---
abstract: 'We study an extended QCD model in $2D$ obtained from QCD in $4D$ by compactifying two spatial dimensions and projecting onto the zero-mode subspace. This system is found to induce a dynamical mass for transverse gluons – adjoint scalars in $QCD_2$, and to undergo a chiral symmetry breaking with the full quark propagators yielding non-tachyonic, dynamical quark masses, even in the chiral limit. We construct the hadronic color singlet bound-state scattering amplitudes and study quark-antiquark bound states which can be classified in this model by their properties under Lorentz transformations inherited from $4D$.'
title: 'Quark-Antiquark Bound States in an Extended $QCD_2$ Model[^1]'
---

[Pedro Labra[ñ]{}a, Jorge Alfaro]{}\
Facultad de Física, Pontificia Universidad Católica de Chile.\
plabrana@puc.cl, jalfaro@puc.cl\
\
St.Petersburg State University and INFN, Sezione di Bologna.\
andrianov@bo.infn.it

We study a QCD reduced model in $2D$ which can be formally obtained from QCD in $4D$ by means of a classical dimensional reduction from $4D$ to $2D$ and neglecting heavy K-K (Kaluza-Klein) states. Thus only zero-modes in the harmonic expansion in compactified dimensions are retained. As a consequence, we obtain a two dimensional model with some resemblances of the real theory in higher dimension, that is, in a natural way adding boson matter in the adjoint representation to $QCD_2$ [@light; @Alfaro:2003yy]. The latter fields being scalars in $2D$ reproduce transverse gluon effects [@adjoint]. Furthermore this model has a richer spinor structure than just $QCD_2$ giving a better resolution of scalar and vector states which can be classified by their properties inherited from $4D$ Lorentz transformations. The model is analyzed in the light cone gauge and using large $N_c$ limit. The contributions of the extra dimensions are controlled by the radiatively induced masses of the scalar gluons as they carry a piece of information of transverse degrees of freedom. We consider their masses as large parameters in our approximations yet being much less than the first massive K-K excitation. This model might give more insights into the chiral symmetry breaking regime of $QCD_4$. Namely, we are going to show that the inclusion of solely lightest K-K boson modes catalyze the generation of quark dynamical mass and allows us to overcome the problem of tachyonic quarks present in $QCD_2$.

We start with the $QCD$ action in $(3+1)$ dimensions for one flavor (extension to more flavors is straightforward): $$S_{QCD}= \int d^4x \left[-\frac{1}{2{\tilde g}^2}tr(G_{\mu\nu}^2)
+ {\bar \Psi}\,(i\gamma^\mu \, D_\mu - m)\,\Psi \right].$$

Follow the scheme of [@Alfaro:2003yy] we proceed to make a dimensional reduction of $QCD$, at the classical level, from $4D$ to $2D$. For this we consider the coordinates $x_{2,3}$ being compactified in a 2-Torus, respectively the fields being periodic on the intervals ($0\leq x_{2,3}\leq L=2\pi R$). Next we assume $L$ to be small enough in order to get an effective model in $2D$ dimensions. Then by keeping only the zero K-K modes, we get the following effective action in $2D$, after a suitable rescaling of the fields: $$\begin{aligned}
\label{Lmodelo2D}
S_2 &=& \int \!d^2x\,\,tr \!\left[-\frac{1}{2}F_{\mu\nu}^2+ (D_\mu
\phi_1)^2 + (D_\mu \phi_2)^2  \right] +
 {\bar \psi}_1\,(i\gamma^\mu \, D_\mu - m)\,\psi_1  \nonumber\\
%
&+& {\bar \psi}_2\,(i\gamma^\mu \, D_\mu - m)\,\psi_2 -
i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_1\,\psi_2 +
{\bar \psi}_2\,\gamma^5\,\phi_1\,\psi_1\right) \\
%
&-& i\frac{g}{\sqrt{N_c}}\left({\bar \psi}_1\,\gamma^5\,\phi_2\,\psi_1 -
{\bar \psi}_2\,\gamma^5\,\phi_2\,\psi_2\right) +
\frac{g^2}{N_c}\,tr[\phi_1,\phi_2]^2\,,
\nonumber\end{aligned}$$ where we have defined the coupling constant of the model $g^2= N_c\,{\tilde g}^2/L^2$. We expect [@Coleman1] the infrared mass generation for the two-dimensional scalar gluons $\phi_i$. To estimate the masses of scalar gluons $\phi_i$ we use the Schwinger-Dyson equations as self-consistency conditions, we get: $$M^2=\frac{2 N_c {\tilde g}^2}{L^2} \int^\Lambda \!\!
\frac{d^2p}{(2\pi)^2} \,\frac{1}{p^2+M^2} \,
 = \frac{N_c {\tilde g}^2\,\Lambda^2}{8\pi^3}\,\,
 \log\frac{\Lambda^2+M^2}{M^2}\,, \label{glumass}$$

![Inhomogeneous Bethe-Salpeter equation for quark-antiquark scattering amplitude $T_5$.[]{data-label="F1"}](Diagramas3b.eps){width="90.00000%"}

thus $M^2$ brings an infrared cutoff as expected. We notice that the gluon mass remains finite in the large-$N_c$ limit if the QCD coupling constant decreases as $1/N_c$ in line with the perturbative law of $4D$ QCD. We adopt the approximation $M \ll \Lambda\simeq 1/R$ to protect the low-energy sector of the model and consider the momenta $|p_{0,1}| \sim M$. Thereby we retain only leading terms in the expansion in $p^2/\Lambda^2$ and $M^2/\Lambda^2$, and also neglect the effects of the heavy K-K modes in the low-energy Wilson action. We observe that the limit $M \ll \Lambda$ supports consistently both the fast decoupling of the heavy K-K modes and moderate decoupling of scalar gluons [@Alfaro:2003yy], the latter giving an effective four-fermion interaction different from [@Burkardt]. Also allow us to define the “heavy-scalar” expansion parameter $A=g^2/(2\pi\,M^2)=1/log \frac{\Lambda^2}{M^2} \ll 1$. Now we proceed to the study of bound states of quark-antiquark. In our reduction we have four possible combinations of quark bilinears to describe these states with valence quarks: $(\psi_1\,{\bar \psi}_1),(\psi_1\,{\bar \psi}_2),$ $
(\psi_2\,{\bar \psi}_1),(\psi_2\,{\bar \psi}_2 )$. We need to compute the full quark-antiquark scattering amplitude $T$ in the different channels. As an example we are going to show the computing of $T_5$ which correspond to the scattering ($q_1 + \bar{q}_2 \longrightarrow q_1 + \bar{q}_2$). It satisfies, the equation given graphically in , in the large $N_c$ limit and in ladder exchange approximation (non-ladder contribution are estimated to be of higher order in the $A$ expansion). Notice that in the equation for $T_5$ the amplitude $T_8$ appears, which correspond to the process $q_2 + \bar{q}_1 \longrightarrow q_1 + \bar{q}_2$. This means that the equations for $T_5$ and $T_8$ are coupled. In Figure \[F1\] all internal fermion lines correspond to dressed quark propagators which were determined in [@Alfaro:2003yy] by using the large $N_c$ limit and the one-boson exchange approximation. There are two kind of solutions for the dressed quark propagators. In one solution, the perturbative one, we have tachyonic quarks in the chiral limit as in $QCD_2$ and our model could be interpreted as a perturbation from the result of chiral QCD in $2D$. But we are not allowed to consider that possibility because the spectrum for the lowest $q {\bar q}$ bound states becomes imaginary if one takes into account the scalar field exchange. The second solution, the non-perturbative one, supports non-tachyonic quarks with masses going to zero, in the chiral limit and also yield real masses for the $q {\bar q}$ bound states.

![$q\bar{q}$ scattering $T_5$ in the color-singlet channel.[]{data-label="F2"}](Diagramas4.eps){width="90.00000%"}

Having found the full quark propagator we proceed to solve the inhomogeneous Bethe-Salpeter equation for $T_5$ and $T_8$. We obtain for $T_5$ : $${\displaystyle T_5}\!
{\textstyle \stackrel{\alpha \beta, \gamma \delta ~~~~~~} {(q,q';p)}}
 =
-i\frac{g^2}{N}\,\frac{\gamma^{\alpha \delta}_-\,\gamma^{\beta
\gamma}_-}{(q-q')^2_-} - i\frac{g^2}{N}\,\frac{\sigma^{\alpha
\gamma}_3\,\sigma^{\beta \delta}_3}{[(q-q')^2 - M^2+i\epsilon]} \,+$$ $$+\, {\displaystyle \sum_j}\,\frac{i}{\big(p^2-\bar{m}^2_j + i\epsilon \big)}\,
\Theta^{\alpha\gamma}_j(q;p)\,\Theta^{\beta\delta}_j(q';p)
 +\, {\displaystyle \sum_k}\,\frac{i}{\big(p^2-\tilde{m}^2_k +i\epsilon \big)}\,
\Lambda^{\alpha\gamma}_k(q;p)\, \Lambda^{\beta\delta}_k(q';p)\, .$$

There are no continuum states in the quark-antiquark amplitude– only bound states at $p^2=\bar{m}^2$ and $p^2=\tilde{m}^2$, whose residue yield the bound state wave functions $\Theta(p,r)$ and $\Lambda(p,r)$. These bound states have a direct interpretation in term of Dirac Bilinears of the theory in $4D$ [@Alfaro:2003yy]. In particular, the pseudoscalar $4D$ states are related with $\Lambda_k(r;p)$ which could be interpreted as the vertex: (quark)-(antiquark)-($4D$ pseudoscalar meson). The masses of the bound states and the vertex functions are fixed by the solution of the homogeneous Bethe-Salpeter equation which, in our model, yield to a eigenvalue problem, generalization of the integral equation found by ’t Hooft [@thooft1] in $QCD_2$. For example the following eigenvalue integral equation determine the mass spectrum of the pseudoscalar bound states (in the chiral limit): $$\label{n1} \tilde{m}^2 \, \phi(x) = \, -\frac{g^2}{\pi}\,
\int^{1}_{0} \! dy \, \frac{{\bf P}}{(x - y)^{2}}\,\, \phi(y) -\,
A\,\frac{\Sigma^{2}_{0}}{x^{1-\beta}} \int^{1}_{0} \! dy  \,
\frac{\phi(y)}{(1 - y)^{1-\beta}}$$ $$- A\,\frac{\Sigma^{2}_{0}}{(1 - x)^{1-\beta}}
\int^{1}_{0} \! dy  \, \frac{\phi(y)}{y^{1-\beta}} +
A\,\frac{\Sigma^{2}_{0}}{[(1 - x)\,x]^{1-\beta}} \int^{1}_{0} \!
dy \, \phi(y) + A\,\Sigma^{2}_{0}\, \int^{1}_{0} \! dy  \,
\frac{\phi(y)}{[(1 - y)\,y]^{1-\beta}} ,$$ where $\beta = A/2$. To explore solutions to Eq.(\[n1\]) we examine small $A$. Evidently Eq.(\[n1\]) does not mix even and odd functions with respect to the symmetry $x \Longleftrightarrow
1 - x$. On the other hand the ground state should be an even function. When inspecting the wave function end-point asymptotics from the integral equation (\[n1\]) one derives the following even function as a ground state solution for $A \rightarrow 0$ limit: $\phi_0(x)=(4x[1-x])^{\frac{A}{2}}-\frac{1}{\pi}\,(4x[1-x])^{\frac{1}{2}}$. This is basically a non-perturbative result that differs from ’t Hooft solution in the $A \rightarrow 0$ limit, giving $p^2=0$, as we would expect from spontaneous chiral symmetry breaking in $4D$. For the other massive states we are unable to find analytic solutions, as happens with equation, but we could estimate them working with the Hamiltonian matrix elements $\tilde{m}^2(\phi,\phi)=(\phi,H\phi)$ and using the regular perturbation theory, starting from solutions ($A=0$). In Table \[1\] we show lowest values of mass spectra for scalar and pseudoscalar states.

Discussions {#discussions .unnumbered}
===========

Quantum Chromodynamics at low energies has been decomposed by means of dimensional reduction from $4D$ to $2D$ and a low energy effective model in $2D$ has been derived. In this model we did an explicit analysis of meson bound states by solving the inhomogeneous and the homogeneous Bethe-Salpeter equations, in the large $N_c$ limit and in the ladder exchange approximation. We found that the $2D$ model has four types of bound states which can be classified by their properties under Lorentz transformations inherited from $4D$. The $4D$ pseudoscalar and scalar sectors of the theory were analyzed and in the quiral limit a massless solution for the pseudoscalar ground state was found. We interpreted this solution as the “pion” of the model.\
Also, in solving the Bethe-Salpeter equations we found a solution to the full quark propagators yielding non-tachyonic dynamical quark masses, in contrast to what happen in $QCD_2$.

  -------------- --------
  Pseudoscalar   Scalar
  0              974
  1286           1531
  1741           1929
  2166           2356
  -------------- --------

Table \[1\]: Some values for the masses of bound states in $[MeV]$,\
           where we have taken $\frac{g}{\sqrt{\pi}}=267 [MeV]$ and $A=0.22$.

**Acknowledgments:** P.L. thanks the organizers of the V-SILAFAE. The work of A.A. is partially supported by RFBR Grant and the Program “Universities of Russia: Basic Research”. The work of P.L. is supported by a Conicyt Ph. D. fellowship (Beca Apoyo Tesis Doctoral). The work of J.A. is partially supported by Fondecyt \# 1010967.

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[^1]: alk given at 5th Latin American Symposium on High Energy Physics , Lima Per[ú]{},
