Broken symmetries and lattice gauge theory (I): LGT, a theoretical femtoscope for non-perturbative strong dynamics Leonardo Giusti University of Milano-Bicocca L. Giusti – GGI School 2016 - Firenze January 2016 – p. 1/13
Quantum Chromodynamics (QCD) QCD is the quantum field theory of strong interactions in Nature. Its action [Fritzsch, Gell-Mann, Leutwyler 73; Gross, Wilczek 73; Weinberg 73] g S [ A, ¯ ψ i , ψ i ; g, m i , θ ] is fixed by few simple principles: g ∗ SU(3) c gauge (local) invariance ∗ Quarks in fundamental representation g 2 ψ i = u, d, s, c, b, t ∗ Renormalizability Present experimental results compatible with θ = 0 It is fascinating that such a simple action and few parameters [ g, m i ] can account for the variety and richness of strong-interaction physics phenomena L. Giusti – GGI School 2016 - Firenze January 2016 – p. 2/13
Asymptotic freedom The renormalized coupling constant is scale dependent µ d dµ g = β ( g ) g ( µ ) µ − → and QCD is asymptotically free [ b 0 > 0 ] [Gross, Wilczek 73; Politzer 73] β ( g ) = − b 0 g 3 − b 1 g 5 + . . . The theory develops a fundamental scale � � � g ( µ ) b 0 g 3 − b 1 1 1 1 − dg β ( g ) + � − b 1 / 2 b 2 − 0 e 0 b 2 2 b 0 g 2( µ ) e b 0 g 2 ( µ ) � 0 g Λ = µ which is a non-analytic function of the coupling constant at g 2 = 0 . Quantization breaks scale invariance at m i = 0 L. Giusti – GGI School 2016 - Firenze January 2016 – p. 3/13
Perturbative corner: hard processes [PDG 2014] Processes where the relevant energy scale is ❙ ✿❀ ❁ ❂ ❃ ❄ ❅ ❆ ❊ ❋ ● ✌ ❇ ✢ ✤ ✴ ✫✬ ✯ ✰ ✱ µ ≫ Λ can be studied by pert. expansion ✖ ✗ ✘ ✕ ✏ ✏ ✎ ✌ ❉ ▲✢ ❇ ✥❈ ✫✬ ✬ ✯ ✰ ✱ ❉✽ ✾ ✳ ✌ ✏ ✴ ✫✬ ✯ ✰ ✱ ❍ ✌ ✢ ✣✤ ✥✧✢ ★ ✩ ☛ ✪ ✎ ✢ ✫✬ ✯ ✰ ✱ ✵✑ ✔ Ð ✲ ✌ ✌ ✳ ✌ ✏ ✴ ✷ ✴ ✸ ✢ ✡✌ ✴ ✫ ✹✺ ✻ ✼ ✬ ✬ ✯ ✰ ✱ ❊ ln(ln( µ 2 ❩ ✡☛☞ ✌ ✍ ✎ ✏ ✫✬ ✯ ✰ ✱ ❏❑ ▼ α s ( µ )= g 2 ( µ ) Λ 2 )) 1 1 − b 1 ✡✡ ♣■ ✳ ✌ ✏ ✴ ✫✬ ✯ ✰ ✱ = + ... ✵✑ ✓ 4 πb 0 ln ( µ 2 b 2 ln( µ 2 4 π Λ 2 ) Λ 2 ) 0 ✵✑ ✒ ◗ �✁ ✦ ✭✂ ✮ ✄ ☎ ✆ ✝ ✝ ✶ ✞ ✟ ☎ ✆ ☎ ☎ ☎ ✠ s ③ ✒ ✒ ✵ ✒ ✵✵ ✒ ✵✵✵ ✗ ✙ ✚✛✜ ❪ An example is given by σ ( e + e − → hadrons ) R = e + σ ( e + e − → µ + µ − ) γ � α s ( µ ) � � � 2 1 + α s ( µ ) i Q 2 = 3 � i · + C 2 + · · · π π e − Experimental results significantly prove the logarithmic dependence in µ/ Λ predicted by perturbative QCD L. Giusti – GGI School 2016 - Firenze January 2016 – p. 4/13
Scale of the strong interactions By comparing these measurements to theory 1 / Λ ∼ 1 fm = 10 − 15 m Λ ∼ 0 . 2 GeV At these distances the dynamics of QCD is non-perturbative A rich spectrum of hadrons is observed at these energies. Their properties such as M n = b n Λ need to be computed non-perturbatively The theory is highly predictive: in the (interesting) limit m u,d,s = 0 and m c,b,t → ∞ , for instance, dimensionless quantities are parameter-free numbers L. Giusti – GGI School 2016 - Firenze January 2016 – p. 5/13
Light pseudoscalar meson spectrum I I 3 S Meson Quark Mass Octet compatible with SSB pattern Content ( GeV ) u ¯ π + d SU (3) L × SU (3) R → SU (3) L+R 1 1 0 0.140 π − 1 -1 0 d ¯ u 0.140 √ ( d ¯ π 0 1 0 0 d − u ¯ u ) / 2 0.135 and soft explicit symmetry breaking 1 1 K + 2 +1 u ¯ s 0.494 m u , m d ≪ m s < Λ 2 1 2 - 1 K 0 2 +1 d ¯ s 0.498 1 2 - 1 K − 2 -1 s ¯ u 0.494 K 0 s ¯ 1 1 2 -1 d 0.498 2 m u , m d ≪ m s = ⇒ m π ≪ m K 0 0 0 η cos ϑη 8 − sin ϑη 0 0.548 η ′ 0 0 0 sin ϑη 8 + cos ϑη 0 0.958 A 9 th pseudoscalar with m η ′ ∼ O (Λ) √ ( d ¯ η 8 = d + u ¯ u − 2 s ¯ s ) / 6 √ ( d ¯ η 0 = d + u ¯ u + s ¯ s ) / 3 − 10 ◦ ϑ ∼ L. Giusti – GGI School 2016 - Firenze January 2016 – p. 6/13
QCD action and its (broken) symmetries QCD action for N F = 2 , M † = M = diag ( m, m ) � � � d 4 x ψDψ + ¯ ¯ ψ R M † ψ L + ¯ S = S G + ψ L Mψ R , D = γ µ ( ∂ µ − iA µ ) SU(3) c × SU(2) L × SU(2) R × U(1) L × U(1) R ×R scale For M = 0 chiral symmetry (dim. transm., chiral anomaly) � 1 ± γ 5 � ψ R , L → V R , L ψ R , L ψ R , L = ψ 2 SU(3) c × SU(2) L × SU(2) R × U(1) B=L+R (Spont. Sym. Break.) Chiral anomaly: measure not invariant SSB: vacuum not symmetric SU(3) c × SU(2) L+R × U(1) B (Confinement) Breaking due to non-perturbative dynamics. Precise quantitative tests are being made SU(2) L+R × U(1) B on the lattice L. Giusti – GGI School 2016 - Firenze January 2016 – p. 7/13
Lattice QCD: action [Wilson 74] QCD can be defined on a discretized space- time so that gauge invariance is preserved a Quark fields reside on four-dimensional lattice, L the gauge field U µ ∈ SU (3) resides on links The Wilson action for the gauge field is S G [ U ] = β � 1 − 1 �� � � � 3 ReTr U µν ( x ) x + µ + ν 2 x + ν x µ,ν where β = 6 /g 2 and the plaquette is U † ν ( x ) µ ) U † ν ) U † U µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ( x + ˆ ν ( x ) x + µ x U µ ( x ) Popular discretizations of fermion action: Wilson, Domain-Wall-Neuberger, tmQCD L. Giusti – GGI School 2016 - Firenze January 2016 – p. 8/13
Lattice QCD: path integral [Wilson 74] The lattice provides a non-perturbative definition of QCD. The path integral at finite spacing and volume is mathematically well defined (Euclidean time) � ψ i Dψ i e − S [ U, ¯ DUD ¯ ψ i ,ψ i ; g,m i ] Z = Nucleon mass, for instance, can be extracted from the behaviour of a suitable two-point correlation function at large time-distance O N ( y ) � = 1 � ψ i Dψ i e − S O N ( x ) ¯ � O N ( x ) ¯ DUD ¯ → R N e − M N | x 0 − y 0 | O N ( y ) − Z For small gauge fields, the pert. expansion differs from usual one for terms of O ( a ) p ′ g � ′ � = − igT a γ µ − i µ ) a + O ( a 2 ) 2 ( p µ + p p Consistency of lattice QCD with standard perturbative approach is thus guaranteed L. Giusti – GGI School 2016 - Firenze January 2016 – p. 9/13
Lattice QCD: universality Continuum and infinite-volume limit of Lattice QCD is the non - perturbative definition of QCD Details of the discretization become irrelevant in the continuum limit, and any reasonable lattice formulation tends to the same continuum theory M N ( a ) = M N + c N a + . . . L. Giusti – GGI School 2016 - Firenze January 2016 – p. 10/13
Lattice QCD: universality Continuum and infinite-volume limit of Lattice QCD is the non - perturbative definition of QCD Details of the discretization become irrelevant in the continuum limit, and any reasonable lattice formulation tends to the same continuum theory M N ( a ) = M N + d N a 2 + . . . By a proper tuning of the action and operators, convergence to continuum can be accelerated without introducing extra free-parameters [Symanzik 83; Sheikholeslami Wohlert 85; Lüscher et al. 96] Finite-volume effects are proportional to exp( − M π L ) at asymptotically large volumes L. Giusti – GGI School 2016 - Firenze January 2016 – p. 10/13
Numerical lattice QCD: machines Correlation functions at finite volume and finite lattice spacing can be computed by Monte Carlo techniques exactly up to statistical errors [Galileo – CINECA] Look at quantities not accessible to experiments: ∗ quark mass dependence ∗ volume dependence ∗ unphysical quantities Σ , χ, . . . for understanding... L. Giusti – GGI School 2016 - Firenze January 2016 – p. 11/13
Numerical lattice QCD: machines Typical lattice parameters: ( a Λ) 2 ∼ 0 . 25% a = 0 . 05 fm L = 3 . 2 fm = ⇒ M π L ≥ 4 , M π ≥ 0 . 25 GeV # points = 2 25 ∼ 3 . 4 · 10 7 V = 2 L × L 3 Monte Carlo algorithms integrate over 10 7 – 10 9 SU(3) link variables [Galileo – CINECA] A typical cluster of PCs: ∗ Standard CPUs [Intel, AMD] ∗ Fast connection [40Gbit/s] Lattice partitioned in blocks which are distributed over the nodes ( 256 × 16 a good example) Data exchange among nodes minimized thanks to the locality of the action L. Giusti – GGI School 2016 - Firenze January 2016 – p. 11/13
Numerical lattice QCD: algorithms Extraordinary algorithmic progress over the last 30 years, keywords: ∗ Hybrid Monte Carlo (HMC) [Della Morte et al. 05] Duane et al. 87 ∗ Multiple time-step integration Sexton, Weingarten 92 ∗ Frequency splitting of determinant Hasenbusch 01 ∗ Domain Decomposition Lüscher 04 ∗ Mass preconditioning and rational HMC Urbach et al 05; Clark, Kennedy 06 ∗ Deflation of low quark modes Lüscher 07 ∗ Avoiding topology freezing Lüscher, Schaefer 12 Light dynamical quarks can be simulated. Chiral regime of QCD is accessible Algorithms are designed to produce exact results up to statistical errors L. Giusti – GGI School 2016 - Firenze January 2016 – p. 12/13
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