Mesons, Baryons and Waves in the Baby Skyrmion Model 1. Introduction

Home , Skyrmion

DTP-96/17

November 28, 1996

Mesons, Baryons and Waves in the Baby Skyrmion Mo del

A. Kudryavtsev

B. Piette,

and

W.J. Zakrzewski

Department of Mathematical Sciences

University of Durham, Durham DH1 3LE, England

also at ITEP, Moscow, Russia

E-Mail: [email protected] [email protected] [email protected]

ABSTRACT

We study various classical solutions of the baby-Skyrmion mo del in (2 + 1) dimen-

sions. We p oint out the existence of higher energy states interpret them as resonances

of Skyrmions and anti-Skyrmions and study their decays. Most of the discussion in-

volves a highly exited Skyrmion-like state with winding numb er one which decays into

an ordinary Skyrmion and a Skyrmion-anti-Skyrmion pair. We also study wave-like

solutions of the mo del and show that some of such solutions can b e constructed from

the solutions of the sine-Gordon equation. We also show that the baby-Skyrmion has

non-top ological stationary solutions. We study their interactions with Skyrmions.

1. Intro duction.

In previous pap ers bytwo of us (BP and WJZ) [1-3] some hedgehog-like solutions

of the so called baby-Skyrmion mo del were studied. It was shown there that the mo del

has soliton-like top ologically stable static solutions (called baby-Skyrmions) and that

these solitons can form b ound states. The interaction b etween the solitons was studied

in detail and it was shown that the long distance force between 2 baby-Skyrmions

dep ends on their relative orientation.

To construct these soliton solutions, one must use a radially symmetric ansatz

(hedgehog con guration) and reduce the equation for the soliton to an ordinary dif-

ferential equation. This equation admits more solutions than those describ ed in [2]

and [3], and, as we will show, they corresp ond to exited states or resonances made

out of b oth Skyrmions and anti-Skyrmions. We will also show that the mo del admits

some solutions in the form of non-linear waves.

The (2 + 1)-dimensional baby-Skyrmion eld theory mo del is describ ed by the La-

grangian density

1 k

~ ~ ~ ~ ~ ~ ~

L = F ) (1 ~n) : (1:1) @ )(@ @ @ @ (@

2 4 1

Here ( ; ; ) denotes a triplet of scalar real elds which satisfy the constraint

1 2 3

2 t i

= 1; (@ @ = @ @ @ @ ). As mentioned in [1-3] the rst term in (1.1) is the

t i

familiar Lagrangian density of the pure S mo del. The second term, fourth order

in derivatives, is the (2+1) dimensional analogue of the Skyrme-term of the three-

[4]

dimensional Skyrme-mo del . The last term is often referred to as a p otential term.

The last two terms in the Lagrangian (1.1) are added to guarantee the stability of a

[5]

Skyrmion .

The vector ~n = (0; 0; 1) and hence the p otential term violates the O (3)-rotational

iso-invariance of the theory. The state 1 is the vacuum state of the theory.

As in [1,2] we x our units of energy and length by setting F = k = 1 and cho ose

2 2

=0:1 for our numerical calculations. The choice =0:1 sets the scale of the energy

distribution for a basic Skyrmion.

As usual we are interested mainly in eld con gurations for which the p otential

energy at in nity vanishes as only they can describ e eld con gurations with nite

total energy. Therefore we lo ok for solutions of the equation of motion for the eld

which satis es

lim (~x; t)=~n (1:2)

jxj!1

for all t. This condition formally compacti es the physical space to a 2-sphere S

2 2

and so all maps from S to S are characterised by the integer-valued degree of this

iso

map (the top ological charge).

The analytical formula for this degree is

~ ~ ~ ~

deg []= (@ @ )d x: (1:3)

1 2

This degree is a homotopyinvariant of the eld and so it is conserved during the

time evolution.

The Euler-Lagrange equation for the Lagrangian L (1.1) is

~ ~ ~ ~ ~ ~ ~

@ ( @ @ (@ @ )) = ~n: (1:4)

One simple solution of (1.4) is given by (~x; t) = ~n. This solution is of degree

zero and describ es the vacuum con guration. Another simple solution of (1.4) is

evidently = ~n. It is also of degree zero and may be considered as a false vacuum

con guration.

2. Static Skyrmions solutions

Some static solutions of the equation of motion (1.3) were discussed in [1,2]. An

imp ortant class of static solutions of the equation of motion consists of elds which

are invariant under the group of simultaneous spatial rotations by an angle 2 [0; 2 ]

and iso-rotations by n , where n is a non-zero integer. Such elds are of the form 2

0 1

sin f (r ) cos (n )

@ A

(~x)= sin f (r ) sin(n ) ; (2:1)

cos f (r )

where (r;) are p olar co ordinates in the (x; y )-plane and f is a function which satis es

certain b oundary conditions which will be sp eci ed b elow. Such elds are analogues

of the hedgehog eld of the Skyrme mo del and were studied in [3] for di erentvalues

of .

The function f (r ), the analogue of the pro le function of the Skyrme mo del, has

to satisfy

f (0) = m ; m 2 Z (2:2)

for the eld (2.1) to b e regular at the origin. To satisfy the b oundary condition (1.2)

we set

lim f (r )=0: (2:3)

r!1

Then solutions of the equation of motion which satisfy (2.2) and (2.3) describ e elds

for which the total energy is nite. Moreover, the degree of the elds (2.1) is

deg []=[cos f (1) cos f (0)] : (2:4)

For elds which satisfy the b oundary conditions (2.2) and (2.3) and which thus

corresp ond to nite energy con gurations, we get from (2.4)

deg []=[1(1) ] : (2:5)

The elds of the form (2.1) which are stationary p oints of the static energy func-

tional V , the time indep endent part of L in (1.1), must satisfy the Euler-Lagrange

equation for f

2 2

2 2 2 0 2

n sin f n sin f n f sin f cos f n sin f cos f

00 0 2

(r + )f +(1 + )f r sin f =0: (2:6)

r r r r

In [1,2] it was shown that the solutions of (2.6) for n =1;2 and m =1, corresp ond to

the absolute minima of the energy functional of degree 1 and 2 resp ectively. Any eld

obtained by translating and iso-rotating the solution corresp onding to n = m =1 was

called a baby Skyrmion. As this eld con guration has a top ological charge deg []=1

we call it a \baryon" and denote it by the symbol B . A solution corresp onding to

n = 1 and m =1 is then an anti-Skyrmion or an antibaryon B .

Clearly, there exist also solutions of the equation of motion which satisfy the

b oundary conditions (2.2) at the origin but which, at in nity, b ehave as

lim f (r )=l :

r!1

(2:7)

l 2 Z: 3

For such elds, if l is o dd, the solutions (2.6) di er from B -Skyrmions and b elong

to a class of solutions with in nite energies.

It is also worth mentioning that the asymptotic b ehaviour of these 1 energy

Skyrmion-like solutions di ers from the asymptotic b ehaviour of B -Skyrmions. Con-

sider, e.g., the case m =1;l =0;n =1 (B-Skyrmion). Then the asymptotic b ehaviour

of this solution is given by [2]:

e (2:8) f (r ) K (r )

1 r !1

where K (x) is the mo di ed Bessel function. On the other hand the asymptotic

b ehaviour of the 1 energy anti-Skyrmion-like solution corresp onding to (n =1;m =

2;l =1) is given by

f (r ) + sin(r + ); (2:9)

where C and are constants. As the energy of these solutions is in nite we will not

discuss them further in this pap er and so we will always assume that l =0 in what

follows.

3.5 2 3 E

2.5 1.5 E 2

1 1.5 Q

1 Q 0.5 0.5

0 0 0 2 4 6 8 10 0 2 4 6 8 10

r r

Figure 1.a : Energy and Top ological Figure 1.b : Energy and Top ological

charge density for the hedgehog solu- charge density for the hedgehog solu-

tion : n=1, m=1 tion : n=2, m=1

We have solved (2.6) for di erent values of n and m using a sho oting metho d

with the appropriate b oundary condition (2.2) (2.3) . Wehave determined the pro le

function f (r ) and the energy and top ological density pro les for each of these solutions.

The total energies for our solutions are as follows: 4

n n m 1 2 3 4

1 1.5642 5.011 10.030 16.492

2 2.9359 7.725 14.185 22.233

3 4.4698 10.555 18.350 27.819

4 6.1145 13.465 22.633 33.395

When m = 1, the top ological charge is given by n, and each con guration is a

sup erp osition of n Skyrmions. We know from [2] that only the rst two are stable.

Figure 1, shows the pro les of the energy and of the top ological density of the states

n =1 and n =2.

When m =2, the top ological charge is 0 and, as can be seen from Figure 2, these

con gurations corresp ond to a sup erp osition of n Skyrmions and n anti-Skyrmions

where the Skyrmions form a ring surrounding the anti-Skyrmion at the centre. We

have integrated separately b oth the p ositive and negative parts of the top ological

charge, and have found them to be n and n resp ectively, thus justifying our interpretation.

5 10 4 E 3

5 E 2

0 0 Q Q -1

-2

-5 0 2 4 6 8 10 0 2 4 6 8 10

r r

Figure 2.a : Energy and Top ological Figure 2.b : Energy and Top ological

charge density for the hedgehog solu- charge density for the hedgehog solu-

tion : n=1, m=2 tion : n=2, m=2

When m = 3, the top ological charge is given by n, and each con guration is a

sup erp osition of 2n Skyrmions and n anti-Skyrmions (Fig. 3). The con guration is

made of 3 layers, with n Skyrmions at the centre, n anti-Skyrmions in the middle,

and n Skyrmions in the outside ring. 5 10 20

E 15 E 5 10

Q 5

Q 0 0

0 2 4 6 8 10 0 2 4 6 8 10

r r

Figure 3.a : Energy and Top ological Figure 3.b : Energy and Top ological

charge density for the hedgehog solu- charge density for the hedgehog solu-

tion : n=1, m=3 tion : n=2, m=3

Note that when m is larger than 1 the energy of the con guration is larger than the

energy of nm baby Skyrmions. This indicates that such con gurations are unstable

and we will now analyse their decay mo des.

3. Exited meson M (n =1;m=2)

1;2

Let us lo ok rst at the eld which corresp onds to n = 1; m = 2. From (2.4) we

see that the top ological charge of this eld con guration is zero. So we can call this

state, a \meson" or a coherent \meson cloud". We note from Figure 2.a that, likeaB

Skyrmion, this solution corresp onds to a radially symmetric extended con guration

with the maximum of the energy density at the origin (r =0).

As the total energy (5:011) exceeds the sum of the masses of a B -Skyrmion and a

B -Skyrmion (2 x 1.5642), this con guration must be unstable.

The distribution of the top ological charge density shows more structure (Fig 2.a.)

We note that the top ological charge density is negative for small r and that it changes

sign at r = r = 2. At this p oint the pro le function f (r ) = , thus the solution

cr cr

corresp onding to the M state lo oks like a Skyrmion surrounded byananti-Skyrmion

1;2

eld. Of course the total top ological charge is zero. We see that the solution still lo oks

like an extended but lo calised con guration. However, although the top ological charge

density suggests that the Skyrmion is at the origin the pro le function there is given by

f =2 and so, from this p oint of view, resembles more the vacuum than a Skyrmion.

This suggests a p ossible interpretation of the M state in terms of Skyrmions and

1;2

anti-Skyrmions of the usual or 1 energy typ e. Whatever the interpretation, the state

is mesonic in nature. Moreover, lo oking at the eld con guration we note that most 6

of its changes takes place around those p oints in the (x; y ) plane where r = r i.e.

where f = . It is remarkable that everywhere along this circle = 1. This circle

is actually the region of instability when the solution M is excited by a non-radial

1;2

p erturbation of small amplitude.

In fact, it is not dicult to demonstrate that at each p oint on the circle r = r

one can create \hedgehog-like" extended ob jects using only in nitesimal p erturba-

tions. Of course, if these extended ob jects were to corresp ond to Skyrmions and

anti-Skyrmions then to conserve the top ological charge they will have to be created

in pairs and we would exp ect them to app ear as so on as we p erturb the initial con-

guration. Tocheck for such a b ehaviour wehave p erturb ed the initial con guration

corresp onding to the M -state. Our p erturbation was in the form of small excess

1;2

of kinetic energy centered around two p oints in the (x,y)-plane chosen symmetrically

with resp ect to r =0. Wehave found that, indeed, the state split into a Skyrmion and

anti-Skyrmion pair of the B -typ e. Their relative orientation was such that the force

between them was repulsive. After their creation the Skyrmion and the anti-Skyrmion

moved in opp osite directions from the centre.

We then exp erimented with p erturbing the initial state by di erent p erturbations

and we observed di erent decay mo des. When the applied p erturbation was not

symmetric with resp ect to r =0 (but was close to b eing symmetrical) the M state

1;2

decayed into a Skyrmion and an anti-Skyrmion, which then rotated in their internal

space so that their relative orientation made them to attract each other. They then

collided into each other and decayed into waves.

We thus conclude that the solution M of (2.6) is indeed a saddle p oint in the

1;2

space of eld con gurations. The state can b e thought of as a resonance of a Skyrmion

and an anti-Skyrmion and it has di erent decay mo des when p erturb ed. It decays

either into a B B -pair or into light \mesons".

It would be interesting to see whether it is p ossible to create the M state in a

1;2

head-on collision of a Skyrmion and an anti-Skyrmion. We are planning to come back

to this question in a future pap er.

4. Exited B baryon (n =1;m =3)

1;3

This is a baryon-typ e state as its top ological charge is one. The pro les of the

energy and of the top ological charge densities for this state are shown in Figure 3a.

Lo oking at the density of the top ological charge we see that this state may b e consid-

ered as the usual B -Skyrmion surrounded byananti-Skyrmion and a further Skyrmion

ring. The con guration has an energy larger than the energy of its constituents and

is thus unstable.

Wehave p erformed several simulations lo oking at the decay pro ducts of this state.

When we used a p erturbation symmetric with resp ect to the origin the con guration

decayed in 2 Skyrmions and 1 anti-Skyrmion, the anti-Skyrmion staying at the ori-

gin, while the two Skyrmions moved in opp osite directions. When the p erturbation 7

was not symmetric, the Skyrmions and the anti-Skyrmion were able to change their

relative orientations and one of the Skyrmions collided with the anti-Skyrmion and

decayed into waves.

5. Mesons made from dibaryons and anti-dibaryons

Let us now discuss various prop erties and decay mo des of exited meson-like states

made out of dibaryons (a b ound state of 2 B -Skyrmions) and antidibaryons. We will

concentrate our attention on the n =2 and m =2 state but our discussion generalises

easily to other states. The top ological charge of the (n =2, m=2) con guration (ie

its baryon numb er) is zero, so this con guration is a meson-like state from the p oint

of view of our classi cation. As n =2 the con guration lo oks like a dibaryon near the

origin, i.e. near r =0. From Figure 2.b, we see that it corresp onds to a dibaryon at the

origin surrounded byananti-dibaryon ring. The b order b etween these two regions of

opp osite top ological charge is, again, very well de ned and is situated along the circle

of radius r =3, (f(r )=). So this ring is again the region of instability. Applying

cr cr

di erenttyp es of p erturbations to our (n =2, m=2) solution, wehave observed three

di erent decay mo des for this state:

(

B + B + B + B

M ! : (5:1)

2;2

B + B + waves

waves

The pictures of the energy density for the rst decay mo des is shown in Figure 4.

Figure 4 : Top ological charge density

for the M ! B + B + B + B decay mo de.

2;2

6. Other exited mesonic and baryonic states

i) Let us lo ok rst at the state (n = 1;m = 4). This state, again, is mesonic with

deg [] = 0. The energy and top ological charge pro les are shown in Figure 5. 8

The state is more complicated as its energy density exhibits additional maxima

and minima. In fact, the state lo oks as if it were a coherent state of Skyrmions

and anti-Skyrmions, with rings of di erent radia o ccupied by elds of alternating

top ological charge. This is clearly seen from the top ological charge density plots;

moreover, the total top ological charge in each ring is +1,-1,+1 and -1 resp ectively

(going out from r =0). Thus we denote this state as M meson.

1;4

20 E E 5 10

0 0 Q Q

-5 -10 0 2 4 6 8 10 0 5 10 15

r r

Figure 5.a : Energy and Top ological Figure 5.b : Energy and Top ological

charge density for the hedgehog solu- charge density for the hedgehog solu-

tion : n=1, m=4 tion : n=2, m=4

Figure 6 : Top ological charge density

for the M ! B + B + B + B + B + B decay

2;3

mo de. 9

ii) Another interesting state is that of M (n = 2;m = 3). Its top ological charge is

2;3

2. Hence this state can be thought of as an exited state of the dibaryon M .

2;1

Its energy is clearly quite large and the state represents a coherent mixture of 4

Skyrmions and 2 anti-Syrmions and so it can decay into di erent channels. One

example of its decay mo de is shown in Figure 6. We see in this picture that the

decay pro ducts consist of two outgoing Skyrmions and two anti-Skyrmions with

the remaining 2 Skyrmions at rest at the origin. The fact that the decay pro ducts

involve one di-Skyrmion at the origin (slightly excited) is due to the attraction

between two Skyrmions, as mass (M ) < 2 mass(B ), see our table in section 2.

2;1

iii) Another interesting state is M which corresp onds to (n =2;m =4) and, as such, is

2;4

a highly excited meson state. Its energy and top ological charge pro les are shown

in Figure 5b.

Clearly, the list of new solutions may b e continued further by taking larger values

of n and m. Of course, having calculated some of their pro le functions we see that as

one increases m their energies increase (quite rapidly). All these higher energy states

can be treated as coherent states of some number of Skyrmions and anti-Skyrmions

and are unstable. Under suitable p erturbations they will decayinto various channels

involving Skyrmions and anti-Skyrmions with some Skyrmions and anti-Skyrmions

annihilating into pure waves.

7. Plane wave solutions

Let us now lo ok at wave-like solutions of our mo del. Recall that such wave-

[6] [7]

like solutions have already b een studied in for a slightly di erent version

of a \skyrme-like" (2+1) dimensional mo del (the p otential term of that mo del was

di erent.)

What typ e of waves can we nd for the Lagrangian in the form (1.1) ? To lo ok for

plane wave-like solutions, we seek solutions which do not dep end on one variable, say

y . However, as so on as we imp ose this condition we note that the Skyrme-like term

vanishes for these eld con gurations. So the discussion is not that dicult and can

be p erformed analytically.

First, we lo ok for solutions of the equation of motion for the eld in the form:

=(sin f cos ; sin f sin ; cos f ); (7:1)

where f = f (x; t) and = (x; t). In terms of f and the Lagrangian takes the form

2 2

L =(1=2) d x (@ f@ f + ((1 cos(2f ))=2)@ @ (1 cos(f )) : (7:2)

To go further we consider the case when the phase is constant. In this case the

eld f satis es

@ @ f + sin(f )=0; (7:3)

which is the sine-Gordon equation. So we see that when = const the wave solutions

of the Lagrangian (1.1) are given by the solutions of the sine-Gordon equation.

Moreover, solutions of (7.3) with small amplitude may be considered as ordinary

plane waves with the disp ersion relation given by

2 2 2

! = + k : (7:4)

Of all nite energy solutions of (7.3) (one-dimensional case) the most imp ortant,

and p erhaps the b est studied, is the solution of the kink typ e:

h i

x x vt

f (x; t)=4atan exp ( ) : (7:5)

(1 v )

When translated to our case we note that when x changes from 1 to +1, the

vector moves along the meridional cross-section of the S -sphere and returns to

iso

the same p oint. Moreover, this is true for any xed time t.

Another solution of (7.3) , called breather, is also well known and has b een studied

[8]

by many p eople. Its form is

2 1=2

sin(! (t t )) (1 ! )

(7:6) f (x; t)=4atan

2 1=2

cosh((1 ! ) (x x ))

where ! is the frequency of the internal breather oscillations.

These 2typ es of solitonic waves are in nite front lines (the solutions dep end only

on one variable, x). Unfortunately, once imb edded into S , the target space for our

mo del, the extra degrees of freedom make these waves unstable. For the kink solution

this is not surprising as the \lo op" around the meridian can easily \slip" on one side

of the sphere, thus decreasing the p otential energy.

To see this, let us take the con guration

0 1

cos sinf

@ A

(~x )= sin cos (1 cosf ) ; (7:7)

1 cos (1 cosf )

where f is given by (7.5) with v set to 0, and where is a function which dep ends

only on y and which go es to 0 when y go es to in nity. This con guration describ es

the kink (7.5) (7.1) p erturb ed lo cally in x and y , and it \displaces" the kink from the

meridian of the 2-sphere onto a lo op of a smaller radius on S . The energy density

for this static con guration is given by

2 2

2 2 2 2

(sin sin f +(1cos f ) ) E = dxdy + cos f

y x

(7:8)

2 2 2 2 2 2 2

(1cos f ) cos + cos (1 cos f ) :

+k f

y x 11

To prove that for some appropriate choice of , the energy decreases, we compute the

change of energy induced by a non-zero

E =E ( =0)E( )

f f

2 2 2

2 2 2 2 2

(sin k (1 cosf ) cos ) (1 + sin cos ) = dxdy 2sin

y y

2 2

(7:9)

+ sin (1 cos f )

h i

2 2 2 2

2 2 2 2 2

dxdy 2sin (sin k cos ) (1 + sin ) + sin (1 cos f )

y y

where we have used the fact that for (7.5) f =2sin(f=2). Given (y ) satisfying the

asymptotic b ehaviour imp osed, we can always stretch it by p erforming the dilation

y ! ay to make small enough to make the two negative terms in (7.9) smaller than

the p ositive terms. This proves the instability of the sin-Gordon kink wave when

imb edded into the S mo del.

We have p erformed some numerical simulations for b oth typ es of waves and have

indeed observed their instability. In b oth cases, as so on as some region of the wave

is p erturb ed, the wave collapses around this p oint, emitting radiation. The collapse

front then propagates rapidly along the solitonic wave destroying it completely.

[7]

In our previous work , we have studied the scattering prop erties of plane waves

and skyrmions. The skyrmions studied in [7] were di erent (the p otential was dif-

ferent) so we have rep eated our analysis for the mo del studied here and have found

no ma jor di erence. Like in the previous case, the skyrmion absorbs a section of the

wave and starts moving after the collision.

Wehave also studied the scattering b etween a skyrmion and the sine-Gordon front

waves. We observed that when the skyrmion is close to a sine-Gordon wave (breather

or kink) it triggers the wave collapse and, as a result, there is no real scattering

between these two ob jects.

While p erforming simulations with large amplitude breather waves we have ob-

served the formation of a radially symmetric breather-like soliton. This solution

lo oks very similar to the pulsons observed in [9]. We discuss some of its prop erties

in the next section.

8. Non-top ological Solitons

In the previous section wehave describ ed our studies of waves in the baby-skyrmion

mo del in which we observed that the plane wave solutions which are given by the

solutions of the (1+1) dimensional sin-Gordon equation are unstable.

However, we when welooked at the decay of some breather frontwaves we encoun-

tered something rather unexp ected. Instead of decaying into waves that dissipate like

kink solutions they pro duced a radially symmetric breather-like eld con guration

which app eared to b e relatively stable. By lo oking at the time evolution of this eld

con guration, as pro duced by our simulation, we were able to conclude that: 12

- the eld con guration is radially symmetric.

- up to a global rotation of S , the solution \lives" in the ; plane of the target

1 3

space (S ).

To study this eld con guration further we make the following ansatz:

=(sin f (r;t);0;cosf (r;t)): (8:1)

The lagrangian density then b ecomes

L = dr r (@ f@ f @ f@ f (1 cos (f )): (8:2)

t t r r

and the equation reduces to the radial sin-Gordon equation:

f f + sin (f )=0: (8:3)

tt rr

This equation has already b een studied in [9], where it was shown that it has time

dep endent solutions similar to a breather, but which radiate their energy and slowly

die out. The authors of [9] decided to call such con gurations pulsons.

In [10] we have also shown that there exist stable time dep endent solutions. The

radial eld con gurations of (8.3) radiate relatively quickly when their amplitude

of oscillation is relatively small, that is when the value of f never b ecomes larger

than =2 at the origin (these are the pulsons studied in [9].) When the amplitude of

oscillation is larger than =2 the breather-like con guration slowly radiates its energy

and asymptotically reduces the amplitude of oscillation to =2 and and settles at a

p erio d of oscillation T 20:5 (when =0:1). By trial and error we have found that

2 r r

f (r; 0)=4atan C exp atan( )

K K

(8:4)

(r; 0)=0

with K =10 and C = tan(=8) is a good initial condition for this metastable pseudo-

breather solution.

As in the case of plane wave solutions wehavetocheck that once they are imb edded

into S they are still stable. We have indeed checked this numerically and we have

found that the solution describ ed by (8.1) and (8.3) is indeed stable in the S mo del.

The energy of this new breather-like solution is given by

E 3:97

which means that it is 2:5 heavier than the baby-skyrmion. Moreover, its top ological

charge density is identically zero but it has enough energy to decay into a skyrmion

anti-skyrmion pair. In what follows we shall refer to these eld con gurations as

pseudo-breathers. 13

In practice, it is very dicult to have a eld con guration of a pseudo-breather.

However, we can nd approximate eld con gurations which still radiate energy and

asymptotically reach the stable (or p erhaps only metastable) con guration con g-

uration of the pseudo-breather. The excess of energy over the nal con guration

can then be seen as an excitation energy which is slowly radiated away. During any

scattering pro cess solitons tend to exchange or radiate some energy. When the ex-

citation energy is large enough, the outgoing pseudo-breather-like con guration may

have enough energy to evolveinto the stable pseudo-breather eld; otherwise, it ends

up with less energy than the metastable con guration and progressively dies out.

The scattering prop erties of pseudo-breathers are quite interesting. When the

pseudo-breathers are imb edded into the baby-skyrmion mo del the eld con gurations

have an extra degree of freedom corresp onding to their orientation inside the ;

1 2

plane. When two pseudo-breathers are set at rest near each other, the force b etween

them dep ends very much on their relative orientation: when they are parallel to each

other and oscillate in phase, they attract each other, overlap and form a new structure

which app ears to be an exited pseudo-breather. This pseudo-breather then slowly

radiates away its energy. The non-top ological nature of pseudo-breathers means that

they can indeed merge to form a new structure of the same typ e.

If the two pseudo-breathers are anti parallel, i.e. if they oscillate completely out

of phase, then the force between them is repulsive. When the two pseudo-breathers

have a di erent orientation they slowly rotate themselves until they b ecome parallel;

then they move towards each other and form an exited pseudo-breather structure.

50 50

40 40

30 30

20 20

10 10

Y 0 Y 0

-10 -10

-20 -20

-30 -30

-40 -40

-50 -50 -40 -20 0 20 40 -40 -20 0 20 40

X X

Figure 7.a : Pseudo-Breather scatter- Figure 7.b : Pseudo-Breather scatter-

ing. Impact Parameter: 10, v =0:2 ing. Impact Parameter: 15, v =0:2

When two pseudo-breathers are senttowards each other with some kinetic energy, 14

the scattering is more interesting. Dep ending on the initial sp eed or the scattering

impact parameter, they either merge into a single pseudo-breather or they undergo

a forward scattering. The details of these scattering prop erties are given in [10]. In

Figure 7 we show two such tra jectories. For b oth scatterings the initial sp eed was

0:2, the only di erence b eing the values of the impact parameter.

9. Pseudo-Breather-Skyrmion Scattering

As we have seen, the baby-skyrmion mo del has two di erent typ es of extended

solutions. The skyrmions are top ological solitons which are very stable, while the

pseudo-breathers are time-p erio dic solitons which can slowly decay if they are p er-

turb ed to o much. It is very unusual to have a mo del that exhibits two such di erent

stationary structures and so it is interesting to analyse how they interact with each other.

Total Energy : t = 0.000000

Min = 0.000009 Max = 0.165958

Figure 8 : Skyrmion and pseudo-breather at rest.

When a skyrmion and a pseudo-breather are put at rest next to each other the 15

overall interaction between them makes the skyrmion slowly move away from the

pseudo-breather-like con guration while the pseudo-breather lo oses some of its energy

faster than when it is placed there by itself.

To scatter a skyrmion with a pseudo-breather wehave placed the pseudo-breather

soliton at rest, and we have sent the skyrmion towards it. We have p erformed this

scattering for di erent orientations of the pseudo-breather, for di erent values of the

impact parameter and for a few di erent sp eeds. In each case, the pseudo-breather

was initially lo cated at the origin while the skyrmion always started from (x ;y )

0 0

where x is the initial p osition along the x axis and y is the impact parameter. The

0 0

pseudo-breather was oscillating in the ( ; ) plane along the direction (cos ( ); sin ( )).

1 2

In Figure 8 we show the initial condition corresp onding to a baby-skyrmion next to

a pseudo-breather. We note that the skyrmion is much more spiky than the breather.

Our numerical results are summarised in the following 4 tables.

v n y0 15 10 7.5 5 3.5 2.5 1.25

0.2 5 18 31 39 58 67 -108

0.3 6 26 16 15 36 35 -24

0.4 4 7 11 0 -9 -8

Table 1.a : Impact parameter and sp eed dep endance of the scattering angle ( =0, x =20).

v n y0 15 10 7.5 5 3.5 2.5 1.25

0.2 8 19 27 30 21 21 90

0.3 6 16 24 23 21 29 23

0.4 5 20 21 20 18 19 4

Table 1.b : Impact parameter and sp eed dep endance of the scattering angle ( = =2, x = 20).

v n y0 15 10 7.5 5 3.5 2.5 1.25

0.2 7 18 29 32 52 62 -166

0.3 7 14 15 2.5 22 45 -27

0.4 3 8 8 15 14 24 -43

Table 1.c : Impact parameter and sp eed dep endance of the scattering angle ( = =4, x = 20).

v n x0 15.2 16.2 17.2 18.2 19.2 20.2

0.2 38 33 33 38 31 21

0.4 15 16 17 20 22 19

Table 2: Scattering angle as a function of the initial distance ( = =2;y =2:5).

The amount of energy lost by the pseudo-breather during the scattering is larger

when the overlap between the skyrmion and the breather, b oth in time and space,

increases. In some cases the pseudo-breather is completely destroyed by the collision.

The oscillation of the pseudo-breather makes the interaction time dep endent, and, as 16

a result, it is dicult to extract a simple pattern from the tables of scattering angles.

10. Conclusions

We have shown that the baby-skyrmion mo del has many interesting classical so-

lutions in addition to the skyrmion solitons. The rst class of solutions describ e

exited states of skyrmions and anti-skyrmions which are unstable with resp ect to

p erturbations.

The second class of solutions involves non-top ological stationary stationary eld

con gurations which are p erio dic in time. They are relatively stable but can be

destroyed if they are p erturb ed to o much.

ACKNOWLEDGEMENTS

One of the authors (AK) thanks The Department for Mathematical Science of

University of Durham for hospitality during his visit. This visit was supp orted by

INTAS grant 93-633 and partly by grants INTAS-CNRS1010-CT93-0023 and RFFR-

95-02-04681.

We want to thank R.S. Ward for helpful comments.

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