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Excited hyperons in the skyrme model
Volume 227, number 3,4
PHYSICS LETTERS B
31 August 1989
EXCITED HYPERONS IN THE SKYRME MODEL K. D A N N B O M , E.M. N Y M A N and D.O. R I S K A Department of Physics, University of Helsinki, SF-O0170 Helsinki, Finland Received 30 May 1989
In the Callan-Klebanov model, hyperons are bound states of skyrmeons and kaons. The model predicts, in addition to the wellestablished octet and decuplet, a definite set of excited states, among them the experimentally established A ( 1405 ), which have no analogues in the non-strange sector. Experimental confirmations of the remaining ones of these states would establish the bound-state model as the correct approach towards hyperon spectroscopy. In its simplest version the model also predicts some exotic states. These may however disappear once the repulsive interactions between the kaons are taken into account.
A m o n g observed hyperon resonances, there are several cases where experimental determination of the quantum numbers has not yet been achieved. In these cases systematic theoretical studies are important. A great deal o f theoretical work in this field has been based on generalized versions of the M I T bag model, incorporating effective two-body interactions between the quarks [ 1 - 5 ] . The C a l l a n - K l e b a n o v model, in which the hyperons are f o r m e d as b o u n d states of kaons and non-strange skyrmeons [6,7] represents a different d y n a m i c a l model, which is a natural generalization o f the topological soliton ( S k y r m e ) model for the baryons [8] to the hyperons. The C a l l a n - K l e b a n o v model and its vector meson generalizations [ 9-11 ] have already been shown to give a good description o f the stable strange hyperons [ 7,10-12 ]. It also leads to very reasonable magnetic m o m e n t s and coupling constants [ 13,14 ]. Although the structures o f strange and nonstrange baryons are totally different, the static properties o f the stable particles are nevertheless found to approximately fulfil SU ( 3 ) and/or quark-model relationships. The most interesting aspects o f the C K model are those predictions which are not even a p p r o x i m a t e l y the same as in SU (3) or the quark model with three quarks. Some o f these are consequences o f the fact that the C K model corresponds to the limit o f an infinite n u m b e r o f colours, and care is required in order not to confuse these with genuine predictions o f 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
the theory. The model actually predicts, at least in its simplest form, some exotic stable particles, which cannot be constructed out o f three quarks. The exotic states occur in the least reliable sectors o f the model, however, and would probably not be b o u n d if all correction terms were i n c o r p o r a t e d into the calculations. On the other hand, resonances where the kaon is in an excited orbital state do exist and lack nonstrange analogues altogether. As these particles do not form the customary S U ( 3 ) group representations (e.g. octets a n d decuplets), they are clearly the most interesting ones both theoretically and experimentally. The quantized collective rotation o f the system transforms the isospin o f the b o u n d kaons into spin. Thus, the C K model is counterintuitive in that the kaons, although bosons, at first a p p e a r to contribute half-integer spin and no isospin to the b o u n d state. The spin o f the soliton must equal its isospin and is quantized to integer or half-integer values d e p e n d i n g on the n u m b e r of b o u n d kaons. Thus, the quantization condition for the rotational state o f the skyrmeon depends on the n u m b e r o f kaons, such that the c o m b i n e d contribution due to the presence o f an extra kaon in the end is the expected one. Strangeness is carried by the kaons alone, while all isospin is transfered to the rotational motion o f the system. The spin o f the hyperon is formed as the sum o f the total angular m o m e n t u m o f the kaons and the spin o f the soliton. 291
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PHYSICS LETTERS B
In the simplest version of the model, kaon-kaon interactions are ignored along with a number of other complications. In this approximation, any number of kaons could be bound to the same skyrmeon. The complete model has, however, a built-in kaon-kaon repulsion, which limits the number of bound kaons. A simple estimate of this interaction between the kaons indicate that at most three kaons will be bound, the K - K repulsion preventing the binding of a fourth one (see below). Thus, the possible values of strangeness will be the same as in the quark model. The accuracy of the model is believed to deteriorate - presumably due to finite-No effects - towards high angular momenta, such that I=J=~ is the largest spin-isospin value one expects to be observed (the A3~-resonance). States with isospin above ~3 are consequences of the rigid-rotator and large-N~, approximations. These states could be eliminated already by improving upon the rigid-rotor approximation [ 15 ]. It appears reasonable to assume that centrifugal and Coriolis effects (which are ignored) would be strong enough to prevent the binding of kaons to the highest existing rotational state. There are in the model two possible bound states for the kaon [6,7] (and perhaps some resonances). As a consequence of the Wess-Zumino interaction, the lowest state occurs in the P-wave, the other bound state being an S-state with an excitation energy of some 200 MeV. The model thus implies an excitation spectrum of an atomic type. Low-lying states of the system are obtained by combining the angular momentum of the kaon or kaons and the possible angular momenta of the soliton in all possible ways. Excited states are those in which one or more of the kaons are in the S-state. These occur in groups which are split by the rotational energy. The A(1405) is identified with a member of such a group, and thus has no analogue in the nonstrange sector. This is the most striking new aspect of the model, distinguishing it from all conventional approaches, where the A ( 1405 ) obviously would have to be a member of an SU (3) representation which also would contain nonstrange particles. The expression for the energy of a hyperon in the bound-state model is the following [ 7 ]:
292
E(np,
ns, J, I,
31 August 1989
JK)M+npcop + nscos
1
+ ~ [cJ(J+ l )+ (1-c)I(I+ l ) +C(C--1)JK(JK+I)]
.
(1)
Here np is the number of kaons in the ground state (the P-state) and ns the number in the S-state. The energies of these states are denoted cop and COs, respectively. The total strangeness S of the hyperon is obtained from the total number of kaons, np + ns = ISI. In ( 1 ) J is the total spin of the hyperon, I the isospin and JK the combined angular momentum of the kaons. For those hyperons in which all kaons are in the same orbital state one has JK = ½ISI due to the requirement of total permutation symmetry of the kaonic state. In eq. ( 1 ) M denotes the mass of the SU (2)-soliton and £2 its moment of inertia, while c is a linear combination of radial matrix elements which depends on the kaon numbers np and ns. If the spinand isospin projection of the soliton is achieved by means of the usual adiabatic rigid rotation [ 16 ], the mass and moment of inertia can be determined from the masses of the nucleon and A33 resonances and one has M=
5raN --ma 4 '
f2=
3 2(ma --mN) "
(2)
From the empirical mass values one obtains M = 866 MeV and f2=0.0051 MeV-~. For a single kaon in the P-state the radial matrix element c in eq. ( 1 ) is [ 11,17 ] f Cp = 1 - - -8COp ~-
drrZlkp(r)
X{COS20[I--[--
12
4~(0'2q-e2 F ~
3 2e2F~[O,2f4 ~,
sin20~l "-7-,,1_]
c°s-~05 - 1)
+sin 0 ( 0 " + ~ ) ] }
.
(3)
Here kv(r) is the radial wavefunction, given in ref. [11 ]. The function O(r) is the usual chiral profile function for the skyrmeon and F~ and e are the usual
Volume 227, number 3,4
PHYSICS LETTERS B
ward by a phenomenological constant of 73 MeV. This adjusts the energy of the A ground state to its empirical value and facilitates the comparison of the theoretical rotational splittings to the empirical ones. We shall return to a discussion of various mechanisms which might generate such a shift below. Considering first the splittings of the S = - 1 states, which in fact have been studied before [ 7,10-12 ], we observe that those states where the kaon is bound in the P-state are positive-parity states which are identified with the ground-state (octet) A and Z particles as well as the Z ( 1385 ) which belongs to the spin 53 decuplet. The splittings between these S = - 1 states are of rotational origin and agree well with the empirical values. When the kaon is bound in the S-state one obtains a set of excited negative-parity states. In the absence of all rotational splittings these states would, using the parameters given above, all be 210 MeV higher than the ground state. As the c-parameter is larger for the S-state kaons, the rotational splittings are slightly different. The lowest state is a ½- state with no isospin. It lies below the I~N threshold and is identified with the A ( 1 4 0 5 ) . The mass is in good agreement,
parameters in the lagrangean density of the Skyrme model [ 16 ]. For a kaon in the S-state the radial matrix c in ( 1 ) is obtained from a slightly different formula [ 11 ]: ~4O9s f d r r 2 l k s ( r ) l 2
cs=l-
0 4 (0,2 + sin 2 0"]] × {2 sin2 5 [ 1 + e--~z ~ ] J 3
e2F~
[0'2(4sin20-1)
__ sin 0 ( 0 " + ~ ) 1 }
.
31 August 1989
(4,
Here ks (r) is the radial wavefunction of the S-state. In table 1 we show the hyperon spectrum for S = - 1, - 2, - 3. Only states with the usual octet and decuplet quantum numbers have been included. We shall discuss the exotic states separately below. The energy values in table 1 were obtained from eq. ( 1 ) with o)p=150 MeV, ~Os=360 MeV, cp=0.6 and Cs=0.9 [11 ]. As we are mainly concerned with the energy splittings, the spectrum has been shifted up-
Table 1 Spectrum of hyperon resonances with nonexotic quantum numbers in the bound state model. Masses are given in MeV/c 2. S
np
ns
Ix
I
IP
E+ 73
Exp.
-1
1 1
0 0
½ ½
0 1
1 0
0 l
½ ~
1 0
½+ ½+ ~+ ~-
1116 1196 1376 1360
A(Il16) Z(1193) Z(1385) A(1405)
0
1
½
1
½
1380
0
1
½
1
~-
1664
Z(1670)
2 2 1 1
0 0 1 1
1 1 0 1
½ ~ ½ ½
½+ ~+ ½½-
E(1318) E(1530)
~(1672)
--2
-3
1
1
1
½
~
0 0
2 2
1 1
½ ½
~+ 3+
1266 1446 1524 1486 1711 1716 1986
3 2 2 1 1 0
0 1 1 2 2 3
~ ½ ~ ½ ~ ~
0 0 0 0 0 0
~+ ½3½+ ~+ ~-
1449 1618 1693 1789 1976 2023
~(1820)
293
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the calculated value being low by only 40 MeV. This state does not fit into the quark model, and calculations typically obtain masses which are too high by several hundred MeV. Further, no corresponding lowlying ½- nucleon resonances have been seen. According to the CK model, the A (1405) belongs to a set of negative-parity states with strangeness at least - 1. As seen in table 1, the other ones are ½- and 3-states with unit isospin. The lowest experimentally well-established negative-parity state with I = 1 is a ~ state at 1670 MeV, in good agreement with the prediction o f the present theory. There are some candidates below this energy, however, and one of them, perhaps the Z (1480), will now have to be a ½state, to be identified with the state listed at 1380 MeV in table 1. While the existence of the A ( 1 4 0 5 ) is already a major success of the theory, the experimental confirmation of its isospin- 1 analogues at somewhat larger masses is a crucial test of the theory. In the Skyrme model, the only states in the S = 0 sector are the nucleon and the A. This agrees with experiments in that a low-energy region below some 500 MeV excitation energy is indeed free of excited states. For S = - 1 this means that the states in table 1 should be the only ones below about 1600 MeV. (The ~state observed at 1520 MeV can probably be accounted for as a resonance in the C K model. ) The lowest S = - 2 two-kaon states in the model provide a good description o f the cascade particles and the lowest cascade resonances -~ ( 1530 ). The calculated energies are near the experimental ones, on the low side. As above, these states are predicted to have partners with the opposite parity, obtained by putting one kaon in the S-state. As the two kaons are now in different spatial states, the requirement of symmetry is less stringent, and theory thus, is in this case predicts an extra ½- state. Experimentally, the only well-established ~, resonance with negative parity is the ~- ~ ( 1 8 2 0 ) resonance. A corresponding state is predicted in table 1 at 1711 MeV, again in resonable agreement, but slightly on the low side. While no ½- resonance has yet been established, the E ( 1 6 9 0 ) and perhaps the -=(1620) resonances may correspond to the ones predicted in this channel. When both kaons are in the S-state a group of higher excited states are formed, where the combinatorics and quantum numbers are the same as in the ground state. One therefore obtains a repetition of the earlier 294
31 August 1989
results at a higher energy. Candidates exist at approximately the right masses, e.g., the cascade resonances ~ ( 1 9 5 0 ) and E ( 2 0 3 0 ) , but their quantum numbers are not known. In this energy region one expects intruder states which are consequences of the structure of the skyrmeon itself. In the S = - 3 channel, the g2 emerges at an energy which, continuing the trend, is lower than the measured value. Excited states of the ~2- are formed by raising either one or more kaons to the S state. It is perhaps not surprising that the low-lying spectrum is the same as for a bag containing three strange quarks (see ref. [17] ). The third and fourth excited states in our table have the opposite ordering compared to the bag-model calculation. Experimentally, very little is known about the S = - 3 states. One might identify the ~ (2250) with one of the predicted states, such as the ~3+ state calculated at 1976 MeV. A structure has also been seen at 2380 MeV [ 18 ]. In the spectrum in table 1 the energy levels of the S = - 1 one kaon states are well fit, while the calculated masses of S = - 2 two-kaon states are low by amounts in the 50-100 MeV range. The mass of the three-kaon ground state comes out low by about 220 MeV. This behaviour signals the presence of a repulsive interaction between the kaons. Simple perturbative estimate of terms o f fourth order in the kaon field give extra energies of the order of 50 MeV S = - 2 . For S = - 3 the repulsion would be three times larger, while a fourth kaon would no longer be bound. If these estimated turn out to be quantitatively accurate, the entire spectrum would be in a very good agreement with experiments. As pointed out in ref. [ 7 ], terms of fourth order in the kaon field suffer from operator ordering ambiguities. Classically, such contributions are well defined, however, and it would obviously be within the spirit o f the model to use estimates based on classical approximations to the kaon fields. If this is done, the fourth-order terms give repulsive contributions to the self-energies of the kaons and would account for some (but not all) of the 73 MeV shift of the calculated energy levels in table 1. There is also a contribution, quadratic in the angular velocity of the soliton, which can approximately be incorporated as a reduction of the m o m e n t of inertia. For the S-state this term was evaluated exactly in ref. [ 11 ] and found to decrease the moment of inertia by ( 12-14) % for the value (2).
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PHYSICS LETTERS B
The consequent upward shift of the S-state is 13-16 MeV. Although these corrections are not sufficient to account for all the discrepancy between the calculated and observed A-mass, the remaining error is well within the accuracy expected of the model. The spectrum in table 1 contains only those states which satisfy the r u l e / + ½[SI ~ ~. These are the states that can be formed of three quarks with the up, down and strange flavours. The spectrum produced by eq. ( 1 ) does, however, also contain exotic states which do not satisfy the SU (3) multiplet rule. No exotic ground-state particles are generated for S = - 1. For S = - 2 one could obtain an exotic state with I = ~_, 3 corresponding to a b o u n d state of the Aresonance and two kaons (the selection rule forbids the b i n d i n g of a single kaon to this state). We argued above that such a state is actually outside the range of applicability of the model. For S = - 3 there exists the possibility of b i n d i n g the kaons to I = 1 soliton. Taking eq. ( 1 ) seriously, this state would actually lie below the f~-. As it would only decay by weak interactions, such a particle would certainly have been seen, and we need arguments why it need not exist. Considering first corrections within the model, we observe that the energy of the exotic I = 1, S = - 3 state would be increased, without affecting the predicted energies of the non-exotic states, by an isospin-dep e n d e n t three-body interaction between the kaons and the skyrmeon. Such an interaction is present in the model and might remove this problem. It would also reduce the b i n d i n g of two kaons to the isospin 3 skyrmeon m e n t i o n e d above. Finally, we have already reported estimates which indicate that states with ISI > 3 are not bound. As pointed out, there exist with the model reasons to assume that the exotic states are at higher energies than those given by eq. ( 1 ). All of these require going beyond the customary second order in the expansion of the kaon fields. One should remember, however, that the exotic states could simply be artifacts of the
31 August 1989
large-No expansion. It is not clear that the breakdown of this approximation necessarily has precursors in the form of excessive correction terms in the orders considered in the present work. The crucial test of the C a l l a n - K l e b a n o v b o u n d state model for the hyperons will be the experimental confirmation of the remaining ones of the predicted negative-parity states.
References [ 1] N. Isgur and G. Karl, Phys. Rev. D 19 (1979) 2653. [2] K.T. Chao, N. Isgur and G. Karl, Phys. Rev. D 23 ( 1981 ) 155. [3] J.M. Richard, Phys. Len. B 100 ( 1981 ) 515. [4] A.T. Aerts and L. Heller, Phys. Rev. D 23 ( 1981 ) 185. [5] A.T. Aerts and L. Heller, Phys. Rev. D 25 (1982) 1365. [6] C.G. Callanand I. Klebanov,Nucl. Phys. B 262 (1985) 365. [7] C.G. Callan, K. Hornbostel and I. Klebanov, Phys. Lett. B 202 (1988) 269. [8] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127. [9] N.N. Scoccola, H. Nadeau, M.A. Nowak and M. Rho, Phys. Lett. B 201 (1988) 425. [ 10] J.P. Blaizot, M. Rho and N.N. Scoccola, Phys. Len. B 209 (1988) 27. [ 11 ] K. Dannbom, U. Biota and D.O. Riska, The hyperons as bound states in the Skyrme model, University of Helsinki preprint HU-TFT-88-42 ( 1988). [12]N.N. Scoccola, D.P. Min, H. Nadeau and M. Rho, The strangeness problem: an SU(3) skyrmion with vector mesons, Universit~it Regensburg preprint TPR-89-10 (1989). [13] E.M. Nyman and D.O. Riska, Magnetic moments of the hyperons in the topological soliton model, University of Helsinki preprint HU-TFT-88-50 ( 1988). [ 14] E.M. Nyman, Phys. Len. B 224 (1989) 21. [ 15 ] J.P. Blaizotand G. Ripka, Non-lineardistortions of rotating chiral solitons, Saclay preprint SphT/87-181 ( 1988). [ 16] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [ 17 ] J. Kunz and P.J. Mulders, Bound state approach for strange dibaryons in the Skyrme model, preprint NIKHEF-P-15 (1988). [ 18] A.T. Aerts and L. Heller, Los Alamos National Laboratory preprint LA-UR-88-3353 ( 1988).
295
PHYSICS LETTERS B
31 August 1989
EXCITED HYPERONS IN THE SKYRME MODEL K. D A N N B O M , E.M. N Y M A N and D.O. R I S K A Department of Physics, University of Helsinki, SF-O0170 Helsinki, Finland Received 30 May 1989
In the Callan-Klebanov model, hyperons are bound states of skyrmeons and kaons. The model predicts, in addition to the wellestablished octet and decuplet, a definite set of excited states, among them the experimentally established A ( 1405 ), which have no analogues in the non-strange sector. Experimental confirmations of the remaining ones of these states would establish the bound-state model as the correct approach towards hyperon spectroscopy. In its simplest version the model also predicts some exotic states. These may however disappear once the repulsive interactions between the kaons are taken into account.
A m o n g observed hyperon resonances, there are several cases where experimental determination of the quantum numbers has not yet been achieved. In these cases systematic theoretical studies are important. A great deal o f theoretical work in this field has been based on generalized versions of the M I T bag model, incorporating effective two-body interactions between the quarks [ 1 - 5 ] . The C a l l a n - K l e b a n o v model, in which the hyperons are f o r m e d as b o u n d states of kaons and non-strange skyrmeons [6,7] represents a different d y n a m i c a l model, which is a natural generalization o f the topological soliton ( S k y r m e ) model for the baryons [8] to the hyperons. The C a l l a n - K l e b a n o v model and its vector meson generalizations [ 9-11 ] have already been shown to give a good description o f the stable strange hyperons [ 7,10-12 ]. It also leads to very reasonable magnetic m o m e n t s and coupling constants [ 13,14 ]. Although the structures o f strange and nonstrange baryons are totally different, the static properties o f the stable particles are nevertheless found to approximately fulfil SU ( 3 ) and/or quark-model relationships. The most interesting aspects o f the C K model are those predictions which are not even a p p r o x i m a t e l y the same as in SU (3) or the quark model with three quarks. Some o f these are consequences o f the fact that the C K model corresponds to the limit o f an infinite n u m b e r o f colours, and care is required in order not to confuse these with genuine predictions o f 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
the theory. The model actually predicts, at least in its simplest form, some exotic stable particles, which cannot be constructed out o f three quarks. The exotic states occur in the least reliable sectors o f the model, however, and would probably not be b o u n d if all correction terms were i n c o r p o r a t e d into the calculations. On the other hand, resonances where the kaon is in an excited orbital state do exist and lack nonstrange analogues altogether. As these particles do not form the customary S U ( 3 ) group representations (e.g. octets a n d decuplets), they are clearly the most interesting ones both theoretically and experimentally. The quantized collective rotation o f the system transforms the isospin o f the b o u n d kaons into spin. Thus, the C K model is counterintuitive in that the kaons, although bosons, at first a p p e a r to contribute half-integer spin and no isospin to the b o u n d state. The spin o f the soliton must equal its isospin and is quantized to integer or half-integer values d e p e n d i n g on the n u m b e r of b o u n d kaons. Thus, the quantization condition for the rotational state o f the skyrmeon depends on the n u m b e r o f kaons, such that the c o m b i n e d contribution due to the presence o f an extra kaon in the end is the expected one. Strangeness is carried by the kaons alone, while all isospin is transfered to the rotational motion o f the system. The spin o f the hyperon is formed as the sum o f the total angular m o m e n t u m o f the kaons and the spin o f the soliton. 291
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In the simplest version of the model, kaon-kaon interactions are ignored along with a number of other complications. In this approximation, any number of kaons could be bound to the same skyrmeon. The complete model has, however, a built-in kaon-kaon repulsion, which limits the number of bound kaons. A simple estimate of this interaction between the kaons indicate that at most three kaons will be bound, the K - K repulsion preventing the binding of a fourth one (see below). Thus, the possible values of strangeness will be the same as in the quark model. The accuracy of the model is believed to deteriorate - presumably due to finite-No effects - towards high angular momenta, such that I=J=~ is the largest spin-isospin value one expects to be observed (the A3~-resonance). States with isospin above ~3 are consequences of the rigid-rotator and large-N~, approximations. These states could be eliminated already by improving upon the rigid-rotor approximation [ 15 ]. It appears reasonable to assume that centrifugal and Coriolis effects (which are ignored) would be strong enough to prevent the binding of kaons to the highest existing rotational state. There are in the model two possible bound states for the kaon [6,7] (and perhaps some resonances). As a consequence of the Wess-Zumino interaction, the lowest state occurs in the P-wave, the other bound state being an S-state with an excitation energy of some 200 MeV. The model thus implies an excitation spectrum of an atomic type. Low-lying states of the system are obtained by combining the angular momentum of the kaon or kaons and the possible angular momenta of the soliton in all possible ways. Excited states are those in which one or more of the kaons are in the S-state. These occur in groups which are split by the rotational energy. The A(1405) is identified with a member of such a group, and thus has no analogue in the nonstrange sector. This is the most striking new aspect of the model, distinguishing it from all conventional approaches, where the A ( 1405 ) obviously would have to be a member of an SU (3) representation which also would contain nonstrange particles. The expression for the energy of a hyperon in the bound-state model is the following [ 7 ]:
292
E(np,
ns, J, I,
31 August 1989
JK)M+npcop + nscos
1
+ ~ [cJ(J+ l )+ (1-c)I(I+ l ) +C(C--1)JK(JK+I)]
.
(1)
Here np is the number of kaons in the ground state (the P-state) and ns the number in the S-state. The energies of these states are denoted cop and COs, respectively. The total strangeness S of the hyperon is obtained from the total number of kaons, np + ns = ISI. In ( 1 ) J is the total spin of the hyperon, I the isospin and JK the combined angular momentum of the kaons. For those hyperons in which all kaons are in the same orbital state one has JK = ½ISI due to the requirement of total permutation symmetry of the kaonic state. In eq. ( 1 ) M denotes the mass of the SU (2)-soliton and £2 its moment of inertia, while c is a linear combination of radial matrix elements which depends on the kaon numbers np and ns. If the spinand isospin projection of the soliton is achieved by means of the usual adiabatic rigid rotation [ 16 ], the mass and moment of inertia can be determined from the masses of the nucleon and A33 resonances and one has M=
5raN --ma 4 '
f2=
3 2(ma --mN) "
(2)
From the empirical mass values one obtains M = 866 MeV and f2=0.0051 MeV-~. For a single kaon in the P-state the radial matrix element c in eq. ( 1 ) is [ 11,17 ] f Cp = 1 - - -8COp ~-
drrZlkp(r)
X{COS20[I--[--
12
4~(0'2q-e2 F ~
3 2e2F~[O,2f4 ~,
sin20~l "-7-,,1_]
c°s-~05 - 1)
+sin 0 ( 0 " + ~ ) ] }
.
(3)
Here kv(r) is the radial wavefunction, given in ref. [11 ]. The function O(r) is the usual chiral profile function for the skyrmeon and F~ and e are the usual
Volume 227, number 3,4
PHYSICS LETTERS B
ward by a phenomenological constant of 73 MeV. This adjusts the energy of the A ground state to its empirical value and facilitates the comparison of the theoretical rotational splittings to the empirical ones. We shall return to a discussion of various mechanisms which might generate such a shift below. Considering first the splittings of the S = - 1 states, which in fact have been studied before [ 7,10-12 ], we observe that those states where the kaon is bound in the P-state are positive-parity states which are identified with the ground-state (octet) A and Z particles as well as the Z ( 1385 ) which belongs to the spin 53 decuplet. The splittings between these S = - 1 states are of rotational origin and agree well with the empirical values. When the kaon is bound in the S-state one obtains a set of excited negative-parity states. In the absence of all rotational splittings these states would, using the parameters given above, all be 210 MeV higher than the ground state. As the c-parameter is larger for the S-state kaons, the rotational splittings are slightly different. The lowest state is a ½- state with no isospin. It lies below the I~N threshold and is identified with the A ( 1 4 0 5 ) . The mass is in good agreement,
parameters in the lagrangean density of the Skyrme model [ 16 ]. For a kaon in the S-state the radial matrix c in ( 1 ) is obtained from a slightly different formula [ 11 ]: ~4O9s f d r r 2 l k s ( r ) l 2
cs=l-
0 4 (0,2 + sin 2 0"]] × {2 sin2 5 [ 1 + e--~z ~ ] J 3
e2F~
[0'2(4sin20-1)
__ sin 0 ( 0 " + ~ ) 1 }
.
31 August 1989
(4,
Here ks (r) is the radial wavefunction of the S-state. In table 1 we show the hyperon spectrum for S = - 1, - 2, - 3. Only states with the usual octet and decuplet quantum numbers have been included. We shall discuss the exotic states separately below. The energy values in table 1 were obtained from eq. ( 1 ) with o)p=150 MeV, ~Os=360 MeV, cp=0.6 and Cs=0.9 [11 ]. As we are mainly concerned with the energy splittings, the spectrum has been shifted up-
Table 1 Spectrum of hyperon resonances with nonexotic quantum numbers in the bound state model. Masses are given in MeV/c 2. S
np
ns
Ix
I
IP
E+ 73
Exp.
-1
1 1
0 0
½ ½
0 1
1 0
0 l
½ ~
1 0
½+ ½+ ~+ ~-
1116 1196 1376 1360
A(Il16) Z(1193) Z(1385) A(1405)
0
1
½
1
½
1380
0
1
½
1
~-
1664
Z(1670)
2 2 1 1
0 0 1 1
1 1 0 1
½ ~ ½ ½
½+ ~+ ½½-
E(1318) E(1530)
~(1672)
--2
-3
1
1
1
½
~
0 0
2 2
1 1
½ ½
~+ 3+
1266 1446 1524 1486 1711 1716 1986
3 2 2 1 1 0
0 1 1 2 2 3
~ ½ ~ ½ ~ ~
0 0 0 0 0 0
~+ ½3½+ ~+ ~-
1449 1618 1693 1789 1976 2023
~(1820)
293
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the calculated value being low by only 40 MeV. This state does not fit into the quark model, and calculations typically obtain masses which are too high by several hundred MeV. Further, no corresponding lowlying ½- nucleon resonances have been seen. According to the CK model, the A (1405) belongs to a set of negative-parity states with strangeness at least - 1. As seen in table 1, the other ones are ½- and 3-states with unit isospin. The lowest experimentally well-established negative-parity state with I = 1 is a ~ state at 1670 MeV, in good agreement with the prediction o f the present theory. There are some candidates below this energy, however, and one of them, perhaps the Z (1480), will now have to be a ½state, to be identified with the state listed at 1380 MeV in table 1. While the existence of the A ( 1 4 0 5 ) is already a major success of the theory, the experimental confirmation of its isospin- 1 analogues at somewhat larger masses is a crucial test of the theory. In the Skyrme model, the only states in the S = 0 sector are the nucleon and the A. This agrees with experiments in that a low-energy region below some 500 MeV excitation energy is indeed free of excited states. For S = - 1 this means that the states in table 1 should be the only ones below about 1600 MeV. (The ~state observed at 1520 MeV can probably be accounted for as a resonance in the C K model. ) The lowest S = - 2 two-kaon states in the model provide a good description o f the cascade particles and the lowest cascade resonances -~ ( 1530 ). The calculated energies are near the experimental ones, on the low side. As above, these states are predicted to have partners with the opposite parity, obtained by putting one kaon in the S-state. As the two kaons are now in different spatial states, the requirement of symmetry is less stringent, and theory thus, is in this case predicts an extra ½- state. Experimentally, the only well-established ~, resonance with negative parity is the ~- ~ ( 1 8 2 0 ) resonance. A corresponding state is predicted in table 1 at 1711 MeV, again in resonable agreement, but slightly on the low side. While no ½- resonance has yet been established, the E ( 1 6 9 0 ) and perhaps the -=(1620) resonances may correspond to the ones predicted in this channel. When both kaons are in the S-state a group of higher excited states are formed, where the combinatorics and quantum numbers are the same as in the ground state. One therefore obtains a repetition of the earlier 294
31 August 1989
results at a higher energy. Candidates exist at approximately the right masses, e.g., the cascade resonances ~ ( 1 9 5 0 ) and E ( 2 0 3 0 ) , but their quantum numbers are not known. In this energy region one expects intruder states which are consequences of the structure of the skyrmeon itself. In the S = - 3 channel, the g2 emerges at an energy which, continuing the trend, is lower than the measured value. Excited states of the ~2- are formed by raising either one or more kaons to the S state. It is perhaps not surprising that the low-lying spectrum is the same as for a bag containing three strange quarks (see ref. [17] ). The third and fourth excited states in our table have the opposite ordering compared to the bag-model calculation. Experimentally, very little is known about the S = - 3 states. One might identify the ~ (2250) with one of the predicted states, such as the ~3+ state calculated at 1976 MeV. A structure has also been seen at 2380 MeV [ 18 ]. In the spectrum in table 1 the energy levels of the S = - 1 one kaon states are well fit, while the calculated masses of S = - 2 two-kaon states are low by amounts in the 50-100 MeV range. The mass of the three-kaon ground state comes out low by about 220 MeV. This behaviour signals the presence of a repulsive interaction between the kaons. Simple perturbative estimate of terms o f fourth order in the kaon field give extra energies of the order of 50 MeV S = - 2 . For S = - 3 the repulsion would be three times larger, while a fourth kaon would no longer be bound. If these estimated turn out to be quantitatively accurate, the entire spectrum would be in a very good agreement with experiments. As pointed out in ref. [ 7 ], terms of fourth order in the kaon field suffer from operator ordering ambiguities. Classically, such contributions are well defined, however, and it would obviously be within the spirit o f the model to use estimates based on classical approximations to the kaon fields. If this is done, the fourth-order terms give repulsive contributions to the self-energies of the kaons and would account for some (but not all) of the 73 MeV shift of the calculated energy levels in table 1. There is also a contribution, quadratic in the angular velocity of the soliton, which can approximately be incorporated as a reduction of the m o m e n t of inertia. For the S-state this term was evaluated exactly in ref. [ 11 ] and found to decrease the moment of inertia by ( 12-14) % for the value (2).
Volume 227, number 3,4
PHYSICS LETTERS B
The consequent upward shift of the S-state is 13-16 MeV. Although these corrections are not sufficient to account for all the discrepancy between the calculated and observed A-mass, the remaining error is well within the accuracy expected of the model. The spectrum in table 1 contains only those states which satisfy the r u l e / + ½[SI ~ ~. These are the states that can be formed of three quarks with the up, down and strange flavours. The spectrum produced by eq. ( 1 ) does, however, also contain exotic states which do not satisfy the SU (3) multiplet rule. No exotic ground-state particles are generated for S = - 1. For S = - 2 one could obtain an exotic state with I = ~_, 3 corresponding to a b o u n d state of the Aresonance and two kaons (the selection rule forbids the b i n d i n g of a single kaon to this state). We argued above that such a state is actually outside the range of applicability of the model. For S = - 3 there exists the possibility of b i n d i n g the kaons to I = 1 soliton. Taking eq. ( 1 ) seriously, this state would actually lie below the f~-. As it would only decay by weak interactions, such a particle would certainly have been seen, and we need arguments why it need not exist. Considering first corrections within the model, we observe that the energy of the exotic I = 1, S = - 3 state would be increased, without affecting the predicted energies of the non-exotic states, by an isospin-dep e n d e n t three-body interaction between the kaons and the skyrmeon. Such an interaction is present in the model and might remove this problem. It would also reduce the b i n d i n g of two kaons to the isospin 3 skyrmeon m e n t i o n e d above. Finally, we have already reported estimates which indicate that states with ISI > 3 are not bound. As pointed out, there exist with the model reasons to assume that the exotic states are at higher energies than those given by eq. ( 1 ). All of these require going beyond the customary second order in the expansion of the kaon fields. One should remember, however, that the exotic states could simply be artifacts of the
31 August 1989
large-No expansion. It is not clear that the breakdown of this approximation necessarily has precursors in the form of excessive correction terms in the orders considered in the present work. The crucial test of the C a l l a n - K l e b a n o v b o u n d state model for the hyperons will be the experimental confirmation of the remaining ones of the predicted negative-parity states.
References [ 1] N. Isgur and G. Karl, Phys. Rev. D 19 (1979) 2653. [2] K.T. Chao, N. Isgur and G. Karl, Phys. Rev. D 23 ( 1981 ) 155. [3] J.M. Richard, Phys. Len. B 100 ( 1981 ) 515. [4] A.T. Aerts and L. Heller, Phys. Rev. D 23 ( 1981 ) 185. [5] A.T. Aerts and L. Heller, Phys. Rev. D 25 (1982) 1365. [6] C.G. Callanand I. Klebanov,Nucl. Phys. B 262 (1985) 365. [7] C.G. Callan, K. Hornbostel and I. Klebanov, Phys. Lett. B 202 (1988) 269. [8] T.H.R. Skyrme, Proc. R. Soc. A 260 ( 1961 ) 127. [9] N.N. Scoccola, H. Nadeau, M.A. Nowak and M. Rho, Phys. Lett. B 201 (1988) 425. [ 10] J.P. Blaizot, M. Rho and N.N. Scoccola, Phys. Len. B 209 (1988) 27. [ 11 ] K. Dannbom, U. Biota and D.O. Riska, The hyperons as bound states in the Skyrme model, University of Helsinki preprint HU-TFT-88-42 ( 1988). [12]N.N. Scoccola, D.P. Min, H. Nadeau and M. Rho, The strangeness problem: an SU(3) skyrmion with vector mesons, Universit~it Regensburg preprint TPR-89-10 (1989). [13] E.M. Nyman and D.O. Riska, Magnetic moments of the hyperons in the topological soliton model, University of Helsinki preprint HU-TFT-88-50 ( 1988). [ 14] E.M. Nyman, Phys. Len. B 224 (1989) 21. [ 15 ] J.P. Blaizotand G. Ripka, Non-lineardistortions of rotating chiral solitons, Saclay preprint SphT/87-181 ( 1988). [ 16] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552. [ 17 ] J. Kunz and P.J. Mulders, Bound state approach for strange dibaryons in the Skyrme model, preprint NIKHEF-P-15 (1988). [ 18] A.T. Aerts and L. Heller, Los Alamos National Laboratory preprint LA-UR-88-3353 ( 1988).
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