*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:2206.14833 .
• 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

The Zee-Babu model extends the Standard Model (SM) by two complex scalars, k-- and h-. Neither carries color or weak isospin but both are charged under weak hypercharge. k-- and h- carry the electric charges Q_k=-2 and Q_h=-1, respectively, and both are assigned lepton number L=+2. This is normalized such that SM leptons carry L=+1.

In terms of the SM Lagrangian L_SM}, the Lagrangian of the Zee-Babu model LZB is

\begin{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mathcal{L}_{\rm ZB} = \mathcal{L}_{\rm SM} + \mathcal{L}_{\rm Kin.} + \mathcal{L}_{\rm Yuk.} + \mathcal{L}_{\rm ZB\ scalar}
 + \delta\mathcal{L}_{\nu}
 \ .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{align}

The kinetic part of the Lagrangian for $k$ and $h$ is given by the following covariant derivatives

\begin{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \mathcal{L}_{\rm Kin.} = (D_\mu k)^\dagger (D^\mu k) + (D_\mu h)^\dagger (D^\mu h), 
  \quad\text{with}\quad
  D_\mu = \partial_\mu +i g_Y \hat{Y} B_\mu\ .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{align}

Here, the weak hypercharge operator is normalized such that the electromagnetic charge operator is Q=T+Y and Y_k = -2 (Y_h = -1).

The Yukawa part describes the coupling of SM leptons to k-- and h-. It is given by

\begin{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mathcal{L}_{\rm Yuk.} & \ ~  = 
 f_{ij}\ \overline{\tilde{L}^i} L^j h^\dagger
 +
 g_{ij}\ \overline{(e_R^c)^i} e_R^j k^\dagger + \text{H.c.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{align}

The scalar potential for k-- and h-, including couplings to the SM Higgs doublet Phi, is given by

\begin{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \mathcal{L}_{\rm ZB\ scalar} &=\ 
\tilde{m}_k^2 k^\dagger k  +\ \tilde{m}_h^2 h^\dagger h\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\ \lambda_k (k^\dagger k)^2\ +\ \lambda_{h} (h^\dagger h)^2\
+\ \lambda_{hk} (k^\dagger k)(h^\dagger h)\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\
&
+\ \left(\mu_{\not L}\ h h k^\dagger + \text{H.c.}\right)\
+\ \lambda_{kH} (k^\dagger k) \Phi^\dagger \Phi\ 
+\ \lambda_{hH} (h^\dagger h) \Phi^\dagger \Phi 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{align}

After EWSB, the physical masses of k-- and h- are, respectively,

\begin{align}
m_k^2 = \tilde{m}^2_k + \frac{\lambda_{kH}}{2}v^2
\quad\text{and}\quad 
m_h^2 = \tilde{m}^2_h + \frac{\lambda_{hH}}{2}v^2
\ .
\end{align}

The parameter mu has mass dimension GeV and the h-h-k vertex violates lepton number conservation.

Light neutrino masses are generated at two loops. They are described by \delta\mathcal{L}_\nu. To phenomenologically parameterize the Lagrangian, neutrinos are assumed to be massless in the UFO. This allows all f and g to be taken independently from one another.

• Note that k^dagger = k++ and h^dagger = h+.

• Note that this model permits lepton flavor violation and lepton number violation.

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_ZeeBabu_NLO.tgz: Standalone NLO UFO folder. Assumes massless bottom quark, massless charged leptons, diagonal CKM.

• ​SM_ZeeBabu_XLO.tgz: Standalone LO UFO folder. Assumes massless bottom quark, massless charged leptons, diagonal CKM.

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

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

Alternative UFO (massive charged leptons)

• ​SM_ZeeBabu_MassiveLeptons_NLO.tgz: Standalone NLO UFO folder. Assumes massless bottom quark, three massive charged leptons, diagonal CKM.

• ​SM_ZeeBabu_MassiveLeptons_XLO.tgz: Standalone LO UFO folder. Assumes massless bottom quark, three massive charged leptons, diagonal CKM.

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

Model parameters / inputs

• For instructions on using the SM_ZeeBabu UFOs, see R. Ruiz, ​arXiv:2206.14833

• In addition to SM parameters, the flagship SM_ZeeBabu_NLO UFO model contains two particles and 19 free parameters:
  • k-- (PID=61) and h- (PID=38) with antiparticles k++ (PID=-61) and h+ (PID=-38)
  • Two masses: mhZB, mkZB. Default values are 300 GeV and 500 GeV, respectively.
  • Two widths: whZB, wkZB. Default values are 1 GeV.
  • One dimensionful parameter: muZB. Default value is 1 TeV.
  • Five real scalar couplings: lamhZB, lamkZB, lamhZBkZB, lamhZBH, lamkZBH. Default values are 1.
  • Six real Yukawa couplings for l-v-h: gee, gemu, getau, gmumu, gmutau, gtautau. Default values are 1.
  • Three real Yukawa couplings for l-l-k: femu, fetau, fmutau. Default values are 1.

Validation

• The model file was checked and validated against partonic results in [ 5 ]
• The cross sections of Table 2 of [ 4 ] can be reproduced 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:2206.14833 [hep-ph]

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