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Short range interactions of Chern-Simons vortices
Physics Letters B 320 (1994) 69-73 North-Holland
PHYSICS LETTERS B
Short range interactions of Chern-Simons vortices* Jacek Dziarmaga Jagelloman Umverstty, Instttute of Physics, Reymonta 4, 30-059 Krakdw, Poland
Received 14 September 1993; rewsed manuscript received 19 November 1993 Editor: P.V. Landshoff We apply the adiabatic approximation to investigate the short range interactions of relativistic serf-dual ChernSimons vortices. Unlike in the Abelian Hlggs model the result of the head-on collision of n vortices is the backward scattering. As the initial velocity rises the scattering angle tends to n/n. We also show the existence of the state in which the n vortices are rotating around the center of mass. The peculiarities are due to the nonzero charge of the Chern-Simons vortices.
Recently there have been some studies of the dynamics and interactions of the Chern-Simons-Higgs solitons with the help of Manton's idea of the adiabatic approximation [1 ]. In [2] the authors show that in the limit of the very slow motion and very large separations of the self-dual Chern-Simons vortices [3-6] the dynamics of the solitons is described by the effective Lagrangian n
Left = ½ E
•, "j + 2nx~-/ d ~-~Arg(~p-~q), g"¢;¢*'
p = 1
(1)
p>q
where ~'s denote the positions o f the n vortices identified with the positions of the zeros of the Higgs field. Arg (~p - ~q) is an angle at which the q-th vortex can be seen from the position of the p-th vortex and gzj is the flat metrics. The parameter K originates from the full field theoretical Lagranglan [ 3-6 ] L=
1 ,,.,, 1 :tee A ~ & A~ + Du~DU q3* - ~-: [@2 ( kbl2 - 1 ) 2
,
(2) where/t, v = 0, 1, 2 and D u = 0 u - iA u. The effective Lagrangian contains the total time derivative of the sum o f the relative angles between the positions of the vortices which is known to lead upon quantisation to the fractional statistics [7]. So far as it is a * Research supported by the grant KBN 2P302 049 05.
total derivative the term has no influence on the classical equations of motion. The classical dynamics is governed by the first term in (1) and it is the dynam. ics of free particles. The derivation of the above effective Lagrangian [2] is based on a set of assumptions which can be expected to be satisfied for the configuration of sufficiently separated vortices. Our aim is to apply Manton's prescription to the case of very small separations of vortices. Similar analysis for the vortices in the critically coupled Abelian Higgs model [ 8,9 ] leads to the conclusion that the head-on collision of n vortices results in the scattering by an angle o f n / n . We show that this is not the case for the CS vortices, because they are composites of the magnetic flux and the electric charge. When the vortices do overlap the interaction between the magnetic field and the electric charges completely dominates the scattering patterns in the limit of the very small velocities. For larger velocities we can expect the influence of the magnetic field on the scattering angle to become less significant because a particle with higher velocity moves along a circle of larger radius. The area in which the magnetic field is nonzero is finite. The arc o f the circle o f larger radius and finite length better approximates a straight line. Thus for larger velocities the effects o f the magnetic field become less and less significant and we can expect n / n scattering, characteristic for noncharged vortices, to be restored although we do not
0370-2693/94/$ 07.00 © 1994 ElsevaerScience B.V All rights reserved. SSD! 0370-2693 (93)E1477-F
69
Volume 320, number 1,2
PHYSICS LETTERS B
6 January 1994
know whether the result of the adiabatic approximation can be extended to this range of velocities. To start our derivation we have to repeat the initial step from [2]. Let us rewrite the Lagrangian (2) as
A = eoAo(r) + a(r,O).
L = (Ot[¢l) 2 + xBOtx + ½xe'JOtA,Aj
ar = ~
From eqs. (9), (10) and eq. (5) upon linearisation we obtain the set of equations
Oa
xg
+ 1¢12(A 0
Otz) 2
21¢12 + /.¢2B2
(
-v D'¢D'¢*+4-~
ao -
1 ¢ 2( ¢ 2
+~1
I I I -
1 )2)
,
(3)
where we have introduced ¢ = I¢1 dz. We can eliminate A° as an auxiliary field, Ao -
xB 21¢12
OtZ.
(4)
(10)
1 Oh
+ r o---~'
(11)
Oh . Or
(12)
10a
r O0
From this set of equations we can calculate the components of a once h and a are already known. Eq. (6) and the gauge condition, O,A~, linearised in the fluctuations, lead to 1 - _1 r Ooar = to--42 f2(1 _ 2 f Z ) h , Orao + raO
(13)
In Manton's approximation the fields fulfil static equations at every moment of time,
Orar + --ar +
(D1 + iDz)~ = 0,
(5)
Substitution of eqs. ( 11 ), (12) into eqs. (13), (14) results in the following set of equations:
2 ¢2(¢2 B + ~-2 I -1) =0,
(6)
vZh + ~ 2 f e ( 1 - 2 f Z ) h = 0,
(15)
V2a = 0.
(16)
where we have assumed that the topological index of the configuration is positive. With the help of eqs. (4)-(6) the Lagrangian (3) reduces to
Leff = _fdZx[(Ot[¢l) 2 + tcBOtz + l~ce'JOtAzAj],
(7) up to a term which is constant in the given topological sector. The fields in the above Lagrangian have to be understood as functionals of some collective coordinates. Later we will establish a connection between the coordinates and parameters of the static solution. A problem with a direct application of Manton's idea arises because we do not know the explicit form of the general static solution. Fortunately in the case of very small separations of vortices we are able to work out an approximate form of the solution by investigation of small deformations of the static axially symmetric n-vortex solution to eqs. (5), (6)
¢(r,O) = f (r) e 'n°,
,4 = eoAo(r).
(8)
Let us disturb the background field:
x=nO+a(r,O), 70
I¢[= f ( r ) [ l
+h(r,O)],
1
1
r
r
Ooao = 0.
(14)
The functions h and a can be Fourier-transformed in 0. As it was shown in [4] the highest Fourier modes of h compatible with the regularity of ¢ at r = 0 are the n-th order modes:
h(r,O) = H(r)[21cos(nO) +,~2sin(n0)],
where 2's are small real parameters and H is such a solution of the equation ( ~
~ +rdr
n~ ) -~ H +
fz(1-2f2)H=O
(18)
that it vanishes at infinity and is normalised so that 1
H(r),-~ 7
as
r~0.
(19)
Once we have specified h we have to choose a in such a way as to avoid singularities of the gauge fields, eqs. (11), (12),
a(r,O) = K(r)[22cos(nO) - 2 1 s i n ( n O ) ] , (9)
(17)
where K ( r ) = 1/r n.
(20)
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PHYSICS LETTERS B
Putting together eqs. (17), (20) we can calculate that up to the linear terms in 2 = 2 ~ + i22 and for small r the disturbed Higgs field looks like
c) ,,~ fo(z n - 2),
(21)
where we have taken into account that f (r) ~,, j~r" and we have made an identification z = x ~ + i x 2. From the above formula we can directly see that the zeros of the Higgs field have been moved to the n-th roots of 2 (up to linear terms in 2). We can denote these roots as R e l{°+p='/n) where p = 0 ..... (n - 1). This field configuration corresponds to the uniform splitting of n vortices from their coincident position in the center of mass frame. This field is identical with that obtained for Abelian Hxggs vortices [8,9] and for solitons in the CP x model [ 10 ] yet it does not in general result in the scattering by n/n as will be shown below. From now on we will use the 2's as collective coordinates. By direct inspection we can see that the third term in the effective Lagrangian (7) vanishes. If we take into account that the LHS of eq. (13) is equal to the fluctuation of the magnetic field the first and the second term of the effective Lagrangian can be shown to be equal to Left = Mn [j~2 ..1_A2cb2] _ tcQn [ A 2 ~ ] ,
(22)
where we have introduced 2 = A e ~°, and the two coefficients are oO
M, = ~ / r d r H 2 ( r ) f 2 (r) ,
(23)
0 oo
Qn = _~4~ f rdr(2f2 - 1 ) f Z K ( r ) H ( r ) .
(24)
. 1
o
We have to mention that the rotation of co from 0 to 2z~ corresponds to the rotation of the field configuration by an angle o f 2rein so it has no effect because of the discrete Zn symmetry of the field. From the resulting effective Lagrangian Left = 112 + A 2 ~ 2 -
g'2nA2(o,
On = X--~7~ ,
(25)
we obtain an equation o f motion for complex 2: )[ = ig2n2.
(26)
6 January 1994
The above Lagrangian is the one for the planar-motion of a charged particle in an external magnetic field perpendicular to the plane. That is what was to be qualitatively expected because of the CS vortices being composites of the magnetic flux and the electric charge. The solution to eq. (26) is a motion along a circle in the complex plane, 2(t) = A + B e ia"t,
(27)
where A and B are some complex constants. In particular 2(t) = A e l~nt corresponds to the rotation of the n vortices around the center of mass. They are kept on this trajectory by the interaction of the magnetic field with their charges so such a state is possible only when the cores of the vortices do overlap. The kinetic energy of this solution is proportional to A 2. I f the radiative corrections are not negligible they will result in the shrinking of the configuration to a very small value of A but in this limit we can expect the adiabatic approximation to be exact and the radiative corrections to be small. In this sense we can say that this state is stable. To discuss the scattering of vortices we have to make an additional construction because we do not know how vortices interact at medium separations. We will make a rough approximation that at the distances larger than say R0 the vortices move along straight lines (in x t, x 2 coordinates) while inside of this circle the dynamics is governed by the above effective Lagrangians. Extensive numerical studies of the moduli space approximation in the Abelian-Higgs model [ 11,12 ] show that the dynamics of vortices at the medium separations smoothly interpolates between that of the short range interactions and that characteristic for distant vortices. On the other hand using our physical interpretation of the short range effects, the strength of the magnetic interaction can be expected to rise smoothly from zero to its full short range value as the cores of vortices and at the same time their charges and fluxes become more and more overlapping. I f this interpretation is correct than we can expect our rough approximation to be qualitatively good. Let us restrict to the case o f the head-on collision. The incoming trajectory of any of the vortices is a straight line (see the figure). Once the vortex has entered the circle with the magnetic field its trajectory 71
Volume 320, number 1,2
PHYSICS LETTERS B
6 January 1994
small x we can expect that the z~/n scattering angle is almost attained before the adiabatic approximation stops to work. These are qualitative results which we think to be worth of further numerical investigation. Let us rewrite the effective Lagrangian (25) with the help of the physical coordinates of the zeros of the Higgs field
k'
1
Fig. 1. The head-on collision of n vortices in 21,22 coordinates. The four trajectories of a chosen vortex correspond to four different initial velocities rising from Vl to v4. For a very small velocity the vortex bounces backwards. When the velocity is large enough the trajectory approximately turns from +4 to - 2 and the vortices scatter by an angle of ~r/n.
is curved by the Lorentz force until the circle is left and the motion is continued along a straight line. The lower is the energy of the collision the smaller is the radius o f the arc along which the vortex moves (in 21, 22 coordinates) and the scattering angle depends on the initial velocity (see the figure). For the very small velocities the vortices bounce backwards. In this limit the adiabatic approximation becomes exact. From the numerical studies [ 11,12 ] we know that nevertheless it works well in the collisions of Abelian vortices up to about 0.3 of the velocity of light. If it is as good for Chern-Simons vortices as for the Abelian-Higgs vortices than we can extend our results to a significant range o f velocities. So as the velocity is rising the scattering angle tends to lr/n as long as the adiabatic approximation is good. This tendency could be especially clearly observed in a numerical simulation o f the head-on collision when the parameter re is small. From eq. (7), where only the first two terms give nonzero contribution, we can see that for small tc the first term which gives rise to the kinetic term in eq. (22) is large as compared to the second one which is the origin of the charge-flux interaction. Thus for 72
Leff = R z ( n - 1 ) (/~ 2 + R 2 0 2) -
1QnR2nO.
(28) n The first part of the Lagrangian is the same as obtained in [ 9 ] for the configuration of n Abehan vortices and is responsible for their scattering by an angle of n/n while the second one is the remnant of the statistical interactions derived in [2] and present in eq. (1). We can expect the general form o f the effective Lagrangian valid for any separations of vortices to be Left = F ( R ) ( / ~ 2 + R 2 ~ 2 ) - £ 2 ( R ) O .
(29)
where F and £2 are functions of R such that the effective Lagrangian interpolates between the form of eq. (1) and that of eq. (28). The two functions have to be determined numerically. The function £2 tends to zero when the distance between vortices shrinks to zero and thanks to it the singularity in the definition of O does not matter. The vanishing of F in the same limit is due to the fact that the motion of the zeros of the Higgs field in the limit of very small separations has little influence on such physical characteristics as the energy density. One of the reasons for the interest in the ChernSimons-Higgs model is that this theory with its multisoliton solutions is a candidate for a field-theoretical realisation of anyon systems. From general considerations [7 ] it is known that only charged vortices with zero width (singular distribution of magnetic flux) can be expected to be an ideal realisation of anyons. U p o n quantisation, because of a nonzero thickness of the Chern-Simons vortices, we can expect some new quantum effects in addition to those of fractional statistics. What we have investigated in this letter is the classical dynamics of vortices when their cores overlap. The effects of short range interactions are the more significant statistically the larger is the width of a separate vortex and our classical results can give us an idea of the expected additional quantum effects. The semiclassical quantisation of the mentioned periodic solution can lead to a characteristic discrete
Volume 320, number 1,2
PHYSICS LETTERS B
spectrum of energy (see [13], p. 690). The spectrum should be similar to that of a charged particle in an external magnetic field. We think this topic to be worth o f further analysis. I would like to thank H. Arocl~ for discussions and reading o f the manuscript and W.J. Zakrzewski for illuminating remarks on the scattering of solitons. Note added. After this work was completed I have found the paper [ 14], which contains qualitative discussion of some similar issues.
References [1]N.S. Manton, Phys. Lett. B 110 (1982) 54; B 154 (1985) 397.
6 January 1994
[2] S.K. Kiln and H. Min, Phys. Lett. B 281 (1992) 81. [3] R. Jackiw and E.J. Weinberg, Phys. Rev. Lett. 64 (1990) 2334. [4] R. Jackiw, K. Lee and E.J. Weinberg, Phys. Rev. D 42 (1990) 3488. [5] C. Lee, K. Lee and H. Min, Phys. Lett. B 252 (1990) 79. [6] J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230. [7] S. Forte, Rev. Mod. Phys. 64 (1992) 193. [8] P.J. Ruback, Nucl. Phys. B 296 (1988) 669. [9 ] J. Dziarmaga, Jagellonian Univ. preprint TPJU-12/93. [10]A. Kudryavtsev, B. Piette and W.J. Zakrzewski, Durham Umv. preprint DTP-93/11. [11] P.J. Ruback and E.P.S. Shellard, Phys. Lett. B 209 (1988) 262. [12] E. Myers, C. Rebbl and R. Strilka, Phys. Rev. D 45 (1992) 1355. [13] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681. [14]Y. Kim and K. Mln, preprint hep-th/9211035 or CERN-TH.6701/92.
73
PHYSICS LETTERS B
Short range interactions of Chern-Simons vortices* Jacek Dziarmaga Jagelloman Umverstty, Instttute of Physics, Reymonta 4, 30-059 Krakdw, Poland
Received 14 September 1993; rewsed manuscript received 19 November 1993 Editor: P.V. Landshoff We apply the adiabatic approximation to investigate the short range interactions of relativistic serf-dual ChernSimons vortices. Unlike in the Abelian Hlggs model the result of the head-on collision of n vortices is the backward scattering. As the initial velocity rises the scattering angle tends to n/n. We also show the existence of the state in which the n vortices are rotating around the center of mass. The peculiarities are due to the nonzero charge of the Chern-Simons vortices.
Recently there have been some studies of the dynamics and interactions of the Chern-Simons-Higgs solitons with the help of Manton's idea of the adiabatic approximation [1 ]. In [2] the authors show that in the limit of the very slow motion and very large separations of the self-dual Chern-Simons vortices [3-6] the dynamics of the solitons is described by the effective Lagrangian n
Left = ½ E
•, "j + 2nx~-/ d ~-~Arg(~p-~q), g"¢;¢*'
p = 1
(1)
p>q
where ~'s denote the positions o f the n vortices identified with the positions of the zeros of the Higgs field. Arg (~p - ~q) is an angle at which the q-th vortex can be seen from the position of the p-th vortex and gzj is the flat metrics. The parameter K originates from the full field theoretical Lagranglan [ 3-6 ] L=
1 ,,.,, 1 :tee A ~ & A~ + Du~DU q3* - ~-: [@2 ( kbl2 - 1 ) 2
,
(2) where/t, v = 0, 1, 2 and D u = 0 u - iA u. The effective Lagrangian contains the total time derivative of the sum o f the relative angles between the positions of the vortices which is known to lead upon quantisation to the fractional statistics [7]. So far as it is a * Research supported by the grant KBN 2P302 049 05.
total derivative the term has no influence on the classical equations of motion. The classical dynamics is governed by the first term in (1) and it is the dynam. ics of free particles. The derivation of the above effective Lagrangian [2] is based on a set of assumptions which can be expected to be satisfied for the configuration of sufficiently separated vortices. Our aim is to apply Manton's prescription to the case of very small separations of vortices. Similar analysis for the vortices in the critically coupled Abelian Higgs model [ 8,9 ] leads to the conclusion that the head-on collision of n vortices results in the scattering by an angle o f n / n . We show that this is not the case for the CS vortices, because they are composites of the magnetic flux and the electric charge. When the vortices do overlap the interaction between the magnetic field and the electric charges completely dominates the scattering patterns in the limit of the very small velocities. For larger velocities we can expect the influence of the magnetic field on the scattering angle to become less significant because a particle with higher velocity moves along a circle of larger radius. The area in which the magnetic field is nonzero is finite. The arc o f the circle o f larger radius and finite length better approximates a straight line. Thus for larger velocities the effects o f the magnetic field become less and less significant and we can expect n / n scattering, characteristic for noncharged vortices, to be restored although we do not
0370-2693/94/$ 07.00 © 1994 ElsevaerScience B.V All rights reserved. SSD! 0370-2693 (93)E1477-F
69
Volume 320, number 1,2
PHYSICS LETTERS B
6 January 1994
know whether the result of the adiabatic approximation can be extended to this range of velocities. To start our derivation we have to repeat the initial step from [2]. Let us rewrite the Lagrangian (2) as
A = eoAo(r) + a(r,O).
L = (Ot[¢l) 2 + xBOtx + ½xe'JOtA,Aj
ar = ~
From eqs. (9), (10) and eq. (5) upon linearisation we obtain the set of equations
Oa
xg
+ 1¢12(A 0
Otz) 2
21¢12 + /.¢2B2
(
-v D'¢D'¢*+4-~
ao -
1 ¢ 2( ¢ 2
+~1
I I I -
1 )2)
,
(3)
where we have introduced ¢ = I¢1 dz. We can eliminate A° as an auxiliary field, Ao -
xB 21¢12
OtZ.
(4)
(10)
1 Oh
+ r o---~'
(11)
Oh . Or
(12)
10a
r O0
From this set of equations we can calculate the components of a once h and a are already known. Eq. (6) and the gauge condition, O,A~, linearised in the fluctuations, lead to 1 - _1 r Ooar = to--42 f2(1 _ 2 f Z ) h , Orao + raO
(13)
In Manton's approximation the fields fulfil static equations at every moment of time,
Orar + --ar +
(D1 + iDz)~ = 0,
(5)
Substitution of eqs. ( 11 ), (12) into eqs. (13), (14) results in the following set of equations:
2 ¢2(¢2 B + ~-2 I -1) =0,
(6)
vZh + ~ 2 f e ( 1 - 2 f Z ) h = 0,
(15)
V2a = 0.
(16)
where we have assumed that the topological index of the configuration is positive. With the help of eqs. (4)-(6) the Lagrangian (3) reduces to
Leff = _fdZx[(Ot[¢l) 2 + tcBOtz + l~ce'JOtAzAj],
(7) up to a term which is constant in the given topological sector. The fields in the above Lagrangian have to be understood as functionals of some collective coordinates. Later we will establish a connection between the coordinates and parameters of the static solution. A problem with a direct application of Manton's idea arises because we do not know the explicit form of the general static solution. Fortunately in the case of very small separations of vortices we are able to work out an approximate form of the solution by investigation of small deformations of the static axially symmetric n-vortex solution to eqs. (5), (6)
¢(r,O) = f (r) e 'n°,
,4 = eoAo(r).
(8)
Let us disturb the background field:
x=nO+a(r,O), 70
I¢[= f ( r ) [ l
+h(r,O)],
1
1
r
r
Ooao = 0.
(14)
The functions h and a can be Fourier-transformed in 0. As it was shown in [4] the highest Fourier modes of h compatible with the regularity of ¢ at r = 0 are the n-th order modes:
h(r,O) = H(r)[21cos(nO) +,~2sin(n0)],
where 2's are small real parameters and H is such a solution of the equation ( ~
~ +rdr
n~ ) -~ H +
fz(1-2f2)H=O
(18)
that it vanishes at infinity and is normalised so that 1
H(r),-~ 7
as
r~0.
(19)
Once we have specified h we have to choose a in such a way as to avoid singularities of the gauge fields, eqs. (11), (12),
a(r,O) = K(r)[22cos(nO) - 2 1 s i n ( n O ) ] , (9)
(17)
where K ( r ) = 1/r n.
(20)
Volume 320, number 1,2
PHYSICS LETTERS B
Putting together eqs. (17), (20) we can calculate that up to the linear terms in 2 = 2 ~ + i22 and for small r the disturbed Higgs field looks like
c) ,,~ fo(z n - 2),
(21)
where we have taken into account that f (r) ~,, j~r" and we have made an identification z = x ~ + i x 2. From the above formula we can directly see that the zeros of the Higgs field have been moved to the n-th roots of 2 (up to linear terms in 2). We can denote these roots as R e l{°+p='/n) where p = 0 ..... (n - 1). This field configuration corresponds to the uniform splitting of n vortices from their coincident position in the center of mass frame. This field is identical with that obtained for Abelian Hxggs vortices [8,9] and for solitons in the CP x model [ 10 ] yet it does not in general result in the scattering by n/n as will be shown below. From now on we will use the 2's as collective coordinates. By direct inspection we can see that the third term in the effective Lagrangian (7) vanishes. If we take into account that the LHS of eq. (13) is equal to the fluctuation of the magnetic field the first and the second term of the effective Lagrangian can be shown to be equal to Left = Mn [j~2 ..1_A2cb2] _ tcQn [ A 2 ~ ] ,
(22)
where we have introduced 2 = A e ~°, and the two coefficients are oO
M, = ~ / r d r H 2 ( r ) f 2 (r) ,
(23)
0 oo
Qn = _~4~ f rdr(2f2 - 1 ) f Z K ( r ) H ( r ) .
(24)
. 1
o
We have to mention that the rotation of co from 0 to 2z~ corresponds to the rotation of the field configuration by an angle o f 2rein so it has no effect because of the discrete Zn symmetry of the field. From the resulting effective Lagrangian Left = 112 + A 2 ~ 2 -
g'2nA2(o,
On = X--~7~ ,
(25)
we obtain an equation o f motion for complex 2: )[ = ig2n2.
(26)
6 January 1994
The above Lagrangian is the one for the planar-motion of a charged particle in an external magnetic field perpendicular to the plane. That is what was to be qualitatively expected because of the CS vortices being composites of the magnetic flux and the electric charge. The solution to eq. (26) is a motion along a circle in the complex plane, 2(t) = A + B e ia"t,
(27)
where A and B are some complex constants. In particular 2(t) = A e l~nt corresponds to the rotation of the n vortices around the center of mass. They are kept on this trajectory by the interaction of the magnetic field with their charges so such a state is possible only when the cores of the vortices do overlap. The kinetic energy of this solution is proportional to A 2. I f the radiative corrections are not negligible they will result in the shrinking of the configuration to a very small value of A but in this limit we can expect the adiabatic approximation to be exact and the radiative corrections to be small. In this sense we can say that this state is stable. To discuss the scattering of vortices we have to make an additional construction because we do not know how vortices interact at medium separations. We will make a rough approximation that at the distances larger than say R0 the vortices move along straight lines (in x t, x 2 coordinates) while inside of this circle the dynamics is governed by the above effective Lagrangians. Extensive numerical studies of the moduli space approximation in the Abelian-Higgs model [ 11,12 ] show that the dynamics of vortices at the medium separations smoothly interpolates between that of the short range interactions and that characteristic for distant vortices. On the other hand using our physical interpretation of the short range effects, the strength of the magnetic interaction can be expected to rise smoothly from zero to its full short range value as the cores of vortices and at the same time their charges and fluxes become more and more overlapping. I f this interpretation is correct than we can expect our rough approximation to be qualitatively good. Let us restrict to the case o f the head-on collision. The incoming trajectory of any of the vortices is a straight line (see the figure). Once the vortex has entered the circle with the magnetic field its trajectory 71
Volume 320, number 1,2
PHYSICS LETTERS B
6 January 1994
small x we can expect that the z~/n scattering angle is almost attained before the adiabatic approximation stops to work. These are qualitative results which we think to be worth of further numerical investigation. Let us rewrite the effective Lagrangian (25) with the help of the physical coordinates of the zeros of the Higgs field
k'
1
Fig. 1. The head-on collision of n vortices in 21,22 coordinates. The four trajectories of a chosen vortex correspond to four different initial velocities rising from Vl to v4. For a very small velocity the vortex bounces backwards. When the velocity is large enough the trajectory approximately turns from +4 to - 2 and the vortices scatter by an angle of ~r/n.
is curved by the Lorentz force until the circle is left and the motion is continued along a straight line. The lower is the energy of the collision the smaller is the radius o f the arc along which the vortex moves (in 21, 22 coordinates) and the scattering angle depends on the initial velocity (see the figure). For the very small velocities the vortices bounce backwards. In this limit the adiabatic approximation becomes exact. From the numerical studies [ 11,12 ] we know that nevertheless it works well in the collisions of Abelian vortices up to about 0.3 of the velocity of light. If it is as good for Chern-Simons vortices as for the Abelian-Higgs vortices than we can extend our results to a significant range o f velocities. So as the velocity is rising the scattering angle tends to lr/n as long as the adiabatic approximation is good. This tendency could be especially clearly observed in a numerical simulation o f the head-on collision when the parameter re is small. From eq. (7), where only the first two terms give nonzero contribution, we can see that for small tc the first term which gives rise to the kinetic term in eq. (22) is large as compared to the second one which is the origin of the charge-flux interaction. Thus for 72
Leff = R z ( n - 1 ) (/~ 2 + R 2 0 2) -
1QnR2nO.
(28) n The first part of the Lagrangian is the same as obtained in [ 9 ] for the configuration of n Abehan vortices and is responsible for their scattering by an angle of n/n while the second one is the remnant of the statistical interactions derived in [2] and present in eq. (1). We can expect the general form o f the effective Lagrangian valid for any separations of vortices to be Left = F ( R ) ( / ~ 2 + R 2 ~ 2 ) - £ 2 ( R ) O .
(29)
where F and £2 are functions of R such that the effective Lagrangian interpolates between the form of eq. (1) and that of eq. (28). The two functions have to be determined numerically. The function £2 tends to zero when the distance between vortices shrinks to zero and thanks to it the singularity in the definition of O does not matter. The vanishing of F in the same limit is due to the fact that the motion of the zeros of the Higgs field in the limit of very small separations has little influence on such physical characteristics as the energy density. One of the reasons for the interest in the ChernSimons-Higgs model is that this theory with its multisoliton solutions is a candidate for a field-theoretical realisation of anyon systems. From general considerations [7 ] it is known that only charged vortices with zero width (singular distribution of magnetic flux) can be expected to be an ideal realisation of anyons. U p o n quantisation, because of a nonzero thickness of the Chern-Simons vortices, we can expect some new quantum effects in addition to those of fractional statistics. What we have investigated in this letter is the classical dynamics of vortices when their cores overlap. The effects of short range interactions are the more significant statistically the larger is the width of a separate vortex and our classical results can give us an idea of the expected additional quantum effects. The semiclassical quantisation of the mentioned periodic solution can lead to a characteristic discrete
Volume 320, number 1,2
PHYSICS LETTERS B
spectrum of energy (see [13], p. 690). The spectrum should be similar to that of a charged particle in an external magnetic field. We think this topic to be worth o f further analysis. I would like to thank H. Arocl~ for discussions and reading o f the manuscript and W.J. Zakrzewski for illuminating remarks on the scattering of solitons. Note added. After this work was completed I have found the paper [ 14], which contains qualitative discussion of some similar issues.
References [1]N.S. Manton, Phys. Lett. B 110 (1982) 54; B 154 (1985) 397.
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[2] S.K. Kiln and H. Min, Phys. Lett. B 281 (1992) 81. [3] R. Jackiw and E.J. Weinberg, Phys. Rev. Lett. 64 (1990) 2334. [4] R. Jackiw, K. Lee and E.J. Weinberg, Phys. Rev. D 42 (1990) 3488. [5] C. Lee, K. Lee and H. Min, Phys. Lett. B 252 (1990) 79. [6] J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230. [7] S. Forte, Rev. Mod. Phys. 64 (1992) 193. [8] P.J. Ruback, Nucl. Phys. B 296 (1988) 669. [9 ] J. Dziarmaga, Jagellonian Univ. preprint TPJU-12/93. [10]A. Kudryavtsev, B. Piette and W.J. Zakrzewski, Durham Umv. preprint DTP-93/11. [11] P.J. Ruback and E.P.S. Shellard, Phys. Lett. B 209 (1988) 262. [12] E. Myers, C. Rebbl and R. Strilka, Phys. Rev. D 45 (1992) 1355. [13] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681. [14]Y. Kim and K. Mln, preprint hep-th/9211035 or CERN-TH.6701/92.
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