Lectures on Chiral Perturbation Theory I. Foundations II. Lattice Applications III. Baryons IV. Convergence Brian Tiburzi RIKEN BNL Research Center Chiral Perturbation Theory I. Foundations Low-energy QCD Mesons Solutions of QCD (courtesy of nature) pseudo-scalar Hyperons Kaons Nucleons Baryons positive parity spin-half Pions Ground state: vacuum • Spectrum (and properties) of low-lying hadrons indicative of symmetries ... and symmetry breaking • ChPT is the tool to study such manifestations in low-energy QCD Nf Massless QCD QCD = ψ + YM ψ = ψ D/ψ i L L L L i i=1 • Take two flavors. These will correspond to up and down quarks. • Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry 1 = (1 γ ) projectors ψL,R = L,R ψ PL,R 2 ∓ 5 P ψD/ψ = ψLD/ψ L + ψRD/ψ R • Left- and right-handed fields do not mix: no chirality changing interaction U(2)L U(2)R ψL LψL ⊗ → ψ (L L + R R) ψ ψ Rψ → P P LR R → R Nf Massless QCD QCD = ψ + YM ψ = ψ D/ψ i L L L L i i=1 • Take two flavors. These will correspond to up and down quarks. • Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry 1 = (1 γ ) projectors ψL,R = L,R ψ PL,R 2 ∓ 5 P ψD/ψ = ψLD/ψ L + ψRD/ψ R • Left- and right-handed fields do not mix: no chirality changing interaction Vector subgroup U(2)L U(2)R U(2)V ψL VψL ⊗ → ψ V ( L + R)ψ = Vψ ψ Vψ → P P L = R = V R → R Chiral Symmetry of Massless QCD Action invariant under U(2) U(2) = U(1) U(1) SU(2) SU(2) L ⊗ R L ⊗ R ⊗ L ⊗ R U(1) : ψ eiθL ψ L L L 1 iθR iθL 1 iθR iθL → ψ (e + e )+ (e e )γ5 ψ U(1) : ψ eiθR ψ → 2 2 − R R → R iθ Vector subgroup θ = θ = θ ψ e ψ U(1)V L R → iθ5γ5 Axial transformation θ = θ = θ ψ [cos θ5 + iγ5 sin θ5]ψ = e ψ − L R 5 → U(1)A Exercise: Consider a non-singlet axial transformation ψi [exp(iφ τ γ 5)]ijψj → · Is there a corresponding symmetry group of the massless QCD action? Chiral Symmetry of Massless QCD Action invariant under U(2) U(2) = U(1) U(1) SU(2) SU(2) L ⊗ R A ⊗ V ⊗ L ⊗ R • Global symmetries lead to classically conserved currents a a a a J5µ = ψγµγ5ψJµ = ψγµψJLµ = ψLγµτ ψL JRµ = ψRγµτ ψR (Regulated) Theory not invariant under flavor-singlet axial transformation e 2π Nf µν Fµν (x) d =2 ∂µJ5µ(x)=∂µJRµ(x) ∂µJLµ(x)= − − αs N GA (x)GA (x) d =4 − 4π f µνρσ µν ρσ Chiral Anomaly U(1) SU(2) SU(2) V ⊗ L ⊗ R The chiral anomaly obstructs chirally invariant lattice regularization of fermions (see Kaplan’s lectures) Fate of Symmetries in Low-Energy QCD U(1) SU(2) SU(2) V ⊗ L ⊗ R • Chiral pairing preferred by vacuum (non-perturbative ground state) Chiral condensate ψψ = ψ ψ + ψ ψ =0 R L L R • Massless quarks can change their chirality by scattering off vacuum condensate U(1) SU(2) SU(2) U(1) SU(2) V ⊗ L ⊗ R −→ V ⊗ V • Spontaneously broken symmetries lead to massless bosonic excitations Nambu-Goldstone Mechanism Broken generators in coset SU(2) SU(2) /SU(2) L ⊗ R V Number of massless particles? ψ ψ = λδ Chiral Condensate iR jL − ji • Choice for vacuum orientation λ R from Vafa-Witten (P) ∈ • After a chiral transformation ψ ψjL Ljj R† ψ ψj L = λ(LR†)ji iR → ii iR − SU(2) SU(2) SU(2) L ⊗ R −→ V • Describe Goldstone fluctuations of vacuum state with fields δji Σji(x)=δji + ... Σ SU(2) SU(2) /SU(2) → ∈ L ⊗ R V iθ (x) τ iθ (x) τ [L(x)R†(x)] =[e L · e− R · ] θ = θ ji ji L − R 2iφ Σ=e2iφ/f =1+ + ... Transformation properties f Σ LΣR† Σ V ΣV † φ VφV† → → → Exercise: Determine the discrete symmetry properties of the Goldstone modes from the coset’s transformation. Dynamics of Goldstone Bosons: Chiral Lagrangian 2iφ Tr φ =0 1 π0 π+ Σ=e2iφ/f =1+ + ... √2 The Pions φ = 1 0 f φ† = φ π− π − √2 • Build chirally invariant theory of coset field 2 Σ LΣR† f → Σ†Σ=1 = Tr ∂µΣ∂µΣ† Σ† RΣ†L L 8 → • Expand about v.e.v. to uncover Gaußian fluctuations 1 2 1 0 0 + 2 = Tr (∂ φ∂ φ)+ (1/f )= ∂ π ∂ π + ∂ π−∂ π + (1/f ) L 2 µ µ O 2 µ µ µ µ O 3 massless modes • Non-linear theory: interactions between multiple pions at “higher orders” Can treat systematically... Including Quark Masses • We began with massless QCD. Quarks have mass, Higgs makes two very light • Chiral symmetry of action is only approximate: explicit symmetry breaking ∆ = m ψ ψ = m ψ ψ + ψ ψ Lψ q i i q iR iL iL iR i i SU(2) SU(2) SU(2) m /Λ 1 L ⊗ R −→ V q QCD • Need to map ∆ onto ChPT operators breaking symmetry in same way Lψ ∆ = m λ Tr Σ+Σ† Leff − q Σ LΣR† Comments: not chirally invariant → Σ† RΣ†L† → new dimensionful parameter λ 2 included only linear quark mass term mq Perturbing about chiral limit f 2 Chiral Lagrangian χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − 1 8mqλ 1 • Expand up to quadratic order = 4m λ + Tr (∂ φ∂ φ)+ Tr (φφ) Lχ − q 2 µ µ f 2 2 2 2 Pion mass mπ =8mqλ/f • Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom ∂ log ZQCD ( +mq ψψ) ZQCD[mq,...]= e− x ··· = ψψ D··· − ∂mq Low-energy degrees of freedom Effective field theory χ(Σ; mq ) Z [m ,...]= Σ e− x L ZQCD[mq,...] ZχPT[mq,...] χPT q D ≡ Matching From before: ∂ log ZχPT ψψ = = λ Tr Σ + Σ† = 2Nf λ ψiRψjL = λδji − ∂mq − − − λ = λ f 2 Chiral Lagrangian χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − 1 8mqλ 1 • Expand up to quadratic order = 4m λ + Tr (∂ φ∂ φ)+ Tr (φφ) Lχ − q 2 µ µ f 2 2 2 2 2 2 Pion mass m =8mqλ/f f m =2m ψψ (Gell-Mann Oakes Renner) π π q| | • Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom ∂ log ZQCD ( +mq ψψ) ZQCD[mq,...]= e− x ··· = ψψ D··· − ∂mq Low-energy degrees of freedom Effective field theory χ(Σ; mq ) Z [m ,...]= Σ e− x L ZQCD[mq,...] ZχPT[mq,...] χPT q D ≡ Matching From before: ∂ log ZχPT ψψ = = λ Tr Σ + Σ† = 2Nf λ ψiRψjL = λδji − ∂mq − − − λ = λ f 2 ChPT χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − • Quadratic fluctuations are the approximate Goldstone bosons of SChSB • Quartic terms describe interactions m λ 1 q φ4 (φ∂ φ)2 ∼ f 4 ∼ f 2 µ • Higher-order interactions renormalize lower-order terms m λ 1 m λ m λ ∆m2 q q Λ2 + q log Λ2 +finite π ∼ f 4 k2 + m2 ∼ f 2 f 2 k π power-law divergence logarithmic divergence Absorb in renormalized mass, Renormalization requires new or just use dimensional regularization operator in chiral Lagrangian • ChPT is non-renormalizable (needing infinite local terms to renormalize) 1). low-energy theory, so who cares? 2). must be able to order terms in terms of relevance “power counting” f 2 Power Counting χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − • Leading-order Lagrangian in expansion in derivatives and quark mass (p2) ∂ pmp2 Low-energy dynamics of pions O µ ∼ q ∼ 1 2 2 4 Propagator p− Vertices ∂ ∂ ,m p Loop integral p k2 + m2 ∼ µ µ q ∼ ∼ π k 4L 2I+2V General Feynman diagram: L loops, I internal lines, V vertices p − ∼ 2L+2 Euler formula L = I V +1 p − ∼ • Loop expansion: one loop graphs require only ( p 4 ) counterterms O Two loop graphs? Exercise: What happens to the power-counting argument in d = 2, 6 dimensions? Do the results surprise you? Why didn’t I ask about d = 3, 5? ( p 4 ) Chiral Lagrangian O • Construct chirally invariant terms out of coset Σ LΣR† → Σ† RΣ†L† → 2 E.g. ( p ) Lagrangian ∂ Σ ∂ Σ† O µ µ • Construct terms that break chiral symmetry in the same way as mass term Simplification: add external scalar field to QCD action ∆ = ψ sψ + ψ s†ψ L L R R L Make the scalar transform to preserve chiral symmetry s LsR† → Giving the scalar a v.e.v. breaks chiral symmetry just as a mass s = m + q ··· 2 E.g. ( p ) Lagrangian Σs† + sΣ† O ( p 4 ) Chiral Lagrangian O Σ LΣR† s LsR† Also impose Euclidean invariance, C, P, T → → Σ† RΣ†L† s† Rs†L† → → 2 2 2 E.g. [Tr Σs† sΣ† ] m [Tr Σ Σ† ] =0 − → q − Easy to generate terms. Care needed to find minimal set. 2 4 = L1[Tr ∂µΣ∂µΣ† ] + L2Tr ∂µΣ∂ν Σ† Tr ∂µΣ∂ν Σ† L 2 mqλ (mqλ) 2 + L3 Tr ∂µΣ∂µΣ† Tr Σ+Σ† + L4 [Tr Σ+Σ† ] f 2 f 4 Lj low-energy constants = Gasser-Leutwyler coefficients, dimensionless { } N.B. these are not Gasser-Leutwyler’s coefficients Complete set of counterterms needed to renormalize one-loop ChPT Additional terms necessary when coupling external fields... Exercise: Determine the effects of strong isospin breaking m = m on the chiral u d Lagrangian.