Constraints from CFT three point functions. Heraklion-Crete, - PowerPoint PPT Presentation

Constraints from CFT three point functions. Heraklion-Crete, 24-04-2014 work in progress with Z. Komargodski, A. Parnachev and A. Zhiboedov Manuela Kulaxizi Universit´ e Libre de Bruxelles

Introduction Why interested in Conformal Field Theories (CFTs) ? • CFTs serve as endpoints of renormalization group flows. • Interesting low energy dynamics for several real world physical systems. e.g. condensed matter, quark gluon plasma, physics beyond the standard model, ... • They are in principle more accessible to study. Non-perturbative techniques available: e.g. conformal bootstrap techniques, AdS/CFT,...

Introduction Correlation functions of spin- ℓ primary operators with conformal dimension ∆ are highly constrained: � � I a 1 b 1 ( x 12 ) · · · I a ℓ b ℓ ( x 12 ) O a 1 ··· a ℓ ( x 1 ) O b 1 ··· b ℓ ( x 2 ) = C ( ℓ, ∆) x 2∆ 12 I ab ( x ) = η ab − 2 x a x b x 2 Unitarity implies C ( ℓ, ∆) ≥ 0 . Choose a basis such that: C ( ℓ, ∆) = 1 and �O ( x 1 ) O ′ ( x 2 ) � = 0 for O � = O ′ .

Introduction Further constraints from the two-point functions of descendants : ∆ ≥ d 2 − 1 , l = 0 ∆ ≥ l + d − 2 , l ≥ 1 Example: A scalar field Φ( x ) of dimension ∆ in d = 4 . 1 � Φ( x )Φ(0) � = x 2∆ ≥ 0 Consider the two point function of the descendant ∂ 2 x Φ( x ) . � � 1 ∆ 2 − 1 0 ≤ ∂ 2 x ∂ 2 x � Φ( x )Φ(0) � ∼ C x 2∆+4 ⇒ ∆ > 1 [Ferrara, Gatto, Grillo][Mack]

Introduction Constraints from two-point functions. What about three point functions? A conformal field theory is characterized by: • Its spectrum. A set of primary operators O ( ℓ, ∆) with conformal di- mensions ∆ . • Three point functions of these operators, which are fixed by conformal incariance up to a few constant parameters. e.g. λ OOO ′ ∆ = [ O ] , ∆ ′ = [ O ′ ] �O ( x 1 ) O ( x 2 ) O ′ ( x 3 ) � = , x 2∆ − ∆ ′ x ∆ 23 x ∆ 12 31

Introduction I µν,ρσ ( x ) � T µν ( x ) T ρσ (0) � = C T x 2 d � T µν ( x 3 ) T ρσ ( x 2 ) T τκ ( x 1 ) � = AJ µνρστκ ( x ) + BK µνρστκ ( x ) + CM µνρστκ ( x ) x d 12 x d 13 x d 23 • Ward Identites relate C T with A , B , C C T = ( d − 1)( d + 2) A − 2 B − 4( d + 1) C d ( d + 2) The coefficients of the conformal anomaly on a curved manifold c , a in four spatial dimensions, are directly re- lated to A , B , C via c ∝ C T and a ∝ 13 A− 2 B− 40 C . 8

Introduction Non-perturbative approach widely used in recent years; The Conformal Bootstrap : Unitarity + Crossing Symmetry of four-point functions. Example: The four point function of scalar operators with the ope � � O ( x 1 ) O ( x 2 ) O ( x 3 ) O ( x 4 ) = � � � O , O ′ λ O λ ′ O I ( x 2 ) O ′ J ( x 4 ) = O C I ( x 12 , ∂ 2 ) C J ( x 34 , ∂ 4 ) � g ℓ, ∆ ( u, v ) λ 2 = O x 2 d 12 x 2 d 34

Introduction • The conformal block : � � g ℓ, ∆ ( u, v ) ≡ x 2 d 12 x 2 d O I ( x 2 ) O ′ J ( x 4 ) 34 C I ( x 12 , ∂ 2 ) C J ( x 34 , ∂ 4 ) • The conformal cross ratios u ≡ x 2 12 x 2 v ≡ x 2 14 x 2 34 23 , x 2 13 x 2 x 2 13 x 2 24 24 Symmetry under the exchange x 1 ↔ x 3 leads to the crossing relation : � u � d � � λ 2 λ 2 O g ℓ, ∆ ( u, v ) = O g ℓ, ∆ ( u, v ) v O O Similar relations from the ope in different channels.

Introduction The conformal bootstrap approach consists in solving these equations. It is a powerful but numerical technique. • It is useful to gather all possible analytic results. Constraints can be obtained analytically for a special class of three-point functions: � O ( ℓ ) T µν O ( ℓ ) � They are obtained from the requirement of positivity of the energy flux �E ( � n ) � ≥ 0

Outline • Energy flux operator; review. • Non-conserved operators; Example: vector operator and constraints. • Connection to Deep Inelastic Scattering (DIS): Some puzzles. • Conclusions and Open Questions

Energy Flux Operator Definition: The energy flux operator E ( � n ) per unit angle measured through a very large sphere of radius r is � n i T 0 r →∞ r d − 2 n i ) E ( � n ) = lim i ( t, r � dt � n i is a unit vector specifying the position on S d − 2 where energy measurements may take place. Integrating over all angles yields the total energy flux at large distances.

Energy Flux Operator Consider the normalized energy flux one-point function � Ψ |E ( � n ) | Ψ � � Ψ | Ψ � There are several possibilities for the states | Ψ � : • Conserved currents � d d xe − iqx J µ ǫ µ | 0 � | Ψ � = � d d xe − iqx T µν ǫ µν | 0 � | Ψ � = • Generic primary � d d xe − iqx O a 1 ··· a l ǫ a 1 ··· a l | 0 � | Ψ � =

Energy Flux Operator • Rotational symmetry fixes the form of the energy flux one–point function up to two independent parame- ters. n ) � T ij = � ǫ ∗ ik T ik E ( � n ) ǫ lj T lj � �E ( � = � ǫ ∗ ik T ik ǫ lj T lj � � � � � �� ǫ ∗ | ǫ ij n i n j | 2 1 2 E il ǫ lj n i n j = 1 + t 2 − + t 4 − d 2 − 1 ǫ ∗ ǫ ∗ Ω d − 2 d − 1 ij ǫ ij ij ǫ ij Here t 2 , t 4 are arbitrary constants. By construction, they can be related to A , B , C . [Hofman, Maldacena]

Energy Flux Operator Demand positivity of the energy flux one point function, i.e. , �E ( � n ) � ≥ 0 . The positivity of the energy flux imposes constraints on t 2 , t 4 : 1 2 C G ( A , B , C ) ≡ 1 − d 2 − 1 t 4 ≥ 0 d − 1 t 2 − 1 2 d 2 − 1 t 4 + t 2 C V ( A , B , C ) ≡ 1 − d − 1 t 2 − 2 ≥ 0 1 d 2 − 1 t 4 + d − 2 2 C S ( A , B , C ) ≡ 1 − d − 1 t 2 − d − 1( t 2 + t 4 ) ≥ 0 Constraints are saturated by free theories of bosons, � � d fermions and 2 − 1 -form fields in even dimensions. [Hofman, Maldacena][Zhiboedov]

Bounds on t 2 , t 4 . Parameter space t 2 , t 4 of a consistent CFT. Values out- side the triangle are forbidden. Example: CFT d = 4 dimensions 1 3 ≤ a c ≤ 31 18

Bounds for generic operators Natural generalization; non-conserved currents. • ope coefficients depend on marginal couplings - non-perturbative results. • Ising model; Indications it contains an infinite number of almost conserved currents. [Komagordski, Zhiboedov] [Fitzpatrick, Kaplan, Poland, Simmons-Duffin] The simplest case to consider: vector operator.

Bounds for a vector operator Generic form of the energy flux for vector operators � � � q 0 n | 2 ǫ 2 | � 1 ǫ. � � �E� = 1 + a 2 + ǫ 2 + g (∆)( ǫ 0 ) 2 − ǫ 2 + g (∆)( ǫ 0 ) 2 Ω d − 2 d − 1 � � � ǫ 0 ( � n ) + c.c. ǫ. � + a 4 ǫ 2 + g (∆)( ǫ 0 ) 2 � • The denominator is fixed by the two point function: � � ( ǫ ∗ .ǫ ) − 2∆ − d ( ǫ ∗ .q )( ǫ.q ) ( q 2 ) ∆ − d � ( ǫ ∗ . O ( − q )) ( O ( q ) .ǫ ) � ∝ 2 2 q 2 ∆ − 1 • g (∆) = ∆+1 − d ≥ 0 , saturated at the unitarity bound. ∆ − 1 • Additional parameter due to non-conservation, a 4 .

Bounds for a vector operator Polarization choices: n = 0 • � ǫ. � a 2 ≤ d − 1 Constraint identical to the case of conserved current. n � = 0 . • � ǫ. � n = (0 , 0 , 1) and ǫ µ = ( ǫ 0 , 0 , 0 , 1) . Choose ǫ 0 which Set � minimizes the energy flux ⇒ ǫ 0 = f ( a 2 , a 4 ) . a 2 g (∆) ≤ 1 + d − 2 4 d − 1 a 2

Bounds for a vector operator Bounds on a 2 , a 4 from the positivity of the energy flux. 2 H D - 1 L a 4 D - 2 2.0 1.5 1.0 0.5 a 2 - 2 - 1 1 2 Which theory corresponds to the cusp?

Bounds for generic operators • What about other operators? – For spin s ≥ 2 the number of structures in the three point function increases linearly with spin: # of possible structures = 3 s [Zhiboedov] – Ward Identities lead to 3 s − 1 structures. – The number of independent parameters in the en- ergy flux expression grows at least like s 2 . 1-1 correspondence with the parameters in the three point function is lost. e.g.: For s = 2 there are five independent parame- ters in the ope but seven in the energy flux. [work in progress]

Positivity of the energy flux The positivity of the energy flux is a reasonable assump- tion - is there a proof? • Proof known for free theories. • Remarkable evidence from holography: Bounds realized earlier holographically. The arena: Black holes in Lovelock gravity, a special class of higher derivative theories, a � = c . The principle: Causality of the retarded propagator at finite temperature T in the limit of large momenta, i.e. , ω, | q | >> T . [Brigante, Liu, Myers, Shenker, Yaida] [Myers, Buchel] [Hofman]

Positivity of the energy flux The energy flux positivity constraints are related to causality in the gravity language. Can we see some- thing similar in field theory? Guide from the AdS/CFT analysis: • Consider the Fourier transform of the two–point func- tion of the stress energy tensor at finite temperature. k • Focus on large momenta, small temperatures T ≫ 1 . • Three independent polarizations; each polarization yields a different set of constraints.