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Vishnu Jejjala,a Michael Kavic,b
Djordje Minic,b Chia Tze,b
IHES, Le Bois-Marie, 35, route des
Chartres, F-91440 Bures sur Yvette, France
b Department of Physics, Virginia Tech
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
 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
 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
 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
 Computation of the spectrum of baryons
 Mn are mass constituents given by
 Computation of scattering amplitudes
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.
 Glueball masses are a sum of their constituents.
• m is the ‘t Hooft coupling
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.