Current quarks, constituent quarks, and symmetries of resonance decays

Nuclear Physics B61 (1973) 205-229. North-Holland Publishing Company

CURRENT

QUARKS, CONSTITUENT

SYMMETRIES OF RESONANCE

QUARKS, AND DECAYS

A.J.G. HEY *, J.L. ROSNER ** and J. WEYERS CERN, Geneva

Received 2 May 1973 Abstract: The transformation between "current" quarks and "constituent" quarks recently suggested by Melosh is examined with respect to its predictions for pionic decays of reso-,

nances. It implies the use of SU (6)W for classifying particle states but not for describing decay processes. Instead, pion emission proceeds v~ ALz = 0, + 1, where L is the internal ("quark") orbital angular momentum. This decay symmetry is called SU (6)w (ALz = 0, + 1). It is proven equivalent for any decay A ~ B + ~r (where A, B are arbitrary q~ or qqcl hadrons) to the 3Po quark-pair creation model for such decays, as formulated by Micu,

Colglazier, Petersen and Rosner. The roles of final orbital angular momenta ! and of SU(3) × SU(3) subgroups of SU(6) W are also discussed, and some new predictions are made for decays of meson resonances below 1700 MeV.

1. Introduction The quark model has met with remarkable success in hadronic physics, both as a "constituent" model for hadrons [1 | and as a model for their weak and electromagnetic currents [2, 3]. The two quark pictures are associated with two different SU(6) algebras o f operators. The algebra SU(6)W,strong introduced by Lipkin and Meshkov [4] seems appropriate for the classification o f the lowest lying hadrons: they appear to fall into simple irreducible representations o f this symmetry. Thus, this "constituent" picture places the observed mesons in 1 and 35 dimensional representations and the observed baryons in 56 and 70 dimensional representations o f SU(6)W,strong. A different SU(6) algebra introduced by Dashen and Gell-Mann [5], which one may call SU(6)W,eun~nts, is obeyed by the so-caUed " g o o d " charges in the infinite momentum limit [6]. These charges are the integrals over local densities which are directly or indirectly measurable in weak and electromagnetic processes. Moreover, since the isovector axial charge is included in thi~ algebra, and pionic decays o f resonances A -~ B + 7r are related by PCAC to the matrix element o f this axial charge, one * Research supported by a Royal Society European Fellowship. Alfred P. Sloan Foundation Fellow, 1971-3. Permanent address: School of Physics and Astronomy, University of Minnesota, Minneapolis.

**

206

A.Z G. Hey et aL, Quarks an~ symmetries of resonance decays

exoects that SU(6)W,eurr~ts may provide a description of some purely hadronic processes as well. Conceptually, constituent quarks and current quarks, and hence their respective SU(6)w algebras, are not necessarily identical. The success of CVC means that the SU(3) subalgebras are identical, but an attempt to identify particle states with pure representations of SU(6)W,curtentsleads to a number of difficulties. Notable among these are the prediction that the Adler-Weisberger relations [7] should be well-saturated with the lowest ½+ and ~+ baryon resonances (which they are not [7, 8]), and that anomalous magnetic moments of baryons should vanish (which they do not) [5]. Moreover, although a proton is well-classified as a state of three "constituent" quarks, it looks considerably more complicated when probed with currents. In such a description (based on deep inelastic lepton scattering [9]) the proton seems to contain additional quark-antiquark pairs and perhaps some neutral "glue" as well [31. Hence, although physically observable transitions are best described in the language of SU(6)W,currents, the fact that the states themselves are complicated mixtures of representations of this group [10] has been a substantial obstacle up to now to concrete applications of this symmetry. Recently Melosh [11 ] has constructed the transformation between the two SU(6)W algebras in the free quark model. He finds that this tLansformation V, acting on the axial charge F i5, produces a transformed axial charge F i5 - V-1 Fi5 V with relatively simple properties in the constituent quark picture. Abstracting from this "free quark" property, several authors [12, 13] have assumed that a similarly simple F i5 exists in the interacting theory as well. The matrix elements of F ~5 between pure states of SU(6)W,suong (corresponding to our idealization of the physical particles) are then relatively straightforward to compute, and imply a number of relations among pionic decays of resonances. In this paper we examine the general structure of such predictions. We fred that the SU(6)W,snong introduced in ref. [4], while perhaps appropriate for classifying particle states, does not follow for collinear processes such as resonance decays. This "wrong" symmetry will be referred to as "naive" SU(6)w. Instead, in the present scheme, some relatively simple selection rules govern pionic transitions between states classified according to SU(6)W,st~ong.In terms of an abstractly defined internal orbital angular momentum L z, these transitions turn out to involve only ALz = 0, +- 1. This "possibly right" decay symmetry will thus be referred to as SU(6) w (AL z = 0, +- 1). We find that for any hadronic decay A ~ B + 7r, the symmetry SU(6) w (ALz = 0, -+ I), motivated by the work of ref. [I I ], is equivalent to the successful phenomenological "3P0" model of Micu [14], Colglazier and Rosner [15] and Petersen and Rosner [16]. The phenomenological prescription [14-19] known as "/-broken SU(6)w" [16, 17] is found to be equivalent to SU(6)w (AL z = 0, + 1) for many cases of physical interest, but a special case of this symmetry (and hence essentially unmotivated on theoretical grounds) in general. We have also derived the predictions of the chiral SU(3) X SU(3)snong subalgebra of SU(6)W,sttong, and dis-

A.J.G. Hey et aL, Quarksand symmetriesof resonance decays

207

cuss the relation to other symmetries in some specific mesonic cases. Various phenomenological applications to decays of the mesons in the " R " region (mass around 1700 MeV) are mentioned. The paper is organized as follows. Sects. 2 and 3 are devoted, respectively, to the constituent and current quark approaches to SU(6) w as applied to hadronic vertices. These sections, as well as the beginnin~ of sect. 4 which deals with the explicit expressions for decay amplitudes, are expository in nature and follow closely the treatments of refs. [11-13]. Then, in the remainder of sect. 4, we discuss in some detail all possible pionic decays of meson resonances up to L = 2. This section includes predictions of interest to experimentalists. Sect. 5 treats the equivalences and differences among the various above-mentioned pictures of hadronic decays. Finally, in sect. 6, we summarize our results and indicate where further work is needed.

2. Constituent quarks Particle states at rest appear to be well described by the quark model with orbital excitations. By taking all mesons as quark-antiquark pairs and all baryons as three quarks one can then understand a number of features of the hadron spectrum. The symmetry associated with the above classification is U(6) × U(6) X 0(3). The first U(6) refers to quarks and the second to antiquarks, while the 0(3) describes the orbital angular momentum L. A number of multiplets of this group are either entirely filled or well on their way to being so. There is strong evidence for mesonic states belonging to (6, ~ , L/) = 0 - , 1+ and 2 - and for the baryonic multiplets ((56, 1), L P = 0+), ((70, 1), L P = 1 - ) and ((56, 1), L p-- 2+). Other mesonic and baryonic multiplets may exist roughly degenerate with the L = 2 levels just mentioned. Clearly this U(6) × U(6) × 0(3) symmetry, which we shall call the rest symmetry, has to be badly violated in resonance decays since it forbids, e.g., A --) Nz or p -} zrr. This is, of course, not surprising since decay amplitudes involve moving particles for which the rest symmetry should not hold. Lipkin and Meshkov [201 pointe¢ out that the reflection and rotation invariance properties of collinear processes, such as two-body decays, lead to invariance under "It-spin" or SU(2) w. For a particle moving along the ~-a~is, the generators of It-spin may be written:

hYx=PintSx,

~=PintSy,

Wz=Sz,

(2.1)

where Pint is the intrinsic parity of the system. For a single free quark movinl~ in the ~-dlrection one would thus write the SU(2) w generators as follows:

wx= o x,

wz= 1

(2.2)

Combining SU(2) w with SU(3) one is thus led to consider SU(6)W, faong as the natural candidate for this "relativistic spin symmetry". In a constituent quark model, the generators of this algebra would then be written:

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A.J.G. Hey et al., Quarks and symmetries o f resonance decays

xi

Xi

Xi

q,q

%

Xi

%

q,q

(2.3)

%

Xi

%

i xi % G = q,-~

-

2

2"

These expressions for the generators clearly commute with ~q,Ft, °~z, which, for free quarks, is the generator of Lorentz boosts in the ~-direction, This last property of SU(6)W,mong for free quarks suggests that perhaps the physical SU(6)W,mong might be an appropriate algebra for particles moving along the ~.axis, and, in partic. ular, for decay amplitudes or for collinear processes. It should be pointed out, however, that this is an assumption which is not really well motivated. It allows us to classify moving states knowing their rest frame classification. In the rest frame both quark spin and W-spin are well defined and one finds that for quarks : antiquarks :

~

= S- ;

Wz

= Sz ,

(2.4) Wx,y = - Sx, y ,

where S stands for ordinary quark spin. These relations follow trivially from flq = +q and ~ = - ~. As is well known, one finds from eqs. (2.4) the so-called W-S flip [2!], e.g.,

but

Ip(Jz=l)>

~ IW=I, Wz=I)

Ip(Yz=0))

~ Ile=O, W z = O ) ,

while In ( J = 0))

~-IW=I,

,

(2.5)

Wz=O) .

Under Lorentz boosts in the ~ direction, L does not remain a good quantum number but L z will continue to be conserved (as will Wz, of course). Even though SU(6) w appears to be an approximate symmetry for the classification of one particle (moving) states it need not be a symmetry of the whole I-Iilbert space. An illustrative example of a similar situation is found in the case of the hydrogen atom. The energy levels of the H atom possess an extra degeneracy beyond that of rotational invariance: states of different angular momentum but the same principal quantum number have the same energy. It may be shown that the generators of an 0(4) symmetry commute with the Hamiltonian of the H atom [22]: however, the scattering of bound states does not possess this symmetry. Thus O (4) is only useful for analyzing the spectrum of isolated states and not for the dynamics of particle interactions. If, however, ono does assume SU(6)W,strong for collinear processes such as threepoint functions (~na'ive" SU(6)w), one has separate conservation of L z and Wz. It is precisely this restriction which appears to be violated in nature [15, 16, 18].

A . Z G. Hey et aL, Quarks and symmetries o f resonance decays

209

Hence, we can only conclude that the generators of SU(6)W,strong approximately commute with the strong interaction Hamiltonian when sandwiched between oneparticle states, and that SU(6)W,stamg is not, even approximately, a vertex symmetry. For this reason phenomenological models were proposed several years ago with the intention of relaxing SU(6)W,strong symmetry somewhat. It was shown by Carlitz and Kislinger [23] that one can construct SU(6) w invariant vertices by working with states of quark spin but allowing an additional quark-antiquark pair to be produced with L z = 0, P = C = +, and internal quantum numbers of an SU(3) singlet. This pair shares its quarks among the two outgoing particles in the decay. On the basis of this picture, it was guessed that a useful modification of naive SU(6)W would result from allowing the pair to have L z = + 1 as well as L z = 0 [151. This model has come to be called the "3P0 picture", [14, 16] since the q~ pair is assumed to have all the quantum numbers of the vacuum. Actually the name is somewhat misleading, since 3P0 implies a definite relation between L z = -+ 1 and L z = O. In contrast, the model assumes the relative admixtures of these two states to be arbitrary. In various practical applications [14-19] it has been shown that the 3P0 picture is equivalent to breaking only those naive SU(6) w relations which link different final relative orbital momenta l between the two resonance decay products. Those naive SU(6) w predictions referring to a given l are preserved. This scheme is the "1 broken SU(6)w" referred to in the introduction. As we shall show, the simplest place where l broken SU(6) w is only a special case of (and not equivalent to) the 3P0 picture is in L = 2 ~- L = 1 + pion decays. Two subgroups of SU(6)W,strong have also been applied to hadron decays. These are the coplanar [15, 24] and chiral [12, 13] versions of U(3) × U(3). In the picturesque language of constituent quarks, the possibility of transverse motion of these quarks inside a hadron suggests that one studies the subset of the generators (2.3) commuting both with a Lorentz boost along the ~-axis and a transverse boost (say a x in the free quark model). This gives rise to the coplanar [U(3) X U(3)] ao first discussed by Dashen and Gell-Mann [25]. This symmetry was explicitly applie~I'to decays in ref. [24] and was found to be weaker than the 3P0 picture but consistent with it. The chiral version of lff(3) X U(3)strong generated by Wz and Wz is somewhat better motivated theoretically since in any interacting quark model these generators should always commute with boosts along the £,-axis. Furthermore, from the current quark point of view, as will be seen in the next section, the chiral U(3) X U(3)eurmats is well established while the existence of tensor currents, which can be constructed in a current quark model and used to extend the chiral algebra to an SU(6)W,eurrmt s algebra, requires an additional assumption. Hence it is of some interest to study the implications and the breaking pattern of the chiral SU(3) X SU(3)strong subalgebra of SU(6)W,snong.

A.J.G. Hey et aL, Quarksand symmetries of resonance decays

210

3. Current quarks To discuss current-quark algebraic structures let us start from the well-known chiral SU(3) X SU(3) algebra [2] of vector and axial vector charges Fi

(t) =f d3x~io(x)

,

Fi5(t) =f d3x ~ ioS(x)

(3.1)

(3.2)

,

where the current densities ff bli (x) and 9ri5 (x) are well defined measurable physical ~ ,, operators in weak and electromagnetic transitions. In a "current.quark model, one would write these vector and axial vector current densities as

9:i~(x) =-~ (x) 7 u ~ h i q(x) ,

(3.3)

9riS~(x) = ~ (x) 3,~,75~ Xi q(x) .

(3.4)

From these expressions and the canonical anticommutation relations of the quark fields one derives the equal time commutators:

[Fi(t),F'(t)l =ifijkFk(t) ;

[Fi(O, FiS(t)] =ifijkFkS(t) (3.5)

IF ;s (0, F ;s (01 -- i~/k ~ k ( 0 . This SU(3) X SU(3) algebraic structure is then assumed for physical vector and axial vector charges independently of any underlying quark structure. As is well known, this algebra leads, with the help of PCAC, to the Adier-Weisberger sum rules. Insofar as they have been tested these are remarkably successful. In view of this, one can attempt, following Dashen and Gell-Mann, to enlarge the algebra. With the equal time commutation relations of the quark model one can clearly define a U(12) algebra generated by 144 charges defined as integrals over the densities ~(x) r ½ xi q (x), where r is one of the 16 Dirac covariants. All these current densities are in principle directly or indirectly measurable: fo/example, although nothing in nature seems to be coupled directly to a tensor current 9 ri~ V (x), represented in the quark model by • ~(x) our ~1 )~! q(x), this current nevertheless appears on the right.hand side of commutators of more directly observable operators, e.g., in [St! (x), a v ~r!5(y)lx,~---va. In this sense all 144 current densities could be meaningful p"hysical operators pVro'-~ vided, of course, that the algebraic structure of the current quark model can be abstracted that far. It is important to realize that contrary to the situation in the constituent quark approach one has here an algebraic structure which is generated by well defined operators: they are integrals over local current densities and their Lorentz transfer-

A.J.G. Hey et aL, Quarks and symmetries of resonance decays

211

mation properties are given "by deffmition". Let us go in particular to the infinite momentum frame. As usual we define as "good" operators those charges whose matrix elements do not vanish when taken between t'mite mass states with infinite momentum along, say, the 2 direction [6]. There are 35 independent good operators * whose quarkjnodel expressions are given by:

F i (t) =fd3x { <,O

=

F (t) =f

= f d3xq+(x)O%½Xiq(x) ,

Fy (t)

=f d3x

l(x)

F~(t)

=f d3x V: 5(x) = f a3xq+(X)Oz½X'q(x) .

(3.6)

= f d3xq+(x)#oy ~xiq(x) ,

These charges generate at infinite momentum an SU(6) algebra which we will call SU(6)W,eunents. The algebraic similarity between these F ~ or "current-quark generators" and the W~ or "constituent quark generators" defined in the previous section is now obvious. It is, however, extremely important not to confuse or to identify too quickly the two sets of generators of these SU(6) w algebras. The F~'s are explicitly integrals over local expressions with well defined Lorentz transformation properties, but they have not been assumed to commute with the strong interaction Hamiltonian. The l¢~a's, on the other hand, are supposed to commute approximately with the strong interaction Hamiltonian at least between one-particle states but there is no a priori reason why they should be intezrals over local measurable densities. (In fact, as we shall see below, one finds that even in the free quark model [11] the F~'s do not commute with the "strong interaction" Hamiltonian and that the W~'s are not local charges.) Hence one has two sets of possibly distinct ooerators which generate the same SU(6)w algebraic structure. A fundamental question is of course the relation, if any, between the two sets of generators. From the successes of the conserved vector current hypothesis (CVC) we are forced to identify the W i and the F i, i.e., the generators of ordinary SU(3) of strong interactions and the integrals of the time components of the vector currents. To identify the full sets of generators of SU(6)W,strong and SU(6)W,eurrents would, however, be incorrect. A strong objection to such an identification comes from the Adler-Weisberger relation which, when saturated with resonances necessarily implies representation mixing, i.e., particles which belong to different SU(6)W,strong representations are connected by the axial charge and thus at least a piece of their wave function must lie in the same representation of SU(6)W,currents. One could argue, perhaps, that if the AdleroWeisberger sum rule is saturated with resonances (N, A) belonging to the same SU(6)W,strong multiplet, the result obtained for -GA/Gv, ~, is not too far removed from the correct * Properties of good currents and indeed the Melosh transformation itself may be reformulated on a light-like surface. This is perhaps a more exact expression of the infinite m o m e n t u m limit.

A.J.G. Hey et hi., Quarks and symmetries of resonance decays

212

value. Dashen and Gell-Mann [5] have shown, however, that in the same approximation, i.e., classifying the nucleon octet and A(1238) decimet in a 56 of SU(6)W,currents (as well as in a 56 of SU(6)W,mong) the anomalous magnetic moments of all ½+ octet and {+ decimet are predicted to vanish. This is clearly unacceptable. One arrives at the conclusion that there must be sizeable representation mixing and thus that the generators of the two SU(6)W algebras cannot be identified. This being the case one may try to relate the two algebras by a unitary transformation. In fact there is no compelling argument in favour of such an assumption: it is motivated by the feeling that the similarity in these algebraic structures is a fundamental property of the hadronic world. If to the quantum numbers corresponding to particle classifications under either SU(6)W, one adds all necessary quantum numbers to obtain two bases of the Hilbert space, a unitary transformation relating one of these bases to the other must exist. This does not imply that the generators of the two SU(6)w algebras are, by themselves, related through a unitary transformation but it makes such an assumption somewhat more plausible. Furthermore Melosh [11 ] showed recently that in the free quark model such a unitary transformation does in fact exist. We will postulate that such a transformation exists in general and write (3.7)

W i~= V F i V -1 ,

where V is unitary and will be called the Melosh transformation. This transformation V thus expresses the general idea that hadrons can be simultaneously described as simple constituent quark states where strong interactions are concerned and as rather complicated structures when currents are involved. It systematizes in a compact way all SU(6)w cun~*, representation mixing for all hadrons. To make this last point 0 clear let us recall' that - W~ and F~0 act like quark spin operators for constituent and current quarks respectively. Both operators commute with Jz and thus make it possible to define "orbital" angular momenta [26] by

and

(3.8) = j

_FO

.

We may then classify states according to their SU(6) w quantum numbers and angulax momentum. For example the SU(6)W,slzon8classification of the lowest lying mesons would be 35 + 1, Lz(W ) = 0 while the first excited states would be 35 + 1, L z ( W ) = - 1, O, + 1. The Melosh transformation V must obviously be invariant under rotations around the g-axis i.e., [V, Jz] = 0. Let us suppose that [V, F ° I * 0 so that F O and WO are different operators. A state with a simple constituent quark spin structure will then in general be expected to correspond to a mixture of different current quark spin states. Similarly, while the constituent orbital angular momentum L z (W) will be simple, one will have a mixture of current orbital angular momenta L z (F) since Jz = WO + L z ( W ) = FzO + L z ( F ) ' and Jz commutes with the Melosh transformation.

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A.J.G. Hey et aL, Quarks and symmetries of resonance decays

To conclude this section on current quarks we recall the explicit construction of the Melosh transformation in the free quark model. Such a model is defined by the Hamiltonian Hfree = f

d3x q+(x) (- ia. a +tim }q ( x )

(3.9)

.

The generators of SU(6)W,currents are given, in this model, by the expressions appearing in eqs. (3.6). They do not commute with Hfr~ and hence cannot be identified with the generators of SU(6)w,s~ong. In this sense, the free quark model, although trivial, exhibits some of the complexity of the full problem. As shown by Melosh with ["free = exp (i Yftee) ,

and

(3.10)

one finds, through eq. (3.7), the following set of operators:

Wtx,f~ = F'x,fr~+ f d3xq+(x) "~1{ 1+1~ "Ix.aim i % -~ kdq(x), wiy,fme z,free

=

Fiy,fte¢ fd3xq~(x) K1 [ 1 +1 I¢ +

k

= Fz,f~e +

d3x q÷(x) -~

1 +r

(3.11) v~.Oi In

m

i "Ys~ ~x q(x), m

where

These transformed operators can be shown to commute with Hfroe and are thus appropriate as SU(6)W,stmng generators. Some of the properties of the Melosh transformation are worth pointing out. The first remarkable property of gfre¢ i$ that it is non-local in the transverse directions and hence, as shown explicitly by eqs. (3. I I), the symmetries of the Hamfltonian are generated by non-local operators! Introducing the (current) quark spin and orbital angular momentum operators, Zi,fi~

= f d3x q+(X) ~otq(x) , (3.12)

Li,~.

= i,ij k f d3x xk q+(~) ~q(x) ,

214

A.J.G. Hey et al., Quarks and symmetries of resonance decays

we can describe the properties of Yfreeunder SU(6)W,euxtents X 0(2) as follows: Yfi~eis the SU(3) singlet member of a 35 with A J z = 0, A~ z,ftee = + 1. Although Yftee only changes L z,free by + 1 the presence of the power series in (¥x. a J_)~m ira. plies that Yfree can mix essentially all total angular momenta. Let us note also that the W/ have simple algebraic properties under SU(6)w,cun~nts × 0(2): they transform as the sum of a 35 with ASz = AL z = 0 and of a 35 with ASz = - ALz = + 1.

4. The model and its appfications to meson resonances 4.1. The SU(6)W ( A L z = O, + 1) model

Given the existence of a unitary transformation between constituent quarks and current quarks we may now proceed, with the help of PCAC, to a "theory" of pionic transitions. As hadrons do seem to fall into irreducible representations of SU(6)W,s~on~, this algebra is an approximate symmetry of the strong interaction hamiltonian (at least for one-particle states). This does not imply that SU(6)W,mong can be considered as an approximate symmetry in the whole Hilbert space. Indeed, as discussed in sect. 2 one finds that SU(6)W,strong is totally unacceptable as a vertex symmetry. On the other hand, the PCAC relation

a~ ~7~i5 = c ~i . . . (i. = 1,

8),

(4.1)

allows us to write pionic three-point functions as follows: (B ~'lA ) "" m 2 - m k c

(BIFiS]A) .

(4.2)

Thus, PCAC reduces those hadronic three-point functions involving at least one pseudoscalar meson to the matrix elements of the axial charge between one-particle states for which SU(6)W,mon~ is expected to be a good approximate symmetry. But the axial charges F i5 are of course generators of SU(6)W,cunent s and they are assumed to be related by the (unitary) Melosh transformation to the SU(6)W,stzong generators. Hence any assumption about the Melosh transformation V will reflect itself in pionic decays of hadrons. Let us formalize this connection by defining the Melosh transformation as the unitary transformation of the Hilbert space which relates the constituent quark representation (i.e., in SU(3) X SU(3)st~ong or in SU(6)W,mong ) of a hadron state h to its current quark representation (i.e., in

SU(3) X S U ( 3 ) ~ t ~ or in SU(6)W,cua~ts): [h; current quark ) = Vih; constituent quark ) . The matrix elements of the axial charges F i5, given by (h'; current quark IF i5 [h; current quark) , can then be rewritten as

(4.3)

A.Z G. Hey et aL, Quarks and symmetries of resonance decays

(h'; constituent quark J V-1 FiS VIh; constituent quark) .

215

(4.4)

As discussed in the previous section, in the free quark model, where Vfreewas explicitly constructed by Melosh, one finds that the transformed axial charge has a remarkably simple algebraic structure: it transforms, under SU(3) X SU(3)s~ong as the sum of the ((8, 1)0, 0) - {(1,8)0,0)and ((3, ~ ) 1 , - 1) - ((g, 3)_1, 1} representations *. Equivalently, the transformed axial charge transforms, under SU(6)W,s~ong, as the sum of the {35, L z = 0) and ~35, L z = .-+ 1) representations. We will assume in this paper that these simple algebraic properties of the transformed free quark axial charge can be extended to the physical axial charge. This is of course an extremely strong assumption: it can be somewhat motivated by the following argument. As long as the Melosh transformation is bilinear in quark fields these algebraic properties of the axial charge are in fact the most general ones. Indeed, for the axial charges belonging to SU(3) octets with &/z = 0, a Melosh transformation bilinear in quarks which is an SU(3) sing,let and commutes with Jz, can only transform them to oetet~ with z ~ z = 0 and ASz = + 1. This is equivalent to the SU(3) X SU(3)strong or SU(6)W, strong transformation properties of the free transformed axial charge. The question then is of course why V should be bilinear in quarks. We do not have any compelling argument in favour of this assumption except that it is the simplest way to connect two remarkable properties of hadronic physics: both currents and hadronic states seem to behave in some sense like free quark currents and free quark states respectively. A bilinear Melosh transformation allows us to connect these two "free quark" approximations. Before using eq. (4.4) to compute pionic decays of hadrons we still have to connect real hadronic states to free constituent quark states. Although we know of no satisfactory prescription to make this connection, especially since we do not know the strong interaction Hamiltonian, it seems reasonable to assume that physical oneparticle states can be constructed by acting with some operator U on free quark states [11 ]. The success of SU(6)W,s~rong or its subgroup SU(3) X SU(3)strong in classifying hadronic states can then be understood if U is assumed to transform as an SU(6)W,s~rongor SU(3) X SU(3)strong singlet. To summarize, our model for pionic decays is based on the following assumptions: (i) PCAC; (ii) "free constituent quark" classification of hadrons in SU(6)W,strong or SU(3) X SU(3)sttong (U transforms as a singlet); (iii) V - 1 F i51I transforms as the sum of the {(8, 1) 0, 0) - {(1, 8)0, 0) and

* We have used the standard notation {(A, B) S , L z ) to label irreducible representations of SU(3) X SU(3) at inf'mite momentum along ~ e ~-axis, corresponding to helicity S z + L z and to total quark spin Sz.

216

A.J.G. Hey et al., Quarks and symmetries of resonance decays

{(3,3)1, - 1} - ((-3, 3)_1, 1) representations of SU(3) X SU(3)stror~ or equivalently as the sum of the {35, L z --- O) and {35, L z = - 1~ representations of SU(6)W,mong. In the remainder of this section we will refer to our model as the SU(6)W (AL z = O, --- 1) model. 4.2. Classification according to U(3J X U(3} and SU(6J w

We shall need the reduction of the (6, 6), L states of U(6) X U(6) X 0 ( 3 ) to combinations of U(3) X U(3) X O(2)Lz or SU(6)W X O(2)Lz states, Once we reduce the states of given quark spin S and z component S z, the extension to states of arbitrary L will be trivial. The relation between quark-spin and W-spin states was already pointed out in sect. 2. In the discussion of (U(3) X U(3))Sz we need only recall that Ozq = Szq while ezq = - Szq. Then table I a shows the necessary relations between the rest symmetry, chiral U(3) X U(3), and SU(6)w classifications. For simplicity we consider only octets, avoiding the discussion of nonet couplings except in the context of phbnomenological interpretations. States of arbitrary L with given helicity are then expressed in terms of those in table 1a v/a Clebsch-Gordan coefficients,

IJSL;k>=

~ (SSzLLzlJX)ISSz>. Sz+Lz=h The JPC values of the specific states considered here are shown in table lb *

(4.5)

Table 1 (a) Classification of ISSz) states of (6, 6) according to U(3) X U(3) and SU(6)w t

S= I,S z=

1

Sz --- 0 sz

= -1

s=O, Sz= o

u(3) x U(3)

su(6) w

(3,3)1 [(8, 1)o + (1, 8)0 ]/~/2

(8, 3)t -(8, 1)o

(~, 3)_x

-(8, 3)_1

[(8, 1 ) - (1,8)o1/~/2

-(8, 3)0

(b) dPC of states considered here

S=I L=0

S=0 1-1~,

0 -÷

L=I

2~,

1--

l*-

L=2

3--, 2--, 1--

2-÷

t SU(6)W representations are written as [<~SU(3),2h/+ 1] Wz *

Footnote: see next page.

A.Z G. Hey et al., Quarks and symmetries of resonance decays

217

Table 2 L = 1 .-,L --- 0 h=1

Process

oo a~.=1

el ak= 1

2 ++--* 1 - - -

+ 3/16

- 1/4x/2

1++ ~ 1 - -

+ 3/16

1/4x/2

1 + - --* 1 - -

h=0 2++ ~ 0 - +

0

1/4

oo ah=o

aOl h=O

+ (1/8K/3

- 1/2~/6

1++ ~ 1 - -

0

1+ - -* 1 - -

+3/8x/2

0 ++ --* 0 - +

-(1/8)x/~

1/2x/2 0 - 1/2~/3

Dictionary SU(6) w (ALz = 0, -+ 1): aLfLi = aLfLi for all h I broken SU(6)w: equivalent to SU(6) W (AL z = 0, + 1) as - - - - a ~ + -Ivr~ a"'°1 (t = 0)

a d --a ~ ' - ] x / ~ a °1 (l = 2) '~Naive" SU(6)W: a Ol = 0, a s = ad

4.3. Predictions for specific processes T h e r e d u c e d m a t r i x e l e m e n t s within a given set o f decays L - , L ' + 7r m " chiral " U ( 3 ) X U ( 3 ) m a y be labelled by the initial L~z, the final L f , and the total helicity ?,. In a d d i t i o n , for r e d u c e d m a t r i x e l e m e n t s o f three octets (8, 1) --, (8, 1) ® (8, 1) and (1, 8) --, (1, 8) ® (1, 8), the relative phase o f the t w o depends on the p r o d u c t o f the charge parities o f A ( L ) , B ( L ' ) and 7r. In practice for the cases we consider, this leads to an e x t r a degree o f f r e e d o m in the Liz = Lfz = X = 1 amplitudes, w h i c h will be expressed v ~ a p a r a m e t e r #. Since chiral S U ( 3 ) X S U ( 3 ) is the weakest o f the s y m m e t r i e s we consider, our tables will give its predictions, w i t h a simple " d i c t i o n a r y " at the end o f e a c h table for c o n v e r t i n g to the stronger symmetries. tfLl The r e d u c e d m a t r i x e l e m e a t s in chiral S U ( 3 ) × S U ( 3 ) , called a ~ z z, are normalized so that in the S U ( 6 ) w (&L z = 0, -+ 1) limit

LfL i LfL i axZ Z = a z z ,

(4.6)

* It is necessary to take some care L~ choosing a consistent set of conventions for the behav~our of ((U(3) × U(3)Sz, L z } representations under charge conjugation and parity. For example, one {(8, 1)o - (1, 8)o, O) representation represents a pion with C = + 1 and another, different ((8, 1)o - (1, 8)o, 0) representation an helicity zeroB with C= - 1.

A.Z G. Hey et eL, Quarksand symmetries of resonance decays

218 Table 3 L=2~L=0 h=

1

Process

a~°__l

a°hl=l

3 - - .-+ 1 - -

- 1/2x/10

+ (1]4) 3x/~

2--.-+ 1--

- 1/4x/2

-(1/16)x/3

2- + ~ 1 - I--~

0

1--

x=o

3 - - --} 0 - + 2--

--+ 1 - -

- 3/8x/2 -

a~°__0

a~,~=0

- (l/4).,v/~

+ 3/4x/10

0

-

2 - + -~ 1 - -

- 1/4

1--

+ 1/2x/10

~ 0 -+

(3/16)x/~

- 1/4x/I0

3/8 0

+ (3/4)~

Dwtionary SU(6)W (ALz = 0, +- 1) : a~ fLi = aLfLi for all k l b r o k ~ SU(6)W_~ ot : equivalent to SU(6)W (ALz = 0, +- 1)

a,,,~a

+(])x/~a

a~-= a °° - ~ a "~ "Naive" SU(6)W

(l=l)

(l = 3) : a °1 = 0, ap = af

except for reduced chiral S U ( 3 ) X S U ( 3 ) m a t r i x elements which vanish in this limit = 0). The helicity amplitudes gx for A ( L ) ~ B ( L ' ) + Ir in S U ( 6 ) w (AL z = 0, +- 1) are:

gh =

~ ~A(~,--Liz)(SAX--Li L L i i j A x ) f L z, L zi

X ~B(~k--L~) ( s B x - L f L

' Lfz [/B~) (4.7)

(8,2wB+1)(8,3) X (wBx-

(8,2wA+l

,,

B n A

Lfz 1 L ~ - Liziw A ~,- L i) aLfzLiz .

The first two Clebsch-Gordan coefficients in eq. (4.7) are those c o m i n g from eq. (4.5), The n e x t t e r m is an S U ( 3 ) scalar factor (a Clebsch-Gordan coefficient o f SU(6) [27]). The i n d e x ot1 denotes s y m m e t r i c or a n t i s y m m e t f i c coupling o f 35 -+ 35 @ 35; it is chosen so that the SU(2) scalar factor (Clebsch-Gordan coeffi-

A.J.G. Hey et at, Quarks and symmetries o f resonance decays

219

Table 4 L=I--*L=I k = 2

Process

a~_- 2

2 ++ ~ 2 ++

0 a~o=l

k = 1

11

1o ak= 2

- 1/4

0

a~=2

aIl k=l

atO

- 3/8x/~

k=l

2 ++ --, 2 ++

- 1/8

- M2

2 + + -~ 1 ++

- 1/8

+ #/2

2 ++ ~ 1+ -

0

1 ++ ~ 2 ++

- 1/8

+ p/2

1 ++ ~ 1++

- 1/8

- p/2

1++ ~ 1 + -

0

+ 1/4X/'2

+ 3/16

1+-~

0

- 1/4x/2

+ 3/16

1+ - ~ 1++

0

+ 1/4X/'2"

+ 3/16

1+ - - ~ 1+ -

0

-p.

2++

- 1/4x/2

: o 2 ++ ~ 1 ++

0

2 ++ -+ 1+ -

0 + 3/8X/'2

0

4'--0

a' 0

- 1/4x/3

+

- 1/2x/~

1 ++ ~ 2 ++

0 + 3/16

0

(1/4),vT~

-(1/8)x~

0

- 1/4x/r3

+ (1/4)x/~

0

- 1/2x/6

1+ - ~ 2++

- 1/2x/6

0

-o/8x/~ -(t/s).,/T

1+ - ~ 0 ++

+ 1/4x/3

0

- (1/4)4:K

1 ++ ~ 0++

0 ++ "-' 1++

0

0 ++ ~ 1+ -

- 1/2x/6

+ 1/4x/3

0

-(i/s),.~ - (1/4)4YK

Dwtionary SU(6) w (AL z = O, + 1) : 4 fLi = a LfLi for all ?~ ; p = 0 ! brr ken S U ( 6 ) w :equivalent to S U ( 6 ) w (AL z = 0, ~ 1) 0 O0 10 -~p a - (3/~/~) a ~ (two independent ~ ' ~-a 11 + ( 3 / 2 x / ~ ) a l ° J 1= 1 combinations) , ~ f ~ a OO- a II + (3/x/r2) a 10 q = 3) ,, .. ,, Natve SU(6)W :a 10 = O ; a f = a0p - a p1 cient of SU(3) [28]) denotes the proper d - type or f-

t y p e c o u p l i n g as d i c t a t e d

b y t h e p r o d u c t o f c h a r g e p a r i t i e s o f A , B a n d n:

e(A) e(B)e(~)

=+

:

o~2 = " d - t y p e "

e ( A ) e ( 8 ) e (~)

= -

•

o~2 = " f - t y p e "

;

(4.8) .

A . Z G. Hey et aL, Quarks and symmetries o f resonance decays

220 Table 5 L=2-~L=I = 2 Process

a °° h=2

a 11

3 - - - + 2 ++

0

2----+2 ++

0

2 - + - ~ 2 ++

0 O0

h= 1

a01 h=2

+(l/8)x/]"

0

0

-1]4x/3"

+(l]8)x/~

0

0

+112,q~

0

0

0

+1/4

1o

11

10

ak= 1

ak= 1

ak= 1

3 - - - - * 2 ++

+3/8,~

-px/~~

+ 1/4,~

-1/2,~

-l/4x/~

3 - - - * - 1 ++ 3---~ I+-

+318,~

+ U ~ + 1]4X/~

- l ] 4 X fS-I/2~1"0

-112x/~ 0

+ 114x~

0

2--'--> 2 ++

+3]16x/2

+ p/,,,/r~

+ 118

+ 1]8,~

+

2--~

1++

+3]16x/2

-#]x/~

-118

+ 1]8x/3

-1/4x[6

2--~1

m"

-I]4x/~

1]4,v/6

+-

0

~(l]16)x,/~-1]4,¢~

0

+1/4~/3"

2 - + -~ 2 ++

0

+3/16

0

+ 1/4~f2

0

2 - + ~ 1++

0

-3/16

0

+ 1/4w/'2

0

2 -+ ~ I +-

0

-p

0

0

0

+1/8x/'5 -.u~,~'~ -1/8.~r5 -(3/16)x/~ -1/4,,4'10

+(1/8)x/~ +(118)~/~ 0

-(1/4)~ +(1/4).V/~"~ -(1/4)x/3-~

~°~o

a[Lo

alo h=O

:I h=O

~12 k=O

3---+1 ++ 3---+1 +-

0 +(3/8)~

+ 3/8x/5 0

-(1[2)~ 0

0 -I/2,¢r5

0 0

2---*2 ++

o

+(I18)3,4T]~ o

+1/2,j~

o

2 - - - - * 0 ++

0

+(1/8X/'3"

0

-1/2x/~

0

2-+~2

+(1/8)x/3-

0

+ 1/2,~

0

0

++

+w,,/3]~

2 - + ~ 0 ++

-(1]8)~/3"~ 0

+ 1/2~v/3

0

0

1 - - - ~ 1 ++

0

+(3/8),4~

+1/2w:5

0

0

1 - - - - * 1+ -

-3/8x/5"

0

0

-(1/2)x/~

0

Dictionary SU(6) W (AL z = O, ± 1) ! broken S U ( 6 ) w 01_

"|2

: a~hfLi = a LfLi for all h ; p -- 0

t~-~

a~ --- a O0 - (~Vr~) a I a ~ ~ a l l - ( ~ / 2 ) a°l o i a . -~ atm - (21~/r3) a I 1 + ~ff~a 1 _ w/-~-aOl 2 'd~laive" S U ( 6 ) w : a°l=a let= a I = 0 0

12

a~= 1

=o

_

01

12 a~=2

a~= 1

1 - - - , 2 ++ +3/16x/]"0 1----+1 ++ +3]16.V/~ 1---+ 1+- 0

10

a~=2

h=2

1

a s - a d + ad,4~

.

_

0

(two independent l = 2 combinations) (l = 4) ; 1

,ag - a d - (2/~/3) a d

A.J.G. Hey et al., Quarks and symmetries o f resonance decays

221

In the isoscalar factors, we shall take the specific isomultiplets A=B=K (the I = ~, Y= + I states) for convenience. The phases ~A and ~B arise from the relation between quark spin and W-spin states (see, e.g., eq. (2.5)], and are ~A,B(sz)

= + 1

: Sz=+ l

=- 1

; otherwise

;

(4.9)

Finally, the last Clebsch-Gordan coefficient in eq. (4.7) expresses W-spin conservation. Note that the value of Wz associated with the pion (I¢= 1) need not be zero. _ f i_ It is zero only in the case of "naive" SU(6)w, where A L z - L z - L z O. The tables are: table 2 : L = l -+ L = O ; t a b l e 3 : L = 2 -~ L = O ; table4:L=l~L=l ; table5: L=2-rL=I . We shall discuss the relations among various symmetries implied by these results in the next section. Meanwhile it is helpful to point out where comparisons with data may be made. 4. 4. A p p l i c a t i o n s t o e x p e r i m e n t

The case o f L = 1 -+L = 0 decays has already been studied in a formalism equivalent to SU(6) w (AL z = 0, + 1) [15]. The ALz = -+ 1 transitions were found to be dominant, as measured in the B -~ w~r decay. Recently these fits are being repeated with kinematic factors appropriate to the use of PCAC [29]; qualitative features are expected to remain the same. Present information about L = 2 -~ L = 0 decays allows one to check some predictions of table 3. It is useful to rewrite these predictions in terms of the 1 = 1 and 1 = 3 amplitudes ap and a F (see the "dictionary" at the bottom of the table). One then obtains the results shown in table 6. We quote here only partial width predictions. No decays (2 -+ -* 1 - - ) have been observed as yet which would allow for comparison with helicity amplitude ratio predictions (in contrast to the case B -* corr.) In constructing table 6 we have used the fact that table 3 refers to "K" ~ "K" ~r. These values are then converted to specific processes using SU(3) and (where necessary) a nonet ansatz for couplings. A convenient table for performing such conversions in general is given in ref. [24]. The experimental predictions have been obtained by using the decay width V(g(1700) ~ 2 ;r) = 40 MeV, an approximate value suggested by a recent analysis [30] of the CERN-Munich data. The masses of decaying states are all taken at the idealized values of 1700 MeV for the purpose of barrier factor calculations. We use zero-radius forms for these factors, as suggested by the successful application of SU(3) to the known L = 2 ~ L = 0 decays [31]. In fact, the interested reader is referred to that reference for a quotation of many SU(3) related results which will not be mentioned here,.and is urged to recalculate any partial widths on the basis of exact masses and various barrier factors. The, results quoted are somewhat sensi-

222

A.J.G. Hey et al., Quarks and symmetries of resonance decays

Table 6 Specific partial width predictions for L = 2 -~ L = 0 + ,r Process type 3--~ 3--

Specific decay

1 - - + lr -'- 1 - -

g ~ to,r

~ a)

F, MeV b)

2 Cp

c~

0

'1 i-~ 1 3-g

+ "n"

0..,3 ~ prr

0

3 - - - - , 0 - + + *r

g-~ nn

0

g~KK

0

A_ 140 ~ 280

m'1--6

s----6 11

~

J-s0

1 is--6 1 9"-6

0

3---,0 -+ +n 2--~

1 - - +*r

p (2--)

2 - + -..',-1 - - + Ir

--',. to'tr

A3 .-+ p~

1 - - ~ 1 - - +*r

p, (L = 2).~ felt c)

1--~0

p'(L=2)"+~'~" c)

-+ +n

12 (28) 36

40 (input) 5.8 (3) 8 25

0

No~s. a) F1 = Pl M ~ (p/po) -(21+1) where M R = mass of the resonance, p is the final c.m. three-momentum, and P0 is some convenient scale factor whose choice of course determines the magnitude of a~ b) Predictions based on assuming F (g ~ Irlr) = 40 MeV, (ref. [30]). Experimental numbers, shown in parentheses, are based on Graham and Yoon, (ref. [31]). Predictions based on a PCAC kinematic factor are somewhat different and are discussed in the text. c) Predictions for the 1 - - states assume no mLxing with the L = 0 multiplet ("radial excitation") expected near the same mass.

tive to these choices. F o r example, the k i n e m a t i c factor arising from the use o f PCAC is such that p(m 2 - m2) 2 PI(A ~ B,r)=

m2

a 2 c2 ,

(4.10)

where p is the final c.m. t h r e e - m o m e n t u m , a ! are the reduced m a t r i x elements (ap a n d a F for table 6, as defined in table 3), and the squares o f the clare shown in table 6. Use o f eq. (4.10) increases the prediction for r (g ~ o~rr) to 26 MeV a n d that for V(co 3 ~ art) to 80 MeV. These values are closer to the fit o f G r a h a m and Y o o n , ref. [31], b u t in our o p i n i o n the data do n o t y e t allow one to decide conclusively in favour o f such a k i n e m a t i c factor in all cases. Note, for example, that the simple zero-radius f-wave barrier factor agrees with S U ( 3 ) for the ratio g ~ K i n g ~ mr, as it seems to do in a n u m b e r o f other cases. The prediction for A 3 ~ prr (l = 3) is notable. N o evidence for this decay has been f o u n d so far either in l = 1 or l = 3. F o r L = 1 ~ L = 1 decays (table 4) o n e expects one f-wave amplitude a n d two i n d e p e n d e n t p-wave amplitudes in I b r o k e n S U ( 6 ) w. A partial wave d e c o m p o s i t i o n o f the helicity amplitudes for the various processes, shows however, that there are n o p u r e l y f-wave decays. Hence simple tests analogous to those implied b y table 3

A.J.G. Hey et aL, Quarks and symmetries o f resonance decays

223

are not possible. Various processes which are purely p-wave depend on different combinations of p-wave amplitudes. However, one does have (in SU(6) w (AL z = 0, -+ 1) or l broken SU(6)W ) the following relations: F [B -~ rtN (980) rt]

(4.11)

=4_

r [D(n_likemixing~ A 1 rt] and

F [B -~ A 1 rt]

r" [D(,7-~emixmg) ~ rtN (980) rt]

=3 .

(4.12)

In both equations F stands for the partial width with barrier factors removed. Since the B is so close to the A lrt threshold, this leads to a rather small physical width for B --* Alrt (a few MeV, based on taking "~ 20 MeV for D -* rtNrr). However, the strong intrinsic BA lrt coupling should be observable in B exchange processes, such as ~r-p -+ A°n. To date we know of no firm evidence for this process, however. Table 5 contains a number of predictions for the s-wave partial widths in decays that involve higher partial waves as well. The most useful of these can be written as:

=

F~--o [A3 (2 -+) -~ fortl

= 3a~/800,

(4.14)

r'/= 0 [p'(1--,L=

=

a2/240 ,

(4.15)

=

a2/160.

(4.16)

2)-~Alrt ]

r/=o[°Y(1--,L=2)~Bzr]

as2/200 ,

(4.13)

P'/=0 [P (2--) ~ A2rt ]

If the A 3 is a resonance, fort is its dominant decay mode. Then

rt__0 [p(2--) -* A2rt]

-_ ~4 ,

(4.17)

rt=o IA3 ( 2 - + ) ~ fort] implies a considerable value for the numerator. We hesitate to quote a predicted partial width since it is not clear whether the whole l = 0 fo n bump usually called A 3 is indeed resonant. It is interesting that there does seem to be evidence for a IrA2 enhancement around 1700 MeV [28]. A partial wave analysis of this effect, for example in the channel rt- + p -~ 0 - ( 2 - - ) + p

½

rt0 L.+ K - K o

(4.18)

L rt--rt+

could be performed using multi-particle spectrometers such as Omega at CERN.

224

A.Z G. Hey et aL, Quarks and symmetries o f resonance decays

There are two d-wave amplitudes in I broken SU(6)W and three in SU(6) w (AL z = 0, + 1). In l broken SU(6)W (but not in SU(6) w (AL z = 0, -+ 1)!) the following relations hold:

F

-*B I

-

do

,

(4.19)

F"[A 3 (2-+) -* D(8) 7r]

-

,-~o (al)2

,

(4.20)

F [p ( 2 - - ) - + n N (980) IrI

-

4ot (aid)2 ,

(4.21)

P" [p' ( 1 - - , L = 2) -+ A2rr ]

-

1{o (a0d)2 ,

(4.22)

F [p ( 2 - - ) -+ Allr ]

= ~

(a°) 2 ,

(4.23)

P" [r~ (2-*) "+ ItN (980) rr]

= ~

(a0) 2

(4.24)

A selection rule, which holds in SU(6)W (AL z = 0, + 1) but not in chiral U(3) X U(3) says (4.25)

A 3 ~-B~r . Normally this decay would be allowed in a d-wave (X = 1 only).

5. Equivalences 5.1. Role o f 3 p 0 model

As mentioned in sect. 4, the SU(6) w (AL z = 0, +- I) model is equivalent to the 3P 0 model. What is meant by this equivalence is that there is a one-to-one correspondence between independent parameters of both models. Strictly speaking one could argue that the two models cannot be identical since in the SU(6) w (AL z = 0, +- 1) model each helicity amplitude for a specific decay channel A -+ B + lr is proportional to m 2 - m~, because of PCAC, while this factor is not present in the usual formulation of the ~ P o model. However, since the 3P 0 model is essentially a phenomenological prescription to break SU(6)W,strong symmetry for vertices, one can always include any function of the external masses in its definition. The proof of the equivalence between the two models in the above sense is straight. forward and not very illuminating. Let us consider the decay amplitudes A ( t , ) -+ B ( / / )

+

,

"K" -~"K"

+~ .

A.J.G. Hey et al., Quarks and symmetries of resonance decays

225

Helicity amplitudes for these decays in the SU(6) w (ALz = 0, + 1) model can be rewritten as

Lz

(8'2WB+1)(8'3)

(8'2Wa+l)al

a2

X { aLzLz (wB?~ - L z 10l wAx - L z) (sA?~ - L z L L z IJA?,) (sB?~- L z L ' L z ]JB?~) + a~z+l L z ( w B X - - L z -- 1 11 I wA?~--Lz) X (sA?~-LzLLziJA~)(sBX-Lz

- 1 L' Lz+ IlJB?~)

+ aLz-1 Lz (wB?~_L z + 1 1 - 11 w A ~ - - L z) X ( s A X - L z L LzlJAh) (SBX--Lz + 1 L ' L z - 1 IJB?~)} ,

(5.1)

ere we have absorbed the factor m 2 - m 2 into the reduced matrix elements

Lz/.

In the 3P0 model, on the other hand, the same helicity amplitudes are parametrized as follows

Lz

(8'2sB+1)(8'3)

(8'2SA+1) ~1

7r

A t~2

X { b LzLz (sB?~ - L z 10 IsA?~- L z ) ( s A ~ - L z L LzIjA?~) (sB?~- L z L'LzIjBx) + b Lz+lLz (sB?~- L z - 1 11 IsA?~- L z )

(5.2)

× ( s A ~ - Lz L L zlJA~) ( s B ~ - L z - 1 L ' L z + 1 IJB?~) + bLZ-1Lz ( s B ~ _ L z + 1 1 - 1 1 s A X - L z ) X ( s A X - - L z L LzIJA~) (SB~ - L z + 1 L ' L z - 11JB~,)} . It is then simple to show that the coefficients of the aLfLiz reduced matrix elements are proportional to the coefficients of the bL[L~ reduced matrix elements and that the proportionality factor is independent o f L z. One finds that with the correspondences aLZLZ ~ bLzLz . '

aLZ + - 1 Lz _..>_

4 3V'2

bLz + - 1 Lz

(5.3) '

226

A.Z G. Hey et al., Quarks and symmetrtes o f resonance decays

or aLzLz + bLzLz .

(5.4)

L z ± 1 Lz __>_ 3____~bLz ±1 L z

'

4

'

the helicity amplitudes gx a n d g x are identical. The correspondence of eq. (5.3) is to be used when the coupling A(quark spin triplet state) ~ B (quark spin triplet state) + 7r , is d-type in SU(3) (i.e., L + L ' even) while eq. (5.4) is to be used when the same coupling is f-type in SU(3) (i.e., L + L ' odd). This thus completes the formal proof that the SU(6) w (AL z = 0, z 1) model and the 3P 0 model are identical up to mass factors. For the case of baryonic decays, the quark spin and W-spin of the baryons are the same. The two pictures are manifestly identical for pionic decays in this case. 5.2. ROle o f l broken S U [ 6 ) w

We shall use here the fact that the 3P0 picture and SU(6) w (AL z = 0, + 1) have been proven identical for A(L) --> B(L') + 7r. Then let l' be the vector difference of L and L'. One can label amplitudes either by l' and l or by L~ and L f. The relation between the two descriptions is, br,1 = ~

i zf LzL

( L L zi l' L zf-

L z1 [L' L f ) (l' L~ - L z i 1 Lzi _ Lfzll O) b LfzLlz "

(5.5)

Note that in the second coefficient the fact that the 3P0 pair has L = 1 has been used. This restriction would not be apparent in the SU(6) w (AL z = 0, + 1) language. In naive SU(6)W one has only L zf - L zi - AL z = 0. The amplitudes al,= l all vanish in this case since the second Clebsch-Gordan coefficient is (I010110) = 0. Moreover, in naive SU(6)w, the amplitudes at,=l± 1,l for various l have relations among them. These are discarded in l broken SU(6) w, and the amplitudes ar=l±l, l treated as free parameters. On the other hand, one never acqmres amplitudes of the form a/,=tt. Such amplitudes are expected to be present in the 3P0 picture, however, and thus give rise to additional degrees of freedom. Exceptions occur in two cases: (i) for L = L', one has bLizLfz = bLfzLiz. The sum over Llz and Lzf in eq. (5.5) then yields no net contribution to ar=l, l even in SU(6) w (AL z = 0, + 1); (ii) for L = 0 (L' = 0) one has only l' = L' (l' = L) and since l = L' + 1 (L -+ 1) by parity conservation, the amplitude ar=l, I cannot occur. Thus, for decays A(L) ~ B(L') + ~r, ! broken SU(6) w is always a special case of the 3P 0 picture (and hence of SU(6) w (AL z = 0, +- 1)). It coincides with the latter only when (i) L = L' or (ii) L or L ' is zerO. The last case applies to all previous phenomenological studies [14-19].

A.Z G. Hey et al., Quarks and symmetries o f resonance decays

227

Table 7 Comparison of U(3) X U(3) relations for L = 1 --*L = 0 + ~rdecays Coplanar

gl (2) +gl (1 ++) = x/2go (1 +-) gl (2) = (I/2),4'3g o (2)

"~"291 (1 +-) = go (1++) ~-2go (2) - go (0) = ,v/3go (I +-) a) Chiral

gl (2) + "q~gx (1 +-) = gl (1++) go (2) + ~-2g o (0) = - ",/3go (l'H') ,v/2go (2) - go (0) -- ,v/~go (I +-)

a)

SU(6) w gal. z = + 1) or aPo picture or I broken SU(6)W b) gl (2) = (31N/'2)D

go (IH') = ~

(,.7- D )

go (2) = ,v/6D

gl (1++) -- "vf2(S+I2D)

go (o) = - , ~ s

go ( : - ) = s + 20

gl (1+-) = S - D Notes a) Relation common to coplanar and chiral U(3) X U(3) b) D and S are conveniently normalized d-wave and s-wave amplitudes. An equivalent parametrization in terms of zkLz = 0, ± 1 amplitudes is given in table 2. 5.3. S U ( 3 ) X S U ( 3 } subgroups

The predictions o f tables 2 - 5 for chiral SU(3) X SU(3) relate only amplitudes of a given helicity to one another. If one decomposes helicity states directly into representations of SU(6) w X O(2)Lz, as we have done, we obtain SU(6)W (AL z = 0,

~1). There is another way o f obtaining the latter symmetry from the former. In the decays L = 1 -->L = 0 + Ir (for mesons) if one adjoins the predictions of the coplanar U(3) X U(3) [24] to those of chiral U(3) X U(3) [13] one obtains those of SU(6) w (AL z = 0, -+ 1) [151. To see this explicitly we compare in table 7 the predictions o f the two U(3) X U(3) symmetries, expressed in terms o f relations for helicity amplitudes. The coplanar symmetry links together predictions o f chiral symmetry for different helicity amplitudes in just such a way as to give SU(6) w (AL z = 0, -+ 1). This relation should be true in general. If we form the algebra containing both the coplanar generators and the ehiral generators, it closes upon SU(6) w. The selection rules for pionic transitions are, moreover, the same as in SU(6) w (Z~Lz = 0, -+ 1).

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A.J.G. Hey et al., Quarks and symmetries of resonance decays

Since the pion remains bilinear in quarks, it must belong to a 35, while the z2tLz properties have already been deFmed at the lower symmetry level. If SU(6)w (ZkLz = 0, + 1) turns out not to hold, one may thus be able to pin the blame either on coplanar or chiral SU(3) X SU(3), with the other symmetry continuing to hold. Table 7 forms the basis for one such exercise (which is not, however, demanded by present data, since both SU(3) X SU(3)'s seem valid).

6. Conclusions

In this paper we have explored some consequences of abstracting certain algebraic properties of the Melosh transformation which relates the generators of SU(6)W,sUong and SU(6)W,eurrents. When PCAC is used to relate pionic decays of resonances to oneparticle matrix elements of the axial charge, one obtains a model for the SU(6)w properties of these pionic transitions. With this model we have understood why a naive application of SU(6) w to vertices should fail whenever ZkLz :/: 0 transitions are possible. Furthermore, this PCAC model has been shown equivalent to the 3P0 model for decays, which has had some phenomenological success. We note, however, that the PCAC model is only formulated to apply to vertices involving a pseudoscalar meson: the 3P0 model has also been applied to transitions involving vector mesons [32]. The application of the Melosh transformation to such decay processes is an interesting open question. Another phenomenological version of SU(6) w, l broken SU(6)w, has been shown to differ in general from SU(6) w (z3&z = O, ± 1) although in many simple cases the two prescriptions give the same results. The application of the Melosh transformation thus provides a theoretical basis for understanding the successes of the 3P0 model. The implications of such a transformation for photoproduction processes, vector dominance, and for current induced processes in general, are presently under investigation. We thank S.P. De Alwis for helpful discussions concerning the light-plane formulation of the quark transformation, H.J. Melosh for sending a copy of his thesis, and F.J. Gilman and M. Kugler for useful conversations and correspondence.

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