Lattice QCD with Wilson fermions Leonardo Giusti CERN - Theory Group CERN L. Giusti – Paris March 2005 – p.1/31 Outline Spontaneous symmetry breaking in QCD Quark mass dependence of pion masses and decay constants Fermions on a lattice: the doubling problem Wilson fermions Chiral Ward identities and additive mass renormalization A new algorithm for full QCD simulations: SAP First dynamical simulations with light quarks Results for pion masses and decay constants L. Giusti – Paris March 2005 – p.2/31 Quantum Chromo Dynamics (QCD) The Euclidean QCD Lagrangian inv. under SU(3) color gauge group (formal level) 4 1 θ SQCD = d x Tr Fµν Fµν + i Tr Fµν F~µν + ¯ D + M − 2g2 16π2 Z h i h i h i ff 1 a a Fµν @µAν @ν Aµ + [Aµ; Aν ] F~µν µνρσFρσ Aµ = A ≡ − ≡ 2 µ T D = γµ @µ + Aµ q1; : : : ; q M diag m1; : : : ; m f g ≡ Nf ≡ f Nf g ˘ ¯ For M = 0 the action is invariant under the global group U(N ) U(N ) f L × f R L VL L ¯L ¯LV y L;R = P ! ! L 1 γ5 R VR R ¯R ¯RV y P = ! ! R 2 When the theory is quantized the chiral anomaly breaks explicitly the subgroup U(1)A For the purpose of this lecture we can put θ = 0 For the rest of this lecture we will assume that heavy quarks have been integrated out and we will focus on the symmetry group SU(3) SU(3) L × R L. Giusti – Paris March 2005 – p.3/31 Light pseudoscalar meson spectrum Octet compatible with SSB pattern I I3 S Meson Quark Mass Content (MeV) SU(3) SU(3) SU(3) + ¯ L × R ! L+R 1 1 0 π ud 140 1 -1 0 π− du¯ 140 and soft explicit symmetry breaking 1 0 0 π0 (dd¯ uu¯)=p2 135 − ———————————————————- m ; m m < Λ 1 1 + u d s QCD 2 2 +1 K us¯ 494 1 1 0 2 - 2 +1 K ds¯ 498 1 1 2 - 2 -1 K− su¯ 494 1 1 0 ¯ 2 2 -1 K sd 498 ———————————————————- mu; md ms = mπ mK ) 0 0 0 η cos ϑη8 + sin ϑη0 547 0 0 0 η0 sin ϑη0 + cos ϑη8 958 th − A 9 pseudoscalar with mη (ΛQCD) 0 ∼ O η8 = (dd¯+ uu¯ 2ss¯)=p6 − η0 = (dd¯+ uu¯ + ss¯)=p3 # 11 ' − ◦ L. Giusti – Paris March 2005 – p.4/31 Vector and Axial Ward Identities By grouping the generators of the SU(3) SU(3) group in the ones of the vector L × R subgroup SU(3)L+R plus the remaining axial generators a a a @µ V (x) = ¯(x) T ; M (x) δ µ O O − V;xO ˙ ¸ D h i E D E a a a @µ A (x) = ¯(x) T ; M γ5 (x) δ µ O O − A;xO ˙ ¸ D n o E D E where currents and densities are defined to be a a a a V ¯γµT A ¯γµγ T µ ≡ µ ≡ 5 Sa ¯T a P a ¯γ T a ≡ ≡ 5 Ward identities encode symmetry properties of the theory, and they remain valid even in presence of spontaneous symmetry breaking L. Giusti – Paris March 2005 – p.5/31 Spontaneous chiral symmetry breaking in QCD By choosing the interpolating operator = P a(0) the AWI reads O a a a a 1 @µ A (x)P (0) = ¯(x) T ; M γ5 (x) P (0) δ(x) ¯ µ − 3 ˙ ¸ D n o E ˙ ¸ In the chiral limit a a @µA (x)P (0) = 0 x = 0 h µ i 6 and by using Lorentz invariance and power counting xµ Aa (x)P a(0) = c x = 0 h µ i (x2)2 6 Integrating by parts the AWI in a ball of radius r a a 3 ¯ dsµ(x) Aµ(x)P (0) = x =r h i − 2 h i Zj j which implies a a 3 xµ @µA (x)P (0) = ¯ x = 0 h µ i − 4π2 h i (x2)2 6 L. Giusti – Paris March 2005 – p.6/31 Pseudoscalar mesons as Goldstone bosons of the theory If ¯ = 0 the relation h i 6 a a 3 xµ @µA (x)P (0) = ¯ x = 0 h µ i − 4π2 h i (x2)2 6 implies that the current-density correlation function is long-ranged The energy spectrum does not have a gap and the correlation function has a particle pole at zero momentum (Goldstone theorem) In the chiral limit ¯ = 0 implies the presence of 8 Goldstone bosons h i 6 identified with the 8 pseudoscalar light mesons [π; : : : ; ; K; : : : ; η] Previous relations lead to a a 0 A P ; pµ = pµ F h j µj i which in turn implies that interactions among peudoscalar mesons vanish for p2 = 0 L. Giusti – Paris March 2005 – p.7/31 Quark mass dependence of the pseudoscalar mesons When M = 0 (and for simplicity in the degenerate case M = m11) 6 1 2m P a(x)P a(0) = ¯ h µ i 3 h i Z and therefore for m 0 ! ¯ M 2 = M 2 = 2 m h i P − 3F 2 It is possible to build an effective theory of QCD with 8 light pseudoscalar mesons as fundamental degrees of freedom In particular for pions, it predicts the following functional forms for masses and decay constants at NLO M 2 M 2 = M 2 1 + log(M 2/µ2 ) π 32π2F 2 π ff 2 M 2 2 Fπ = F 1 log(M /µ ) − 16π2F 2 F ff L. Giusti – Paris March 2005 – p.8/31 p.9/31 – 2005 ch Mar is ar P – a Giusti L. ff techniques i 0) ) + µ ν + µ ; x lo x x ) ( U x ( ( y µν 2 Car ) µ U (x) O x µ ) ( aA U + y x ν ; ) U − x ) U Monte ( 1 ( ν 1 x is µν ' + via O + ν (x) U ) ) y x + ν x h ( x U U ely ( ( y µ r µ T U theor ) U 1 6 YM µ S y) − − + turbativ e 1 x U ( theor ν ef ef ef cd cd cd D a ¢ ¢ ¢ 7¢88 7 E¢FF E S¢TT S a¢bb a U ang–Mills ) X Y Z x non-per x,µ<ν ¢ !¢"" ! 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K¢LLM KM Y¢ZZ[ Y[ the -¢../¢001 -/1 ) computed x ) QCD ¥¢¦¦§¢¨¨ ¥§ '¢(()¢** ') +¢,,- +- ( or ¢ ;¢<< ; I¢JJ I W¢XX W f x 1 ( fields ¥ ¥ ' ' of O ¢ 9¢::; 9; G¢HHI GI U¢VVW UW h µν YM U S action functions g g g g g g g g g g g g g g g gauge L Wilson small gularization re or The F Correlation Lattice Lattice regularization of QCD Given a generic massive Dirac operator D(x; y) and the corresponding action S = ¯(x)D(x; y) (x) q1; : : : ; q F ≡ Nf x;y X n o the functional integral is defined to be Z = δUδ δ ¯ exp S S {− YM − Fg R By integrating over the Grassman fields, a generic Euclidean corr. function is 1 S O1(x1)O2(x2) = δU e− YM DetD [O1(x1)O2(x2)] h i Z Wick Z For vector gauge theories and positive masses, Det D is real and positive Correlation functions can be computed non-perturbatively via Monte Carlo techniques L. Giusti – Paris March 2005 – p.10/31 Naive discretization of the Dirac operator The naive gauge invariant discretization of the Dirac operator is Sin(pa) 1 D = γµ ∗ + µ + m 2 rµ r ˘ ¯ − π/a π/a where (a is the lattice spacing) p 1 µ (x) = Uµ(x) (x + aµ^) (x) r a − h i 1 (x) = (x) U y(x aµ^) (x aµ^) rµ∗ a − µ − − h i In the free case and in the Fourier basis (p¯µ = sin(pµa)=a) 1 iγµp¯µ + m D~ − (p) = − p¯2 + m2 there are 15 extra poles (doublers)! L. Giusti – Paris March 2005 – p.11/31 Nielsen-Ninomiya theorem Nielsen Ninomiya 81 The following properties cannot hold simultaneously for free fermions on the lattice: 1. D~ (P ) is an analytic periodic function of pµ with period 2π=a 2 2. For pµ π=a D~ (P ) = iγµpµ + (ap ) O 3. D~ (P ) is invertible at all non-zero momenta (mod 2π=a) 4. D anti-commute with γ5 (for m = 0) (1) is needed for locality, (2) and (3) ensures the correct continuum limit Chiral symmetry in the continuous form (4) must be broken on the lattice Physics essence: if action invariant under standard chiral sym. = no chiral anomaly ) L. Giusti – Paris March 2005 – p.12/31 Wilson fermions Wilson's proposal is to add an irrelevant operator to the action 1 0 DW = γµ( ∗ + µ) a ∗ µ + m 2 rµ r − rµr ˘ ¯ which breaks chiral symmetry explicitly (SU(3)L+R vector symmetry preserved!) The Wilson term a µ removes the doubler poles. In the free case rµ∗ r 0 1 iγµp¯µ + m (p) 0 0 a 2 D~ − (p) = − m (p) m + p^ p¯2 + m0(p)2 ≡ 2 2 pµa where p^µ = sin a 2 “ ” At the classical level Wilson term is irrelevant, it gives vanishing contributions for a 0 ! L. Giusti – Paris March 2005 – p.13/31 Axial Ward identities for Wilson fermions Bochicchio et al. 85 By performing a non-singlet axial rotation in the functional integral a a 0 a a @µ A (x) = ¯(x) T ; M γ5 (x) + X (x) δ h µ Oi h Oi h Oi − h xOi n o At the classical level the operator X a(x) vanishes for a 0.