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 $O_{uG}$

\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 $O_{u\phi}$

\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:

\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 $c_i\frac{m_t^2}{\Lambda2}$.

### 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

\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

\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

\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

\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

\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

\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:

\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:

\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:

\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

\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

\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.