In quantum field theory, the Lagrangian density describes the dynamics of a field and its interactions. The Lagrangian is typically divided into a kinetic energy term and a potential energy term. The kinetic energy term corresponds to the terms in the Lagrangian that involve derivatives of the field, while the potential energy term corresponds to the terms that depend on the field itself.
The absence of a mass term in the kinetic energy term of the Lagrangian for certain fields is related to the nature of those fields. Fields that do not possess an intrinsic mass, such as the electromagnetic field or the gluon field in quantum chromodynamics (QCD), do not have a mass term in their kinetic energy.
This absence of a mass term arises from the underlying symmetries and gauge invariance of the theory. Gauge symmetry is a fundamental principle in many quantum field theories, and it requires that the gauge fields (such as the electromagnetic or gluon fields) be massless. Introducing a mass term for these fields would break the gauge symmetry of the theory.
On the other hand, fields that do have an intrinsic mass, such as scalar fields (e.g., the Higgs field) or massive fermionic fields, do have mass terms in their potential energy. The presence of these mass terms arises from the interactions of the field with the Higgs field or other mechanisms that generate mass.
It's worth noting that the Higgs mechanism itself provides a way for particles to acquire mass. In this mechanism, the Higgs field interacts with certain fields, giving them mass. This includes not only the Higgs boson but also other elementary particles that acquire mass through their interactions with the Higgs field.
In summary, the absence of a mass term in the kinetic energy of certain fields in quantum field theory is a consequence of the symmetries and gauge invariance of the theory. Fields that do not possess an intrinsic mass, due to gauge symmetry requirements, do not have a mass term in their kinetic energy, while fields with an intrinsic mass have mass terms in their potential energy.