In quantum field theory, the Lagrangian density describes the dynamics of fields, and it is composed of the kinetic and potential energy terms. The kinetic energy term characterizes the field's behavior, while the potential energy term represents interactions and the potential energy associated with the field.
The absence of a mass term in the kinetic energy part of the Lagrangian is a consequence of the nature of relativistic field theories, which aim to incorporate special relativity. In special relativity, the energy of a particle is related to its mass and momentum by the famous equation E^2 = (mc^2)^2 + (pc)^2, where m is the mass, p is the momentum, and c is the speed of light.
For a field theory to be Lorentz invariant (invariant under Lorentz transformations), the Lagrangian density should be a Lorentz scalar. This means that the kinetic energy term should be quadratic in the field derivatives to have the correct Lorentz transformation properties. Introducing a mass term into the kinetic energy would break this Lorentz symmetry.
Instead, the mass term appears in the potential energy part of the Lagrangian density. The potential energy term typically takes the form of a self-interaction potential for the field, which determines how the field interacts with itself or with other fields. The mass term in the potential energy contributes to the mass of the particle associated with the field. It is through the potential energy term that particles acquire mass in quantum field theory.
In summary, the absence of a mass term in the kinetic energy of field Lagrangians in quantum field theory is due to the requirement of Lorentz invariance. The mass term, which contributes to the mass of the particle, appears in the potential energy part of the Lagrangian density.