Study of momentum space scalar amplitudes in AdS spacetime

141

We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

Section: Five-point Treesupporting

confidence: 81%

“…the Schwinger-Keldysh formalism [33][34][35][36][37][38][39]. 5 For further work on AdS/CFT in momentum space see [60][61][62][63][64][65][66][67][68].…”

Section: Jhep11(2020)073mentioning

confidence: 99%

CFT unitarity and the AdS Cutkosky rules

Meltzer

Sivaramakrishnan

2020

141

“…Furthermore, solving the special conformal Ward identity is in general hard as they are complicated second order differential equations in momentum variables. For specific higher point correlators limited progress has been made [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Despite the difficulties it is desirable to understand CFT in momentum space.…”

Section: Introductionmentioning

confidence: 99%

Momentum space spinning correlators and higher spin equations in three dimensions

Jain

John

Malvimat

2020

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator $$ \overline{\psi}\psi $$ ψ ¯ ψ is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $$ \overline{\psi}\psi $$ ψ ¯ ψ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

“…Of course, in a CFT the correlation function in position space already offers a crossing symmetric observable which also admits an OPE decomposition, and the bootstrap approach has prospered in 2d [3], and in higher d [4,5]. Still, conformal field theories in momentum space have recently drawn increasing interest [6][7][8][9][10][11][12][13][14][15][16][17][18]. While the four-point function of primary operators in momentum space has been analyzed before [8,9], it is important to emphasize that this object does not admit a conformal block decomposition for a generic choice of the kinematics.…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…This formula can be easily checked against the known 4d and 6d blocks. 18 It would be nice to prove this formula for any d. This should be possible using the diagonal limit of conformal blocks studied in [47].…”

Section: Partials Waves From Conformal Blocksmentioning

confidence: 99%

A scattering amplitude in Conformal Field Theory

Gillioz

Meineri

Penedones

2020

We define form factors and scattering amplitudes in Conformal Field Theory as the coefficient of the singularity of the Fourier transform of time-ordered correlation functions, as p2 → 0. In particular, we study a form factor F(s, t, u) obtained from a four-point function of identical scalar primary operators. We show that F is crossing symmetric, analytic and it has a partial wave expansion. We illustrate our findings in the 3d Ising model, perturbative fixed points and holographic CFTs.