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 h0| T {φ(x1)φ(x2)φ(x3)φ(x4)} |0i “scattering amplitudes” unitarity → Wightman functions that satisfy both crossing symmetry h0| φ(x1)φ(x2)φ(x3)φ(x4) |0i and unitarity? |P {z } = h0|φ(x1)φ(x2)|OihO|φ(x3)φ(x4)|0i 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 h0| T {φ(x1)φ(x2)φ(x3)φ(x4)} |0i “scattering amplitudes” → Wightman functions that satisfy both crossing symmetry h0| φ(x1)φ(x2)φ(x3)φ(x4) |0i and unitarity? |P {z } = h0|φ(x1)φ(x2)|OihO|φ(x3)φ(x4)|0i 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 h0| φ(x1)φ(x2)φ(x3)φ(x4) |0i and unitarity? |P {z } = h0|φ(x1)φ(x2)|OihO|φ(x3)φ(x4)|0i 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 h0| T {φ(x1)φ(x2)φ(x3)φ(x4)} |0i 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 h0| T {φ(x1)φ(x2)φ(x3)φ(x4)} |0i unitarity → Wightman functions h0| φ(x1)φ(x2)φ(x3)φ(x4) |0i |P {z } = h0|φ(x1)φ(x2)|OihO|φ(x3)φ(x4)|0i 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 h0| T {φ(x1)φ(x2)φ(x3)φ(x4)} |0i “scattering amplitudes” unitarity → Wightman functions that satisfy both crossing symmetry h0| φ(x1)φ(x2)φ(x3)φ(x4) |0i and unitarity? |P {z } = h0|φ(x1)φ(x2)|OihO|φ(x3)φ(x4)|0i 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) |φ1, p1i ∼ C12O |O, p1 + p2i O
LSZ reduction formula
Why do scattering amplitude satisfy unitarity relations? They are Wightman functions:
2 2 lim (p − m ) h0| T {φ1(p1) ··· φn(pn)} |0i 2 2 1 1 p1→m1 2 2 = lim (p − m ) h0| T {φ2(p2) ··· φn(pn)}φ1(p1) |0i 2 2 1 1 p1→m1
≡ h0| T {φ2(p2) ··· φn(pn)} |φ1, 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) |φ1, p1i ∼ C12O |O, p1 + p2i O
LSZ reduction formula
Why do scattering amplitude satisfy unitarity relations? They are Wightman functions:
2 d/2−∆1 lim (p ) h0| T {φ1(p1) ··· φn(pn)} |0i 2 1 p1→0
2 d/2−∆1 = lim (p ) h0| T {φ2(p2) ··· φn(pn)}φ1(p1) |0i 2 1 p1→0
≡ h0| T {φ2(p2) ··· φn(pn)} |φ1, 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 ) h0| T {φ1(p1) ··· φn(pn)} |0i 2 1 p1→0
2 d/2−∆1 = lim (p ) h0| T {φ2(p2) ··· φn(pn)}φ1(p1) |0i 2 1 p1→0
≡ h0| T {φ2(p2) ··· φn(pn)} |φ1, 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) |φ1, p1i ∼ C12O |O, 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 h0| φ1(p1)φ2(p2)φ3(p3) |0i = 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 ∞ ∞ ∆1−∆2+∆3 ∆1+∆2+∆3−d X X 2 2 F (x, y) = n+m n+m xnym. 4 d−2 d−2 n=0 m=0 n!m! ∆1 − 2 n ∆3 − 2 m
analyticity of F4 ⇔ operator product expansion
Conformal correlators in momentum space
The Wightman 2-point function:
h0| φ(p0)φ(p) |0i =Θ( p0)Θ(p2) δd(p0 + p)( p2)∆−d/2 | {z } | {z } only support for 1 for free field physical momenta p2
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 6 / 10 Conformal correlators in momentum space
The Wightman 2-point function:
h0| φ(p0)φ(p) |0i =Θ( p0)Θ(p2) δd(p0 + p)( p2)∆−d/2 | {z } | {z } only support for 1 for free field physical momenta p2 The Wightman 3-point function:
0 2 0 2 d h0| φ1(p1)φ2(p2)φ3(p3) |0i = 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 ∞ ∞ ∆1−∆2+∆3 ∆1+∆2+∆3−d X X 2 2 F (x, y) = n+m n+m xnym. 4 d−2 d−2 n=0 m=0 n!m! ∆1 − 2 n ∆3 − 2 m
analyticity of F4 ⇔ operator product expansion Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 6 / 10 The “CFT amplitude”
4 ! Y 2 d/2−∆φ M(s, θ) ≡ lim (−pi − i) h0| T {φ(p1)φ(p2)φ(p3)φ(p4)} |0i p2→0 i=1 i
2 s: center-of-mass energy s = (p1 + p2) 1 2 θ: scattering angle t = − 2 s(1 − cos θ) = (p1 + p3) Conformal block expansion Sum over primary operators O with scaling dimension ∆ and spin `:
Γ ∆ − d−2 Γ(∆ + `)Γ(∆ + ` − 1) d/2−2∆φ X 2 2 M(s, θ) = s λφφO 2` ∆+` 4 O | {z } 2 Γ 2 Γ(∆ − 1) OPE coeff. h i iπ(2∆φ−∆+`) × e − 1 × g∆,`(cos θ) | {z } | {z } polynomial of deg. ` zeroes when ∆=2∆φ+2n MG 1807.07003
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 7 / 10 2 1 19 2 [φ ], with ∆[φ2] = 2∆φ + 3 + 162 + ... T µν and other double-trace operators of the form [φ∂`φ] 1 2 ∆[φ∂`φ] = 2∆φ + ` − 9`(`+1) + ... More operators at order 3 and higher
g 1 1 at the fixed point where (4π)2 = 3 + ..., with ∆φ = 1 − 2 + ...
Operators contributing to the conformal block expansion:
An example: φ4 theory in d = 4 − dimensions
g + + + + ...
g 1h i = −ig 1 − log(−s − i) + log(−t − i) + log(−u − i) + ... (4π)2 2
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 8 / 10 2 1 19 2 [φ ], with ∆[φ2] = 2∆φ + 3 + 162 + ... T µν and other double-trace operators of the form [φ∂`φ] 1 2 ∆[φ∂`φ] = 2∆φ + ` − 9`(`+1) + ... More operators at order 3 and higher
Operators contributing to the conformal block expansion:
An example: φ4 theory in d = 4 − dimensions
g + + + + ...
(4π)2 1 (sin θ)2 = −i sd/2−2∆φ + log − iπ 2 + ... 3 6 4
g 1 1 at the fixed point where (4π)2 = 3 + ..., with ∆φ = 1 − 2 + ...
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 8 / 10 T µν and other double-trace operators of the form [φ∂`φ] 1 2 ∆[φ∂`φ] = 2∆φ + ` − 9`(`+1) + ... More operators at order 3 and higher
An example: φ4 theory in d = 4 − dimensions
g + + + + ...
(4π)2 1 (sin θ)2 = −i sd/2−2∆φ + log − iπ 2 + ... 3 6 4
g 1 1 at the fixed point where (4π)2 = 3 + ..., with ∆φ = 1 − 2 + ...
Operators contributing to the conformal block expansion: 2 1 19 2 [φ ], with ∆[φ2] = 2∆φ + 3 + 162 + ...
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 8 / 10 More operators at order 3 and higher
An example: φ4 theory in d = 4 − dimensions
g + + + + ...
(4π)2 1 (sin θ)2 = −i sd/2−2∆φ + log − iπ 2 + ... 3 6 4
g 1 1 at the fixed point where (4π)2 = 3 + ..., with ∆φ = 1 − 2 + ...
Operators contributing to the conformal block expansion: 2 1 19 2 [φ ], with ∆[φ2] = 2∆φ + 3 + 162 + ... T µν and other double-trace operators of the form [φ∂`φ] 1 2 ∆[φ∂`φ] = 2∆φ + ` − 9`(`+1) + ...
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 8 / 10 An example: φ4 theory in d = 4 − dimensions
g + + + + ...
(4π)2 1 (sin θ)2 = −i sd/2−2∆φ + log − iπ 2 + ... 3 6 4
g 1 1 at the fixed point where (4π)2 = 3 + ..., with ∆φ = 1 − 2 + ...
Operators contributing to the conformal block expansion: 2 1 19 2 [φ ], with ∆[φ2] = 2∆φ + 3 + 162 + ... T µν and other double-trace operators of the form [φ∂`φ] 1 2 ∆[φ∂`φ] = 2∆φ + ` − 9`(`+1) + ... More operators at order 3 and higher
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 8 / 10 Crossing symmetries s ↔ t and s ↔ u relate the physical region (s > 0, t < 0, u < 0) to unphysical regions ⇒ requires analytic continuation ⇒ outside the domain of convergence of the s-channel OPE! Sufficient to apply the LSZ reduction formula to 3 operators, 2 keeping the 4th momentum off-shell, with s + t + u = p4:
3 ! Y 2 d/2−∆φ M(s, t, u) ≡ lim (−pi − i) h0| T {φ(p1)φ(p2)φ(p3)φ(p4)} |0i p2→0 i=1 i
2 With p4 < 0 (space-like), there is an “Euclidean” region s, t, u < 0 in which all 3 OPE channels converge
Crossing symmetry and momentum-space bootstrap
Can we use this CFT amplitude to bootstrap a theory? Crossing symmetry t ↔ u is trivially satisfied by each individual conformal block: M(s, cos θ) = M(s, − cos θ)
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 9 / 10 Sufficient to apply the LSZ reduction formula to 3 operators, 2 keeping the 4th momentum off-shell, with s + t + u = p4:
3 ! Y 2 d/2−∆φ M(s, t, u) ≡ lim (−pi − i) h0| T {φ(p1)φ(p2)φ(p3)φ(p4)} |0i p2→0 i=1 i
2 With p4 < 0 (space-like), there is an “Euclidean” region s, t, u < 0 in which all 3 OPE channels converge
Crossing symmetry and momentum-space bootstrap
Can we use this CFT amplitude to bootstrap a theory? Crossing symmetry t ↔ u is trivially satisfied by each individual conformal block: M(s, cos θ) = M(s, − cos θ) Crossing symmetries s ↔ t and s ↔ u relate the physical region (s > 0, t < 0, u < 0) to unphysical regions ⇒ requires analytic continuation ⇒ outside the domain of convergence of the s-channel OPE!
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 9 / 10 Crossing symmetry and momentum-space bootstrap
Can we use this CFT amplitude to bootstrap a theory? Crossing symmetry t ↔ u is trivially satisfied by each individual conformal block: M(s, cos θ) = M(s, − cos θ) Crossing symmetries s ↔ t and s ↔ u relate the physical region (s > 0, t < 0, u < 0) to unphysical regions ⇒ requires analytic continuation ⇒ outside the domain of convergence of the s-channel OPE! Sufficient to apply the LSZ reduction formula to 3 operators, 2 keeping the 4th momentum off-shell, with s + t + u = p4:
3 ! Y 2 d/2−∆φ M(s, t, u) ≡ lim (−pi − i) h0| T {φ(p1)φ(p2)φ(p3)φ(p4)} |0i p2→0 i=1 i
2 With p4 < 0 (space-like), there is an “Euclidean” region s, t, u < 0 in which all 3 OPE channels converge
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 9 / 10 Future applications: Extraction of the CFT data in Banks-Zaks-type theories Litim, Sannino ’14 Antipin, Mojaza, Sannino ’11 + several other CP3 members A conformal bootstrap in momentum space?
Conclusions and outlook
Our results: Conformal correlation functions have power-law divergences 2 in the limit pi → 0: Q 2 ∆φ−d/2 hφ(p1)φ(p2)φ(p3)φ(p4)i ≈ (−pi − i) M(s, t, u) i The “amplitude” M admits a conformal block expansion, i.e. it can be expressed in terms of conformal data {∆i, λijk} Generalization of the concept of amplitude with one off-shell leg, leading to a set of non-trivial crossing equations
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 10 / 10 Conclusions and outlook
Our results: Conformal correlation functions have power-law divergences 2 in the limit pi → 0: Q 2 ∆φ−d/2 hφ(p1)φ(p2)φ(p3)φ(p4)i ≈ (−pi − i) M(s, t, u) i The “amplitude” M admits a conformal block expansion, i.e. it can be expressed in terms of conformal data {∆i, λijk} Generalization of the concept of amplitude with one off-shell leg, leading to a set of non-trivial crossing equations
Future applications: Extraction of the CFT data in Banks-Zaks-type theories Litim, Sannino ’14 Antipin, Mojaza, Sannino ’11 + several other CP3 members A conformal bootstrap in momentum space?
Marc Gillioz (EPFL) Scattering amplitudes in CFT May 21, 2019 10 / 10