DTP-96/17 November 28, 1996 Mesons, Baryons and Waves in the Baby Skyrmion Mo del 1 A. Kudryavtsev B. Piette, and W.J. Zakrzewski Department of Mathematical Sciences University of Durham, Durham DH1 3LE, England 1 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 2 1 k 2 ~ ~ ~ ~ ~ ~ ~ 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 2 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. 3 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 2 for all t. This condition formally compacti es the physical space to a 2-sphere S ph 2 2 and so all maps from S to S are characterised by the integer-valued degree of this iso ph map (the top ological charge). The analytical formula for this degree is Z 1 2 ~ ~ ~ ~ deg []= (@ @ )d x: (1:3) 1 2 4 ~ 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 2 ~ ~ ~ ~ ~ ~ ~ @ ( @ @ (@ @ )) = ~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 2 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 n ~ deg []=[cos f (1) cos f (0)] : (2:4) 2 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) n m ~ deg []=[1(1) ] : (2:5) 2 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) 2 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]: r r e (2:8) f (r ) K (r ) 1 r !1 2r where K (x) is the mo di ed Bessel function. On the other hand the asymptotic 1 b ehaviour of the 1 energy anti-Skyrmion-like solution corresp onding to (n =1;m = 2;l =1) is given by C f (r ) + sin(r + ); (2:9) p r 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.