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April 10, 2006

2+1 D Yang-Mills at Large-N

A while back, I pointed to an announcement by Leigh, Minic and Yelnikov, proposing a solution to 2+1 D Yang-Mills at large-NN. Well, the long version of their paper has appeared.

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) =g YM 2N2πJ aδδJ a+Ω ab(z,w)δδJ a(z)δδJ b(w)+12g YM 2¯J a¯J a \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 Ω ab(z,w)=NπD w ab𝒢(wz) \Omega^{a b}(z,w) = \frac{N}{\pi} D^{a b}_w \mathcal{G}(w-z) Here D=πN[J,] D = \partial - \frac{\pi}{N} [J, \cdot] and 𝒢(wz)\mathcal{G}(w-z) is the ordinary Green’s function, satisfying ¯ z𝒢=δ (2)(z)\overline{\partial}_z\mathcal{G} = \delta^{(2)}(z).

They take as an ansatz that the ground state wave functional takes the form Ψ 0(J)=exp(π2Nm 2tr¯JK(Δm 2)¯J) \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=g YM 2N2πm= \frac{g_{\text{YM}}^2 N}{2\pi} and Δ={D,¯}/2\Delta = \{D,\overline{\partial}\}/2, for some kernel K(Δm 2)K\left(\frac{\Delta}{m^2}\right). Formally, they then expand K(L)= n=0 c nL n K(L) = \sum_{n=0}^\infty c_n L^n and find that the Schrœdinger equation, Ψ 0=0\mathcal{H}\Psi_0=0 is equivalent to a Riccati equation, 12LddL(L 2K)+LK 2+1=0 -\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)=1LJ 2(4L)J 1(4L) 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 Ψ 0\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-00 glueballs that I discussed previously, they also produce predictions for the spectrum of higher-spin glueball states.

1 The trick is to write K(L)=4f(x)ddx(f(x)/x) K(L)= -\frac{4}{f(x)}\frac{d}{d x} (f(x)/x) with x=4Lx=4\sqrt{L}, which converts the (nonlinear) Riccati equation into the (linear) Bessel equation, f+1xf+(12x 2)f=0 f'' +\frac{1}{x} f' +\left(1-\frac{2}{x^2}\right) f =0

Posted by distler at April 10, 2006 8:02 PM

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Re: 2+1 D Yang-Mills at Large-N

Its certainly an interesting paper. There are certain parts that get a little hazy to me, for instance, its not clear to me why they restrict to quadratic order in solving the Schroedinger eqn (presumably b/c they want to restrict themselves to dealing with simple local ‘probing’ operators like tr(dbar j dbar j).

They feel this is morally equivalent to some sort of expansion alla string theory, where higher order terms end up probing spatial extent. But, I don’t quite see why that has to be the case, and you might worry that theres a lot of physics in there thats getting chopped off.

Still the numbers are eerily close to lattice results

Posted by: Haelfix on April 12, 2006 5:14 AM | Permalink | Reply to this

Re: 2+1 D Yang-Mills at Large-N

Why isn’t this filtered out:

Didn’t you explain that there is some basic filtering?

Posted by: Michael on April 14, 2006 10:17 AM | Permalink | Reply to this

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