1. Lattice QCD 2007Near Light Cone QCD Near Light Cone QCD On The Lattice H.J. Pirner, D. Grünewald E.-M. Ilgenfritz, E. Prokhvatilov Partially funded by the EU project EU RII3-CT-2004-50678 and the GSI

2. Lattice QCD 2007Near Light Cone QCD Outline Why Near-Light-Cone coordinates ? Hamiltonian formulation: –Continuum Hamiltonian –Lattice Hamiltonian Trial Wavefunctional Ansatz: –Strong Coupling Solution –Weak Coupling Solution –Optimization Simple Operator expectation values Conclusions

3. Lattice QCD 2007Near Light Cone QCD Why Near Light Cone QCD ? Near light cone Wilson loop correlation functions determine the dipole-dipole cross section in QCD. By taking into account the hadronic wave functions one obtains hadronic cross sections

4. Lattice QCD 2007Near Light Cone QCD Allow quantization on space like surfaces Near light-cone coordinates Near light-cone theory is a limit of equal time theory, Related to it by Lorentz boost

5. Lattice QCD 2007Near Light Cone QCD Near light-cone coordinates spatial distance ( ) scalar product Light Cone time axis along which the system evolves

6. Lattice QCD 2007Near Light Cone QCD Motivation Near light-cone QCD has a nontrivial vacuum Near light-cone coordinates are a promising tool to investigate high energy scattering on the lattice: NLC make high lab frame momenta accessible on the lattice with small

7. Lattice QCD 2007Near Light Cone QCD Euclidean Lattice Gauge Theory near the Light Cone ? Action remains complex -> sign problem Possible way out: Hamiltonian formulation Sampling of the ground state wavefunctional with guided diffusion quantum Monte-Carlo

8. Lattice QCD 2007Near Light Cone QCD Continuum Hamiltonian and momentum Perform Legendre transformation of the Lagrange density for : Then, the Hamiltonian is given by Linear momentum operator term disturbs Diffusion Monte Carlo The Hamiltonian has to be supplemented by Gauss law

9. Lattice QCD 2007Near Light Cone QCD It is sufficient to consider Constants of motion Translation operator (obtained via the energy-momentum tensor)

10. Lattice QCD 2007Near Light Cone QCD Hamiltonian only depends on the product Effective equal time theory with strong anisotropy Light cone limit due to strong boost related to The lattice Hamiltonian Derivation of the lattice Hamiltonian from the path integral formulation with the transfer matrix-method (Creutz Phys. Rev. D 15, 1128):

11. Lattice QCD 2007Near Light Cone QCD Effective Lattice Hamiltonian Adding the translation Operator one obtains the effective lattice Hamiltonian With only quadratic momenta

12. Lattice QCD 2007Near Light Cone QCD Strong Coupling Solution Strong coupling wavefunctional Energy density in the strong coupling limit: Product state of single plaquette excitations LC limit: transversal dynamics decouple

13. Lattice QCD 2007Near Light Cone QCD Weak Coupling Solution Weak coupling wavefunctional formed with magnetic fields Energy density in the weak coupling limit: Gaussian wavefunctional LC limit: longitudinal magnetic fields become unimportant

14. Lattice QCD 2007Near Light Cone QCD Variational optimization Trial wavefunctional Restrict to product of single plaquette wavefunctionals. Choose plaquettes which respect the Z(2) symmetry of the original Hamiltonian Optimize the energy density with respect to and The expectation values are computed via the prob. measure Produced by a standard local heatbath algorithm

15. Lattice QCD 2007Near Light Cone QCD Optimal and Strong coupling limit is reproduced (solid line) One sees the effect of the phases associated with the center symmetry

16. Lattice QCD 2007Near Light Cone QCD as translation operator : How close is to the exact generator of lattice translations ? (important for the applicability of QDMC) For every purely real valued wave- functional we have which follows from partial integration and is consistent with an exact eigen- state Look at the second moment Fluctuations of are always less than 1% of the total energy around its mean value equal to zero and may be neglected in realistic computations

17. Lattice QCD 2007Near Light Cone QCD Wilson Loop Expectation Values Trace of phase transition Assume constant string tension independent of orientation Possibility to extract realistic anisotropic lattice constants

18. Lattice QCD 2007Near Light Cone QCD Lattice spacings the transversal lattice constant is varying with the boost parameter SIGNAL! Should introduce two different couplings for the longitudinal and transversal part of the Hamiltonian three couplings which can be tuned in such a way that

19. Lattice QCD 2007Near Light Cone QCD Conclusions: Near light cone coordinates are a promising tool to calculate high energy scattering on the lattice Euclidean path integral as well as Diffusion Quantum Monte Carlo treatments of the theory are not possible due to complex phases during the update process An effective lattice Hamiltonian, however, avoiding this problem can be derived Simple trial wavefunctionals have been constructed for strong and weak coupling and optimized variationally Work is in progress to correct for unwanted dependences of the lattice constants on the near light cone parameter, then a calculation of a dipole- plaquette cross section is feasible

20. Lattice QCD 2007Near Light Cone QCD

21. Lattice QCD 2007Near Light Cone QCD

22. Lattice QCD 2007Near Light Cone QCD Equal time theories: Covariance matrix weakly off-diagonal Product of single site wavefunctionals suitable Decreasing Correlations among longitudinally separated plaquettes become increasingly important LC Limit Each plaquette is equally correlated with every other longitudinally separated plaquette Trial wavefunctional (Restrict to product of single site wavefunctionals)

23. Lattice QCD 2007Near Light Cone QCD There are two equivalent ways to extract the string tension for quarks which are seperated along the 2-axes Lorentz invariance time is not a special coordinate heavy quark potential may be extracted from spacelike Wilson loops, too Renormalization From these one can extract Euclidean LGT: String tension by expectation values of extended timelike Wilson loops Hamiltonian dynamics Lagrangean dynamics The string tension is obtained by fitting the exponential fall off of Wilson loops elongated along the long side n and the short side m to

24. Lattice QCD 2007Near Light Cone QCD 1st order phase transition in accordance with the Ehrenfest classification Phase transition Energy density for fixed optimal as a function of for different values of trafo corresponds to single minimum turns into two degenerate minima which differ by a trafo Analytic estimate (strong coupling) By choosing fixed we are able to decrease without crossing the critical line