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A bstract We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar N = 4 super-Yang-Mills theory from recent QCD results, obtaining G ^ 0 4 = − 300 ζ 7 − 256 ζ 2 ζ 5 − 384 ζ 3 ζ 4 . This value agrees with a previous numerical result to within 0.2%. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.

A bstract We study scattering amplitudes in massless non-abelian gauge theory where all outgoing gluons have positive helicity. It has been argued recently by Costello that for a particular fermion representation (8 fundamentals plus one antisymmetric-tensor representation in SU( N )) the one-loop amplitudes vanish identically. We show that this vanishing leads to previously-observed identities among one-loop color-ordered partial amplitudes. We then turn to two loops, where Costello has computed the all-plus amplitudes for this theory, as rational functions of the kinematics for any number of gluons using the celestial chiral algebra (CCA) bootstrap. We show that in dimensional regularization, these two-loop amplitudes are not rational, and they are not even finite as ϵ → 0. However, the finite remainder for four gluons agrees with the formula by Costello. In addition, we provide a mass regulator for the infrared-divergent loop integrals; with this regulator, the CCA bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to compute the double-trace terms in the theory at two loops for an arbitrary number of gluons.

A bstract We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N = 4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q ¯ differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N 3 LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. We also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms.

A bstract We bootstrap the three-point form factor of the chiral part of the stress­tensor supermultiplet in planar N = 4 super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, “extended-Steinmann-like” conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as gg → Hg and γ * → q q ¯ g . We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar N = 4 three-point form factor can be placed in an even smaller space of functions, with no independent ζ values at weights two and three.

A bstract We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planar N = 4 supersymmetric Yang-Mills theory at six, seven, and eight loops, using boundary data from the form factor operator product expansion. This may represent the highest perturbative order to which multi-variate quantities in a unitary four-dimensional quantum field theory have been computed. In computing this form factor, we observe and employ new restrictions on pairs and triples of adjacent letters in the symbol. We provide details about the function space required to describe the form factor through eight loops. Plotting the results on various lines provides striking numerical evidence for a finite radius of convergence of perturbation theory. By the principle of maximal transcendentality, our results are expected to give the highest weight part of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD through eight loops. These results were also recently used to discover a new antipodal duality between this form factor and a six-point amplitude in the same theory.

A bstract We compute the six-particle maximally-helicity-violating (MHV) amplitude in planar N = 4 super-Yang-Mills theory at eight loops, using antipodal duality and the recently computed eight-loop three-point form factor for the chiral stress energy tensor multiplet. Antipodal duality maps the form factor symbol to the amplitude symbol on a two-dimensional parity-preserving surface in the three-dimensional amplitude kinematics. There are remarkably few ambiguities in lifting from two to three dimensions, nor in promoting the symbol to a function. The amplitude passes many tests, including near-collinear, multi-Regge, factorization, self-crossing and origin limits. These checks also constitute a validation of antipodal duality at eight loops.

A bstract The soft emission factor is a central ingredient in the factorization of generic n -particle gauge theory amplitudes with one soft gluon in the external state. We present the complete two-loop soft factor, capturing the leading power behavior in the soft-gluon momentum. At two loops, the color structure and the kinematic dependence of the soft factor become nontrivial as the soft gluon can couple to three hard partons for the first time (tripole terms). The nontrivial kinematic dependence of the tripole terms is of uniform, maximal transcendental weight, and can be expressed (in a “Euclidean” region) in terms of single-valued harmonic polylogarithms. Our results are consistent with the behavior of the recently computed symbol of the two-loop five-particle amplitude in N = 4 super-Yang-Mills theory. In the limit where the outgoing soft gluon is also collinear with an incoming hard parton, potentially dangerous factorization-violating terms can arise.

A bstract We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N = 4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.

A bstract We investigate color-kinematics duality for gauge-theory amplitudes produced by the pure nonabelian Yang-Mills action deformed by higher-dimension operators. For the operator denoted by F 3 , the product of three field strengths, the existence of color-kinematic dual representations follows from string-theory monodromy relations. We provide explicit dual representations, and show how the double-copy construction of gravity amplitudes based on them is consistent with the Kawai-Lewellen-Tye relations. It leads to the amplitudes produced by Einstein gravity coupled to a dilaton field ϕ , and deformed by operators of the form ϕR 2 and R 3 . For operators with higher dimensions than F 3 , such as F 4 -type operators appearing at the next order in the low-energy expansion of bosonic and superstring theory, the situation is more complex. The color structure of some of the F 4 operators is incompatible with a simple color-kinematics duality based on structure constants f abc , but even the color-compatible F 4 operators do not admit the duality. In contrast, the next term in the α ′ expansion of the superstring effective action — a particular linear combination of D 2 F 4 and F 5 -type operators — does admit the duality, at least for amplitudes with up to six external gluons.

A bstract We consider four-point integrals arising in the planar limit of the conformal “fishnet” theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-point equations are found to match known two-cut singular equations arising in matrix models, enabling us to obtain a concise parametric expression for the free-energy density in terms of complete elliptic integrals. Interestingly, the latter depends non-trivially on the fishnet aspect ratio and differs from a scaling formula due to Zamolodchikov for large periodic fishnets, suggesting a strong sensitivity to the boundary conditions. We also find an intriguing connection between the saddle-point equation and the equation describing the Frolov-Tseytlin spinning string in AdS 3 × S 1 , in a generalized scaling combining the thermodynamic and short-distance limits.