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General three-spinor wave functions and the relativistic quark model
ANNALS
OF PHYSICS
93, 125-151 (1975)
General Three-Spinor
Wave
Functions and the Relativistic
Quark
Model
A. B. HENRIQUES, B. H. KELLETT, Department
of Natural
Philosophy,
AND R. G. MOORHOUSE
University
of Glasgow,
Glasgow,
Scotland
Received March 11, 1975
The complete set of three particle relativistic spinors is constructed in a set of evidently covariant forms classified into irreducible representations of the permutation group, S, , for use in the three quark model of baryons. A few of the various types of three quark spinor functions are illustrated by evaluating their electromagnetic current interactions using the four dimensional oscillator models of Feynman, Kislinger, and Ravndal and Lipes for the spatial part of the wave function: one of our types gives markedly better results for nucleon electromagnetic form factors than the type corresponding to the original models of these authors, but problems, largely associated with the four dimensional oscillator, remain in the electromagnetic transitions to higher baryon resonances. 1. INTR~OUCTI~N
The problem of constructing a covariant formulation of the phenomenologically successfulnonrelativistic quark model has attracted considerable attention in recent years. An important feature of this enterprise is the treatment of the spin structure of composite systemsconsisting of two or three spin i-quarks. The general form of the two-spinor wave function was given many years ago by Goldstein [l], and these forms have been widely used for the quark-antiquark mesonic system [2-41. In this paper, we consider the relativistic spin structure of the three-quark baryonic system. The general form of the three-spinor wave function can be developed as a natural extension of the two-spinor system, but the situation is considerably complicated by the necessity for maintaining the symmetry properties of the wave function under quark interchange. To this end we express the general covariant spinor basisstatesas irreducible representations of the permutation group of three objects S, . Wave functions with the desired symmetry properties, definite parity, and definite total spin, can then be constructed by combining the spinor part with the center of mass (c.m.) momentum, which is a symmetric singlet under S, , and the two independent relative momenta, which form a mixed symmetry doublet under S, . Previous attempts to formulate a relativistic quark model for baryons [5-91 have 125 Copyright All rights
Q 1975 by Academic Press, Inc. of reproduction in any form reserved.
126
HENRIQUES,
KELLETT
AND
MOORHOUSE
treated the spin structure in a rather arbitrary way. The models of Refs. [5-71 are based on the requirement that the wave function reduce to the usual SU(6) positive energy (Pauli) spinor form in the hadron rest frame. Such a form excludes, inter alia, any dependence on the internal quark momenta and such momentum dependence may be important for obtaining agreement with experiment, as has been claimed by Le Yaouanc et al. [9]. It is clearly desirable to investigate other possibilities, though using the most general relativistic spin structure does not guarantee phenomenological success, since this finally depends on having a realistic dynamical model. To get some feeling for the use of the general spin functions in practice, we investigate the simple harmonic oscillator model of Feynman, Kislinger, and Ravndal [5] and Lipes [7]. It becomes clear on comparison with experiment for the baryon-electromagnetic current interaction, that even with the generalized spinor structure, this simple interaction has quantitative inadequacies comparable to those of the nonrelativistic quark model. Although this latter model has good qualitative agreement with photoproduction experiments [for the (56,0+) and (70, l-) multiplets], one would have hoped for some improvement in models claiming to be relativistic1 On the other hand, we do find that the undesirable form factors obtained in previous work are not a necessary consequence of the harmonic oscillator interaction, so the way is open for a more realistic treatment of the problem using a sum of two-body oscillator potentials in a spin-dependent equation, such as the Bethe-Salpeter equation. Technically, such a program is very complicated because of the large number of independent spinor forms available for the construction of the individual wave functions; thus, we restrict ourselves in this paper to the development of the formalism, and make some general comments, leaving the detailed dynamical applications for future work. In Section 2 we develop the general form of the wave function. The generalized Fierz transformation for determining the symmetry under quark interchange is described in Section 3. This allows the classification of the general forms into irreducible representations of S, and the construction of wave functions with definite spin, parity, and symmetry properties. In Section 4 the simple oscillator model of [5] and [7] is used to investigate the interaction with the electromagnetic current and we detail the behavior of the nucleon form factors, with particular reference to the connection with earlier work. The 4(1236) wave function, and the rival transition are considered in Section 5, and we make some general comments on the electromagnetic transitions to higher resonances in the simple model. Some technical details and a complete list of the possible covariant spinor wave functions, classified according to the representations of S, , are given in the Appendix. 1 Note, however, that in the nonrelativistic quark model the magnetic moment of the proton is an arbitrary parameter of the theory; an achievement of the model of [5] with respect to baryons is the (approximately) correct prediction of this number.
QUARK
MODEL
WAVE
2. THE THREE-SPINOR
FUNCTIONS
WAVE
127
FUNCTIONS
A relativistic spin J-particle is described by a four-component Dirac spinor, whether the particle is free or bound. In the single particle case, this spinor is the solution of the Dirac equation with an appropriate potential. When two or more spin &-particles mutually interact to form a composite system, the general wave function has the form of a product of the Dirac spinors for the individual constituents. Goldstein [l] has shown that for the two-fermion case, it is convenient to rewrite this direct product of spinors as an outer product, and expand the resulting 4 x 4 matrix representation in terms of the complete set of 16 Dirac matrices2 1, yS , y“, y5yUand c?’ = @(y”y” - r”r“). In this representation, operators acting on the second quark are represented by postmatrix multiplication by the transpose of the operator. The two-fermion wave function is written (2.1) Xao= *ol*BT where #a is the individual quark spinor, and
%3~xus’= XtYa4%3r)7
(2.2)
where O,,, is a general matrix operator for the second spinor. It is convenient to introduce the matrix A = C-ly,(= y1y3 in our notation), where C is the usual charge conjugation matrix. Then AyuTA-l = yu and A-l = AT = A+ = -A.
(2.3)
If we then introduce
Eq. (2.2) becomes
&4as~ = AdAOTA-%~
(2.5)
and the operator (AOTA-l) does not involve the transpose of Dirac matrices. The matrix +aocan then be expanded in terms of the complete set of Dirac matrices.3 2 Our metric and notation for the Dirac matrices are given in the appendix. In what follows we shall use Greek letters p. Y, X, etc., as Lorentz indices; Greek letters from the beginning of the alphabet, 01,fl, y, etc., as spinor indices; and the Latin letters a, b, c as quark labels. a In the qualk-antiquark system for mesons, it is convenient to use the charge conjugation matrix C rather than A, since then starting from the two quark system, 4 is the quark antiquark wave function as it stands. 595/93/l-2-9
128
HENRIQUES,
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AND
MOORHOUSE
For the three-quark system we wish to develop a similar generalized matrix representation for the direct product of three spinors. In general, this is achieved by selecting any two-quark subsystem as above and then taking the direct product with the third quark. There are clearly three ways in which one can select the two quark subsystem, and the transformation between the matrix representations corresponding to different initial choices determines the permutation symmetry properties of the composite wave function. We will label the quarks a, b, and c, and represent the (b, c) subsystem in the matrix form (2.4). The further four independent degrees-of-freedom associated with quark a could be incorporated by multiplying &,, by an arbitrary spinor basis, such as the four free-particle Dirac spinors in the rest frame, but it is more convenient to introduce a redundant basis at this stage. For the first quark we write raa~@), with the reduced single spinor basis I,V+) and I/J-), which in the hadron rest frame takes the form: (2.6)
and I’,,, is a general 4 x 4 matrix that can be expanded in terms of the Dirac matrices as usual. The general three-quark wave function is then written y&A = WMb4,Y We adopt the convention Eq. (2.7) so that
*
(2.7)
that the order of the spinor indices on Y is governed by YBW = w,
(b&Y
(2.8)
2
and so on. The Lorentz transformation properties of YUBYare those of the product of three spinors. Each spinor receives the boost denoted by the usual spinor operator S(A). Now ASyA) A-l = S-l(A), (2.9) so that
Y& = cw) rw4)ad
(W) $Q.*(W) WY4
43, .
(2.10)
The combined wave function therefore transforms as the product of a tensor of rank determined by the number of free Lorentz indices on I’ and q5,and a spinor $. For example, yU~Fi)(yUA)By transforms as a spin-up or spin-down Dirac spinor for do not * (f) or $(-), respectively. Since the permutation group transformations change the Lorentz transformation properties, it is convenient to subdivide the
QUARK
MODEL
WAVE
FUNCTIONS
129
general forms (2.7) according to the number of free Lorentz indices. We shall refer to those with no free indices as being of zero rank, and so on. Clearly, nothing new in terms of the spinor structure occurs above the fourth rank, since the metric tensor guV and the Levi-Civita tensor Euvr\pdo not form part of the spinor basis. The matrices r and $ involve only Dirac matrices, but in the formation of an actual physical wave function of definite spin and parity, the free indices are contracted with four-vector momenta, or polarization four-vectors. The individual quarks have momenta pa , pb , and pc , respectively, which we separate into c.m. momentum P and relative momentap and q according to p=P,+PP,+P,, P = Pb + PC - 2Pa >
(2.11)
rl = (3Y (PC - Pb), and the inverse
(2.12)
The momentum independent spinors are normalized conventionally as a product of three quark spinors: (2.13)
for each individual tensor component. Care is necessary,however, in normalizing linear combinations of wave functions of given rank, since the different tensor components are not necessarilyorthogonal. This point is discussedin the Appendix. States of definite parity can also be constructed. For a state of parity qp , we have
where p” = (p. , -p) etc.
130
HENRIQUES, KELLETT
AND MOORHOUSE
3. SYMMETRY PROPERTIES OF THE WAVE FUNCTION
By construction, the permutation symmetry of the three-spinor wave function reduces to the index symmetry of !P+, . We can define the generators of the permutation group S, by QaaKov = Yom, 7 (3.1) where Qab indicates the operation of interchanging quarks a and b, or more generally, of interchanging the quarks associated with the first two indices of UC, whatever their particular labels. Similarly we define Qbc and Qnc . The interpretation of the group as an index symmetry allows the result
Qac= QbcQnbQbc .
(3.2)
Thus, we need consider only two operators. The operator QbCon the state (2.7) is trivial, since A, yJ, and YJ are antisymmetric, while 757&A and uGVAare symmetric. Interchange of quarks a and b is more complicated since we interchange indices on two independent sets of Dirac matrices. However, the explicit spinor #a* in Eq. (2.7) is irrelevant for this interchange, and the point of the redundant basis for quark a becomes clear when it is realized that the action of Qob in Eq. (2.7) corresponds simply to the standard spinor reordering or Fierz transformation, familiar from crossing in nucleon-nucleon scattering for example [lo]. The Fierz transformation matrix is readily obtained from the completeness relation for the Dirac matrices
- (YsYJua~ biY”)so’ + t(%“)cm,(~93B~l~
(3.3)
A group of three objects can have four types of symmetry under permutations. These are the irreducible representations of the group S, , and comprise the completely symmetric S, the two mixed symmetry types 01and /3, and the completely antisymmetric state A. Acting on the singlet S, Qab = Qbc = 1, and on the singlet A, Qab = Qb, = - 1. The mixed symmetry states do not transform independently, and form a doublet, defined by the transformation matrices.
(3.4)
As an example, it is readily verified that the c.m. momentum
P defined above is a
QUARK
MODEL
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131
FUNCTIONS
symmetric singlet, while the relative momenta belong to the mixed symmetry doublet (g) . To illustrate the application of these general principles to the classification of wave functions of the form (2.7), we shall look in some detail at the zero rank spinor forms. We have five possible covariants of each parity in this case. For positive parity, these are the conventional p-decay covariants used in nucleonnucleon scattering s = (l)&
(43Y
p = (Y5La’ 6%43V v = bAd
w-4,,
A = -bwu)aa~ T = ~(4m~
(3.5) (Y&‘A)B~
(u““A)m
where we omit the common #n, , which is irrelevant for our present purpose. If we distinguish the covariants of Y,,, , which are those of Eq. (3.5) with the indices cy. and p interchanged, by a prime, Eq. (3.3) leads to the crossing matrix
where use has been made of the y-matrix identities given in the Appendix. Noticing that A and T are even under Qhe, while the rest are odd, it is straightforward to deduce that the combinations ;A and
___ form normalized
(3.7)
[S + PI
i
q&y
PS - 2p
+ VI
!
doublets under S, , while ~
1 2(6)‘/”
[V - s + P]
(3.8)
is a normalized antisymmetric singlet. The negative parity covariants can be obtained from Eq. (3.5) by replacing #carby
132
HENRIQUES,
KELLETT
AND
MOORHOUSE
(y5& for instance. This does not change the transformation and Qb, . Thus, for
properties
under Qab
we have the same multiplet structure as in Eqs. (3.7) and (3.8). In the Appendix we list the covariants for each rank and give their symmetry properties in terms of the irreducible representations of the permutation group. The individual covariants can be interpreted as generalized Rarita-Schwinger wave functions; thus, the spin associated with a given form depends on the free Lorentz indices as well as the isolated spinor for quark a. Further independent covariants of lower spin may be formed by contracting the Lorentz indices with P, p, and q. The overall symmetry properties are preserved by combining definite symmetry states according to the rules of the permutation group. These are listed in the Appendix. Not all the covariants formed in this way from the spinor states of the Appendix will be independent. Because the original three-spinor composite system has only 43 = 64 independent degrees-of-freedom, combinations with the available momenta in all possible ways can give at most 64 independent basis states for each definite spin, parity, and symmetry type. Each of these independent states can be multiplied by an arbitrary Lorentz scalar function of the momenta. The redundancy of possible states is a reflection of the spurious freedom introduced by the treatment of the third quark, so that once the symmetry and spin properties have been established, the covariants can be reduced to a simpler form. This must in fact be done so that the individual covariants can be normalized correctly. The possibility of spinor small components playing an important role in the relativistic situation means that the spectrum of possible states is very much richer than in the nonrelativistic (NR) three-spinor system. Combining the spinor covariants with the momenta, we can form states of any spin and parity in each of the four symmetry types. These possibilities, of course, may be restricted in a particular dynamical model, and in a realistic case the number of possible basis states of a given spin may be considerably less than the theoretical maximum. In the oscillator model of [5], for example, the eigenstates of the Hamiltonian are each of definite order in the internal momenta, so that the NR spectrum is reproduced with only a few additional specifically relativistic states. In particular, the parity doublets have negative norm.
QUARK MODEL WAVE FUNCTIONS
133
The full symmetric quark model wave functions are constructed by combining the spinor wave functions with the standard SU(3) wave functions according to the symmetry rules given in the Appendix. The spin and orbital angular momentum wave functions for the excited states of the quark model, in principle, are already contained in the development of the spinor states of the appendix with the momenta, but in practice it is simpler to construct the orbital angular momentum states separately, and combine them with the most general spin * and $-spinor states with the usual Clebsch-Gordan coefficients. In this way the quark model (70, L = I-) and (56, L = 2-l-), etc., and whatever other states are demanded by the dynamics, are readily constructed.
4. THE NUCLEONELASTIC FORM FACTORSIN A SIMPLEMODEL
To illustrate the use of the formalism developed in the preceding sections in a definite dynamical model, in this and the following section we investigate the electromagnetic properties of the baryons in a simple relativistic oscillator quark model. The most fully developed relativistic harmonic oscillator model for the baryons in the literature is that usedby Feynman et al. [5] and Lipes [7]. This model has the advantage, as mentioned above, that the ground-state oscillator wave function depends on the relative momenta only through an overall Gaussian function, and apart from this Gaussian, the excited states are homogeneous functions of p and q. Consequently, the number of possible independent spinor covariants at each level is severely restricted. Even so, there is considerably more freedom than is allowed for by Feynman et al. and Lipes, and we show that the spinor forms used by these authors are in fact equivalent, corresponding to just one possibility from several, and better results for the form factors in particular are achieved if one makes a different choice. In brief, starting from the harmonic oscillator equation [5], {3(Pa2 +
Pb2+ PC?+ hQ2K~a - x0)3f (Xb - x,)2 + (xc - x,)2] + C] Y = 0 (4.1)
and making a minimal electromagnetic coupling, assuming quark additivity, we deduce the electromagnetic current operator? (4.2) for an incoming photon of momentum k. We assume(frame-dependent) boundary 4X, x, andy are the coordinatesconjugateto P,
p,
and y, respectively.
134
HENRIQUES,
KELLETT
AND
MOORHOUSE
conditions relevant to the four-dimensional oscillator without a Wick rotation [l 11. The ground state solution of Eq. (4.1) in momentum space is then [6, 71:
1g) = -(&)2exp -
&
[2 (2)
- p2 + 2 (%j”
- qq.
(4.3)
For the excited states, the troublesome time-like excitations are avoided by using P” and q” - (P q/A43 P”, rather than p” and q” themselves. This postpones, rather than solves, the basic difficulty of interpreting the time-like excitations, which can have imaginary mass in this model, and we assume that the violation of unitarity is not serious for low lying states. The ground state of the NR quark model allows just three independent spin states: a mixed symmetry doublet of spin 4, and a symmetric singlet of spin 8. However, because of the relativistic presence of spinor small components, the ground state of the above model has rather more spin states. These are most readily specified by considering the constituent spinor composition in the rest frame. For all upper component spinors we have the NR states, but overall positive parity also allows states with one large and two small component constituent spinors. In the helicity 8 configuration these give a mixed symmetry doublet of spin 8 and a further symmetric singlet of spin +. For spin &, the mixture of upper and lower component spinors allows a further three mixed-symmetry doublets, and a symmetric and an antisymmetric singlet of spin 4. These spin 4 singlets, and the spin + doublet are relativistic states with no NR analogue and they do not have any place in the standard SU(6) classification of resonant states. In a symmetric quark model they give rise to a kinematically allowed additional .P = #+ octet, a &+ decuplet, and a ++ SU(3) singlet. There are certainly more independent states than
p” - (Pp/M2)
the spectrum
requires and the superfluous
states must be forbidden
by spin and
chirality-dependent dynamics giving rise, in some approximation, to selection operators multiplying the simple four-dimensional oscillator. We do not concern ourselves with such possible dynamics here. Now consider which combinations of the independent states might be possible candidates for the known physical baryons. There are four independent spinor forms for the spin &+ octet of the symmetric quark model (56, L = O+). These come from the two zero-rank mixed-symmetry doublets, and by reducing the higher order doublets, contracted with the total momentum P”, to their simplest form, we find the two remaining independent forms:
(4.4)
QUARK
MODEL
WAVE
FUNCTIONS
135
;A i
-&
12s - 2p
-1 JA i
(4.5)
(4.6)
(4.7)
where These quark uf = initial
we indicate the required momentum contractions in a obvious notation. doublets correspond to helicity 5s according to whether the spinor for a in the barryon rest frame is chosen to be 4t-b) = ($), or z/-j = (“i), where (3 and u- = Q) . To be definite, we choose the kinematics such that the state is at rest, and the incident photon propagates in the $2 direction: P” = (M, O), P’u = (E, 0, 0, k3), k” = (v, k) = (v, 0, 0, k3).
The helicity +$ cy.symmetry 1 2(6)‘/”
part of (4.4) for example, in the rest frame is %“P+Yo>(~~v~
(following the abbreviated notation of the Appendix), and the corresponding spinor for the final state is (1/2(6)l/3 T(P’) = (1/4(6)lj2) ~,&(+)(P’)(u”~). The nucleon wave function for this spinor doublet then has the form
where j S,), and I S,), are the SU(3) nucleon wave functions. The matrix element of the current (4.2) between nucleon states of the form (4.8) is readily calculated, and the charge and magnetic form factors can be read off immediately as the
136
HENRIQUES,
KELLETT
AND
MOORHOUSE
coefficients of co and io . Ex k/2M, respectively, where EUis the photon polarization vector. We find G,*(P)
= +$-
G(k2),
G,*(k2) = 3G(k7, (4.9)
WV2) = - -!f? 2M2 G(k2), GMn(k2) = -2G(k2), where G(k2) = (1 - &)-”
exp [$
(I - $$)-l].
(4.10)
Apart from the neutron electric form factor, these results are the sameas those of Lipes after his ad hoc neglect of an overall factor [l - (k2/4M2)]. The form factors show an asymptotic dipole behavior, and the average experimental shape is satisfactorily fitted for a value [7] Q = 0.78 GeV2. However, the old difficulty of fitting the charge radius in a harmonic oscillator model is not completely solved. For this value of Q, Eq. (4.9) gives (rD2)lj2 = 0.56 fermi, and to reproduce the experimental 0.814 fermi would require a > 0.44 GeV2, which is inconsistent with the observed level spacing of about 1 GeV2. The neutron charge radius, on the other hand, is independent of fin, and Eq. (4.9) gives (rn2j1J2 ‘v 0.36 fermi, which is in good agreement with the experimental result [12] of about 0.35 fermi. We recover the standard SU(6) result pp/f*?l = -$, but asymptotically, the ratio p”GE”(kz)/GMfl(k2) + 2, instead of being less than one [13]. The comparison of these results with the unmanipulated results of [5] and [7] with exactly the same dynamical model, however, shows the importance of the spinor structure in such calculations. We can understand the origin of the additional factors obtained in the earlier work by looking at the alternative spinor forms. A boosted elementary spinor has the form
where K = [(E + M)/2M]l/“, so that for composite spinor functions constructed as the direct product of three elementary spinors, asin [5], a factor K3 automatically occurs. Similarly, the factor (1 - k2/4M2), dropped by Lipes, is just this additional
QUARK
MODEL
WAVE
137
FUNCTIONS
factor of K2, which does not arise with the spinor structure (4.4), since the boost factor K2 associated with the (b, c) subsystem is absorbed by the contraction of the tensor character of this subsystem with the corresponding tensor operating on quark a. The wave functions used in [S] and [7] arise naturally in our scheme as the first rank doublet of Eq. (4.6). In the rest frame we have 1 - M[p*‘(y”) 2(2)1/Z [A, -t A,], P@ = M$&‘(yO
+ r”$P(l)] + 1).
Now, since $(y” + 1) = (i i), this is equivalent to taking only spinors with upper components (uO*), in the original direct product wave function. Similarly, A, -t A, gives only large component spinors. Thus, Eq. (4.6) corresponds essentially to a boosted NR spinor as used in [5], and since M#(r” + 1) ---f $@ + A4), we recognize also the spin $ wave function used by Lipes5 Using this doublet instead of (4.4) in the form factor calculation, the extra factor (1 - k2/4M2) over the results (4.9) appears as expected, and this time G,“(k2) = 0. The difference between the spinor structures (4.4) and (4.6) can be seen to consist in the role of the spinor small components. Analyzing the doublet (4.4) as above, we find it consists of equal amounts of upper and lower spinors (uo*) and (&) . The structure of the doublet (4.5) is more complicated, and it does not give as good results for the form factors. We find G,y(k2)
= [+fj;
1 iI’
k” - - 4M2
1G(k2)
G,,n(k2) = 2G(k2) GETA
= - &
(4.12) G(k2)
GM”l(k2) = -G(k2) By itself, this is phenomenologically bad and severely breaks SU(6) in the result ,up/pn = -2, but clearly a small admixture of Eq. (4.5) with (4.4) could reduce pp from 3 to 2.79, bringing pL” closer to - 1.91 at the same time, at the price of compromising the asymptotic dipole form for G Ep. However, such effects are beyond the scope of the oversimplified dynamics of Eq. (4.1). 5 Note in passing that this relativistic form higher symmetry schemes of Salam, Delburgo,
(4.6) is exactly and Strathdee
that suggested [14] and Sakita
in the relativistic and Wali [15].
138
HENRIQUES, KELLETT
AND MOORHOUSE
The final independent spinor form for the nucleon, the second-rank spinor of Eq. (4.7), is the most complicated. It gives the form factors 2(M2 - k2)
WW) = [ 2M2 _ k2 (1 - $&) G,p(k2) = 2 (1 - $1
+ $&]
G(k2)
G(k2) (4.13)
W(k2)
= -!f12M2 G(k2)
G,“(k2) = - (1 - &)
G(k2)
which are like Eq. (4.12) and are not satisfactory by themselves.Thus, in the simple model of (4.1) it would seemthat the spinor form (4.4) should dominate the nucleon wave function, although the dynamics do not favor this choice over any other. The new relativistic spin Qstatesof the four-dimensional oscillator model are the symmetric singlet 6(2)‘112M [3& - 4 + 4 - 41, pLL and the antisymmetric singlet -L[V - s + P] 2(6)112 we have no interpretation for resulting &+ decuplet and singlet, and do not consider these states further.
5. PHOTOPRODUCTION
MATRIX
ELEMENTS
An important application of any model for the interaction of quarks with currents is the calculation of photoproduction matrix elements. In this section, we consider the d-proton electromagnetic transition matrix element in somedetail and briefly discussthe transitions to higher quark model excited states. The spin # wave function of definite helicity is readily constructed from our general spinor covariants by contracting a permutation symmetric representation with one free Lorentz index with the polarization vectors & of the Appendix. With the possibility of forms involving P”, there are a number of covariants to consider. With the interaction (4.1), the A belongsto the ground state (56, L = O+)
QUARK
of the harmonic in the previous Appendix must replacing b by The independent
MODEL
WAVE
139
FUNCTIONS
oscillator, and there are just the two independent forms mentioned section. To obtain positive parity states, the covariants of the be multiplied by ys . As discussed in Section 3, we achieve this by (r5#>, , and this will be indicated by a suffix 5 on the covariants. covariants with one free Lorentz index we choose for the A are 1 4(6)lj2 [3A, 6(9’l”
- A, t A, - AsILL,
M [24 + 24
These covariants are correctly normalized we also have the mixed symmetry doublet
- B&m py-
when contracted
3
(5.1) (5.2) with 6~~. For Jp = $+
(5.3)
but this specifically relativistic state does not correspond to any known resonance, and will not be considered further at this stage. The helicity g state is obtained in the rest frame by choosing #L+’ = (U& and contracting with [email protected] helicity 4 state is rather more complicated since, as is usual with Rarita-Schwinger wave functions, we can get helicity 4 from (#(+))U cZLL or (4(-j), E+u. Because of the difficulties mentioned in the Appendix, these states are not necessarily orthogonal, so in general the combination corresponding to 1 1, +> must be constructed by hand. However, in practice for the covariants above, the correctly normalized helicity f state is obtained by using the usual ClebschGordan coefficients (5.4) The orthogonal
combination (5.5)
either vanishes, or simply reproduces the symmetric $+ singlet (4.14). The rest frame constituent spinor structure of the covariants (5.1) and (5.2) contains mixtures of large and small components. Nevertheless, an appropriate combination can be found that involves only spinors of the form (“,) , so with this combination, and the proton doublet (4.6) we can reproduce the results of [5] and
140
HENRIQUES,
KELLETT
AND
MOORHOUSE
[7]. However, the results of the previous section suggest that the dominant form of the nucleon wave function is given by the zero rank doublet (4.4), so we shall calculate the matrix elements using this form for the proton. We find that the photoproduction matrix elements for the A of both helicity + and $ vanish for A wave functions of the form (5.1). For the form (5.2), and the proton form (4.4), we find for helicity $0 A 3,2 = -2(3)1’2
KZ 2
(MA - M,),
(5.6)
A
and for helicity $ A, = -2KI(M,
- M,)
(5.7)
where
K = [(E, + M,)/2M,P2, and I=
(MA2 - kf,‘)’ J&MA2 + MD’)
i
In Table I, the numerical
1*
values of these matrix elements (with Q = 0.78 GeV2 as TABLE
I
Comparison of the Helicity Amplitudes for d Photoproduction Calculated in This Work with Other Quark Model Calculations and Experiment (GeV-l12 x 1O-5)
This calculation; Eqs. (5.6) and (5.7) with Q = 0.78 GeV2
-113
-85
Feynman et al. [5] Lipes [7]
-187 -148
-108
Walker (NR model, [17]) Experiment 1161
-85 -103
-178 -261
i 1
-142 i 1
determined from the fit to the nucleon form factors [7]) are compared with experiment [16], and the results of [5], [7], and the nonrelativistic calculation of Walker [17], which is based on the NR oscillator quark models of Copely, Karl, and Obryk [IS] and Faiman and Hendry [19]. We immediately seethat Eqs. (5.6) and (5.7) give rather worse agreement with experiment than the nonrelativistic calcula-
QUARK
MODEL
WAVE
FUNCTIONS
141
tion.6 In the helicity 3 amplitude, the factor M,/M, is particularly unfortunate, and it is interesting to observe this SU(6)-breaking factor arising from the relativistic spin structure. But a large part of the suppression over the NR model arises from the gaussian factor I of Eq. (5.8) which is different from the frame-dependent Gaussian factor of the NR model. This factor is directly associated with the harmonic oscillator model of equation (4.1) and occurs for all possible choices of spin structure. Thus, the general structure of the matrix elements (5.6) and (5.7) is an invariant feature of this interaction. The numerical coefficients may be slightly different for other basic covariants, and the factor MD/M, does not always occur, but there is no way in which such kinematical considerations can account for the factor 2 discrepancy between the calculated values and experiment. We can calculate the photoproduction matrix elements for the excited quark states in the same way, using the most general spin 4 and $ state in conjunction with states of definite orbital angular momentum and symmetry constructed from the relative momenta p and 4. Ey and large, the formal structure is similar to previous models, but the suppression arising from the factor 1 becomes progressively more pronounced as the mass of the resonance increases.7 The results are consequently uninteresting and will not be detailed here. The situation is only marginally improved by increasing 52, since even for Sz = co, the basic d amplitudes are too small. Apart from an ad hoc modification of (5.8) in the manner of [5], there is no way in which this simple oscillator interaction can give an adequate description of the photoproduction of nucleon resonances.
6.
DISCUSSION
The conclusions we have reached using the simplest harmonic oscillator model are rather discouraging, but we feel that these results must be kept in perspective. We have chosen to illustrate the application of our general spinor forms for the composite fermion wave functions in terms of the four-dimensional oscillator interaction (4.1) because it is simple and familiar, rather than because we consider it to be a realistic approximation to quark dynamics. The basic difficulties in the quantitative comparison with the experimental photoproduction results can be seen to arise from the nature of this interaction (as indeed similar difficulties arise 6 It should be remembered that although [5] and [7] use the interaction (4.1), in addition to using different spinor wave functions, they both use an arbitrary overall factor. For the transition results of Table I, however, this makes a difference of less than 5 ‘A. Moreover, there is an arithmetic error in Lipes paper, and the value given in Table I is the correct result for his model (with Q = 0.78 GeP). 7 See the result of [7] for the F15 (1688). Since Lipes retains the factor I of Eq. (5.8) his higher mass transition matrix elements are unduly suppressed.
142
HENRIQUES,
KELLETT
AND
MOORHOUSE
from the three-dimensional oscillator in the NR model), and are thus not fundamental. These difficulties were bypassed in [5] by the use of an ad hoc factor F, which was supposed also to cover error due to the possibly inadequate treatment of spin. We have shown that the different allowed spin structures do not improve the quantitative comparison with the photoproduction data. Our more general treatment of the spin dependence has indeed shown that the four-dimensional oscillator model can give agreement with the observed nucleon form factors, but the spinor functions required lead to somewhat worse agreement with the photoproduction results than the spinor wave functions of [5] and [7]. The wave functions used in [5] and [7] are the only ones that reduce to the nonrelativistic form in the baryon rest frame (they neglect any involvement of internal motion of the quarks with the spinors), and this is possibly an argument for choosing them (as a first approximation anyway) in preference to more general forms, since the NR quark model in some sense explains the spectrum, and gives at least a qualitative agreement with the photoproduction results. The NR model of course does not explain the electromagnetic form factors, but one can reasonably argue that these depend on the detailed short distance behavior of the interquark potential, which is not necessarily harmonic, even though the resonance spectrum suggests a harmonic long range part to the potential. This issue remains unresolved, and therefore, we would see our work as a necessary preparation for a more realistic treatment of the three-quark system. The spinor formalism that we have developed is completely general and it provides the necessary kinematical framework for any future model. Although the interaction (4.1) is clearly inadequate, the harmonic oscillator itself is not necessarily ruled out as the basic potential in a different wave equation. Because of the size of the general spin basis, the general solution of a spin dependent equation, such as the Bethe-Salpeter equation, is very complicated, but work along these lines is at present under way. APPENDIX
1. Dirac Matrix We use the metric tensor g,, convention for Dirac matrices Mechanics (McGraw-Hill, New Civita tensor @“~Ois defined by y5
=
Identities
defined by g,, = -g,, = 1, gij == 0 for i # j. The follows Bjorken and Drell, Relativistic Quantum York, 1964). The completely antisymmetric Levic”lz3 = - l oIz3= 1. Then y5
=
iyOy1y2y3
1
QUARK
MODEL
WAVE
143
FUNCTIONS
In constructing states of definite spin it is convenient to project out particular Lorentz components in the rest frame for instance. To this end we define the polarization vectors c&U = f &
(0, 1, fi,
0)
EQU= (0, 0, 0, -1). Then we define
for any four-vector
f such as y, p, or q.
2. Combinations of States of Definite Permutation Symmetry Defining the symmetry 01,/3, or A, we have
of the states 1 1) and 12) by the appropriate
I 1)sIV, = I >s
I 1)s 13, = I >,
I 1j.s I 2h = I h?
I 1)s I QA = I )A
I I)‘4 I 2)s = I )A
I l)A I ax
-I l>A I a3 = I >, &p
= Ih
I 1)A 19‘4 = I >s
[I I>.%I 3, + I 1h I a31 = I >s
-&-
1-I l>, I 3, + I 1h3I %I = i h
+
[I l>a I% + I I), I2hl = I h
-(2;1/” [-I I>, I% + I 1x9 I2),1 = I )A * 595/93/r-Z-10
label S,
144
HENRIQUES,
KELLETT
AND
MOORHOUSE
3. Spinor Covariants We list the possible spinor covariants in order according to the number of free Lorentz indices, and give the combinations with definite symmetry. Only states of one parity are given for each rank. The opposite parity states can be obtained trivially as described in Section 3. The covariants are not individually normalized, but they are chosen to be orthogonal for a particular choice of Lorentz components in a given rank. However, different Lorentz components are not necessarily orthogonal. The definite symmetry combinations are normalized to fl for each Lorentz component, but, the overall normalization for the 1 3 i2 combinations of components relevant for definite spin states must be obtained independently because of the possible cross terms. For brevity, we introduce an abbreviated notation by omitting the spinor indices, the spinor #Jfor quark a, and the matrix A. The matrix referring to the (b, c) subsystem is distinguished by enclosing it in brackets. Thus the zero-rank covariants of Eq. (3.5) become s = l(1) p = Y&J v = YWW)
(A.1)
A = -Y5Yu(Y5YU”) T = ~cr,,(u~*~) which
transform
under the permutation
and an antisymmetric
group as two doublets
singlet 1 -2(6)1/z [V - s + P].
64.3)
First Rank
(-4.4)
QUARK
transforming
a symmetric
MODEL
WAVE
FUNCTIONS
145
as three doublets
(A.3
singlet
1
___ 6(2)‘/” and an antisymmetric
[3&
- A, + A, - 431
(‘4.6)
singlet
12(6j1j2 [A, - A, + A, + 44
(A-7)
Second Rank Symmetric
in the Lorentz indices: 68)
giving a doublet (A-9) and a symmetric
singlet
--!6(2)112 Antisymmetric
W32
+ &I
(A.lO)
in the Lorentz indices:
(A.1 1)
146
HENRIQUES,
KELLETT
AND
MOORHOUSE
giving three doublets
A2(6)li2 P, + & + &,I i
L2(2)‘/” LB, + &I
+ LB, - & + Be1 1
4(3)‘/” [B, - B, + Bs + 2421
--!-4(3)1/Z LB, + B, - B, - 2&l
I
(A.12)
- ; LB4+ B, - &,I
i two symmetric singlets
12(6)lj2 [B, + B, - B, + B,,]
(A.13) --!4(3)1/Z [2B, + 2B, - B,,] and an antisymmetric
singlet
L2(6)112 [B+,- B, + 4, - &I
(A. 14)
Third Rank
Antisymmetric
in Lorentz indices:
(A.15)
QUARK
MODEL
WAVE
147
FUNCTIONS
giving three doublets
giving two doublets
-J-6(2)‘/” [Co - Cl1 + G,l 1 ‘Qj)W
‘lo
-J-2(6)1/2 [Cl, - Cl, + Cl01 1 2(2)‘/”
G4
(A.20)
148
HENRIQUES,
KELLETT
AND
MOORHOUSE
and four symmetric singlets
(A.21)
Fourth Rank It is convenient to introduce some intermediate
covariants
QUARK
The orthogonal
covariants
MODEL
WAVE
149
FUNCTIONS
are then given by
D, = R1 - Rz - 2R, + R, - R, - 2R, f .$[S,+ -S,+
t S,+ -&+I
- 2w1
D, = R, + R, + R, + R, - &S,+ + S,+ - 2S,+ + S,+ + S,+ - 2&j+] + D, = R, - Rz + R, - R4 + R, - R, + S,- -S,-
iw,
f Se-
D, = R, - Rz - 2R, - R, + R, + 2R, + $$S,- -S,-]
- $[S,- -S,-]
+ Se-
D, = R, + R, - R, - R, - &[S,- + S- - 2S,- + S,- + S,-] D, = T, + T, i- Ts + Ta + U, - U, f U, + U,
(A.23)
D, = Ts + T, f T7 A- Ts + U, - U, + h, + UI, D, = Tl + T, - T3 - T4 - U, + U, + U, - U, D, = T5 + T, - T7 - T, - U, + U, + U,, - U,, D,, = Tl - T, - T, + T4 - U, $ U, - U, + U, D,, = T5 - T6 - T7 i- T8 - U, t U, - U,, + U,,
!
giving three doublets
------20, 1
+ D, - 30, + 3D,]
12(3)lj2
p-----20, '
- 2D4 + 3D,]
12(3)‘/”
(A.24) 1 ~12(3)li2 1203 + 04 + 30, + 30111 ; DIO
i and five symmetric
singlets 1
1
4(3)'/" Dl '
-[2D, 12(6)lj2
+ D4 - 30, - 6D,], (A.25)
------[2D3 1 12(6)lj2
- 2D4 - 6D,],
1 ~ [2D, + D, + 30, - 60111. 12(6)1/2
150
HENKIQUES, KELLETT
Three of the combinations identities
AND MOORHOUSE
of covariants suggested by Eq. (A.22) satisfy the
which, together with the possible covariants E~~,J(& and •~~~,r~(l), form a set with the zero rank spinor structure multiplied by the Levi-Civita tensor, so do not give anything essentially new. For the second and higher ranks there are clearly also many possible covariants obtained from combining the metric tensor with lower rank covariants, but again these have been omitted as adding nothing essentially new. ACKNOWLEDGMENT The first author would like to acknowledge Foundation.
a research grant from the Calouste Gulbenkian
REFERENCES 1. J. S. GOLDSTEIN, Phys. Rev. 91 (1953), 1516. 2. C. H. LLEWELLYN-SMITH, Ann. Phys. (N.Y.) 3. M. B~HM, H. Joos, AND M. KRAMMER, Nucl.
53 (1969), 521. Phys. B51 (1973), 397; Nucl. Phys. B69 (1974),
349; DESY Report DESY 73/20 (1973). 4. P. J. WALTERS, A. M. THOMSON, AM) F. D. GAULT, J. Phys. A7 (1974), 1681. 5. R. P. FEYNMAN, M. KISLINGER, AND F. RAVNDAL, Phys. Rev. D 3 (1971), 2706. 6. K. F~JIMURA, T. KOBAYASHI, AND M. NAMIKI, Prog. Them. Phys. 44 (1970), 193. 7. R. G. LIPES, Phys. Rev. D 5 (1972), 2849. 8. B. H. KELLETT, Ann. Phys. (N.Y.) 87 (1974), 60. 9. A. LE YAOUANC, L. OLIVER, 0. PBNE, AND J. C. RAYNAL, Phys. Rev. D 9 (1974), 2636. 10. B. H. KELLEIT, Nuovo Cimento 5tiA (1968), 1003. 11. M. A. GONZALES AND P. J. S. WATSON, Nuovo Cimento 12A (1972), 889. 12. R. WILSON, Proceedings of the 5th Internat. Symp. on Electron and Photon Interactions
High Energies, Cornell, 1971. 13. W. BARTEL, F. W. Btiss~~, W. R. DIX, R. FELST, D. HARMS, H. KREHBIEL, J. MCELROY, J. MEYER AND G. WEBER, Nucl. Phys. B58 (1973), 429.
P. E. KIJHLMANN,
at
QUARK
14. 15. 16. 17.
A. B. R. R. at 18. L. 19. D.
MODEL
WAVE
FUNCTIONS
151
SALAM, R. DELBOURGO, AND J. STRATHDEE, Proc. Roy. Sot. Ser. A. 284 (1965), 146. SAKITA AND K. C. WALI, Phys. Rec.. 139 (1965), B1355. G. M~RHOUSE, H. OBERLACK, AND A. H. ROSENFELD, Phys. Rev. D 9 (1974), 1. L. WALKER, Proceedings of the 4th Internat. Symp. on Electron and Photon Interactions High Energies, Daresbury, 1969. A. COPLEY, G. KARL, AND E. OBRYK, Nucl. Phys. B 13 (1969), 303. FAIMAN AND A. W. HENDRY, Phys. Reo. 180 (1969), 1572.
OF PHYSICS
93, 125-151 (1975)
General Three-Spinor
Wave
Functions and the Relativistic
Quark
Model
A. B. HENRIQUES, B. H. KELLETT, Department
of Natural
Philosophy,
AND R. G. MOORHOUSE
University
of Glasgow,
Glasgow,
Scotland
Received March 11, 1975
The complete set of three particle relativistic spinors is constructed in a set of evidently covariant forms classified into irreducible representations of the permutation group, S, , for use in the three quark model of baryons. A few of the various types of three quark spinor functions are illustrated by evaluating their electromagnetic current interactions using the four dimensional oscillator models of Feynman, Kislinger, and Ravndal and Lipes for the spatial part of the wave function: one of our types gives markedly better results for nucleon electromagnetic form factors than the type corresponding to the original models of these authors, but problems, largely associated with the four dimensional oscillator, remain in the electromagnetic transitions to higher baryon resonances. 1. INTR~OUCTI~N
The problem of constructing a covariant formulation of the phenomenologically successfulnonrelativistic quark model has attracted considerable attention in recent years. An important feature of this enterprise is the treatment of the spin structure of composite systemsconsisting of two or three spin i-quarks. The general form of the two-spinor wave function was given many years ago by Goldstein [l], and these forms have been widely used for the quark-antiquark mesonic system [2-41. In this paper, we consider the relativistic spin structure of the three-quark baryonic system. The general form of the three-spinor wave function can be developed as a natural extension of the two-spinor system, but the situation is considerably complicated by the necessity for maintaining the symmetry properties of the wave function under quark interchange. To this end we express the general covariant spinor basisstatesas irreducible representations of the permutation group of three objects S, . Wave functions with the desired symmetry properties, definite parity, and definite total spin, can then be constructed by combining the spinor part with the center of mass (c.m.) momentum, which is a symmetric singlet under S, , and the two independent relative momenta, which form a mixed symmetry doublet under S, . Previous attempts to formulate a relativistic quark model for baryons [5-91 have 125 Copyright All rights
Q 1975 by Academic Press, Inc. of reproduction in any form reserved.
126
HENRIQUES,
KELLETT
AND
MOORHOUSE
treated the spin structure in a rather arbitrary way. The models of Refs. [5-71 are based on the requirement that the wave function reduce to the usual SU(6) positive energy (Pauli) spinor form in the hadron rest frame. Such a form excludes, inter alia, any dependence on the internal quark momenta and such momentum dependence may be important for obtaining agreement with experiment, as has been claimed by Le Yaouanc et al. [9]. It is clearly desirable to investigate other possibilities, though using the most general relativistic spin structure does not guarantee phenomenological success, since this finally depends on having a realistic dynamical model. To get some feeling for the use of the general spin functions in practice, we investigate the simple harmonic oscillator model of Feynman, Kislinger, and Ravndal [5] and Lipes [7]. It becomes clear on comparison with experiment for the baryon-electromagnetic current interaction, that even with the generalized spinor structure, this simple interaction has quantitative inadequacies comparable to those of the nonrelativistic quark model. Although this latter model has good qualitative agreement with photoproduction experiments [for the (56,0+) and (70, l-) multiplets], one would have hoped for some improvement in models claiming to be relativistic1 On the other hand, we do find that the undesirable form factors obtained in previous work are not a necessary consequence of the harmonic oscillator interaction, so the way is open for a more realistic treatment of the problem using a sum of two-body oscillator potentials in a spin-dependent equation, such as the Bethe-Salpeter equation. Technically, such a program is very complicated because of the large number of independent spinor forms available for the construction of the individual wave functions; thus, we restrict ourselves in this paper to the development of the formalism, and make some general comments, leaving the detailed dynamical applications for future work. In Section 2 we develop the general form of the wave function. The generalized Fierz transformation for determining the symmetry under quark interchange is described in Section 3. This allows the classification of the general forms into irreducible representations of S, and the construction of wave functions with definite spin, parity, and symmetry properties. In Section 4 the simple oscillator model of [5] and [7] is used to investigate the interaction with the electromagnetic current and we detail the behavior of the nucleon form factors, with particular reference to the connection with earlier work. The 4(1236) wave function, and the rival transition are considered in Section 5, and we make some general comments on the electromagnetic transitions to higher resonances in the simple model. Some technical details and a complete list of the possible covariant spinor wave functions, classified according to the representations of S, , are given in the Appendix. 1 Note, however, that in the nonrelativistic quark model the magnetic moment of the proton is an arbitrary parameter of the theory; an achievement of the model of [5] with respect to baryons is the (approximately) correct prediction of this number.
QUARK
MODEL
WAVE
2. THE THREE-SPINOR
FUNCTIONS
WAVE
127
FUNCTIONS
A relativistic spin J-particle is described by a four-component Dirac spinor, whether the particle is free or bound. In the single particle case, this spinor is the solution of the Dirac equation with an appropriate potential. When two or more spin &-particles mutually interact to form a composite system, the general wave function has the form of a product of the Dirac spinors for the individual constituents. Goldstein [l] has shown that for the two-fermion case, it is convenient to rewrite this direct product of spinors as an outer product, and expand the resulting 4 x 4 matrix representation in terms of the complete set of 16 Dirac matrices2 1, yS , y“, y5yUand c?’ = @(y”y” - r”r“). In this representation, operators acting on the second quark are represented by postmatrix multiplication by the transpose of the operator. The two-fermion wave function is written (2.1) Xao= *ol*BT where #a is the individual quark spinor, and
%3~xus’= XtYa4%3r)7
(2.2)
where O,,, is a general matrix operator for the second spinor. It is convenient to introduce the matrix A = C-ly,(= y1y3 in our notation), where C is the usual charge conjugation matrix. Then AyuTA-l = yu and A-l = AT = A+ = -A.
(2.3)
If we then introduce
Eq. (2.2) becomes
&4as~ = AdAOTA-%~
(2.5)
and the operator (AOTA-l) does not involve the transpose of Dirac matrices. The matrix +aocan then be expanded in terms of the complete set of Dirac matrices.3 2 Our metric and notation for the Dirac matrices are given in the appendix. In what follows we shall use Greek letters p. Y, X, etc., as Lorentz indices; Greek letters from the beginning of the alphabet, 01,fl, y, etc., as spinor indices; and the Latin letters a, b, c as quark labels. a In the qualk-antiquark system for mesons, it is convenient to use the charge conjugation matrix C rather than A, since then starting from the two quark system, 4 is the quark antiquark wave function as it stands. 595/93/l-2-9
128
HENRIQUES,
KELLETT
AND
MOORHOUSE
For the three-quark system we wish to develop a similar generalized matrix representation for the direct product of three spinors. In general, this is achieved by selecting any two-quark subsystem as above and then taking the direct product with the third quark. There are clearly three ways in which one can select the two quark subsystem, and the transformation between the matrix representations corresponding to different initial choices determines the permutation symmetry properties of the composite wave function. We will label the quarks a, b, and c, and represent the (b, c) subsystem in the matrix form (2.4). The further four independent degrees-of-freedom associated with quark a could be incorporated by multiplying &,, by an arbitrary spinor basis, such as the four free-particle Dirac spinors in the rest frame, but it is more convenient to introduce a redundant basis at this stage. For the first quark we write raa~@), with the reduced single spinor basis I,V+) and I/J-), which in the hadron rest frame takes the form: (2.6)
and I’,,, is a general 4 x 4 matrix that can be expanded in terms of the Dirac matrices as usual. The general three-quark wave function is then written y&A = WMb4,Y We adopt the convention Eq. (2.7) so that
*
(2.7)
that the order of the spinor indices on Y is governed by YBW = w,
(b&Y
(2.8)
2
and so on. The Lorentz transformation properties of YUBYare those of the product of three spinors. Each spinor receives the boost denoted by the usual spinor operator S(A). Now ASyA) A-l = S-l(A), (2.9) so that
Y& = cw) rw4)ad
(W) $Q.*(W) WY4
43, .
(2.10)
The combined wave function therefore transforms as the product of a tensor of rank determined by the number of free Lorentz indices on I’ and q5,and a spinor $. For example, yU~Fi)(yUA)By transforms as a spin-up or spin-down Dirac spinor for do not * (f) or $(-), respectively. Since the permutation group transformations change the Lorentz transformation properties, it is convenient to subdivide the
QUARK
MODEL
WAVE
FUNCTIONS
129
general forms (2.7) according to the number of free Lorentz indices. We shall refer to those with no free indices as being of zero rank, and so on. Clearly, nothing new in terms of the spinor structure occurs above the fourth rank, since the metric tensor guV and the Levi-Civita tensor Euvr\pdo not form part of the spinor basis. The matrices r and $ involve only Dirac matrices, but in the formation of an actual physical wave function of definite spin and parity, the free indices are contracted with four-vector momenta, or polarization four-vectors. The individual quarks have momenta pa , pb , and pc , respectively, which we separate into c.m. momentum P and relative momentap and q according to p=P,+PP,+P,, P = Pb + PC - 2Pa >
(2.11)
rl = (3Y (PC - Pb), and the inverse
(2.12)
The momentum independent spinors are normalized conventionally as a product of three quark spinors: (2.13)
for each individual tensor component. Care is necessary,however, in normalizing linear combinations of wave functions of given rank, since the different tensor components are not necessarilyorthogonal. This point is discussedin the Appendix. States of definite parity can also be constructed. For a state of parity qp , we have
where p” = (p. , -p) etc.
130
HENRIQUES, KELLETT
AND MOORHOUSE
3. SYMMETRY PROPERTIES OF THE WAVE FUNCTION
By construction, the permutation symmetry of the three-spinor wave function reduces to the index symmetry of !P+, . We can define the generators of the permutation group S, by QaaKov = Yom, 7 (3.1) where Qab indicates the operation of interchanging quarks a and b, or more generally, of interchanging the quarks associated with the first two indices of UC, whatever their particular labels. Similarly we define Qbc and Qnc . The interpretation of the group as an index symmetry allows the result
Qac= QbcQnbQbc .
(3.2)
Thus, we need consider only two operators. The operator QbCon the state (2.7) is trivial, since A, yJ, and YJ are antisymmetric, while 757&A and uGVAare symmetric. Interchange of quarks a and b is more complicated since we interchange indices on two independent sets of Dirac matrices. However, the explicit spinor #a* in Eq. (2.7) is irrelevant for this interchange, and the point of the redundant basis for quark a becomes clear when it is realized that the action of Qob in Eq. (2.7) corresponds simply to the standard spinor reordering or Fierz transformation, familiar from crossing in nucleon-nucleon scattering for example [lo]. The Fierz transformation matrix is readily obtained from the completeness relation for the Dirac matrices
- (YsYJua~ biY”)so’ + t(%“)cm,(~93B~l~
(3.3)
A group of three objects can have four types of symmetry under permutations. These are the irreducible representations of the group S, , and comprise the completely symmetric S, the two mixed symmetry types 01and /3, and the completely antisymmetric state A. Acting on the singlet S, Qab = Qbc = 1, and on the singlet A, Qab = Qb, = - 1. The mixed symmetry states do not transform independently, and form a doublet, defined by the transformation matrices.
(3.4)
As an example, it is readily verified that the c.m. momentum
P defined above is a
QUARK
MODEL
WAVE
131
FUNCTIONS
symmetric singlet, while the relative momenta belong to the mixed symmetry doublet (g) . To illustrate the application of these general principles to the classification of wave functions of the form (2.7), we shall look in some detail at the zero rank spinor forms. We have five possible covariants of each parity in this case. For positive parity, these are the conventional p-decay covariants used in nucleonnucleon scattering s = (l)&
(43Y
p = (Y5La’ 6%43V v = bAd
w-4,,
A = -bwu)aa~ T = ~(4m~
(3.5) (Y&‘A)B~
(u““A)m
where we omit the common #n, , which is irrelevant for our present purpose. If we distinguish the covariants of Y,,, , which are those of Eq. (3.5) with the indices cy. and p interchanged, by a prime, Eq. (3.3) leads to the crossing matrix
where use has been made of the y-matrix identities given in the Appendix. Noticing that A and T are even under Qhe, while the rest are odd, it is straightforward to deduce that the combinations ;A and
___ form normalized
(3.7)
[S + PI
i
q&y
PS - 2p
+ VI
!
doublets under S, , while ~
1 2(6)‘/”
[V - s + P]
(3.8)
is a normalized antisymmetric singlet. The negative parity covariants can be obtained from Eq. (3.5) by replacing #carby
132
HENRIQUES,
KELLETT
AND
MOORHOUSE
(y5& for instance. This does not change the transformation and Qb, . Thus, for
properties
under Qab
we have the same multiplet structure as in Eqs. (3.7) and (3.8). In the Appendix we list the covariants for each rank and give their symmetry properties in terms of the irreducible representations of the permutation group. The individual covariants can be interpreted as generalized Rarita-Schwinger wave functions; thus, the spin associated with a given form depends on the free Lorentz indices as well as the isolated spinor for quark a. Further independent covariants of lower spin may be formed by contracting the Lorentz indices with P, p, and q. The overall symmetry properties are preserved by combining definite symmetry states according to the rules of the permutation group. These are listed in the Appendix. Not all the covariants formed in this way from the spinor states of the Appendix will be independent. Because the original three-spinor composite system has only 43 = 64 independent degrees-of-freedom, combinations with the available momenta in all possible ways can give at most 64 independent basis states for each definite spin, parity, and symmetry type. Each of these independent states can be multiplied by an arbitrary Lorentz scalar function of the momenta. The redundancy of possible states is a reflection of the spurious freedom introduced by the treatment of the third quark, so that once the symmetry and spin properties have been established, the covariants can be reduced to a simpler form. This must in fact be done so that the individual covariants can be normalized correctly. The possibility of spinor small components playing an important role in the relativistic situation means that the spectrum of possible states is very much richer than in the nonrelativistic (NR) three-spinor system. Combining the spinor covariants with the momenta, we can form states of any spin and parity in each of the four symmetry types. These possibilities, of course, may be restricted in a particular dynamical model, and in a realistic case the number of possible basis states of a given spin may be considerably less than the theoretical maximum. In the oscillator model of [5], for example, the eigenstates of the Hamiltonian are each of definite order in the internal momenta, so that the NR spectrum is reproduced with only a few additional specifically relativistic states. In particular, the parity doublets have negative norm.
QUARK MODEL WAVE FUNCTIONS
133
The full symmetric quark model wave functions are constructed by combining the spinor wave functions with the standard SU(3) wave functions according to the symmetry rules given in the Appendix. The spin and orbital angular momentum wave functions for the excited states of the quark model, in principle, are already contained in the development of the spinor states of the appendix with the momenta, but in practice it is simpler to construct the orbital angular momentum states separately, and combine them with the most general spin * and $-spinor states with the usual Clebsch-Gordan coefficients. In this way the quark model (70, L = I-) and (56, L = 2-l-), etc., and whatever other states are demanded by the dynamics, are readily constructed.
4. THE NUCLEONELASTIC FORM FACTORSIN A SIMPLEMODEL
To illustrate the use of the formalism developed in the preceding sections in a definite dynamical model, in this and the following section we investigate the electromagnetic properties of the baryons in a simple relativistic oscillator quark model. The most fully developed relativistic harmonic oscillator model for the baryons in the literature is that usedby Feynman et al. [5] and Lipes [7]. This model has the advantage, as mentioned above, that the ground-state oscillator wave function depends on the relative momenta only through an overall Gaussian function, and apart from this Gaussian, the excited states are homogeneous functions of p and q. Consequently, the number of possible independent spinor covariants at each level is severely restricted. Even so, there is considerably more freedom than is allowed for by Feynman et al. and Lipes, and we show that the spinor forms used by these authors are in fact equivalent, corresponding to just one possibility from several, and better results for the form factors in particular are achieved if one makes a different choice. In brief, starting from the harmonic oscillator equation [5], {3(Pa2 +
Pb2+ PC?+ hQ2K~a - x0)3f (Xb - x,)2 + (xc - x,)2] + C] Y = 0 (4.1)
and making a minimal electromagnetic coupling, assuming quark additivity, we deduce the electromagnetic current operator? (4.2) for an incoming photon of momentum k. We assume(frame-dependent) boundary 4X, x, andy are the coordinatesconjugateto P,
p,
and y, respectively.
134
HENRIQUES,
KELLETT
AND
MOORHOUSE
conditions relevant to the four-dimensional oscillator without a Wick rotation [l 11. The ground state solution of Eq. (4.1) in momentum space is then [6, 71:
1g) = -(&)2exp -
&
[2 (2)
- p2 + 2 (%j”
- qq.
(4.3)
For the excited states, the troublesome time-like excitations are avoided by using P” and q” - (P q/A43 P”, rather than p” and q” themselves. This postpones, rather than solves, the basic difficulty of interpreting the time-like excitations, which can have imaginary mass in this model, and we assume that the violation of unitarity is not serious for low lying states. The ground state of the NR quark model allows just three independent spin states: a mixed symmetry doublet of spin 4, and a symmetric singlet of spin 8. However, because of the relativistic presence of spinor small components, the ground state of the above model has rather more spin states. These are most readily specified by considering the constituent spinor composition in the rest frame. For all upper component spinors we have the NR states, but overall positive parity also allows states with one large and two small component constituent spinors. In the helicity 8 configuration these give a mixed symmetry doublet of spin 8 and a further symmetric singlet of spin +. For spin &, the mixture of upper and lower component spinors allows a further three mixed-symmetry doublets, and a symmetric and an antisymmetric singlet of spin 4. These spin 4 singlets, and the spin + doublet are relativistic states with no NR analogue and they do not have any place in the standard SU(6) classification of resonant states. In a symmetric quark model they give rise to a kinematically allowed additional .P = #+ octet, a &+ decuplet, and a ++ SU(3) singlet. There are certainly more independent states than
p” - (Pp/M2)
the spectrum
requires and the superfluous
states must be forbidden
by spin and
chirality-dependent dynamics giving rise, in some approximation, to selection operators multiplying the simple four-dimensional oscillator. We do not concern ourselves with such possible dynamics here. Now consider which combinations of the independent states might be possible candidates for the known physical baryons. There are four independent spinor forms for the spin &+ octet of the symmetric quark model (56, L = O+). These come from the two zero-rank mixed-symmetry doublets, and by reducing the higher order doublets, contracted with the total momentum P”, to their simplest form, we find the two remaining independent forms:
(4.4)
QUARK
MODEL
WAVE
FUNCTIONS
135
;A i
-&
12s - 2p
-1 JA i
(4.5)
(4.6)
(4.7)
where These quark uf = initial
we indicate the required momentum contractions in a obvious notation. doublets correspond to helicity 5s according to whether the spinor for a in the barryon rest frame is chosen to be 4t-b) = ($), or z/-j = (“i), where (3 and u- = Q) . To be definite, we choose the kinematics such that the state is at rest, and the incident photon propagates in the $2 direction: P” = (M, O), P’u = (E, 0, 0, k3), k” = (v, k) = (v, 0, 0, k3).
The helicity +$ cy.symmetry 1 2(6)‘/”
part of (4.4) for example, in the rest frame is %“P+Yo>(~~v~
(following the abbreviated notation of the Appendix), and the corresponding spinor for the final state is (1/2(6)l/3 T(P’) = (1/4(6)lj2) ~,&(+)(P’)(u”~). The nucleon wave function for this spinor doublet then has the form
where j S,), and I S,), are the SU(3) nucleon wave functions. The matrix element of the current (4.2) between nucleon states of the form (4.8) is readily calculated, and the charge and magnetic form factors can be read off immediately as the
136
HENRIQUES,
KELLETT
AND
MOORHOUSE
coefficients of co and io . Ex k/2M, respectively, where EUis the photon polarization vector. We find G,*(P)
= +$-
G(k2),
G,*(k2) = 3G(k7, (4.9)
WV2) = - -!f? 2M2 G(k2), GMn(k2) = -2G(k2), where G(k2) = (1 - &)-”
exp [$
(I - $$)-l].
(4.10)
Apart from the neutron electric form factor, these results are the sameas those of Lipes after his ad hoc neglect of an overall factor [l - (k2/4M2)]. The form factors show an asymptotic dipole behavior, and the average experimental shape is satisfactorily fitted for a value [7] Q = 0.78 GeV2. However, the old difficulty of fitting the charge radius in a harmonic oscillator model is not completely solved. For this value of Q, Eq. (4.9) gives (rD2)lj2 = 0.56 fermi, and to reproduce the experimental 0.814 fermi would require a > 0.44 GeV2, which is inconsistent with the observed level spacing of about 1 GeV2. The neutron charge radius, on the other hand, is independent of fin, and Eq. (4.9) gives (rn2j1J2 ‘v 0.36 fermi, which is in good agreement with the experimental result [12] of about 0.35 fermi. We recover the standard SU(6) result pp/f*?l = -$, but asymptotically, the ratio p”GE”(kz)/GMfl(k2) + 2, instead of being less than one [13]. The comparison of these results with the unmanipulated results of [5] and [7] with exactly the same dynamical model, however, shows the importance of the spinor structure in such calculations. We can understand the origin of the additional factors obtained in the earlier work by looking at the alternative spinor forms. A boosted elementary spinor has the form
where K = [(E + M)/2M]l/“, so that for composite spinor functions constructed as the direct product of three elementary spinors, asin [5], a factor K3 automatically occurs. Similarly, the factor (1 - k2/4M2), dropped by Lipes, is just this additional
QUARK
MODEL
WAVE
137
FUNCTIONS
factor of K2, which does not arise with the spinor structure (4.4), since the boost factor K2 associated with the (b, c) subsystem is absorbed by the contraction of the tensor character of this subsystem with the corresponding tensor operating on quark a. The wave functions used in [S] and [7] arise naturally in our scheme as the first rank doublet of Eq. (4.6). In the rest frame we have 1 - M[p*‘(y”) 2(2)1/Z [A, -t A,], P@ = M$&‘(yO
+ r”$P(l)] + 1).
Now, since $(y” + 1) = (i i), this is equivalent to taking only spinors with upper components (uO*), in the original direct product wave function. Similarly, A, -t A, gives only large component spinors. Thus, Eq. (4.6) corresponds essentially to a boosted NR spinor as used in [5], and since M#(r” + 1) ---f $@ + A4), we recognize also the spin $ wave function used by Lipes5 Using this doublet instead of (4.4) in the form factor calculation, the extra factor (1 - k2/4M2) over the results (4.9) appears as expected, and this time G,“(k2) = 0. The difference between the spinor structures (4.4) and (4.6) can be seen to consist in the role of the spinor small components. Analyzing the doublet (4.4) as above, we find it consists of equal amounts of upper and lower spinors (uo*) and (&) . The structure of the doublet (4.5) is more complicated, and it does not give as good results for the form factors. We find G,y(k2)
= [+fj;
1 iI’
k” - - 4M2
1G(k2)
G,,n(k2) = 2G(k2) GETA
= - &
(4.12) G(k2)
GM”l(k2) = -G(k2) By itself, this is phenomenologically bad and severely breaks SU(6) in the result ,up/pn = -2, but clearly a small admixture of Eq. (4.5) with (4.4) could reduce pp from 3 to 2.79, bringing pL” closer to - 1.91 at the same time, at the price of compromising the asymptotic dipole form for G Ep. However, such effects are beyond the scope of the oversimplified dynamics of Eq. (4.1). 5 Note in passing that this relativistic form higher symmetry schemes of Salam, Delburgo,
(4.6) is exactly and Strathdee
that suggested [14] and Sakita
in the relativistic and Wali [15].
138
HENRIQUES, KELLETT
AND MOORHOUSE
The final independent spinor form for the nucleon, the second-rank spinor of Eq. (4.7), is the most complicated. It gives the form factors 2(M2 - k2)
WW) = [ 2M2 _ k2 (1 - $&) G,p(k2) = 2 (1 - $1
+ $&]
G(k2)
G(k2) (4.13)
W(k2)
= -!f12M2 G(k2)
G,“(k2) = - (1 - &)
G(k2)
which are like Eq. (4.12) and are not satisfactory by themselves.Thus, in the simple model of (4.1) it would seemthat the spinor form (4.4) should dominate the nucleon wave function, although the dynamics do not favor this choice over any other. The new relativistic spin Qstatesof the four-dimensional oscillator model are the symmetric singlet 6(2)‘112M [3& - 4 + 4 - 41, pLL and the antisymmetric singlet -L[V - s + P] 2(6)112 we have no interpretation for resulting &+ decuplet and singlet, and do not consider these states further.
5. PHOTOPRODUCTION
MATRIX
ELEMENTS
An important application of any model for the interaction of quarks with currents is the calculation of photoproduction matrix elements. In this section, we consider the d-proton electromagnetic transition matrix element in somedetail and briefly discussthe transitions to higher quark model excited states. The spin # wave function of definite helicity is readily constructed from our general spinor covariants by contracting a permutation symmetric representation with one free Lorentz index with the polarization vectors & of the Appendix. With the possibility of forms involving P”, there are a number of covariants to consider. With the interaction (4.1), the A belongsto the ground state (56, L = O+)
QUARK
of the harmonic in the previous Appendix must replacing b by The independent
MODEL
WAVE
139
FUNCTIONS
oscillator, and there are just the two independent forms mentioned section. To obtain positive parity states, the covariants of the be multiplied by ys . As discussed in Section 3, we achieve this by (r5#>, , and this will be indicated by a suffix 5 on the covariants. covariants with one free Lorentz index we choose for the A are 1 4(6)lj2 [3A, 6(9’l”
- A, t A, - AsILL,
M [24 + 24
These covariants are correctly normalized we also have the mixed symmetry doublet
- B&m py-
when contracted
3
(5.1) (5.2) with 6~~. For Jp = $+
(5.3)
but this specifically relativistic state does not correspond to any known resonance, and will not be considered further at this stage. The helicity g state is obtained in the rest frame by choosing #L+’ = (U& and contracting with [email protected] helicity 4 state is rather more complicated since, as is usual with Rarita-Schwinger wave functions, we can get helicity 4 from (#(+))U cZLL or (4(-j), E+u. Because of the difficulties mentioned in the Appendix, these states are not necessarily orthogonal, so in general the combination corresponding to 1 1, +> must be constructed by hand. However, in practice for the covariants above, the correctly normalized helicity f state is obtained by using the usual ClebschGordan coefficients (5.4) The orthogonal
combination (5.5)
either vanishes, or simply reproduces the symmetric $+ singlet (4.14). The rest frame constituent spinor structure of the covariants (5.1) and (5.2) contains mixtures of large and small components. Nevertheless, an appropriate combination can be found that involves only spinors of the form (“,) , so with this combination, and the proton doublet (4.6) we can reproduce the results of [5] and
140
HENRIQUES,
KELLETT
AND
MOORHOUSE
[7]. However, the results of the previous section suggest that the dominant form of the nucleon wave function is given by the zero rank doublet (4.4), so we shall calculate the matrix elements using this form for the proton. We find that the photoproduction matrix elements for the A of both helicity + and $ vanish for A wave functions of the form (5.1). For the form (5.2), and the proton form (4.4), we find for helicity $0 A 3,2 = -2(3)1’2
KZ 2
(MA - M,),
(5.6)
A
and for helicity $ A, = -2KI(M,
- M,)
(5.7)
where
K = [(E, + M,)/2M,P2, and I=
(MA2 - kf,‘)’ J&MA2 + MD’)
i
In Table I, the numerical
1*
values of these matrix elements (with Q = 0.78 GeV2 as TABLE
I
Comparison of the Helicity Amplitudes for d Photoproduction Calculated in This Work with Other Quark Model Calculations and Experiment (GeV-l12 x 1O-5)
This calculation; Eqs. (5.6) and (5.7) with Q = 0.78 GeV2
-113
-85
Feynman et al. [5] Lipes [7]
-187 -148
-108
Walker (NR model, [17]) Experiment 1161
-85 -103
-178 -261
i 1
-142 i 1
determined from the fit to the nucleon form factors [7]) are compared with experiment [16], and the results of [5], [7], and the nonrelativistic calculation of Walker [17], which is based on the NR oscillator quark models of Copely, Karl, and Obryk [IS] and Faiman and Hendry [19]. We immediately seethat Eqs. (5.6) and (5.7) give rather worse agreement with experiment than the nonrelativistic calcula-
QUARK
MODEL
WAVE
FUNCTIONS
141
tion.6 In the helicity 3 amplitude, the factor M,/M, is particularly unfortunate, and it is interesting to observe this SU(6)-breaking factor arising from the relativistic spin structure. But a large part of the suppression over the NR model arises from the gaussian factor I of Eq. (5.8) which is different from the frame-dependent Gaussian factor of the NR model. This factor is directly associated with the harmonic oscillator model of equation (4.1) and occurs for all possible choices of spin structure. Thus, the general structure of the matrix elements (5.6) and (5.7) is an invariant feature of this interaction. The numerical coefficients may be slightly different for other basic covariants, and the factor MD/M, does not always occur, but there is no way in which such kinematical considerations can account for the factor 2 discrepancy between the calculated values and experiment. We can calculate the photoproduction matrix elements for the excited quark states in the same way, using the most general spin 4 and $ state in conjunction with states of definite orbital angular momentum and symmetry constructed from the relative momenta p and 4. Ey and large, the formal structure is similar to previous models, but the suppression arising from the factor 1 becomes progressively more pronounced as the mass of the resonance increases.7 The results are consequently uninteresting and will not be detailed here. The situation is only marginally improved by increasing 52, since even for Sz = co, the basic d amplitudes are too small. Apart from an ad hoc modification of (5.8) in the manner of [5], there is no way in which this simple oscillator interaction can give an adequate description of the photoproduction of nucleon resonances.
6.
DISCUSSION
The conclusions we have reached using the simplest harmonic oscillator model are rather discouraging, but we feel that these results must be kept in perspective. We have chosen to illustrate the application of our general spinor forms for the composite fermion wave functions in terms of the four-dimensional oscillator interaction (4.1) because it is simple and familiar, rather than because we consider it to be a realistic approximation to quark dynamics. The basic difficulties in the quantitative comparison with the experimental photoproduction results can be seen to arise from the nature of this interaction (as indeed similar difficulties arise 6 It should be remembered that although [5] and [7] use the interaction (4.1), in addition to using different spinor wave functions, they both use an arbitrary overall factor. For the transition results of Table I, however, this makes a difference of less than 5 ‘A. Moreover, there is an arithmetic error in Lipes paper, and the value given in Table I is the correct result for his model (with Q = 0.78 GeP). 7 See the result of [7] for the F15 (1688). Since Lipes retains the factor I of Eq. (5.8) his higher mass transition matrix elements are unduly suppressed.
142
HENRIQUES,
KELLETT
AND
MOORHOUSE
from the three-dimensional oscillator in the NR model), and are thus not fundamental. These difficulties were bypassed in [5] by the use of an ad hoc factor F, which was supposed also to cover error due to the possibly inadequate treatment of spin. We have shown that the different allowed spin structures do not improve the quantitative comparison with the photoproduction data. Our more general treatment of the spin dependence has indeed shown that the four-dimensional oscillator model can give agreement with the observed nucleon form factors, but the spinor functions required lead to somewhat worse agreement with the photoproduction results than the spinor wave functions of [5] and [7]. The wave functions used in [5] and [7] are the only ones that reduce to the nonrelativistic form in the baryon rest frame (they neglect any involvement of internal motion of the quarks with the spinors), and this is possibly an argument for choosing them (as a first approximation anyway) in preference to more general forms, since the NR quark model in some sense explains the spectrum, and gives at least a qualitative agreement with the photoproduction results. The NR model of course does not explain the electromagnetic form factors, but one can reasonably argue that these depend on the detailed short distance behavior of the interquark potential, which is not necessarily harmonic, even though the resonance spectrum suggests a harmonic long range part to the potential. This issue remains unresolved, and therefore, we would see our work as a necessary preparation for a more realistic treatment of the three-quark system. The spinor formalism that we have developed is completely general and it provides the necessary kinematical framework for any future model. Although the interaction (4.1) is clearly inadequate, the harmonic oscillator itself is not necessarily ruled out as the basic potential in a different wave equation. Because of the size of the general spin basis, the general solution of a spin dependent equation, such as the Bethe-Salpeter equation, is very complicated, but work along these lines is at present under way. APPENDIX
1. Dirac Matrix We use the metric tensor g,, convention for Dirac matrices Mechanics (McGraw-Hill, New Civita tensor @“~Ois defined by y5
=
Identities
defined by g,, = -g,, = 1, gij == 0 for i # j. The follows Bjorken and Drell, Relativistic Quantum York, 1964). The completely antisymmetric Levic”lz3 = - l oIz3= 1. Then y5
=
iyOy1y2y3
1
QUARK
MODEL
WAVE
143
FUNCTIONS
In constructing states of definite spin it is convenient to project out particular Lorentz components in the rest frame for instance. To this end we define the polarization vectors c&U = f &
(0, 1, fi,
0)
EQU= (0, 0, 0, -1). Then we define
for any four-vector
f such as y, p, or q.
2. Combinations of States of Definite Permutation Symmetry Defining the symmetry 01,/3, or A, we have
of the states 1 1) and 12) by the appropriate
I 1)sIV, = I >s
I 1)s 13, = I >,
I 1j.s I 2h = I h?
I 1)s I QA = I )A
I I)‘4 I 2)s = I )A
I l)A I ax
-I l>A I a3 = I >, &p
= Ih
I 1)A 19‘4 = I >s
[I I>.%I 3, + I 1h I a31 = I >s
-&-
1-I l>, I 3, + I 1h3I %I = i h
+
[I l>a I% + I I), I2hl = I h
-(2;1/” [-I I>, I% + I 1x9 I2),1 = I )A * 595/93/r-Z-10
label S,
144
HENRIQUES,
KELLETT
AND
MOORHOUSE
3. Spinor Covariants We list the possible spinor covariants in order according to the number of free Lorentz indices, and give the combinations with definite symmetry. Only states of one parity are given for each rank. The opposite parity states can be obtained trivially as described in Section 3. The covariants are not individually normalized, but they are chosen to be orthogonal for a particular choice of Lorentz components in a given rank. However, different Lorentz components are not necessarily orthogonal. The definite symmetry combinations are normalized to fl for each Lorentz component, but, the overall normalization for the 1 3 i2 combinations of components relevant for definite spin states must be obtained independently because of the possible cross terms. For brevity, we introduce an abbreviated notation by omitting the spinor indices, the spinor #Jfor quark a, and the matrix A. The matrix referring to the (b, c) subsystem is distinguished by enclosing it in brackets. Thus the zero-rank covariants of Eq. (3.5) become s = l(1) p = Y&J v = YWW)
(A.1)
A = -Y5Yu(Y5YU”) T = ~cr,,(u~*~) which
transform
under the permutation
and an antisymmetric
group as two doublets
singlet 1 -2(6)1/z [V - s + P].
64.3)
First Rank
(-4.4)
QUARK
transforming
a symmetric
MODEL
WAVE
FUNCTIONS
145
as three doublets
(A.3
singlet
1
___ 6(2)‘/” and an antisymmetric
[3&
- A, + A, - 431
(‘4.6)
singlet
12(6j1j2 [A, - A, + A, + 44
(A-7)
Second Rank Symmetric
in the Lorentz indices: 68)
giving a doublet (A-9) and a symmetric
singlet
--!6(2)112 Antisymmetric
W32
+ &I
(A.lO)
in the Lorentz indices:
(A.1 1)
146
HENRIQUES,
KELLETT
AND
MOORHOUSE
giving three doublets
A2(6)li2 P, + & + &,I i
L2(2)‘/” LB, + &I
+ LB, - & + Be1 1
4(3)‘/” [B, - B, + Bs + 2421
--!-4(3)1/Z LB, + B, - B, - 2&l
I
(A.12)
- ; LB4+ B, - &,I
i two symmetric singlets
12(6)lj2 [B, + B, - B, + B,,]
(A.13) --!4(3)1/Z [2B, + 2B, - B,,] and an antisymmetric
singlet
L2(6)112 [B+,- B, + 4, - &I
(A. 14)
Third Rank
Antisymmetric
in Lorentz indices:
(A.15)
QUARK
MODEL
WAVE
147
FUNCTIONS
giving three doublets
giving two doublets
-J-6(2)‘/” [Co - Cl1 + G,l 1 ‘Qj)W
‘lo
-J-2(6)1/2 [Cl, - Cl, + Cl01 1 2(2)‘/”
G4
(A.20)
148
HENRIQUES,
KELLETT
AND
MOORHOUSE
and four symmetric singlets
(A.21)
Fourth Rank It is convenient to introduce some intermediate
covariants
QUARK
The orthogonal
covariants
MODEL
WAVE
149
FUNCTIONS
are then given by
D, = R1 - Rz - 2R, + R, - R, - 2R, f .$[S,+ -S,+
t S,+ -&+I
- 2w1
D, = R, + R, + R, + R, - &S,+ + S,+ - 2S,+ + S,+ + S,+ - 2&j+] + D, = R, - Rz + R, - R4 + R, - R, + S,- -S,-
iw,
f Se-
D, = R, - Rz - 2R, - R, + R, + 2R, + $$S,- -S,-]
- $[S,- -S,-]
+ Se-
D, = R, + R, - R, - R, - &[S,- + S- - 2S,- + S,- + S,-] D, = T, + T, i- Ts + Ta + U, - U, f U, + U,
(A.23)
D, = Ts + T, f T7 A- Ts + U, - U, + h, + UI, D, = Tl + T, - T3 - T4 - U, + U, + U, - U, D, = T5 + T, - T7 - T, - U, + U, + U,, - U,, D,, = Tl - T, - T, + T4 - U, $ U, - U, + U, D,, = T5 - T6 - T7 i- T8 - U, t U, - U,, + U,,
!
giving three doublets
------20, 1
+ D, - 30, + 3D,]
12(3)lj2
p-----20, '
- 2D4 + 3D,]
12(3)‘/”
(A.24) 1 ~12(3)li2 1203 + 04 + 30, + 30111 ; DIO
i and five symmetric
singlets 1
1
4(3)'/" Dl '
-[2D, 12(6)lj2
+ D4 - 30, - 6D,], (A.25)
------[2D3 1 12(6)lj2
- 2D4 - 6D,],
1 ~ [2D, + D, + 30, - 60111. 12(6)1/2
150
HENKIQUES, KELLETT
Three of the combinations identities
AND MOORHOUSE
of covariants suggested by Eq. (A.22) satisfy the
which, together with the possible covariants E~~,J(& and •~~~,r~(l), form a set with the zero rank spinor structure multiplied by the Levi-Civita tensor, so do not give anything essentially new. For the second and higher ranks there are clearly also many possible covariants obtained from combining the metric tensor with lower rank covariants, but again these have been omitted as adding nothing essentially new. ACKNOWLEDGMENT The first author would like to acknowledge Foundation.
a research grant from the Calouste Gulbenkian
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at
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