SM + Heavy N at NLO in QCD

Implementation Author

Richard Ruiz

• University of Durham (IPPP)
• richard.ruiz AT durham.ac.uk

In collaboration with Daniel Alva and Tao Han, and Celine Degrande, Olivier Mattelear, and Jessica Turner.

Model Description

This effective 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 [ 1 ]

\begin{equation}
\mathcal{L}_{\rm Type~I} = \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}

The mixing matrix elements Vlk parametrize the mixing between the (active) lepton flavor eigenstate l and the heavy mass eigenstate Nk that corresponds to mass mNk. The precise values of Vlk are model dependent and, in principle, are coupled to the mass and widths of the heavy neutrinos. Therefore, some care by the user is required. However, 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 within. 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

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

• ​heavyN.nb: Mathematica notebook file that loads FR and model files. Allows user to run quick sanity checks (optional) and outputs UFO file.

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

• ​Seesaw_TypeI_withUFO.tgz: Same as Seesaw_TypeI.tgz but with UFO folder generated using restrictions massless.rst (only massive fermions are the top quark and three Majorana neutrinos) and diagonalCKM.rst (CKM matrix set to unity).

• ​Seesaw_TypeI_UFO.tgz: Standalone UFO folder as packaged in Seesaw_TypeI_withUFO.tgz.

Notes

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

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

~/Path $ tar -zxvf Seesaw_TypeI.tgz

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

~/Path $ tar -zxvf Seesaw_TypeI_WithUFO.tgz

• 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 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" comparisons can be performed with [ 2 ] and 3 ].

Validation

• The model file was validated at LO in [ 2 ]; see Tables 2 and 5 for further validation checks.
• The model file was validated at NLO in [ 3 ]; 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}
  

List of analyses that have used the model file

• Please email to update this space.

References

• Please cite [ 1 ] for model and [ 2-3 ] for the FR/UFO model files/validation.

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

[2] C. Degrande, O. Mattelear, R. Ruiz, J. Turner, ​arXiv:1602.00000 [hep-ph]

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