Photon propagator for inflation in the general covariant gauge. (English) Zbl 07917321
Summary: Photon propagator for power-law inflation is considered in the general covariant gauges within the canonical quantization formalism. Photon mode functions in covariant gauges are considerably more complicated than their scalar counterparts, except for the special choice of the gauge-fixing parameter we call the simple covariant gauge. We explicitly construct the position space photon propagator in the simple covariant gauge, and find the result considerably more complicated than its scalar counterpart. This is because of the need for explicitly inverting the Laplace operator acting on the scalar propagator, which results in Appell’s fourth function. Our propagator correctly reproduces the de Sitter and flat space limits. We use this propagator to compute two simple observables: the off-coincident field strength-field strength correlator and the energy-momentum tensor, both of which yield consistent results. As a spinoff of our computation we also give the exact expression for the Coulomb gauge propagator in power-law inflation in arbitrary dimensions.
MSC:
| 83Cxx | General relativity |
| 81Txx | Quantum field theory; related classical field theories |
| 83Fxx | Relativistic cosmology |
Software:
DLMFReferences:
| [1] | Bezrukov, FL; Shaposhnikov, M., The Standard Model Higgs boson as the inflaton, Phys. Lett. B, 659, 703, 2008 · doi:10.1016/j.physletb.2007.11.072 |
| [2] | George, DP; Mooij, S.; Postma, M., Effective action for the Abelian Higgs model in FLRW, JCAP, 11, 043, 2012 · doi:10.1088/1475-7516/2012/11/043 |
| [3] | Bezrukov, F., The Higgs field as an inflaton, Class. Quant. Grav., 30, 2013 · Zbl 1277.83003 · doi:10.1088/0264-9381/30/21/214001 |
| [4] | George, DP; Mooij, S.; Postma, M., Quantum corrections in Higgs inflation: the real scalar case, JCAP, 02, 024, 2014 · doi:10.1088/1475-7516/2014/02/024 |
| [5] | Obata, I.; Miura, T.; Soda, J., Dynamics of Electroweak Gauge Fields during and after Higgs Inflation, Phys. Rev. D, 90, 2014 · doi:10.1103/PhysRevD.90.045005 |
| [6] | Bezrukov, F.; Rubio, J.; Shaposhnikov, M., Living beyond the edge: Higgs inflation and vacuum metastability, Phys. Rev. D, 92, 2015 · doi:10.1103/PhysRevD.92.083512 |
| [7] | George, DP; Mooij, S.; Postma, M., Quantum corrections in Higgs inflation: the Standard Model case, JCAP, 04, 006, 2016 · doi:10.1088/1475-7516/2016/04/006 |
| [8] | Ema, Y.; Jinno, R.; Mukaida, K.; Nakayama, K., Violent Preheating in Inflation with Nonminimal Coupling, JCAP, 02, 045, 2017 · doi:10.1088/1475-7516/2017/02/045 |
| [9] | Adshead, P.; Giblin, JT; Weiner, ZJ, Non-Abelian gauge preheating, Phys. Rev. D, 96, 2017 · doi:10.1103/PhysRevD.96.123512 |
| [10] | Sfakianakis, EI; van de Vis, J., Preheating after Higgs Inflation: Self-Resonance and Gauge boson production, Phys. Rev. D, 99, 2019 · doi:10.1103/PhysRevD.99.083519 |
| [11] | Adshead, P.; Giblin, JT; Grutkoski, R.; Weiner, ZJ, Gauge preheating with full general relativity, JCAP, 03, 017, 2024 · Zbl 07868062 · doi:10.1088/1475-7516/2024/03/017 |
| [12] | Pajer, E.; Peloso, M., A review of Axion Inflation in the era of Planck, Class. Quant. Grav., 30, 2013 · Zbl 1277.83008 · doi:10.1088/0264-9381/30/21/214002 |
| [13] | Adshead, P.; Giblin, JT; Scully, TR; Sfakianakis, EI, Gauge-preheating and the end of axion inflation, JCAP, 12, 034, 2015 · doi:10.1088/1475-7516/2015/12/034 |
| [14] | Adshead, P.; Giblin, JT; Scully, TR; Sfakianakis, EI, Magnetogenesis from axion inflation, JCAP, 10, 039, 2016 · doi:10.1088/1475-7516/2016/10/039 |
| [15] | Domcke, V.; Mukaida, K., Gauge Field and Fermion Production during Axion Inflation, JCAP, 11, 020, 2018 · Zbl 1527.83122 · doi:10.1088/1475-7516/2018/11/020 |
| [16] | Gorbar, EV; Schmitz, K.; Sobol, OO; Vilchinskii, SI, Gauge-field production during axion inflation in the gradient expansion formalism, Phys. Rev. D, 104, 2021 · doi:10.1103/PhysRevD.104.123504 |
| [17] | Caravano, A.; Komatsu, E.; Lozanov, KD; Weller, J., Lattice simulations of Abelian gauge fields coupled to axions during inflation, Phys. Rev. D, 105, 2022 · doi:10.1103/PhysRevD.105.123530 |
| [18] | Figueroa, DG; Lizarraga, J.; Urio, A.; Urrestilla, J., Strong Backreaction Regime in Axion Inflation, Phys. Rev. Lett., 131, 2023 · doi:10.1103/PhysRevLett.131.151003 |
| [19] | B. Ratra, Cosmological ‘seed’ magnetic field from inflation, Astrophys. J. Lett.391 (1992) L1 [INSPIRE]. |
| [20] | L. Parker, Particle creation in expanding universes, Phys. Rev. Lett.21 (1968) 562 [INSPIRE]. · Zbl 0186.58603 |
| [21] | L. Parker, Quantized fields and particle creation in expanding universes. 1, Phys. Rev.183 (1969) 1057 [INSPIRE]. · Zbl 0186.58603 |
| [22] | L. Parker, Quantized fields and particle creation in expanding universes. 2, Phys. Rev. D3 (1971) 346 [INSPIRE]. · Zbl 0186.58603 |
| [23] | V.F. Mukhanov and G.V. Chibisov, Quantum Fluctuations and a Nonsingular Universe, JETP Lett.33 (1981) 532 [INSPIRE]. |
| [24] | A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett.30 (1979) 682 [INSPIRE]. |
| [25] | E.O. Kahya and R.P. Woodard, Charged scalar self-mass during inflation, Phys. Rev. D72 (2005) 104001 [gr-qc/0508015] [INSPIRE]. |
| [26] | E.O. Kahya and R.P. Woodard, One Loop Corrected Mode Functions for SQED during Inflation, Phys. Rev. D74 (2006) 084012 [gr-qc/0608049] [INSPIRE]. |
| [27] | T. Prokopec, N.C. Tsamis and R.P. Woodard, Two Loop Scalar Bilinears for Inflationary SQED, Class. Quant. Grav.24 (2007) 201 [gr-qc/0607094] [INSPIRE]. · Zbl 1133.83341 |
| [28] | Chen, X.; Wang, Y.; Xianyu, Z-Z, Standard Model Background of the Cosmological Collider, Phys. Rev. Lett., 118, 2017 · doi:10.1103/PhysRevLett.118.261302 |
| [29] | F. Fabián González and T. Prokopec, Renormalization group approach to scalar quantum electrodynamics on de Sitter, arXiv:1611.07854 [INSPIRE]. |
| [30] | Chen, X.; Wang, Y.; Xianyu, Z-Z, Standard Model Mass Spectrum in Inflationary Universe, JHEP, 04, 058, 2017 · Zbl 1378.85005 · doi:10.1007/JHEP04(2017)058 |
| [31] | Popov, FK, Debye mass in de Sitter space, JHEP, 06, 033, 2018 · Zbl 1395.83095 · doi:10.1007/JHEP06(2018)033 |
| [32] | Glavan, D.; Rigopoulos, G., One-loop electromagnetic correlators of SQED in power-law inflation, JCAP, 02, 021, 2021 · Zbl 1484.83120 · doi:10.1088/1475-7516/2021/02/021 |
| [33] | T. Prokopec, O. Tornkvist and R.P. Woodard, Photon mass from inflation, Phys. Rev. Lett.89 (2002) 101301 [astro-ph/0205331] [INSPIRE]. |
| [34] | T. Prokopec, O. Tornkvist and R.P. Woodard, One loop vacuum polarization in a locally de Sitter background, Annals Phys.303 (2003) 251 [gr-qc/0205130] [INSPIRE]. · Zbl 1028.81043 |
| [35] | T. Prokopec and R.P. Woodard, Vacuum polarization and photon mass in inflation, Am. J. Phys.72 (2004) 60 [astro-ph/0303358] [INSPIRE]. · Zbl 1074.83011 |
| [36] | T. Prokopec and E. Puchwein, Photon mass generation during inflation: de Sitter invariant case, JCAP04 (2004) 007 [astro-ph/0312274] [INSPIRE]. |
| [37] | Prokopec, T.; Tsamis, NC; Woodard, RP, Stochastic Inflationary Scalar Electrodynamics, Annals Phys., 323, 1324, 2008 · Zbl 1151.81030 · doi:10.1016/j.aop.2007.08.008 |
| [38] | Prokopec, T.; Tsamis, NC; Woodard, RP, Two loop stress-energy tensor for inflationary scalar electrodynamics, Phys. Rev. D, 78, 2008 · doi:10.1103/PhysRevD.78.043523 |
| [39] | A.A. Starobinsky, Stochastic de sitter (inflationary) stage in the early universe, Lect. Notes Phys.246 (1986) 107 [INSPIRE]. |
| [40] | A.A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the De Sitter background, Phys. Rev. D50 (1994) 6357 [astro-ph/9407016] [INSPIRE]. |
| [41] | M.S. Turner and L.M. Widrow, Inflation Produced, Large Scale Magnetic Fields, Phys. Rev. D37 (1988) 2743 [INSPIRE]. |
| [42] | E.A. Calzetta, A. Kandus and F.D. Mazzitelli, Primordial magnetic fields induced by cosmological particle creation, Phys. Rev. D57 (1998) 7139 [astro-ph/9707220] [INSPIRE]. |
| [43] | M. Giovannini and M.E. Shaposhnikov, Primordial magnetic fields from inflation?, Phys. Rev. D62 (2000) 103512 [hep-ph/0004269] [INSPIRE]. |
| [44] | T. Prokopec, Cosmological magnetic fields from photon coupling to fermions and bosons in inflation, astro-ph/0106247 [INSPIRE]. |
| [45] | A.-C. Davis, K. Dimopoulos, T. Prokopec and O. Tornkvist, Primordial spectrum of gauge fields from inflation, Phys. Lett. B501 (2001) 165 [astro-ph/0007214] [INSPIRE]. · Zbl 0977.83100 |
| [46] | Durrer, R.; Neronov, A., Cosmological Magnetic Fields: Their Generation, Evolution and Observation, Astron. Astrophys. Rev., 21, 62, 2013 · doi:10.1007/s00159-013-0062-7 |
| [47] | B.L. Hu and K. Shiokawa, Wave propagation in stochastic space-times: Localization, amplification and particle creation, Phys. Rev. D57 (1998) 3474 [gr-qc/9708023] [INSPIRE]. |
| [48] | Dalvit, DAR; Mazzitelli, FD; Molina-Paris, C., One loop graviton corrections to Maxwell’s equations, Phys. Rev. D, 63, 2001 · doi:10.1103/PhysRevD.63.084023 |
| [49] | Leonard, KE; Prokopec, T.; Woodard, RP, Covariant Vacuum Polarizations on de Sitter Background, Phys. Rev. D, 87, 2013 · doi:10.1103/PhysRevD.87.044030 |
| [50] | Leonard, KE; Woodard, RP, Graviton Corrections to Vacuum Polarization during Inflation, Class. Quant. Grav., 31, 2014 · Zbl 1287.83050 · doi:10.1088/0264-9381/31/1/015010 |
| [51] | Glavan, D.; Miao, SP; Prokopec, T.; Woodard, RP, Electrodynamic Effects of Inflationary Gravitons, Class. Quant. Grav., 31, 2014 · Zbl 1300.83026 · doi:10.1088/0264-9381/31/17/175002 |
| [52] | Wang, CL; Woodard, RP, Excitation of Photons by Inflationary Gravitons, Phys. Rev. D, 91, 2015 · doi:10.1103/PhysRevD.91.124054 |
| [53] | Glavan, D.; Miao, SP; Prokopec, T.; Woodard, RP, Graviton Loop Corrections to Vacuum Polarization in de Sitter in a General Covariant Gauge, Class. Quant. Grav., 32, 2015 · Zbl 1327.83115 · doi:10.1088/0264-9381/32/19/195014 |
| [54] | Wang, CL; Woodard, RP, One-loop quantum electrodynamic correction to the gravitational potentials on de Sitter spacetime, Phys. Rev. D, 92, 2015 · doi:10.1103/PhysRevD.92.084008 |
| [55] | Glavan, D.; Miao, SP; Prokopec, T.; Woodard, RP, One loop graviton corrections to dynamical photons in de Sitter, Class. Quant. Grav., 34, 2017 · Zbl 1368.83034 · doi:10.1088/1361-6382/aa61da |
| [56] | Miao, SP; Prokopec, T.; Woodard, RP, Scalar enhancement of the photon electric field by the tail of the graviton propagator, Phys. Rev. D, 98, 2018 · doi:10.1103/PhysRevD.98.025022 |
| [57] | Glavan, D.; Miao, SP; Prokopec, T.; Woodard, RP, Explaining large electromagnetic logarithms from loops of inflationary gravitons, JHEP, 08, 195, 2023 · Zbl 07749070 · doi:10.1007/JHEP08(2023)195 |
| [58] | Planck collaboration, Planck 2018 results. X. Constraints on inflation, Astron. Astrophys.641 (2020) A10 [arXiv:1807.06211] [INSPIRE]. |
| [59] | F. Lucchin and S. Matarrese, Power Law Inflation, Phys. Rev. D32 (1985) 1316 [INSPIRE]. |
| [60] | D. La and P.J. Steinhardt, Extended Inflationary Cosmology, Phys. Rev. Lett.62 (1989) 376 [Erratum ibid.62 (1989) 1066] [INSPIRE]. |
| [61] | Koksma, JF; Prokopec, T., Fermion Propagator in Cosmological Spaces with Constant Deceleration, Class. Quant. Grav., 26, 2009 · Zbl 1170.83495 · doi:10.1088/0264-9381/26/12/125003 |
| [62] | Bilandzic, A.; Prokopec, T., Quantum radiative corrections to slow-roll inflation, Phys. Rev. D, 76, 2007 · doi:10.1103/PhysRevD.76.103507 |
| [63] | Janssen, TM; Miao, SP; Prokopec, T.; Woodard, RP, The Hubble Effective Potential, JCAP, 05, 003, 2009 · doi:10.1088/1475-7516/2009/05/003 |
| [64] | Herranen, M.; Markkanen, T.; Tranberg, A., Quantum corrections to scalar field dynamics in a slow-roll space-time, JHEP, 05, 026, 2014 · doi:10.1007/JHEP05(2014)026 |
| [65] | Miao, SP; Woodard, RP, Fine Tuning May Not Be Enough, JCAP, 09, 022, 2015 · doi:10.1088/1475-7516/2015/09/022 |
| [66] | Markkanen, T., Light scalars on cosmological backgrounds, JHEP, 01, 116, 2018 · Zbl 1384.83071 · doi:10.1007/JHEP01(2018)116 |
| [67] | Fröb, MB, One-loop quantum gravitational backreaction on the local Hubble rate, Class. Quant. Grav., 36, 2019 · Zbl 1476.83044 · doi:10.1088/1361-6382/ab10fb |
| [68] | Miao, SP; Park, S.; Woodard, RP, Ricci Subtraction for Cosmological Coleman-Weinberg Potentials, Phys. Rev. D, 100, 2019 · doi:10.1103/PhysRevD.100.103503 |
| [69] | Kyriazis, A.; Miao, SP; Tsamis, NC; Woodard, RP, Inflaton effective potential for general ϵ, Phys. Rev. D, 102, 2020 · doi:10.1103/PhysRevD.102.025024 |
| [70] | Katuwal, S.; Miao, SP; Woodard, RP, Inflaton effective potential from photons for general ϵ, Phys. Rev. D, 103, 2021 · doi:10.1103/PhysRevD.103.105007 |
| [71] | Lima, WCC, Graviton backreaction on the local cosmological expansion in slow-roll inflation, Class. Quant. Grav., 38, 2021 · Zbl 1510.83031 · doi:10.1088/1361-6382/abfaeb |
| [72] | Glavan, D.; Prokopec, T., When tadpoles matter: one-loop corrections for spectator Higgs in inflation, JHEP, 10, 063, 2023 · Zbl 07774668 · doi:10.1007/JHEP10(2023)063 |
| [73] | Glavan, D.; Marunović, A.; Prokopec, T.; Zahraee, Z., Abelian Higgs model in power-law inflation: the propagators in the unitary gauge, JHEP, 09, 165, 2020 · doi:10.1007/JHEP09(2020)165 |
| [74] | Glavan, D.; Prokopec, T., Photon propagator in de Sitter space in the general covariant gauge, JHEP, 05, 126, 2023 · Zbl 07701941 · doi:10.1007/JHEP05(2023)126 |
| [75] | Glavan, D.; Prokopec, T., Even the photon propagator must break de Sitter symmetry, Phys. Lett. B, 841, 2023 · Zbl 1523.83078 · doi:10.1016/j.physletb.2023.137928 |
| [76] | N.C. Tsamis and R.P. Woodard, A Maximally symmetric vector propagator, J. Math. Phys.48 (2007) 052306 [gr-qc/0608069] [INSPIRE]. · Zbl 1144.81417 |
| [77] | B. Allen and T. Jacobson, Vector Two Point Functions in Maximally Symmetric Spaces, Commun. Math. Phys.103 (1986) 669 [INSPIRE]. · Zbl 0632.53060 |
| [78] | Fröb, MB; Higuchi, A., Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincaré patch of de Sitter space, J. Math. Phys., 55, 2014 · Zbl 1294.81135 · doi:10.1063/1.4879496 |
| [79] | Glavan, D., Photon quantization in cosmological spaces, Phys. Rev. D, 109, 2024 · doi:10.1103/PhysRevD.109.085014 |
| [80] | Youssef, A., Infrared behavior and gauge artifacts in de Sitter spacetime: The photon field, Phys. Rev. Lett., 107, 2011 · doi:10.1103/PhysRevLett.107.021101 |
| [81] | Rendell, N., Large-distance behavior of the massless vector two-point function in de Sitter spacetime, Int. J. Mod. Phys. D, 27, 1843005, 2018 · doi:10.1142/S0218271818430058 |
| [82] | N.A. Chernikov and E.A. Tagirov, Quantum theory of scalar fields in de Sitter space-time, Ann. Inst. H. Poincaré A Phys. Theor.9 (1968) 109 [INSPIRE]. · Zbl 0162.57802 |
| [83] | T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A360 (1978) 117 [INSPIRE]. |
| [84] | F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark eds., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge (2010). · Zbl 1198.00002 |
| [85] | F.W.J. Olver et al. eds., NIST Digital Library of Mathematical Functions, Release 1.1.10 of 2023-06-15, https://dlmf.nist.gov/. · Zbl 1019.65001 |
| [86] | J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys.2 (1961) 407 [INSPIRE]. · Zbl 0098.43503 |
| [87] | K.T. Mahanthappa, Multiple production of photons in quantum electrodynamics, Phys. Rev.126 (1962) 329 [INSPIRE]. · Zbl 0117.23906 |
| [88] | P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 1, J. Math. Phys.4 (1963) 1 [INSPIRE]. · Zbl 0126.24401 |
| [89] | P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 2, J. Math. Phys.4 (1963) 12 [INSPIRE]. · Zbl 0126.24401 |
| [90] | L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz.47 (1964) 1515 [INSPIRE]. |
| [91] | R.D. Jordan, Effective Field Equations for Expectation Values, Phys. Rev. D33 (1986) 444 [INSPIRE]. |
| [92] | E. Calzetta and B.L. Hu, Closed Time Path Functional Formalism in Curved Space-Time: Application to Cosmological Back Reaction Problems, Phys. Rev. D35 (1987) 495 [INSPIRE]. |
| [93] | J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc.739 (2004) 3 [hep-ph/0409233] [INSPIRE]. |
| [94] | D. Glavan and T. Prokopec, A pedestrian introduction to non-equilibrium QFT, https://webspace.science.uu.nl/ proko101/LecturenotesNonEquilQFT.pdf. |
| [95] | L.H. Ford and L. Parker, Infrared Divergences in a Class of Robertson-Walker Universes, Phys. Rev. D16 (1977) 245 [INSPIRE]. |
| [96] | B. Allen, Vacuum States in de Sitter Space, Phys. Rev. D32 (1985) 3136 [INSPIRE]. |
| [97] | B. Allen and A. Folacci, The Massless Minimally Coupled Scalar Field in De Sitter Space, Phys. Rev. D35 (1987) 3771 [INSPIRE]. |
| [98] | Janssen, TM; Prokopec, T., Regulating the infrared by mode matching: A Massless scalar in expanding spaces with constant deceleration, Phys. Rev. D, 83, 2011 · doi:10.1103/PhysRevD.83.084035 |
| [99] | Janssen, T.; Prokopec, T., A graviton propagator for inflation, Class. Quant. Grav., 25, 2008 · Zbl 1136.83040 · doi:10.1088/0264-9381/25/5/055007 |
| [100] | I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, seventh edition, Elsevier/Academic Press (2007) [INSPIRE]. · Zbl 1208.65001 |
| [101] | A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series, Volume 2: Special Functions, Gordon and Breach (1986). · Zbl 0733.00004 |
| [102] | Janssen, TM; Miao, SP; Prokopec, T.; Woodard, RP, Infrared Propagator Corrections for Constant Deceleration, Class. Quant. Grav., 25, 2008 · Zbl 1255.83151 · doi:10.1088/0264-9381/25/24/245013 |
| [103] | V. Moncrief, Space-Time Symmetries and Linearization Stability of the Einstein Equations. 2, J. Math. Phys.17 (1976) 1893 [INSPIRE]. |
| [104] | J.M. Arms, Linearization stability of gravitational and gauge fields, J. Math. Phys.20 (1979) 443 [INSPIRE]. |
| [105] | A. Higuchi, Quantum linearization instabilities of de Sitter space-time. 1, Class. Quant. Grav.8 (1991) 1961 [INSPIRE]. |
| [106] | A. Higuchi, Quantum linearization instabilities of de Sitter space-time. 2, Class. Quant. Grav.8 (1991) 1983 [INSPIRE]. |
| [107] | N.C. Tsamis and R.P. Woodard, The structure of perturbative quantum gravity on a De Sitter background, Commun. Math. Phys.162 (1994) 217 [INSPIRE]. · Zbl 0809.53084 |
| [108] | D. Marolf, Refined algebraic quantization: Systems with a single constraint, gr-qc/9508015 [INSPIRE]. · Zbl 0887.46050 |
| [109] | Marolf, D.; Morrison, IA, Group Averaging of massless scalar fields in 1+1 de Sitter, Class. Quant. Grav., 26, 2009 · Zbl 1159.83353 · doi:10.1088/0264-9381/26/3/035001 |
| [110] | Marolf, D.; Morrison, IA, Group Averaging for de Sitter free fields, Class. Quant. Grav., 26, 2009 · Zbl 1181.83059 · doi:10.1088/0264-9381/26/23/235003 |
| [111] | Miao, SP; Tsamis, NC; Woodard, RP, Transforming to Lorentz Gauge on de Sitter, J. Math. Phys., 50, 2009 · Zbl 1372.81127 · doi:10.1063/1.3266179 |
| [112] | Gibbons, J.; Higuchi, A., Removing the Faddeev-Popov zero modes from Yang-Mills theory in spacetimes with compact spatial sections, Phys. Rev. D, 91, 2015 · doi:10.1103/PhysRevD.91.024006 |
| [113] | Altas, E.; Kilicarslan, E.; Tekin, B., Einstein-Yang-Mills theory: gauge invariant charges and linearization instability, Eur. Phys. J. C, 81, 648, 2021 · doi:10.1140/epjc/s10052-021-09460-7 |
| [114] | S.N. Gupta, Theory of longitudinal photons in quantum electrodynamics, Proc. Phys. Soc. A63 (1950) 681 [INSPIRE]. · Zbl 0040.42403 |
| [115] | K. Bleuler, A new method of treatment of the longitudinal and scalar photons, Helv. Phys. Acta23 (1950) 567 [INSPIRE]. · Zbl 0040.42404 |
| [116] | Falceto, F., Canonical Quantization of the Electromagnetic Field in Arbitrary ξ-Gauge, Annales Henri Poincaré, 25, 517, 2024 · Zbl 1540.81172 · doi:10.1007/s00023-022-01259-w |
| [117] | Cotaescu, II; Crucean, C., The quantum theory of the free Maxwell field on the de Sitter expanding universe, Prog. Theor. Phys., 124, 1051, 2010 · Zbl 1213.83141 · doi:10.1143/PTP.124.1051 |
| [118] | Belokogne, A.; Folacci, A., Stueckelberg massive electromagnetism in curved spacetime: Hadamard renormalization of the stress-energy tensor and the Casimir effect, Phys. Rev. D, 93, 2016 · doi:10.1103/PhysRevD.93.044063 |
| [119] | Belokogne, A.; Folacci, A.; Queva, J., Stueckelberg massive electromagnetism in de Sitter and anti-de Sitter spacetimes: Two-point functions and renormalized stress-energy tensors, Phys. Rev. D, 94, 2016 · doi:10.1103/PhysRevD.94.105028 |
| [120] | D.M. Capper and M.J. Duff, Trace anomalies in dimensional regularization, Nuovo Cim. A23 (1974) 173 [INSPIRE]. |
| [121] | L.S. Brown and J.P. Cassidy, Stress Tensor Trace Anomaly in a Gravitational Metric: General Theory, Maxwell Field, Phys. Rev. D15 (1977) 2810 [INSPIRE]. |
| [122] | S.L. Adler, J. Lieberman and Y.J. Ng, Regularization of the Stress Energy Tensor for Vector and Scalar Particles Propagating in a General Background Metric, Annals Phys.106 (1977) 279 [INSPIRE]. |
| [123] | L.S. Brown and J.P. Cassidy, Stress Tensors and their Trace Anomalies in Conformally Flat Space-Times, Phys. Rev. D16 (1977) 1712 [INSPIRE]. |
| [124] | R. Endo, Gauge Dependence of the Gravitational Conformal Anomaly for the Electromagnetic Field, Prog. Theor. Phys.71 (1984) 1366 [INSPIRE]. · Zbl 1046.83505 |
| [125] | Y.A. Brychkov, Higher derivatives of the Bessel functions with respect to the order, Integral Transform. Spec. Funct.27 (2016) 566. · Zbl 1351.33005 |
| [126] | S.O. Rice, On contour integrals for the product of two Bessel functions, Quart. J. Math.1 (1935) 52. · JFM 61.0398.02 |
| [127] | W.N. Bailey, Some Infinite Integrals Involving Bessel Functions, Proc. Lond. Math. Soc.2-40 (1936) 37. · Zbl 0013.30701 |
| [128] | Y.A. Brychkov and N. Saad, On some formulas for the Appell function F_4(a, b; c, c^′; w, z), Integral Transform. Spec. Funct.28 (2017) 629. · Zbl 1376.33004 |
| [129] | H. Exton, On the system of partial differential equations associated with Appell’s function F_4, J. Phys. A28 (1995) 631. · Zbl 0856.33012 |
| [130] | J.L. López, P. Pagola and E. Pérez Sinusía, Asymptotics of the first Appell functionF1with large parameters, Integral Transform. Spec. Funct.24 (2013) 715. · Zbl 1280.33013 |
| [131] | J.L. López, P. Pagola and E.P. Sinusía, Asymptotics of the first Appell functionF1with large parameters II, Integral Transform. Spec. Funct.24 (2013) 982. · Zbl 1282.33022 |
| [132] | J.L. López et al., in progress. |
| [133] | K. Peeters, Introducing Cadabra: A symbolic computer algebra system for field theory problems, hep-th/0701238 [INSPIRE]. · Zbl 1196.68333 |
| [134] | Peeters, K., A Field-theory motivated approach to symbolic computer algebra, Comput. Phys. Commun., 176, 550, 2007 · Zbl 1196.68333 · doi:10.1016/j.cpc.2007.01.003 |
| [135] | K. Peeters, Cadabra2: computer algebra for field theory revisited, J. Open Source Softw.3 (2018) 1118 [INSPIRE]. |
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