== The effective field theory for single Higgs and double Higgs production beyond leading order ==
=== Authors ===

Long-Bin Chen
   * Guangzhou University
   * chenlb_AT_gzhu_dot_edu_dot_cn

Hai Tao Li
   * LANL
   * haitaoli_AT_lanl_dot_gov

Hua-Sheng Shao
   * LPTHE
   * huasheng_dot_shao_AT_lpthe_dot_jussieu_dot_fr

Jian Wang
   * Shandong University
   * j_dot_wang_AT_sdu_dot_edu_dot_cn

=== References ===

* L.-B. Chen, H. T. Li, H.-S. Shao and J. Wang, [https://arxiv.org/abs/1909.06808 arXiv:1909.06808], [https://arxiv.org/abs/1912.13001 arXiv:1912.13001].

=== Description of the model ===

In the infinite top-quark mass limit, the interaction between the Higgs field and the gluon field can be effectively described by

[[latex($\mathcal{L}_{\rm eff}=-\frac{1}{4}(C_h \frac{h}{v}-C_{hh}\frac{h^2}{2v^2})G^a_{\mu\nu}G^{a~\mu\nu}$)]]

where [[latex($G^a_{\mu\nu}$)]] is the gluon field strength tensor.

The NLO UFO model based on [[latex($\mathcal{L}_{\rm eff}+\mathcal{L}_{\rm SM}$)]] can be downloaded from [/raw-attachment/wiki/NLOModels/HEFT_DH_UFO.tar.gz here].

The analytical expressions for [[latex($R_2$)]] Feynman rules can be found in appendix B of [https://arxiv.org/abs/1912.13001 arXiv:1912.13001].

=== An example in MadGraph5_aMC@NLO ===

In order to handle mixed-order senarios, one has to use the version 3.0.1+ of MadGraph5_aMC@NLO.

In version 3.0.X, the NLO QCD correction to di-Higgs+jet at [[latex($\mathcal{O}(\alpha_s^3)$)]] is generated by

{{{
> import model HEFT_DH_UFO
> generate p p > h h j / t t~ QCD=3 QED=2 NP=2 [QCD] QCD^2<=6 QED^2<=4 NP^2<=4
}}}

In versions 3.1.0+, the generation syntax has been changed [http://amcatnlo.web.cern.ch/amcatnlo/co.htm here]. It is generated by two processes, i.e.,
{{{
> import model HEFT_DH_UFO
> generate p p > h h j / t t~ QCD=3 QED=2 NP=2 [QCD] QCD^2<=6 QED^2<=4 NP^2<=4
}}}
and
{{{
> import model HEFT_DH_UFO
> generate p p > h h j / t t~ QCD^2==8 QED^2<=4 NP^2<=4
}}}
The second part is the pure interference term at tree level but contributing to [[latex($\mathcal{O}(\alpha_s^4)$)]].