A curvature bound from gravitational catalysis. Riccardo Martini - PowerPoint PPT Presentation

A curvature bound from gravitational catalysis. Riccardo Martini Friedrich-Schiller-Universit¨ at Jena Based on a joint work with H. Gies: [arXiv:1802.02865] May 15th, 2018 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 1 / 22

Outline Introduction and motivations 1 Framework 2 D = 3 3 D = 4 4 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 2 / 22

Outline Introduction and motivations 1 Framework 2 D = 3 3 D = 4 4 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 3 / 22

Chiral symmetry Chiral transformation acts indipendently on the right and left components of fermions: � → e i θ L ψ L ψ L − U ( N f ) R × U ( N f ) L ⇒ β ( λ k ) → e i θ R ψ R ψ R − A mass term is not chiral invariant λ k It can be used to define a chiral condensate m eff ≃ � ¯ ψψ � The condensate represents the order parameter Riccardo Martini (FSU) Curvature Bound May 15th, 2018 4 / 22

Gravitational catalysis ❼ Gravitational catalysis indicates the breaking of chiral symmetry due to the presence of a curved background [Buchbinder and Kirillova, 1989; Sachs and Wipf, 1994; Elizalde, Leseduarte, Odinstov and Sil’nov, 1996 ]. ❼ In negatively curved spacetimes it can be understood as an effective dimensional reduction of the long range dynamics of fermionic modes from D + 1 to 1 + 1 dimensions [Gorbar, 2009] . β ( λ k ) ❼ The fixed point structure of sys- tems undergoing gravitational catal- λ k ysis was studied [Scherer and Gies, 2012; Gies and Lippoldt, 2013] . Riccardo Martini (FSU) Curvature Bound May 15th, 2018 5 / 22

Light fermions Under the assumption of chiral symmetry breaking being triggered by quantum gravity one would expect a consequent mass gap comparable to the Planck mass [Eichhorn and Gies, 2011] . � � � � � Can we use gravitational catalysis to constrain quantum gravity? Riccardo Martini (FSU) Curvature Bound May 15th, 2018 6 / 22

Outline Introduction and motivations 1 Framework 2 D = 3 3 D = 4 4 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 7 / 22

Bosonization The action for our model reads: ¯ � λ + ψ a γ µ ψ a � 2 ψ a γ µ γ 5 ψ a � 2 �� � �� � S [ ¯ ¯ ¯ ¯ ψ, ψ ] = ψ / ∇ ψ + − . (1) 2 x By means of Fierz identities we re organize the interaction as ( V ) + ( A ) = − 2[( S N ) − ( P N )] (2) where ψ a ψ b ) 2 = ( ¯ ( S N ) ( ¯ ψ a ψ b )( ¯ ψ b ψ a ) , = ψ a γ 5 ψ b ) 2 = ( ¯ ( P N ) ( ¯ ψ a γ 5 ψ b )( ¯ ψ b γ 5 ψ a ) , = (3) obtaining a NJL-type of action: � ψ a ψ b � 2 ψ a γ 5 ψ b � 2 �� � �� � S [ ¯ ¯ ∇ ψ − ¯ ¯ ¯ ψ / ψ, ψ ] = λ + − . (4) x Riccardo Martini (FSU) Curvature Bound May 15th, 2018 8 / 22

Bosonization Making use of the chiral projectors P L = 1 − γ 5 P R = 1 + γ 5 , , 1 = P L + P R , (5) 2 2 and the following auxiliary fields, satisfying: φ ab = − 2¯ λ ¯ ( φ † ) ab = − 2¯ λ ¯ ψ b R ψ a ψ b L ψ a L , R λ = 2¯ ¯ λ + , (6) = we can implement the Hubbard-Stratonovich trick and map ⇒ our model to a Yukawa-type interaction as: ∇ + P L ( φ † ) ab + P R φ ab ] ψ b + 1 L ( φ, ¯ ψ, ψ ) = ¯ ψ a [ / λ tr( φ † φ ) . (7) 2¯ It is clear that the nonzero components of � φ ab � are connected to the dynamically generated fermion mass. Riccardo Martini (FSU) Curvature Bound May 15th, 2018 9 / 22

Mean field analysis assuming breaking pattern, φ ab = φ 0 δ ab and introducing Schwinger proper time T we finally write: � ∞ U ( φ ) = N f 0 + N f dT e − φ 2 ∇ 2 T . λφ 2 0 T Tr e / (8) 2¯ 2 T 0 with: ∇ 2 T = Tr K ( x , x ′ ; T ) =: K T , / Tr e (9) and the heat kernel obeying: T → 0 + K ( x , x ′ ; T ) = δ ( x − x ′ ) ∂ 2 K , ∂ T K = / ∇ lim . (10) √ g Riccardo Martini (FSU) Curvature Bound May 15th, 2018 10 / 22

RG analysis ❼ In order to investigate the flow of the potential we introduce a propertime regulator function f k : f k = e − ( k 2 T ) p . (11) 1 ❼ Thus, at some given average scale k IR ∼ T : √ 1/k IR � Λ U k IR = U Λ − dk ∂ k U k (12) k IR together with N f λ Λ φ 2 U Λ = 0 , (13) 2¯ � ∞ T e − φ 2 ∂ k U k = N f dT 0 T ∂ k f k K T . 2 0 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 11 / 22

SSB The fermion mass is generated by breaking a U ( N f ) R × U ( N f ) L symmetry. We focus on a second order phase transition mechanism. We check the curvature of the 100 potential in the origin of the field space. U'' ( 0 )> 0 50 The condition U ′′ (0) = 0 will 1 2 3 4 5 k 2 be a function of the ratio IR R U'' ( 0 )< 0 - 50 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 12 / 22

Outline Introduction and motivations 1 Framework 2 D = 3 3 D = 4 4 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 13 / 22

D = 3 Given κ 2 = D ( D − 1) = | R | | R | the heat kernel can be expressed as: 6 1 1 + 1 � � K D =3 2 κ 2 T = , (14) T 3 3 2 T 8 π 2 and the effective potential can be computed analytically: � 1 U k IR = − N f − 1 − k IR + N f 2 − 3 � � 3 � 2 φ 2 ( φ 2 0 + k 2 2 k IR φ 2 0 − k 3 IR ) 0 IR ¯ ¯ λ cr λ Λ 4 π 12 π − N f 16 π κ 2 �� � φ 2 0 + k 2 IR − k IR . (15) At criticality, in order to avoid chiral symmetry breaking the curvature needs to satisfy: κ 2 ⇒ | R | ≤ 24 k 2 ≤ 4 = (16) IR k 2 IR Riccardo Martini (FSU) Curvature Bound May 15th, 2018 14 / 22

Outline Introduction and motivations 1 Framework 2 D = 3 3 D = 4 4 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 15 / 22

D = 4 � 1 � κ = − N f φ 2 − 1 0 κ 2 , � �� 0 + κ 2 A − 12 N f ξ k IR φ 2 U k IR k IR ; p (17) � ¯ ¯ 2 � φ 2 λ cr λ Λ 0 � � 1 Γ 1 + k 2 (4 π ) 2 1 − 1 (4 π ) 2 κ 2 + 2 κ 2 p � � IR ¯ A ∼ − Γ √ π , λ cr = � , (18) � p k IR 1 − 1 Λ 2 Γ p U'' ( 0 ) IR A 0.04 0.02 k IR k IR k IR κ κ κ κ κ 0.5 1.0 1.5 2.0 - 0.02 - 0.04 κ - 0.06 κ κ κ - 0.08 Analytical Numerics Riccardo Martini (FSU) Curvature Bound May 15th, 2018 16 / 22

Curvature bound ❼ For ξ k IR = 0, in order to avoid gravitationally catalyzed chiral symmetry breaking we have: κ κ � � p =2 ≤ 1 . 8998 , p →∞ ≤ 1 . 5757 . (19) � � k IR � k IR � ❼ Allowing different values for ξ k IR , the bound is shifted: U'' (0) U'' ( 0 ) IR k IR 100 ξ IR = - 2 50 ξ IR = - 1 κ κ ξ IR = 0 k IR k IR 10 4 0.1 1 10 100 1000 ξ IR = 1 - 50 ξ IR = 2 - 100 - 150 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 17 / 22

Constraining quantum gravity in asymptotic safety The background metric is a solution of the semiclassical equations of motion R R µν ( � g � k ) = ¯ Λ k � g µν � k = ⇒ k 2 = 4 λ ∗ , (20) at UV f.p. where ¯ Λ k is the scale dependent cosmological constant and λ ∗ its UV fixed point value. The presence of fermionic d.o.f. drags the cosmological constant UV fixed points towards negative values. Identifying k IR with the coarse graining scale k of asymptotic safety is possible to study a bound on the number of matter d.o.f.: κ 2 = | λ ∗ | with λ ∗ = λ ∗ ( N S , N f , N V ) . (21) k 2 3 IR Riccardo Martini (FSU) Curvature Bound May 15th, 2018 18 / 22

Constraining quantum gravity in asymptotic safety Given d g = N S − 4 N V + 2 N f , d λ = N S + 2 N V − 4 N f we can parametrize the space of fixed point of a matter-gravity system: d λ 100 R > 0 PF 30 d g - 30 - 20 - 10 10 20 SM + N f - 100 - 200 p = ∞ Gravitational catalysis - 300 p = 2 - 400 Plot based on the results from [Biemans, Platania and Saueressig, 2017] . Riccardo Martini (FSU) Curvature Bound May 15th, 2018 19 / 22

Comparing different techniques N f,gc PF SM+ N f MSSM+ N f one-loop approx. (type IIa) 17.58 35.97 20.3 [Codello, Percacci and Rahmede, 2009] background-field approximation 8.21 26.5 no FP [Don´ a, Eichhorn and Percacci, 2014] RG flow on foliated spacetimes 9.27 27.67 10.01 [Biemans, Platania and Saueressig, 2017] dynamical FRG 48.7 [Meybohm, Pawlowski and Reichert, 2016] Riccardo Martini (FSU) Curvature Bound May 15th, 2018 20 / 22

Higher dimensions In absence of new operators and in the limit p → ∞ , the bound in D = 6 results in κ � p →∞ ≤ 1 . 0561 . (22) � k IR � 1 1 σ σ 0 0 1.4 For D odd the decreasing behavior 1.2 is clear: σ 1.0 D √ π 1 � � D − 1 0.8 k IR ≤ 1 κ σ 0 ≡ � � 0.6 D Γ ( D − 2) 2 0.4 0.2 D D 5 10 15 20 Riccardo Martini (FSU) Curvature Bound May 15th, 2018 21 / 22