PHYSICAL REVIEW D VOLUME 35, NUMBER 9

A S = 1 chiral Lagrangians revisited

Eugene Golowich Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts 01003

(Received 27 October 1986)

Previous work on the contributions of quartic chiral Lagrangians to kaon and hyperon nonlepton- ic decays is extended by considering an expanded set of such operators at tree level. Despite the ap- parent larger operator basis, the extended model gives a rather clear pattern of prediction.

I. INTRODUCTION

Our intent in this paper is to first identify and then study the structure of a certain operator-basis dependence present in chiral-Lagrangian descriptions of A S = l , AI = f processes.' We consider in turn K ~ T ' and hype- ron2 nonleptonic decays. For the latter, the correspon- dence between baryons and the fields appearing in the chiral Lagrangian is established via the Skyrme solution.

The hl= f K 3~ amplitudes have been extensively studied experimentally, and a reasonably good data base exists. In all, there are four independent K 3.rr transitions: K + + T + T + T - , T ~ T ~ . ~ ~ + and ~~-.rr~.rr~a~,.rr+ .rr-TO. The A1 = f amplitude can be parametrized with quantities a,b,c,d as'

where X, Y are standard kinematical variables

with

The experimental values for a,b,c,d are"

The point to note about c and d, which parametrize the dependence of AlI2 on quadratic terms in X and Y, is not that they are small in the given units but rather that they are unambiguously nonzero. This provides a significant hint regarding construction of the A I = f K 3~ amplitude based on the method of chiral Lagrangians.

An important contribution to the description of kaon decays with the chiral-Lagrangian approach was made by ~ r o n i n . ~ His analysis was based on the AS = 1 quadratic Lagrangian

1 MAY 1987

with

M = exp( ih+/F,) ,

where M behaves as M-ULMUR-' under the chiral transformations of SU(3IL X SU( 3 I R , 4 represents field operators of the light pseudoscalar mesons, F,=93.5 MeV, and ~ r L " L ~ = 2 6 ~ ~ defines our normalization of the Gell-Mann matrices (Ref. 5 ) . L o is the only AS = 1 La- grangian which is quadratic in momenta and transforms as ( 8,, l R ) under chiral transformations. The coupling constant go is not determined by the dictates of chiral symmetry. However its value can be fixed phenomenolog- ically from K - 2 ~ decays:

With go known, L o can be used to provide a description of AI= K - 2 ~ decay, and one finds

where the subscript "quad" refers to the quadratic deriva- tive structure of L o . At the time of Cronin's analysis the degree of agreement for parameters a and b (15% and 30% too small, respectively) was deemed reasonable, and the null predictions for c and d were consistent with the existing lack of experimental signals for these quantities. There was no reason to push the calculation any further.

However, with time the quality of K - 3 ~ data im- proved. Donoghue, Golowich, and Holstein (DGH) pointed out that the presence of nonzero c and/or d terms in the K-3.n amplitude requires the use of quartic chiral Lagrangians (i.e., those with four derivatives) in the phenomenology of K-377 decays.' A tedious but straightforward analysis of AS= 1 quartic Lagrangians demonstrated that such operators fall into either of two classes-those which give c =d = O and those which do not. It is useful in this context to define

which is a singlet under right-handed transformations, behaves as X, - ULX UL - ' under left-handed transfor-

.c" mations, and obeys X, = -X,. DGH showed that to ob- tain nonzero values for c,d requires the presence of four

2764 1987 The American Physical Society

3 5 - A S = 1 CHIRAL LAGRANGIANS REVISITED 2765

factors of X,. The particular choices

and

L 2 =g2Tr( h&pXvXpXv)

=g2Tr( h 6 a , ~ a , & f t a w ~ a v ~ t ) (11)

were adopted, and several phenomenological approaches to fix g l and g 2 were employed.6 Specifically, fitting g , and g 2 to c and d defines fit 1,

whereas an overall least-squares procedure defines fit 2,

There is not much difference in the quality of these fits. In either case the inclusion of the quartic Lagrangians is seen to have two salutory effects, nonzero values for c,d, and improved values for a,b.

There are two reasons for continuing the study of these AS = 1 chiral Lagrangians. Despite the success of the work just described, it is nonetheless not unique. There are other operators, not considered by DGH, belonging to the class having nonzero c and d. On a purely phenomenological level, could a different choice of La- grangians lead to an even better fit? This point is ad- dressed in the following section. Perhaps of greater signi- ficance is the disturbing conclusion reached in Ref. 2 that employing the Skyrme solution in Lagrangian Lo, L ,, L 2 yields a poor description of hyperon decays. Could a dif- ferent operator set provide an acceptable description? We turn to this point in Sec. I11 and summarize our work in Sec. IV. Throughout we shall maintain the procedure of Refs. 1 and 2 by computing K3a decay amplitudes at the tree level and hyperon decay amplitudes to lowest order in the semiclassical quantization of soliton solutions.

11. K -377 DECAY

As just mentioned, we are interested in the class of AS = 1 chiral Lagrangians which contain four factors of the operator X?. A particularly useful identity in the search for additional operators is

TrX, = 0 (14)

whose proof is left to the reader. In addition to the Lagrangian L 1 , L 2 of Eqs. (10) and

(1 1) there is one additional AS = 1, A1 = operator which involves a single trace:

Lagrangians containing various combinations of commu-

tators or anticommutators of the X, can be expressed in terms of L 1 , L 2 , L 3 .

There are three distinct AS = 1, A1 = Lagrangians which contain pairs of traces. Because of Eq. (141, there are no such operators with three traces. These new La- grangians are

L 4 =g4Tr (h&pXv)Tr (X~Xv)

= g 4 ~ r ( h 6 a , ~ a f i t ) ~ r ( a p M a v M t ) , (16)

L 5 =g5Tr(h&,X~)Tr(XvXv)

and

Thus there are in all six AS = 1 Lagrangians which con- tain four factors of X, and so can contribute to the x2, y2 kinematical dependence of Eq. (1).

A lengthy calculation of K 377- amplitudes induced by the Lagrangians Lo , . . . , L 6 yields the results [in the no- tation of Eq. ( 1 )]

A comforting check of these results arises from separately computing each of the four K 377 amplitudes, and finding internal consistency of the equations.

The decay amplitudes just described are "direct emis- sion" processes, as depicted in Fig. l(a) . Equations (19c) and (19d) show that indeed they contribute to the terms quadratic in the kinematic variables. However there is another procedure, via the pole diagrams of Figs. l(b) and l(c) for generating such terms. The idea is to introduce the desired kinematical behavior with strong-interaction Lagrangians as follows. Analogous to the AS = 1 La- grangians L I , . . . , L 6 we define

2766 EUGENE GOLOWICH 3 5 -

There is no analog of L 6 due to Eq. (14), and the would- be analog of L, is identical to E 2 . One can then use the strong Lagrangians E l ,z2,E4,Es along with the AS = 1 Lagrangian L o of Eq. (5) to induce K + 3 a transitions from pole diagrams. Upon doing so we find that the am- plitudes associated with E , , z 2 , z 4 , E , have precisely the kinematic dependence of the amplitudes induced by their AS = 1 counterparts L ,L2 ,L4 ,L ,. Hence they can be ab- sorbed into a redefinition of the (initially undetermined) couplings g l ,g2 ,g4,gs . Finally, there are no pole terms in which the weak transition is induced by the AS = 1 opera- tors L ', . . . , L 6 because such Lagrangians cannot contri- bute to the one-particleione-particle amplitudes of Figs. I(b) and I(c).

Equations (1 9a)-( 19d) together with the corresponding experimental values of Eq. (4) form the basis of our phenomenological analysis. First observe that the La- grangian L, does not contribute to the K 3 a decays. The reason is that the first factor in L , gives rise to the field operator of a KL meson, so that the three final-state pion fields must come from the second factor. However the latter is proportional to an SU(3) d symbol, and this has zero overlap with the three-pion sector.

It would be of real interest if anv one of the set L , , . . . , L , were "magically" able to reproduce the exper- imental c and d values. Unfortunately this cannot happen as can be seen from the following. Any one of the L ,, L 3 , or L would imply c =d, but these are known to differ, ( c /d ),,,,-.O. 3. The use of L 4 would be particular- ly poor in that c,d would be predicted to have opposite phase and c /d > 1. Both predictions are false. While L 2 passes the test for predicting the relative magnitude of c,d in reasonable fashion, it incorrectly assigns them op- posite phase. Thus, the empirical situation cannot be the simplest possible one in this operator basis.

However, there does turn out to exist a rather clean inference of this extended model. Clearly, the Lagrang- i a n s ' L l , L 3 , and L , yield equivalent descriptions of the K 377 data. We delay explication of this to Sec. IV. It is therefore convenient to employ the combination

Then from Eqs. (19a)-(19d) we see that the quantities a ,c depend only on the combination g +g2 +2g4, whereas the pair b,d are functions only of g -2g2 -g4. Upon fitting these combinations of parameters to c,d, we have predic- tions for a,b, viz.,

and

That is, even in the extended model ( L o , . . . , L 6 ) there exists a unique correction to the lowest-order parameters a ,b from the quadratic coefficients c,d. Numerically, we find for Eq. (22a)

and for Eq. (22b)

The discussion directly above takes the K 3 a phenomenology of this model as far as it can go, and several conclusions follow. First, to obtain the most suc- cessful content of the model requires using any combina- tion of g , , g3, or g, along with either of g2 or g4 . This is in fact the situation with the DGH calculation' who used g l and g 2 , and explains why the right-hand sides of Eqs. (23a) and (23b) give the same values for a ,b as do the fit-1 values of Eq. (12). In other words, DGH (fortuitously) did as well as even the extended model would allow.

To what extent is the K 377 data successfully reproduced by the extended model? The predicted correlation be- tween a and c of Eq. (22a) is seen in Eq. (23a) to be in agreement within experimental errors. However the correlation between b and d of Eq. (22b) is off by several standard deviations as evidenced by Eq. (23b). At this point it is impossible to know if additional theoretical in- put is needed or whether the fault lies with either the cen- tral values or estimated errors of the K 377 data. Further discussion of this issue is delayed to our summary in Sec. IV.

Finally, in purely numerical terms of fitting procedures we find the ranges of values g +g, +2g, = -0.16 x lop8 (fitting to c ) or -0 .14X10-~ (fitting to a ) and

-2g2 -g4 = -0.53 x (fitting to d l or -0.44 X lop8 (fitting to b).

X E @ .+ 111. HYPERON S-WAVE DECAY

The motivation for employing soliton solutions of the AS= l chiral Lagrangian in hyperon decay is to describe

( c 1 kaon and hyperon sectors with a common parameter set. For the sake of clarity, we shall review some of the

F I G . 1. K 377 decay amplitudes. Both direct-emission (a) and relevant background, and refer the reader to the literature pole contributions (b) and (c) contribute to K3.r; transitions as in for a more detailed account. K + +T+T+T-, The SU(3) chiral soliton can be written in the form7

3 5 - A S = 1 CHIRAL LAGRANGIANS REVISITED

where the radial profile F ( r ) is given in Ref. 8. This soli- ton is a solution of the strong-interaction sector; the weak interactions considered here are a tiny perturbation on it. To quantize the soliton we introduce collective coordi- nates

where A ( t ) is a time-dependent SU(3 ) matrix. To incor- porate pseudoscalar mesons in a chirally acceptable manner, we extend this solution to9

where

The quantum soliton is typically considered within the so-called semiclassical approximation. This means that, in view of the fact that it represents a slowly rotating sys- tem, one expands the theory in the number of time deriva- tives acting on the soliton. To lowest order, which we em- ploy here, there are no time derivatives. There is no analogous statement regarding space derivatives; any trun- cation of these is a dynamical assumption.

The decay amplitudes for the hyperon nonleptonic tran- sition

are

which become, in the limit of small momentum,

where M = (M'+ M ) / 2 . We shall restrict our analysis to computation of the S-wave amplitudes A only. The effec- tive chiral Lagrangian which induces the AS = 1 decay is found by inserting expression (26) into the operators Lo, . . . , L 6 , keeping terms which contain one pion field, and finally performing the spatial integration. We obtain, for .rro emission,

where

is an SU(3 ) representation matrix element and I is the sum of radial integrals:

In Eq. (32) we use g = g , +g3 +2g5 as in Eq. (21) and, with u = 2eF,r, define

The factor 2eF, which serves to cast the integrals of Eq. (33) in dimensionless form is found in Ref. 8 to have the numerical value 0.752. We have computed the integrals 10,1,2,4 by first numerically solving the soliton nonlinear differential equations8

The S-wave amplitudes are obtained upon evaluating the operator of Eq. (30) between initial ant1 final states of the decay process. It is convenient to us<: an SU(3) f , d parametrization for the set of S-wave aml)litudes.'O The experimental values are (Ref. 1 1 ) f -9.2 x lo- ' , f / d - -2 , where our phase convention here differs from that of Refs. 2 and 10. The D 7 8 ( A ) factor of Eq. (29) im- plies that for the soliton model f / d = - 3, in reasonable agreement with the experimental value. This result is a consequence of the semiclassical approximation to leading order and is independent of the number of spatial deriva- t i v e ~ . ' ~ The prediction of our extended model for f is

It is instructive to observe the effect of the go term which arises from the quadratic Lagrangian L o . Calling the pre- diction fo , we find from Eqs. ( 7 ) , (32) , ant1 (34) the value fo-4.8X l o p 8 .

Equation (32) displays the dependence of the hyperon amplitudes on the coupling constants g, ( i =O,. . . ). The quartic Lagrangians contribute to the hyperon sector much like they do to the kaon sector, that is, in terms of g, g 2 , and g,. Since the kaon amplitudes yield informa- tion on the combinations g +g2 +2g4 and g -2g2 -g4 , it seems reasonable to use Eq. (34) for the -f amplitude to supply a third relation involving these quantities. Thus, we are led to the phenomenological equation set

To solve these equations, we compute the inverse to the matrix M of coefficients. This is, of course, inversely proportional to detM,

detM =3(11 + I 2 - I 4 )=0.001 (36)

as determined from the numerical values i r ~ Eq. (32) . The unexpectedly tiny value of detM arises f r o ~ n cancellations between the various soliton integrals. Evidently there is a nearby zero.

It is clear that Eq. (36) precludes a succ:essful applica- tion of the chiral-Lagrangian model considtxed in this pa- per. The values of g,g2,g4 implied by Eq. (35) would be

2768 EUGENE GOLOWICH 35 -

substantially larger than would be expected for a normal perturbative expansion having go as a first approximation, g, g2 , and g 4 at the next level, and so on. It seems natural to place the "blame" for this result on the hyperon sector inasmuch as the K 377 analysis of DGH in Ref. 1 yields a sensible set of couplings.

Since the usual procedure10 for computing P-wave hyperon decay amplitudes employs the S-wave amplitudes in a pole model, there is little reason to extend our calcu- lation in that direction. Although it was pointed out in Ref. 2 that the soliton amplitudes contain a momentum- dependent nonpole term, it is hard to believe that it could overcompensate a disasterous pole contribution, particu- larly in view of the fact that when the empirical S-wave amplitudes are used in the pole model, agreement to within a factor of 2 to P-wave data is attained. Therefore, although we have computed these nonpole contributions in the extended model, we shall not list them here. Just as a technical point however it turns out that L 6 , which con- tributes to neither K 377 nor hyperon S-wave amplitudes, does have a nonzero matrix element for P waves. It is not the zero operator.

IV. CONCLUSION

To summarize, we searched for the set of AS = 1, A1 = chiral Lagrangians containing four factors of X , and found six. Using this expanded operator basis, we then computed K 377 and S-wave hyperon amplitudes. There were some surprises.

For the K 3 7 ~ system it happens that the four kinematic coefficients a,b,c,d are expressible in terms of three in- dependent couplings g2 , g,, and g = g l +g2 +2g5 such that a,c and separately b,d depend on common linear combinations of these couplings. Thus, if c,d are used to constrain g2, g,, and g, the quartic contributions to a,b are then uniquely determined. Although the modification

-

'J. F. Donoghue, E. Golowich, and B. R. Holstein, Phys. Rev. D 30, 587 (1984).

2J. F. Donoghue, E. Golowich, and Y.-C. R. Lin, Phys. Rev. D 32, 1733 (1985); 33, 2728(E) (1986).

3T. J. Devlin and J. 0. Dickey, Rev. Mod. Phys. 51, 237 (1979); W. T. Ford et al., Phys. Lett. 38B, 335 (1972).

,J. Cronin, Phys. Rev. 161, 1483 (1967). 5For reviews of the chiral-Lagrangian approach, see, e.g., S.

Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969); H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/ Cummings, Menlo Park, CA, 1984).

61n Ref. 1, these Lagrangians were referred to as L, and L,,

of b induced by d improves matters, the result is several standard deviations outside the experimental range. Still, the overall agreement between the tree-level chiral La- grangian description and experiment is fairly impressive. Additional A1 = f and/or A1 = data would be wel- come.

The K 3 7 ~ data place two constraints on g2, g,, and g. A third constraint comes from S-wave hyperon ampli- tudes; these turn out to depend on the same three cou- plings. In this approach, chiral soliton solutions are used to represent the hyperons. The resulting fit is a disaster. Cancellations between integrals over the soliton radial profile render values for g2, g,, and g which are unac- ceptably large. This signals a breakdown of the assump- tion that the number of spatial derivatives in the soliton sector can be truncated at four. Taken together with the evident inability of the soliton solution to generate a satis- factory NN repulsive core,13 it suggests the need for fresh insights.

Instead of being awash in a sea of adjustable parameters we have found a relatively clean set of results. Contribut- ing to these were occurrences such as the absence of con- tributions from L6 , discussed already in Sec. 11, and the equivalence of g l , g2 , and g 5 in both kaon and hyperon sectors. Although there are several technical reasons for this, the basic idea is that even though Lagrangians L I , . . . ,L6 are indeed independent, the restriction to K 3 7 ~ and hyperon decay sectors is a very severe one. As a result, we do not uniquely determine g , , . . . ,g6 . This is actually fortunate because consequently, we are able to re- veal more general aspects of the chiral-Lagrangian description.

ACKNOWLEDGMENT

The research described here was supported in part by the National Science Foundation.

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