Anomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006

Transcription

1 Anomaly Kenichi KONISHI University of Pisa College de France, 14 February 2006 Abstract Symmetry and quantization U A (1) anomaly and π 0 decay Origin of anomalies Chiral and nonabelian anomaly Anomally and nonabelian dynamics Anomaly cancellation as consistency

2 ANOMALIES 1 Symmetries and Quantum mechanics Quantization, Symmetries, Phase Factor - three melodies of theoretical physics of the 20th century (C.N. Yang, Paris TH 2002) Anomalies are consequences of Quantum mechanics and Symmetries; Anomaly as a certain symmetry, modified by quantum effects, in a subtle and precise way; Consequences Calculable physical effects; Nontrivial consistency conditions

3 ANOMALIES 2 Symmetries in Nature Classic examples (crystalographic symmetries, left-right symmetry of certain biological systems, etc) Symmetries and conservation laws: Homogeneity in spacetime (E and p i conservation); isotropy of space (angular momentum); Internal symmetries Isospin (p, n as the two quantum states of the same particle); Unitary symmetries and the quark model (u, d, s quarks); Important discrete symmetries (parity; time reversal, CP, CPT, etc.) Symmetries in Nambu-Goldstone mode (spontaneously broken symmetries), i.e., massless pions SU L (2) SU R (2) U V (1) U A (1) SU isospin (2) U V (1) (1)

4 ANOMALIES 3 Symmetries and interactions (gauge principle) Charge conservation from the invariance for the electron field ψ(x) e iα ψ(x); Require invariance under α = α(x µ ): L = 1 4 F µν + i ψ(x) γ µ ( µ iea µ (x)) ψ(x), F µν = µ A ν ν A µ (quantum electrodynamics). More general groups (G = SU(2), SU(3), etc.): Yang-Mills theories Standard model of strong and electroweak interactions (Fritsch, Gell-Mann, Glashow, Weinberg, Salam) G = SU C (3) SU L (2) U Y (1) (2)

5 ANOMALIES 4 Elementary particles in Nature Quarks and their Charges ( ul d L ), Quarks SU(3) SU L (2) U Y (1) U EM (1) ( cl s L ), ( tl b L ) ( ) u R, c R, t R d R, s R, b R Table 1: The primes indicate that the mass eigenstates are different from the states transforming as multiplets of SU L (2) U Y (1). They are linearly related by Cabibbo-Kobayashi-Maskawa mixing matrix.

6 ANOMALIES 5 Leptons and Their Charges ( ) ν e L, e L Leptons SU L (2) U Y (1) U EM (1) ( ν µ L µ L ), ( ) ν τ L τ L 2 1 ( 0 1 ) e R, µ R, τ R Table 2: The primes indicate again that the mass eigenstates are different from the states transforming as multiplets of SU L (2) U Y (1), as required by the observed neutrino oscillations.

7 ANOMALIES 6 Higgs doublet Higgs doublet SU L (2) U Y (1) U EM (1) ( φ + φ 0 ) 2 1 ( 1 0 ) Table 3: Higgs doublet scalars and their charges Gauge Boson Masses photon gluons W ± (GeV) Z (GeV) ± ± Table 4:

8 ANOMALIES 7 Chirality Fundamental particles come as chiral fermion (e L, e R, ν L ) (Feynman, Gell-Mann) Dirac fermion ψ L + ψ R Parity violation in (electro-) weak interactions (Fig.) Chiral symmetry: quarks u L, d L, u R, d R G F = SU L (2) SU R (2) U L (1) U R (1) G F Spontaneously Broken to (Pions: π ±, π 0 ) G manifest = SU isospin (2) U B (1)

9 ANOMALIES 8 Chiral fermion p Spin Figure 1:

10 ANOMALIES 9 Left - Right Symmetry (Spontaneously) Broken Left - Right Symmetry OK for World Each Person is Left-Right Symmetric Figure 2:

11 ANOMALIES 10 Invariance under ψ(x) e iγ5α ψ ( x) γ 5 = U(1) (axial) Anomaly ( ), ψ(x) = ( ψr (x) ψ L (x) ) µ J µ 5 = 0 + e2 16π 2F F µν µν = e2 8π 2E H, J F µν = 1 2 ɛµνρσ F ρσ ; ɛ 0123 = 1. µ 5 = ψiγ 5 γ µ ψ The three point function (Fig.) G µ;λ,ν T J µ 5 (x)j λ (y)j ν (z) (3) with a linearly divergent one-loop contribution, satisfies: µ x G µ;λ,ν 0, if y λ Gµ;λ,ν = ν z G µ;λ,ν = 0. (4) (Fukuda, Miyamoto, Steinberger ( 49), Schwinger ( 51), Adler, Bell, Jackiw ( 68))

12 ANOMALIES 11 Triangle anomaly and π 0 γγ Figure 3: vs Γ(π 0 2γ) = α2 64π 3 m 3 π F 2 π N 2 c [ (2 3 ) 2 + ( 1 3) 2 ] MeV, Γ exp (π 0 2γ) = τ MeV.

13 ANOMALIES 12 Sutherland-Veltman s theorem Figure 4:

14 ANOMALIES 13 Origin of the Anomaly Locality (and Causality) of fundamental interactions: fact of Nature (String theory?) Divergences Neccessity of renormalization. (renormalizable field theories: paradigm of modern particle physics) (Tomonaga, Dyson, Feynman, Schwinger) 1. Regularize each operator (precise procedure for removing divergences); 2. Fit the value of certain (conveniently chosen) reference quantities to experimental data (e.g., e); 3. Remove the regularization; 4. If the procedure works to all orders, the theory is renormalizable 5. Well defined predictions order by order of perturbation Axial current operator ( point-splitting method) i ψ(x) γ 5 γ µ ψ(x) lim ɛ 0 i ψ(x + ɛ 2 ) γ5 γ µ exp[ i e c x+ɛ/2 x ɛ/2 dx µ A µ ] ψ(x ɛ 2 )

15 ANOMALIES ɛ divergences when inserted in a graph; 2. O(ɛ) contriibution from the string bit; 3. Compute the finite, ɛ 1 ɛ piece; 4. ɛ 0 5. Axial anomaly: µ J µ 5 = e2 F 16π 2 µν F µν. Other regularizations (with the same result) 1. Momentum space subtractions (BPHZ); 2. Dimensional regularization; 3. Pauli-Villars (regulator fields with big mass); 4. Functional integral method (Fujikawa) e Γ(A) = Dψ D ψ e R dx ψiγd Aψ, A 5 µ A 5 µ µ α(x) ψ(x) = e i γ5 α(x) ψ (x) J = e 2 i R dx α(x) P n ψ n (x) γ5 ψ n (x) ψn(x) γ 5 ψ n (x) = 0 Regular. e2 16π 2F µν µν F n

16 ANOMALIES 15 Chiral anomaly / nonabelian anomaly e Γ(A) = Dψ D ψ e R dx ψiγd Aψ. Gauge transformation If the theory is invariant, A µ A µ D µ v, Γ(A) Γ(A) + D µ v = µ v + ig[a µ, v], dx v a δγ(a) D µ δ A a. µ δγ(a) δ v a = 0,... D µ J a µ = 0, J a µ = δγ(a) δ A a µ Actually, the variation is nonvanishing (anomaly!) 1 δ vl Γ(A) = 24π 2 d 4 x Tr v(x) ɛ λµαβ λ (A µ α A β A µa α A β ) 1 = 24π 2 d 4 x Tr v(x) d(ada A3 ) TrT a {T a T c } = d abc (r)

17 ANOMALIES 16 Cancellation of anomalies in the standard model Standard model (Eq.2) is chiral. Anomalies potentially break gauge invariance. Algebraic cancellations! (Fig.) U Y (1) 3 vertex: [ ( ) ( ) 3 ( ] + [ ( 1) 3 3) 3 2 ( 2) 3 ] = U Y (1) SU L (2) SU L (2) vertex: q L u R d R (ν, e) L e R ( ) = 1 1 Quark-lepton universality (Cabibbo angle, KM mixing, CP violation)

18 ANOMALIES 17 U Y (1) U Y (1) U Y (1) U Y (1) SU L (2) SU L (2) U Y (1) SU C (3) SU C (3) Figure 5:

19 ANOMALIES 18 Wess-Zumino consistency condition, Witten s quantization The anomaly as a variation of the effective action (Wess-Zumino) (δ v1 δ v2 δ v2 δ v1 ) Γ(A) = δ [v1,v 2 ] Γ(A). U Nambu-Goldstone bosons of SU(3) SU(3) SU(3) SU(3), π 5 (SU(3)) Z,... n Z (n = N c = 3 actually) Local action on the boundary of S 4 R 4 Γ eff = n dσ i1 i 2...i 5 ω i1 i 2...i 5 n Γ(U), ω i1 i 2...i 5 = i 240 π 2 Tr (U 1 i1 U)(U 1 i2 U)... (U 1 i5 U), Γ eff Γ eff (U, A) e.g. π 0 γγ. n = N c = 3.

20 ANOMALIES 19 t Hooft s anomaly matching condition Proton uud; neutron udd; Quarks and leptons: are they truly elementary? Or composite of even more fundamental constituents (preons)? Suppose some strong gauge forces G strong confine preons into quarks, e.g., q P P P If P r of the flavor group G F (representation) then q r r r. Fermions are in chiral (not left-right symmetric) representations of G F Introduce hypothetical, weakly coupled gauge bosons of G F, introducing if necessary hypothetical leptons to cancel the anomaly;

21 ANOMALIES 20 Nontrivial conditions on possible composite modes preons d abc (r) = quarks d abc (r q ) Dynamical result (number and types of light composite fermions) must obey the algebraic conditions Deep connections between the dynamics and global symmetry

22 ANOMALIES 21 Anomaly Matching Elementary Fermions Composite Fermions Figure 6:

23 ANOMALIES 22 Anomaly matching condition and π 0 γγ (i) Both consequence of axial (or chiral) anomaly; (ii) The same anomaly content in the high-energy and low-energy theories; (iii) The anomaly singularity at q 2 = 0 in the F.T. of the three point function such as (3); (iv) Singularity either due to massless Goldstone boson (G F spontaneously broken); due to pairs of massless fermions (unbroken G F : t Hooft); or both (supersymmetric theories)

24 ANOMALIES 23 Anomaly and dynamics of QCD U A (1) anomaly in the presence of nonabelian gauge interactions µ J µ 5 = 0. + g2 32π 2Ga µν G µν a, J µ 5 = qiγ 5γ µ q G a µν = µ Aa ν ν Aa µ + f abc A b µa c ν π 2 (SU(2)) = Z nonperturbative effects (instantons); G a µν gluons of QCD Solution of the U(1) problem Prof. Veneziano s Lecture on the U(1)-Problem G a µν electroweak W boson tensor Particle productions ( B 0, L 0 ).

25 ANOMALIES 24 Summary Anomalies are a subtle modification of symmetry by quantum effects; Precise predictions (e.g., π 0 γγ); Locality of fundamental interactions (renormalization theory); Chiral structure of the world; Standard model of the strong and electroweak interactions based on G = SU C (3) SU L (2) U Y (1) quantum-mechanically consistent; Quark-lepton universality: origin? (Grand unification? String theory?) Dynamical effects on the right hand side of µ J µ 5 = e2 F 16π 2 µν F µν?

26 ANOMALIES 25 Instantons Prof. Veneziano s coming lectures My next seminar (21st February)