PHYSICAL REVIEW D VOLUME 26, NUMBER 11 1 DECEMBER 1982

Strange-baryon spectroscopy through Bethe-Salpeter approach under harmonic confinement

D. S. Kulshreshtha, A. N. Mitra, and I. Santhanam Department of Physics, University of Delhi, Delhi 110007, India

(Received 16 February 1982)

The unequal-quark-mass problem of excited A and Z states is treated through a mul- tichannel generalization of a relativistic Bethe-Salpeter treatment of the simpler (equal- mass) problem of N and A as well as meson states recently found to give excellent fits to the corresponding spectra. The present fits to the A and B masses are fully in tune with the quality of the NL and AL results, as evidenced by the highly consistent values of the appropriate universal function F ( M ) representing the central (masd2 for each N super- multiplet, calculated with the same reduced spring constant and quark masses as em- ployed earlier for N L , AL, and meson states. These F ( M ) values now include the one- gluon-exchange corrections which help in improving the F ( M ) regularities over the pure harmonic-oscillator prediction. The F ( M ) representation also brings out rather succinctly a modest symmetry-breaking trend ( - 5 - 10 % ) at the collective supermultiplet level with little scatter among individual members.

I. INTRODUCTION

It has recently been pointed out' that the intrin- sically relativistic character of light-quark systems makes the Bethe-Salpeter (BS) framework a more realistic starting point than, e.g., the Schrodinger equation with (v /c12 corrections. The dominance of the confining region for such systems necessi- tates a conscious choice of the corresponding ker- nel and, in the absence of a theoretical consensus on the confinement mechanism, the harmonic os- cillator2 represents the simplest possibility (subject to its consistency with the data). Recently such a formalism has been developed1 and applied3 to qH mesons and nonstrange qqq baryons (N,A) to gen- erate spin- and momentum-dependent corrections to the corresponding spectra with a high degree of success. The baryon mass formula is conveniently represented by a relation of the form3

where F ( M ) is a known function of the actual baryonic mass, the reduced spring constant (G), and the quark mass (m4). Its significance is that of a central ( m a d 2 , common to all members of a given N supermultiplet, after correction for spin- and momentum-dependent effects and normalized to a unit spring c o n ~ t a n t . ~ I t is directly calculable from the experimental masses, thus facilitating a direct test of its universal supermultiplet character as well as its unit-spacing property (AF = 1, for

AN = 1).3,4 A comparison of the F ( M ) values for N and A states up to L = 5 shows a remarkable de- gree of consistency with the expected universal character of this function4 thus providing an in- direct yet convincing check on the goodness of fit to the (N,A) masses achieved by the BS model.

In view of this unexpected success of the BS model for nonstrange-baryon spectroscopy (involv- ing equal-mass quark constituents), it is obviously of interest to explore the corresponding predictions for strange (48) baryon states as well. While the general equations for qqq states with unequal-mass kinematics have been written down earlier,' their applications to the present (A,2) case involves much more algebra than for (N,A) states. Nevertheless, the calculations are a straightforward generalization of the earlier equal-mass formalism for the evaluation of the universal F ( M ) function for A and Z states and have been made with the same input values of the fundamental parameters (G,m,) as employed in the work of Mitra and Santhanam (MS) .~ The results seem to exhibit a remarkable consistency of the F ( M ) values of (A,Z) states with those of the corresponding (N ,A) states, thus strengthening the conclusion of the (ex- pected) flavor independence and consequent univer- sality of the F ( M ) function3 on a wider scale (uds) than hitherto achieved (on a mere ud basis).3r4 The consistency seems to improve with the inclusion of one-gluon-exchange (QCD) effects.

Because of the greater amount of (albeit straight-

26 - 3131 @ 1982 The American Physical Society

3132 D. S. KULSHRESHTHA, A. N. MITRA, AND I. SANTHANAM - 26

forward) algebra involved in the present case, most of the calculational details analogous to MS (Ref. 3) need not be repeated here, except for recording the essential outline together with the characteristi- cally new features of unequal-mass kinematics, so as to keep the present paper reasonably self- contained. Section I1 summarizes the principal features of the master equation for qqq states (m 1#m2 =m3) and the method of extraction of F ( M ) as an eigenvalue problem arising out of its n X n matrix generalization (3 5 n 5 5) of the earlier equal-mass case. The QCD effect has now been

generalized to include the Coulomb interaction as well, instead of assuming the latter effect to be im- bedded in the harmonic-oscillator (h.0.) term,5 a procedure adopted ear~ier .~ The principal steps thereof are relegated to two appendices (A and B). Section 111 is devoted to a presentation of the nu- merical F ( M ) results for (A,Z) states, together with a comparison of a few reference cases taken from (N,A) results3 now extended to include the Coulomb effects as well. Absolute mass predic- t i o n ~ , ~ on the other hand, have not been attempted in this paper for reasons discussed in Sec. 111.

11. BS EQUATION AND MASS FORMULA FOR qqq STATE

Following closely the procedure of MS (Ref. 3) for a reduction of the general qqq equation1 the "master" equation for m 1 #m 2 = m 3 with the harmonic term alone takes the form

1 - 7 [ a ( l - ~ ~ m o - ~ ) m ~ ~ ~ + ( ~ ~ ~ { -&ft2)+( f ~ , i j 2 - B , f , 2 ) ] $ ( ~ , r j )

I where some of the obvious symbols are defined as The "mixed" angular momenta are defined by

2 A=gm12 m2-2- 1, p=(3m l/mo)1/2A . (2.4)

M is the baryon mass, and the quantity 5 is the re- duced spring constant as defined in M S , ~ but is now related to the w, parameter for m l#m2 by7

3wqq2( 12)=w qq -2( 12)=m12z2 . (2.5)

The internal variables g, ij are defined in terms of the individual momenta pi ( i = 1,2,3) by

while their canonical coordinates ii,J are given by

The orbital and spin angular momenta appearing in (2.1) are defined as follows: -+ - - L g = i i x ( , L , = J x ? j , (2.8)

t where a g i , agi are the normalized creation and an- nihilation operators associated with { motio_n (and likewise for 71 motion). Of these two only M is relevant for the mass determinatio_n of a baryonic state of given N excitation while N connects two such states differing by two units of N excitation.

There are now two distinct spring constants whose ingredients in Eq. (2.1) are defined as

follows:

- -

32 X,, = m 1 2 2 ~ 7 + j ~ 2 r n l m 2 m o o 2 ~ a ~ , (2.16)

jj - 1 - 2 7 - 3 W mlm2M/m0, (2.17)

At= $ m 0 ( m o + m 2 ) / m l m 2 m ~ ~ ~ ,

~ , = + m ~ ~ - ~ + im2-2+ +m02ml-1m12-2 ,

8 ~ a ~ = = m ~ ~ m ~ - ~ ( m ~ ~ ~ + 2 m ~ m ~ ) - ~

~[m12~m2-~+2(m1~+m~~)m~-~m12-~l , 8Ma, =mo2(m12+m22)/m12m22m12 .

In terms of these quantities we have

26 - STRANGE-BARYON SPECTROSCOPY THROUGH BETHE. . . 3133

flg,,=4m02m 12-2(26,,Bg,, )'I2 . (2.18) and

The "inverse radii" PC,,, which are defined by Q = -2- pi 2 +mj-'mk-'qjk2)fij2+(iej)

appear in the exponential (f, 7j+FI') dependence of the and whose method of evaluation has already been ground-state function 4o as described in M S . ~

Equation (2.1) in general involves a multicom- ponent wave function, the different components be- ing connected by the various spin and angular momentum operators generated by the BS dynam- ics with an input h.0. kernel. It is easily checked that these mixing operators vanish for m 1 =m2 since in that case h = p =0, leading to the simpli- fied description given in MS (Ref. 3) for non- strange baryons.

Finally the quantity H in Eq. (2.1) involves the counterparts of QB,Qi for equal-mass kinematiq3 which arise out of the expectation values of opera- tors of the form3

* - Q ~ ~ = ~ ~ ~ ~ ~ v ~ ~ ~ + ~ C / ~ ~ - V ~ , + ~ - - ~ ~ ~ T ~ ~ (2.20)

I

After the evaluation, H is of the form

where a,b,c are (dimensionless) functions of m l ,m2 and3

t Nc is the eigenvalue of the operator ayag , and likewise for N,. In the equal-mass case, the per- mutation symmetry of a given qqq state demand Nc = N, = N. The corresponding assumption for m l#m2, while no longer an exact statement, is nevertheless quite reasonable for a description of these correction terms and leads to the simplified picture

The two independent modes ( 6 , ~ ) of excitation characterized by the two Feynman-Kislinger- ~ a v n d a l - t ~ ~ e ' spring constants flL, are in formal analogy to the nonrelativistic results of Isgur and ~ a r l , ~ though our f16,, incorporate relativistic ef- fects though their M dependence. To calculate the mass spectrum in the unequal-mass case, it is first necessary to define a suitable matrix generalization of the F(M) function of M S . ~ For this purpose the basis functions of A,Z states of J =L & may be taken for L > 0 as follows [omitting the SU(3) fac- tors]:

Z(L odd): X"$;, X";I, X'$i , (A')

A(L even): X1$:, X"$t ,XS$t, X1$; , (B)

Z(L even): X"$;, XS$;, X'$L ,xl'+;, ~5); .

These functions are not quite the same as those used in Ref. (3, but serve the purposes of diago- nalization equally well. The difference in the structue of odd- versus even-L cases arises from our neglect of possible couplings to ( 56, odd) states1' where experimental states are at best highly

3 obscure. For J =L + , states, the matrix size is at

3134 D. S. KUESHRESHTHA, A. N. MITRA, AND I. SANTHANAM

most 2 x 2 (generally 1 X 1). For even-L excita- $Spx l .$-2hj'' .sll tions, the effective FKR-type is essen- tially (Rg~1,)"~ while for odd L, it is Rg or 0, ac- -+hS.(25-c,). (2.32) cording to whether the overall excitation is c-like or 7-like, respectively. The F ( M ) matrix is now of The elements of the Q ̂ matrix for A and Z states the form of J =L + are listed in Appendix A, along with

~ - ~ m ~ ~ 4RKij some formal expressions for the normalized qqq Fij(M) = 6ij + (2.30) wave functions. Similar expressions hold for

Ri ( n i f i j J = L - f states as well. The rest of the K matrix

where Ri is the spring constant for the ith basis is diagonal.

state and Kij =Kji is the i j element of the operator Before effecting the diagonalization of (2.30), it

defined by is necessary to take account of the one-gluon-

2 - - 3 " exchange corrections, now including the Coulombic K=HN-(3+k) ( ? J . S - ? ) + Q , (2.31) part as well (see Sec. I). For this purpose the basis

TABLE I. E(M) values for high-trajectory iA,8) states (high J for moderate M) with (i) a , = O (pure h.o.), (ii) a,=0.30, (iii) a,=0.64. Some standard N,A cases are included for calibration. For notation of states see PDG, Ref. 11.

FiM) No. Baryon N,L, J MZ a, =0.0 a, =0.30 a, =0.64

26 - STRANGE-BARYON SPECTROSCOPY THROUGH BETHE. . . 3135

states of A,Z are taken the same as in (A,A1,B,B'). This will give rise to the modification

where the calculation of the second term is out- lined in Appendix B. Note that while the h.0. term (2.30) is already "diagonal" for m = m,, the QCD term is not so, even for m, =m2, so that (even apart from the Coulomb terms which are now included) one should expect corrections to the results obtained earlier3 for FQcD.

111. RESULTS AND DISCUSSION

The input values for the basic parameters i3 and m, are the same as those employed in M S , ~ and (in GeV units) are given by

The diagonalization of the F ( M ) matrix in each case has been achieved with experimental M values

corresponding to spin-parity assignments in accor- dance with the latest Particle Data Group (PDG) tables." The choice of the "true" F ( M ) eigenvalue (out of the multiplicity of solutions so obtained for a given A or Z state) has been governed simply by the following consideration: For each member of a given N supermultiplet, there must exist at least one eigenvalue whose "matching" with the F ( M ) values for other members of the same N must be visually evident. This indeed is the most natural expectation if the logic behind the F ( M ) represen- tation (which has already received powerful numer- ical support from the equal-mass cases of mesons and baryons324 ) is to make physical sense. This matching condition can in practice be rapidly facil- itated through certain "reference points" such as the F ( M ) values for a few unique cases which are governed by 1 X 1 matrices (e.g., 56 A states). The diagonalization has been carried out with and without the FQcD terms described in Appendix B. The values chosen for this purpose are

TABLE 11. F ( M ) values for lower trajectory states (low J for high MI, with (i) a, = O (pure h.o.1, (ii) a, =0.30, (iii) a, =0.64. For notation see PDG, Ref. 1 1 .

F ( M ) No. Baryon N,L,J M 2 a, =O.O a, =0.3 a, = 0.64

3136 D. S. KULSHRESHTHA, A. N. MITRA, AND I. SANTHANAM - 26

The former value had already been employed3'4 for equal-mass kinematics governing nonstrange baryons, while the latter has recently been suggest- edI2 for a fit to several other hadronic data.

The F ( M ) values are summarized in Tables I and I1 where the different cases are listed accord- ing to the following scheme. Table I contains mostly those states which are characterized by re- latively high J values for comparatively low masses (M). Table I, contains both even-(L) and odd-L cases arranged in this order. Table I1 contains a few remaining cases of low J but generally high mass (including radial excitations), which figure in the PDG tables." The three main columns in each table correspond to the values a, =O.O (pure h.o., a, =0.3, and a, =0.64, respectively. Since the N,A-type cases had been extensively discussed ear- lier, these tables contain mostly (A,2) cases, but a few standard N,A cases are included in context in order to facilitate a more direct comparison of the F ( M ) values for both strange and nonstrange cases.

Now, for the ( A,&) states there is expected to be an extra symmetry-breaking effect (at the spin level) arising out of a tendency of the two "like" quarks to be in a spin-singlet state. This effect is expected to be particularly marked for L =0, and amounts to the following replacement in Eq. (2.1):

Both tables seem to bring out at the (uds) level the qualitative features of the F ( M ) representation noticed earlier3'4 at the (ud) level of flavors only. These include (i) considerable reduction of "scatter" in the F ( M ) values for different members of a given N supermultiplet, despite considerable (and often "chaotic") differences in their actual M~ values, and (ii) fulfillment of the equal-spacing rule (AF = 2) for successive Regge recurrences not only for individual species (N,B,A,Z), but also for their collective N-supermultiplet structures (with a very small spread). This trend is quite visible separately for even- and odd-N states, though there seems to be a small breaking of 56-70 symmetry between these two (as noticed already for nonstrange baryons4).

A quantitative difference between Tables I and I1 is in the generally higher value of F ( M ) for the former, as compared to the latter (despite the op- posite nature of the actual mass variations). This could well be indicative of another type of symme- try breaking (distinct from 3-70 breaking) as re-

vealed from the present analysis of the data in terms of the P ( M ) representation. It is of course quite another matter as to how to account for these two types of symmetry-breaking features. Presum- ably these require additional dynamical inputs hitherto missed in our BS formulation. So far the present analysis has merely served to focus atten- tion on these two distinct types of symmetry- breaking effects as a fairly general trend after iron- ing out many loca1 irregularities due to mass varia- tions of individual members of a given supermul- tiplet, as a result of its theoretical emphasis on the F ( M ) function.

In general the universality of the F ( M ) function, together with its AF =2 rule for alternating N values, seems to be so well satisfied on the whole that this theoretical feature can be profitably em- ployed as a possible criterion for determining the N-supermultiplet class of certain states, whose quantum numbers are less certain. Thus, accord- ing to this rule Diji(1940) seems to belong rather unambiguously to N = 3, L = 1. A similar classifi- cation seems to hold for Fo5(21 10) with N =4, L =2. More interestingly, by this very criterion, the S;;I,SG states which along with their non- strange partners S ;, D ;; have been traditionally assigned the status of low-J satellites of L = 1, N = 1 spin-quartet states, now seem to admit the alternative interpretation of N = 3, L = 1, J =L = - spin-doublet assignments. This alter- native, albeit unorthodox, interpretation helps obvi- ate the problem of experimentally identifying spin- quartet states of unusually low J by the relatively insensitive methods of partial-wave analysis, ar- gand diagrams, etc. A particularly interesting case of this type is PG(1910) for which a (70,0f) as- signment was noticed significantly to improve its F ( M ) value? in preference to its 56 statust3 with J =L - (L =2) presumably based on the tradi- tional ansatz of "small" spin-orbit couplings.

The overall pattern of F ( M ) regularity seems to be somewhat improved over the simple h.0. pattern (a, =0) if anything, by including the QCD effects. This is particularly noticeable through an inspec- tion of certain high-mass cases like D 13 (1670), P;'i5(191 5 ) , F2,(2030) which look much less "out of step" with aS#O than with a, =O. Most of the other cases which look already "good" at the pure h.o. level (a, =0) seem to maintain their "good- ness" for a,+O as well. As for a, =0.30 versus a, =0.64, the higher value seems to be slightly favored. This is evidenced from a graphic compar- ison of the results of Table I in Fig. 1 for even-L

STRANGE-BARYON SPECTROSCOPY THROUGH BETHE.. . 3137

EVEN-L STATES

7.0

6 .O

3 .0

2.0 O D D - L STATES

FIG. 1. Plots of F(M) versus N (total quantum num- ber) for the high-trajectory states listed in Table I. For a,=O and a,=0.64, the F(M) values are shown as one unit less and one unit more, respectively, than those given in Table I. Plots (a) and (b) are for even-L and odd-L states, respectively.

and odd-L cases, respectively (some points are indeed so close together for a, =0.64 that they are adequately represented by a single symbol).

For a possible choice between the two listed values of a,(0.30, 0.64), we have been guided by still another consideration, viz., the role of the QCD terms has been essentially one of filling the gap between the standard h.o. value of the zero- toint,energy (ZPE) for F ( M ) for qqq states (viz., i + ~ = 3 ) and the actual ZPE value for F ( M ) as deduced from the data. From the earlier investiga-

t i ~ n ~ ' ~ where the Coulomb term had been specifi- cally excluded from the QCD corrections (on the understanding that it was effectively imbedded in the confining interaction5 in the original De Rujula et al. l 4 spirit), it had appeared that the ZPE shyrt- fall in F ( M ) for ud-flavored states was nearly units for q4 states and about 2 units for qqq states. An inspection of Tables I and I1 now shows that this shortfall is greatly reduced when the effect of the Coulomb term on F ( M ) is taken into account. In particular, for a, ~ 0 . 6 4 , the ZPE value is now almost 2 unit suggesting an overall shortfall of one unit only. The same value of a,( = 0.64) has also been found t~ reduce the ZPE short-fall for q 4 states from , units without the Coulomb term to almost 1 unit15 with the Coulomb term. These q 4 and qqq results taken together (with the higher a, value) are desirable insofar as they indicate a uni- form ZPE short-fall of one unit for both q4 and qqq states when their F ( M ) values are deduced from the data, thus lending some crediability to the conjecture that the same constant background of one unit might also persist for the F ( M ) values for other hadron states with more quarks, in par- ticular for baryonium ( q q a states.I5

The fact that this higher value a, =0.64, which has been claimed to be favored by other data as we11,12 brings about this pleasant feature apart from producing a slightly better F ( M ) regularity in Tables I and I1 (and Fig. 1) would seem to give it a theoretical advantage over a lower value, say a, =0.3, which would yield different ZPE short- falls for different hadronic states. This last feature of a constant ZPE shortfall of one unit for all had- ron states has recently been made use of15 in predicting absolute masses of qqq7 states by calcu- lating the corresponding F( M) functions with the inclusion of Coulomb effects for a, =0.64. Al- though the BS model employed for the purpose is merely an extensioni5 of the one developed in Refs. 1 and 3, specific information regarding the precise amount of ZPE shortfall is much more crucial here for a nontrivial mass prediction of qqqqp states, in the absence of reliable experimental knowledge thereof, than, e.g., is the case with q4 or qqq states (which abound in data).

Despite these attractive features, the model as presented has obvious limitations, insofar as some prominent symmetry-breaking effects especially at the flavor level remain unaccounted for. These are particularly noticeable from Table 11 which shows the F ( M ) values for the doubly and triply strange baryons as considerably out of step with the main

3138 D. S. KULSHRESHTHA, A. N. MITRA, AND I. SANTHANAM

sequence. The same problem seems to be in evi- dence (but in the opposite numerical direction) for Ac,Zc states as well. On the other hand, even for these cases, certain advantages of the F ( M ) repre- sentation are still noticeable, viz. the considerably smaller F ( M ) variations in the pairs (Ac,Zc) as well as (Z,Z*), compared to their M 2 variations.

The other symmetry-breaking effects discussed in the earlier paragraphs of this section are of a more global nature, especially a small but distinct, 56-70 breaking, which the F ( M ) representation has -- quite clearly revealed as a definite trend (rather than as aberrations of individual states). While these features no doubt bring out once again the advantage of the F ( M ) representation, they nevertheless serve to focus attention on the direc- tions in which the theory still needs improvenlent. Since these symmetry-breaking effects have not yet been formulated in the model, we have refrained in this paper from offering absolute mass predictions for different baryonic states (which can at best be done at the cost of a couple of extra ad hoc parameters). At this stage, the F ( M ) values, which represent the most compact language of our BS model, seem to represent the most succinct form of comparison with the data, revealing both the strong and weak points of the theory. Several oth- er applications, as well as attempts to incorporate certain symmetry-breaking effects within the BS model, are currently under way.

APPENDIX A

We first collect the relevant set of normalized qqq spatial functions for arbitary orbital and radial excitations an! then list the matrix elements Qij of the operator Q of Eq. (2.32) for A,Z states of J =L + 3 with a serial order defined as in Eqs. (A), (A') (B), (B') of the text.

Using the notation of Refs. 9 and 3 for states of S,M1, and M" symmetry the maximally stretched states of given L ( = 21, 21 + 1, 21 + 2 ) defined by

Here ac- is the destruction operator for a 6 excita- tion with me = + 1, etc. Thus

Then it is easy to show that

~ ~ ~ - ' = 2 ' ( 1 ! ) , fi21+1-1=(1 +1)1'22'(1!) ,

R21+2-2=(1!)2221+1(1 +1)(1+2) . For radial (56,0+) or (70,0+) type excitations, it is convenient to define the operators

A5=agag, A, =aTiavi

and their Hermitian conjugates, in terms of which a (56,O) radial state is given by

&=NJA:+A;)~ I O ) .

Then using the result

t [AE,A~]=4Nt+6, Nt=agiati

and its q counterpart, it is easy to show that

( 0 A ; A ~ / 0)= (2 r + I ) !

Whence one obtains the result

~ , - ~ = ( n ! ) ~ 2 ~ " - ' ( n -+ l ) (n +2) . For a (70,0+)-type radial state, viz.

$ ~ ; $ ~ = @ n ( ~ l + ~ ; ) n [ 2 a & a ; i ; ~ : - ~ ~ ] 0 ) ,

one obtains similarly

(fin )1~=3(n !)22n-1(n +4)! .

Normalized states of J =L + can be easily constructed in terms of these stretched orbital functions. For example, an unstretched (70,L) state of 8, with J =& + is given by

and so on. Using such representations, it is possi- ble to ca lcu la~ rapidly all the matrix elements of the operator Q.

The results for A and Z states of J =L + are as follows:

L =21+1: Define

A states:

~ , , = ; h ( 1 + 2 ) , Q l 2 = - f p ( l + 1 ) ,

26 - STRANGE-BARYON SPECTROSCOPY THROUGH BETHE. . .

Z states:

~ ~ ] = - f h , Q12=hhi(21+1), 1

Q13=-Tp(1+ l ) , Q 2 2 = h ( l + l ) ,

Q23=-2phl ( l+1) , ~ 3 3 = f h ( 2 1 + 3 ) .

L =21+2: Define

2 f l 2 ~ ( 1 + 1)(1+2) ,

glr+2/2(41 +7)-'12 . A states:

~ ~ ~ = - ~ ~ ~ = ~ 4 4 = f h ( l + 2 ) ,

Q33 = Q I ~ = Q I ~ = Q ~ ~ = O 7

Q23=hgl(41+5), ~ 2 4 = - 4 1 . f i 3

Q34 = -4pfrgi

2 states: 2 Q ~ ~ = - Q ~ ~ = Q ~ = - ~ A ( ~ + 2 ) ,

Q22=Q55=O 3 Q13=Qz3=0,

The corresponding matrix elements for J =L - as well as radial states (available with the authors) are omitted for brevity.

APPENDIX B

We outline the procedure for evaluating the one-gluon-exchange corrections (termed QCL) to the qqq spectrum in the unequal-mass case, on the lines of M S , ~ but now including the Coulombic part as well. In the same normalization as for the F ( M ) matrix of the text, the QCD matrix is of the form

FQcD=C (fQCD(23)) 9 (B1) 123

where, in the (23) representation,

I while the (1 2) and (1 3) terms are more involved. m23 '=3m1+m2 , 2ml -ml+m2 , Evaluation of the expectation value of the (23) (B3) operator (B2) is achieved through a straightforward mo ++(ml+m2) use of the various wave functions listed in Appen- and call the resultant quantity FbcD (23). Then dix A. The matrix elements that survive in this the complete expression for (B 1) is given, to a fair- case are ly good approximation, by

The details (which are available with the authors) are analogous to the results of MS (Appendix B), and are omitted for brevity. As for the (12) and (133 terms, these are now much more difficult to evaluate exactly. However, an inspection of the m, structure of these operators, together with the perturbative character of these effects, suggests the following recipe for their evaluation: In the matrix element of (B2), regarded as a function of m23(=2m2) and mo( r m l +2m2), make the re- placements

FQcn z F Q c ~ ( 2 3 ) + 2 F h c ~ ( 2 3 ) . iB4)

We emphasize here that this last procedure is not formally necessary but greatly simplifies the alge- bra without introducing any serious error.

To bring these numbers in line with those of I;,(M) of the text, the quantity flB-' appearing in (B2) must be simultaneously replaced with

flB-l-(fljflj)-]I2 , (B5)

where fli and 0, are the values of the FKR-type spring constants appropriate to the nature of the two states i and j in accordance with the prescrip- tion given in the text.

3 140 D. S. KULSHRESHTHA, A. N. MITRA, AND I. SANTMANAM 26

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