Standard Model + an array of color-sextet fields

This page documents the implementation in FeynRules of a minimal addition to the Standard Model featuring complex scalars and Dirac fermions in the six-dimensional (sextet, 6) representation of SU(3).
The defining characteristic of this model is the coupling of these sextets to a gluon and some quark, which is possible because the direct product of the fundamental (3) and adjoint (8) representations of SU(3) contains a sextet:

\begin{align*}
\boldsymbol{3} \otimes \boldsymbol{8} = \boldsymbol{3} \oplus \boldsymbol{\bar{6}} \oplus \boldsymbol{15}.
\end{align*}

This is the first public implementation of this color structure in FeynRules, but despite its novelty it is achievable using existing color-algebraic objects and requires no hard coding.

Contact information

Taylor Murphy
murphy.1573@…

The Ohio State University
Department of Physics

Based on ​a paper written in collaboration with L. M. Carpenter and T. M. P. Tait.

Model description and FeynRules implementation

Color-sextet fields are highly relevant to the ongoing LHC program, as they can be (at minimum) copiously pair-produced in proton-proton collisions and potentially enjoy a range of couplings to SM fields that can result in interesting collider signatures.
One family of possible couplings exists between color sextets and a quark plus a gluon.
These couplings are implied in principle by the existence of an invariant combination of the 3, 6, and 8 of SU(3). A FeynRules implementation of such a contraction, however, requires explicit knowledge of the associated generalized Clebsch-Gordan coefficients

\begin{align*}
\mathrm{\textsf{\textsl{J}}}^{\,s\, ia}\ \ \ \text{with}\ \ \ s \in \{1,\dots,6\},\ i \in \{1,2,3\},\ a \in \{1,\dots,8\},
\end{align*}

a set of six 3 x 8 matrices. Essential properties of these Clebsch-Gordan coefficients and their explicit form (in the basis where the generators of the fundamental representation of SU(3) are half the Gell-Mann matrices) are provided in Appendix A of the accompanying paper.

FeynRules has previously been extended to include analogous group-theoretical objects corresponding to the two possible contractions of two SU(3) fundamentals; i.e., the totally antisymmetric contraction with a third fundamental and the symmetric contraction with a sextet.
These color structures have been explored ​in the literature, and their implementations in FeynRules are documented ​here and ​here on the Model Database. We summarize the existing Clebsch-Gordan coefficients below.

\begin{center}
\begin{tabular}{c || c| c| c}
SU(3) invariant & Notation & Properties & \textsf{FeynRules} syntax\\[0.83ex]
\hline
\hline
\rule{0pt}{3.5ex}$\boldsymbol{3} \otimes \boldsymbol{3} \otimes \boldsymbol{3}$ & $\mathrm{\textsf{\textsl{L}}}^{ijk}$ & $= \frac{1}{\sqrt{2}}\,\epsilon^{ijk}$ & \texttt{K3[i,j,k]}\\[0.83ex]
\hline
\rule{0pt}{3.5ex}$\boldsymbol{3} \otimes \boldsymbol{3} \otimes \boldsymbol{\bar{6}}$ & $\mathrm{\textsf{\textsl{K}}}_s^{\ \,ij}$ & $i \leftrightarrow j$ symmetric & \texttt{K6[s,i,j]}\\[0.83ex]
\end{tabular}
\end{center}

These coefficients are useful because they can be contracted in a specific way with the generators of the fundamental representation of SU(3) to produce the novel coefficients we seek. The specific relations we exploit are

\begin{align*}
    \mathrm{\textsf{\textsl{J}}}^{\,s\, ia} = -\mathrm{i} \sqrt{2}\, \mathrm{\textsf{\textsl{L}}}^{ijk}\, [\mathrm{\textsf{\textsl{t}}}_{\boldsymbol{3}}^a]_j^{\ \,l} \bar{\mathrm{\textsf{\textsl{K}}}}{}^s_{\ \,lk}\ \ \ \text{and}\ \ \ \bar{\mathrm{\textsf{\textsl{J}}}}{}_{s\, ai} = \mathrm{i} \sqrt{2}\, \mathrm{\textsf{\textsl{K}}}_s^{\ \,kl}\,[\mathrm{\textsf{\textsl{t}}}_{\boldsymbol{3}}^a]_l^{\ \, j} \bar{\mathrm{\textsf{\textsl{L}}}}_{\,ijk},
\end{align*}

where bars over Clebsch-Gordan coefficients denote Hermitian conjugation.
Implementing the novel color invariant in this way produces a valid FeynRules model, and can furthermore produce a Universal FeynRules Output (UFO) module that, once lightly modified, can be used as input for e.g. MadGraph5_aMC@NLO.

It is straightforward to show that there are no renormalizable operators with this color structure if the color triplet and octet are SM fields.
We therefore implement a small selection of higher-dimensional (effective) operators coupling color-sextet fermions and scalars to a gluon, an up- or down-type quark, and possibly other fields pursuant to Lorentz invariance. These operators are described below.

Sextet fermions

The fermion sector of this model implementation has the following Lagrange density:

\begin{multline*}
    \mathcal{L} \supset \bar{\Psi}_q(\mathrm{i} D_{\mu}\gamma^{\mu} - m_{\Psi_q})\Psi_q\\ + \frac{1}{\Lambda_{\Psi_q}}\,[\kappa_q^I\,\mathrm{\textsf{\textsl{J}}}^{\,s\,ia}\,(\,{\mkern 2mu\overline{\mkern-4mu q^{\text{c}}_{\text{R}}\mkern-4mu}\mkern 2mu}_{Ii}\,\sigma^{\mu\nu}\Psi_{qs})\, G_{\mu\nu\,a} + \text{H.c.}]\\+ \frac{1}{\Lambda_{\Psi_{qB}}^3}\,[\kappa_{qB}^I\,\mathrm{\textsf{\textsl{J}}}^{\,s\,ia}\,(\,{\mkern 2mu\overline{\mkern-4mu q^{\text{c}}_{\text{R}}\mkern-4mu}\mkern 2mu}_{Ii}\,\Psi_{qs})\, B^{\mu\nu}\,G_{\mu\nu\,a} + \text{H.c.}].
\end{multline*}

The exotic fields can couple either to up-type quarks (q = u) or to down-type quarks (q = d).
The sextet weak hypercharge differs in each case in order to preserve SM U(1) invariance.
We allow for different couplings and effective cutoffs for each operator (and for each fermion-quark combination, with I being a quark flavor index).
The third line contains an interesting operator coupling sextets to a quark, a gluon, and a photon or Z boson.

The couplings in these operators are implemented as follows in the model file:

\begin{center}
\begin{tabular}{c || c| c}
Parameter & Description & \textsf{FeynRules} syntax\\[0.83ex]
\hline
\hline
\rule{0pt}{3.5ex}$\kappa_u^I\,\Lambda_{\Psi_u}^{-1}$ & Up-type quark coupling & \texttt{CFu[I]}\\[0.83ex]
\hline
\rule{0pt}{3.5ex}$\kappa_d^I\,\Lambda_{\Psi_d}^{-1}$ & Down-type quark coupling & \texttt{CFd[I]}\\[0.83ex]
\hline
\rule{0pt}{3.5ex}$\kappa_{uB}^I\,\Lambda_{\Psi_{uB}}^{-3}\,$ & Up-type quark + $B$ coupling & \texttt{CFBu[I]}\\[0.83ex]
\hline
\rule{0pt}{3.5ex}$\kappa_{dB}^I\,\Lambda_{\Psi_{dB}}^{-3}\,$ & Down-type quark + $B$ coupling & \texttt{CFBd[I]}\\[0.83ex]
\end{tabular}
\end{center}

Finally, it is worth noting that the Hermitian-conjugate operators are written explicitly in the model file in order to preserve correct color flow.

Sextet scalars

The scalar sector of this model implementation has the following Lagrange density:

\begin{multline*}
    \mathcal{L} \supset (D_{\mu}\Phi_q)^{\dagger}D^{\mu}\Phi_q - m_{\Phi_q}^2 \Phi_q^{\dagger}\Phi_q\\ + \frac{1}{\Lambda^2_{\Phi_q}}\,[\lambda^{XI}_q \mathrm{\textsf{\textsl{J}}}^{\,s\, ia}\,\Phi_{qs}\,(\,{\mkern 2mu\overline{\mkern-4mu q^{\text{c}}_{\text{R}}\mkern-4mu}\mkern 2mu}_{Ii}\,\sigma^{\mu\nu}\ell_{\text{R}X})\, G_{\mu\nu\,a} + \text{H.c.}].
\end{multline*}

We again have scalars coupling to both up- and down-type quarks.
In this case, simultaneous Lorentz and gauge invariance requires a SM lepton (in the absence of other exotic fields).
For full generality, these couplings are therefore in both quark and lepton generation space.

The couplings in these operators are implemented as follows in the model file:

\begin{center}
\begin{tabular}{c || c| c}
Parameter & Description & \textsf{FeynRules} syntax\\[0.83ex]
\hline
\hline
\rule{0pt}{3.5ex}$\lambda_u^{XI}\,\Lambda_{\Phi_u}^{-2}$ & Up-type quark + lepton coupling & \texttt{CSu[X,I]}\\[0.83ex]
\hline
\rule{0pt}{3.5ex}$\lambda_d^{XI}\,\Lambda_{\Phi_d}^{-2}$ & Down-type quark + lepton coupling & \texttt{CSd[X,I]}\\[0.83ex]
\end{tabular}
\end{center}

Hermitian conjugation is again made explicit in the model file.

Model Files

These files are confirmed to work with the indicated software as of October 22, 2021.

• FeynRules model file; tested with the FeynRules ​SM implementation against FeynRules version 2.3.43
• Example notebook; tested against Wolfram Mathematica version 12.0
• UFO module; tested against MadGraph5_aMC@NLO version 3.2.0

We reiterate that the attached UFO has been modified after being generated by FeynRules so that certain color structures are understood correctly by MG5_aMC.
The specific modification involves replacing some totally antisymmetric symbols by their conjugates ("bars") to enable the correct flow of SU(3) fundamental/antifundamental indices.