Comment on theππγContribution to the Absorptive Amplitude forKL→μμ¯

The contribution of the $\pi \pi \gamma$ intermediate state to the absorptive part of the $K_L\to \mu \overline{\mu }{}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ amplitude is reevaluated by allowing for more general dependence of the $K_L\to \pi \pi \gamma$ invariant amplitude on the relevant Mandelstam variables than that considered in previous work. Allowance is also made for the corrections to the $K_L\to \pi \pi \gamma$ rate, which arise due to the dependence of the experimental detection efficiency on the nature of the photon spectrum. For a wide range of variation of the behavior of the $K_L\to \pi \pi \gamma$ amplitude, it is found that the $\pi \pi \gamma$ contribution to the rate of $K_L\to \mu \overline{\mu }$ decay (relative to that of the $2\gamma$ state) is less than 2-4%, if the pion electromagnetic form factor may be represented by ${(1-\frac{s}{{m_{\rho }}^2})}^{-n}$, $n\le 3$. Some pathological situations are also considered; in these cases the contribution can still be bounded by ten or fifteen percent as overestimated upper bounds.