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\(\kappa\)-Poincaré invariant orientable field theories at one-loop. (English) Zbl 1409.83070

Summary: We consider a family of \(\kappa\)-Poincaré invariant scalar field theories on 4-d \(\kappa\)-Minkowski space with quartic orientable interaction, that is for which \(\phi\) and its conjugate \({\phi}^{\dagger}\) alternate in the quartic interaction, and whose kinetic operator is the square of a \(U_{\kappa}\)(iso(4))-equivariant Dirac operator. The formal commutative limit yields the standard complex \(\phi^4\) theory. We find that the 2-point function receives UV linearly diverging 1-loop corrections while it stays free of IR singularities that would signal occurrence of UV/IR mixing. We find that all the 1-loop planar and non-planar contributions to the 4-point function are UV finite, stemming from the existence of the particular estimate for the propagator partly combined with its decay properties at large momenta, implying formally vanishing of the beta-functions at 1-loop so that the coupling constants stay scale-invariant at 1-loop.

MSC:

83C45 Quantization of the gravitational field
83C65 Methods of noncommutative geometry in general relativity
81T15 Perturbative methods of renormalization applied to problems in quantum field theory

References:

[1] S. Majid and H. Ruegg, Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett.B 334 (1994) 348 [hep-th/9405107] [INSPIRE]. · Zbl 1112.81328 · doi:10.1016/0370-2693(94)90699-8
[2] J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoï, q-deformation of Poincaré algebra, Phys. Lett.B 264 (1991) 331 [INSPIRE].
[3] J. Lukierski, A. Nowicki and H. Ruegg, New quantum Poincaré algebra and κ-deformed field theory, Phys. Lett.B 293 (1992) 344 [INSPIRE]. · Zbl 0834.17022 · doi:10.1016/0370-2693(92)90894-A
[4] S. Doplicher, K. Fredenhagen and J.E. Roberts, The quantum structure of space-time at the Planck scale and quantum fields, Commun. Math. Phys.172 (1995) 187 [hep-th/0303037] [INSPIRE]. · Zbl 0847.53051 · doi:10.1007/BF02104515
[5] S. Doplicher, K. Fredenhagen and J.E. Roberts, Space-time quantization induced by classical gravity, Phys. Lett.B 331 (1994) 39 [INSPIRE]. · doi:10.1016/0370-2693(94)90940-7
[6] H.-J. Matschull and M. Welling, Quantum mechanics of a point particle in (2 + 1)-dimensional gravity, Class. Quant. Grav.15 (1998) 2981 [gr-qc/9708054] [INSPIRE]. · Zbl 1063.83534
[7] L. Freidel and E.R. Livine, 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett.96 (2006) 221301 [Bulg. J. Phys.33 (2006) 111] [hep-th/0512113] [INSPIRE]. · Zbl 1228.83047
[8] V.G. Drinfeld, Quantum groups, in Proc. Int. Cong. Math., volumes 1, 2, Berkeley, CA, U.S.A. (1986), AMS, Providence, RI, U.S.A. (1987), pg. 798 [J. Sov. Math.41 (1988) 898] [Zap. Nauchn. Semin.155 (1986) 18] [INSPIRE]. · Zbl 0667.16003
[9] L.A. Takhtadzhyan, Lectures on quantum groups, Nankai Lectures on Mathematical Physics, M.-L. Ge and B.-H. Zhao eds., World Scientific, Singapore (1989).
[10] J. Lukierski, κ-deformations: historical developments and recent results, J. Phys. Conf. Ser.804 (2017) 012028 [arXiv:1611.10213] [INSPIRE].
[11] G. Amelino-Camelia, Doubly special relativity, Nature418 (2002) 34 [gr-qc/0207049] [INSPIRE]. · Zbl 1070.83500
[12] G. Amelino-Camelia, G. Gubitosi, A. Marciano, P. Martinetti and F. Mercati, A no-pure-boost uncertainty principle from spacetime noncommutativity, Phys. Lett.B 671 (2009) 298 [arXiv:0707.1863] [INSPIRE]. · doi:10.1016/j.physletb.2008.12.032
[13] J. Kowalski-Glikman, Introduction to doubly special relativity, in Planck scale effects in astrophysics and cosmology, Lect. Notes Phys.669 (2005) 131 [hep-th/0405273] [INSPIRE].
[14] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, The principle of relative locality, Phys. Rev.D 84 (2011) 084010 [arXiv:1101.0931] [INSPIRE]. · Zbl 1263.83064
[15] G. Gubitosi and F. Mercati, Relative locality in κ-Poincaré, Class. Quant. Grav.30 (2013) 145002 [arXiv:1106.5710] [INSPIRE]. · Zbl 1273.83066 · doi:10.1088/0264-9381/30/14/145002
[16] G. Amelino-Camelia, V. Astuti and G. Rosati, Relative locality in a quantum spacetime and the pregeometry of κ-Minkowski, Eur. Phys. J.C 73 (2013) 2521 [arXiv:1206.3805] [INSPIRE]. · doi:10.1140/epjc/s10052-013-2521-8
[17] A. Agostini, G. Amelino-Camelia, M. Arzano and F. D’Andrea, Action functional for κ-Minkowski noncommutative spacetime, hep-th/0407227 [INSPIRE]. · Zbl 1051.83003
[18] A. Agostini, G. Amelino-Camelia and F. D’Andrea, Hopf algebra description of noncommutative space-time symmetries, Int. J. Mod. Phys.A 19 (2004) 5187 [hep-th/0306013] [INSPIRE]. · Zbl 1078.81036 · doi:10.1142/S0217751X04020919
[19] A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano and R.A. Tacchi, Generalizing the Noether theorem for Hopf-algebra spacetime symmetries, Mod. Phys. Lett.A 22 (2007) 1779 [hep-th/0607221] [INSPIRE]. · Zbl 1143.83317 · doi:10.1142/S0217732307024280
[20] G. Amelino-Camelia and M. Arzano, Coproduct and star product in field theories on Lie algebra noncommutative space-times, Phys. Rev.D 65 (2002) 084044 [hep-th/0105120] [INSPIRE].
[21] M. Dimitrijević, L. Jonke, L. Möller, E. Tsouchnika, J. Wess and M. Wohlgenannt, Deformed field theory on κ space-time, Eur. Phys. J.C 31 (2003) 129 [hep-th/0307149] [INSPIRE]. · Zbl 1032.81529 · doi:10.1140/epjc/s2003-01309-y
[22] M. Dimitrijević, F. Meyer, L. Möller and J. Wess, Gauge theories on the κ-Minkowski space-time, Eur. Phys. J.C 36 (2004) 117 [hep-th/0310116] [INSPIRE]. · Zbl 1191.81204 · doi:10.1140/epjc/s2004-01887-0
[23] M. Dimitrijević, L. Jonke and L. Möller, U(1) gauge field theory on κ-Minkowski space, JHEP09 (2005) 068 [hep-th/0504129] [INSPIRE]. · doi:10.1088/1126-6708/2005/09/068
[24] M. Dimitrijević, L. Jonke and A. Pachol, Gauge theory on twisted κ-Minkowski: old problems and possible solutions, SIGMA10 (2014) 063 [arXiv:1403.1857] [INSPIRE]. · Zbl 1295.81125
[25] A. Borowiec and A. Pachol, κ-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev.D 79 (2009) 045012 [arXiv:0812.0576] [INSPIRE].
[26] A. Pachoł and P. Vitale, κ-Minkowski star product in any dimension from symplectic realization, J. Phys.A 48 (2015) 445202 [arXiv:1507.03523] [INSPIRE]. · Zbl 1327.53111
[27] S. Meljanac, A. Samsarov, M. Stojic and K.S. Gupta, κ-Minkowski space-time and the star product realizations, Eur. Phys. J.C 53 (2008) 295 [arXiv:0705.2471] [INSPIRE]. · Zbl 1189.81115
[28] S. Meljanac and A. Samsarov, Scalar field theory on κ-Minkowski spacetime and translation and Lorentz invariance, Int. J. Mod. Phys.A 26 (2011) 1439 [arXiv:1007.3943] [INSPIRE]. · Zbl 1214.81285 · doi:10.1142/S0217751X11051536
[29] E. Harikumar, T. Jurić and S. Meljanac, Electrodynamics on κ-Minkowski space-time, Phys. Rev.D 84 (2011) 085020 [arXiv:1107.3936] [INSPIRE].
[30] S. Meljanac, A. Samsarov, J. Trampetic and M. Wohlgenannt, Scalar field propagation in the ϕ4κ-Minkowski model, JHEP12 (2011) 010 [arXiv:1111.5553] [INSPIRE]. · Zbl 1306.81305 · doi:10.1007/JHEP12(2011)010
[31] F. Mercati and M. Sergola, Pauli-Jordan function and scalar field quantization in κ-Minkowski noncommutative spacetime, Phys. Rev.D 98 (2018) 045017 [arXiv:1801.01765] [INSPIRE].
[32] T. Poulain and J.-C. Wallet, κ-Poincaré invariant quantum field theories with KMS weight, Phys. Rev.D 98 (2018) 025002 [arXiv:1801.02715] [INSPIRE].
[33] H. Grosse and M. Wohlgenannt, On κ-deformation and UV/IR mixing, Nucl. Phys.B 748 (2006) 473 [hep-th/0507030] [INSPIRE]. · doi:10.1016/j.nuclphysb.2006.05.004
[34] M. Dimitrijević, L. Möller and E. Tsouchnika, Derivatives, forms and vector fields on the κ-deformed Euclidean space, J. Phys.A 37 (2004) 9749 [hep-th/0404224] [INSPIRE]. · Zbl 1073.81053
[35] D. Williams, Crossed products of C*-algebras, Math. Surv. Monogr.134, AMS, Providence, RI, U.S.A. (2007). · Zbl 1119.46002
[36] B. Durhuus and A. Sitarz, Star product realizations of κ-Minkowski space, J. Noncommut. Geom.7 (2013) 605 [arXiv:1104.0206] [INSPIRE]. · Zbl 1282.46063 · doi:10.4171/JNCG/129
[37] M. Matassa, On the spectral and homological dimension of κ-Minkowski space, arXiv:1309.1054 [INSPIRE]. · Zbl 1283.83028
[38] M. Matassa, A modular spectral triple for κ-Minkowski space, J. Geom. Phys.76 (2014) 136 [arXiv:1212.3462] [INSPIRE]. · Zbl 1283.83028 · doi:10.1016/j.geomphys.2013.10.023
[39] P. Vitale and J.-C. Wallet, Noncommutative field theories on Rλ3: toward UV/IR mixing freedom, JHEP04 (2013) 115 [arXiv:1212.5131] [INSPIRE]. · Zbl 1342.81641 · doi:10.1007/JHEP04(2013)115
[40] A. Géré, P. Vitale and J.-C. Wallet, Quantum gauge theories on noncommutative three-dimensional space, Phys. Rev.D 90 (2014) 045019 [arXiv:1312.6145] [INSPIRE].
[41] P. Martinetti, P. Vitale and J.-C. Wallet, Noncommutative gauge theories on Rθ2as matrix models, JHEP09 (2013) 051 [arXiv:1303.7185] [INSPIRE]. · Zbl 1342.81309 · doi:10.1007/JHEP09(2013)051
[42] T. Jurić, T. Poulain and J.-C. Wallet, Closed star product on noncommutative R3and scalar field dynamics, JHEP05 (2016) 146 [arXiv:1603.09122] [INSPIRE]. · Zbl 1388.83538
[43] T. Jurić, T. Poulain and J.-C. Wallet, Involutive representations of coordinate algebras and quantum spaces, JHEP07 (2017) 116 [arXiv:1702.06348] [INSPIRE]. · Zbl 1380.83095
[44] A. Géré, T. Jurić and J.-C. Wallet, Noncommutative gauge theories on Rλ3: perturbatively finite models, JHEP12 (2015) 045 [arXiv:1507.08086] [INSPIRE]. · Zbl 1387.81266
[45] J.-C. Wallet, Exact partition functions for gauge theories on Rλ3, Nucl. Phys.B 912 (2016) 354 [arXiv:1603.05045] [INSPIRE]. · Zbl 1349.81173 · doi:10.1016/j.nuclphysb.2016.04.001
[46] J. Kustermans, KMS-weights on C*-algebras, funct-an/9704008. · Zbl 0998.46025
[47] F. Combes, Poids sur une C*-algèbre (in French), J. Math. Pures Appl.47 (1968) 57. · Zbl 0165.15401
[48] M. Takesaki, Theory of operator algebras I, Encyclopaedia Math. Sci.124, Springer, Berlin Heidelberg, Germany (2002). · Zbl 0990.46034
[49] M. Takesaki, Theory of operator algebras II, Encyclopaedia Math. Sci.125, Springer, Berlin Heidelberg, Germany (2003). · Zbl 1059.46031
[50] M. Takesaki, Theory of operator algebras III, Encyclopaedia Math. Sci.126, Springer, Berlin Heidelberg, Germany (2003). · Zbl 1059.46032
[51] A. Connes and C. Rovelli, Von Neumann algebra automorphisms and time thermodynamics relation in general covariant quantum theories, Class. Quant. Grav.11 (1994) 2899 [gr-qc/9406019] [INSPIRE]. · Zbl 0821.46086
[52] F. Vignes-Tourneret, Renormalization of the orientable non-commutative Gross-Neveu model, Annales Henri Poincaré8 (2007) 427. · Zbl 1133.81058 · doi:10.1007/s00023-006-0312-6
[53] A. de Goursac and J.-C. Wallet, Symmetries of noncommutative scalar field theory, J. Phys.A 44 (2011) 055401 [arXiv:0911.2645] [INSPIRE]. · Zbl 1208.81192
[54] A. de Goursac, A. Tanasa and J.-C. Wallet, Vacuum configurations for renormalizable non-commutative scalar models, Eur. Phys. J.C 53 (2008) 459 [arXiv:0709.3950] [INSPIRE]. · Zbl 1189.81213 · doi:10.1140/epjc/s10052-007-0465-6
[55] A. de Goursac, J.-C. Wallet and R. Wulkenhaar, Noncommutative induced gauge theory, Eur. Phys. J.C 51 (2007) 977 [hep-th/0703075] [INSPIRE]. · Zbl 1189.81215 · doi:10.1140/epjc/s10052-007-0335-2
[56] J.-C. Wallet, Noncommutative induced gauge theories on Moyal spaces, J. Phys. Conf. Ser.103 (2008) 012007 [arXiv:0708.2471] [INSPIRE]. · doi:10.1088/1742-6596/103/1/012007
[57] H. Grosse, H. Steinacker and M. Wohlgenannt, Emergent gravity, matrix models and UV/IR mixing, JHEP04 (2008) 023 [arXiv:0802.0973] [INSPIRE]. · Zbl 1246.81162 · doi:10.1088/1126-6708/2008/04/023
[58] A. de Goursac, J.-C. Wallet and R. Wulkenhaar, On the vacuum states for noncommutative gauge theory, Eur. Phys. J.C 56 (2008) 293 [arXiv:0803.3035] [INSPIRE]. · Zbl 1189.81214 · doi:10.1140/epjc/s10052-008-0652-0
[59] F. D’Andrea, Remarks on the geometry of κ-Minkowski space, J. Math. Phys.47 (2006) 062105 [hep-th/0503012] [INSPIRE]. · Zbl 1112.58009 · doi:10.1063/1.2204808
[60] A. Connes and H. Moscovici, Type III and spectral triples, in Traces in number theory, geometry and quantum fields, Aspects Math.E 38, Vieweg, Wiesbaden, Germany (2008), pg. 57 [math.OA/0609703]. · Zbl 1159.46041
[61] F. D’Andrea, Remarks on the geometry of κ-Minkowski space, J. Math. Phys.47 (2006) 062105 [hep-th/0503012] [INSPIRE]. · Zbl 1112.58009 · doi:10.1063/1.2204808
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