= A Kaluza-Klein Gluon Model =

== Corresponding Authors ==

* Elizabeth Drueke (Michigan State University)
* Reinhard Schwienhorst (Michigan State University)
* Natascia Vignaroli (Michigan State University)

== Other Contributors ==

* Joseph Nutter (Michigan State University)
* Devin G. E. Walker (SLAC National Accelerator Laboratory)
* Jiang-Hao Yu (The University of Texas at Austin)
* R. Sekhar Chivukula (Michigan State University)
* Elizabeth H. Simmons (Michigan State University)

== Description of the Model ==

Colored vector bosons from new strong dynamics, Kaluza-Klein gluons or KKg’s (G*) in a dual 5D picture, have been searched for mainly in the t-tbar channel.  The analysis in [http://arxiv.org/abs/1409.7607v2 1409.7607v2] analyzes the tc decay as depicted below:
[[Image(KKg.png)]]
The benchmark adopted here is a simple renormalizable model of an
extended color gauge sector, which realizes next-to-minimal flavor violation (NMFV). In this model, the third generation quarks couple differently than the light quarks under an extended 
{{{
#!latex
$SU(3)_1 \times SU(3)_2$
}}}
color gauge group.  The mixing between light and third generation quarks is induced by the interactions of all three generation quarks with a set of new heavy vector-like quarks.  The model reproduces the CKM mixing and generates flavor-changing neutral currents (FCNCs) from non-standard interactions.  Due to the specific structure of the model, dangerous FCNCs are naturally suppressed and a large portion of the model parameter space is allowed by the data on meson mixing process and on 
{{{
#!latex
$b \to s\gamma$.
}}}
The model has the color gauge structure
{{{
#!latex
$SU(3)_1 \times SU(3)_2$
}}}
The extended color symmetry is broken down to 
{{{
#!latex
$SU(3)_C$
}}}
by the (diagonal) expectation value,
{{{
#!latex
$\langle \Phi \rangle \propto u \cdot {\cal I}$,
}}}
of a scalar field Phi which transforms as a 
{{{
#!latex
$(\bf 3, \bar{3})$
}}}
under the color gauge structure.  It is assumed that color gauge breaking occurs at a scale much higher than the electroweak scale, u>>v.

Breaking the color symmetry induces a mixing between the 
{{{
#!latex
$SU(3)_1$ \rm{and} $SU(3)_2$
}}}
gauge fields
{{{
#!latex
$A^{1}_{\mu}$ \rm{and} $A^{2}_{\mu}$,
}}}
which is diagonalized by a rotation determined by 
{{{
#!latex
$\cot\omega = \frac{g_1}{g_2} \qquad g_s = g_1 \sin\omega = g_2 \cos\omega$,
}}}
where g_s is the QCD strong coupling and g_1, g_2 are the SU(3)_1 and SU(3)_2 gauge couplings, respectively.  The mixing diagonalization reveals two color vector boson mass eigenstates: the mass-less SM gluon and a new massive color-octet vector boson G* given by
{{{
#!latex
$G^{*}_{\mu}=\cos\omega A^{1}_{\mu} - \sin\omega A^{2}_{\mu} \qquad M_{G^{*}} = \frac{g_s u}{\sin\omega \cos\omega}.$
}}}
In the NMFV model, the third generation quarks couple differently than the light quarks under the extended color group.  
{{{
#!latex
$g_L=(t_L, b_L),$ \rm{ } $t_R,$ \rm{ and } $b_R,$
}}}
as well as a new weak-doublet of vector-like quarks, transform as 
{{{
#!latex
$({\bf 3,1})$
}}}
under the color gauge group, while the light generation quarks are charged under SU(3)_2 and transform as
{{{
#!latex
$({\bf 1,3})$
}}}
The G* interactions with the color currents associated with SU(3)_1 and SU(3)_2 are given by
{{{
#!latex
$g_s \left(\cot\omega J^{\mu}_1 - \tan\omega J^{\mu}_2 \right)G^{*}_{\mu}.$
}}}


The G* can be produced at the LHC by quark-antiquark fusion determined by the G* coupling to light quarks 
{{{
#!latex
$g_s \tan\omega$
}}}
Gluon-gluon fusion production is forbidden at tree level by SU(3)_C gauge invariance. 

The G* decay widths are:
{{{
#!latex
$\Gamma[G^{*} \to t\bar t] = \frac{g^2_s}{24\pi} M_{G^{*}}\cot^2\omega \sqrt{1-4 \frac{m^2_t}{M^2_{G^{*}}}} (1+2\frac{m^2_t}{M^2_{G^{*}}}),$ \newline
$\Gamma[G^{*} \to b\bar b] = \frac{g^2_s}{24\pi} M_{G^{*}}\cot^2\omega,$ \newline
$\Gamma[G^{*} \to j j] = \frac{g^2_s}{6\pi} M_{G^{*}}\tan^2\omega.$
}}}
Additionally, the NMFV flavor structure of the model generates a G* to tc flavor violating decay with rate
{{{
#!latex
$\Gamma[G^{*} \to t_L \bar c_L]=\Gamma[G^{*} \to c_L \bar t_L]\simeq \left(V_{cb}\right)^2 \frac{g^2_s}{48\pi} M_{G^{*}} \left( \cot\omega+\tan\omega \right)^2,$
}}}
where 
{{{
#!latex
$V_{cb}=0.0415$
}}}
is the CKM matrix element. Note here that G* FCNCs are induced by the mixing among left-handed quarks generated by the exchange of heavy vector-like quarks. This mixing is controlled by the 3x3 matrices U_L and D_L in the up- and down-quark sectors, respectively. In particular, the 
{{{
#!latex
$G* \to tc$
}}}
flavor violating decay is controlled by the 
{{{
#!latex
$(U_L)_{23}$
}}}
element. The CKM mixing matrix is given by 
{{{
#!latex
$V_{CKM}=U^{\dagger}_L D_L$.
}}}
At first order in the mixing parameters, 
{{{
#!latex
$(U_L)_{23}\equiv V_{cb} - (D_L)_{23}$.
}}}
The non-diagonal elements of D_L are strongly constrained by the data on 
{{{
#!latex
$b\to s \gamma$. 
}}}
So
{{{
#!latex
$(D_L)_{23}$
}}}
is thus forced to be small and, as a consequence, 
{{{
#!latex
$(U_L)_{23}\simeq V_{cb}$.
}}}
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 KKg (place in Cards/)
* [attachment:run_card.dat run_card]: for generation of 500 GeV KKg (place in Cards/)
* [attachment:kkg_FV.zip kkg_FV]: the model and parameter cards for specific mass generations

== Generation specifics ==
   
In [http://arxiv.org/abs/1409.7607v2 1409.7607v2], the samples were generated with the KKg 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 file for 500 GeV mass included above.  The specific generations run were 
{{{
p p > kkg > b~ c l- vl~ @1
p p > kkg > b c~ l+ vl @2
}}}
To generate the settings for a specific KKg mass, param_card.dat in the generation directory to the card of the appropriate mass in the param_cards directory (included as part of the model zip file).