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Strangeness analog resonances
ANNALS
OF PHYSICS:
66, 138-757 (1971)
Strangeness
Analog
Resonances*
A. K. KERMAN* Massachusetts
Institute
of Technology,
Cambridge,
Massachusetts
AND H. J. LIPKIN Weizmann Institute Argonne National
of Science, Laboratory,
Rehovoth, Argonne,
Israel, Illinois
and 60439
ReceivedSeptember28, 1970 Strangenessanalog resonancesin hypernucleiare defined by analogy to the well-known isobaric resonances.Their relationship to the Sakata SU(3) model is discussed.Estimates are made for their energies and total widths. It is suggested that the excited states of JF and dN14 seenin stopped Kaon experiments are examples.
I. INTRODUCTION Amos de-Shalit believed in the unity of high-energy and low-energy nuclear physics. He was interested in the use of elementary particles as tools for understanding nuclear structure and also in the application to high-energy physics of methods and techniques developed for nuclear structure. Because of this he was one of the founders of a series of conferences on high-energy physics and nuclear structure. His students were encouraged to work in this area and made pioneering contributions
in the application
of methods of nuclear spectroscopy to hyper-
nuclei and in the development of reaction models applicable to both nuclei and elementary particles. He would have been excited by the experimental discovery of continuum excited states in hypernuclei [l] and enthusiastic about theoretical descriptions of these states via both the approaches developed by nuclear physicists for isobaric analog states and the symmetry approaches used in particle physics. We feel that he would have agreed with our view that this is the beginning of an exciting new * This work is supported in part through funds provided by the Atomic Energy Commission under Contract AT(30-l)-2098. t Work done while the author was visiting Argonne National Laboratory. 738
STRANGENESS
ANALOG
RESONANCES
739
program of theoretical and experimental work which should lead to a better understanding of both hypernuclear structure and the &nucleon interaction. Continuum hypernuclear excited states have been observed [I] in the reactions K- + Cl2 -
T- + /Q2* + 7~- + p + nBil,
(l.la)
K- + N14 -
T- + AN14* ---zv- + p + /1c13,
(I.lb)
induced by kaons stopped in emulsions. The pion spectra showed peaks indicating the existence of excited states in the accompanying hypernuclei with an excitation energy of about 10 MeV. The decay of the excited state by proton emission provided a characteristic signature which identified the event in the emulsion. Other decay modes, which are probably much more frequent, where not identified. New and interesting information is obtainable from investigations of continuum hypernuclear states. In particular, direct comparison is possible between states of nuclei and hypernuclei having the same configurations. In these excited states, one can study what happens when a A is changed into a nucleon without otherwise changing the wavefunction and therefore can study directly the difference between &nucleon and nucleon-nucleon interactions. All present discussion of hypernuclear spectra are based on data for groundstate configurations. These place the /l in a well-defined s orbit attached to a nucleus in a well-defined state. A new aspect in the spectroscopy of excited states can be seen in the example of ,C 12. From the point of view of ground-state hypernuclear spectroscopy, 11C12is a fl added to P. The ground-state configuration had the /l in an s state. One naturally thinks of an excited configuration in which the fl is in a p state. However, this nucleus can also be excited by leaving the fl in the s state and exciting one of the s-shell nucleons into the p shell. At this point we should no longer consider the rl and the nucleon as different entities, each in a well-defined state. Excited states can be linear combinations of excitations of the types discussed above. Both excitations are produced by raising a particle from the s shell to the p shell; one raises a fl, the other raises a nucleon. These excitations can be compared with excitations produced in ordinary nuclei by exciting a neutron or a proton from one shell to another. When both neutrons and protons are in the same shell, states with only a neutron or only a proton excited are not considered separately. The two excitations are mixed by the residual two-body interactions. The eigenfunctions are excitations that have a well-defined isospin. In hypernuclei there exists the analogous possibility of forming a state that is a particular linear combination of a /l excitation and a nucleon excitation to satisfy a symmetry condition that is a generalization of isospin. The relevant symmetry is not the SU(3) symmetry called the eightfold way, which is useful in particle physics. It is a different SU(3) group related to the old Sakata model for elementary particles [2]. These symmetries are discussed in detail below.
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KERMAN
AND
LIPKIN
Excited hypernuclear configurations that lie in the continuum and decay by emission of particles are best discussed as “elementary excitations” which depend upon the particular production mechanism, and are not exact eigenstates of the Hamiltonian. This is analogous to the study of ordinary nuclear continuum states. For example, stripping and pickup experiments study single-particle excitations produced by adding or removing a single particle from a particular nucleus. Photonuclear experiments study excitations produced by the absorption of a photon; a particular linear combination of particle-hole excitations. Isobaricanalog excitations are produced by charge-exchange reactions in which a neutron in a nucleus is transformed into a proton without otherwise changing the wavefunction [3]. Photonuclear and isobaric-analog excitations are collective because any nucleon in the nucleus can absorb the y ray or any neutron in the nucleus can be transformed into a proton. The excitation mechanism chooses a particular linear combination of these components. The coherence of the different components in the collective excitation gives rise to an enhanced cross section for the process, and the excited state is sometimes called a giant resonance. Although these collective states are not eigenstates of the Hamiltonian and decay into other states in the continuum, their particular coherence properties can give them a comparatively long lifetime. They are then observed as resonances with comparatively narrow widths in plots of certain cross sections as a function of energy. In hypernuclear spectroscopy, single-A excitations cannot easily be studied without beams of hyperons or hypernuclei which can transfer a /1 to a nuclear target. However, excited hypernuclear states can be produced by incident kaons either stopped or in beams. The basic process is a strangeness exchange in which the incident kaon becomes a pion and changes a nucleon into a d. This is directly analogous to the charge-exchange process which gives rise to isobaric-analog states. Since any nucleon in the nucleus can be transformed into a L’I, collective coherence effects can give an enhanced cross section for the particular state in which a nucleon is transformed into a (1 without changing the wave function. This state resembles an isobaric-analog state, which differs from “parent” state only by the transformation of a neutron into a proton without changing the wavefunction. We therefore call these strangeness analog states. Just as in the case of isobaricanalog states, they are the linear combination of d-particle and nucleon-hole excitations which preserve the permutation symmetry of the target state. The observation of these states as strangeness analog resonances will exhibit the difference between the d-nucleon and nucleon-nucleon interactions in the same manner that the isobaric-analog resonances exhibit the difference between neutronnucleon and proton-nucleon interactions. The excitation of such strangeness analog resonances has been predicted [4] for
STRANGENESS ANALOG
741
RESONANCES
such reactions as (1. l), and we suggest that the observed continuum excited states in Cl2 and N14 are indeed these resonances. This suggestion is supported by the relatively large cross section (a large fraction of the observed events showed this state) and by crude estimates of the excitation energy and width in Section II. Symmetry plays an important role in analog states, If isospin were a perfect symmetry, changing a neutron into a proton in a nucleus by operating on the wavefunction with an isospin step operator should give another degenerate nuclear state in the same isospin multiplet. Because isospin symmetry is broken, this isobaric-analog state is not an exact eigenstate of the nuclear Hamiltonian and is not degenerate with the initial state. The symmetry-breaking electromagnetic interaction shifts the energy of the state by the Coulomb energy introduced in changing a neutron into a proton. The state is shifted into the continuum where it has a finite width and lifetime and decays into other states. The isospin step operator thus connects a discrete bound state with an unbound continuum state, even though both states are members of the same isospin multiplet and both are discrete and degenerate in the isospin symmetry limit. For the strangeness analog state, consider a symmetry limit with identical A-nucleon and nucleon-nucleon interactions, so that changing a nucleon into a A does not change interactions and hence produces a state degenerate with the initial state. Both states would be members of a supermultiplet of some higher symmetry expressing the equivalence of nucleon-nucleon and A-nucleon interactions. The breaking of the symmetry by the difference between nucleon-nucleon and Anucleon interactions shifts the strangeness analog state to higher energies at which it is in the continuum and has a width. If the width were small enough, experimental measurements of the shifts and widths of strangeness analog states would give new information about the difference between the A-nucleon and nucleon-nucleon interactions.
II.
EXCITATION
ENERGIES AND WIDTHS
OF STRANGENESS ANALOG
STATES
We now attempt to estimate the excitation energy and width of a strangeness analog state on the basis of information available from properties of nuclei and hypernuclei. Our estimates are only qualitative, with the aim of establishing the accessibility of these states to experiment-i.e., with the aim of showing that they are not too high nor too broad to be observed. We do not in this paper attempt to give precise quantitative predictions. When more experimental data become available, better calculations can give information on the A-nucleon interaction. The simplest shell-model argument would take the average particle-hole excitation energy for the energy of the analog state. This is incorrect because it neglects the residual interactions which split the particle-hole states. The particle-hole
742
KERMAN AND LIPKIN
n
P
n
A
P
n
FIG. 1. Two examples of d-particle proton-hole states that go into a strangeness analog state. Since the average potential for the A has the same shape as that for a proton, all such states have the same energy in first approximation.
excitations that form the analog are nearly degenerate in the single-particle approximation since the A particle is always in the same shell as the nucleon hole (as shown schematically in Fig. 1). The residual interaction removes the degeneracy. Linear combinations of these states become eigenstates and have their energies shifted up or down, depending on their permutation symmetry and on the form of the interaction. The analog state is produced by changing a neutron into a A without changing the wavefunction; the A thus remains antisymmetrized with respect to all of the neutrons even though this is no longer required by the Pauli principle. The particular state with this property has equal coefficients for all the particle-hole components. Other hypernuclear states have different permutation symmetries, such that the A can be in symmetric states with respect to some of the neutrons. Which of these states lies lowest depends on the nature of the A-nucleon interaction. The experimental data [5] indicate that the AN interaction is stronger in the antisymmetric state (5) than in the symmetric state (3S). The analog state, which has a A in an antisymmetric state with respect to all neutrons, has a stronger attractive two-body interaction and should lie lower than other states having different permutation symmetries. This is exactly opposite to the isobaric analog case in which a neutron becomes a proton while remaining antisymmetrized with respect to all the neutrons. The totally antisymmetric isobaric analog state lies higher than the other particle-hole excitations because the nucleon-nucleon interaction is stronger in the symmetric state. Thus the inclusion of residual interactions suggests that the strangeness analog state should lie lower than the particle-hole excitation of Fig. 1. In the p-shell nuclei such as C12, this is indeed in the neighborhood of 10 MeV. The excitation energy of the strangeness analog state above the corresponding hypernuclear ground state can be more closely estimated by the following independent argument. The parent of the analog state differs from the analog state only by having a nucleon instead of the A, but otherwise has the same wavefunction. The ground state of the hypernucleus differs from the parent of the analog state by having one less nucleon in the highest orbit and a A in the lowest s orbit. The ground state of the hypernucleus can therefore be transformed into the
STRANGENESS
ANALOG
743
RESONANCES
strangeness analog state in three steps: (1) removing the A from the s orbit, (2) adding a nucleon into the highest orbit to make the parent of the strangeness analog state, and (3) changing a nucleon into a il without changing the wavefunction. The excitation energy is the sum of the changes in binding energy in the above three steps. (The effects of the A-nucleon mass difference cancel in taking the difference in energy between the analog state and the hypernuclear ground state and is therefore disregarded.) Figure 2 shows plots of the energy levels relevant to this calculation for two cases. The removal of the A from the lowest s state of the hypernucleus costs the A binding energy BA . The addition of the nucleon in the outermost unfilled orbit gains the nucleon separation energy B, for the parent nucleus. The transformation of the nucleon into the A without changing the wavefunction loses an amount proportional to the difference d V,, between the depths of the A and nucleon wells. In Cl2 (Fig. 2a), the neutron separation energy is 19 MeV; and the A binding energy is around 11 MeV for hypernuclei in this mass region. This would give 10 MeV for the excitation energy of the analog state with about 18 MeV difference between the effective A and nucleon well depths. This is not far from what we expect this difference to be, and (considering the roughness of the calculation) the result is again in agreement with the excitation energy of the experimentally observed continuum state in P. A(Z-I,
(Z-I,A-I
A I*
I+P
13~~23
E*=30
l--l
AtZ-I,A)
FIG. 2. Energy-level nucleus.
diagram for strangeness analog states; (a) in C12, and (b) in a heavy
In a heavy nucleus the excitation energy would be expected to be higher because the A binding energy is greater and the nucleon separation energy much lower. Taking a A binding energy of 23 MeV, a nucleon separation energy of 8 MeV, and 15 MeV for the difference between the A and nucleon potentials gives a value
744
KERMAN AND LIPKIN
of 30 MeV for the excitation energy of the strangeness analog state in a heavy nucleus. Figure 2b shows this for the case of the analog state produced by changing a proton into a 4. The analog states produced by changing a neutron into a (1 in a heavy nucleus are more complicated because of isospin considerations. These are discussed in detail in Section III. However, the result of 30 MeV excitation is qualitatively correct also for this case. A qualitative estimate of the width of the strangeness analog state is obtained by comparing the symmetry-breaking interaction with similar interactions in nuclear physics that give rise to widths for nuclear continuum states. The symmetrybreaking interaction is conveniently divided into two components: (1) the difference the average interaction introduces between the A and the nucleon potential wells, and (2) the difference between the Ll-nucleon and nucleon-nucleon residual interactions, The effect of the difference between the two wells can be qualitatively estimated by comparison with isobaric-analog states, for which the relevant symmetrybreaking mechanism is the difference between the neutron and proton potential wells-i.e., the Coulomb interaction. A rough picture indicates that the difference between a A-nucleus and nucleon-nucleus potential is qualitatively comparable to the Coulomb potential for a heavy nucleus. Thus the amount that the difference in the potential contributes to the width of the strangeness analog state should be qualitatively of the order of the widths of isobaric analog states, except for barrier penetration effects. The difference between the A-nucleon and nucleon-nucleon residual interactions is about one-half as big as the residual nucleon-nucleon interaction itself and is of the same range [5]. The nucleon-nucleon residual interaction is responsible for the spreading of collective particle-hole states in complex nuclei. Since the strangeness analog state is a collective particle-hole excitation, the contribution of the residual interaction to its width should be less than one-fourth of the largest widths of nuclear collective particle-hole states. Neither of these estimated contributions to the widths should be taken seriously as a quantitative prediction. However, both indicate that the states should be observable with resonable widths if-they occur at a low excitation energy. These arguments can be refined to give more precise predictions.
III. EXCITATION
AND COLLECTIVE
FEATURES
OF STRANGENEW
ANALOG
STATES
The collective features associated with the excitation of the strangeness analog state can be seen as follows. The reaction K-
+ Cl2 + 7r- + AC12
(3.la)
STRANGENESS
ANALOG
RESONANCES
745
can be considered as a result of the elementary interaction K-+n+w+A
(3.lb)
taking place on any one of the six neutrons in C 12. We first assume for simplicity that the ground state of Cl2 is well described by a single-particle shell model in which each of the nucleons is in a well-defined single-particle orbit. Consider those states of AC12, denoted by 1nC12, OI), which differ from the ground state of Cl2 only by having a neutron in the state 01 replaced by a A with the same wavefunction. In the independent-particle shell model, the A orbit has the same angular-momentum quantum numbers and angular dependence as the neutron orbit but the radial wavefunctions can be different. We neglect this difference at the present stage; it contributes to the width of the analog state and to an associated shift of its energy. The transition matrix for exciting a particular state I nC12, a> in the reaction (3.la) is approximately given by the product of the transition matrix element for the elementary process (3.lb) and a form factor F(q), where q is the momentum transfer; i.e., (T-; JT2, 011T I K-; P)
w (ml
1 T 1K-n) F(q).
(3.2)
The strangeness analog state / S) can be defined (a more general definition is given below in Eq. (3.5)) as the normalized sum of all the states / nC12, a). That is, (3.3a) where the sum is over all neutron states OLand N is the number of neutrons. The transition matrix element for producing the analog state (3.3a) is thus (r,
S 1F / K-C12) = dr(w~‘I
1F / K-n> F(q).
(3.3b)
The factor v’JV appearing on the right side of Eq. (3.3b) is the typical enhancement factor characteristic of these collective states and gives an enhancement by a factor N in the cross section. In the case of isobaric analog states, the enhancement factor is (N - 2). The above derivation presents a qualitative picture of the collectivity and the enhancement of the strangeness analog state, but avoids two points which need further elucidation. (1) The analog state (3.3a) is implicitly assumed to be an approximate stationary state of the Hamiltonian. (2) The independent-particle picture of the nucleus neglects the effect of correlations. We now consider these points in more detail.
746
KERMAN
AND
LIPKIN
(1) All of the above discussion would still be correct if the analog state were a linear combination of eigenstates of the Hamiltonian with energies spread uniformly over an energy range of 1 BeV. However, it would be irrelevant, as no enhancement would be observed in the production cross section for this state; the state would not be observed as a resonance. To be observed it must be concentrated in a small energy region. If it is expanded in eigenfunctions of the Hamiltonian, a large part of the wavefunction must be contained in a set of eigenfunctions within a very small energy region. The same condition can be expressed using the complementary time description rather than the energy description. One can say that the analog state (3.3a) is the only one created in the reaction (3.la) at time t = 0. The decay in time of this analog state can then be followed. If the lifetime of the analog state is comparatively long, it will have a narrow width and will be observed as a line in the production experiment. (2) Our definition (3.3a) of 1S) keeps correlations introduced by the Pauli principle because the initial ground-state wavefunction is antisymmetrized and the symmetric combination (3.3a) preserves this antisymmetry. A more general formulation which does not require the independent-particle model involves the use of second quantization. Let a,?, b,+, and c,+ be creation operators for creating a proton, neutron, and /1, respectively, in the state a:. Then ) ,Q2, a) = c,+b, 1C?“). In this notation
(3.4)
the strangeness analog state can be written (3.5)
The sum over all states 01 defines the strangeness analog state in a general formulation that no longer depends on the validity of the independent-particle shell model. The symmetric operator clearly preserves the symmetry of the state and Eq. (3.5) is equivalent to Eq. (3.3a) for the case of the independent-particle model. Equation (3.5) defines the state 1S) for any many-body wavefunction in just the way required to give the enhancement relation between the matrix elements (3.3b). The expression (3.5) resembles the operation of an isospin generator on a nucleus. Instead of changing a neutron into a proton and leaving it in the same state, it changes a neutron into a (1. In direct analogy to the isospin group, one can define an SU(2) group in which neutrons and As are changed into one another. This group has been called U spin [6] in the context of the Sakata model for elementary particles.
STRANGENESS
ANALOG
747
RESONANCES
The isobaric analog state / A) for a given parent state 1 n) is given by [7] IA> =
(3.6a)
T+ I 7~).
Z&-T
The state (3.5) is analogously written in the U-spin formulation 1sj = &
u- / P),
(3.6b)
where the U-spin generators are defined by the relations U- = 1 c,+ba ,
(3.7a)
U+ = c b,+c, ,
(3.7b)
U, = $ C (b,+b, -
c,+c,)
(3.7c)
The particular state (3.6b) will be an approximate stationary state of the Hamiltonian to the extent that U spin is a good symmetry for these states. The validity of this symmetry is considered in detail in Section V. From Eqs. (3.2), (3.4), and (3.7a), we see that the transition matrix element for the production of any state 4 is proportional to the matrix element of the operator U- between the parent state and 4; i.e., (T-, cj 1F 1K-P’)
= (4 I U- I Cl2)(.rr-fl = fd( r-/l
/ F I K-n> F(q)
17 / K-n) F(q),
(3.8)
where the enhancement factor is
f* = (4 I u- I C12>. The sum of the transition satisfy the sum rule T I(r,
(3.9)
strengths to a complete set of states 4 is seen to
4 I F 1K-P)j2
= T fd2 l(rA
I F I K-n)12 [F(q)12
= N l(r-~‘I j 9 j K-n)j2 [F(q)]“.
(3.10a) (3.10b)
The factor N simply expresses the possibility that the strangeness exchange reaction can occur on any one of the N neutrons in the target. The enhancement factor f+ thus describes the extent to which the total strength is concentrated in the particular state ~5.
748
KERMAN
AND
LIPKIN
In similar manner, one can define analog states in which a proton is changed into a II. These states would be produced in a reaction of combined charge and strangeness exchange, e.g., a reaction in which the incident K- becomes a no. For these states, operators analogous to (3.7) can be defined. These operators, called V-spin operators in the Sakata model, are
V+= COLcata, ,
(3.1 la)
V- = C a,+c, ,
(3.11b)
V, = * C (c,+c, - aeta,).
(3.1 lc)
m
01
The corresponding enhancement factor for the production combined charge- and strangeness-exchange reaction is then h = (9 I v+ I C12).
of a state I# in the (3.12)
All this discussion is clearly general and applies to the production of strangeness analog states on any target, not necessarily C12. If the parent nucleus is a self-conjugate nucleus such as C12, N14, 016, etc., the U-spin and V-spin strangeness analog states form a pair of states belonging to the same isodoublet and thus have very similar properties. If the parent state is a heavy nucleus with a neutron excess, the U-spin and V-spin analog states have different isospin properties. The V-spin analog state produced by changing a proton into a (1 has a larger neutron excess than the parent state and therefore its isospin is larger by l/2 than that of the parent state. It is an isospin eigenstate since the A has zero isospin and does not contribute to the isospin of the system. The U-spin analog state produced by changing a neutron into a A has more complicated isospin properties because the neutron hole produced in the parent state can be either in the neutron excess or in the core. The resulting nucleus has a neutron excess smaller than that of the parent nucleus, and the isospin of the neutron hole can be coupled to the isospin of the parent in two ways giving an isospin either greater than or less than the isospin of the parent. The U-spin analog state therefore is a linear combination of two isospins with coefficients depending upon the action of the U-spin generator on the particular state in the Sakata supermultiplet. The neutron or U-spin analog state should be split into its two components having pure isospin, since the isospin splitting due to symmetry energy is appreciable. The strangeness-exchange reaction should thus produce two separated peaks rather than a single peak corresponding to a state without a well-defined isospin. The details of this splitting are discussed in Sections IV and V.
STRANGENESS ANALOG
IV. STRANGENESS ANALOG
749
RESONANCES
STATES IN THE SAKATA
MODEL
The octet model of unitary symmetry, which has been successful in particle physics, is not useful for nuclear and hypernuclear physics. The principal reason is the (12 mass difference of 80 MeV, which is small on the energy scale of particle physics but very large on the energy scale of nuclear binding energies. Hypernuclei observed in nature are known to contain a A and not a Z since a Z could decay into a II and provide an excitation energy of 80 MeV to the nucleus. The octetmodel description of hypernuclei would classify them in states containing linear combinations of As and Es. The old Sakata triplet model [2] attempted to create all hadron states out of the basic npfl triplet. This model is no longer relevant to hadron physics and has been superseded by the octet. However, the complex nuclei and hypernuclei having strangeness 0 and -1 are indeed composed only of members of the basic Sakata triplet. It is therefore useful to consider hypernuclear spectroscopy from the point of view of the Sakata model by using the triplet rather than the octet model of unitary symmetry. A similar situation exists in the well-established applications of the Wigner supermultiplet SU(4) group in particle and nuclear physics. Particle physicists classify the nucleon in the 20-dimensional representation of SU(4), which contains the d (3-3 resonance) as well as the nucleon. Nuclear physicists classify the nucleon in the four-dimensional representation which contains only the nucleon in its four spin and charge states. The 300-MeV A-N mass difference can be considered small in some sense in particle physics and one can consider symmetry transformations which change nucleons into As. In nuclear physics it is clear that nuclei contain nucleons with only a tiny admixture of the A. (For a crude estimate of this sort of mixing in the deuteron see Ref. 8.) Triplet unitary symmetry assumes that the cl-nucleon interaction is the same as the nucleon-nucleon interaction. Although this symmetry is a very poor approximation for the two-body system (in contrast to charge independence, which is very good and violated only by small electromagnetic contributions), we believe that the Sakata symmetry may be useful for the many-particle states having strangeness 0 and - 1 (the states that so far have been found to be relevant to hypernuclear spectroscopy). For these states the primary symmetry breaking is due to the difference between the nucleon-nucleus and the L-nucleus interactions rather than the two-body interactions; i.e., the validity of the shell-model description of complex nuclei allows us to replace the major portion of the two-body interaction by an average field in which the single nucleon or hyperon moves. The radius of the baryon-nucleus interaction potential is determined by the size of the nucleus and is independent of whether the baryon considered is a nucleon or a (1. The latter affects only the depth of the well. Thus the major breaking of
750
KERMAN
AND
LIPKIN
the Sakata symmetry for these states results from the difference in the well depths seen by the nucleon and the fl. As we have seen in Section II, this gives a large diagonal contribution to the energy of a nucleus or hypernucleus, but may not change the wavefunction too much from that given under the assumption of Sakata SU(3) symmetry. The symmetry-breaking effects that would mix in other SU(3) representations and destroy the purity of the Sakata state are smaller effects depending on differences in the shape of the neutron and d wells and differences in the residual two-body interaction. The SU(3) algebra used in the Sakata model is well known. The algebra is generated by the nine operators of isospin, U spin, and V spin. Only eight operators are independent because of the relation To + Uo + Vo = 0
(4-l)
The isospin, U-spin, and V-spin operators satisfy angular-momentum commutation rules among themselves. The commutation relations that hold between different spins are (4.2a) K’+, v+l = T-2
w+ 9v-1 = 0,
(4.2b)
[V, ) u-1 = +JJ- )
(4.2~)
[Uo, V+l = iv+.
(4.2d)
Other relations are obtainable from these by taking Hermitean conjugates and cyclic permutations of T, U, and V. The parent state from which the analog state is produced is a nuclear ground state containing Z protons and N neutrons. This state 1 n) is a simultaneous eigenfunction of T,, , U, , and V. ; i.e., T, I n> = ii@ - N) I n>,
(4.3a)
uo I r> = W I 71.>, v, 17i-) = -$Z 1 7r),
(4.3b) (4.3c)
where we have used the conventions of particle physics for the eigenvalues of T (which are listed in Table I). TABLE I The Eigenvalues oft, , uO , and v,, for the Single Baryon States in the Sakata Model ?I
to RI 00
P
1 42 0
-*
4 0
A 0 1 &
STRANGENESS
ANALOG
RESONANCES
751
Since nuclei contain no As, it follows that
u+I r> = 0,
(4.4a)
v- I 7r) = 0.
(4.4b)
In all cases of practical interest, 1 rr> is a low-lying nuclear state which has a neutron excess. We shall neglect isospin violation and assume that it is an eigenstate of isospin on the edge of the isospin multiplet. Therefore T- 1 n) = 0.
(4.4c)
The isospin of the state is given by Tz
N-z
2 Using the above properties of these operators, we can calculate useful expectation values, namely (9~ j V-V+ I T) = (7~ 1 [V-, (T I V-T-U(n [ V-T-T+V+
I rr) = (rr I [V-,
V+] 17~) = -2(37 / V, I 77) = Z, [T-,
U-11 I n-> = -Z,
I z-) = Z(2T + 1) = Z(N - Z + 1).
(4Sa) (4Sb) (4.k)
The simplest strangeness analog state is the V-spin analog state obtained by changing a proton into a A, i.e., the state /S,)
=
“;g;
)
where the normalization factor is given by Eq. (4.5a). This state is an isospin eigenstate with the eigenvalue T + &. The corresponding state generated from the parent state by a U-spin generator which changes a neutron into a fI is
However, this state is not an isospin eigenstate if N is greater than Z and would therefore not be interesting in itself. We therefore separate this state into its two components S, and S, having isospins T + & and T - 4, respectively. The state S, is the isobaric analog of the V-spin analog state / S,), namely [ s,;
= d:;
“yji
= --T+v,
z/Z(N
IL,
- Z + 1)
(4.8a)
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KERMAN
AND
LIPKIN
The other state S, , obtained by the Schmidt orthogonalization the states (4.7) and (4.8a), is ,s
)
<
=
[U-
+
~+V+@T
procedure from
+ 111I r>
d(N +1)2Tl(2T+1)-
.
(4.8b)
This state can be seen to be normalized and to be orthogonal to the state (4.8a). We can now calculate the enhancement factors f for the production of these analog states in a strangeness-exchange reaction. These are given by the matrix elements between the parent state and the analog state of the operator V+ for the V-spin analog states and of U- for the U-spin analog states. Using the definitions (4.6), (4.7) and (4.8) and the expectation values (4.5), we obtain
f, = (S,1v+177)= (n ’ g+ ’ r, = fl (n- I --= V-T-U1/(2T+l)z
f>=(&-lU-I~)=
=
I rr)
tN + ‘1 (2T2;
V. THE VALIDITY
OF
-(2Tz+
1)’
1) ’
(4.9b)
(4.9c)
SAKATA SYMMETRY; CONCLUSIONS
We now examine the breaking of the Sakata SU(3) symmetry in detail and formalize the qualitative arguments, given in the Section IV, which indicate that this breaking is small. By analogy with the case of isospin, we write the two-body force Wij as the sum of a symmetry-conserving term and a symmetry-violating term and use U-spin and V-spin operators in the violating term. However, our formulation differs from the isospin case in that we are considering only nuclei containing no As and hypernuclei containing a single A. We therefore need not consider the A-A interaction and can express a two-body interaction in a form that holds only for the nucleon-nucleon and nucleon-A interactions. We therefore need only a constant term and a term that is linear in the U-spin and V-spin operators in our expansion of the two-body force. A quadratic term would be needed, as in isospin, if we also considered the A-A interaction. We write
STRANGENESS
ANALOG
RESONANCES
153
where Wij is the two-body interaction for a baryon pair (which can be either &nucleon or nucleon-nucleon), Wz! is the interaction between particles i and j if they are both nucleons, u ,,i, u,,j, v,,~,and vOjare the eigenvalues of the u- and v-spin operators for particles i and j, and d wij is the difference between the nucleonnucleon and nucleon-/I interactions, i.e., (5.2a) We neglect isospin-symmetry breaking. This can be included in the formulation by adding the usual terms depending on the isospin generators. Equation (5.1) is seen to give the correct values for the nucleon-nucleon and nucleon-/l interaction since (uol + zlo2- vol - vo2 - 1) 1nn) = 0,
(5.2b)
(uol + urJ2- Vll - VI2 -
(5.2~)
1) / nn> = -8 1nA).
The total interaction for the nuclear or hypernuclear system is w = 3 c wij i#j
(5.3)
As in the isobaric case [7], the symmetry-breaking effects depend upon the commutation relations between the Hamiltonian and the U-spin or V-spin generator which creates the analog state from the parent. We neglect the symmetry breaking in the kinetic energy terms due to the cl-nucleon mass difference as a smaller effect and consider the commutator with the interaction. The relevant commutators are
w;-’ -
[u_ , w] = & 1 (u-i + u-j) A W, i#j
W?’ = [V+ , W] = $ 1 (u+i + t)+j) A wij, i#j
(5.4b)
where the total U spin and V spin are defined as the sums of the II and v spins of the individual particles-i.e., u=p
z
v = c vi,
(5.5a)
754
KERMAN
and we have used the commutation
AND
LIPKIN
relations
[ix ) (uoi- v,i)] = $Li,
(5.5c)
[V+ ) (uoi - v,i)] = gv+i.
(5.5d)
For simplicity we consider first the V-spin analog state that is an isospin eigenstate. Since the energy of this state is just the expectation value of the Hamiltonian, we see from Eqs. (4.6) and (4.4b) that the binding energy difference dE, between the analog state and the parent state is simply expressed in terms of commutators of V-spin operators with the interaction; i.e.,
dE, = CTI IV-, w, V+lI r> = _ CTI iv- ~z%+lI n> z = 4 (3-r1 c (2):+ voj)Awij I Tr). i#j This is again analogous to the isospin case except that here the parent has good V spin and there is no correction to the double commutator [7]. If particle i is a proton, the expression is (~~~vg~dw~~~~)=g~(~~(w~~j j
W$)pT).
(5.7a)
If particle i is a neutron, the analog of the left side of Eq. (5.7a) vanishes, since voi = 0 for a neutron. The right side of (5.7a) is just the difference between the average fields on a proton and on a A. The summation over i in Eq. (5.6) and the inclusion of the two u. terms give a factor 22 which cancels these factors in the denominator to give the result
LIE, = --(A V,,)
m +25 MeV,
(5.7b)
where d V,, is the difference in the interaction of a proton in the nucleus and that of a A with the same wavefunction. In a single-particle model
(5.7c) where the q& are the single-particle wavefunctions. The result (5.7) is in agreement with our previous simple observation that the energy shift of the analog state relative to the parent state is just the difference between the depths of the potentials seen by a nucleon and a A. The energies of the two U-spin analog states are easily obtained from the energy of the V-spin analog state. Since I S,) is the isobaric analog of ] S,), its energy
STRANGENESS ANALOG
RESONANCES
755
differs from that of / S,) by the Coulomb energy dE,,,, , as shown in Fig. 3. The state I S,) is then lower by the usual nuclear symmetry energy.
FIG. 3. Schematicrepresentationof the strangenessanalog states and the transitions from the
parent. The decay widths are also expressible by use of the commutators (5.4). The dominant term in the amplitude for the decay of the analog state to any state @ is given by [7] by (5.8) There are two kinds of decay mechanisms: (1) the escape of a single particle, in which case the state @ has a particle ejected from the nucleus; and (2) damping through compound-nucleus states. We first consider L! escape. The relevant matrix element resembles Eq. (5.7a) except that it is an off-diagonal matrix element between the state r and the state @, and the operator is V+ instead of V”. As in the case of Eq. (5.7a), the summation over j gives the average field due to all the other particles, as seen by particle i. The escape of the A is therefore due to the change in the average field seen by the particle as it changes from a proton to a (1. The diagonal matrix element of this change in average field gives the energy shift as in Eq. (5.7b). The off-diagonal matrix elements describe the escape of the fl from the nucleus. In the single-particle model, the fl escape amplitude is given by (5.9) The average field appearing in (5.7) and (5.9) can be compared with the Coulomb field responsible for the shift and width of isobaric-analog states. The average symmetry-breaking field in the hypernuclear case is of the same order of magnitude as the Coulomb field in the isobaric case, even though the relevant parameter
756
KERMAN AND LIPKIN
characterizing the long-range Coulomb interaction in Z/137. Because of its long range, its average field strength can be comparable to the strength of the field produced by the hypernuclear symmetry-breaking interaction, which is short range. The additional barrier-penetration factor (present in the isobaric case and absent in the hypernuclear case) must also be considered, but this is not a very large factor. All other escapes involve single two-body matrix elements which are not summed over particles to give an average field. For these the only terms in the expression (5.4b) that contribute to the matrix element (5.8) are those in which the pair zj is uniquely defined as consisting of the proton that is changed to a (1 and the nucleon that escapes. Since these two-body matrix elements are not summed, they should be less important than /.I escape and should give a smaller contribution to the width. The damping contributions cannot be calculated without some detailed model, for which collective states may be important. However, we expect that the damping mechanism is similar to that for the damping of other collective particle-hole states. Since the operator appearing in expression (5.8) is only the difference between d-nucleon and nucleon-nucleon interactions while in the nuclear case it is the full nuclear interaction, this may not be large. When experimental data become available, the values of the widths of the analog states will give us specific information about the strength .4 W of the difference between the nucleon-nucleon and d-nucleon interactions.
ACKNOWLEDGMENT We should like to acknowledge stimulating S. Okubo, M. Peshkin, and J. P. Schiffer.
discussions with A. R. Bodmer,
D. Kurath,
REFERENCES 1. D. H. DAVIS AND J. SACTON, in “Proceedings of International Conference on Hypernuclear Physics,” (A. R. Bodmer and L. A. Hyman, Eds.), p. r59, Argonne National Laboratory 1969; see also R. H. DALITZ, p. 708. 2. S. SAKATA,
Progr.
Theor.
Phys.
16 (1956),
686.
3. J. D. ANDERSON, C. WONG, AND J. W. MCCLURE, Phy& Rev. 126 (1962), 2170; R. A. MADSEN, in “Nuclear Isospin.” (J. D. Anderson, S. D. Bloom, J. Cemy, and W. W. True, Eds.), p. 315, Academic Press, New York/London, 1969. 4. H. J. LIPKIN, Phys. Rev. Lets 14 (1965), 18; H. FESHBACH AND A. K. KERMAN, in “Preludes in Theoretical Physics,” (A. de-Shalit, H. Feshbach, and L. Van Hove, Eds.), p. 260, NorthHolland, Amsterdam, 1966; L. KISSLINGER, Phy.s. Rev. 157 (1967), 1358.
STRANGENESS
ANALOG
RESONANCES
757
5. A. R. BODMER AND J. W. MURPHY, Nucl. Phys. 64 (1965), 593; A. R. BODMER, in “High Energy Physics and Nuclear Structure,” (G. Alexander, Ed.), p. 60, North-Holland, Amsterdam 1967; A. GAL, J. M. SOPER,AND R. H. DALITZ, Ann. Phys. New York (1971). 6. H. J. LIPKIN, “Lie Groups for Pedestrians,” North-Holland, Amsterdam, 1965. 7. A. K. KERMAN, in “Nuclear Isospin,” (J. D. Anderson, S. D. Bloom, J. Cerny, and W. W. True, Eds.), p. 315, Academic Press, New York/London, 1969. 8. A. K. KERMAN AND L. S. KISSLINGER, Phys. Rev. 180 (1969), 1483.
OF PHYSICS:
66, 138-757 (1971)
Strangeness
Analog
Resonances*
A. K. KERMAN* Massachusetts
Institute
of Technology,
Cambridge,
Massachusetts
AND H. J. LIPKIN Weizmann Institute Argonne National
of Science, Laboratory,
Rehovoth, Argonne,
Israel, Illinois
and 60439
ReceivedSeptember28, 1970 Strangenessanalog resonancesin hypernucleiare defined by analogy to the well-known isobaric resonances.Their relationship to the Sakata SU(3) model is discussed.Estimates are made for their energies and total widths. It is suggested that the excited states of JF and dN14 seenin stopped Kaon experiments are examples.
I. INTRODUCTION Amos de-Shalit believed in the unity of high-energy and low-energy nuclear physics. He was interested in the use of elementary particles as tools for understanding nuclear structure and also in the application to high-energy physics of methods and techniques developed for nuclear structure. Because of this he was one of the founders of a series of conferences on high-energy physics and nuclear structure. His students were encouraged to work in this area and made pioneering contributions
in the application
of methods of nuclear spectroscopy to hyper-
nuclei and in the development of reaction models applicable to both nuclei and elementary particles. He would have been excited by the experimental discovery of continuum excited states in hypernuclei [l] and enthusiastic about theoretical descriptions of these states via both the approaches developed by nuclear physicists for isobaric analog states and the symmetry approaches used in particle physics. We feel that he would have agreed with our view that this is the beginning of an exciting new * This work is supported in part through funds provided by the Atomic Energy Commission under Contract AT(30-l)-2098. t Work done while the author was visiting Argonne National Laboratory. 738
STRANGENESS
ANALOG
RESONANCES
739
program of theoretical and experimental work which should lead to a better understanding of both hypernuclear structure and the &nucleon interaction. Continuum hypernuclear excited states have been observed [I] in the reactions K- + Cl2 -
T- + /Q2* + 7~- + p + nBil,
(l.la)
K- + N14 -
T- + AN14* ---zv- + p + /1c13,
(I.lb)
induced by kaons stopped in emulsions. The pion spectra showed peaks indicating the existence of excited states in the accompanying hypernuclei with an excitation energy of about 10 MeV. The decay of the excited state by proton emission provided a characteristic signature which identified the event in the emulsion. Other decay modes, which are probably much more frequent, where not identified. New and interesting information is obtainable from investigations of continuum hypernuclear states. In particular, direct comparison is possible between states of nuclei and hypernuclei having the same configurations. In these excited states, one can study what happens when a A is changed into a nucleon without otherwise changing the wavefunction and therefore can study directly the difference between &nucleon and nucleon-nucleon interactions. All present discussion of hypernuclear spectra are based on data for groundstate configurations. These place the /l in a well-defined s orbit attached to a nucleus in a well-defined state. A new aspect in the spectroscopy of excited states can be seen in the example of ,C 12. From the point of view of ground-state hypernuclear spectroscopy, 11C12is a fl added to P. The ground-state configuration had the /l in an s state. One naturally thinks of an excited configuration in which the fl is in a p state. However, this nucleus can also be excited by leaving the fl in the s state and exciting one of the s-shell nucleons into the p shell. At this point we should no longer consider the rl and the nucleon as different entities, each in a well-defined state. Excited states can be linear combinations of excitations of the types discussed above. Both excitations are produced by raising a particle from the s shell to the p shell; one raises a fl, the other raises a nucleon. These excitations can be compared with excitations produced in ordinary nuclei by exciting a neutron or a proton from one shell to another. When both neutrons and protons are in the same shell, states with only a neutron or only a proton excited are not considered separately. The two excitations are mixed by the residual two-body interactions. The eigenfunctions are excitations that have a well-defined isospin. In hypernuclei there exists the analogous possibility of forming a state that is a particular linear combination of a /l excitation and a nucleon excitation to satisfy a symmetry condition that is a generalization of isospin. The relevant symmetry is not the SU(3) symmetry called the eightfold way, which is useful in particle physics. It is a different SU(3) group related to the old Sakata model for elementary particles [2]. These symmetries are discussed in detail below.
740
KERMAN
AND
LIPKIN
Excited hypernuclear configurations that lie in the continuum and decay by emission of particles are best discussed as “elementary excitations” which depend upon the particular production mechanism, and are not exact eigenstates of the Hamiltonian. This is analogous to the study of ordinary nuclear continuum states. For example, stripping and pickup experiments study single-particle excitations produced by adding or removing a single particle from a particular nucleus. Photonuclear experiments study excitations produced by the absorption of a photon; a particular linear combination of particle-hole excitations. Isobaricanalog excitations are produced by charge-exchange reactions in which a neutron in a nucleus is transformed into a proton without otherwise changing the wavefunction [3]. Photonuclear and isobaric-analog excitations are collective because any nucleon in the nucleus can absorb the y ray or any neutron in the nucleus can be transformed into a proton. The excitation mechanism chooses a particular linear combination of these components. The coherence of the different components in the collective excitation gives rise to an enhanced cross section for the process, and the excited state is sometimes called a giant resonance. Although these collective states are not eigenstates of the Hamiltonian and decay into other states in the continuum, their particular coherence properties can give them a comparatively long lifetime. They are then observed as resonances with comparatively narrow widths in plots of certain cross sections as a function of energy. In hypernuclear spectroscopy, single-A excitations cannot easily be studied without beams of hyperons or hypernuclei which can transfer a /1 to a nuclear target. However, excited hypernuclear states can be produced by incident kaons either stopped or in beams. The basic process is a strangeness exchange in which the incident kaon becomes a pion and changes a nucleon into a d. This is directly analogous to the charge-exchange process which gives rise to isobaric-analog states. Since any nucleon in the nucleus can be transformed into a L’I, collective coherence effects can give an enhanced cross section for the particular state in which a nucleon is transformed into a (1 without changing the wave function. This state resembles an isobaric-analog state, which differs from “parent” state only by the transformation of a neutron into a proton without changing the wavefunction. We therefore call these strangeness analog states. Just as in the case of isobaricanalog states, they are the linear combination of d-particle and nucleon-hole excitations which preserve the permutation symmetry of the target state. The observation of these states as strangeness analog resonances will exhibit the difference between the d-nucleon and nucleon-nucleon interactions in the same manner that the isobaric-analog resonances exhibit the difference between neutronnucleon and proton-nucleon interactions. The excitation of such strangeness analog resonances has been predicted [4] for
STRANGENESS ANALOG
741
RESONANCES
such reactions as (1. l), and we suggest that the observed continuum excited states in Cl2 and N14 are indeed these resonances. This suggestion is supported by the relatively large cross section (a large fraction of the observed events showed this state) and by crude estimates of the excitation energy and width in Section II. Symmetry plays an important role in analog states, If isospin were a perfect symmetry, changing a neutron into a proton in a nucleus by operating on the wavefunction with an isospin step operator should give another degenerate nuclear state in the same isospin multiplet. Because isospin symmetry is broken, this isobaric-analog state is not an exact eigenstate of the nuclear Hamiltonian and is not degenerate with the initial state. The symmetry-breaking electromagnetic interaction shifts the energy of the state by the Coulomb energy introduced in changing a neutron into a proton. The state is shifted into the continuum where it has a finite width and lifetime and decays into other states. The isospin step operator thus connects a discrete bound state with an unbound continuum state, even though both states are members of the same isospin multiplet and both are discrete and degenerate in the isospin symmetry limit. For the strangeness analog state, consider a symmetry limit with identical A-nucleon and nucleon-nucleon interactions, so that changing a nucleon into a A does not change interactions and hence produces a state degenerate with the initial state. Both states would be members of a supermultiplet of some higher symmetry expressing the equivalence of nucleon-nucleon and A-nucleon interactions. The breaking of the symmetry by the difference between nucleon-nucleon and Anucleon interactions shifts the strangeness analog state to higher energies at which it is in the continuum and has a width. If the width were small enough, experimental measurements of the shifts and widths of strangeness analog states would give new information about the difference between the A-nucleon and nucleon-nucleon interactions.
II.
EXCITATION
ENERGIES AND WIDTHS
OF STRANGENESS ANALOG
STATES
We now attempt to estimate the excitation energy and width of a strangeness analog state on the basis of information available from properties of nuclei and hypernuclei. Our estimates are only qualitative, with the aim of establishing the accessibility of these states to experiment-i.e., with the aim of showing that they are not too high nor too broad to be observed. We do not in this paper attempt to give precise quantitative predictions. When more experimental data become available, better calculations can give information on the A-nucleon interaction. The simplest shell-model argument would take the average particle-hole excitation energy for the energy of the analog state. This is incorrect because it neglects the residual interactions which split the particle-hole states. The particle-hole
742
KERMAN AND LIPKIN
n
P
n
A
P
n
FIG. 1. Two examples of d-particle proton-hole states that go into a strangeness analog state. Since the average potential for the A has the same shape as that for a proton, all such states have the same energy in first approximation.
excitations that form the analog are nearly degenerate in the single-particle approximation since the A particle is always in the same shell as the nucleon hole (as shown schematically in Fig. 1). The residual interaction removes the degeneracy. Linear combinations of these states become eigenstates and have their energies shifted up or down, depending on their permutation symmetry and on the form of the interaction. The analog state is produced by changing a neutron into a A without changing the wavefunction; the A thus remains antisymmetrized with respect to all of the neutrons even though this is no longer required by the Pauli principle. The particular state with this property has equal coefficients for all the particle-hole components. Other hypernuclear states have different permutation symmetries, such that the A can be in symmetric states with respect to some of the neutrons. Which of these states lies lowest depends on the nature of the A-nucleon interaction. The experimental data [5] indicate that the AN interaction is stronger in the antisymmetric state (5) than in the symmetric state (3S). The analog state, which has a A in an antisymmetric state with respect to all neutrons, has a stronger attractive two-body interaction and should lie lower than other states having different permutation symmetries. This is exactly opposite to the isobaric analog case in which a neutron becomes a proton while remaining antisymmetrized with respect to all the neutrons. The totally antisymmetric isobaric analog state lies higher than the other particle-hole excitations because the nucleon-nucleon interaction is stronger in the symmetric state. Thus the inclusion of residual interactions suggests that the strangeness analog state should lie lower than the particle-hole excitation of Fig. 1. In the p-shell nuclei such as C12, this is indeed in the neighborhood of 10 MeV. The excitation energy of the strangeness analog state above the corresponding hypernuclear ground state can be more closely estimated by the following independent argument. The parent of the analog state differs from the analog state only by having a nucleon instead of the A, but otherwise has the same wavefunction. The ground state of the hypernucleus differs from the parent of the analog state by having one less nucleon in the highest orbit and a A in the lowest s orbit. The ground state of the hypernucleus can therefore be transformed into the
STRANGENESS
ANALOG
743
RESONANCES
strangeness analog state in three steps: (1) removing the A from the s orbit, (2) adding a nucleon into the highest orbit to make the parent of the strangeness analog state, and (3) changing a nucleon into a il without changing the wavefunction. The excitation energy is the sum of the changes in binding energy in the above three steps. (The effects of the A-nucleon mass difference cancel in taking the difference in energy between the analog state and the hypernuclear ground state and is therefore disregarded.) Figure 2 shows plots of the energy levels relevant to this calculation for two cases. The removal of the A from the lowest s state of the hypernucleus costs the A binding energy BA . The addition of the nucleon in the outermost unfilled orbit gains the nucleon separation energy B, for the parent nucleus. The transformation of the nucleon into the A without changing the wavefunction loses an amount proportional to the difference d V,, between the depths of the A and nucleon wells. In Cl2 (Fig. 2a), the neutron separation energy is 19 MeV; and the A binding energy is around 11 MeV for hypernuclei in this mass region. This would give 10 MeV for the excitation energy of the analog state with about 18 MeV difference between the effective A and nucleon well depths. This is not far from what we expect this difference to be, and (considering the roughness of the calculation) the result is again in agreement with the excitation energy of the experimentally observed continuum state in P. A(Z-I,
(Z-I,A-I
A I*
I+P
13~~23
E*=30
l--l
AtZ-I,A)
FIG. 2. Energy-level nucleus.
diagram for strangeness analog states; (a) in C12, and (b) in a heavy
In a heavy nucleus the excitation energy would be expected to be higher because the A binding energy is greater and the nucleon separation energy much lower. Taking a A binding energy of 23 MeV, a nucleon separation energy of 8 MeV, and 15 MeV for the difference between the A and nucleon potentials gives a value
744
KERMAN AND LIPKIN
of 30 MeV for the excitation energy of the strangeness analog state in a heavy nucleus. Figure 2b shows this for the case of the analog state produced by changing a proton into a 4. The analog states produced by changing a neutron into a (1 in a heavy nucleus are more complicated because of isospin considerations. These are discussed in detail in Section III. However, the result of 30 MeV excitation is qualitatively correct also for this case. A qualitative estimate of the width of the strangeness analog state is obtained by comparing the symmetry-breaking interaction with similar interactions in nuclear physics that give rise to widths for nuclear continuum states. The symmetrybreaking interaction is conveniently divided into two components: (1) the difference the average interaction introduces between the A and the nucleon potential wells, and (2) the difference between the Ll-nucleon and nucleon-nucleon residual interactions, The effect of the difference between the two wells can be qualitatively estimated by comparison with isobaric-analog states, for which the relevant symmetrybreaking mechanism is the difference between the neutron and proton potential wells-i.e., the Coulomb interaction. A rough picture indicates that the difference between a A-nucleus and nucleon-nucleus potential is qualitatively comparable to the Coulomb potential for a heavy nucleus. Thus the amount that the difference in the potential contributes to the width of the strangeness analog state should be qualitatively of the order of the widths of isobaric analog states, except for barrier penetration effects. The difference between the A-nucleon and nucleon-nucleon residual interactions is about one-half as big as the residual nucleon-nucleon interaction itself and is of the same range [5]. The nucleon-nucleon residual interaction is responsible for the spreading of collective particle-hole states in complex nuclei. Since the strangeness analog state is a collective particle-hole excitation, the contribution of the residual interaction to its width should be less than one-fourth of the largest widths of nuclear collective particle-hole states. Neither of these estimated contributions to the widths should be taken seriously as a quantitative prediction. However, both indicate that the states should be observable with resonable widths if-they occur at a low excitation energy. These arguments can be refined to give more precise predictions.
III. EXCITATION
AND COLLECTIVE
FEATURES
OF STRANGENEW
ANALOG
STATES
The collective features associated with the excitation of the strangeness analog state can be seen as follows. The reaction K-
+ Cl2 + 7r- + AC12
(3.la)
STRANGENESS
ANALOG
RESONANCES
745
can be considered as a result of the elementary interaction K-+n+w+A
(3.lb)
taking place on any one of the six neutrons in C 12. We first assume for simplicity that the ground state of Cl2 is well described by a single-particle shell model in which each of the nucleons is in a well-defined single-particle orbit. Consider those states of AC12, denoted by 1nC12, OI), which differ from the ground state of Cl2 only by having a neutron in the state 01 replaced by a A with the same wavefunction. In the independent-particle shell model, the A orbit has the same angular-momentum quantum numbers and angular dependence as the neutron orbit but the radial wavefunctions can be different. We neglect this difference at the present stage; it contributes to the width of the analog state and to an associated shift of its energy. The transition matrix for exciting a particular state I nC12, a> in the reaction (3.la) is approximately given by the product of the transition matrix element for the elementary process (3.lb) and a form factor F(q), where q is the momentum transfer; i.e., (T-; JT2, 011T I K-; P)
w (ml
1 T 1K-n) F(q).
(3.2)
The strangeness analog state / S) can be defined (a more general definition is given below in Eq. (3.5)) as the normalized sum of all the states / nC12, a). That is, (3.3a) where the sum is over all neutron states OLand N is the number of neutrons. The transition matrix element for producing the analog state (3.3a) is thus (r,
S 1F / K-C12) = dr(w~‘I
1F / K-n> F(q).
(3.3b)
The factor v’JV appearing on the right side of Eq. (3.3b) is the typical enhancement factor characteristic of these collective states and gives an enhancement by a factor N in the cross section. In the case of isobaric analog states, the enhancement factor is (N - 2). The above derivation presents a qualitative picture of the collectivity and the enhancement of the strangeness analog state, but avoids two points which need further elucidation. (1) The analog state (3.3a) is implicitly assumed to be an approximate stationary state of the Hamiltonian. (2) The independent-particle picture of the nucleus neglects the effect of correlations. We now consider these points in more detail.
746
KERMAN
AND
LIPKIN
(1) All of the above discussion would still be correct if the analog state were a linear combination of eigenstates of the Hamiltonian with energies spread uniformly over an energy range of 1 BeV. However, it would be irrelevant, as no enhancement would be observed in the production cross section for this state; the state would not be observed as a resonance. To be observed it must be concentrated in a small energy region. If it is expanded in eigenfunctions of the Hamiltonian, a large part of the wavefunction must be contained in a set of eigenfunctions within a very small energy region. The same condition can be expressed using the complementary time description rather than the energy description. One can say that the analog state (3.3a) is the only one created in the reaction (3.la) at time t = 0. The decay in time of this analog state can then be followed. If the lifetime of the analog state is comparatively long, it will have a narrow width and will be observed as a line in the production experiment. (2) Our definition (3.3a) of 1S) keeps correlations introduced by the Pauli principle because the initial ground-state wavefunction is antisymmetrized and the symmetric combination (3.3a) preserves this antisymmetry. A more general formulation which does not require the independent-particle model involves the use of second quantization. Let a,?, b,+, and c,+ be creation operators for creating a proton, neutron, and /1, respectively, in the state a:. Then ) ,Q2, a) = c,+b, 1C?“). In this notation
(3.4)
the strangeness analog state can be written (3.5)
The sum over all states 01 defines the strangeness analog state in a general formulation that no longer depends on the validity of the independent-particle shell model. The symmetric operator clearly preserves the symmetry of the state and Eq. (3.5) is equivalent to Eq. (3.3a) for the case of the independent-particle model. Equation (3.5) defines the state 1S) for any many-body wavefunction in just the way required to give the enhancement relation between the matrix elements (3.3b). The expression (3.5) resembles the operation of an isospin generator on a nucleus. Instead of changing a neutron into a proton and leaving it in the same state, it changes a neutron into a (1. In direct analogy to the isospin group, one can define an SU(2) group in which neutrons and As are changed into one another. This group has been called U spin [6] in the context of the Sakata model for elementary particles.
STRANGENESS
ANALOG
747
RESONANCES
The isobaric analog state / A) for a given parent state 1 n) is given by [7] IA> =
(3.6a)
T+ I 7~).
Z&-T
The state (3.5) is analogously written in the U-spin formulation 1sj = &
u- / P),
(3.6b)
where the U-spin generators are defined by the relations U- = 1 c,+ba ,
(3.7a)
U+ = c b,+c, ,
(3.7b)
U, = $ C (b,+b, -
c,+c,)
(3.7c)
The particular state (3.6b) will be an approximate stationary state of the Hamiltonian to the extent that U spin is a good symmetry for these states. The validity of this symmetry is considered in detail in Section V. From Eqs. (3.2), (3.4), and (3.7a), we see that the transition matrix element for the production of any state 4 is proportional to the matrix element of the operator U- between the parent state and 4; i.e., (T-, cj 1F 1K-P’)
= (4 I U- I Cl2)(.rr-fl = fd( r-/l
/ F I K-n> F(q)
17 / K-n) F(q),
(3.8)
where the enhancement factor is
f* = (4 I u- I C12>. The sum of the transition satisfy the sum rule T I(r,
(3.9)
strengths to a complete set of states 4 is seen to
4 I F 1K-P)j2
= T fd2 l(rA
I F I K-n)12 [F(q)12
= N l(r-~‘I j 9 j K-n)j2 [F(q)]“.
(3.10a) (3.10b)
The factor N simply expresses the possibility that the strangeness exchange reaction can occur on any one of the N neutrons in the target. The enhancement factor f+ thus describes the extent to which the total strength is concentrated in the particular state ~5.
748
KERMAN
AND
LIPKIN
In similar manner, one can define analog states in which a proton is changed into a II. These states would be produced in a reaction of combined charge and strangeness exchange, e.g., a reaction in which the incident K- becomes a no. For these states, operators analogous to (3.7) can be defined. These operators, called V-spin operators in the Sakata model, are
V+= COLcata, ,
(3.1 la)
V- = C a,+c, ,
(3.11b)
V, = * C (c,+c, - aeta,).
(3.1 lc)
m
01
The corresponding enhancement factor for the production combined charge- and strangeness-exchange reaction is then h = (9 I v+ I C12).
of a state I# in the (3.12)
All this discussion is clearly general and applies to the production of strangeness analog states on any target, not necessarily C12. If the parent nucleus is a self-conjugate nucleus such as C12, N14, 016, etc., the U-spin and V-spin strangeness analog states form a pair of states belonging to the same isodoublet and thus have very similar properties. If the parent state is a heavy nucleus with a neutron excess, the U-spin and V-spin analog states have different isospin properties. The V-spin analog state produced by changing a proton into a (1 has a larger neutron excess than the parent state and therefore its isospin is larger by l/2 than that of the parent state. It is an isospin eigenstate since the A has zero isospin and does not contribute to the isospin of the system. The U-spin analog state produced by changing a neutron into a A has more complicated isospin properties because the neutron hole produced in the parent state can be either in the neutron excess or in the core. The resulting nucleus has a neutron excess smaller than that of the parent nucleus, and the isospin of the neutron hole can be coupled to the isospin of the parent in two ways giving an isospin either greater than or less than the isospin of the parent. The U-spin analog state therefore is a linear combination of two isospins with coefficients depending upon the action of the U-spin generator on the particular state in the Sakata supermultiplet. The neutron or U-spin analog state should be split into its two components having pure isospin, since the isospin splitting due to symmetry energy is appreciable. The strangeness-exchange reaction should thus produce two separated peaks rather than a single peak corresponding to a state without a well-defined isospin. The details of this splitting are discussed in Sections IV and V.
STRANGENESS ANALOG
IV. STRANGENESS ANALOG
749
RESONANCES
STATES IN THE SAKATA
MODEL
The octet model of unitary symmetry, which has been successful in particle physics, is not useful for nuclear and hypernuclear physics. The principal reason is the (12 mass difference of 80 MeV, which is small on the energy scale of particle physics but very large on the energy scale of nuclear binding energies. Hypernuclei observed in nature are known to contain a A and not a Z since a Z could decay into a II and provide an excitation energy of 80 MeV to the nucleus. The octetmodel description of hypernuclei would classify them in states containing linear combinations of As and Es. The old Sakata triplet model [2] attempted to create all hadron states out of the basic npfl triplet. This model is no longer relevant to hadron physics and has been superseded by the octet. However, the complex nuclei and hypernuclei having strangeness 0 and -1 are indeed composed only of members of the basic Sakata triplet. It is therefore useful to consider hypernuclear spectroscopy from the point of view of the Sakata model by using the triplet rather than the octet model of unitary symmetry. A similar situation exists in the well-established applications of the Wigner supermultiplet SU(4) group in particle and nuclear physics. Particle physicists classify the nucleon in the 20-dimensional representation of SU(4), which contains the d (3-3 resonance) as well as the nucleon. Nuclear physicists classify the nucleon in the four-dimensional representation which contains only the nucleon in its four spin and charge states. The 300-MeV A-N mass difference can be considered small in some sense in particle physics and one can consider symmetry transformations which change nucleons into As. In nuclear physics it is clear that nuclei contain nucleons with only a tiny admixture of the A. (For a crude estimate of this sort of mixing in the deuteron see Ref. 8.) Triplet unitary symmetry assumes that the cl-nucleon interaction is the same as the nucleon-nucleon interaction. Although this symmetry is a very poor approximation for the two-body system (in contrast to charge independence, which is very good and violated only by small electromagnetic contributions), we believe that the Sakata symmetry may be useful for the many-particle states having strangeness 0 and - 1 (the states that so far have been found to be relevant to hypernuclear spectroscopy). For these states the primary symmetry breaking is due to the difference between the nucleon-nucleus and the L-nucleus interactions rather than the two-body interactions; i.e., the validity of the shell-model description of complex nuclei allows us to replace the major portion of the two-body interaction by an average field in which the single nucleon or hyperon moves. The radius of the baryon-nucleus interaction potential is determined by the size of the nucleus and is independent of whether the baryon considered is a nucleon or a (1. The latter affects only the depth of the well. Thus the major breaking of
750
KERMAN
AND
LIPKIN
the Sakata symmetry for these states results from the difference in the well depths seen by the nucleon and the fl. As we have seen in Section II, this gives a large diagonal contribution to the energy of a nucleus or hypernucleus, but may not change the wavefunction too much from that given under the assumption of Sakata SU(3) symmetry. The symmetry-breaking effects that would mix in other SU(3) representations and destroy the purity of the Sakata state are smaller effects depending on differences in the shape of the neutron and d wells and differences in the residual two-body interaction. The SU(3) algebra used in the Sakata model is well known. The algebra is generated by the nine operators of isospin, U spin, and V spin. Only eight operators are independent because of the relation To + Uo + Vo = 0
(4-l)
The isospin, U-spin, and V-spin operators satisfy angular-momentum commutation rules among themselves. The commutation relations that hold between different spins are (4.2a) K’+, v+l = T-2
w+ 9v-1 = 0,
(4.2b)
[V, ) u-1 = +JJ- )
(4.2~)
[Uo, V+l = iv+.
(4.2d)
Other relations are obtainable from these by taking Hermitean conjugates and cyclic permutations of T, U, and V. The parent state from which the analog state is produced is a nuclear ground state containing Z protons and N neutrons. This state 1 n) is a simultaneous eigenfunction of T,, , U, , and V. ; i.e., T, I n> = ii@ - N) I n>,
(4.3a)
uo I r> = W I 71.>, v, 17i-) = -$Z 1 7r),
(4.3b) (4.3c)
where we have used the conventions of particle physics for the eigenvalues of T (which are listed in Table I). TABLE I The Eigenvalues oft, , uO , and v,, for the Single Baryon States in the Sakata Model ?I
to RI 00
P
1 42 0
-*
4 0
A 0 1 &
STRANGENESS
ANALOG
RESONANCES
751
Since nuclei contain no As, it follows that
u+I r> = 0,
(4.4a)
v- I 7r) = 0.
(4.4b)
In all cases of practical interest, 1 rr> is a low-lying nuclear state which has a neutron excess. We shall neglect isospin violation and assume that it is an eigenstate of isospin on the edge of the isospin multiplet. Therefore T- 1 n) = 0.
(4.4c)
The isospin of the state is given by Tz
N-z
2 Using the above properties of these operators, we can calculate useful expectation values, namely (9~ j V-V+ I T) = (7~ 1 [V-, (T I V-T-U(n [ V-T-T+V+
I rr) = (rr I [V-,
V+] 17~) = -2(37 / V, I 77) = Z, [T-,
U-11 I n-> = -Z,
I z-) = Z(2T + 1) = Z(N - Z + 1).
(4Sa) (4Sb) (4.k)
The simplest strangeness analog state is the V-spin analog state obtained by changing a proton into a A, i.e., the state /S,)
=
“;g;
)
where the normalization factor is given by Eq. (4.5a). This state is an isospin eigenstate with the eigenvalue T + &. The corresponding state generated from the parent state by a U-spin generator which changes a neutron into a fI is
However, this state is not an isospin eigenstate if N is greater than Z and would therefore not be interesting in itself. We therefore separate this state into its two components S, and S, having isospins T + & and T - 4, respectively. The state S, is the isobaric analog of the V-spin analog state / S,), namely [ s,;
= d:;
“yji
= --T+v,
z/Z(N
IL,
- Z + 1)
(4.8a)
752
KERMAN
AND
LIPKIN
The other state S, , obtained by the Schmidt orthogonalization the states (4.7) and (4.8a), is ,s
)
<
=
[U-
+
~+V+@T
procedure from
+ 111I r>
d(N +1)2Tl(2T+1)-
.
(4.8b)
This state can be seen to be normalized and to be orthogonal to the state (4.8a). We can now calculate the enhancement factors f for the production of these analog states in a strangeness-exchange reaction. These are given by the matrix elements between the parent state and the analog state of the operator V+ for the V-spin analog states and of U- for the U-spin analog states. Using the definitions (4.6), (4.7) and (4.8) and the expectation values (4.5), we obtain
f, = (S,1v+177)= (n ’ g+ ’ r, = fl (n- I --= V-T-U1/(2T+l)z
f>=(&-lU-I~)=
=
I rr)
tN + ‘1 (2T2;
V. THE VALIDITY
OF
-(2Tz+
1)’
1) ’
(4.9b)
(4.9c)
SAKATA SYMMETRY; CONCLUSIONS
We now examine the breaking of the Sakata SU(3) symmetry in detail and formalize the qualitative arguments, given in the Section IV, which indicate that this breaking is small. By analogy with the case of isospin, we write the two-body force Wij as the sum of a symmetry-conserving term and a symmetry-violating term and use U-spin and V-spin operators in the violating term. However, our formulation differs from the isospin case in that we are considering only nuclei containing no As and hypernuclei containing a single A. We therefore need not consider the A-A interaction and can express a two-body interaction in a form that holds only for the nucleon-nucleon and nucleon-A interactions. We therefore need only a constant term and a term that is linear in the U-spin and V-spin operators in our expansion of the two-body force. A quadratic term would be needed, as in isospin, if we also considered the A-A interaction. We write
STRANGENESS
ANALOG
RESONANCES
153
where Wij is the two-body interaction for a baryon pair (which can be either &nucleon or nucleon-nucleon), Wz! is the interaction between particles i and j if they are both nucleons, u ,,i, u,,j, v,,~,and vOjare the eigenvalues of the u- and v-spin operators for particles i and j, and d wij is the difference between the nucleonnucleon and nucleon-/I interactions, i.e., (5.2a) We neglect isospin-symmetry breaking. This can be included in the formulation by adding the usual terms depending on the isospin generators. Equation (5.1) is seen to give the correct values for the nucleon-nucleon and nucleon-/l interaction since (uol + zlo2- vol - vo2 - 1) 1nn) = 0,
(5.2b)
(uol + urJ2- Vll - VI2 -
(5.2~)
1) / nn> = -8 1nA).
The total interaction for the nuclear or hypernuclear system is w = 3 c wij i#j
(5.3)
As in the isobaric case [7], the symmetry-breaking effects depend upon the commutation relations between the Hamiltonian and the U-spin or V-spin generator which creates the analog state from the parent. We neglect the symmetry breaking in the kinetic energy terms due to the cl-nucleon mass difference as a smaller effect and consider the commutator with the interaction. The relevant commutators are
w;-’ -
[u_ , w] = & 1 (u-i + u-j) A W, i#j
W?’ = [V+ , W] = $ 1 (u+i + t)+j) A wij, i#j
(5.4b)
where the total U spin and V spin are defined as the sums of the II and v spins of the individual particles-i.e., u=p
z
v = c vi,
(5.5a)
754
KERMAN
and we have used the commutation
AND
LIPKIN
relations
[ix ) (uoi- v,i)] = $Li,
(5.5c)
[V+ ) (uoi - v,i)] = gv+i.
(5.5d)
For simplicity we consider first the V-spin analog state that is an isospin eigenstate. Since the energy of this state is just the expectation value of the Hamiltonian, we see from Eqs. (4.6) and (4.4b) that the binding energy difference dE, between the analog state and the parent state is simply expressed in terms of commutators of V-spin operators with the interaction; i.e.,
dE, = CTI IV-, w, V+lI r> = _ CTI iv- ~z%+lI n> z = 4 (3-r1 c (2):+ voj)Awij I Tr). i#j This is again analogous to the isospin case except that here the parent has good V spin and there is no correction to the double commutator [7]. If particle i is a proton, the expression is (~~~vg~dw~~~~)=g~(~~(w~~j j
W$)pT).
(5.7a)
If particle i is a neutron, the analog of the left side of Eq. (5.7a) vanishes, since voi = 0 for a neutron. The right side of (5.7a) is just the difference between the average fields on a proton and on a A. The summation over i in Eq. (5.6) and the inclusion of the two u. terms give a factor 22 which cancels these factors in the denominator to give the result
LIE, = --(A V,,)
m +25 MeV,
(5.7b)
where d V,, is the difference in the interaction of a proton in the nucleus and that of a A with the same wavefunction. In a single-particle model
(5.7c) where the q& are the single-particle wavefunctions. The result (5.7) is in agreement with our previous simple observation that the energy shift of the analog state relative to the parent state is just the difference between the depths of the potentials seen by a nucleon and a A. The energies of the two U-spin analog states are easily obtained from the energy of the V-spin analog state. Since I S,) is the isobaric analog of ] S,), its energy
STRANGENESS ANALOG
RESONANCES
755
differs from that of / S,) by the Coulomb energy dE,,,, , as shown in Fig. 3. The state I S,) is then lower by the usual nuclear symmetry energy.
FIG. 3. Schematicrepresentationof the strangenessanalog states and the transitions from the
parent. The decay widths are also expressible by use of the commutators (5.4). The dominant term in the amplitude for the decay of the analog state to any state @ is given by [7] by (5.8) There are two kinds of decay mechanisms: (1) the escape of a single particle, in which case the state @ has a particle ejected from the nucleus; and (2) damping through compound-nucleus states. We first consider L! escape. The relevant matrix element resembles Eq. (5.7a) except that it is an off-diagonal matrix element between the state r and the state @, and the operator is V+ instead of V”. As in the case of Eq. (5.7a), the summation over j gives the average field due to all the other particles, as seen by particle i. The escape of the A is therefore due to the change in the average field seen by the particle as it changes from a proton to a (1. The diagonal matrix element of this change in average field gives the energy shift as in Eq. (5.7b). The off-diagonal matrix elements describe the escape of the fl from the nucleus. In the single-particle model, the fl escape amplitude is given by (5.9) The average field appearing in (5.7) and (5.9) can be compared with the Coulomb field responsible for the shift and width of isobaric-analog states. The average symmetry-breaking field in the hypernuclear case is of the same order of magnitude as the Coulomb field in the isobaric case, even though the relevant parameter
756
KERMAN AND LIPKIN
characterizing the long-range Coulomb interaction in Z/137. Because of its long range, its average field strength can be comparable to the strength of the field produced by the hypernuclear symmetry-breaking interaction, which is short range. The additional barrier-penetration factor (present in the isobaric case and absent in the hypernuclear case) must also be considered, but this is not a very large factor. All other escapes involve single two-body matrix elements which are not summed over particles to give an average field. For these the only terms in the expression (5.4b) that contribute to the matrix element (5.8) are those in which the pair zj is uniquely defined as consisting of the proton that is changed to a (1 and the nucleon that escapes. Since these two-body matrix elements are not summed, they should be less important than /.I escape and should give a smaller contribution to the width. The damping contributions cannot be calculated without some detailed model, for which collective states may be important. However, we expect that the damping mechanism is similar to that for the damping of other collective particle-hole states. Since the operator appearing in expression (5.8) is only the difference between d-nucleon and nucleon-nucleon interactions while in the nuclear case it is the full nuclear interaction, this may not be large. When experimental data become available, the values of the widths of the analog states will give us specific information about the strength .4 W of the difference between the nucleon-nucleon and d-nucleon interactions.
ACKNOWLEDGMENT We should like to acknowledge stimulating S. Okubo, M. Peshkin, and J. P. Schiffer.
discussions with A. R. Bodmer,
D. Kurath,
REFERENCES 1. D. H. DAVIS AND J. SACTON, in “Proceedings of International Conference on Hypernuclear Physics,” (A. R. Bodmer and L. A. Hyman, Eds.), p. r59, Argonne National Laboratory 1969; see also R. H. DALITZ, p. 708. 2. S. SAKATA,
Progr.
Theor.
Phys.
16 (1956),
686.
3. J. D. ANDERSON, C. WONG, AND J. W. MCCLURE, Phy& Rev. 126 (1962), 2170; R. A. MADSEN, in “Nuclear Isospin.” (J. D. Anderson, S. D. Bloom, J. Cemy, and W. W. True, Eds.), p. 315, Academic Press, New York/London, 1969. 4. H. J. LIPKIN, Phys. Rev. Lets 14 (1965), 18; H. FESHBACH AND A. K. KERMAN, in “Preludes in Theoretical Physics,” (A. de-Shalit, H. Feshbach, and L. Van Hove, Eds.), p. 260, NorthHolland, Amsterdam, 1966; L. KISSLINGER, Phy.s. Rev. 157 (1967), 1358.
STRANGENESS
ANALOG
RESONANCES
757
5. A. R. BODMER AND J. W. MURPHY, Nucl. Phys. 64 (1965), 593; A. R. BODMER, in “High Energy Physics and Nuclear Structure,” (G. Alexander, Ed.), p. 60, North-Holland, Amsterdam 1967; A. GAL, J. M. SOPER,AND R. H. DALITZ, Ann. Phys. New York (1971). 6. H. J. LIPKIN, “Lie Groups for Pedestrians,” North-Holland, Amsterdam, 1965. 7. A. K. KERMAN, in “Nuclear Isospin,” (J. D. Anderson, S. D. Bloom, J. Cerny, and W. W. True, Eds.), p. 315, Academic Press, New York/London, 1969. 8. A. K. KERMAN AND L. S. KISSLINGER, Phys. Rev. 180 (1969), 1483.