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A Highest loop order of experimental relevance?

  1. Sep 12, 2017 #21

    ohwilleke

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    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.

    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.

    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.
     
    Last edited: Sep 12, 2017
  2. Sep 13, 2017 #22

    Urs Schreiber

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    Thanks. Might you have a pointer to a theorist saying this? Might there be a printed record of this saying?
     
  3. Sep 13, 2017 #23

    Urs Schreiber

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    Thanks for all the detailed additional pointers! That's useful.
     
  4. Sep 13, 2017 #24

    Vanadium 50

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    I don't think there are any, because there are counterexamples to these "rules of thumb." I would say the rules of thumb are, a) each higher order makes a contribution smaller than the previous order, and b) an estimate of the uncertainty due to uncalculated higher orders is the variation in the quantity you are calculating on the renormalization scale (since that's unphysical, at infinite order it must be zero).

    The counterexample I am thinking of is heavy flavor production. The NLO contributions are about the same size as the LO contributions, and the scale dependence is actually worse at NLO than LO.
     
  5. Sep 13, 2017 #25

    mfb

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    Theorists wouldn't calculate higher orders if they wouldn't expect this.
    I'm not aware of explicit statements like this, but the whole idea of calculating higher orders is based on that expectation.
     
  6. Sep 13, 2017 #26

    Urs Schreiber

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    Isn't it a little more subtle than this makes it sound? Because as the order goes to infinity, the perturbation series is not to be expected to approximate the true physical value, but to be infinitely far from it! The perturbation series of realistic QFTs is expected to be divergent (this goes back to Dyson 52.)

    Now for a general divergent formal power series, even summing up the first few terms makes no particular sense. But since we may assume that the perturbation series is the Taylor expansion of an actual smooth function (namely the non-perturbative theory) it is plausible to expect that it is, even if not convergent, an "asymptotic series". If so, this sort of guarantees that the first few terms (depending on "how small Plnack's constant really is") give a good approximation, but it still means that beyond that the series will diverge arbitrarily far from the desired physical value.
     
    Last edited: Sep 13, 2017
  7. Sep 13, 2017 #27

    Urs Schreiber

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    All right, thanks for finally saying that this is your reasoning!
     
  8. Sep 13, 2017 #28

    Vanadium 50

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    I dunno. It's a counterexample, after all.
     
  9. Sep 13, 2017 #29

    Urs Schreiber

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    Ah, sorry about this, now I see that I misread what you wrote, actually you are saying precisely what I am after. Okay so your hint is:

    Could you point me to a good reference for this?
     
  10. Sep 13, 2017 #30

    Urs Schreiber

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    By the way, the discussion here reminds me of the following quotes from the (very nice) review of asymptotic perturbation series theory in Suslov 05 (please take this in a good spirit, I don't mean to bug anyone):

     
  11. Sep 14, 2017 #31

    Vanadium 50

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    The original NLO paper was (Paulo) Nason, (Sally) Dawson, and (R. Keith) Ellis, around 1989. It builds on a paper a few years earlier by (John) Collins, (Dave) Soper and (Jack) Smith where they derive the relevant factorization theorems. Matteo Cacciari was giving talks about LO, NLO and the state of the art about ten years ago; if you find a conference proceedings by him that references one or both of the above papers, that's probably as good as you are going to get in one place.
     
  12. Sep 14, 2017 #32

    Urs Schreiber

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    Thanks. Maybe slide 12 in
    • Matteo Cacciari: "(Theoretical) review of heavy quark production" BNL 14/12/2005 (pdf)
    has the kind of statement that you are referring to.
     
  13. Sep 14, 2017 #33

    Vanadium 50

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    I think the slides as a whole give a reasonable view of the heavy flavour state of the art. Slide 5 is a motivation for NNLO (and why N3LO may play only a minor role).
     
  14. Sep 14, 2017 #34

    Urs Schreiber

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    Right, sorry, I meant slide 12 (I was pointing somebody else to slide 5 for another reason, and mixed up the numbers when writing here).

    I am trying to pinpoint the statement which you were referring to above when you wrote:

     
  15. Sep 16, 2017 at 5:20 PM #35

    Vanadium 50

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    I'm sorry, but that's a little unfair. "Here's an article I found - why can't I find a statement you made in it?"

    I think I did a pretty good job of pointing you in the right direction, but it may well be that a single document that has everything you want doesn't exist. But if a literature search needs to be done, I don't think I am the one who needs to do it.
     
  16. Sep 18, 2017 at 2:57 AM #36

    Urs Schreiber

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    There is some misunderstanding here. But never mind.
     
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