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Infrared gluons in the stochastic quantization approach

Takuya Saito （ Kochi), Nakagawa Yoshiyuki (Osaka), Nakamura Atsushi (Hiroshima), Toki Hiroshi (Osaka). Contents Introduction Method: Stochastic gauge fixing Gluon propagators Numerical results Summary. Infrared gluons in the stochastic quantization approach. Introduction(1).

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Infrared gluons in the stochastic quantization approach

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  1. Takuya Saito（Kochi), Nakagawa Yoshiyuki(Osaka), Nakamura Atsushi(Hiroshima), Toki Hiroshi(Osaka) Contents Introduction Method: Stochastic gauge fixing Gluon propagators Numerical results Summary Infrared gluons in the stochastic quantization approach Lattice2008

  2. Introduction(1) • Confinement • Quarks and gluons are basic quantities of QCD. In ultraviolet region, the perturbative QCD works well but in the confining region, some non-perturbative modes dominates hadron physics. • Infrared physics of QCD: Confinement, Chiral symmetry breaking; these non-perturbative phenomena are deeply related to infrared singularities of QCD. • Infrared (transverse) gluon propagators • If confinement exists, one can expects that a transverse gluon propagator has an infinite mass, and will vanish in the IR limit. • On the other hands, the ghost propagator diverges in the IR limit. • We can find many lattice studies for these in many references; however, there are no distinctive signals, particularly for gluons. Lattice2008

  3. Introduction(2) === Some difficultiesfor lattice calculations for gluons === • Numerical difficulty : • Finite volume size effect; the infrared physics requires large lattices. • Gauge fixing computation on the large lattices is very hard, time-consuming simulations if we use the iterative gauge fixing. • Conceptual difficulty: • Lattice configuration can not be gauge-fixed uniquely due to Gribov ambiguity. • We expect that the Gribov copy configuration will fade the infrared physics we are interested in. • Gribov copy problem is not fully understood now. Lattice2008

  4. Introduction(3) === Aim in this study === • Calculations of the gluon propagator in the stochastic quantization with the Coulomb gauge • This method has some advantage: • We do not use the iterative gauge fixing method. • Gauge configurations go to the Gribov region automatically. • Gauge parameter is easy to change. • Measure of the transverse gluon propagators • Transverse gluon propagator is a physical quantity. • We expect that the gluon propagator in the infrared limit will be suppressed with an infinite effective masses. This means gluons are confining. Lattice2008

  5. Method(1) === Stochastic quantization with the gauge fixing === Langevin equation for the gauge theory with the gauge fixing ( a la Zwanziger) Virtual time for the hypothetical stochastic process Gauge parameter Stochastic Gauge fixing ：D.Zwanziger,Nucl.Phys.B192(1981) Gaussian white noise Lattice2008

  6. Method(2) === Stochastic quantization on the lattice === Driving force Gauge rotation Lattice generalization of stochastic gauge fixing：A.Nakamura and M. Mizutani, Vistas in Astronomy (Pergamon Press,1993), vol.37 p.305. , M. Mizutani and A.Nakamura, Nucl. Phys. B (Proc.Suppl.)34(1994),253. Lattice2008

  7. Method(3) === Conceptual reason for using SGF === • Conceptual reason • Gauge copy problemGauge configurations not fixed completely on the non-perturbative lattice calculation • Gauge fixing term of SGF • It makes gauge configurations go to the Gribov region. • This term works as an attractive driving force. • More effective approach Lattice2008

  8. Langevin steps Method(4) === Practical reason for using SGF === ～ Monte Carlo Quantization～ • Practical reason • For a gauge fixing, we don’t use any iterative methods and so there is no critical slowing down of this algorithm. It is a great advantage for large lattice simulation with gauge fixing. • Changing a gauge parameter is easier than the iterative methods. Monte Carlo Steps Gauge rotations ～ Stochastic Quantization ～ Lattice2008

  9. Coulomb gauge QCD === basic issues === • Hamiltonian of Coulomb gauge QCD • A transverse part makes a physics gluon field. • A source term makes a color-Coulomb instantaneous (confining ) potential among quarks, causing by a singular eigenvalue of F.P. • No negative norm : A physical interpretation is very clear. Lattice2008

  10. Gluon propagators(1) === General form in the perturbative region === General form of gluon propagators For free case, we have If adding an anomalous dimension, we have Lattice2008

  11. Gluon propagators(2) === Assumptions in the non-perturbative region === Mandlestamhypothesise ( if the confining potential is linear ) Gluon propagator with an effective mass Gluon propagator vanishes in the IR limit Lattice2008

  12. Gluon propagators(3) === Gluon propagators on the lattice === Gauge field on the lattice in this calculation Fourier transform Gluon correlators ( we’ll measure ) Lattice2008

  13. Numerical parameters Quenched Wilson action simulations with hypercubic lattices Simulation parameters Lattice2008

  14. Numerical result (1) === Volume dependence at beta=6.0 === Flat in the IR region, but not suppressed. Not diverge in the IR region. All the data are on the same line. For largest volume (64)4=(6.4fm) 4 Lattice2008

  15. Numerical result (2) === Volume dependence at beta=5.7 === Flat in the IR region, but not suppressed. Not diverge in the IR region. All the data are on the same line. For largest volume (32)4=(5.4fm) 4 Lattice2008

  16. Numerical result (3) === α-parameter dependence at beta=5.7 === In the UV region, small variation with α In the IR region, large change with α? For smallest α, we got better result. Lattice2008

  17. Summary • We try to calculate gluon propagators in the confinement region in the stochastic gauge fixing method with the Coulomb gauge. • For this new calculation, we need more information and arguments. • We find sign of an infrared suppression of gluon propagators. • Larger physical volume ? • We find that the infrared gluons are strongly affected by variation of alpha-gauge parameter. • Why ? • We need investigation of the lowest eigenvalue of FP operator, the relation of the sharp gauge, etc. Lattice2008

  18. Method(5) === Disadvantage for using SGF === Langevin step dependence Lattice2008

  19. Gauge fixing term Gauge fixing term  α-paramter small, dτ  small  more computation time Lattice2008

  20. Numerical results of Gluon propagators Volume dependence , beta dependence , alpha parameter dependence Lattice2008

  21. Numerical results (1) Lattice2008

  22. Numerical results (1) Lattice2008

  23. Lattice2008

  24. 　クーロンゲージQCDにおけるハミルトニアン 　クーロンゲージQCDにおけるファデーフポボフ 　グルーオン伝播関数の時間成分 クーロンゲージQCD 遅延部分 瞬間力部分 JPS２００６S

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