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= {{{SMWeinberg}}}: The Standard Model + The Dimension Five Weinberg Operator at NLO in QCD =

=== Contact Author ===
Richard Ruiz
* Universite Catholique de Louvain
* richard.ruiz AT uclouvain.be

In collaboration with
Benjamin Fuks, Jonas Neundorf, Krisztian Peters, and Matthias Saimpert


For additional instructions and examples on using the {{{SMWeinberg}}} UFO libraries, see B. Fuks, et al, arXiv:2012.abcde

* If using the {{{SMWeinberg}}} UFO, please cite [ [#Fuks 1] ] along with the appropriate {{{FeynRules}}} and generator papers. 


== Model Description ==

=== Synopsis ===
This set of UFO libraries enables one to simulate the Weinberg operator in high-energy scattering and resonant decay processes up to next-to-leading order in QCD.


=== Full Description ===
This model file is constructed in the context of the Standard Model (SM) Effective Field Theory (SM EFT), and extend the SM Lagrangian by gauge-invariant operators up to dimension {{{$d=5$}}}. In the canonical representation the Lagrangian is given by
{{{ 
#!latex 
\begin{equation}
\mathcal{L} = \mathcal{L}_{\rm SM} + \mathcal{L}_{5} + \mathcal{O}(\Lambda^{-1})
\end{equation}
}}}
The first term is the Standard Model Lagrangian. The second is the Weinberg operator
{{{ 
#!latex 
\begin{equation}
 \mathcal{L}_5 = \frac{C_5^{\ell\ell'}}{\Lambda} \big[\Phi\!\cdot\! \overline{L}^c_{\ell }\big]
    \big[L_{\ell'}\!\!\cdot\!\Phi\big],
\end{equation}
}}}
where 
{{{$\Lambda$}}} is the EFT cutoff scale, 
{{{$C^{\ell\ell'}_5$}}} is the flavor-dependent Wilson coefficient in the flavor basis,
and the the SU(2)-invariant product {{{$\Phi\cdot \overline{L^c} = \Phi^i\varepsilon_{ij} \overline{L^{cj}}$}}} is fixed by {{{$\varepsilon_{12}=1$}}}.


The UFO file models the Weinberg operator by exploiting the observation [ [#Fuks 1] ] that an intermediate current of massless neutrinos {{{$(\nu_\ell \nu_{\ell'}^c)$}}} can be an unphysical Majorana neutrino with mass {{{$m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$}}}. 



 In the mass basis, the heavy Majorana neutrinos' kinetic and mass terms are
{{{ 
#!latex 
\begin{equation}
\mathcal{L}_{N} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N,
\end{equation}
}}}

and its interactions with the Weak gauge and Higgs bosons are given by
{{{ 
#!latex 
\begin{eqnarray}
\mathcal{L}_{N~\text{Int}} = 
&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_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}\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} P_{L}\nu_\ell
+{\rm H.c.}
\end{eqnarray}
}}} 
... 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. 



== QCD Corrections ==
The above Lagrangian with Goldstone boson couplings and in the Feynman Gauge was implemented into FeynRules  2.3.36. QCD UV renormalization and R2 rational counter terms are extracted using NLOCT 1.02 and FeynArts 3.11. Feynman rules were collected into a single UFO, available below. In the UFO file, five massless quarks are assumed as are zero off-diagonal CKM matrix entries. For additional details, see [ [#Fuks 1] ].


These Feynman rules 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.

 * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_NLO_UFO.tgz SM_HeavyN_NLO_UFO.tgz]: Standalone UFO folder. Assumes massless bottom quark and tau lepton, diagonal CKM.

 






== Download and Unpack ==
 * To download any of the packages and unpack via the terminal, use the commands:

==== NLO UFO ====
~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_NLO_UFO.tgz

~/Path $ tar -zxvf SM_HeavyN_NLO_UFO.tgz

==== LO UFO ====
~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_FilesWithUFO.tgz

~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.tgz



== Notes ==
 * For instructions on using the {{{SMWeinberg}}} UFO, see B. Fuks, et al, arXiv:2020.abcde 

 * This model contains seven external parameters:
  * One effective field theory scale: {{{Lambda}}}
  * Six Wilson coefficients: {{{Cee,Cem,Cet,Cmm,Cmt,Ctt}}}

 * This model contains 15 free parameters:
  * One Majorana neutrino mass: {{{mN1}}} with default value 49.1 GeV
  * One width: {{{wN1}}} with default value zero
  * '''Note''': {{{Cll}}} are restricted to be real in the model file.
  * '''Note''': External parameters must be tuned to reproduce [ [#Fuks 1] ].

* For Majorana file, particle identification (PID) codes for N1 follow standard HEP MCPID codes: 9900012


  

== Validation ==
* The model file was validated at NLO in [ [#Fuks 1] ]; see Table 1 for validation checks 

== Studies that have used the above model files ==
Please email to update this space.
* ...

== References ==
* For studies employing the {{{SMWeinberg}}} UFO, please cite [ 1 ].

[=#Fuks] [1] B. Fuks, J. Neundorf, K. Peters, R. Ruiz and M. Saimpert,
''To Appear,'' arXiv:2012.abcde [hep-ph]

[=#Fuks] [2] B. Fuks, J. Neundorf, K. Peters, R. Ruiz and M. Saimpert,
''Majorana Neutrinos in Same-Sign $W^\pm W^\pm$ Scattering at the LHC: Breaking the TeV Barrier,'' arXiv:2011.02547 [hep-ph]