Changes between Version 4 and Version 5 of SMWeinberg


Ignore:
Timestamp:
Dec 18, 2020, 7:23:15 AM (4 years ago)
Author:
Richard Ruiz
Comment:

Lagrangian update

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  • SMWeinberg

    v4 v5  
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    33
    4 = {{{SMWeinberg}}}: The Standard Model + The Dimension Five Weinberg Operator at NLO in QCD =
     4= {{{SMWeinberg}}}: The Standard Model + The Weinberg Operator at NLO in QCD =
    55
    66=== Contact Author ===
    … …  
    2525
    2626=== Full Description ===
    27 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 canonical representation the Lagrangian is given by
     27This 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
    2828{{{
    2929#!latex
    … …  
    3232\end{equation}
    3333}}}
    34 The first term is the Standard Model Lagrangian. The second is the Weinberg operator
     34The first term is the Standard Model Lagrangian. The second term {{{L5}}} is the Weinberg operator
    3535{{{
    3636#!latex
    … …  
    4242where
    4343{{{Lambda}}} is the EFT cutoff scale [GeV],
    44 {{{#!latex
    45 $C^{\ell\ell'}_5$
    46 }}}
    47 {{{#!latex $C^{\ell\ell'}_5$}}}
    48  is the flavor-dependent Wilson coefficient in the flavor basis,
    49 and the the SU(2)-invariant product {{{$\Phi\cdot \overline{L^c} = \Phi^i\varepsilon_{ij} \overline{L^{cj}}$}}} is fixed by {{{$\varepsilon_{12}=1$}}}.
     44{{{Cll}}} is the flavor-dependent Wilson coefficient [dimensionless],
     45{{{Phi}}} is the SM Higgs doublet with vev {{{v}}}, and {{{L}}} is the SM lepton doublet of flavor {{{l,l'}}}.
    5046
    5147
    52 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 {{{$m_{\ell \ell'}=C^{\ell\ell'}_5 v^2/\Lambda$}}}.
    53 
    54 
    55 
    56  In the mass basis, the heavy Majorana neutrinos' kinetic and mass terms are
     48A 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
    5749{{{
    5850#!latex
    5951\begin{equation}
    60 \mathcal{L}_{N} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N,
     52 m_{\ell\ell'} = C^{\ell\ell'}_5 v^2 / \Lambda.
    6153\end{equation}
    6254}}}
    6355
    64 and its interactions with the Weak gauge and Higgs bosons are given by
     56
     57In practice, the Lagrangian term {{{L5}}} is given by
     58{{{
     59#!latex
     60\begin{equation}
     61\mathcal{L}_{5} = \frac{1}{2}\overline{N} i\!\not\!\partial N - \frac{1}{2}m_{N} \overline{N}N + \mathcal{L}_{Int.},
     62\end{equation}
     63}}}
     64which describes a single (unphysical) Majorana neutrino {{{N}}} of mass
     65{{{
     66#!latex
     67\begin{equation}
     68 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,
     69\end{equation}
     70}}}
     71that couples to electroweak bosons through the interactions (in standard notation)
    6572{{{
    6673#!latex
    6774\begin{eqnarray}
    68 \mathcal{L}_{N~\text{Int}} =
    69 &-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k}\gamma^{\mu}P_{L}\ell^{-}
     75\mathcal{L}_{Int.} =
     76&-&\frac{g}{\sqrt{2}} W_{\mu}^{+}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\ell^{-}
    7077+{\rm H.c.}
    7178\\
    72 &-&\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
     79&-&\frac{g}{2\cos\theta_W}Z_{\mu}\sum_{\ell=e}^{\tau} \overline{N}\gamma^{\mu}P_{L}\nu_\ell
    7380+{\rm H.c.}
    7481\\
    75 &-&\frac{g m_N}{2 M_W}         h \sum_{k=1}^{3}\sum_{\ell=e}^{\tau} \overline{N_k} P_{L}\nu_\ell
     82&-&\frac{g m_N}{2 M_W}         h \sum_{\ell=e}^{\tau} \overline{N} P_{L}\nu_\ell
    7683+{\rm H.c.}
    7784\end{eqnarray}
    7885}}}
    79 ... 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.
    8086
     87The new external parameters of the {{{SMWeinberg}}} UFO are the six Wilson coefficients and the effective field theory cutoff scale.
    8188
    8289