Heptagon functions and seven-gluon amplitudes in multi-Regge kinematics

We compute all 2 → 5 gluon scattering amplitudes in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory in the multi-Regge limit that is sensitive to the non-trivial (“long”) Regge cut. We provide the amplitudes through four loops and to all logarithmic accuracy at leading power, in terms of single-valued multiple polylogarithms of two variables. To obtain these results, we leverage the function-level results for the amplitudes in the Steinmann cluster bootstrap. To high powers in the series expansion in the two variables, our results agree with the recently conjectured all-order central emission vertex used in the Fourier-Mellin representation of amplitudes in multi-Regge kinematics. Our results therefore provide a resummation of the Fourier-Mellin residues into single-valued polylogarithms, and constitute an important cross-check between the bootstrap approach and the all-orders multi-Regge proposal.

Conformal correlators as simplex integrals in momentum space

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Lorentzian inversion and anomalous dimensions in Mellin space

In this note, we derive a Mellin space version of the Lorentzian inversion formula for CFTs by explicitly integrating over the cross-ratios in d = 2 and d = 4 spacetime dimensions. We use the simplicity of the Mellin representation of Witten diagrams and the double discontinuity to find the OPE coefficients and anomalous dimensions of double- trace primaries in large N CFTs to order $$ \frac{1}{N^4} $$ 1 N 4 . We find that our results match analytically at order $$ \frac{1}{N^2} $$ 1 N 2 , and numerically at order $$ \frac{1}{N^4} $$ 1 N 4 with existing literature.

EXISTENCE OF ASYMPTOTIC EXPANSIONS IN NONCOMMUTATIVE QUANTUM FIELD THEORIES

Starting from the complete Mellin representation of Feynman amplitudes for noncommutative vulcanized scalar quantum field theory, introduced in a previous publication, we generalize to this theory the study of asymptotic behaviors under scaling of arbitrary subsets of external invariants of any Feynman amplitude. This is accomplished in both convergent and renormalized amplitudes.

ASYMPTOTIC EXPANSIONS OF FEYNMAN AMPLITUDES IN A GENERIC COVARIANT GAUGE

We show in this paper how to construct Symanzik polynomials and the Schwinger parametric representation of Feynman amplitudes for gauge theories in a generic covariant gauge. The complete Mellin representation of such amplitudes is then established in terms of invariants (squared sums of external momenta and squared masses). From the scaling of the invariants by a parameter we extend for the present situation a theorem on asymptotic expansions, previously proven for the case of scalar field theories, valid for both ultraviolet or infrared behaviors of Feynman amplitudes.