Conformal 3-point functions and the Lorentzian OPE in momentum space. (English) Zbl 1470.81053
Summary: In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell \(F_4\) type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T05 | Axiomatic quantum field theory; operator algebras |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 81T11 | Higher spin theories |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 53Z05 | Applications of differential geometry to physics |
Software:
DLMFReferences:
| [1] | Polyakov, AM, Conformal symmetry of critical fluctuations, JETP Lett., 12, 381-383 (1970) |
| [2] | Polyakov, AM, Conformal symmetry of critical fluctuations, Pisma Zh. Eksp. Teor. Fiz., 12, 538 (1970) |
| [3] | Schreier, EJ, Conformal symmetry and three-point functions, Phys. Rev. D, 3, 980-988 (1971) |
| [4] | Osborn, H.; Petkou, AC, Implications of conformal invariance in field theories for general dimensions, Ann. Phys., 231, 311-362 (1994) · Zbl 0795.53073 |
| [5] | Erdmenger, J.; Osborn, H., Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B, 483, 431-474 (1997) · Zbl 0925.81340 |
| [6] | Giombi, S.; Prakash, S.; Yin, X., A note on CFT Correlators in three dimensions, JHEP, 07, 105 (2013) · Zbl 1342.81492 |
| [7] | Costa, MS; Penedones, J.; Poland, D.; Rychkov, S., Spinning conformal correlators, JHEP, 11, 071 (2011) · Zbl 1306.81207 |
| [8] | Maldacena, J.; Zhiboedov, A., Constraining conformal field theories with a higher spin symmetry, J. Phys. A, 46, 214011 (2013) · Zbl 1339.81089 |
| [9] | Zhiboedov, A.: A note on three-point functions of conserved currents. arXiv:1206.6370 [hep-th] |
| [10] | Elkhidir, E.; Karateev, D.; Serone, M., General three-point functions in 4D CFT, JHEP, 01, 133 (2015) |
| [11] | Fortin, J-F; Prilepina, V.; Skiba, W., Conformal two-point correlation functions from the operator product expansion, JHEP, 04, 114 (2020) · Zbl 1436.81063 |
| [12] | Fortin, J.-F., Prilepina, V., Skiba, W.: Conformal three-point correlation functions from the operator product expansion. arXiv:1907.08599 [hep-th] · Zbl 1436.81063 |
| [13] | Karateev, D.; Kravchuk, P.; Simmons-Duffin, D., Weight shifting operators and conformal blocks, JHEP, 02, 081 (2018) · Zbl 1387.81323 |
| [14] | Karateev, D.; Kravchuk, P.; Simmons-Duffin, D., Harmonic analysis and mean field theory, JHEP, 10, 217 (2019) · Zbl 1427.81137 |
| [15] | Pappadopulo, D.; Rychkov, S.; Espin, J.; Rattazzi, R., OPE convergence in conformal field theory, Phys. Rev. D, 86, 105043 (2012) |
| [16] | Hogervorst, M.; Rychkov, S., Radial coordinates for conformal blocks, Phys. Rev. D, 87, 106004 (2013) |
| [17] | Rychkov, S.; Yvernay, P., Remarks on the convergence properties of the conformal block expansion, Phys. Lett. B, 753, 682-686 (2016) · Zbl 1367.81127 |
| [18] | Mukhametzhanov, B.; Zhiboedov, A., Analytic Euclidean bootstrap, JHEP, 10, 270 (2019) · Zbl 1427.81142 |
| [19] | Rattazzi, R.; Rychkov, VS; Tonni, E.; Vichi, A., Bounding scalar operator dimensions in 4D CFT, JHEP, 12, 031 (2008) · Zbl 1329.81324 |
| [20] | Rychkov, S.: EPFL lectures on conformal field theory in D \(\ge 3\) dimensions. Springer Briefs in Physics (2016). arXiv:1601.05000 [hep-th] |
| [21] | Simmons-Duffin, D.: The conformal bootstrap. In: Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, USA, June 1-26, 2015, pp. 1-74 (2017). arXiv:1602.07982 [hep-th] · Zbl 1359.81165 |
| [22] | Poland, D.; Rychkov, S.; Vichi, A., The conformal bootstrap: theory, numerical techniques, and applications, Rev. Mod. Phys., 91, 1, 15002 (2019) |
| [23] | Poland, D.; Rychkov, S.; Vichi, A., The conformal bootstrap: theory, numerical techniques, and applications, Rev. Mod. Phys., 91, 015002 (2019) |
| [24] | Bautista, T.; Godazgar, H., Lorentzian CFT 3-point functions in momentum space, JHEP, 01, 142 (2020) · Zbl 1434.81094 |
| [25] | Coriano, C.; Delle Rose, L.; Mottola, E.; Serino, M., Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynman’s master integrals, JHEP, 07, 011 (2013) · Zbl 1342.81138 |
| [26] | Bzowski, A.; McFadden, P.; Skenderis, K., Implications of conformal invariance in momentum space, JHEP, 03, 111 (2014) · Zbl 1406.81082 |
| [27] | Bzowski, A.; McFadden, P.; Skenderis, K., Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies, JHEP, 03, 066 (2016) |
| [28] | Coriano, C.; Maglio, MM; Mottola, E., TTT in CFT: trace identities and the conformal anomaly effective action, Nucl. Phys. B, 942, 303-328 (2019) · Zbl 1415.81077 |
| [29] | Bzowski, A.; McFadden, P.; Skenderis, K., Renormalised 3-point functions of stress tensors and conserved currents in CFT, JHEP, 11, 153 (2018) · Zbl 1404.81219 |
| [30] | Isono, H.; Noumi, T.; Shiu, G., Momentum space approach to crossing symmetric CFT correlators, JHEP, 07, 136 (2018) · Zbl 1395.81228 |
| [31] | Bzowski, A.; McFadden, P.; Skenderis, K., Renormalised CFT 3-point functions of scalars, currents and stress tensors, JHEP, 11, 159 (2018) · Zbl 1404.83115 |
| [32] | Corianò, C.; Maglio, MM, The general 3-graviton vertex \((TTT)\) of conformal field theories in momentum space in \(d = 4\), Nucl. Phys. B, 937, 56-134 (2018) · Zbl 1402.81199 |
| [33] | Isono, H.; Noumi, T.; Takeuchi, T., Momentum space conformal three-point functions of conserved currents and a general spinning operator, JHEP, 05, 057 (2019) · Zbl 1416.81157 |
| [34] | Isono, H.; Noumi, T.; Shiu, G., Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension, JHEP, 10, 183 (2019) · Zbl 1427.81135 |
| [35] | Maldacena, JM; Pimentel, GL, On graviton non-Gaussianities during inflation, JHEP, 09, 045 (2011) · Zbl 1301.81147 |
| [36] | Creminelli, P.; Noreña, J.; Simonović, M., Conformal consistency relations for single-field inflation, JCAP, 1207, 052 (2012) |
| [37] | Kundu, N.; Shukla, A.; Trivedi, SP, Constraints from conformal symmetry on the three point scalar correlator in inflation, JHEP, 04, 061 (2015) · Zbl 1388.81729 |
| [38] | Arkani-Hamed, N., Maldacena, J.: Cosmological collider physics. arXiv:1503.08043 [hep-th] |
| [39] | Kundu, N.; Shukla, A.; Trivedi, SP, Ward identities for scale and special conformal transformations in inflation, JHEP, 01, 046 (2016) · Zbl 1388.83929 |
| [40] | Shukla, A.; Trivedi, SP; Vishal, V., Symmetry constraints in inflation, \( \alpha \)-vacua, and the three point function, JHEP, 12, 102 (2016) · Zbl 1390.83477 |
| [41] | Arkani-Hamed, N., Baumann, D., Lee, H., Pimentel, G.L.: The cosmological bootstrap: inflationary correlators from symmetries and singularities. arXiv:1811.00024 [hep-th] · Zbl 1436.85001 |
| [42] | Sleight, C., A Mellin space approach to cosmological correlators, JHEP, 01, 090 (2020) · Zbl 1434.81117 |
| [43] | Sleight, C.; Taronna, M., Bootstrapping inflationary correlators in Mellin space, JHEP, 02, 098 (2020) · Zbl 1435.81174 |
| [44] | Gillioz, M.; Lu, X.; Luty, MA, Scale anomalies, states, and rates in conformal field theory, JHEP, 04, 171 (2017) · Zbl 1378.81117 |
| [45] | Gillioz, M.; Lu, X.; Luty, MA, Graviton scattering and a sum rule for the C anomaly in 4D CFT, JHEP, 09, 025 (2018) · Zbl 1398.83107 |
| [46] | Gillioz, M., Momentum-space conformal blocks on the light cone, JHEP, 10, 125 (2018) · Zbl 1402.83089 |
| [47] | Belitsky, AV; Hohenegger, S.; Korchemsky, GP; Sokatchev, E.; Zhiboedov, A., From correlation functions to event shapes, Nucl. Phys. B, 884, 305-343 (2014) · Zbl 1323.81084 |
| [48] | Kravchuk, P.; Simmons-Duffin, D., Light-ray operators in conformal field theory, JHEP, 11, 102-236 (2018) · Zbl 1404.81234 |
| [49] | Kologlu, M., Kravchuk, P., Simmons-Duffin, D., Zhiboedov, A.: Shocks, Superconvergence, and a stringy equivalence principle. arXiv:1904.05905 [hep-th] |
| [50] | Kologlu, M., Kravchuk, P., Simmons-Duffin, D., Zhiboedov, A.: The light-ray OPE and conformal colliders. arXiv:1905.01311 [hep-th] |
| [51] | Hofman, DM; Maldacena, J., Conformal collider physics: energy and charge correlations, JHEP, 05, 012 (2008) |
| [52] | Cordova, C.; Maldacena, J.; Turiaci, GJ, Bounds on OPE coefficients from interference effects in the conformal collider, JHEP, 11, 032 (2017) · Zbl 1383.81196 |
| [53] | Cordova, C.; Diab, K., Universal bounds on operator dimensions from the average null energy condition, JHEP, 02, 131 (2018) |
| [54] | Chowdhury, SD; David, JR; Prakash, S., Constraints on parity violating conformal field theories in \(d=3\), JHEP, 11, 171 (2017) · Zbl 1383.81194 |
| [55] | Afkhami-Jeddi, N.; Hartman, T.; Kundu, S.; Tajdini, A., Shockwaves from the operator product expansion, JHEP, 03, 201 (2019) · Zbl 1414.83020 |
| [56] | Afkhami-Jeddi, N.; Kundu, S.; Tajdini, A., A conformal collider for holographic CFTs, JHEP, 10, 156 (2018) · Zbl 1402.83037 |
| [57] | Meltzer, D., Higher spin ANEC and the space of CFTs, JHEP, 07, 001 (2019) · Zbl 1418.81076 |
| [58] | Belin, A.; Hofman, DM; Mathys, G., Einstein gravity from ANEC correlators, JHEP, 08, 032 (2019) · Zbl 1421.83010 |
| [59] | Hofman, DM, Higher derivative gravity, causality and positivity of energy in a UV complete QFT, Nucl. Phys. B, 823, 174-194 (2009) · Zbl 1196.81266 |
| [60] | Hartman, T.; Kundu, S.; Tajdini, A., Averaged null energy condition from causality, JHEP, 07, 066 (2017) · Zbl 1380.81327 |
| [61] | Córdova, C.; Shao, S-H, Light-ray operators and the BMS algebra, Phys. Rev. D, 98, 12, 125015 (2018) |
| [62] | Katz, E.; Khandker, ZU; Walters, MT, A conformal truncation framework for infinite-volume dynamics, JHEP, 07, 140 (2016) · Zbl 1390.81702 |
| [63] | Fitzpatrick, AL; Kaplan, J.; Katz, E.; Vitale, LG; Walters, MT, Lightcone effective Hamiltonians and RG flows, JHEP, 08, 120 (2018) · Zbl 1396.81143 |
| [64] | Erramilli, RS; Iliesiu, LV; Kravchuk, P., Recursion relation for general 3D blocks, JHEP, 12, 116 (2019) · Zbl 1431.81137 |
| [65] | Mack, G., Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys., 53, 155 (1977) |
| [66] | Gillioz, M.; Lu, X.; Luty, MA; Mikaberidze, G., Convergent momentum-space OPE and bootstrap equations in conformal field theory, JHEP, 03, 102 (2020) · Zbl 1435.81073 |
| [67] | NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, release 1.0.23 of 2019-06-15. http://dlmf.nist.gov/. Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds.) |
| [68] | Simmons-Duffin, D., Projectors, shadows, and conformal blocks, JHEP, 04, 146 (2014) · Zbl 1333.83125 |
| [69] | Dolan, F.A., Osborn, H.: Conformal partial waves: further mathematical results. arXiv:1108.6194 [hep-th] · Zbl 1097.81735 |
| [70] | Costa, MS; Hansen, T.; Penedones, J.; Trevisani, E., Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP, 07, 018 (2016) · Zbl 1388.81798 |
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