Vortex -- radiation interactions in 4 theory -...
Transcript of Vortex -- radiation interactions in 4 theory -...
-
Introduction Vortices Interaction Numerical results
Vortex radiation interactions in 4 theory
rpd Lukcs
MTA KFKI Research Institute for Nuclear and Particle Physics
Austria-Croatia-Hungary Triangle Workshop onStrong Interactions in Quantum Field Theory,
Frstenfeld, 16 April, 2009.
Collaborators: Pter Forgcs, Tomasz Romanczukiewicz
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Introduction
Motivation: negative radiation pressure in the vortexradiationinteraction
(Video: Tomasz Romanczukiewicz) The vortex starts moving towards the source
Lukcs . Vortex Scattering
amplitude.aviMedia File (video/avi)
-
Introduction Vortices Interaction Numerical results
Outline
1 Introduction
2 Vortices
3 Vortex radiation interactionsThe perturbation problem of the vortexCross sections and force
4 Numerical results
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Vortexek
(Figure source: Vilenkin and Shellard: Cosmic strings. . . , CUP, 2000)
We are working with a 2 + 1D scalarfield theory (z-translation invariantconfigurations in 3 + 1D).
L = V (, ) ,
with V (, ) = ( 1)2. Weexamine solutions of the form
(r , ) = f (r)ein .
Eqn. for the vortex profile:
f 1r
f +n2
r2f + 2f (f 2 1) = 0
Asymptotic behaviour of the solution:
f (r) f (n)rn (r 0) and f (r) 1 n2
4r2(r ) .
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Vortex profile
(r , ) = f (r)ein
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Vortices in Physical Theories I
Cosmic StringsA vortex is a slice of an idealized (straight) cosmic stringFormation of vortices: during phase transitionPhysical background: field theoryDescription of the dynamics
field theoryNambuGoto action from field theorystring network simulationsstring formation
Observation: based on the effects of cosmic strings on CMBRs
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Vortices in Physical Theories II
Vortex strings in superfluid heliumA vortex is a slice of an idealized (straight) vortex stringDescription of superfluids: Gross-Pitaevskii equation:
i~
t=
( ~
2
2m + V0||2
) ,
which is the non-relativistic limit of 4 theory.Forces acting on the vortex
Magnus forceacoustic drag (interaction with incoming radiation)transversal force
The physical backgrounds of the two examples differ significantly.The mathematical descriptions of the vortices are very similar.
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results Perturbations Force
The perturbation problem of the vortex I
Field equations:+ 2( 1) = 0
We add perturbations to the vortex solution: + ,
+ 2(2 1)+ 22 = 0
Partial wave expansion:
=
`=
ei(n+`)(eits+` + e
its`)
Let =
r(s+` , s` )
T . (D` 2f 2
2f 2 D`
) = 2
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results Perturbations Force
The perturbation problem of the vortex II
with
D` = d2
dr2+
(n + `)2 1/4r2
+ 2(2f 2 1)
Properties of the vortex perturbation problem:asymptotically decoupled modes: a` = 12 (s
+` + s
` ),
g` = 12 (s+` s
` )
at the origin the two modes are mixing; the coupling is 1/r2
Asymptotic solution of the scattering problem = +eit + eit with
=12r
`
ei(n+`)i`[
(ha`,hg`)S`
(ag
)+ (ha`,h
g`)
(ag
)]+ =
12r
`
ei(n`)i`[
(ha`,hg`)S`(
ag
)+ (ha`,hg`)
(ag
)].
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results Perturbations Force
Cross sections
The cross sections expressed with the S-matrix elements: usualformulae. Transition amplitudes:
fa,ga,g() =1
2ka,gei
4
`=
ei(n+`)(S11,22 1) ,
fag,ga() =1
2kg,aei
4
`=
ei(n+`)S21 ,
This yields the total cross sections
aa,gg =1
ka,g
`=
|S(`)11,22 1|2 ,
ag,ga =1
kg,a
`=
|S(`)21 |2 .
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results Perturbations Force
Force
The force can be calculated from the momentum balance T = 0:
Px = Fx =
Rd(Txx cos+ Txy sin) ,
averaging over a period: Px = Fx = 1T t+T/2
tT/2 Fx (t)dt . The part of
the stressenergy tensor quadratic in the perturbations:
T (2) = +
g{
()2()2 ()2()2 2(2 1)},
thus
Fx = 4 Re`
[(a,g)S`
(ka
kg
)S`+1
(ag
) (a,g)
(ka
kg
)(ag
)].
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Numerical results I
Obtaining the vortex profile: (r , ) = f (r)ein: to solve the scatteringproblem, a good precision solution of the vortex profile equation
f 1r
f +n2
r2f + 2f (f 2 1) = 0
is needed (esp. for large vals of `). The method used: collocations(COLSYS).
f (r) f (n)rn (r 0)
f (r) 1 n2
4r2(r ) .
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Numerical results II
To calculate the force, we solve the scattering problem of the coupled2-component Schrdinger-operator.
The method of solution: RK oninterpolated backgound
Negative force in the case of thea modePositive force in the case of theg mode (acoustic drag)
Explanation : scattering from a modeof mass 4 into a massless one;surplus momentum behind the vortex
Further possibilities: calculating the transverse (Iordanskii) force.(D` 2f 2
2f 2 D`
) = 2
with D` = d2
dr2 +(n+`)21/4
r2 + 2(2f2 1).
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Numerical results III
Partial wave components of the force for 2 = 6.5, 9 and 11.5:
force dominated by components with moderate `
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Properties of the scattering problem
(n `)2
r2terms in the diagonal AharonovBohm effect
( integer flux; tricky partial wave summation)analytical approximation for large `partial wave components of the force vanish rapidly for large `: (ka/kg)`, ka < kg .contribution of modes for ` = 7 . . . 7 provides a very goodapproximation for a wide range of in this setting: no transversal force, unlike for moving superfluidvortices (Iordanskii force)
Lukcs . Vortex Scattering
-
Introduction Vortices Interaction Numerical results
Conclusion
For the massive incoming wave mode the vortex is pulledtowards the sourceExplanation: scattering of a massive mode into a massless one;surplus momentum behind the vortexSimilar effect: negative radiation pressure for 1+1 D kinksPossible applications: superfluid vortices, cosmic stringsTodo: generalization for moving vortices, both in fluid andvacuum
Lukcs . Vortex Scattering
IntroductionVorticesVortex -- radiation interactionsThe perturbation problem of the vortexCross sections and force
Numerical results