Loop e ects in QED in space with a black hole

Transcription

1 Loop e ects in QED in space with a black hole Slava Emelyanov Institute of Theoretical Physics Karlsruhe Institute of Technology viacheslav.emelyanov@kit.edu DESY Theory Workshop Hamburg, 28th September /22

2 Overview Schwarzschild black hole with quantum fields. Schwarzschild-black-hole evaporation. Fermion- and vector-field propagator Radiative corrections in QED. Photon self-energy. Electron/positron self-energy One-loop e ects induced by small black holes. How small?. Reminder: Hot neutral e e + plasma. Debye-like screening. Modified photon dispersion relation. Modified electron dispersion relation Concluding remarks 2/22

3 Schwarzschild black hole with quantum fields. Sketch of Schwarzschild-black-hole evaporation Inward-negative-energy ow R Outward-positive-energy ow \Black-hole atmosphere" W.G. Unruh, Phys. Rev. D15, 365(1977) P. Candelas, Phys. Rev. D21, 2185(1980) S.B. Giddings, Phys. Lett. B754, 39(2016) 3/22

4 Schwarzschild black hole with quantum fields. Fermion- and vector-field propagator Black hole evaporate through quantum fields m Field propagators get a correction describing the energy flux 4/22

5 Schwarzschild black hole with quantum fields. Fermion- and vector-field propagator: Electron/positron field Under the gravitational collapse lack Hole fermion propagator in near- & far-horizon region becomes i(/p+m) p 2 m 2 +i" {z } empty space =) i( /p+m) p 2 m p (/p+m) g R +i" e p0 p (p0 2 m 2 ) 1 2 n +1 {z } space with a black hole 5/22

6 Schwarzschild black hole with quantum fields. Fermion- and vector-field propagator: Photon field Under the gravitational collapse lack Hole photon propagator in near- & far-horizon region becomes i µ k 2 +i" {z } empty space =) i µ k 2 +i" 4 2 g R k µ e k 0 1 (k k 0n) {z } space with a black hole 6/22

7 Schwarzschild black hole with quantum fields. Fermion- and vector-field propagator: Parameters g R and The parameters g R and can be shown to be given by g R 27r 2 H 16 ( +1/R 2, R r H, 4/r 2 H, R r H, and ( = 1 1, R rh, T H 2, R r H, where T H is the Hawking temperature parameter. V.A. Emelyanov, Annalen der Physik, (2017) 7/22

8 Radiative corrections in QED. Vacuum polarization tensor The classical Maxwell equation changes due to radiative corrections. In momentum space, it reads k 2 µ k µ k + µ (k) A (k) = 0, where vacuum polarization tensor is pictorially given by i µ (k) = µ k + µ k +2 µ k +O 3 8/22

9 Radiative corrections in QED. Vacuum polarization tensor The classical Maxwell equation changes due to radiative corrections. In momentum space, it reads k 2 µ k µ k + µ (k) A (k) = 0, where vacuum polarization tensor is pictorially given by i µ (k) = µ k + µ k +2 µ k +O 2 9/22

10 Radiative corrections in QED. Vacuum polarization tensor The vacuum polarization tensor can be represented as follows: µ (k) = T (k 0, k)p µ + L (k 0, k)q µ, where P µ & Q µ are projections being orthogonal to each other. The photon propagator then aquires the following form: G µ (k 0, k) = ip µ k 2 T (k 0, k)+i" + iq µ k 2 L (k 0, k)+i" H.A. Weldon, Phys. Rev. D26, 1394(1982) 10/22

11 Radiative corrections in QED. Electron/positron self-energy Loop diagrams in case of the fermion propagator leads to i /p m+i" =) i /p m (/p)+i", where, at the leading order of the approximation, it holds that i (/p) = p +O 2 M.E. Peskin & D.V. Schroeder, Introduction to QFT (1995) 11/22

12 Small black holes: One-loop e ects. How small? A black hole is said to be small if T H m e In this case, the electron/positron field can be considered as being e ectively massless (hard-thermal-loop approximation). In other words, small black holes have a mass in the following range: g. M g, where the lower bound is to have a quasi-equilibrium approximation. 12/22

13 Small black holes: One-loop e ects. Reminder: Isotropic neutral electron-positron plasma at T m e r D p γ M. Le Bellac, Thermal Field Theory (1996) 13/22

14 Small black holes: One-loop e ects. Debye-like screening: Spacetime with an evaporating black hole r R lack Hole r) R r H q L R 14/22

15 Small black holes: One-loop e ects. Debye-like screening: Spacetime with an evaporating black hole Detector lack Hole r) q L R 15/22

16 Small black holes: One-loop e ects. Debye-like screening: Spacetime with an evaporating black hole Detector r R lack Hole r) q R r H L R 16/22

17 Small black holes: One-loop e ects. Debye-like screening: Spacetime with an evaporating black hole In the far-from-horizon region, a point-like charge q at rest has the following (modified due to L (0, k) 6= 0) electrostatic potential: (r) q 4 r e r/r D with r D 2 m, where m e2 TL 2 TH 2 rh R 2 R 2. V.A. Emelyanov, Nucl. Phys. B919, 110(2017) Nucl. Phys. B921, 796(2017) 17/22

18 Small black holes: One-loop e ects. Modified photon dispersion relation Far horizon region Near horizon region lack Hole r H R R r H 18/22

19 Small black holes: One-loop e ects. Modified photon dispersion relation Due to T (k 0, k) 6= 0inthelimit k!k 0, the photon propagator gets its pole shifted, namely ( k0 2 k 2 = m /R 2 128, R r H, 1/rH 2, R r H, where R is a distance to the black-hole centre. V.A. Emelyanov, Nucl. Phys. B919, 110(2017) Nucl. Phys. B921, 796(2017) arxiv:hep-th/ /22

20 Small black holes: One-loop e ects. Modified electron dispersion relation The pole structure of the electron propagator is also modified: ( p0 2 p 2 me /R 2 256, R r H, 1/rH 2, R r H, where R is a distance to the black-hole centre. 20/22

21 Concluding remarks Is the Debye-like screening e ect testable? There are two problems: It is far from clear whether small black holes exist in nature. If existent, it is unclear how long one needs to wait to have at least one of these in the neighbourhood of earth at the distance not larger than roughly 10 3 km. 21/22

22 Concluding remarks Is the Einstein causality violated in the near-horizon region? In general, a negative mass-squared term implies either instability or causality violation. The instability is caused by modes with k applem 1/r H, but our approximation holds only for modes with k 1/r H. It means that there is a natural IR cuto being of order 1/r H. A posteriori we know that modes with momentum higher or equal than local space-time curvature is immaterial for particle physics. Thus, it seems that locality is violated near the event horizon of evaporating black holes. 22/22