Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010
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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

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

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Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Setup

z

Shake

the

Solenoid!

B

y

x


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Setup

z

e+

Pop!

e-

e-

B

e+

y

x


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Setup

  • Effective theory of magnetic flux tube: Alford and Wilczek (1989).

  • Bosonic or fermionic quantum electrodynamics (QED)

Bosonic QED

Fermionic QED


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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]


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Moving frames φ results

  • Rate carries periodic dependence on eФ

  • Non-relativistic


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Moving frames ψ results

  • Rate carries periodic dependence on eФ

  • Non-relativistic


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Relativistic

eФ << 1

φ

φ*


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Small AB phase: eФ << 1

φ

φ*

  • Valid for any flux tube trajectory.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Small AB phase: eФ << 1

φ

φ*


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Small AB phase: eФ << 1

φ

φ*

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


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Small AB phase results


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

eФ<<1: Total Power

v0 = 1

v0 = 0.1

v0 = 0.001

  • Similar plot for bosons.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

eФ<<1, Ω>>m, v0~1, kz=0, kxy=k’xy


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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.


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

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


Yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 20101

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]


N body problem in gr

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?


N body problem in gr1

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


N body problem in gr2

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


N body problem in gr3

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)


Motivation i

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)


Motivation i1

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


Motivation ii

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).


Why quantum field theory

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:


Dynamics action

Dynamics: Action

  • GR: Einstein-Hilbert

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


Dynamics action1

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


Dynamics action2

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.


Dynamics action3

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.


Dynamics action4

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.


Dynamics action5

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


Perturbation theory

Perturbation Theory

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


Perturbation theory1

Perturbation Theory

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

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


Dimensional analysis

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


Dimensional analysis1

Dimensional Analysis

  • Lowest order effective action

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

  • Virial theorem


Dimensional analysis2

Dimensional Analysis

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


Dimensional analysis3

Dimensional Analysis

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


Dimensional analysis4

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


Dimensional analysis5

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


Superposition

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


Newtonian gravity n 2 body problem

Newtonian Gravityn = 2 Body Problem


O v c 2 1 pn n 3 body problem

O[(v/c)2]: 1 PNn = 3 Body Problem

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams


O v c 2 1 pn n 3 body problem1

O[(v/c)2]: 1 PNn = 3 Body Problem

Relativistic corrections

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams


O v c 2 1 pn n 3 body problem2

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


O v c 2 1 pn n 3 body problem3

O[(v/c)2]: 1 PNn = 3 Body Problem

  • Non-linearities of GR

  • Three body force

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams


O v c 2 1 pn n 3 body problem4

O[(v/c)2]: 1 PNn = 3 Body Problem

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

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams


O v c 4 2 pn n 4 body problem 2 body diagrams

O[(v/c)4]: 2 PNn = 4 Body Problem2 body diagrams

No graviton

vertices

Graviton

vertices


O v c 4 2 pn n 4 body problem 3 body diagrams

O[(v/c)4]: 2 PNn = 4 Body Problem3 body diagrams

No graviton

vertices

Graviton

vertices


O v c 4 2 pn 4 body diagrams

O[(v/c)4]: 2 PN4 Body Diagrams

Graviton

vertices

No graviton

vertices


O v c 4 2 pn n 4 body problem

O[(v/c)4]: 2 PNn = 4 Body Problem


O v c 4 2 pn n 4 body problem1

O[(v/c)4]: 2 PNn = 4 Body Problem

Work in progress (with Roman Buniy)


Third order o v c 6 beyond newton

Third Order – O[(v/c)6]Beyond Newton


O v c 6 3 pn feynman diagrams 2 distinct bodies

O[(v/c)6]: 3 PN Feynman diagrams2 (distinct) bodies


O v c 6 3 pn feynman diagrams 3 distinct bodies i of ii

O[(v/c)6]: 3 PN Feynman diagrams3 (distinct) bodies: I of II


O v c 6 3 pn feynman diagrams 3 distinct bodies ii of ii

O[(v/c)6]: 3 PN Feynman diagrams3 (distinct) bodies: II of II


O v c 6 3 pn feynman diagrams 4 distinct bodies i of ii

O[(v/c)6]: 3 PN Feynman diagrams4 (distinct) bodies: I of II


O v c 6 3 pn feynman diagrams 4 distinct bodies ii of ii

O[(v/c)6]: 3 PN Feynman diagrams4 (distinct) bodies: II of II


O v c 6 3 pn feynman diagrams 5 distinct bodies

O[(v/c)6]: 3 PN Feynman diagrams5 (distinct) bodies


The road ahead

The road ahead

  • 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.


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