# Effects of Light Scalar Mesons in eta -> 3pi decay Page: 2 of 17

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proposed new effects. Doubts about whether an unreasonably large value was assumed in [30] were expressed in [31].

These doubts were confirmed [32] using the work of the present paper. Still another reason for the interest in the

effects of the scalars in r7 -+ 37 is to provide an orientation for the discussion of the apparently puzzling r' -* 3 decays

in which light scalar mesons can be reasonably expected to have very large effects. We will give only a preliminary

discussion of this process here.

In section II we give a brief historical outline of treatments of ri - 37 decay based on chiral symmetry. A number

of well known ambiguities in the analysis are briefly described.

Our calculation is based on the tree level treatment of a chiral Lagrangian containing pseudoscalars, vectors and a

postulated nonet of light scalars. Since the calculation is somewhat complicated, it seems to us helpful to present the

results in a series of steps. First, in section III we give the results of using a Lagrangian containing only pseudoscalars

with minimal symmetry breaking terms.

To this Lagrangian we add, in section IV, the scalar mesons. It will be seen that the individual scalar diagrams

are quite large but there is a lot of cancellation so that the net effect is not at all dominant. However the scalars

do, as desired, increase the predicted decay rate in a noticeable way. Next, the effect of adding some derivative type

symmetry breakers for the pseudoscalars is described in section V. This doesn't much change the overall rate but

modifies the somewhat delicate cancellations so that the scalars end up making a larger percentage contribution than

before. In low energy calculations of this sort one always may expect some contributions from the vector mesons.

This is discussed in section VI where it is shown that, although there is a new type of diagram the vectors do not

produce a big change in the previous results.

Section VII contains a discussion of the results and directions for further work. For the convenience of readers,

material describing the chiral Lagrangian used is brought together in Appendix A. Similarly the detailed expression

for the decay amplitude is given in Appendix B.

II. HISTORICAL BACKGROUND ON THE r 4 37r DECAY

The study of ? - 3w has turned out to be surprisingly complicated and correspondingly important for understanding

the non-perturbative (low energy) structure of QCD. Chiral dynamics in various forms has been the basic tool. Since

the process violates G-parity it was initially assumed to be of electromagnetic nature, mediated by an effective photon

exchange operator proportional to the product of two electromagnetic currents. The old "current algebra" approach

had previously predicted the KL - w+,-o spectrum shape [33] to be

1 - 2E (1)

where m is the KL mass and Eo the energy of the 70 in the KL rest frame. This shape, which is in reasonable

agreement with experiment, resulted from the vanishing commutator of the axial charge transforming like a n+ with

the appropriate product of two weak currents. When Sutherland [34] repeated this type of calculation for ij -+ w+lr-hrO

with the product of two electromagnetic currents he found that the decay amplitude was actually zero (to this leading

order). Thus the sj -4 3w decay did not seem to be mediated by a virtual photon emission and reabsorption. In fact,

it was found [35] that a quark scalar density operator with the -XI = 1 property proportional to

fu - dd (2)

would give a non-zero result for the decay rate. A more detailed treatment [36] showed that the quark density operator

gave the same spectrum for rj -* +7-rO as in Eq.(1) with m the ij mass in this case. Such a result is in fairly good

agreement with experiment. The scalar density interaction in Eq.(2) was recognized [37] to be the fundamental

up-down quark mass difference generated by the Higgs meson in the electroweak theory.

However, the predicted rates of the 77 -+ 7+7--O and T/ -* 3w0 modes (both the ratios and the absolute values)

did not agree well with experiment at that time. Some years later, after more precise experiments, the ratio of the

rates for w+-7O to 3w0 modes stabilized around the value expected from isospin invariance. On the other hand the

absolute rate has only recently stabilized to a value notably larger than that predicted by theory. The theory behind

the current algebra results could be economically presented in the framework of an effective chiral Lagrangian. For

most low energy processes where the scheme could be expected to work, the tree level computation did produce results

within 25 % or so of experiment. Thus the relatively poor prediction for r) -+ 3w at tree level is somewhat surprising.

An improvement was obtained by Gasser and Leutwyler [29] who carried the computation of the chiral Lagrangian

amplitude to one loop level. Since the non-linear chiral Lagrangian is non-renormalizable, this required the addition

of new counterterms. Their finite parts were new parameters which could be mostly determined from other processes.

They obtained the result F(77 -+ +r- ) = 160 + 50 eV which may be compared with the present experimental

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Abdel-Rehim, Abdou; Black, Deirdre; Fariborz, Amir H. & Schechter, Joseph. Effects of Light Scalar Mesons in eta -> 3pi decay, article, October 1, 2002; Newport News, Virginia. (digital.library.unt.edu/ark:/67531/metadc741604/m1/2/: accessed November 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.