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Quantum gravitational corrections for spinning particles. (English) Zbl 1390.83028
J. High Energy Phys. 2016, No. 10, Paper No. 51, 46 p. (2016); erratum ibid. 2016, No. 11, Paper No. 176, 1 p. (2016).
Summary: We calculate the quantum corrections to the gauge-invariant gravitational potentials of spinning particles in flat space, induced by loops of both massive and massless matter fields of various types. While the corrections to the Newtonian potential induced by massless conformal matter for spinless particles are well known, and the same corrections due to massless minimally coupled scalars [S. Park and R. P. Woodard, Classical Quantum Gravity 27, No. 24, Article ID 245008, 10 p. (2010; Zbl 1206.83080)], massless non-conformal scalars [A. Marunović and T. Prokopec, “Antiscreening in perturbative quantum gravity and resolving the Newtonian singularity”, Phys. Rev. D (3) 87, No. 10, Article ID 104027, 14 p. (2013; doi:10.1103/physrevd.87.104027)] and massive scalars, fermions and vector bosons [D. Z. Freedman and A. Van Proeyen, Supergravity. Cambridge: Cambridge University Press (2012; Zbl 1245.83001)] have been recently derived, spinning particles receive additional corrections which are the subject of the present work. We give both fully analytic results valid for all distances from the particle, and present numerical results as well as asymptotic expansions. At large distances from the particle, the corrections due to massive fields are exponentially suppressed in comparison to the corrections from massless fields, as one would expect. However, a surprising result of our analysis is that close to the particle itself, on distances comparable to the Compton wavelength of the massive fields running in the loops, these corrections can be enhanced with respect to the massless case.

##### MSC:
 83C10 Equations of motion in general relativity and gravitational theory
##### Keywords:
effective field theories; models of quantum gravity
DLMF; pAQFT
Full Text:
##### References:
 [1] Donoghue, JF, General relativity as an effective field theory: the leading quantum corrections, Phys. Rev., D 50, 3874, (1994) [2] Burgess, CP, Quantum gravity in everyday life: general relativity as an effective field theory, Living Rev. Rel., 7, 5, (2004) · Zbl 1070.83009 [3] Heisenberg, W.; Euler, H., Consequences of dirac’s theory of positrons, Z. Phys., 98, 714, (1936) · JFM 62.1002.03 [4] H. Euler, Über die Streuung von Licht an Licht nach der Diracschen Theorie (in German), Ann. Phys. (Leipzig)26 (1936) 398. · JFM 62.1634.03 [5] Weinberg, S., Phenomenological Lagrangians, Physica, A 96, 327, (1979) [6] Weinberg, S., Effective gauge theories, Phys. Lett., B 91, 51, (1980) [7] Radkowski, AF, Some aspects of the source description of gravitation, Ann. Phys., 56, 319, (1970) [8] Schwinger, JS, Sources and gravitons, Phys. Rev., 173, 1264, (1968) [9] Duff, MJ, Quantum corrections to the Schwarzschild solution, Phys. Rev., D 9, 1837, (1974) [10] Capper, DM; Duff, MJ; Halpern, L., Photon corrections to the graviton propagator, Phys. Rev., D 10, 461, (1974) [11] Capper, DM; Duff, MJ, The one loop neutrino contribution to the graviton propagator, Nucl. Phys., B 82, 147, (1974) [12] Donoghue, JF, Leading quantum correction to the Newtonian potential, Phys. Rev. Lett., 72, 2996, (1994) [13] Muzinich, IJ; Vokos, S., Long range forces in quantum gravity, Phys. Rev., D 52, 3472, (1995) [14] Hamber, HW; Liu, S., On the quantum corrections to the Newtonian potential, Phys. Lett., B 357, 51, (1995) [15] Akhundov, AA; Bellucci, S.; Shiekh, A., Gravitational interaction to one loop in effective quantum gravity, Phys. Lett., B 395, 16, (1997) [16] M.J. Duff and J.T. Liu, Complementarity of the Maldacena and Randall-Sundrum pictures, Class. Quant. Grav.18 (2001) 3207 [Phys. Rev. Lett.85 (2000) 2052] [hep-th/0003237] [INSPIRE]. · Zbl 1369.83089 [17] Duff, MJ; Liu, JT, Complementarity of the Maldacena and Randall-Sundrum pictures, Class. Quant. Grav., 18, 3207, (2001) · Zbl 0993.83043 [18] I.B. Khriplovich and G.G. Kirilin, Quantum power correction to the Newton law, J. Exp. Theor. Phys.95 (2002) 981 [Zh. Eksp. Teor. Fiz.95 (2002) 1139] [gr-qc/0207118] [INSPIRE]. [19] I.B. Khriplovich and G.G. Kirilin, Quantum long range interactions in general relativity, J. Exp. Theor. Phys.98 (2004) 1063 [Zh. Eksp. Teor. Fiz.125 (2004) 1219] [gr-qc/0402018] [INSPIRE]. · Zbl 1097.83516 [20] N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev.D 68 (2003) 084005 [Erratum ibid.D 71 (2005) 069904] [hep-th/0211071] [INSPIRE]. [21] N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev.D 67 (2003) 084033 [Erratum ibid.D 71 (2005) 069903] [hep-th/0211072] [INSPIRE]. [22] Satz, A.; Mazzitelli, FD; Alvarez, E., Vacuum polarization around stars: nonlocal approximation, Phys. Rev., D 71, 064001, (2005) [23] Park, S.; Woodard, RP, Solving the effective field equations for the Newtonian potential, Class. Quant. Grav., 27, 245008, (2010) · Zbl 1206.83080 [24] Marunovic, A.; Prokopec, T., Time transients in the quantum corrected Newtonian potential induced by a massless nonminimally coupled scalar field, Phys. Rev., D 83, 104039, (2011) [25] Marunovic, A.; Prokopec, T., Antiscreening in perturbative quantum gravity and resolving the Newtonian singularity, Phys. Rev., D 87, 104027, (2013) [26] R.E. Kallosh and I.V. Tyutin, The equivalence theorem and gauge invariance in renormalizable theories, Yad. Fiz.17 (1973) 190 [Sov. J. Nucl. Phys.17 (1973) 98] [INSPIRE]. [27] Lam, Y-MP, Equivalence theorem on bogolyubov-parasiuk-Hepp-Zimmermann renormalized Lagrangian field theories, Phys. Rev., D 7, 2943, (1973) [28] B.R. Holstein and A. Ross, Spin effects in long range gravitational scattering, arXiv:0802.0716 [INSPIRE]. [29] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys., B 425, 217, (1994) · Zbl 1049.81644 [30] H. Elvang and Y.-T. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE]. · Zbl 1332.81010 [31] L.J. Dixon, A brief introduction to modern amplitude methods, in Proceedings, 2012 European School of High-Energy Physics (ESHEP 2012$$)$$, La Pommeraye Anjou France June 6-19 2012, pg. 31 [arXiv:1310.5353] [INSPIRE]. · Zbl 1334.81001 [32] E. Witten, Quantum gravity in de Sitter space, in Strings 2001: International Conference, Mumbai India January 5-10 2001 [hep-th/0106109] [INSPIRE]. · Zbl 1054.83013 [33] Bousso, R., Cosmology and the S-matrix, Phys. Rev., D 71, 064024, (2005) [34] Wang, CL; Woodard, RP, One-loop quantum electrodynamic correction to the gravitational potentials on de Sitter spacetime, Phys. Rev., D 92, 084008, (2015) [35] Park, S.; Prokopec, T.; Woodard, RP, Quantum scalar corrections to the gravitational potentials on de Sitter background, JHEP, 01, 074, (2016) · Zbl 1388.83138 [36] Fröb, MB; Verdaguer, E., Quantum corrections to the gravitational potentials of a point source due to conformal fields in de Sitter, JCAP, 03, 015, (2016) [37] Bardeen, JM, Gauge invariant cosmological perturbations, Phys. Rev., D 22, 1882, (1980) [38] J. Lense and H. Thirring, Über den Einfluß der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie (in German), Phys. Z.19 (1918) 156 [INSPIRE]. · JFM 46.1317.01 [39] Ciufolini, I.; Pavlis, EC, A confirmation of the general relativistic prediction of the lense-Thirring effect, Nature, 431, 958, (2004) [40] Everitt, CWF; etal., Gravity probe B: final results of a space experiment to test general relativity, Phys. Rev. Lett., 106, 221101, (2011) [41] Iorio, L.; Lichtenegger, HIM; Ruggiero, ML; Corda, C., Phenomenology of the lense-Thirring effect in the solar system, Astrophys. Space Sci., 331, 351, (2011) · Zbl 1209.83002 [42] Lalak, Z.; Pokorski, S.; Wess, J., Spin 1/2 particle in gravitational field of a rotating body, Phys. Lett., B 355, 453, (1995) · Zbl 0997.83505 [43] Tomboulis, E., 1/N expansion and renormalization in quantum gravity, Phys. Lett., B 70, 361, (1977) [44] Hartle, JB; Horowitz, GT, Ground state expectation value of the metric in the 1/N or semiclassical approximation to quantum gravity, Phys. Rev., D 24, 257, (1981) [45] Schwinger, JS, Brownian motion of a quantum oscillator, J. Math. Phys., 2, 407, (1961) · Zbl 0098.43503 [46] L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz.47 (1964) 1515 [Sov. Phys. JETP20 (1965) 1018] [INSPIRE]. [47] Chou, K-C; Su, Z-B; Hao, B-L; Yu, L., Equilibrium and nonequilibrium formalisms made unified, Phys. Rept., 118, 1, (1985) [48] Jordan, RD, Effective field equations for expectation values, Phys. Rev., D 33, 444, (1986) [49] DeWitt, BS, Quantum theory of gravity. 2. the manifestly covariant theory, Phys. Rev., 162, 1195, (1967) · Zbl 0161.46501 [50] Kluberg-Stern, H.; Zuber, JB, Renormalization of non-abelian gauge theories in a background field gauge. 1. Green functions, Phys. Rev., D 12, 482, (1975) [51] I. Ya. Arefeva, L.D. Faddeev and A.A. Slavnov, Generating functional for the S matrix in gauge theories, Theor. Math. Phys.21 (1975) 1165 [Teor. Mat. Fiz.21 (1974) 311] [INSPIRE]. [52] Abbott, LF, The background field method beyond one loop, Nucl. Phys., B 185, 189, (1981) [53] Hollands, S.; Wald, RM, Conservation of the stress tensor in interacting quantum field theory in curved spacetimes, Rev. Math. Phys., 17, 227, (2005) · Zbl 1078.81062 [54] Capper, DM, On quantum corrections to the graviton propagator, Nuovo Cim., A 25, 29, (1975) [55] Martín, R.; Verdaguer, E., Stochastic semiclassical fluctuations in Minkowski space-time, Phys. Rev., D 61, 124024, (2000) [56] Ford, LH; Woodard, RP, Stress tensor correlators in the Schwinger-Keldysh formalism, Class. Quant. Grav., 22, 1637, (2005) · Zbl 1073.83023 [57] Fröb, MB, Fully renormalized stress tensor correlator in flat space, Phys. Rev., D 88, 045011, (2013) [58] Abramo, LRW; Brandenberger, RH; Mukhanov, VF, The energy-momentum tensor for cosmological perturbations, Phys. Rev., D 56, 3248, (1997) [59] Nakamura, K., Second-order gauge invariant cosmological perturbation theory: Einstein equations in terms of gauge invariant variables, Prog. Theor. Phys., 117, 17, (2007) · Zbl 1125.83016 [60] Fröb, MB, The Weyl tensor correlator in cosmological spacetimes, JCAP, 12, 010, (2014) [61] M. Mathisson, Neue Mechanik materieller Systemes (in German), Acta Phys. Polon.6 (1937) 163 [Gen. Rel. Grav.42 (2010) 1011] [INSPIRE]. [62] Papapetrou, A., Spinning test particles in general relativity. 1, Proc. Roy. Soc. Lond., A 209, 248, (1951) · Zbl 0044.22801 [63] A. Trautman, Lectures on general relativity, King’s college, London U.K. (1958), published for the Brandeis Univ. Summer Inst. Theoretical Physics 1964, A. Trautmann, F.A.E. Pirani and H. Bondi eds., Prentice-Hall, Englewood Cliffs U.S.A. (1965) [Gen. Rel. Grav.34 (2002) 721] [INSPIRE]. [64] Tulczyjew, WM, Motion of multipole particles in general relativity theory, Acta Phys. Polon., 18, 393, (1959) · Zbl 0097.42402 [65] B. Tulczyjew and W.M. Tulczyjew, On multipole formalism in general relativity, in Recent Developments in General Relativity, Pergamon Press, New York U.S.A. (1962), pg. 465. · Zbl 0097.42402 [66] Taub, AH, Motion of test bodies in general relativity, J. Math. Phys., 5, 112, (1964) · Zbl 0119.23901 [67] Dixon, WG, A covariant multipole formalism for extended test bodies in general relativity, Nuovo Cim., 34, 317, (1964) · Zbl 0124.22201 [68] Ohashi, A., Multipole particle in relativity, Phys. Rev., D 68, 044009, (2003) · Zbl 1244.83007 [69] Steinhoff, J., Canonical formulation of spin in general relativity, Annalen Phys., 523, 296, (2011) · Zbl 1218.83018 [70] Blanchet, L., Post-Newtonian theory and the two-body problem, Fundam. Theor. Phys., 162, 125, (2011) · Zbl 1213.83034 [71] J. Frenkel, Die Elektrodynamik des rotierenden Elektrons (in German), Z. Phys.37 (1926) 243 [INSPIRE]. · JFM 52.0960.06 [72] F.A.E. Pirani, On the physical significance of the Riemann tensor, Acta Phys. Polon.15 (1956) 389 [Gen. Rel. Grav.41 (2009) 1215] [INSPIRE]. · Zbl 1178.83050 [73] Deriglazov, AA; Ramírez, WG, Lagrangian formulation for mathisson-papapetrou-tulczyjew-Dixon (MPTD) equations, Phys. Rev., D 92, 124017, (2015) [74] A.A. Deriglazov and W.G. Ramírez, Ultra-relativistic spinning particle and a rotating body in external fields, arXiv:1511.00645 [INSPIRE]. [75] H. Stephani, E. Herlt, M. MacCullum, C. Hoenselaers and D. Kramer, Exact solutions of Einstein’s field equations, 2\^{}{nd} ed., Cambridge University Press, Cambridge U.K. (2003). · Zbl 1057.83004 [76] Osborn, H.; Shore, GM, Correlation functions of the energy momentum tensor on spaces of constant curvature, Nucl. Phys., B 571, 287, (2000) · Zbl 1028.81510 [77] G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE]. [78] Penedones, J., Writing CFT correlation functions as AdS scattering amplitudes, JHEP, 03, 025, (2011) · Zbl 1301.81154 [79] Fitzpatrick, AL; Kaplan, J.; Penedones, J.; Raju, S.; Rees, BC, A natural language for AdS/CFT correlators, JHEP, 11, 095, (2011) · Zbl 1306.81225 [80] Marolf, D.; Morrison, IA, The IR stability of de Sitter QFT: results at all orders, Phys. Rev., D 84, 044040, (2011) [81] Hollands, S., Correlators, Feynman diagrams and quantum no-hair in de Sitter spacetime, Commun. Math. Phys., 319, 1, (2013) · Zbl 1278.81136 [82] Marolf, D.; Morrison, IA; Srednicki, M., Perturbative S-matrix for massive scalar fields in global de Sitter space, Class. Quant. Grav., 30, 155023, (2013) · Zbl 1273.83008 [83] Korai, Y.; Tanaka, T., Quantum field theory in the flat chart of de Sitter space, Phys. Rev., D 87, 024013, (2013) [84] Becchi, C.; Rouet, A.; Stora, R., Renormalization of the abelian Higgs-kibble model, Commun. Math. Phys., 42, 127, (1975) [85] Becchi, C.; Rouet, A.; Stora, R., Renormalization of gauge theories, Annals Phys., 98, 287, (1976) [86] Kugo, T.; Ojima, I., Manifestly covariant canonical formulation of Yang-Mills field theories. 1. the case of Yang-Mills fields of Higgs-kibble type in Landau gauge, Prog. Theor. Phys., 60, 1869, (1978) · Zbl 1098.81591 [87] S. Weinberg, The quantum theory of fields, volume 2: modern applications, Cambridge University Press, Cambridge U.K. (2005). · Zbl 1069.00007 [88] Barnich, G.; Brandt, F.; Henneaux, M., Local BRST cohomology in gauge theories, Phys. Rept., 338, 439, (2000) · Zbl 1097.81571 [89] Duetsch, M.; Fredenhagen, K., Causal perturbation theory in terms of retarded products and a proof of the action Ward identity, Rev. Math. Phys., 16, 1291, (2004) · Zbl 1084.81054 [90] Hollands, S., Renormalized quantum Yang-Mills fields in curved spacetime, Rev. Math. Phys., 20, 1033, (2008) · Zbl 1161.81022 [91] K.A. Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory, Ph.D. thesis, Universität Hamburg, Hamburg Germany (2011) [arXiv:1111.5130] [INSPIRE]. [92] Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys., 317, 697, (2013) · Zbl 1263.81245 [93] Rejzner, K., Remarks on local symmetry invariance in perturbative algebraic quantum field theory, Annales Henri Poincaré, 16, 205, (2015) · Zbl 1306.81137 [94] M.B. Fröb, J. Holland and S. Hollands, All-order bounds for correlation functions of gauge-invariant operators in Yang-Mills theory, arXiv:1511.09425 [INSPIRE]. [95] Gracia-Bondía, JM; Gutiérrez-Garro, H.; Várilly, JC, Improved Epstein-glaser renormalization in x-space versus differential renormalization, Nucl. Phys., B 886, 824, (2014) · Zbl 1325.81117 [96] NIST Digital Library of Mathematical Functions webpage, http://dlmf.nist.gov. [97] S. Weinberg, The quantum theory of fields, volume 1: foundations, Cambridge University Press, Cambridge U.K. (2005). · Zbl 1069.81500 [98] D.Z. Freedman and A. van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012). · Zbl 1245.83001 [99] Burns, D.; Pilaftsis, A., Matter quantum corrections to the graviton self-energy and the Newtonian potential, Phys. Rev., D 91, 064047, (2015) [100] D. Burns. private communication, (2016). [101] F.W.J. Olver, Asymptotics and special functions, A.K. Peters, Wellesley U.S.A. (1997). · Zbl 0982.41018 [102] N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Plante and P. Vanhove, Light-like scattering in quantum gravity, arXiv:1609.07477 [INSPIRE]. · Zbl 1390.83057 [103] R.P. Geroch and J.H. Traschen, Strings and other distributional sources in general relativity, Phys. Rev.D 36 (1987) 1017 [Conf. Proc.C 861214 (1986) 138] [INSPIRE]. [104] Balasin, H.; Nachbagauer, H., Distributional energy momentum tensor of the Kerr-Newman space-time family, Class. Quant. Grav., 11, 1453, (1994) [105] Kawai, T.; Sakane, E., Distributional energy-momentum densities of Schwarzschild space-time, Prog. Theor. Phys., 98, 69, (1997) [106] Balasin, H., Distributional energy momentum tensor of the extended Kerr geometry, Class. Quant. Grav., 14, 3353, (1997) · Zbl 0904.53063 [107] N.R. Pantoja and H. Rago, Energy-momentum tensor valued distributions for the Schwarzschild and Reissner-Nordström geometries, gr-qc/9710072 [INSPIRE]. [108] Iliopoulos, J.; Tomaras, TN; Tsamis, NC; Woodard, RP, Perturbative quantum gravity and newton’s law on a flat Robertson-Walker background, Nucl. Phys., B 534, 419, (1998) · Zbl 1041.83507
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