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Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal Field Theory in Momentum space Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre Research STA G Centre

  1. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal Field Theory in Momentum space Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre Research STA G Centre Oxford University 3 March 2015 Kostas Skenderis Conformal Field Theory in Momentum space

  2. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Outline 1 Introduction Scalar 2-point functions 2 Scalar 3-point functions 3 Tensorial correlators 4 5 Conclusions Kostas Skenderis Conformal Field Theory in Momentum space

  3. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Introduction ➢ Conformal invariance imposes strong constraints on correlation functions. ➢ It determines two- and three-point functions of scalars, conserved vectors and the stress-energy tensor [Polyakov (1970)] ... [Osborn, Petkou (1993)] . For example, �O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) � c 123 = | x 1 − x 2 | ∆ 1 +∆ 2 − ∆ 3 | x 2 − x 3 | ∆ 2 +∆ 3 − ∆ 1 | x 3 − x 1 | ∆ 3 +∆ 1 − ∆ 2 . ➢ It determines the form of higher point functions up to functions of cross-ratios. Kostas Skenderis Conformal Field Theory in Momentum space

  4. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Introduction These results (and many others) were obtained in position space. This is in stark contrast with general QFT were Feymnan diagrams are typically computed in momentum space. While position space methods are powerful, typically they provide results that hold only at separated points ("bare" correlators). are hard to extend beyond CFTs The purpose of this work is to provide a first principles analysis of CFTs in momentum space. Kostas Skenderis Conformal Field Theory in Momentum space

  5. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Introduction Momentum space results were needed in several recent applications: ➢ Holographic cosmology [McFadden, KS](2010)(2011) [Bzowski, McFadden, KS (2011)(2012)] [Pimentel, Maldacena (2011)][Mata, Raju,Trivedi (2012)] [Kundu, Shukla,Trivedi (2014)] . ➢ Studies of 3d critical phenomena [Sachdev et al (2012)(2013)] Kostas Skenderis Conformal Field Theory in Momentum space

  6. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions References Adam Bzowski, Paul McFadden, KS Implications of conformal invariance in momentum space 1304.7760 Adam Bzowski, Paul McFadden, KS Renormalized scalar 3-point functions 15xx.xxxx Adam Bzowski, Paul McFadden, KS Renormalized tensor 3-point functions 15xx.xxxx Kostas Skenderis Conformal Field Theory in Momentum space

  7. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal invariance Conformal transformations consist of dilatations and special conformal transformations. Dilatations δx µ = λx µ , are linear transformations, so their implications are easy to work out. Special conformal transforms, δx µ = b µ x 2 − 2 x µ b · x , are non-linear, which makes them difficult to analyse (and also more powerful). The corresponding Ward identities are partial differential equations which are difficult to solve. Kostas Skenderis Conformal Field Theory in Momentum space

  8. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal invariance In position space one overcomes the problem by using the fact that special conformal transformations can be obtained by combining inversions with translations and then analyzing the implications of inversions. In momentum space we will see that one can actually directly solve the special conformal Ward identities. Kostas Skenderis Conformal Field Theory in Momentum space

  9. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal Ward identities These are derived using the conformal transformation properties of conformal operators. For scalar operators: ∆ 1 /d ∆ n /d ∂x ′ ∂x ′ � � � � �O 1 ( x ′ 1 ) · · · O n ( x ′ � � � � �O 1 ( x 1 ) · · · O n ( x n ) � = · · · n ) � � � � � ∂x ∂x � � � � x = x 1 x = x n For (infinitesimal) dilatations this yields   n n ∂ � � x α  �O 1 ( x 1 ) . . . O n ( x n ) � . 0 = ∆ j + j  ∂x α j j =1 j =1 In momentum space this becomes   n n − 1 ∂ � � p α  �O 1 ( p 1 ) . . . O n ( p n ) � , 0 = ∆ j − ( n − 1) d − j  ∂p α j j =1 j =1 Kostas Skenderis Conformal Field Theory in Momentum space

  10. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Special conformal Ward identity For (infinitesimal) special conformal transformations this yields  � n � ∂ ∂ � 2∆ j x κ j + 2 x κ j x α − x 2  �O 1 ( x 1 ) . . . O n ( x n ) � 0 = j j  ∂x α ∂x jκ j j =1 In momentum space this becomes 0 = K µ �O 1 ( p 1 ) . . . O n ( p n ) � ,  � n − 1 � 2(∆ j − d ) ∂ ∂ ∂ ∂ ∂ K µ = � − 2 p α + ( p j ) κ j  ∂p κ ∂p α ∂p κ ∂p α  ∂p jα j j j j j =1 Kostas Skenderis Conformal Field Theory in Momentum space

  11. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Special conformal Ward identities ➢ To extract the content of the special conformal Ward identity we expand K µ is a basis of linear independent vectors, the ( n − 1) independent momenta, K κ = p κ 1 K 1 + . . . + p κ n − 1 K n − 1 . ➠ Special conformal Ward identities constitute ( n − 1) differential equations. Kostas Skenderis Conformal Field Theory in Momentum space

  12. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Conformal Ward identities ➢ Poincaré invariant n -point function in d ≥ n spacetime dimensions depends on n ( n − 1) / 2 kinematic variables. ➢ Thus, after imposing ( n − 1) + 1 conformal Ward identities we are left with n ( n − 1) − n = n ( n − 3) 2 2 undetermined degrees of freedom. ➢ This number equals the number of conformal ratios in n variables in d ≥ n dimensions. ➠ It is not known however what do the cross ratios become in momentum space. Kostas Skenderis Conformal Field Theory in Momentum space

  13. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Outline 1 Introduction Scalar 2-point functions 2 Scalar 3-point functions 3 Tensorial correlators 4 5 Conclusions Kostas Skenderis Conformal Field Theory in Momentum space

  14. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Outline 1 Introduction Scalar 2-point functions 2 Scalar 3-point functions 3 Tensorial correlators 4 5 Conclusions Kostas Skenderis Conformal Field Theory in Momentum space

  15. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Scalar 2-point function ➢ The dilatation Ward identity reads � d − ∆ 1 − ∆ 2 + p ∂ � 0 = � O 1 ( p ) O 2 ( − p ) � ∂p ➠ The 2-point function is a homogeneous function of degree (∆ 1 + ∆ 2 − d ) : � O 1 ( p ) O 2 ( − p ) � = c 12 p ∆ 1 +∆ 2 − d . where c 12 is an integration constant. Kostas Skenderis Conformal Field Theory in Momentum space

  16. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Scalar 2-point function ➢ The special conformal Ward identity reads K = d 2 dp 2 − 2∆ 1 − d − 1 d 0 = K� O 1 ( p ) O 2 ( − p ) � , p dp ➢ Inserting the solution of the dilatation Ward identity we find that we need ∆ 1 = ∆ 2 Kostas Skenderis Conformal Field Theory in Momentum space

  17. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Scalar 2-point function The general solution of the conformal Ward identities is: � O ∆ ( p ) O ∆ ( − p ) � = c 12 p 2∆ − d . ➢ This solution is trivial when ∆ = d 2 + k, k = 0 , 1 , 2 , ... because then correlator is local, � O ( p ) O ( − p ) � = cp 2 k → � O ( x 1 ) O ( x 2 ) � ∼ � k δ ( x 1 − x 2 ) ➢ Let φ 0 be the source of O . It has dimension d − ∆= d/ 2 − k . The term φ 0 � k φ 0 has dimension d and can act as a local counterterm. Kostas Skenderis Conformal Field Theory in Momentum space

  18. Introduction Scalar 2-point functions Scalar 3-point functions Tensorial correlators Conclusions Position space [Petkou, KS (1999)] In position space, it seems that none of these are an issue: C �O ( x ) O (0) � = x 2∆ This expression however is valid only at separated points, x 2 � = 0 . Correlation functions should be well-defined distributions and they should have well-defined Fourier transform. Fourier transforming we find: � d − 2∆ x 2∆ = π d/ 2 2 d − 2∆ Γ � � d d x e − i p · x 1 2 p 2∆ − d , Γ(∆) This is well-behaved, except when ∆ = d/ 2 + k , where k is a positive integer. Kostas Skenderis Conformal Field Theory in Momentum space

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