SM + Heavy N at NLO in QCD

Contact Author

Richard Ruiz

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

In collaboration with: Daniel Alva and Tao Han [ 1 ]; Celine Degrande, Olivier Mattelear, and Jessica Turner [ 2 ]; and Silvia Pascoli and Cedric Weiland [ 3 ].

Model Description

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 [ 4 ]

\begin{equation}
\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{N} + \mathcal{L}_{N~\text{Int.}}
\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.

Dirac

The Dirac variant of the HeavyN model file contains the same interaction Lagrangian as the Majorana case. The heavy Dirac neutrinos' kinetic and mass terms are

\begin{equation}
\mathcal{L}_{N} = \overline{N_k} i\!\not\!\partial N_k - m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3.
\end{equation}

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

Majorana

• ​SM_HeavyN_NLO_UFO.tgz: Standalone UFO folder.

• ​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.

Dirac

• ​SM_HeavyN_Dirac_NLO_UFO.tgz: Standalone NLO UFO folder

• ​SM_HeavyN_Dirac_LO_UFO.tgz: Standalone LO UFO folder.

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

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

• This model contains 15 free parameters:
  • Three masses: mN1, mN2, mN3. Defaults are 300 GeV, 500 GeV, and 1 TeV, respectively.
  • Three widths: WN1, WN2, WN3. Defaults are 0.303 GeV, 1.50 GeV, and 12.3 GeV, respectively.
  • Nine real (no CP violation) mixing parameters: Vlk for l = e, mu, tau and k = 1,2,3. Default values are Vlk = Identity(3x3), i.e., Ve1 = Vmu2 = Vta3 = 1 and all others zero.
  • Note: VlN are restricted to be real in the model file.
  • Note: Default parameters are set so "out-of-the-box" checks can be made with [ 1 ] and [ 2 ].

• For Majorana file, particle identification (PID) codes for N1,...,N3, follow standard HEP MCPID codes: 9900012, 9900014, 9900016

• For Dirac file, to avoid conflict with Pythia8, where the above PIDs are reserved for Majorana fields, the nonstandard HEP MCPID codes for N1,...,N3 are:9990012, 9990014, 9990016

Validation

• The model file was validated at LO in [ 1 ]; see Tables 2 and 5 for further validation checks.
• The model file was validated at NLO in [ 2 ]; see Table 1 for further validation checks.
• For very large masses, this model has been constructed to satisfy the Goldstone Equivalence Theorem:

  \begin{eqnarray}
  \text{BR}(N_i \rightarrow W^+ \ell^-) &=& \text{BR}(N_i \rightarrow W^- \ell^+) =
  \\ 
  \text{BR}(N_i \rightarrow Z \nu_\ell + Z \overline{\nu_\ell} ) &=&
  \text{BR}(N_i \rightarrow h \nu_\ell + h \overline{\nu_\ell} ) = 25\%.
  \end{eqnarray}
  

Studies that have used the above model files

Please email to update this space.

• Golling, et al., Physics at a 100 TeV pp collider: beyond the Standard Model phenomena, ​arXiv:1606.00947
• CMS, Search for heavy neutral leptons in events with three charged leptons in proton-proton collisions at 13 TeV, ​arXiv:1802.02965 [hep-ex]
• CMS, Search for heavy Majorana neutrinos in same-sign dilepton channels in proton-proton collisions at 13 TeV, ​arXiv:1806.10905 [hep-ex]

References

• Please cite [ 4 ] for the model, [ 1-2 ] for the Majorana FR/UFO files, and [ 2-3 ] for the Dirac FR/UFO files.

[1] D. Alva, T. Han, R. Ruiz, Heavy Majorana neutrinos from $W\gamma$ fusion at hadron colliders, JHEP 1502, 072 (2015), ​arXiv:1411.7305 [hep-ph]

[2] C. Degrande, O. Mattelear, R. Ruiz, J. Turner, Fully-Automated Precision Predictions for Heavy Neutrino Production Mechanisms at Hadron Colliders, PRD 94, 053002 (2016), ​arXiv:1602.06957 [hep-ph]

[3] S. Pascoli, R. Ruiz and C. Weiland, Safe Jet Vetoes, PLB 786, 106 (2018), ​arXiv:1805.09335 [hep-ph]

[4] A. Atre, T. Han, S. Pascoli and B. Zhang, The Search for Heavy Majorana Neutrinos, JHEP 0905, 030 (2009), ​arXiv:0901.3589 [hep-ph]