(adsbygoogle = window.adsbygoogle || []).push({}); It is also important to note that the relevance is different in different parts of the Standard Model. Electroweak processes converge much more quickly than QCD processes. Urs Schreiber said: ↑What is the highest loop order in standard model scattering computations that still contributes a measurable effect seen in past and present particle collider experiments?

In other words, to which order are loop corrections necessary for accounting for observed high energy physics?

I expect it is order-1 for most computations and order-2 in some rare cases, and not higher. Or are there notable exceptions? What would be a good reference to check this?

As noted in previous posts in this thread, state of the art QED calculations get you to 13 orders of magnitude precision results with five loops and experimental measurements can measure QED observables to comparable degrees of precision. Two loop calculations in QED are accurate to about a factor of about 10^{-3}(better than four loop calculations in QCD).

Leading order results in QCD are accurate to a factor of 2. An NLO result in QCD (i.e. 2-loop) gets you a result with single digit percentage precision, an NNLO result in QCD (i.e. 3-loop) gets you a result with 1% precision or so (or worse, e.g. this NNLO result has 3% precision), and an NNNLO result in QCD (i.e. 4-loop), which is about as good as it gets for most purposes in QCD, gets you to perhaps 0.2% to 0.5% precision (for example here). This may be generous. As noted in a 2015 review paper:

As of 2016, five loop calculations in QCD were vaporware for anything other than the beta function of the QCD coupling constant. For example, the dilepton invariant mass distribution has been measured for masses between 15 and 2000 GeV, covering more than 8 orders of magnitude in cross section. NNLO QCD predictions, together with modern PDF sets and including higher-order electroweak and QED final-state radiation corrections, describe the data to within 5–10% over this large range, whereas NLO predictions show larger deviations, unless matched to a parton shower.

In QCD, the limiting factor is primarily computational capacity because QCD calculations converge much, much more slowly with higher order terms having more relevance to the outcome. Even if you could do 10-loop calculations in QCD (which is far beyond any reasonable near term realm of possibility, but extrapolating how much gain has been obtained from additional loops so far) the precision you get would be comparable to perhaps 3-loop calculations in QED.

We can do measurements of observables in QCD that are thousands and even millions of times as precise as our theoretical computations in some cases. For example, we can compute the proton mass from first principals in QCD to more than 1% precision but less than 0.1% precision (it was about 2% in 2008), but can measure it experimentally to eleven or twelve significant digits. (The far more obscure bottom lambda baryon is still described experimentally to five significant digits, while Delta and Omega baryon masses are computed from first principles to about 0.2% precision.).

On the other hand, there are other observables (like the strength of the QCD coupling constant at high energies also reviewed here) where the precision of the experimental measurements can be as bad as low single digit percentages. Poor precision in the measurement of this experimental constant is another reason that QCD calculations are so imprecise.

Similarly, while in principle, you can compute parton distribution functions in QCD from first principles, in practice, in the real world, physicists always use experimental data fitted to hypothetical functional relations motivated by but not rigorously derived from pure QCD equations and fundamental constants. (The handbook for beginners in the subject from CERN available online runs to something like a 194 pages of detailed discussion of the sourcing of current PDF data sources and the fine points of the pros and cons of different sources.)

Weak force calculations are in between, mostly because imprecision in experimental measurements of the relevant physical constants limits the accuracy of any weak force calculation to about five significant digits (see the Particle Data Group data on W bosons and Z boson) no matter how many loops you take theoretical calculation out to, with any additional precision being spurious. This is the main area in the Standard Model where you wouldn't do calculations to as many loops as you had the computational capacity to do in a reasonable time because more precise theoretical calculations wouldn't be relevant. Also, weak force calculations often have some QCD influences and the recommended practice in considering what are primarily weak force calculations is to include at least NLO (2-loop) QCD calculations even though QCD contributions are a minor second order factor.

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# A Highest loop order of experimental relevance?

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