SMWeinberg: The Standard Model + The Weinberg Operator at NLO in QCD

Contact Author

Richard Ruiz

• Universite Catholique de Louvain
• richard.ruiz AT uclouvain.be

In collaboration with Benjamin Fuks, Jonas Neundorf, Krisztian Peters, and Matthias Saimpert

For additional instructions and examples on using the SMWeinberg UFO libraries, see B. Fuks, et al, ​arXiv:2012.abcde

• If using the SMWeinberg UFO, please cite [ 1 ] along with the appropriate FeynRules and generator papers.

Model Description

Synopsis

The SMWeinberg UFO libraries allows one to simulate the Weinberg operator in high-energy scattering and resonant decay processes up to next-to-leading order in QCD when used in conjunction with event generators, like MadGraph5_aMC@NLO.

Full Description

This model file works in the context of the Standard Model (SM) Effective Field Theory (SM EFT), where the SM Lagrangian is extended by gauge-invariant operators up to dimension d=5. In the standard representation, i.e., the Warsaw basis, the Lagrangian is given by

\begin{equation}
\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{5} + \mathcal{O}(\Lambda^{-1})
\end{equation}

The first term is the Standard Model Lagrangian. The second term L5 is the Weinberg operator

\begin{equation}
 \mathcal{L}_5 = \frac{C_5^{\ell\ell'}}{\Lambda} \big[\Phi\!\cdot\! \overline{L}^c_{\ell }\big]
    \big[L_{\ell'}\!\!\cdot\!\Phi\big],
\end{equation}

where Lambda is the EFT cutoff scale [GeV], Cll is the flavor-dependent Wilson coefficient [dimensionless], Phi is the SM Higgs doublet with vev v, and L is the SM lepton doublet of flavor l or l'.

A novelty of this implementation is the fact that under certain conditions [ 1 ], the intermediate propagation of light Majorana neutrinos (\nu_l \nu^c_l') can be modeled as an unphysical Majorana neutrino with mass

\begin{equation}
 m_{\ell\ell'} = C^{\ell\ell'}_5 v^2 / \Lambda.
\end{equation}

In practice, the Lagrangian term L5 is given by

\begin{equation}
\mathcal{L}_{5} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N + \mathcal{L}_{Int.},
\end{equation}

which describes a single (unphysical) Majorana neutrino N of mass

\begin{equation}
 m_{N} = \left\vert C^{ee}_5+C^{e\mu}_5+C^{e\tau}_5+C^{\mu\mu}_5+C^{\mu\tau}_5+C^{\tau\tau}_5 \right\vert v^2 / \Lambda,
\end{equation}

that couples to electroweak bosons through the interactions (in standard notation)

\begin{eqnarray}
\mathcal{L}_{Int.} = 
&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\ell^{-}
+{\rm H.c.}
\\
&-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\nu_\ell
+{\rm H.c.}
\\
&-&\frac{g m_N}{2 M_W}         h \sum_{\ell=e}^{\tau} \overline{N} P_{L}\nu_\ell
+{\rm H.c.}
\end{eqnarray}

The new external parameters of the SMWeinberg UFO are the six Wilson coefficients and the effective field theory cutoff scale.

The Standard Model Lagrangian

This UFO employs version 1.4.7 of the SM Lagrangian sm.fr as implemented into FeynRules by Christensen, Duhr, and Fuks. Numerical inputs for the SM are set to the global averages reported in the 2020 PDG.

QCD Corrections

The above Lagrangian with Goldstone boson couplings and in the Feynman Gauge was implemented into FeynRules 2.3.36. QCD UV renormalization and R2 rational counter terms are extracted using NLOCT 1.02 and FeynArts 3.11. Feynman rules were collected into a single UFO, available below. In the UFO file, five massless quarks are assumed as are zero off-diagonal CKM matrix entries. For additional details, see [ 1 ].

These Feynman rules permit tree-level calculations at LO and NLO in QCD and loop-induced calculations at LO in QCD using MadGraph_aMC@NLO.

Model Files

Note: The only difference between NLO and LO libraries is the presence of additional (effective) O(a_s) Feynman rules. By definition the NLO libraries can compute tree-level processes at LO precision. See Attachments at the bottom of the page for all available files.

• ​SMWeinbergNLO.tgz : Standalone NLO in QCD UFO file. Assumes nf=5 massless quarks, massless tau lepton, diagonal CKM.

• ​SMWeinbergNLO_4FS.tgz : Standalone NLO in QCD UFO file. Assumes nf=4 massless quarks, massive tau lepton, diagonal CKM.

• ​SMWeinbergXLO.tgz : Standalone QCD UFO file. Assumes nf=5 massless quarks, massless tau lepton, diagonal CKM.

• ​SMWeinbergXLO_4FS.tgz : Standalone UFO file. Assumes nf=4 massless quarks, massive tau lepton, diagonal CKM.

• ​weinberg_NLO.fr : FeynRules model file for Weinberg operator

• ​sm.fr : FeynRules model file for the SM

• ​smWeinberg_NLO_public.nb : Mathematica notebook for generating UFO files

Download instructions

• To download any of the packages and unpack via the terminal, use the commands:

~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergNLO.tgz

~/Path $ tar -zxvf SMWeinbergNLO.tgz

In MadGraph, import using the command

~/Path $ mv SMWeinbergNLO ~/path_to_madgraph/models

MG5_aMC>import model SMWeinbergNLO

Notes

• For instructions on using the SMWeinberg UFO, see B. Fuks, et al, ​arXiv:2020.abcde

• This model contains seven external parameters in addition to those in the SM:
  • One effective field theory scale: Lambda with default value 200 TeV.
  • Six Wilson coefficients: Cee,Cem,Cet,Cmm,Cmt,Ctt with various default values.
  • Note: Cll are restricted to be real in the model file but can be negative.
  • Note: External parameters must be tuned to reproduce [ 1 ].

• This model contains two internal parameters:
  • One Majorana neutrino mass: mN1 with default value 2.4 GeV
  • One Majorana neutrino width: wN1 with default value zero

• Particle identification (PID) codes for N1 follow standard HEP MCPID codes: 9900012

Validation

• The model file was validated at NLO in [ 1 ]; see Table 1 for validation checks
• The model file was checked against the HeavyN UFO ​https://feynrules.irmp.ucl.ac.be/wiki/HeavyN in [ 2 ].

Studies that have used the above model files

Please email to update this space.

• ...

References

• For studies employing the SMWeinberg UFO, please cite [ 1 ].

[1] B. Fuks, J. Neundorf, K. Peters, R. Ruiz and M. Saimpert, Probing the Weinberg Operator at Colliders, ​arXiv:2012.abcde [hep-ph]

[2] B. Fuks, J. Neundorf, K. Peters, R. Ruiz and M. Saimpert, Majorana Neutrinos in Same-Sign $W^{\pm W}\pm$ Scattering at the LHC: Breaking the TeV Barrier, ​arXiv:2011.02547 [hep-ph]