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Exotic baryon number B=2 states in the SU(2) skyrme model
Volume 168B, number 4
PHYSICS LETTERS
13 March 1986
EXOTIC BARYON NUMBER B = 2 STATES IN T H E SU(2) S K Y R M E M O D E L H. W E I G E L , B. S C H W E S I N G E R and G. H O L Z W A R T H FB7 Physik, Universiti~t-Gesamthochschule Siegen, D-5900 Siegen, Fed. Rep. Germany R e c e i v e d 30 O c t o b e r 1985; revised m a n u s c r i p t received 24 D e c e m b e r 1985
A specific a n s a t z for h i g h e r b a r y o n n u m b e r s yields B = 2 c o n f i g u r a t i o n s w h i c h lead to b o u n d T = 0, j,o = 0 + a n d slightly u n b o u n d T = 0, JP = 1 + solitons a f t e r q u a n t i z a t i o n of the r o t a t i o n a l m o t i o n . D u e to the small size o f the r e s u l t i n g soliton it is difficult to i n t e r p r e t the T = 0, JP = I + s t a t e as a d e u t e r o n .
The complex non-linear structure o f the SU(2) Skyrme lagrangian has so far prohibited a complete analysis of different soliton solutions. Generally, the structure o f the chiral fields considered U = exp(i~ • n )
(1)
is closely related to the original hedgehog ansatz = ix(r)
(2)
found by Skyrme [1]. For baryon number B = 1 the hedgehog leads to the lowest possible mass but for B > 1 the situation is rather unclear. In the B = 2 sector the ans/itze considered were aimed to extract information on b a r y o n - b a r y o n interactions in the Skyrme model. So the following types o f field configurations have been examined: (i) Products o f two B = 1 hedgehogs centered at different points [ 1 - 4 ] . In this case the lowest possible mass amounts to twice the B = 1 mass when the two solitons are far apart. (ii) A B = 2 hedgehog configuration where the chiral angle X rotates twice as fast from X(~) = 0 to X(0) = 2Tr as all space is covered [1,2]. The minimum energy o f this configuration is as large as three B = 1 masses. (iii) General B = 2 configurations were obtained numerically [5,6] under the constraint o f axial symmetry. Here a parameter related to the "distance" between the two B = 1 "constituents" was held fixed. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
For small "distances" this configuration leads to an infinite repulsion of the solitons whereas for large distances twice the B = 1 mass results as lowest energy. Finally, in ref. [7] the extension of the SU(2) Skyrme model to SU(3) is treated and B = 2 configurations basically of type (i) are considered. Strangeness S = - 2 solitons are then shown to have energies as low as two B = 1 masses even for small "separations" of the two sofitons. Here we present a specific B = 2 ansatz where the additional winding number of the fields is obtained by doubling the twist in the isovector fields n rather than in the chiral angle X (case (ii)):
7t (r) = it 2(r)x2(r) ,
(3a)
with I cos(ntp) sin(0)'
it n(r, O, ~) = / sin(n~) sin(0) /
(cos(0)
(3b)
and Xn(oo) = O, Xn(O) = 7r. It is a simple exercise to verify that ansatz (3a), (3b) leads to a spherical baryon number density
B ° (r) = - ( 1 / 2 4 n 2 ) e o u v p X tr [(U~uU+)(U'dvU+)(U~ P U+)] = -(n/21r 2) [X'n(r)/r 2 ] sin2Xn (r),
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PHYSICS LETTERS
corresponding to total b a r y o n n u m b e r B = n. F r o m the Skyrme lagrangian
'E
13 March 1986
BO(x) 0.5
i1~\\
L = g1f n2 a[" tr(3uU+i)uU ) d3r
+
0.4.
,2f tr{[(~uU, avU) _ (avU, a.U) ]
X [C)uU+OvU) - (OvU~)uU)] } d3r
(5)
the equations o f m o t i o n for static soliton configurations ×n (A = 16e2 /(Xof~r) 2, x = r/xo) follow as
/ /',\
//,,\
0.3-
0.2-
" 2 XnXn" 0 = x 2Xn. . .+. z-XXn +~A(n 2 + 1 ) sin 0.1
- sin ×n cos ×n [(n2 + I) + (An2/x 2) sin2× n - - ~A (n 2
+ 1)(X~) 2]
.
(6)
The resulting chiral angle and the b a r y o n n u m b e r density o f the n = 2 solution are displayed in figs. 1 and 2. Most n o t a b l y , the a s y m p t o t i c behaviour o f the chiral angle at large distances follows a power law r - S w i t h s = ~ + ( n 2 + ~)1/2. This differs from the usual hedgehog behaviour r - 2 w h e n n > 1 b e c a u s e the pion fields are no longer pure dipole fields. F o r
Xo(x)
3.0
2.5
2.0
1,5
1.0
)
",...2 215
5
10
15
Fig. 1. The chiral angle Xn(X) for n = 1 (dashed line) and n : 2 (full line) as a function of x = r/xo. For the parameters of ref. [8] x o = 0.28 fm. 322
X
is
x
Fig. 2. The normalized baryon number density (l/n) B°n(x) for n = 1 (dashed line) and n = 2 (full line) as a function of x = r/xo. For the parameters of ref. [8] xo = 0.28 fm.
the same reason we find for small r: ×(r) = Ir - a r s - 1 This gives rise to a dip in the b a r y o n density at r = 0 w h e n n > 1 which is, however, hidden in fig. 2 because of an additional factor of 47rx 2 there. Since the chiral angle o f the n = 2 solution rises m o n o t o n o u s l y from zero to ~r the b a r y o n density resembles a situation where two B = 1 solitons sit on t o p o f each other. This differs from the o n i o n like configuration (ii) where one B = 1 soliton seems to be wrapped a r o u n d one located at the center (see refs. [5,6] for more details). Relative to the n = 1 case we observe an increase in size which for the parameters used here (those o f ref. [8]) results in a dilatation b y a factor o f roughly 1.6. The mass o f the resulting n = 2 soliton is slightly higher t h a n twice the n = 1 mass
Mn= 2 = 2 . 1 4 M n = 1 , 0.5
5.0
(7)
signalizing that the n = 2 soliton will disintegrate i n t o two separated n = 1 solitons. Since the energy difference o f these two configurations a m o u n t s to 14% o f the soliton mass o n l y it is conceivable that a n o n spherical n = 2 soliton, Xn = Xn( r, O, ~) can lead to b i n d i n g for n = 2. The results presented so far are n o t m u c h more
Volume 168B, number 4
PHYSICS LETTERS
than a mere curiosity because they might not bear any relevance to a two fermion system. Some of the B = 2 configurations (i)--(iii) cited earlier obviously are related to two fermions, In the remainder of this note we will investigate the kind of states which can be built from this n = 2 soliton. We do this b y projection onto states of good spin, isospin and parity from the n = 2 soliton via the semiclassical quantization first applied to the Skyrme soliton by Adkins, Nappi and Witten [8] To this end we introduce potentially timedependent isospin rotations I and rotations in ordinary space R such that
(r, R , 10 = lit 2(r')x2(r') ,
13 March 1986
Since the inertia _8123 1 ((Xn) 2 + [(n2+l)/x2]sin2Xn}) (gn-~nf~xo f ( 1 + ~A
X sin2XnX2dx ,
2 031~A f ~n4Xn dx , A n -_8~Trf~rX
(14)
are independent of the Euler angles parametrizing I and R the quantization of spin and isospin is straightforward giving Trot n=2 = ( ~ O 2 _ ~ A 2 ) - 1 j 2 + ( 2 ( 9 2 _ ~ A 2 ) - l T 2 + [(2(92 _ 8A2)-1 _ (2(9 2 _ ~A2)-1
r' = R - l r ,
(8)
represents a soliton configuration degenerate with the one having I = R = 1 in case of time independent rotations. The time dependence of the two rotations gives rise to kinetic energy contributions
Ln
T rot
= --n
(9)
-- Mn,
Trot = ~(gn [( 602 + 6°I) + (c°3 + n~23)2 n
-- ~-An [~(n2 + 3)(602 + 6o2) + n2(co 3 + n a 3 ) 2 + ~( n4 + 3)( ~ 2 + ~ i ) + 26n1(6°1~1 + 6°2122)] •
(10) The ~2i(coi) are the components of the angular velocities for rotations (isorotations) with respect to the body-fixed axes: (1-11)ab=--eabcCOc .
(11)
The spin j/bf and isospin T/bf components relative to the body-fixed axes are obtained via (see also ref. [9])
Tbf = aTrot/~coi.
(12)
Due to the special structure of the ansatz where a rotation about the 3-axis is equivalent to an isorotation about the same axis by n times the corresponding angleJ bf and T bf are constrained to J ~ f = - n T be .
Of course, body-fixed and laboratory spins J (isospins T) are related to each other by rotations J = R jbf (T = ITbf). The eigenfunctions of T rot may be given in terms of the rotation matrices [ 10] as
= {[(2T + 1 ) ( 2 / + 1)] 1/2/81r2}
+ ~(n 2 + 3 ) ( ~ 2 + ~ 2 ) + 2~n1(6o1~1 + 6o2~22)1
dbf= _~Trot/~2i,
(15)
(I, R ITK , JM, L )
with
(R-lk)ab=eabc~c,
-- (~(92 -- L~2A2 ) - 1 ] (Tbf)2.
X DTL(I)DJM_2L(R).
(16)
What is the parity of the states given in eq. (16)? The parity transformation on Uis performed by the operations
PU(r,t)=U+(-r,t)=exp[-i~'lt(--r,t)]
,
(17)
taking into account the pseudoscalar nature of the pion fields. For our special ansatz eq. (8) the parity transformation is identical to an isospin rotation around the body-fixed 3-axis by an angle of 7r
I3(zr) =
0
-1
0
0
=I~-l(rr).
(18)
Thus under parity transformations the collective wavefunction transforms according to
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PHYSICS LETTERS
P(I, R ITK, JM, L ) = {[(2T+ 1)(~
+ 1)] ~/2/8~r2)
X D T L ( I I ' 3 l(rr)) D J _ 2L(R) = e-irrL(I, R ITK,JM, L ) .
(19)
Integer spin and due to eq. (13) integer isospin are needed for physical eigenvalues P = -+1 o f parity. The same is required by the soliton quantization prescriptions of Finkelstein and Rubinstein [1 ] and o f Witten [12]. We are now ready to give the sequence o f states obtained from the quantized rotational motion o f the soliton in eq. (3) (table 1). The rotational energy eq. (15) is always positive and the lowest state possible has the quantum numbers T = O,J p = 0 +. In the Skyrme model the rotational energy o f two nucleons separated by a large distance is just half o f the energy difference between d e l t a - i s o b a r and nucleon. So comparing the energy o f the T = 0, JP = 0 + state above with the energy of two separated nucleons Mnucl we find
E(T--" 0 , J P = 0 +) - 2Mnucl =Mn= 2 - 2Mn= 1 - 3/(4On= 1 - 4 A n = l ) .
(20)
Numerically the last term is around - 1 5 0 MeV for parameter sets which reproduce the n u c l e o n - d e l t a mass splitting [2,8,9]. The difference Mn= 2 - 2Mn= l = 0.14Mn= 1 is below 150 MeV for Mn= 1 < 1 GeV indicating that
Table 1 Isospin T, spin, parity JP and energies (relative to two isolated nucleons) for the lowest rotational excitations of the B = 2 soliton. The parameters of reL [8] are used here. The asterisk indicates that the corresponding quantum numbers are not realized in the two-nucleon system. T
JP
E(T, JP) - 2 Mnucl
[MeV]
324
0 0
0+ * 1+
-30 46
1 1 1
0+ 1 +* 2-
90 166 183
13 March 1986
the T = O,J P = 0 + state is bound if the parameters are adjusted to give M n = 1 < 1 GeV and if zero.point oscillations d o not upset the whole picture. However, a T = 0 , J P = 0 + state is not accessible from the two-nucleon system which is constrained to (_)L+S+T = --1. Such a state can also n o t be constructed from six quarks in an s-orbit because the s p i n - i s o s p i n wavefunction ~ ST must be conjugate to the color wavefunction ~ C. mq ST does n o t contain S = 0, T = 0 [13]. The occurrence of this state here is due to the fact that the SU(2) Skyrme model allows for the B = 1 solitons also to be bosons [11,12] (any B = 2 configuration can be continuously deformed into a configuration where two B = 1 solitons are infinitely separated, and there the T = 0, jl' = 0 + state can only be constructed from bosons). The next rotational excitations in the sequence considered are accessible from the two-nucleon system. Numerically, with the parameters adopted here [8] the rotational energies follow I r ° t 2 = [60 MeV] T 2 + [38 M e V I J 2 - [135 MeV] (T~f) 2 .
(21)
76 MeV above the previous state, i.e. slightly unbound we find a T = O,J P = 1+ excitation which carries the quantum numbers of the deuteron. Clearly only minor changes in the parameters would suf. flee to make this state bound too. We have to remember, however, that the size of the object here is roughly 1.6 times the nucleon size whereas the deuteron RMS radius experimentally is 2.6 ¢r2~1/2 [14]. The " "nucl interpretation of this state as a deuteron therefore would be highly questionable. Another 44 MeV higher we find as next excitation a T = 1, JP = 0 + also accessible from the two-nucleon system. The difference in interaction energy between two nucleons with T = O,J P = 1+ relative to T = 1, J / ' = 0 + is, however, an order o f magnitude smaller than the energy difference obtained here. This once again points to the difficulty o f interpreting these states as two-nucleon states. Table 1 displays the five lowest rotational excitations. Notably, the first negative parity state only shows up at rather high energies. Due to P = exp(-i*rL) and the constraint j b f = _ 2 T b f it must have at least J=2. In conclusion we think we have presented a curious
Volume 168B, number 4
PHYSICS LETTERS
and exotic B = 2 soliton solution in the SU(2) Skyrme model. The resulting states might very well be artifacts o f the model, i.e. o f the large number of colors limit. However, the mere existence o f such solutions urges a cautious examination o f n u c l e o n - n u c l e o n interactions from the Skyrme model at short distances. This is so because some o f the small, almost bound states obtained here carry quantum numbers accessible from the two-nucleon system. The main problem in this respect seems to be the definition o f "distance" between two baryons in the Skyrme model once they have merged. Using the points where U = - 1 as proposed b y Skyrme [1] and as used in ref. [6] would imply, that the B = 2 configurations presented here involve two baryons at the same point. We are grateful to A. Hayashi for valuable help.
References [ 1 ] T.H .R. Skyrme, Proc. R. Soc. London 260 (1961) 127 ; Nucl. Phys. 31 (1962) 556.
13 March 1986
[2] A. Jackson, A.D. Jackson and V. Pasquier, Nucl. Phys. A432 (1984) 567. [3] R. Vinh Mau, M. Lacombe, B. Loiseau, W.N. Cottingham and P. Lisboa, Phys. Lett. 150B (1985) 259. [4] H. Yabu and K. Ando, Kyoto University preprint KUNS789. [5] M. Kutschera and C.J. Pethick, Nucl. Phys. A440 (1985) 670. [6] H.M. Sommermann, H.W. Wyld and C.J. Pethick, Phys. Rev. Lett. 55 (1985) 476. [7] A_P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers and A. Stern, Phys. Rev. Lett. 52 (1984) 887. [8] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [9] Ch. Hajduk and B. Schwesinger, Phys. Left. 145B (1984) 171. [10] A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975). [11 ] D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762. [12] E. Witten, Nucl. Phys. B223 (1983) 422. [13] M. Harvey, Nucl. Phys. A352 (1981) 301. [14] R.C. Barrett and D.F. Jackson, Nuclear sizes and nuclear structure (Clarendon, Oxford, 1977).
325
PHYSICS LETTERS
13 March 1986
EXOTIC BARYON NUMBER B = 2 STATES IN T H E SU(2) S K Y R M E M O D E L H. W E I G E L , B. S C H W E S I N G E R and G. H O L Z W A R T H FB7 Physik, Universiti~t-Gesamthochschule Siegen, D-5900 Siegen, Fed. Rep. Germany R e c e i v e d 30 O c t o b e r 1985; revised m a n u s c r i p t received 24 D e c e m b e r 1985
A specific a n s a t z for h i g h e r b a r y o n n u m b e r s yields B = 2 c o n f i g u r a t i o n s w h i c h lead to b o u n d T = 0, j,o = 0 + a n d slightly u n b o u n d T = 0, JP = 1 + solitons a f t e r q u a n t i z a t i o n of the r o t a t i o n a l m o t i o n . D u e to the small size o f the r e s u l t i n g soliton it is difficult to i n t e r p r e t the T = 0, JP = I + s t a t e as a d e u t e r o n .
The complex non-linear structure o f the SU(2) Skyrme lagrangian has so far prohibited a complete analysis of different soliton solutions. Generally, the structure o f the chiral fields considered U = exp(i~ • n )
(1)
is closely related to the original hedgehog ansatz = ix(r)
(2)
found by Skyrme [1]. For baryon number B = 1 the hedgehog leads to the lowest possible mass but for B > 1 the situation is rather unclear. In the B = 2 sector the ans/itze considered were aimed to extract information on b a r y o n - b a r y o n interactions in the Skyrme model. So the following types o f field configurations have been examined: (i) Products o f two B = 1 hedgehogs centered at different points [ 1 - 4 ] . In this case the lowest possible mass amounts to twice the B = 1 mass when the two solitons are far apart. (ii) A B = 2 hedgehog configuration where the chiral angle X rotates twice as fast from X(~) = 0 to X(0) = 2Tr as all space is covered [1,2]. The minimum energy o f this configuration is as large as three B = 1 masses. (iii) General B = 2 configurations were obtained numerically [5,6] under the constraint o f axial symmetry. Here a parameter related to the "distance" between the two B = 1 "constituents" was held fixed. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
For small "distances" this configuration leads to an infinite repulsion of the solitons whereas for large distances twice the B = 1 mass results as lowest energy. Finally, in ref. [7] the extension of the SU(2) Skyrme model to SU(3) is treated and B = 2 configurations basically of type (i) are considered. Strangeness S = - 2 solitons are then shown to have energies as low as two B = 1 masses even for small "separations" of the two sofitons. Here we present a specific B = 2 ansatz where the additional winding number of the fields is obtained by doubling the twist in the isovector fields n rather than in the chiral angle X (case (ii)):
7t (r) = it 2(r)x2(r) ,
(3a)
with I cos(ntp) sin(0)'
it n(r, O, ~) = / sin(n~) sin(0) /
(cos(0)
(3b)
and Xn(oo) = O, Xn(O) = 7r. It is a simple exercise to verify that ansatz (3a), (3b) leads to a spherical baryon number density
B ° (r) = - ( 1 / 2 4 n 2 ) e o u v p X tr [(U~uU+)(U'dvU+)(U~ P U+)] = -(n/21r 2) [X'n(r)/r 2 ] sin2Xn (r),
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Volume 168B, number 4
PHYSICS LETTERS
corresponding to total b a r y o n n u m b e r B = n. F r o m the Skyrme lagrangian
'E
13 March 1986
BO(x) 0.5
i1~\\
L = g1f n2 a[" tr(3uU+i)uU ) d3r
+
0.4.
,2f tr{[(~uU, avU) _ (avU, a.U) ]
X [C)uU+OvU) - (OvU~)uU)] } d3r
(5)
the equations o f m o t i o n for static soliton configurations ×n (A = 16e2 /(Xof~r) 2, x = r/xo) follow as
/ /',\
//,,\
0.3-
0.2-
" 2 XnXn" 0 = x 2Xn. . .+. z-XXn +~A(n 2 + 1 ) sin 0.1
- sin ×n cos ×n [(n2 + I) + (An2/x 2) sin2× n - - ~A (n 2
+ 1)(X~) 2]
.
(6)
The resulting chiral angle and the b a r y o n n u m b e r density o f the n = 2 solution are displayed in figs. 1 and 2. Most n o t a b l y , the a s y m p t o t i c behaviour o f the chiral angle at large distances follows a power law r - S w i t h s = ~ + ( n 2 + ~)1/2. This differs from the usual hedgehog behaviour r - 2 w h e n n > 1 b e c a u s e the pion fields are no longer pure dipole fields. F o r
Xo(x)
3.0
2.5
2.0
1,5
1.0
)
",...2 215
5
10
15
Fig. 1. The chiral angle Xn(X) for n = 1 (dashed line) and n : 2 (full line) as a function of x = r/xo. For the parameters of ref. [8] x o = 0.28 fm. 322
X
is
x
Fig. 2. The normalized baryon number density (l/n) B°n(x) for n = 1 (dashed line) and n = 2 (full line) as a function of x = r/xo. For the parameters of ref. [8] xo = 0.28 fm.
the same reason we find for small r: ×(r) = Ir - a r s - 1 This gives rise to a dip in the b a r y o n density at r = 0 w h e n n > 1 which is, however, hidden in fig. 2 because of an additional factor of 47rx 2 there. Since the chiral angle o f the n = 2 solution rises m o n o t o n o u s l y from zero to ~r the b a r y o n density resembles a situation where two B = 1 solitons sit on t o p o f each other. This differs from the o n i o n like configuration (ii) where one B = 1 soliton seems to be wrapped a r o u n d one located at the center (see refs. [5,6] for more details). Relative to the n = 1 case we observe an increase in size which for the parameters used here (those o f ref. [8]) results in a dilatation b y a factor o f roughly 1.6. The mass o f the resulting n = 2 soliton is slightly higher t h a n twice the n = 1 mass
Mn= 2 = 2 . 1 4 M n = 1 , 0.5
5.0
(7)
signalizing that the n = 2 soliton will disintegrate i n t o two separated n = 1 solitons. Since the energy difference o f these two configurations a m o u n t s to 14% o f the soliton mass o n l y it is conceivable that a n o n spherical n = 2 soliton, Xn = Xn( r, O, ~) can lead to b i n d i n g for n = 2. The results presented so far are n o t m u c h more
Volume 168B, number 4
PHYSICS LETTERS
than a mere curiosity because they might not bear any relevance to a two fermion system. Some of the B = 2 configurations (i)--(iii) cited earlier obviously are related to two fermions, In the remainder of this note we will investigate the kind of states which can be built from this n = 2 soliton. We do this b y projection onto states of good spin, isospin and parity from the n = 2 soliton via the semiclassical quantization first applied to the Skyrme soliton by Adkins, Nappi and Witten [8] To this end we introduce potentially timedependent isospin rotations I and rotations in ordinary space R such that
(r, R , 10 = lit 2(r')x2(r') ,
13 March 1986
Since the inertia _8123 1 ((Xn) 2 + [(n2+l)/x2]sin2Xn}) (gn-~nf~xo f ( 1 + ~A
X sin2XnX2dx ,
2 031~A f ~n4Xn dx , A n -_8~Trf~rX
(14)
are independent of the Euler angles parametrizing I and R the quantization of spin and isospin is straightforward giving Trot n=2 = ( ~ O 2 _ ~ A 2 ) - 1 j 2 + ( 2 ( 9 2 _ ~ A 2 ) - l T 2 + [(2(92 _ 8A2)-1 _ (2(9 2 _ ~A2)-1
r' = R - l r ,
(8)
represents a soliton configuration degenerate with the one having I = R = 1 in case of time independent rotations. The time dependence of the two rotations gives rise to kinetic energy contributions
Ln
T rot
= --n
(9)
-- Mn,
Trot = ~(gn [( 602 + 6°I) + (c°3 + n~23)2 n
-- ~-An [~(n2 + 3)(602 + 6o2) + n2(co 3 + n a 3 ) 2 + ~( n4 + 3)( ~ 2 + ~ i ) + 26n1(6°1~1 + 6°2122)] •
(10) The ~2i(coi) are the components of the angular velocities for rotations (isorotations) with respect to the body-fixed axes: (1-11)ab=--eabcCOc .
(11)
The spin j/bf and isospin T/bf components relative to the body-fixed axes are obtained via (see also ref. [9])
Tbf = aTrot/~coi.
(12)
Due to the special structure of the ansatz where a rotation about the 3-axis is equivalent to an isorotation about the same axis by n times the corresponding angleJ bf and T bf are constrained to J ~ f = - n T be .
Of course, body-fixed and laboratory spins J (isospins T) are related to each other by rotations J = R jbf (T = ITbf). The eigenfunctions of T rot may be given in terms of the rotation matrices [ 10] as
= {[(2T + 1 ) ( 2 / + 1)] 1/2/81r2}
+ ~(n 2 + 3 ) ( ~ 2 + ~ 2 ) + 2~n1(6o1~1 + 6o2~22)1
dbf= _~Trot/~2i,
(15)
(I, R ITK , JM, L )
with
(R-lk)ab=eabc~c,
-- (~(92 -- L~2A2 ) - 1 ] (Tbf)2.
X DTL(I)DJM_2L(R).
(16)
What is the parity of the states given in eq. (16)? The parity transformation on Uis performed by the operations
PU(r,t)=U+(-r,t)=exp[-i~'lt(--r,t)]
,
(17)
taking into account the pseudoscalar nature of the pion fields. For our special ansatz eq. (8) the parity transformation is identical to an isospin rotation around the body-fixed 3-axis by an angle of 7r
I3(zr) =
0
-1
0
0
=I~-l(rr).
(18)
Thus under parity transformations the collective wavefunction transforms according to
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PHYSICS LETTERS
P(I, R ITK, JM, L ) = {[(2T+ 1)(~
+ 1)] ~/2/8~r2)
X D T L ( I I ' 3 l(rr)) D J _ 2L(R) = e-irrL(I, R ITK,JM, L ) .
(19)
Integer spin and due to eq. (13) integer isospin are needed for physical eigenvalues P = -+1 o f parity. The same is required by the soliton quantization prescriptions of Finkelstein and Rubinstein [1 ] and o f Witten [12]. We are now ready to give the sequence o f states obtained from the quantized rotational motion o f the soliton in eq. (3) (table 1). The rotational energy eq. (15) is always positive and the lowest state possible has the quantum numbers T = O,J p = 0 +. In the Skyrme model the rotational energy o f two nucleons separated by a large distance is just half o f the energy difference between d e l t a - i s o b a r and nucleon. So comparing the energy o f the T = 0, JP = 0 + state above with the energy of two separated nucleons Mnucl we find
E(T--" 0 , J P = 0 +) - 2Mnucl =Mn= 2 - 2Mn= 1 - 3/(4On= 1 - 4 A n = l ) .
(20)
Numerically the last term is around - 1 5 0 MeV for parameter sets which reproduce the n u c l e o n - d e l t a mass splitting [2,8,9]. The difference Mn= 2 - 2Mn= l = 0.14Mn= 1 is below 150 MeV for Mn= 1 < 1 GeV indicating that
Table 1 Isospin T, spin, parity JP and energies (relative to two isolated nucleons) for the lowest rotational excitations of the B = 2 soliton. The parameters of reL [8] are used here. The asterisk indicates that the corresponding quantum numbers are not realized in the two-nucleon system. T
JP
E(T, JP) - 2 Mnucl
[MeV]
324
0 0
0+ * 1+
-30 46
1 1 1
0+ 1 +* 2-
90 166 183
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the T = O,J P = 0 + state is bound if the parameters are adjusted to give M n = 1 < 1 GeV and if zero.point oscillations d o not upset the whole picture. However, a T = 0 , J P = 0 + state is not accessible from the two-nucleon system which is constrained to (_)L+S+T = --1. Such a state can also n o t be constructed from six quarks in an s-orbit because the s p i n - i s o s p i n wavefunction ~ ST must be conjugate to the color wavefunction ~ C. mq ST does n o t contain S = 0, T = 0 [13]. The occurrence of this state here is due to the fact that the SU(2) Skyrme model allows for the B = 1 solitons also to be bosons [11,12] (any B = 2 configuration can be continuously deformed into a configuration where two B = 1 solitons are infinitely separated, and there the T = 0, jl' = 0 + state can only be constructed from bosons). The next rotational excitations in the sequence considered are accessible from the two-nucleon system. Numerically, with the parameters adopted here [8] the rotational energies follow I r ° t 2 = [60 MeV] T 2 + [38 M e V I J 2 - [135 MeV] (T~f) 2 .
(21)
76 MeV above the previous state, i.e. slightly unbound we find a T = O,J P = 1+ excitation which carries the quantum numbers of the deuteron. Clearly only minor changes in the parameters would suf. flee to make this state bound too. We have to remember, however, that the size of the object here is roughly 1.6 times the nucleon size whereas the deuteron RMS radius experimentally is 2.6 ¢r2~1/2 [14]. The " "nucl interpretation of this state as a deuteron therefore would be highly questionable. Another 44 MeV higher we find as next excitation a T = 1, JP = 0 + also accessible from the two-nucleon system. The difference in interaction energy between two nucleons with T = O,J P = 1+ relative to T = 1, J / ' = 0 + is, however, an order o f magnitude smaller than the energy difference obtained here. This once again points to the difficulty o f interpreting these states as two-nucleon states. Table 1 displays the five lowest rotational excitations. Notably, the first negative parity state only shows up at rather high energies. Due to P = exp(-i*rL) and the constraint j b f = _ 2 T b f it must have at least J=2. In conclusion we think we have presented a curious
Volume 168B, number 4
PHYSICS LETTERS
and exotic B = 2 soliton solution in the SU(2) Skyrme model. The resulting states might very well be artifacts o f the model, i.e. o f the large number of colors limit. However, the mere existence o f such solutions urges a cautious examination o f n u c l e o n - n u c l e o n interactions from the Skyrme model at short distances. This is so because some o f the small, almost bound states obtained here carry quantum numbers accessible from the two-nucleon system. The main problem in this respect seems to be the definition o f "distance" between two baryons in the Skyrme model once they have merged. Using the points where U = - 1 as proposed b y Skyrme [1] and as used in ref. [6] would imply, that the B = 2 configurations presented here involve two baryons at the same point. We are grateful to A. Hayashi for valuable help.
References [ 1 ] T.H .R. Skyrme, Proc. R. Soc. London 260 (1961) 127 ; Nucl. Phys. 31 (1962) 556.
13 March 1986
[2] A. Jackson, A.D. Jackson and V. Pasquier, Nucl. Phys. A432 (1984) 567. [3] R. Vinh Mau, M. Lacombe, B. Loiseau, W.N. Cottingham and P. Lisboa, Phys. Lett. 150B (1985) 259. [4] H. Yabu and K. Ando, Kyoto University preprint KUNS789. [5] M. Kutschera and C.J. Pethick, Nucl. Phys. A440 (1985) 670. [6] H.M. Sommermann, H.W. Wyld and C.J. Pethick, Phys. Rev. Lett. 55 (1985) 476. [7] A_P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers and A. Stern, Phys. Rev. Lett. 52 (1984) 887. [8] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B228 (1983) 552. [9] Ch. Hajduk and B. Schwesinger, Phys. Left. 145B (1984) 171. [10] A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975). [11 ] D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762. [12] E. Witten, Nucl. Phys. B223 (1983) 422. [13] M. Harvey, Nucl. Phys. A352 (1981) 301. [14] R.C. Barrett and D.F. Jackson, Nuclear sizes and nuclear structure (Clarendon, Oxford, 1977).
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