Quantum field theory is the framework that combines quantum mechanics and special relativity by describing particles as excitations of underlying fields. It is important because it forms the basis of modern particle physics and helps us to understand fundamental forces, elementary particles, and many phenomena in high-energy and condensed matter physics.

However, the intuitive formulation of this framework produces ultraviolet divergences when fields are evaluated at arbitrarily short distances, or equivalently at high momenta. Thus, renormalisation is necessary to have finite predictions. These divergences do not simply signal a failure of the theory; rather, they show that the parameters and operators appearing in the Lagrangian are not directly the finite physical quantities measured at a given scale. The renormalisation procedure reorganises the theory by expressing bare quantities in terms of finite, scale-dependent renormalised quantities. The dependence on the renormalisation scale then becomes physically useful: it describes how the effective description of the theory changes when we probe different energy or length scales.

Composite operators provide an additional and important layer of this problem. A composite operator is a local product of fields and derivatives. Since the fields are multiplied at the same spacetime point, correlation functions with composite operator insertions contain new short-distance singularities which are not removed by the renormalisation of the elementary fields and other quantities alone. Therefore, composite operators require their own renormalisation. In general, a renormalised composite operator is not simply proportional to a single bare operator, but can mix with all operators with the same quantum numbers and compatible dimension. This operator mixing is naturally described by a renormalisation matrix. We determine this renormalisation matrix with a perturbative approach.

From this renormalisation matrix we calculate an important object: the anomalous-dimension matrix that describes how composite operators evolve under changes of scale. Diagonalising this matrix identifies special linear combinations of operators which have definite scaling behaviour. These eigenoperators are the natural objects near a fixed point, where correlation functions obey power laws. Their scaling dimensions determine whether perturbations are relevant, irrelevant, or marginal, and they encode universal information about the theory. Thus, the computation of anomalous dimensions of composite operators is not only a technical renormalisation step but also a way to extract the physical spectrum of eigenoperators and the critical behaviour of the theory.

We renormalise some composite operators in the phi-cubed and phi-four theories. Studying these scalar theories is interesting because they are simple but powerful toy models that reveal important features of quantum field theory. We perform direct perturbative computations up to next-to-leading order, which in the cases considered mostly amounts to one-loop computations. We identify the corresponding diagrams of correlation functions with composite operator insertions and organise the required Z factors into renormalisation matrices. We compute the anomalous dimension matrices associated with these renormalisations and, by diagonalising the renormalisation flow, we determine these new operators.