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Dirac states from the 't Hooft model

Paul Hoyer

TL;DR

This work demonstrates that in QCD$_2$ at $N_c\to\infty$ the dynamics of a light quark bound to a heavy one reduces to the Dirac equation with a linear confining potential $V(x)=V'|x|$ in any bound-state frame. By analyzing the bound-state equation and its boost properties, the author shows that in the heavy-quark limit the light-quark sector satisfies the Dirac equation with mass $m_1$ and potential $V$, and that the bound-state and Dirac problems are linked via a well-defined boost to a boosted coordinate $u=\gamma x$. The analytic solutions are expressed in terms of confluent hypergeometric functions, with discrete bound-state masses arising from regularity conditions in the bound-state sector, while the Dirac equation with a linear potential yields a continuous spectrum. A numerical example in the rest frame corroborates the Dirac limit, showing $M-m_2\to M_D$ and convergence of wavefunctions as $m_2$ grows, supporting the frame-independent Dirac dynamics of the light quark. The findings offer a clear bridge between bound-state formalisms and the Dirac description, with potential extensions to $D=3+1$ via a relativistic bound-state framework.

Abstract

The dynamics of a light fermion bound to a heavy one is expected to be described by the Dirac equation with an external potential. The potential breaks translation invariance, whereas the bound state momentum is well defined. Boosting the bound state determines the frame dependence of the light fermion dynamics. I study the Dirac limit of QCD$_2$ in the limit of $N_c \to \infty$. The light quark wave function turns out to be independent of the frame of the bound state, up to an irrelevant Lorentz contraction. The discrete bound state spectrum determines corresponding discrete energies of the Dirac equation, which for a linear potential allows a continuous spectrum.

Dirac states from the 't Hooft model

TL;DR

This work demonstrates that in QCD at the dynamics of a light quark bound to a heavy one reduces to the Dirac equation with a linear confining potential in any bound-state frame. By analyzing the bound-state equation and its boost properties, the author shows that in the heavy-quark limit the light-quark sector satisfies the Dirac equation with mass and potential , and that the bound-state and Dirac problems are linked via a well-defined boost to a boosted coordinate . The analytic solutions are expressed in terms of confluent hypergeometric functions, with discrete bound-state masses arising from regularity conditions in the bound-state sector, while the Dirac equation with a linear potential yields a continuous spectrum. A numerical example in the rest frame corroborates the Dirac limit, showing and convergence of wavefunctions as grows, supporting the frame-independent Dirac dynamics of the light quark. The findings offer a clear bridge between bound-state formalisms and the Dirac description, with potential extensions to via a relativistic bound-state framework.

Abstract

The dynamics of a light fermion bound to a heavy one is expected to be described by the Dirac equation with an external potential. The potential breaks translation invariance, whereas the bound state momentum is well defined. Boosting the bound state determines the frame dependence of the light fermion dynamics. I study the Dirac limit of QCD in the limit of . The light quark wave function turns out to be independent of the frame of the bound state, up to an irrelevant Lorentz contraction. The discrete bound state spectrum determines corresponding discrete energies of the Dirac equation, which for a linear potential allows a continuous spectrum.

Paper Structure

This paper contains 9 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Dirac equation from $\mathbf{q{\bar{q}}}$ bound states
  3. Rest frame
  4. CM momentum $P>0$
  5. Solutions of the Bound State and Dirac equations
  6. The Bound State Equation
  7. The Dirac equation
  8. A numerical example
  9. Discussion

Figures (2)

  • Figure 1: Approach of the $P=0$ bound state wave function (\ref{['3.4']}) to the Dirac wave function (\ref{['3.9']}) with increasing $m_2$. All units are in terms of $V'=1$ and $m_1=0.14$. Left:$M-m_2$ (\ref{['2.7']}) tending to $M_D$=1.532. Right:${\mathrm{Re}}(\phi_1-\varphi_D)$ at $x=4.0$.
  • Figure 2: Comparison of the bound state wave function (\ref{['2.6']}) ($m_1=0.14,\ m_2=20,\ M-m_2=1.551$) and the Dirac wave function (\ref{['3.9']}) $(m=0.14,\ M_D=1.532)$. All units are in terms of $V'=1$. Left:${\mathrm{Re}}\,\phi_1(x)$ (blue) and ${\mathrm{Re}}\,\varphi_D(x)$ (brown). Right:${\mathrm{Re}}\,\phi_0(x)$ (blue) and $-{\mathrm{Re}}\,\chi_D(x)$ (brown). The imaginary parts agree at a similar level.