Asymptotic Series in QFT: What to do when all "trustworthy" terms are known?

Question

In my Introduction to QFT lecture, we quantized a Klein-Gordon Field and as a toy model we looked at $\phi^3$ theory. For this toy model we expanded the $S = U(-\infty, \infty)$ operator in a series (using Wick's theorem) and then where able to compute observables like scattering cross-sections and decay widths.

I have read a little about the non-existence of the operator $S$ and some other mathematical problems concerning QFT. And for now I tend to accept most of these problems and learn more about them later. But there is one point that really bothers me. I have read that the series expansion of the $S$-matrix is actually "asymptotic" hence the radius of convergence is $0$. The argument what usually follows is that up to some finite order the series still approximates the actual solution$^1$. Okay lets assume that this is actually the case. What is the future of the Standard Model of Particle Physics when all "trustworthy" terms are known? Or even worse: when do we know that the terms are not trustworthy anymore? What if the theory differs from a really precise experiment in the future -- can we distinguish new physics from effects that come from the divergence of the expansion?

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$^1$ Is there actually a proof of this statement?

Answer 1

This is only a problem when we're trying to use a perturbation series as a substitute for a nonperturbative definition of the theory.

For many quantum field theories, including QED and QCD, we have perfectly healty nonperturbative definitions based on lattice QFT. In lattice QFT, the challenge of finding a nonperturbative definition is replaced by the challenge of proving the existence of a nontrivial continuum limit, but that usually doesn't matter for physics because most QFTs aren't meant to be Theories of Everything. When people say things like "QED doesn't exist," they're referring to the non-existence of a nontrivial continuum limit. That's an issue for aesthetics, but it's not a problem for physics.

As of today, I don't think anybody has explicitly given a nonperturbative definition of non-abelian chiral gauge theories like the (various) Standard Models. The obstruction seems to be a matter of difficulty/creativity rather than impossibility, though. A nonperturbative definition of anomaly-free abelian chiral gauge theory was given over two decades ago in arXiv:hep-lat/9811032 using lattice QFT, and an impressive argument for the existence of a nonperturbative definition of the (various) Standard Models was reported in arXiv:1809.11171, again based on lattice QFT.

At worst, maybe a nonperturbative definition of the Standard Model requires viewing it as a low-energy approximation to a Theory of Everything. I think that's an unlikely scenario, but even if it turns out to be true, it still solves the problem.

For perspective: According to estimates reviewed in another answer, we are not even close to having calculated all of the trustworty terms in the Standard Model perturbation series. Discrepancies with experiment are (happily!) already starting to emerge, and they're probably not due to the nonconvergence of the series. Still, a nonperturbative definition would have other benefits, including clarity: mathematically-clear definitions can be a great source of intuition. The paper arXiv:1705.06728 is one of many examples of this benefit.