Global scale WI The WI of Weyl transformations Gravity Conclusions The Ward identity of scale transformations Roberto Percacci SISSA, Trieste Functional Renormalization Heidelberg, March 10, 2017
Global scale WI The WI of Weyl transformations Gravity Conclusions References Based on: R.P . and G.P . Vacca, “The background scale Ward identity in quantum gravity” Eur.Phys.J. C77 (2017) no.1, 52 arXiv:1611.07005 [hep-th] and work in progress with G. P . Vacca and V. Skrinjar.
Global scale WI The WI of Weyl transformations Gravity Conclusions Scale transformations ds 2 = g µν dx µ dx ν Rescaling of lenghts δ s = ǫ s can be realized either by δ x µ = ǫ x µ or δ g µν = 2 ǫ g µν . Choose the latter.
Global scale WI The WI of Weyl transformations Gravity Conclusions Anomalous scale invariance Scalar field in external metric with action S ( φ ; g µν ) . Probe behavior under scale transformations. δ ǫ g µν = 2 ǫ g µν − d − 2 δ ǫ Φ = ǫ Φ 2 Assume δ ǫ S = 0 Scale invariance broken in the quantum theory.
Global scale WI The WI of Weyl transformations Gravity Conclusions The cutoff term � d d x √ g φ R k φ . ∆ S k ( φ ; g µν ) = R k = k 2 r ( y ) ; y = ∆ / k 2 Since δ ǫ ∆ = − 2 ǫ ∆ . we have δ ǫ R k = − 2 ǫ k 2 yr ′ . On the other hand ∂ t R k = 2 k 2 r − 2 k 2 yr ′ , so δ ǫ R k = − 2 ǫ R k + ǫ∂ t R k . The cutoff term transforms as δ ǫ ∆ S k ( φ ; g µν ) = ǫ 1 � d d x √ g φ ∂ t R k φ . 2
Global scale WI The WI of Weyl transformations Gravity Conclusions The EAA � � � � d d x J Φ W k ( J ; g µν ) = log ( d Φ) Exp − S − ∆ S k + � d d x J ϕ − ∆ S k ( ϕ ) Γ k ( ϕ ; g µν ) = − W k ( J ; g µν ) + where ϕ = � φ � = δ W k δ J
Global scale WI The WI of Weyl transformations Gravity Conclusions The Ward identity � d d x J � δ ǫ Φ � δ ǫ W k = −� δ ǫ ∆ S k � + � δ 2 W k − 1 δ J δ J + 1 d d x √ g δ W k δ W k � = ǫ 2 Tr ∂ t R k δ J ∂ t R k 2 δ J � + d − 2 � d d x J δ W k , 2 δ J � d d x J δ ǫ ϕ − δ ǫ ∆ S k ( ϕ ) δ ǫ Γ k ( ϕ ) = − δ ǫ W k + δ 2 W k ǫ 1 = 2 Tr ∂ t R k δ J δ J � δ 2 Γ k � − 1 ǫ 1 k dR k = 2 Tr δϕδϕ + R k dk = ǫ∂ t Γ k
Global scale WI The WI of Weyl transformations Gravity Conclusions Meaning δ ǫ Γ k = ǫ∂ t Γ k thus defining ǫ = δ ǫ − ǫ k d δ E dk we have δ E Γ k = 0
Global scale WI The WI of Weyl transformations Gravity Conclusions The trace anomaly � � � δ Γ k − d − 2 ϕδ Γ k = ǫ k d Γ k d d x ǫ 2 g µν δ g µν 2 δϕ dk dx √ g � T µµ � k = d − 2 � � dx ϕδ Γ k δϕ + k d Γ k 2 dk
Global scale WI The WI of Weyl transformations Gravity Conclusions Local transformations Assume S is invariant under Weyl transformations δ ǫ g µν = 2 ǫ ( x ) g µν − d − 2 δ ǫ Φ = ǫ ( x )Φ 2
Global scale WI The WI of Weyl transformations Gravity Conclusions Plan If we can write cutoff actions that are invariant under extended transformations where the fields transform as above and also δ k = − ǫ ( x ) k then we will find the same msWI as before. Note that k cannot be constant!
Global scale WI The WI of Weyl transformations Gravity Conclusions Weyl calculus Introduce a dilaton field χ and define flat abelian gauge field b µ = − χ − 1 ∂ µ χ transforming as δ b µ = ∂ µ ǫ . For scalar field φ of weight w D µ φ = ∂ µ φ − wb µ φ More generally Γ µλν = Γ µλν − δ λ ˆ µ b ν − δ λ ν b µ + g µν b λ is invariant under local Weyl transformations, hence for a tensor of weight w D µ t = ˆ ∇ µ t − wb µ t is diffeomorphism and Weyl covariant.
Global scale WI The WI of Weyl transformations Gravity Conclusions Cutoff terms Replacing ∇ µ by D µ the cutoff terms now satisfy � dx ǫ k δ δ ǫ ∆ S k = δ k ∆ S k
Global scale WI The WI of Weyl transformations Gravity Conclusions Local ERGE �� δ 2 Γ k � − 1 � δ k δ Γ k δ k = 1 δ k δ R k δφδφ + R k 2 STr δ k
Global scale WI The WI of Weyl transformations Gravity Conclusions Modified Weyl WI � dx ǫ k δ Γ k δ ǫ Γ k = δ k � T µµ � k ( x ) = d − 2 ϕ ( x ) δ Γ k δϕ ( x ) + k ( x ) δ Γ k 2 δ k ( x )
Global scale WI The WI of Weyl transformations Gravity Conclusions Note: where is the RG? Assume u = k /χ is constant. Think of the EAA as Γ k ( φ ; g µν , χ ) = Γ u ( φ ; g µν , χ ) It satisfies � δ 2 Γ k � − 1 u d Γ k du = 1 u δ R u δϕδϕ + R k 2 Tr du R. P ., New J. Phys. 13 125013 (2011) arXiv:1110.6758 [hep-th] A. Codello, G. D’Odorico, C. Pagani, R. P ., Class. Quant. Grav. 30 (2013), arXiv:1210.3284 [hep-th] C. Pagani, R. P . Class. Quant. Grav. 31 (2014) 115005, arXiv:1312.7767 [hep-th]
Global scale WI The WI of Weyl transformations Gravity Conclusions Turn on gravity Definition of EAA requires a background split g µν = ¯ g µν + h µν Two generalizations “physical” scale transformation δ ǫ g µν = 2 ǫ g µν “background” scale transformations δ ǫ g µν = 0
Global scale WI The WI of Weyl transformations Gravity Conclusions Split symmetry Bare action is invariant under g µν = ¯ g µν + h µν δ ¯ g µν = ǫ µν , δ h µν = − ǫ µν . but the EAA Γ k ( h ; ¯ g ) is not. Same with exponential split g e X ; X ρν = ¯ g ρσ h σν . g = ¯ ad X g − 1 δ ¯ δ ¯ e ad X − 1 ¯ g µν = ǫ µν , δ X = − g .
Global scale WI The WI of Weyl transformations Gravity Conclusions Plan Write the anomalous Ward identity for the split symmetry or a subgroup thereof Solve it to eliminate from the EAA a number of fields equal to the number of parameters of the transformation Write the flow equation for the EAA depending on the remaining variables
Global scale WI The WI of Weyl transformations Gravity Conclusions Transformations Here I will consider the case of a rescaling of the background δ ¯ g µν = 2 ǫ ¯ g µν d δ µν h , h = h ⊥ + ¯ gh ⊥ = 0 Define h µν = h T µν + 1 � ¯ � h with dx In exponential parametrization g µν is left invariant provided δ h T µν = 0 δ h ⊥ = 0 δ ¯ h = − 2 d ǫ (Note δ h T µν = 2 ǫ h T µν )
Global scale WI The WI of Weyl transformations Gravity Conclusions Gauge fixing S GF = 1 � d d x g F µ Y µν F ν , � ¯ 2 α ∇ ρ h ρµ − β + 1 F µ = ¯ ¯ ∇ µ h d δ F µ = 0 � ¯ � ¯ To compensate δ g = d ǫ g , choose Y µν = ¯ d − 2 g µν . 2 ¯ ∆ Since δ ¯ ∆ = − 2 ǫ ¯ ∆ , we have δ S GF = 0.
Global scale WI The WI of Weyl transformations Gravity Conclusions Ghost action � µ , C µ ; ¯ d d x µ Y µν ∆ FP νρ C ρ S gh ( C ∗ � g C ∗ ¯ g µν ) = e X = L η g = L η ¯ g e X + ¯ g L η e X . δ ( Q ) g δ ( Q ) ¯ ¯ g = 0 ; η η � � C X ) ρµ + 1 + β ∆ FP µν C ν = ¯ ( δ ( Q ) δ ρµ tr ( δ ( Q ) ∇ ρ C X ) d ad X � g + L C e X e − X � δ ( Q ) g − 1 L C ¯ ¯ C X = e ad X − 1 g + L C X + 1 g − 1 L C ¯ g − 1 L C ¯ g , X ] + O ( C X 2 ) ¯ 2 [¯ =
Global scale WI The WI of Weyl transformations Gravity Conclusions Ghosts Choosing δ C µ = 0 . δ C ∗ µ = 0 , one has δ ∆ FP µν C ν = 0 and again δ S gh = 0 Finally � � g B µ Y µν B ν ¯ S aux = dx If δ B µ = 0 then δ S aux = 0
Global scale WI The WI of Weyl transformations Gravity Conclusions Cutoff terms g µν ) = 1 � g h µν R k ( ¯ ∆ S k ( h µν ; ¯ d d x � ∆) h νµ ¯ 2 � ∆ S gh µ R gh k ( ¯ k ( C ∗ µ , C µ ; ¯ d d x � g C ∗ ∆) C µ ¯ g µν ) = � ∆ S aux d d x � g B ∗ g µν R aux ( ¯ ( B µ ; ¯ ¯ µ ¯ g µν ) = ∆) B ν k k R k = c k d r ( y ) R gh k ( ¯ ∆) = c gh k d r ( y ) ( ¯ R aux ∆) = c aux k d − 2 r ( y ) k and y = ¯ ∆ / k 2
Global scale WI The WI of Weyl transformations Gravity Conclusions Transformations As before δ R k = ǫ ( − d R k + ∂ t R k ) . 1 � � � δ ∆ S k ( h µν ; ¯ d d x � h T µν ∂ t R k h T νµ + h ∂ t R k h ¯ g µν ) = 2 ǫ g � d d x � ¯ − 2 d ǫ g R k h , � k C µ . δ ∆ S gh µ ∂ t R gh µ , C µ ; ¯ d d x k ( C ∗ � g C ∗ ¯ g µν ) = ǫ � δ ∆ S aux d d x � g µν ∂ t R aux ( B µ ; ¯ g B µ ¯ ¯ g µν ) = ǫ B ν . k k
Global scale WI The WI of Weyl transformations Gravity Conclusions The generating functionals � � e W k ( j T µν , j , J µ ∗ , J µ ;¯ g µν ) ( dhdC ∗ dCdB ) Exp = − S − S GF − S gh − ∆ S k − ∆ S gh k − ∆ S aux k � � µ + J µ C µ � � d d x ν h T µν + jh + J µ ∗ C ∗ + j T µ Γ k ( h T µ , C µ ; ¯ ν , j , J µ µν , h , C ∗ ∗ , J µ ; ¯ g µν ) = − W k ( j T µ g µν ) � � µ + J µ C µ � d d x ν h T µν + jh + J µ ∗ C ∗ + j T µ − ∆ S k − ∆ S gh k − ∆ S aux k
Global scale WI The WI of Weyl transformations Gravity Conclusions The msWI δ ǫ Γ k = ǫ∂ t Γ k Under finite transformations Γ k ( h T µν , h ⊥ , ¯ g µν ) = Γ Ω − 1 k ( h T µν , h ⊥ , ¯ µ , C µ ; ¯ µ , C µ ; Ω 2 ¯ h , C ∗ h − 2 d log Ω , C ∗ g µν )
Recommend
More recommend
