= [[span(style=color: #FF0000, '''*New*''' )]] {{{SM_ZeeBabu}}}: The Standard Model + Zee-Babu scalars at NLO in QCD [[span(style=color: #FF0000, '''*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 [ [#Zee1 1], [#Zee2 2] ] and Babu  [ [#Babu 3] ].
* If using any of the UFOs, please cite the companion paper [ [#Ruiz 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
{{{ 
#!latex 
\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
{{{
#!latex 
\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
{{{
#!latex 
\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
{{{
#!latex 
\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,
{{{ 
#!latex 
\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 [ [#Ruiz 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) ====
 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/SM_ZeeBabu_NLO.tgz SM_ZeeBabu_NLO.tgz]: Standalone NLO UFO folder. Assumes massless bottom quark, massless charged leptons, diagonal CKM.

 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/SM_ZeeBabu_XLO.tgz SM_ZeeBabu_XLO.tgz]: Standalone LO UFO folder. Assumes massless bottom quark, massless charged leptons, diagonal CKM.

 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/zeebabu_NLO_public.fr zeebabu_NLO_public.fr]: Main model file. Relies on sm.fr (default FR model file) being declared first. 

 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/zeebabu_NLO_public.nb 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) ====
 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/SM_ZeeBabu_MassiveLeptons_NLO.tgz SM_ZeeBabu_MassiveLeptons_NLO.tgz]: Standalone NLO UFO folder. Assumes massless bottom quark, three massive charged leptons, diagonal CKM.

 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/ZeeBabu/SM_ZeeBabu_MassiveLeptons_XLO.tgz 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

== 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: V''lk'' for ''l'' = ''e, mu, tau'' and ''k'' = 1,2,3. Default values are V''lk'' = 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 [ [#Alva 1] ] and [ [#Degrande 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 [ [#Gunion 99] ]
* The cross sections of Table 2 of [ [#Ruiz 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 [ [#Zee1 1], [#Zee2 2] ] and Babu  [ [#Babu 3] ].
* If using any of the UFOs, please cite the companion paper [ [#Ruiz 4] ].

[=#Zee1] [1] A. Zee, ''Charged Scalar Field and Quantum Number Violations,'' Phys. Lett. '''B161''' (1985) 141, https://inspirehep.net/literature/214241

[=#Zee2] [2] A. Zee, ''Charged Scalar Field and Quantum Number Violations,'' Nucl. Phys. '''B264''' (1986) 99, 
https://inspirehep.net/literature/218115

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

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

[=#Gunion] [99] J.F. Gunion, C. Loomis, K.T. Pitts, ``Searching for Doubly-Charged Higgs Bosons at Future Colliders,'' arXiv:hep-ph/9610237