SM + W' and W' at NLO in QCD

Contact Author

Benjamin Fuks

• UPMC Univ. Paris, LPTHE
• fuks AT lpthe.jussieu.fr

Richard Ruiz

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

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 [ 3 ]

\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}^q_{\rm NP} =
    - \frac{g}{\sqrt{2}} \sum_{i,j} \Big[
       \overline{u}_i V_{ij}^{\rm CKM}~W_\mu^{\prime+}~\gamma^\mu
       \Big(\kappa_L^{q} P_L + \kappa_R^{q}P_R\Big)~ d_j + \text{H.c.}\Big]
\\
    - \frac{g}{\cos\theta_W} \sum_{q=u,d}\sum_i \Big[
      \overline{q}_i ~Z_\mu^\prime \gamma^\mu
      \Big(\zeta_{L}^{q}P_L + \zeta_{R}^{q}P_R\Big)q_i\Big] \ ,
+{\rm H.c.}
\end{eqnarray}

$i$ and $j$ denote flavor indices, $P_{L/R}=\frac12 (1\mp\gamma_5)$ and are the usual left/right-handed chirality projectors, $V^{{\rm CKM}$ is the Cabbibo-Kobayashi-Maskawa (CKM) matrix, and $g$ and $\theta_W$ are the weak coupling constant and mixing angle respectively. We choose coupling normalizations facilitating the mapping to the reference SSM Lagrangian ${\cal L}_{\rm SSM}$~\cite{Altarelli:1989ff}. The real-valued quantities $\kappa_{L,R}}q$ and $\zeta_{L,R}^q$

\begin{eqnarray}
  \mathcal{L}^\ell_{\rm NP} =
   - \frac{g}{\sqrt{2}}\sum_i\Big[
       \overline{\nu}_{\ell_i} W_\mu^{\prime+}~\gamma^\mu
        \kappa_L^{\ell} P_L\ell_i^- + \text{H.c.}\Big]
\\
    - \frac{g}{\cos\theta_W} \sum_{f=\ell,\nu_\ell}\sum_i \Big[
      \overline{f}_i ~Z_\mu^\prime \gamma^\mu
      \Big(\zeta_{L}^{f}P_L + \zeta_{R}^{f}P_R\Big)f_i\Big] \ .
+{\rm H.c.}
\end{eqnarray}

The quantities $\kappa_L^{\ell$ are real-valued and serve as normalizations for leptonic coupling strengths. As no right-handed neutrinos are present in the SM, the corresponding right-handed leptonic new physics couplings are omitted ($\zeta_R}\nu = \kappa_R^{\ell = 0$). We assume that leptonic interactions are flavor diagonal.}

\begin{eqnarray}
  \zeta_{R,L}^{f} = \zeta_{Z_{\rm SSM}}^f \Big( g_V^f \pm g_A^f\Big)
  \qquad\text{with}\qquad
  g_V^f = \frac{1}{2}T_L^{3,f} - Q^f\sin^2\theta_W
  \qquad\text{and}\qquad
  g_A^f=-\frac{1}{2}T_L^{3,f},
\end{eqnarray}

In the canonical SSM, the overall normalizations are further trivially fixed as

\begin{eqnarray}
 \kappa^{q,\ell}_L=1, \qquad
 \kappa_R^q=0,
 \qquad\text{and}\qquad
 \zeta_{Z_{\rm SSM}}^{f}=1 \ .
\end{eqnarray}

QCD Corrections

The above Lagrangian 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, loop-induced calculations at LO in QCD, and NLO matched to NNLL(Veto) resummation using MadGraph_aMC@NLO.

Model Files

• ​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.

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/HeavyN/SM_HeavyN_NLO_UFO.tgz

~/Path $ tar -zxvf SM_HeavyN_NLO_UFO.tgz

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

~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.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 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 ].

Validation

• The model file was validated at NLO in [ 2 ] by comparing against Standard Model predictions and $W'$ predictions from [ #Sullivan ], as well as private LO and NLO calculations. See Table III of [ 2 ] for representative total widths that have been checked against analytic computation.

List of studies that have used the model file

• Please email to update this space.

References

• Please cite [ ] for the model file, [] for NLO corrections, and [] for automated veto resummation.

[] B. Fuks, R. Ruiz, A Comprehensive Framework for Studying $W'$ and $Z'$ Bosons at Hadron Colliders with Automated Jet Veto Resummation, ​arXiv:1701.xxxx [hep-ph]

[] Z. Sullivan, Fully differential $W'$ production and decay at next-to-leading order in QCD, PRD66, 075011 (2002), ​arXiv:hep-ph/0207290.

[] T. Becher, R. Frederix, M. Neubert and L. Rothen, Automated NNLL $+$ NLO resummation for jet-veto cross sections, EPJC75, no. 4, 154 (2015) ​arXiv:1412.8408 [hep-ph].