PHYSICAL REVIEW D VOLUME 36, NUMBER 11 1 DECEMBER 1987

Addendum to "AS = 1 chiral Lagrangians revisited"

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

(Received 23 July 1987)

It is shown that the number of independent four-derivative quartic chiral Lagrangians which can contribute to the quadratic Dalitz-plot parameters of K3a is four.

In a recent paper,1 the operator set

was defined, where X, = ( ~ , M ) M + and M is the usual field matrix behaving as M + U L M U R - ' under chiral SU(3), X SU( 3!R. The quartic operators in (11, along with the quadratic operator 0, = Tr( AJ,Xp ), were used to analyze K 3 n and hyperon S-wave decay amplitudes. For the kaon processes, M has meson content M =exp( ih .4 /FT ), where 4 represents the light pseu- doscalar mesons. For hyperon amplitudes, M becomes the meson-perturbed SU(3) chiral soliton M = UA (t)I;,A - ' ( t ) ~ , where A ( t ! is a time-dependent SU(3) matrix, U =exp( ih .# /2F, ) , and 2, is an SU(3) chiral soliton. The soliton sector is treated in the semi- classical approximation; to leading order, no time derivates occur.

Reference 1 may be considered an extension of earlier in which the only quartic operators considered

were 0, and 0 2 . The basic thrust of Ref. 1 is purely phenomenological: to confront kaon and hyperon data with the expanded operator set. Note that quartic A S = 1 operators proportional to cwVaB are not present because they contribute to neither K 3 n nor (in light of the semiclassical approximation) to hyperon amplitudes.

Our purpose here is to point out that the quartic operators of Eq. (1) are not totally independent., The Cayley-Hamilton theorem can be used to reduce the di- mension of this basis from six to four. For example, it is technically straightforward to show

We can use Eq. (2) together with an analogous relation with X,,-+h, to eliminate operators 0, and 0, via

Thus, in Ref. 1, the quartic part of the A S = 1 Lagrang- ian ~ , 6 = , g i 0, becomes instead z!=, G, 0, , where from Eq. (3) above we have G I =gl + g 5 +fg6 , G2=g2-g5 ++g6, G3 = g 3 +g,, G,=g,+g, -+g6. In short, it is as if g 5 and g, vanish and g ,+Gi ( i = 1 , . . . , 4 ) . Howev- er, the results of the analysis are unchanged. In particu- lar, Eq. (35) of Ref. 1 now appears as

where G = G , + G, and we have used I, = I, + I2 to eliminate I,. The matrix of coefficients still has a zero determinant, so the K3r-hyperon equation set remains degenerate.

To proceed beyond the work described both here and in Refs. 1 and 2, there remain, in principle, mass and loop effects to consider. However, operators with mass terms would necessarily be of higher order than quartic (in order to generate the desired Dalitz-plot structure in K 3 n ) and, thus, lie outside the scope of the present ap- proach. Loop corrections would, for reasons of con- sistency, need to be computed for soliton as well as for meson amplitudes.

We thank Eduardo de Rafael for his comments.

I E . Golowich, Phys. Rev. D 35, 2764 (1987). 3J. F. Donoghue, E. Golowich, and Y.-C. R. Lin, Phys. Rev. D 2 ~ . F. Donoghue, E. Golowich, and B. R. Holstein, Phys. Rev. 32, 1733 (1985); 33, 2728(E) (1986).

D 30, 587 (1984). 4 ~ . de Rafael (private communication).

a 1987 The American Physical Society