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 ￿→ ￿ i￿i￿ i￿R ￿ ￿ − 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. At what order does the pion isospin multiplet split? Simplest one-loop computation: Chiral Condensate

∂ log ZχPT ψψ = = λ Tr Σ + Σ† = 2Nf λ “Tree-Level” ￿ ￿ − ∂mq − ￿ ￿ − ￿ ￿ 4 2 Σ+Σ† =2 φ + One Loop − f 2 ··· 4λ 12λ 1 ∆ ψψ =+ 3 G (0) = ￿ ￿ f 2 × π f 2 k2 + m2 ￿k π m2 1 µ2 π γ + log 4π + log +1 Dimensionally regulated integral −(4π)2 ￿ − E m2 ￿ π ￿ 2 4 λ mqλ 2 (p ) L3 Tr ∂µΣ∂µΣ† Tr Σ+Σ† +2L4 [Tr Σ+Σ† ] f 2 f 4 O 2 2 “Tree-Level” ￿ ￿ ￿ ￿ mqλ ￿ mπ ￿ = 32 L4 =4 L4 λ f 4 f 2 Final result: Chiral Logarithm 2 2 2 3 mπ µ mπ ψψ = 4λ 1+ log +1 (µ) d 3 2 2 2 L4 µ2 = ￿ ￿ − (4πf) m − f 2 L4 2 ￿ ￿ π ￿ ￿ dµ 16π f 2 = Tr ∂ Σ∂ Σ† m λ Tr Σ+Σ† Two-Flavor ChPT L2 8 µ µ − q ￿ ￿ ￿ ￿ ￿￿ • Leading and next-to-leading order Lagrangian in isospin limit mu = md

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 • Compute quark mass￿ dependence￿ of￿ chiral condensate,￿ pion mass,￿ pion-pion￿ scattering, ..., in terms of a few low-energy constants ψψ = A [1 + B m (log m + C )] ￿ ￿ 0 0 q q 0 2 mπ = A1 mq [1 + B1 mq (log mq + C1)] I=2 aππ = A2√mq [1 + B2 mq (log mq + C2)]

• Further applications: electroweak properties of pions require external fields Incorporating External Fields in ChPT

• Start by incorporating external gauge fields in QCD

ψ = ψLD/ L ψL + ψRD/ R ψR L E.g. external vector field L = R = Qe µ µ Aµ (DL)µ = ∂µ + ig Gµ + iLµ (D ) = ∂ + ig G + iR Local invariance [SU(2) ] [SU(2) ] R µ µ µ µ L ⊗ R

ψL L(x)ψL Lµ L(x)LµL†(x)+i[∂µL(x)]L†(x) −→ and −→ ψR R(x)ψR R R(x)R R†(x)+i[∂ R(x)]R†(x) −→ µ −→ µ µ • Then incorporate external gauge fields in ChPT

Σ L(x)ΣR†(x) Need covariant derivative DµΣ L(x)[DµΣ]R†(x) → → D Σ=∂ Σ+iL Σ iΣR† µ µ µ − µ Leading-order chiral Lagrangian [with external fields counted as ( p ) ] O f 2 2 = Tr DµΣDµΣ† mqλ Tr Σ+Σ† *Additional operators L 8 − at higher orders ￿ ￿ ￿ ￿ ￿￿ W µ π π µ + νµ What is f? νµ →

+ + Pion weak decay ∆ = W −J J = uLγµdL L µ µL µL Strong part factorizes into QCD matrix element 0 J + π(p￿ ) = ip f (the rest you learned how to compute in QFT) ￿ | µL| ￿ µ π 2 2 2 pion decay constant GF 2 2 2 mµ Γπ µ+ν = f m mπ Vud 1 µ π µ 2 fπ = 132 MeV → 8π | | ￿ − mπ ￿

1 τ + = (τ 1 + iτ 2) ChPT current matches the QCD current 2 2 a ∂ χ f a f a JµL = La = Tr iτ Σ∂µΣ† + = Tr (τ ∂µφ)+ ∂Lµ ￿ 4 ··· 2 ··· ￿Lµ=0 ￿ ￿ ￿ ￿ + ￿ 0 JµL π(p￿ ) = ipµ (f + ) 2 2 ￿ | | ￿ ··· Dimensionless power counting p /Λχ 2 2 Λχ =2√2πf 1.2 GeV mπ/Λχ fπ = f [1 + Bmq (log mq + C)] ∼ Exercise:

The masses of hadrons are modified by electromagnetism.

Construct all leading-order electromagnetic mass operators by promoting the electric charge matrix to fields transforming under the chiral group. (Notice that no photon fields will appear in the electromagnetic mass operators because there are no external photon lines.) Which pion masses are affected by the leading-order operators? Finally give an example of a next-to-leading order operator, or find them all.