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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

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.

Setup

z

Shake

the

Solenoid!

B

y

x

Setup

z

e+

Pop!

e-

e-

B

e+

y

x

Setup

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

Bosonic QED

Fermionic QED

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.

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]

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.

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.

Moving frames φ results

- Rate carries periodic dependence on eФ
- Non-relativistic

Moving frames ψ results

- Rate carries periodic dependence on eФ
- Non-relativistic

Relativistic

eФ << 1

φ

φ*

Small AB phase: eФ << 1

φ

φ*

- Valid for any flux tube trajectory.

Small AB phase: eФ << 1

φ

φ*

Small AB phase: eФ << 1

φ

φ*

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

Small AB phase results

eФ<<1: Total Power

v0 = 1

v0 = 0.1

v0 = 0.001

- Similar plot for bosons.

eФ<<1, Ω>>m, v0~1, kz=0, kxy=k’xy

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.

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.

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.

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

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]

- System of n ≥ 2 gravitationally bound compact objects:
- Planets, neutron stars, black holes, etc.

- What is their effective gravitational interaction?

- 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

- 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

- 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)

- 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)

- 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

- 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).

- 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:

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

- 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

- 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.

- 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.

- 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.

- GR: Einstein-Hilbert

- For non-rotating compact objects, up to O[(v/c)8], only minimal terms -Ma∫dsa needed

- Expand GR and point particle action in powers of graviton fields hμν …

- Expand GR and point particle action in powers of graviton fields hμν …

- ∞ terms just from Einstein-Hilbert and -Ma∫dsa.

- … 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

- Lowest order effective action

- Schematically, conservative part of effective action is a series:

- Virial theorem

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

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

- 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

- 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

- 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

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

Relativistic corrections

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

Gravitational

1/r2 potentials

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

- Non-linearities of GR
- Three body force

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

- Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits.

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

No graviton

vertices

Graviton

vertices

No graviton

vertices

Graviton

vertices

Graviton

vertices

No graviton

vertices

Work in progress (with Roman Buniy)

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