Short answer: we do not know.
Our goal in this blog is to show that divergences in Feynman integrals are an artifact of the approximation method which is used. A proper approximation method does not need any regularization nor renormalization.
We believe that divergences in Feynman integrals are not an indication of new physics at high energies or short distances. Thus, we believe that the concept of an effective field theory is wrong in this case.
However, there is a much harder mathematical problem in nonlinear differential equations in general. It is almost impossible to prove anything about them. Coupled fields typically form a nonlinear differential equation.
Sergiu Klainerman and his coauthors have been able to prove existence theorems for solutions in certain cases. Typically, one assumes that the fields differ from flat very little (small data or small perturbation), and one can then show the existence of a solution for at least a short period of time.
Sidenote: In this blog we have the hypothesis that Einstein equations do not have solutions for any realistic case, because the equations are too "strict". They are not flexible, like the equations of a rubber sheet are.
Demetrios Christodoulou and Sergiu Klainerman in 1994 were able to prove the stability of the Minkowski space under "small" perturbations. We believe that under large perturbations, no solution exists. Sidenote ends.
If it turns out impossible to prove the existence and smoothness of solutions for coupled field equations, is that an indication of new physics lurking at very short distances?
No. Even if we discretize physics at very short distances, we still face the problem of proving that large amounts of extreme frequencies cannot appear in a physical process. How to prove that if we input visible light into a physical process, no 1 GeV quanta are produced? The proof may be even harder if we assume that physics is discrete at the Planck length.
What about discretizing physics at many scales, like in quantum mechanics? A wave is represented by a quantum whose size depends on the frequency? That is a better idea, but we face the problem of accelerating emitters whose wave does not seem to conform to simple quantization.
It may be that nonlinear differential equations in nature are well-behaved. We just lack the mathematical ability to prove that. Then there is no need for new physics at all.
This discussion has similarity to various incompleteness theorems proved by Kurt Gödel. For example, we cannot prove the consistency of the Peano arithmetic. Some day, a correct proof of 0 = 1 may appear in a mathematical journal.
Is Gödel's incompleteness an indication of new physics lurking behind the veils? Maybe the universe is a finite state machine and no true Peano arithmetic is needed at all?
It may be that the Peano arithmetic is consistent. Then there is no need for new physics in that case.
The philosophy of effective field theories is that at short distances some new physical law saves us from mathematical uncertainty. The problem in this philosophy is that we cannot know if our inability to prove something is an indication of any physical problem.
If there were mathematical certainty that our theory is inconsistent at short distances, then we would know that there must be new physics. For example, mini black holes of the Planck mass certainly present challenges to field theories.
Conclusions
It may be that coupled field equations are well-behaved in general. We just cannot prove that. Then mathematical problems in them do not require any new physics at short distances, or at any distance.
The problem of divergences in Feynman integrals is probably much easier to solve than the general problem of well-behavedness of nonlinear differential equations.
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