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Action principle for OPE. (English) Zbl 1380.81187
Summary: We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by C. Behan [(2017; arXiv:1709.03967)] using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.

81T08 Constructive quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
81T17 Renormalization group methods applied to problems in quantum field theory
Full Text: DOI
[1] Bashmakov, V.; Bertolini, M.; Raj, H., On non-supersymmetric conformal manifolds: field theory and holography · Zbl 1383.81180
[2] Behan, C., Conformal manifolds: ODEs from opes · Zbl 1388.81630
[3] Costa, M. S.; Penedones, J.; Poland, D.; Rychkov, S., Spinning conformal blocks, J. High Energy Phys., 1111, (2011) · Zbl 1306.81148
[4] Costa, M. S.; Hansen, T.; Penedones, J.; Trevisani, E., Radial expansion for spinning conformal blocks, J. High Energy Phys., 1607, (2016) · Zbl 1390.81501
[5] Dimock, J., The renormalization group according to Balaban III. convergence, Ann. Henri Poincaré, 15, 11, 2133, (2014) · Zbl 1302.81139
[6] J. Dimock, The renormalization group according to Balaban - II. Large fields, AIP No. 9, https://doi.org/10.1063/1.4821275. · Zbl 1284.81217
[7] Dimock, J., The renormalization group according to Balaban, I. small fields, Rev. Math. Phys., 25, 7, (2013) · Zbl 1275.81068
[8] Efremov, A. N.; Guida, R.; Kopper, C., Renormalization of SU(2) Yang-Mills theory with flow equations, J. Math. Phys., 58, 9, (2017) · Zbl 1442.81051
[9] El-Showk, S.; Paulos, M. F.; Poland, D.; Rychkov, S.; Simmons-Duffin, D.; Vichi, A., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D, 86, (2012)
[10] Feldman, J.; Magnen, J.; Rivasseau, V.; Seneor, R., A renormalizable field theory: the massive Gross-Neveu model in two-dimensions, Commun. Math. Phys., 103, 67, (1986) · Zbl 0594.58060
[11] Fröb, M. B.; Holland, J.; Hollands, S., All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory, J. Math. Phys., 57, (2016) · Zbl 1356.81167
[12] Fröb, M. B.; Holland, J., All-order existence of and recursion relations for the operator product expansion in Yang-Mills theory
[13] Gadde, A., In search of conformal theories
[14] Gawedzki, K.; Kupiainen, A., Gross-Neveu model through convergent perturbation expansions, Commun. Math. Phys., 102, 1, (1985)
[15] Glimm, J.; Jaffe, A., Quantum physics, a functional integral point of view, (1987), Springer
[16] Guida, R.; Magnoli, N., All order IR finite expansion for short distance behavior of massless theories perturbed by a relevant operator, Nucl. Phys. B, 471, 361, (1996) · Zbl 1003.81514
[17] Guida, R.; Kopper, Ch., All-order uniform momentum bounds for the massless \(\phi^4\) theory in four dimensional Euclidean space
[18] R. Guida, Ch. Kopper, All-order uniform momentum bounds for the massless \(\phi_4^4\) field theory, in preparation.
[19] Holland, J.; Hollands, S., Associativity of the operator product expansion, J. Math. Phys., 56, 12, (2015) · Zbl 1429.81041
[20] Hollands, S.; Kopper, C., The operator product expansion converges in perturbative field theory, Commun. Math. Phys., 313, 257, (2012) · Zbl 1332.81119
[21] Holland, J.; Hollands, S.; Kopper, C., The operator product expansion converges in massless \(\varphi_4^4\)-theory, Commun. Math. Phys., 342, 2, 385, (2016) · Zbl 1337.81092
[22] Holland, J.; Hollands, S., Recursive construction of operator product expansion coefficients, Commun. Math. Phys., 336, 3, 1555, (2015) · Zbl 1429.81042
[23] Hollands, S., Quantum field theory in terms of consistency conditions. I. general framework, and perturbation theory via Hochschild cohomology, SIGMA, 5, (2009)
[24] Hollands, S.; Wald, R. M., Axiomatic quantum field theory in curved spacetime, Commun. Math. Phys., 293, 85, (2010) · Zbl 1193.81076
[25] Holland, J.; Hollands, S., Operator product expansion algebra, J. Math. Phys., 54, (2013) · Zbl 1287.81085
[26] Hollands, S., Talk at “wolfhard Zimmermann memorial symposium”, Munich (May 2017), available at
[27] Hollands, S., Talk at “in memoriam rudolf haag”, (September 2016), Hamburg, available at
[28] Huang, Y.-Z.; Kong, L., Full field algebras, Commun. Math. Phys., 272, (2007)
[29] Keller, G.; Kopper, Ch., Perturbative renormalization of composite operators via flow equations. 2. short distance expansion, Commun. Math. Phys., 153, 245, (1993) · Zbl 0793.47061
[30] Karateev, D.; Kravchuk, P.; Simmons-Duffin, D., Weight shifting operators and conformal blocks · Zbl 1387.81323
[31] Kravchuk, P., Casimir recursion relations for general conformal blocks · Zbl 1387.81324
[32] Kravchuk, P.; Simmons-Duffin, D., Counting conformal correlators · Zbl 1387.81325
[33] Lüscher, M., Operator product expansions on the vacuum in conformal quantum field theory in two space-time dimensions, Commun. Math. Phys., 50, 23, (1976)
[34] Mack, G., Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys., 53, 155, (1977)
[35] Osborn, H., Conformal blocks for arbitrary spins in two dimensions, Phys. Lett. B, 718, 169, (2012)
[36] Pappadopulo, D.; Rychkov, S.; Espin, J.; Rattazzi, R., OPE convergence in conformal field theory, Phys. Rev. D, 86, (2012)
[37] Polchinski, J., Renormalization and effective Lagrangians, Nucl. Phys. B, 231, 269, (1984)
[38] Rychkov, S., EPFL lectures on conformal field theory in \(D \geq 3\) dimensions · Zbl 1365.81007
[39] Schomerus, V.; Sobko, E.; Isachenkov, M., Harmony of spinning conformal blocks, J. High Energy Phys., 1703, (2017) · Zbl 1377.81182
[40] Wetterich, Ch., Exact evolution equation for the effective potential, Phys. Lett. B, 301, 90, (1993)
[41] Wilson, K. G., Non-Lagrangian models of current algebra, Phys. Rev., 179, 1499, (1969)
[42] Zimmermann, W., Normal products and the short distance expansion in the perturbation theory of renormalizable interactions, Ann. Phys., 77, 570, (1973)
[43] Zimmermann, W., Composite operators in the perturbation theory of renormalizable interactions, (Lect. Notes Phys., vol. 558, (2000)), 77, 244, (1973)
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