Problem with source term in qft

[DrDu]

I have a problem understanding the effect of source terms in qft.
I am interested in understanding how a static point source will interact with a
massless field.
Let me consider a simple example of a linear chain of masses coupled by some springs.
In the continuum limit, the Lagrangian of that model will only depend on the derivatives of the displacement of the masses and not on the absolute displacements. In qft such a field is called "massless". Now a scalar source term e.g. $$\delta(x) \phi(x)$$ corresponds to a force acting on one mass point on the chain. Obviously, this will lead to constant acceleration of the system (or of ever growing parts of it). The Hamiltonian will fail to be bounded from below.
Apparently, adding a source term leads to a catastrophe. This will happen also for the discrete chain, so it isn't a problem of renormalization.
I guess this is what is called an infrared catastrophe as it involves the excitation of waves of longer and longer wavelength. How is this controlled in qft?

 

[DrDu]

I shall try to give an answer myself: Probably by demanding that the integral over space of the source density vanishes, i.e., considering only over-all neutral systems.
However, e.g. in qft people write down Lagrangians for the interaction of electrons and photons without caring about the positive counter-charges.