Do Weak Interactions Become Strong at High Energy?

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Weak coupling theories in a semiclassical approximation lead to rapidly growing cross-sections coming from multiparticle production. In these lectures, I discuss two circumstances where such an approximation provides a sensible starting point: B + L violation in the standard electroweak theory and high order estimates in λφ4. Even though a rapidly growing cross-section may obtain for a limited energy range, such behaviour eventually must level off. Such a leveling off is hinted at by studying corrections to the semiclassical limit, and a general formalism for performing these corrections, due to Khlebnikov, Rubakov and Tyniakov, is explained. Because no closed form of the correction exists, however, the situation remains unclear. For the λφ4 theory, because of its Borel summability, one can show that the total cross-section at weak coupling is exponentially small. For the electroweak theory, however, a similar strong assertion cannot be made. Nevertheless, unitarity arguments, emphasized particularly by Zakharov and by Maggiore and Shifman, strongly hint that also B + L violation in the standard model will not grow strong at high energy, reaching at most a cross-section of order \(\frac{1}{s}{{e}^{{ - 2\pi /{{\alpha }_{\omega }}}}}\). Some of the caveats surrounding these conclusions are emphasized.