Musings

The main previously-missing detail which is filled in here (perhaps making sense of my previous discussion) is how exactly they propose to regularize the Karabali-Nair Hamiltonian (see Appendix A of their paper) $$\mathcal{H}= \frac{g_{\text{YM}}^2 N}{2\pi}\int J^a \frac{\delta}{\delta J^a} +\doubleintegral \Omega^{a b}(z,w) \frac{\delta}{\delta J^a(z)}\frac{\delta}{\delta J^b(w)} +\frac{1}{2 g_{\text{YM}}^2}\int \overline{\partial}J^a\overline{\partial}J^a$$ where $$\Omega^{a b}(z,w) = \frac{N}{\pi} D^{a b}_w \mathcal{G}(w-z)$$ Here $$D = \partial - \frac{\pi}{N} [J, \cdot]$$ and $\mathcal{G}(w-z)$ is the ordinary Green’s function, satisfying $\overline{\partial}_z\mathcal{G} = \delta^{(2)}(z)$.

They take as an ansatz that the ground state wave functional takes the form $$\Psi_0(J)= \exp\left(-\frac{\pi}{2N m^2}\int\tr \overline{\partial}J\, K\left(\frac{\Delta}{m^2}\right)\, \overline{\partial}J\right)$$ where $m= \frac{g_{\text{YM}}^2 N}{2\pi}$ and $\Delta = \{D,\overline{\partial}\}/2$, for some kernel $K\left(\frac{\Delta}{m^2}\right)$. Formally, they then expand $$K(L) = \sum_{n=0}^\infty c_n L^n$$ and find that the Schrœdinger equation, $\mathcal{H}\Psi_0=0$ is equivalent to a Riccati equation, $$-\frac{1}{2L}\frac{d}{d L} (L^2 K) +L K^2 + 1 =0$$ which is solved¹ by a ratio of Bessel functions. $$K(L) = \frac{1}{\sqrt{L}}\frac{J_2(4\sqrt{L})}{J_1(4\sqrt{L})}$$ (There’s a constant of integration that is fixed by demanding that $\Psi_0$ be normalizable. This choice also, as they argue, yields the correct asymptotically-free UV behaviour and IR behaviour corresponding to confinement and a mass-gap.)

In addition to the predictions for the spectrum of spin-$0$ glueballs that I discussed previously, they also produce predictions for the spectrum of higher-spin glueball states.