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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?

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

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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.

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    $\begingroup$ @AlmostClueless It depends on what you mean by "the problem." I thought the question was motivated by statements about non-existence of $S$ and other issues like that, so I pointed to lattice QFT as a way of making things well-defined. Doing calculations in a reasonable amount of time given current computing resources is a different problem, and you're right: our ability to finish calculations in a reasonable amount of time still relies on perturbation theory in many cases. Is the question about what we would do computationally if both lattice QFT and perturbation theory were too cumbersome? $\endgroup$
    – Chiral Anomaly
    Commented Jun 13, 2021 at 17:18
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    $\begingroup$ @AlmostClueless If you're asking about the precision/accuracy of the theory itself, then lattice QFT is plenty good enough. I mean, for any given model (like QCD), we can make the lattice spacing so fine (say $10^{-1000}$ times the Planck scale) that it won't have any noticeable effect in real experiments. Other limitations of the model, like the fact that QCD doesn't account for gravity, will rear their heads long before lattice artifacts do. But again, this is separate from the issue of finite computational resources -- our ability to finish the calculations in a reasonable amount of time. $\endgroup$
    – Chiral Anomaly
    Commented Jun 13, 2021 at 17:27
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    $\begingroup$ @AlmostClueless A transient comment asked for a good primer... I assume the requirements are (1) you can download it for free; (2) it's well-written, so you can spend your time learning instead of decrypting; (3) it introduces lattice QFT as a way of defining QFT nonperturbatively, instead of focusing only on computational/numerical issues; and (4) it doesn't assume that you're already an expert in other formulations of QFT. Those are tough requirements, but I'll look. :) $\endgroup$
    – Chiral Anomaly
    Commented Jun 13, 2021 at 18:10
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    $\begingroup$ @AlmostClueless Sorry I've been slow to respond... I haven't spent much time looking yet, but I haven't forgotten. The best book I've found on the subject is Montvay and Münster's Quantum Fields on a Lattice, but "best" is relative. It doesn't satisfy (1) at all, and it only partly satisfies (4). But it's pretty good at (2) and halfway-decent at (3), at least for scalar fields and gauge fields. (It over-complicates the approach to fermion fields, IMO.) Finding a better intro is still on my to-do list, but it might take a while. My standards are high. $\endgroup$
    – Chiral Anomaly
    Commented Jun 17, 2021 at 15:37
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    $\begingroup$ @AlmostClueless For a concise introduction to the basic idea of lattice gauge theory, chapter 4 in Tong's Lectures on Gauge Theory looks pretty good. $\endgroup$
    – Chiral Anomaly
    Commented Jun 26, 2021 at 22:01

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