ALRM_general: A general implementation of the alternative left-right symmetric model

Contact Information

Implementation author: Benjamin Fuks

• LPTHE / Sorbonne U.
• fuks@…

In collaboration with Mariana Frank and Özer Özdal. See ​arXiv:1911.12883 [hep-ph]. This implementation is the most general one of the model.

Model Description and FeynRules Implementation

In the alternative left-right symmetric model, the SU(2)_R partner of the right-handed up-quark u_R is an exotic down-type quark d′_R (instead of the SM right-handed down-type quark d_R), and the SU(2)_R partner of the right-handed charged lepton e_R is an exotic neutral lepton, the scotino n_R (instead of the right-handed neutrino ν_R). The right-handed neutrino ν_R and down-type quark d_R remain singlets under both the SU(2)_L and SU(2)_R groups. To preserve the left-right symmetry, the model includes an SU(2)_L singlet scotino n_Land down-type quark d'_L. The model is moreover invariant under a global symmetry U(1)_S and a gauged B-L symmetry.

The gauge and global symmetry SU(2)_R × U(1)_B−L × U(1)_S is first broken to the hypercharge through an SU(2)_R doublet of scalar fields χ_R charged under U(1)_S. The model moreover includes an SU(2)_L counterpart χ_L, insensitive to U(1)_S, to maintain the left-right symmetry. The electroweak symmetry is then broken down to electromagnetism by means of a bidoublet of Higgs fields charged under both SU(2)_L and SU(2)_R and without any B − L quantum numbers.

The model Lagrangian includes, in addition to gauge-invariant kinetic terms for all fields, a Yukawa Lagrangian generating masses for all fields,

\begin{eqnarray*}
  \mathcal{L}_{\rm Y} &=& \bar Q_L {\bf \hat Y}^u \hat\phi^\dag Q_R
     - \bar Q_L {\bf \hat Y}^d \chi_L d_R
     - \bar Q_R {\bf \hat Y}^{d'}\chi_R d'_L
     - \bar L_L {\bf \hat Y}^e \phi L_R
     + \bar L_L {\bf \hat Y}^\nu \hat\chi_L^\dag \nu_R \\&&
     + \bar L_R {\bf \hat Y}^n \hat\chi_R^\dag n_L + {\rm h.c.} \ ,
\end{eqnarray*}

and the Higgs potential is given by

\begin{eqnarray*}
  V_{\rm H}  &= &
    -\mu_1^2 {\rm Tr} \big[\phi^\dag \phi\big]
     -\mu_2^2 \big[\chi_L^\dag \chi_L + \chi_R^\dag \chi_R\big]
     + \lambda_1 \big({\rm Tr}\big[\phi^\dag \phi\big]\big)^2
     + \lambda_2\ (\phi\!\cdot\!\hat\phi)\ (\hat\phi^\dag\!\cdot\!\phi^\dag)
  \\&&
     + \lambda_3 \Big[\big(\chi_L^\dag \chi_L\big)^2 +
           \big(\chi_R^\dag\chi_R\big)^2\Big]
     + 2 \lambda_4\ \big(\chi_L^\dag \chi_L\big)\ \big(\chi_R^\dag\chi_R\big)
  \\&&
     + 2 \alpha_1 {\rm Tr} \big[\phi^\dag \phi\big]
            \big[\chi_L^\dag \chi_L + \chi_R^\dag \chi_R\big]
     + 2 \alpha_2 \big[ \big(\chi_L^\dag \phi\big) \big(\chi_L\phi^\dag\big)      +
            \big(\phi^\dag \chi_R^\dag\big)\ \big(\phi\chi_R\big)\big]
  \\&&
     + 2 \alpha_3 \big[ \big(\chi_L^\dag \hat\phi^\dag\big)\
            \big(\chi_L\hat\phi\big) + \big(\hat\phi\chi_R^\dag\big)\
            \big(\hat\phi^\dag\chi_R\big)\big]
     + \kappa \big[\chi_L^\dag \phi \chi_R + \chi_R^\dag\phi^\dag\chi_L\big]      \  .
\end{eqnarray*}

We refer to ​arXiv:1912.NNNNN [hep-ph] for more detailed information, in particular on the minimisation of the potential and the diagonalisation of the gauge eigenbasis.

The above Lagrangian was implemented in the unitarity gauge into FeynRules 2.3.35. Feynman rules were collected into a single UFO model files that permits tree-level calculations at LO using MadGraph_aMC@NLO and dark matter computations using MadDM. MicrOMEGAs can also be used for testing the cosmology of the model, relying on the generated CalcHEP model file.

The model contains 16 free parameters (on top of the Standard Model ones):

• the ratio of the bidoublet to the SU(2)_L triplet vacuum expectation values tanβ (tb, block SMINPUTS, entry 5);
• the SU(2)_R coupling constant g_R (gR, block SMINPUTS, entry 6);
• the square root of the sum of the squared bidoublet and SU(2)_R vevs v’ (vevp, block SMINPUTS, entry 7);
• the potential parameters λ₂ (lam2), λ₃(lam3), α₁(alp1), α₂(alp2), α₃(alp3), and κ (kap) (block HPOTINPUTS);
• the masses of the neutrino (PDG 12, 14, 16), scotino (PDG 6000012, 6000014, 6000016) and exotic down quark (PDG 6000001, 6000003, 6000005) fields (in the LH block MASS);
• the 8 inputs dictating the two CKM matrices (block CKMBLOCK);
• the 8 inputs dictating the two PMNS matrices (block PMNSBLOCK).

All other parameters are derived quantities. We refer to ​arXiv:1911.12883 [hep-ph] for more information.

Model Files

• FeynRules model file;
• UFO model file;
• CalcHEP model file.