A new method of calculating the running coupling constant --- numerical results --- Etsuko Itou (YITP, Kyoto University) Lattice of William.

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Presentation transcript:

A new method of calculating the running coupling constant --- numerical results --- Etsuko Itou (YITP, Kyoto University) Lattice of William and Mary

Numerical simulation was carried out on the vector supercomputer NEC SX-8 in YITP.

1.Introduction  Recently, it is suggested that there can exist a conformal fixed point in large flavor QCD using the running coupling in Schroedinger Functional scheme.  It is important to confirm this result using an independent method.  We develop a new scheme ("Wilson loop scheme") for the running coupling constant.  We carry out a quenched QCD test of our scheme.

Outline 1. Introduction 2. Basic idea –summary of the method- 3. Simulation parameters 4. Simulation details 5. Results 6. Conclusion

1.Basic idea –summary of the method-  fix the free parameter in the renormalization condition  take the continuum limit  is the scale which defines the running coupling constant of step scaling We choose the renormalization scheme: renormalized coupling

To take the continuum limit, we have to set the scale “ ”. It corresponds to tuning to keep a certain input physical parameter constant. How to take the continuum limit Examples of input physical parameters: Sommer scale, Note: available only for low energy scale Alpha collaboration (Nucl.Phys. B544 (1999) , S. Capitani et. al.) step scaling in Schroedinger functional scheme Choose as a constant input, is an output. Our choice in this quenched QCD test Choose or Sommer scale as inputs, are outputs.

2. Simulation parameters  pseudo-heatbath algorithm and Over-relaxation  # of gauge configurations 100  periodic b.c. and twisted b.c. (’t Hooft,1979) lattice  parameter sets of the lattice size and bare coupling to keep the input physical quantities constant (Today’s talk)

Set1Set2Set3Set4Set5 betaL 0 /a (s=1) L 0 /a (s=2) betaL 0 /a (s=1) L 0 /a (s=2) betaL 0 /a (s=1) L 0 /a (s=2) betaL 0 /a (s=1) L 0 /a (s=2) betaL 0 /a (s=1) L 0 /a (s=1.5) L 0 /a (s=2) (8) (8) (8) (8) (8) (10) (10) (10) (10) (10) Ref 1 : Set1-4 (Nucl.Phys. B544 (1999) , S. Capitani et. al.) Ref 2 : Set5 (Nucl.Phys. B535 (1998) , M. Guagnelli et. al.) is constant for each column. (Ref.1) Sommer scale is a constant. (Ref.2) High energy Low energy Parameter sets of the lattice size and bare coupling Set 1 Set 2 In this test, we study the step scaling in our scheme.

3.Simulation details We define the renormalized coupling constant in our scheme: is estimated by calculating the Creutz ratio. Renormalized coupling in “Wilson loop scheme”

 Smearing of link variables  Interpolation of the Creutz ratios  Extrapolation to the continuum limit of the running coupling APE Smearing of the link variables Definition: smearing level : n smearing parameter: Technical steps

definition : nr=0.25r=0.30r=0.35 n=1L 0 /a >10L 0 /a >8.3L 0 /a >7.1 n=2L 0 /a >18L 0 /a >15L 0 /a >12.8 n=3L 0 /a >26L 0 /a >21.6L 0 /a >18.5 Table: The lower bound for L 0 /a  Discretization error should be controlled larger r  Noise (statistical error) should be small smaller r or higher n  Oversmearing should be avoided n should be smaller than R/2, Conditions for good choice of r and n Oversmearing for n=1,2 Optimal choice!!

Interpolation of the Creutz ratios Fit function : Fit ranges : To obtain the value of the Creutz ratios for noninteger R, we have to interpolate them. Ex) L 0 /aR+1/2R minR max

Extrapolation to the continuum limit of the running coupling Fit function: Set 1

4.Results Set1

Set2 The parameter set to give step scaling in SF scheme also gives step scaling in our scheme!

Set1,2 Set3

Set1 - 3 Set4

Set1 -4 Set5

1 loop MC

1 loop 2 loop MC

5.Conclusion  We calculate the running coupling of quenched QCD in “Wilson loop scheme”.  The number of gauge configurations is only 100, however, we have shown that smearing drastically reduces the statistical error.  We found there is a window for the parameters (r,n) which both the statistical and discretization errors are under control.  This method is promising. We will investigate the large flavor QCD using this new renormalization scheme.