Local Observables for the Renormalized Mass in the Quartic Scalar Lattice Quantum Field Theory II: With Constant External Sources

We examine the behavior of the local position-space observables for the renormalized mass parameter, that were introduced in a previous paper, but in our case here extended to the case in which we have the presence in the theory of finite and non-zero external sources which are constant over the whole Euclidean lattice. We also introduce and test a new set of local observables, which eliminate the small dependence of the local observables on the momentum-space observables through the residue of the pole of the propagator, which was still present in the previous set of local observables.

We verify that the original local observables still work in the presence of non-zero external sources, when compared with the results obtained from the momentum-space propagator. In fact, we verify that in the presence of non-zero external sources all the local observables previously defined give the correct results for the renormalized mass parameter both in the symmetric phase and in the broken-symmetric phase. Further, we verify that the newly defined local observables work equally well, and therefore we establish the complete independence of the local position-space observables from the momentum-space measurements, thus eliminating the need for Fourier transforms.

Finally, we determine how the two main properties of the momentum-space propagator, namely the renormalized mass parameter and the residue of the propagator, depend on the value of the external sources. We find that the renormalized mass increases with the external source, while the residue of the propagator is not affected too much. We show that the changes in the renormalized mass parameter due to the external sources are largest near the critical point. Deep in the symmetric phase these changes decrease, until we reach the free theory, where they vanish altogether. Deep in the broken-symmetric phase they also decrease and seem to approach zero.