Intro. Strat. Tests Results Concl. Determination of topological charge following several definitions and preliminary results of χ t in N f = 1 + 2 Julien Frison , Ryuichiro Kitano, Nori Yamada KEK 34rd International Symposium on Lattice Field Theory Lattice’16 - Southampton - July 25th, 2016
Intro. Strat. Tests Results Concl. Introduction 1 The strong CP problem today Instanton contribution to the mass Topology on the lattice Topology ambiguity or mass ambiguity? Strategy 2 Objective Ensembles Tests on topological charge determination 3 Gradient flow at large flow time Continuum limit and universality Topological Charge Density Correlator Preliminary results in N f = 1 + 2 4 Spectrum and PCAC masses ( m u , χ t ) plot Conclusion 5
Intro. Strat. Tests Results Concl. The strong CP problem today Why is there no θ F ˜ F term in the Lagrangian? Trivial solution: m u e i θ = 0 Other popular solution: Peccei-Quinn mechanism (axion) m u = 0 solution New lattice computations make m MS = 0 very unlikely u Is m u = 0 physically defined without massless pion? Is perturbative MS really what we need? Non-perturbative contributions make this solution ill-defined What latticists should really check is whether χ physical = 0 t
Intro. Strat. Tests Results Concl. Instanton contribution to the mass ’t Hooft vertex Instanton computation [Dine:1410.8505] [Creutz:0711.2640] u u R L 10 m u � MeV � 1 I 0.1 s d R L ms dR s L 0.01 0.6 0.8 1.0 1.2 1.4 md 1 � Ρ 0 � GeV �
Intro. Strat. Tests Results Concl. Topology on the lattice On the lattice, Q is ill-defined too! Only defined on smooth configurations How arbitrary are the definitions? Are some better than others? Bosonic versus fermionic definitions Does continuum limit trivially remove ambiguity? Even with Wilson fermions? On Q or on � Q 2 � ?
Intro. Strat. Tests Results Concl. Topology ambiguity or mass ambiguity? Mass and topology are related through Ward identities Earlier works have tried to make both definitions compatible [Bochicchio’84-85-86] In general, arbitrary definitions will break singlet Ward identities at finite lattice spacing, and χ t ( m u = 0) = 0 is not guaranteed. In N f = 2 + 1, χ t ( m u = 0) = 0 has been empirically checked, ∝ � m − 1 agreeing with ChPT prediction χ − 1 t What in N f = 1 + smthg ? “SU(1) ChPT” makes no sense.
Intro. Strat. Tests Results Concl. Objective We want to determine χ t at m PCAC = 0 u In N f = 1 + 2, where m d = m physical so that the ’t Hooft s vertex effect is amplified Only m u will be taken close to zero We use Wilson-like fermions to study the worst scenario We choose parameters similar to BMW HEX2 N f = 2 + 1 ensembles
Intro. Strat. Tests Results Concl. Ensembles N f = 2 + 1 Ensemble (cross-check) β = 3 . 31 L¨ uscher-Weisz w/ HEX2 Clover ( a ∼ 0 . 116 fm ), = − 0 . 04, 16 3 × 32 m bare = − 0 . 07, m bare s ud N f = 1 + 2 Ensembles = − 0 . 04, 16 3 × 32 m bare = − 0 . 07 , − 0 . 093 , − 0 . 09756, m bare u ds A larger volume and a finer lattice are both being generated Other Ensembles Many quenched ensembles have been used for tests, either generated for this project or for another project
Intro. Strat. Tests Results Concl. Gradient flow at large flow time Fixed fermionic topology (finite temperature) 3 x4, β =3.2, Q=-2 Topological charge Topological charge 24 3 x4, β =3.2, Q=-2, Symanzik flow 3 x4 β =3.2 Q=-2 24 24 2 3 2 2 1 1 1 0 0 0 Q 5Li Q 5Li Q 5Li -1 -1 -1 -2 -2 -2 -3 -3 -3 -4 -4 -4 0 2 4 6 8 10 0 5 10 0 0.2 0.4 0.6 0.8 1 Symanzik flow time Iwasaki flow time DBW2 flow time Remark: c 1 increases both stability and convergence speed ( n c = (3 − 15 c 1 ) τ [Alexandrou:1509.04259]) Main ensembles Topological charge Topological charge Topological charge history N f =1+2 Clover N f =2+1 clover ensemble N f =1+2 Clover 20 20 20 20 cfg 180 cfg 190 Iwasaki flow cfg 200 Wilson flow cfg 210 Symanzik flow cfg 220 10 10 10 10 cfg 230 Q 5Li Q 5Li 0 0 0 Q 0 -10 -10 -10 -10 -20 -20 -20 -20 0 2 4 6 8 10 0 0 2 2 4 4 6 6 8 8 10 10 40 60 80 100 120 140 Iwasaki(X)/Symanzik(sq) flow time Configuration number Iwasaki flow time
Intro. Strat. Tests Results Concl. Continuum limit and universality Topological susceptibility L=10,12,16 V phys ~cst β = 2 . 256 Sym Iwa DBW2 c − 24 1 Sym X 0.922 0.914 0.908 Symanzik Iwa X X 0.961 0.948 22 Iwasaki DBW2 X X X 0.984 DBW2 - c 1 X X X X c − 20 1 β = 2 . 37 Sym Iwa DBW2 c − 1 Sym X 0.985 0.969 0.954 18 2 > Iwa X X 0.989 0.976 <Q DBW2 X X X 0.989 16 c − X X X X 1 14 β = 2 . 556 Sym Iwa DBW2 c − 1 Sym X 0.981 0.974 0.977 12 Iwa X X 0.994 0.991 DBW2 X X X 0.998 10 c − X X X X 2.2 2.3 2.4 2.5 2.6 1 β Quenched ensembles at fixed physical volume Strong correlations at finest ensemble Nevertheless individual Q values almost never agree/plateau The closer the c 1 the stronger the correlation
Intro. Strat. Tests Results Concl. Topological Charge Density Correlator Topological charge density correlator Topological charge density correlator 3 x32, various Iwasaki flow times N f =1+2 zero-temperature m u =-0.0093 16 3 x32, various Iwasaki flow times N f =1+2 zero-temperature m u =-0.0093 16 1 0.8 t=0 t=0 t=0.1 t=0.1 t=0.2 t=0.2 0.6 t=0.3 t=0.3 0 t=0.4 t=0.4 t=0.6 t=0.6 t=0.8 t=0.8 ∆ ln|<q(x,t)q(0,t)>| t=1.0 t=1.0 <q(x,t)q(0,t)> 0.4 t=1.3 t=1.3 t=1.6 t=1.6 -1 t=2.0 t=2.0 t=2.5 t=2.5 t=3.0 0.2 t=3.0 t=5.0 t=5.0 t=10 t=10 -2 0 -0.2 -3 0 5 10 15 20 0 5 10 15 20 25 2 2 x x
Intro. Strat. Tests Results Concl. Spectrum and PCAC masses Hadron spectrum 3 x32 Hadron spectrum N f =1+2 m l =-0.093 16 Meson masses 3 x32 1 N f =1+2 m f =-0.09756 16 light-light (unphysical) 1 strange-light strange-strange light-light (PQuenched) M π for N f =2+1 light-light vector strange-light 0.8 M K for N f =2+1 1 0.8 strange-strange M π for N f =1+2 light-light vector M K for N f =1+2 0,8 BMW central value for M π 0.6 0.6 M Eff MEff M Eff 0,6 0.4 0.4 0,4 0.2 0.2 0,2 0 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t t t PCAC mass 3 x32, m bare =(-0.07,-0.04) PCAC masses Clover 16 PCAC masses 3 x32 3 x32 N f =1+2 m l =-0.093 16 N f =1+2 m l =-0.09756 16 0.1 Nf 1+2 LL 0.1 0.1 Nf 1+2 HH Nf 1+2 HL light-light (unphysical) light-light (PQuenched) Nf 2+1 LL strange-light Nf 2+1 HH strange-light 0.08 strange-strange 0.08 strange-strange 0.08 Nf 2+1 HL d 4 <A 4 P>/2<PP> 0.06 0.06 0.06 M Eff M Eff 0.04 0.04 0.04 0.02 0.02 0.02 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t t t
Intro. Strat. Tests Results Concl. ( m u , χ t ) plot Topological susceptibility 3 x32) in N f =1+2 ( β =3.31, 16 90 Iwasaki flow DBW2 flow 80 70 2 > 60 <Q 50 40 30 0 0.005 0.01 0.015 0.02 0.025 PCAC = m HL PCAC - m HH PCAC /2 m u
Intro. Strat. Tests Results Concl. We suggest that the m u = 0 solution to the strong CP problem should be assessed in terms of χ t and not m u we have presented a strategy to estimate or bound the mistake the PCAC method could make We have presented preliminary results in N f = 1 + 2 Unfortunately we have not been able to explore much of the expensive Index ( D ov ) approach We have large statistical errors for the moment We need lighter quarks, finer ensembles, and probably larger volumes Investigating m u ∼ 0 ( χ t ∼ 0) may require specific methods (see hep-lat/1606.07175)
Intro. Strat. Tests Results Concl. Thanks for your attention!
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