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Abstract
This thesis describes an experimental study of the decay $K\sp+\to\mu\sp+\nu\sb{\mu}\gamma,$ which can proceed via two distinct diagrams. The first, called Inner Bremsstrahlung, is simply a radiative version of the familiar $K\sp+\to\mu\sp+\nu\sb{\mu}$ decay. Its branching ratio can be readily calculated in the Standard Model, and it has been well measured. The second diagram, called structure dependent, involves the emission of a photon from internal hadronic states, and is therefore much more difficult to calculate.
Observing decays due to this diagram would give important information on the structure of the $K\sp+$ and test models of low-energy QCD such as Chiral Perturbation Theory. Structure dependent decays may be distinguished from Inner Bremsstrahlung by the different spectra they produce, but to date, no structure dependent components have yet been observed. The previous best limit on the "SD$\sp+$" component was $BR({\rm SD}\sp+)\ <\ 3.0\ \times\ 10\sp{-5}.$</p>
Using the E787 experiment at Brookhaven National Laboratory, which was designed to search for $K\sp+\to\pi\sp+\nu\={\nu},$ we have observed a structure dependent component in $K\sp+\to\mu\sp+\nu\sb{\mu}\gamma$ for the first time. Our branching ratio measurement is $BR({\rm SD}\sp+)$ = (1.331 $\pm$ 0.120 $\pm$ 0.181) $\times\ 10\sp{-5}$ where the first error is statistical and the second systematic. This corresponds to a measurement of the associated form factors of $\vert F\sb{v}\ +\ F\sb{A}\vert\ =\ 0.165\ \pm\ 0.007\ \pm\ 0.011.$
