The four-loop N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 SYM Sudakov form factor

We present the Sudakov form factor in full color N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 supersymmetric Yang- Mills theory to four loop order and provide uniformly transcendental results for the relevant master integrals through to weight eight.


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The remainder of this paper is organized as follows. In section 2 we define the Sudakov form factor that we consider and review the known reduced integrand at 4 loops. In section 3 we describe our calculation of the relevant master integrals to transcendental weight 8. In section 4 we give the result for the Sudakov form factor. In section 5 we conclude.

Reduced integrand
The Sudakov form factor in N = 4 SYM that we consider in this paper is defined as where the expectation value of a local length-two operator is computed between the vacuum and a state with 2 on-shell scalar particles. The Lorentz scalars φ a 12 carry subscripts corresponding to the 6 representation of the R-symmetry group SU(4) R and a superscript corresponding to the adjoint representation of the gauge group SU(N c ). The overall normalization N is chosen such that the tree level contribution is normalized to 1.
The definitions of the master integrals in ref. [46] can be mapped to just the 10 integral families (complete sets of propagators) shown in table 1. The integrals are then linear combinations of four-loop Feynman integrals where f =A,. . . ,J labels the family and q 2 = −1. Depending on the topology, up to 12 indices ν i in eq. (2.6) are positive and correspond to actual denominators of the integrand; some of the remaining indices may be negative to denote irreducible numerators. We note that one family covers in general more than one trivalent graph, family A for example covers all planar graphs. The propagator denominators D i follow Minkowskian conventions and depend implicitly on the integral family f . We provide expressions for the master integrals I (n i ) i in terms of integrals in these families in the supplementary material of this paper. We note that integration-by-parts reductions allow to remove any reference to family I, which was used to map I (2) p,2 . For some topologies, only a subset of the irreducible integrals enters JHEP01(2022)091 and I (26) 7 , appear in eq. (2.4). Analytical results for the master integrals in eq. (2.4) have been given through to weight 6 in [10]. Here, we present their analytical calculation through to weight 8 as required for the finite part of the Sudakov form factor.

Master integrals to weight eight
We employ two different methods to evaluate the master integrals: the direct integration of finite integrals and the method of differential equations with an auxiliary scale.
In principle, all topologies but two have been shown to be linearly reducible [47,48] and are thus accessible to direct integrations based on the Feynman parametric representation.

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Moreover, the only two topologies which have not yet been proven to be linearly reducible after a change of variables in the Feynman parametric representation were dealt with in ref. [45] using the method of differential equations. In order to perform parametric integrations, we select a basis of finite integrals [49][50][51][52][53] with Reduze 2 [54]. Here, the finite integrals are typically defined in 6 − 2 dimensions and involve higher powers of the propagators ("dots"). The basis change is computed with the private code Finred based on [55][56][57][58][59][60]. For some topologies, it is necessary to perform variable changes in the Feynman parametric representation to find a linearly reducible integration order. We then employ the program HyperInt [61] to expand the Feynman integral around = 0 and integrate the expansion coefficients. In this way, we solved a subset of the master integrals through to weight 8 with HyperInt. Depending on the integral, however, we found that the computing resources required to compute the relevant orders can be prohibitive, such that we resorted to the method of differential equations in many cases.
The method of differential equations [62][63][64] is a powerful technique to solve Feynman integrals with non-trivial dependence on the kinematics, see e.g. ref.
[65, section E.8] for a recent review. While our integrals have only a trivial dependence on the kinematics, the method becomes applicable by considering vertex integrals with two off-shell and one massless leg [66] instead. The differential equations in the auxiliary parameter then connect the sought after vertex integrals with two massless legs (x = 0) with propagator type integrals (x = 1) known from refs. [67,68], see ref. [8] for more details. We employ Fire 6 [69] and LiteRed [59,70] to find the differential equations in x for some initial choice of basis. Subsequently, we apply the method of refs. [71,72] as implemented in Libra [73] to bring the system in form [74]. At this point, we are forced to introduce algebraic extensions x 1 = √ x, x 2 = x − 1/4, and x 3 = 1/x − 1/4 in order to secure an -form of the differential system. The complete alphabet sufficient for all families consists of the letters appearing in the derivatives with respect to x. In particular, the letters involving x 1 , x 2 , x 3 are required for topology (26) in figure 1, while the topologies (12) and (25) contain those involving x 1 , x 3 . It turns out that each iterated integral in the results for master integrals contains at most one of x 1 , x 2 , x 3 , so it is always possible to rationalize the weights by passing to the corresponding letter. Note that the differential equations approach allows one to construct uniform transcendentality (UT) bases of one-scale integrals. Indeed, the column of asymptotic coefficients c 0 at x = 0 is expressed via the column of coefficients c 1 at x = 1 as (see eq. (28) of ref. [8]) Here the associator U 01 is UT by construction. The column of boundary constants to pull from L 1 an overall, rational in , factor which can be determined by examining the simplest non-zero entry of column c 1 (this simplest entry is always known exactly in terms of a product of Γ functions). So, the column of boundary constants at x = 0, i.e.
However, there is one obstacle here. The column c 0 contains not only naive limits (obtained by setting x = 0 under the integral sign), which correspond to one-scale integrals, but also the asymptotic coefficients in front of non-integer powers of x. Thus, in general, each entry of C 0 is expressed not only via one-scale integrals, but also via some asymptotic coefficients in front of non-integer powers of x. This can be fixed by quasi-diagonalizing (reducing to Jordan normal form) the residue, A 0 , at x = 0 of the matrix on the right-hand side of the differential system in -form. Since the fractional powers of x in the asymptotics are in one-to-one correspondence with eigenspaces of A 0 , the Jordan normal form of A 0 necessarily has a block-diagonal structure with blocks corresponding to different fractional powers of x. The matrix L 0 also acquires the same block-diagonal structure. Then, those entries of C 0 which correspond to a block with integer powers of x are expressed solely via one-scale integrals. Since the matrix L 0 is invertible by construction, it is easy to establish, that the number of such entries is sufficient to furnish a basis. Let us demonstrate this approach on the example of the two integrals presented in figure 2, where we use a dot to indicate a squared propagator. The differential system for those two integrals has the form d dx Note that this sector has no non-zero subsectors. We construct the transformation j = T J with j = (j 1 , j 2 ), J = (J 1 , J 2 ), and which reduces the system to an -form. The factor f ( ) will be fixed later to secure uniform transcendentality of C 0 and C 1 . We have d dx Using Libra, we find the following relation between asymptotic coefficients at x = 0 and x = 1: where [j k ] y α denotes the coefficient in front of y α in y → 0 asymptotics of j k . In principle, also different choices are possible (e.g.
with appropriate modifications of L 0 and L 1 . For our choice the matrices L 0 and L 1 have the form ) (3.10) The associator reads At the point x = 1 we have only "naive" limits, so It is easy to see that for f ( ) = is uniformly transcendental. Then C 0 can be computed from C 0 = U 01 C 1 and is also uniformly transcendental. On the other hand, we have We see that each entry of C 0 is a linear combination of "naive" limit constant [j 1 ] x 0 , which corresponds to a specific on-shell vertex integral, and of the constant [j 1 ] x 2−3 , which corresponds to a contribution of some non-trivial region in x → 0 asymptotics. Thus the comparison to C 0 = U 01 C 1 does not allow for the extraction of is a transformation diagonalizing S 0 in eq. (3.7), we obtain from eq. (3.14)C

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and the comparison toC 0 = Q −1 U 01 C 1 immediately provides us a result for [j 1 ] x 0 . The first entry ofC 0 is expressed via on-shell vertex integral. Since Q is rational numeric matrix, C 0 remains UT and we have achieved our goal.
In this way, we obtain UT bases for the vertex integrals with two massless legs through to weight 9, written in terms of multiple polylogarithms G with argument 1 and indices {0, ±1, ±i √ 3, e ±iπ/3 , e ±2iπ/3 , e ±iπ/3 /2}. Employing the PSLQ algorithm [75], these results can be expressed in terms of regular multiple zeta values.
We computed many integral coefficients in both approaches (direct integrations and differential equations), which allowed us to cross-check a substantial fraction of our results analytically. To facilitate the checks of our results, we expressed all master integrals in terms of finite integrals, which we define allowing also for higher dimensions and/or additional dots. We determined all finite integrals to the required order in needed for complete weight 8 information, which occasionally involved also weight 9 contributions. We also employed Fiesta [76] for numerical checks of many integrals. By performing these checks directly for finite integrals defined in 6 − 2 dimensions, we were able to achieve a typical relative agreement of 10 −4 or better with modest run times.
Using the one-, two-and three-loop results of [35] and our four-loop result (4.1), we find for the finite remainders of the logarithm of the form factor We observe that the subtraction of exponentiated terms leads to somewhat simpler rational prefactors in the finite remainder (4.12) compared to the weight 8 terms of (4.1).

Conclusions
We presented the analytical calculation of the Sudakov form factor in N = 4 supersymmetric Yang-Mills theory to four loop order. To solve the master integrals to weight 8, we employed direct parametric integrations and the method of differential equations with an auxiliary scale.
To the best of our knowledge, this is the first time that a form factor has been computed to four-loop order in full-color Yang-Mills theory, and we hope that our explicit results are helpful in further studies of formal and phenomenological theories at high perturbative orders. The master integrals entering the present calculation form a subset of the most complicated integrals needed for general massless three-point functions with one off-shell leg. Our methods allow us to calculate also the remaining master integrals, providing the last missing building block for the calculation of the massless corrections to the quarkphoton vertex and the effective gluon-Higgs vertex in four-loop Quantum Chromodynamics.