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* Two-loop helicity amplitudes for $V+$jet production including axial-vector couplings to higher orders in $\epsilon$ *
* Thomas Gehrmann, Petr Jakubčík, Cesare Carlo Mella, Nikolaos Syrrakos, Lorenzo Tancredi *
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The ancillary files to this article include, always for the two channels Vggg and Vgqq:
- projectors onto independent tensor structures
- relations between helicity coefficients and form factors
- helicity amplitudes in the decay region extended to weight 6
- finite remainders of helicity amplitudes in decay kinematics and continued to all production regions
The files are in Mathematica-readable format.
1) Projectors
The files Projectors_Vggg.m and Projectors_Vgqq.m contain the inverse of the matrix M^P as in Eq. (3.15). We set m_V^2 = 1.
The form factors of the parity-even tensor structures F[i] are followed by the form factors of the parity-odd tensor structures G[i].
The coefficient of the j-th tensor structure in the i-th projector is given by the element in the i-th row and j-th column of this matrix.
2) Helicity Coefficients
The files HelicityCoeffs_Vggg.m and HelicityCoeffs_Vgqq.m contain the relations between the helicity coefficients
alpha[1], ... , alpha[3],..., delta[1], ..., delta[3] as in Eqs. (6.15), (6.16), (6.19), (6.20) in this paper.
For the process Vgqq, they corresond to Eqs. (5.8), (5.9), (5.12), (5.13) in Ref. [1]. We set m_V^2 = 1.
3) Extended helicity amplitudes
In the directory AmplitudesW6, we provide the independent helicity amplitudes in the decay region with Minkowski kinematics,
extended to transcendental weight 6, UV-renormalized but before IR subtraction. The files are organized by process, loop order
and helicity coefficients as defined in Eqs. (6.13), (6.14), (6.17), (6.18) of this paper for Vggg,
and in Eqs. (5.6), (5.7) and (5.10), (5.11) of Ref. [1] for Vgqq:
- Vggg/Vgqq
- alpha1.m, alpha2.m, alpha3.m, beta1.m, beta2.m, beta3.m, gamma1.m, gamma2.m, gamma3.m, delta1.m, delta2.m, delta3.m
Each file contains both non-singlet and pure-singlet contributions to the given helicity coefficient up to weight 6 and up to 2 loops.
In the case of Vggg, we drop the tag "PS".
Each file is a list of replacement rules in the format
coefficient[index, loops, (non-)singlet] -> expression,
eg. alpha[1, 0, "NS"] -> 0
is the tree-level expression for the non-singlet contribution to the helicity coefficient alpha_1.
In these expressions, we set m_V^2 = 1, y = s13, z = s23.
4) Analytic continuation of amplitudes
In the directory FiniteRemainders, we provide the finite remaiders of the independent helicity amplitudes in the decay region,
and continued to all relevant production regions as explained in Section 6.3 of this paper. The structure is as follows:
- Vggg
- decay.m V_L -> g_+(p1) + g_+(p2) + g_+(p3) AND V_L -> g_+(p1) + g_-(p2) + g_-(p3)
- mmP.m g_-(p1) + g_-(p3) -> V_R + g_+(p2)
- mpM.m g_-(p1) + g_+(p3) -> V_R + g_-(p2)
- ppP.m g_+(p1) + g_+(p3) -> V_R + g_+(p2)
- pmP.m g_+(p1) + g_-(p3) -> V_R + g_+(p2)
- Vgqq
- decay.m V_L -> qb_R(p1) + q_L(p2) + g_+(p3) AND V_L -> qb_R(p1) + q_L(p2) + g_-(p3)
- lmL.m qb_L(p1) + g_-(p3) -> V_R + qb_L(p2)
- lpL.m qb_L(p1) + g_+(p3) -> V_R + qb_L(p2)
- rmR.m q_R(p1) + g_-(p3) -> V_R + q_R(p2)
- rpR.m q_R(p1) + g_+(p3) -> V_R + q_R(p2)
- lrP.m qb_L(p1) + q_R(p3) -> V_R + g_+(p2)
- lrM.m qb_L(p1) + q_R(p3) -> V_R + g_-(p2)
Each file is a list of replacement rules in the format
coefficient[index, loops, (non-)singlet] -> expression,
eg. alpha[1, 0, "NS"] -> 0
is the tree-level expression for the non-singlet contribution to the helicity coefficient alpha_1. In the case of Vggg, we drop the tag "PS".
Some files contain only the coefficients alpha, beta, others gamma, delta, based on which of the two independent helicity amplitudes
from the decay region was continued. Expressions in the production regions are in the variables v = 1/y and u = -z/y, where y = s13, z = s23.
We denote M_V^2 with q2.