wiki:ZeeBabu

Version 8 (modified by Richard Ruiz, 2 years ago) ( diff )

updated Lagrangian (finished)

*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

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

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 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.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)

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

  • 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

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

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