TY - JOUR

T1 - Building bases of loop integrands

AU - Bourjaily, Jacob L.

AU - Herrmann, Enrico

AU - Langer, Cameron

AU - Trnka, Jaroslav

PY - 2020/11/23

Y1 - 2020/11/23

N2 - We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of 'power-counting' for multi-loop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all 'leading-(transcendental-)weight' contributions of any four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent all scattering amplitude integrands in the theory - for generic helicities and arbitrary multiplicity.

AB - We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of 'power-counting' for multi-loop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all 'leading-(transcendental-)weight' contributions of any four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent all scattering amplitude integrands in the theory - for generic helicities and arbitrary multiplicity.

KW - Scattering Amplitudes

KW - 1

KW - N Expansion

KW - Gauge Symmetry

KW - DIFFERENTIAL-EQUATIONS

KW - SCATTERING-AMPLITUDES

KW - FEYNMAN DIAGRAMS

KW - TREE AMPLITUDES

KW - REDUCTION

KW - UNITARITY

KW - RENORMALIZATION

KW - REGULARIZATION

U2 - 10.1007/JHEP11(2020)116

DO - 10.1007/JHEP11(2020)116

M3 - Journal article

VL - 2020

JO - Journal of High Energy Physics (Online)

JF - Journal of High Energy Physics (Online)

SN - 1126-6708

IS - 11

M1 - 116

ER -