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.
• For studies of lepton number violation in $W^\pm W^\pm$ scattering, please also consider citing [ 2 ].

Model Description

The UFO file models the Weinberg operator for high-energy scattering and decays 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 $m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$.

Majorana

This effective/simplified model extends the Standard Model (SM) field content by introducing three right-handed (RH) neutrinos, which are singlets under the SM gauge symmetry (no color, weak isospin, or weak hypercharge charges). Each RH neutrino possesses one RH Majorana mass. After electroweak symmetry breaking, the Lagrangian with three heavy Majorana neutrinos Ni (for i=1,2,3) is given by [ 5 ]

\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. In the mass basis, i.e., after mixing with active neutrinos, the heavy Majorana neutrinos' kinetic and mass terms are

\begin{equation}
\mathcal{L}_{N} = \frac{1}{2}\overline{N_k} i\!\not\!\partial N_k - \frac{1}{2}m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3,
\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}V_{\ell 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}V_{\ell 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}V_{\ell k}^{*}P_{L}\nu_\ell
+{\rm H.c.}
\end{eqnarray}

Neutrino masses (mNk) and mixing parameters (Vlk) between heavy mass eigenstate and (active) flavor eigenstates 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. The lepton number- and flavor-violating interactions of the Lagrangian allow for modeling of the Type I, Inverse, and Linear seesaw mechanisms at both lepton, hadron, and lepto-hadron colliders.

QCD Corrections

The above Lagrangian with Goldstone boson couplings and in the Feynman Gauge was implemented into FeynRules 2.3.10. QCD renormalization and R2 rational counter terms were determined using NLOCT 1.02 and FeynArts 3.8. 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 [ 2 ] and references therein. These additions 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.

• ​SM_HeavyN_Gen3Mass_NLO.tgz: Standalone UFO folder. Assumes massive bottom quark and tau lepton, diagonal CKM.

• ​SM_HeavyN_CKM_AllMasses_LO.tgz: Standalone UFO folder. Assumes LO in QCD, with CKM elements (in radians), and all particle masses. Majorana equivalent of SM_HeavyN_Dirac_CKM_Masses_LO_UFO.

• ​heavyN.fr: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere.

• ​heavyN_NLO.nb: Mathematica notebook file that generates UFO file from FeynRules model files. Allows user to also run quick sanity checks (optional) on model.

• ​SM_HeavyN_Files.tgz: Standalone package containing heavyN.fr, heavyN_NLO.nb, massless.rst (default FR file), diagonalCKM.rst (default FR file), and sm.fr (default FR file).

• ​SM_HeavyN_FilesWithUFO.tgz: Combination of SM_HeavyN_NLO_UFO.tgz and SM_HeavyN_Files.tgz.

Download and Unpack

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

Majorana NLO

~/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

Majorana LO

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

~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.tgz

Dirac NLO

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

~/Path $ tar -zxvf SM_HeavyN_Dirac_NLO_UFO.tgz

Dirac LO

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

~/Path $ tar -zxvf SM_HeavyN_Dirac_LO_UFO.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 of Majorana neutrinos, please consider citing [ 5 ] for the Lagrangian and [ 1-2 ] for the Majorana FR/UFO files.
• For studies of heavy Dirac neutrinos, please also consider citing [ 4 ].

[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]