= A Kaluza-Klein Gluon Model =

== Authors ==

* Elizabeth Drueke (Michigan State University)
* Joseph Nutter (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)

== 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/pdf/1409.7607v2.pdf 1409.7607v2] analyzes the tc decay as depicted below:
[[Image(wiki:KKg.png)]]
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 vector0like 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
$b \to \gamma$.
}}}
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.

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 and 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
$g_s \left(\cot\omega J^{\mu}_1 - \tan\omega J^{\mu}_2 \right)G^{*}_{\mu}.$
}}}


G*'s form an extended color group and 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 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 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
$b\to s \gamma$. \rm{So } $(D_L)_{23}$
}}}
is thus forced to be small and, as a consequence, 
{{{
#!latex
$(U_L)_{23}\simeq V_{cb}$.
}}}

== Note ==

Need to reread and make sure everything is the same as paper.