 Maximum laboratory deflection in relativistic elastic scattering

ID: maximum-laboratory-deflection-in-relativistic-elastic-scattering

m_{1}

elastically scattering from a stationary target of mass

m_{2} < m_{1}

has the displayed maximum laboratory angle. In

c = 1

units, let its incident energy be

E > m_{1}

and

s = m_{1}^{2} + m_{2}^{2} + 2 m_{2} E

. Boosting the outgoing four-momentum from the center-of-momentum frame gives

tan θ = sin φ / [Γ (cos φ + B)]

, with

Γ = (E + m_{2}) / s

and

B = (m_{1}^{2} + m_{2} E) / [m_{2} (E + m_{2})] > 1

. The maximum at

cos φ = - 1/ B

has

tan θ_{m a x} = 1/ [Γ B^{2} - 1] = m_{2} / m_{1}^{2} - m_{2}^{2}

. Equality is allowed; a strict upper bound would exclude a realizable outgoing direction. The angular cap is independent of incident energy.

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