SM + W' and Z' at NLO in QCD

Contact Authors

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 the massive vector fields W'^+/- and Z' bosons, which are electrically charged and neutral, respectively. To remain model independence, couplings to SM gauge bosons and scalars are omitted.

Following Refs. [ 1, 2 ], the Lagrangian parameterizing the new vector bosons' couplings to up-type and down-type quark fields u_i and d_j is given by

\begin{eqnarray}
\mathcal{L}^q  =
    - \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}

Here, $i$ and $j$ denote flavor indices, P_L/R are the usual left/right-handed chirality projectors, V^CKM is the 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 Sequential Standard Model (SSM) Lagrangian ${\cal L}_{\rm SSM}$ [ 3 ]. The real-valued quantities $\kappa_{L,R}^q$ and $\zeta_{L,R}^q$ serve as overall normalization of the new interactions relative to the strength of the weak coupling constant.

The interactions involving charged lepton $\ell$ and massless neutrino $\nu_\ell$ fields are parametrized by

\begin{eqnarray}
\mathcal{L}^\ell =
   - \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 ($\zeta_R^\nu = \kappa_R^\ell = 0$).

From our general Lagrangians, the SSM limit is obtained by imposing the coupling strengths to be equal to the SM weak couplings up to an overall normalization factor:

\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}

NLO in 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 [ 4 ] 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 [ 4 ] by comparing against Standard Model predictions and W^' predictions from [ 5 ], as well as private LO and NLO calculations. See Table III of [ 4 ] 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 [3,4] for the model file, [4,5] for NLO corrections, and [6] for automated veto resummation.

[1] S. Gopalakrishna, T. Han, I. Lewis, Z.-G. Si and Y. F. Zhou, Chiral Couplings of $W'$ and Top Quark Polarization at the LHC, PRD82, 115020 (2010) ​arXiv:1008.3508 [hep-ph]

[2] T. Han, I. Lewis, R. Ruiz and Z.-G. Si, Lepton Number Violation and $W^\prime$ Chiral Couplings at the LHC, PRD87, no. 3, 035011 (2013) ​arXiv:1211.6447 [hep-ph]

[3] G. Altarelli, B. Mele and M. Ruiz-Altaba, Searching for New Heavy Vector Bosons in $p \bar{p}$ Colliders, Z. Phys. C45, 109 (1989), ​doi:10.1007/BF01556677

[4] 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]

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

[6] 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]