== 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.

== Beyond-SM Operators with the Top Quark ==

### Motivation

This model is a reimplementation of the model behind the following paper:
[@frederix2009top]. The paper looks at top pair invariant mass distribution as
a window for new physics by studying the effects that various s-channel
resonance would exert. The original model was implemented in `MadGraph4`. Here
we provide a reimplementation in the `FeynRules`-`MadGraph5` toolset.

The model is not restricted to use only for studying the top pair invariant
mass distribution as will be seen below. For each newly implemented particle we
will discuss how it couples to the Standard Model particles, how the model was
validated against [@frederix2009top] and other previous studies, and whether
there are any constraints on the versions of `FeynRules` and `MadGraph5` to be
used.

In addition, the new model files provide the width of the particles (there is
no need for them to be computed separately). Also, the constraint on the
particle masses were lifted (the previous version provided certain couplings
only for certain mass ranges, and the couplings themselves were expressed only
as series expansions). The new model provides the exact expressions for all
masses.

### Spin Zero, Color Singlet Particle 

The name used in [@frederix2009top] for this resonance is `S0` for "color
[S]inglet, spin [Zero]". It is coupled only to the top with different couplings
for the left and for the right top. The effective vertex of gluon fusion through
a top loop is explicitly given in the Lagrangian as well.

The coupling to the top operator is

{{{
#!latex
\begin{equation}
\mathcal{L}_{S_0 t}\; =\;
c_{s0scalar}\, \frac{m_t}{v} S_0\, \bar{t}.t \;
+ \; i\, c_{s0axial}\, \frac{m_t}{v} S_0\, \bar{t}.\gamma^5.t.
\end{equation}
}}}

The gluon fusion effective operator must be added explicitly because it is a
beyond-tree-level effect. In general, such an operator takes the form

{{{
#!latex
\begin{equation}
\mathcal{L}_{G\,fusion\,scalar\,S_0}\; =\;
-\frac{1}{4} c_{s0fusion\,scalar} S_0 \; FS(G)_{\mu \nu}^a \; FS(G)^{\mu \nu a}
\end{equation}
}}}

or

{{{
#!latex
\begin{equation}
\mathcal{L}_{G\,fusion\,axial\,S_0}\; =\;
-\frac{1}{4} c_{s0fusion\,axial} S_0 \; FS(G)_{\mu \nu}^a \; \widetilde{FS}(G)^{\mu \nu a}
\end{equation}
}}}

where $FS(G)$ is the field strength for the gluon field and ~ denotes a dual field.

By comparing the vertices produces by these operators to the result of the
integrated top loop we get

{{{
#!latex
\begin{equation}
c_{s0fuison\,scalar} = -c_{s0scalar} \frac{g_s^2}{12 \pi^2 v} \;
f_S\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
\end{equation}
\begin{equation}
c_{s0fuison\,axial} = -c_{s0axial} \frac{g_s^2}{8 \pi^2 v} \;
f_A\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
\end{equation}
}}}

with

{{{
#!latex
\begin{equation}
f_S(t) = 
  \begin{cases} 
    \frac{3}{2} t \left(1 + \frac{1}{4} \left(t - 1\right) \left(\log{\left(\frac{\sqrt{1 - t} + 1}{1 - \sqrt{1 - t}}\right)} - i \pi\right)^2\right) & t \leq 1 \\
    \frac{3}{2} t \left(1 + \left(1 - t\right) \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2\right) & 1 \leq t.
  \end{cases}
\end{equation}
}}}

and

{{{
#!latex
\begin{equation}
f_A(t) = 
  \begin{cases} 
    - \frac{t}{4} \left(\log{\left(\frac{\sqrt{1 - t} + 1}{1 - \sqrt{1 - t}}\right)} - i \pi\right)^2 & t \leq 1 \\
    t \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2 & 1 \leq t.
  \end{cases}
\end{equation}
}}}

With appropriate branch cuts in the complex plane these expressions are
actually the same when $\arcsin$ is expressed in terms of $\log$. The
integration of the top loop was verified with the `FeynCalc` package and the
notebook is provided together with the models. 

Finally, given this Lagrangian the width of the new particles is:

{{{
#!latex
\begin{equation} 
W_{S_0}\;=\;
\frac{3 m_t^2  m_{S_0}}{8 \pi v^2} \sqrt{1 - \frac{4 m_t^2}{m_{S_0}^2}} \left(-\frac{4 m_t^2}{m_{S_0}}
c_{s0scalar}^2 + \left(c_{s0axial}^2 +
c_{s0scalar}^2\right)\right)
\end{equation}
}}}

#### Validation of the Model

The first step is to compare the old and the new implementations through the
`standalone` mode. However this is complicated by the fact that certain
parameters in the old model are to be evaluated at each point in phase space,
which the `standalone` mode does not permit. A short patch is provided in the
annex with an explanation of the necessary changes.

After the application of the patch, the model was validated against the old
implementation in `standalone` mode. The decay width and the cross-section in
various processes was validated as well, after taking into account the
differences at runtime between `MadGraph4` and `MadGraph5`.

However the old model is only for heavy $S_0$ particles ($m_{S_0}>2m_t$). The
changes permitting work with light $S_0$ particles:

- correct calculation of the width when decay to top pair is impossible
- correct expression for the effective gluon fusion vertex

were not major and were validated using the build-in tools in `FeynRules` and
`MadGraph5`. Moreover studies for such light particles are probably of minor
interest.

### Spin Zero, Color Octet Particle 

The name for this resonance is `O0` for "color
[O]ctet, spin [Zero]". Like `S0` it is coupled only to the top with different couplings
for the left and for the right top and there is an effective vertex of gluon fusion through
a top loop is explicitly given in the Lagrangian as well.

The operators are:

{{{
#!latex
\begin{equation}
\mathcal{L}_{O_0 t}\; =\;
c_{o0scalar}\, \frac{m_t}{v} O_0^a\, \bar{t}.T^a.t \;
+ \; i\, c_{o0axial}\, \frac{m_t}{v} O_0^a\, \bar{t}.\gamma^5.T^a.t.
\end{equation}

\begin{equation}
\mathcal{L}_{G\,fusion\,scalar\,O_0}\; =\;
-\frac{1}{4} c_{o0fusion\,scalar} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; FS(G)^{\mu \nu c}
\end{equation}

\begin{equation}
\mathcal{L}_{G\,fusion\,axial\,O_0}\; =\;
-\frac{1}{4} c_{o0fusion\,axial} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; \widetilde{FS}(G)^{\mu \nu c}
\end{equation}
}}}

where $S_{SU3}^{abc}$ is the completely symmetric tensor and where
$c_{o0fusion\,scalar}$ and $c_{o0fusion\,axial}$ are the same as for `S0` with
coupling and masses appropriately substituted.

Again, given this Lagrangian the width of the new particles is:

{{{
#!latex
\begin{equation} 
W_{O_0}\;=\;
\frac{m_t^2  m_{O_0}}{16 \pi v^2} \sqrt{1 - \frac{4 m_t^2}{m_{O_0}^2}} \left(-\frac{4 m_t^2}{m_{O_0}}
c_{o0scalar}^2 + \left(c_{o0axial}^2 +
c_{o0scalar}^2\right)\right)
\end{equation}
}}}

which is $\frac{1}{6}$ times the expression for $W_{S_0}$ with appropriately
substituted couplings and masses.

#### Validation of the Model

As with `S0` a patch is necessary before one can proceed with validation in the
`standalone` mode. The model was validated against the old implementation in
that mode, as well as in `MadEvent` mode: both the decay width and the
cross-sections of various processes were checked.

The new model permits the use of light `O0` unlike the old implementation for
`MadGraph4`. As in the case of `S0` this part was validated only through the
build-in tools in `FeynRules` and `MadGraph5`.

### Spin One, Color Singlet Particle

The name for this resonance is `S1`. It has both vector and axial couplings to
all quarks and leptons. It is used mostly for a "model-independent" vector
boson ($Z^\prime$). For convenience the Lagrangian has exactly the same form as
the part of the Standard Model Lagrangian that governs the coupling of the SM Z
to the fermions. In addition to that each coupling is parametrized by coupling
constant with default value of 1.

- `s1uleft` for the coupling to up, charm and top left quarks;
- `s1dleft` for the coupling to down, strange and bottom left quarks;
- `s1uright` and `s1dright` for the corresponding right quarks;
- `s1eleft` for the left electron, muon and tau-lepton;
- `s1eright` for the right charged leptons;
- `s1nu` for the neutrinos.

For example the coupling to neutrinos is

{{{
#!latex
\begin{equation}
\mathcal{L}_{S_{1}\nu}\;=\;
c_{s1nu}\;
\frac{e}{2\sin{\theta_W}\cos{\theta_W}}\; S_1^\mu\;
\underset{f=e,mu,tau}{\sum}\bar{L}_2^f.\gamma_\mu.L_2^f
\end{equation}
}}}

where $\theta_W$ is
the Weinberg angle, $e$ is the electric coupling constant, $L$ is the leptonic
doublet and $L_2$ is its second component.

The width of the particle is calculated and provided in the model as well.

#### Validation of the Model

Besides the basic correctness tests provided by `FeynRules` and `MadGraph5` the
`S1` model was verified against the original `MadGraph4` model.

In `standalone` mode both models produce the same differential cross-section
withing machine precision. In `MadEvent` mode the decay width is the same in
both cases. When accounting for the differences at runtime in `MadGraph4` and
`MadGraph5` the cross sections of the various tested processes are the same as
well.

### Spin One, Color Octet Particle

The name for this resonance is `O1`. The need for a `FeynRules` version of it
is what originally caused the request for reimplementation of the whole model.
This field lives in the same representation of the gauge group as the gluons.
It is used to represent color vector particle (coloron) or an color axial
particle (axigluon).

The Lagrangian is of the form

{{{
#!latex
\begin{equation}
\mathcal{L}_{O_1}\; =\;
\sum_i c_i g_s O_1^{\mu a} \; \bar{q_i}.\gamma_\mu.T^a.q_i
\end{equation}
}}}
where $i$ goes over right and left handedness of the up and down quarks of each
generation. $T$ is the representation of the SU3 group generators and $g_s$ is
the strong coupling constant.

The width of the particles is calculated and provided in the model as well.

#### Validation of the Model

Similar models are discussed in [@choudhury2007top] and [@antunano2008top].
Their results confirm both the width and the differential cross-section
calculated in the `FeynRules` model.

Another `FeynRules` model is available that implements axigluons in
[@falkowski2012axigluon]. It produces the same vertices, however it differs in
that it provides for a mixing between the axigluons and the gluons.

The original `MadGraph4` model gives the same results in the `standalone`
configuration. Both the decay width and the cross section of top pair
production were checked as well. Well accounting for the differences in the
`MadGraph4` and `MadGraph5` runtime they produce the same results. Details are
provided in the annex.


#### Technical Constraints

During the implementation of this model a bug in the canonicalization routines
of `FeynRules` was encountered. Whenever a tensor contraction expression is
passes through `FeynRules` it needs to get into a canonical form (in order to
permit equality checks, pattern matching and simplifications) before the
canonical quantization is executed. The symmetric tensor for the SU3 group was
not taken into account in this canonicalization. Benjamin Fuks graciously and
quickly fixed the issue, however for the model to work correctly at least
`FeynRules 1.7.178` or later is necessary.

## General Technical Constraints

### Required Versions

As was mentioned above, the minimal version of `FeynRules` in which the models
are guaranteed to work is `1.7.178`.

Moreover, there is a disaccord between the formats for saving models in the
current versions of `MadGraph5` and `FeynRules`. It should be fixed in the next
versions, however if a runtime error message concerning undefined Goldstone
bosons is raised by `MadGraph` it can be quickly fixed by manually modifying
the offending lines in `particles.py`. It can be done automatically with the
following command:

`perl -pi -e 's/goldstone/GoldstoneBoson/g' ./models/topBSM_UFO/particles.py`.

### Setting Mass Ranges

The calculation of the widths of different particles (especially `S0` and `O0`)
as well as the effective couplings for gluon fusion vertices changes
qualitatively if the mass of the particle passes over or under two times the mass
of the top. This is implemented in `FeynRules` with a delayed rewrite rule,
however `MadGraph5` does not permit such branching. Hence if the need arises to
change the mass of these particles it is important to change it from
`FeynRules` and not from `MadGraph5`.

# Annex

## Patching the `standalone` Mode

In `standalone` mode couplings are evaluated only once, before generating a
random phase space point at which to evaluate the matrix element. This does not
permit testing some of the more complicated models like the original
implementation of the `S0` and `O0` particles. 

As a workaround for this issue, one can modify the code so that `setparam` is
called after each generation of random phase space points. A patch that does
this automatically is provided with the models.