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Symmetries of the Strong Interaction

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Symmetries of the Strong Interaction Symmetries of the strong interaction H. Leutwyler University of Bern DFG-Graduiertenkolleg ”Quanten- und Gravitationsfelder” Jena, 19. Mai 2009 H. Leutwyler – Bern Symmetries of the strong interaction – p. 1/33 QCD with 3 massless quarks As far as the strong interaction goes, the only difference between the quark flavours u,d,...,t is the mass mu, md, ms happen to be small For massless fermions, the right- and left-handed components lead a life of their own Fictitious world with mu =md =ms =0: QCD acquires an exact chiral symmetry No distinction between uL,dL,sL, nor between uR,dR,sR Hamiltonian is invariant under SU(3) SU(3) ⇒ L× R H. Leutwyler – Bern Symmetries of the strong interaction – p. 2/33 Chiral symmetry of QCD is hidden Chiral symmetry is spontaneously broken: Ground state is not symmetric under SU(3) SU(3) L× R symmetric only under the subgroup SU(3)V = SU(3)L+R Mesons and baryons form degenerate SU(3)V multiplets ⇒ and the lowest multiplet is massless: Mπ± =Mπ0 =MK± =MK0 =MK¯ 0 =Mη =0 Goldstone bosons of the hidden symmetry H. Leutwyler – Bern Symmetries of the strong interaction – p. 3/33 Spontane Magnetisierung Heisenbergmodell eines Ferromagneten Gitter von Teilchen, Wechselwirkung zwischen Spins Hamiltonoperator ist drehinvariant Im Zustand mit der tiefsten Energie zeigen alle Spins in dieselbe Richtung: Spontane Magnetisierung Grundzustand nicht drehinvariant ⇒ Symmetrie gegenüber Drehungen gar nicht sichtbar Symmetrie ist versteckt, spontan gebrochen Goldstonebosonen in diesem Fall: "Magnonen", "Spinwellen" Haben keine Energielücke: ω 0 für λ → → ∞ Nambu realisierte, dass auch in der Teilchenphysik Symmetrien spontan zusammenbrechen können H. Leutwyler – Bern Symmetries of the strong interaction – p. 4/33 Chiral symmetry QCD with three massless quarks is invariant under G = SU(3) SU(3) L× R SU(3) has 8 parameters Symmetry under Lie group with 16 parameters ⇒ 16 conserved “charges” ⇒ V V Q1,... ,Q8 (vector currents, R+L) A A Q1,... ,Q8 (axial currents, R-L) H. Leutwyler – Bern Symmetries of the strong interaction – p. 5/33 Chiral symmetry QCD with three massless quarks is invariant under G = SU(3) SU(3) L× R SU(3) has 8 parameters Symmetry under Lie group with 16 parameters ⇒ 16 conserved “charges” ⇒ V V Q1,... ,Q8 (vector currents, R+L) A A Q1,... ,Q8 (axial currents, R-L) Charges commute with the Hamiltonian: V A [Qi ,H0]=0 [Qi ,H0] = 0 Representation of charges in terms of conserved µ µ currents, such as uγ d and uγ γ5d: V 3 0 A 3 0 Q1 = d x uγ d, Q1 = d x uγ γ5d H. Leutwyler – Bern Z Z Symmetries of the strong interaction – p. 5/33 Symmetry properties of the ground state Vafa and Witten 1984: state of lowest energy is invariant under the vector charges QV 0 = 0 i | i Axial charges ? QA 0 =? i | i H. Leutwyler – Bern Symmetries of the strong interaction – p. 6/33 Two alternatives for axial charges A 0 = 0 Qi | i Wigner-Weyl realization of G ground state is symmetric 0 0 = 0 h | qR qL | i ordinary symmetry spectrum contains parity partners degenerate multiplets of G A 0 = 0 Qi | i 6 Nambu-Goldstone realization of G ground state is asymmetric 0 0 = 0 h | qR qL | i 6 “order parameter” spontaneously broken symmetry spectrum contains Goldstone bosons degenerate multiplets of SU(3)V ⊂ G = G SU(3)R × SU(3)L H. Leutwyler – Bern Symmetries of the strong interaction – p. 7/33 Goldstone bosons QCD chooses the Nambu-Goldstone mode: QA 0 = 0 i | i 6 Immediate consequence: H QA 0 = QA H 0 = 0 for i = 1,..., 8 0 i | i i 0 | i Spectrum must contain 8 states QA 0 ,...,QA 0 ⇒ 1 | i 8 | i with E = 0, spin 0, negative parity, octet of SU(3)V "Goldstone bosons" Indeed, the 8 lightest mesons do have these quantum numbers: + 0 + 0 0 π , π , π−,K ,K , K¯ ,K−,η All other one-particle states must form degenerate multiplets of SU(3)V H. Leutwyler – Bern Symmetries of the strong interaction – p. 8/33 Side remark: mathematics used is slippery Argument given for the occurrence of Goldstone bosons is not quite water tight: 0 QA QA 0 = d3xd3y 0 A0(x) A0(y) 0 h | i k | i Z h | i k | i 0 A0(x) A0(y) 0 only depends on ~x ~y h | i k | i − 0 QA QA 0 is proportional to the volume of the ⇒ h | i k | i universe, QA 0 = | i | i| ∞ Rigorous proof of Goldstone theorem given later H. Leutwyler – Bern Symmetries of the strong interaction – p. 9/33 Quark condensate uR uL dR uL sR uL qR qL = uR dL dR dL sR dL uR sL dR sL sR sL Transforms like (3, 3)¯ under SU(3) SU(3) L × R If the ground state were symmetric, the matrix 0 qR qL 0 would have to vanish, because it singles out ah direction| | i in flavour space 0 qR qL 0 is referred to as the “quark condensate”, quantitativeh | | i measure of the strength of spontaneous symmetry breaking, “order parameter” 0 q q 0 is the analog of magnetization h | R L | i H. Leutwyler – Bern Symmetries of the strong interaction – p. 10/33 Quark condensate Ground state is invariant under SU(3)V 0 q q 0 is proportional to unit matrix ⇒ h | R L | i 0 u u 0 = 0 d d 0 = 0 s s 0 h | R L | i h | R L | i h | R L | i 0 u d 0 = . = 0 h | R L | i H. Leutwyler – Bern Symmetries of the strong interaction – p. 11/33 Quark masses Real world = paradise 6 In reality, the multiplets are split and the lightest mesons are not massless m , m , m = 0 u d s 6 Quark masses break chiral symmetry allow the left to talk to the right Chiral symmetry broken in two ways: ⇒ spontaneously 0 q q 0 = 0 h | R L | i 6 explicitly m , m , m = 0 u d s 6 H. Leutwyler – Bern Symmetries of the strong interaction – p. 12/33 Quark masses Only the diagonal vector currents are strictly conserved in QCD: Nu, Nd, Ns, Nc, Nb, Nt baryon number, electric charge, strangeness, charm,→ ... It so happens that mu, md, ms are small H has an approximate SU(3) SU(3) symmetry ⇒ QCD L× R Masses of the light quarks enter the Hamiltonian via HQCD = H0 + H1 3 H = d x mu uu + md dd + ms ss 1 Z { } H0 describes u,d,s as massless, c,b,t as massive H is invariant under SU(3) SU(3) 0 L× R H. Leutwyler – Bern Symmetries of the strong interaction – p. 13/33 Pattern of light quark masses With the discovery of QCD, the mass of the quarks became an unambiguous concept: quark masses occur in the Hamiltonian of the theory. First crude estimate within QCD relied on a model for the wave functions of π, K, ρ, based on SU(6) (spin-flavour-symmetry) 2 1 FπMπ 2(mu + md) = 5 MeV, ms 135 MeV 3FρMρ ≃ ≃ “Is the quark mass as small as 5 MeV ?” 1974 Not very different from the pattern found within the Nambu-Jona-Lasinio model (1961) or the one obtained from sum rules by Okubo (1969) H. Leutwyler – Bern Symmetries of the strong interaction – p. 14/33 Notion of quark mass Quarks do not occur in isolation Mass not directly observable like mass of electron Quark mass is a theoretical notion ⇒ Bare masses in the Hamiltonian cannot serve to define mu, md, ms, need to be renormalized Quark masses depend on renormalization convention Generally accepted convention: compare values in MS scheme (based on dimensional regularization) In this scheme, the quark masses depend on the running scale µ, much like the coupling constant gs Estimates quoted in the 2008 edition of the PDG tables: 1(m + m ) = 2.5 5 MeV, m = 70 130 MeV 2 u d ÷ s ÷ refer to µ=2 GeV H. Leutwyler – Bern Symmetries of the strong interaction – p. 15/33 Pattern of light quark masses Difference between mu and md ? e.m. self energy: proton > neutron M > M cannot be due to the e.m. interaction ⇒ n p Mn > Mp must be due to md > mu Isospin is not a symmetry of the strong interaction ! ⇒ Gasser & L. 1975 In fact a very strong breaking appears to be needed: m 2.5 MeV, m 5 MeV PDG 2008 u ≃ d ≃ H. Leutwyler – Bern Symmetries of the strong interaction – p. 16/33 Crude picture for mu,md,ms m 2.5 MeV, m 5 MeV, m 100 MeV u ≃ d ≃ s ≃ mu and md are very different mu and md are small compared to ms “constituent masses” / Lagrangian of QCD ∈ H. Leutwyler – Bern Symmetries of the strong interaction – p. 17/33 Approximate symmetries are natural in QCD Why is isospin such a good quantum number ? (a) Dimensional transmutation, logarithmic divergences of perturbation theory physics ∈ QCD has an intrinsic scale ⇒ (b) m m scale of QCD, not m + m d − u ≪ ≪ u d Why is the eightfold way a decent approximate symmetry ? m m scale of QCD s − u ≪ Isospin is an even better symmetry because m m m m d − u ≪ s − u m m m , m , m scale of QCD u ≪ s ⇒ u d s ≪ Masses of the light quarks represent perturbations ⇒ Can neglect these in a first approximation H. Leutwyler – Bern Symmetries of the strong interaction – p. 18/33 Quark masses as perturbations HQCD = H0 + H1 3 H = d x mu uu + md dd + ms ss 1 Z { } Expansion in Perturbation series powers of mu, md, ms ⇐⇒ in powers of H1 H0 treats π,K,η as massless, H1 gives them a mass H.
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