Effective Left-Right Symmetric Model at NLO in QCD

Contact Author

Richard Ruiz

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

In collaboration with Olivier Mattelaer and Manimala Mitra [ 1 ], and Manimala Mitra, Darren Scott, and Michael Spannowsky [ 2 ].

Model Description

This effective model extends the Standard Model (SM) field content by introducing heavy Left-Right symmetric gauge bosons W_R^+/-, Z_R with masses M_WR and M_ZR, as well as three heavy Majorana neutrinos N_i with mass m_Ni. The Lagrangian is given by,

\begin{equation}
\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{\rm LRSM~Kin.} + \mathcal{L}_{\rm LRSM~Int.}
\nonumber
\end{equation}

The first term is the Standard Model (SM) 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}_{\rm LRSM~Kin.} \ni \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.
\nonumber
\end{equation}

The W_R, Z_R, couplings to fermions is separated into three pieces:

\begin{equation}
\mathcal{L}_{\rm LRSM~Int.} = \mathcal{L}_{W_R-q-q'} + \mathcal{L}_{W_R-\ell-\nu/N} + 
\mathcal{L}_{Z_R-f-f}
\end{equation}
\nonumber

The W_R chiral coupling to quarks are

\begin{equation}
 \mathcal{L}_{W_R-q-q'} = \frac{-\kappa_R^q g}{\sqrt{2}}\sum_{i,j=u,d,\dots}\overline{u}_i V_{ij}^{\rm CKM'}~W_{R \mu}^+ \gamma^\mu P_R~ d_j + \text{H.c.}
\nonumber
\end{equation}

V_ij^CKM' is the right-handed (RH) Cabbibo-Kobayashi-Masakawa (CKM) matrix and is related to the SM CKM matrix. g=\sqrt{4\pi\alpha_{\rm EM}(M_Z)}/\sin\theta_W is the SM Weak coupling constant and k_R^q is an overall normalization for the W_R interaction strength. For leptons, the W_R coupling and leptonic mixing is parametrized by [ 3, 4 ],

\begin{equation}
 \mathcal{L}_{W_R-\ell-\nu/N} = \frac{-\kappa_R^\ell g}{\sqrt{2}}
 \sum_{\ell=e,\mu,\tau}
 \sum_{m'=1}^3  \overline{N_{m'}} Y_{\ell m'}
~W_{R \mu}^+ \gamma^\mu P_R~ \ell^-+\text{H.c.}
\nonumber
\end{equation}

For chiral fermion f, the Z_R neutral currents are parameterized by

\begin{equation}
 \mathcal{L}_{Z_R-f-f} = \frac{-\kappa_{R}^f g}{\sqrt{1 - \left(1/\kappa_{R}^f\right)^2 \tan^2\theta_W}}\sum_{f=u,e,\dots}
 \overline{f} Z_{R \mu} \gamma^\mu \left(g_{L}^{Z_R,f}P_L + g_{R}^{Z_R,f}P_R\right)f.
\nonumber
\end{equation}

k_R^f are the same k_R^q,l as for W_R. The chiral couplings g_L,R^Z_R,f are combinations of electric and isospin charges. For more details, see [ 1 ].

QCD Corrections

The SM Lagrangian with Goldstone boson couplings in the Feynman Gauge and the above Lagrangian in the Unitary 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 SM CKM and RH CKM' matrix entries. For additional details, see [ 1 ]. These additions permit tree-level calculations at LO and NLO in QCD and loop-induced calculations at LO in QCD using MadGraph_aMC@NLO, HERWIG, and SHERPA.

Model Files

• EffLRSM_NLO.tgz: Standalone NLO in QCD-accurate UFO folder.

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

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

• EffLRSMnlo_Files.tgz: Standalone package containing effLRSM.fr, effLRSMnlo.nb, diagonalMixing.rst, massless.rst (default FR file), and sm.fr (default FR file).

• EffLRSMnlo_FilesWithUFO.tgz: Combination of EffLRSMnlo_Files.tgz and EffLRSM_NLO.tgz.

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

~/Path $ tar -zxvf EffLRSM_NLO.tgz

~/Path $ tar -zxvf EffLRSMnlo_FilesWithUFO.tgz

Notes

• Important: Default parameters are set so "out-of-the-box" checks can be made with [ 1 ] and [ 2 ].

• Aside from the SM inputs, this model contains 28 free parameters:
  • Five masses: MWR, MZR, mN1, mN2, mN3. Defaults are 3 TeV, 5.07 TeV, 173.3 GeV, 10¹² GeV, and 10¹⁴ GeV, respectively.
  • Five widths: WWR, WZR, WN1, WN2, WN3. Defaults are 84.3 GeV, 114 GeV, 2.12^-8 GeV, 100 GeV, and 100 GeV, respectively.
  • Nine real (no CP violation) mixing parameters for RH quark mixing: VCKMR[u,d] for u = up (1), charm (2), top (3) and d = down (1), strange (2), bottom (3). Default values are VCKMR = Identity(3x3), i.e., VCKMR[1,1] = VCKMR[2,2] = VCKMR[3,3] = 1 and all others zero.
  • Nine real (no CP violation) mixing parameters for RH leptonic mixing: YlN[l,N] for l = e (1), mu (2), tau (3) and N = N1 (1), N2 (2), N3 (3). Default values are YlN = Identity(3x3), i.e., YlN[1,1] = YlN[2,2] YlN[3,3] and all others zero.

• Default model settings assumes five massless quarks, massless light neutrinos, and massless charged leptons. This is easily modifiable by regenerating UFO files.

Validation

• Partial and total widths for W_R, Z_R, and N₁ are checked against exact analytic calculations. See [ 1 ] for more information.
• The model file was checked against exact NLO W_R predictions from [ 2 ] and references within [ 1 ].

List of studies that have used the model file

• Please email to update this space.

References

• Please cite [ 1 ] for the model file and [ 2 ] for neutrino jets.

[1] O. Mattelaer, M. Mitra, R. Ruiz, Automated Neutrino Jet and Top Jet Predictions at NLO+PS Matching in Effective Left-Right Symmetric Models, ​arXiv:1610.XXXXX [hep-ph]

[2] M. Mitra, D. Scott, and M. Spannowsky, Neutrino Jets from High-Mass W_R Gauge Bosons in TeV-Scale Left-Right Symmetric Models, ​arXiv:1607.03504 [hep-ph]

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

[4] T. Han, I. Lewis, R. Ruiz, Z.-G. Si, Lepton Number Violation and W' Chiral Couplings at the LHC, PRD87, 035011 (2013)