Conformal two-point correlation functions from the operator product expansion. (English) Zbl 1436.81063
Summary: We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in [J.-F. Fortin and W. Skiba, “A recipe for conformal blocks”, Preprint, arXiv:1905.000361; “New methods for conformal correlation functions”, Prperint, arXiv:1905.00434]. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.
MSC:
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 83A05 | Special relativity |
| 22E15 | General properties and structure of real Lie groups |
| 62P35 | Applications of statistics to physics |
Keywords:
operator product expansion (OPE); conformal field theory; conformal and W symmetry; field theories in higher dimensions; Lorentz representationSoftware:
CFTs4DReferences:
| [1] | J.-F. Fortin and W. Skiba, A recipe for conformal blocks, arXiv:1905.00036 [INSPIRE]. |
| [2] | J.-F. Fortin and W. Skiba, New Methods for Conformal Correlation Functions, arXiv:1905.00434 [INSPIRE]. |
| [3] | F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE]. · Zbl 1097.81734 |
| [4] | F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE]. · Zbl 1097.81735 |
| [5] | F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE]. · Zbl 1097.81735 |
| [6] | R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP12 (2008) 031 [arXiv:0807.0004] [INSPIRE]. · Zbl 1329.81324 |
| [7] | S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys.76 (1973) 161 [INSPIRE]. |
| [8] | A.M. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz.66 (1974) 23 [Sov. Phys. JETP39 (1974) 9] [INSPIRE]. |
| [9] | 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]. |
| [10] | P.A.M. Dirac, Wave equations in conformal space, Annals Math.37 (1936) 429 [INSPIRE]. · Zbl 0014.08004 |
| [11] | G. Mack, Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory, Commun. Math. Phys.53 (1977) 155 [INSPIRE]. |
| [12] | R. Rattazzi, S. Rychkov and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev.D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE]. · Zbl 1206.81116 |
| [13] | D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP05 (2011) 017 [arXiv:1009.2087] [INSPIRE]. · Zbl 1296.81067 |
| [14] | M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE]. · Zbl 1306.81207 |
| [15] | D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP04 (2014) 146 [arXiv:1204.3894] [INSPIRE]. · Zbl 1333.83125 |
| [16] | T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP10 (2016) 141 [arXiv:1601.07904] [INSPIRE]. · Zbl 1390.83114 |
| [17] | D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A Proof of the Conformal Collider Bounds, JHEP06 (2016) 111 [arXiv:1603.03771] [INSPIRE]. · Zbl 1388.81048 |
| [18] | T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP07 (2017) 066 [arXiv:1610.05308] [INSPIRE]. · Zbl 1380.81327 |
| [19] | N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP12 (2017) 049 [arXiv:1610.09378] [INSPIRE]. · Zbl 1383.83024 |
| [20] | G.F. Cuomo, D. Karateev and P. Kravchuk, General Bootstrap Equations in 4D CFTs, JHEP01 (2018) 130 [arXiv:1705.05401] [INSPIRE]. · Zbl 1384.81094 |
| [21] | P. Cvitanović, Group theory: Birdtracks, Lie’s and exceptional groups, Princeton University Press, Princeton U.S.A. (2008) and online at http://press.princeton.edu/titles/8839.html. · Zbl 1152.22001 |
| [22] | M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP02 (2015) 151 [arXiv:1411.7351] [INSPIRE]. · Zbl 1388.53102 |
| [23] | M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP07 (2016) 018 [arXiv:1603.05551] [INSPIRE]. · Zbl 1388.81798 |
| [24] | D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight Shifting Operators and Conformal Blocks, JHEP02 (2018) 081 [arXiv:1706.07813] [INSPIRE]. · Zbl 1387.81323 |
| [25] | M.S. Costa and T. Hansen, AdS Weight Shifting Operators, JHEP09 (2018) 040 [arXiv:1805.01492] [INSPIRE]. · Zbl 1398.81207 |
| [26] | F. Rejon-Barrera and D. Robbins, Scalar-Vector Bootstrap, JHEP01 (2016) 139 [arXiv:1508.02676] [INSPIRE]. |
| [27] | H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys.231 (1994) 311 [hep-th/9307010] [INSPIRE]. · Zbl 0795.53073 |
| [28] | R.F. Streater and A.S. Wightman, PCT, spin and statistics, and all that, Advanced Book Classics, volume 231, Addison-Wesley, Redwood City U.S.A. (1989). · Zbl 0704.53058 |
| [29] | E.S. Fradkin and M.Y. Palchik, Conformal quantum field theory in D-dimensions, Mathematics and its Applications, volume 376, Springer (1996). · Zbl 0896.70001 |
| [30] | S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics, Springer, Cham Switzerland (2016), [arXiv:1601.05000] [INSPIRE]. |
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