Standard Model Effective Theory at Next-to-Leading-Order in QCD

Céline Degrande, Gauthier Durieux, Fabio Maltoni, Ken Mimasu, Eleni Vryonidou & Cen Zhang

A complete implementation of the SMEFT compatible with NLO QCD predictions.

The implementation is based on the Warsaw basis of operators and includes all degrees of freedom consistent with the following symmetry assumptions:

• CP-conservation.
• U(2)_Q x U(2)_u x U(3)_d x U(3)_L x U(3)_e flavor symmetry.

The CKM matrix is approximated as a unit matrix. The flavor symmetry imposes that only the top quark is massive. The model therefore implements the 5-flavor scheme for PDFs. The bosonic operators are implemented as in the Warsaw basis employing the M_Z, M_W, G_F scheme of Electroweak input parameters.

The Standard Model input parameters that need to be specified are:

M_Z, M_W, G_F, M_H, M_t, α_S(M_Z)

The fermionic degrees of freedom (2 & 4 fermion operators) are defined according to the common standards and prescriptions established by the LHC TOP WG for the EFT interpretation of top-quark measurements at the LHC (see the ​dim6top page for more information). This model has been validated at LO with the dim6top implementation.

A new coupling order, NP, is added to the model for the SMEFT interactions. It is assigned through the universal cutoff parameter, Lambda, which takes a default value of 1 TeV^-2 and can be modified along with the Wilson coefficients in the param card.

The ​definitions.pdf document specifies the operators definitions, normalisations and coefficient names in the UFO model

Usage notes

Restriction cards

Because of the mixture of LO/NLO compatible operators included in the model, restriction cards must be used to access the SMEFT interactions.

Default loading of the model

 > import model SMEFTatNLO_U2_2_U3_3_cG_4F_LO_UFO

will load the pure SM without any effective operators.

The LO restriction card should be used when importing the model for LO generation:

 > import model SMEFTatNLO_U2_2_U3_3_cG_4F_LO_UFO-LO

For NLO QCD generation, the NLO restriction card should be used when importing the model:

 > import model SMEFTatNLO_U2_2_U3_3_cG_4F_LO_UFO-NLO

This invokes a restricted set of operators for which the required counterterms are implemented.

Coupling orders

We recommend specifying the full QCD, QED and NP orders for process generation.

For example:

 > generate p p > t t~ QCD=2 QED=0 NP=2 [QCD]

Generates the NLO QCD top pair production process including the QCD-induced SM and the SMEFT contributions.

Excluding operators

We recommend avoiding setting values of Wilson coefficients to 0 when computing at NLO using MadGraph5_aMC@NLO.

Operators should either be removed explicitly with restriction cards or set to a very small non-zero value, e.g., 1e-5

Model Files

• ​SMEFTatNLO_U2_2_U3_3_cG_4F_LO_UFO.tgz : archive of UFO model - 4 fermion and c_G operators at LO, U(2)² x U(3)³ flavor symmetry

Version updates

• 2018/12/20 - v.0.1: First version upload, 4F and c_G operators at LO pending validation; a few minor convention tweaks required to match dim6top exactly. decays.py missing.
• 2019/04/03 - v.0.1: Added definitions.pdf document and uploaded a new version with a fix for restrict_default.dat
• 2019/08/12 - v.0.1: Uploaded a new version matching dim6top operator conventions, also some bugfixes and gs normalisation for otG