A scattering amplitude in Conformal Field Theory

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Section: Jhep01(2021)192mentioning

confidence: 99%

Section: Jhep01(2021)192mentioning

confidence: 99%

Conformal correlators as simplex integrals in momentum space

Bzowski

McFadden

Skenderis

2021

“…While an understanding of CFTs in momentum space is desirable especially because of their applicability in the context of cosmology [7][8][9][10][11][12][13][14][15][16], it is far less developed compared to its position space counterpart. One of the major obstacles is the lack of momentum space analogue of position space cross-ratios [12,17,18].…”

Section: Introductionmentioning

confidence: 99%

Constraining momentum space correlators using slightly broken higher spin symmetry

Jain

John

Malvimat

2021

In this work, building up on [1] we present momentum space Ward identities related to broken higher spin symmetry as an alternate approach to computing correlators of spinning operators in interacting theories such as the quasi-fermionic and quasi-bosonic theories. The direct Feynman diagram approach to computing correlation functions is intricate and in general has been performed only in specific kinematic regimes. We use higher spin equations to obtain the parity even and parity odd contributions to two-, three- and four-point correlators involving spinning and scalar operators in a general kinematic regime, and match our results with existing results in the literature for cases where they are available.One of the interesting facts about higher spin equations is that one can use them away from the conformal fixed point. We illustrate this by considering mass deformed free boson theory and solving for two-point functions of spinning operators using higher spin equations.

“…Meanwhile, the bootstrap approach has also been applied in cosmology [10][11][12], which motivates momentum space considerations [13,14] for CFT correlators generally [15][16][17][18][19][20][21][22][23][24][25][26] which now also have Lorentzian analogues [27,28], with applications [29].…”

Section: Introductionmentioning

confidence: 99%

ANEC in λϕ4 theory

Bautista¹,

Casarin²,

Godazgar³

2021

Motivated by the goal of applying the average null energy condition (ANEC) to renormalisation group flows, we calculate in λϕ4 theory the expectation value of the ANEC operator in a particular scalar state perturbatively up to third order in the quartic coupling and verify the expected CFT answer. The work provides the technical tools for studying the expectation value of the ANEC operator in more interesting states, for example tensorial states relevant to the Hofman-Maldacena collider bounds, away from critical points.