Implications of conformal invariance in momentum space
By Adam Bzowski, Paul McFadden, Kostas Skenderis. 2013
281 citations recorded by the source corpus.
A bstractWe present a comprehensive analysis of the implications of conformal invariance for 3-point functions of the stress-energy tensor, conserved currents and scalar operators in general dimension and in momentum space. Our starting point is a novel and very effective decomposition of tensor correlators which reduces their computation to that of a number of scalar form factors. For example, the most general 3-point function of a conserved and traceless stress-energy tensor is determined by only five form factors. Dilatations and special conformal Ward identities then impose additional conditions on these form factors. The special conformal Ward identities become a set of first and second order differential equations, whose general solution is given in terms of integrals involving a product of three Bessel functions (‘triple-K integrals’). All in all, the correlators are completely determined up to a number of constants, in agreement with well-known position space results. In odd dimensions 3-point functions are finite without renormalisation while in even dimensions non-trivial renormalisation in required. In this paper we restrict ourselves to odd dimensions. A comprehensive analysis of renormalisation will be discussed elsewhere.This paper contains two parts that can be read independently of each other. In the first part, we explain the method that leads to the solution for the correlators in terms of triple-K integrals while the second part contains a self-contained presentation of all results. Readers interested only in results may directly consult the second part of the paper.
Subfields: Astronomy and Astrophysics, Biomedical Engineering, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/a3a55c1eb2d13dfad41dc4f0fd12ead2e65f451b
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Citations in the corpus
- The Cosmological Bootstrap: Spinning correlators from symmetries and factorization (2020, background): “In [56], the expression for a three-point function involving a conserved current and two ∆ = 1 scalars is given.”
- Late-time structure of the Bunch-Davies de Sitter wavefunction (2014, methodology): “Motivated by our bulk analysis, we chose a slightly different cutoff procedure from [58], where we instead cut the integral off at some small z = zc.”
- The cosmological bootstrap: inflationary correlators from symmetries and singularities (2018, unknown): “Just like Poincaré symmetry completely fixes three-particle scattering amplitudes, conformal symmetry determines the functional form of three-point functions [84, 85] (see [32] for a detailed analysis in momentum space).”
- Bootstrapping inflationary correlators in Mellin space (2019, methodology): “where kt = k1 + k2 + k3, which matches the result in [104].”
- A Mellin space approach to cosmological correlators (2019, background): “The exchange The constraints of Conformal Ward identities on Conformal Structures in Fourier space have been studied in [6, 20–30].”
- Building a boostless bootstrap for the bispectrum (2020, background): “ortunately, in this particular setting, we were not able to x the last parameter D 1 without appealing to de Sitter boosts23. To account for boost invariance, the condition to be imposed is (see e.g. [66]) (K a K b)B (k 1;k 2;k 3) = 0; (4.29) for a6= b, a;b= 1;2;3, where K a= @2 @k2 a + 4 k a @ @k a”
- The cosmological bootstrap: weight-shifting operators and scalar seeds (2019, methodology): “(See [13–34] for other studies on conformal correlators in momentum space.”
- Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies (2015, methodology): “For tensorial operators an additional term appears, see [7].”
- Holography at finite cutoff with a T2 deformation (2018, background): “ihJi(y′)Jm(y)i+ σ2 λ2(d−2)/d * δFij(y′) δAl(x) δFij(y′) δAm(y) +! . (7.4) This flow equation simplifes in momentum space, where Lorentz invariance forces the two point function of Ji to be of the form [47] hJl(k)Jm(−k)i = C(λ,k)πlm, πlm = δlm − klkm k2 , (7.5) 27 with C a function of λ and k that com”