== Top Quark Decay to a Higgs and a Light Quark Operator ==

=== Motivation ===

Neutral Flavor Changing couplings are absent in the Standard Model at tree
level. Moreover, at next-to-leading order they are supressed by the GIM
mechanism. Therefore a detection of such processes would be a strong hint at
new physics. Here we focus Neutral Flavor Change mediated by the Higgs boson
following [@zhang2013top].

The lowest dimensional operators compatible with the symmetries of the Standard
Model are the following six-dimensional operators (for a comprehensive list of
all six-dimensional operators compatible with Standard Model symmetries consult
[@grzadkowski2010dimension]):

- chromomagnetic operator $O_{uG}$

{{{
#!latex
\begin{equation}
\begin{matrix}
O^{1,3}_{uG} = y_t g_s (\bar{q} \sigma^{\mu\nu} T^a t) \bar{\phi} G^a_{\mu\nu}; \\
 \\
O^{3,1}_{uG} = y_t g_s (\bar{Q} \sigma^{\mu\nu} T^a u) \bar{\phi} G^a_{\mu\nu};
\end{matrix}
\end{equation}
}}}

- dimension-six Yukawa interaction $O_{u\phi}$

{{{
#!latex
\begin{equation}
\begin{matrix}
O^{1,3}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{q} t) \bar{\phi}; \\
 \\
O^{3,1}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{Q} u) \bar{\phi};
\end{matrix}
\end{equation}
}}}

- To each (1,3) operator corresponds a (3,1) operator where the flavors are
  reversed.

- To each operator (e.g. (1,3)) corresponds another where the up quark is
  exchanged for a charm quark (e.g. (2,3)).

- The hermitian conjugates of the above-mentioned operators contributing with
  the opposite chirality.

Where we denoted:

- $\phi$ is the Higgs doublet;
- $Q$ and $q$ are respectively the 1st (or 2nd) and the 3th left-handed quark
  doublet;
- $u$ (or $c$) and $t$ are the right-handed quarks;
- $\bar{\phi} = i \sigma^2 \phi$
- $y_t = \sqrt{2}\frac{m_t}{v}$ the top quark Yukawa coupling.

The complete Lagrangian takes the form:

{{{
#!latex
\begin{equation}
\mathcal{L}_{eff} = \mathcal{L}_{SM} + \sum_i \frac{c_i O_i}{\Lambda^2},
\end{equation}
}}}

where $\Lambda$ is the new physics energy scale, $O_i$ is for the various
six-dimensional operators in consideration and  $c_i$ are relative couplings.

The normalizations for the six-dimensional operators were chosen such that for
any new SM-like vertices the ratio of the new couplings to the SM couplings is
of the form $c_i\frac{m_t^2}{\Lambda^2}$. 

=== Implementation and Validation ===

The implementation is a straightforward transcription of the Lagrangian into
`FeynRules` format as no new fields need to be defined. 

The model was validated using the build-in checks in `FeynRules` and
`MadGraph5`. Moreover the decay widths were confirmed through `MadGraph5` and
compared to the analytical results.