= {{{SMWeinberg}}}: The Standard Model + The 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 === The {{{SMWeinberg}}} UFO libraries allows one to simulate the Weinberg operator in high-energy scattering and resonant decay processes up to next-to-leading order in QCD when used in conjunction with event generators, like {{{MadGraph5_aMC@NLO}}}. === Full Description === This model file works in the context of the Standard Model (SM) Effective Field Theory (SM EFT), where the SM Lagrangian is extended by gauge-invariant operators up to dimension ''d=5''. In the standard representation, i.e., the Warsaw basis, 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 term {{{L5}}} 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 [GeV], {{{Cll}}} is the flavor-dependent Wilson coefficient [dimensionless], {{{Phi}}} is the SM Higgs doublet with vev ''v'', and {{{L}}} is the SM lepton doublet of flavor {{{l}}} or {{{l'}}}. A novelty of this implementation is the fact that under certain conditions [ [#Fuks 1] ], the intermediate propagation of light Majorana neutrinos {{{(\nu_l \nu^c_l')}}} can be modeled as an unphysical Majorana neutrino with mass {{{ #!latex \begin{equation} m_{\ell\ell'} = C^{\ell\ell'}_5 v^2 / \Lambda. \end{equation} }}} In practice, the Lagrangian term {{{L5}}} is given by {{{ #!latex \begin{equation} \mathcal{L}_{5} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N + \mathcal{L}_{Int.}, \end{equation} }}} which describes a single (unphysical) Majorana neutrino {{{N}}} of mass {{{ #!latex \begin{equation} m_{N} = \left\vert C^{ee}_5+C^{e\mu}_5+C^{e\tau}_5+C^{\mu\mu}_5+C^{\mu\tau}_5+C^{\tau\tau}_5 \right\vert v^2 / \Lambda, \end{equation} }}} that couples to electroweak bosons through the interactions (in standard notation) {{{ #!latex \begin{eqnarray} \mathcal{L}_{Int.} = &-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\ell^{-} +{\rm H.c.} \\ &-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\nu_\ell +{\rm H.c.} \\ &-&\frac{g m_N}{2 M_W} h \sum_{\ell=e}^{\tau} \overline{N} P_{L}\nu_\ell +{\rm H.c.} \end{eqnarray} }}} The new external parameters of the {{{SMWeinberg}}} UFO are the six Wilson coefficients and the effective field theory cutoff scale. == The Standard Model Lagrangian == This UFO employs version 1.4.7 of the SM Lagrangian {{{sm.fr}}} as implemented into FeynRules by Christensen, Duhr, and Fuks. Numerical inputs for the SM are set to the global averages reported in the 2020 PDG. == 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. See ''Attachments'' at the bottom of the page for all available files. * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergNLO.tgz SMWeinbergNLO.tgz ]: Standalone NLO in QCD UFO file. Assumes nf=5 massless quarks, massless tau lepton, diagonal CKM. * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergNLO_4FS.tgz SMWeinbergNLO_4FS.tgz ]: Standalone NLO in QCD UFO file. Assumes nf=4 massless quarks, massive tau lepton, diagonal CKM. * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergXLO.tgz SMWeinbergXLO.tgz ]: Standalone QCD UFO file. Assumes nf=5 massless quarks, massless tau lepton, diagonal CKM. * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergXLO_4FS.tgz SMWeinbergXLO_4FS.tgz ]: Standalone UFO file. Assumes nf=4 massless quarks, massive tau lepton, diagonal CKM. * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/weinberg_NLO.fr weinberg_NLO.fr ]: FeynRules model file for Weinberg operator * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/sm.fr sm.fr ]: FeynRules model file for the SM * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/smWeinberg_NLO_public.nb smWeinberg_NLO_public.nb ]: Mathematica notebook for generating UFO files == Download instructions == * To download any of the packages and unpack via the terminal, use the commands: {{{~/Path $ wget https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/SMWeinberg/SMWeinbergNLO.tgz}}} {{{~/Path $ tar -zxvf SMWeinbergNLO.tgz}}} In !MadGraph, import using the command {{{~/Path $ mv SMWeinbergNLO ~/path_to_madgraph/models}}} {{{MG5_aMC>import model SMWeinbergNLO}}} == Notes == * For instructions on using the {{{SMWeinberg}}} UFO, see B. Fuks, et al, arXiv:2020.abcde * This model contains seven external parameters in addition to those in the SM: * One effective field theory scale: {{{Lambda}}} with default value 200 TeV. * Six Wilson coefficients: {{{Cee,Cem,Cet,Cmm,Cmt,Ctt}}} with various default values. * '''Note''': {{{Cll}}} are restricted to be real in the model file but can be negative. * '''Note''': External parameters must be tuned to reproduce [ [#Fuks 1] ]. * This model contains two internal parameters: * One Majorana neutrino mass: {{{mN1}}} with default value 2.4 GeV * One Majorana neutrino width: {{{wN1}}} with default value zero * 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 * The model file was checked against the HeavyN UFO [https://feynrules.irmp.ucl.ac.be/wiki/HeavyN] in [ [#Fuks 2] ]. == 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, ''Probing the Weinberg Operator at Colliders,'' 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]