THE RENORMALIZATION GROUP AND WEYL-INVARIANCE

Transcription

1 THE RENORMALIZATION GROUP AND WEYL-INVARIANCE Asymptotic Safety Seminar Series 2012 [ Based on arxiv: ] GIULIO D ODORICO with A. CODELLO, C. PAGANI and R. PERCACCI 1

2 Outline Weyl-invariance & Weyl geometry Free Matter in external gravity Effective Average Action (EAA) Interacting matter Dynamical gravity 2

3 Weyl-invariance Scale transformation g μν Ω 2 g μν Scaling dimension defined through: maps one theory to another S(g μν,ψ a,g i )=S(Ω 2 g μν, Ω w a ψ a, Ω w i g i ) Weyl transformation Ω=Ω(x) weights Introduce a dilaton as a local mass scale: μ χ(x) χ Ω 1 χ One can construct a (nonmetric) connection and a Weyl-covariant derivative: ˆΓ μ λ ν =Γ μ λ ν δ λ μb ν δ λ ν b μ + g μν b λ D μ t = ˆ μ t wb μ t pure gauge field b μ = χ 1 μ χ 3

4 This defines the (integrable) Weyl geometry on the manifold. Curvatures are then built as usual: [D μ,d ν ]v ρ = R μν ρ σ v σ If every parameter in the theory is rewritten as a dilaton coupling g i = χ wiĝ i and all derivatives and curvatures are replaced by Weyl-covariant derivatives and curvatures, we obtain a Weyl-invariant action (Stückelberg trick). The final recipe amounts to: Ŝ(g μν,χ,ψ a, ĝ i )=S(χ 2 g μν,χ w a ψ a,χ w i g i ) 4

5 Free Matter - Standard Measure Conformally coupled scalar field in external gravitational field Metric in field space: Action: G(φ, φ )=μ 2 S S (g μν,φ)= 1 2 G ( φ, Δ(0) Weyl covariance of the Laplacian means: dx gφφ From the gaussian integral we obtain the Effective Action (EA): μ 2 ) φ 2 φ Δ (0) = 2 + d 2 4(d 1) R Ω=1+ω Δ (0) Ω 2 g =Ω 1 d 2 Δ (0) g Ω d 2 1 δ ω Δ (0) = 2ωΔ (0) 5

6 Γ I (φ, g μν )=S S (φ, g μν )+ 1 ( ) Δ (0) 2 Tr log μ 2 different for fermion field Δ (1/2) = 2 + R 4 A Maxwell field gives a contribution: S M (A μ,g μν )+ 1 2 Tr log ( Δ (1) μ 2 ) Tr log ( Δ (gh) μ 2 ) These two operators are not Weyl-covariant (d=4): ρ (gh) δ ω Δ (gh) = 2ωΔ (gh) 2 ν ω ν δ ω Δ (1) μ ν = 2ωΔ (1) μ ν +2 μ ω ν 2 ν ω μ 2 μ ν ω ρ (1) 6

7 The Trace Anomaly Under an infinitesimal Weyl transformation: δ ω Γ I = dx δγi 2ωg μν = δg μν dx gω T μ μ From the proper-time representation of the EA for a Weyl-covariant operator, and the Heat Kernel expansion, one easily derives the relation: δ ω Γ I = 1 dx gωb (4π) d/2 d (Δ) The coefficients can be computed in dimension 2 and 4 giving: T μ μ = c 24π R c = n S + n D 2D 7 T μ μ = cc 2 ae 1 a = 360(4π) 2 (n S +11n D +62n M ) 1 c = 120(4π) 2 (n S +6n D +12n M ) 4D

8 The trace anomaly is a manifestation of the dependence on the mass scale: dx g Tμ μ = μ d 1 dμ 2 Tr log Δ μ 2 In the spin 1 case we have Weyl noncovariant operators, giving an additional contribution 1 2 Trρ(1) e tδ (1) Trρ (gh) e tδ(gh) One can see that these contributions cancel in the traces, giving the correct result δ ω Γ I = 1 (4π) 2 d 4 x g [ ] b 4 (Δ (1) ) 2b 4 (Δ (gh) ) 8

9 Free Matter - New Measure One can construct Weyl-invariant metrics in field space replacing the mass scale with the dilaton: G S (φ, φ )= d 4 x gχ 2 φφ, G D ( ψ, ψ )= d 4 x g 1 2 χ[ ψψ + ψ ψ], G M (A, A )= d 4 x gχ 2 A μ g μν A ν Likewise define covariant operators: O S = χ 2 Δ (0), O D = χ 2 Δ (1/2), O Mμ ν = χ 2 g μσ (Δ (1)) σν, O gh = χ 2 Δ (gh) 9

10 This will also define a new EA: Γ II = S + 1 Tr log O 2 While the standard EA is anomalous, δ ω Γ I 0 the new one is Weyl-invariant by construction: δ ω Γ II = dx gω ) (2 δγii g μν δγii δg μν δχ χ =0 10

11 Wess-Zumino Action The Wess-Zumino action is defined through the variation of the (standard) EA under a finite Weyl transformation: Γ I (g Ω ) Γ I (g) =Γ WZ (g, μω) Important because all the dilaton-dependence in the Weyl-invariant EA comes from the Wess-Zumino term: Γ II (g, χ) =Γ I (g)+γ WZ (g, χ) Also, from the relation dx g δγ WZ δg μν 2ωg μν =0 (g,χ=μ) we see that the Weyl-invariant EA reproduces the trace anomaly in the gauge in which the dilaton is constant 11

12 Wess-Zumino action from integration of the anomaly One-parameter family of Weyl transformations: Ω(t) :[0, 1] [1, Ω] g(t) μν = g Ω(t) μν Γ WZ (g μν, Ω) = 1 0 dt dx δγ δg μν δg(t) μν = g(t) 1 0 dt dx g(t) Tμ μ 1 dω k Ω(t) dt For instance: 2D Γ WZ (g μν,μe σ )= c 24π d 2 x g ( Rσ σ 2 σ ) Γ WZ (g μν,μe σ )= d 4 x g { cc 2 σ a [(E 23 R ) ]} σ +2σΔ 4 σ Δ 4 = 2 +2R μν μ ν 2 3 R μ R μ 4D 12

13 Effective Average Action Meaning of the cutoff k in the Weyl-invariant approach: The cutoff must be allowed to be a generic non-negative function on spacetime. We will take it to be proportional to the dilaton. Cutoff term: k 2 /χ 2 ΔS k (g μν,φ)= 1 2 u2 n a 2 nr ( λn ) eigenvalues of the Weyl covariant operator O 13

14 The EAA: k dγ k dk = 1 2 Tr [ δ 2 (Γ k +ΔS k ) δφδφ ] 1 k d dk δ 2 ΔS k δφδφ is in the two cases: R k (Δ) = k 2 r(δ/k 2 ) k dγi k ) (Δ/k 2 )r (Δ/k 2 ) dk =Trr(Δ/k2 (Δ/k 2 )+r(δ/k 2 ) k is a scale u dγii u ) (O/u 2 )r (O/u 2 ) du =Trr(O/u2 (O/u 2 )+r(o/u 2 ) k is a field 14

15 The c-anomaly in d=2 General curvature-squared truncation nonlocal form factor Γ I k = d 2 x g [a k + b k R + Rc k (Δ)R]+O ( R 3) Beta function of c has been computed in [Codello, Ann.Phys. 325 (2010) 1727] UV c k x IR Optimized cutoff

16 When the form factor admits a series expansion: c k (Δ) = 1 k 2 n=1 c n k 2n Δ n conformal variation of the EAA gives the k-dependent trace anomaly: T μ μ I k = 2 g g μν δγ I u δg μν = 4c( Δ)R 2 k 2 n=0 n 1 k=1 c n ( 1 Δk R )( 1 Δ n k R ) c k (Δ) = 1 c( Δ) Δ Δ =Δ/k 2 and likewise for the Weyl-invariant EAA: Tμ μ II u = 2 δγ II u g μν = 4c g δg μν ( O u 2 ) R 2 u 2 χ 2 n=1 n 1 k=1 ( )( ) u 2k u 2(n k) c n O k R O n k R 16

17 4c x k Asymptotic value IR Exponential cutoff Mass cutoff Optimized cutoff x k 2 UV 17

18 The c-anomaly in d=4 Curvature squared expansion: Γ I k = d 4 x g [ a k + b k R + Rf R,k (Δ)R + C μνρσ f C,k (Δ)C μνρσ + O(R 3 ) ] goes to 0 goes to one-loop EA Γ I = (4π) 2 d 4 x g n S +6n D +12n M 120 C μνρσ log ( Δ k 2 0 ) C μνρσ 0.03 Deser-Schwimmer action f C,k x shape of the form-factor towards the IR 18

19 Another action that generates the full anomaly is the Riegert action: W (g μν )= d 4 x g 1 8 (E 23 R ) Δ 1 4 [ 2cC 2 a (E 23 )] R + a 18 R2 BUT: wrong flat spacetime limit of T T-correlator. By using the WZ action in the relation previously found Γ I (g μν )=Γ II (g μν,χ) Γ WZ (g μν,χ) we reobtain the Riegert action, e.g., from the equation of motion of the dilaton. Other choices give Riegert + Weyl-invariant terms One can compute the Weyl-invariant EAA for the c-term. In the IR limit this gives Γ II (g μν,χ)= (4π) 2 d 4 x g n S +6n D +12n M 120 C μνρσ log ( O u 2 0 ) C μνρσ 19

20 Interacting Matter For free matter, the trace anomaly is zero in flat space For interacting matter, with Weyl-invariant interactions of the form S int (g μν, Ψ a )=λ d 4 x g L int one has in flat space: Two points of view: I. Weyl invariance = zero beta functions. dimensionless d 4 xω T μ μ = δ ω S int = β λ = k dλ dk d 4 xωβ λ L int II. Change RG prescription: λ λ(u) u = k/χ Then δ ω S int =0 even if β λ 0 20

21 The two terms in the frg flow equation have the transformation properties: field of weight w δ 2 (Γ k +ΔS k ) δφδφ k d dk δ 2 ΔS k δφδφ Ω d w δ2 (Γ k +ΔS k ) δφδφ Ω d w k d dk δ 2 ΔS k δφδφ Ω w Ω w, Since the beta functional is Weyl invariant, if we start from an initial condition which is Weyl invariant, the EA will be Weyl invariant as well. 21

22 Dynamical Gravity Usual Background Field Method, with background-split for metric and dilaton. We chose a constant dilaton background. Covariant measure for gravitons and gravity ghosts introduced exactly as before. Einstein-Hilbert truncation: S(g, χ) = Beta-functions agree with the known ones: u dz du = 1 4π 2 (23 + 2n M n D ) u 2 u dλ du = 2+n S +2n M 4n D 32π 2 Z 2 u 2 d 4 x g [ u 2 16λZ [ λz 2 χ 4 1 ] 12 Zχ2 R n M n D 2+n S +2n M 4n D ] R = R 6χ 1 χ Zχ 2 = 12 16πG λ = 2π 9 GΛ Λ =Λ/k 2 =6λZ/u 2 G = Gk 2 =3u 2 /4πZ 22

23 General solution of this system: boundary conditions Gravitational fixed point corresponds to large-u behaviour (note that Z is redundant): At a FP, the dilaton decouples. Z(u) =Z n M n D 8π 2 u 2, λ(u) = π2 ((2 + n S +2n M 4n D )u π 2 Z 2 0λ 0 ) 2(8π 2 Z 0 +(23+2n M n D )u 2 ) 2 λ(u) λ = π2 (2 + n S +2n M 4n D ) 2(23 + 2n M n D ) 2 Z(u) Z u 2 = n M n D 8π 2 u 2 The RG trajectory corresponding to the FP is the one with the boundary condition for Z set to zero. In this case, we find a vanishing EA. 23

24 Conclusions With a dilaton one can construct a Weyl invariant measure and EA This can be generalized to interacting matter using the EAA, and to dynamical gravity This construction clarifies the meaning of Weyl invariance in an RG context It helps clarifying some ambiguities for the trace anomaly in d=4 24

25 THANK YOU 25