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Near Light Cone QCD On The Lattice

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. Outline. Why Near-Light-Cone coordinates ? Hamiltonian formulation: Continuum Hamiltonian Lattice Hamiltonian

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Near Light Cone QCD On The Lattice

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  1. 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 Near Light Cone QCD

  2. 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 Near Light Cone QCD

  3. 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 Near Light Cone QCD

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

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

  6. 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 Near Light Cone QCD

  7. 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 Near Light Cone QCD

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

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

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

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

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

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

  14. 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 Near Light Cone QCD

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

  16. 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 Near Light Cone QCD

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

  18. 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 Near Light Cone QCD

  19. 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 Near Light Cone QCD

  20. Near Light Cone QCD

  21. Near Light Cone QCD

  22. 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) Near Light Cone QCD

  23. Renormalization • Euclidean LGT: String tension by expectation values of extended timelike Wilson loops • Lorentz invariance time is not a special coordinate heavy quark potential may be extracted from spacelike Wilson loops, too • 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 • There are two equivalent ways to extract the string tension for quarks which are seperated along the 2-axes • From these one can extract Near Light Cone QCD

  24. 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 • By choosing fixed we are able to decrease without crossing the critical line Phase transition • 1st order phase transition in accordance with the Ehrenfest classification • Analytic estimate (strong coupling) Near Light Cone QCD

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