1. Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010 Don’t Shake That Solenoid Too Hard: Particle Production From Aharonov-Bohm K. Jones-Smith, H. Mathur, and T. Vachaspati, Phys. Rev. D 81:043503,2010 Y.-Z.Chu, H. Mathur, and T. Vachaspati, Phys. Rev. D 82:063515,2010

2. Aharonov and Bohm (1959) Quantum Mechanics: Quantum Mechanics: Vector potential A μ is not merely computational crutch but indispensable for quantum dynamics of charged particles. Vector potential A μ is not merely computational crutch but indispensable for quantum dynamics of charged particles. (2010) Quantum Field Theory: (2010) Quantum Field Theory: Spontaneous pair production of charged particles just by shaking a thin solenoid. Spontaneous pair production of charged particles just by shaking a thin solenoid.

3. Setup B x y z ShaketheSolenoid!

4. B x y ze+e- e+e-Pop! Setup

5. Setup Effective theory of magnetic flux tube: Alford and Wilczek (1989). Effective theory of magnetic flux tube: Alford and Wilczek (1989). Bosonic or fermionic quantum electrodynamics (QED) Bosonic or fermionic quantum electrodynamics (QED) Bosonic QED Fermionic QED

6. I: Time dependent H The gauge potential A μ around a moving solenoid is time-dependent. The gauge potential A μ around a moving solenoid is time-dependent. Hamiltonian of QFT is explicitly time- dependent: H i = ∫d 3 x A μ J μ. Hamiltonian of QFT is explicitly time- dependent: H i = ∫d 3 x A μ J μ. Zero particle state (in the Heisenberg picture) at different times not the same vector – i.e. particle creation occurs. Zero particle state (in the Heisenberg picture) at different times not the same vector – i.e. particle creation occurs.

7. II: Aharonov-Bohm Interaction However thin the flux tube is, particle production will occur – purely quantum process. However thin the flux tube is, particle production will occur – purely quantum process. Pair production rate contains topological aspect. Pair production rate contains topological aspect. B ABQM QM: Amp. for paths belonging to different classes will differ by exp[i(integer)eФ] QM: Amp. for paths belonging to different classes will differ by exp[i(integer)eФ]

8. However thin the flux tube is, particle production will occur – purely quantum process. However thin the flux tube is, particle production will occur – purely quantum process. Pair production rate contains topological aspect. Pair production rate contains topological aspect. B Expect: Pair production rate to have periodic dependence on AB phase eФ. Expect: Pair production rate to have periodic dependence on AB phase eФ. II: Aharonov-Bohm Interaction ABQM

9. Moving frames Expand scalar operator φ and Dirac operator ψ in terms of the instantaneous eigenstates of the Hamiltonian of the field equations. Expand scalar operator φ and Dirac operator ψ in terms of the instantaneous eigenstates of the Hamiltonian of the field equations. i.e. Solve the mode functions for the stationary solenoid problem and shift them by ξ, location of moving solenoid. i.e. Solve the mode functions for the stationary solenoid problem and shift them by ξ, location of moving solenoid.

10. Moving frames φ results Rate carries periodic dependence on eФ Rate carries periodic dependence on eФ Non-relativistic Non-relativistic

11. Moving frames ψ results Rate carries periodic dependence on eФ Rate carries periodic dependence on eФ Non-relativistic Non-relativistic

12. Relativistic eФ << 1 φ*φ*φ*φ*φ

13. Small AB phase: eФ << 1 φ*φ*φ*φ*φ o Valid for any flux tube trajectory.

14. Small AB phase: eФ << 1 φ*φ*φ*φ*φ

15. φ*φ*φ*φ*φ Spins of e + e - anti-correlated along direction determined by their momenta and I + x I -. Spins of e + e - anti-correlated along direction determined by their momenta and I + x I -.

16. Small AB phase results

17. eФ<<1: Total Power o Similar plot for bosons. v 0 = 0.001 v 0 = 0.1 v 0 = 1

18. eФ >m, v 0 ~1, k z =0, k xy =k’ xy

19. Cosmic String Loops o Motivated by astrophysical observations of anomalous excess of e+e-, some recent particle physics models involve mixing of photon U(1) vector potential with some “dark” sector (spontaneously broken) U(1)’ vector potential. o This allows emission of electrically charged particle-antiparticle pairs from cosmic strings via the AB interaction. o Consider: Kinky and cuspy loops.

20. eФ << 1: Kinky Cosmic String Loop o Like solenoid, there are infinite # of harmonics o Both bosonic and fermionic emission show similar linear-in-N behavior. o Cut-off determined by finite width of cosmic string loop. Kinky loop configuration

21. eФ<<1: Cuspy Cosmic String Loop o Like solenoid, there are infinite # of harmonics o Both bosonic and fermionic emission show similar constant-in-ℓ behavior. o Cut-off determined by finite width of cosmic string loop. Cuspy loop configuration

22. Gravitational “AB” interaction Spacetime metric far from a straight, infinite cosmic string is Minkowski, with a deficit angle determined by string tension. Spacetime metric far from a straight, infinite cosmic string is Minkowski, with a deficit angle determined by string tension. Shaking a cosmic string would generate a time-dependent metric and induce gravitationally induced production of all particle spieces. Shaking a cosmic string would generate a time-dependent metric and induce gravitationally induced production of all particle spieces. Scalars, Photons, Fermions Scalars, Photons, Fermions

23. The N-Body Problem in General Relativity from Perturbative (Quantum) Field Theory Y.-Z.Chu, Phys. Rev. D 79: 044031, 2009 arXiv: 0812.0012 [gr-qc] Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010

24. System of n ≥ 2 gravitationally bound compact objects: System of n ≥ 2 gravitationally bound compact objects: Planets, neutron stars, black holes, etc. Planets, neutron stars, black holes, etc. What is their effective gravitational interaction? What is their effective gravitational interaction?

25. Compact objects ≈ point particles Compact objects ≈ point particles n-body problem: Dynamics for the coordinates of the point particles n-body problem: Dynamics for the coordinates of the point particles Assume non-relativistic motion Assume non-relativistic motion GR corrections to Newtonian gravity: an expansion in (v/c) 2 GR corrections to Newtonian gravity: an expansion in (v/c) 2 Nomenclature: O[(v/c) 2Q ] = Q PN

26. Note that General Relativity is non-linear. Note that General Relativity is non-linear. Superposition does not hold Superposition does not hold 2 body lagrangian is not sufficient to obtain n-body lagrangian 2 body lagrangian is not sufficient to obtain n-body lagrangian Nomenclature: O[(v/c) 2Q ] = Q PN

27. n-body problem known up to O[(v/c) 2 ]: n-body problem known up to O[(v/c) 2 ]: Einstein-Infeld-Hoffman lagrangian Einstein-Infeld-Hoffman lagrangian Eqns of motion used regularly to calculate solar system dynamics, etc. Eqns of motion used regularly to calculate solar system dynamics, etc. Precession of Mercury’s perihelion begins at this order Precession of Mercury’s perihelion begins at this order O[(v/c) 4 ] only known partially. O[(v/c) 4 ] only known partially. Damour, Schafer (1985, 1987) Damour, Schafer (1985, 1987) Compute using field theory? (Goldberger, Rothstein, 2004) Compute using field theory? (Goldberger, Rothstein, 2004)

28. Solar system probes of GR beginning to go beyond O[(v/c) 2 ]: Solar system probes of GR beginning to go beyond O[(v/c) 2 ]: New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions? New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions? ASTROD, LATOR, GTDM, BEACON, etc. ASTROD, LATOR, GTDM, BEACON, etc. See e.g. Turyshev (2008) See e.g. Turyshev (2008)

29. n-body L eff gives not only dynamics but also geometry. n-body L eff gives not only dynamics but also geometry. Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses Metric can be read off its L eff Metric can be read off its L eff

30. Gravitational wave observatories may need the 2 body L eff beyond O[(v/c) 7 ]: Gravitational wave observatories may need the 2 body L eff beyond O[(v/c) 7 ]: LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[10 4 ] orbital cycles. LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[10 4 ] orbital cycles. GW detection: Raw data integrated against theoretical templates to search for correlations. GW detection: Raw data integrated against theoretical templates to search for correlations. Construction of accurate templates requires 2 body dynamics. Construction of accurate templates requires 2 body dynamics. Currently, 2 body dynamics known up to O[(v/c) 7 ], i.e. 3.5 PN Currently, 2 body dynamics known up to O[(v/c) 7 ], i.e. 3.5 PN See e.g. Blanchet (2006). See e.g. Blanchet (2006).

31. Starting at 3 PN, O[(v/c) 6 ], GR computations of 2 body L eff start to give divergences – due to the point particle approximation – that were eventually handled by dimensional regularization. Starting at 3 PN, O[(v/c) 6 ], GR computations of 2 body L eff start to give divergences – due to the point particle approximation – that were eventually handled by dimensional regularization. Perturbation theory beyond O[(v/c) 7 ] requires systematic, efficient methods. Perturbation theory beyond O[(v/c) 7 ] requires systematic, efficient methods. Renormalization & regularization Renormalization & regularization Computational algorithm – Feynman diagrams with appropriate dimensional analysis. Computational algorithm – Feynman diagrams with appropriate dimensional analysis. QFTOffers:

32. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line

33. –M∫ds describes structureless point particle –M∫ds describes structureless point particle GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line

34. Non-minimal terms encode information on the non-trivial structure of individual objects. Non-minimal terms encode information on the non-trivial structure of individual objects. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line

35. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line Coefficients {c x } have to be tuned to match physical observables from full description of objects. Coefficients {c x } have to be tuned to match physical observables from full description of objects. E.g. n non-rotating black holes. E.g. n non-rotating black holes.

36. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line Point particle approximation gives us computational control. Point particle approximation gives us computational control. Infinite series of actions truncated based on desired accuracy of theoretical prediction. Infinite series of actions truncated based on desired accuracy of theoretical prediction.

37. For non-rotating compact objects, up to O[(v/c) 8 ], only minimal terms -M a ∫ds a needed For non-rotating compact objects, up to O[(v/c) 8 ], only minimal terms -M a ∫ds a needed GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line

38. Expand GR and point particle action in powers of graviton fields h μν … Expand GR and point particle action in powers of graviton fields h μν …

39. ∞ terms just from Einstein-Hilbert and -M a ∫ds a. ∞ terms just from Einstein-Hilbert and -M a ∫ds a.

40. … but some dimensional analysis before computation makes perturbation theory much more systematic … but some dimensional analysis before computation makes perturbation theory much more systematic The scales in the n-body problem The scales in the n-body problem r – typical separation between n bodies. r – typical separation between n bodies. v – typical speed of point particles v – typical speed of point particles r/v – typical time scale of n-body system r/v – typical time scale of n-body system

41. Lowest order effective action Lowest order effective action Schematically, conservative part of effective action is a series: Schematically, conservative part of effective action is a series: Virial theorem Virial theorem

42. Look at Re[Graviton propagator], non- relativistic limit: Look at Re[Graviton propagator], non- relativistic limit:

44. n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as With n (w) world line terms -M a ∫ds a, With n (w) world line terms -M a ∫ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as

45. n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of -M a ∫ds a with χ powers of velocities scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as =1 for classical problem Q PN With n (w) world line terms -M a ∫ds a, With n (w) world line terms -M a ∫ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as Know exactly which terms in action & diagrams are necessary. Know exactly which terms in action & diagrams are necessary.

46. Limited form of superposition holds Limited form of superposition holds At Q PN, i.e. O[(v/c) 2Q ], max number of distinct point particles in a given diagram is Q+2 At Q PN, i.e. O[(v/c) 2Q ], max number of distinct point particles in a given diagram is Q+2 1 PN, O[(v/c) 2 ]: 3 body problem 1 PN, O[(v/c) 2 ]: 3 body problem 2 PN, O[(v/c) 4 ]: 4 body problem 2 PN, O[(v/c) 4 ]: 4 body problem 3 PN, O[(v/c) 6 ]: 5 body problem 3 PN, O[(v/c) 6 ]: 5 body problem Every Feynman diagram scales as Every Feynman diagram scales as

48. 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions

49. 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Relativistic corrections

50. 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Gravitational 1/r 2 potentials

51. 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Non-linearities of GR Non-linearities of GR Three body force Three body force

52. 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits. Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits.

53. No graviton vertices Gravitonvertices

54. vertices Gravitonvertices

55. vertices Gravitonvertices

57. Work in progress (with Roman Buniy)

65. N-body problem for rotating masses, multi-pole moments, gravitational radiation, etc. N-body problem for rotating masses, multi-pole moments, gravitational radiation, etc. Higher PN computation: Higher PN computation: Different field variables: ADM, Kol- Smolkin-Kaluza-Klein. Different field variables: ADM, Kol- Smolkin-Kaluza-Klein. Different gravitational lagrangian: Bern-Grant. Different gravitational lagrangian: Bern-Grant. Are there recursion relations for off- shell gravitational amplitudes? Are there recursion relations for off- shell gravitational amplitudes? Software development. Software development.