QCD Simulations at Realistic Quark Masses: Probing the Chiral Limit - PowerPoint PPT Presentation

QCD Simulations at Realistic Quark Masses: Probing the Chiral Limit G. Schierholz Deutsches Elektronen-Synchrotron DESY – QCDSF Collaboration –

Special mention: M. G¨ ockeler, T. Hemmert, R. Horsley, Y. Nakamura, D. Pleiter, P.E.L. Rakow, W. Schroers, T. Streuer, H. St¨ uben and J. Zanotti

Objective Solve QCD and probe the limits of the Standard Model · · · – Λ QCD resp. α s ( Q 2 ) • Parameters of QCD – Quark masses – θ angle • QCD in the wider world – CKM matrix • How does QCD work ? – Hadron structure – Spectroscopy • Fundamental properties – χ SB – Confinement · · · in concert with Exp & Phen

Problem: Chiral Extrapolation Recently ChPT O ( p 4 ) Stat. error � 5% 68 . 3% CL Need to reduce (scale) error to a few %

Outline Lattice Simulations Pion Sector Nucleon Sector Miscellaneous Conclusions & Outlook

Lattice Simulations

Action N f = 2 S = S G + S F 1 − 1 “ ” X S G = β 3Re Tr U µν ( x ) x,µ<ν n X ψ ( x ) ψ ( x ) − κ ¯ ¯ ψ ( x ) U † µ ( x − ˆ µ )[1 + γ µ ] ψ ( x − ˆ S F = µ ) x µ ) − 1 o − κ ¯ 2 κ c SW g ¯ ψ ( x ) U µ ( x )[1 − γ µ ] ψ ( x + ˆ ψ ( x ) σ µν F µν ( x ) ψ ( x ) � ∂ µ A imp = 2 m q P µ Clover Fermions

Advantages • Local • Transfer matrix • O ( a ) improved • Flavor symmetry Prerequisite to making contact with SU (2) ChPT – Finite size corrections – Chiral extrapolation – Determination of low-energy constants • Fast to simulate

Cost of Simulation 1000 Configurations ∝ L 4 . 8 ( m π /m ρ ) − 3 . 6 ( r 0 /a ) 0 . 9 uscher, Urbach et al., · · · Hasenbusch, QCDSF, L¨

Compared to · · · Clark

Parameters N f = 2 ← a = 0 . 065 fm ← a = 0 . 077 fm � 24 3 48 � 32 3 64 + 40 3 64 For gauge field sampling we use ‘ordinary’ HMC algorithm with Hasenbusch integration + 3 time scales

Obstructions ? 2 m π 1 st order transition 2 O(a ) m q 2 m π Aoki phase 2 m O(a ) q

← cold start ← hot start

Landscape « − 1 3 „ 2 m π ( L ) = 1 + 0 L 2 2 . 837 Minimal pion mass : Leutwyler 2 f 2 4 πf 2 0 L 3 Hasenfratz & Niedermayer

Effect of Unquenching ? Vector Ward Identity ? χ top ≡ � Q 2 � = Σ m q V 2 “ 1 “ 1 2 ” 2 ” 2 ” 2 “ = + D¨ urr χ ∞ χ top Σ m q top

Pion Sector

Pion Mass Raw data NLO 1 + 1 » – m 2 PS = m 2 2 x ˆ l 3 + O ( x 2 ) 0 m PS − m PS ( L ) x h I (2) X = − mPS ( λ ) m PS 2 λ | � n |� =0 i + xI (4) mPS ( λ ) Colangelo, D¨ urr & Haefeli m 2 m 2 0 ↑ 0 =2Σ m q , x = , λ = m PS | � n | L 16 π 2 f 2 0 NPRen l i =ln Λ 2 ˆ i m 2 No 1 st order phase transition or Aoki phase ! 0

I (2) mPS ( x ) = − B 0 ( x ) „ − 55 l 1 + 8 l 2 − 5 « I (4) 18 + 4¯ ¯ ¯ l 3 − 2¯ B 0 ( x ) mPS ( x ) = l 4 3 2 „ 112 − 8 l 1 − 32 « ¯ ¯ B 2 ( x ) + S (4) + l 2 mPS ( x ) 9 3 3 mPS ( x ) = 13 3 g 0 B 0 ( x ) − 1 3 (40 g 0 + 32 g 1 + 26 g 2 ) B 2 + · · · S (4) l i = ln Λ 2 B 0 ( x ) = 2 K 1 ( x ) , B 2 ( x ) = 2 K 2 ( x ) /x , ¯ i m 2 PS Λ i , g i from hep-lat/05030142

FS corrected Corrections r 0 f 0 = 0 . 179(2) , r 0 Λ 3 = 1 . 82(7)

Pion Decay Constant FS corrected Corrections f PS − f PS ( L ) x h i h i l 4 + O ( x 2 ) I (2) fPS ( λ ) + xI (4) 1 + x ˆ X f PS = f 0 = fPS ( λ ) f PS λ | � n |� =0 r 0 f 0 = 0 . 179(2) r 0 Λ 4 = 3 . 32(6) f PS ← NPRen

I (2) fPS ( x ) = − 2 B 0 ( x ) − 7 l 1 + 4 „ « I (4) 9 + 2¯ ¯ l 2 − 3¯ B 0 ( x ) fPS ( x ) = l 4 3 „ 112 − 8 l 1 − 32 « ¯ ¯ B 2 ( x ) + S (4) + l 2 fPS ( x ) 9 3 3 fPS ( x ) = 1 6 (8 g 0 − 13 g 1 ) B 0 ( x ) − 1 3 (40 g 0 − 12 g 1 − 8 g 2 − 13 g 3 ) B 2 + · · · S (4) Colangelo, D¨ urr & Haefeli

Partially Quenched m PS ≡ m SS PS → m AB f PS ≡ f SS PS → f AB PS , PS , A, B ∈ { V, S | V � = S } 0.015 κ =0.1355 κ =0.1359 0.010 0.005 0.000 0.015 κ =0.1362 κ =0.13632 0.010 0.005 0.000 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 ξ ξ ! f V S ln m V V 2 − m V V 2 ξ = m V V 2 R 1 ˆ PS PS PS PS R ≡ = = − + 1 , m SS 2 m SS 2 m SS 2 m SS 2 q 8(4 πr 0 f 0 ) 2 m SS 2 f V V PS f SS PS PS PS PS PS PS Sharpe

Nucleon Sector

Nucleon Mass FS corrected Corrections g 0 r 0 f 0 = 0 . 179(2) A = 1 . 15 r 0 = 0 . 45(3) fm r 0 m 0 = 2 . 00 c 1 /r 0 = − 0 . 43

" PS − 3 g 0 2 “ g 0 2 3 − c 2 ” m N = m 0 − 4 c 1 m 2 m 3 A A PS + e 1 ( µ ) − 32 πf 2 64 π 2 f 2 m 0 2 0 0 # 3 g 0 2 “ g 0 2 3 g 0 2 ln m PS ” m 4 m 5 PS + O ( m 6 A A A − − 8 c 1 + c 2 + 4 c 3 PS + PS ) 32 π 2 f 2 256 πf 2 0 m 2 m 0 µ 0 0 Z ∞ m N − m N ( L ) = − 3 g 0 2 A m 0 m 2 „q « 0 z 2 + m 2 X PS m 2 dzK 0 PS (1 − z ) | � n | L 16 π 2 f 2 0 0 | � n |� =0 " # − 3 m 4 (2 c 1 − c 3 ) K 1 ( m PS | � n | L ) K 2 ( m PS | � n | L ) X + O ( m 5 PS + c 2 PS ) 4 π 2 f 2 m PS | � n | L ( m PS | � n | L ) 2 0 | � n |� =0

Axial Coupling Preliminary ↓ 1.4 1.4 1.2 1.2 g A g A 1.0 1.0 =5.20 =5.20 =5.25 =5.25 0.8 0.8 =5.29 =5.29 =5.40 =5.40 0.0 0.16 0.32 0.48 0.64 0.8 0.0 0.16 0.32 0.48 0.64 0.8 m 2 [GeV 2 ] m 2 [GeV 2 ] χ PT O ( p 3 ) 68 . 3% CL

Miscellaneous

Rho Mass Not FS corrected

Delta Mass Not FS corrected

Conclusions & Outlook

• Simulations at pion masses of O (300) MeV with • Improvement of algorithms Wilson-type fermions feasible now • Increase ofcomputing power • Extrapolation to chiral limit and infinite volume FS corrections surprisingly well greatly improved described by ChPT • First meaningful lattice determination of low energy constants : Preliminary ! r 0 f 0 Λ 3 Λ 4 0.45(3) fm 79(5) MeV 0.80(5) GeV 1.46(10) GeV • Major investment in FS corrections (including partially quenched data) and δ expansion needed