The beam or power gain of a radio antenna is nothing but directional gain, the gain that varies toward different direction written as $G(\theta,\phi)$. It is normalized to be $1$ at the center of the field of view of an antenna.

So beam can be defined as the direction-dependent gain of a directional antenna with respect to the gain of an isotropic antenna:

$$ G(\theta,\phi) = 10 \log_{10} \frac{G}{G_i} $$

where $G_i$ is the gain of an isotropic antenna. This value is given in units of decibel (dB). By definition, the gain integrated over the surface of a sphere has to be unity:

$$ \int_{sphere} G \ d\Omega = 1 $$

which gives rise to the definition of beam solid angle:

$$ \Omega_A = \frac{4\pi}{G_0} $$

where $G_0$ is the peak gain. For example, the gain pattern or radiation pattern of a short dipole

$$ G = \frac{3}{2} \sin^2\theta = G_0 \sin^2\theta $$

which can be derived from the formula of power for a dipole antenna. Here $G_0=3/2 = 1.5 = 1.76$ dB is the peak gain because this would be the gain at $\theta=90^\circ$.