Vortex -- radiation interactions in 4 theory - UNIGRAZphysik.uni-graz.at/itp/siq/talks/lukacs.pdf · Vortex – radiation interactions in ... Vortex – radiation interactions

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

Motivation: negative radiation pressure in the vortexradiationinteraction

(Video: Tomasz Romanczukiewicz) The vortex starts moving towards the source


amplitude.aviMedia File (video/avi)


Outline

1 Introduction

2 Vortices

3 Vortex radiation interactionsThe perturbation problem of the vortexCross sections and force

4 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 ) .



Vortex profile

(r , ) = f (r)ein



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



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.


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



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

)].



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 .



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

)].



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 ) .



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).



Numerical results III

Partial wave components of the force for 2 = 6.5, 9 and 11.5:

force dominated by components with moderate `



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)



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


IntroductionVorticesVortex -- radiation interactionsThe perturbation problem of the vortexCross sections and force

Numerical results

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Vortex -- radiation interactions in 4 theory - UNIGRAZphysik.uni-graz.at/itp/siq/talks/lukacs.pdf · Vortex – radiation interactions in ... Vortex – radiation interactions