The SMEFT gauge fixed using the background field method.

Author: Tyler Corbett

corbett.t.s*gmail.com

Niels Bohr International Academy, Niels Bohr Institute

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Description

Following ​arXiv:1803.08001 we implement the Standard Model Effective Field Theory (SMEFT) dimension six operators gauge fixed using the background field method (see e.g. ​Abbott, Introduction to the Background Field Method). The implementation of the model is elaborated in ​arXiv:2010.15852, and is verified in both this paper and ​arXiv:2010.08451 where the one-loop Ward identities are checked for UV divergent two-point functions and the Tadpole.

The model includes all SMEFT operators at dimension six as well as dimension-eight operators of Classes 2, 3, 4, and 7. Fermionic operators are introduced with no assumptions on flavor and the Yukawa operators and class 5 operators are diagonalized using standard rotations of the Left and Right-handed fermionic fields, resulting in meaningful goldstone-fermion couplings which also incorporate the CKM matrix for charged goldstone interactions. Another appeal of the model is that it incorporates field and mass redefinitions coming from the SMEFT to order 1/Lambda⁴. This is to the author's knowledge the only package incorporating these effects. As dimension-six-squared effects are frequently included in the literature without incorporating these redefinitions this tool provides an excellent opportunity to study the implications of incomplete calculations. This is discussed in the paper outlining the implementation, ​arXiv 2010.15852. The Class 15 leptonic operators as defined in ​arXiv 2005.00059 have also been included.

Usage

This model may be used to derive the Feynman rules in the Feynrules interface as well as for obtaining a model file for usage with Formcalc and Feynarts. The ancillary files demonstrate usage in Feynrules and in the Feynarts and Formcalc workflow in order to obtain the two-point and tadpole amplitudes.

This model is not designed for use with simulation packages such as MadGraph5, and usage of the the UFO output of Feynrules is not recommended.

Description of Attachments

1. SMEFT_BGFM_1.1.zip includes only the files required to derive the Feynman rules and no ancillary files.

2. (note: the .fr files have not been updated in this attachment to v1.1 and should be separately added and used with the mathematica notebooks contained in these files) SMEFT_BGFM_wAnc.zip includes the files required to derive the Feynman rules, an example Mathematica notebook which selects only n-point rules and creates the Feynarts model file or outputs the rules to the screen, and a directory which derives two-point functions and the Tadpole. This directory, "SMEFT6_3and4pts," includes:
   • SMEFT6_3and4pts.gen and SMEFT6_3and4pts.mod, Feynarts model files including only the 3 and 4 point vertices for the SMEFT at dimension-six.
   • rulesrearrange.nb a Mathematica notebook that reads in the Feynarts model file and simplifies the rules, greatly speeding up the performance of Formcalc.
   • ampprint.nb a Mathematica notebook that reads in the simplified Feynarts model file and derives the two-point functions and the Tadpole. It also uses Package-X to derive the UV-divergent part of these amplitudes and prints the desired Wilson coefficient dependence of the amplitudes to screen.
   • note: the .fr files have not been updated in this attachment to v1.1 and should be separately added and used with the mathematica notebooks contained in these files

Usage of the ancillary files requires the user to have Feynarts, Formcalc, and Package-X installed in Mathematica and some basic understanding of the packages to run.

3. SMEFT_BGFM.zip is the 1.0 version of the Feynrules files and is kept for backtesting both by the creator and users.

Change Log

v1.1 7 Oct 2021: Added Cl7 and Cl15 operators at dimension eight. The covariant derivative was rewritten in terms of Q_e g2bar, gzbar, ebar, and sz for clearer Feynman rules. Fixed barred fermion mass substitutions.

v1.0 29 Nov 2020: The first version.

References

1. T. Corbett, The Feynman Rules for the SMEFT in the Background Field Gauge, ​arXiv:2010.15852.
   Paper introducing the model and its implementation as well as verifying the SMEFT Class 2, 3, and 4 operator squared contributions to the SMEFT Ward Identities.

2. T. Corbett, M. Trott, One loop verification of SMEFT Ward Identities, ​arXiv:2010.08451.
   Paper verifying SMEFT dimension-six Ward Identities.