== 2 Gravitons minimally coupled to Standard Model ==

=== Author ===
Tran Quang Loc
- Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge
- loctranq@gmail.com

=== Description of the model & references ===
Gravitons minimally coupled to Standard Model (SM) is described by the theory of SM with weak gravitational perturbations of Minkowski space, in which graviton is suppressed by a factor of two over Planck scale (denoted as kappa). 

In this implementation, we start from the generic Lagrangian and expand the metric in linearised theory up to second-order expansion. The bosonic sector is trivially expanded while the spinor sector makes use of newly introduced vielbein fields and torsion-free spin connections to construct the covariant derivative. 

At first-order, the model can be used to produce the interactions including 1 graviton minimally coupled to SM, i.e. light-by-light scattering in [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.091602/ Phys.Rev.Lett. 127 (2021) 9, 091602]: Katsuki Aoki, Tran Quang Loc, Toshifumi Noumi, Junsei Tokuda, ''Is the Standard Model in the Swampland? Consistency Requirements from Gravitational Scattering''

At second-order, the model is useful for processes with 2 gravitons minimally coupled to SM vertices.
=== Model files & extensions ===
- Main FeynRules file for the 2 gravitons coupled to full SM: [https://feynrules.irmp.ucl.ac.be/attachment/wiki/2GravitonsSM/2GravitonsSM.fr/ 2Gravitons.fr].
- Example of a Mathematica® notebook loading the model and the parameters: [https://feynrules.irmp.ucl.ac.be/attachment/wiki/2GravitonsSM/2GravitonsSM.fr/ 2Gravitons.nb].
- A model for fourth-order formalism is being developed.

=== Validation ===

At first order, the model has reproduced the results from:

 
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.091602/ Phys.Rev.Lett. 127 (2021) 9, 091602]: Katsuki Aoki, Tran Quang Loc, Toshifumi Noumi, Junsei Tokuda, ''Is the Standard Model in the Swampland? Consistency Requirements from Gravitational Scattering''

[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.125020/ Phys. Rev. D 103, 125020 (2021)]: Lasma Alberte, Claudia de Rham, Sumer Jaitly, Andrew J. Tolley, ''QED positivity bounds''

The model has also been cross-checked with [http://feynrules.irmp.ucl.ac.be/wiki/Spin2/ Spin 2 simplified model]. We found complete agreement.

At second order, the model has been checked with Graviton + Photon -> Graviton + Photon process. Its amplitudes satisfy crossing symmetric properties and find complete agreement to ones generated by manual inputs of Feynman rules.