A beam machining head for beam cutting of a workpiece, with an interface for an energy beam source (14; 14, 201) for generating a focused machining energy beam (15; 206) selected from a particle beam source, a fuel fluid beam source, a plasma beam source and/or a source for electromagnetic radiation; having an exit opening for the machining energy beam bounded by an opening edge; an optical detector unit for recording at least one image of an electromagnetic radiation emitted from the workpiece through the exit opening into the beam machining head (10; 100; 200), which is induced in the workpiece by the machining energy beam and is a process light; and a monitoring unit connected to the optical detector unit in a data-transmitting manner for monitoring a positional relationship between a centre of the emitted electromagnetic radiation and the exit opening, wherein said monitoring unit has a first determination module for determining at least one position (180; 182) of the exit opening in the at least one image; a second determination module for determining at least one position of the centre of the emitted electromagnetic radiation in the at least one image; and a third determination module for determining the positional relationship between the at least one position of the centre of the emitted electromagnetic radiation and the at least one position (180; 182) of the exit opening; characterized in that the second determination module is configured to determine the position of a centre of highest energy density of the emitted radiation as the position of the centre of the emitted electromagnetic radiation. with an interface for an energy beam source (14; 14, 201) for generating a focused machining energy beam (15; 206) selected from a particle beam source, a fuel fluid beam source, a plasma beam source and/or a source for electromagnetic radiation; having an exit opening for the machining energy beam bounded by an opening edge; an optical detector unit for recording at least one image of an electromagnetic radiation emitted from the workpiece through the exit opening into the beam machining head (10; 100; 200), which is induced in the workpiece by the machining energy beam and is a process light; and a monitoring unit connected to the optical detector unit in a data-transmitting manner for monitoring a positional relationship between a centre of the emitted electromagnetic radiation and the exit opening, wherein said monitoring unit has a first determination module for determining at least one position (180; 182) of the exit opening in the at least one image; a second determination module for determining at least one position of the centre of the emitted electromagnetic radiation in the at least one image; and a third determination module for determining the positional relationship between the at least one position of the centre of the emitted electromagnetic radiation and the at least one position (180; 182) of the exit opening; characterized in that the second determination module is configured to determine the position of a centre of highest energy density of the emitted radiation as the position of the centre of the emitted electromagnetic radiation.