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December 12, 2005

2+1 D Yang Mills

Leigh, Minic and Yelnikov have a very interesting announcement of new results on 2+1 D Yang Mills Theory. Using a formalism pioneered by Karabali and Nair, they compute the glueball spectrum analytically at large-NN. The result is expressed in terms of zeroes of the Bessel function.

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


M(z,z¯)M(z,z¯)h (z¯),M (z,z¯)h(z)M (z,z¯) 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)h(z) is an SL(N,)SL(N,\mathbb{C})-valued holomorphic function.

The Yang-Mills Hamiltonian, =tr(g YM 2E 2+1g YM 2B 2) \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”, JJ (and its conjugate momentum), where J=NπHH 1 J = \frac{N}{\pi} \partial H H^{-1} is gauge-invariant, and transforms inhomogeneously JhJh 1+Nπhh 1 J\to h J h^{-1} + \frac{N}{\pi} \partial h h^{-1} under (1). ¯J\overline{\partial}J transforms homogeneously, as does the “covariant derivative”, D=πN[J,] D= \partial - \frac{\pi}{N} [J,\cdot]

Leigh et al claim to have found the ground state wave functional1, Ψ 0(J)=exp(π2Nm 2tr¯JK(L)¯J) \Psi_0(J)= \exp\left(-\frac{\pi}{2N m^2}\int\tr \overline{\partial}J\, K(L)\, \overline{\partial}J\right) where m=g YM 2N2πm= \frac{g_{\text{YM}}^2 N}{2\pi} and L=D¯m 2L=\frac{D\overline{\partial}}{m^2}. K(L)K(L) is given 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})}

In the large-NN 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, J ν1(z)J ν(z)=2νz+2z n=1 1z 2j ν,n 2 \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 ν,n 2j^2_{\nu,n} are the ordered zeroes of J ν(z)J_\nu(z), one has K 1(k)=12 n=1 M n 2k 2+M n 2,M n=j 2,nm/2 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 ++J^{PC}=0^{++} glueballs couple to tr(¯J¯J)\tr(\overline{\partial}J\overline{\partial}J) tr(¯J¯J)(x)tr(¯J¯J)(y)(K˜ 1(xy)) 2=(142π|xy| n=1 (M n) 3/2e M n|xy|) 2 \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 ++0^{++} glueballs are given as sums M n+M n=(j 2,n+j 2,n)m/2M_{n}+M_{n'}= (j_{2,n}+j_{2,n'}) m/2.

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

Update (12/13/2005):

Georg von Hippel (the new proprietor of Life on the Lattice) has some interesting comments about this paper, including the following observation.

For large zz, J ν(z)2πzcos(zπν2π4) J_\nu(z)\sim \sqrt{\frac{2}{\pi z}} \cos\left(z-\frac{\pi\nu}{2}-\frac{\pi}{4}\right) and hence, for n1n\gg 1, the zeroes of the Bessel function that enter into the formulæ for the masses of the 0 ++0^{++} and 0 0^{--} glueballs above go like2 j 2,n(n+3/4)πfor n1 j_{2,n}\sim (n+3/4)\pi\quad \text{for }\, n\gg 1 So, for large nn, one has a large number (𝒪(n)\mathcal{O}(n)) of nearly-degenerate 0 ++0^{++} glueballs of mass (n+3/2)πm/2\sim (n+3/2)\pi m/2, corresponding to different ways of partitioning nn into a pair of positive integers, n 1,n 2n_1,n_2. And, similarly, for the 0 0^{--} glueballs.

1 Here and below, everything that looks like a 2-form has been converted to a scalar by dividing by the area element, i2dzdz¯\frac{i}{2} dz\wedge d\overline{z}.

2 Actually, you don’t need to go to very large nn. For n4n\geq4, the deviation from this simple linear formula is, at most, a 1% effect.

Posted by distler at December 12, 2005 12:51 AM

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Re: 2+1 D Yang Mills

I know what i want to tell u has nothing to do with the subject of the note.
but please help me as matematician.
are likely all the facts that lies behind the “hidden messages” of the bible?
i dont mean about faith..i mean about the statistics of the that subjects.

Posted by: bonhamled on December 18, 2005 1:52 PM | Permalink | Reply to this

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