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Gap Domain Wall Fermions

Gap Domain Wall Fermions. USQCD SFW, BU February 2008. DWF as a tool for lattice QCD. m f. m 0. m 0. <= 5 th direction =>. =. T. T. T. T. T. T. T. T. T. detD. b. T. T. T. b. -. +. F. Ls. =. b. T. b. -. +. =. T. transfer.

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Gap Domain Wall Fermions

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  1. Gap Domain Wall Fermions USQCD SFW, BU February 2008 P. Vranas, LLNL

  2. DWF as a tool for lattice QCD mf m0 m0 <= 5th direction => = T T T T T T T T T detD b ...... T T T b - + F Ls = b T b - + = T transfer Matrix with corresponding Hamiltonian H Chiral symmetry is restored as: e-k0Ls where k0 is the smallest eigenvalue of H H(m0) = c5DWilson(-m0) P. Vranas, LLNL

  3. Instantons a Lattice dislocations Zeroes of H(m0) and instantons R. Edwards, U. Heller, R. Narayanan, NPB 535 (1998) 403. Instantons > a Inactive Larger Brillouin zones Quenched QCD a-1 = 2 GeV * P. Vranas, LLNL

  4. At strong coupling Quenched QCD a-1 = 1 GeV R. Edwards, U. Heller, R. Narayanan, NPB 535 (1998) 403. P. Vranas, LLNL

  5. Gap Domain Wall Fermions • Improve DWF in the region 1 GeV < a-1 < 2 GeV. • Since the problem occurs when H(m0) is small multiply the Botzman weight with: det[H(m0)] = det[D(-m0)] • This is the same as inserting Wilson fermions with heavy mass in the supercritical region (for example m0 = 1.9). I will use 2 flavors. • This will forbid zero crossings at m0 and therefore enlarge the gap and reduce the residual mass. • It will suppress instantons with size near the lattice spacing which are a lattice artifact (dislocations). • Must check that the added Wilson fermions: - have hadron spectrum above the cutoff and are therefore irrelevant. - do not break parity (Aoki phase). - allow crossings due to instantons/anti-instantons with sizes > a (active topology). P. Vranas, LLNL

  6. Numerical simulations with quenched DWF and GDWF P. Vranas, LLNL

  7. Quenched DWF, GDWF scale matching • DWF data (diamonds) are from [RBC, PRD 69 (2004) 074502]. • Matching is better than 5%. • Use the rho to set the scale. P. Vranas, LLNL

  8. GDWF DWF a-1 = 1 GeV a-1 = 1.4 GeV a-1 = 2 GeV P. Vranas, LLNL

  9. Eigenvalue distribution • Distribution of the 100 smallest eigenvalues from 110 independent configurations • Here a-1 = 1.4 GeV DWF GDWF P. Vranas, LLNL

  10. The heavy Wilson flavors • The pion (diamonds), rho (squares) and nucleon (stars) masses for 2 flavor dynamical Wilson flavors with mass = -1.9. The straight line marks the cutoff P. Vranas, LLNL

  11. The residual mass a-1 = 1.0 GeV a-1 = 1.4 GeV mf = 0.02 and m0 = 1.9 DWF a-1 = 2.0 GeV DWF GDWF P. Vranas, LLNL

  12. About that pion a-1 = 1.0 GeV a-1 = 1.4 GeV Ls= 16 m0 = 1.9 a-1 = 2.0 GeV GDWF P. Vranas, LLNL

  13. Pions in the rough seas I “imitate” a dynamical simulation with: Ls = 24, mf = 0.005, V = 16 x 32, m0 = 1.9, beta=4.6. I Find: • a-1 = 1.356(75) GeV evaluated at mf . • mres = 0.00064(4) which is about 10% of mf. • Finally: mpion = 140(40) MeV at mf = 0.005. P. Vranas, LLNL

  14. Algorithmic and computational costs • Simple to implement as an extension of Wilson and DWF: Add the two force terms and the two HMC Hamiltonians. • For a 2 flavor DWF simulation it is an additional cost of 2 Dirac operators and therefore an additional 1/Ls cost. For Ls = 24 this is about 5%. P. Vranas, LLNL

  15. Net topology change • GDWF may reduce the net-topology sampling of the traditional HMC because it forbids smooth deformations of topological objects. The eigenvalue flow can not cross m0. • This is only an algorithmic issue. We need algorithms that can tunnel between topological sectors. • The same problem occurs in QCD anyway with or without GDWF. The QCD topological sectors are separated by energy barriers that become infinitely high as we approach the continuum. We have not been to small enough coupling yet in QCD to see the phenomenon. • GDWF resemble continuous QCD in this way even more. • In many cases net-topology change is not important provided one uses a large enough volume (see H. Leutwyler, A. Smilga, Phys. Rev D 46 (1992) 5607). • It is important to see crossings in the larger-instanton regime since they confirm a topologically active vacuum. The net index may be fixed but the appearance/disappearance of instantons/anti-instantons is a property of the QCD vacuum and has to be there. Obviously then for large enough volumes cluster decomposition ensures correct physics. • For research related to theta one has to address the algorithmic issue of net-topology sampling in any case. P. Vranas, LLNL

  16. GDWF: V=163 x 32, b=4.6, a-1 = 1.4 GeV P. Vranas, LLNL

  17. Conclusion • GDWF work very well in the 1 < a < 2 GeV range. • Residual masses ~ 0.001 at Ls=16 (10x faster than DWF) • Topology is held fixed. This is a general problem brought to us early by GDWF. It must be addressed. P. Vranas, LLNL

  18. P.M. Vranas, NATO Workshop, (2000) 11, Dubna, Russia, hep-lat/0001006. P.M. Vranas, hep-lat/0606014. T. Izubuchi, C. Dawson, Nucl. Phys. B (Proc. Suppl.) {\bf 106} (2002) 748. H. Fukaya, Ph.D. Thesis, Kyoto University, 2006, hep-lat/0603008. H. Fukaya et. al. hep-lat/0607020. H. Fukaya, S. Hashimoto,T. Kaneko, N. Yamada: Latt06, Chiral Symmetry 3. P. Vranas, LLNL

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