The polynomial form of the scattering equations is anH-basis

Abstract

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gröbner basis, but circumvents the heavy computation of the latter. As an example, we apply the H-basis to prove the conjecture that the dual basis of the polynomial scattering equations must contain one constant term.

Derivation of Feynman rules for higher order poles using cross-ratio identities in CHY construction

“…Among investigations from various directions [6][7][8][9][10][11][12][13][14][15][16][17][18], the socalled integration rule method inspired by string theory is one of the most efficient and systematic approaches [17][18][19]. This approach replies only on the CHY integrands, without mentioning the solutions of scattering equations and the integration measure.…”

“…Results (4.7) and (4.8) are, in fact, two possible Feynman rules for the triple pole. 12 It is worth to emphasize that when exchanging the ordering of the sum, the i∈Λ\{1} j∈Λ\{n} s ij produces (2p B · p C ). Also, changing of summation ordering is allowed because we have fixed the first gauge choice 1 ∈ Λ, n ∈ Λ for all splittings of ABCD .…”

“…For the second gauge choice [1, n; n; Λ], we get 12) where the rule (4.8) has been used. For the third gauge choice [m; n; m; Λ], the thing is a little bit complicated: since we have fixed the subset A to be the one containing node 1, we need to consider two different cases.…”

Understanding the cancelation of double poles in the Pfaffian of CHY-formulism

“…Some of the methods borrow the ideas from computational algebraic geometry, by use of Vieta formula [16], elimination theory [17][18][19], companion matrix [20], Bezoutian matrix [21]. Based on the polynomial form [22] of scattering equations, polynomial reduction techniques are also introduced in this problem [23,24]. In [25,26], differential operators are applied to the evaluation of CHY-integrals.…”

A combinatoric shortcut to evaluate CHY-forms

“…A direct evaluation of the CHY form for the MHV tree amplitude is given in [39]. Algebraic geometry based methods -the companion matrix method [40], Bezoutian matrix method [41], and polynomial reduction techniques [42,43] -are employed to evaluate the CHY expressions, without solving the underlying scattering equations.…”