Intro Coupling Masses Summary Fundamental Parameters of QCD from the Lattice Hubert Simma Milano Bicocca, DESY Zeuthen GGI Firenze, Feb 2007 Introduction Coupling Masses Summary and Outlook Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary QCD Lagrangian and Parameters N f L QCD ( g 0 , m 0 ) = 1 � ψ ( i � D − m ( f ) 4 F µν F µν + 0 ) ψ f =1 parameters ( RGI ) Experiment Predictions � �� � � �� � � �� � Λ QCD F π ξ ˆ m π M = ( M u + M d ) / 2 F B L QCD ( g 0 , m 0 ) m K + M s B B ⇒ = m D M c . . . m B M b Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary 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 )+ · · · g 2 ⇒ α MS ( µ ) ≡ 4 π (depends on Φ, choice of µ ≈ q , and order of PT) MS ⇒ m MS ( µ ) (may require additional assumptions, e.g. QCD sum rules) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Renormalisation At low energies: Simulation at finite lattice spacing a S W = 1 � � � ψ x ( D W + m ( f ) tr (1 − U p ) + 0 ) ψ x g 2 0 p x f Hadronic scheme ( am H ) m exp = lim H a ( g 0 ) a → 0 depending on choice of m H , and on N f ratios m H ′ / m H (to be kept at physical values) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Renormalization Group and Λ -Parameter RGE for mass-independent scheme: g ≡ g ( µ ) µ∂ g = β ( g ) ∂µ ¯ g → 0 g 2 + b 2 ¯ g 4 + . . . g 3 � � ∼ − ¯ b 0 + b 1 ¯ ◮ exact equation for “integration constant” Λ � �� � g � 0 e − 1 / 2 b 0 g 2 exp 1 1 b 0 g 3 − b 1 Λ = µ ( b 0 g 2 ) − b 1 / 2 b 2 − d g β ( g ) + b 2 0 g 0 Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Renormalization Group and Λ -Parameter RGE for mass-independent scheme: g ≡ g ( µ ) µ∂ g = β ( g ) ∂µ ¯ g → 0 g 2 + b 2 ¯ g 4 + . . . g 3 � � ∼ − ¯ b 0 + b 1 ¯ ◮ exact equation for “integration constant” Λ � �� � g � 0 e − 1 / 2 b 0 g 2 exp 1 1 b 0 g 3 − b 1 Λ = µ ( b 0 g 2 ) − b 1 / 2 b 2 − d g β ( g ) + b 2 0 g 0 ◮ trivial scheme dependence α a = α b + c ab α 2 b + O ( α 3 b ) ⇒ Λ a / Λ b = e c ab / (4 π b 0 ) ◮ use a suitable physical coupling (scheme) and non-perturbative β ( g ) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Connecting Hadronic and High-Energy Physics Problem: Large scale differences a − 1 ≫ µ PT ≫ µ H ≫ L − 1 Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Connecting Hadronic and High-Energy Physics Solution: Intermediate Renormalisation Scheme Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary A Collaboration Project LPHA Use Schr¨ odinger 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 , ...) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary A LPHA Collaboration . . . initiated through key work and ideas of M. L¨ uscher et. al Univ. Bern S. D¨ urr 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. M¨ unster P. Fritzsch, J. Heitger MPI M¨ unchen P. Weisz Univ. Roma II P. Dimopoulos, R. Frezzotti, M. Guagnelli, A. Vladikas Univ. Southampton A. J¨ uttner Univ. Wuppertal F. Knechtli http://www-zeuthen.desy.de/alpha/ Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Definition of Schr¨ odinger 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 ) C’ � Z SF ( C , C ′ ) = e − Γ( η ) = L e − S ( η ) fields time 0 C space (LxLxL box with periodic b.c.) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Definition of Schr¨ odinger 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 ) C’ � Z SF ( C , C ′ ) = e − Γ( η ) = L e − S ( η ) fields time ◮ renormalised coupling � ∂ Γ( η ) k � 0 ≡ � g 2 ∂η SF ( L ) � C η =0 space (LxLxL box with periodic b.c.) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Definition of Schr¨ odinger 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 ) C’ � Z SF ( C , C ′ ) = e − Γ( η ) = L e − S ( η ) fields time ◮ renormalised coupling � ∂ Γ( η ) k � 0 ≡ � g 2 ∂η SF ( L ) � C η =0 space ◮ mass-independent scheme (LxLxL box with periodic b.c.) m PCAC = 0 ◮ renormalisation scale µ = 1 / L Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Properties of Schr¨ odinger 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) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Step Scaling Function (SSF) ◮ “discrete” β -function σ ( g 2 ( L )) ≡ g 2 (2 L ) ◮ 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 ) Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary SSF for N f = 2 u=3.3340 u=2.4792 u=2.0142 u=1.7319 u=1.5031 u=1.1814 u=0.9793 Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Simulation Parameters of SSF u ≡ g 2 ( L ) ( g 0 , a / L ) → Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Simulation Parameters of SSF Repeat for decreasing a / L = 1 / 6 , 1 / 8 , . . . → continuum limit Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Precision test of the continuum extrapolation ⇒ procedure of continuum limit (with NP improved SF) is safe Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Conversion of SSF to Beta Function by solving � σ ( u ) d x − 2 ln 2 = √ x β ( √ x ) u with parametrised SSF ◮ clear effect of N f ◮ strong deviation from 3-loop PT for α SF ≥ 0 . 25 ◮ without indication from within PT Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Running of α N f = 2, NP + PT, SF scheme Experiment + PT, MS scheme error bars smaller than symbol size [Bethke 2000] Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Matching to Hadronic Scheme ◮ SSF yields precise Λ L max (e.g. 7 % on Λ) N f = 0 , u max = 3 . 48 : ln (Λ MS L max ) = − 0 . 84(8) ln (Λ MS L max ) = − 0 . 40(7) N f = 2 , u max = 4 . 61 : ◮ For Λ in MeV need scale from aF K (or aF π ) � � · F exp a K Λ = (Λ L max ) lim L max ( aF K ) g 0 → 0 � �� � � �� � SF large V 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 Hubert Simma Fundamental Parameters of QCD from the Lattice
Intro Coupling Masses Summary Setting the Scale by r 0 ◮ Currently need to use r 0 ≈ 0 . 5 fm � �� r 0 � a 1 Λ = (Λ L max ) L max a 0 . 5 fm 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) Hubert Simma Fundamental Parameters of QCD from the Lattice
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