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

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3

value [38] r (ri - 7-+- o)e = 267 25 eV. The extra effects included involve both the implicitly integrated-out

heavier meson exchanges and partial unitarization to one loop order. One might expect a two loop calculation in

the chiral perturbation scheme to be valuable but this may involve too many unknown parameters at the present

stage- A dispersion approach using the Gasser-Leutwyler result as a subtraction gave an improved estimate [39]

F (--* w+7-wo) = 209 20 eV, which still seems too small.

A possible source of ambiguity arises from the determination of the coefficient of the driving scalar density interaction

in Eq.(2). This is determined from the K0 - K+ mass difference, which in turn has two components

m2(K0) - n2(K+) = [m2(K0) - m2(K+)] kmass + [m2(K0) - m2(K+)]j (3)

corresponding to the quark mass differences and the virtual photon emission and reabsorption diagrams respectively.

The latter is given in the chiral limit by m2(w0) - m2(w+) according to Dashen's theorem [40] and the reasonable

assumption that the photon contribution saturates the pion mass difference. A number of authors [41] have argued

that there are important corrections to Dashen's theorem which have the effect of boosting the rj -* 37 decay rate.

If one questions Dashen's theorem it is natural to also question Sutherland's result, which deals with the direct

electromagnetic contribution to 77 -> 37 . An investigation of this point yielded [42] the estimate that there was only

about a 2% change arising from this, although it decreased rather than raised the rate.

Still another point which may repay further investigation concerns the possible subtleties arising from 77 - rj' mixing.

An understanding of the 17' --+ 3w process, for example. might clarify this point. This process has been treated by

some authors [43], [44] in the literature but has received only a fraction of the attention given 7 37 .

In the present paper we will focus on learning more about the putative nonet of light scalar mesons by studying

their contribution to r< -* 3w.

II. CHIRAL SYMMETRY RESULTS TO LOWEST ORDER

For comparison, we first present the well-known results when only the terms present in the lowest order chiral

Lagrangian of pseudoscalars are kept.

FO = F Tr (OUOuUt) + 6'Tr [M (U + Ut)] + ln detY) (4)

where the last term [see Eq.(A14) and comments there] supplies mass to the SU(3) singlet state and Mk4 is defined in

(A11). Fitting LLO to the experimental masses determines 6' = Frn/8.

The j - w+7o amplitude receives, in this approximation, contributions from diagrams (a), (b) and (c) of Fig.1,

which are given in Eq.(Bl) (with the non leading corrections deleted). To a reasonable approximation which displays

the key dependences these sum up to the lowest order result for the 7 -+ 7r+r-ora amplitude

Mo+-(E1, E2, E3) 1 l6t ycosoP(1 - ,). (5)

Here El is the 7r' energy in the 7 restframe and y is the dimensionless parameter in Eq.(A11) which measures the

isospin violation in the quark mass matrix. Assuming Dashen's theorem, Eq.(4) yields

8S'y = F; (mo - m2, - m1 o + m+), (6)

which allows us to solve for y. Furthermore Q, is the "nonstrange-strange" pseudoscalar mixing angle defined in

Eq.(A15); it is generally taken to be about 371. It is related to the "octet-singlet" angle, 0 by

cos9P - cosO - vhsino ()

Then Eq.(5) agrees with Eq. (1.14) of [29] except that they neglected 77 - 7j' mixing by replacing cosO, -+ 1/0' in

what was denoted the current algebra formula. The matrix element for l - 3r0 is given in general byMooo = Mo+-(EiE2,Es) + Mo+-(E2,E E3)+Mo+-(E3,E2,E). 8

(8)

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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/m1/3/: accessed November 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.