Changes between Version 1 and Version 2 of 368sextets


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Timestamp:
Oct 21, 2021, 7:02:12 PM (3 years ago)
Author:
TaylorMurphy
Comment:

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  • 368sextets

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    1 ...
     1= Standard Model + an array of color-sextet fields =
     2
     3This 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).[[BR]]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:
     4{{{
     5#!latex
     6\begin{align*}
     7\boldsymbol{3} \otimes \boldsymbol{8} = \boldsymbol{3} \oplus \boldsymbol{\bar{6}} \oplus \boldsymbol{15}.
     8\end{align*}
     9}}}
     10This 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.
     11
     12
     13== Contact information
     14
     15Taylor Murphy[[BR]]murphy.1573@osu.edu
     16
     17//The Ohio State University//[[BR]]//Department of Physics//
     18
     19Based on [https://arxiv.org/abs/1911.12883 a paper] written in collaboration with L. M. Carpenter and T. M. P. Tait.
     20
     21== Model description and FeynRules implementation
     22
     23Color-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.[[BR]]One family of possible couplings exists between color sextets and a quark plus a gluon.[[BR]]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//
     24{{{
     25#!latex
     26\begin{align*}
     27\mathrm{\textsf{\textsl{J}}}^{\,s\, ia}\ \ \ \text{with}\ \ \ s \in \{1,\dots,6\},\ i \in \{1,2,3\},\ a \in \{1,\dots,8\},
     28\end{align*}
     29}}}
     30a 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.
     31
     32!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.[[BR]]These color structures have been explored [[https://arxiv.org/abs/0909.2666|in the literature]], and their implementations in !FeynRules are documented [[https://feynrules.irmp.ucl.ac.be/wiki/Triplets|here]] and [[https://feynrules.irmp.ucl.ac.be/wiki/Sextets|here]] on the Model Database. We summarize the existing Clebsch-Gordan coefficients below.
     33{{{
     34#!latex
     35\begin{center}
     36\begin{tabular}{c || c| c| c}
     37SU(3) invariant & Notation & Properties & \textsf{FeynRules} syntax\\[0.83ex]
     38\hline
     39\hline
     40\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]
     41\hline
     42\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]
     43\end{tabular}
     44\end{center}
     45}}}
     46These 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
     47{{{
     48#!latex
     49\begin{align*}
     50    \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},
     51\end{align*}
     52}}}
     53where bars over Clebsch-Gordan coefficients denote Hermitian conjugation.[[BR]]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.
     54
     55It is straightforward to show that there are no renormalizable operators with this color structure if the color triplet and octet are SM fields.[[BR]]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.
     56
     57=== Sextet fermions ===
     58
     59The fermion sector of this model implementation has the following Lagrange density:
     60{{{
     61#!latex
     62\begin{multline*}
     63    \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}\,\Phi_{qs})\, B^{\mu\nu}\,G_{\mu\nu\,a} + \text{H.c.}].
     64\end{multline*}
     65}}}
     66The exotic fields can couple either to up-type quarks (//q// = //u//) or to down-type quarks (//q// = //d//). We allow for different couplings and effective cutoffs for each operator (and for each fermion-quark combination, with //I// being a quark flavor index).[[BR]]The third line contains an interesting operator coupling sextets to a quark, a gluon, and a photon or //Z// boson.
     67
     68The couplings in these operators are implemented as follows in the model file:
     69{{{
     70#!latex
     71\begin{center}
     72\begin{tabular}{c || c| c}
     73Parameter & Description & \textsf{FeynRules} syntax\\[0.83ex]
     74\hline
     75\hline
     76\rule{0pt}{3.5ex}$\kappa_u^I\,\Lambda_{\Psi_u}^{-1}$ & Up-type quark coupling & \texttt{CFu[I]}\\[0.83ex]
     77\hline
     78\rule{0pt}{3.5ex}$\kappa_d^I\,\Lambda_{\Psi_d}^{-1}$ & Down-type quark coupling & \texttt{CFd[I]}\\[0.83ex]
     79\hline
     80\rule{0pt}{3.5ex}$\kappa_{uB}^I\,\Lambda_{\Psi_{uB}}^{-3}\,$ & Up-type quark + $B$ coupling & \texttt{CFBu[I]}\\[0.83ex]
     81\hline
     82\rule{0pt}{3.5ex}$\kappa_{dB}^I\,\Lambda_{\Psi_{dB}}^{-3}\,$ & Down-type quark + $B$ coupling & \texttt{CFBd[I]}\\[0.83ex]
     83\end{tabular}
     84\end{center}
     85}}}
     86Finally, it is worth noting that the Hermitian-conjugate operators are written explicitly in the model file in order to preserve correct color flow.
     87
     88=== Sextet scalars ===
     89
     90The scalar sector of this model implementation has the following Lagrange density:
     91{{{
     92#!latex
     93\begin{multline*}
     94    \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.}].
     95\end{multline*}
     96}}}
     97In this case, simultaneous Lorentz and gauge invariance requires a SM lepton (in the absence of other exotic fields).\\
     98For full generality, these couplings are therefore in both quark and lepton generation space.
     99
     100The couplings in these operators are implemented as follows in the model file:
     101{{{
     102#!latex
     103\begin{center}
     104\begin{tabular}{c || c| c}
     105Parameter & Description & \textsf{FeynRules} syntax\\[0.83ex]
     106\hline
     107\hline
     108\rule{0pt}{3.5ex}$\lambda_u^{XI}\,\Lambda_{\Phi_u}^{-2}$ & Up-type quark + lepton coupling & \texttt{CSu[I]}\\[0.83ex]
     109\hline
     110\rule{0pt}{3.5ex}$\lambda_d^{XI}\,\Lambda_{\Phi_d}^{-2}$ & Down-type quark + lepton coupling & \texttt{CSd[I]}\\[0.83ex]
     111\end{tabular}
     112\end{center}
     113}}}
     114
     115Hermitian conjugation is again made explicit in the model file.
     116
     117== Model Files
     118
     119These files are confirmed to work with the indicated software as of October 22, 2021.
     120
     121* !FeynRules model file; tested against !FeynRules version 2.3.43
     122* Example //Mathematica// notebook; tested against Wolfram //Mathematica// version 12.0
     123* UFO module; tested against MadGraph5_aMC@NLO version 3.2.0
     124
     125We reiterate that the attached UFO has been modified after being generated by FeynRules so that certain color structures are understood correctly by MG5_aMC.[[BR]]The specific modification involves replacing some totally antisymmetric symbols by their conjugates ("bars") to enable the correct flow of SU(3) fundamental/antifundamental indices.