Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010

1. Don’t Shake That Solenoid Too Hard: Particle Production From Aharonov-Bohm Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010 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: • Vector potential Aμ is not merely computational crutch but indispensable for quantum dynamics of charged particles. • (2010) Quantum Field Theory: • Spontaneous pair production of charged particles just by shaking a thin solenoid.

3. Setup z Shake the Solenoid! B y x

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

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

6. I: Time dependent H • The gauge potential Aμ around a moving solenoid is time-dependent. • Hamiltonian of QFT is explicitly time-dependent: Hi = ∫d3x AμJμ. • 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. • Pair production rate contains topological aspect. B AB QM • QM: Amp. for paths belonging to different classes will differ by exp[i(integer)eФ]

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

9. Moving frames • 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.

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

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

12. Relativistic eФ << 1 φ φ*

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

14. Small AB phase: eФ << 1 φ φ*

15. Small AB phase: eФ << 1 φ φ* • 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 v0 = 1 v0 = 0.1 v0 = 0.001 • Similar plot for bosons.

18. eФ<<1, Ω>>m, v0~1, kz=0, kxy=k’xy

19. Cosmic String Loops • 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. • This allows emission of electrically charged particle-antiparticle pairs from cosmic strings via the AB interaction. • Consider: Kinky and cuspy loops.

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

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

22. Gravitational “AB” interaction • 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. • Scalars, Photons, Fermions

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

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

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

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

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

28. Motivation I • 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? • ASTROD, LATOR, GTDM, BEACON, etc. • See e.g. Turyshev (2008)

29. Motivation I • n-body Leff 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 • Metric can be read off its Leff

30. Motivation II • Gravitational wave observatories may need the 2 body Leff beyond O[(v/c)7]: • LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[104] orbital cycles. • GW detection: Raw data integrated against theoretical templates to search for correlations. • Construction of accurate templates requires 2 body dynamics. • Currently, 2 body dynamics known up to O[(v/c)7], i.e. 3.5 PN • See e.g. Blanchet (2006).

31. Why (Quantum) Field Theory • Starting at 3 PN, O[(v/c)6], GR computations of 2 body Leff 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. • Renormalization & regularization • Computational algorithm – Feynman diagrams with appropriate dimensional analysis. QFT Offers:

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

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

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

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

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

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

38. Perturbation Theory • Expand GR and point particle action in powers of graviton fields hμν …

39. Perturbation Theory • Expand GR and point particle action in powers of graviton fields hμν … • ∞ terms just from Einstein-Hilbert and -Ma∫dsa.

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

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

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

43. Dimensional Analysis • Look at Re[Graviton propagator], non-relativistic limit:

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

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

46. Superposition • Every Feynman diagram scales as • 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 • 1 PN, O[(v/c)2]: 3 body problem • 2 PN, O[(v/c)4]: 4 body problem • 3 PN, O[(v/c)6]: 5 body problem

47. Newtonian Gravityn = 2 Body Problem

48. O[(v/c)2]: 1 PNn = 3 Body Problem Einstein-Infeld-Hoffman d-spacetime dimensions 2 body diagrams 3 body diagrams

49. O[(v/c)2]: 1 PNn = 3 Body Problem Relativistic corrections Einstein-Infeld-Hoffman d-spacetime dimensions 2 body diagrams 3 body diagrams

50. O[(v/c)2]: 1 PNn = 3 Body Problem Gravitational 1/r2 potentials Einstein-Infeld-Hoffman d-spacetime dimensions 2 body diagrams 3 body diagrams