A conformal dispersion relation: correlations from absorption
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Received: March 10, 2020 Accepted: July 27, 2020 Published: September 1, 2020
Dean Carmia and Simon Caron-Huotb ´ Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Rte de la Sorge, BSP 728, CH-1015 Lausanne, Switzerland b Department of Physics, McGill University, 3600 Rue University, Montr´eal, QC Canada H3A 2T8
a
E-mail: [email protected], [email protected] Abstract: We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theoryindependent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block. Keywords: Conformal Field Theory, Field Theories in Higher Dimensions ArXiv ePrint: 1910.12123
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)009
JHEP09(2020)009
A conformal dispersion relation: correlations from absorption
Contents 1
2 Preliminaries 2.1 Review of amplitude dispersion relation 2.1.1 Dispersion relation from the Froissart-Gribov formula 2.2 Review of CFT kinematics 2.2.1 CFT dispersion relation from Lorentzian inversion formula
3 3 5 6 8
3 Computing the CFT dispersion relation kernel 3.1 Performing the ∆ integration in d = 2 3.1.1 The contact term KC 3.1.2 The bulk term KB 3.2 Main result from Legendre PPPQ sum 3.3 Match with d = 4 and differential equation 3.3.1 Contact term 3.3.2 Bulk term 3.4 Differential equation for unequal scaling dimensions
9 10 11 12 13 15 15 15 16
4 Direct proof of dispersion relation 4.1 Contour deformation trick 4.1.1 Why two variables? 4.2 Convergence and subtractions 4.2.1 Subtracted dispersion relation 4.2.2 Keyhole contour near cross-channel singularity
19 20 23 23 24 24
5 Checks and discussion 5.1 Numerical check for generalized free fields 5.2 Holographic correlators 5.3 An integral relation between conformal blocks 5.4 3D Ising model and analytic functionals
25 26 26 28 29
6 Conclusion
31
A Identities for spin sums
32
B Inverted block from harmonic function when a = 0, b =
–i–
1 2
33
JHEP09(2020)009
1 Introduction
1
Introduction
t
1
Z
G (z, z¯) =
dwdwK(z, ¯ z¯, w, w)dDisc[G(w, ¯ w)] ¯
(1.1)
0
where we separate the t and u channel contributions and a possible finite sum of nonnormalizable blocks (see section 4.2.1): G(z, z¯) = G t (z, z¯) + G u (z, z¯) + (non-norm.) .
(1.2)
The input dDisc[G(z, z¯)] represents the double-discontinuity of the correlator around z¯ = 1, defined below,
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