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Technicolor

MSEL = 50, 51
ISUB =
149 $\mathrm{g}\mathrm{g}\to \eta_{\mathrm{tc}}$ (obsolete)
191 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \rho_{\mathrm{tc}}^0$ (obsolete)
192 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \rho_{\mathrm{tc}}^+$ (obsolete)
193 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \omega_{\mathrm{tc}}^0$ (obsolete)
194 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{f}_k \overline{\mathrm{f}}_k$
195 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{f}_k \overline{\mathrm{f}}_l$
361 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{W}^+_{\mathrm{L}} \mathrm{W}^-_{\mathrm{L}} $
362 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{W}^{\pm}_{\mathrm{L}} \pi^{\mp}_{\mathrm{tc}}$
363 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \pi^+_{\mathrm{tc}} \pi^-_{\mathrm{tc}}$
364 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma \pi^0_{\mathrm{tc}} $
365 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \gamma {\pi'}^0_{\mathrm{tc}} $
366 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{Z}^0 \pi^0_{\mathrm{tc}} $
367 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{Z}^0 {\pi'}^0_{\mathrm{tc}} $
368 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{W}^{\pm} \pi^{\mp}_{\mathrm{tc}}$
370 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^{\pm}_{\mathrm{L}} \mathrm{Z}^0_{\mathrm{L}}$
371 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^{\pm}_{\mathrm{L}} \pi^0_{\mathrm{tc}}$
372 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \pi^{\pm}_{\mathrm{tc}} \mathrm{Z}^0_{\mathrm{L}} $
373 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \pi^{\pm}_{\mathrm{tc}} \pi^0_{\mathrm{tc}} $
374 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \gamma \pi^{\pm}_{\mathrm{tc}} $
375 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{Z}^0 \pi^{\pm}_{\mathrm{tc}} $
376 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^{\pm} \pi^0_{\mathrm{tc}} $
377 $\mathrm{f}_i \overline{\mathrm{f}}_j \to \mathrm{W}^{\pm} {\pi'}^0_{\mathrm{tc}}$
381 $\mathrm{q}_i \mathrm{q}_j \to \mathrm{q}_i \mathrm{q}_j$
382 $\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{q}_k \overline{\mathrm{q}}_k$
383 $\mathrm{q}_i \overline{\mathrm{q}}_i \to \mathrm{g}\mathrm{g}$
384 $\mathrm{f}_i \mathrm{g}\to \mathrm{f}_i \mathrm{g}$
385 $\mathrm{g}\mathrm{g}\to \mathrm{q}_k \overline{\mathrm{q}}_k$
386 $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$
387 $\mathrm{f}_i \overline{\mathrm{f}}_i \to \mathrm{Q}_k \overline{\mathrm{Q}}_k$
388 $\mathrm{g}\mathrm{g}\to \mathrm{Q}_k \overline{\mathrm{Q}}_k$

Technicolor (TC) uses strong dynamics instead of weakly-coupled fundamental scalars to manifest the Higgs mechanism for giving masses to the $\mathrm{W}$ and $\mathrm{Z}$ bosons. In TC, the breaking of a chiral symmetry in a new, strongly interacting gauge theory generates the Goldstone bosons necessary for electroweak symmetry breaking (EWSB). Thus three of the technipions assume the rôle of the longitudinal components of the $\mathrm{W}$ and $\mathrm{Z}$ bosons, but other states can remain as separate particles depending on the gauge group: technipions ( $\pi_{\mathrm{tc}}$), technirhos ( $\rho_{\mathrm{tc}}$), techniomegas ( $\omega_{\mathrm{tc}}$), etc.

No fully-realistic model of strong EWSB has been found so far, and some of the assumptions and simplifications used in model-building may need to be discarded in the future. The processes represented here correspond to several generations of development. Processes 149, 191, 192 and 193 should be considered obsolete and superseded by the other processes 194, 195 and 361-377. The former processes are kept for cross-checks and backward compatibility. In section [*] it is discussed how processes 71-77 can be used to simulate a scenario with techni-$\rho$ resonances in longitudinal gauge boson scattering.

Process 149 describes the production of a spin-0 techni-$\eta$ meson (particle code KF = 3000331), which is an electroweak singlet and a QCD colour octet. It is one of the possible techni-$\pi$ particles; the name `techni-$\eta$' is not used universally in the literature. The techni-$\eta$ couples to ordinary fermions proportional to fermion mass. The dominant decay mode is therefore $\t\overline{\mathrm{t}}$, if kinematically allowed. An effective $\mathrm{g}\mathrm{g}$-coupling arises through an anomaly, and is roughly comparable in size with that to $\b\overline{\mathrm{b}}$. Techni-$\eta$ production at hadron colliders is therefore predominantly through $\mathrm{g}\mathrm{g}$ fusion, as implemented in process 149. In topcolor-assisted technicolor (discussed below), particles like the techni-$\eta$ should not have a predominant coupling to $\t $ quarks. In this sense, the process is considered obsolete.

(The following discussion borrows liberally from the introduction to Ref. [Lan99a] with the author's permission.) Modern technicolor models require walking technicolor [Hol81] to prevent large flavor-changing neutral currents and the assistance of topcolor (TC2) interactions that are strong near 1 TeV [Nam88,Hil95,Lan95] to provide the large mass of the top quark. Both additions to the basic technicolor scenario [Wei79,Eic80] tend to require a large number $N_D$ of technifermion doublets to make the $\beta$-function of walking technicolor small. They are needed in TC2 to generate the hard masses of quarks and leptons, to induce the right mixing between heavy and light quarks, and to break topcolor symmetry down to ordinary colour. A large number of techni-doublets implies a relatively low technihadron mass scale [Lan89,Eic96], set by the technipion decay constant $F_T \simeq F_\pi/\sqrt{N_D}$, where $F_\pi = 246$ GeV.

The model adopted in PYTHIA is the `Technicolor Straw Man Model' (TCSM) [Lan99a,Lan02a]. The TCSM describes the phenomenology of color-singlet vector and pseudoscalar technimesons and their interactions with SM particles. These technimesons are expected to be the lowest-lying bound states of the lightest technifermion doublet, $(T_U, T_D)$, with components that transform under technicolor ${\bf SU(N_{TC})}$ as fundamentals, but are QCD singlets; they have electric charges $Q_U$ and $Q_D=Q_U-1$. The vector technimesons form a spin-one isotriplet $\rho_{\mathrm{tc}}^{\pm,0}$ and an isosinglet $\omega_\mathrm{tc}$. Since techni-isospin is likely to be a good approximate symmetry, $\rho_{\mathrm{tc}}$ and $\omega_\mathrm{tc}$ should be approximately mass-degenerate. The pseudoscalars, or technipions, also comprise an isotriplet $\Pi_\mathrm{tc}^{\pm,0}$ and an isosinglet $\Pi_\mathrm{tc}^{0 \prime}$. However, these are not mass eigenstates. In this model, they are simple, two-state mixtures of the longitudinal weak bosons $W_L^\pm$, $Z_L^0$ -- the true Goldstone bosons of dynamical electroweak symmetry breaking in the limit that the $\bf SU(2) \otimes U(1)$ couplings $g,g'$ vanish -- and mass-eigenstate pseudo-Goldstone technipions $\pi_\mathrm{tc}^\pm, \pi_\mathrm{tc}^0$:

\begin{displaymath}
\vert\Pi_\mathrm{tc}\rangle = \sin\chi \thinspace \vert
W_L...
...e}\thinspace \vert\pi_\mathrm{tc}^{0 \prime}\rangle\ + \cdots,
\end{displaymath} (138)

where $\sin\chi = F_T/F_\pi \ll 1$, $\chi'$ is another mixing angle and the ellipsis refer to other technipions needed to eliminate the TC anomaly from the $\Pi_\mathrm{tc}^{0 \prime}$ chiral current. These massive technipions are also expected to be approximately degenerate.

The coupling of technipions to quarks and leptons are induced mainly by extended technicolor (ETC) interactions [Eic80]. These couplings are proportional to fermion mass, except for the case of the top quark, which has most of its mass generation through TC2 interactions. The coupling to electroweak gauge boson pairs vanishes at tree-level, and is assumed to be negligible. Thus the ordinary mechanisms for producing Higgs-like bosons through enhanced couplings to heavy fermions or heavy gauge bosons is absent for technipions. In the following, we will concentrate on how technipions decay once they are produced. Besides coupling to fermions proportional to mass (except for the case of top quarks where the coupling strength should be much less than $m_\t $), the $\pi_\mathrm{tc}^{0 \prime}$ can decay to gluon or photon pairs through technifermion loops. However, there may be appreciable $\pi_\mathrm{tc}^0$- $\pi_\mathrm{tc}^{0 \prime}$ mixing [Eic96]. If that happens, the lightest neutral technipions are ideally-mixed $\bar T_U T_U$ and $\bar T_D T_D$ bound states. To simulate this effect, there are separate factors $C_{\pi_\mathrm{tc}^0\to \mathrm{g}\mathrm{g}}$ and $C_{\pi_\mathrm{tc}^{0 \prime}\to \mathrm{g}\mathrm{g}}$ to weight the $\pi_\mathrm{tc}$ and $\pi_\mathrm{tc}^{0 \prime}$ partial widths for $\mathrm{g}\mathrm{g}$ decays. The relevant technipion decay modes are $\pi_\mathrm{tc}^+\rightarrow \t\bar \b , \c\bar \b , \u\bar \b $, $\c\bar \mathrm{s}$, $\c\bar \d $ and $\tau^+ \nu_\tau$; $\pi_\mathrm{tc}^0\rightarrow \t\bar\t , \b\bar \b $, $\c\bar\c $, and $\tau^+\tau^-$; and $\pi_\mathrm{tc}^{0 \prime}\rightarrow \mathrm{g}\mathrm{g}$, $\t\bar\t , \b\bar \b $, $\c\bar\c $, and $\tau^+\tau^-$. In the numerical evaluation of partial widths, the running mass (see PYMRUN) is used, and all fermion pairs are considered as final states. The decay $\pi_\mathrm{tc}^+\to \mathrm{W}^+\b\bar\b $ is also included, with the final-state kinematics distributed according to phase space (i.e. not weighted by the squared matrix element). The $\pi_\mathrm{tc}$ couplings to fermions can be weighted by parameters $C_\c $, $C_\b $, $C_\t $ and $C_\tau$ depending on the heaviest quark involved in the decay.

The technivector mesons have direct couplings to the technipion interaction states. In the limit of vanishing gauge couplings $g,g' = 0$, the $\rho_{\mathrm{tc}}$ and $\omega_\mathrm{tc}$ coupling to technipions are:

$\displaystyle \rho_{\mathrm{tc}}$ $\textstyle \rightarrow$ $\displaystyle \Pi_\mathrm{tc} \Pi_\mathrm{tc} = \cos^2 \chi\thinspace (\pi_\mat...
..._\mathrm{tc}) + \sin^2 \chi \thinspace (\mathrm{W}_L \mathrm{W}_L) \thinspace ;$  
$\displaystyle \omega_\mathrm{tc}$ $\textstyle \rightarrow$ $\displaystyle \Pi_\mathrm{tc} \Pi_\mathrm{tc} \Pi_\mathrm{tc} =
\cos^3 \chi \thinspace (\pi_\mathrm{tc}\pi_\mathrm{tc}\pi_\mathrm{tc}) + \cdots \thinspace .$ (139)

The $\rho_{\mathrm{tc}}\to\pi_\mathrm{tc}\pi_\mathrm{tc}$ decay amplitude, then, is given simply by
\begin{displaymath}
{\cal M}(\rho_{\mathrm{tc}}(q) \rightarrow \pi_A(p_1) \pi_B(...
...al C}_{AB}
\thinspace \epsilon(q)\cdot(p_1 - p_2) \thinspace ,
\end{displaymath} (140)

where the technirho coupling $\alpha_{\rho_{\mathrm{tc}}}\equiv g_{\rho_{\mathrm{tc}}}^2/4\pi = 2.91(3/N_{TC})$ is scaled naïvely from QCD ($N_{TC}=4$ by default) and ${\cal C}_{AB} = \cos^2\chi$ for $\pi_\mathrm{tc}\pi_\mathrm{tc}$, $\sin\chi \cos\chi$ for $\pi_\mathrm{tc}\mathrm{W}_L$, and $\sin^2\chi$ for $\mathrm{W}_L \mathrm{W}_L$. While the technirho couples to $\mathrm{W}_L \mathrm{W}_L$, the coupling is suppressed. Technivector production will be addressed shortly; here, we concentrate on technivector decays.

Walking technicolor enhancements of technipion masses are assumed to close off the channel $\omega_\mathrm{tc}\rightarrow \pi_\mathrm{tc}\pi_\mathrm{tc}\pi_\mathrm{tc}$ (which is not included) and to kinematically suppress the channels $\rho_{\mathrm{tc}}\rightarrow \pi_\mathrm{tc}\pi_\mathrm{tc}$ and the isospin-violating $\omega_\mathrm{tc}\rightarrow \pi_\mathrm{tc}\pi_\mathrm{tc}$ (which are allowed with appropriate choices of mass parameters). The rates for the isospin-violating decays $\omega_\mathrm{tc}\rightarrow \pi^+_A \pi^-_B =
\mathrm{W}^+_L \mathrm{W}^-_L$, $\mathrm{W}^\pm_L \pi_\mathrm{tc}^\mp $, $\pi_\mathrm{tc}^+\pi_\mathrm{tc}^-$ are given by $\Gamma(\omega_\mathrm{tc}\rightarrow \pi^+_A \pi^-_B) = \vert\epsilon_{\rho\omega}\vert^2 \thinspace
\Gamma(\rho_{\mathrm{tc}}^0\rightarrow \pi^+_A \pi^-_B)$ where $\epsilon_{\rho\omega}$ is the isospin-violating $\rho_{\mathrm{tc}}$- $\omega_\mathrm{tc}$ mixing. Based on analogy with QCD, mixing of about $5\%$ is expected. Additionally, this decay mode is dynamically suppressed, but it is included as a possibility. While a light technirho can decay to $\mathrm{W}_L \pi_\mathrm{tc}$ or $\mathrm{W}_L \mathrm{W}_L$ through TC dynamics, a light techniomega decays mainly through electroweak dynamics, $\omega_\mathrm{tc}\rightarrow \gamma\pi_\mathrm{tc}^0$, $Z^0\pi_\mathrm{tc}^0$, $W^\pm \pi_\mathrm{tc}^\mp $, etc., where $\mathrm{Z}$ and $\mathrm{W}$ may are transversely polarized. Since $\sin^2\chi \ll 1$, the electroweak decays of $\rho_{\mathrm{tc}}$ to the transverse gauge bosons $\gamma,\mathrm{W},\mathrm{Z}$ plus a technipion may be competitive with the open-channel strong decays.

Note, the exact meaning of longitudinal or transverse polarizations only makes sense at high energies, where the Goldstone equivalence theorem can be applied. At the moderate energies considered in the TCSM, the decay products of the $\mathrm{W}$ and $\mathrm{Z}$ bosons are distributed according to phase space, regardless of their designation as longitudinal $\mathrm{W}_L/\mathrm{Z}_L$ or ordinary transverse gauge bosons.

To calculate the rates for transverse gauge boson decay, an effective Lagrangian for technivector interactions was constructed [Lan99a], exploiting gauge invariance, chiral symmetry, and angular momentum and parity conservation. As an example, the lowest-dimensional operator mediating the decay $\omega_\mathrm{tc}(q) \rightarrow \gamma(p_1) \pi_\mathrm{tc}^0(p_2))$ is $(e/M_V)\thinspace F_{\rho_{\mathrm{tc}}} \cdot \widetilde{F}_\gamma \thinspace
\pi_\mathrm{tc}^0$, where the mass parameter $M_V$ is expected to be of order several 100 GeV. This leads to the decay amplitude:

\begin{displaymath}
{\cal M}(\omega_\mathrm{tc}(q) \rightarrow \gamma(p_1) \pi_\...
...on_\mu(q) \epsilon^*_\nu(p_1) q_\lambda
p_{1\rho} \thinspace .
\end{displaymath} (141)

Similar expressions exist for the other amplitudes involving different technivectors and/or different gauge bosons [Lan99a], where the couplings are derived in the valence technifermion approximation [Eic96,Lan99]. In a similar fashion, decays to fermion-antifermion pairs are included. These partial widths are typically small, but can have important phenomenological consequences, such as narrow lepton-antilepton resonances produced with electroweak strength.

Next, we address the issue of techniparticle production. Final states containing Standard Model particles and/or pseudo-Goldstone bosons (technipions) can be produced at colliders through two mechanisms: technirho and techniomega mixing with gauge bosons through a vector-dominance mechanism, and anomalies [Lan02] involving technifermions in loops. Processes 191, 192 and 193 are based on $s$-channel production of the respective resonance [Eic96] in the narrow width approximation. All decay modes implemented can be simulated separately or in combination, in the standard fashion. These include pairs of fermions, of gauge bosons, of technipions, and of mixtures of gauge bosons and technipions. Processes 194, 195 and 361-377, instead, include interference, a correct treatment of kinematic thresholds and the anomaly contribution, all of which can be important effects, but also are limited to specific final states. Therefore, several processes need to be simulated at once to determine the full effect of TC.

Process 194 is intended to more accurately represent the mixing between the $\gamma^*$, $\mathrm{Z}^0$, $\rho_{\mathrm{tc}}^0$ and $\omega_{\mathrm{tc}}^0$ particles in the Drell-Yan process [Lan99]. Process 195 is the analogous charged channel process including $\mathrm{W}^{\pm}$ and $\rho_{\mathrm{tc}}^{\pm}$ mixing. By default, the final-state fermions are $\mathrm{e}^+\mathrm{e}^-$ and $\mathrm{e}^{\pm} \nu_{\mathrm{e}}$, respectively. These can be changed through the parameters KFPR(194,1) and KFPR(195,1), respectively (where the KFPR value should represent a charged fermion).

Processes 361-368 describe the pair production of technipions and gauge bosons through $\rho_{\mathrm{tc}}^0/\omega_{\mathrm{tc}}^0$ resonances and anomaly contributions. Processes 370-377 describe pair production through the $\rho_{\mathrm{tc}}^{\pm}$ resonance and anomalies. It is important to note that processes 361, 362, 370, 371, 372 include final states with only longitudinally-polarized $\mathrm{W}$ and $\mathrm{Z}$ bosons, whereas the others include final states with only transverse $\mathrm{W}$ and $\mathrm{Z}$ bosons. Again, all processes must be simulated to get the full effect of the TC model under investigation. All processes 361-377 are obtained by setting MSEL = 50.

The vector dominance mechanism is implemented using the full $\gamma$-$\mathrm{Z}^0$- $\rho_{\mathrm{tc}}$- $\omega_\mathrm{tc}$ propagator matrix, $\Delta_0(s)$, including the effects of kinetic mixing. With the notation ${\cal M}^2_V = M^2_V - i \sqrt{s} \thinspace \Gamma_V(s)$ and $\Gamma_V(s)$ the energy-dependent width for $V = \mathrm{Z}^0,\rho_{\mathrm{tc}},\omega_\mathrm{tc}$, this matrix is the inverse of

\begin{displaymath}
\Delta_0^{-1}(s) =\left(\begin{array}{cccc}
s & 0 & s f_{\ga...
...cal M}^2_{\omega_\mathrm{tc}}
\end{array}\right) \thinspace .
\end{displaymath} (142)

The parameters $f_{\gamma\rho_{\mathrm{tc}}} = \xi$, $f_{\gamma\omega_\mathrm{tc}} = \xi \thinspace (Q_U + Q_D)$, $f_{\mathrm{Z}\rho_{\mathrm{tc}}} = \xi \thinspace \cot 2\theta_W $, and $f_{\mathrm{Z}\omega_\mathrm{tc}} = - \xi \thinspace
(Q_U + Q_D) \tan\theta_W $, and $\xi = \sqrt{\alpha/\alpha_{\rho_{\mathrm{tc}}}}$ determine the strength of the kinetic mixing, and are fixed by the quantum numbers of the technifermions in the theory. Because of the off-diagonal entries, the propagators resonate at mass values shifted from the nominal $M_V$ values. Thus, while users input the technihadron masses using PMAS values, these will not represent exactly the resulting mass spectrum of pair-produced particles. Note that special care is taken in the limit of very heavy technivectors to reproduce the canonical $\gamma^*/\mathrm{Z}^*\to\pi_\mathrm{tc}^+\pi_\mathrm{tc}^-$ couplings. In a similar fashion, cross sections for charged final states require the $\mathrm{W}^\pm$- $\rho_{\mathrm{tc}}^\pm $ matrix $\Delta_{\pm}$:
\begin{displaymath}
\Delta_{\pm}^{-1}(s) =\left(\begin{array}{cc} s - {\cal M}^2...
...M}^2_{\rho_{\mathrm{tc}}^\pm } \end{array}\right) \thinspace ,
\end{displaymath} (143)

where $f_{\mathrm{W}\rho_{\mathrm{tc}}} = \xi/(2\sin\theta_W )$.

By default, the TCSM Model has the parameters $N_{TC}$= 4, $\sin\chi$ = $1\over 3$, $Q_U$ = $4 \over 3$, $Q_D=Q_U-1$ = $1\over 3$, $C_\b =C_\c =C_\tau$= 1, $C_\t $= $m_\b /m_\t $, $C_{\pi_\mathrm{tc}}$= $\textstyle {4\over{3}}$, $C_{\pi_\mathrm{tc}^0\to \mathrm{g}\mathrm{g}}$=0, $C_{\pi_\mathrm{tc}^{0 \prime}\to \mathrm{g}\mathrm{g}}$=1, $\vert\epsilon_{\rho\omega}\vert$ = 0.05, $F_T = F_\pi \sin\chi$ = $82$ GeV, $M_{\rho_{\mathrm{tc}}^\pm }= M_{\rho_{\mathrm{tc}}^0} = M_{\omega_\mathrm{tc}}$ = $210$ GeV, $M_{\pi_\mathrm{tc}^\pm } = M_{\pi_\mathrm{tc}^0} = M_{\pi_\mathrm{tc}^{0 \prime}}$ = $110$ GeV, $M_{V} = M_{A}$ = $200$ GeV. The techniparticle mass parameters are set through the usual PMAS array. Parameters regulating production and decay rates are stored in the RTCM array in PYTCSM. This concludes the discussion of the electroweak sector of the strawman model.

In the original TCSM outlined above, the existence of top-color interactions only affected the coupling of technipions to top quarks, which is a significant effect only for higher masses. In general, however, TC2 requires some new and possibly light coloured particles. In most TC2 models, the existence of a large $\t\bar\t $, but not $\b\bar\b $, condensate and mass is due to ${\bf SU(3)_1}\otimes {\bf U(1)_1}$ gauge interactions which are strong near 1 TeV. The ${\bf SU(3)_1}$ interaction is $\t $-$\b $ symmetric while ${\bf U(1)_1}$ couplings are $\t $-$\b $ asymmetric. There are weaker ${\bf SU(3)_2}\otimes {\bf U(1)_2}$ gauge interactions in which light quarks (and leptons) may [Hil95], or may not [Chi96], participate. The two ${\bf U(1)}$'s must be broken to weak hypercharge ${\bf U(1)_Y}$ at an energy somewhat higher than 1 TeV by electroweak-singlet condensates. The full phenomenology of even such a simple model can be quite complicated, and many (possibly unrealistic) simplifications are made to reduce the number of free parameters [Lan02a]. Nonetheless, it is useful to have some benchmark to guide experimental searches.

The two TC2 ${\bf SU(3)}$'s can be broken to their diagonal ${\bf SU(3)}$ subgroup by using technicolor and ${\bf U(1)_1}$ interactions, both strong near 1 TeV. This can be explicitly accomplished [Lan95] using two electroweak doublets of technifermions, $T_1= (U_1,D_1)$ and $T_2 =
(U_2,D_2)$, which transform respectively as $(3,1,N_{TC})$ and $(1,3,N_{TC})$ under the two colour groups and technicolor. The desired pattern of symmetry breaking occurs if ${\bf SU(N_{TC})}$ and ${\bf U(1)_1}$ interactions work together to induce electroweak and ${\bf SU(3)_1}\otimes {\bf SU(3)_2}$ non-invariant condensates $\langle \thinspace \bar U_{iL} U_{jR} \rangle$ and $\langle \thinspace
\bar D_{iL} D_{jR} \rangle$, $(i,j = 1,2)\thinspace $. This minimal TC2 scenario leads to a rich spectrum of colour-nonsinglet states readily accessible in hadron collisions. The lowest-lying ones include eight `colorons', $V_8$, the massive gauge bosons of broken topcolor ${\bf SU(3)}$; four isosinglet $\rho_{\mathrm{tc}8}$ formed from $\bar T_i T_j$ and the isosinglet pseudo-Goldstone technipions formed from $\bar T_2 T_2$. In this treatment, the isovector technipions are ignored, because they must be pair produced in $\rho_{\mathrm{tc}8}$ decays, and such decays are assumed to be kinematically suppressed.

The colorons are new fundamental particles with couplings to quarks. In standard TC2 [Hil95], top and bottom quarks couple to ${\bf SU(3)_1}$ and the four light quarks to ${\bf SU(3)_2}$. Because the ${\bf SU(3)_1}$ interaction is strong and acts exclusively on the third generation, the residual $V_8$ coupling can be enhanced for $\t $ and $\b $ quarks. The coupling $g_a=g_c\cot\theta_3$ for $\t $ and $\b $ and $g_a=-g_c\tan\theta_3$ for $\u ,\d ,\c ,\mathrm{s}$, where $g_c$ is the QCD coupling and $\cot\theta_3$ is related to the original $\g_1$ and $g_2$ couplings. In flavor-universal TC2 [Chi96] all quarks couple to ${\bf SU(3)_1}$, not ${\bf SU(3)_2}$, so that colorons couple equally and strongly to all flavors: $g_a=g_c\cot\theta_3$.

Assuming that techni-isospin is not badly broken by ETC interactions, the $\rho_{\mathrm{tc}8}$ are isosinglets labeled by the technifermion content and colour index $A$: $\rho_{11}^{A}, \rho_{22}^{A}, \rho_{12}^{A}, \rho_{12'}^{A}$. The first two of these states, $\rho_{11}$ and $\rho_{22}$, mix with $V_8$ and $g$. The topcolor-breaking condensate, $\langle \bar T_{1L} T_{2R} \rangle
\neq 0$, causes them to also mix with $\rho_{12}$ and $\rho_{12'}$. Technifermion condensation also leads to a number of (pseudo)Goldstone boson technipions. The lightest technipions are expected to be the isosinglet ${\bf SU(3)}$ octet and singlet $\bar T_2 T_2$ states $\pi_{22}^{A}$ and $\pi_{22}^0$.

These technipions can decay into either fermion-antifermion pairs or two gluons; presently, they are assumed to decay only into gluons. As noted, walking technicolor enhancement of technipion masses very likely close off the $\rho_{\mathrm{tc}8}\rightarrow
\pi_\mathrm{tc}\pi_\mathrm{tc}$ channels. Then the $\rho_{\mathrm{tc}8}$ decay into $\mathrm{q}\bar\mathrm{q}$ and $\mathrm{g}\mathrm{g}$. The rate for the former are proportional to the amount of kinetic mixing, set by $\xi_\mathrm{g}= {g_c / {g_{\rho_{\mathrm{tc}}}}}$. Additionally, the $\rho_{22}$ decays to $\mathrm{g}\pi_{22}^{0,A}$.

The $V_8$ colorons are expected to be considerably heavier than the $\rho_{\mathrm{tc}8}$, with mass in the range 0.5-1 TeV. In both the standard and flavor-universal models, colorons couple strongly to $\bar T_1 T_1$, but with only strength $g_c$ to $\bar T_2 T_2$. Since relatively light technipions are $\bar T_2 T_2$ states, it is estimated that $\Gamma(V_8 \rightarrow \pi_\mathrm{tc}\pi_\mathrm{tc}) = {\cal O}(\alpha_c)$ and $\Gamma(V_8 \rightarrow g\pi_\mathrm{tc}) = {\cal O}(\alpha_c^2)$. Therefore, these decay modes are ignored, so that the $V_8$ decay rate is the sum over open channels of

\begin{displaymath}
\Gamma(V_8 \rightarrow \mathrm{q}_a \bar \mathrm{q}_a) = {\a...
...) \thinspace
\left(s - 4m_a^2\right)^{1\over{2}} \thinspace ,
\end{displaymath} (144)

where $\alpha_a = g^2_a/4\pi$.

The phenomenological effect of this techniparticle structure is to modify the gluon propagator in ordinary QCD processes, because of mixing between the gluon, $V_8$ and the $\rho_{\mathrm{tc}8}$'s. The $\mathrm{g}$-$V_8$-$\rho_{11}$-$\rho_{22}$-$\rho_{12}$-$\rho_{12'}$ propagator is the inverse of the symmetric matrix

\begin{displaymath}
D^{-1}(s) = \left(\begin{array}{cccccc}
s & 0 & s \thinspace...
...2'} & s - {\cal M}^2_{12'} \\
\end{array}\right) \thinspace .
\end{displaymath} (145)

Here, ${\cal M}^2_V = M^2_V - i \sqrt{s} \thinspace \Gamma_V(s)$ uses the energy-dependent widths of the octet vector bosons, and the $\xi_{\rho_{ij}}$ are proportional to $\xi_\mathrm{g}$ and elements of matrices that describe the pattern of technifermion condensation. The mixing terms $M^2_{ij,kl}$, induced by $\bar T_1 T_2$ condensation are assumed to be real.

This extension of the TCSM is still under development, and any results should be carefully scrutinized. The main effects are indirect, in that they modify the underlying two-parton QCD processes much like compositeness terms, except that a resonant structure is visible. Similar to compositeness, the effects of these colored technihadrons are simulated by setting ITCM(5) = 5 for processes 381-388. By default, these processes are equivalent to the 11, 12, 13, 28, 53, 68, 81 and 82 ones, respectively. The last two are specific for heavy-flavour production, while the first six could be used to describe standard or non-standard high-$p_{\perp}$ jet production. These six are simulated by MSEL = 51. The parameter dependence of the `model' is encoded in $\tan\theta_3$ (RTCM(21)) and a mass parameter $M_8$ (RTCM(27)), which determines the decay width $\rho_{22}\to \mathrm{g}\pi_{22}$ analogously to $M_V$ for $\omega_\mathrm{tc}\to\gamma\pi_\mathrm{tc}$. For ITCM(2) equal to 0 (1), the standard (flavor universal) TC2 couplings are used. The mass parameters are set by the PMAS array using the codes: $V_8$ (3100021), $\pi_{22}^{1}$ (3100111), $\pi_{22}^{8}$ (3200111), $\rho_{11}$ (3100113), $\rho_{12}$ (3200113), $\rho_{21}$ (3300113), and $\rho_{22}$ (3400113). The mixing parameters $M_{ij,kl}$ take on the (arbitrary) values $M_{11,22}=100$ GeV, $M_{11,12}=M_{11,21}=M_{22,12}=150$ GeV, $M_{22,21}=75$ GeV and $M_{12,21}=50$ GeV, while the kinetic mixing terms $\xi_{\rho_{ij}}$ are calculated assuming the technicolor condensates are fully mixed, i.e. $\langle T_i\bar T_j \rangle \propto {1 / \sqrt{2}}$.


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Stephen Mrenna 2007-10-30