Strange dibaryons in the simplified skyrme model

Nuclear Physics A532 (1991) 708-714 North-Holland

STRANGE DIBARYONS IN THE SIMPLIFIED SKYRME MODEL Nils DALARSSON Department of Theoretical Physics, Royal Institute of Technology, S-10044 Stockholm, Sweden

Received 27 February 1991 Abstract: We show that the bound-state approach to the strange dibaryons in the Skyrme model can be considerably simplified by omitting the Skyrme stabilizing term proportional to e-2 and using the regularized version of the quantum stabilization proposed by Mignaco and Wulck . The strange dibaryons are obtained assystems consisting ofbound antikaons in a (variational) axially symmetric SU(2)-skyrmion background field with baryon number B> 1 .

1. Introduction

Callan and Klebanov ') showed that a fairly good description of the hyperon spectrum in the Skyrme model 2) is obtained, if the hyperons are treated as bound kaon-soliton systems. Callan, Hornbostel and Klebanov (CHK) 3) successfully completed this program. The basic idea of their model is to treat strangeness separately from the isospin in the Skyrme model, assuming that the vacuum is approximately SU(3) symmetric i.e. FK - F., . The strange baryons are generated by binding kaons in the field of "rotating" SU(2) solitons. Since there is no static field associated with the strangeness quantum number, it is essential in this picture that there exist bound states in the kaon-soliton complex that give rise to hyperons. CHK showed that such bound states do exist. A remarkable property of the kaons in the CHK mode: is that after quantization they look like s-quarks, due to topological effects. This leads to a spectroscopy ofhyperons quite similar to that of quark models. In the CHK approach the kaon-soliton field is written in the form U=(U

7r

)'/2UK(UV)'/2,

(1 .1)

where U,r is a SU(3)-extension of the usual SU(2) skyrmion field used to describe the nucleon spectrum 4), and UK is the field describing the kaons U,r =exp(2iF -,,'Aj7r'), UK=

exp (2iF-'AQK a ) ,

j=1,2,3, a =4, 5, 6, 7 .

(1 .2) (1 .3)

The A-matrices are the familiar SU(3) matrices . In this paper we apply the CHK approach to the baryon number n sector of the simplified Skyrme model 5), with the stabilizing term proportional to e-2 omitted, using the quantum stabilization method proposed by Mignaco and Wulck (MW), generalized to the baryon number n > 1 sector and regularized using a modified radial ansatz 6 ), in order to avoid the difficulties with the soliton boundary conditions g) . We derive the results for the rotational energies of the arbitrary baryon 0375-9474/91/$03 .50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

N. Dalarsson / Strange dibaryons

709

number n > 1 states, but the primary objective of the present study is the dibaryon states with n = 2. This paper consists of four sections . In sect. 2 the effective interaction for the bound kaon-soliton system is derived. In sect. 3 the rotational energies of the strange dibaryons are calculated . In sect. 4 we make the final conclusions . 2. The effective interaction The lagrangian density for a bound kaon-soliton system in the simplified Skyrme model is given by Y= i6F2,r Tra~,Ua'`U + + 4g F~(m~+2mK) Tr (U+ U+ -2) +24V F2V (mV- mK) Tr A8 (U+ U+) , (2.1) where m,r and MK are pion and kaon masses, respectively, and u.~ 0 2 312 [ 0 K U. = 1 , UK = exp i . (2.2) + 0 K 0 F ] In (2.2) u,, is the SU(2)-skyrmion field for which we choose the variational ansatz discussed in ref. 7) with static radial modification phase function V = V(r) discussed in ref. 6) . Compared to the usual n =1 case, this ansatz has a different twist in the isovector field zr rather than a different boundary condition for the chiral phase function F = F(r), i.e. sin 0 cos nçb u,, = exp [ër - zrF(r) + i(p (r)] , Tr = sin 0 sin nç , cos 0 (2 .3) 2 dF 2 2+2n(2n-1)F2-(n2+1)sin2F r LV =(2rR+R2) dr dr where F(r) is a radial function which (in the absence of the Skyrme stabilizing term and for m, = 0) satisfies the non-linear differential equation d r2 d F = 2n(2n -1)F (2.4) dr ( dr with the boundary conditions F(0) = -v and F(oo) = 0 and where R is a dimensional scale parameter. The presence ofthe static radial modification phase Sp = p(r) ensures the existence of the solution F(r) = -a(r/ R + 1) -2n with proper soliton boundary conditions. The two-dimensional vector K in (2.2) is the kaon-doublet [K+J K= K o K

K+ = [K -

O] .

(2.5)

In addition to the simplified Skyrme model action obtained using the lagrangian density (2.1), the Wess-Zumino action in the form iN, - s elvaßy S = -2 dx Tr [ U+aN,UU + a,UU +a a UU+aß UU+a,,U] (2.6) 240~r J must be included into the total action of a kaon-soliton system. In (2.6) N, is the number of colours in the underlying QCD.

N. Dalarsson / Strange dibaryons

10

Substituting (1.1), with U,, and UK defined by (2.2), into the total action of the kaon-soliton system, expanding UK to second order in the kaon fields (2.5) and using the following ansatz for stationary antikaon states _ t) _ e-"~kp(r),r . K(r, t) =z Iank(r, (2 .7) ITnX,

where X is a two-component spinor, we obtain the effective interaction-lagrangian density for the kaon-soliton system in the lowest bound state (see ref. 5)) dk+ iA(r)(k + k - k+k) - k+[MK+ Veff(r)]k, r dr

k+k+ dk+ where

F`' n'+ 1 1 [(+!! ' (dF ` 1 . +2n(2n -1) , + cos4 2F, V~a(r) = r2 ) dr A(r)

®

N, sin'` FdF . - 2~'F_ r' dr

(2 .8)

(2.9)

(2.10)

e hamiltonian density corresponding to the lagrangian density (2.8) is given by

= +k + k+ ll - Y,

(2.11)

where H+ = aY/ak = k+ + iA(r)k + ,

(2.12)

11=ate/ak+ =k-U(r)k .

(2 .13)

Substituting (2.7), (2.11) and (2.12) into (2.10) we obtain

+ - dk+ dk dr dr

-iA(r)(k+H-II+k)+k+[ m2+ Ve,,(r)]k

+k+A`'(r)k.

(2.14)

From (2.14) we obtain the radial wave equation for the lowest bound-state antikaon wave function uo = rkp(r) in the form d z uo (2.15) vefif(r)uo+[ca t- mK+2wA(r)]uo=0, dr where w is the lowest bound-state energy, and the antikaon modes are normalized according to 81r (2 drr2[w+À(r)]k**(r)kp(r)= 1 .

10

.l6)

If the differential equation (2.4) is solved and the phase F(r) is known it is possible to solve (2.15) with (2.9)-(2.10) and find the lowest bound-state energy w.

N. Dalarsson / Strange dibaryons

3. The rotational energies of the strange dibaryons

In order to calculate the spectrum of the strange dibaryons we must take into account the rotational modes of the soliton ' ). We consider the spatial (R) and isospin (A) rotations, where the corresponding angular velocities do and ß with respect to the body fixed axes are given by (R- ~ aoR)ab = Eabc k s

(3 .1)

A- ' ao A = 2iß ' T,

where Eabe (a, b, c = l, 2, 3) is the totally antisymmetric tensor with and the soliton fields are rotated according to

(3 .2) E, 23 =

l . The kaon

K -o- RAK = (RA-r - = ^ .A- ')Ak,

(3.3)

u,, = RAuA-' .

(3.4)

Substituting (2.2) with (2.3), (2.5), (3.3) and (3 .4) into (2.1) and keeping in the expansion of the kaon field only the two eigenmodes in (2.7) with coefficients a, and a2 for the up and down spinor X, respectively, we obtain the following expression for the total lagrangian of the kaon-soliton system L=L(u,)+L(K)-N- ß+(1-c) X C(NA

+ N2ß2)- (NI 61 + N2C 2)sn1 +N3(03 - nCi3)]

+ 211Cf1+f2+4(n2+3)(C11+~2)-2(~jf1+6202)sni+(03-nCl3)2 with

x

,f2 = 37rF-'7r I dr r2 sin-' F, .ro

c =1-3â~

0

dr r-'kp(r) COs-' 2Fkp ( r) ,

where N is the spin of the antikaon modes given by N -- 2ai Tiiai

]

(3.5) (3 .6) (3 .7)

(3.8)

and â ;(âi ) are creation (annihilation) operators of antikaon modes. The quantization of the rotational energy leads to ff~ot-

1 (J2--13)+I -' -I3+c-'(T-' -T3)+(I3+cT3)2 2,f2 n 24+ 3 +2c(I - T - I3 T3 ) ,

(3.9)

where J is the total angular momentum of the kaon-soliton system, I is the collective angular momentum of the rotating n > 1 soliton and T is the angular momentum

N. Dalarsson / Strange dibaryons

712

of the kaons. In the special case when n =1, i .e. for the spherically symmetric s soliton, (3.9) reduces to the well-known form ) 1 (3 (I+cT) 2 , n =1 . .10) HT®,= 212 The eigenfunctions of Htat are product states of the spatial and isospin rotation matrices with the kaon eigenstates, given by Jbf= 111 Ibf= _ _nQ, T3) ra Q Tz, ~ 13 ; J' J3 ; ®

t)

'H J,

gâr`

(3 .11)

where we introduced the angular momenta .I bf and Ibf relative to body-fixed axes as Ibf = aL jbf = aL (3 .12) a&' aft ' such that the axial symmetry of the classical solution gives rise to the constraint

J3 f = -n(I3f+ T3) .

(3.13)

For non-strange dibaryons (T = ;ISI = 0), the states ofthe form I I, 13 ; J, J3 ; 13 f= Q, Jbf = -nQ, T = 0) are the eigenstates of H,,,,, and the rotational energy is given by 3 4 I [ 20 n 2 +3

4n 2 n 2 +3

For strange dibaryons (T = ?ISI > 0), the eigenstates are obtained after the diagonalization of H,,i . For dibaryons with strangeness S=-1 the isospin of the bound antikaon is T = â and the hamiltoniian is a 2 x 2 matrix. If the basis states are II, I; ; J, J3 ; Ibf=QT ;, Jbf=-nQ, T3 = t2), the rotational energy is given by ,~t=

4n2 1 [ ,4 J(J+1)+I(I+1)Q2+c2 20 n 2 +3 n 2 +3 + c [ Q- 1 +4c j 21l I +1)-Q2+4]1/

[I(I+l)-Q2+â]1/2

(3 .15)

For dibaryons with strangeness S = -2 the isospin of the bound antikaons must be T =1, since the antikaons are bosons, and the rotational hamiltonian has the form of a 3 x 3 matrix . If the basis states are chosen aE 11, 13 ; J, J3 ; I3f= Q, Jbf = -nQ, T3 = 0) and 11, 13 ; J, J3 ; 13f = Q ~ l, J3f = - nQ, T3 =_-i:1), ordered according to increasing T3 values, the rotational energy is given by __ [ 4 3 J(J+l)+I(1+1)-n+3 Q2+c2 °` -2D n 2(Q-1)+c [21(I+1)-2Q(Q-1)]1/2 0 1 i2 +2[2I(I+1)-2Q(Q-1)] ~ c [2I(I+l)-2Q(Q+1)]'/2 0 [2I(I+1)-2Q(Q+1)]`i2 -2(Q+1)+c

Er

(3.16)

N. Dalarsson / Strange dibaryons

71 3

In (3.14)-(3 .16) the values of I and J are subject to constraints I , II3fl and J' IJ 3fI" For n = 2 the parity of the states is P = (-)Q and Q must be an integer. We now apply the quantum stabilization method due to MW, generalized to the baryon number n > 1 sector and regularized using the modified radial ansatz 6), to the energy of the non-strange dibaryon (3.14). Thus we minimize the energy of the Skyrme soliton only, and not the energy of the kaon-soliton system as a whole. The MW method gives the following expression for the moment of inertia (3.6) occurring in (3.14)-(3 .16)

1 13(4 )2 1 [ 4 b J(J+1)+I(I+1)- 4n2 4F, 2 ~r ab n 2 +3 n 2 +3 Q2 with a=2

b=

10 0

dx

1

dx

)

+2n(2n - 1)F2

Il

dx 3x2 sin'` F(x),

Il

3/4

(3.17)

(3-18) (3.19)

where x = r/ R is a dimensionless variable. Thus the rotational energies of dibaryons in the simplified CHK approach are given by eqs. (3.14)-(3.16), where the moment of inertia dl is given by eq. (3.17) together with eqs. (3 .18) and (3.19). The numerical results for E,,ot in MeV for some dibaryon states (n = 2) in the S = -1 and S = -2 sector, for N,, = 3 and using the rotational energy of the NN state in ref.') as the base value, are given in table l. The predicted dibaryon massses are in qualitative agreement with the values calculated with the complete CHK model (including the Skyrme stabilizing term proportional to e-2) in ref. 7). The states with n = 2 are classified and compared with the corresponding quark model states in ref. 7) and we do not discuss their classification in the present paper. The differences between our results and those given in ref.') can be explained by the considerable simplification of the employed model, where only the soliton energy (and not the energy of the kaon-soliton system as a whole) is minimized TABLE 1 The rotational energies of some strange dibaryons in MeV Dibaryon

I

S

E,o,

NN AN IN AA £A

0,1 i

0 -1 -1 -2 -2 -2

(147) 101 189 27 117 213

3 2

0 1 2

Erot

[ref. 7)] 147 90 165 33 108 183

71 4

N. Dalarsson / Strange dibaryons

and the dynamical term in the rotational perturbation is only evaluated to first order . s in ref. 7) the pion mass m.. was assumed to be zero. . Conclusions The present paper shows a possibility to use the Skyrme model, or more precisely model, for calculation ofthe dibaryon spectrum without use of the Skyrme the C stabilizing term, proportional to e -`', which makes the practical calculations very lengthy and tedious. The results are in qualitative agreement with those of the complete CHK model (with the stabilizing term included) . No attempt to calculate the rotational energy of NN state has been made. It was simply adopted as a base value. Furthermore, the pion mass was assumed to be zero (m,, = 0). e erences 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

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