Absolute X-distribution and self-duality

Motivation

Various models of QCD vacuum predict that it is dominated by excitations that are predominantly self-dual or anti self-dual. In this work we look at the tendency for self-duality in the case of pure-glue SU(3) gauge theory using the overlap-based definition of the field-strength tensor. To gauge this tendency, we use the absolute X-distribution method which is designed to quantify the dynamical tendency for polarization for arbitrary random variables that can be decomposed in a pair of orthogonal subspaces.

Ensembles

$$ \def\fm {\,{\tt fm}} \def\MeV {\,{\tt MeV}} \def\GeV {\,{\tt GeV}} \def\degC{\,{^\circ{\tt C}}} \def\degK{\,{\tt K}} $$

Ensemble	Size	Volume	$$N_\text{config}$$ dmatrix	$$N_\text{config}$$ eigenmodes	Lattice spacing	Iwasaki $$\beta$$	Location
$$E_1$$	$$8^4$$	$$(1.32\fm)^4$$	0?	100	$$0.165\fm$$	$$2.295$$	dlx:scratch/overlap_eigensystem
$$E_2$$	$$12^4$$	$$(1.32\fm)^4$$	400	97(3)	$$0.110\fm$$	$$2.53$$	dlx:scratch/overlap_eigensystem dlx:scratch/topo12/dmatrices
$$E_3$$	$$16^4$$	$$(1.32\fm)^4$$	200	99(1)	$$0.0825\fm$$	$$2.725$$	dlx:scratch/overlap_eigensystem dlx:scratch/topo16/dmatrices
$$E_8$$	$$20^4$$	$$(1.32\fm)^4$$	80	0	$$0.066\fm$$	$$2.892$$(?)	dlx:scratch/topo20/dmatrices
$$E_4$$	$$24^4$$	$$(1.32\fm)^4$$	39(1)	96(4)	$$0.055\fm$$	$$3.0375$$	dlx:scratch/overlap_eigensystem dlx:scratch/topo24/dmatrices
$$E_7$$	$$32^4$$	$$(1.32\fm)^4$$	19(1)	0	$$0.041\fm$$	$$3.278$$(?)	dlx:scratch/topo32
$$E_5$$	$$16^4$$	$$(1.76\fm)^4$$	0	99(1)	$$0.110\fm$$	$$2.53$$	dlx:scratch/overlap_eigensystem
$$E_6$$	$$32^4$$	$$(1.76\fm)^4$$	20	20	$$0.055\fm$$	$$3.0375$$	qcd.phys.gwu.edu:scratch/topo32new
$$E_9$$	$$12^3\times6$$	$$(1.32\fm)^3\times0.66\fm$$	200	100	$$0.110\fm$$	$2.53$	pyramid:scratch/iwasaki_configs/conf126 gpucluster:scratch/iwasaki_configs/conf126

The number of configurations in each ensemble is given by the number of dmatrix files computed. The number in parentheses in the $$N_\text{config}$$ column is the number of corrupted dmatrix files in that ensemble. The $$\beta$$ parameter determination is discussed here. The values that are marked with a question mark are inferred using the lattice spacing, whereas the other values are based on the following e-mail exchange with Jianbo Zhang:

Date: Thu, 04 Mar 2004 09:11:43 +1030
From: Jianbo Zhang <jzhang@physics.adelaide.edu.au>
To: Ivan Horvath <horvath@pa.uky.edu>
Subject: Re: 8^4 configs

Hi Ivan,

   I used the table which Frank. Lee give me,  the table presents the
\beta value and corresponding lattice spacing.    and using string
tension.   You can ask Frank Lee
for details.

  for 8^4  , I use 16 processors ( 2,2,2,2),   \beta = 2.295
  for 12^4, Iuse 27 processors ( 1,3,3,3),    \beta = 2.53
  for 16^4, Iuse  64 processors ( 2,2,4,4) ,   \beta = 2.725
  for 24^4, I use  108 processors ( 2,2,6,6),   \beta = 3.0375
BTW,   12^4 50 configurations have finished and 16^4 now have 20 configs.

Jianbo

To do

Some of the configurations seem corrupt. For now we will remove them from the analysis but they need to be fixed. The following configurations have problems:

• dmatrix g3232iwa041_364.Dmatrix from ensemble $E_7$. The total charge for this dmatrix file is -8.5689925884e-01, which is significantly different from an integer value. For the other dmatrices the difference is of the order $$O(10^{-4})$$. Running the get_fmunu_components_find0 code on this dmatrix reports 16477 exact zero entries for $$F_{\mu\nu}$$ which is most likely due to the fact that these entries are corrupted. We need to identify the position of these entries and re-run the calculation of the dmatrix for these points.
• dmatrix g2424iwa055_188.Dmatrix from ensemble $E_4$ has charge -2.5969878024e+00 and get_fmunu_components_find0 reports two exact zeros. No other configuration in this ensemble has a charge so different from an integer and exact zeros for $$F_{\mu\nu}$$.

Papers

• A. Alexandru and I. Horváth, How Self-Dual is QCD? , Physics Letters B (2011) arXiv:1110.2762.
• A. Alexandru, T. Draper, I. Horvath, and T. Streuer, The Analysis of Space-Time Structure in QCD Vacuum II: Dynamics of Polarization and Absolute X-Distribution, Annals of Physics (2011) arXiv:1009.4451.
• A. Alexandru, T. Draper, I. Horvath, and T. Streuer, Absolute Measure of Local Chirality and the Chiral Polarization Scale of the QCD Vacuum, PoS LATTICE2010 (2010) 082, arXiv:1010.5474.
• T. Draper, A. Alexandru, Y. Chen, S.-J. Dong, I. Horvath, et. al., Improved measure of local chirality, Nucl.Phys.Proc.Suppl. 140 (2005) 623–625, hep-lat/0408006.
• C. Gattringer, Testing the self-duality of topological lumps in SU(3) lattice gauge theory, Phys. Rev. Lett. 88 (2002) 221601, hep-lat/0202002.
• I. Horvath et. al., Local chirality of low-lying Dirac eigenmodes and the instanton liquid model, Phys. Rev. D66 (2002) 034501, hep-lat/0201008.
• I. Horvath, N. Isgur, J. McCune, and H. B. Thacker, Evidence against instanton dominance of topological charge fluctuations in QCD, Phys. Rev. D65 (2002) 014502, hep-lat/0102003.