Causality in nonlocal gravity Stefano Giaccari Holon Institute of - PowerPoint PPT Presentation

Causality in nonlocal gravity Stefano Giaccari Holon Institute of Technology, Holon 10th Mathematical Phyisics Meeting Belgrade, 9-14 September, 2019 based on work in collaboration with Pietro Don` a, Leonardo Modesto, Les� law Rachwa� l and Yiwei Zhu

Introduction Renormalizability and unitarity are requiremets that can hardly be reconciled within a consistent theory of quantum gravity. Einstein-Hilbert gravity is non-renormalizable, but, if we include infinitely many counterterms, it is pertubatively unitary. Renormalizable higher-derivative theories of gravity (e.g. Stelle’s quadratic theory) can be attained, but are generically expected to be non-unitary. Recently, [Camanho, Edelstein, Maldacena, Zhiboedov,’16] it has been argued that higher-derivative corrections to the 3-graviton coupling in a weakly coupled theory of gravity are constrained by causality. Stefano Giaccari Causality in nonlocal gravity 2 / 18

Weakly nonlocal gravity We consider the model d D x √− g [ R + G µν γ ( � ) R µν + V ( R )] , S g = 2 � (1) κ 2 D where ( σ ≡ ℓ 2 Λ ) γ ( � ) = e H ( σ � ) − 1 . (2) � exp H ( z ) is asymptotically polynomial γ � D exp H ( z ) → | z | γ +N+1 for | z | → + ∞ , 2 , (3) with 2N + 4 = D even or 2N + 4 = D odd + 1, to guarantee the locality of counterterms. V ( R ) ∼ O ( R 3 ), but quadratic in the Ricci tensor, is a local potential containing at most 2 γ + 2 N + 4 derivatives. Stefano Giaccari Causality in nonlocal gravity 3 / 18

Super-renormalizability and finiteness By standard power-counting � I � � � 1 p 2 γ + D � V δ D ( K ) Λ 2 γ ( L − 1) ( d D p ) L p 2 γ + D we get the degree of divergence ω ( G ) ≡ D even − 2 γ ( L − 1) and ω ( G ) ≡ D odd − (2 γ + 1)( L − 1). if γ > ( D odd − 1) / 2, no divergences! if γ > D even / 2, only 1-loop divergences ! Some terms in V ( R ) can be used as “killers” of the 1-loop divergences. For example, in D = 4, the two terms s 1 R 2 � γ − 2 ( R 2 ) , s 2 R µν R µν � γ − 2 ( R ρσ R ρσ ) , (4) modify the beta-functions for R and R 2 µν by a contribution linear in s 1 and s 2 , making it possible to have them vanishing. The killer terms should be in general at least quadratic in the Ricci tensor. Stefano Giaccari Causality in nonlocal gravity 4 / 18

Perturbative unitarity In the harmonic gauge ( ∂ µ h µν = 0) P (2) P (0) O − 1 ≈ k 2 e H ( k 2 / Λ 2 ) − ( D − 2) k 2 e H ( k 2 / Λ 2 ) . (5) No ghosts appear if H ( z ) are entire functions with no poles. The usual analytic continuation from Euclidean to Minkowski cannot be performed due to thebehavior at infinity of exp H , but [Modesto,Briscese,2018], [Pius,Sen,2016] still the ordinary Cutkosky rules can be derived and it is possible to prove at all perturbative levels the unitarity relation � T ab − T ∗ ba = i T ∗ cb T ca (6) c Stefano Giaccari Causality in nonlocal gravity 5 / 18

Quadratic gravity The most general gravity action quadratic in the curvatures is d D x √− g � � 0 R 2 + γ ′ � S g = − 2 κ − 2 2 R 2 R + γ ′ µν + γ 4 GB , (7) D Major advantages GB gives no contribution to the propagator for any D (neither to the vertices in D = 4, being topological ) expanding around a flat background ( R (0) = R (0) µν = R (0) µνρσ = 0), vertices are greatly simplified by the relationships √− g (1) = R (1) = R (1) µν = 0 valid for on-shell legs. Three level amplitudes with all external legs on graviton shell are calculable by standard techniques Stefano Giaccari Causality in nonlocal gravity 6 / 18

Scattering amplitudes for Stelle’s theory Born approximation four graviton scattering amplitudes in the center-of-mass reference frame, s = 4 E 2 , t = − 2 E 2 (1 − cos θ ) and u = − 2 E 2 (1 + cos θ ) A (++ , ++) = A s (++ , ++) + A t (++ , ++) + A u (++ , ++) + A contact (++ , ++) � � − 2 1 E 2 = − 2 i sin 2 θ , κ 2 4 The amplitude doesn’t have the expected UV behavior ∼ E 4 and is the same as the one determined in Einstein gravity by dimensional analysis and symmetry arguments. This is the result of non-trivial cancellation between the massive poles in the propagator and the three-graviton vertices and between the contact and exchange diagrams. Our result is consistent with the fact that in the absence of the Einstein term we are left with scale invariant terms whose contribution to amplitudes for dimensionless particles is vanishing. Stefano Giaccari Causality in nonlocal gravity 7 / 18

For D > 4 the Gauss-Bonnet term contributes the vertices A D =5 (++ , ++) 1 + 8 E 2 (3( γ 0 − γ 4 ) + ( γ 2 + 4 γ 4 )) � � 16 E 6 γ 2 � � = − i 2 1 4 (1 − 4 E 2 ( γ 2 + 4 γ 4 )) [3 + 4 E 2 (16( γ 0 − γ 4 ) + 5( γ 2 + 4 γ 4 ))] − 2 E 2 sin 2 θ κ 2 5 A D =6 (++ , ++) 1 + 8 E 2 (3( γ 0 − γ 4 ) + ( γ 2 + 4 γ 4 )) � � 8 E 6 γ 2 � � = − i 2 1 4 (1 − 4 E 2 ( γ 2 + 4 γ 4 )) [1 + 2 E 2 (10( γ 0 − γ 4 ) + 3( γ 2 + 4 γ 4 ))] − 2 E 2 sin 2 θ κ 2 6 In D > 4 the expected linear term in γ 4 is absent due to a non trivial cancellation between contact and exchange diagrams. In D > 4 the dependence on γ ′ 0 and γ ′ 2 is due to the fact that in exchange diagrams the massive poles cannot cancel with the three-graviton vertices of GB. This is associated to the dependence on arbitrary power of E in the IR. Stefano Giaccari Causality in nonlocal gravity 8 / 18

Scattering amplitudes for weakly nonlocal gravity If γ ′ 0 = γ ′ 0 ( � ), γ ′ 2 = γ ′ 2 ( � ) and γ 4 = 0, � 9 t ( s + t ) 9 9 � A s (++ , ++) = − 2 κ − 2 s 2 + ( s + 2 t )2 � s 2 γ 0( s ) � − + γ 2( s ) + , (8) 4 8 s 32 8 � s 3 − 5 s 2 t − st 2 + t 3 � ( s + t )2  1 A t (++ , ++) = − 2 κ − 2  − 4 s 3 t 8 � 2 s 4 − 10 s 3 t − s 2 t 2 + 4 st 3 + t 4 � ( s + t )2 t 2( s + t )4  1 1  , + γ 2( t ) + γ 0( t ) (9) s 4 s 4 16 8 � s 3 − 5 s 2 u − su 2 + u 3 � ( s + u )2  1 A u (++ , ++) = − 2 κ − 2  − 4 s 3 u 8 � 2 s 4 − 10 s 3 u − s 2 u 2 + 4 su 3 + u 4 � ( s + u )2 u 2( s + u )4  1 1 + γ 2( u ) + γ 0( u )  , (10) s 4 s 4 16 8  s 4 + s 3 t − 2 st 3 − t 4 1 9 9 A contact(++ , ++) = − 2 κ − 2 � s 2 + ( s + 2 t )2 � s 2 γ 0( s )  − − γ 2( s ) − 4 s 3 4 32 8 � 2 s 4 − 10 s 3 t − s 2 t 2 + 4 st 3 + t 4 � ( s + t )2 t 2( s + t )4 1 1 − γ 2( t ) − γ 0( t ) s 4 s 4 16 8 � 2 s 4 − 10 s 3 u − s 2 u 2 + 4 su 3 + u 4 � ( s + u )2 u 2( s + u )4  1 1  . − γ 2( u ) − γ 0( u ) (11) s 4 s 4 16 8 The cancellation of poles occurs separately in each channel A (++ , ++) = A (++ , ++) EH . (12) Stefano Giaccari Causality in nonlocal gravity 9 / 18

A field redefinition theorem Given two actions S ′ ( g ) and S ( g ) such that S ′ ( g ) = S ( g ) + E i ( g ) F ij ( g ) E j ( g ) , (13) where F ij can contain derivatives and E i = δS/δg i , there exist a field redefinition g ′ i = g i + ∆ ij E j ∆ ij = ∆ j i , (14) such that, perturbatively in F and to all orders in powers of F , we have the equivalence S ′ ( g ) = S ( g ′ ) . (15) The theorem states the equivalence of the two theories only perturbatively in F . In particular the two theories are clearly different if S ′ ( g ) has additional poles wrt S ( g ′ ). The theorem in particular applies to tree-level amplitudes whenever the external legs are on the mass-shell shared by the two theories. Stefano Giaccari Causality in nonlocal gravity 10 / 18

Implications for higher derivative theories Any higher derivative gravity theory which can be recast in the form S ′ ( g ) = S EH ( g ) + R µν ( g ) F µν,ρσ ( g ) R ρσ ( g ) . (16) µνρσ , with F µν,ρσ = g µν g ρσ γ 0 ( � ) + g µρ g νσ γ 2 ( � ) + ˜ V ( R , Ric , Riem , ∇ ) shares the same n -graviton on-shell tree-level amplitude as S EH ( g ). If we neglect finite contributions to the quantum effective action, this result can be applied to all finite weakly nonlocal theories with γ 4 ( � ) = 0 and killers of the kind R 2 � γ − 2 ( R 2 ) and R µν R µν � γ − 2 ( R ρσ R ρσ ). It also applies to 1-loop super-renormalizable theories in D = 4, while in general terms giving non vanishing contribution will be generated in renormalizable and super-renormalizable theories for D ≥ 6. More in general the theorem applies whenever the finite contributions to the quantum effective action can be cast in such a way as to be at least quadratic in the scalar curvature and Ricci tensors. = ⇒ Higher derivatives terms contain crucial physical information about the UV behavior, but, at least in some cases, look quite elusive observationally. Stefano Giaccari Causality in nonlocal gravity 11 / 18