1. Infrared gluons in the stochastic quantization approach Lattice20081 Contents 1.Introduction 2.Method: Stochastic gauge fixing 3.Gluon propagators 4.Numerical results 5.Summary Takuya Saito （ Kochi), Nakagawa Yoshiyuki (Osaka), Nakamura Atsushi (Hiroshima), Toki Hiroshi (Osaka)

2. Introduction(1) Lattice20082 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.

3. Introduction(2) Lattice20083 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. === Some difficulties for lattice calculations for gluons ===

4. Introduction(3) Lattice20084 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. === Aim in this study ===

5. Method(1) Lattice20085 === Stochastic quantization with the gauge fixing === Stochastic Gauge fixing ： D.Zwanziger,Nucl.Phys.B192(1981) Langevin equation for the gauge theory with the gauge fixing ( a la Zwanziger) Virtual time for the hypothetical stochastic process Gauge parameter Gaussian white noise

6. Method(2) Lattice20086 === Stochastic quantization on the lattice === 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. Driving force Gauge rotation

7. Method(3) Lattice20087 === Conceptual reason for using SGF === Conceptual reason Gauge copy problem Gauge configurations not fixed completely on the non- perturbative lattice calculation Gauge fixing term of SGF 1.It makes gauge configurations go to the Gribov region. 2.This term works as an attractive driving force. 3.More effective approach

8. Method(4) Lattice20088 === Practical reason for using SGF === 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 ～ Monte Carlo Quantization ～ Gauge rotations ～ Stochastic Quantization ～ Langevin steps

9. Coulomb gauge QCD Lattice20089 === 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.

10. Gluon propagators(1) Lattice200810 === General form in the perturbative region === General form of gluon propagators For free case, we have If adding an anomalous dimension, we have

11. Gluon propagators(2) Lattice200811 === Assumptions in the non-perturbative region === Mandlestam hypothesise ( if the confining potential is linear ) Gluon propagator with an effective mass Gluon propagator vanishes in the IR limit

12. Gluon propagators(3) Lattice200812 === Gluon propagators on the lattice === Gauge field on the lattice in this calculation Fourier transform Gluon correlators ( we’ll measure )

13. Numerical parameters Lattice200813 Quenched Wilson action simulations with hypercubic lattices Simulation parameters

14. Numerical result (1) Lattice200814 === 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

15. Numerical result (2) Lattice200815 === 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

16. Numerical result (3) Lattice200816 === 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.

17. Summary Lattice200817 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.

18. Method(5) Lattice200818 === Disadvantage for using SGF === Langevin step dependence

19. Lattice200819 Gauge fixing term Gauge fixing term  α-paramter  small, dτ  small  more computation time

20. Lattice200820 Numerical results of Gluon propagators Volume dependence, beta dependence, alpha parameter dependence

21. Lattice200821 Numerical results (1)

22. Lattice200822 Numerical results (1)

23. Lattice200823

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