Scattering Amplitudes in Conformal Field Theory Marc Gillioz Theoretical Particle Physics Laboratory, EPFL, Lausanne, Switzerland work in progress with Marco Meineri and Jo~aoPenedones Origin of Mass 2019 Odense May 21, 2019 Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 1 / 10 Yet simple enough that it can (sometimes) be completely solved ! even in the strong-coupling regime Examples of CFTs: Free theories The Ising and O(N) models in 3 dimensions N = 4 supersymmetric Yang-Mills QCD-like theories in the conformal window No particles, so what does it mean to speak of scattering amplitudes? Conformal field theory: motivations Conformal field theory is: Omniscient in physics, in the low- and high-energy regimes of QFT ! related to QFT by conformal perturbation theory Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 2 / 10 Examples of CFTs: Free theories The Ising and O(N) models in 3 dimensions N = 4 supersymmetric Yang-Mills QCD-like theories in the conformal window No particles, so what does it mean to speak of scattering amplitudes? Conformal field theory: motivations Conformal field theory is: Omniscient in physics, in the low- and high-energy regimes of QFT ! related to QFT by conformal perturbation theory Yet simple enough that it can (sometimes) be completely solved ! even in the strong-coupling regime Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 2 / 10 No particles, so what does it mean to speak of scattering amplitudes? Conformal field theory: motivations Conformal field theory is: Omniscient in physics, in the low- and high-energy regimes of QFT ! related to QFT by conformal perturbation theory Yet simple enough that it can (sometimes) be completely solved ! even in the strong-coupling regime Examples of CFTs: Free theories The Ising and O(N) models in 3 dimensions N = 4 supersymmetric Yang-Mills QCD-like theories in the conformal window Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 2 / 10 Conformal field theory: motivations Conformal field theory is: Omniscient in physics, in the low- and high-energy regimes of QFT ! related to QFT by conformal perturbation theory Yet simple enough that it can (sometimes) be completely solved ! even in the strong-coupling regime Examples of CFTs: Free theories The Ising and O(N) models in 3 dimensions N = 4 supersymmetric Yang-Mills QCD-like theories in the conformal window No particles, so what does it mean to speak of scattering amplitudes? Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 2 / 10 Some analytical methods: Large charge expansions Hellerman, Orlando, Reffert, Watanabe '15 Alvarez-Gaume, Loukas et al. '17, Monin, Pirtskhalava, Rattazzi, Seibold '17 Large spin expansion Alday '16, Cuomo, de la Fuente et al. '17 Light-cone bootstrap and OPE inversion formulæ Caron-Huot '17 Simmons-Duffin, Stanford, Witten '17, Mukhamedzhanov, Zhiboedov '18 The conformal bootstrap ∆O ­ Solving conformal field theory 5.5 based on symmetry only. 5 4.5 Success story, but: 4 mostly numerical 3.5 no positive results in d ≥ 4 3 for non-SUSY theories 2.5 limited reach close to 2 ∆ mean free theory 1 1.2 1.4 1.6 1.8 Poland, Simmons-Duffin, Vichi 1109.5176 Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 3 / 10 The conformal bootstrap ∆O ­ Solving conformal field theory 5.5 based on symmetry only. 5 4.5 Success story, but: 4 mostly numerical 3.5 no positive results in d ≥ 4 3 for non-SUSY theories 2.5 limited reach close to 2 ∆ mean free theory 1 1.2 1.4 1.6 1.8 Poland, Simmons-Duffin, Vichi 1109.5176 Some analytical methods: Large charge expansions Hellerman, Orlando, Reffert, Watanabe '15 Alvarez-Gaume, Loukas et al. '17, Monin, Pirtskhalava, Rattazzi, Seibold '17 Large spin expansion Alday '16, Cuomo, de la Fuente et al. '17 Light-cone bootstrap and OPE inversion formulæ Caron-Huot '17 Simmons-Duffin, Stanford, Witten '17, Mukhamedzhanov, Zhiboedov '18 Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 3 / 10 A difficulty with Lorentzian CFT 2 ingredients of the bootstrap: crossing symmetry ! time-ordered products Can we define h0j T fφ(x1)φ(x2)φ(x3)φ(x4)g j0i \scattering amplitudes" unitarity ! Wightman functions that satisfy both crossing symmetry h0j φ(x1)φ(x2)φ(x3)φ(x4) j0i and unitarity? |P {z } = h0jφ(x1)φ(x2)jOihOjφ(x3)φ(x4)j0i O Scattering amplitudes and conformal field theory? Conformal field theory in momentum space: translation symmetry is trivial (≡ momentum conservation) special conformal transformations realized non-linearly ) difficult disconnected diagrams naturally separated (mean free field theory has a trivial S-matrix) Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 4 / 10 ! time-ordered products Can we define h0j T fφ(x1)φ(x2)φ(x3)φ(x4)g j0i \scattering amplitudes" ! Wightman functions that satisfy both crossing symmetry h0j φ(x1)φ(x2)φ(x3)φ(x4) j0i and unitarity? |P {z } = h0jφ(x1)φ(x2)jOihOjφ(x3)φ(x4)j0i O Scattering amplitudes and conformal field theory? Conformal field theory in momentum space: translation symmetry is trivial (≡ momentum conservation) special conformal transformations realized non-linearly ) difficult disconnected diagrams naturally separated (mean free field theory has a trivial S-matrix) A difficulty with Lorentzian CFT 2 ingredients of the bootstrap: crossing symmetry unitarity Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 4 / 10 Can we define \scattering amplitudes" ! Wightman functions that satisfy both crossing symmetry h0j φ(x1)φ(x2)φ(x3)φ(x4) j0i and unitarity? |P {z } = h0jφ(x1)φ(x2)jOihOjφ(x3)φ(x4)j0i O Scattering amplitudes and conformal field theory? Conformal field theory in momentum space: translation symmetry is trivial (≡ momentum conservation) special conformal transformations realized non-linearly ) difficult disconnected diagrams naturally separated (mean free field theory has a trivial S-matrix) A difficulty with Lorentzian CFT 2 ingredients of the bootstrap: crossing symmetry ! time-ordered products h0j T fφ(x1)φ(x2)φ(x3)φ(x4)g j0i unitarity Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 4 / 10 Can we define \scattering amplitudes" that satisfy both crossing symmetry and unitarity? Scattering amplitudes and conformal field theory? Conformal field theory in momentum space: translation symmetry is trivial (≡ momentum conservation) special conformal transformations realized non-linearly ) difficult disconnected diagrams naturally separated (mean free field theory has a trivial S-matrix) A difficulty with Lorentzian CFT 2 ingredients of the bootstrap: crossing symmetry ! time-ordered products h0j T fφ(x1)φ(x2)φ(x3)φ(x4)g j0i unitarity ! Wightman functions h0j φ(x1)φ(x2)φ(x3)φ(x4) j0i |P {z } = h0jφ(x1)φ(x2)jOihOjφ(x3)φ(x4)j0i O Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 4 / 10 Scattering amplitudes and conformal field theory? Conformal field theory in momentum space: translation symmetry is trivial (≡ momentum conservation) special conformal transformations realized non-linearly ) difficult disconnected diagrams naturally separated (mean free field theory has a trivial S-matrix) A difficulty with Lorentzian CFT 2 ingredients of the bootstrap: crossing symmetry ! time-ordered products Can we define h0j T fφ(x1)φ(x2)φ(x3)φ(x4)g j0i \scattering amplitudes" unitarity ! Wightman functions that satisfy both crossing symmetry h0j φ(x1)φ(x2)φ(x3)φ(x4) j0i and unitarity? |P {z } = h0jφ(x1)φ(x2)jOihOjφ(x3)φ(x4)j0i O Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 4 / 10 A version of the LSZ reduction formula can be applied to conformal d correlators of operators with scaling dimensions ∆ < 2 The subtlety in CFT: no Fock-space factorization of the Hilbert space ! but state/operator correspondence instead: P φ2(p2) jφ1; p1i ∼ C12O jO; p1 + p2i O LSZ reduction formula Why do scattering amplitude satisfy unitarity relations? They are Wightman functions: 2 2 lim (p − m ) h0j T fφ1(p1) ··· φn(pn)g j0i 2 2 1 1 p1!m1 2 2 = lim (p − m ) h0j T fφ2(p2) ··· φn(pn)gφ1(p1) j0i 2 2 1 1 p1!m1 ≡ h0j T fφ2(p2) ··· φn(pn)g jφ1; p1i finite! Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 5 / 10 The subtlety in CFT: no Fock-space factorization of the Hilbert space ! but state/operator correspondence instead: P φ2(p2) jφ1; p1i ∼ C12O jO; p1 + p2i O LSZ reduction formula Why do scattering amplitude satisfy unitarity relations? They are Wightman functions: 2 d=2−∆1 lim (p ) h0j T fφ1(p1) ··· φn(pn)g j0i 2 1 p1!0 2 d=2−∆1 = lim (p ) h0j T fφ2(p2) ··· φn(pn)gφ1(p1) j0i 2 1 p1!0 ≡ h0j T fφ2(p2) ··· φn(pn)g jφ1; p1i finite! A version of the LSZ reduction formula can be applied to conformal d correlators of operators with scaling dimensions ∆ < 2 Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 5 / 10 LSZ reduction formula Why do scattering amplitude satisfy unitarity relations? They are Wightman functions: 2 d=2−∆1 lim (p ) h0j T fφ1(p1) ··· φn(pn)g j0i 2 1 p1!0 2 d=2−∆1 = lim (p ) h0j T fφ2(p2) ··· φn(pn)gφ1(p1) j0i 2 1 p1!0 ≡ h0j T fφ2(p2) ··· φn(pn)g jφ1; p1i finite! A version of the LSZ reduction formula can be applied to conformal d correlators of operators with scaling dimensions ∆ < 2 The subtlety in CFT: no Fock-space factorization of the Hilbert space ! but state/operator correspondence instead: P φ2(p2) jφ1; p1i ∼ C12O jO; p1 + p2i O Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 5 / 10 The Wightman 3-point function: 0 2 0 2 d h0j φ1(p1)φ2(p2)φ3(p3) j0i = Θ(−p1)Θ(p1)Θ(p3)Θ(p3) δ (p1 + p2 + p3) 2 ∆1−d=2 2 ∆3−d=2 2 2 (p1) (p3) p1 p3 × λ123 F4 ; 2 (∆1−∆2+∆3)=2 2 2 (p2) p2 p2 with Appell's double hypergeometric