Effects of Light Scalar Mesons in eta -> 3pi decay Page: 4 of 17
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The widths are then, defining 0+- = L(- 7+-rio) and rooo = r(n - 3qO):
ro+- = 64 mJ dEidE2Mo+-4 ,
1ooo = dE1dE2MN00o 12 (9)
Using CLO, with parameters determined as described above, we get the tree-level results [from the first terms in
each of Ma'i' in Eqs.(B1)]
ro+- = 106eV,
1700 = 1.40. (10)
These may be compared with the experimental results 
(o+-)eXpt = 267 25eV,
( 1000 ) = 1.40 0.01, (11)
which demonstrate the disagreement with experiment for the overall rates in the simplest model. However the width
ratio has about the correct magnitude. The related energy spectrum is also about the correct magnitude. The squared
matrix element is usually described by quantities a, b and c defined from
IMoK+- oc (1 +aY+bY2 +cX2...), (12)
with X = (E2 - E3) and Y = 3 (E1 - 7mn,) - 1. In the present paper we shall not take into account the
(not completely negligible) kinematic 77 - ;+ mass difference. See  for a discussion of this point. The predictions
from this simple model, a v -1 and b ~ 0.25 are similar to the experimental results  aexp = -1.19 + 0.07 and
bexp = 0.19 0.11 with c=0.
It is of some interest to also give the predictions for the q' - 3, decay process at tree level using the simple
Lagrangian Eq.(4). It just is necessary (see Appendix B) to replace cosO, by sinO, and m, by m,, in Eq.(5) to get
for the n' -* 0+i- matrix element:
11+-(El, E2, Es) 1 6 sin,(1 - 2Ei.). (13)
This leads to the predictions for the 0' modes:
Lo +- = 497 eV,
Fooo = 562 eV. (14)
The experimental results are given as 
(o+-)Cxp < 10- 4
(rooo)exp = 315 56 eV. (15)
Only the 370 mode has really been measured; its width is smaller than predicted in the simple model just presented.
One would, of course. expect better agreement for the low energy process i - 3; for which chiral perturbation theory
should be more clearly reliable- In the present paper we shall just make a few remarks on this more complicated
It may also be worthwhile to give a rough estimate of the corrections to the rates corresponding to violations of
Dashen's Theorem mentioned earlier. If we parameterize the electromagnetic contribution to the K+ - Ko mass
(mo - mi ) = f (mo - mn'), (16)
where f = 1 corresponds to Dashen's Theorem, we would find by using Eq.(3) that the s7, 7' -4 3r rates predicted for
1LO should be multiplied by
(mno - m% ) - f(M1.o - m7+) ( 17)
(M2 - mK_ ) - (MI - m2,)j
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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/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.