b.m.a.g. piette, b.j. schroers and w.j. zakrzewski- dynamics of baby skyrmions

8/3/2019 B.M.A.G. Piette, B.J. Schroers and W.J. Zakrzewski- Dynamics of baby Skyrmions

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ELSEVIER Nuclear Physics B 439 (1995) 205-235

N U C L E A RP H Y S I C S B

Dynamics of baby SkyrmionsB . M . A . G . P i e t t e 1, B . J . S c h r o e r s 2 , W . J . Z a k r z e w s k i 3

Department of Mathematical Sciences, South Road, Durham DH1 3LE, United KingdomReceived 4 Octob er 1994; accepted 30 Decem ber 1994

A b s t r a c tBab y Skyrmions are topological sol i tons in a (2+ I)-d im en sio na l field theory w hich resemblesthe Skyrme model in important respects . We apply some of the techniques and approximationscom mo nly used in discussions of the Skyrme m odel to the dynamics o f baby Skyrmions anddirectly test them against numerical simulations. Specifically we study the effect of spin on theshape o f a s ingle b aby Skyrmion, the dependence of the forces between two bab y Skyrmions onthe baby Skyrmions ' relat ive orientat ion and the forces between two baby Skyrmions when one

of them is spinning.

1 . I n t r o d u c t i o n

T h e g o a l o f th i s p a p e r i s to s t u d y th e d y n a m i c s o f s o li to n s in a ( 2 + l ) - d i m e n s i o n a lv e r s i o n o f t h e S k y r m e m o d e l . B y a s o l i t o n w e m e a n a l o c a l i s e d , f i n i t e - e n e r g y s o l u -t i o n o f a n o n - l i n e a r f i e l d t h e o r y . T h e S k y r m e m o d e l i s a n o n - l i n e a r f i e l d t h e o r y f o rp i o n s i n 3 + 1 d i m e n s i o n s w i t h s o l i to n s o l u ti o n s c a l le d S k y r m i o n s [ 1 ] . S u i t a b l y q u a n -t is e d , S k y r m i o n s a r e m o d e l s f o r p h y s i c a l b a r y o n s . S k y r m e ' s th e o r y i s n o n - i n te g r a b l ea n d t h e r e f o r e p r o g r e s s i n u n d e r s ta n d i n g S k y r m i o n d y n a m i c s h a s d e p e n d e d o n n u m e r -i c a l s i m u l a t i o n s , a p p r o x i m a t i o n s c h e m e s o r a c o m b i n a t i o n o f b o th . T h i s a p p r o a c h h a sb e e n q u i t e s u c c e s s f u l i n t h e s t u d y o f s ta t ic s o l i t o n s o lu t i o n s in S k y r m e ' s th e o r y [ 2 , 3 ] .H o w e v e r , th e i n t e r a c t iv e d y n a m i c s o f t w o S k y r m i o n s , w h i c h o n e n e e d s t o u n d e r s ta n di n o r d e r to e x t r a c t th e S k y r m e m o d e l ' s p r e d i c t io n s f o r th e n u c le a r t w o - b o d y p r o b l e m ,i s m o r e d i f f i c u l t t o d e s c r i b e . C e r t a i n s c a t t e r i n g p r o c e s s e s o f t w o S k y r m i o n s h a v e b e e n

l E-mail: [email protected] E-m ail: b.j.schroers@durham .ac.uk.3 E-m ail: w.j.zakrzewski@ durham.ac.uk.0550-3213 /95/$09.50 (~) 1995 Elsevie r Scien ce B.V. All figh ts reservedSSDI 05 50 - 321 3 ( 9 5 ) 0 00 1 1 -9


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206 B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235s i m u l a t ed num er i ca l l y [ 4 ] , bu t t he va r i e t y o f pos s i b l e i n i t i a l cond i t i ons t ha t one cou l dcons i de r i s s o g r ea t t ha t i t s e em s i m po s s i b l e t o ge t an ove r a l l p ic t u r e o f the s ca t t e r i ngbeha v i ou r f r om j u s t a f ew p r oces s e s . On t he o t he r hand , va r i ous app r ox i m a t i ons havebeen us ed wh i ch t yp i ca l l y i nvo l ve t runca t i ng t he f i e ld t heo r y wi t h i n f i n i t e ly m an y de -g r e e s o f f r e e d o m t o a f i n it e - d im e n s i o n a l d y n a m i c a l s y s t e m [ 5 ] . S o m e a p p r o x i m a t i o n shave becom e wi de l y accep t ed wi t hou t , howeve r , hav i ng been t e s t ed aga i ns t num er i ca ls i m u l a t i ons o f t he f u l l t heo r y . He r e we wi l l app l y m any o f t he concep t s and app r ox i -m a t i o n s d e v e l o p e d i n t h e S k y r m e m o d e l t o o u r m o d e l a n d d i r e c tl y c o m p a r e t h e m w i thnum er i ca l s i m u l a t i ons .

Our s o l i t ons a r e exponen t i a l l y l oca l i s ed i n s pace , a p r ope r t y s ha r ed by Skyr m i onsw h e n t h e p h y s i c a l p i o n m a s s i s i n c l u d e d i n t h e S k y r m e m o d e l . I n o u r m o d e l a s o l i t o nhas a f i xed s i z e bu t a r b i t r a r y pos i t i on and o r i en t a t i on . I n t wo s pa t i a l d i m ens i ons t h i sc o r r e s p o n d s t o t h r e e d e g r e e s o f f r e e d o m , t w o f o r t h e s o l i t o n ' s p o s i t i o n a n d o n e a n g l et o de s c r i be i t s o r i en t a t i on . Th i s s hou l d aga i n be com par ed wi t h Skyr m i ons , wh i ch havea de f i n i t e s i z e and s i x deg r ee s o f f r e edom , t h r ee g i v i ng i t s pos i t i on i n s pace and t h r eepa r am e t r i s i ng i t s o r i en t a t i on . The l ong - r ange i n t e r ac t i on behav i ou r o f ou r s o l i t ons r e -s em bl e s t ha t o f Sky r m i ons i n i m por t an t r e s pec t s : i n bo t h ca s e s t he a s ym pt o t i c f o r ce sbe t ween t wo s o l i t ons depend on t he i r s epa r a t i on and t he i r r e l a t i ve o r i en t a t i on and a r eo f t h e d i p o l e - d i p o l e t y p e . F u r t h e r m o r e , t h e r e i s a b o u n d s t a t e o f t w o s o l i t o n s w h o s eene r gy d i s t r i bu t i on i s m ax i m a l on a r i ng i n bo t h ca s e s . When t he s o l i t ons a r e o r i en t a t eds o t ha t t he f o r ce s a r e m os t a t t r a c t i ve and t hen r e l ea s ed f r om r e s t t hey s ca t t e r t h r ought he t o r o i da l co n f i gu r a t i on and em er ge a t 90 deg r ee s r e l a t i ve to t he i n i ti a l d i r ec t ion o ft he i r m o t i on .

Becaus e o f a l l t he s e s i m i l a r i t i e s we ca l l ou r s o l i t ons baby Skyr m i ons . Th i s t e r mh a s b e e n u s e d q u i t e w i d e l y t o d e s c r i b e s o l i t o n s i n 2 + 1 d i m e n s i o n s w h i c h r e s e m b l eSkyr m i ons i n c e r t a i n r e s pec t s . Howeve r , i n a l l t he m ode l s s t ud i ed s o f a r t he m om en t o fi ne r t i a f o r t he r o t a t i on o f a s i ng l e s o l i t on i s i n f i n i t e , s o t ha t t he r o t a t i ona l deg r ee s o ff r eedom a r e no t dynam i ca l l y r e l evan t . Ye t t he r o t a t i ona l deg r ee s o f f r e edom a r e c r uc i a li n Sk yr m i on dynam i cs . I n t h i s pape r we a r e t he r e f o r e pa r t i cu l a rl y in t e r e s t ed i n t hos ea s p ec t s o f o u r m o d e l w h i c h d e p e n d o n t h e s o l it o n s ' o r i e n ta t io n .

Af t e r a d e s c r i p t i on o f ou r m ode l and a qu i ck r ev i ew o f it s s ta t ic s o l u t i ons , d i s cus s ed i nde t a i l i n [ 6 ] and [ 7 ] , we f o cus on the f o l l owi ng ques t i ons . Ar e t he r e exac t s o l u t ions o ft h e f i e l d e q u a t i o n s r e p r e s e n t i n g s p i n n i n g b a b y S k y r m i o n s ? H o w d o e s a b a b y S k y r m i o nc h a n g e i t s s h a p e w h e n i t s p i n s ? H o w d o t h e a s y m p t o t i c f o r c e s b e t w e e n t w o b a b yS k y r m i o n s d e p e n d o n t h e i r r e la t iv e o ri e n t a ti o n ? W h a t a r e t h e f o r c e s b e t w e e n t w o b a b yS k y r m i o n s w h e n o n e o f t h e m i s s p in n i n g ?

W h i l e t h e q u e s ti o n s w e i n v e st ig a t e h e r e a r e la r g e ly m o t i v a te d b y t h e ( 3 + l ) - d i m e n -s i ona l Sky r m e m ode l we s hou l d em phas i s e t ha t ou r m ode l i s a l s o o f i n t e r e s t i n t he l a r gea n d g r o w i n g a r e a o f ( 2 + 1 ) - d i m e n s i o n a l s o l it o n p h e n o m e n o l o g y . T h e s o l i to n s in o u rm o de l hav e a nu m b er o f p r ope r t i e s wh i ch a r e nove l i n th i s con t ex t .


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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235 2072 . T h e m o d e l

T h e b a s i c f i e ld o f o u r m o d e l i s a m a p~ b :M 3 ~ S i2so, (2 .1 )

w h e r e M 3 i s t h r e e -d i m e n s i o n a l M i n k o w s k i s p a c e w it h t h e m e t r ic d i a g ( 1 , - 1 , - 1 ) . W es e t th e s p e e d o f l i g h t to 1 a n d w r i t e e l e m e n t s o f M 3 a s ( t , x ) , w h e r e x i s a 2 - v e c t o rw i t h c o o r d i n a t e s x i , i = 1 , 2 , a l s o s o m e t i m e s d e n o t e d b y x a n d y . W e u s e t h e n o ta t i o nx ~ , a = 0 , 1 , 2 , t o r e f e r t o b o t h t h e t i m e an d sp a t i a l co m p o n en t s o f ( t , x ) , so t = x . T h et a rg e t sp ac e S~iso i s t h e 2 - sp h e re o f u n i t r ad i u s em b ed d ed i n eu c l i d ean 3 - sp a ce w i t h t h eR i e m a n n i a n m e t r i c i n d u c e d b y t h a t e m b e d d i n g . H e r e t h e s u ff i x " i s o " i s u s e d t o e m p h a s i s et h e a n a l o g y w i t h t h e t a r g e t s p a c e i n t h e S k y r m e m o d e l , w h i c h i s o f t e n r e f e r r e d t o a si so - sp ac e . T h e f i e ld ~ b i s a s ca l a r f i e l d w i t h t h r ee co m p o n e n t s ~ b a, a = 1 , 2 , 3, s a t i s fy i n gthe c on st r ain t ~b. ~ = ~bl + ~b~ + ~b~ = 1 fo r al l x E M 3. O ur La gr an gi an den si ty, thes t a t ic p a r t o f w h i ch w a s f ir s t co n s i d e red i n [6 ] an d d i s cu s sed fu r t h e r in [ 7 ] , i s). = F C7c,#~ . 3 '* dp - T ( ? ,~ t~ x ,9 3d~ ) ( 3 '~ dp c~ J~ ) - / z 2 ( 1 - n . ~b) ,

( 2 . 2 )w h e re n = (0 , 0 , 1 ) an d ? ,~ = 3 / 3 x " . T h e co n s t an t s F , K an d / z a r e f r ee p a ram e t e r s :F h a s t h e d i m e n s i o n e n e r g y a n d K a n d 1 / / z h a v e t h e d i m e n s i o n l e n g th . I t i s u s e f u l tot h i n k o f F a n d K a s n a t u ra l u n i t s o f e n e r g y a n d l e n g t h r e s p e c ti v e l y a n d o f 1 / / z a s as e c o n d l e n g t h s c a l e i n o u r m o d e l . H e r e w e f i x o u r u n i t s o f e n e r g y a n d l e n g t h b y s e t t i n gF = K = 1 . W e h av e a l r ead y se t t h e sp eed o f li g h t t o 1 , so w e u se g eo m e t r i c u n i t s inw h i c h a l l p h y s i c a l q u a n t i t i es a r e d i m e n s i o n l e s s . T h u s w e c a n n o t s e t / z t o 1 b y a c h o i c eo f u n i t s , an d w e w i l l f i x i t s v a l u e l a t e r a f t e r w e h av e d i s cu s sed i t s s i g n i f i c an ce . N o t ea l s o t h a t P l a n c k ' s c o n s t a n t h w i l l b e s o m e n u m b e r , b u t n o t n e c e s s a r i l y e q u a l t o 1 . T h ef i r s t t e r m i n ( 2 . 2 ) i s f a m i l i a r f r o m o - m o d e l s w h o s e s o l i t o n s o l u t i o n s h a v e b e e n s t u d i e de x t e n s i v e l y [ 8 ] . T h e s e c o n d t e r m , f o u r t h o r d e r i n d e ri v a ti v e s , i s t h e a n a l o g u e o f th eS k y r m e t e r m i n t h e u s u a l S k y r m e m o d e l . I n [ 7 ] w e e x p l a i n e d i n m o r e p r e c i s e t e r m s i nw h i c h s e n s e t h i s a n a l o g y h o l d s b y a p p e a l i n g t o a g e n e r a l g e o m e t r i c f r a m e w o r k f o r t h eS k y r m e m o d e l d u e t o M a n t o n [ 9 ] . F i n a ll y , t h e la s t t e r m d o e s n o t c o n t a i n a n y d e r i v a ti v e sa n d i s o f t e n r e f e r r e d t o s i m p l y a s a p o t e n t i a l . I n t h r e e ( s p a t i a l ) d i m e n s i o n s t h e S k y r m et e rm i s n ece s sa ry fo r t h e ex i s t en ce o f so l i t o n so l u t i o n s b u t t h e i n c l u s i o n o f a p o t en t i a li s o p t i o n a l f r o m t h e m a t h e m a t i c a l p o i n t o f v ie w . P h y s i ca l ly , h o w e v e r , a p o t e n t i a l o f ac e r t a in f o r m i s r e q u i r e d t o g iv e t h e p io n s a m a s s [ 1 0 ] . B y c o n t ra s t , i n tw o d i m e n s i o n s ap o t e n t i a l m u s t b e i n c l u d e d i n t h e a b o v e L a g r a n g i a n i n o r d e r t o o b t a i n s o l i t o n s o l u t i o n s .

T o s e e t h i s , a n d t o u n d e r s t a n d t h e L a g r a n g i a n L = f d 2 x bet t e r , we wr i t e i t i n theu su a l f o r m L = T - V . H e re T i s t h e k i n e t i c en e rg y

= [ 4 , . 4 , + ( 4 , a , , , ) . ( 4 , d 2 x , ( 2 . 3 )J


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B.M.A.G. Piette et al . /Nu clear Physics B 439 (1995) 205-235 209o n e c h e c k s t h a t t h e l i n e a r i s e d e q u a t i o n f o r t p i s t h e m a s s i v e K l e i n - G o r d o n e q u a t i o n

( [ ] + / z 2 ) ~ = O , ( 2 .1 0 )w h e r e [ ] = 0 a 0 a i s th e w a v e o p e r a t o r i n 2 + 1 d i m e n s i o n s . I n t h e l a n g u a g e o f p e r t u r h a t iv eq u a n t u m f i e l d t h e o r y , w h e r e o n e q u a n t i s e s s m a l l f l u c t u a t i o n s a r o u n d t h e v a c u u m , t h esca l a r f i e ld s ~ l and ~o2 the r e f o r e co r r e spo nd to s ca l a r pa r t i c le s o r meson s o f m ass h / z.W e n o w u n d e r s t a n d / z w e l l e n o u g h t o f i x i t s v a l u e . O u r c h o i c e i s a g a i n d i c t a t e d b y t h ed e s i r e t o r e p r o d u c e i m p o r t a n t f e a tu r e s o f th e S k y r m e m o d e l . T h e r e , a s i n r e a l n u c l e a rp h y s i c s , t h e C o m p t o n w a v e l e n g t h o f t h e p i o n i s o f t h e s a m e o r d e r a s ( i n f a c t , s l i g h t l yl a r g e r t h a n ) t h e s i z e o f a S k y r m i o n . T h u s , w e w a n t t o t u n e ] z i n o u r m o d e l s o t h a t t h ee n e r g y d i s t r i b u t i o n o f i t s b a s i c s o l i t o n s o l u t i o n i s c o n c e n t r a t e d i n a r e g i o n o f d i a m e t e r

1 / / z . B y t ri a l a n d e r r o r w e f in d t h a t th i s i s t h e c a s e w h e n w e s e t / x 2 = 0 . 1 , w h i c h w ed o f o r t h e r e s t o f t h i s p a p e r .

F o r l a t e r u s e w e n o t e a c o n s e r v a t i o n l a w t h a t ca n b e r e a d o f f i m m e d i a t e l y f r o m t h ee q u a t i o n o f m o t i o n ( 2 . 8 ) . T a k i n g th e s c a l a r p r o d u c t w i t h n o n b o t h s id e s o f th e e q u a t io nwe f i nd t ha t t he cu r r en t

n . ~ x O , ~b - ( n . 0 ~ $ ) ( O # ~b ~b 0 , ~ $ ) ( 2 . 1 1 )h a s v a n i s h i n g d i v e r g e n c e . T h e s y m m e t r y t h a t l e a d s t o t h i s c o n s e r v a t i o n l a w i s S O ( 2 )r o t a t i o n s o f t h e f i e ld q ~ a b o u t n , w h i c h c a n b e w r i t te n i n t e r m s o f a n a n g l e X a s

(& l , ~b2 , ~b3 ) ~ (co s X $1 + s in X~b2, - s in X~bl + cos X t~ , t~3 ) . (2 . 12 )W e c a l l s u c h a t r a n s f o r m a t i o n a n i s o - r o t a t i o n ; t h e c o r r e s p o n d i n g c o n s e r v e d q u a n t i t y i sc a l l e d i s o s p i n a n d d e n o t e d b y I :

1 = f n . d p x d p + ( n . O i d p) (O i dp .d p x ~ p ) d 2 x . ( 2 . 1 3 )F o r a s y s t e m a t i c d is c u s s i o n o f t h e s y m m e t r i e s o f o u r m o d e l w e r e f e r t h e r e a d e r to [ 7 ] ,h u t h e r e w e n e e d o n l y n o t e t h a t b o t h t h e L a g r a n g i a n L a n d t h e d e g r e e ( 2 . 6 ) a r e in v a r i a n tu n d e r s i m u l t a n e o u s r e f l e c t i o n s i n s p a c e a n d i s o - s p a c e

P x : ( x , y ) ~-~ ( - x , y ) and (~bl ,~b2 , t~3) ~ ( - - ~ 1 , ~ 2 , t ~ 3 ) . ( 2 . 1 4 )L a t e r w e w i l l a l s o m a k e u s e o f t h e i n v a r i a n c e o f b o t h t h e d e g r e e a n d t h e L a g r a n g i a nu n d e r t h e c o m b i n a t i o n o f P x wi th a r o t a t i on by z r i n bo th space and i so - space :

P y : ( x , y ) ~ -, ( x , - y ) a n d ( t~ l ,q ~ 2 , t~ 3 ) ~ ( ~1 , - - q~E , t~3 ) ( 2 . 15 )3 . S t a t i c s o l u t i o n s r e v i s i t e d

T i m e - i n d e p e n d e n t s o l u ti o n s o f th e e q u a t io n s o f m o t i o n ( 2 . 8 ) , w h i c h a r e s t a t io n a r yp o i n t s o f t h e p o t e n t i a l e n e r g y f u n c t i o n a l V ( 2 . 4 ) , w e r e s t u d i e d i n d e t a i l i n [ 7 ] . W ebr i e f l y r eca l l t hose r e su l t s wh ich a r e r e l evan t he r e . An impor t an t c l a s s o f s t a t i c so lu t i onso f t h e e q u a t i o n s o f m o t i o n c o n s i s t s o f f i e l d s w h i c h a r e i n v a r i a n t u n d e r t h e g r o u p o f


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21 0 B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235s i m u l t aneous s pa t i al r o t a t i ons by s om e ang l e a E [ 0 ,2~r ) and is o - r o t a t i ons by - n t ~ ,whe r e n i s a non - ze r o i n t ege r . Such f i e l d s a r e o f t he f o r m

f s in f ( r ) c o s ( n 0 - X )q ~ ( x ) = t s i n f ( r ) s i n ( n 0 - X ) ) , ( 3 . 1 )\ c o s f ( r )

wh e r e ( r , 0 ) a r e po l a r coo r d i na t e s i n t he x - p l ane and f i s a f unc t i on s a t i s f y i ng ce r t a inbou nda r y con d i t i ons t o be s pec i f i ed be l ow. The a ng l e g i s a l so a r b i tr a r y , bu t f i el d s wi t hd i f f e r en t X a r e r e l a t ed by an i s o - r o t a t ion and t he r e f o r e degene r a t e in ene r gy . Thu s w econc en t r a t e on t he s t anda r d f i e ld s whe r e g = 0 . Such f i e l d s a r e t he ana l ogue o f t hehed geh og f i e ld s i n t he Skyr m e m o de l and w e r e a ls o s tud i ed i n [ 6 ] f o r d i f f e r en t va l ue so f / z 2.

Th e f unc t i on f , wh i ch i s c a ll ed t he p r o f i l e f unc t i on , ha s to s a t i s fyf ( 0 ) = m ~ ' , m C Z , ( 3 .2 )

f o r t he f i e l d ( 3 .1 ) t o be r egu l a r a t t he o r i g i n ; t o s a t i s f y t he bounda r y cond i t i on ( 2 .5 )w e s e t

l im f ( r ) = 0 . ( 3 . 3 )r---~OOH er e w e w i l l on l y be i n t e r e s t ed i n p r o f il e f unc t i ons whe r e m = I ( t he s i t ua t i on f o rgene r a l m i s d i s cus s ed i n [ 7 ] ) . O ne t hen f i nds t ha t the deg r ee o f t he f i e ld ( 3 .1 ) i s

d e g [ q l ] = n . ( 3 . 4 )Fo r a fi e l d o f t he f o r m ( 3 .1 ) t o be a s t a ti ona r y po i n t o f t he ene r gy f un c t i ona l V , f ha st o s a t i s f y t he Eu l e r - Lag r ange equa t i on

( r + n s i ~ z f ) f " 1 n2 sin2r------5 ~ + n 2 f ' s i n f c s f )n z s in f co s f r /x 2 s in f = 0 .r (3.5)

I t was s ho wn i n [ 7 ] t ha t t he hed geho g f ie l d s ( 3 .1 ) wi t h n = 1 and n = 2 and p r o f i l ef unc t i ons s a t i s f y i ng t he equa t i on above f o r t hos e va l ue s o f n a r e t he abs o l u t e m i n i m aof V am ongs t a l l f i e l d o f deg r ee 1 and 2 r e s pec t i ve l y . We wr i t e q~o) and ~ ( 2 ) f o rt hos e f i e ld s i n s t anda r d i s o - o r i en t a t i on , i.e . w i t h X = 0 i n ( 3 .1 ) , and d eno t e the i r p r o f il ef u n c t i o n s b y f ( 1 ) and f t z ) . T r ans l a t i ons in phys i ca l s pace and i s o - ro t a t i ons ac t non -t r iv ia l ly on ~( 1 ) and q~(2~, s o t he r e i s a t h r ee - d i m en s i ona l f am i l y o f m i n i m a o f t heene r gy f unc t i ona l f o r bo t h n = 1 and n = 2 . We ca l l any f i e l d ob t a i ned by t r ans l a t i ngand i s o - r o t a t i ng t he f ield ~ ( l ) a b a b y S k y r m i o n . B a b y S k y r m i o n s a r e t h e b as ic s o l i to n so f o u r m o d e l ; a s p r o m i s e d i n t h e i n t ro d u c t i o n t h e y h a v e t h r ee d e g r e es o f f re e d o m : t w ot r ans la t i ona l and on e r o t a ti ona l . Fo r t he t o t al ene rgy , o r m as s , o f a baby Sky r m i o n wef in d 1 . 5 6 4 . 4 ~ r ; t h e e n e r g y d e n s i ty i s ro t a t io n a l l y s y m m e t r ic a n d p e a k e d a t t h e b a b y


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B.M.A.G. Pierre et al./Nu clear Physics B 439 (1995) 205-235 211S k y r m i o n ' s c e n t r e ( w h e r e ( ~ ( l) = ( 0 , 0 , - 1 ) ) . T h e f i e ld & (2 ) ( a n d a ll t h o s e o b t a i n e d b yt r an s l a ti n g a n d r o t a t i n g it ) m a y b e t h o u g h t o f a s a b o u n d s ta te o f t w o b a b y S k y r m i o n sand i s de s c r i bed i n de t a i l i n [ 7 ] . The ene r gy dens i t y i s aga i n r o t a t i ona l l y s ym m et r i c bu tpeak ed a t a d i s t ance r ~ 1 .8 f r om t he cen tr e . Th e m as s is 2 .936 .47 r .I n s ubs equen t s ec t i ons we wi l l be i n t e r e s t ed i n t he f i e l d o f a s p i nn i ng baby Skyr m i ona n d t h e f o r c e s b e t w e e n t w o b a b y S k y r m i o n s . W e r e c a l l s o m e s i m p l e o b s e r v a t i o n c o n -ce r n i ng t he a s y m pt o t i c behav i ou r o f & ( 1 ) f r om [ 7 ] wh i ch a r e the ba s is o f a r em ar k ab l ya c c u r a t e m o d e l f o r t h e d y n a m i c a l p h e n o m e n a w e w i l l t h e n e n c o u n t e r . F o r l a r g e r , a n dhence s m a l l f , t he equa t i on ( 3 .5 ) f o r n = l s i m p l i f i e s t o t he m od i f i ed Bes s e l equa t i on

f " + l f ' - ( - ~ + lx 2) f = O . ( 3 . 6 )r

A s o l u t i on o f t h i s equa t i on wh i ch t ends t o z e r o a t r = oo i s t he m od i f i ed Bes s e l f unc t i onK1 ( / z r ) . Thu s , t he p r o f i le f unc t i on f ( l ) ( wh i ch s a t is f ie s ( 3 .5 ) f o r n = 1 ) i s p r opo r t i ona lt o K1 f o r l a r ge r and we can wr i t e

p /~f O ) ( r ) ,-~ ~--~K1 (b t r ) , (3 . 7)wh e r e p i s a cons t an t wh i ch w e wi l l in t e r p r e t f u r t he r be l ow. S i nce the m od i f i ed Bes s e lf u n c t i o n h a s t h e a s y m p t o t i c b e h a v i o u r

K1 ( txr) ~ 21 ~re - ~ r 1 + O ( 3 .8 )t h e l e a d in g t e r m i n a n a s y m p t o t ic e x p a n s io n o f f ( l ) i s p r o p o r t i o n a l t o e-~r/~/-~, w h i c hs hows i n pa r t i cu l a r tha t t he ( po t e n t i a l ) ene r gy d i s t r ibu t i on o f t he f i e l d 4~ l ) i s expo nen -t i a ll y l oca l i sed . M os t o f t he ana l y t i c a l r e s u l ts i n t h i s pape r a r e ba s ed o n t he obs e r va t i on ,m ade i n [ 71 , t ha t the a s ym pt o t i c f i el d ~ ( l ) o f 4~ 1 ) ( de f i ned a s i n ( 2 . 9 ) ) c an be i n te r -p r e t ed i n t e r m s o f d i po l e f i e ld s . Th i s c an be s een a s f o ll ows . Fo r la r ge r we app r ox i m a t es i n f ( l ) ,,~ f ( l ) a n d c o s f ( 1 ) ,~ 1 a n d , u s in g t h e a s y m p t o t ic ex p r e s s io n ( 3 . 7 ) , w e w r i tethe f i e ld q~( l ) as

/ c o s (O - x )~ ( ~ ) ( x ) P ~= ~ K l ( l z r ) ~ l n ( 0 X ) ) ( 3 . 9)

O r , i n t r o d u c i n g t h e o r t h o g o n a l v e c t o r sP l = P ( c s x , s i n x ) , P2 = P ( - s i n x , c s x ) ( 3 . 1 0 )

and $c = x/ r , w e c a n w r i t eq~(al) ( x ) / . t p 1= 2 7 r a ' Y C K l ( t zr ) = - 2 -~ p a ' V K ( I ~ r ) ' a = 1 , 2 . ( 3 .1 1 )


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212 B.M.A.G. Piet te et al . /Nuclear Physics B 439 (1995) 205-235Howe ve r , s inc e the Gre e n func t ion o f the s t at ic K le in -Gordon e qua t ion i s K o ( b r r ) ,

( A - / z 2 ) K 0 ( / z r ) = - 2 7 r S ( 2 ) ( x ) , ( 3.1 2 )w e h a v e

(Z I - /Z2)~(al) (X ) = P a " XTt~(2)(X), a = 1 , 2 . (3 .13 )Th i s e qua t ion l e a ds to the in t e rp re ta t ion o f the a sym pto t i c f i e ld ~ (1 ) a s the f i e ld p roduc e dby a doub le t o f o r thogon a l d ipo le s , one fo r e a c h o f the c ompon e n t s tp l 1) a nd ~ 1 ) , i n al ine a r f ie ld the o ry , na m e ly K le in -Gord on the o ry . The s t r e ng th o f the d ipo le c a n h e c a l -c u l at e d f r o m t h e a s y m p t o t ic f o rm o f f O ) . O n e f in d s , b y n u m e r i c al l y s o l v in g E q . ( 3 . 5 ) ,

p = 2 4 . 1 6 . ( 3 . 1 4 )Onc e th i s s ing le numbe r i s c a l c u la t e d f rom the non- l ine a r e qua t ion (3 .5 ) muc h c a n bed e d u c e d a b o u t t h e d y n a m i c s o f b a b y S k y r m i o n s u s i n g o n l y t h e l in e a ri s ed e q u a t io n s o fm o t i o n .

4. Spinn ing bab y SkyrmionsHow doe s a so l i ton in two o r th re e d ime ns ions c ha nge i t s sha pe whe n i t sp ins? Wha t

i s the in t e ra c t ive dyn a m ic s o f several so l i tons whe n som e o f the m a re sp inn ing? F romthe po in t o f v i e w o f pa r t i c le phys ic s the se a re ve ry na tu ra l que s t ions to a sk . Ye t the re a resu rp r i s ing ly fe w so l i ton mode l s in wh ic h the se que s t ions ha s be e n a dd re sse d se r ious lya nd e ve n fe w e r in wh ic h sa t i s fa c to ry a nswe rs ha ve be e n foun d . Th i s i s pa r t ly be c a use theq u e s t io n s d o n o t m a k e s e n s e in s o m e o f t h e m o s t p o p u la r m o d e l s . In t h e m u c h s t u d ie da be l i a n H igg s m ode l [ 12 ] , fo r e xa mple , t he so l i ton so lu t ions , c a ll e d vo r ti c e s, a re fu l lyc ha ra c te ri se d by the i r pos i t ion a nd ha ve no ro ta t iona l de g re e s o f f r e e dom. L um ps in theC P ~ mo de l , on the o the r h a nd , c a n ha ve a n a rb i tr a ry o r ie n ta t ion , bu t the mom e n t o fine r t i a a ssoc ia te d w i th c ha n ge s in the o r i e n ta t ion is in f in i t e so tha t the ro ta t iona l de g re eo f f r e e dom i s dyna m ic a l ly f roz e n ou t . The re i s a m od i f i c a t ion o f the C P l mo de l [ 13 ]in wh ic h the so l i tons , c a l l ed Q- lumps , ne c e ssa r ily sp in , bu t s ing le so l iton so lu t ions ha veinf in i te energ y an d in con f igura t ion s of severa l so l i tons a l l so l i tons have to ro ta te w i ththe sa m e a ngu la r f r e que nc y . Thus , the e f fe c t o f r e l a t ive ro ta t ion c a nno t be inve s t iga te d .

I n t h e S k y r m e m o d e l , s p i n 1 / 2 q u a n t u m s t a te s o f a s i n g l e S k y r m i o n a re m o d e l s f o rphys ic a l nuc le ons , so the que s t ion o f sp inn ing Sky rmions ha s na tu ra l ly a t tr a c te d a lo t o fa t t e n tion . In the f ir s t pa pe r o n th i s sub je c t [ 14 ] , i t wa s a ssum e d tha t a Sk yrm ion wou ldro ta t e w i thou t c ha ng ing i t s sha pe , a nd , to ob ta in the qua n tum s ta t e s c o r re spond ing tothe pro ton , neu tron and th e A-resonance, i t was quan t ised as a r ig id body. A l tho ug h i twa s qu ic k ly po in te d ou t tha t the cl a ss ic a l f r e que nc y a t wh ic h a Sky rmion w ou ld ha ve toro ta te to have sp in 1 /2 i s so la rge tha t cent r i fuga l and re la t iv is t ic e f fec ts a re important ,the re se e m s to be no qua n t i t a t ive a na lys i s o f the se e f fe c t s in the l i te ra tu re . One re a sonfo r th i s is t ha t , in th re e d ime ns ions , a ro ta t ing Sky rmion on ly ha s a x ia l sym m e t ry (a bo u tt h e a x is o f r o ta t i o n ) a n d i s n o t o f t h e S O ( 3 ) s y m m e t r ic h e d g e h o g f o r m o f t h e s t a ti c


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B . M . A . G .P i e r r e t a l . /N u c l e a rP h y s i c sB 4 3 9 (1 9 9 5 )2 0 5 - 2 3 5 213so lu t ion . Thu s , to f ind the e xa c t fo rm o f a sp inn ing Skyrm ion one ne e ds to so lve c oup le dnon- l ine a r pa r t i a l d i f f e re n t i a l e qua t ions s im i la r to the one s s tud ie d in [ 1 5 ] .

In ou r two -d im e ns iona l mo de l , by c on t ra s t , the re a re so lu t ions de sc r ib ing sp inn ingb a b y S k y r m i o n s w h i c h a r e o f t h e h e d g e h o g f o r m ( 3 . 1 ) . T h u s w e c a n s t u d y t h e e f f e c to f ro ta t ion on the so l i ton ' s sha pe , i t s ma ss a nd i t s mome n t o f ine r t i a by s imp ly so lv ingord ina ry d i f fe re n t i a l e qua t ions . I t fo l lows f rom the "p r inc ip le o f symme t r i c c r i t i c a l i ty "[ 16 ] tha t we c a n f ind t ime -de pe n de n t so lu t ions o f the f i e ld e qua t ions (2 . 8 ) by m a k ingt h e t i m e - d e p e n d e n t h e d g e h o g a n s a tz

s in f ( r ) c o s ( 0 - t ot ) )d p ~ ( t , x ) = s i n f ( r ) s i n ( 0 - t ot ) , ( 4 .1 )

c os f ( r )wh ere to , an a rb i t ra ry rea l nu mb er , can be in te rpre ted as the f ie ld ' s angular f requenc y andf sa t is f i es the bounda ry c ond i t ions (3 .3 ) a nd (3 .2 ) w i th m = 1 . The n the f ie ld (4 .1 )i s a so lu t ion o f the Eu le r -La g ra nge e qua tion (2 . 8 ) i f f sa ti s fi e s the Eu le r -La gra ngee qua t ion ob ta ine d f rom the re s t r i c t ion o f the La gra ng ia n L to f i e ld s o f the fo rm (4 .1 ) .Exp l i c i t ly th i s i s t he o rd ina ry d i f fe re n t ia l e qua t ion

( r + ( 1 - - t o 2 r ) s i n 2 f ) f ' '+ ( 1 - ( t o 2 + ~ ) s i n 2 f + ( 1 - t o 2 r ) f ' s i n f c o s f ) f '- ( 1 - t o 2 r ) s i n f c o s f - r l z 2 s i n f = O . (4 .2 )

W e c a ll so lu t ions o f the fo rm (4 .1 ) sp inn ing ba by Skyrmions . The to ta l e ne rgy o f asp inn ing ba by Skyrmion de pe nds on to a nd i s g ive n by

M ( t o ) = r / r ( f ' 2 + ( t o 2 + ~ ) ( l + f ' 2 ) s i n 2 f + 2 1 z 2 ( 1 - c o s f ) ) d r . (4 .3 )M (0 ) i s ju s t t he ma ss o f a ba by Skyrm ion c a lc u la te d e ar l ie r a nd sha l l he nc e fo r th bede no te d M0 . To s tud y the de pe nde n c e o f M a nd the f i e ld ~ '~ o n to we ne e d to d i s t ingu i sht w o r e g im e s , t o < / z a n d to > / z . T h i s c an b e s e en f r o m t h e b e h a v i ou r o f f f o r la r g e r .In th i s l imi t f i s sma l l a nd the e qua t ion (4 .2 ) s imp l i fi e s to

f " + l f ' - ( ~ + ( l ' t 2 - t o 2 ) ) f (4 .4 )Thus , fo r to < /z , f de c a ys e xpon e n t i a l ly fo r l a rge r w h i l e fo r to > / z i t i s o sc i l la to ry .

I t i s i n t e re s t ing to c om pa re the a sy mp to t i c f i eld o f a sp inn ing ba by S kyrm ion w i th thef i eld p rodu c e d b y a d oub le t o f sc a la r d ipo le s . W e the n ne e d to c ons ide r t ime -de pe nde n td ipo le s a nd no te he re tha t the e qua t ion fo r sc a lar f i e ld s q~a p roduc e d by a ro ta t ing pa i r o fo r t h o g o n a l d i p o l e s P a ( t ) , a = 1 ,2 , i s t he Kle in -G ordon e qua t ion w i th a t ime -de pe nd e n td ipo le sou rc e t e rm:

(I--] + /~ 2)q ~a( t , x ) = - P a ( t ) V t s ( E ) ( x ) . ( 4 . 5 )


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2 1 4 B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235I f t he d i po l e s r o t a t e un i f o r m l y a t cons t an t angu l a r ve l oc i t y ca , t hen ~0 = - ca2 pa andt he above equa t i on can be s o l ved exp l i c i t l y i n t e r m s o f Bes s e l f unc t i ons . We wi l l dot h i s be l ow f o r t he t wo r eg i m es ca < I x and ca > I x and com par e t he r e s u l t s wi t h t hea s ym pt o t i c f o r m o f d~ ' .4 . 1 . T h e c a s e c a < t x

I f ca < # , t he Eq . ( 4 .4 ) i s aga in t he m od i f i ed Bes s e l equa t i on o f f ir s t o r de r . Asbe f o r e we a r e i n t e r e s t ed in a s o l u t i on f wh i ch i s expon en t i a l l y s m a l l f o r l a r ge r. Su cha s o l u t i on is a s ym pt o t i c a l l y p r op o r t i ona l to t he m o d i f i ed Bes s e l f unc t i on K 1 ( K r ) , w h e r eK = ~ / I X 2 _ 0 . ) 2 , i .e.

K pf ~ ~- -~K] (K r) ( 4 . 6 )

f o r s o m e c o n s t a n t p . T h e n t h e a s y m p t o t i c f ie l d po, o f t h e r o t a t in g b a b y S k y r m i o n ( 4 . 1 )wi t h t ha t p r o f i l e f unc t i on i s

p K ( CO S( 0 - - c a t) )q ~ ' ( t , X ) = ~ - - ~ K l ( K r ) s i n ( 0 - - o J t) . ( 4 .7 )

0T h u s , d e f in i n g t h e t i m e - d e p e n d e n t d i p o l e m o m e n t s

P l = p ( c o s c a t , s i n c a t) a n d t72 = p ( - s i n c a t , c o s c a t) , ( 4 . 8 )t he f ir s t t wo com pon en t s o f ~po, c an be wr i t t en

1~ ( t , x ) = - - P a ( t ) V K o ( K r ) , a = 1 2 . ( 4 . 9 )- 2~r

One checks ea s i l y t ha t t he s e f i e l d s s a t i s f y t he l i nea r equa t i on ( 4 .5 ) . Thus , j u s t a s i n t hes ta t ic c a s e, t h e a s y m p t o t ic f o r m o f th e r o t a ti n g h e d g e h o g s o l u ti o n ( 4 . I ) m a y b e t h o u g h to f a s b e i n g p r o d u c e d b y a r o t a t in g p a i r o f o r th o g o n a l d i p o le s .

W e have s o l ved t he r ad i a l equa t i on ( 4 .2 ) f o r va r ious va l ue s o f ca < IX. I n F i g . 1 wep l o t t he c o r r e s po nd i ng p r o f i l e f unc t i ons . As ca app r oaches IX f r om be l ow, t he s o l i t on ' se n e r g y d i s t r i b u t i o n b e c o m e s m o r e a n d m o r e s p r e a d o u t , w h i c h o n e m a y i n t e r p r e t a s acen t r i f uga l e f f ec t . No t e , how eve r , t ha t t he i n i ti a l g r ad i en t o f t he p r o f i l e f unc t i ons va r i e sve r y l i t t l e and t ha t m os t o f t he change occu r s i n t he t a i l , wh i ch i s we l l de s c r i bed by t hem od i f i ed Bes s e l f unc t i on . Th i s i s t he f i r s t i nd i ca t i on t ha t one can unde r s t and m any o ft he f ea t u r e s o f a s p i nn i ng baby Skyr m i on i n t e r m s o f t he f i e l d i n t he a s ym pt o t i c r eg i onw h e r e i t ( a p p r o x i m a t e l y ) o b e y s l i n e a r e q u a t i o n s .

N e x t w e w a n t t o u n d e rs t a n d t h e d e p e n d e n c e o f a s p i n n in g b a b y S k y r m i o n ' s m a s s o ni t s f r equency . Rec a l l i ng t he a s ym pt o t i c f o r m u l a ( 3 .8 ) we s ee t ha t t he ene r gy d i s tr i bu t i ono f a baby Sk yr m i on s p i nn i ng a t c a < IX i s exponen t i a l ly l oca l i s ed and t ha t t he t o t a l m as sM (ca ) i s f in i t e . A t the c r i t ica l ang ular ve lo c i ty ca = IX, how ever , the Eq. (4 .4) i s so lve dby f = 1 / r ; t h e c o r r e s p o n d i n g b a b y S k y r m i o n i s t h u s o n l y p o w e r - l a w l o c a l i s e d a n d


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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235 215

3 .5

3

2 .5

2

1 .5

1

0 .5

5 1 0 1 5 2 0r

2 5 3 0

Fig. 1. Profile functions for a static ba by Skyrm ion (bottom curve) and a spinning ba by Skyrmion withangular frequency (from top to bottom ) 0.316,0.3 and 0.2.i t s mass i s i n f i n i t e . Fo r l a t e r u se we no t e t ha t t he d ive r gen t pa r t o f t he i n t eg r a l i n t hef o r m u l a ( 4 . 3 ) i s

/ r ( to 2 s in 2 f + 2/ .t2( 1 - co s f ) ) d r . ( 4 . 1 0 )TD u e t o t h e s p r e a d i n g o f t h e e n e r g y d e n s i t y as to a p p r o a c h e s / . t o n e n e e d s t o in t e g r a t et o e v e r l a rg e r v al u e s o f r w h e n c o m p u t i n g M ( t o ) n u m e r i c a ll y u s i n g th e f o r m u l a ( 4 . 3 ) .I t is then m or e p r ac t i c a l t o f i nd cons t an t s r 0 and C such t ha t f o r r > r 0 t he p r o f i l ef u n c t i o n i s w e l l a p p r o x i m a t e d b y C e x p ( - K r ) / v / 7 . T h e n o n e i n t e g r a t e s t h e e n e r g yd e n s i t y n u m e r i c a l l y f r o m 0 t o r 0 a n d p e r f o r m s th e r e m a i n i n g i n t eg r a l a n a ly t ic a l ly , u s in gs i n 2 f ,,m f 2 and cos f ~ 1 - f 2 / 2 . W e p l o t t h e f u n c ti o n M ( t o ) i n F ig . 2 a . I t g r o w sr a p i d l y a s to - - , / . t w h i c h i s c o n s i s t e n t w i t h o u r e a r l ie r o b s e r v a t io n t ha t M ( # ) i s i n fi n it e .

A f u r t h e r q u a n t i t y o f i n t e r e s t i s t h e c o n s e r v e d c h a r g e I d i s c u s s e d a t t h e e n d o fS e c t i o n 2 . F o r f i el d s o f t h e h e d g e h o g f o r m , w h e r e s p a t i a l r o t a t io n s a n d i s o - r o t a ti o n s a r ee q u i v a l e n t , w e c a n i n t e rp r e t I a s t h e a n g u l a r m o m e n t u m o r s p in o f t h e fi e ld $ a n d w edeno te i t by J i n t h i s con t ex t . One f i nds t ha t

J ( t o ) = t o . 2 ~ f r s in 2 f ( 1 + f t 2 ) d r . ( 4 . 1 1 )T h e q u a n t it y

A ( t o ) = J ( t o ) / t o ( 4 . 1 2 )m a y b e i n t e r p r e t e d a s a m o m e n t o f in e r ti a . S i n c e a s p in n i n g b a b y S k y r m i o n c h a n g e si t s s h a p e a s t o v a r i e s , t h e c o r r e s p o n d i n g m o m e n t o f i n e r t i a c h a n g e s , t o o . A s m e n t i o n e d


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2 1 6 B.M.A.G. P ie t te e t a l . /N uc lea r Phys ic s B 439 (1995) 205-2 35

3. 0 ( a )

2.5

2. 0

1.5 f0 .05 0 ,1 0 .15 0 .2 0 .25 0 .3 0 .353 ! ( b )

2 .5 ++ + +2 . 0 +

+

i

J

+++

+ ++

++

F ig . 2 . (a ) The m as s M (to) o f a s p inning baby Skyrm ion in unit s o f 4r as a funct ion o f the angular frequency .(b ) Mas s -sp in re la t ions hip for a s p inning baby Skyrm ion. The cros ses m ark pa irs (M , J ) ca lculated v ia (4 . 3 )and (4 . 11 ) for the s am e va lue o f to; both M and J are p lo tted in uni ts o f 4~r . The s o l id l ine i s a p lo t o f thefunction M of J (4 . 15 ) ; here , too , M and J are p lo t ted in uni ts o f 4r .

a b o v e t h i s e f f e c t i s c u s t o m a r i l y i g n o r e d i n t h e S k y r m e m o d e l . T o c h e c k t h e v a l i d i ty o ft h i s a p p r o x i m a t i o n i n o u r m o d e l w e d e f i n e .4 0 = A ( 0 ) , f o r l at er u se . N u m e r i c a l l y w ef i n d A o = 2 z r . 7 . 5 5 8

W e h a v e c a l c u l a t ed J ( t o ) f o r v a r i o u s v a l u e s o f to < / z u s i n g t h e s a m e t e c h n i q u e a sd e s c r i b e d a b o v e f o r t h e c o m p u t a t i o n o f M ( t o ) . L i k e M , J d i v e r g es a s to ~ / z f r o m


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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235 2 1 7b e l ow , b u t o n e c h e c k s t h a t o n l y t h e t e r m

27r / r sin 2 f d r ( 4 . 1 3 )O .i s a d i ve r gen t i n t eg r a l when t o = / z . The d i ve r gence i s o f t he s am e o r de r a s t ha t i n( 4 .1 0 ) and , com par i ng coe f f i c i en t s, one i s le ad to t he a s ym pt o t i c f o r m u l a

M ~ N + ~ J ( 4 . 1 4 )f o r s o m e c o n s t a n t N . I n F i g . 2 b w e p lo t th e p r e c i se re l a ti o n b e t w e e n M ( t o ) a n d J ( t o ) .C l ea r l y t he g r aph i s we l l de s c r i bed by t he l i nea r f o r m u l a ( 4 .14 ) a l r eady f o r qu i t e s m a l lva l ue s o f t o . No t e t ha t a l i nea r f o r m u l a o f t h i s f o r m ho l ds exactly f o r Q - l u m p s [ 1 3 ] ,wh e r e t he cons t an t N i s the Q - l um ps t opo l og i ca l cha r ge . One m ay i n t e r p r e t i t i n wo r ds a s

" m a s s o f a sp i n n i n g so l i to n = c o n s ta n t + m e s o n m a s s x a n g u l a r m o m e n t u m " .F o r s m a l l to t h e m a s s M ( t o ) d e p e n d s q u a d r a ti c a ll y o n J ( w ) a s o n e w o u l d e x p e c t f o rt he r o t a t i on o f a r i g i d body . I t i s i n s t r uc t i ve t o com par e ou r exac t r e s u l t s wi t h t henon- r e l a t i v i s t i c r i g i d body f o r m ul a

j 2AT/= M0 -1- - - (4 . 15 )2 A 0 "We p l o t t he g r aph o f AT/ ( J ) i s F i g . 2b a s we l l . No t e t ha t t he r i g i d body f o r m ul a i son l y a goo d app r ox i m a t i on t o t he t r ue m as s - s p i n r e l a t i on f o r s m a l l s p i n s and s m a l l m as sd i f fe r e n c es M ( t o ) - M . T h i s o b s e rv a ti o n m i g h t b e r e le v a n t f o r b a ry o n p h e n o m e n o l o g y int he Sk yr m e m ode l . The r e a non - r e l a t iv i s t i c f o r m u l a li ke ( 4 .1 5 ) i s u s ed t o c a l cu l a t e t het heo r e t i c a l p r ed i c t i on s f o r t he m as s e s o f t he nuc l eons and t he A- r e s onance . Ho wev e r , t henuc l eon m as s i s abou t 10% l a r ge r t han t he m as s o f a Sky r m i on and t he , t i s abou t 40%heav i e r t han a Sky r m i on . Our ca l cu l a t i ons i nd i ca t e t ha t t he f o r m u l a ( 4 .15 ) i s a poo rapp r ox i m a t i on f o r a r e l a t i ve m as s d i f f e r ence as l a rge a s 40% and t ha t i t w i ll gene r a l l yg i ve t oo l a r ge a va l ue f o r t he m as s o f a s p i nn i ng s o l i t on a t a g i ven angu l a r m om en t um .

4.2. The case to > IXW hen to > / z t he Eq . ( 4 .4 ) i s t he ( un m o d i f i ed ) Bes s e l equa t i on o f f ir s t o r de r , a l l

s o l u t io n s o f w h i c h a re o s c i ll a to r y f o r l a rg e r . T h u s , i n t e rm s o f k = ~ - / z 2 w e c a nwr i t e t he a s ym pt o t i c f o r m o f t he s o l u t i on i n t e r m s o f t he Bes s e l f unc t i ons o f f i r s t ands e c o n d k i n d

Jl (kr) . . . .an d

Y1 (kr ) . . . .

/ - 5dJo (kr) ~ ~[--7. s i n ( k r - 1 7 r )k dr v ~ r/ - - 31 dI'~ (kr) ~,, - V ~ r c o s ( k r - 1r).k dr

B ot h t he s e f unc t i ons m a y o ccu r , s o we wr i t e t he a s ym pt o t i c f o r m ~o ' as

( 4 . 1 6 )

( 4 . 1 7 )


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218 B.M.A.G . P ierre e t a l . /N uc lea r Phys i c s B 439 (1995) 205 -23 5

2 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

~ . + + . ~ . * ~ . . . . , , ~ _ . t : : : : , , , J + + + . a, + + + + . h~ ,~ .. ,~ ., ,. ., .. ~ + . t : : : : : ; , + ~ ~ ,~ + ~ - . . ~ . J , ~ . , , . + . ~ - _ + + . t : : : ; : : : ; T T ~ ' ~ '. + + + ) * . ~ . * ~ - , ~ . , * . h ~ . , + . . ~ : : : : : : . . . . , ~ . ~ # . ~ . ~. ~ . +. *. ~. *~ .~ .h ,m ., +. ~ ~ - ~ . . . . . . . r r . . . .

~ ' p . ~ l, . * . * .m . , + ., * . m . * ,~ ~ ; : : : : : S . . . . + ~ ' ~ " ~ '~ ' ', + + + . . . . . . . . . . - . . . . . . . ~ . ~ . . , e ~ - - , v . ~ ' ~ , ~ * , ~ , ~ "

. . . . . l I I I I ' " ' " - " t I I l l l l " ' " l l l i i . . . . .

~ + + + + + + + + + + + + ~ t : : : ; ; + + ~ . . . . . . ~ . . + + + + + + + + + + -, + + + + + + . + . + ~ : : ; ~ + . ~ , - ~ . ~ . . ~ + + + + + + + ., + + + + + + . + + + + + + + + + + + + + ,~ + + + + . + + + + + + + + + + + ~ ,. + + + + + + . * * + + + + . + + + + + + . + . ,- 2 0 " ~ . . . . . . . . . . t . . . . . . . . . . . . ::1 . . . . ~ . . . . . . . I :: . . . . . . . . . . . . .

- 2 0 - 1 0 0 1 0 2 0F i g . 3 . P l o t o f t h e f ie l d o f a b a b y S k y r m i o n w i t h in i t ia l a n g u l a r f r e q u e n c y t o = 0 . 5 a t t i m e t = 1 0 . A t e v e r yl a t ti c e s i t e i n p h y s i c a l s p a c e w e p l o t a n a r r o w o f u n i t l e n g t h w h o s e d i r e c ti o n i s t h a t o f ( ~ , ~ ) ( w e i d e n t i fyt h e 1 - a n d 2 - a x e s i n t h e ta r ge t s p a c e S 2 w i t h t h o s e i n p h y s i c a l s p a c e ) . A t t h e h e a d o f t h e a r r ow w e p u t a" + " i f ~ 3 i s p o s i t i v e a n d a " x " i f ~ 3 i s n e g a t i v e . I f ( ~ + 0 ~ ) < 2 x 1 0 - 4 n o ar ro w i s p l o t te d . T h u s t h ev a c u u m i s r e p r e s e n t e d s i m p l y b y a " + " .

k / c o s ( 0 - t o t)~ ( t , x ) = - ~ ( p Y l ( k r ) + q J l (k r ) ) k ~ i n ( 0 - t o t ) ) , ( 4 . 1 8 )

w h e r e q a n d p a r e c o n s t a n t s w h o s e m e a n i n g w e w i l l e x p l a i n la te r . A h e d g e h o g f ie l dw i t h t h e a s y m p t o t i c f o r m ( 4 . 1 8 ) h a s in f in i te e n er g y ( 4 . 3 ) a n d is t h u s r a th e r u n p h y s i c a l .N e v e r t h e l e s s t h i s s o l u t i o n h a s a n a t u r a l i n t e r p r e ta t i o n i n t e r m s o f th e d i p o l e m o d e l w h i c hw e w i l l g i v e b e l o w .

F i rs t, h o w e v e r , w e w a n t t o s t u d y th e ti m e e v o l u t i o n o f a b a b y S k y r m i o n w h i c h i sg i v e n a n i n i t i a l a n g u l a r v e l o c i t y t o > / z . T o i n v e s t ig a t e t h i s q u e s t i o n w e h a v e s o l v e dt h e f ie l d e q u a t i o n s ( 2 . 8 ) n u m e r i c a l l y w i t h i n i ti a l v a l u e s ~ b ( t = 0 ) = ~ ( 1 ) a n d ~ ( t =0 ) = - t o n ~ ( l ) . f o r t o = 0 . 5 a n d to = 0 . 9 . T h e g r i d f o r o u r s i m u l a t i o n s i s a s q u a r eo f 2 5 0 2 5 0 p o i n t s , e x t e n d i n g i n b o t h th e x a n d y d i r e c t io n f r o m - 2 5 t o 2 5 . A t t h eb o u n d a r y w e s e t t h e fi e ld t o t h e v a c u u m v a l u e n a n d a b s o r b a n y i n c i d e n t k i n e t ic e n e r g y .F o r b o t h o f t h e in i t ia l v a l u e s o f t o w e f in d th a t t h e b a b y S k y r m i o n r a d i a te s . I n F i g . 3w e d i s p l a y t h e f ie l d o f th e b a b y S k y r m i o n w h o s e i n it ia l a n g u l a r v e l o c i t y w a s t o = 0 .5 ,1 0 u n i t s o f t i m e a f te r th e s t ar t o f t h e s i m u l a t i o n . T h e p i c t u r e c le a r l y s h o w s t h e s o r t o fs p ir a l p a t te r n w h i c h i s fa m i l ia r f r o m d i p o l e r a d i a t i o n i n li n e a r r e l a t iv i s t ic f i e l d t h e o r i e s[171.


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B . M . A .G . P i e tt e e t a l . / N u c l e a r P h y s i c s B 4 3 9 (1 9 9 5 ) 2 0 5 - 2 3 5 219

3 .2

3

2 . 8

2 . 6

uJ 2 .4

2 .2

2

1 .8

1 .6 0 5 0 0 1 0 0 0t

1 5 ~ 2 ~ 0

Fig. 4. Total energ y of spinning baby Skyrm ions with initial angu lar frequencies 0.9 (to p) and 0.5 (bot tom )in units of 4~r.T h e r a d i a t i o n c a r r i e s a w a y b o t h e n e r g y a n d a n g u l a r m o m e n t u m . A s a r e s u l t t h e b a b y

S k y r m i o n s l o w s d o w n u n t i l th e a n g u l a r v e lo c i ty h a s d ro p p e d t o a v a l u e b e l o w / z i n b o t hs i m u l a t i o n s . F i g . 4 s h o w s h o w t h e t o t a l e n e r g y f o r t h e t w o s i m u l a t i o n s d e c r e a s e s w i t ht i m e . W e h a v e a l s o c h e c k e d t h a t a ft e r 1 0 0 0 u n i t s o f t i m e t h e f ie l d h a s s e t tl e d d o w n t o au n i f o r m l y r o t a ti n g f ie l d o f t h e f o r m ( 4 . 1 ) , a n d w e h a v e e x t r ac t e d t h e a n g u l a r f re q u e n c y .I n t he s imu la t i on whe r e i n i t i a l l y t o = 0 . 5 we now f ind t o ~ 0 . 28 and i n t he s imu la t i onw he r e i n i t i a l l y to = 0 . 9 w e now f ind to ~ 0 . 3 .

I t is i n s t r u c ti v e t o i n t e r p re t b o t h t h e s o l u t io n o f t h e h e d g e h o g f o r m a n d t h e n u m e r i c a l l yf o u n d s o l u t i o n w i t h t h e s p i r a l p a tt e r n s h o w n i n F i g. 3 i n t e r m s o f th e d i p o l e m o d e l . F o rth i s pu r pose i t i s be s t t o combine t he a sympto t i c f i e ld s q~ l and ~2 i n to t he complex f i e ld

= ~Pl + i~p2 . The n , t he K le in - G or do n equa t ion ( 4 . 5 ) , w i th d ipo l e m om en t s g iven by( 4 . 8 ) ( w h e r e n o w to > / z ) , c a n b e w ri tt e n

( [ -q d - /z2 ) t2B( t , X ) = - p e - i t ~ t (01 + i02 )t~ (2) (x ) . ( 4 . 1 9 )T o s o l v e t h i s e q u a t i o n w e s e p a r a t e t h e t i m e d e p e n d e n c e i n t h e f o r m

( t , x ) = e - i ~ t g ( x ) , ( 4 .2 0 )so t ha t g ha s t o s a t i s f y t he s t a t i c equa t ion

( A - t- k 2 ) g ( x ) = p ( 0 1 q - i 0 2 ) ~ ( 2 ) ( x ) , ( 4 . 2 1 )w i t h k a s d e f in e d a b o v e . N e x t w e n e e d s u i ta b l e G r e e n f u n c t i o n s G o f t h e H e l r n h o l t ze q u a t i o n i n t w o d i m e n s i o n s , n o r m a l i s e d s o t h a t

( A + k 2 ) G ( k r ) = t ~ ( 2 ) ( x ) . ( 4 . 2 2 )


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220 B.M.A.G. Piette et al . /Nuc lear Physics B 439 (1995) 205 -235A so lu t ion wh ic h de sc r ibe s a n ou tgo ing wa ve a t i n f in ity c a n be e xp re sse d in t e rms o fthe f i rs t Ha nke l fun c t ion

G + ( k r ) = 14i "'0t4(1) k r ) "~ - i ~ 8 ~ r e i ( k r - ~ r / 4 ) , (4 .23 )a nd a so lu t ion wh ic h de sc r ibe s a n inc om ing wa ve a t i n f in i ty i s g ive n in t e rms o f thes e c o n d H a n k e l f u n c t i o n

= ,,~ i 8 ~ k ri //0(2 ( k r ) e - i ( k r - l r / 4 )G - ( k r ) ~ . (4 .24 )A pa r t i c u la r re a l so lu t ion o f (4 .22 ) i s thus g ive n by

l ( G + ( k r ) + G - ( k r ) ) = Yo (k r ) , ( 4 . 2 5 )bu t to ob ta in the ge ne ra l so lu t ion we shou ld a dd a n a rb i tr a ry mu l t ip l e o f the re a l so lu t iono f t h e h o m o g e n e o u s H e l m h o l t z e q u a t io n

{ ( G + ( k r ) - G - ( k r ) ) = { J o ( k r ) . ( 4 .26 )Thus , c onve r t ing to po la r c oo rd ina te s

i OC?l q- ic92 = ei ( ff-~ d- r - ~ ) ( 4 . 2 7 )a nd c hoos ing the r e a l Gre e n func t ion Y o ( k r ) + ( q / 4 p ) J o ( k r ) f o r s o m e r e al n u m b e r qw e o b t a in a s o l u t i o n o f ( 4 . 1 9 )

kqbr t , x ) = - - ~ ( P J 1 ( kr ) + qY1 ( kr ) ) e i ( O-~ t ) , ( 4 . 2 8 )wh ic h i s ju s t t he a symp to t i c fo rm ~o , (4 .18) o f the he dge hog f i e ld wr i t t e n in c omple xno ta t ion . Thus the he dge hog sp inn ing a t t o > /z r e p re se n t s a so lu t ion w i th a r a d ia t ionf i e ld tha t c ons i s t s o f bo th inc oming a nd ou tgo ing ra d ia t ion . Th i s i s the phys ic a l o r ig ino f the in f in i t e e ne rgy o f the he dge hog so lu t ion .

I t i s no t d i f f i c u l t t o gue ss wh ic h Gre e n func t ion w i l l l e a d to a so lu t ion o f (4 .19 )d i sp la y in g the sp i ra l pa tt e rn obse rve d in ou r s im u la t ion o f the sp inn ing ba by Skyrm ion .Co ns ide r the so lu t ion c ons t ruc te d f rom the Gre e n func t ion G + . Us ing

d H ( D ( k r ) = - k H ~ l ) ( k r ) ( 4 . 2 9 )d rt ha t so lu t ion is

p ~ k e i (k r + O - n -Tr /4 ) . ( 4 . 3 0 )~ + ( t , X ) = i e i ( O - ~ t ) H O ) ( k r ) ~ 8 ~rrRe me mbe r ing tha t the r e a l a nd ima g ina ry pa r t o f ~+ shou ld be in t e rp re te d a s the f i r s ttwo c ompone n t s o f the a sympto t i c f i e ld +p+ o f a sp inn ing ba by Skyrmion we se e tha t ,fo r su f f i c i e n t ly l a rge r a nd a f ixe d va lue o f t , t he d i re c tion o f tp+ i s c ons ta n t a long


17/31

B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235 221the sp i ra ls k r = - 0 . This i s p re c i se ly the sp i ra l pa t t e rn we obse rve d in the nume r ic a ls i m u l a t io n o f a b a b y S k y r m i o n s p in n i n g w i t h to > / ~ . T h e d i p o l e p i c tu r e s h o w s t h a t itc a n be a c c oun te d fo r in t e rms o f the G re e n fun c t ion G + .

The d ipo le p ic tu re c a n be u se d to ma ke se nse o f ma ny o f the qua l i t a t ive p rope r t i e so f a sp inn ing ba by Skyrmion . In p r inc ip le one c ou ld a l so c he c k whe the r the e ne rgyloss th roug h ra d ia t ion p lo tt e d in F ig . 4 c a n be qua n t i t a t ive ly mo de l l e d in t e rms o fd ipo le r a d ia t ion . Howe ve r , t he c e n t r i fuga l e f fe c t s in sp inn ing ba by Skyrmions c ha ngethe d ipo le s t r e ng th , wh ic h the re fo re de pe nds on the ba by Sk yrm ion ' s a ngu la r f r e que nc y .Th i s c ons ide ra b ly c ompl ic a te s the c a lc u la t ions a nd we the re fo re ha ve no t pu rsue d th i spa th .

5 . T h e d i p o l e m o d e l f o r th e i n te r a c ti o n o f b a b y S k y r m i o n sIn Se c t ion 3 we sa w tha t a ba by Sky rmion a c ts l i ke the sou rc e o f a doub le t o f sc a la r

d ipo le f ie ld s . In [7 ] i t wa s show n , a ssuming a c e r t a in supe rpos i t ion p roc e dure fo r we l l -se pa ra te d ba b y S kyrm ions , tha t a ba by S kyrm ion a lso re a c ts to the f i e ld o f a d i s t a n t ba bySky rmio n l ike a doub le t o f sc a la r d ipo le s . The re i s a s imi la r c o r re sponde nc e be twe e n asupe rpos i t ion p roc e dure a nd a l ine a r mode l fo r the fo rc e s be twe e n so l i tons in the Skyrm emo de l . In [ 18 ] i t wa s show n tha t the p roduc t a nsa tz in the Skyrm e mo de l (w i thou t thep ion ma ss t e rm) l e a ds to the sa me fo rc e s be twe e n we l l - se pa ra te d mov ing a nd sp inn ingSkyrmions a s a s imp le d ipo le mode l fo r Skyrmions , p rov ide d re l a t iv i s t i c c o r re c t ionssuc h a s r e t a rda t ion e f fe c ts a re inc lude d in bo th a pp rox ima t ions . In th i s se c t ion w e w i l ligno re suc h re l a t iv i s t i c e f fe c t s a nd de sc r ibe the d ipo le mode l fo r s lowly mov ing ba byS k y r m i o n s .

Thu s c ons ide r two we l l - se pa rate d ba by Sky rmions , the f i rs t c e n t re d a t Rl a nd ro ta t e dre la tive to the s t a nda rd he dge ho g ~ (1 ) by a n a ng le X l a nd the se cond c e n t re d a t R2 a ndro ta t e d by a n a ng le X2. F rom [7 ] we kn ow tha t , a t la rge se para tion , the l e a d ing te rm inthe po te n t i a l de sc r ib ing the in t e ra c tion o f two ba b y S kyrm ions is the in t e ra c t ion e ne rgyof two doub le t s o f sc a la r d ipo le s in the p la ne , one s i tua te d a t RI a nd the o the r a t R2suc h tha t [RI - R2[ i s la rge c ompa re d to 1 / / z . The d ipo le mom e n ts o f the fi r st d ipo lea re Pa , a = 1 ,2 , whe re

P l = P ( c s x l , s i n x 1 ) , P2 = P ( - s i n x l , c o s x l ) (5 .1 )a n d t h e d i p o l e m o m e n t s o f t h e s e c o n d a r e qa, a = 1 ,2 , whe re

q l = p ( c o s x 2 , s i n x 2 ) , q2 = P ( - s i n x 2 , c o s x 2 ) (5 .2 )The n , i f t he d ipo le s P l a nd q l a nd the d ipo le s P2 a nd q2 in te ra c t v i a a sc a lar f i e ldobe y ing the Kle in -Gordon e qua t ion w i th ma ss / . t , t he in t e ra c t ion e ne rgy be twe e n thed o u b l e t s P a a n d qa is

1W = E ~ (Pa" ) ( q a " V ) K o ( / z R ) , ( 5. 3)a=l,2


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222 B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235w h e r e R -- R~ - / / 2 a n d R = I R I . I n t r oduc i ng a ls o t he r e l a t ive ang l e - - Xl - X2 onef inds

p2 p2/z2W ( , R ) = - - c o s~ b A K o (I zR ) = K o ( t z R ) c o s ~ b . ( 5 . 4 )77" 77"

I n t he la s t s tep w e have u s ed ( 3 .1 2 ) and have om i t t ed t he 6 ( 2 ) - f unc t ion t e r m becaus ewe a r e on l y i n t e r e s ted in l a r ge s epa ra t i ons R > 1 / / z .

T h u s w e o b t a i n t h e f ir st p r e d i c t io n o f t h e d i p o l e m o d e l . T h e f o r c e b e t w e e n t w o b a b ySky r m i o ns depends on t he i r re l a t ive o r i en ta t i on . I n pa r ti cu la r , tw o bab y Sky r m i on s in t hes am e o r i en t a t i on r epe l e ach o t he r ; i f one i s r o t a t ed re l a t i ve t o t he o t he r by 90 t he r e a r eno s t a ti c f o r ce s ; i f on e i s r o t a t ed r e l a ti ve t o t he o t h e r by 18 0 t he f o r ce s a r e a t tr a c ti ve .T h e f o r c e s al w a y s a c t a l o n g t h e l in e j o i n i n g t h e t w o b a b y S k y r m i o n s , b u t in a d d i t i o nt he r e i s a t o r que wh i ch t ends t o r o t a t e t he r e l a t i ve ang l e t o 18 0 . I n ana l ogy wi t h t het e r m i no l og y us ed i n d i s cus s i ng Sk yr m i on dynam i cs we ca l l t h i s con f i gu r a t ion the m o s ta t t ra c t i ve cha nne l .

To ob t a i n a m or e quan t i ta t i ve p i c t u r e w e m us t t ake i n t o accoun t t he m as s and t hem o m e n t o f i n e r ti a o f t h e b a b y S k y r m i o n s . W e a s s u m e t h a t th e r o t a ti o n s o f t h e i n d iv i d u a lbaby Sk yr m i ons a r e s u f f ic i en t ly s l ow s o t ha t we can ap p r ox i m a t e the f unc t i ons M ( w )and A ( w ) b y t h e c o n s ta n t s M 0 a n d A 0. H e n c e o u r m o d e l f o r t h e a s y m p t o t i c d y n a m i c so f t w o b a b y S k y r m i o n s h a s t h e L a g r a n g i a n

1 2 1 . 2 1 . 2L d i p o l e = I M o R I 2 + ~MoR2 + ~Aox1 + ~AoX2 W ( ~ , R ) . ( 5 . 5 )

I n f a c t t h e c e n t re o f m a s s p o s i t i o n S = (R 1 + R 2 ) / 2 a n d t h e an g l e X = (X 1 + X 2 ) / 2a r e c y c l i c a l c o o r d i n a t e s a n d d e c o u p l e f r o m t h e r e m a i n i n g c o o r d i n a t e s . T h u s w e w o r ki n th e c e n t r e o f m a s s f r a m e a n d s e t X = 0 a n d S = 0 . M o r e o v e r w e c a n i n t r o d u c epo l a r coo r d i na t e s ( R , ~b) f o r t he r e l a t i ve pos i ti on ve c t o r R . T hen ~b i s a l s o a cyc l i c a lcoo r d i na t e and i t is cons i s t en t t o s e t ~ = ~b = 0 . Then we ob t a i n t he dynam i ca l s y s t emw i t h e q u a t i o n s o f m o t i o n

p2/.t2Ao~J = - - K o ( t z R ) s in~h,'IT2 3 M 0/~ = P - /' t~ K l ( / z R ) c o s . ( 5 . 6 )7r

Thes e equa t i ons can ea s i l y be s o l ved num er i ca l l y , and i n t he nex t s ec t i on we wi l lcom par e t he s o l i t on t r a j ec t o r i e s p r ed i c t ed by t hem wi t h t hos e ca l cu l a t ed f r om t he f u l lf i e ld equa t i ons ( 2 . 8 ) . Som e reade r s m a y t hen f ind i t u s e f u l t o t h ink o f Eqs . ( 5 .6 ) i nt e r m s o f th e c o u p l e d m o t i o n o f a p e n d u l u m a n d a p o i n t p ar ti c le . M o r e p r e c i se l y th e a n g l e~k m ay be t ho ugh t o f a s cha r ac t e ri s i ng t he angu l a r pos i t i on o f a phys i ca l pend u l um . T hent he f ir s t equa t i on i n ( 5 .6 ) s pec i fi e s t he t o r que ac t i ng on t he pendu l um : i t van i s he s a t t hes t ab l e equ i l i b r i um po i n t ~, = 7 r and t he uns t ab l e equ i l i b r i um po i n t = 0 and i s m a x i m a lw h e n ~b = ~ ' / 2 . M o r e o v e r t h e s t re n g t h o f t h e to r q u e d e p e n d s o n t h e " e x t e r n a l " p a ra m e t e rR and dec r ea s e s w i t h i nc r ea s i ng R . The s econd equa t i on i n ( 5 .6 ) c an be i n t e r p r e t ed a st he equa t i on f o r t he l i nea r m o t i on o f a po i n t pa r t i c l e wi t h pos i t i on R . The f o r ce ac t i ng


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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995 ) 205-235 223on t he pa r t i c l e depends on i t s pos i t i on , i t s s t r eng t h dec r ea s i ng wi t h i nc r ea s i ng R , bu ti s a l s o con t r o l l ed by t he " ex t e r na l " pa r am e t e r ~ /,. W hen ~O = 7 r ( the pen du l u m ' s s t ab l eequ i l i b r i um ) t he f o r ce i s a tt r a c ti ve , t end i ng t o dec r ea s e R ; w hen ~p = 0 ( t he pe ndu l um ' suns t ab l e equ i l i b r i um ) t he f o r ce i s r epu l s i ve , t end i ng t o i nc r ea s e R .

6 . N u m e r i c a l s i m u l a t i o n s

Al l t he s i m u l a t i ons o f t he f i e l d equa t i ons t o be d i s cus s ed i n t h i s s ec t i on t ake p l aceon a s qua r e g r i d o f 250 x 250 po i n t s , ex t end i ng i n bo t h t he x and y d i r ec t i on f r om- 2 5 t o 2 5 . T h e i n i ti a l c o n f ig u r a t io n s a r e c o n s tr u c t e d f ro m t w o b a b y S k y r m i o n fi el d sus i ng t he s upe r po s i t i on p r oc edu r e r e f e r r ed t o above ; thus , t he s e con f i gu r a t i ons can b echa r ac t e r i s ed by g i v i ng t he i nd i v i dua l baby Skyr m i ons ' pos i t i ons and o r i en t a t i ons . Thebaby Sky r m i ons ' c en t r e o f m as s pos i t i on and t he ove r a ll i s o - o r i en t a t ion a r e i m m a t e r i a lf o r t he dynam i cs , and w e t ake t he f o r m e r t o be a t t he g r i d ' s o r i g i n and us ua l l y choos e t hel a t te r s o t ha t the i nd i v i dua l baby Sky r m i o ns ' o r i en t a t i ons a r e equa l and oppo s i t e . Fo r e achs i m u l a t i on we wi l l s p ec i f y t he i n i ti a l r e la t i ve pos i t ion and t he i n i t ia l r e l a ti ve o r i en t a ti on ,deno t ed ~ /'0 . W e w i l l a l s o cons i de r i n i t ia l cond i t i ons whe r e bo t h baby Sky r m i on s haves om e no n- ze r o i n i ti a l ve loc i t y . W e t hen wor k i n the cen t r e o f m as s f r am e , s o t ha t t heb a b y S k y r m i o n s ' v e l o c it i es a re e q u a l a n d o p p o s i te , a n d w e i n c l u d e t h e e f f ec t o f L o r e n t zcon t r ac t i on i n ou r i n i t i a l con f i gu r a t i on . Un l e s s s pec i f i ed o t he r wi s e we wi l l cons i de ri n i t i a l ve l oc i t i e s a l ong t he x - ax i s , s o gene r i ca l l y one baby Skyr m i on i s i n i t i a l l y i n t heh a l f p l a n e x > 0 w i th v e l o c i t y ( - v , 0 ) a n d t h e o t h e r i n th e h a l f p l a n e x < 0 w i t hve l oc i t y ( v ,0 ) , wh e r e 0 ~< v < 1 .6 . 1. S c a t t e r i n g f r o m r e s t

Suppos e t he t wo baby Skyr m i ons a r e i n i t i a l l y a t r e s t and we l l - s epa r a t ed , one cen t r eda t ( 1 0 , 0 ) a n d t h e o t h e r a t ( - 1 0 , 0 ) . W e h a v e c a l cu l a te d th e t i m e e v o l u t io n o f t h eco r r e s p ond i ng f i e ld con f i gu r a t i on f o r a va r i e t y o f in i t ia l o r i en t a t ions 0 C [ 0 , z r] andi n F i g . 5a we p l o t R / 2 , t he s epa r a t i on o f e i t he r baby Skyr m i on f r om t he cen t r e , a s af u n c t i o n o f t i m e . W e o n l y s h o w t h e t i m e e v o l u t i o n u n t i l t h e b a b y S k y r m i o n s c o l l i d e -t he ac t ua l co l l i s ion w i ll be d i s cus s ed i n t he nex t s ec t i on . The q ua l i ta t i ve f ea t u r e s o f t het i m e e v o l u t i o n c a n e a s i l y b e u n d e r s t o o d i n t e rm s o f t h e d i p o l e m o d e l .

Co ns i de r f o r exam pl e t he m o t i on w hen ~b0 = 7 r. The n t he b aby Sky r m i on s a re a l r eadyi n the m os t a t t ra c t i ve chann e l and r em a i n t he r e ; henc e t he r e i s an a t t r a c ti ve fo r ce be t weent he i r c en t r e s t h r oug hou t the i r i n t e r ac ti on . The p l o t o f t he ac t ua ll y obs e r ved t i m e evo l u t i ono f R / 2 s h o w s e x a c t l y s u c h a n a c c e l e ra t in g m o t i o n . W h e n 0 = ~ / 2 t h e in i ti a l t o r q u e ism ax i m a l , bu t t he i n i t i a l f o r ce van i s he s . Thus t he r e l a t i ve o r i en t a t i on ~p s wi ngs t h r ought he a t t ra c t i ve channe l - - ~ r, a t wh i ch p o i n t t he f o r ce be t ween t he baby Sky r m i on s i sm ax i m a l l y a t tr a c t ive , bu t t hen ov e r s hoo t s and app r oaches ~b -- 3w / 2 . W hi l e ~b i s c l o s et o t h i s va l ue t he f o r ce be t ween t he baby Skyr m i ons i s aga i n ve r y s m a l l o r z e r o , s ot ha t we expec t t he s epa r a t i on pa r am e t e r R t o be a l i nea r f unc t i on o f t i m e he r e . Th i s i s


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224 B.M.A.G. Piet te et al . /Nuc lear Physics B 439 (1995) 205-23 5

( a )12

10

8

6

4

2

i i i i i i /2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0

t18oo

12 ( b )10

8

6

4

2

i i i i iO 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1~ w 1 4 00

tFig. 5. (a ) Relative motion of two baby Skyrmions released from rest. R / 2 is plotted as a function of timefor, from top to bot tom, ~bo = 0.275- ~r, 0.3 . I t, 0.4. Ir, 0.5- 7r and ~bo = ~r. (b) Prediction of the dipole modelfor the relative motion of two baby Skyrmions released from rest. R / 2 (top ) and ~ (bott om) as a functionof time for ~bo = 0.28855.1r.

pr ec i s e l y w ha t we s ee i n F i g . 5a . W hen ~P0 i s dec r ea s ed f u r t he r t he f o r ce be t we en t heb a b y S k y r m i o n s i s i n i ti a ll y r e p u ls i v e. T h e b a b y S k y r m i o n s m o v e a p a rt b u t a t t h e s a m et i m e ~b i nc r ea s e s s o t ha t s om e t i m e l a t e r t he baby Skyr m i ons a r e i n t he m os t a t t r a c t i vechanne l . The f o r ce i s now a t t r a c t i ve and t he baby Skyr m i ons app r oach each o t he r aga i n .As t he r e l a t i ve o r i en t a t i on o s c il l a te s t he baby Sky r m i on s expe r i ence a l te r na t i ng a t t ra c t i veand r epu l s i ve f o r ce s and t hus pe r f o r m t he o s c i l l a t o r y m o t i on m os t c l e a r l y s een i n t he


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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995 ) 205-235 225t r a j ec t o r y f o r 0 = 0 .3 z r. F i na l l y t he baby Sky r m i on s ge t t r apped i n t he a t t ra c t i vechanne l and co l l i de .

When ~b0 i s decreased fur ther the in i t i a l r epuls ive force increases and as a resu l t thebaby Skyr m i ons ' s epa r a t i on m ay i n i t i a l l y i nc r ea s e s o r ap i d l y t ha t t he a t t r a c t i ve f o r cewh i ch t he ba by Sky r m i o ns expe r i enc e on ce t hey a r e i n the a t tr a c ti ve channe l i s t oowe ak t o i nve r t t he i r r e l a t ive ve l oc i ty . Th e baby Sky r m i on s t hen e s cape t o in f i n it y , wh i chi n ou r s i m u l a t i ons m eans t ha t t hey h i t t he bounda r y o f t he g r i d . We have i nves t i ga t edb o u n d a r y e f f e c t s b y s e n d i n g a s i ng l e b a b y S k y r m i o n t o w a r d s t h e b o u n d a r y w i th v e l o c i t yv = 0 . 1 a n d f i n d w e a k r e p u l s i v e f o r c e s w h e n t h e b a b y S k y r m i o n i s a p p r o x i m a t e l y 8un i t s away f r om t he bounda r y . The r e l evan t bounda r y f o r t he p r e s en t s i m u l a t i on i s a tx = 2 5 , s o w e i n t e r p r e t s i m u l a t i o n s w h e r e R / 2 becom es l a r ge r t han 15 a s "e s cape t oinf in i ty" . This h app ens for ~k0 ~< 7r /4 . In o ur s im ula t ions the smal les t v a lue o f ~0 f orwh i ch t he bab y Sky r m i o ns u l t i m a t e l y co l l i de is ~b0 = 0 .2 75 . z r , f o r wh i ch t he t r a j e c t o r yi s a l s o s hown i n F i g . 5a .

W e have a l s o s o l ved Eqs . ( 5 .6 ) num er i ca l l y f o r a r ange o f in i t ia l va l ue s 0 , a l way ss e tt in g R ( 0 ) = 2 0 , R ( 0 ) = ~ ( 0 ) = 0 . C o m p a r i n g t h e s e s o l u ti o n s w i t h t h e c o r r e s p o n d i n gt r a j ec t o r i e s f ound i n ou r s i m u l a t i ons o f t he f u l l f i e l d equa t i ons we f i nd qua l i t a t i veag r eem en t i n a l l c a se s , bu t quan t i t a ti ve ag r eem e n t on l y f o r t he f i rs t pa r t o f t he t r a j e c to r i e s( t yp i ca l l y 0 ~< t < 100 ) . M or eove r , t he depen denc e o f t he s o l u t ions o f ( 5 .6 ) on ~P0 i sexa c t ly as f ou nd in th e f i e ld theory . T her e i s a c r i t ica l v a lue ~bc such tha t fo r ~b0 E (~bc , 0] ,R ( t ) t ends to in f i n i ty a s t ~ c~ , and f o r 0 E ( c ,7 r ] , R ( t ) a p p r o a c h e s z e r o ( w h e r et he equa t i ons a r e s i ngu l a r ) a f t e r a num ber o f o s c i l l a t i ons wh i ch becom es a r b i t r a r i l yla rge as ~b0 ~ ~bc . I t s num er ica l va lue i s ~bc ~ 0 .28 85 45, w hich shou ld be co m pare d w i tht he va l ue 0 .275 f o und i n t he f ie l d t heo r y . I n F ig . 5b w e s how R and a s a f unc t i on o ft i m e f o r 0 = 0 .28 8 55 . ~ - . The qua l i t a t i ve f ea t u r e s o f t he i n t e r p l ay be t ween t he angu l a rm o t i on ( ~b ) and t he l i nea r m o t i on ( R) d i s cus s ed ea r l i e r a r e c l ea r l y i l l u s t r a t ed .6 .2 . H e a d - o n c o l l i s i o n s

The d i po l e m ode l on l y de s c r i be s t he a s ym pt o t i c pa r t o f t he t r a j e c t o r i e s d i s cus s ed s of ar . W e h a v e s e en , h o w e v e r , th a t f o r 0 > 0 . 2 7 5 . 7 r t h e tw o b a b y S k y r m i o n s u l t im a t e l yad j us t t he i r r e l a t i ve o r ien t a t i on s o t ha t t hey a r e i n t he a t tr a c t ive channe l , and co l l i de headon . I n t he nex t s e t o f s i m u l a t i ons w e s t udy t he ou t co m e o f s uch a head - o n co l l i s i on i nthe a t t rac t ive c han nel for a var ie ty o f d i f fe rent in i t ia l speeds . W e f ix ~P0 = 7r and p lac et h e b a b y S k y r m i o n s a t ( 7 . 5 , 0 ) a n d ( - 7 . 5 , 0 ) , g i v in g t h em i ni ti al v e lo c it ie s ( - v , 0 )and (v ,0 ) respec t ive ly , whe re 0 .1 ~< v ~< 0 .6 .

I n a l l t he s i m u l a t i ons t he baby Skyr m i ons m er ge i n t o t he r i ng - l i ke s t r uc t u r e o f t he2 - s o l i t on s o l u t i on and em er ge a t f i gh t ang l e s t o t he i r i n i t i a l d i r ec t i on o f m o t i on . Af t e rt he s ca t t e r i ng t hey m ove away f r om each o t he r wi t h t he i r r e la t i ve o r ien t a t i on st il l in t hea t t ra c t i ve channe l . Th i s i s t he 90 s ca t t e ri ng t ha t i s now a f am i l ia r and appa r en t l y gene r i cf ea t u r e o f t opo l og i ca l s o l i t on dynam i cs . Howeve r , i n ou r m ode l t h i s s ca t t e r i ng p r oces si s a cco m p an i ed by t he em i s s i on o f a l a rge am o un t o f r ad i a t ion . I n F i g . 6 w e s how t heene r gy d i s t r i bu t i on i m m ed i a t e l y a f t e r t he co l l i s i on wi t h v = 0 .6 : t he r i ngs o f r ad i a t i on


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8 . 1 4 3 7 0

O . m

Fig. 6. Total energy density shortly afte r the head-on collision of two ba by Skyrmions with initial speedsv = 0.6.a r e c l e a r l y v i s i b l e . T h e r a d i a t i o n c a r r i e s a w a y s o m u c h e n e r g y t h a t t h e b a b y S k y r m i o n so n l y " e s c a p e t o i n f i n i t y " ( d e f i n e d a s a b o v e ) f o r v / > 0 . 4 6 . F o r s m a l l e r i n i t i a l v e l o c i t i e st h e a t t r a c t i v e f o r c e s b e t w e e n t h e b a b y S k y r m i o n s a f t e r t h e c o l l i s i o n p u l l t h e m b a c ka n d t h e y p e r f o r m a n o t h e r h e a d - o n c o l l i s io n , a g a i n s c a t t e r in g t h r o u g h 9 0 a n d e m e r g i n ga long the i r i n i t i a l d i r ec t i on o f m o t io n and i n t he a t t r a c t ive channe l . The s econd co l l i s i oni s a g a in a c c o m p a n i e d b y t h e e m i s s i o n o f r a d i at i o n , s o t h e s o l i to n s t r a v el le s s f a r t h a na f t e r t he i r f i r s t co l l i s i on be f o r e t hey t u r n r ound . Th i s p r oces s i s r epea t ed , bu t now them o t i o n r e m a i n s c l o s e t o t h e r i n g - l ik e 2 - s o l i to n s o l u t i o n a t al l t im e s . T h e i n d i v id u a l b a b yS k y r m i o n s a r e n o l o n g e r d i s t i n c t , a n d t h e m o t i o n l o o k s l i k e a n o s c i l l a t o r y e x c i t a t i o n o ft h e 2 - s o l i to n . T h e e m i s s i o n o f r a d i a ti o n , h o w e v e r , c o n t i n u e s u n t i l t h e k i n e ti c e n e r g y h a sv i r t u a l l y d i s a p p e a r e d . T h e f i n a l c o n f i g u r a t i o n i s n u m e r i c a l l y i n d i s t i n g u i s h a b l e f r o m t h e2 - so l i t on so lu t i on .

T h e s c a t t e r i n g p r o c e s s d e s c r i b e d a b o v e i s m u c h m o r e r a d i a t i v e t h a n a n y o b s e r v e di n p r e v i o u s s i m u l a t i o n s o f l u m p s c a t t e ri n g i n t h e C P 1 m o d e l [ 8 ] o r i n o t h e r t w o -d i m e n s i o n a l v e r s i o n s S k y r m e m o d e l s w i t h p o t e n t ia l s d i f f er e n t f r o m o u r s [ 11 ] . A t f ir s ts i gh t t h i s i s su r p r i s i ng : t he r ad i a t i on i n ou r mode l i s mass ive whe r ea s t he r e a r e mass l e s sr a d i a t i o n m o d e s i n a l l t h e c o m p a r a b l e m o d e l s m e n t i o n e d a b o v e . H o w e v e r , o u r m o d e li s a l s o t h e o n l y o n e i n w h i c h t h e r e a r e s t r o n g a t t r a c t i v e f o r c e s . M o r e o v e r t h e s e f o r c e sa r e sho r t - r anged , so t ha t t he po t en t i a l ene r gy f unc t i ona l V ( 2 . 4 ) ha s a l a r ge g r ad i en ta t c o n f i g u r a t i o n s c o n s i s t i n g o f t w o n e a r b y b a b y S k y r m i o n s . T h i s m e a n s t h a t t h e t i m ee v o l u t i o n o f a c o n f i g u r a t io n i n t h e v i c i n i t y o f t h o s e p o i n t s i n t h e c o n f i g u r a t i o n s p a c e i s


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B.M.A.G. Piette et al./Nuclear P hysics B 439 (1995) 205-235 227no t ad i aba t i c . I t f o l l ows i n pa r t i cu l a r t ha t t he ad i aba t i c o r m odu l i s pace app r ox i m a t i onp r op os ed f o r t he Sky r m e m ode l i n [ 5 ] i s no t s u it ab l e f o r de s c r i b i ng s o l it on co l l i s ionsi n o u r m o d e l .

I n t he nex t s e t o f s i m u l a t i ons we keep t he i n i t i a l ve l oc i t y v f i xed a t 0 .5 bu t va r y t hei n i t i a l r e l a t i ve o r i en t a t i on ~0 be t ween 0 and 7 r . I n i t i a l l y , t he baby Skyr m i ons a r e aga i np l a c e d a t ( 7 . 5 , 0 ) a n d ( - 7 . 5 , 0 ) . T o u n d e r s ta n d t h e e n s u i n g i n te r a c ti o n p r o c e s s es i t i su s e f u l t o n o t e t he s y m m et r i e s o f t he in i t ia l cond i t i ons . Reca l l ing t ha t i n ou r conv en t i onst he baby Skyr m i ons have equa l and oppos i t e i n i t i a l o r i en t a t i ons we f i r s t obs e r ve t ha tt he i n i t i a l con f i gu r a t i ons a r e i nva r i an t unde r t he r e f l e c t i on Px ( 2 . 1 4 ) . S i n c e P x is as ym m et r y o f t he Lag r ang i an , t he co n f i gu r a t i ons wi l l r em a i n i nva r ian t unde r t ha t ope r a t i ond u r i n g t h e i r t i m e e v o l u t io n . H e n c e , a s s u m i n g t ha t t he baby Skyr m i ons s epa r a t e a f t e r t heco l l i s i on , we can deduce t ha t t hey m us t s epa r a t e a l ong e i t he r t he x - ax i s o r t he y - ax i s .I n o t he r w or ds , i f s ca t te r i ng t ake s p l ace , i t m us t be s ca t te r i ng t h r ough e i t he r 0 ( i .e .t r i v i a l ) , 90 o r 18 0 . W e a l s o n o t e t ha t , i f t he baby Sky r m i ons s epa r a t e a l ong t he y - ax i s ,t h e r e q u i r e m e n t o f i n v a r ia n c e u n d e r P x a l l ows on l y t w o pos s i b i l it i e s f o r t he i r i nd iv i dua lo r i en t a t i ons : e i t he r t he s t anda r d o r i en t a t i on whe r e , u s i ng t he conven t i ons o f F i g . 3 , t hef i el d s po i n t r ad i a l l y ou t wa r ds , o r t he s t anda r d o r i en t a t i on r o t a t ed by r , wh e r e t he f i e ld spo i n t r ad i a l l y i nwar ds . Thus , a f t e r t he s ca t t e r i ng t he baby Skyr m i ons e i t he r have t hes am e o r i en t a t i on and a r e i n t he m os t r epu l s i ve channe l o r t he i r o r ien t a t i ons d i f fe r by 7 r,i n wh i ch ca s e t hey a r e i n t he m os t a t t r a c t i ve channe l .

When 0 = 0 or ~b0 = 7r the in i t i a l conf igura t ions a re addi t iona l ly invar ian t under ,respec t ive ly , the re f lec t ion Py ( 2 . 1 5 ) o r t h e c o m b i n a t i o n o f Py wi t h an i s o - r o t a t i on byo r. I t f o l l ow s t ha t t he con f i gu r a t ions a f t e r t he i n t e r ac ti on m us t h ave t he s am e i nva r iances .I n pa r t i cu l a r f o r 0 = 7 r we can p r ed i c t pu r e l y on t he ba s i s o f s ym m et r y t ha t , i f t hebaby Sky r m i o ns s ca t t e r t h r oug h 90 , t hey m u s t be equ i d i s tan t f r om t he o r i g i n a f t e r t hes ca t t e r i ng and t hey m us t be i n t he m os t a t t r a c t i ve channe l . Of cou r s e , t h i s i s p r ec i s e l ywh a t w e ob s e r ved i n ou r p r ev i ous s i m u l a ti on . Fo r t he o t he r " s pec i a l " i n it ia l con f i gu r a t i on ,wh e r e ~b0 = 0 , we f ind , how eve r , t ha t t he baby Sky r m i on s s ca t t e r t h r ough 18 0 . M or ep r ec i s e l y t hey head t owar ds each o t he r and s l ow down un t i l t hey com e t o a ha l t a t as epa r a t i on R ~ 3 . Then t hey t u r n r ound and e s cape t o i n f i n i t y a l ong t he l i ne o f i n i t i a lapp r oach . The r e l a t i ve o r i en t a t i on ~b does no t change a t a l l du r i ng t h i s p r oce s s . How,wi t h i n t he cons t r a i n t s i m pos ed by t he s ym m et r i e s de s c r i bed above , does t he s ca t t e r i ngin te rp ola te be tw een 90 sca t t e r ing and 180 sc a t t e r ing as w e v ary ~P0 f ro m 7r to 0?

I n ou r s i m u l a t i ons w e f ind tha t t he r e i s an i n t e rva l 190 = [ ~ / 6 , 7 r ] s uch t ha t t he bab ySky r m i on s s ca t t e r t h r oug h 90 f o r tP0 E 190 and a s m a l l e r i n t e rva l i 18 0 = [ 0 , ~ / 2 0 ] s ucht ha t t hey s ca t t e r t h r ough 18 0 f o r 0 E 118 . How eve r , a s 0 ~ r / 6 t he baby Sky r m i on sem er ge f r om t he 90 - s ca t t e r i ng wi t h d i f f e r en t s peeds : t he baby Skyr m i on m ov i ng i n t hepos i t i ve y - d i r ec t i on m ov es f a s t e r than t he o ne m o v i ng i n t he nega t i ve y - d i r ec t ion . Them o m e n t u m b a l a n c e i s r e s t o r e d b y r a d i a t i o n w h i c h i s e m i t t e d p r e d o m i n a n t l y a l o n g t h enega t i ve y - ax i s . Th i s p r oce s s i s p r e s en t ed s chem a t i ca ll y i n F i g . 7b .

W he n '0 i s in the re m ain ing in te rva l I eapt = ( I r / 2 0 , ~ r / 6 ) t h e tw o b a b y S k y r m i o n sdo no t s epa r a t e a f t e r the co l l i s ion bu t f o r m t he o s c i l l a t o r i ly exc i t ed s t at e o f t he 2 - s o l i t ona l r eady encoun t e r ed i n t he p r ev i ous s e t o f s i m u l a t i ons . Howeve r , t h i s t i m e t he exc i t ed


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228 B.M.A.G. Pie t te e t a l . /N uc l ear Phys ic s B 439 (1995) 205 -23 5

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B.M.A.G. Piette et al./Nuclear Physics B 439 (1995) 205-235

I c l

229

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xFig . 7 . Head on col l i s ions . Left page: ( a ) S ke t c h o f ba by S ky r m i on ve l oc it i e s a nd r a d ia t ion e m i t t e d s ho r t lya f te r s ca t t e r ing wi th ~o = ~ ' . (b) Ske tch of baby Skyrmion ve loc i t i e s and radia t ion emi t t ed shor t ly a f t e rs c a t t e r i ng w i t h ~ o = , r / 6 . This page: (c ) Ske tch o f 2-sof i ton ve loc i ty and radia tion emi t t ed shor t ly a f t e rsca t t e r ing wi th ~0 E I capt. (d ) Ske tch o f baby Skyrm ion ve loc i t i e s shor t ly a f te r s ca t t e r ing wi th ~o = O.


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230 B.M.A.G. Piette et al./Nuclear Physics B 43 9 (1995 ) 205-2352- s o l i t on a s a w ho l e m oves i n t he pos i t i ve y - d i rec t i on , and r ad i a t i on i s em i t t ed i n t heopp os i t e d i r ec t i on t o r e s t o r e t he m o m en t u m ba l ance . Th i s p r oce s s i s s ke t ched i n F ig . 7c .

The no t i on o f t he m os t a t t r a c t i ve channe l i s u s e f u l f o r s um m ar i s i ng ou r r e s u l t s . Fo r~0 E 190 U I eap t t he baby Sky r m i o ns m e r ge and s ca t te r t h r ou gh 90 ; t he c l o s e r t he t wobaby Skyr m i ons a r e i n i t i a l l y t o be i ng i n t he m os t a t t r a c t i ve channe l t he l e s s r ad i a t i on i sem i t t ed i n t he s ca t t e ri ng p r oces s . Fo r ~0 E 190 t he baby Sky r m i on s t he r e f o r e e s cape t oi n f i n it y a f t e r t he co l l i s i on ; f o r 0 E I eapt t hey f o r m a co i nc i den t con f i gu r a t i on wh i ch wei n t e r p r e t a s an a s ym m et r i c a l l y de f o r m ed 2 - s o l i t on s o l u t i on . The i nc r ea s ed r ad i a t i on andi ts a s y m m et r y i s due t o t ha t de f o r m a t i on . F i na ll y , f o r ~b0 E 118 t he r e i s no t enou gh t i m ef o r t he baby Sk yr m i ons ' r e l a t ive o r i en t a t ion t o ad j us t be f o r e t he co l l i s ion , and a s a r e s u l tt he f o r ce be t ween t hem i s a l ways r epu l s i ve . We have no t d i s cus s ed nega t i ve va l ue s f o r~0 s epa r a t e l y he r e becaus e a s ca t t e r i ng p r oces s wi t h g i ven nega t i ve ~0 i s r e l a t ed t o t hep r o c e s s e s w i t h - 0 b y th e r e f le c t io n P y .Cl ea r l y t he p r ec i s e va l ue s o f t he bounda r i e s o f t he i n t e r va l s 9 0 , l e a p t and i 18 0 de -pend on t he baby Skyr m i ons ' i n

b.m.a.g. piette, b.j. schroers and w.j. zakrzewski- dynamics of baby skyrmions

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