Loop Corrections and Renormalisation
Up to now, we have only considered the lowest-order calculations of cross sections, for which it is sufficient to consider tree-level diagrams that do not contain any loops. Loop integrals typically give rise to infinities, which can be regularised by considering, for example, a cutoff in momentum space, as was discussed in Chapter 7. Another possibility of regularising the theory is by discretising space-time, amounting to a lattice formulation; see Equation (7.5). In both of these cases there exists a maximal energy (equivalent to a minimal distance). The parameters, like the coupling constants, masses and field renormalisation constants, will depend on this cutoff parameter, generically denoted by an energy or a distance a = 1/ . How to give a physical definition of the mass in terms of the full propagator and why field renormalisation is necessary was discussed in Chapter 9. For the renormalisation of the coupling constant, it is best to define the physical coupling constant in terms of a particular scattering process, as that is what can be measured in experiment. Alternatively, as these are strongly related, the physical couplings can be defined in terms of an amputated 1P I n-point function, with prescribed momenta assigned to the external lines, all proportional to an energy scale called µ . As an example consider the self-interacting scalar field, with a four-point coupling λ [see, for example, Equation (21.2)]. We define the physical four-point coupling constant in terms of the 1P I four-point function with the momenta on the amputated lines set to some particular value proportional to µ (the precise choice is not important for the present discussion). It is clear that this gives a function λreg(λ, µ, ). The dependence on other coupling constants and the mass parameters is left implicit.