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Vishnu Jejjala,a Michael Kavic,b Djordje Minic,b Chia Tze,b a IHES, Le Bois-Marie, 35, route des Chartres, F-91440 Bures sur Yvette, France b Department of Physics, Virginia Tech Introduction With the new Large Hadron Collider (LHC) becoming operational in the near future, our understanding of quantum chromodynamics (QCD) is essential in analyzing the data to be collected. One area in which we lack understanding is in the nonperturbative effects of the theory. Understanding the non-perturbative dynamics of YangMills theory will bring us one step closer to this goal. Currently, lattice gauge theory has made great progress in understanding QCD, however, analytic understanding has not come as far. In our present work, we consider Yang-Mills theory in (2+1) and (3+1) dimensions with a large number of colors. The primary focus of our research is to construct the spectrum of gauge-invariant glueball states. In the 2+1 case, we use a Hamiltonian approach proposed by Karabali, Kim, and Nair (1997) in which the theory is rewritten in terms of gauge-invariant “corner” variables. Using this approach, analytic computations can be done. In the 3+1 case, the Karabali, Kim, and Nair formalism is extended from 2+1 to 3+1 using corner variables (Bars 1978). This extension allows us to compute our results in 3+1 using the same physical insight and analytic tools as in the 2+1 case. Vacuum Wave Functional In 2+1 dimensions, we take as the vacuum wavefunctional ansatz Comparison With Lattice (2+1) 0++ states The Scrödinger equation becomes (2+1) 0– states where E0 is a divergent vacuum energy The kernel equation is (2+1) 2++ and 2-+ states which has a general solution in terms of Bessel functions Only one solution is normalizable and has the correct asymptotics in the UV and IR limits and it is given by Summary of Results (3+1) J++ states Determined a new non-trivial form of the vacuum wavefunctional by solving the Schrodinger equation for (2+1) and (3+1) Yang-Mills theory. Computed glueball mass spectrum in (2+1) and (3+1) Yang-Mills theory. The 0++ glueball mass in (2+1) is statistically indistinguishable from the lattice results! UV: IR: In 3+1 dimensions, we take as the vacuum wave functional ansatz Computed string tension to within 1% of lattice result. Note that in the 2+1 case (3+1) J-+ states The Schrödinger equation with this wave functional yields the kernel equation Background We begin with (2+1) Yang-Mills theory. The Hamiltonian for the system is Again, only one normalizable solution with the correct asymptotics is found QCD String where E is conjugate to A in the temporal gauge A0 = 0. We choose the dynamics fields to be We quantize the system using Observable quantities and physical states must be gauge-invariant as a consequence of Gauss’ law By calculating the expectation value of a large spatial Wilson loop, the string tension is determined to be The Bessel function is essentially sinusoidal, so its zeros are evenly spaced (better for large n) Thus, the predicted spectrum has approximate degeneracies This agrees within 1% of the lattice result The spectrum is organized into bands concentrated around a given level Mass Spectrum •Background Independent Matrix Theory • We parameterize the gauge fields by • M transforms linearly under gauge transformations Glueball states may found by computing the equal-time correlators of gaugeinvariant probe operators with the correct JPC quantum numbers Preliminary counting suggests that there is an approximate (in the sense that the degeneracies are not exact) Hagedorn spectrum of states We believe that this is a basic manifestation of the QCD string For example, 0++ is probed using Tr (B2): We can expand the kernel using the formula Future Prospects • Gauge-invariant variables are constructed using • The volume measure of the configuration space is given by a hermitian Wess-Zumino-Witten action • The volume of the configuration space is finite. • Let us introduce the current • In terms of the current, the Yang-Mills Hamiltonian is Extension of method to include fundamental fermions (QCD) and other types of matter to get Application to Yang-Mills theories at finite temperature Computation of the spectrum of baryons Mn are mass constituents given by 2+1: Computation of scattering amplitudes 3+1: where 2,n are the ordered zeros of J2 and 3,n are the ordered zeros of J2. At large separation distances, we find contributions of single particle poles • Extension to supersymmetric and superconformal gauge theories Condensed matter and Statistical Mechanics application: 3D Ising model, High-Tc superconductivity, etc. with Glueball masses are a sum of their constituents. • m is the ‘t Hooft coupling References R. G. Leigh, D. Minic and A. Yelnikov, Phys. Rev. Lett. 96:222001 (2006); hep-th/0512111. R. G. Leigh, D. Minic and A. Yelnikov, hep-th/0604060. L. Freidel, R. G. Leigh and D. Minic, Phys. Lett. B641:105-111 (2006); hep-th/0604184. L. Freidel, hep-th/0604185. L. Freidel, R. G. Leigh, D. Minic and A. Yelnikov, hep-th/0801.1113.