Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations

The IR-divergent scattering amplitudes of N=4 supersymmetric Yang-Mills theory can be regulated in a variety of ways, including dimensional regularization and massive (or Higgs) regularization. The IR-finite part of an amplitude in different regularizations generally differs by an additive constant at each loop order, due to the ambiguity in separating finite and divergent contributions. We give a prescription for defining an unambiguous, regulator-independent finite part of the amplitude by factoring off a product of IR-divergent"wedge"functions. For the cases of dimensional regularization and the common-mass Higgs regulator, we define the wedge function in terms of a form factor, and demonstrate the regularization independence of the n-point amplitude through two loops. We also deduce the form of the wedge function for the more general differential-mass Higgs regulator, although we lack an explicit operator definition in this case. Finally, using extended dual conformal symmetry, we demonstrate the link between the differential-mass wedge function and the anomalous dual conformal Ward identity for the finite part of the scattering amplitude.


Introduction and overview
Unraveling the pattern of soft and collinear divergences in scattering amplitudes is a critical endeavor to advance our understanding of gauge theories in general and to assist in concrete computations for collider phenomenology, e.g., in massless gauge theories such as Quantum Chromodynamics (QCD). These studies have a long history (see e.g. the early review [1]) and have contributed to our knowledge of the universal infrared (IR) structure of gauge theory amplitudes. Based on the concepts of soft and collinear factorization, non-abelian exponentiation, and the study of collinear limits, significant information about scattering amplitudes is available to all orders in perturbation theory. For precision predictions at modern colliders, especially within QCD and including higher order quantum corrections, these insights are of great practical importance [2,3].
Quite generally, factorization implies the separation of scales in a given scattering reaction, i.e. the process-dependent hard scale q 2 from those governing the soft and collinear limit, defined for instance by the masses m i of the scattering particles with q 2 ≫ m 2 i or by the regions of soft momenta. Note that the soft and collinear singularities of massless particles (gauge bosons) strictly require the definition of a regulator, which is conveniently performed in D = 4 − 2ǫ dimensions. As an immediate consequence of factorization, evolution equations emerge, which depend on the kinematics of the specific process and on the chosen regulator. Their solution leads to non-abelian exponentiation, a result which also arises from an effective field theory formulation based on the ultra-violet (UV) renormalization properties of effective operators and their anomalous dimensions; see e.g. ref. [4]. Moreover, for scattering amplitudes in gauge theories, the underlying factorization imposes strong constraints on the anomalous dimensions and the all-order structure of the IR singularities [5][6][7][8].
In the present paper we will specialize our investigations in a number of ways. First, we choose to work in the N = 4 supersymmetric Yang-Mills (SYM) theory, which is the simplest non-abelian gauge theory in four dimensions due to the vanishing of the four-dimensional β function. In our study, we are concerned with form factors and scattering amplitudes in this theory, which allows us to study their IR singularities without interference from UV divergences. Second, we will be working in the so-called planar limit and, for scattering amplitudes A n of n external particles, we will assume color ordering. Our main focus is on the study of different kinematical regimes, i.e. scattering amplitudes of massless and massive particles and associated form factors, using different regulator schemes.
The general property of factorization prompts us to ask whether one can delineate a welldefined finite part of A n independent of the chosen IR regularization. While reasoning along these lines has already been employed in the derivation of radiative corrections for heavy-quark hadroproduction at two loops in QCD [9,10] (see also refs. [11,12]), this issue is more generally related to the important question whether physical observables in theories with massless particles are independent of the regulator 1 ; see e.g. the discussion in ref. [14]. To that end, in this paper we specifically compare dimensional and massive regularizations schemes for n-particle scattering amplitudes A n in N = 4 SYM theory, an ideal testing ground for these questions due to the simplicity of its loop expansion. 2 In-depth studies of the latter may lead to new insights for gauge theories with massive particles which will eventually also be of interest for applications in collider phenomenology.
Let us start thus, for simplicity, with maximally-helicity-violating (MHV) scattering amplitudes. Factorization implies that the color-ordered amplitude A n = A tree n M n of n particles can be written as (see e.g. ref. [16]) M n = S n × J n × H n . (1.1) Here S n and J n are "soft" and "jet" functions, respectively, and H n is an IR-finite "hard function". In general, M n and H n are vectors in a space of possible color structures, and S n is a matrix. In the planar limit, S n is proportional to the identity matrix, and one can combine S n and J n into a product of "wedge" functions W (s i−1,i ) that depend only on two adjacent particles i − 1 and i of the color-ordered amplitude [17] As will be detailed below, the factorization (1.2) holds not only in dimensional regularization, but also in cases where masses are used to partially or fully regulate the IR divergences. The wedge functions W (q 2 ) satisfy renormalization group equations which imply that they exponentiate. The factorization (1.2) fixes the hard function H n only up to finite pieces, but with a suitable definition of the wedge function, we suggest that can be used to define a regulator-independent finite part of the amplitude. For regulators that leave the external particles massless, such as dimensional regularization in D = 4 − 2ǫ dimensions or the common-mass Higgs regulator described below, each wedge has half the IR divergences of a Sudakov form factor Φ(q 2 ) (see e.g. refs. [18,19]), so it is natural to define W (q 2 ) = Φ(q 2 ) [3,17]. We show in this paper that, with this definition, log H n is identical for both of these regulators through two-loop order.
We also analyze a refined version of the Higgs regulator with differential masses, described below. In this case, the external particles have distinct masses, and so the wedge function cannot simply be defined in terms of a Sudakov form factor. Instead, we define the one-loop wedge function in terms of a certain IR-divergent triangle diagram, and then use extended dual conformal invariance to extend this to an all-loop expression for the sum of wedge functions. With this choice for the IR-divergent wedge function, we establish that the IR-finite hard function log H n takes precisely the same form for the differential-mass Higgs regulator as for the commonmass regulator. We lack, however, an explicit operator definition for the wedge function in this case.
For N = 4 SYM theory, the regulator-independent hard function log H n takes the simple form due to the conjectured duality between the finite part of the MHV scattering amplitudes and the (UV renormalized) expectation value of certain cusped Wilson loops (see refs. [20,21] for reviews). In eq. (1.4), γ(a) is the cusp anomalous dimension [22], for which a prediction to all orders in the coupling constant a = g 2 N/(8π 2 ) exists [23], and D(a) and C(a) are kinematic-independent functions. The amplitude is expressed as a function of the dual or region momenta x µ i , which are defined by The first three terms on the r.h.s. of eq. (1.4), whose kinematical dependence is determined solely by the one-loop contribution H (1) n (x 2 ij ), constitute the ABDK/BDS ansatz [17,24]. The a priori undetermined remainder function R n (x 2 ij , a) contains the only non-trivial, i.e. loop-dependent, kinematical dependence. Equation (1.4) follows from a conformal Ward identity for the dual Wilson loop [25,26]. The first term on the r.h.s. of eq. (1.4) provides a particular solution to this Ward identity. The remainder function R n is the general homogeneous solution to the Ward identity, and depends only on dual conformal crossratios, which take the form x 2 ij x 2 mn /(x 2 im x 2 jn ). Due to the absence of dual conformal cross-ratios for n = 4 and n = 5, the remainder functions R 4 and R 5 vanish and therefore the corresponding hard functions log H 4 and log H 5 are completely determined by their one-loop value and the kinematicindependent functions γ(a), D(a), and C(a). For n ≥ 6, dual conformal cross-ratios can be built, and the remainder function is known to be non-zero starting from two loops and n = 6 external particles [27,28]. Its higher-loop and higher-point form is under intense investigation; see e.g. refs. [29][30][31][32][33].
The planar MHV n-point amplitude for N = 4 SYM theory has been studied using dimensional regularization and also using an alternative massive IR regulator 3 [38][39][40][41][42]. The latter is motivated by the AdS/CFT correspondence and consists of computing scattering amplitudes on the Coulomb branch of N = 4 SYM theory, i.e. giving a non-trivial vacuum expectation value to some of the scalars. One can achieve a situation where the propagators on the perimeter of any loop diagram are massive, thereby regulating the IR divergences. The simplest case, the "common mass Higgs regulator" in which only one mass m is introduced, corresponds to the breaking of the U(N + M) gauge group to U(N) ×U(M), with fields in the adjoint representation of U(M) remaining massless.
In the more general "differential-mass Higgs regulator", one breaks the gauge group further to U(N) × U(1) M , thereby introducing various masses m i , i = 1, . . . , M. Fields in the adjoint of the broken U(M), which appear as external states in the scattering amplitudes, now have nonzero masses |m i − m i+1 | = 0. We use a decomposition of the one-loop MHV n-point amplitude into a sum of IR-divergent triangle diagrams and IR-finite six-dimensional box integrals to define the sum of one-loop wedge functions as 4 in the uniform small mass limit (i.e., m i = α i m, with α i fixed and m → 0). The one-loop hard function H (1) n (x 2 ij ) is then expressed in terms of IR-finite quantities, and thus is manifestly regulator-independent.
A key point is that the massive regulator is closely connected to dual conformal symmetry. The Higgs masses can be interpreted within the AdS/CFT duality as the radial coordinates in a T -dual AdS 5 space. While the isometries of this space yield the usual dual conformal transformations for zero masses, they define a different realization of this symmetry for finite masses, dubbed "extended dual conformal symmetry" [38]. Since no further regulator is needed in the massive setup, the extended dual conformal symmetry is expected to be an exact symmetry of the planar amplitudes. Recently, it was shown that tree-level amplitudes on the Coulomb branch of N = 4 SYM and also all cuts of planar loop amplitudes do indeed have this extended symmetry [43,44]. Together with the expected cut-constructibility of N = 4 SYM this then proves the extended dual conformal symmetry property conjectured in ref. [38].
What we wish to emphasize is that, while planar amplitudes have extended dual conformal symmetry, the wedge functions and regulator-independent hard functions separately do not. This is not surprising, since extended dual conformal transformations act on the masses m i (as well as the dual variables x i ), whereas the hard function is, by definition, independent of the masses in the uniform small mass limit. Nevertheless, extended dual conformal symmetry can be used to determine the all-loop structure of the IR divergences of scattering amplitudes in the case of the differential-mass Higgs regulator. Assuming the validity of eq. (2.10) for MHV scattering amplitudes, together with eq. (1.6), we obtain the following expression for the IR-divergent pieces in the differential-mass setup valid for uniform small masses.
Having deduced the form of the IR divergences of the amplitude for the differential-mass regulator, we turn the argument around and use eq. (1.7) together with extended dual conformal symmetry to deduce the anomalous dual conformal Ward identity, from which the all-loop result (1.4) follows. Hence, a derivation of eq. (1.7) from first principles would constitute a proof of eq. (1.4) without having to rely on the scattering amplitude/Wilson loop duality. It would be very interesting to understand the origin of eq. (1.7) from a renormalization group approach. A first step could be to find a suitable operator definition for the wedge function in the differential-mass regulator case. We leave these questions for future work. This paper is organized as follows. In section 2, we review the form of color-ordered MHV scattering amplitudes in planar N = 4 SYM in dimensional regularization and in the massive regularization of ref. [38]. In section 3 we discuss factorization and exponentiation properties of scattering amplitudes and form factors. We propose a definition, involving Sudakov form factors, for a regulator-independent hard function that can be computed from the IR-divergent scattering amplitudes. We review the result for the form factors up to two loops in dimensional regularization, and compute the analogous quantities to two-loop order in the massive regularization. We then show that the (logarithm of the) hard function defined earlier is the same in both cases. In section 4, we discuss a more general differential-mass Higgs regularization, and compute the IR-divergent terms of the one-loop amplitude in this regularization. We then use extended dual conformal symmetry to derive the all-loop form of the IR-divergent terms, and discuss their relation to the dual conformal Ward identity. Section 5 contains our conclusions, and two appendices contain technical details used in the paper.

Review of MHV amplitudes in N = 4 SYM
In this section, we briefly review the form of color-ordered MHV amplitudes in planar N = 4 SYM theory in different regularization schemes.
The all-loop-order n-point amplitude is given by the tree-level amplitude times a helicityindependent function M n , which we expand in the 't Hooft parameter where ǫ = (4 − D)/2. Loop-level amplitudes are UV-finite but suffer from IR divergences which can be regulated using either dimensional regularization in D dimensions, or a Higgs regulator in four dimensions. We discuss each of these in turn.

Dimensional regularization of amplitudes
In dimensional regularization, the n-point amplitude takes the form [17] log The momentum dependence of the amplitude is expressed in terms of dual variables x i defined via x i −x i+1 = p i , where p i are the momenta of the external states; we also define The terms on the first line of eq. (2.2) are IR-divergent and are specified in terms of the cusp and collinear anomalous dimensions [22] γ The terms on the second line of eq. (2.2) are IR-finite and are determined by the finite part of the one-loop amplitude as well as the constants [17] f and a remainder function R n (x 2 ij , a) potentially contributing beginning at two loops. The original proposal by Bern, Dixon, and Smirnov [17] conjectured that eq. (2.2) holds with R n (x 2 ij , a) = 0. Explicit calculations bore this out for n = 4 (through four loops) [45] and n = 5 (through two loops) [46], but the two-loop calculation for n = 6 [27,28,47] revealed the necessity for a non-constant function R 6 (x 2 ij , a). Explicit expressions for eq. (2.5) are given in ref. [17]. For n = 4 and n = 5, they are

Higgs regularization of amplitudes
The four-, five-, and six-point functions have also been computed [38][39][40][41] using the common-mass Higgs regulator described in the introduction. These amplitudes exhibit an exponentiation similar to eq. (2.2) which motivated the following analog for Higgs-regulated n-point amplitudes [38,40] log (1) The terms on the first line of eq. (2.10) are IR-divergent. The cusp anomalous dimension (2.3) is independent of regularization scheme, but the analog of the collinear anomalous dimension is given byG The terms on the second line of eq. (2.10) are IR-finite and are determined by the finite part of the one-loop amplitudeF as well as the constants [40]f and a remainder functionR n (x 2 ij , a). As in the case of dimensional regularization, the remainder function vanishes for four-and five-point amplitudes. For n = 6, it was shown [41] that the two-loop remainder functionR (2) 6 (x 2 ij ) in the Higgs-regulated amplitude is precisely equal to its value R (2) 6 (x 2 ij ) in dimensional regularization, and this is expected to hold generally. The one-loop amplitudes may be evaluated to show [40] F (1) 4 3 Defining a regulator-independent IR-finite amplitude Comparing the known expressions for Higgs-regulated amplitudes (2.10) with those for dimensionallyregulated ones (2.2), one observes that the IR-finite parts of the amplitudes are equal in both regularizations, up to constants. In this section, we make the connection more precise by introducing a regulator-independent expression for the finite part of the amplitude. In a planar theory, the factorization (see e.g. ref. [16]) of color-ordered amplitudes takes the specific form [17] is an IR-divergent "wedge function" depending only on (p i−1 + p i ) 2 and resulting from the exchange of soft gluons in the wedge between the (i − 1)th and ith external particles, and H n (x 2 ij ) is an IR-finite hard function. With a suitable definition for W ( to define the IR-finite part of the amplitude. The forms of both M n and W will depend on the specific regulator, but we will find that log H n is regulator-independent.

Dimensional regularization of the form factor
In dimensional regularization, the wedge function can be defined as the square root of the gluon form factor [3,17]. Form factors in N = 4 SYM have been studied at strong coupling [48,49], at one loop [50] and at two loops [51,52], while three-loop results can be inferred from the respective QCD computations [53][54][55][56][57][58] using the principle of maximal transcendentality; see e.g. ref. [59].
In N = 4 SYM we can equivalently use the form factor for scalars φ I coupling to the operator The operator O IJ belongs to the stress-energy multiplet of N = 4 SYM and is not UV renormalized. This form factor has been computed to two loops in dimensional regularization [51] Φ(q 2 ) = 1 + aq  fig. 3.) The explicit expressions for these integrals given in appendix A reveal that the form factor exponentiates to two-loop order Equation (3.6) can be promoted to all orders in perturbation theory since the momentum dependence of Φ is governed by an evolution equation (see e.g. refs. [18,19,53,60]). In D = 4 −2ǫ dimensions, the following factorization ansatz holds: All dependence of Φ on the hard momentum q 2 , which is taken to be Euclidean (q 2 > 0 in our mostly-plus metric convention) here and in the sequel, rests inside the function G(q 2 /µ 2 , a, ǫ). The latter is finite in four dimensions and can be considered as a suitable continuation of the collinear anomalous dimension (2.4) to D = 4 − 2ǫ dimensions. K(a, ǫ) on the other hand serves as a pure counterterm. Renormalization group invariance of Φ yields which exhibits scale dependence on dimensional grounds and vanishes in the IR for D = 4 − 2ǫ with ǫ < 0. Using eq. (3.9) and exploiting the fact that K has no explicit scale dependence, which allows one to express it entirely through γ(a), one arrives at the following all-order expression for Φ log Φ(q 2 ) = 1 2 where the explicit solution of eq. (3.8) for the counterterm function K has been used K(a, ǫ) = − 1 2 µ 2 0 dλ λ γ(ā(λ, ǫ)) . (3.11) The double poles 1/ǫ 2 at each loop order are generated by the two λ-and ξ-integrations over γ, while the single poles in ǫ arise from the outer ξ-integration over G. Explicit computation, e.g. along the lines of refs. [17,53], yields where the boundary condition for G has been chosen as which is consistent with eq. (3.6). G 0 (a) is given in eq. (2.4) and φ(a) can be read off from eq. (3.6) as 14) The exponentiation of eq. (3.10) proceeds trivially with the help of the boundary condition for Φ in D dimensions, i.e., Φ(q 2 = 0) = 1, which is implicit also in our choice for G(1, a, ǫ) in eq. (3.13). Note also that the all-order result for the form factor in eq. (3.10) applies literally to theories with less supersymmetry, e.g., to QCD. There, the coupling constantā has to be read as the strong coupling α s continued to D-dimensions and the respective QCD expressions for the anomalous dimensions γ(a) and G 0 (a) are related to eqs. (2.3) and (2.4) by the principle of maximal transcendentality. Moreover, G(1, a, ǫ) admits a further decomposition [19] into three terms: a universal (spin-independent) eikonal anomalous dimension, (twice) the coefficient of the δ(1 − x)-term in the collinear evolution kernel, and a process-dependent term accounting for the running coupling in the coefficient function of the hard scattering. The latter is proportional to the QCD β-function and is, of course, absent in N = 4 SYM. We now introduce the wedge function as announced above. Defining W (q 2 ) = Φ(q 2 ), we see from eq. (3.12) that allowing us to rewrite it in its final form as with C(a) given by eq. (2.7). Using renormalization group arguments similar to the derivation of eq. (3.10) for the form factor Φ, it is obvious that log W , log M n , and therefore also log H n in eq. (3.19) can be expressed to all orders via (double-)integrals over the respective anomalous dimensions; see e.g. refs. [3,16].

Higgs regularization of the form factor
Now we turn to the study of the Higgs-regulated N = 4 SYM form factor Φ(q 2 ). We will assume that it is given by the same scalar integrals as in fig. 1 except that some of the internal legs are now massive Φ(q 2 ) = 1 + aq There are several different mass assignments for the two-loop integrals (see fig. 2), as can be seen from the double-line representation in fig. 3. We have computed these integrals (see appendix c1 c2 d A) and they reveal that the Higgs-regulated form factor exponentiates to two-loop order The all-loop-order generalization of eq. (3.22) relies on the same factorization ansatz discussed before and the separation of scales, i.e. q 2 ≫ m 2 , so that the momentum dependence of Φ is described by the evolution equation [1,61]  The exponentiation of eq. (3.26) requires further matching conditions for Φ to be obtained from explicit ℓ-loop computations. A few comments are in order here. First, matching to fixed-order computations could, in principle, also impose the conditionG = G, i.e. demand that the collinear anomalous dimensions coincide. This would proceed at the expense of a non-zero result forK (1, a). Next, the Higgsregulated form factor is finite, so that eq. (3.26) can be evaluated in four dimensions. In theories with broken supersymmetry, e.g. QCD with massive quarks, collinear singularities are regulated by the heavy quark masses, whereas all soft gluon divergences require dimensional regularization. In such a case, the analogous functions K and G have a clear physical interpretation and are independent (see e.g. ref. [61]). For example, the (electric) form factor of a massive quarkanti-quark pair in QCD is known to two loops [62,63], and the analogs of collinear anomalous dimensions naturally coincide in this case, i.e.G = G.
In the case of the common-mass Higgs regulator (so that the external states are massless), the wedge function can again be defined as the square root of the form factor (3.29) so that and γ(a) andG 0 (a) were given in eqs. withC(a) given by eq. (2.14).
In complete analogy to the previous discussion, it is evidently possible to exploit renormalization group properties to provide expressions for log W , log M n , and hence logH n in eq. (3.34) in terms of (double-)integrals over the anomalous dimensions similar to eq. (3.26).

Comparison of regulators
By comparing the results of the last two subsections, we can see that log H n (x 2 ij ) as we have defined it in eq. (3.2) is a good candidate for a regularization-independent IR-finite quantity characterizing the planar MHV n-point amplitude. The one-loop hard functions are identical in both dimensional and Higgs regularization (cf. eqs. (2.17), (3.18), and (3.33))  .35)). The regulator independence of C(a) was previously observed in ref. [40]. If eq. (3.37) holds to all loops, then the regulator independence of the four-and five-point hard functions necessarily follows. For n = 6 at two loops, agreement between the remainder function in dimensional regularization and massive regularization was observed in ref. [41], and this agreement is expected for all n. We thus expect the hard function log H n to be regularization-independent for all n-point functions, that is for dimensional and Higgs regularizations.

Differential-mass Higgs regulator
In section 3, we defined an IR-finite hard function log H n for the n-point amplitude, and showed (through two loops) that it has the same form (including constants) for dimensional regularization and for a common-mass Higgs regulator. In this section, we generalize our discussion to a more general class of regulators, viz., the Higgs regulator with arbitrary distinct masses. This is also interesting from the point of view of collider phenomenology. In QCD, amplitudes with different masses have been considered to two loops for the heavy-to-light transitions, i.e., the (axial)-vector form factor with one massive and one massless quark [64][65][66][67]. Also, electroweak logarithms in four-fermion processes at high energy arising from loop corrections with massive W -and Z-gauge bosons have been considered to two loops (see e.g. ref. [68]). Recall that breaking the U(N + M) symmetry of N = 4 SYM theory to U(N) × U(1) M by assigning distinct vacuum expectation values to one of the scalar fields results in non-zero masses |m i − m j | for the off-diagonal adjoint fields and distinct masses m i for the internal propagators of the scalar integrals that characterize loop amplitudes. (In fact, extended dual conformal invariance requires the freedom to vary the masses.) One can then define a differential-mass Higgs regulator by taking all the masses m i to zero. More precisely, if m i = α i m, the "uniform small mass limit" is defined as the limit m → 0 with α i held fixed. Regulator independence means that the result does not depend on the choice of α i . i i+1 j j+1 Figure 4: Two-mass-easy diagram corresponding to the integral I 2me ij .
Because the external legs of the n-point amplitude now have distinct masses |m i − m i+1 |, it is no longer possible to define the wedge function W (x 2 i−1,i+1 ) as the square root of a form factor as we did in sec. 3. In fact, it is not obvious what the operational definition of W (x 2 i−1,i+1 ) for the differential-mass Higgs regulator should be, and we leave this question to the future.
For now, we adopt a different approach by decomposing the one-loop n-point amplitude into an IR-divergent and a manifestly regulator-independent IR-finite piece, and then defining the one-loop wedge function in terms of the former. The extended dual conformal invariance of the BDS ansatz then allows us to generalize this to an all-loop wedge function.

One-loop amplitude with differential-mass Higgs regulator
As is well known, the one-loop MHV n-point amplitude in dimensional regularization can be written as a sum of two-mass-easy (and one-mass 5 ) scalar box integrals [69,70] (4.1) We will assume that the amplitude on the Coulomb branch is given, at least up to O(m 2 ), by the same set of integrals, with the mass configuration dictated by dual conformal symmetry. The two-mass-easy diagram in fig. 4 corresponds to the integral Later, we will take the uniform small mass limit, so henceforth we drop all mass dependence from the numerators, as those terms would only contribute at O(m 2 ).
It is known that one can decompose eq. (4.2) into a sum of IR-divergent triangle integrals and an IR-finite six-dimensional integral (see e.g. ref. [71]) and with x a and x b the two solutions of the equations x 2 0i = x 2 0,i+1 = x 2 0j = x 2 0,j+1 = 0. In appendix B of this paper, we review the derivation of this decomposition using twistor methods. Since the six-dimensional integral (4.5) is IR-finite, it is independent of which IR-regulator we employ to regulate the n-point amplitude. Therefore in the decomposition of the differential-mass Higgsregulated amplitude a natural candidate for the regulator-independent hard function is Moreover, the sum of the one-loop wedge functions in eq. (4.6) will then be given by the sum of triangle diagrams (4.4) that contribute to the n-point amplitude. Most of the triangle diagrams in this sum cancel, leaving This suggests the following one-loop expression for the wedge function 6 although we could in principle add a contribution that vanishes upon summing over i. Equation (4.9) reduces to our previous definition W (q 2 ) = Φ(q 2 ) when m i−1 = m i = m i+1 . Evaluating eq. (4.9) in the uniform small mass limit, we obtain Substituting eq. (4.10) into eq. (4.6) and using the identity Li 2 (1 −z) + Li 2 (1 −z −1 ) + 1 2 log 2 z = 0, we finally obtain the differential-mass Higgs-regulated n-point amplitude where, as discussed above, H n (x 2 ij ) is IR-finite and regulator-independent; in particular, it does not depend on α i in the uniform small mass limit m → 0, where m i = α i m with α i fixed.
Note that although M n has extended dual conformal invariance, the decomposition into H (1) n and the specific IR-divergent pieces in eq. (4.11) breaks this symmetry. This is not surprising, as triangle integrals manifestly violate dual conformal symmetry.

Higher loops
In the previous subsection, we derived an expression (4.10) for the one-loop wedge function valid for the differential-mass Higgs regulator. We now use the extended dual conformal invariance of the amplitude and the BDS ansatz to derive the explicit form of the wedge function at higher loops.
Recall that extended dual conformal invariance implies [38] that the amplitude can only be a function of (4.12) For the common-mass Higgs regulator, this reduces to u ij = m 2 /x 2 ij . Hence, assuming the validity of the all-loop expression (2.10) for the common-mass Higgs-regulated amplitude, its unique generalization is obtained by replacing x 2 ij with m 2 /u ij everywhere to obtain where we have also used eqs. (3.33) and (3.36). In the uniform small mass limit, we can neglect (m i − m j ) 2 relative to x 2 ij in eq. (4.12) so that u ij becomes m i m j /x 2 ij yielding The apparent dependence ofR n on m i is illusory since the mass dependence cancels out in the dual conformal cross ratios on whichR n only depends, so thatR n (m 2 x 2 ij /m i m j , a) =R n (x 2 ij , a). There does, however, remain some dependence on m i in H (1) n .
Applying the same reasoning as above to the one-loop amplitude, we obtain Now since eqs. (4.11) and (4.15) are both valid expressions for the differentially-regulated oneloop amplitude, we deduce that Substituting eq. (4.16) into eq. (4.14) and using eq. (3.35), we obtain 7 where now only the terms in the sum on the first line depend on the regulator, while the pieces on the second line are all regulator-independent. Recalling that we deduce from eq. (4.17) that the all-order expression for the sum of wedge functions in differential-mass Higgs regularization is 8 and the regulator-independent IR-finite piece is, as before where we have dropped all the tildes. We observe again that, although the amplitude log M n has extended dual conformal invariance, the separate terms in the decomposition (4.18) do not.

Relation to anomalous dual conformal Ward identity
In the previous section, we obtained the all-loop expression (4.19) for the sum of wedge functions by assuming eq. (2.10). In this section, we show inversely that eq. (2.10) follows from eq. (4.19). Therefore, it would be interesting to have a first-principles derivation of eq. (4.19).
The n-point amplitude has exact extended dual conformal symmetry, and so is annihilated by the generator of dual special conformal transformationŝ In ref. [38], it was suggested that the IR-divergent properties of the Higgs-regulated amplitude provides a relation between this exact Ward identity and the anomalous dual conformal Ward identity for the IR-finite part of the n-point amplitude that was originally derived in a Wilson loop context [25,26]. We will see that this is indeed the case.
As we have seen, the n-point amplitude can be written as Using the expression (4.19) for the sum of the wedge functions, one can easily show that which by virtue of eq. (4.1) implies that But log H n is regulator-independent, i.e., has no dependence on m i in the uniform small mass limit, so the m-dependent pieces inK µ drop out when acting on log H n and we have which is precisely the anomalous dual conformal Ward identity [25,26]. This in turn implies eq. (2.10).
We thus see that the decomposition of the amplitude into contributions which separately do not possess extended dual conformal invariance was necessary to obtain the anomalous dual conformal Ward identity for the finite (regulator-independent) part of the amplitude.

Discussion
In this paper, we have given a prescription for defining an unambiguous, regulator-independent IR-finite part of the MHV n-point scattering amplitude in planar N = 4 SYM theory. This prescription involves the definition of an IR-divergent wedge function associated with a pair of adjacent external legs of the amplitude. The IR-finite part of the amplitude is then defined as the quotient of the n−point amplitude by the product of wedge functions, cf. eq. (1.3).
For regulators that leave the external legs massless (e.g., dimensional regularization or the common-mass Higgs regulator), the wedge function can be naturally defined in terms of a form factor Φ which has the same IR-divergences. Computation of this form factor in dimensional regularization and in the common-mass Higgs regularization through two loops shows that the IR-finite part of the amplitude is identical for these two regularizations. For the more general differential-mass Higgs regulator, which gives (small) masses to the external legs, a wedge function that results in a regulator-independent hard function can still be calculated, but an operator definition in this case is still lacking.
We remark that the idea of defining a regulator-independent finite hard function can also be applied to other objects, e.g. Wilson loops. This is particularly interesting in the context of the Wilson loop/scattering amplitudes duality, since the two objects have different types of divergences, viz., UV and IR divergences respectively. Although these divergences are related, defining hard functions for both objects could be useful for stating the duality in a regulatorindependent way.
There exist in the literature other procedures for removing the IR divergences of scattering amplitudes. For example, for non-MHV amplitudes, one can define an IR-finite ratio function [72] by factoring out the entire MHV amplitude, using the universality of IR divergences, i.e. that they do not depend on the helicity configuration. Another example involves MHV amplitudes with n ≥ 6 external legs. Since the four-and five-point amplitudes (or, equivalently, Wilson loops) are known up to kinematic-independent functions, they can be used to remove the divergences of higher-point amplitudes by defining suitable ratios [73]. This latter procedure preserves dual conformal symmetry.
The hard functions defined in this paper are not dual conformal invariant; they have the advantage, however, of allowing us to study the n = 4 and n = 5 cases as well. In particular, it would be interesting to understand better the systematics of how the BES equation [23] for γ(a) arises from the loop expansion of the four-point amplitude.
The breaking of dual conformal invariance by the hard function also implies an intimate connection between the anomalous dual conformal Ward identity it satisfies and the IR divergences (wedge functions) of differential-mass regulated amplitudes. A first-principles derivation of the latter would therefore be most interesting.
Finally, we believe that, although our investigation has been specialized to N = 4 SYM theory, the insight into the interplay between regulator, kinematics, and soft and collinear momentum configurations applies to many gauge field theories, including those with broken supersymmetry, such as QCD, and also to electroweak radiative corrections in the Standard Model.

Acknowledgments
It is a pleasure to thank Z. Bern, J. Drummond, and G. Korchemsky for interesting discussions. All of the authors also wish to express their gratitude to the Kavli Institute for Theoretical Physics, where part of this work was carried out during the "The Harmony of Scattering Amplitudes" program in spring 2011. This research was supported in part by the European-Union funded network LHCPhenoNet contract No. PITN-GA-2010-264564, as well as the National Science Foundation under Grant Nos. PHY05-51164 and PHY07-56518.

A Results for one-and two-loop integrals
In this appendix, we list the results for various massless and massive three-point integrals that contribute to the form factors computed in this paper. We use the mostly-plus metric, the propagators are of the form k 2 +m 2 , and the measure of each internal loop momentum is multiplied by a factor of (µ 2 e −γ ) ǫ /(iπ d/2 ). The massless integrals shown in fig. 1 are dimensionally-regulated, giving rise to the following Laurent expansions [74][75][76] where q 2 = x 2 13 . The integrals shown in fig. 2 use a common-mass Higgs regulator, and can be evaluated to givẽ

B Decomposition of the 2me box integral
In this appendix, we derive the decomposition of the 2me box integral into triangle integrals and an IR-finite six-dimensional integral that was used in sec. 4.1.
We begin by rewriting the dual conformal invariant integral of equation (4.2) in terms of momentum twistors [77]; see ref. [78] for a pedagogical introduction to this topic. A point x i in dual space corresponds to a (projective) line Z A i−1 Z B i in momentum twistor space. The invariant x 2 ij can be expressed as where the twistor four-bracket is We introduce the infinity bitwistor I AB , which when contracted with Z C α Z D β gives the two-bracket Finally, we introduce a modified mass-regulated four-bracket α, β, γ, δ i ≡ α, β, γ, δ + m 2 i α, β γ, δ .
We observe that the integrands in eq. (B.12) contain "magic" numerators, which render the resulting integrals IR-finite. Hence we have dropped the mass dependence in the denominator. One can show that this integral is in fact equivalent to the scalar two-mass easy integral in six dimensions [71] I 2me x 2 0i x 2 0,i+1 x 2 0j x 2 0,j+1 . (B.14) Since this integral is IR-finite, it is manifestly independent of which IR regulator is used to regulate the amplitude. In particular, it has no dependence on α i in the uniform small mass limit of the differential-mass Higgs regulator introduced in sec. 4.