*New* SM_ZeeBabu: The Standard Model + Zee-Babu scalars at NLO in QCD *New*

Contact Author

Richard Ruiz

• Institute of Nuclear Physics Polish Academy of Science (IFJ PAN)
• richard.physics AT gmail.com

Usage resources

• For detailed instructions and examples on using the SM_ZeeBabu UFO libraries, see R. Ruiz ​arXiv:2207.qwerty .
• See Validation section below for additional information

Citation requests

• For the Lagrangian, please cite the original papers by Zee [ 1, 2 ] and Babu [ 3 ].
• If using any of the UFOs, please cite the companion paper [ 4 ].

Model Description

This effective/simplified 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 [ 6 ]

\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}_{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}

Neutrino masses (mNk) and mixing parameters (Vlk) between heavy mass eigenstate and (active) flavor eigenstates are taken to be independent, phenomenological parameters. This allows for maximum flexibility and model independence when calculating rates. Therefore, some care is required by the user. 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 2.3.36. QCD renormalization and R2 rational counter terms were determined using NLOCT 1.02 and FeynArts 3.11. Feynman rules were collected into a single UFO, available below. In the default UFO, five massless quarks are assumed as well as a CKM matrix equal to the identity matrix. For additional details, see [ 4 ] and references therein. 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

Note: The only difference between NLO and LO libraries is the presence of additional (effective) O(a_s) Feynman rules. By definition the NLO libraries can compute tree-level processes at LO precision.

Default UFO (massless charged leptons)

• ​SM_HeavyN_NLO_UFO.tgz: Standalone UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.

• ​SM_HeavyN_Gen3Mass_NLO.tgz: Standalone UFO folder. Assumes massive bottom quark and tau lepton, diagonal CKM.

• ​SM_HeavyN_CKM_AllMasses_LO.tgz: Standalone UFO folder. Assumes LO in QCD, with CKM elements (in radians), and all particle masses. Majorana equivalent of SM_HeavyN_Dirac_CKM_Masses_LO_UFO.

• ​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.

Alternative UFO (massive charged leptons)

• ​SM_HeavyN_Dirac_NLO_UFO.tgz: Standalone NLO UFO libraries

• ​SM_HeavyN_Dirac_LO_UFO.tgz: Standalone LO UFO libraries.

• ​SM_HeavyN_Dirac_CKM_Masses_LO_UFO.tgz: Standalone LO UFO libraries, assuming non-diagonal CKM matrix, six quark masses, and three charged lepton masses

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

Download and Unpack

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

Massless Leptons NLO

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

~/Path $ tar -zxvf SM_ZeeBabu_NLO.tgz

Massless Leptons LO

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

~/Path $ tar -zxvf SM_ZeeBabu_XLO.tgz

Massive Leptons NLO

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

~/Path $ tar -zxvf SM_ZeeBabu_MassiveLeptons_NLO.tgz

Massive Leptons LO

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

~/Path $ tar -zxvf SM_ZeeBabu_MassiveLeptons_XLO.tgz

Notes

• For instructions on using the HeavyN UFO, see C. Degrande, et al, ​arXiv:1602.06957 and S. Pascoli, et al, ​arXiv:1812.08750

• The flagship HeavyN UFO 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 ].

• For the Majorana file, particle identification (PID) codes for N1,...,N3, follow standard HEP MCPID codes: 9900012, 9900014, 9900016

• For the Dirac file, to avoid conflict with Pythia8, where the above PIDs are reserved for Majorana fields, the nonstandard HEP MCPID codes for N1,...,N3 are:9990012, 9990014, 9990016

• For the vSMEFT file, 10 additional model parameters are introduced:
  • One EFT cutoff scale Lambda in units of GeV.
  • Nine Wilson coefficients coupling N_k to l: CeN1, CeN2, CeN3, CmuN1, CmuN2, CmuN3, CtaN1, CtaN2, CtaN3
  • Note: Default parameters are set such that Lambda=1000 (GeV), CeN1=CmuN2=CtaN3=1, and all other coefficients are zero

Validation

• The model file was checked and validated against partonic results in [ 99 ]
• The cross sections of Table 2 of [ 4 ] can be generated using the script available from ​https://gitlab.cern.ch/riruiz/public-projects/-/tree/master/ZeeBabu_LHC_Update/LHC_XSec_Table2

Studies that have used the above model files

Please email to update this space.

• ...

References

• For the Lagrangian, please cite the original papers by Zee [ 1, 2 ] and Babu [ 3 ].
• If using any of the UFOs, please cite the companion paper [ 4 ].

[1] A. Zee, Charged Scalar Field and Quantum Number Violations, Phys. Lett. B161 (1985) 141, ​https://inspirehep.net/literature/214241

[2] A. Zee, Charged Scalar Field and Quantum Number Violations, Nucl. Phys. B264 (1986) 99, ​https://inspirehep.net/literature/218115

[3] K. Babu, Model of 'Calculable' Majorana Neutrino Masses, Phys. Lett. B203 (1988) 132, ​https://inspirehep.net/literature/22952

[4] R. Ruiz, Doubly Charged Higgs Boson Production at Hadron Colliders II: A Zee-Babu Case Study, ​arXiv:2207.qwerty [hep-ph]

[99] J.F. Gunion, C. Loomis, K.T. Pitts, Searching for Doubly-Charged Higgs Bosons at Future Colliders, ​arXiv:hep-ph/9610237