters are treated in just the same way. In this manner, we find for the webfunction in Eq. (9), the form w (cid:0) α s (1 /λσ, ε ) , λσµ , ε (cid:1) = n − (cid:88) e =1 n − e (cid:88) e =1 e (cid:89) j =1 (cid:90) λλ j − dλ j e (cid:89) k =1 (cid:90) σσ k − dσ k δ ( λ e − λ ) δ ( σ e − σ ) × (cid:34) − δ ( λ e − − λ ) X ( λ )( n ) e ,e ( { λ j } , { σ k } ) − δ ( σ e − − σ ) X ( σ )( n ) e ,e ( { λ j } , { σ k } ) (35)+ δ ( λ e − − λ ) δ ( σ e − − σ ) X ( λσ )( n ) e ,e ( { λ j } , { σ k } ) + W ( n ) e ,e ( { λ j } , { σ k } ) (cid:35) . Once web diagrams are summed over at any order, this form is gauge invariant, and producesthe same cusp integrand for finite lines as for infinite lines. This is because the infinitesimalgauge variation of a product of Wilson lines as in Eq. (1) produces a ghost propagator endingon the ends of the lines, which vanishes when those lines are taken to infinity [44]. Even ifthe ends of the lines are at finite distances, the prescription to discard the upper limit oftotal derivatives automatically removes these gauge variations. When the end points, whichgeneralize E end in Eq. (24) in our discussion above, are at finite distances, however, we mustkeep these terms and combine them with the remainder of the diagrams of the graph toderive the full, gauge-invariant result. IV. APPLICATIONS TO POLYGON LOOPS
The above reasoning leads to a number of interesting results for polygonal closed Wil-son loops [11–13]. These amplitudes also exponentiate in perturbation theory in terms of16ebs [11]. To this observation we may apply once again the lack of subdivergences for webs.Generic diagrams for quadrilateral loops are shown in Figs. 4 and 5. In Fig. 4, for example,the a th vertex of the polygon represents a cusp vertex that connects two Wilson lines, ofvelocity β a − and β a , with β ≡ β .Exponentiation in coordinate space implies that the logarithm of a polygon P is a sumof the web configurations illustrated by the figures,ln P = (cid:88) cusps a W a + (cid:88) sides { a +1 ,a } W a +1 ,a + W plane . (36)The first terms organize webs associated entirely with one of the cusps of the polygon,constructed in terms of the coordinate webs identified above. Because each edge is of finitelength, we must now retain the additional gauge-variant terms associated with the end-pointcontributions ( E end above), which are to be combined with gauge-variant end points fromwebs connecting three or four sides to derive a gauge-invariant result. The cancellation ofsubdivergences in webs implies that after a sum over diagrams, only the cusp poles and asingle, overall collinear singularity survives [11, 38]. There remains a finite contribution fromwebs that connect all four (or in general more) of the Wilson lines, and these are representedby the final term in (36).Evidently, the single-cusp contribution, W a ( β a , β a − ) has the same gauge-invariant inte-grand as for the finite Wilson lines in Eq. (10), in terms of the lengths L a of the sides of thepolygon, between vertices a and a + 1 W a ( β a , β a − , L a , L a − ) = (cid:90) L a dλ a λ a (cid:90) − L a − dσ a σ a w ( α s (1 /λ a σ a , ε ) , ε ) . (37)The web function w for the cusp can depend only on the scalar products of the velocities,and we may assume for simplicity that these are all of the same order. X a X a − σ a β a − X a + λ a β a w. . . ...FIG. 4. A single-cusp web W a , in the sum of Eq. (36). on the ends of the lines, which vanishes when those lines are taken to infinity [37]. Even ifthe ends of the lines are at finite distances, the prescription to discard the upper limit oftotal derivatives automatically removes these gauge variations. When the end-points, whichgeneralize E end , Eq. (24) in our discussion above, are at finite distances, however, we mustkeep these terms and combine them with the remainder of the diagrams of the graph toderive the full, gauge invariant result. IV. APPLICATIONS TO POLYGON LOOPS
The above reasoning leads to a number of interesting results for polygonal closed Wilsonloops [10, 11, 13]. These amplitudes also exponentiate in perturbation theory in terms ofwebs [10]. To this observation we may apply once again the lack of subdivergences for webs.Generic diagrams for quadrilateral loops are shown in Figs. 4 and 5. In Fig. 4, for example,the a th vertex of the polygon represents a cusp vertex that connects two Wilson lines, ofvelocity β a − and β a , with β ≡ β .Exponentiation in coordinate space implies that the logarithm of a polygon P is a sumof the web configurations represented by the figures,ln P = cusps a W a + sides { a +1 ,a } W a +1 ,a + W plane . (36)The first terms organize webs associated entirely with one of the cusps of the polygon,constructed in terms of the coordinate webs identified above. Because each edge is of finitelength, we must now retain the additional gauge variant terms associated with the end-pointcontributions ( E end above), which are to be combined with gauge-variant end-points fromwebs connecting three or four sides to derive a gauge-invariant result. The cancellation of16 FIG. 4. A single-cusp web W a , in the sum of Eq. (36). L a η , with L a thelength of this side, and η a typical distance of vertices in the web from the side. As a result,the general form of the W a +1 ,a in Eq. (36) is W a +1 ,a ( L a ) = (cid:90) L a dηη w a +1 ,a ( α s ( ηL a , ε )) , (38)for a function w a +1 ,a ( α s ), where we assume all the sides are of a similar length. Finally, forthe diagrams in which the web is stretched out between more than three sides of a polygon(in this case, the web is connected to all four sides of the quadrilateral), W plane , the onlyscale available is the area of the quadrilateral, and these web contributions are an expansionin the coupling evaluated at the inverse area, with finite coefficients.The two-loop diagrams for all of these topologies were computed in [11]. We note that inthe results quoted there, the cusp anomalous dimension does not appear until all diagramsof the topologies of W a and W a +1 ,a are combined. Following the prescription for the webintegrand given above, however, the two-loop cusp is associated entirely with the diagramsdressing a single corner, W a , precisely because the gauge-variant end-point contributions E end of Eq. (24) are not included in that object. For polygons, these gauge-variant termsat two loops, or any order, cancel contributions from the two-cusp contributions W a +1 ,a ,which also give rise to gauge-variant terms that cancel those from planar diagrams. Thesegauge-variant terms contain subdivergences in general. The complete result, of course, isgauge invariant and corresponds at two loops to the full calculation in Refs. [10] and [11].subdivergences in webs implies that after a sum over diagrams, only the cusp poles and asingle, overall collinear singularity survives [10, 36]. There remains a finite contribution fromwebs that connect all four (or in general more) of the Wilson lines, and these are representedby the final term in (36).Evidently, the single-cusp contribution, W a ( β a , β a − ) has the same gauge invariant inte-grand as for the finite Wilson lines in Eq. (10), in terms of the lengths L a of the sides of thepolygon, between vertices a and a + 1 W a ( β a , β a − , L a , L a − ) = L a dλ a λ a − L a − dσ a σ a w ( α s (1 /λ a σ a , ε ) , ε ) . (37)The web function w for the cusp can depend only on the scalar products of the velocities,and we may assume for simplicity that these are all of the same order.The two-cusp contributions connect three sides, and the only available singular configu-ration is when all lines in the web are parallel to the side between the two adjacent vertices.The only invariants on which the web can then depend are of the form L a λ , with L a thelength of this side, and λ a typical distance of vertices in the web from the side. As a result,the general form of the W a +1 ,a in Eq. (36) is W a +1 ,a = L a dλλ w a +1 ,a ( α s ( λL a , ε )) , (38)for a function w a +1 ,a ( α s ). Finally, for the planar diagrams, in which the web is stretchedout over the dimensions of the polygon (quadrilateral in this case), W plane , the only scaleavailable is the area of the quadrilateral, and these web contributions are an expansion inthe coupling evaluated at the inverse area, with finite coefficients. (a) X a X a +1 ... ... . . . (b) ... ... . . .. . . FIG. 5. (a) A ‘side’ web W a +1 ,a in of Eq. (36), in this case associated with the lightlike side between X a and X a +1 . (b) A web that contributes to W plane in Eq. (36). (a) subdivergences in webs implies that after a sum over diagrams, only the cusp poles and asingle, overall collinear singularity survives [10, 36]. There remains a finite contribution fromwebs that connect all four (or in general more) of the Wilson lines, and these are representedby the final term in (36).Evidently, the single-cusp contribution, W a ( β a , β a − ) has the same gauge invariant inte-grand as for the finite Wilson lines in Eq. (10), in terms of the lengths L a of the sides of thepolygon, between vertices a and a + 1 W a ( β a , β a − , L a , L a − ) = L a dλ a λ a − L a − dσ a σ a w ( α s (1 /λ a σ a , ε ) , ε ) . (37)The web function w for the cusp can depend only on the scalar products of the velocities,and we may assume for simplicity that these are all of the same order.The two-cusp contributions connect three sides, and the only available singular configu-ration is when all lines in the web are parallel to the side between the two adjacent vertices.The only invariants on which the web can then depend are of the form L a λ , with L a thelength of this side, and λ a typical distance of vertices in the web from the side. As a result,the general form of the W a +1 ,a in Eq. (36) is W a +1 ,a = L a dλλ w a +1 ,a ( α s ( λL a , ε )) , (38)for a function w a +1 ,a ( α s ). Finally, for the planar diagrams, in which the web is stretchedout over the dimensions of the polygon (quadrilateral in this case), W plane , the only scaleavailable is the area of the quadrilateral, and these web contributions are an expansion inthe coupling evaluated at the inverse area, with finite coefficients. (a) X a X a +1 ... ... . . . (b) ... ... . . .. . . FIG. 5. (a) A ‘side’ web W a +1 ,a in of Eq. (36), in this case associated with the lightlike side between X a and X a +1 . (b) A web that contributes to W plane in Eq. (36). (b) FIG. 5. (a) A “side” web W a +1 ,a in Eq. (36), in this case associated with the lightlike side between X a and X a +1 . (b) A web that contributes to W plane in Eq. (36). dd ln µ P ren = − (cid:88) a Γ cusp ( α s ( µ )) ln( µ L a L a − β a · β a − ) − Γ co (cid:0) α s ( µ ) (cid:1) , (39)where the L a and µ -dependence of the first term is characteristic of cusps with lightlike Wil-son lines [17], and where the second term, Γ co was called the collinear anomalous dimensionin Ref [10]. Aside from overall factors associated with the number of sides of the polygon, thecollinear anomalous dimension for the quadrilateral is identical to G eik in Eq. (32), exceptfor the coefficient of ζ , which differs due to extra diagrams that connect three sides of thequadrilateral.Polygons of this sort have been studied in the context of a duality to scattering amplitudesin conformal theories [11, 12]. Here, we consider a four-sided polygon that projects to asquare in the x /x plane, with side X , as in Figs. 4–5. In four dimensions, the loop startsat the origin, travels along the plus- x direction for a “time” X = X , then changes directionto x for time X , and then moves backwards in time and space, first in the x direction,then x , back to the origin. We can now use the coordinates x and x to define parameters λ a and σ a for each of the cusp integrals W a in Eq. (37), σ = − x , λ = x ,σ = x − X , λ = x ,σ = x − X , λ = X − x ,σ = − x , λ = X − x . (40)In this notation, we can add the four cusp web integrals of Eq. (37), to get a single integralover x and x . The web functions, of course, depend on the particular forms of λ and σ above. We find (cid:88) a =1 W a ( β a , β a − ) = (cid:90) X dx (cid:90) X dx ( X − x )[( X − x ) w + x w ] + x [ x w + ( X − x ) w ] x ( X − x ) x ( X − x ) , (41)where w a ≡ w ( α s ( λ a ( x , x ) σ a ( x , x ))). For a conformal theory, all dependence on the σ a and λ a is in the denominators and we can sum over a to get a result in terms of a constantweb function w . Changing variables to y a = 1 − x a /X , we derive the unregularized formfound from the analysis of extremal two-dimensional surfaces embedded in a five-dimensionalbackground in [12], (cid:88) a =1 W a ( β a , β a − ) = (cid:90) − dy (cid:90) − dy w (1 − y )(1 − y ) , (42)19o which we should add the collinear and finite multi-cusp contributions of Fig. 5. V. CONCLUSIONS
We have found that when the massless cusp is analyzed in coordinate space, it is naturallywritten as the exponential of a two-dimensional integral. The integrand, a web function,depends on the single invariant scale through the running of the coupling, which for a theorythat is conformal in four dimensions agrees with strong-coupling results [12, 13, 43]. Thisagreement extends to aspects of closed, polygonal Wilson loops. These results do not relyon a planar limit [16], but it is natural to conjecture that for large N c the integral may takeon an even more direct interpretation in terms of surfaces for nonconformal theories.In QCD, of course, our explicit knowledge of the web function is limited to the first fewterms in the perturbative series, which run out of predictive power as the invariant distanceincreases. The integral forms derived above, however, hold to all orders in perturbationtheory, and may point to an interpolation between short and long distances. ACKNOWLEDGMENTS
We thank G. P. Korchemsky and B. van Rees for helpful discussions. This work wassupported by the National Science Foundation, Grants No. PHY-0969739 and No. PHY-1316617. [1] I. Bialynicki-Birula, Bull. Acad. Polon. Sci. , 135 (1963);S. Mandelstam, Phys. Rev. , 1580 (1968).[2] C. N. Yang, Phys. Rev. Lett. , 445 (1974);A. M. Polyakov, Phys. Lett. B , 477 (1978);L. Susskind, Phys. Rev. D , 2610 (1979).[3] K. G. Wilson, Phys. Rev. D , 2445 (1974).[4] G. P. Korchemsky, G. Marchesini, Phys. Lett. B313 , 433-440 (1993).[5] G. P. Korchemsky and G. F. Sterman, Nucl. Phys. B , 415 (1995) [hep-ph/9411211].[6] A. V. Belitsky, Phys. Lett. B , 307 (1998) [hep-ph/9808389].
7] R. Kelley, M. D. Schwartz, R. M. Schabinger and H. X. Zhu, Phys. Rev. D , 045022 (2011)[arXiv:1105.3676 [hep-ph]].[8] E. Laenen, G. F. Sterman and W. Vogelsang, Phys. Rev. D , 114018 (2001) [hep-ph/0010080].[9] I. O. Cherednikov, T. Mertens, P. Taels and F. F. Van der Veken, Int. J. Mod. Phys. Conf.Ser. , 1460006 (2014) [arXiv:1308.3116 [hep-ph]].[10] I. A. Korchemskaya, G. P. Korchemsky, Phys. Lett. B287 , 169-175 (1992).[11] J. M. Drummond, G. P. Korchemsky and E. Sokatchev, Nucl. Phys. B , 385 (2008)[arXiv:0707.0243 [hep-th]];J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, Nucl. Phys. B , 52 (2008)[arXiv:0709.2368 [hep-th]].[12] L. F. Alday and J. M. Maldacena, JHEP , 064 (2007) [arXiv:0705.0303 [hep-th]];L. F. Alday and J. Maldacena, JHEP , 068 (2007) [arXiv:0710.1060 [hep-th]].[13] L. F. Alday and R. Roiban, Phys. Rept. , 153 (2008) [arXiv:0807.1889 [hep-th]].[14] Y. -T. Chien, M. D. Schwartz, D. Simmons-Duffin and I. W. Stewart, Phys. Rev. D , 045010(2012) [arXiv:1109.6010 [hep-th]].[15] B. Basso, A. Sever and P. Vieira, Phys. Rev. Lett. , 091602 (2013) [arXiv:1303.1396 [hep-th]].[16] G. ’t Hooft, Nucl. Phys. B , 461 (1974).[17] G. P. Korchemsky, A. V. Radyushkin, Nucl. Phys. B283 , 342-364 (1987).[18] E. Laenen, K. J. Larsen and R. Rietkerk, arXiv:1410.5681 [hep-th].[19] N. Kidonakis, G. Oderda and G. F. Sterman, Nucl. Phys. B , 365 (1998) [hep-ph/9803241].[20] C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D , 054022 (2002) [hep-ph/0109045].[21] A. Mitov, G. F. Sterman and I. Sung, Phys. Rev. D , 094015 (2009) [arXiv:0903.3241[hep-ph]].[22] M. Beneke, P. Falgari and C. Schwinn, Nucl. Phys. B , 414 (2011) [arXiv:1007.5414 [hep-ph]].[23] A. Ferroglia, M. Neubert, B. D. Pecjak and L. L. Yang, JHEP , 062 (2009)[arXiv:0908.3676 [hep-ph]].[24] N. Kidonakis, Phys. Rev. D , 114030 (2010) [arXiv:1009.4935 [hep-ph]].
25] E. Gardi, E. Laenen, G. Stavenga and C. D. White, JHEP , 155 (2010) [arXiv:1008.0098[hep-ph]].[26] R. Kelley and M. D. Schwartz, Phys. Rev. D , 045022 (2011) [arXiv:1008.2759 [hep-ph]].[27] T. T. Jouttenus, I. W. Stewart, F. J. Tackmann and W. J. Waalewijn, Phys. Rev. D ,114030 (2011) [arXiv:1102.4344 [hep-ph]].[28] E. Gardi, J. M. Smillie and C. D. White, JHEP , 088 (2013) [arXiv:1304.7040 [hep-ph]].[29] E. Gardi, arXiv:1401.0139 [hep-ph].[30] R. A. Brandt, F. Neri and M. -a. Sato, Phys. Rev. D , 879 (1981). arXiv:1302.6765 [hep-th].[31] I. A. Korchemskaya and G. P. Korchemsky, Nucl. Phys. B , 127 (1995) [hep-ph/9409446].[32] J. G. M. Gatheral, Phys. Lett. B , 90 (1983);J. Frenkel and J. C. Taylor, Nucl. Phys. B , 231 (1984);G. Sterman, in “Perturbative Quantum Chromodynamics”, D. W. Duke and J. F. Owens ed.,AIP Conf. Proc. , 22 (American Inst. of Phys., 1981);A. A. Vladimirov, Phys. Rev. D , 066007 (2014) [arXiv:1406.6253 [hep-th]].[33] V. S. Dotsenko and S. N. Vergeles, Nucl. Phys. B , 527 (1980).[34] C. F. Berger, arXiv:hep-ph/0305076;C. F. Berger, Phys. Rev. D , 116002 (2002) [arXiv:hep-ph/0209107].[35] L. Magnea and G. Sterman, Phys. Rev. D , 4222 (1990).[36] S. Catani, Phys. Lett. B , 161 (1998) [hep-ph/9802439];G. Sterman and M. E. Tejeda-Yeomans, Phys. Lett. B , 48 (2003) [arXiv:hep-ph/0210130];Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D , 085001 (2005) [arXiv:hep-th/0505205].[37] L. J. Dixon, L. Magnea and G. Sterman, JHEP , 022 (2008) [arXiv:0805.3515 [hep-ph]].[38] O. Erdo˘gan and G. Sterman, arXiv:1411.4588 [hep-ph].[39] G. Date, doctoral thesis, UMI-83-07385.[40] O. Erdo˘gan, Phys. Rev. D , 085016 (2014) [Erratum-ibid. D , 089902 (2014)][arXiv:1312.0058 [hep-th]].[41] G. T. Bodwin, Phys. Rev. D , 2616 (1985) [Erratum-ibid. D , 3932 (1986)];J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B , 104 (1985); , 833 (1988).[42] J. C. Collins, D. E. Soper and G. Sterman, in “Perturbative Quantum Chromodynamics”, A.H.Mueller, ed., Adv. Ser. Direct. High Energy Phys. , 1 (World Scientific, 1988) [arXiv:hep- h/0409313];J. Collins, “Foundations of Perturbative QCD” (Cambridge Univ. Pr., 2011).[43] M. Kruczenski, JHEP , 024 (2002) [hep-th/0210115].[44] G. ’t Hooft, Nucl. Phys. B , 173 (1971);G. ’t Hooft and M. J. G. Veltman, NATO Adv. Study Inst. Ser. B Phys. , 177 (1974).[45] A. Mitov, G. Sterman, I. Sung, Phys. Rev. D82 , 096010 (2010). [arXiv:1008.0099 [hep-ph]].
Appendix A: Two-loop Integrals1. The 3-scalar integral
To evaluate the the 3-scalar term in Eq. (24), we integrate over the position of the three-gluon vertex after combining the denominators by Feynman parametrization. Introducingthe Feynman parameters α and α , the 3-scalar contribution is given by E s = − N g ( ε ) (cid:90) ∞ dλ dσ (cid:90) d − ε y Γ(3 − ε )Γ (1 − ε ) × (cid:90) dα (cid:90) − α dα (1 − α − α ) − ε α − ε α − ε [ − y + 2 α (1 − α − α ) λσ + i(cid:15) ] − ε , (A1)where y ≡ x − α λβ − (1 − α − α ) σβ . The integral over y is straightforward after doinga clockwise Wick rotation, E s = − N g ( ε ) (cid:18) − iπ − ε − ε Γ(1 − ε )Γ (1 − ε ) (cid:19) (cid:90) ∞ dλ dσ ( λσ ) − ε × (cid:90) dα (cid:90) − α dα (1 − α − α ) − ε α − ε α − ε . (A2)The integrals over Feynman parameters α , α now factor from the integrals over eikonalparameters λ, σ . After a change of variables η ≡ α / (1 − α ), they can be integrated inde-pendently, (cid:90) dα α − ε (1 − α ) ε − (cid:90) dη η ε − (1 − η ) ε − = 1 ε Γ(1 − ε )Γ(1 + ε ) . (A3)In Eq. (A2), this gives the scaleless λ, σ integral times a constant with a double pole in ε ,given in Eq. (25). 23 . The end-point term We now return to the λ = Λ end-point contribution from the second term on the right-hand side of Eq. (22), which vanishes in the Λ → ∞ limit for any fixed values of the vertex x µ .If we integrate over x µ first, however, we get a singular contribution, associated with therenormalization of a Wilson line of finite length. It cancels in the gauge-invariant polygonsdiscussed in Sec. IV, and extensively in Refs. [10, 11]. After the x µ integral, we have E end = − N g ( ε ) (cid:18) − iπ − ε − ε Γ(1 − ε )Γ (1 − ε ) (cid:19) (cid:90) Σ0 dσσ − ε (cid:90) Λ0 dλ (A4) × (cid:90) dα (cid:90) − α dα α ε − (1 − α − α ) − ε α − ε [ α Λ + (1 − α − α ) λ ] − ε . Changing variables to η = α / (1 − α ), we find a form that is easy to evaluate, E end = − N g ( ε ) (cid:18) − iπ − ε − ε Γ(1 − ε )Γ (1 − ε ) (cid:19) (cid:90) Σ0 dσσ − ε (A5) × (cid:90) dα α ε − (cid:90) dη (1 − η ) − ε η − ε (cid:90) Λ0 dλ [ η Λ + (1 − η ) λ ] − ε = (cid:16) α s π (cid:17) C A C F (cid:0) πµ ΛΣ (cid:1) ε ε (cid:2) Γ(1 − ε ) Γ(1 − ε ) Γ(1 + ε ) − Γ (1 − ε ) (cid:3) . If we add this result to the expressions found by integrating the σ and λ integrals of E s ,Eq. (25) and E psepse