Abstract
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.
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H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1 (1955) 205 [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part III: higher dimensional amplitudes, JHEP 12 (2019) 040 [arXiv:1708.06765] [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques, and applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals, JHEP 07 (2013) 011 [arXiv:1304.6944] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
C. Corianò and M.M. Maglio, On some hypergeometric solutions of the conformal Ward identities of scalar 4-point functions in momentum space, JHEP 09 (2019) 107 [arXiv:1903.05047] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal n-point functions in momentum space, Phys. Rev. Lett. 124 (2020) 131602 [arXiv:1910.10162] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators, JHEP 07 (2018) 136 [arXiv:1805.11107] [INSPIRE].
H. Isono, T. Noumi and G. Shiu, Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension, JHEP 10 (2019) 183 [arXiv:1908.04572] [INSPIRE].
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space, JHEP 01 (2020) 142 [arXiv:1908.04733] [INSPIRE].
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space, Commun. Math. Phys. 379 (2020) 227 [arXiv:1909.00878] [INSPIRE].
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The cosmological bootstrap: inflationary correlators from symmetries and singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The cosmological bootstrap: weight-shifting operators and scalar seeds, arXiv:1910.14051 [INSPIRE].
C. Sleight, A Mellin space approach to cosmological correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping inflationary correlators in Mellin space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
S. Albayrak, C. Chowdhury and S. Kharel, Study of momentum space scalar amplitudes in AdS spacetime, Phys. Rev. D 101 (2020) 124043 [arXiv:2001.06777] [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Scale anomalies, states, and rates in conformal field theory, JHEP 04 (2017) 171 [arXiv:1612.07800] [INSPIRE].
M. Gillioz, X. Lu and M.A. Luty, Graviton scattering and a sum rule for the c anomaly in 4D CFT, JHEP 09 (2018) 025 [arXiv:1801.05807] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Harmonic analysis and mean field theory, JHEP 10 (2019) 217 [arXiv:1809.05111] [INSPIRE].
R.S. Erramilli, L.V. Iliesiu and P. Kravchuk, Recursion relation for general 3d blocks, JHEP 12 (2019) 116 [arXiv:1907.11247] [INSPIRE].
V. Goncalves, Skeleton expansion and large spin bootstrap for ϕ3 theory, arXiv:1809.09572 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
M. Gary, S.B. Giddings and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev. D 80 (2009) 085005 [arXiv:0903.4437] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Scalar 3-point functions in CFT: renormalisation, β-functions and anomalies, JHEP 03 (2016) 066 [arXiv:1510.08442] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Renormalised 3-point functions of stress tensors and conserved currents in CFT, JHEP 11 (2018) 153 [arXiv:1711.09105] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Renormalised CFT 3-point functions of scalars, currents and stress tensors, JHEP 11 (2018) 159 [arXiv:1805.12100] [INSPIRE].
M. Gillioz, Momentum-space conformal blocks on the light cone, JHEP 10 (2018) 125 [arXiv:1807.07003] [INSPIRE].
G. Mack, Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys. 53 (1977) 155 [INSPIRE].
P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part I. Cross-ratio space, JHEP 05 (2020) 137 [arXiv:2001.08778] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
A. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Analytic Euclidean bootstrap, JHEP 10 (2019) 270 [arXiv:1808.03212] [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
S. Rychkov and P. Yvernay, Remarks on the convergence properties of the conformal block expansion, Phys. Lett. B 753 (2016) 682 [arXiv:1510.08486] [INSPIRE].
M. Gillioz, X. Lu, M.A. Luty and G. Mikaberidze, Convergent momentum-space OPE and bootstrap equations in conformal field theory, JHEP 03 (2020) 102 [arXiv:1912.05550] [INSPIRE].
G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
J. Penedones, J.A. Silva and A. Zhiboedov, Nonperturbative Mellin amplitudes: existence, properties, applications, JHEP 08 (2020) 031 [arXiv:1912.11100] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
M. Hogervorst, H. Osborn and S. Rychkov, Diagonal limit for conformal blocks in d dimensions, JHEP 08 (2013) 014 [arXiv:1305.1321] [INSPIRE].
G.P. Korchemsky, Exact scattering amplitudes in conformal fishnet theory, JHEP 08 (2019) 028 [arXiv:1812.06997] [INSPIRE].
T. Okuda and J. Penedones, String scattering in fiat space and a scaling limit of Yang-Mills correlators, Phys. Rev. D 83 (2011) 086001 [arXiv:1002.2641] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, JHEP 02 (2019) 162 [arXiv:1803.10233] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II: natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
M.F. Paulos, Analytic functional bootstrap for CFTs in d > 1, JHEP 04 (2020) 093 [arXiv:1910.08563] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, A basis of analytic functionals for CFTs in general dimension, arXiv:1910.12855 [INSPIRE].
D. Carmi and S. Caron-Huot, A conformal dispersion relation: correlations from absorption, JHEP 09 (2020) 009 [arXiv:1910.12123] [INSPIRE].
L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
G. Sommer, Present state of rigorous analytic properties of scattering amplitudes, Fortsch. Phys. 18 (1970) 577 [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
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Gillioz, M., Meineri, M. & Penedones, J. A scattering amplitude in Conformal Field Theory. J. High Energ. Phys. 2020, 139 (2020). https://doi.org/10.1007/JHEP11(2020)139
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DOI: https://doi.org/10.1007/JHEP11(2020)139