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On the fractional-parentage expansions of color-singlet six-quark states in a cluster model
Nuclear Physics A352 (1981) 301-325 Not
to
be reproduced by photoprint or
@ North-Holland
microfilm
ON THE FRACTIONAL-PARENTAGE SINGLET SIX-QUARK STATES
without
written
Publishing Co., Amsterdam permission
from
the publisher
EXPANSIONS OF COLORIN A CLUSTER MODEL
M. HARVEY Chalk River Nuclear Laboratories,
(AECL),
Chalk River, Ont., Canada
KOJ lJ0
Received 16 July 1979 (Revised 5 September 1980) Abstract: The classification is given of observable (color-singlet) antisymmetric six-quark states in a cluster model with definite orbital and isospin-spin symmetries (“symmetry basis”). A number of “hidden color” states that cannot be represented in terms of free baryons is deduced. Tables of coefficients of fractional parentage (cfp) are recorded which allow the determination of an effective baryon-baryon interaction from a given non-relativistic quark-quark interaction. It is shown that the cfp method can be used to calculate the transformation between the states of the symmetry basis and those of the “physical” basis of di-baryons (N*, NA and A*) and hidden-color states.
1. Introduction Considerable success has been claimed for understanding the properties of observed baryons and their excited resonances in ternis of configurations of three constituent considered
quarks [cf. refs. 1*2)an d re f erences contained therein]. The quarks are to be elementary fermions with spin 3 and to come in several flavors; the (N) and A are supposed
nucleon
to consist primarily
of two types of flavored
quark (u
and d) the SU(2) symmetry of which is interpreted as isospin. In addition to the above properties, and an orbital structure, quarks have a new degree of freedom called color. In the early developments of the quark model 3P4)the color property was seen to be t,he savior for the Fermi character of the quarks: in the “ground state” of each baryon species it is the color state that carries the necessary antisymmetry (Young tableau [f] = [ill]) of the triplet of quarks. There must thus be at least three different colors and recent inclusive e’e- experiments 5, provide evidence for only three. All observed baryons thus belong to the singlet representation of the SU(3) color group. Current theoretical studies 1z6-8) suggest that it is through the color property that quarks interact strongly: color is to the strong interaction and quantum chromodynamics (QCD) as electric charge is to the electromagnetic interaction and quantum electrodynamics (QED). The strong interaction is thus supposed to derive 7z8)from the exchange between the colored quarks of strong (vector) “photons” - “the gluons”. Although the precise properties of the gluon interaction have still to be February
1981
301
302
M. Harvey / Fracrional-parentage
worked out, the resulting non-relativistic action is expected to have the form rq6)
expansions
two-body,
effective, quark-quark
inter-
(14
i
Here ai is the vector of generating operators for the SU(3) color group operating on the ith particle with the properties -$
(Ai
for color antisymmetric
pairs [f]= = [l l]
* Aj)=
i
+ 4 for color symmetric pairs [f& = [Z] ,
(lb)
The lack of strong experimental evidence for the observation of free quarks suggests that, within a baryon having only antisymmetric pairs of color states, quarks are strongly confined i.e. the function -$0 (rij) increases (positively) with distance analogous to a harmonic oscillator potential. It follows from eq. (lb) that pairs of quarks in symmetric colored states are repeZled with a strength that grows with distance. According to QCD therefore objects with color (like quarks) will interact strongly but objects without color (like leptons) will not interact strongly. Since all observed baryons are color singlets they behave like colorless objects at large distances and hence will WI interact strongly. At intermediate distances however the color distribution within baryons should reveal itself and some remnant of the strong interaction between the constituent quarks should emerge as an effective interaction between baryons. This baryon-baryon interaction is seen in analogy with the molecular interaction between neutral atoms arising from the underlying electromagnetic interactions between the constituent nuclei and electrons. It is of interest to explore this remnant of the strong color interaction in the case of nucleons to see how many of the characteristics of the nucleon-nucleon force can be understood in terms of what is believed known about the quark-quark interaction. It is also of interest to see whether the known characteristics of the nucleon-nucleon force place restrictions on the possible effective quark-quark interaction. Any such information on the quark system is considered desirable in view of the large amount of supposition used in setting up the model necessitated by the lack of observation of a free quark. Of particular interest would be to understand how the meson-exchange character of nuclear forces emerges as the remnant of the strong interaction between constituent quarks. The purpose of this paper is first to detail the mathematical framework of six-quark “cluster” structures in terms of which some aspects of the effective baryon-baryon interaction can be calculated given a quark-quark interaction. The second purpose is to show how the mathematical formalism can be used to reveal the quark structure of di-baryon states. The contents of this paper are used in ref. 9>in an analysis of the nucleon-nucleon force in the singlet-triplet fdeuteron) and tripletsingIet channels.
M. Harvey / Fractional-parentage
We restrict
ourselves
flavors. The two triplets two confining experimental
potentials observation
to six-quark
cluster
of the six quarks
303
expansions
systems
are restricted
made
up from
the u and d
to the lowest s-states
(s+) of
at distances 2 = *$X from the origin. The lack of direct of color “charges” suggests that not just the free baryons
but all observable structures are SU(3) color singlets. We thus restrict our study of six-quark systems to those having color Young symmetry [222]. In sect. 2 we give the allowed set of antisymmetric six-quark structures with the above restrictions on orbital, color and flavor spaces. The structures are given with definite orbital, isospin (T) and spin (S) symmetries and will be referred to as the “symmetry” basis. Within this basis set of states are to be found pairs of free baryons. In our case these are restricted to N*, NA and A2 di-baryon systems because of the restriction to just the u and d flavored quarks. The number of states of a given T and S character in the di-baryon systems is often less than the number with the same T and S in the symmetry basis. This difference yields a number of “hidden-color” six-quark states that cannot be separated into free color-less baryons. We shall refer to this set of di-baryon and hidden-color states as the “physical” basis. In sect. 3 we discuss the fractional parentage reduction of states in the symmetry basis and tabulate the required coefficients for the six-quark cluster problem. We discuss here also the computation of Clebsch-Gordan (CG) coefficients for the symmetric group and again tabulate those features required for the quark calculation. The general details of the fractional parentage reduction we consider well documented in ref. lo). S ome details for the CG coefficients for the symmetric group are given in ref. 11) b u t are amplified in appendix B. In section. 4 we show an application of the tabulations of sect. 3 in the determination of the transformation coefficients between the symmetry basis and the physical basis. A summary of this work and others is given in sect. 5. We have tried to use a uniform notation throughout of these,
this paper and for ease of referencing together
with some standard
2. Classification
definitions,
have collected in appendix
of six-quark
the most important A.
states
Antisymmetric six-quark states can be constructed by coupling an orbital function of Young symmetry [flo to a color-isospin-spin function of dual symmetry (cf. appendix A) [&-Ts = [f]. The [f] cTs label can be considered as the classification with respect to the SU(12) group of transformations within the 3-color, 2-isospin and 2-spin spaces. We are interested in that class of six-quark states that contain pairs of free baryons (N and A) in their ground states. Since the lowest state of a free baryon has orbital symmetry [f]o = [3] it follows that we need only consider six-quark orbital states with symmetry [flo = [3] X [3] = [6] + [5 l]+ [42] + [33]. The sub-classification of [fcrs of SU(12) to symmetries of SU(3)(color) and SU(4) (isospin-spin) = SU(4) (TS) follow from the standard methods of finding inner products. We consider the SU(3) (color) symmetry [f]= = [222] as being the only one of physical relevance since
304
M. Harvey / Fruct~~~al-~are~fageex~a~~iv~s TABLE 1 Classification of six-quark, antisymmetric states with [222] color symmetry Orbital
SU(4)( TTS)
E61 WI
r331* [421* E3211
(01) (10) (12) (21) (03) (30) (00)(02) (20) (11)2 (12) (21) (13) (31) (22) (01) (i0) (02) (20) (ll)* (12) (21)
[511*
(01) (10) (12) (21) (23) W (11) (22)
[333* [4111 [2211] c3211 PI* r421* [2221 [3111]
(01) (10) (01) (10) (01) (10) (OI)(10) (00) (11) (00) (02) (00) (11) (00) (11)
[421
D31
SU(2)( T) x SU(Z)(S)
(12) (21) (03) (30)
(11) (12) (21) (22) (02) (20) (ll? (12) (21) (22) (33) (20) (11)2 (12) (21) (13) (31) (22)
The asterisks pick out the isospin-spin W(4) symmetries which contain di-baryon states in the cluster model and the right hand column lists the ( TS) subclassifications with respect to SU(Z)( T) X SU(Z)(S)
this is the only color singlet state of six particles. In table 1 we list those [& symmetries which go with each [f& symmetry ([flC-[222] being understood) to form a totally antisymmetric state and show the further SU(2)(T) xSU(2)(S) subclassification. The free N and A both have SU(4) (TS) symmetry [f]rs = [3], hence pairs of nucleons and deltas will only be found in six-quark states with [flTs = [3] x [3] = [6]+ [Sl] + [42] + [33]. These particular symmetries are shown by an asterisk in table 1. We note that these latter details have been given in ref. 17)but include them here for later reference. In table 2a we list the SU(2)( T) x SU(2)(S) subclassification for pairs of N and/or A treating them as elementary fermions. The difference in the number of states for a given (7’S) in table 1 for the asterisked SU(4) symmetries and table 2a yields the number of hidden-color states as listed in table 2b [ref. ‘)I. All states of the SU(4) TABLE 2 Classification of di-baryon (a) and hidden-color (b) six-quark states from the SU(4) symmetries with asterisks in table 1
(TS) a
N2 A2 NA
b
cc
(00) (01) (10) (11) (00) (01) (02) (03) (10) (11) (12) (13) (20) (21) (22) (23) (30) (31) (32) (33) (ll)* (12)’ (21)2 (22)* (00) (01) (10) (ll)* (12)2 (21)* (22) (03) (30) (02) (20) (13) (31)
h4. Harvey / Fractional-parentage
30s
expansions
symmetries in table 1 without asterisks are also “hidden-color” representation. 3. Structure of six-quark states in the symmetry In sect. 2 we discussed the classi~cation of antis~metric symmetry basis which, more explicitly, could be written
ul,cflo[f’lc[f’lTs~~s)= {{J/6(f){u,(f’)r6(f”Ts))r)rl61
states in this cluster
basis
six-quark states in a
*
(2)
Here IJ~, QZ6and jrs refer respectively to orbital, color and isospin-spin six-quark functions of the designated Young symmetries f, f’ and f’. (We maintain throughout the notation f, f’ and f’ for orbital, color and isospin-spin symmetries respectively). The curly brackets ( } refer to Clebsch-Gordan coupling with respect to the symmetric group S6 with implied summation over the Yamanouchi labels “1. The tilde over f indicates the dual symmetry (cf. appendix A) and the whole state !Pe is antisymmetric with respect to interchange of any pair of particles, i.e. has Young symmetry [16]. For practical calculations it is more convenient to rewrite the state !Pe in the equivalent form
~~([~l~r~l=~l=~~s~= ~{~~(~)~~(~)}~~~(~ ~W[Pl *
(3)
Here the combined orbital-color space has symmetry F and this, together with the isospin-spin space of dual symmetry, forms the antisymmetric state. This latter representation brings the structure closer to the form used in nuclear spectroscopy lo) where the structure of the nucleus is written in terms of its constituent nucleons. The only additional complexity here is the appearance of the color state and the orbital-color symmetry coupling. Following the conventions of nuclear spectroscopy lo), the isospin-spin states will be defined in terms of the adjoint representation (cf. appendix A) in which case all the Sg CG coefficients for the coupling, to an antisymmetric state, of the orbital-color state of symmetry? with the isospin-spin state of symmetryf” have the value Jl/n, where nr is the dimension of the representation p of S6. Thus, in a more expanded form, we may write ~6([~~0[~1=~~1~~~s~
=
J
;
,-,z,
I ,I
s(fYf’y’lf;‘~)~6(fY)~6(fY’)f6(f”Y”TS),
where S( / ) is an Sg CG coefficient and Y, Y’, Y” denote allowed Yamanouchi labels iosl’ ) for the corresponding Young tableaux f, f’, f”. It is well known in general that, with two antisymmetric states Pm, &, of II particles, matrix elements of a symmetric two-body operator (e.g. a two-body interaction) may be written
306
M. Harvey j ~rQc~~onaf-pare~fageexpansions
The r.h.s. of eq. (5) can be determined of sums of products of anti-symmetric the last pair (@z) e.g.
by expanding the states P” and @” in terms states of the first n -2 particles (!Pe2) and
Here ty and p denote the different possible structures for f~- 2 and the remaining 2 particles in P”, and Pap is the coefficient of fractional parentage (cfp). With a similar expansion for @“, the matrix element in eq. (5) can be written
$l(n - 1) t: PupPa?~~t~;l-z(~)l?lii;t-*(~‘))(~*tP)IVn-l,”l~2(~)). mrpg
In order for this procedure to be carried out with the six-quark states of eq. (4) it is thus necessary to write each function J16, %&and r6 in terms of their two-body cfp expansions with the possible states I,!Q(LY), Y&(cu’)and r&“) of the first four particles and &(p), c&3’) and r#“) of the last pair. Two-body isospin-spin cfp reduction. We shall write the two-body cfp reduction for the isospin-spin state in the form r&“([p”q”]y”)TS)
=
(f;~~qa”T”S” : [ts])}f”[ p”q”]TS) C U”7v“fS x~~(f~~‘~,~y”ff”T”S”~~*(~t~]) .
(8)
Here the Yamanouchi label Y’ of eq. (4) has been written Y” = [p’q’]y’ (see appendix A for this and other notation). Following ref. lo) we use the diagonalized Young-Yamanouchi-Rutherford (d-YYR) representation in which the last pair have definite symmetry (121 or [ll]) implied by the [ ] brackets: thus for the symmetry [Z] the (ts) labels for yz are restricted to (00) or (11) whereas for the antisymmetric pair (ts) = (01) or (10). In eq. (8) the cfp’s for r6 are given by the notation ( 1) ). Summation is over all possible isospin-spin labels T’S’ for fourparticle states in I’,, all possible labels ts of the remaining two particles and any other label (Y”necessary to completely define the four-particle state. Note that the cfp is independent of the label y’. The method for calculating the two-body cfp’s for an isospin-spin state of definite symmetry is well documented in ref. lo) and will not be repeated here. Unfortunately not all the cfp coefficients for the asterisked SU(4) symmetries in table 1 have been tabulated. We have thus computed all the required cfp’s according to the conventions of ref. lo) and list them in table 3. We include the [33]rs = [2%] symmetry for completeness although the reduction for this is also given in ref. lo). Two-body color cfp reduction. We write the two-body cfp reduction for the color state in the form
M. Harvey / Fractional-parentage TABLE
expansions
307
3
Isospin-spin two-body fractional parentage coefficients of eq. (8) for six-particle states [41: PI [61
(11 11)
(11 33)
(33 11)
[511
(11 11)
(11 33)
(33 11)
(33 33)
(55 11)
(55 33)
(55 11)
(55 33)
[41: PI
(33 33) -1
(13)
-1
(31) (33) (35) (53) (55)
J32 72
-4” 72
Jft
Ji Jg -4
_J25
Ji' J$
Jf JQ 1
(57) (75)
1 [41:[111
[511
(11 13)
(13)
J$
(31)
(11 31)
(33 13)
-J$
-Jf J; J;
(33 31)
(55 13)
(55 31)
-Jf
(33) (35) (53)
J;
(55)
Jf -Ji J; Jf
(57)
-Ji 1
(75)
1 [31:2]
I511
(13 11)
(13)
-J”
(31) (33) (35) (53) (55) (57) (75)
27
(13 33)
(31 11)
(31 33)
(33 11) (cont.on
-J& -J& J’6 72
Ji%
_J’2
27 J’6 72
J80 216
J24 72
nextpage)
308
M. Harvey / Fractional-parentage
expansions
TABLE 3 (continued)
Pll
(33 33)
(13)
4%
(31)
JB
(33)
4%
(35) (53)
_J307.16 -JJo 216
(55)
(35 11)
[31:2] (cont.) (35 33) (53 11)
(53 33)
G J?i
J&.
-J$
J& 216 J45
-J&S
-JQ
JO 216
JR -JA
J45 216
-Ji
1
(57)
1
(75)
[421
(11 11)
(11)
J$
(11 33)
[41: El (33 11) (33 33)
(55 11)
(55 33)
-J$
(33,)
-J&
J&
(332)
J32
J’o8
312
Ji?j
-J& _J25
_JM
312
312
312
JE5 J&
(35) (53)
-J’o16 -J’o16 1
(37) (73) (15) (51) (55)
[421
(33 11)
(33,)
[3l1:[21 (13 11)
(13 33)
(31 11) (cont. below)
-1
(11) (33,)
(33 33)
J’o
_J’5 -J&
39
_J&
-J& -J&j
(35) (53)
J’ 15.50 _J/‘6 40 -Jd40
(37) (73) (15) (51) (55)
[421
(31 33)
(35 11)
[3 11: [21 (cont.) (35 33) (53 11)
(53 33)
(331) (332) (cont. on next page)
M. Harvey / Fractional-parentage
expansions
309
TABLE 3 (continued)
r421
(31 33)
(35 11) J++
(35)
[31]:[2] (cont.) (35 33) (53 11) -J&j
(53)
-J&
(37)
-1
(53 33) -J&
4%
-J&j -1
(73) -J$
(15)
-J; -Ji
(51)
-Ji
(55)
[31]:[11] [421
(11) (331) (332) (35)
(33 13)
(33 31)
-Jgj
-JN
-J&l
-JZ
(13 13)
(13 31)
(31 13) (cont.below)
-J&
-J&
-J$ J48 130
-J$j
J48 130
-J&
(53) (37) (73)
-4
(15)
(51) (55)
[421
(31 31)
(11)
-J$
(331) (332) (35) (53) (37) (73) (15) (51) (55)
1421
(11) (33)l (33)z
(35 13)
-4% -JS 130
J&
[31]:[11] (35 31)
(cont.) (53 13)
(53 35)
-J& -J’
J&
J&
130
J&
-1
-J$
(11 11)
-1
J; -Jf (11 33)
(33 11)
-JB52 -J/64 195
JA
P2l:Pl (33 33)
(15 11)
(15 33)
(51 11)
(51 33)
J$
-J:
-JzO
_J’o8 195
-JZ
(35)
J$
(53)
J$
J&
-JO 195 -Jf
-4 1
(37) (cont. on next page)
J& -JO19.5
310
M. Harvey / Fractional-parentage TABLE
[421
(11 11)
(11 33)
(33 11)
expansions
3 (continued)
[22] : [2] (cont.) (33 33) (15 11)
(15 33)
(51 11)
(51 33) 1
(73) (15)
-./$
(51)
-J$
(13)
-J$
1
(55)
[33]=[E]
-J$
(11 13)
(33 13)
(15 13)
[22]:[11] (51 13) (11 31)
J3
(15 31)
JT%
-J&
-Jfr
(31)
(33 31)
-J&
-J&
-4
(35)
(15 31)
J$ J4
(53)
-J;
1
(17)
1
(71)
[33]=[2%] (13) (31) (35) (53) (17) (71)
(13 11)
(31 11)
(35 11)
(53 11)
[311: [21 (13 33)
J” 54
J” 54
(31 33)
-J& -J&
J&
J& JB
J&
(33 33) J3Q
54
(35 33)
-J$
J30
J; J&
(53 33)
-J$
-J& -Jft
-J$ -J&
-1 -1
In each block the six-particle symmetry f’ is given on the left together with the possible (2T+ 1 2S + 1) labels. The four and two particle symmetries (v6eC,f,]:[ 1)are given above the tables toegether with their possible (2 T” + 1 2s” + 1 2t + 1 2s + 1) labels.
where LX’and p’ denote the distribution of the three colors in the first four quarks for the state Ce4and the last pair of quarks in the state c respectively. The other notations are similar to those for isospin-spin states of the previous subsection. The only six-quark color state to be considered has [f’] = [222] in which case there are only two types of cfp expansions with [&I = [22] and a symmetric ([2]) last pair or [fL3]= [211] and an antisymmetric ([ll]) last pair. Both sets of cfp coefficients could be deduced from ref. lo) which lists the orbital coefficients for the nuclear p-shell; it is sufficient to identify the three p-components with the three colors. In practical calculations, because of the assumed color-charge symmetry of quarkquark forces, it is only necessary to know whether the last pair of quarks is color symmetric or antisymmetric to use the property in eq. (lb). Thus in practice all that is
M. Harvey / Fractional-parentage
usually required of the color cfp’s is their normalization 1 (f;ya'
a’p’
311
expansions
property
(10)
: [P’llIf’CP’4’l~2 = 1*
Two-body orbital cfp reduction. In a similar manner to the isospin-spin and color states of the two previous subsections, we write the two-body cfp reduction for orbital states in the form +6(f([p(I]y))
=N6(f)
5
(f&
: [P]l}f[ps])~,4(fp4y()~2([P])
.
(11)
The six particles of the orbital states in the cluster model are assumed to occupy two types of orbitals a = s+ and b = s- with a configuration a3b3 in which three particles are in the state a and three in the state b. The summation over (Yand p is for the three different types of distributions of particles in the four-particle state G4 and the remaining pair in ~35~; namely (Y= a3b and B = b2, (Y= a2b2 and p = ab, or (Y= ab3 and p = a2. In table 4 we list the cfp coefficients for the four different orbital symmetries noted in table 1. In this table the bar or tilde over the state of the last pair denote a symmetric or antisymmetric combination, respectively. In eq. (11) we include a normalization factor N6(f) to take into account the possible non-orthogonality of states a=s+ and b=s_. The cfp’s are defined in the limit where (alb) =0 i.e. iV6([f])1. The general form for the normalization function Ns[f] for the cluster configuration a3b3 is given later in eq. (16). TABLE Orbital
4
two-body fractional parentage coefficients of eq. (11) for the state with structure a3b3 with the symmetries on the left of each block
[41:[111
[41: El a3b3
a3b : b2
ab3 : a2
a3b3
a3b : b2
ab3: a*
[511 [421 [331
+Ji
-Ji
a2b2
: ab
a2b2 : ab
a2b2
:i&
[31]:[11] a2b2: ab
[311: r21
j
[421 I331
a2b2
:&
1
J+
0
[22:[11] a2b2 : & 0 1
0
The possible four-particle block with their symmetries.
0 1
_$ P21: PI
a3b3
-J$
and two-particle
structures
are listed above each
312
M. Harvey / F~a~t~~nal-parentage
expansions
Clebsch-Gordan coefficients of S6. The one remaining obstacle to writing down the
complete cfp expansion of states in eq. (4) is the determination of the ClebschGordan coefficient S(fYf’Y’lp?‘) of S6. The problem for calculating these coefficients has largely been avoided in nuclear physics and so we give more details in appendix B. There we note how, for the d-YYR representation, a CG coefficient for S6 can be factorized in terms of a CG coefficient for Sq and a matrix k With the notation of appendix A we can write ~(f[P~lYf’[P’~‘lY’lf7’~~lY”)
In what follows we show that all that is required in a practical calculation is knowledge of the Z?-matrices. The method for calculating these functions is detailed in appendix B and the results needed for the six-quark cluster problem are listed in table 5. Matrix elements of two-body operators with totally anti-symmetric states. With the substitution of eqs. (8), (9), (11) and (12) into eq. (4), an antisymmetric six-quark state can now be written in the form
= N(f) c ~~(f[Wlf~P’4’lPI~P14”lf
x (f&
: Ml~fb4l)(fhI’~ ‘: [/3’]~}f’[p’q’])(f;~~,J-T”:
[ts]j(f’[p”q”]TS)
x 1Yq~~fpql*~Cf~~,~l,~‘Cf~~~,~~l~s~”~”~~22(~Pl~~~~B’I~Y~~~~I~ 9
(13)
with summation over the labels [pq], [p’q’], [p”q”], cy,[@I, a‘, [@‘I, T’S” and [ts]. We recall that f, f, f” etc. are symmetries in the orbital, color and isospin-spin spaces, respectively. The antisymmetric (unnormalized) state of four particles has the form Wh~~f~l,~Efb~,~lc~‘~f~“q”l*S~“~“~
x ~~(f,ycr)Ce4(f~,,,y’tu’)T4(f~“,-y”T”S”)
.
(14)
Although it is not explicitly obvious from eq. (13), the values of the coefficients are such that the only two-body states (62, c and y that are possible are those for which the combined symmetry is antisymmetric ([ll]). To calculate the matrix elements of two-body operators it is now necessary only to apply eq. (7) with the cfp coefficient P explicitly defined now in terms of products of the known cfp’s in the orbital, color and isospin-spin spaces, the known K matrices
M. Harvey / Fractional-parentage TABLE
Elements
of the matrices
expansions
5
f(f[pqlf’[p’q’]lf’[p”q”])
[6112221
[61D221 -11 33 [222]
33
[2211]
D221
[21111]
[22111
-J&
1
x557 Jf
A421 wi --
_
12 33
J30
22 33
Jzi
zz
J” 54
-;”
i5 (m
24
23
0
11 33
22
[511 Pw 12 33
J$
24
--
P221
Tlz
1
11 33
313
J&
J$
22.33
r331 r2221 IT.55
JB
-J"27
-
22.55
[331 [2221 i2.33
f and f’ are given in the first line above each block and [pq] and [ p’q’] pairs are given in the second line (the bar denoting a symmetric pair and the tilde an antisymmetric pair). Each row is labelled by [f’] [p”q”].
M. Ziarmy / ~racfio~al-parenfag~ extensions
314
and the normalization coefficients NJfl: The evaluation of the overlaps between antisymmetric four-particle states simplifies considerably with the use of the general properties of orthogonality between states of different symmetry, orthogonality between states of the same symmetry but different Yamanou~hi labels, the independence of the normalization of a state to the Yamanouchi label and the normalization property for the ~lebsch-Gordan coefficients 1 s(f~,y~~,,,y’1~,,,,,y”“)*= 1 . YY’
Applying the above properties we may write the overlaps of antisymmetric particle states (with a simplified notation) w4mo~rflcdflll =
four-
TSfYRTS)IY4([~]o(Y[f?]c~‘[$l]lSCYnl
s~~~p~~~-s~~~~~~“~~~~s~~~~~~~l~~I~~~fl~~ , (1%
where the orbital overlap is for any one of the states (any chosen Yamanouchi label) of the [f] representation having structure (Ywith the state of the same symmetry having structure E. If the constituent single-particle orbitals a and b are orthonormal then the final orbital overlap yields ~5,~s.Wowever with a = s+ and b = s- in the cluster model the orthogonality of a and b is only established at large distances. For intermediate distances, which are important for the calculation of baryon-baryon forces, the orthogonality of a and b must not be assumed. The values for the orbital overlaps for the symmetries of the cluster model, and for the possible four-particle configurations, are listed in table 6 in terms of the basic overlap m = (ajb). Using tables 4 and 6 the normalization functions Ne(f) can be computed to be ~~([6])=rl+9~2+9~4+~6]-1’z, N&51])=[l+3m2-3m4-m6]-1~2, N6([42])=[1-m2-m4+m6]-‘~2,
(16)
Ar6([33])=[1-3m2+3m4-m6]-*‘2. One-body fractional parentage
expansion.
expansion of a six-quark antisymmetric
x (fp@: PllfPMfbsa'
The one body fractional parentage state in the symmetry basis can be written
: P’l}f’p’),(f,“-T”S”:~4j}~p”TS)ls
x ly,([f,lotn[fb,lcQ’ffp”ITST”SII)~)1(P)C(Pf)Y(3~~,
with the five-particle state P5 written in analogy to that of the four-particle state eq. (14). Summation is over all possible symmetries and other quantum numbers of the
315
M. Harvey / Fractional-parentage expansions TABLE
6
Overlaps between four-particle states with the designated symmetry and orbital structures (Here m = (alb).) r41
a’b
a2b2
a3b a2b2 ab’
l+3m2 J6m(l +m’) m’(3 + m2)
J6m(l+ m2) 1 +4mz+??14 J6m(l+ m2)
a3b
a2b2
r311 a3b a*b2 ab3
1-m’ -J2m(lm’) -m2(1 -m2)
WI
a2b2
a2b2
ab3 mZ(3+m2) d6m(l+m2) 1+3m2 ab3 -m2(1 -m2) -J2m(l -m*) 1-m’
-J2m(l-m2) l-m4 -J2m(l-m*)
(1 -m’)*
five-particle systems. The required cfp’s for the isospin-spin, orbital spaces and the K-matrices are listed in tables 7,8 and 9, respectively. The overlap matrices for the possible five-body states from the cluster structure (a3b3) are given in table 10. Limits asX -+ 0. It is instructive to demonstrate the structures of the various orbital symmetry states J/&f]) = N&f]&?)f as the two confining potentials merge (X + 0)
TABLE 7 As in table 3 for the one-body, isospin-spin coefficients of fractional parentage
[61
22
PI 44
66
22
24
42
[51]
22
PI 44
66
22
24
42
13 13 31 33 35 53
1 1
55 57 75
$
r411 44
46
64
44
46
64
r411
-V$ $ J; _J’636
-J,s 1
-4 -J$
-4%
-J&
_J2” 36
_-J&j -J&
_-JZ36 -yfi
J$ 1 I
VI&i -1
-4% Jk -1
316
M. Harvey / Fractional-parentage
expansions
TABLE 7 (eontinued~
[42] 11
22
-V’& Jrn 468
[42]
22
11
3%
35 53 37 73
0 ._J?” 468
0 _Jrn 468
[411
44
46
64
J& -4
Jfs
-4
J$
24
-1
-J1”
_&585
-J&
.p?& 585 $3 25
15
_J’“6 2.5
51 55 [33]
42
1
331 332 35 53
331
24
22
24
42
-J& J&L 585
[321
o -.,/I.@ ,585 &
Jii
Jlr? 25
26
-J&
62
-J&
1
-J&
-Jgj
42
44
1 -J&
1 [321
44
i.e. s++ s_*s. Writing m == 1 -E* deduce from eq. (16) the following limits
26
62
and noting that E + 0 as X+ 0, we
NSj[6]+ l/x%, N6[51]-+ l/x&, (17) Na[42]-, l/x’%*, N6[333-+ l/xk3. In the case of [6] symmetry, there is no problem with either the configuration (s$s!)~G~ or the normalization coefficient as X + 0 and we can immediately see that &[6]+ s6.
M. Harvey / Fractional-parentage
expansions
TABLE 8 As in table 4 for the one-body,
orbital coefficients of fractional-parentage [51 a2b3 : a
a3b2 : b
a3b3
44 J:
[61 [511
[321 a3b2 : b
a2b3 : a
[421
Jf
-4;
[331
Jf
a3b3
a3b2 : b
a3b3
J; 1411
[511
a2b3 : a -J;
$2
[421
J$
TABLE 9 As in table 5 for the K(fpf’p’lf”p”)
matrices
[‘A~[2221 1 [222]
3 1
3
[42]. [222]
[222]
3
D11111 {:
13
23
J$
J$
-.g
:g
[33]. [222] 23 P61 L22111
1 1 _ ,l
6 {;
[511’[222] 13
23
_;
-2
0
L22111
{:
317
318
M. Harvey / Fractional-parentage expansions TABLE 10 As in table 6 but for five-particle states
a3b2 a2b3
1+6m2+3m4 3m+6m3+m5
a2b3
l+m*-2m4 -2m+m3+m5
-2m+m3+m5 l+m2-2m4
a3b2
[321 a3b2 a2b3
3m+6m3+m5 1+6m2+3m4
a3b2
[411 a3b2 a2b3
azb3
a3b2
[Sl
a2b3
1-2m2+m4 -m+2m3-m5
-m+2m3-m5 l-2m2+m4
For the (51) symmetry the configuration (s?s!)~~~J vanishes as X+0 and the normalization coefficient becomes infinite (as the inverse first order in F). Taking the limit by differentiating the configuration and the inverse of iV6[5 l] with respect to X we find ~6[51]‘3(s:s:s3+S:S2SI)/J12E~_,s5p~, (18) where we denote the differential by a prime e.g. S’CCpo. Similarly for the $6[42] state a double derivative has to be performed to deduce the limiting structure rj16[42]+3(2s+s:%3+6&:&‘+2&s~)/2&E’*. (19) + s4p; . (Here
we drop
terms
like s:s:s?+s5sff
which
vanish
in the limit,
for the [42]
symmetry, since one cannot have more than four like particles in such a symmetry.) It follows naturally now that 46[33] + s3pz. It is interesting to note, in the derivation of eq. (19), terms like s+s?asp~ and S:S: a s*po. These two terms are components of the excited positive and negativeparity nucleon resonances N** and N*, respectively *). We comment on the significance
of this fact in the summary. 4. The physical basis
A direct application of the cfp expansions given in the last section is to the derivation of the transformation matrices between the symmetry basis and the physical basis for a given TS. The physical basis is defined to contain all of those di-baryon states which, in the cluster configuration s”,s!, for large separation of the two confining potentials [i.e. (s+ls_) + 01, can be represented in terms of pairs of known (colorless) free baryons e.g. N*, NA, A*. The physical basis also contains the remaining (orthogonal) states, whose asymptotic triplets of quarks remain colorful, and which we refer to as hidden-color states.
A4 Harvey / Fractional-parentage
expansions
319
The above definition of di-baryon states implies that, in a two-body cfp expansion, when the last pair of particles have orbital structure & = sl or s!, their color state c must only be antisymmetric. In the case of the (7’S) = (01) six-quark states, for example, the cfp expansion of the state with [f& = [42] and [flTS = [Sl] (see table 1) has this property but it is not shared by the two states having [f]o = [I;] and [f10 = [42] both with [f& = [33]. It is possible to take a linear combination of these latter two states however to yield an allowed di-baryon state. The orthogonal combination defines the hidden-color (23) = (01) state of table 2b. To determine the N or A character of the di-baryon states we again refer to the cfp expansion. If the last pair of quarks having
In this paper we have recorded (table 1) the classification of six-quark states in a cluster model in which triplets of quarks occupy the lowest orbit of a confining potential. We have deduced (table 2) the number of “hidden-color” states that
TABLE 11 Transformation coefficients between the physical basis states (with CC denoting a hidden-color state) shown on the left of each block, and the symmetry basis states whose structure is identified above each block by the orbital symmetry [fl and isopsin-spin symmetry {f3; each block is for the designated (TS) pairs
T=l T=O
S=O
T=O
S=O
S=l
N2
A2 cc T=l
S=l
r511142I
$2 j!; [5lW&
[42X331 [42x511
[3311421 WI161 _J36
-J& -JE +J$
JZ 0
r51114212 EWWh
J241 1053
_JJS1053
-J" 1053
_J468 1053
-J101053
JL1053
JA?C 1053
JLE 1053
0
0
0
NA
_JzO ,053
J256 1053
_J468 1053
S=2 S=l
T=2
S=2 -A2 NA
d$ 0 [6X331
J$ 0
$ [42l{W
rsl1142) [WW JZO 4.5
&
PWJI -J&. 0
_J’645 -&
cc
J36 45
jy
j!;
[421m
0 0
1
0 0 0
0 0
[WI511
G
[5111421
[3311421
[4211511 0 1
0 0
0
S=3 S=l
S=2 S=O
WIWI
S=3 S=2 A2
T=O [33]{42}T=3
S=3 S=O
-J$ J$
A2 cc T=2 T=3
[=I{@
-A2 EA cc,
T=l T=3 T=O T=2
[WWI,
N2
cc2
T=l T=2
[61{331
WIWI 1
[6X33
+J$ -J+
A2 6Q
T=3
S=3 A2
142lWI
W1{6) 1
M. Harvey / Fractional-parentage
expansions
321
cannot be represented in terms of two free baryons from the asterisked SU(4) symmetries of table 1. In sect. 3 we showed how the method of fractional parentage expansions could be used to calculate energies in a six-quark cluster model and recorded in tables 3 to 10 the coefficients required to apply the technique. Finally in sect. 4 we showed an application of the technique by deducing the transformation (table 11) between the “symmetry basis”, in which quark states have definite orbital and isospin-spin symmetry, to the “physical basis”, in which observable di-baryon states are identified. With the cfp techniques it is possible to take a two-body quark-quark interaction, deduce the effective interaction between free baryons and study how this is modified by interaction with other di-baryon channels and by the hidden-color states. Such a calculation is reported in the accompanying paper 9). This paper already provides clues, however, as to the problems and difficulties that lie ahead. In sect. 3 we discussed the limiting structures of the cluster states as the confining potentials merge. There we showed for example how the states of [42] orbital symmetry contain within their structure terms like N*N* and N**N with N* and N** the excited negative and positive-parity nucleon resonances, respectively. One cannot expect to reproduce nuclear forces therefore in a quark model unless the model can simultaneously reproduce the N” and N** spectra. In this regard Isgur and Karl ‘) have demonstrated the importance of the tensor force for the relative N* spectrum and this provides a clue as to the possible origin of the tensor effects in the deuteron. However it is also necessary to use confining potentials in which the N* and N”” spectra are placed in their correct position relative to the N-and such potentials have yet to be found 9). With the tables in this paper however any proposed quark-quark force can quickly be translated into an effective nucleon-nucleon interaction. A few papers now exist in the published literature that consider six-quark structures but none, to our knowledge, treats them by the cfp cluster method suggested here or has shown the close relationship among the N*, NA and A* and hidden-color systems. Thus Jaffe ‘**r3) has examined six-quark “exotics” (spherical s6 states of [6] symmetry) and deduced their energies. Matveev and Sorba i4) question whether the deuteron can be treated as a six-quark system and deduce, from unitary symmetries for the (TS) = (01) channel, the limited expansion of the spherical s6 state into the N*, A* and hidden-color (6Q) channels [see our table 11, the first column for (TS) = (Ol)]. Matveev and Sorba i5) have since extended their analysis to spherical states of nine and twelve quarks. The extension to six-quark, spherical states but with more than two flavors has been made by Hogaasen and Sorba 16). The quark structure of the deuteron has also been invoked by Neudatchin and collaborators r7-19): arguing that the hard core of the nucleon-nucleon potential implies a node in the relative wave function of the nucleons ‘“), they deduce that the dominant quark orbital symmetry must be [42] with an isospin-spin symmetry of [S l] because of the structure of quark-quark forces. From our table 11 ((TS) = (01))
322
M. Harvey / Fractional-parentage expansions
however we see that their conclusions are not entirely correct. The states of f42] orbital symmetry do indeed dominate the nucleon-nucleon channel (by 88.8%) but only 44.4% has the [51] isospin-spin symmetry: the remaining 44.4% has the lower [33] isospin-spin symmetry. A few attempts have been made to deduce an effective baryon-baryon force from the remnant of the quark-quark interaction. The feeling seems to be that such first attempts should describe only the shorter range parts of the NN force, leaving the description of the long-range part to 7r-meson exchange or, equivalently, to complex quark exchange mechanisms. Such an approach is probably justified in view of the difficulty the quark model has to account for the r-meson mass. Liberman *l) has considered the non-relativistic approach and is perhaps the closest to our own 9, but he only considers the NN channel with no NA, A* or hidden-color channel coupling. de Tar 22*23 ) approaches the problem relativistically in the MIT bag model but, again, only considers the NN system in isolation. The analysis of these papers and results of our own calculations we leave to the accompanying paper 9). It is a pleasure to thank Paul Lee, Faqir Khanna, Nathan Isgur and Gabriel Karl for discussions and correspondence on the quark model.
Appendix A DEFINITIONS AND NOTATION
Notation f
Y y = (P4Y)
y = b?lY
1
f, f’, f’ or K Y’, Y” etc.
- a Young tableau for a symmetry of S,, the symmetry group of n particles. - a Yamanouchi label l”,ll) for the Young symmetry f. - a Yamanouchi label in which the it th particle is in row p of f, the (n - 1)th particle in row 4 and the remaining particles have a Yamanouchi distribution y. The labelling refers to the standard Young-Yamanouchi representation (cf. below). - a Yamanouchi label for the diagonalized Young-YamanouchiRutherford representation (cf. below) in which the nth and (n - 1)th particle has definite (but unspecified) symmetry. Where the symmetry is to be specified we write p4 orEto denote the symmetric or antisymmetric pairing. - a Young tableau of S,_i (Snw2)derived from the tableau f of S, by the removal of a square in the pth (and qth) row (s). - the tableau of the dual symmetry to f obtained by interchanging rows and columns. - when used in the same expression refer to orbital, color and isospin-spin spaces respectively.
M. Harvey / Fractional-parentage
323
expansions
Definitions (i) Standard Young-Yamanouchi
representation losll) for the transposition P,,,_l in the symmetry f is defined as follows for the three possible types of Yamanouchi labelling:
case (a) for p = f.j Kn-Ilf(PqYN = +llf(PqY)) 7 case (b) for p > q with Y = (pqy) existing but not Y = (qpy) Pn.“-llf(PqYN
=
-llf(pqy)),
case (c) for p > q with Y = (Pqy) and (qpy) existing R.“-llf(PqYN=
P",n-llf(qPY))
- i
J/L’-1 jP(
If(pqy))+-
JyTT = -j--JImIY))+;
IfhPY)) 7
If(PqYN~
Here p is the axial distance (number of lines crossed counting along rows and columns) between the nth particle and (n - 1)th particle in the Young tableau fi In general we write (A.1)
p,,,-IIf(PqY)) = 4q If(PqYN+&7lf(PqYN 9
with a’, and p’, taking the above values depending on the situation [e.g. p’, = 0 in f cases (i) and (ii)]. Note that ((~fp~)‘+@I’,,)’ = 1 and a’,, = -%P* (ii) Adjoint Young- Yumunouchi representation lo) for the transposition P,,,-l in the symmetry f is the same as that for the standard representation except that all the PL coefficients are negative. (iii) Diagonalised Young-Yamanouchi-Rutherford representation lo) is such that the last pair of particles have definite symmetry. In terms of the standard representation we may write for the three cases: case (a) IfGP41YN’If(P4Y)) case (b) If(rtGlYN= If(PqYN9 case (c) (P >q)lf(ElY))=
(~If(4PY))+JCL_llf(PqY)))lJ2CL,
lf([~lY)>=(JCL-~If(qPY))-JCL+~If(P4Y)))l~~
-
324
M. Harvey / Fractional-parentage
expansions
In general we write
If(CP41Y))=Y~4IN~~Y))+S~4IP(CIPY)),
(A.3
with the above definitions for y and 6 depending on the situation,
Appendix B CLEBSCH-GORDAN
COEFFICIENTS
FOR THE SYMMETRIC
GROUP
S,
With the notation of appendix A, a state Ip’ Y") can be constructed from two sets of states IfY) and If’ Y’) by the Clebsch-Gordan expansion If’Y”> = 2, s(fYfY’lf’y”)lfY)lf’Y’>,
03.1)
where S(fY f Y'/f"Y") is the Clebsch-Gordan (CG) coefficient. Hammermesh [ref. I’), section 7.141 has shown how the CG coefficients for S,, can be related to those for S,_, by a matrix K. With Y = (pqy) (cf. appendix A) we may write
S(f(pqYlf'(p'q'Y')lf'(p"q"Y")) =K(fpf'p'lf'p")S(f,(qY)f~,(q'Y')lf,N~(q"Y")) ~Kz(f(pq)f'(p'q'>lf"(p"q"))s(~~~Y~~~~,Y'l~~~,~~Y~) * 03.2) The first equality is as given by Hammermesh. The second equality follows by a second application of the theorem relating the CG for S,-i to those for Sn_2 and the definition ~2(f(pqlf’(p’q’)lf”(P”q”)
= K(fpf’p’lf”P”)K:(f~~~4’lf,“~q”)
*
(8.3)
Hammermesh also shows, with the notation of appendix A, that the K-matrices may be calculated using the following relationships: K(fpf'p'lf"p")K(f,q'l~~,,q~)(~~~~~,,
-*&f)
+K(fp~q'lf'p").lu(~~~,p'J~~"q~)~~~~,*,
-r-k;r(f4f’P’lf”Pf’)K(~~P~~,q’~~~~‘q”)~~~~~*, +K(f4f’4’lf”4”)K(f,P’l~~~,q”)~~~~~, =
(with a! and p defined in A-1) and the orthonormal c
03.4)
K(fpf'p'lf'q")K(f,q'l~~,,p")~~,~,,
K(fpf'p'lf'p")K(fpf'p'l~~")6(f,N,,~~,,)=
condition s(f”~)s(p”p”>
.
(B.3
PP'
The above expressions allow the determination of K relating the CG of S, to S,-i once the K relating the CG of S,_r to 5L-2 are known. Since (trivially~
M. Harvey / Fractional-parentage K(
11 11111) = 1, all K-matrices
can be determined
325
expansions
by iteration.
The KZ matrices
being products of K-matrices (cf. eq. (B.3) are thus also determined. The determination of the K-matrices above is for the standard Yamanouchi representation. We define a K-matrix for the diagonalised Yamanouchi-Rutherford
representation
YoungYoung-
by
~(f~P4lf~P’4’llf’~~“~“I)
with ~PpqK~(f(~q)f)(~‘4’)P(~“4) and the y and S defined The elements
= K~(f(~)fr(~‘q’)]fl(p”q”))
etc.
in eq. (A.2).
of the K-matrices
required
in the quark-cluster
problem
are listed in
table 5. Note that the K-matrices have the property that all elements are zero unless the product of the symmetries of [ pq] and [ $4’1 is equal to the symmetry for [ ,“,“I. The K-matrices relate the CG coefficients of S, to those for Sn_2 in the d-YYR representation. References 1) A. de Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147 2) N. Isgur and G. Karl, Phys. Lett. 72B (1977) 109; Phys. Rev. D18 (1978) 4187; Phys. Rev. D19 3) 4) 5) ‘5) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
(1979) 2653; R. Koniuk and N. Isgur, Phys. Rev. D21(1980) 1868 O.W. Greenberg, Phys. Rev. Lett. 13 (1964) 598 H. Fritz and M. Gell-Mann, in Scale and conformal symmetry in hadron physics, ed. R. Gatto (Wiley, New York, 1973) J.E. Augustin et al., Phys. Rev. Lett. 34 (1975) 764 T. de Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 S. Weinberg, Phys. Rev. Lett. 31 (1973) 494 D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343 M. Harvey, Nucl. Phys. A352 (1981) 326 J.P. Elliott, J. Hope and H.A. Jahn, Phil. Trans. Roy. Sot. 246 (1953) 241 M. Hammermesh, in Group theory, second printing (Addison-Wesley, Reading, Mass., 1964) R.L. Jaffe, Phys. Rev. D15 (1977) 281 R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195 V.A. Matveev and P. Sorba, Nuovo Cim. Lett. 20 (1977) 435 V.A. Matveev and P. Sorba, Nuovo Cim. 45 (1978) 257 H. Hogaasen and P. Sorba, Nucl. Phys. B150 (1979) 427 V.G. Neudatchin, I.T. Obukhovsky, V.I. Kukulin and N.F. Golovanova, Phys. Rev. Cl1 (1975) 128 V.G. Neudatchin, Yu.F. Smirnov and R. Tamagaki, Prog. Theor. Phys. 58 (1977) 1072 Yu.F. Smirnov, I.T. Gbukhouskii, V.G. Neudatchin and R. Tamagaki, SOV. J. Nucl. Phys. 27 (3) (1978) 456 R. Tamagaki, Rev. Mod. Phys. 39 (1967) 629 D. Liberman, Phys. Rev. D16 (1977) 1542 C. de Tar, Phys. Rev. D17 (1978) 323 C. de Tar, preprint, Univ. of Utah, HEP 78/l
to
be reproduced by photoprint or
@ North-Holland
microfilm
ON THE FRACTIONAL-PARENTAGE SINGLET SIX-QUARK STATES
without
written
Publishing Co., Amsterdam permission
from
the publisher
EXPANSIONS OF COLORIN A CLUSTER MODEL
M. HARVEY Chalk River Nuclear Laboratories,
(AECL),
Chalk River, Ont., Canada
KOJ lJ0
Received 16 July 1979 (Revised 5 September 1980) Abstract: The classification is given of observable (color-singlet) antisymmetric six-quark states in a cluster model with definite orbital and isospin-spin symmetries (“symmetry basis”). A number of “hidden color” states that cannot be represented in terms of free baryons is deduced. Tables of coefficients of fractional parentage (cfp) are recorded which allow the determination of an effective baryon-baryon interaction from a given non-relativistic quark-quark interaction. It is shown that the cfp method can be used to calculate the transformation between the states of the symmetry basis and those of the “physical” basis of di-baryons (N*, NA and A*) and hidden-color states.
1. Introduction Considerable success has been claimed for understanding the properties of observed baryons and their excited resonances in ternis of configurations of three constituent considered
quarks [cf. refs. 1*2)an d re f erences contained therein]. The quarks are to be elementary fermions with spin 3 and to come in several flavors; the (N) and A are supposed
nucleon
to consist primarily
of two types of flavored
quark (u
and d) the SU(2) symmetry of which is interpreted as isospin. In addition to the above properties, and an orbital structure, quarks have a new degree of freedom called color. In the early developments of the quark model 3P4)the color property was seen to be t,he savior for the Fermi character of the quarks: in the “ground state” of each baryon species it is the color state that carries the necessary antisymmetry (Young tableau [f] = [ill]) of the triplet of quarks. There must thus be at least three different colors and recent inclusive e’e- experiments 5, provide evidence for only three. All observed baryons thus belong to the singlet representation of the SU(3) color group. Current theoretical studies 1z6-8) suggest that it is through the color property that quarks interact strongly: color is to the strong interaction and quantum chromodynamics (QCD) as electric charge is to the electromagnetic interaction and quantum electrodynamics (QED). The strong interaction is thus supposed to derive 7z8)from the exchange between the colored quarks of strong (vector) “photons” - “the gluons”. Although the precise properties of the gluon interaction have still to be February
1981
301
302
M. Harvey / Fracrional-parentage
worked out, the resulting non-relativistic action is expected to have the form rq6)
expansions
two-body,
effective, quark-quark
inter-
(14
i
Here ai is the vector of generating operators for the SU(3) color group operating on the ith particle with the properties -$
(Ai
for color antisymmetric
pairs [f]= = [l l]
* Aj)=
i
+ 4 for color symmetric pairs [f& = [Z] ,
(lb)
The lack of strong experimental evidence for the observation of free quarks suggests that, within a baryon having only antisymmetric pairs of color states, quarks are strongly confined i.e. the function -$0 (rij) increases (positively) with distance analogous to a harmonic oscillator potential. It follows from eq. (lb) that pairs of quarks in symmetric colored states are repeZled with a strength that grows with distance. According to QCD therefore objects with color (like quarks) will interact strongly but objects without color (like leptons) will not interact strongly. Since all observed baryons are color singlets they behave like colorless objects at large distances and hence will WI interact strongly. At intermediate distances however the color distribution within baryons should reveal itself and some remnant of the strong interaction between the constituent quarks should emerge as an effective interaction between baryons. This baryon-baryon interaction is seen in analogy with the molecular interaction between neutral atoms arising from the underlying electromagnetic interactions between the constituent nuclei and electrons. It is of interest to explore this remnant of the strong color interaction in the case of nucleons to see how many of the characteristics of the nucleon-nucleon force can be understood in terms of what is believed known about the quark-quark interaction. It is also of interest to see whether the known characteristics of the nucleon-nucleon force place restrictions on the possible effective quark-quark interaction. Any such information on the quark system is considered desirable in view of the large amount of supposition used in setting up the model necessitated by the lack of observation of a free quark. Of particular interest would be to understand how the meson-exchange character of nuclear forces emerges as the remnant of the strong interaction between constituent quarks. The purpose of this paper is first to detail the mathematical framework of six-quark “cluster” structures in terms of which some aspects of the effective baryon-baryon interaction can be calculated given a quark-quark interaction. The second purpose is to show how the mathematical formalism can be used to reveal the quark structure of di-baryon states. The contents of this paper are used in ref. 9>in an analysis of the nucleon-nucleon force in the singlet-triplet fdeuteron) and tripletsingIet channels.
M. Harvey / Fractional-parentage
We restrict
ourselves
flavors. The two triplets two confining experimental
potentials observation
to six-quark
cluster
of the six quarks
303
expansions
systems
are restricted
made
up from
the u and d
to the lowest s-states
(s+) of
at distances 2 = *$X from the origin. The lack of direct of color “charges” suggests that not just the free baryons
but all observable structures are SU(3) color singlets. We thus restrict our study of six-quark systems to those having color Young symmetry [222]. In sect. 2 we give the allowed set of antisymmetric six-quark structures with the above restrictions on orbital, color and flavor spaces. The structures are given with definite orbital, isospin (T) and spin (S) symmetries and will be referred to as the “symmetry” basis. Within this basis set of states are to be found pairs of free baryons. In our case these are restricted to N*, NA and A2 di-baryon systems because of the restriction to just the u and d flavored quarks. The number of states of a given T and S character in the di-baryon systems is often less than the number with the same T and S in the symmetry basis. This difference yields a number of “hidden-color” six-quark states that cannot be separated into free color-less baryons. We shall refer to this set of di-baryon and hidden-color states as the “physical” basis. In sect. 3 we discuss the fractional parentage reduction of states in the symmetry basis and tabulate the required coefficients for the six-quark cluster problem. We discuss here also the computation of Clebsch-Gordan (CG) coefficients for the symmetric group and again tabulate those features required for the quark calculation. The general details of the fractional parentage reduction we consider well documented in ref. lo). S ome details for the CG coefficients for the symmetric group are given in ref. 11) b u t are amplified in appendix B. In section. 4 we show an application of the tabulations of sect. 3 in the determination of the transformation coefficients between the symmetry basis and the physical basis. A summary of this work and others is given in sect. 5. We have tried to use a uniform notation throughout of these,
this paper and for ease of referencing together
with some standard
2. Classification
definitions,
have collected in appendix
of six-quark
the most important A.
states
Antisymmetric six-quark states can be constructed by coupling an orbital function of Young symmetry [flo to a color-isospin-spin function of dual symmetry (cf. appendix A) [&-Ts = [f]. The [f] cTs label can be considered as the classification with respect to the SU(12) group of transformations within the 3-color, 2-isospin and 2-spin spaces. We are interested in that class of six-quark states that contain pairs of free baryons (N and A) in their ground states. Since the lowest state of a free baryon has orbital symmetry [f]o = [3] it follows that we need only consider six-quark orbital states with symmetry [flo = [3] X [3] = [6] + [5 l]+ [42] + [33]. The sub-classification of [fcrs of SU(12) to symmetries of SU(3)(color) and SU(4) (isospin-spin) = SU(4) (TS) follow from the standard methods of finding inner products. We consider the SU(3) (color) symmetry [f]= = [222] as being the only one of physical relevance since
304
M. Harvey / Fruct~~~al-~are~fageex~a~~iv~s TABLE 1 Classification of six-quark, antisymmetric states with [222] color symmetry Orbital
SU(4)( TTS)
E61 WI
r331* [421* E3211
(01) (10) (12) (21) (03) (30) (00)(02) (20) (11)2 (12) (21) (13) (31) (22) (01) (i0) (02) (20) (ll)* (12) (21)
[511*
(01) (10) (12) (21) (23) W (11) (22)
[333* [4111 [2211] c3211 PI* r421* [2221 [3111]
(01) (10) (01) (10) (01) (10) (OI)(10) (00) (11) (00) (02) (00) (11) (00) (11)
[421
D31
SU(2)( T) x SU(Z)(S)
(12) (21) (03) (30)
(11) (12) (21) (22) (02) (20) (ll? (12) (21) (22) (33) (20) (11)2 (12) (21) (13) (31) (22)
The asterisks pick out the isospin-spin W(4) symmetries which contain di-baryon states in the cluster model and the right hand column lists the ( TS) subclassifications with respect to SU(Z)( T) X SU(Z)(S)
this is the only color singlet state of six particles. In table 1 we list those [& symmetries which go with each [f& symmetry ([flC-[222] being understood) to form a totally antisymmetric state and show the further SU(2)(T) xSU(2)(S) subclassification. The free N and A both have SU(4) (TS) symmetry [f]rs = [3], hence pairs of nucleons and deltas will only be found in six-quark states with [flTs = [3] x [3] = [6]+ [Sl] + [42] + [33]. These particular symmetries are shown by an asterisk in table 1. We note that these latter details have been given in ref. 17)but include them here for later reference. In table 2a we list the SU(2)( T) x SU(2)(S) subclassification for pairs of N and/or A treating them as elementary fermions. The difference in the number of states for a given (7’S) in table 1 for the asterisked SU(4) symmetries and table 2a yields the number of hidden-color states as listed in table 2b [ref. ‘)I. All states of the SU(4) TABLE 2 Classification of di-baryon (a) and hidden-color (b) six-quark states from the SU(4) symmetries with asterisks in table 1
(TS) a
N2 A2 NA
b
cc
(00) (01) (10) (11) (00) (01) (02) (03) (10) (11) (12) (13) (20) (21) (22) (23) (30) (31) (32) (33) (ll)* (12)’ (21)2 (22)* (00) (01) (10) (ll)* (12)2 (21)* (22) (03) (30) (02) (20) (13) (31)
h4. Harvey / Fractional-parentage
30s
expansions
symmetries in table 1 without asterisks are also “hidden-color” representation. 3. Structure of six-quark states in the symmetry In sect. 2 we discussed the classi~cation of antis~metric symmetry basis which, more explicitly, could be written
ul,cflo[f’lc[f’lTs~~s)= {{J/6(f){u,(f’)r6(f”Ts))r)rl61
states in this cluster
basis
six-quark states in a
*
(2)
Here IJ~, QZ6and jrs refer respectively to orbital, color and isospin-spin six-quark functions of the designated Young symmetries f, f’ and f’. (We maintain throughout the notation f, f’ and f’ for orbital, color and isospin-spin symmetries respectively). The curly brackets ( } refer to Clebsch-Gordan coupling with respect to the symmetric group S6 with implied summation over the Yamanouchi labels “1. The tilde over f indicates the dual symmetry (cf. appendix A) and the whole state !Pe is antisymmetric with respect to interchange of any pair of particles, i.e. has Young symmetry [16]. For practical calculations it is more convenient to rewrite the state !Pe in the equivalent form
~~([~l~r~l=~l=~~s~= ~{~~(~)~~(~)}~~~(~ ~W[Pl *
(3)
Here the combined orbital-color space has symmetry F and this, together with the isospin-spin space of dual symmetry, forms the antisymmetric state. This latter representation brings the structure closer to the form used in nuclear spectroscopy lo) where the structure of the nucleus is written in terms of its constituent nucleons. The only additional complexity here is the appearance of the color state and the orbital-color symmetry coupling. Following the conventions of nuclear spectroscopy lo), the isospin-spin states will be defined in terms of the adjoint representation (cf. appendix A) in which case all the Sg CG coefficients for the coupling, to an antisymmetric state, of the orbital-color state of symmetry? with the isospin-spin state of symmetryf” have the value Jl/n, where nr is the dimension of the representation p of S6. Thus, in a more expanded form, we may write ~6([~~0[~1=~~1~~~s~
=
J
;
,-,z,
I ,I
s(fYf’y’lf;‘~)~6(fY)~6(fY’)f6(f”Y”TS),
where S( / ) is an Sg CG coefficient and Y, Y’, Y” denote allowed Yamanouchi labels iosl’ ) for the corresponding Young tableaux f, f’, f”. It is well known in general that, with two antisymmetric states Pm, &, of II particles, matrix elements of a symmetric two-body operator (e.g. a two-body interaction) may be written
306
M. Harvey j ~rQc~~onaf-pare~fageexpansions
The r.h.s. of eq. (5) can be determined of sums of products of anti-symmetric the last pair (@z) e.g.
by expanding the states P” and @” in terms states of the first n -2 particles (!Pe2) and
Here ty and p denote the different possible structures for f~- 2 and the remaining 2 particles in P”, and Pap is the coefficient of fractional parentage (cfp). With a similar expansion for @“, the matrix element in eq. (5) can be written
$l(n - 1) t: PupPa?~~t~;l-z(~)l?lii;t-*(~‘))(~*tP)IVn-l,”l~2(~)). mrpg
In order for this procedure to be carried out with the six-quark states of eq. (4) it is thus necessary to write each function J16, %&and r6 in terms of their two-body cfp expansions with the possible states I,!Q(LY), Y&(cu’)and r&“) of the first four particles and &(p), c&3’) and r#“) of the last pair. Two-body isospin-spin cfp reduction. We shall write the two-body cfp reduction for the isospin-spin state in the form r&“([p”q”]y”)TS)
=
(f;~~qa”T”S” : [ts])}f”[ p”q”]TS) C U”7v“fS x~~(f~~‘~,~y”ff”T”S”~~*(~t~]) .
(8)
Here the Yamanouchi label Y’ of eq. (4) has been written Y” = [p’q’]y’ (see appendix A for this and other notation). Following ref. lo) we use the diagonalized Young-Yamanouchi-Rutherford (d-YYR) representation in which the last pair have definite symmetry (121 or [ll]) implied by the [ ] brackets: thus for the symmetry [Z] the (ts) labels for yz are restricted to (00) or (11) whereas for the antisymmetric pair (ts) = (01) or (10). In eq. (8) the cfp’s for r6 are given by the notation ( 1) ). Summation is over all possible isospin-spin labels T’S’ for fourparticle states in I’,, all possible labels ts of the remaining two particles and any other label (Y”necessary to completely define the four-particle state. Note that the cfp is independent of the label y’. The method for calculating the two-body cfp’s for an isospin-spin state of definite symmetry is well documented in ref. lo) and will not be repeated here. Unfortunately not all the cfp coefficients for the asterisked SU(4) symmetries in table 1 have been tabulated. We have thus computed all the required cfp’s according to the conventions of ref. lo) and list them in table 3. We include the [33]rs = [2%] symmetry for completeness although the reduction for this is also given in ref. lo). Two-body color cfp reduction. We write the two-body cfp reduction for the color state in the form
M. Harvey / Fractional-parentage TABLE
expansions
307
3
Isospin-spin two-body fractional parentage coefficients of eq. (8) for six-particle states [41: PI [61
(11 11)
(11 33)
(33 11)
[511
(11 11)
(11 33)
(33 11)
(33 33)
(55 11)
(55 33)
(55 11)
(55 33)
[41: PI
(33 33) -1
(13)
-1
(31) (33) (35) (53) (55)
J32 72
-4” 72
Jft
Ji Jg -4
_J25
Ji' J$
Jf JQ 1
(57) (75)
1 [41:[111
[511
(11 13)
(13)
J$
(31)
(11 31)
(33 13)
-J$
-Jf J; J;
(33 31)
(55 13)
(55 31)
-Jf
(33) (35) (53)
J;
(55)
Jf -Ji J; Jf
(57)
-Ji 1
(75)
1 [31:2]
I511
(13 11)
(13)
-J”
(31) (33) (35) (53) (55) (57) (75)
27
(13 33)
(31 11)
(31 33)
(33 11) (cont.on
-J& -J& J’6 72
Ji%
_J’2
27 J’6 72
J80 216
J24 72
nextpage)
308
M. Harvey / Fractional-parentage
expansions
TABLE 3 (continued)
Pll
(33 33)
(13)
4%
(31)
JB
(33)
4%
(35) (53)
_J307.16 -JJo 216
(55)
(35 11)
[31:2] (cont.) (35 33) (53 11)
(53 33)
G J?i
J&.
-J$
J& 216 J45
-J&S
-JQ
JO 216
JR -JA
J45 216
-Ji
1
(57)
1
(75)
[421
(11 11)
(11)
J$
(11 33)
[41: El (33 11) (33 33)
(55 11)
(55 33)
-J$
(33,)
-J&
J&
(332)
J32
J’o8
312
Ji?j
-J& _J25
_JM
312
312
312
JE5 J&
(35) (53)
-J’o16 -J’o16 1
(37) (73) (15) (51) (55)
[421
(33 11)
(33,)
[3l1:[21 (13 11)
(13 33)
(31 11) (cont. below)
-1
(11) (33,)
(33 33)
J’o
_J’5 -J&
39
_J&
-J& -J&j
(35) (53)
J’ 15.50 _J/‘6 40 -Jd40
(37) (73) (15) (51) (55)
[421
(31 33)
(35 11)
[3 11: [21 (cont.) (35 33) (53 11)
(53 33)
(331) (332) (cont. on next page)
M. Harvey / Fractional-parentage
expansions
309
TABLE 3 (continued)
r421
(31 33)
(35 11) J++
(35)
[31]:[2] (cont.) (35 33) (53 11) -J&j
(53)
-J&
(37)
-1
(53 33) -J&
4%
-J&j -1
(73) -J$
(15)
-J; -Ji
(51)
-Ji
(55)
[31]:[11] [421
(11) (331) (332) (35)
(33 13)
(33 31)
-Jgj
-JN
-J&l
-JZ
(13 13)
(13 31)
(31 13) (cont.below)
-J&
-J&
-J$ J48 130
-J$j
J48 130
-J&
(53) (37) (73)
-4
(15)
(51) (55)
[421
(31 31)
(11)
-J$
(331) (332) (35) (53) (37) (73) (15) (51) (55)
1421
(11) (33)l (33)z
(35 13)
-4% -JS 130
J&
[31]:[11] (35 31)
(cont.) (53 13)
(53 35)
-J& -J’
J&
J&
130
J&
-1
-J$
(11 11)
-1
J; -Jf (11 33)
(33 11)
-JB52 -J/64 195
JA
P2l:Pl (33 33)
(15 11)
(15 33)
(51 11)
(51 33)
J$
-J:
-JzO
_J’o8 195
-JZ
(35)
J$
(53)
J$
J&
-JO 195 -Jf
-4 1
(37) (cont. on next page)
J& -JO19.5
310
M. Harvey / Fractional-parentage TABLE
[421
(11 11)
(11 33)
(33 11)
expansions
3 (continued)
[22] : [2] (cont.) (33 33) (15 11)
(15 33)
(51 11)
(51 33) 1
(73) (15)
-./$
(51)
-J$
(13)
-J$
1
(55)
[33]=[E]
-J$
(11 13)
(33 13)
(15 13)
[22]:[11] (51 13) (11 31)
J3
(15 31)
JT%
-J&
-Jfr
(31)
(33 31)
-J&
-J&
-4
(35)
(15 31)
J$ J4
(53)
-J;
1
(17)
1
(71)
[33]=[2%] (13) (31) (35) (53) (17) (71)
(13 11)
(31 11)
(35 11)
(53 11)
[311: [21 (13 33)
J” 54
J” 54
(31 33)
-J& -J&
J&
J& JB
J&
(33 33) J3Q
54
(35 33)
-J$
J30
J; J&
(53 33)
-J$
-J& -Jft
-J$ -J&
-1 -1
In each block the six-particle symmetry f’ is given on the left together with the possible (2T+ 1 2S + 1) labels. The four and two particle symmetries (v6eC,f,]:[ 1)are given above the tables toegether with their possible (2 T” + 1 2s” + 1 2t + 1 2s + 1) labels.
where LX’and p’ denote the distribution of the three colors in the first four quarks for the state Ce4and the last pair of quarks in the state c respectively. The other notations are similar to those for isospin-spin states of the previous subsection. The only six-quark color state to be considered has [f’] = [222] in which case there are only two types of cfp expansions with [&I = [22] and a symmetric ([2]) last pair or [fL3]= [211] and an antisymmetric ([ll]) last pair. Both sets of cfp coefficients could be deduced from ref. lo) which lists the orbital coefficients for the nuclear p-shell; it is sufficient to identify the three p-components with the three colors. In practical calculations, because of the assumed color-charge symmetry of quarkquark forces, it is only necessary to know whether the last pair of quarks is color symmetric or antisymmetric to use the property in eq. (lb). Thus in practice all that is
M. Harvey / Fractional-parentage
usually required of the color cfp’s is their normalization 1 (f;ya'
a’p’
311
expansions
property
(10)
: [P’llIf’CP’4’l~2 = 1*
Two-body orbital cfp reduction. In a similar manner to the isospin-spin and color states of the two previous subsections, we write the two-body cfp reduction for orbital states in the form +6(f([p(I]y))
=N6(f)
5
(f&
: [P]l}f[ps])~,4(fp4y()~2([P])
.
(11)
The six particles of the orbital states in the cluster model are assumed to occupy two types of orbitals a = s+ and b = s- with a configuration a3b3 in which three particles are in the state a and three in the state b. The summation over (Yand p is for the three different types of distributions of particles in the four-particle state G4 and the remaining pair in ~35~; namely (Y= a3b and B = b2, (Y= a2b2 and p = ab, or (Y= ab3 and p = a2. In table 4 we list the cfp coefficients for the four different orbital symmetries noted in table 1. In this table the bar or tilde over the state of the last pair denote a symmetric or antisymmetric combination, respectively. In eq. (11) we include a normalization factor N6(f) to take into account the possible non-orthogonality of states a=s+ and b=s_. The cfp’s are defined in the limit where (alb) =0 i.e. iV6([f])1. The general form for the normalization function Ns[f] for the cluster configuration a3b3 is given later in eq. (16). TABLE Orbital
4
two-body fractional parentage coefficients of eq. (11) for the state with structure a3b3 with the symmetries on the left of each block
[41:[111
[41: El a3b3
a3b : b2
ab3 : a2
a3b3
a3b : b2
ab3: a*
[511 [421 [331
+Ji
-Ji
a2b2
: ab
a2b2 : ab
a2b2
:i&
[31]:[11] a2b2: ab
[311: r21
j
[421 I331
a2b2
:&
1
J+
0
[22:[11] a2b2 : & 0 1
0
The possible four-particle block with their symmetries.
0 1
_$ P21: PI
a3b3
-J$
and two-particle
structures
are listed above each
312
M. Harvey / F~a~t~~nal-parentage
expansions
Clebsch-Gordan coefficients of S6. The one remaining obstacle to writing down the
complete cfp expansion of states in eq. (4) is the determination of the ClebschGordan coefficient S(fYf’Y’lp?‘) of S6. The problem for calculating these coefficients has largely been avoided in nuclear physics and so we give more details in appendix B. There we note how, for the d-YYR representation, a CG coefficient for S6 can be factorized in terms of a CG coefficient for Sq and a matrix k With the notation of appendix A we can write ~(f[P~lYf’[P’~‘lY’lf7’~~lY”)
In what follows we show that all that is required in a practical calculation is knowledge of the Z?-matrices. The method for calculating these functions is detailed in appendix B and the results needed for the six-quark cluster problem are listed in table 5. Matrix elements of two-body operators with totally anti-symmetric states. With the substitution of eqs. (8), (9), (11) and (12) into eq. (4), an antisymmetric six-quark state can now be written in the form
= N(f) c ~~(f[Wlf~P’4’lPI~P14”lf
x (f&
: Ml~fb4l)(fhI’~ ‘: [/3’]~}f’[p’q’])(f;~~,J-T”:
[ts]j(f’[p”q”]TS)
x 1Yq~~fpql*~Cf~~,~l,~‘Cf~~~,~~l~s~”~”~~22(~Pl~~~~B’I~Y~~~~I~ 9
(13)
with summation over the labels [pq], [p’q’], [p”q”], cy,[@I, a‘, [@‘I, T’S” and [ts]. We recall that f, f, f” etc. are symmetries in the orbital, color and isospin-spin spaces, respectively. The antisymmetric (unnormalized) state of four particles has the form Wh~~f~l,~Efb~,~lc~‘~f~“q”l*S~“~“~
x ~~(f,ycr)Ce4(f~,,,y’tu’)T4(f~“,-y”T”S”)
.
(14)
Although it is not explicitly obvious from eq. (13), the values of the coefficients are such that the only two-body states (62, c and y that are possible are those for which the combined symmetry is antisymmetric ([ll]). To calculate the matrix elements of two-body operators it is now necessary only to apply eq. (7) with the cfp coefficient P explicitly defined now in terms of products of the known cfp’s in the orbital, color and isospin-spin spaces, the known K matrices
M. Harvey / Fractional-parentage TABLE
Elements
of the matrices
expansions
5
f(f[pqlf’[p’q’]lf’[p”q”])
[6112221
[61D221 -11 33 [222]
33
[2211]
D221
[21111]
[22111
-J&
1
x557 Jf
A421 wi --
_
12 33
J30
22 33
Jzi
zz
J” 54
-;”
i5 (m
24
23
0
11 33
22
[511 Pw 12 33
J$
24
--
P221
Tlz
1
11 33
313
J&
J$
22.33
r331 r2221 IT.55
JB
-J"27
-
22.55
[331 [2221 i2.33
f and f’ are given in the first line above each block and [pq] and [ p’q’] pairs are given in the second line (the bar denoting a symmetric pair and the tilde an antisymmetric pair). Each row is labelled by [f’] [p”q”].
M. Ziarmy / ~racfio~al-parenfag~ extensions
314
and the normalization coefficients NJfl: The evaluation of the overlaps between antisymmetric four-particle states simplifies considerably with the use of the general properties of orthogonality between states of different symmetry, orthogonality between states of the same symmetry but different Yamanou~hi labels, the independence of the normalization of a state to the Yamanouchi label and the normalization property for the ~lebsch-Gordan coefficients 1 s(f~,y~~,,,y’1~,,,,,y”“)*= 1 . YY’
Applying the above properties we may write the overlaps of antisymmetric particle states (with a simplified notation) w4mo~rflcdflll =
four-
TSfYRTS)IY4([~]o(Y[f?]c~‘[$l]lSCYnl
s~~~p~~~-s~~~~~~“~~~~s~~~~~~~l~~I~~~fl~~ , (1%
where the orbital overlap is for any one of the states (any chosen Yamanouchi label) of the [f] representation having structure (Ywith the state of the same symmetry having structure E. If the constituent single-particle orbitals a and b are orthonormal then the final orbital overlap yields ~5,~s.Wowever with a = s+ and b = s- in the cluster model the orthogonality of a and b is only established at large distances. For intermediate distances, which are important for the calculation of baryon-baryon forces, the orthogonality of a and b must not be assumed. The values for the orbital overlaps for the symmetries of the cluster model, and for the possible four-particle configurations, are listed in table 6 in terms of the basic overlap m = (ajb). Using tables 4 and 6 the normalization functions Ne(f) can be computed to be ~~([6])=rl+9~2+9~4+~6]-1’z, N&51])=[l+3m2-3m4-m6]-1~2, N6([42])=[1-m2-m4+m6]-‘~2,
(16)
Ar6([33])=[1-3m2+3m4-m6]-*‘2. One-body fractional parentage
expansion.
expansion of a six-quark antisymmetric
x (fp@: PllfPMfbsa'
The one body fractional parentage state in the symmetry basis can be written
: P’l}f’p’),(f,“-T”S”:~4j}~p”TS)ls
x ly,([f,lotn[fb,lcQ’ffp”ITST”SII)~)1(P)C(Pf)Y(3~~,
with the five-particle state P5 written in analogy to that of the four-particle state eq. (14). Summation is over all possible symmetries and other quantum numbers of the
315
M. Harvey / Fractional-parentage expansions TABLE
6
Overlaps between four-particle states with the designated symmetry and orbital structures (Here m = (alb).) r41
a’b
a2b2
a3b a2b2 ab’
l+3m2 J6m(l +m’) m’(3 + m2)
J6m(l+ m2) 1 +4mz+??14 J6m(l+ m2)
a3b
a2b2
r311 a3b a*b2 ab3
1-m’ -J2m(lm’) -m2(1 -m2)
WI
a2b2
a2b2
ab3 mZ(3+m2) d6m(l+m2) 1+3m2 ab3 -m2(1 -m2) -J2m(l -m*) 1-m’
-J2m(l-m2) l-m4 -J2m(l-m*)
(1 -m’)*
five-particle systems. The required cfp’s for the isospin-spin, orbital spaces and the K-matrices are listed in tables 7,8 and 9, respectively. The overlap matrices for the possible five-body states from the cluster structure (a3b3) are given in table 10. Limits asX -+ 0. It is instructive to demonstrate the structures of the various orbital symmetry states J/&f]) = N&f]&?)f as the two confining potentials merge (X + 0)
TABLE 7 As in table 3 for the one-body, isospin-spin coefficients of fractional parentage
[61
22
PI 44
66
22
24
42
[51]
22
PI 44
66
22
24
42
13 13 31 33 35 53
1 1
55 57 75
$
r411 44
46
64
44
46
64
r411
-V$ $ J; _J’636
-J,s 1
-4 -J$
-4%
-J&
_J2” 36
_-J&j -J&
_-JZ36 -yfi
J$ 1 I
VI&i -1
-4% Jk -1
316
M. Harvey / Fractional-parentage
expansions
TABLE 7 (eontinued~
[42] 11
22
-V’& Jrn 468
[42]
22
11
3%
35 53 37 73
0 ._J?” 468
0 _Jrn 468
[411
44
46
64
J& -4
Jfs
-4
J$
24
-1
-J1”
_&585
-J&
.p?& 585 $3 25
15
_J’“6 2.5
51 55 [33]
42
1
331 332 35 53
331
24
22
24
42
-J& J&L 585
[321
o -.,/I.@ ,585 &
Jii
Jlr? 25
26
-J&
62
-J&
1
-J&
-Jgj
42
44
1 -J&
1 [321
44
i.e. s++ s_*s. Writing m =
26
62
and noting that E + 0 as X+ 0, we
NSj[6]+ l/x%, N6[51]-+ l/x&, (17) Na[42]-, l/x’%*, N6[333-+ l/xk3. In the case of [6] symmetry, there is no problem with either the configuration (s$s!)~G~ or the normalization coefficient as X + 0 and we can immediately see that &[6]+ s6.
M. Harvey / Fractional-parentage
expansions
TABLE 8 As in table 4 for the one-body,
orbital coefficients of fractional-parentage [51 a2b3 : a
a3b2 : b
a3b3
44 J:
[61 [511
[321 a3b2 : b
a2b3 : a
[421
Jf
-4;
[331
Jf
a3b3
a3b2 : b
a3b3
J; 1411
[511
a2b3 : a -J;
$2
[421
J$
TABLE 9 As in table 5 for the K(fpf’p’lf”p”)
matrices
[‘A~[2221 1 [222]
3 1
3
[42]. [222]
[222]
3
D11111 {:
13
23
J$
J$
-.g
:g
[33]. [222] 23 P61 L22111
1 1 _ ,l
6 {;
[511’[222] 13
23
_;
-2
0
L22111
{:
317
318
M. Harvey / Fractional-parentage expansions TABLE 10 As in table 6 but for five-particle states
a3b2 a2b3
1+6m2+3m4 3m+6m3+m5
a2b3
l+m*-2m4 -2m+m3+m5
-2m+m3+m5 l+m2-2m4
a3b2
[321 a3b2 a2b3
3m+6m3+m5 1+6m2+3m4
a3b2
[411 a3b2 a2b3
azb3
a3b2
[Sl
a2b3
1-2m2+m4 -m+2m3-m5
-m+2m3-m5 l-2m2+m4
For the (51) symmetry the configuration (s?s!)~~~J vanishes as X+0 and the normalization coefficient becomes infinite (as the inverse first order in F). Taking the limit by differentiating the configuration and the inverse of iV6[5 l] with respect to X we find ~6[51]‘3(s:s:s3+S:S2SI)/J12E~_,s5p~, (18) where we denote the differential by a prime e.g. S’CCpo. Similarly for the $6[42] state a double derivative has to be performed to deduce the limiting structure rj16[42]+3(2s+s:%3+6&:&‘+2&s~)/2&E’*. (19) + s4p; . (Here
we drop
terms
like s:s:s?+s5sff
which
vanish
in the limit,
for the [42]
symmetry, since one cannot have more than four like particles in such a symmetry.) It follows naturally now that 46[33] + s3pz. It is interesting to note, in the derivation of eq. (19), terms like s+s?asp~ and S:S: a s*po. These two terms are components of the excited positive and negativeparity nucleon resonances N** and N*, respectively *). We comment on the significance
of this fact in the summary. 4. The physical basis
A direct application of the cfp expansions given in the last section is to the derivation of the transformation matrices between the symmetry basis and the physical basis for a given TS. The physical basis is defined to contain all of those di-baryon states which, in the cluster configuration s”,s!, for large separation of the two confining potentials [i.e. (s+ls_) + 01, can be represented in terms of pairs of known (colorless) free baryons e.g. N*, NA, A*. The physical basis also contains the remaining (orthogonal) states, whose asymptotic triplets of quarks remain colorful, and which we refer to as hidden-color states.
A4 Harvey / Fractional-parentage
expansions
319
The above definition of di-baryon states implies that, in a two-body cfp expansion, when the last pair of particles have orbital structure & = sl or s!, their color state c must only be antisymmetric. In the case of the (7’S) = (01) six-quark states, for example, the cfp expansion of the state with [f& = [42] and [flTS = [Sl] (see table 1) has this property but it is not shared by the two states having [f]o = [I;] and [f10 = [42] both with [f& = [33]. It is possible to take a linear combination of these latter two states however to yield an allowed di-baryon state. The orthogonal combination defines the hidden-color (23) = (01) state of table 2b. To determine the N or A character of the di-baryon states we again refer to the cfp expansion. If the last pair of quarks having
In this paper we have recorded (table 1) the classification of six-quark states in a cluster model in which triplets of quarks occupy the lowest orbit of a confining potential. We have deduced (table 2) the number of “hidden-color” states that
TABLE 11 Transformation coefficients between the physical basis states (with CC denoting a hidden-color state) shown on the left of each block, and the symmetry basis states whose structure is identified above each block by the orbital symmetry [fl and isopsin-spin symmetry {f3; each block is for the designated (TS) pairs
T=l T=O
S=O
T=O
S=O
S=l
N2
A2 cc T=l
S=l
r511142I
$2 j!; [5lW&
[42X331 [42x511
[3311421 WI161 _J36
-J& -JE +J$
JZ 0
r51114212 EWWh
J241 1053
_JJS1053
-J" 1053
_J468 1053
-J101053
JL1053
JA?C 1053
JLE 1053
0
0
0
NA
_JzO ,053
J256 1053
_J468 1053
S=2 S=l
T=2
S=2 -A2 NA
d$ 0 [6X331
J$ 0
$ [42l{W
rsl1142) [WW JZO 4.5
&
PWJI -J&. 0
_J’645 -&
cc
J36 45
jy
j!;
[421m
0 0
1
0 0 0
0 0
[WI511
G
[5111421
[3311421
[4211511 0 1
0 0
0
S=3 S=l
S=2 S=O
WIWI
S=3 S=2 A2
T=O [33]{42}T=3
S=3 S=O
-J$ J$
A2 cc T=2 T=3
[=I{@
-A2 EA cc,
T=l T=3 T=O T=2
[WWI,
N2
cc2
T=l T=2
[61{331
WIWI 1
[6X33
+J$ -J+
A2 6Q
T=3
S=3 A2
142lWI
W1{6) 1
M. Harvey / Fractional-parentage
expansions
321
cannot be represented in terms of two free baryons from the asterisked SU(4) symmetries of table 1. In sect. 3 we showed how the method of fractional parentage expansions could be used to calculate energies in a six-quark cluster model and recorded in tables 3 to 10 the coefficients required to apply the technique. Finally in sect. 4 we showed an application of the technique by deducing the transformation (table 11) between the “symmetry basis”, in which quark states have definite orbital and isospin-spin symmetry, to the “physical basis”, in which observable di-baryon states are identified. With the cfp techniques it is possible to take a two-body quark-quark interaction, deduce the effective interaction between free baryons and study how this is modified by interaction with other di-baryon channels and by the hidden-color states. Such a calculation is reported in the accompanying paper 9). This paper already provides clues, however, as to the problems and difficulties that lie ahead. In sect. 3 we discussed the limiting structures of the cluster states as the confining potentials merge. There we showed for example how the states of [42] orbital symmetry contain within their structure terms like N*N* and N**N with N* and N** the excited negative and positive-parity nucleon resonances, respectively. One cannot expect to reproduce nuclear forces therefore in a quark model unless the model can simultaneously reproduce the N” and N** spectra. In this regard Isgur and Karl ‘) have demonstrated the importance of the tensor force for the relative N* spectrum and this provides a clue as to the possible origin of the tensor effects in the deuteron. However it is also necessary to use confining potentials in which the N* and N”” spectra are placed in their correct position relative to the N-and such potentials have yet to be found 9). With the tables in this paper however any proposed quark-quark force can quickly be translated into an effective nucleon-nucleon interaction. A few papers now exist in the published literature that consider six-quark structures but none, to our knowledge, treats them by the cfp cluster method suggested here or has shown the close relationship among the N*, NA and A* and hidden-color systems. Thus Jaffe ‘**r3) has examined six-quark “exotics” (spherical s6 states of [6] symmetry) and deduced their energies. Matveev and Sorba i4) question whether the deuteron can be treated as a six-quark system and deduce, from unitary symmetries for the (TS) = (01) channel, the limited expansion of the spherical s6 state into the N*, A* and hidden-color (6Q) channels [see our table 11, the first column for (TS) = (Ol)]. Matveev and Sorba i5) have since extended their analysis to spherical states of nine and twelve quarks. The extension to six-quark, spherical states but with more than two flavors has been made by Hogaasen and Sorba 16). The quark structure of the deuteron has also been invoked by Neudatchin and collaborators r7-19): arguing that the hard core of the nucleon-nucleon potential implies a node in the relative wave function of the nucleons ‘“), they deduce that the dominant quark orbital symmetry must be [42] with an isospin-spin symmetry of [S l] because of the structure of quark-quark forces. From our table 11 ((TS) = (01))
322
M. Harvey / Fractional-parentage expansions
however we see that their conclusions are not entirely correct. The states of f42] orbital symmetry do indeed dominate the nucleon-nucleon channel (by 88.8%) but only 44.4% has the [51] isospin-spin symmetry: the remaining 44.4% has the lower [33] isospin-spin symmetry. A few attempts have been made to deduce an effective baryon-baryon force from the remnant of the quark-quark interaction. The feeling seems to be that such first attempts should describe only the shorter range parts of the NN force, leaving the description of the long-range part to 7r-meson exchange or, equivalently, to complex quark exchange mechanisms. Such an approach is probably justified in view of the difficulty the quark model has to account for the r-meson mass. Liberman *l) has considered the non-relativistic approach and is perhaps the closest to our own 9, but he only considers the NN channel with no NA, A* or hidden-color channel coupling. de Tar 22*23 ) approaches the problem relativistically in the MIT bag model but, again, only considers the NN system in isolation. The analysis of these papers and results of our own calculations we leave to the accompanying paper 9). It is a pleasure to thank Paul Lee, Faqir Khanna, Nathan Isgur and Gabriel Karl for discussions and correspondence on the quark model.
Appendix A DEFINITIONS AND NOTATION
Notation f
Y y = (P4Y)
y = b?lY
1
f, f’, f’ or K Y’, Y” etc.
- a Young tableau for a symmetry of S,, the symmetry group of n particles. - a Yamanouchi label l”,ll) for the Young symmetry f. - a Yamanouchi label in which the it th particle is in row p of f, the (n - 1)th particle in row 4 and the remaining particles have a Yamanouchi distribution y. The labelling refers to the standard Young-Yamanouchi representation (cf. below). - a Yamanouchi label for the diagonalized Young-YamanouchiRutherford representation (cf. below) in which the nth and (n - 1)th particle has definite (but unspecified) symmetry. Where the symmetry is to be specified we write p4 orEto denote the symmetric or antisymmetric pairing. - a Young tableau of S,_i (Snw2)derived from the tableau f of S, by the removal of a square in the pth (and qth) row (s). - the tableau of the dual symmetry to f obtained by interchanging rows and columns. - when used in the same expression refer to orbital, color and isospin-spin spaces respectively.
M. Harvey / Fractional-parentage
323
expansions
Definitions (i) Standard Young-Yamanouchi
representation losll) for the transposition P,,,_l in the symmetry f is defined as follows for the three possible types of Yamanouchi labelling:
case (a) for p = f.j Kn-Ilf(PqYN = +llf(PqY)) 7 case (b) for p > q with Y = (pqy) existing but not Y = (qpy) Pn.“-llf(PqYN
=
-llf(pqy)),
case (c) for p > q with Y = (Pqy) and (qpy) existing R.“-llf(PqYN=
P",n-llf(qPY))
- i
J/L’-1 jP(
If(pqy))+-
JyTT = -j--JImIY))+;
IfhPY)) 7
If(PqYN~
Here p is the axial distance (number of lines crossed counting along rows and columns) between the nth particle and (n - 1)th particle in the Young tableau fi In general we write (A.1)
p,,,-IIf(PqY)) = 4q If(PqYN+&7lf(PqYN 9
with a’, and p’, taking the above values depending on the situation [e.g. p’, = 0 in f cases (i) and (ii)]. Note that ((~fp~)‘+@I’,,)’ = 1 and a’,, = -%P* (ii) Adjoint Young- Yumunouchi representation lo) for the transposition P,,,-l in the symmetry f is the same as that for the standard representation except that all the PL coefficients are negative. (iii) Diagonalised Young-Yamanouchi-Rutherford representation lo) is such that the last pair of particles have definite symmetry. In terms of the standard representation we may write for the three cases: case (a) IfGP41YN’If(P4Y)) case (b) If(rtGlYN= If(PqYN9 case (c) (P >q)lf(ElY))=
(~If(4PY))+JCL_llf(PqY)))lJ2CL,
lf([~lY)>=(JCL-~If(qPY))-JCL+~If(P4Y)))l~~
-
324
M. Harvey / Fractional-parentage
expansions
In general we write
If(CP41Y))=Y~4IN~~Y))+S~4IP(CIPY)),
(A.3
with the above definitions for y and 6 depending on the situation,
Appendix B CLEBSCH-GORDAN
COEFFICIENTS
FOR THE SYMMETRIC
GROUP
S,
With the notation of appendix A, a state Ip’ Y") can be constructed from two sets of states IfY) and If’ Y’) by the Clebsch-Gordan expansion If’Y”> = 2, s(fYfY’lf’y”)lfY)lf’Y’>,
03.1)
where S(fY f Y'/f"Y") is the Clebsch-Gordan (CG) coefficient. Hammermesh [ref. I’), section 7.141 has shown how the CG coefficients for S,, can be related to those for S,_, by a matrix K. With Y = (pqy) (cf. appendix A) we may write
S(f(pqYlf'(p'q'Y')lf'(p"q"Y")) =K(fpf'p'lf'p")S(f,(qY)f~,(q'Y')lf,N~(q"Y")) ~Kz(f(pq)f'(p'q'>lf"(p"q"))s(~~~Y~~~~,Y'l~~~,~~Y~) * 03.2) The first equality is as given by Hammermesh. The second equality follows by a second application of the theorem relating the CG for S,-i to those for Sn_2 and the definition ~2(f(pqlf’(p’q’)lf”(P”q”)
= K(fpf’p’lf”P”)K:(f~~~4’lf,“~q”)
*
(8.3)
Hammermesh also shows, with the notation of appendix A, that the K-matrices may be calculated using the following relationships: K(fpf'p'lf"p")K(f,q'l~~,,q~)(~~~~~,,
-*&f)
+K(fp~q'lf'p").lu(~~~,p'J~~"q~)~~~~,*,
-r-k;r(f4f’P’lf”Pf’)K(~~P~~,q’~~~~‘q”)~~~~~*, +K(f4f’4’lf”4”)K(f,P’l~~~,q”)~~~~~, =
(with a! and p defined in A-1) and the orthonormal c
03.4)
K(fpf'p'lf'q")K(f,q'l~~,,p")~~,~,,
K(fpf'p'lf'p")K(fpf'p'l~~")6(f,N,,~~,,)=
condition s(f”~)s(p”p”>
.
(B.3
PP'
The above expressions allow the determination of K relating the CG of S, to S,-i once the K relating the CG of S,_r to 5L-2 are known. Since (trivially~
M. Harvey / Fractional-parentage K(
11 11111) = 1, all K-matrices
can be determined
325
expansions
by iteration.
The KZ matrices
being products of K-matrices (cf. eq. (B.3) are thus also determined. The determination of the K-matrices above is for the standard Yamanouchi representation. We define a K-matrix for the diagonalised Yamanouchi-Rutherford
representation
YoungYoung-
by
~(f~P4lf~P’4’llf’~~“~“I)
with ~PpqK~(f(~q)f)(~‘4’)P(~“4) and the y and S defined The elements
= K~(f(~)fr(~‘q’)]fl(p”q”))
etc.
in eq. (A.2).
of the K-matrices
required
in the quark-cluster
problem
are listed in
table 5. Note that the K-matrices have the property that all elements are zero unless the product of the symmetries of [ pq] and [ $4’1 is equal to the symmetry for [ ,“,“I. The K-matrices relate the CG coefficients of S, to those for Sn_2 in the d-YYR representation. References 1) A. de Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147 2) N. Isgur and G. Karl, Phys. Lett. 72B (1977) 109; Phys. Rev. D18 (1978) 4187; Phys. Rev. D19 3) 4) 5) ‘5) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
(1979) 2653; R. Koniuk and N. Isgur, Phys. Rev. D21(1980) 1868 O.W. Greenberg, Phys. Rev. Lett. 13 (1964) 598 H. Fritz and M. Gell-Mann, in Scale and conformal symmetry in hadron physics, ed. R. Gatto (Wiley, New York, 1973) J.E. Augustin et al., Phys. Rev. Lett. 34 (1975) 764 T. de Grand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 S. Weinberg, Phys. Rev. Lett. 31 (1973) 494 D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343 M. Harvey, Nucl. Phys. A352 (1981) 326 J.P. Elliott, J. Hope and H.A. Jahn, Phil. Trans. Roy. Sot. 246 (1953) 241 M. Hammermesh, in Group theory, second printing (Addison-Wesley, Reading, Mass., 1964) R.L. Jaffe, Phys. Rev. D15 (1977) 281 R.L. Jaffe, Phys. Rev. Lett. 38 (1977) 195 V.A. Matveev and P. Sorba, Nuovo Cim. Lett. 20 (1977) 435 V.A. Matveev and P. Sorba, Nuovo Cim. 45 (1978) 257 H. Hogaasen and P. Sorba, Nucl. Phys. B150 (1979) 427 V.G. Neudatchin, I.T. Obukhovsky, V.I. Kukulin and N.F. Golovanova, Phys. Rev. Cl1 (1975) 128 V.G. Neudatchin, Yu.F. Smirnov and R. Tamagaki, Prog. Theor. Phys. 58 (1977) 1072 Yu.F. Smirnov, I.T. Gbukhouskii, V.G. Neudatchin and R. Tamagaki, SOV. J. Nucl. Phys. 27 (3) (1978) 456 R. Tamagaki, Rev. Mod. Phys. 39 (1967) 629 D. Liberman, Phys. Rev. D16 (1977) 1542 C. de Tar, Phys. Rev. D17 (1978) 323 C. de Tar, preprint, Univ. of Utah, HEP 78/l