Quenched staggered eigenvalues Eduardo Follana, Christine Davies, Alistair Hart HPQCD and UKQCD SUPA: Universities of Glasgow and Edinburgh January 2006 Motivation Summary of: Phys. Rev. Lett. 93 (2004) 241601, Phys. Rev. D 72 (2005) 054501, Lat04, Lat05... Study of low-lying (improved) staggered Dirac eigenmodes quenched gauge backgrounds. Results compared with: continuum QCD expectations, Random Matrix Theory. (Constrains A. Hasenfratz’s musings on possible additive quark mass renormalisation [hep-lat/0511021]). Continuum QCD spectral expectations D / anti-Hermitian ⇒ Df / s = i λs f s {D, / γ5 } = 0 ⇒ sp(D) / = {±i λs , λs ∈ R}. ( ±1 if λs = 0, with number n± Chirality: χs = fs† γ5 fs = 0 otherwise Topological charge: Q= R F F̃ ∈ Z Atiyah-Singer index theorem: Q = n+ − n− The staggered spectrum D / anti-Hermitian ⇒ Df / s = i λs f s P {D, / ǫ(x)} = 0 with ǫ(x) = (−1) µ xµ . ⇒ sp(D) / = {±i λs , λs ∈ R}. Continuum limit: SU(4)⊗SU(4) chiral symmetry Expect (Nt = 4)-fold copy of continuum Finite-a: O(a2 ) corrections: Near zero modes: Q × Nt /2 = 2Q positive modes Non-zero modes: Nt = 4-plets, slightly split Do we see this continuum picture? How large are the O(a2 ) corrections? Measuring topological charge No topological stability at finite a. O(a2 ) ambiguity in Q (Improved) staggered: no exact index GW: index, but no exact index theorem Us: 4 gluonic and 1 fermionic method Each agrees with another to within 10%. Simulation details Glue: SU(3), d = 3 + 1, quenched, tadpole improved Symanzik action Quarks: staggered actions Naı̈ve, Asqtad, Hisq(-ish). Lattices: a/fm 0.125 0.093 0.077 size, aL/fm 1.1 1.5 1.9 × × × × × The staggered spectrum Wilson Glue (a ≈ 0.1 fm, V = 163 x 32) 1 FAT7xASQ ASQTAD ONE-LINK CHIRALITY 0.8 0.014 EIGENVALUES 0.012 0.01 0.6 0.008 0.4 0.006 0.2 0.004 0.002 0 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 10 12 14 16 10 12 14 16 Improved Glue (a ≈ 0.093 fm, V = 164) 1 0.09 FAT7xASQ ASQTAD ONE-LINK CHIRALITY 0.8 EIGENVALUES 0.08 0.07 0.06 0.6 0.05 0.04 0.4 0.03 0.2 0.02 0.01 0 0 2 4 6 8 10 12 14 16 2 4 6 8 Improved Glue (a ≈ 0.077 fm, V = 204) 1 0.09 FAT7xASQ ASQTAD ONE-LINK CHIRALITY 0.8 EIGENVALUES 0.08 0.07 0.06 0.6 0.05 0.04 0.4 0.03 0.2 0.02 0.01 0 0 2 4 6 8 10 12 14 16 2 4 6 8 Chirality gap 1 FAT7XASQ 0.8 0.6 0.4 fat7xasqtad, fine lattice 0.8 0.2 0 0.0001 0.7 0.001 0.01 0.1 1 0.6 ASQTAD 0.8 0.5 0.6 0.4 0.4 0.2 0 0.0001 0.3 0.001 0.01 0.1 0.2 1 ONE-LINK 0.1 0.8 0 1e-08 0.6 0.4 0.2 0 0.0001 0.001 0.01 0.1 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Near zero modes and lattice spacing |Q|=1 20 |Q|=2 ONE-LINK ASQTAD FAT7xASQ aλ0 = 0 + O(a2 ) <Λ0>Q / MeV 15 10 5 0 0 0.01 0 0.01 2 (Lattice spacing, a / fm) Near zero modes and volume |Q|=1 |Q|=2 20 ONE-LINK FAT7xASQ √ aλ0 = O(1/ a4 V ) <Λ0>Q / MeV 15 10 5 0 0 0.3 0.6 0 4 0.5 1/(a V) 0.3 / fm -2 0.6 Non-zero modes and lattice spacing Q=0 |Q|=1 |Q|=2 <Λ2>Q / MeV 125 ONE-LINK ASQTAD FAT7xASQ 100 75 50 100 <Λ1>Q / MeV aλs = const. + O(a2 ) 75 50 25 0 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 2 (Lattice spacing, a / fm) 0.005 0.01 0.015 Non-zero modes and volume |Q|=1 Q=0 |Q|=2 <Λ2>Q / MeV ONE-LINK FAT7xASQ aλs = O(1/(a4 V )) 200 100 <Λ1>Q / MeV 0 200 100 0 0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 4 -4 1/(a V) / fm 0.2 0.4 0.6 Intra-quartet splitting and lattice spacing Q=0 |Q|=1 |Q|=2 30 ONE-LINK ASQTAD FAT7xASQ <δΛ1>Q / MeV 20 10 0 0 0.01 0 0.01 0 2 (Lattice spacing, a / fm) 0.01 Conclusions Features of continuum spectrum well reproduced (Also predictions of Random Matrix Theory.) Near-zero and non-zero modes clearly distinguished O(a2 ) corrections small Corrections → 0 as a → 0, V → ∞. How small do we need? e.g. (a = 0.09, L = 20): λ0 ∼ √ (3 + Q) MeV Q̄ = χV = 2.6 ⇒ λ¯0 ∼ 5 MeV ≃ mu,d Suggests any additive quark mass renormalisation also small (also volume dependent?)