Spectrum of quantum chromodynamics in the limit of an infinite number of colors at fixed coupling

Abstract

We study quantum chromodynamics in the limit of an infinite number of colors $N_c$ at fixed coupling $g_s$. On a null-plane lattice, the field theory can be recast as an interacting string model. The resulting dual string has internal degrees of freedom corresponding to the spin of the gluon quanta. The dynamics of these spins is that of an isotropic Heisenberg antiferromagnet. In the limit $N_c\to \infty$, low orders in the dual loop expansion are expected to give a good approximation to the spectrum (modulo effects such as spontaneous symmetry breakdown). We find that, in this limit, all particles lie on linear Regge trajectories. Quarkless gluonic states correspond to closed strings, and the spectrum is Lorentz invariant if the intercept of the leading trajectory is 2. Open strings are terminated by a quark and an antiquark, and the $\rho$ trajectory must have intercept unity. Our limit preserves chiral symmetry: There is a $J^{\mathrm{PC}}=0^{++}$ scalar degenerate with the "pion" $0^{-+}$. Both these particles are tachyons, so chiral symmetry may be spontaneously broken by string interactions. Our $N_c\to \infty$ calculations are similar to zero-temperature expansions in statistical mechanics. We expect them to give a qualitatively correct description of the ordered phase (confined quarks). We suggest that there is a critical number of colors below which confinement is lost. If $N_c=3\mathrm{}$ is close to this critical value (it is hoped, above it), the $\frac{1}{N_c}$ expansion might not give quantitatively accurate results.