Changes between Version 2 and Version 3 of SMWeinberg


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Timestamp:
Dec 15, 2020, 5:02:47 PM (4 years ago)
Author:
Richard Ruiz
Comment:

building UFO page. Refs, Past Studies, Validation done.

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  • SMWeinberg

    v2 v3  
     1'''THIS PAGE IS UNDER CONSTRUCTION. Come back next week'''
     2
     3
    14= {{{SMWeinberg}}}: The Standard Model + The Dimension Five Weinberg Operator at NLO in QCD =
    25
    … …  
    1316
    1417* If using the {{{SMWeinberg}}} UFO, please cite [ [#Fuks 1] ] along with the appropriate {{{FeynRules}}} and generator papers.
    15 * For studies of lepton number violation in {{{$W^\pm W^\pm$}}} scattering, please also consider citing [ [#Fuks 2] ].
    1618
    1719
    1820== Model Description ==
    1921
    20 The UFO file models the Weinberg operator for high-energy scattering and decays 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 {{{$m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$}}}.
     22=== Synopsis ===
     23This 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.
    2124
    2225
    23 === Majorana ===
    24 This effective/simplified model extends the Standard Model (SM) field content by introducing three right-handed (RH) neutrinos, which are singlets under the SM gauge symmetry (no color, weak isospin, or weak hypercharge charges). Each RH neutrino possesses one RH Majorana mass. After electroweak symmetry breaking, the Lagrangian with three heavy Majorana neutrinos ''N''i (for i=1,2,3) is given by [ [#Atre 5] ]
     26=== Full Description ===
     27This 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
    2528{{{
    2629#!latex
    … …  
    2932\end{equation}
    3033}}}
    31 The first term is the Standard Model Lagrangian. In the mass basis, i.e., after mixing with active neutrinos, the heavy Majorana neutrinos' kinetic and mass terms are
     34The first term is the Standard Model Lagrangian. The second is the Weinberg operator
    3235{{{
    3336#!latex
    3437\begin{equation}
    35 \mathcal{L}_{N} = \frac{1}{2}\overline{N_k} i\!\not\!\partial N_k - \frac{1}{2}m_{N_k} \overline{N_k}N_k, \quad k=1,\dots,3,
     38 \mathcal{L}_5 = \frac{C_5^{\ell\ell'}}{\Lambda} \big[\Phi\!\cdot\! \overline{L}^c_{\ell }\big]
     39    \big[L_{\ell'}\!\!\cdot\!\Phi\big],
     40\end{equation}
     41}}}
     42where
     43{{{$\Lambda$}}} is the EFT cutoff scale,
     44{{{$C^{\ell\ell'}_5$}}} is the flavor-dependent Wilson coefficient in the flavor basis,
     45and the the SU(2)-invariant product {{{$\Phi\cdot \overline{L^c} = \Phi^i\varepsilon_{ij} \overline{L^{cj}}$}}} is fixed by {{{$\varepsilon_{12}=1$}}}.
     46
     47
     48The 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 {{{$m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$}}}.
     49
     50
     51
     52 In the mass basis, the heavy Majorana neutrinos' kinetic and mass terms are
     53{{{
     54#!latex
     55\begin{equation}
     56\mathcal{L}_{N} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N,
    3657\end{equation}
    3758}}}
    … …  
    4263\begin{eqnarray}
    4364\mathcal{L}_{N~\text{Int}} =
    44 &-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\ell^{-}
     65&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}\gamma^{\mu}P_{L}\ell^{-}
    4566+{\rm H.c.}
    4667\\
    47 &-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}\gamma^{\mu}P_{L}\nu_\ell
     68&-&\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
    4869+{\rm H.c.}
    4970\\
    50 &-&\frac{g m_N}{2 M_W}         h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}V_{\ell k}^{*}P_{L}\nu_\ell
     71&-&\frac{g m_N}{2 M_W}         h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k} P_{L}\nu_\ell
    5172+{\rm H.c.}
    5273\end{eqnarray}
    5374}}}
    54 Neutrino masses (mNk) and mixing parameters (Vlk) between heavy mass eigenstate and (active) flavor eigenstates 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.
    55 The lepton number- and flavor-violating interactions of the Lagrangian allow for modeling of the Type I, Inverse, and Linear seesaw mechanisms at both lepton, hadron, and lepto-hadron colliders.
     75... 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.
    5676
    5777
    5878
    5979== QCD Corrections ==
    60 The above Lagrangian with Goldstone boson couplings and in the Feynman Gauge was implemented into FeynRules  2.3.10. QCD renormalization and R2 rational counter terms were determined using NLOCT 1.02 and FeynArts 3.8. Feynman rules were collected into a single UFO, available below.
    61 In the UFO file, five massless quarks are assumed as are zero off-diagonal CKM matrix entries.
    62 For additional details, see [ [#Degrande 2] ] and references therein.
    63 These additions permit tree-level calculations at LO and NLO in QCD and loop-induced calculations at LO in QCD using MadGraph_aMC@NLO.
     80The 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] ].
     81
     82
     83These Feynman rules permit tree-level calculations at LO and NLO in QCD and loop-induced calculations at LO in QCD using MadGraph_aMC@NLO.
    6484
    6585
    … …  
    6989 * [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.
    7090
    71  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Gen3Mass_NLO.tgz SM_HeavyN_Gen3Mass_NLO.tgz]: Standalone UFO folder. Assumes massive bottom quark and tau lepton, diagonal CKM.
    72 
    73  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_CKM_AllMasses_LO.tgz SM_HeavyN_CKM_AllMasses_LO.tgz]: Standalone UFO folder. Assumes LO in QCD, with CKM elements (in radians), and all particle masses. Majorana equivalent of SM_HeavyN_Dirac_CKM_Masses_LO_UFO.
    74 
    75 
    76  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/heavyN.fr heavyN.fr]: Main model file. Relies on sm.fr (default FR model file) being declared elsewhere.
    77 
    78  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/heavyN_NLO.nb heavyN_NLO.nb]: Mathematica notebook file that generates UFO file from FeynRules model files. Allows user to also run quick sanity checks (optional) on model.
    79 
    80  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Files.tgz SM_HeavyN_Files.tgz]: Standalone package containing heavyN.fr, heavyN_NLO.nb, massless.rst (default FR file), diagonalCKM.rst (default FR file), and sm.fr (default FR file).
    81 
    82  * [https://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_FilesWithUFO.tgz SM_HeavyN_FilesWithUFO.tgz]: Combination of SM_HeavyN_NLO_UFO.tgz and SM_HeavyN_Files.tgz.
     91 
    8392
    8493
    … …  
    9099 * To download any of the packages and unpack via the terminal, use the commands:
    91100
    92 ==== Majorana NLO ====
     101==== NLO UFO ====
    93102~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_NLO_UFO.tgz
    94103
    95104~/Path $ tar -zxvf SM_HeavyN_NLO_UFO.tgz
    96105
    97 ==== Majorana LO ====
     106==== LO UFO ====
    98107~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_FilesWithUFO.tgz
    99108
    100109~/Path $ tar -zxvf SM_HeavyN_FilesWithUFO.tgz
    101110
    102 ==== Dirac NLO ====
    103 ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Dirac_NLO_UFO.tgz
    104111
    105 ~/Path $ tar -zxvf SM_HeavyN_Dirac_NLO_UFO.tgz
    106 
    107 ==== Dirac LO ====
    108 ~/Path $ wget http://feynrules.irmp.ucl.ac.be/raw-attachment/wiki/HeavyN/SM_HeavyN_Dirac_LO_UFO.tgz
    109 
    110 ~/Path $ tar -zxvf SM_HeavyN_Dirac_LO_UFO.tgz
    111112
    112113== Notes ==
    … …  
    136137
    137138== References ==
    138 * For studies of Majorana neutrinos, please consider citing [ 5 ] for the Lagrangian and [ 1-2 ] for the Majorana FR/UFO files.
    139 * For studies of heavy Dirac neutrinos, please also consider citing [ 4 ].
    140 
     139* For studies employing the {{{SMWeinberg}}} UFO, please cite [ 1 ].
    141140
    142141[=#Fuks] [1] B. Fuks, J. Neundorf, K. Peters, R. Ruiz and M. Saimpert,