Lectures on Chiral Perturbation Theory

Transcription

1 Lectures on Chiral Perturbation Theory I. Foundations II. Lattice Applications III. Baryons IV. Convergence Brian Tiburzi RIKEN BNL Research Center

2 Chiral Perturbation Theory I. Foundations

3 Low-energy QCD Mesons pseudo-scalar Solutions of QCD (courtesy of nature) Hyperons Nucleons Baryons positive parity spin-half Ground state: vacuum Kaons Pions 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

4 Massless QCD L QCD = L ψ + L YM L ψ = Take two flavors. These will correspond to up and down quarks. N f i=1 ψ i D/ ψ i Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry P L,R = 1 2 (1 γ 5) projectors ψ L,R = P L,R ψ ψd/ ψ = ψ L D/ ψ L + ψ R D/ ψ R Left- and right-handed fields do not mix: no chirality changing interaction U(2) L U(2) R L R ψ L Lψ L ψ R Rψ R ψ (LP L + R P R ) ψ

5 Massless QCD L QCD = L ψ + L YM L ψ = Take two flavors. These will correspond to up and down quarks. N f i=1 ψ i D/ ψ i Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry P L,R = 1 2 (1 γ 5) projectors ψ L,R = P L,R ψ ψd/ ψ = ψ L D/ ψ L + ψ R D/ 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 L = R = V ψ L Vψ L ψ R Vψ R ψ V (P L + P R )ψ = Vψ

6 Chiral Symmetry of Massless QCD Action invariant under U(1) L : ψ L e iθ L ψ L U(1) R : ψ R e iθ R ψ R U(2) L U(2) R = U(1) L U(1) R SU(2) L SU(2) R 1 ψ 2 (eiθ R + e iθ L )+ 1 2 (eiθ R e iθ L )γ 5 ψ Vector subgroup θ L = θ R = θ ψ e iθ ψ U(1) V Axial transformation θ L = θ R = θ 5 ψ [cos θ 5 + iγ 5 sin θ 5 ]ψ = e iθ 5γ 5 ψ U(1) A Exercise: Consider a non-singlet axial transformation ψ i [exp(iφ τ γ 5 )] ij ψ j Is there a corresponding symmetry group of the massless QCD action?

7 Chiral Symmetry of Massless QCD Action invariant under U(2) L U(2) R = U(1) A U(1) V SU(2) L SU(2) R Global symmetries lead to classically conserved currents J 5µ = ψγ µ γ 5 ψ J µ = ψγ µ ψ J a Lµ = ψ L γ µ τ a ψ L J a Rµ = ψ R γ µ τ a ψ R (Regulated) Theory not invariant under flavor-singlet axial transformation µ J 5µ (x) = µ J Rµ (x) µ J Lµ (x) = Chiral Anomaly e 2π N f µν F µν (x) d =2 α s 4π N f µνρσ G A µν(x)g A ρσ(x) d =4 U(1) V SU(2) L SU(2) R The chiral anomaly obstructs chirally invariant lattice regularization of fermions (see Kaplan s lectures)

8 Fate of Symmetries in Low-Energy QCD U(1) V SU(2) L SU(2) R Chiral pairing preferred by vacuum (non-perturbative ground state) Chiral condensate ψψ = ψ R ψ L + ψ L ψ R =0 Massless quarks can change their chirality by scattering off vacuum condensate Spontaneously broken symmetries lead to massless bosonic excitations Nambu-Goldstone Mechanism U(1) V SU(2) L SU(2) R U(1) V SU(2) V Broken generators in coset SU(2) L SU(2) R /SU(2) V Number of massless particles?

9 Chiral Condensate ψ ir ψ jl = λδ ji Choice for vacuum orientation λ R from Vafa-Witten (P) After a chiral transformation ψ ir ψ jl L jj R i i ψ i Rψ j L = λ(lr ) ji SU(2) L SU(2) R SU(2) V Describe Goldstone fluctuations of vacuum state with fields δ ji Σ ji (x) =δ ji +... Σ SU(2) L SU(2) R /SU(2) V Σ=e 2iφ/f =1+ 2iφ f +... [L(x)R (x)] ji =[e i θ L (x) τ e i θ R (x) τ ] ji θl = θ R Transformation properties Σ LΣR Σ V ΣV φ VφV Exercise: Determine the discrete symmetry properties of the Goldstone modes from the coset s transformation.

10 Dynamics of Goldstone Bosons: Chiral Lagrangian Σ=e 2iφ/f =1+ 2iφ f +... Tr φ =0 φ = φ The Pions φ = 1 2 π 0 π + π 1 2 π 0 Build chirally invariant theory of coset field Σ LΣR Σ Σ=1 Σ RΣ L L = f 2 8 Tr µ Σ µ Σ Expand about v.e.v. to uncover Gaußian fluctuations L = 1 2 Tr ( µφ µ φ)+o(1/f 2 )= 1 2 µπ 0 µ π 0 + µ π µ π + + O(1/f 2 ) 3 massless modes Non-linear theory: interactions between multiple pions at higher orders Can treat systematically...

11 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 L ψ = m q ψ i ψ i = m q ψir ψ il + ψ il ψ ir i i SU(2) L SU(2) R SU(2) V m q /Λ QCD 1 Need to map L ψ onto ChPT operators breaking symmetry in same way L eff = m q λ Tr Σ+Σ Comments: not chirally invariant new dimensionful parameter included only linear quark mass term Σ LΣR Σ RΣ L λ m 2 q Perturbing about chiral limit

12 Chiral Lagrangian L χ = f 2 8 Tr µ Σ µ Σ m q λ Tr Σ+Σ Expand up to quadratic order L χ = 4m q λ Tr ( µφ µ φ)+ 8m qλ 1 f 2 Tr (φφ) 2 Pion mass m 2 π =8m q λ/f 2 Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom Z QCD [m q,...]= D e x ( +m qψψ) log Z QCD m q = ψψ Low-energy degrees of freedom Z χpt [m q,...]= DΣ e x L χ(σ; m q ) Matching Effective field theory Z QCD [m q,...] Z χpt [m q,...] From before: ψψ = log Z χpt m q = λtr Σ + Σ = 2N f λ ψ ir ψ jl = λδ ji λ = λ

13 Chiral Lagrangian L χ = f 2 8 Tr µ Σ µ Σ m q λ Tr Σ+Σ Expand up to quadratic order L χ = 4m q λ Tr ( µφ µ φ)+ 8m qλ 1 f 2 Tr (φφ) 2 Pion mass m 2 π =8m q λ/f 2 f 2 m 2 π =2m q ψψ (Gell-Mann Oakes Renner) Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom Z QCD [m q,...]= D e x ( +m qψψ) log Z QCD m q = ψψ Low-energy degrees of freedom Z χpt [m q,...]= DΣ e x L χ(σ; m q ) Matching Effective field theory Z QCD [m q,...] Z χpt [m q,...] From before: ψψ = log Z χpt m q = λtr Σ + Σ = 2N f λ ψ ir ψ jl = λδ ji λ = λ

14 ChPT L χ = f 2 8 Tr µ Σ µ Σ m q λ Tr Σ+Σ Quadratic fluctuations are the approximate Goldstone bosons of SChSB Quartic terms describe interactions m qλ f 4 1 f 2 (φ µφ) 2 Higher-order interactions renormalize lower-order terms m 2 π m qλ 1 f 4 k k 2 + m 2 m qλ π f 2 Λ 2 + m qλ log Λ 2 f 2 +finite φ4 power-law divergence Absorb in renormalized mass, or just use dimensional regularization logarithmic divergence Renormalization requires new 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

15 Power Counting L χ = f 2 8 Tr µ Σ µ Σ m q λ Tr Σ+Σ Leading-order Lagrangian in expansion in derivatives and quark mass O(p 2 ) µ p m q p 2 Low-energy dynamics of pions 1 Propagator k 2 + m 2 π p 2 Vertices µ µ,m q p 2 Loop integral k p 4 General Feynman diagram: L loops, I internal lines, V vertices p 4L 2I+2V Euler formula L = I V +1 p 2L+2 Loop expansion: one loop graphs require only Two loop graphs? O(p 4 ) counterterms 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?

16 O(p 4 ) Chiral Lagrangian Construct chirally invariant terms out of coset Σ LΣR Σ RΣ L E.g. O(p 2 ) Lagrangian µ Σ µ Σ Construct terms that break chiral symmetry in the same way as mass term Simplification: add external scalar field to QCD action Make the scalar transform to preserve chiral symmetry L = ψ L sψ R + ψ R s ψ L s LsR Giving the scalar a v.e.v. breaks chiral symmetry just as a mass s = m q + E.g. O(p 2 ) Lagrangian Σs + sσ

17 O(p 4 ) Chiral Lagrangian Also impose Euclidean invariance, C, P, T Σ LΣR s LsR Σ RΣ L s Rs L E.g. [Tr Σs sσ ] 2 m 2 q[tr Σ Σ ] 2 =0 Easy to generate terms. Care needed to find minimal set. L 4 = L 1 [Tr µ Σ µ Σ ] 2 + L 2 Tr µ Σ ν Σ Tr µ Σ ν Σ + L 3 m q λ f 2 Tr µ Σ µ Σ Tr Σ+Σ + L 4 (m q λ) 2 f 4 [Tr Σ+Σ ] 2 {L j } 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 u = m d on the chiral Lagrangian. At what order does the pion isospin multiplet split?

18 Simplest one-loop computation: Chiral Condensate ψψ = log Z χpt m q Σ+Σ =2 4 f 2 φ2 + ψψ =+ 4λ f 2 3 G π(0) = 12λ 1 f 2 k k 2 + m 2 π Dimensionally regulated integral L 3 λ f 2 Tr µ Σ µ Σ Tr Σ+Σ +2L 4 m q λ 2 f 4 [Tr Σ+Σ ] 2 Final result: ψψ = 4λ 1+ 3 m2 π (4πf) 2 = λtr Σ + Σ = 2N f λ m2 π (4π) 2 1 γ E + log 4π + log µ2 m 2 π = 32 L 4 m q λ 2 Chiral Logarithm log µ2 +1 m 2 π f 4 =4 m2 π f 2 L 4 λ m2 π f 2 L 4(µ) +1 Tree-Level One Loop O(p 4 ) Tree-Level µ 2 d dµ 2 L 4 = 3 16π 2

19 Two-Flavor ChPT L 2 = f 2 8 Tr µ Σ µ Σ m q λ Tr Σ+Σ Leading and next-to-leading order Lagrangian in isospin limit m u = m d L 4 = L 1 [Tr µ Σ µ Σ ] 2 + L 2 Tr µ Σ ν Σ Tr µ Σ ν Σ + L 3 m q λ f 2 Tr µ Σ µ Σ Tr Σ+Σ + L 4 (m q λ) 2 f 4 [Tr Σ+Σ ] 2 Compute quark mass dependence of chiral condensate, pion mass, pion-pion scattering,..., in terms of a few low-energy constants ψψ = A 0 [1 + B 0 m q (log m q + C 0 )] m 2 π = A 1 m q [1 + B 1 m q (log m q + C 1 )] a I=2 ππ = A 2 mq [1 + B 2 m q (log m q + C 2 )] Further applications: electroweak properties of pions require external fields

20 Incorporating External Fields in ChPT Start by incorporating external gauge fields in QCD L ψ = ψ L D/ L ψ L + ψ R D/ R ψ R (D L ) µ = µ + ig G µ + il µ (D R ) µ = µ + ig G µ + ir µ E.g. external vector field Local invariance L µ = R µ = QeA µ [SU(2) L ] [SU(2) R ] ψ L L(x)ψ L ψ R R(x)ψ R and L µ L(x)L µ L (x)+i[ µ L(x)]L (x) R µ R(x)R µ R (x)+i[ µ R(x)]R (x) Then incorporate external gauge fields in ChPT Σ L(x)ΣR (x) Need covariant derivative D µ Σ L(x)[D µ Σ]R (x) L 2 = f 2 8 D µ Σ= µ Σ+iL µ Σ iσr µ Leading-order chiral Lagrangian [with external fields counted as O(p) ] Tr Dµ ΣD µ Σ m q λ Tr Σ+Σ *Additional operators at higher orders

21 W µ What is f? π ν µ π µ + ν µ Pion weak decay L = W µ J + µl J + µl = u Lγ µ d L Strong part factorizes into QCD matrix element (the rest you learned how to compute in QFT) Γ π µ+νµ = G2 F 8π f 2 πm 2 µm π V ud 2 1 m2 µ m 2 π 2 0 J + µl π(p ) = ip µ f π pion decay constant f π = 132 MeV ChPT current matches the QCD current JµL a = L χ L a = f 2 µ 4 Tr iτ a Σ µ Σ + = f 2 Tr (τ a µ φ)+ Lµ =0 τ + = 1 2 (τ 1 + iτ 2 ) 0 J + µl π(p ) = ip µ (f + ) f π = f [1 + Bm q (log m q + C)] Dimensionless power counting Λ χ =2 2πf 1.2 GeV p 2 /Λ 2 χ m 2 π/λ 2 χ

22 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.