Abstract
The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.
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Douglas M.R., Nekrasov N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2001)
Wulkenhaar R.: Field theories on deformed spaces. J. Geom. Phys. 56, 108–141 (2006)
Connes A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Dubois-Violette M., Kerner R., Madore J.: Noncommutative differential geometry of matrix algebras. J. Math. Phys. 31, 316 (1990)
Connes A., Lott J.: Particle models and noncommutative geometry. in Recent advances in field theory (Annecy-le-Vieux, 1990). Nucl. Phys. Proc. Suppl. B 18, 29–47 (1991)
de Goursac A., Masson T., Wallet J.C.: Noncommutative ɛ-graded connections. J. Noncommut. Geom. 6, 343–387 (2012)
Cagnache E., Masson T., Wallet J.C.: Noncommutative Yang–Mills–Higgs actions from derivation-based differential calculus. J. Noncommut. Geom. 5, 39 (2011)
Grosse H., Wulkenhaar R.: Renormalisation of phi**4 theory on noncommutative R**4 in the matrix base. Commun. Math. Phys. 256, 305–374 (2005)
Minwalla S., Van Raamsdonk M., Seiberg N.: Noncommutative perturbative dynamics. JHEP 02, 020 (2000)
Rivasseau V., Vignes-Tourneret F., Wulkenhaar R.: Renormalization of noncommutative phi**4-theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006)
Gurau R., Magnen J., Rivasseau V., Vignes-Tourneret F.: Renormalization of non-commutative phi**4(4) field theory in x space. Commun. Math. Phys. 267, 515–542 (2006)
Gurau R., Tanasa A.: Dimensional regularization and renormalization of non-commutative QFT. Annales Henri Poincaré 9, 655–683 (2008)
Tanasa A., Vignes-Tourneret F.: Hopf algebra of non-commutative field theory. J. Noncommut. Geom. 2, 125–139 (2008)
Grosse H., Wulkenhaar R.: The beta-function in duality-covariant noncommutative phi**4 theory. Eur. Phys. J. C 35, 277–282 (2004)
Disertori M., Gurau R., Magnen J., Rivasseau V.: Vanishing of beta function of non commutative phi(4)**4 theory to all orders. Phys. Lett. B 649, 95–102 (2007)
Grosse, H., Wulkenhaar, R.: Self-dual noncommutative phi4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory. arXiv:1205.0465 [math-ph]
Langmann E., Szabo R.J.: Duality in scalar field theory on noncommutative phase spaces. Phys. Lett. B 533, 168–177 (2002)
de Goursac A.: On the origin of the harmonic term in noncommutative quantum field theory. SIGMA 6, 048 (2010)
de Goursac A., Tanasa A., Wallet J.C.: Vacuum configurations for renormalizable non-commutative scalar models. Eur. Phys. J. C 53, 459–466 (2008)
Hounkonnou M.N., Samary D.O.: Twisted Grosse–Wulkenhaar phi*4 model: dynamical noncommutativity and Noether currents. J. Phys. A 43, 155202 (2010)
Ben Geloun J., Hounkonnou M.N.: Energy-momentum tensors in renormalizable noncommutative scalar field theory. Phys. Lett. B 653, 343–345 (2007)
de Goursac A., Wallet J.C.: Symmetries of noncommutative scalar field theory. J. Phys. A Math. Theor. 44, 055401 (2011)
Langmann E., Szabo R.J., Zarembo K.: Exact solution of quantum field theory on noncommutative phase spaces. JHEP 01, 017 (2004)
Vignes-Tourneret F.: Renormalization of the orientable non-commutative Gross–Neveu model. Annales Henri Poincaré 8, 427–474 (2007)
Gurau R., Magnen J., Rivasseau V., Tanasa A.: A translation-invariant renormalizable non-commutative scalar model. Commun. Math. Phys. 287, 275–290 (2009)
Ben Geloun J., Tanasa A.: One-loop β functions of a translation-invariant renormalizable noncommutative scalar model. Lett. Math. Phys. 86, 19–32 (2008)
Magnen J., Rivasseau V., Tanasa A.: Commutative limit of a renormalizable noncommutative model. Europhys. Lett. 86, 11001 (2009)
Blaschke D.N., Gieres F., Kronberger E., Schweda M., Wohlgenannt M.: Translation-invariant models for non-commutative gauge fields. J. Phys. A 41, 252002 (2008)
Blaschke D.N., Rofner A., Schweda M., Sedmik R.I.P.: One-loop calculations for a translation invariant non-commutative gauge model. Eur. Phys. J. C 62, 433–443 (2009)
Pinzul A.: UV/IR mixing as a twisted Poincaré anomaly. J. Phys. A 45, 075401 (2012)
Liang M.L., Jiang Y.: Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane. Phys. Lett. A 375, 1–5 (2010)
de Goursac A., Wallet J.C., Wulkenhaar R.: Noncommutative induced gauge theory. Eur. Phys. J. C 51, 977–987 (2007)
Grosse H., Wohlgenannt M.: Induced gauge theory on a noncommutative space. Eur. Phys. J. C 52, 435–450 (2007)
de Goursac A., Wallet J.C., Wulkenhaar R.: On the vacuum states for noncommutative gauge theory. Eur. Phys. J. C 56, 293–304 (2008)
Blaschke D.N., Grosse H., Schweda M.: Non-commutative U(1) gauge theory on R**4(Theta) with oscillator term and BRST symmetry. Europhys. Lett. 79, 61002 (2007)
Blaschke, D.N., Grosse, H., Kronberger, E., Schweda, M., Wohlgenannt, M.: Loop calculations for the non-commutative U(1) gauge field model with oscillator term. arXiv:0912.3642 [hep-th]
Wulkenhaar, R.: Non-compact spectral triples with finite volume. arXiv: 0907.1351 [hep-th]
Rivasseau V.: From Perturbative to Constructive Renormalization. Princeton University Press, NJ (1991)
Vignes-Tourneret, F.: Renormalisation of non commutative field theories. arXiv: math-ph/0612014
Zimmermann W.: Convergence of Bogolyubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208 (1969)
Bieliavsky P., de Goursac A., Tuynman G.: quantization for Heisenberg supergroup. J. Funct. Anal. 263, 549–603 (2012)
de Goursac, A.: On the Hopf algebra setting of the flat superspace’s deformation. arXiv:1105.2420 [math.QA]
Gurau R., Rivasseau V., Vignes-Tourneret F.: Propagators for noncommutative field theories. Annales Henri Poincaré 7, 1601–1628 (2006)
Grosse H., Vignes-Tourneret F.: Quantum field theory on the degenerate Moyal space. J. Noncommut. Geom. 4, 555–576 (2010)
Grosse, H., Wohlgenannt, M.: Degenerate noncommutativity. arXiv:1201.5982 [hep-th]
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Communicated by Christoph Kopper.
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de Goursac, A. Renormalization of the Commutative Scalar Theory with Harmonic Term to All Orders. Ann. Henri Poincaré 14, 2025–2043 (2013). https://doi.org/10.1007/s00023-012-0226-4
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DOI: https://doi.org/10.1007/s00023-012-0226-4