Broken symmetries and lattice gauge theory (I): LGT, a theoretical femtoscope for non-perturbative strong dynamics

Transcription

1 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 26 - Firenze January 26 p. /3

2 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 ψ i = u,d,s,c,b,t g 2 Renormalizability Present experimental results compatible with θ = 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 26 - Firenze January 26 p. 2/3

3 Asymptotic freedom The renormalized coupling constant is scale dependent µ d dµ g = β(g) and QCD is asymptotically free [b > ] [Gross, Wilczek 73; Politzer 73] µ g(µ) β(g) = b g 3 b g The theory develops a fundamental scale [ Λ = µ [ b g 2 (µ) ] b /2b 2 e 2b g 2 (µ) e g(µ) dg β(g) + b g 3 b b 2 g ] which is a non-analytic function of the coupling constant at g 2 =. Quantization breaks scale invariance at m i = L. Giusti GGI School 26 - Firenze January 26 p. 3/3

4 Perturbative corner: hard processes Processes where the relevant energy scale is µ Λ can be studied by pert. expansion [PDG 24] α s (µ)= g2 (µ) 4π = b 4πb ln( µ2 Λ 2 ) b 2 ln(ln( µ2 Λ 2 )) ln( µ2 Λ 2 ) +... Ð s 3 An example is given by R = σ(e+ e hadrons) σ(e + e µ + µ ) = 3 i Q2 i [ ( ) + 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 L. Giusti GGI School 26 - Firenze January 26 p. 4/3

5 Scale of the strong interactions By comparing these measurements to theory Λ.2 GeV /Λ fm = 5 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 M n = b n Λ need to be computed non-perturbatively The theory is highly predictive: in the (interesting) limit m u,d,s = and m c,b,t, for instance, dimensionless quantities are parameter-free numbers L. Giusti GGI School 26 - Firenze January 26 p. 5/3

6 Light pseudoscalar meson spectrum Octet compatible with SSB pattern I I 3 S Meson Quark Mass Content (GeV) 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 π + u d.4 - π dū.4 π (d d uū)/ K+ u s K d s K sū K s d.498 η cosϑη 8 sinϑη.548 A 9 th pseudoscalar with m η O(Λ) η sinϑη 8 +cosϑη.958 η 8 = (d d+uū 2s s)/ 6 η = (d d+uū+s s)/ 3 ϑ L. Giusti GGI School 26 - Firenze January 26 p. 6/3

7 QCD action and its (broken) symmetries QCD action for N F = 2, M = M = diag(m,m) S = S G + d 4 x { ψdψ + ψr M ψ L + ψ } L Mψ R, D = γ µ ( µ ia µ ) For M = chiral symmetry SU(3) c SU(2) L SU(2) R U() L U() R R scale ( ) ±γ5 (dim. transm., chiral anomaly) ψ R,L V R,L ψ R,L ψ R,L = 2 ψ SU(3) c SU(2) L SU(2) R U() B=L+R Chiral anomaly: measure not invariant SSB: vacuum not symmetric SU(3) c SU(2) L+R U() B (Spont. Sym. Break.) Breaking due to non-perturbative dynamics. Precise quantitative tests are being made on the lattice SU(2) L+R U() B (Confinement) L. Giusti GGI School 26 - Firenze January 26 p. 7/3

8 Lattice QCD: action [Wilson 74] QCD can be defined on a discretized spacetime so that gauge invariance is preserved Quark fields reside on 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 µ,ν [ 3 { } ] ReTr U µν (x) x + ν x + µ + ν where β = 6/g 2 and the plaquette is 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, tmqcd L. Giusti GGI School 26 - Firenze January 26 p. 8/3

9 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 = 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) = Z DUD ψ i Dψ i e S O N (x)ōn(y) R N e M N x y For small gauge fields, the pert. expansion differs from 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 standard perturbative approach is thus guaranteed L. Giusti GGI School 26 - Firenze January 26 p. 9/3

10 Lattice QCD: universality Continuum and infinite-volume limit of Lattice QCD 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+... L. Giusti GGI School 26 - Firenze January 26 p. /3

11 Lattice QCD: universality Continuum and infinite-volume limit of Lattice QCD 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 L. Giusti GGI School 26 - Firenze January 26 p. /3

12 Numerical lattice QCD: machines Correlation functions at finite volume and f inite lattice spacing can be computed by Monte Carlo techniques exactly up to statistical errors Look at quantities not accessible to experiments: [Galileo CINECA] quark mass dependence volume dependence unphysical quantities Σ, χ,... for understanding... L. Giusti GGI School 26 - Firenze January 26 p. /3

13 Numerical lattice QCD: machines Typical lattice parameters: a =.5 fm (aλ) 2.25% L = 3.2 fm = M π L 4, M π.25 GeV V = 2L L 3 #points = Monte Carlo algorithms integrate over 7 9 SU(3) link variables A typical cluster of PCs: [Galileo CINECA] Standard CPUs [Intel, AMD] Fast connection [4Gbit/s] Lattice partitioned in blocks which are distributed over the nodes (256 6 a good example) Data exchange among nodes minimized thanks to the locality of the action L. Giusti GGI School 26 - Firenze January 26 p. /3

14 Numerical lattice QCD: algorithms Extraordinary algorithmic progress over the last 3 years, keywords: Hybrid Monte Carlo (HMC) Duane et al. 87 [Della Morte et al. 5] Multiple time-step integration Sexton, Weingarten 92 Frequency splitting of determinant Hasenbusch Domain Decomposition Lüscher 4 Mass preconditioning and rational HMC Urbach et al 5; Clark, Kennedy 6 Deflation of low quark modes Lüscher 7 Avoiding topology freezing Lüscher, Schaefer 2 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 26 - Firenze January 26 p. 2/3

15 Numerical lattice QCD: algorithms Extraordinary algorithmic progress over the last 3 years, keywords: Hybrid Monte Carlo (HMC) Duane et al [BMW Collaboration 9] Multiple time-step integration Sexton, Weingarten 92 Frequency splitting of determinant Hasenbusch Domain Decomposition Lüscher 4 Mass preconditioning and rational HMC Urbach et al 5; Clark, Kennedy 6 M[MeV] 5 5 p K r K * N L S X D S* O X* experiment width input QCD Deflation of low quark modes Lüscher 7 Avoiding topology freezing Lüscher, Schaefer 2 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 26 - Firenze January 26 p. 2/3

16 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 ne (2M N 3M π ) x y decreases exp. with time-distance of sources. At physical point 2M N 3M π 7 fm Computers L. Giusti GGI School 26 - Firenze January 26 p. 3/3

17 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 L. Giusti GGI School 26 - Firenze January 26 p. 3/3

18 Broken symmetries and lattice gauge theory (II and III): chiral anomaly and the Witten Veneziano mechanism Leonardo Giusti University of Milano-Bicocca L. Giusti GGI School 26 - Firenze January 26 p. /7

19 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.4 - π dū.4 π (d d uū)/ K+ u s K d s K sū K s d.498 η cosϑη 8 sinϑη.548 η sinϑη 8 +cosϑη.958 A 9 th pseudoscalar with m η O(Λ) η 8 = (d d+uū 2s s)/ 6 η = (d d+uū+s s)/ 3 ϑ L. Giusti GGI School 26 - Firenze January 26 p. 2/7

20 QCD action and its (broken) symmetries QCD action for N F = 2, M = M = diag(m,m) S = S G + d 4 x { ψdψ + ψr M ψ L + ψ } L Mψ R, D = γ µ ( µ ia µ ) For M = chiral symmetry SU(3) c SU(2) L SU(2) R U() L U() R R scale ( ) ±γ5 (dim. transm., chiral anomaly) ψ R,L V R,L ψ R,L ψ R,L = 2 ψ SU(3) c SU(2) L SU(2) R U() B=L+R Chiral anomaly: measure not invariant SSB: vacuum not symmetric SU(3) c SU(2) L+R U() B (Spont. Sym. Break.) Breaking due to non-perturbative dynamics. Precise quantitative tests are being made on the lattice SU(2) L+R U() B (Confinement) L. Giusti GGI School 26 - Firenze January 26 p. 3/7

21 Numerical challenge A Monte Carlo computation of χ YM L = V (n + n ) 2 YM is challenging for several reasons L fm and a.8 fm = dim[d] the full matrix not feasible : computing and diagonalizing A standard minimization would require high precision to beat contamination from quasi-zero modes At large V the probability distribution has a width which increases linearly with V P Q = e 2πVχ YM L Q2 2V χ YM L {+O(V )} = computing χ YM L requires very high statistics L. Giusti GGI School 26 - Firenze January 26 p. 4/7

22 Algorithm for zero-mode counting In finite V null prob. for n + and n D + D Simultaneous minimization of Ritz functionals associated to D ± = P ± DP ± P ± = ±γ 5 2 Ritz Ritz to find the gap in one of the sectors Run again the minimization in the sector without gap and count zero modes No contamination from quasi-zero modes n + { L. Giusti GGI School 26 - Firenze January 26 p. 5/7

23 Non-perturbative computation for N c = 3 [Del Debbio et al. 4; Cè et al. 4] With the GW definition a fit of the form r 4 χ YM (a,s) = r 4 χ YM +c (s) a2 gives r 4χYM =.59±.3 r GW s=.4 (DGP 4) GW s=. (DGP 4) By setting the scale F K = 9.6 MeV r 4 χ.6 χ YM = (.85±.5 GeV) 4.4 to be compared with F 2 2N F (M 2 η +M 2 η 2M 2 K ) exp (.8 GeV) (a/r ) 2 The (leading) QCD anomalous contribution to M 2 η supports the Witten Veneziano explanation for its large experimental value L. Giusti GGI School 26 - Firenze January 26 p. 6/7

24 Non-perturbative computation for N c = 3 [Del Debbio et al. 4; Cè et al. 4] With the GW definition a fit of the form r 4 χ YM (a,s) = r 4 χ YM +c (s) a2 gives r 4χYM =.59±.3 r GW s=.4 (DGP 4) GW s=. (DGP 4) Spec. Projector (LP ) Spec. Projector (Cichy et al. 5) Wilson Flow (LP ) Wilson Flow (Chowdhury et al. 4) Wilson Flow (Ce et al. 5) By setting the scale F K = 9.6 MeV r 4 χ.6 χ YM = (.85±.5 GeV) 4.4 With the Wilson flow definition r 4 χym =.54± (a/r ) 2 which corresponds to χ YM = (.8±.4 GeV) 4 From an unsolved problem to a universality test of lattice gauge theory! L. Giusti GGI School 26 - Firenze January 26 p. 6/7

25 How the WV mechanism works? [LG, Petrarca, Taglienti 7; Cè et al. 4] Vacuum energy and charge distribution are π e F(θ) = e iθq, P Q = π dθ 2π e iθq e F(θ) 8 6 Norm Inst Edge Data Their behaviour is a distinctive feature of the configurations that dominate the path integr. Large N c predicts [ t Hooft 74; Witten 79; Veneziano 79] Q 2n con Q 2 N 2n 2 c Various conjectures. For example, dilute-gas instanton model gives [ t Hooft 76; Callan et al. 76;... ] F Inst (θ) = VA{cos(θ) } Q 2n con Q 2 = <Q 4 > con /<Q 2 > Gas of instantons L=.3 fm GPT 7 L=.2 fm GPT 7 Ce et al (a/r ) 2 L. Giusti GGI School 26 - Firenze January 26 p. 7/7

26 How the WV mechanism works? [LG, Petrarca, Taglienti 7; Cè et al. 4] Vacuum energy and charge distribution are π e F(θ) = e iθq, P Q = π dθ 2π e iθq e F(θ) 8 6 Norm Inst Edge Data Their behaviour is a distinctive feature of the configurations that dominate the path integr. Lattice computations give Q 4 con Q 2 =.3 ±. Ginsparg Wilson =.23 ±.5 Wilson-Flow i.e. supports large N c and disfavours a dilute gas of instantons <Q 4 > con /<Q 2 > Gas of instantons L=.3 fm GPT 7 L=.2 fm GPT 7 Ce et al. 5 The anomaly gives a mass to the η thanks to the NP quantum fluctuations of Q (a/r ) 2 L. Giusti GGI School 26 - Firenze January 26 p. 7/7

27 Broken symmetries and lattice gauge theory (IV): spontaneous symmetry breaking and the Banks Casher relation Leonardo Giusti University of Milano-Bicocca L. Giusti GGI School 26 - Firenze January 26 p. /

28 Light pseudoscalar meson spectrum Octet compatible with SSB pattern I I 3 S Meson Quark Mass Content (GeV) 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 π + u d.4 - π dū.4 π (d d uū)/ K+ u s K d s K sū K s d.498 η cosϑη 8 sinϑη.548 A 9 th pseudoscalar with m η O(Λ) η sinϑη 8 +cosϑη.958 η 8 = (d d+uū 2s s)/ 6 η = (d d+uū+s s)/ 3 ϑ L. Giusti GGI School 26 - Firenze January 26 p. 2/

29 QCD action and its (broken) symmetries QCD action for N F = 2, M = M = diag(m,m) S = S G + d 4 x { ψdψ + ψr M ψ L + ψ } L Mψ R, D = γ µ ( µ ia µ ) For M = chiral symmetry SU(3) c SU(2) L SU(2) R U() L U() R R scale ( ) ±γ5 (dim. transm., chiral anomaly) ψ R,L V R,L ψ R,L ψ R,L = 2 ψ SU(3) c SU(2) L SU(2) R U() B=L+R Chiral anomaly: measure not invariant SSB: vacuum not symmetric SU(3) c SU(2) L+R U() B (Spont. Sym. Break.) Breaking due to non-perturbative dynamics. Precise quantitative tests are being made on the lattice SU(2) L+R U() B (Confinement) L. Giusti GGI School 26 - Firenze January 26 p. 3/

30 Banks Casher relation [Banks, Casher 8] 4 The spectral density of D is [Banks, Casher 8; Leutwyler, Smilga 92; Shuryak, Verbaarschot 93] 3 ρ(λ,m) = V δ(λ λ k ) k ρ(λ,m) (π/σ) 2 where... indicates path-integral average λ [GeV] The Banks Casher relation lim lim lim ρ(λ,m) = Σ λ m V π can be read in both directions: a non-zero spectral density implies that the symmetry is broken with a non-vanishing Σ and vice versa. To be compared, for instance, with the free case ρ(λ) λ 3 L. Giusti GGI School 26 - Firenze January 26 p. 4/

31 Banks Casher relation [Banks, Casher 8] The spectral density of D is [Banks, Casher 8; Leutwyler, Smilga 92; Shuryak, Verbaarschot 93] ρ(λ,m) = V δ(λ λ k ) where... indicates path-integral average k ν(λ,m) L=2.5 fm, V=2L Λ [GeV] The number of modes in a given energy interval ν(λ,m) = V Λ Λ dλ ρ(λ,m) ν(λ,m) = 2 ΛΣV +... π grows linearly with Λ, and they condense near the origin with values /V In the free case ν(λ,m) VΛ 4 L. Giusti GGI School 26 - Firenze January 26 p. 4/

32 Numerical computation forn f = 2 [Cichy et al. 3; Engel et al. 4] Twisted-mass QCD [Cichy et al. 3]: a = fm m = 6 47 MeV M = 5 2 MeV M = Λ 2 +m 2 ν a=.85fm m=2mev fit M[MeV] O(a) improved Wilson fermions [Engel et al. 4]: a = fm m = 6 37 MeV Λ = 2 5 MeV ν = 9.(3)+2.7(7)Λ+.22(4)Λ 2 ν m = 2.9 MeV 5 a =.48 fm Λ [MeV] The mode number is a nearly linear function in Λ up to approximatively MeV. The modes do condense near the origin as predicted by the Banks Casher mechanism L. Giusti GGI School 26 - Firenze January 26 p. 5/

33 Numerical computation forn f = 2 [Cichy et al. 3; Engel et al. 4] Twisted-mass QCD [Cichy et al. 3]: a = fm m = 6 47 MeV M = 5 2 MeV M = Λ 2 +m 2 ν a=.85fm m=2mev fit M[MeV] O(a) improved Wilson fermions [Engel et al. 4]: a = fm m = 6 37 MeV Λ = 2 5 MeV ν = 9.(3)+2.7(7)Λ+.22(4)Λ 2 ν m = 2.9 MeV 5 a =.48 fm Λ [MeV] At fixed lattice spacing and at the percent precision, however, data show statistically significant deviations from the linear behaviour of O(%). L. Giusti GGI School 26 - Firenze January 26 p. 5/

34 r Continuum limit [Engel et al. 4] By defining ρ(λ,λ 2,m) = π 2V ν(λ 2 ) ν(λ ) Λ 2 Λ the continuum limit is taken at fixed m, Λ and Λ 2 [Λ = (Λ +Λ 2 )/2] Data are extrapolated linearly in a 2 as dictated by the Symanzik analysis L. Giusti GGI School 26 - Firenze January 26 p. 6/

35 Continuum limit [Engel et al. 4] By defining 4 ρ(λ,λ 2,m) = π 2V ν(λ 2 ) ν(λ ) Λ 2 Λ the continuum limit is taken at fixed m, Λ and Λ 2 [Λ = (Λ +Λ 2 )/2] Data are extrapolated linearly in a 2 as dictated by the Symanzik analysis ρ /3 [MeV] Λ [MeV] m = 2.9 MeV a = L. Giusti GGI School 26 - Firenze January 26 p. 6/

36 Continuum limit [Engel et al. 4] By defining 4 ρ(λ,λ 2,m) = π 2V ν(λ 2 ) ν(λ ) Λ 2 Λ the continuum limit is taken at fixed m, Λ and Λ 2 [Λ = (Λ +Λ 2 )/2] Data are extrapolated linearly in a 2 as dictated by the Symanzik analysis ρ /3 [MeV] Λ [MeV] m = 2.9 MeV a = It is noteworthy that no assumption on the presence of SSB was needed so far The results show that at small quark masses the spectral density is non-zero and (almost) constant in Λ near the origin Data are consistent with the expectations from the Banks Casher mechanism in the presence of SSB. In this case NLO ChPT indeed predicts (N f = 2) { ρ nlo = Σ + mσ [ ( Σm ) ( Λ (4π) 2 F 4 3 l 6 + ln(2) 3ln F 2 µ 2 + g ν m, Λ ) ]} 2 m L. Giusti GGI School 26 - Firenze January 26 p. 6/

37 Spectral density in ChPT [Osborn et al. 99; LG, Lüscher 9; Damgaard, Fukaya 9; Damgaard et al. ; Necco, Shindler ] When chiral symmetry is spontaneously broken, the spectral density can be computed in ChPT. At the NLO ρ nlo (λ,m) = Σ π { + mσ [ ( ( ) Σm λ ]} (4π) 2 F 4 3 l 6 + ln(2) 3ln )+g F 2 µ 2 ν m where g ν (x) is a parameter-free known function The NLO formula has properties which can be confronted against the NP results: at fixed λ no chiral logs are present when m g ν (x) x 3 ln(x) in the chiral limit ρ nlo (λ,m) becomes independent of λ This is an accident of the N f = 2 ChPT theory at NLO [Smilga, Stern 93] the λ dependence of ρ nlo (λ,m) is a known function (up to overall constant). The spectral density is a slowly decreasing function of λ at fixed m L. Giusti GGI School 26 - Firenze January 26 p. 7/

38 Chiral limit [Engel et al. 4] In the chiral limit NLO ChPT predicts ρ to be Λ-independent. By extrapolating to m = [ ρ MS ] /3 = [Σ MS BK (2 GeV)]/3 = 26(6)(8) MeV where the spacing is fixed by introducing a quenched strange quark withf K = 9.6 MeV ρ /3 [MeV] m = 5 a = Λ [MeV] L. Giusti GGI School 26 - Firenze January 26 p. 8/

39 Chiral limit [Engel et al. 4] In the chiral limit NLO ChPT predicts ρ to be Λ-independent. By extrapolating to m = [ ρ MS ] /3 = [Σ MS BK (2 GeV)]/3 = 26(6)(8) MeV where the spacing is fixed by introducing a quenched strange quark with F K = 9.6 MeV ρ /3 [MeV] m = 5 a = Λ [MeV] The distinctive signature of SSB is the agreement between ρ and the slope of M 2 πf 2 π/2 with respect to m in the chiral limit M π 2 /(2m RGI F) On the same set of configurations by fitting the data with NLO (W)ChPT for M π < 4 MeV M π 2 /(4πF) [Σ MS GMOR (2 GeV)]/3 = 263(3)(4) MeV to be compared with the previous result Continuum data m RGI /(4πF) L. Giusti GGI School 26 - Firenze January 26 p. 8/

40 Chiral limit [Engel et al. 4] In the chiral limit NLO ChPT predicts ρ to be Λ-independent. By extrapolating to m = [ ρ MS ] /3 = [Σ MS BK (2 GeV)]/3 = 26(6)(8) MeV where the spacing is fixed by introducing a quenched strange quark with F K = 9.6 MeV ρ /3 [MeV] m = 5 a = Λ [MeV] The distinctive signature of SSB is the agreement between ρ and the slope of M 2 πf 2 π/2 with respect to m in the chiral limit M π 2 /(2m RGI F) On the same set of configurations by fitting the data with NLO (W)ChPT for M π < 4 MeV M π 2 /(4πF) [Σ MS GMOR (2 GeV)]/3 = 263(3)(4) MeV to be compared with the previous result Banks-Casher + GMOR Continuum data m RGI /(4πF) L. Giusti GGI School 26 - Firenze January 26 p. 8/

41 Gell-Mann Oakes Renner relation The spectral density of the Dirac operator in the continuum is at the origin for m = M π 2 /(2m RGI F) The low-modes of the Dirac operator do condense following Banks Casher mechanism M π 2 /(4πF) Banks-Casher + GMOR Continuum data m RGI /(4πF) The rate of condensation agrees with the GMOR relation, and it explains the bulk of the pion mass up to M π 5 MeV The dimensionless ratios [Σ RGI ] /3 /F = 2.77(2)(4), Λ MS /F = 3.6(2) are geometrical properties of the theory. They belong to the category of unambiguous quantities in the two flavour theory that should be used for quoting and comparing results rather than those expressed in physical units They can be directly compared with your preferred approximation/model L. Giusti GGI School 26 - Firenze January 26 p. 9/

42 Summary An impressive global (lattice) community effort to reach a precise quantitative understanding of the behaviour of QCD in the chiral regime from first principles The spectral density of the Dirac operator in the continuum and chiral limits is at the origin. The rate of condensation explains the bulk of the pion mass up to M π 5 MeV M π 2 /(4πF) RGI M π /(2m F) Banks-Casher + GMOR Continuum data m RGI /(4πF) All numerical results for χ YM are consistent with the conceptual progress made over the last decade. A percent precision reached. Universality is at work if χ is (properly) defined on the lattice! The (leading) QCD anomalous contribution to M 2 η supports the Witten Veneziano explanation for its large experimental value r 4 χ GW s=.4 (DGP 4) GW s=. (DGP 4) Spec. Projector (LP ) Spec. Projector (Cichy et al. 5) Wilson Flow (LP ) Wilson Flow (Chowdhury et al. 4) Wilson Flow (Ce et al. 5) (a/r ) 2 L. Giusti GGI School 26 - Firenze January 26 p. /