Anomalies in Exact Renomalization Group

Transcription

1 Anomalies in Exact Renomalization Group at YITP, Jul. 6, 2009 Y. Igarashi, H. Sonoda a, and K. I. Faculty of Education, Niigata University a Department of Physics, Kobe University Anomalies in Exact Renomalization Group, at YITP 1/19

2 The contents Z φ [J] = Dφ exp ( S[φ] J φ). Gradulal integration of higher momentum modes defines a RG flow When a symmetry exits, we find an expression, Σ Λ = 0. Σ Λ = 0 is the cutoff dependent Ward-Takahashi identity. Σ Λ changes along the flow as a composite operator. If anomalous, we find Σ Λ ghost anomaly. We study it in a typical example. Anomalies in Exact Renomalization Group, at YITP 2/19

3 The cutoff function 1 K(p/ Λ) 0 1 p 2 Λ 2 The UV action with the cutoff Λ 0 S[φ; Λ 0 ] = 1 2 φ K 1 0 D φ + S I[φ; Λ 0 ]. Z φ [J] = Dφ exp ( S[φ; Λ 0 ] K0 1 J φ). K 0 (p) K(p/Λ 0 ) d 4 p φ D φ = (2π) 4φA ( p)d AB (p)φ B (p), J φ = d 4 p (2π) 4J A( p)φ A (p) Anomalies in Exact Renomalization Group, at YITP 3/19

4 Introduce a cutoff Λ(< Λ 0 ) with K(p/Λ), and decompose φ A into IR fields Φ A and UV fields χ A : K 0 D 1 = KD 1 + (K K 0 )D 1 Integration over the UV fields gives Z φ [J] = N J Z Φ [J], Z Φ [J] DΦ exp ( S[Φ; Λ] K 1 J Φ ) The Wilson action with the cutoff Λ S[Φ; Λ] 1 2 Φ K 1 D Φ + S I [Φ; Λ] its interaction part S I [Φ; Λ] exp ( S I [Φ; Λ]) Dχ exp [ 1 ] 2 χ (K 0 K) 1 D χ S I [Φ + χ; Λ 0 ]. Anomalies in Exact Renomalization Group, at YITP 4/19

5 The normalization factor N J is given by ln N J = ( )ϵ A 2 J A K 1 0 K 1 (K 0 K) ( D 1) AB JB. Anomalies in Exact Renomalization Group, at YITP 5/19

6 The gradual integration gives a RG flow, or the Polchinski equation Λ Λ S = ( K 1 K ) [ (p) Φ A l ] S (p) p Φ A (p) + 1 ( ) ϵ ( A KD 1 (p) ) [ AB l S r S 2 Φ B ( p) Φ A (p) with the initial condition p l r ] S Φ B ( p) Φ A (p) S[Φ; Λ = Λ 0 ] = S[Φ; Λ 0 ] The functional integration and solving the Polchinski equation are equivallent; two different languages to describe the same. Anomalies in Exact Renomalization Group, at YITP 6/19

7 The composite operator is a useful notion Equivallent definitions for the composite operator O[Φ; Λ] 1. Via the linearized Polchinski equation, with an initial condition at Λ 0 Λ O[Φ; Λ] = D O[Φ; Λ] Λ [ (K 1 D K ) Φ A l Φ A + ( ( )ϵ A KD 1) ( AB l S r Φ B Φ A 1 l r )] 2 Φ B Φ A O p 2. Given an operator O[φ; Λ 0 ] at the UV scale Λ 0, the corresponding IR composite operator O[Φ; Λ] may be constructed as O[Φ; Λ] e SI[Φ;Λ] Dχ O[Φ + χ; Λ 0 ] e 1 2 χ (K 0 K) 1 D χ S I [Φ+χ,φ ;Λ 0 ] 3. The expectation values in the presence of arbitrary sources satisfy O[Φ; Λ] Φ,K 1 J = N 1 J O[φ; Λ 0] φ,k 1 0 J Anomalies in Exact Renomalization Group, at YITP 7/19

8 Consider some transformation φ A φ A = φ A + δ λ φ A, δ λ φ A = δφ A λ = K 0 R A [φ; Λ 0 ] λ. Dφ ( ) K0 1 J δφ + Σ[φ; Λ 0] exp ( S[φ; Λ 0 ] K 1 0 J φ) = 0 where the quantity Σ[φ; Λ 0 ] is given as Σ[φ; Λ 0 ] r S φ AδφA r φ AδφA. Σ[φ, Λ 0 ] is the sum of the change of the original gauge fixed action S[φ; Λ 0 ] and that of the functional measure Dφ δ λ S = r S φ Aδ λφ A, δ λ ln Dφ = ( ) ϵ A r φ Aδ λφ A = r φ AδφA λ. Anomalies in Exact Renomalization Group, at YITP 8/19

9 Σ[φ; Λ 0 ] = 0 implies that the UV action S[φ; Λ 0 ] is invariant under δφ Appropriate to call Σ[φ; Λ 0 ] as the WT operator Let us see how the transformation and the WT operator changes as the scale changes. We have the relation Σ[φ; Λ 0 ] φ, K 1 0 J = K 1 0 J δφ φ, K 1 0 J = J R[φ; Λ 0 ] φ, K 1 0 J = J R[K 0 l J; Λ 0 ] Z φ [J] Using the relation Z φ [J] = N J Z Φ [J], we may rewrite the above for Z Φ [J]. Then, we expect to find a similar relation for the lower scale Λ. Important to note that J acts on N J Anomalies in Exact Renomalization Group, at YITP 9/19

10 To find δφ and Σ at the scale Λ, use (cf. the definition of composite operator) K 1 δφ A Φ,K 1 J = N 1 J K 1 0 δφa φ,k 1 0 J Σ[Φ; Λ] Φ,K 1 J = N 1 J Σ[φ; Λ 0] φ,k 1 0 J As for the transformation K 1 0 δφ = R[φ; Λ 0] N 1 J K 1 0 δφa φ,k 1 0 J = N 1 J RA [K 0 J; l Λ 0 ]Z φ [J] = N 1 J RA [K 0 J; l Λ 0 ]N J Z Φ [J] Note here l J acts on N j, that produces the scale change of the transformation. Equating the above expression with the following K 1 δφ A Φ,K 1 J = N 1 J K 1 0 δφa φ,k 1 0 J = RA [K J; l Λ]Z Φ [J] we find the transformation of the IR fields δφ A [Φ; Λ] = KR A [Φ; Λ] Anomalies in Exact Renomalization Group, at YITP 10/19

11 For the WT operator, we find Therefore The relation Σ[Φ; Λ] Φ,K 1 J = J R[K l J; Λ]Z Φ [J] Σ[Φ; Λ] = r S[Φ; Λ] Φ A δφa r Φ AδΦA Σ[Φ; Λ] Φ,K 1 J = N 1 J Σ[φ; Λ 0] φ,k 1 0 J implies that if the WT operator vanishes at the scale Λ 0, it does at any lower scale. A different way to observe the same property: the WT operator flows as a composite operator Λ Σ[Φ; Λ] = D Σ[Φ; Λ] Λ Anomalies in Exact Renomalization Group, at YITP 11/19

12 Where to find an anomaly? The vanishing of the WT operator implies symmetry: Σ 0 for an anomalous theory. The WT operator Σ evolves as a composite operator. Σ may contain the information of anomaly, that also follows the flow equation. The well-known form of anomay may be observed in the limit of Λ. We will see in an example Σ[Φ; Λ] ghost anomaly as Λ Anomalies in Exact Renomalization Group, at YITP 12/19

13 An example U(1) V U(1) A gauge theory: WT identities and BRS transformations two sets of gauge sector: (A µ, h V, c V, c V ) and (B µ, h A, c A, c A ) S[φ; Λ 0 ] = 1 2 φk 1 0 D φ + S I [φ; Λ 0 ] 1 2 φk 1 0 D φ = p [ K0 1 ψ( p)/pψ(p) + 1 ( 2 A µ(p 2 δ µν p µ p ν )A ν h V ip A + ξ ) V 2 h V + c V ip 2 c V + 1 ( 2 B µ(p 2 δ µν p µ p ν )B ν h A ip B + ξ ) ] A 2 h A + c A ip 2 c A Anomalies in Exact Renomalization Group, at YITP 13/19

14 The BRS transformation for axial gauge symmetry δ A B µ (p) = ik 0 (p)p µ c A (p), δ A c A = ik 0 (p)h A (p), δ A c A (p) = δ A h A (p) = 0 δ A ψ(p) = ie A K 0 (p) γ 5 ψ(p k)c V (k), δ A ψ( p) = iea K 0 (p) k k ψ( p k)c A (k)γ 5, δ A A µ (p) = δ A h V (p) = δ A c V (p) = δ A c V (p) = 0 Similar transformation for vector gauge symmetry. With the following part of the interaction aciton S I [φ; Λ 0 ] p,k ) ψ( p k) (e V /A(k) + e A /B(k)γ 5 ψ(p) we find the contribution of order e 2 relevant for anomalies Anomalies in Exact Renomalization Group, at YITP 14/19

15 [ S I [Φ; Λ] = e2 V ψ( l k) /A(k) (K 0(l) K(l)) /A(q) 2 l,k,q /l +/A(q) (K ] 0(l + k q) K(l + k q)) /A(k) ψ(l q) /l + /k /q e2 A 2 l,k,q [ ψ( l k) /B(k) (K 0(l) K(l)) /l + /B(q) (K 0(l + k q) K(l + k q)) /l + /k /q e [ V e A ψ( l k) 2 l,k,q /B(q) ] /B(k) ψ(l q) /B(k)γ 5 (K 0 (l) K(l)) /l +/A(q) (K 0(l + q k) K(l + k q)) /l + /k /q +/A(k) (K 0(l) K(l)) /l /B(q)γ 5 + /B(q)γ 5 (K 0 (l + q k) K(l + k q)) /l + /k /q /B(k)γ 5 /A(q) ] /A(k) ψ(l q) Anomalies in Exact Renomalization Group, at YITP 15/19

16 Calculate Σ A [Φ; Λ] in the limit of Λ 0 Σ A = ( ie A e 2 K(p)(1 K(p + q + k)) V c A ( q k) p,q,k (p + q + k) 2 [ ] (1 K(p + q))(/p + /q) T r γ 5 (/p + /q + /k)/a(k) (p + q) 2 /A(q) +ie A e 2 K(p)(1 K(p q k)) V c A ( q k) p,q,k (p q k) 2 [ ] ) (1 K(p k))(/p /k) T r γ 5 (/p /q /k)/a(k) (p k) 2 /A(q) ( ) + e V e A, /A /B Similar expression for Σ V [Φ; Λ] Anomalies in Exact Renomalization Group, at YITP 16/19

17 The results in the limits of Λ 0, Λ are summarized as Σ A = i e2 48π 2 Σ V = i2 e Ae 2 V 12π 2 ϵ µνρσ x c A (x) ϵ µνρσ ( F V µν (x)f V ρσ(x) + F A µν(x)f A ρσ(x) ) q,k c V ( q k)k µ B ν (k)q ρ A σ (q) Add the following counter term to the Wilson action S I S I + S c S c = a ϵ µνρσ q,k B µ (k)a ν ( k q)q ρ A σ (q) where a ia a constant to be determined below. Anomalies in Exact Renomalization Group, at YITP 17/19

18 Choose the parameter a as a = e Ae 2 V 6π 2 so that the vector gauge symmetry is preserved Σ V = 0, while, the anomaly in the axial gauge symmetry is shifted to Σ A = i e Ae 2 V 4π 2 ϵ µνρσ i e3 A 12π 2ϵ µνρσ q,k q,k c A ( q k)k µ A ν (k)q ρ A σ (q) c A ( q k)k µ B ν (k)q ρ B σ (q) Anomalies in Exact Renomalization Group, at YITP 18/19

19 Summary Z φ [J] = Dφ exp ( S[φ] J φ). Gradulal integration of higher momentum modes defines a RG flow When a symmetry exits, we find an expression, Σ Λ = 0. For a gauge symmetry, Σ Λ = 0 is the Ward-Takahashi identity (cutoff dependent). Σ Λ changes along the flow as a composite operator. For anomalous symmetry, we have seen Σ Λ ghost anomaly in the limit of Λ. Anomalies in Exact Renomalization Group, at YITP 19/19