Musings

In $A_0=0$ gauge, Karabali and Nair parametrize the gauge field in the plane, using complex coordinates, as $$A = -\partial M M^{-1}+ (M^\dagger)^{-1} \overline{\partial} M^\dagger$$ where $M$ is an $SL(N,\mathbb{C})$ matrix-valued function. The usual gauge symmetry $$A\to g A g^{-1} - d g g^{-1}$$ becomes simply $M\to g M$, so that $H=M^\dagger M$ is gauge-invariant. This parametrization has an additional redundancy

$$M(z,\overline{z})\to M(z,\overline{z}) h^\dagger(\overline{z}),\qquad M^\dagger(z,\overline{z})\to h(z) M^\dagger(z,\overline{z})$$ where $h(z)$ is an $SL(N,\mathbb{C})$-valued holomorphic function.

The Yang-Mills Hamiltonian, $$\mathcal{H} = \int \tr \left(g_{\text{YM}}^2 E^2 + \frac{1}{g_{\text{YM}}^2} B^2 \right)$$ can be written as a (nonlocal, but relatively simple) functional of the “current”, $J$ (and its conjugate momentum), where $$J = \frac{N}{\pi} \partial H H^{-1}$$ is gauge-invariant, and transforms inhomogeneously $$J\to h J h^{-1} + \frac{N}{\pi} \partial h h^{-1}$$ under (1). $\overline{\partial}J$ transforms homogeneously, as does the “covariant derivative”, $$D= \partial - \frac{\pi}{N} [J,\cdot]$$

Leigh et al claim to have found the ground state wave functional¹, $$\Psi_0(J)= \exp\left(-\frac{\pi}{2N m^2}\int\tr \overline{\partial}J\, K(L)\, \overline{\partial}J\right)$$ where $m= \frac{g_{\text{YM}}^2 N}{2\pi}$ and $L=\frac{D\overline{\partial}}{m^2}$. $K(L)$ is given by a ratio of Bessel functions. $$K(L) = \frac{1}{\sqrt{L}}\frac{J_2(4\sqrt{L})}{J_1(4\sqrt{L})}$$

In the large-$N$ limit, this wave functional is essentially a Gaussian, and one can trivially compute glueball masses by studying the falloff, for large separation, of the equal-time 2-point correlation function. From the identity, $$\frac{J_{\nu-1}(z)}{J_\nu(z)} = \frac{2\nu}{z} + 2z \sum_{n=1}^\infty \frac{1}{z^2- j^2_{\nu,n}}$$ where $j^2_{\nu,n}$ are the ordered zeroes of $J_\nu(z)$, one has $$K^{-1}(k) = -\frac{1}{2} \sum_{n=1}^\infty \frac{M_n^2}{k^2 +M_n^2},\quad M_n=j_{2,n} m/2$$ The $J^{PC}=0^{++}$ glueballs couple to $\tr(\overline{\partial}J\overline{\partial}J)$ $$\langle \tr(\overline{\partial}J\overline{\partial}J)(x) \tr(\overline{\partial}J\overline{\partial}J)(y)\rangle \sim \left(\tilde{K}^{-1}(x-y)\right)^2 = \left(-\frac{1}{4\sqrt{2\pi|x-y|}}\sum_{n=1}^\infty (M_n)^{3/2} e^{- M_n|x-y|}\right)^2$$ So the masses of the $0^{++}$ glueballs are given as sums $M_{n}+M_{n'}= (j_{2,n}+j_{2,n'}) m/2$.

Similarly, the $0^{--}$ glueballs couple to $\tr(\overline{\partial}J\overline{\partial}J\overline{\partial}J)$, and so their masses are given as sums $M_{n}+M_{n'}M_{n''}= (j_{2,n}+j_{2,n'}+j_{2,n''}) m/2$

For large $z$, $$J_\nu(z)\sim \sqrt{\frac{2}{\pi z}} \cos\left(z-\frac{\pi\nu}{2}-\frac{\pi}{4}\right)$$ and hence, for $n\gg 1$, the zeroes of the Bessel function that enter into the formulæ for the masses of the $0^{++}$ and $0^{--}$ glueballs above go like² $$j_{2,n}\sim (n+3/4)\pi\quad \text{for }\, n\gg 1$$ So, for large $n$, one has a large number ($\mathcal{O}(n)$) of nearly-degenerate $0^{++}$ glueballs of mass $\sim (n+3/2)\pi m/2$, corresponding to different ways of partitioning $n$ into a pair of positive integers, $n_1,n_2$. And, similarly, for the $0^{--}$ glueballs.