Symmetry breaking in baryons

Volume 74B, number 4, 5 PHYSICS LETTERS 17 April 1978

SYMMETRY BREAKING IN BARYONS ~

Nathan ISGUR 1 Department of Theoretical Physics, University of Oxford, Oxford, England

and

Gabriel KARL Department of Physics, University of Guelph, Guelph, Canada

Received 14 February 1978

Assuming universal confining forces between quarks we discuss a new symmetry breaking mechanism specific to baryons: a pair of orbital states which are degenerate when all three quarks have the same mass splits up when the quarks have different masses. This effect is operative in all low mass negative parity hyperons. The observed inversion of A5/2-(1830) and 1~5/2-(1765) relative to the ground states A1/2÷(1115) and S1/2+(1190) is one consequence of this mechanism. The zero-order strange eigenstates of the confining hamiltonian (without hyperfine interactions) are similar to the states in the approximate mixing schemes of Petersen and Rosner and of Faiman.

There is wide consensus about the principal assumptions on which a (future) theory of hadrons (com- posed of quarks) should be based. According to one of these assumptions the confining forces between quarks are universal: the same for all flavors and spin orientations of quark. In addition to the confining forces there are spin dependent - hyperfine and sp in -o rb i t - couplings between quarks. The mass splitting between the ground state ~21/2 + and A1/2 + is a simple consequence of the hyperfine interactions provided we accept a very plausible assumption of de Rfljula et al. [1] about the "color" magnetic moment of quarks. In this note we discuss a second, new mechanism which splits negative parity ~ and A hyperons and which relies on (and therefore tests) the idea of universal quark confining forces.

We first recall the nonstrange members of the fami- ly of negative parity baryons. These states are inter- preted [2,3] as p-wave excitations of the system of

,7 Research supported in part by the National Research Coun- cil of Canada and the Connaught Fund of the University of Toronto.

I Permanent address: Department of Physics, University of Toronto, Toronto, Canada.

three equal mass nonstrange quarks. There are two degenerate orbital states for each value o f m (= 1, 0, - 1 ) . In one of these states (which we call the 0 state) the p-wave excitation is localized in the pair of particles 1 and 2, while in the other linearly independent state (called the ?,-state) the p-wave excitation is in the relative motion of particle 3 about the centre of mass of particles 1 and 2. In group theoretic language this degeneracy arises because the hamiltonian of the system of three equal mass particles with identical forces in all pairs is invariant under the permutat ion group of the three particles S 3. The lowest L = 1 excitation of this system belongs to the two-dimensional representation of this group. These two orbital wavefunctions ~10m and ~(m are combined with the spin and isospin wavefunctions of the three quarks according to the prescriptions of the symmetric quark model [2] to give the well known 2N, 4N and 2A states; the spin and orbital angular momentum are then coupled to give states of fixed total angular momentum J. Fi- nally, if one diagonalizes the hyperfine interaction in this set of states, the resulting spectrum and mixing angles are in good agreement with experiment [4].

What changes when we turn to the strange counter-

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parts of these states? All we do is replace one of the three non-strange quarks with a strange one; since we assume that the confining forces remain unaltered between all pairs of quarks, the only changes in the hamiltonian are due to the strange quark being heavier than the non-strange quark. Let us first ignore the hyperfine forces to discuss the zero-order eigenstates of the confining forces. It is clear that the hamiltonian has lost the permutat ion group S 3 as its invariance group since the quarks have different masses. As a re-

sult the two previously degenerate orbital states are

n o w split. It is no longer equivalent whether the p- wave excitation is localized in the non-strange pair or in the relative motion of the strange quark with the non-strange pair. While this result is general to any pair potential between quarks (and probably holds even in the bag model , 1 ) it is particularly easy to verify with a harmonic oscillator model.

In a harmonic oscillator model with the same spring constant between all pairs of quarks, the three-quark system is equivalent, after elimination of the centre of mass motion, to two uncoupled harmonic oscillators (the p and k oscillators). These two oscillators have the same frequency provided the three particles have identical masses. If one of the three quarks has a mass different from the other two, then the two oscillators have different frequencies which are related by

co o - co x = coo(1 - X/(2x + 1)/3), (1)

where x - 1 is the ratio of the mass of the odd quark to the mass of one of the other quarks (if the odd quark is strange and the other two non-strange x ~ 0.6). We emphasise that this splitting of the degeneracy of a

+1 In the bag model our p state corresponds to the configuration in which a non-strange quark is in a p-mode while the strange quark and the other non-strange quark are in the Sl/2 mode. Our X state corresponds to the configuration in which a strange quark is in a p-mode while the two non- strange quarks are in the Sl/2 mode. The splitting between these two configurations has been noted earlier by De Grand and Jaffe [10] in the case of the Pl/2 mode, and our estimate (1) agrees in sign and magnitude with the dif- ference of their quoted kinetic energies. Unfortunately the P3/2 mode was not calculated in ref. [10] and this pre- vented the authors from dealing with the J = 5/2 states where this splitting is manifested directly. We thank Dr. Hasenfratz for help with this question and also for an estimate of the splitting in the P3/2 mode which indicates that the same sign is relevant also in this sector.

mixed pair of states has nothing to do with hyperfine interactions. Aside from this split in frequencies, the wavefunctions ~ P and ~ x also suffer modifications from their equal mass limit, but we shall not explicitly deal with those effects here.

There is a simple physical example of a Y~-A splitting which is a direct manifestation of the mechanism just described. The relevant states have JP = 5 / 2 - , and are "fully stretched" in the coupling of the orbital angular momentum L = 1 - to the total quark spin S = 3/2. Experimentally, this particular pair of states A 5 / 2 - ( 1 8 3 0 ) and Y~5/2- (1765) is inverted relative to the pair A1/2 + (1115) and Z 1/2 + (1190) of the ground states. We can easily see that this is a consequence of the splitting of the p and ?` normal modes. The quartet spin wavefunction of the A 5 / 2 - and Y~5/2- is totally symmetric under permutations. The A 5 / 2 - , with zero isospin, is antisymmetric under the exchange of the non-strange quarks and therefore must correspond to the p-orbitally excited state which is also antisymmetric under the same permutation. Similarly we find that the ~25/2-, which has unit isospin, nmst have the ?`- orbitally excited state. These two orbital states are non-degenerate in zeroth order: the ~25/2- and A 5 / 2 - are split before the hyperfine interaction is introduced. It is easy to appreciate qualitatively that the ?,-oscillator, which has higher reduced mass, will have the low- er frequency and thus £ 5 / 2 - should be lighter than A5/2. Quantitatively, using formula (1) with an effec- tive frequency of hco o = 520 MeV (this is the value appropriate from ref. [4]) a n d x = 0.6 we find

M 0 ( A 5 / 2 - ) - M 0 ( ~ 5 / 2 - ) = hcop -- hco x ~ 75 MeV,

(2)

in reasonable agreement with the observed mass differ-

ence [81

A(1830) - 2;(1"165)= (1825 +-- 5) - ( 1 7 7 4 -+ 5) (3)

51 +- 15 MeV.

The effect of the hyperfine interactions, while small, accounts for this discrepancy: when evaluated with harmonic oscillator wavefunctions [5] these interactions reduce the predicted splitting to ~ 50 MeV. The relative ineffectiveness of the hyperfine interaction in this case is due both to these states having identical spin structure and to their being coupled to the highest angular momentum possible.

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While this case offers a clear illustration of the loss of permutat ional symmetry in hyperons (hw o ~ tiwx) the same segregation of zero-order states correspond- ing to p excitations or k excitations also takes place in the other angular momentum sectors JP = 3 / 2 - and 1 /2 - . However, these sectors are each three-dimensional (4p, 2p and 2X in the A sector and 4X, 2 x and

2p in the E sector) and therefore the hyperfine interactions are more important in shifting and mixing these states. We have diagonalized the hyperfine interaction in these sectors using a set of zero-order states as discussed above and found very good agreement with experiment both for masses and for mixing angles. The results are rather bulky and will be discussed elsewhere in detail [5], so we only show in fig. 1 a comparison of the predicted spectrum with the experimental data.

In our discussion above we have tacitly assumed, as has ref. [1], that the overall wavefunction of hyperons is symmetric only under permutations of (equal mass) non-strange quarks. This is analogous to the res- striction of the Pauli principle to "equivalent" elec-

trons in an atom, i.e. those which belong to the same orbitals. Although one can symmetrize with respect to the strange quarks as well, no additional constraints arise. It is easy to verify that the same number of states with the same quantum numbers arise indeed whether one symmetrizes all the quarks or only the non-strange ones.

It is amusing that the hyperon eigenstates of the zero-order mass matrix which arise here simply as a consequence of the universality of quark confining forces are closely related though not identical to the eigenstates of an approximate mixing scheme first proposed by Petersen and Rosner [6] for the A 1 / 2 - sector and later generalized by Faiman [6] as "ideal mixing for baryons"• These authors speculated that the observed hyperon eigenstates are linear combina- tions of SU 6 eigenstates (21, 28 and 48) defined by the requirement that certain of the states decouple from the KN and ETr decay channel. The states which decouple from the KN channel (with single quark emission operators) are precisely the p-excitations. This proper ty follows from the ant isymmetry of ffP

A

>~ 1700 i

" " 1600 o E 1500

140(

20001

1900 I

• . 63z8+ .64"8 - 4 4 2 1 t:fs. 8 . . ~ . . . . 39 ,t|

.90z l+ .43z8 + .0648

• 8 021 + . 6028 - . 0 448 v / / / ~ / z / / / / / / i =

2 2 4 92 I 0 ÷ . 3 3 8- - .21 8

. 8 2 2 8 + 5 4 % - 17Zlo

F///////////A

?

.9948 +.1128 - . 0 421

2 4 2 4

.92z8 - . 3 9 z l - . 0 0 " 8 . . . . ~ . ' z 8 . . . . . .

.92z1+.3928 - .0448

I I I I I

V/J/'//'//////J

I"//././///////

A"/2- y*'/2- A'%- y*%- A*%- z*%-

Fig. 1. Comparison of the predicted and observed spectrum of negative parity strangeness minus one baryons. The predicted com- position of a state is displayed directly above the bar indicating its position. The experimental co mpo sition, when known [9], is given in the most convenient location with respect to the shaded region which indicates its experimental position. The parameters used arex ~ 0.6, m s - m d ~ 280 MeV, hc~p = 520 MeV and ~ - N = 300 MeV.

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under the exchange of the non-strange quarks 1 and 2. A single quark operator must act on quark 3 if a strange meson is emit ted (as in K) but then the re- maining part of the wavefunction ~o is orthogonal to the ground orbital state. On the other hand the )t states do couple to the KN channel. This segregation of states into those that do and those that do not couple to the KN decay channel is valid only when quark hyperfine interactions are neglected, since these interactions mix P and X excitations. In practice, however, the hyperfine interactions preserve the "ideal mixing" to a good approximation. Thus the approximate mixing scheme of Petersen and Rosner and of Faiman receives dynamical justification from the universality

of quark confining forces. Similar and even more striking symmetry breaking

effects take place for higher excitations of a harmonic oscillator system. For example, there are [7] three L = 2 + excited states o f N = 2 which are degenerate for equal mass quarks. These states split up equidis- tantly if one of the quarks has a different mass. The three states which in the degenerate case are a "mixed" pair and a totally symmetric state break into XX, ),p, and pp excitations. The lowest eigenstate is the ?iX

excitation. In summary we have discussed in the context of

the negative parity baryon system a new symmetry breaking mechanism based on flavor independent quark confinement. The underlying assumption of universality o f quark confining forces is supported by

our investigation. At the same time, the symmetry. breaking described here provides a physical mechanism for the approximate mixing schemes previously

proposed in the literature [6].

We thank Jonathan L. Rosner for drawing our at- tention to the mixing schemes of ref. [6]. One of us (N.I.) thanks R.H. Dalitz for hospitali ty in the Depart- ment of Theoretical Physics and for his kind interest

in this work.

References

[ 1 ] A. de Rfijula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147.

[2] O.W. Greenberg, Phys. Rev. Lett. 13 (1964) 598. [3] R.H. Dalitz, in: High energy physics, Ecole d'Et6 de

Physique Th~orique, eds. C. De Witt and M. Jacob (Les Houches, 1965) (Gordon and Breach, New York, 1966).

[4] N. Isgur and G. Karl, Phys. Lett. 72B (1977) 109. [5] N. Isgur and G. Karl, submitted to Phys. Rev. D. [6] W.P. Petersen and J.L. Rosner, Phys. Rev. D6 (1972)

820 (see in particular column 4 of table lI); D. Faiman, Phys. Rev. D15 (1977) 854.

[7] See, for example: G. Karl and E. Obryk, Nucl. Phys. B8 (1968) 609.

[8] G.P. Gopal et al., Nucl. Phys. Bl19 (1977) 362. [9] D. Faiman and D.E. Plane, Nucl. Phys. B50 (1972) 379;

A.J.G. Hey, P.J. Litchfield and R.J. Cashmore, Nucl. Phys. B95 (1975) 516.

[10] T.A. De Grand and R.L. Jaffe, Ann. Phys. 100 (1976) 425.

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Symmetry breaking in baryons