The Quest for Solving QCD: Light Quarks with Twisted Mass Fermions Karl Jansen • Introduction • New Formulations of Lattice Fermions: Overlap contra Twisted Mass Fermions • Dynamical Quarks – Understanding the Phase Structure of Lattice QCD – Breakthrough in Simulation Algorithm • Precision results from N f = 2 dynamical twisted mass fermions • Summary
Quarks are the fundamental constituents of nuclear matter Friedman and Kendall, 1972) f ( x, Q 2 ) � x ≈ 0 . 25 ,Q 2 > 10GeV independent of Q 2 � ( x momentum of quarks, Q 2 momentum transfer) Interpretation (Feynman): scattering on single quarks in a hadron → (Bjorken) scaling
Quantum Fluctuations and the Quark Picture analysis in perturbation theory � � 3 � � 2 � 1 1 − α s ( Q 2 ) α s ( Q 2 ) α s ( Q 2 ) � � 0 dxf ( x, Q 2 ) = 3 − a ( n f ) − b ( n f ) π π π – a ( n f ) , b ( n f ) calculable coefficients deviations from scaling → determination of strong coupling
Examples of quantities computable on the lattice • Moments of structure functions: � x n � = dxx n f ( x ) � lowest moment, � x � : corresponds to average momentum of quark in hadron • Pion decay constant: � 0 | A µ | π ( q ) � = f π q µ ( A µ Axial current, q momentum) • Particle Masses, transition amplitudes, ... ( O Latt − O Exp ) /O Exp
There are dangerous lattice animals → violation of chiral symmentry (exchange of massless left- and right-handed quarks)
Problem to reach physical value of pion mass quenched example: chiral extrapolation of � x � Guagnelli, K.J., Palombi, Petronzio, Shindler, Wetzorke • Schr¨ odinger Functional • combined Wilson and O ( a ) -improved Wilson • controlled – non-perturbative renormalization – continuum limit – finite volume effects – statisitical errors 0.6 • want to reach: 1st cont, 2nd chiral − S − <x> M ( µ = 2 GeV ) m 2 π = 0 . 02 [GeV 2 ] 1st chiral, 2nd cont 0.5 experimental 0.4 0.3 0.2 0.1 2 [GeV 2 ] m π 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2
solution, give up anti-commutation condition with γ 5 : Ginsparg-Wilson relation ⇒ D − 1 γ 5 + γ 5 D − 1 = 2 aγ 5 γ 5 D + Dγ 5 = 2 aDγ 5 D Ginsparg-Wilson relation implies an exact lattice chiral symmetry (L¨ uscher) : for any operator D which satisfies the Ginsparg-Wilson relation, the action S = ¯ ψDψ is invariant under the transformations δ ¯ ψ = ¯ δψ = γ 5 (1 − 1 ψ (1 − 1 2 aD ) ψ , 2 aD ) γ 5 ⇒ almost continuum like behaviour of fermions one local (Hern´ uscher, K.J.) solution: overlap operator D ov (Neuberger) andez, L¨ 1 − A ( A † A ) − 1 / 2 � � D ov = with A = 1 + s − D w ( m q = 0) ; s a tunable parameter, 0 < s < 1
Wilson (Frezzotti, Rossi) twisted mass QCD (Frezzotti, Grassi, Sint, Weisz) D tm = m q + iµτ 3 γ 5 + 1 ∇ µ + ∇ ∗ − a 1 2 ∇ ∗ � � 2 γ µ µ ∇ µ µ quark mass parameter m q , twisted mass parameter µ • m q = m crit → O(a) improvement for hadron masses, matrix elements, form factors, decay constants • det[ D tm ] = det[ D 2 Wilson + µ 2 ] ⇒ protection against small eigenvalues • computational cost comparable to staggered • simplifies mixing problems for renormalization • serious competitor to Ginsparg-Wilson fermions ⋆ based on symmetry arguments ⇒ check how it works in practise Drawback: explicit breaking of isospin symmetry for any a > 0
A first test twisted mass against overlap fermions: how chiral can we go? Bietenholz, Capitani, Chiarappa, Christian, Hasenbusch, K.J., Nagai, Papinutto, Scorzato, Shcheredin, Shindler, Urbach, Wenger, Wetzorke fixed lattice spacing of a = 0 . 125 fm ⇒ twisted mass simulations can reach quarks masses as small as overlap substantially smaller than O(a)-improved Wilson fermions
Scaling of � x � and F PS with twisted mass fermions K.J., M. Papinutto, A. Shindler, C. Urbach, I. Wetzorke 0.4 0.55 pion , m PS =368 MeV κ c κ c PCAC , m PS =368 MeV pion , m PS =728 MeV κ c 0.5 0.35 κ c PCAC , m PS =728 MeV pion , m PS =1051 MeV κ c 0.45 κ c PCAC , m PS =1051 MeV 0.3 <x> MS f PS r 0 0.4 pion , m PS =377 MeV κ c 0.35 0.25 PCAC , m PS =377 MeV κ c pion , m PS =715 MeV κ c 0.3 PCAC , m PS =715 MeV κ c 0.2 pion , m PS =1.03 GeV κ c 0.25 PCAC , m PS =1.03 GeV κ c 0.2 0.15 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 2 2 (a/r 0 ) (a/r 0 ) → O( a 2 ) scaling for two realizations of O(a)-improvement → κ PCAC very small O( a 2 ) effects c → κ pion larger O( a 2 ) effects, late scaling β ≥ 6 c → consistent with theoretical considerations (Frezzotti, Martinelli, Papinutto, Rossi; Sharpe, Wu; Aoki, B¨ ar)
F PS and � x � with twisted mass (S. Capitani, K.J., M. Papinutto, A. Shindler, C. Urbach, I. Wetzorke) 0.5 <x> MS 0.24 ( µ =2 GeV) f PS [GeV] 0.45 0.4 0.22 0.35 0.2 0.3 0.18 0.25 0.2 0.16 Wilson tmQCD at π /2 Wilson tmQCD at π /2 combined Wilson-Clover 0.15 NP O(a) improved Wilson experimental value 0.14 0.1 2 [GeV 2 ] m PS 2 [GeV 2 ] m PS 0.12 0.05 0 0.1 0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Cost comparison T. Chiarappa, K.J., K. Nagai, M. Papinutto, L. Scorzato, A. Shindler, C. Urbach, U. Wenger, I. Wetzorke V, m π Overlap Wilson TM rel. factor 12 4 , 720 Mev 48.8(6) 2.6(1) 18.8 12 4 , 390 Mev 142(2) 4.0(1) 35.4 16 4 , 720 Mev 225(2) 9.0(2) 25.0 16 4 , 390 Mev 653(6) 17.5(6) 37.3 16 4 , 230 Mev 1949(22) 22.1(8) 88.6 timings in seconds on Jump
Dynamical Quarks: The phase structure of lattice QCD Farchioni, Frezzotti, Hofmann, K.J., Montvay, M¨ unster, Rossi, Scorzato,Scholz, Shindler, Ukita, Urbach, Wenger, Wetzorke Let me describe a typical computer simulation:[...] the first thing to do is to look for phase transitions ( G. Parisi ) lattice simulations are done under the assumption that the transition is continuum like • first order, jump in < ¯ ΨΨ > when quark mass m changes sign • pion mass vanishes at phase transition point ⇒ single phase transition line → twisted mass fermions offer a tool to check this
Revealing the generic phase structure of lattice QCD Aoki phase: Ilgenfritz, M¨ uller-Preussker, Sternbeck, St¨ uben → Knowledge of phase structure for a particular formulation of lattice QCD: pre-requisite for numerical simulation
Chiral perturbation theory for the phase transition Sharpe, Wu; Hofmann, M¨ unster; Scorzato; Aoki, B¨ ar In the regime m/ Λ QCD � a Λ QCD mP CAC,χ r m 2 P CAC,χ + µ 2 M = 2 B 0 /ZP Λ R = 4 πF 0 cos ω = r m 2 P CAC,χ + µ 2 ! M 2 m 2 M 2(2 L 86 − L 54) + 4 aM cos ω ( w − ˜ 8 n o M = M + w ) + 0 π 2 log π F 02 32 F 2 Λ2 R ! F 0 + 4 M M fπ = { ML 54 + 4 a cos ω ˜ w } − 16 F 0 π 2 log F 0 Λ2 R " !# F 0 + 4 M M gπ = B 0 /ZP { M (4 L 86 − L 54) + 4 a cos ω (2 ws − ˜ w ) } − 32 F 0 π 2 log Λ2 F 0 R parameters to fit: B 0 /Z P , F 0 , L 86 , L 54 , w , ˜ w
Serious Consequence: minimal pseudo scalar mass 550 β =0.67; a~0.19 fm • Continuum picture not realized 500 • pion does not vanish 450 rather reaches a minimal value m π [MeV] 400 • strength of phase transition depends 350 on lattice spacing a 300 • minimal pion mass depends on strength of phase transition 250 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 m PS vanishes with rate O ( a ) 1/ κ −1/ κ c
Costs of dynamical fermions simulations, the “Berlin Wall” see panel discussion in Lattice2001, Berlin, 2001 � − z π � ( L ) z L ( a ) − z a m π formula C ∝ 1000 configurations with L=2fm m ρ [Ukawa (2001)] 20 z π = 6 15 z L = 5 TFlops × year 10 z a = 7 a -1 = 3 GeV 5 a -1 = 2 GeV 0 0 0.2 0.4 0.6 0.8 1 ↑ ↑ m PS / m V | | | | physical contact to point χ PT (?)
A hypothetical dynamical computation of F π in 2000 for up and down quarks ( N f = 2 ) 0.24 f PS [GeV] 0.22 0.2 0.18 0.16 Wilson tmQCD at π /2 NP O(a) improved Wilson 0.14 2 [GeV 2 ] m PS 0.12 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 quenched ( N f = 0 ) dynamical u and d quarks ( N f = 2 )
European Twisted Mass Collaboration The quest for solving QCD B. Blossier, Ph. Boucaud, P. Dimopoulos, F. Farchioni, R. Frezzotti, V. Gimenez, G. Herdoiza, K. Jansen, V. Lubicz, G. Martinelli, C. McNeile, C. Michael, I. Montvay, M. Papinutto, O. P` ene, J. Pickavance, G.C. Rossi, L. Scorzato, A. Shindler, S. Simula, C. Urbach, U. Wenger
Target setup for N f = 2 maximally twisted Dynamical Quarks • β = 3 . 9 , 5000 thermalized trajectories • simulations at a smaller and a larger lattice spacing at matched pion masses and volumes are in progress • test scaling and perform continuum limit L 3 · T β κ crit aµ a [fm] m π [MeV] 24 3 · 48 3.90 0.160856 0.0040 ≈ 0.095 280 24 3 · 48 3.90 0.160856 0.0064 ≈ 0.095 350 24 3 · 48 3.90 0.160856 0.0100 ≈ 0.095 430 24 3 · 48 3.90 0.160856 0.0150 ≈ 0.095 510
Shift the Berlin Wall and Twist New variant of HMC algorithm (Urbach, Shindler, Wenger, K.J.) (see also SAP (L¨ uscher) and RHMC (Clark and Kennedy) algorithms) • even/odd preconditioning • (twisted) mass-shift (Hasenbusch trick) • multiple time steps 0.25 Tflops · years Urbach et al. Orth et al. 0.2 tm @ β = 3 . 9 – twisted mass at much smaller m PS /m V 0.15 – compatible with (our own) Wilson 0.1 – compatible with staggered 0.05 – compatible with RHMC 0 0 0.5 1 m PS /m V ⇒ 3 algorithms to drive Wilson fermions towards the physical point
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