Fundamental Parameters of QCD from the Lattice

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1 Fundamental Parameters of QCD from the Lattice Milano Bicocca, DESY Zeuthen GGI Firenze, Feb 2007 Introduction Coupling Masses Summary and Outlook

2 QCD Lagrangian and Parameters L QCD (g 0, m 0 ) = 1 4 F µνf µν + ψ(i D m (f ) 0 )ψ N f f =1 Experiment { }} { F π m π m K m D m B L QCD (g 0, m 0 ) = parameters (RGI ) { }} { Λ QCD ˆM = (M u + M d )/2 M s M c + M b Predictions { }} { ξ F B B B.

3 Renormalisation At high energies: PT and MS Φ(q, r) = C 0 (q, r)+c 1 (q, r, µ) α MS (µ)+c 2 (q, r, µ) α 2 MS (µ)+ α MS (µ) g 2 MS 4π (depends on Φ, choice of µ q, and order of PT) m MS (µ) (may require additional assumptions, e.g. QCD sum rules)

4 Renormalisation At low energies: Simulation at finite lattice spacing a S W = 1 g 2 0 tr(1 U p ) + p f x ψ x (D W + m (f ) 0 )ψ x Hadronic scheme m exp H = lim (am H ) a 0 a(g 0 ) depending on choice of m H, and on N f ratios m H /m H (to be kept at physical values)

5 Renormalization Group and Λ-Parameter RGE for mass-independent scheme: g g(µ) µ g µ = β(g) ḡ 0 ḡ 3 { b 0 + b 1 ḡ 2 + b 2 ḡ } exact equation for integration constant Λ { g [ Λ = µ (b 0 g 2 ) b1/2b2 0 e 1/2b 0g 2 1 exp dg β(g) + 1 b 0 g 3 b ] } 1 b0 2g 0

6 Renormalization Group and Λ-Parameter RGE for mass-independent scheme: g g(µ) µ g µ = β(g) ḡ 0 ḡ 3 { b 0 + b 1 ḡ 2 + b 2 ḡ } exact equation for integration constant Λ { g [ Λ = µ (b 0 g 2 ) b1/2b2 0 e 1/2b 0g 2 1 exp dg β(g) + 1 b 0 g 3 b ] } 1 b0 2g trivial scheme dependence α a = α b +c ab α 2 b +O(α3 b ) Λ a/λ b = e c ab/(4πb 0 ) use a suitable physical coupling (scheme) and non-perturbative β(g) 0

7 Connecting Hadronic and High-Energy Physics Problem: Large scale differences a 1 µ PT µ H L 1

8 Connecting Hadronic and High-Energy Physics Solution: Intermediate Renormalisation Scheme

9 A Collaboration LPHA Project Use Schrödinger Functional (SF) as intermediate scheme Calculate relation between low- and high-energy quantities in QCD with N f = 0, 2,... flavors: define and compute NP renormalisation and running implementation and test of Symanzik improvement perform reliable continuum limit verify that systematic errors are under control Not only applicable to fundamental parameters, but also to effective operators (B K,...)

10 A Collaboration LPHA... initiated through key work and ideas of M. Lüscher et. al Univ. Bern S. Dürr CERN M. Della Morte, C. Pena Univ. Colorado R. Hoffmann DESY, Zeuthen D. Guazzini, B. Leder, H.S., R. Sommer Univ. Dublin S. Sint Univ. Edinburgh J. Wennekes Humboldt Univ. Berlin J. Rolf, O. Witzel, U. Wolff NIC, Zeuthen K. Jansen, I. Wetzorke, A. Shindler Univ. Mainz F. Palombi, H. Wittig MIT H. Meyer Univ. Münster P. Fritzsch, J. Heitger MPI München P. Weisz Univ. Roma II P. Dimopoulos, R. Frezzotti, M. Guagnelli, A. Vladikas Univ. Southampton A. Jüttner Univ. Wuppertal F. Knechtli

11 Definition of Schrödinger Functional finite physical volume L 4, T = L Dirichlet b.c. C(η), C (η) at x 0 = 0, T periodic b.c. in space (up to phase θ) Z SF (C, C ) = e Γ(η) = e S(η) fields time L C 0 C space (LxLxL box with periodic b.c.)

12 Definition of Schrödinger Functional finite physical volume L 4, T = L Dirichlet b.c. C(η), C (η) at x 0 = 0, T periodic b.c. in space (up to phase θ) Z SF (C, C ) = e Γ(η) = renormalised coupling Γ(η) η η=0 e S(η) fields k g 2 SF (L) time L 0 C C space (LxLxL box with periodic b.c.)

13 Definition of Schrödinger Functional finite physical volume L 4, T = L Dirichlet b.c. C(η), C (η) at x 0 = 0, T periodic b.c. in space (up to phase θ) Z SF (C, C ) = e Γ(η) = renormalised coupling Γ(η) η η=0 e S(η) fields k g 2 SF (L) time L 0 C C mass-independent scheme m PCAC = 0 renormalisation scale µ = 1/L space (LxLxL box with periodic b.c.)

14 Properties of Schrödinger Functional NP definition in continuum g SF is local (plaquette-like) observable on the lattice spectral gap 1/L allows simulation with massless quarks known perturbative expansion (can use PT for running at very large µ after checking that it coincides with NP running)

15 Step Scaling Function (SSF) discrete β-function σ(g 2 (L)) g 2 (2L) determines NP running u k = g 2( L max /2 k) u 0 = g 2( L max ) computation on the lattice Σ(u, a/l) = σ(u) + O(a/L)

16 SSF for N f = 2 u= u= u= u= u= u= u=0.9793

17 Simulation Parameters of SSF (g 0, a/l) u g 2 (L)

18 Simulation Parameters of SSF Repeat for decreasing a/l = 1/6, 1/8,... continuum limit

19 Precision test of the continuum extrapolation procedure of continuum limit (with NP improved SF) is safe

20 Conversion of SSF to Beta Function by solving 2ln2 = σ(u) u dx xβ( x) with parametrised SSF clear effect of N f strong deviation from 3-loop PT for α SF 0.25 without indication from within PT

21 Running of α N f = 2, NP + PT, SF scheme error bars smaller than symbol size Experiment + PT, MS scheme [Bethke 2000]

22 Matching to Hadronic Scheme SSF yields precise ΛL max (e.g. 7 % on Λ) N f = 0, u max = 3.48 : ln(λ MS L max ) = 0.84(8) N f = 2, u max = 4.61 : ln(λ MS L max ) = 0.40(7) For Λ in MeV need scale from af K (or af π ) ( ) a Λ = (ΛL max ) lim g0 0 L max }{{} SF F exp K (af K ) }{{} largev keeping N f suitable flavoured mass ratios m H /F K fixed. N.B.: standard values of β = 6/g 2 0 may need non-integer a/l max from interpolation of u(g 0, a/l) = u max... N f = 2 simulations with large volumes running on apenext

23 Setting the Scale by r 0 Currently need to use r fm ( ) a (r0 ) 1 Λ = (ΛL max ) L max a 0.5fm e.g. with QCDSF data for r 0 /a (extrapolated to chiral limit) Summary of Λ MS r 0 for different N f N f = 0 N f = 2 N f = 4 N f = 5 SF (ALPHA) 0.60(5) 0.62(6) DIS (NLO) 0.57(8) world av. 0.74(10) 0.54(8)

24 RGI Mass Parameter RGE in mass-independent scheme µ m µ = τ(g) m ḡ 0 ḡ 2 { d 0 + d 1 ḡ 2 + d 2 ḡ } RGI mass (integration constant of RGE) M (f ) = lim µ (2b 0g) d 0/2b 0 m (f ) (µ) = m (f ) (µ) (2b 0 g 2 ) d 0/2b 0 exp { g [ τ(g) dg 0 β(g) d ]} 0 b 0 g

25 RGI Mass Parameter RGE in mass-independent scheme µ m µ = τ(g) m ḡ 0 ḡ 2 { d 0 + d 1 ḡ 2 + d 2 ḡ } RGI mass (integration constant of RGE) M (f ) = lim µ (2b 0g) d 0/2b 0 m (f ) (µ) = m (f ) (µ) (2b 0 g 2 ) d 0/2b 0 exp scale and scheme independent parameter use non-perturbative β(g) and τ(g) { g [ τ(g) dg 0 β(g) d ]} 0 b 0 g

26 Renormalised Quark Mass In a mass-independent scheme m (f ) (µ) = Z m (µa, g 0 ) }{{} m (f ) bare (g 0) flavour independent can solve running once and for all M (f ) M (j) = m(f ) (µ) m (j) (µ) = m (f ) bare (g 0) m (j) bare (g 0) defining m (f ) bare e.g. by PCAC relation µa (f ) µ = 2 m (f ) PCAC P(f ) Z m (µa, g 0 ) = Z A(g 0 ) Z P (g 0, L/a)

27 Definition of Z P in the SF T = L, C = C = 0 θ = 1/2, m = 0 f1 Z P (L) c f P (L/2) = 1 + O(g 2 )

28 SSF for Quark Mass Σ P (u, a/l) Z P(2L) Z P (L) g 2 (L)=u σ P (u) lim a 0 Σ P (u, a/l) Σ P (u,a/l) (σ P (u) 1)/u /2 loop 2/3 loop (a/l) u

29 NP Running of the Quark Mass solve combined recursion for σ P (u) and σ(u) (and PT from L max /2 k to for k = 6) N f = 2, u max = 4.61 : M RGI m(µ) = 1.297(16)

30 Determination of the Quark Masses M (f ) RGI = M RGI m(µ) lim g 0 0 Z m(g 0, aµ) m (f ) 0 (κf, g 0, a/l) Only m (f ) 0 is flavour-dependent, i.e. must be determined by matching (a ratio of) flavoured hadron masses M s : m K M c : m D (so far only N f = 0) M b : m B (after matching to NP renormalised HQET)

31 Charm Quark Mass (N f = 0) large mass renders O(a) improvement essential different definitions of m (c) 0 differ by O(a 2 m 2 c) errors difficult continuum extrapolation M c = 1.65(5) GeV, or m MS c (m c ) = 1.30(3) GeV (N f = 0)

32 Strange Quark Mass (N f = 2) determine reference quark mass m ref, s.t. m PS (m ref, m ref ) = m K from QCDSF data for r 0 /a and am PS determine κ ref (β) at β = 5.2, 5.29, 5.4 computing m(l max ) at κ ref (β) yields M ref = 72(3)(13) MeV (β = 5.4)

33 Strange Quark Mass (cont.) lowest order 3-flavour χpt m 2 K = 1 2 yields for 2 degenerate quarks ( m 2 K + + m 2 K 0 ) = ( ˆM + Ms )B RGI 2M ref = ( ˆM + M s ) use M s / ˆM = 24.4(1.5) [Leutwyler 1996], i.e. M s = 48/25M ref M s = 138(5)(26) MeV, or m MS s (2GeV) = 97(23) MeV

34 Simulation Algoritms Various algorithmic improvements investigated on APE, e.g. Polynomial HMC Hasenbusch trick Multiple time scale integration Trajectory lengths [Frezzotti, Jansen] [Lüscher, Urbach et al.]

35 Main Steps year N f = 0 hep-lat SSF running coupling , NP improvement , Z A, Z V SSF running mass , L ref /r N f = NP improvement , SSF running coupling , Z A, Z V SSF running mass ? L ref F π

36 Computing Resources A Collaboration LPHA is running (almost) exclusively on APE since 1994! significant fraction of APE installation at DESY contribution to APE development (O(25) man years out of ALPHA, QCDSF, NIC) early physics codes for qualification of APEmille/apeNEXT O(80) publications based on numerical results from APE

37 Outlook Challenges: matching to hadronic scheme (Λ in MeV) for N f = 2 heavy quark physics (HQET, M c for N f = 2, f B, M b, B decays,... ) N f > 2... unlikely to be completed on apenext