Lattice QCD: a theoretical femtoscope for non-perturbative strong dynamics

Transcription

1 L. Giusti Napoli May 2010 p. 1/34 Lattice QCD: a theoretical femtoscope for non-perturbative strong dynamics Leonardo Giusti CERN Theory Group and University of Milano-Bicocca CERN

2 L. Giusti Napoli May 2010 p. 2/34 Outline Introduction to (lattice) QCD: Asymptotic freedom and dimensional transmutation Quantum chromodynamics on a lattice Spontaneous symmetry breaking: Banks Casher relation Renormalization of the spectral density Exploratory numerical study Witten Veneziano solution to the U(1) A problem: Definition of the topological susceptibility Non-perturbative computation Conclusions

3 L. Giusti Napoli May 2010 p. 3/34 Quantum chromodynamics (QCD) QCD is assumed to be 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 ψ i = u, d, s, c, b, t g 2 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

4 L. Giusti Napoli May 2010 p. 4/34 Asymptotic freedom Quantization breaks scale invariance at m i = 0 The renormalized coupling constant is scale dependent µ d dµ g = β(g) and QCD is asymptotically free [b 0 > 0] [Gross, Wilczek 73; Politzer 73] µ g(µ) β(g) = b 0 g 3 b 1 g The theory develops a fundamental scale Λ = µ ˆb 0 g 2 1 /2b (µ) b 2 0 e 1 2b 0 g 2 (µ) e R» g(µ) 0 dg 1 β(g) + 1 b 0 g 3 b 1 b 2 0 g which is a non-analytic function of the coupling constant at g 2 = 0

5 L. Giusti Napoli May 2010 p. 5/34 Perturbative corner: hard processes Processes where the relevant energy scale is µ Λ can be studied by perturbative expansion α s (µ) = g2 (µ) 4π = 1 4πb 0 ln( µ2 Λ 2 ) 2 41 b 1 b 2 0 ln(ln( µ2 Λ 2 )) ln( µ2 Λ 2 ) α s (Q) [Bethke 09] July 2009 Deep Inelastic Scattering e + e Annihilation Heavy Quarkonia An example is given by 0.1 QCD α s(μ Z) = ± Q [GeV] R = σ(e+ e hadrons) σ(e + e µ + µ ) = 3 P i Q2 i» 1 + α s(µ) αs (µ) 2 + C π 2 π + e + e γ Experimental results significantly prove the logarithmic dependence in µ/λ predicted by perturbative QCD

6 L. Giusti Napoli May 2010 p. 6/34 Scale of the strong interactions By comparing these measurements to theory Λ 0.2 GeV 1/Λ 1 fm = m At these distances the dynamics of QCD is non-perturbative A rich spectrum of hadrons is observed at these energies. Their properties such as the mass 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

7 L. Giusti Napoli May 2010 p. 7/34 Lattice QCD: action [Wilson 74] QCD can be defined on a discretized space-time so that gauge invariance is preserved Quark fields reside on a four-dimensional lattice, the gauge field U µ SU(3) resides on links L a The Wilson action for the gauge field is S G [U] = β 2 X X x µ,ν»1 13 n o ReTr U µν (x) x + ν x + µ + ν where β = 6/g 2 and the plaquette is defined as U ν (x) U µν (x) = U µ (x) U ν (x + ˆµ) U µ(x + ˆν) U ν(x) x x + µ U µ (x) Popular discretizations of fermion action: Wilson, Domain-Wall-Neuberger, perfect actions, tmqcd

8 L. Giusti Napoli May 2010 p. 8/34 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) Z = Z DUD ψ i Dψ i e S[U, ψ i,ψ i ;g,m i ] Nucleon mass, for instance, can be extracted from the behaviour of a suitable two-point correlation function at large time-distance O N (x)ōn(y) = 1 Z Z DUD ψ i Dψ i e S O N (x)ōn(y) R N e M N x 0 y 0 For small gauge fields, the perturb. expansion differs from the usual one for terms of O(a) g p p = igt a { γ µ i 2 (p µ + p µ )a + O(a2 ) } Consistency of lattice QCD with the standard perturbative approach is thus guaranteed

9 L. Giusti Napoli May 2010 p. 9/34 Lattice QCD: universality Continuum and infinite-volume limit of LQCD is the non-perturbative def inition 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 +...

10 L. Giusti Napoli May 2010 p. 9/34 Lattice QCD: universality Continuum and infinite-volume limit of LQCD is the non-perturbative def inition 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 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

11 L. Giusti Napoli May 2010 p. 10/34 Numerical lattice QCD: machines Correlation functions at f inite volume and f inite lattice spacing can be computed by Monte Carlo techniques exactly up to statistical errors

12 L. Giusti Napoli May 2010 p. 10/34 Numerical lattice QCD: machines Typical lattice parameters: a = 0.05 fm (aλ) % L = 3.2 fm = M π L 4, M π 0.25 GeV V = 2L L 3 #points = Monte Carlo algorithms integrate over SU(3) link variables A typical cluster of PCs: Standard CPUs [AMD, Intel] Fast connection [40Gbit/s] Lattice partitioned in blocks which are distributed over the nodes (128 a good example) Data exchange among nodes minimized thanks to the locality of the action

13 L. Giusti Napoli May 2010 p. 11/34 Numerical lattice QCD: algorithms Extraordinary algorithmic progress over the last 30 years, keywords: [Della Morte et al. 05] Hybrid Monte Carlo (HMC) 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 the low quark modes Lüscher 07 Light dynamical quarks can be simulated (continuum limit still problematic). Chiral regime of QCD is becoming accessible Algorithms are designed to produce exact results up to statistical errors

14 L. Giusti Napoli May 2010 p. 12/34 Lattice QCD: a theoretical femtoscope Lattice QCD is the femtoscope for studying strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look also at quantities not accessible to experiments which may help understanding the underlying mechanisms Femtoscope still rather crude. Often we compute what we can and not what would like to Lattice quantum field theory Observables (probes) An example: the signal-to-noise ratio of the nucleon two-point correlation function Algorithms O N Ō N 2 2 n e (2M N 3M π ) x 0 y 0 decreases exp. with time-distance of sources. At physical point 2M N 3M π 7 fm 1 Computers

15 L. Giusti Napoli May 2010 p. 12/34 Lattice QCD: a theoretical femtoscope Lattice QCD is the femtoscope for studying strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look also at quantities not accessible to experiments which may help understanding the underlying mechanisms Femtoscope still rather crude. Often we compute what we can and not what would like to Lattice quantum field theory Observables (probes) Analogous problem for glueballs in Yang Mills theory solved by decomposing the path integral and by enforcing the global symmetries of the theory into the Monte Carlo [Della Morte, LG 08-10] Algorithms Computers

16 L. Giusti Napoli May 2010 p. 12/34 Lattice QCD: a theoretical femtoscope Lattice QCD is the femtoscope for studying strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look also at quantities not accessible to experiments which may help understanding the underlying mechanisms Femtoscope still rather crude. Often we compute what we can and not what would like to Lattice quantum field theory Observables (probes) A rather general strategy is emerging: design special purpose algorithms which exploit known math. and phys. properties of the theory to be faster Results from first-principles when all syst. uncertainties quantified. This achieved without introducing extra free parameters or dynamical assumptions but just by improving the femtoscope Algorithms Computers

17 L. Giusti Napoli May 2010 p. 13/34 QCD action and its (broken) symmetries QCD action for N f = 3, M = diag(m u, m d, m s ) Z S = S G + n o d 4 x ψdψ + ψmψ, D = γ µ ( µ + ia µ ) For M = 0 chiral symmetry SU(3) c SU(3) L SU(3) R U(1) L U(1) R R scale ψ R,L V R,L ψ R,L ψ R,L = «1 ± γ5 2 ψ (dim. transm., chiral anomaly) Chiral anomaly: measure not invariant SSB: vacuum not symmetric SU(3) c SU(3) L SU(3) R U(1) B=L+R (Spont. Sym. Break.) Gauge symmetry ψ(x) G(x)ψ(x) Confinement: no isolated coloured charge SU(3) c SU(3) L+R U(1) B SU(3) L+R U(1) B (Confinement)

18 L. Giusti Napoli May 2010 p. 13/34 QCD action and its (broken) symmetries QCD action for N f = 3, M = diag(m u, m d, m s ) Z S = S G + n o d 4 x ψdψ + ψmψ, D = γ µ ( µ + ia µ ) SU(3) c SU(3) L SU(3) R U(1) L U(1) R R scale (dim. transm., chiral anomaly) Confinement and SSB due to non-perturbative dynamics SU(3) c SU(3) L SU(3) R U(1) B=L+R The mechanisms are still not know (Spont. Sym. Break.) Today focus on SSB and chiral anomaly SU(3) c SU(3) L+R U(1) B (Confinement) SU(3) L+R U(1) B

19 L. Giusti Napoli May 2010 p. 14/34 Outline Introduction to (lattice) QCD: Asymptotic freedom and dimensional transmutation Quantum chromodynamics on a lattice Spontaneous symmetry breaking: Banks Casher relation Renormalization of the spectral density Exploratory numerical study Witten Veneziano solution to the U(1) A problem: Definition of the topological susceptibility Non-perturbative computation Conclusions

20 L. Giusti Napoli May 2010 p. 15/34 Pseudo Nambu Goldstone bosons in QCD An axial Ward identity of the chiral group is [for simplicity M = diag(m, m, m)] Z ψ 1 ψ 1 = m d 4 x P 12 (x)p 21 (0), P ij = ψ i γ 5 ψ j In the limit m 0 Σ = lim m 0 ψ 1 ψ 1 = 0 = M 2 = 2mΣ F 2 [Gell-Mann, Oakes, Renner 68] where the decay constant is defined as 0 Â12,µ π, p = 2 F π p µ, F = lim m 0 F π

21 L. Giusti Napoli May 2010 p. 16/34 Light pseudoscalar meson spectrum Octet compatible with SSB pattern SU(3) L SU(3) R SU(3) L+R and soft explicit symmetry breaking m u, m d m s < Λ m u, m d m s = m π m K I I 3 S Meson Quark Mass Content (GeV) π + u d π dū π 0 (d d uū)/ K+ u s K0 d s K sū K0 s d η cos ϑη 8 sin ϑη A 9 th pseudoscalar with m η O(Λ) η sin ϑη 8 + cos ϑη η 8 = (d d + uū 2s s)/ 6 η 0 = (d d + uū + s s)/ 3 ϑ 10

22 L. Giusti Napoli May 2010 p. 17/34 SU(2) chiral effective theory [Weinberg 79; Gasser, Leutwyler 84] Chiral effective theory for pions S eff = S 2 eff (U; m, F, Σ) + S4 eff (U; m, F, Σ,Λ i) + encodes spontaneous symmetry breaking For m = 0 pions can interact only if they carry momentum. Expansion in p and m Chiral dynamics parameterized by effective low-energy coupling constants [Colangelo, Gasser, Leutwyler 01; Leutwyler 09] universal band tree, one loop, two loops scalar radius CGL MILC NPLQCD CERN a 2 ETM RBC/UKQCD 0 JLQCD PACS-CS Feng, Jansen & Renner DIRAC NA48 K3π NA48 Ke Weinberg a 0 For instance the pion mass and decay constant at O(p 4 ) are given by M 2 π = M 2 n 1 + M2 M 2 o 32π 2 F 2 ln Λ 2 3, F π = F n1 M2 M 2 o 16π 2 F 2 ln Λ 2 4 Analogous expressions for other quantities such as S-wave ππ scattering lengths a 0 0 and a2 0

23 L. Giusti Napoli May 2010 p. 18/34 Lattice QCD with two light dynamical quarks [Del Debbio, LG, Lüscher, Petronzio, Tantalo 06-08] Chiral regime is becoming accessible to lattice QCD simulations The pion mass squared is found to be a nearly linear function of quark mass up to (0.5 GeV) 2. At smallest masses non-linear correction is 1-3% a= fm, L=2.5 fm Non-Abelian chiral symmetry spontaneously broken as expected M π 2 [GeV 2 ] Compatible with the fact that the bulk of the mass is given by the leading term in standard ChPT Relations dictated by SSB can be verified quantitatively. GMOR is maybe the simplest to start with < GeV m R [GeV] Low-energy constants will finally be determined

24 L. Giusti Napoli May 2010 p. 19/34 Banks Casher relation [Banks, Casher 80] For each gauge configuration 4 D m χ k = (m + iλ k )χ k The spectral density of D is ρ(λ, m) = 1 V X δ(λ λ k ) k ρ(λ,m) (π/σ) where... indicates path-integral average λ [GeV] The Banks Casher relation lim lim lim ρ(λ, m) = Σ λ 0 m 0 V π provides a link between the condensate and the (non-zero) spectral density at the origin. To be compared, for instance, with the free case ρ(λ) λ 3

25 L. Giusti Napoli May 2010 p. 19/34 Banks Casher relation [Banks, Casher 80] For each gauge configuration D m χ k = (m + iλ k )χ k The spectral density of D is ρ(λ, m) = 1 V X δ(λ λ k ) k ν(λ,m) L=2.5 fm, V=2L 4 where... indicates path-integral average Λ [GeV] The number of modes in a given energy interval ν(λ, m) = V Z Λ Λ dλ ρ(λ, m) ν(λ, m) = 2 π ΛΣV +... grows linearly with Λ, and they condense near the origin with values 1/V In the free case ν(λ, m) V Λ 4

26 L. Giusti Napoli May 2010 p. 20/34 Renormalization and continuum limit [LG, Lüscher 09] Instead of the spectral density, consider the spectral sum σ k (m v, m) = V Z ρ(λ, m) dλ (λ 2 + m 2 v )k = a 8k X x 1...x 2k P 12 (x 1 )P 23 (x 2 )... P 2k1 (x 2k ) Integral converges if k 3 The relation between σ k (m v, m) and ρ(λ, m) invertible for every k Renormalization properties of ρ(λ, m) can thus be inferred from those of σ k

27 L. Giusti Napoli May 2010 p. 20/34 Renormalization and continuum limit [LG, Lüscher 09] Instead of the spectral density, consider the spectral sum σ k (m v, m) = V Z ρ(λ, m) dλ (λ 2 + m 2 v )k = a 8k X x 1...x 2k P 12 (x 1 )P 23 (x 2 )... P 2k1 (x 2k ) Corr. functions of pseudoscalar densities at physical distance renormalized by (1/Z m ) 2k At short distance the flavour structure implies P 12 (x 1 )P 23 (x 2 ) C(x 1 x 2 )S 13 (x 1 ) S 13 = ψ 1 ψ 3 where C(x) diverges like x 3 and it is therefore integrable. Analogous argument for all other short-distance singularities. No extra contact terms needed to renormalize σ k

28 L. Giusti Napoli May 2010 p. 20/34 Renormalization and continuum limit [LG, Lüscher 09] Instead of the spectral density, consider the spectral sum σ k (m v, m) = V Z ρ(λ, m) dλ (λ 2 + m 2 v )k = a 8k X x 1...x 2k P 12 (x 1 )P 23 (x 2 )... P 2k1 (x 2k ) Once the gauge coupling and the mass(es) are renormalized, the spectral sum σ k,r (m vr, m R ) = Zm 2k σ mvr k, m «R Z m Z m is ultraviolet finite. Continuum limit universal (if same renormalization conditions are used)

29 L. Giusti Napoli May 2010 p. 20/34 Renormalization and continuum limit [LG, Lüscher 09] Instead of the spectral density, consider the spectral sum σ k (m v, m) = V Z ρ(λ, m) dλ (λ 2 + m 2 v )k = a 8k X x 1...x 2k P 12 (x 1 )P 23 (x 2 )... P 2k1 (x 2k ) The spectral density thus renormalizes as ρ R (λ R, m R ) = Zm 1 ρ λr, m «R Z m Z m For Wilson fermions similar derivation but twisted-mass valence quarks

30 L. Giusti Napoli May 2010 p. 20/34 Renormalization and continuum limit [LG, Lüscher 09] Instead of the spectral density, consider the spectral sum σ k (m v, m) = V Z ρ(λ, m) dλ (λ 2 + m 2 v )k = a 8k X x 1...x 2k P 12 (x 1 )P 23 (x 2 )... P 2k1 (x 2k ) It follows that the mode number is a renormalization-group invariant ν R (Λ R, m R ) = ν(λ, m) and its continuum limit is universal for any value of Λ and m

31 L. Giusti Napoli May 2010 p. 21/34 Numerical computation (I) [LG, Lüscher 09] Lattice details: N f = 2 degenerate quarks Action: O(a)-improved Wilson a = fm V = 2L L 3, L = 1.9,2.5 fm m MS (2GeV) = 0.013,0.026, GeV R ν L=2.5 fm L=1.9 fm a= fm, m=0.026 GeV Λ MS(2GeV) = 0.07,0.085, 0.1,0.115 GeV 10 R Λ R [GeV]

32 L. Giusti Napoli May 2010 p. 21/34 Numerical computation (I) [LG, Lüscher 09] Lattice details: N f = 2 degenerate quarks Action: O(a)-improved Wilson a = fm V = 2L L 3, L = 1.9,2.5 fm m MS (2GeV) = 0.013,0.026, GeV R Λ MS (2GeV) = 0.07,0.085, 0.1,0.115 GeV R Finite volume effects below stat. errors (1.5%). ChPT suggests a fraction of a percent ν a= fm, m=0.026 GeV L=2.5 fm L=1.9 fm Rescaled Λ R [GeV] A nearly linear function up to 0.1 Gev. Qualitative in line with ChPT, but the fact that the linear behaviour extends to such large values of Λ R is rather striking and unexpected

33 L. Giusti Napoli May 2010 p. 21/34 Numerical computation (I) [LG, Lüscher 09] Lattice details: N f = 2 degenerate quarks Action: O(a)-improved Wilson a = fm V = 2L L 3, L = 1.9,2.5 fm m MS (2GeV) = 0.013,0.026, GeV R Λ MS (2GeV) = 0.07,0.085, 0.1,0.115 GeV R Finite volume effects below stat. errors (1.5%). ChPT suggests a fraction of a percent In the effective theory at NLO ν a= fm, m=0.026 GeV L=2.5 fm L=1.9 fm Rescaled Λ R [GeV] ν nlo (Λ, m) = 2ΛΣV π j 1 mσ (4π) 2 F 4» «ΛΣ 3ln F 2 Λ 2 + ln(2) + π 6 2 m m 2 «ff Λ + O Λ 2 corrections of O(10%) for Λ = GeV and m 0.02 GeV. No chiral logs m ln(m)

34 L. Giusti Napoli May 2010 p. 22/34 Numerical computation (II) [LG, Lüscher 09] An effective condensate can be defined as Σ R = π 2V Λ R ν R (Λ R, m R ) GMOR a= fm, L=2.5 fm 0.8 prefactor so that Σ R coincides with Σ at LO A linear extrapolation to the chiral limit yields M π 2 [GeV 2 ] h i 0.1 1/3 Σ MS (2 GeV) < GeV R = 0.276(3)(4)(5) GeV m R [GeV] A clear and consistent picture is emerging. For m R 0.05 GeV the GMOR formula accounts for the bulk of the pion mass. But discretization errors not quantified yet

35 L. Giusti Napoli May 2010 p. 23/34 Why is the symmetry spontaneously broken? Dynamical process not yet known. Studies of low modes can provide important clues α k 1/2 [GeV] 0.12 L = 1.9 fm L = 2.5 fm The Banks Casher mechanism is: 0.10 insensitive to lattice details (universality) 0.08 largely insensitive to dynamical quark effects 0.06 present also in quenched QCD It is tempting to read the relation in the other direction, i.e. chiral symmetry is broken because the low-modes of the Dirac operator condense Eigenvalues of D m D m Σ π = lim lim lim ρ(λ, m) λ 0 m 0 V

36 L. Giusti Napoli May 2010 p. 24/34 Outline Introduction to (lattice) QCD: Asymptotic freedom and dimensional transmutation Quantum chromodynamics on a lattice Spontaneous symmetry breaking: Banks Casher relation Renormalization of the spectral density Exploratory numerical study Witten Veneziano solution to the U(1) A problem: Definition of the topological susceptibility Non-perturbative computation Conclusions

37 L. Giusti Napoli May 2010 p. 25/34 Light pseudoscalar meson spectrum Octet compatible with SSB pattern SU(3) L SU(3) R SU(3) L+R and soft explicit symmetry breaking m u, m d m s < Λ m u, m d m s = m π m K I I 3 S Meson Quark Mass Content (GeV) π + u d π dū π 0 (d d uū)/ K+ u s K0 d s K sū K0 s d η cos ϑη 8 sin ϑη A 9 th pseudoscalar with m η O(Λ) η sin ϑη 8 + cos ϑη η 8 = (d d + uū 2s s)/ 6 η 0 = (d d + uū + s s)/ 3 ϑ 10

38 L. Giusti Napoli May 2010 p. 26/34 The Witten Veneziano mechanism [Witten 79; Veneziano 79] An axial Ward identity of the chiral group is Z d 4 x Q(x)Q(0) = m 1 m 2 Z d 4 x P 11 (x)p 22 (0), Q(x) = 1 h i 32π 2 ǫ µνρσtr F µν (x)f ρσ (x) In the limit N c Z χ = lim Nc d 4 x Q(x)Q(0) = 0 = lim Nc lim m i 0 F 2 M 2 η 2N f = χ Note that for N c : U(1) A is restored η becomes a Nambu Goldstone boson = M η = 0 At first order in 1/N c, Mη 2 = O(N f /N c )

39 L. Giusti Napoli May 2010 p. 26/34 The Witten Veneziano mechanism [Witten 79; Veneziano 79] An axial Ward identity of the chiral group is Z d 4 x Q(x)Q(0) = m 1 m 2 Z d 4 x P 11 (x)p 22 (0), Q(x) = 1 h i 32π 2 ǫ µνρσtr F µν (x)f ρσ (x) In the limit N c Z χ = lim Nc d 4 x Q(x)Q(0) = 0 = lim Nc lim m i 0 F 2 M 2 η 2N f = χ An unambiguous definition of the topological susceptibility is required. Naive definition would diverge as χ = Z d 4 x Q(x)Q(0) 1 a 4

40 L. Giusti Napoli May 2010 p. 27/34 The Witten Veneziano mechanism with Ginsparg Wilson fermions With Ginsparg Wilson fermions the lattice Ward identity is [Neuberger 97; Hasenfratz, Laliena, Niedermayer 98; Lüscher 98; LG, Rossi, Testa, Veneziano 01] X X a 4 Q(x)Q(0) = m 1 m 2 x x a 4 P 11 (x)p 22 (0), Q(x) = 1 h i 2a 3 Tr γ 5 D(x, x) In the limit N c χ = lim X a 4 Q(x)Q(0) = 0 = lim Nc x Nc lim m i 0 F 2 M 2 η 2N f = χ Need to demonstrate that the topological susceptibility suggested by GW fermions χ = X x a 4 Q(x)Q(0) is ultraviolet finite and unambiguously defined

41 L. Giusti Napoli May 2010 p. 28/34 Renormalization and continuum limit [LG, Rossi, Testa 04; Lüscher 04] A chain of Ward identities holds N f = 2 χ = m 1 m 2 X N f = 5 χ = m 1... m 5 X x 1 a 4 P 11 (x 1 )P 22 (0) x 1...x 4 a 16 P 31 (x 1 )S 12 (x 2 )S 23 (x 3 )P 54 (x 4 )S 45 (0) It follows that the topological susceptibility is finite, it is renormalization-group invariant and its continuum limit is universal for any value of m A definition of χ even if the regularization breaks chiral symmetry The limit N c is given by lim χ = lim χ YM Nc Nc and finiteness in Yang Mills theory is proven analogously by introducing pseudofermions

42 L. Giusti Napoli May 2010 p. 29/34 Algorithm for zero-mode counting [LG, Hoelbling, Lüscher and Wittig 02] A Monte Carlo computation of D + D χ YM = 1 V D(n + n ) 2E YM is challenging for several reasons Ritz Ritz L 2 fm and a 0.08 fm = dim[d] In finite V null probability for n + 0 and n 0 Simultaneous minimization of Ritz functionals for D ± = P ± DP ± P ± = 1 ± γ 5 2 n + { 0 to find the gap in one of the sectors and to count the zero modes in the other No contamination from quasi-zero modes

43 L. Giusti Napoli May 2010 p. 30/34 Non-perturbative computation for N c = 3 [Del Debbio, LG, Pica 04] Combined fit of the form [χ 2 dof = 0.73] gives r 4 0χ YM (a, s) = r 4 0χ YM + c 1 (s) a2 r 4 0 χym = ± r 2 0 r 0 4 χl YM (a/r 0 ) 2

44 L. Giusti Napoli May 2010 p. 30/34 Non-perturbative computation for N c = 3 [Del Debbio, LG, Pica 04] Combined fit of the form [χ 2 dof = 0.73] r 4 0χ YM (a, s) = r 4 0χ YM + c 1 (s) a2 gives r0 4χYM = ± By setting the scale F K = 0.113(1) GeV χ YM = (0.191 ± GeV) 4 r 2 0 r 0 4 χl YM (a/r 0 ) 2

45 L. Giusti Napoli May 2010 p. 30/34 Non-perturbative computation for N c = 3 [Del Debbio, LG, Pica 04] Combined fit of the form [χ 2 dof = 0.73] gives r 4 0χ YM (a, s) = r 4 0χ YM + c 1 (s) a2 r 4 0 χym = ± By setting the scale F K = 0.113(1) GeV χ YM = (0.191 ± GeV) 4 to be compared with F 2 2N f (M 2 η + M2 η 2M 2 K ) exp r 2 0 (0.175 GeV)4 r 0 4 χl YM (a/r 0 ) 2

46 L. Giusti Napoli May 2010 p. 30/34 Non-perturbative computation for N c = 3 [Del Debbio, LG, Pica 04] Combined fit of the form [χ 2 dof = 0.73] gives r 4 0χ YM (a, s) = r 4 0χ YM + c 1 (s) a2 r 4 0 χym = ± By setting the scale F K = 0.113(1) GeV χ YM = (0.191 ± GeV) 4 to be compared with F 2 2N f (M 2 η + M2 η 2M 2 K ) exp r 2 0 (0.175 GeV)4 r 0 4 χl YM (a/r 0 ) 2 The (leading) QCD anomalous contribution to M 2 η supports the Witten Veneziano explanation for its large experimental value

47 L. Giusti Napoli May 2010 p. 31/34 Do the instantons play a rôle? [LG, Petrarca, Taglienti 07] Vacuum energy and charge distribution are e F(θ) = e iθq, P Q = Z π π dθ 2π e iθq e F(θ) Norm Inst Edge Data Their behaviour is a distinctive feature of the configurations that dominate the path integral Large N c expansion predicts Q 2n con Q 2 1 N 2n 2 c L=1.30 fm L=1.12 fm Various conjectures. For example, dilute-gas instanton model gives [ t Hooft 74; Callan et al. 76;... ] F Inst (θ) = V A{cos(θ) 1} Q 2n con Q 2 = 1 <Q 4 > con /<Q 2 > (a/r 0 ) 2

48 L. Giusti Napoli May 2010 p. 31/34 Do the instantons play a rôle? [LG, Petrarca, Taglienti 07] Vacuum energy and charge distribution are e F(θ) = e iθq, P Q = Z π π dθ 2π e iθq e F(θ) Norm Inst Edge Data Their behaviour is a distinctive feature of the configurations that dominate the path integral A lattice computation gives Q 4 con Q 2 = 0.30 ± L=1.30 fm L=1.12 fm Witten Veneziano mechanism: the anomaly gives a mass to the η boson thanks to the non-perturbative quantum fluctuations of the topological charge <Q 4 > con /<Q 2 > (a/r 0 ) 2

49 L. Giusti Napoli May 2010 p. 32/34 Conclusions Lattice QCD is a phenomenal theoretical femtoscope to explore strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look at quantities not accessible to experiments that may unveil the the underlying mechanisms of non-perturbative strong dynamics A large variety of physics applications: QCD, flavour physics, beyond Standard Model physics, etc. Thanks to the recent extraordinary conceptual, technical and algorithmic advances the chiral regime of the theory is becoming accessible Today two particularly interesting applications: Banks Casher relation Witten Veneziano mechanism

50 L. Giusti Napoli May 2010 p. 32/34 Conclusions Lattice QCD is a phenomenal theoretical femtoscope to explore strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look at quantities not accessible to experiments that may unveil the the underlying mechanisms of non-perturbative strong dynamics A large variety of physics applications: QCD, flavour physics, beyond Standard Model physics, etc. Condensation of low-modes of the Dirac operator most direct piece of theoretical evidence for SSB α k 1/2 [GeV] L = 1.9 fm L = 2.5 fm 0.06 The rate of condensation explains the bulk of the pion mass up to 0.5 GeV

51 L. Giusti Napoli May 2010 p. 32/34 Conclusions Lattice QCD is a phenomenal theoretical femtoscope to explore strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look at quantities not accessible to experiments that may unveil the the underlying mechanisms of non-perturbative strong dynamics A large variety of physics applications: QCD, flavour physics, beyond Standard Model physics, etc. Quantum fluctuations of the topological charge in Yang Mills theory generate a non-zero value of χ YM Norm Inst Edge Data Its value supports the Witten Veneziano explanation for the large mass of the η

52 L. Giusti Napoli May 2010 p. 32/34 Conclusions Lattice QCD is a phenomenal theoretical femtoscope to explore strong dynamics. Its lenses are made of quantum field theory, numerical techniques and computers It allows us to look at quantities not accessible to experiments that may unveil the the underlying mechanisms of non-perturbative strong dynamics A large variety of physics applications: QCD, flavour physics, beyond Standard Model physics, etc. The femtoscope, however, is still rather crude. There is continuous conceptual and technical progress to empower it LQCD will lead us to a precise quantitative understanding of QCD in the low-energy regime, and to validate the theory to be the one of the strong interactions in Nature

53 L. Giusti Napoli May 2010 p. 33/34 Lattice QCD with two light dynamical quarks [Del Debbio, LG, Lüscher, Petronzio, Tantalo 06-08] Chiral regime is becoming accessible to lattice QCD simulations The pion mass squared is found to be a nearly linear function of quark mass up to (0.5 GeV) 2. At smallest masses non-linear correction is 1-3% a= fm, L=2.5 fm Non-Abelian chiral symmetry spontaneously broken as expected M π 2 [GeV 2 ] Compatible with the fact that the bulk of the mass is given by the leading term in standard ChPT Relations dictated by SSB can be verified quantitatively. GMOR is maybe the simplest to start with < GeV m R [GeV] An example of the potentiality. From a fit to the curve 0.47 Λ GeV to be compared with 0.2 Λ 3 2 GeV [Gasser, Leutwyler 84]

54 L. Giusti Napoli May 2010 p. 34/34 Numerical computation (II) [LG, Lüscher 09] An effective condensate can be defined as Σ R = π 2V Λ R ν R (Λ R, m R ) GMOR a= fm, L=2.5 fm prefactor so that Σ R coincides with Σ at LO A linear extrapolation to the chiral limit yields h Σ MS R (2 GeV) i 1/3 = 0.276(3)(4)(5) GeV The ETM collaboration from an overall fit of the pion mass and decay constant M π 2 [GeV 2 ] < GeV m R [GeV] h Σ MS R (2 GeV) i 1/3 GMOR = 0.270(7) GeV [ETM Coll. 09] A clear and consistent picture is emerging. For m R 0.05 GeV the GMOR formula accounts for the bulk of the pion mass. But discretization errors not quantified yet