Built on

D. Anninos, V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, Cosmological Shapes of Higher-Spin Gravity
Original
1902
Earlier work this paper cites.
H. Isono, T. Noumi, and T. Takeuchi, Momentum space conformal three-point functions of conserved currents and a general spinning operator
Original
1903
Earlier work this paper cites.
C. Corianò and M. M. Maglio, On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
Original
1903
Earlier work this paper cites.
S. Albayrak, C. Chowdhury, and S. Kharel, New relation for AdS amplitudes
Original
1904
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C. Sleight, A Mellin Space Approach to Cosmological Correlators
Original
1906
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C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space
Original
1907
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S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes. Part II. Gravitons
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1908
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H. Isono, T. Noumi, and G. Shiu, Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension
Original
1908
Earlier work this paper cites.
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space
Original
1908
Earlier work this paper cites.
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space
Original
1909
Earlier work this paper cites.
A. Bzowski, P. McFadden, and K. Skenderis, Conformal n n -point functions in momentum space
Original
1910
Earlier work this paper cites.
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds
Original
1910
Earlier work this paper cites.
M. Chernodub, C. Corianò, and M. M. Maglio, Anomalous Gravitational TTT Vertex, Temperature Inhomogeneity, and Pressure Anisotropy
Original
1910
Earlier work this paper cites.
N. Anand, Z. U. Khandker, and M. T. Walters, Momentum space CFT correlators for Hamiltonian truncation
Original
1911
Earlier work this paper cites.
A. E. Lipstein and P. McFadden, Double copy structure and the flat space limit of conformal correlators in even dimensions
Original
1912
Earlier work this paper cites.
M. Gillioz, X. Lu, M. A. Luty, and G. Mikaberidze, Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory
Original
1912
Earlier work this paper cites.
C. Sleight and M. Taronna, The Unique Polyakov Blocks
Original
1912
Earlier work this paper cites.
C. Corianò, M. M. Maglio, and D. Theofilopoulos, Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
Original
1912
Earlier work this paper cites.
S. Bedrosian, Converse of the Star-Mesh Transformation
1961
Earlier work this paper cites.
A. Bzowski, P. McFadden, and K. Skenderis, Holographic predictions for cosmological 3-point functions
Original
1967
Earlier work this paper cites.
A. M. Polyakov, Conformal symmetry of critical fluctuations
1970
Earlier work this paper cites.
K. Symanzik, On Calculations in conformal invariant field theories
1972
Earlier work this paper cites.
M. van Lier and R. Otten, Planarization by transformation
1973
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, Covariant expansion of the conformal four-point function
1973
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion
1973
Earlier work this paper cites.
A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory
1974
Earlier work this paper cites.
C.U.P., 1990
R. Ward and R. Wells, Jr, The Klein correspondence. In Twistor Geometry and Field Theory (Cambridge Monographs on Mathematical Physics, pp. 7-67) · 1990
Earlier work this paper cites.
E. Boos and A. I. Davydychev, A Method of evaluating massive Feynman integrals
1991
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A. I. Davydychev, Recursive algorithm of evaluating vertex type Feynman integrals
1992
Earlier work this paper cites.
Springer, N.Y., 1997
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory · 1997
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Cambridge University Press, 1998
A. Beutelspacher and U. Rosenbaum, Projective Geometry: from foundations to applications · 1998
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2020

Conformal correlators as simplex integrals in momentum space

Bzowski, Adam, McFadden, Paul, Skenderis, Kostas

Understand

We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension.

The solution is given in terms of integrals over $(n-1)$-simplices in momentum space.
The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables.
The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex.

Built on

D. Anninos, V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, Cosmological Shapes of Higher-Spin Gravity
Original
1902
Earlier work this paper cites.
H. Isono, T. Noumi, and T. Takeuchi, Momentum space conformal three-point functions of conserved currents and a general spinning operator
Original
1903
Earlier work this paper cites.
C. Corianò and M. M. Maglio, On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
Original
1903
Earlier work this paper cites.
S. Albayrak, C. Chowdhury, and S. Kharel, New relation for AdS amplitudes
Original
1904
Earlier work this paper cites.
C. Sleight, A Mellin Space Approach to Cosmological Correlators
Original
1906
Earlier work this paper cites.
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space
Original
1907
Earlier work this paper cites.
S. Albayrak and S. Kharel, Towards the higher point holographic momentum space amplitudes. Part II. Gravitons
Original
1908
Earlier work this paper cites.
H. Isono, T. Noumi, and G. Shiu, Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension
Original
1908
Earlier work this paper cites.
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space
Original
1908
Earlier work this paper cites.
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space
Original
1909
Earlier work this paper cites.
A. Bzowski, P. McFadden, and K. Skenderis, Conformal n n -point functions in momentum space
Original
1910
Earlier work this paper cites.
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, and G. L. Pimentel, The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds
Original
1910
Earlier work this paper cites.
M. Chernodub, C. Corianò, and M. M. Maglio, Anomalous Gravitational TTT Vertex, Temperature Inhomogeneity, and Pressure Anisotropy
Original
1910
Earlier work this paper cites.
N. Anand, Z. U. Khandker, and M. T. Walters, Momentum space CFT correlators for Hamiltonian truncation
Original
1911
Earlier work this paper cites.
A. E. Lipstein and P. McFadden, Double copy structure and the flat space limit of conformal correlators in even dimensions
Original
1912
Earlier work this paper cites.
M. Gillioz, X. Lu, M. A. Luty, and G. Mikaberidze, Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory
Original
1912
Earlier work this paper cites.
C. Sleight and M. Taronna, The Unique Polyakov Blocks
Original
1912
Earlier work this paper cites.
C. Corianò, M. M. Maglio, and D. Theofilopoulos, Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
Original
1912
Earlier work this paper cites.
S. Bedrosian, Converse of the Star-Mesh Transformation
1961
Earlier work this paper cites.
A. Bzowski, P. McFadden, and K. Skenderis, Holographic predictions for cosmological 3-point functions
Original
1967
Earlier work this paper cites.
A. M. Polyakov, Conformal symmetry of critical fluctuations
1970
Earlier work this paper cites.
K. Symanzik, On Calculations in conformal invariant field theories
1972
Earlier work this paper cites.
M. van Lier and R. Otten, Planarization by transformation
1973
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, G. Parisi, and R. Gatto, Covariant expansion of the conformal four-point function
1973
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion
1973
Earlier work this paper cites.
A. M. Polyakov, Nonhamiltonian approach to conformal quantum field theory
1974
Earlier work this paper cites.
C.U.P., 1990
R. Ward and R. Wells, Jr, The Klein correspondence. In Twistor Geometry and Field Theory (Cambridge Monographs on Mathematical Physics, pp. 7-67) · 1990
Earlier work this paper cites.
E. Boos and A. I. Davydychev, A Method of evaluating massive Feynman integrals
1991
Earlier work this paper cites.
A. I. Davydychev, Recursive algorithm of evaluating vertex type Feynman integrals
1992
Earlier work this paper cites.
Springer, N.Y., 1997
P. Di Francesco, P. Mathieu, and D. Senechal, Conformal field theory · 1997
Earlier work this paper cites.
Cambridge University Press, 1998
A. Beutelspacher and U. Rosenbaum, Projective Geometry: from foundations to applications · 1998
Earlier work this paper cites.

Similar

E. D’Hoker, D. Z. Freedman, and L. Rastelli, AdS / CFT four point functions: How to succeed at z integrals without really trying
1999
Cited alongside, same era.
E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, Graviton exchange and complete four point functions in the AdS / CFT correspondence
1999
Cited alongside, same era.
hep-th/0106109
E. Witten, Quantum gravity in de Sitter space · 2001
Cited alongside, same era.
A. Strominger, The dS / CFT correspondence
2001
Cited alongside, same era.
A. Strominger, Inflation and the dS / CFT correspondence
2001
Cited alongside, same era.
F. A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion
2001
Cited alongside, same era.