= A Coloron Model =

== Corresponding Authors ==

* Elizabeth Drueke (Michigan State University)
* Reinhard Schwienhorst (Michigan State University)
* Natascia Vignaroli (Michigan State University)
* Devin G. E. Walker (SLAC National Accelerator Laboratory)
* Jiang-Hao Yu (The University of Texas at Austin)

== Other Contributors ==

* Joseph Nutter (Michigan State University)
* R. Sekhar Chivukula (Michigan State University)
* Elizabeth H. Simmons (Michigan State University)

== Description of the Model ==

The 
{{{
#!latex
$SU(3)_1 \times SU(3)_2 \to SU(3)_C$
}}}
breaking induced by the expectation value of the 
{{{
#!latex
({$\bf 3,\bar{ 3}$})
}}}
scalar field Phi generates color-octet and color-singlet scalars. The most general renormalizable potential for Phi is:
{{{
#!latex
$V(\Phi)=-m^2_{\Phi}\text{Tr}(\Phi\Phi^\dagger) -\mu (\text{det }\Phi+\text{H.c.})+\frac{\xi}{2}\left[ \text{Tr}(\Phi\Phi^\dagger) \right]^2+\frac{k}{2}\text{Tr}(\Phi\Phi^\dagger\Phi\Phi^\dagger) \ ,$
}}}
where 
{{{
#!latex
$\text{det } \Phi = \frac{1}{6}\epsilon^{ijk}\epsilon^{i'j'k'}\Phi_{ii'}\Phi_{jj'}\Phi_{kk'} \ ,$
}}}
and where, without loss of generality, one can choose mu > 0. Assuming 
{{{
#!latex
$m^2_\Phi >0$,
}}}
Phi acquires a (positive) diagonal expectation value:
{{{
#!latex
$\langle \Phi \rangle = u \cdot \mathcal{I} \,.$
}}}
The Phi expansion around the vacuum gives:
{{{
#!latex
$\Phi=u+\frac{1}{\sqrt{6}}\left(\phi_R+i\phi_I\right)+\left(G^a_H+iG^a_G\right)T^a \ ,$
}}}
where 
{{{
#!latex
$\phi_R$, $\phi_I$
}}}
are singlets under SU(3)_C Additionally, 
{{{
#!latex
$G^a_G$, $a=1,\dots,8$, 
}}}
are the Nambu-Goldstone bosons associated with the color-symmetry breaking, and
{{{
#!latex
$G^a_H$
}}}
are color octets. 

GH can be produced in pairs through its interactions with gluons:
{{{
#!latex
$\frac{g^2_s}{2}f^{abc}f^{ade}G^b_{\mu}G^{\mu d}G^c_H G^e_H +g_s f^{abc} G^a_{\mu} G^b_H \partial^{\mu} G^c_H \ ,$
}}}
or it can be produced singly via gluon-gluon fusion. This occurs at one-loop order through the cubic interaction
{{{
#!latex
$\frac{\mu}{6} d_{abc} G^a_H G^b_H G^c_H   \,,$
}}}
which arises from the 
{{{
#!latex
$\mu(\det\Phi+\text{H.c.})$
}}}
term in the potential; where 
{{{
#!latex
$d_{abc}$
}}}
is the SU(3) totally symmetric tensor. The single production of GH can be described by the effective coupling
{{{
#!latex
$-\frac{1}{4} C_{ggG} d_{abc} G^a_{\mu\nu} G^{\mu\nu b} G^c_H$
}}}
with
{{{
#!latex
$C_{ggG}=\sqrt{\frac{1}{6}}\frac{\alpha_s}{\pi }\frac{\mu}{M^2_{G_H}}\left(\frac{\pi^2}{9}-1\right) \ .$
}}}
Note that single production is suppressed by a factor 
{{{
#!latex
$(\pi^2/9 -1)^2$,
}}}
which is an accidental suppression factor coming from the loop.
Above the threshold for decays into a single top quark, GH has two main decay modes: the decay into gluons, which occurs at loop-level similar to single coloron production, and the flavor-violating decay into tc. The corresponding rates are:
{{{
#!latex
$\Gamma \left[G_H \to (\bar{c}_L t_R +\bar{t}_R c_L )\right] =\left(V_{cb}\right)^2 \frac{M_{G_H}}{16 \pi} \frac{m^2_t}{u^2}\left(1-\frac{m^2_t}{M^2_{G_H}}\right)^2 \,, $ \newline
$\Gamma \left[G_H \to gg \right]=\frac{5 \alpha^2_s}{1536 \pi^3}\frac{\mu^2}{M_{G_H}}\left(\frac{\pi^2}{9}-1\right)^2 \,.$
}}}
We set u=mu (the stability of the potential forbids mu>u); and consider for simplicity the set of 
{{{
#!latex
$(M_{G_H}, \mu)$ 
}}}
values that give a 50% GH decay into tc and 50% into gg. GH is a very narrow resonance, with a width of the order of 10^-4 GeV. 

Various Feynman Diagrams for GH processes discussed in [http://arxiv.org/abs/1409.7607v2 1409.7607v2] are shown below:

[[Image(Coloron.png)]]

[[Image(Colorong.png)]] [[Image(Colorong2.png)]]

[[Image(ColoronDouble1.png)]] [[Image(ColoronDouble2.png)]]
 
See more details in

* [http://arxiv.org/abs/1409.7607v2 1409.7607v2]
* [http://arxiv.org/abs/1412.3094 1412.3094]

== Model Files ==

* [attachment:proc_card_mg5.dat proc_card]: for generation of 500 GeV coloron (place in Cards/)
* [attachment:run_card.dat run_card]: for generation of 500 GeV coloron (place in Cards/)
* [attachment:Octet-tcgg-new.zip Octet-tcgg]: the model

== Generation specifics ==
   
In [http://arxiv.org/abs/1409.7607v2 1409.7607v2], the samples were generated with the coloron mass as the scale, dsqrt_q2fact1, and dsqrt_q2fact2 in the run_card.dat file.  These samples were also generated without !MadGraph cuts as demonstrated in the run_card.dat for 500 GeV mass included above.  The specific generations run were 
{{{
p p > GH, GH > b c~ l+ vl @1 GHT=1 QED=2
p p > GH, GH > b~ c l- vl~ @2 GHT=1 QED=2
}}}
To generate the settings for a specific coloron mass, use the appropriate model directory contained in the Octet-tcgg zip file.