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SMWeinberg: The Standard Model + The Dimension Five 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

This set of UFO libraries enables one to simulate the Weinberg operator in high-energy scattering and resonant decay processes up to next-to-leading order in QCD.

Full Description

This model file is constructed in the context of the Standard Model (SM) Effective Field Theory (SM EFT), and extend the SM Lagrangian by gauge-invariant operators up to dimension $d=5$. In the canonical representation 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 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, $C^{\ell\ell'}_5$ is the flavor-dependent Wilson coefficient in the flavor basis, and the the SU(2)-invariant product $\Phi\cdot \overline{L^c} = \Phi^i\varepsilon_{ij} \overline{L^{cj}}$ is fixed by $\varepsilon_{12}=1$.

The UFO file models the Weinberg operator by exploiting the observation [ 1 ] that an intermediate current of massless neutrinos $(\nu_\ell \nu_{\ell'}^c)$ can be an unphysical Majorana neutrino with mass $m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$.

In the mass basis, the heavy Majorana neutrinos' kinetic and mass terms are

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

and its interactions with the Weak gauge and Higgs bosons are given by

\begin{eqnarray}
\mathcal{L}_{N~\text{Int}} = 
&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}\gamma^{\mu}P_{L}\ell^{-}
+{\rm H.c.}
\\
&-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}\gamma^{\mu}P_{L}\nu_\ell
+{\rm H.c.}
\\
&-&\frac{g m_N}{2 M_W}         h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k} P_{L}\nu_\ell
+{\rm H.c.}
\end{eqnarray}

... are taken to be independent, phenomenological parameters. This allows for maximum flexibility and model independence when calculating rates. Therefore, some care is required by the user.

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.

• ​SM_HeavyN_NLO_UFO.tgz: Standalone UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.

Download and Unpack

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

NLO UFO

~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_NLO_UFO.tgz

~/Path $ tar -zxvf SM_HeavyN_NLO_UFO.tgz

LO UFO

~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_FilesWithUFO.tgz

~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.tgz

Notes

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

• This model contains seven external parameters:
  • One effective field theory scale: Lambda
  • Six Wilson coefficients: Cee,Cem,Cet,Cmm,Cmt,Ctt

• This model contains 15 free parameters:
  • One Majorana neutrino mass: mN1 with default value 49.1 GeV
  • One width: wN1 with default value zero
  • Note: Cll are restricted to be real in the model file.
  • Note: External parameters must be tuned to reproduce [ 1 ].

• For Majorana file, 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

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, To Appear, ​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]