Quadratic Feynman loop integrands from massless scattering equations

Recently, the Cachazo-He-Yuan (CHY) approach has been extended to the loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman’s. In this paper, we present a new technique that directly produces quadratic propagators identical to Feynman’s from the CHY approach. This paper focuses on the ${\Phi }^3$ theory, but extensions to other theories are briefly discussed. In addition, our proposal has an interesting geometric meaning; we can interpret this new formula as a unitary cut on a higher genus Riemann surface.

Figure 1

${\mathrm{\Phi }}^3$ symmetrized Feynman diagram.

Figure 2

Schematic representation for the new CHY proposal.

Figure 3

Equality among CHY and Feynman integrands.

Figure 4

Color code in CHY graphs.

Figure 5

Correspondence between the ${\mathrm{\Phi }}^3$ Feynman diagrams ($n$-gon symmetrized) and the CHY graphs, up to the partial fraction identity and $\frac{1}{{\ell }^2}$ factor. $S_n$ is the permutation group, and ${\alpha }_i≔\alpha (i)$.

Figure 6

Graph representation for the CHY integrand ${\mathbf{I}}^{\mathrm{CHY}}$ given in (3.1).

Figure 7

Schematic equivalence between the ${\mathrm{\Phi }}^3$ general Feynman diagrams (symmetrized $n$-gon) and the CHY graphs.

Figure 8

CHY-graph representation of ${\mathbf{I}}^{\mathrm{CHY}}$ written in (4.1). All possible nonzero cuts for ${\mathbf{I}}^{\mathrm{CHY}}$.

Figure 9

Graph representation for ${\mathcal{I}}_{\mathrm{sym}}^{2\text{−}\mathrm{gon}\text{−}\mathrm{CHY}}(1,2|[a_1,b_1],[a_2,b_2])$ (up to integration over the moduli space, to wit, $\int d\mu$).

Figure 10

Feynman diagram for the symmetrized bubble.

Figure 11

CHY graph of the ${\mathbf{I}}^{\mathrm{CHY}}$ integrand given in (4.9).

Figure 12

All possible nonzero cuts.

Figure 13

CHY-graph representation for ${\mathcal{I}}_{\mathrm{sym}}^{3\text{−}\mathrm{gon}\text{−}\mathrm{CHY}}(1,2,3|[a_1,b_1],[a_2,b_2])$, up to the integral over the moduli space, i.e., $\int d\mu$.

Figure 14

Feynman diagram for the symmetrized triangle.

Figure 15

CHY graph of the integrand in (4.12).

Figure 16

All possible nonzero cuts.

Figure 17

CHY graph for ${\mathcal{I}}_{4\text{−}\text{triangle}}^{\mathrm{CHY}}(1|2,3|4|[a_1,b_1],[a_2,b_2])$, up to the integral over the moduli space, i.e., $\int d\mu$.

Figure 18

Symmetrized Feynman diagram of the triangle with four particles.

Figure 19

${\mathrm{\Phi }}^3$ Feynman diagram at two loops.

Figure 20

CHY-graph representation for the ${\mathbf{I}}_{2\text{−}\mathrm{loop}}^{\mathrm{CHY}}$ integrand in (5.1).

Figure 21

CHY graph that naively would be able to produce just quadratic Feynman propagators.

Figure 22

Singular cut (tacnode singularity).

Figure 23

Feynman diagram meaning for the new CHY proposal.

Figure 24

Representation of the process to obtain quadratic Feynman propagators at two loops. Sym means a sum over all permutations for each set of external legs.