# A Highest loop order of experimental relevance?

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1. Sep 9, 2017

### Urs Schreiber

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?

2. Sep 9, 2017

### Staff: Mentor

Some Higgs calculations are done at NNNLO to improve the accuracy. The calculations for the electron g-factor work with up to 5 loops (12672 diagrams) but that is not a collider experiment.

In general NNLO is strongly preferred where available, as often the theoretical uncertainties are larger than the experimental ones.

3. Sep 11, 2017

### Urs Schreiber

Thanks, mfb! Just the kind of reply that I was hoping for.

Here is a related question: Given that the perturbation series is non-converging and at best asymptotic, is there a general consensus or idea on the order up to which we have the right to expect the result to improve, before it starts diverging?

4. Sep 11, 2017

### Staff: Mentor

You could argue that we reached this point in low-energy QCD already, where perturbative approaches don't work.

Apart from that, we are far away from divergence. Here is an estimate - the QED contributions stop getting smaller at around n=430, where their contribution is 10-187.
The problem arises earlier in (high-energy) QCD, but still way beyond the reach of calculations.

5. Sep 11, 2017

### Urs Schreiber

Thanks once more.

Now I remember having seen that Physics.SE thread before (It seems I had even commented on it.)

I'd like to check where that estimate of order 430 comes from. The author Diego Mazón indicates that he thinks of it as the ratio $\pi/\alpha\,,$ so probably he has some estimate of small phases in mind? But is that how one should determine this number? I'd think it can be very subtle to determine the point where an asymptotic series starts going bad. In particular it is in general unrelated to the point where the contributions cease to decrease (if they ever do).

(For instance in $\sum_{n = 1}^\infty 1/n$ the contributions keep decreasing, and yet there is no point at which this can be truncated to give an approximation to anything.)

6. Sep 11, 2017

### Staff: Mentor

$\displaystyle \left(\frac {\alpha}{ \pi}\right)^n$ is simply the power in the expansion.
$(2n-1)!! \approx \sqrt{2n!}$ looks like the number of diagrams.

Both up to some numerical prefactors I don't know about.
I'm not aware of mathematical proofs that the series do indeed go towards the physical value in the range where contributions get smaller, but they do.

7. Sep 11, 2017

### Urs Schreiber

Sorry, how do you know that? What's a reference for the claim that you are thinking of here?

8. Sep 11, 2017

### Staff: Mentor

Well, the calculations agree with experimental results in the range accessible so far, and it is what you would generally expect from perturbation theory.

9. Sep 11, 2017

### Urs Schreiber

But that refers to calculations to some very small order, not calculation in the full range where the contributions get smaller. Or does it? Where is that computation to order 430 done?

10. Sep 11, 2017

### Staff: Mentor

We can't do computation to 430 orders, of course, but all these high orders are tiny (and that we know without explicit calculations). The 5th order in QED for g-2 is 6*10-13, and relevant only for electron g-2.

11. Sep 12, 2017

### Urs Schreiber

Thanks for your patience, sorry for being slow. Please bear with me: I gather I am missing one bit of information, which you are taking for granted.
Could you remind me of the order of magnitude of $\text{AbsoluteValue}\left( \text{result}_{\text{experiment}} - \text{result}_{\text{theory}} \right)$
and of its uncertainty for the case at hand? I suppose you are saying that $430 \cdot 6 \cdot 10^{-13}$ is much smaller than both of these?

12. Sep 12, 2017

### Staff: Mentor

That depends on the measurement.

The experimental value for (g-2)/2 is 0.001 159 652 180 73 (28), an absolute uncertainty of 2.8*10-13. Measured here.
I misinterpreted the abstract of the theory paper, the 5-loop contribution number is for g/2 not for g, so 6*10-13 means it is twice the experimental uncertainty. It has a 6% relative uncertainty. The uncertainty on the 4-loop contribution is a bit larger than that (~6*10-14), but still much smaller than the experimental uncertainty.
It is important to consider the 5-loop contribution to compare theory and experiment.

Edit: Missed minus signs in exponents

Last edited: Sep 12, 2017
13. Sep 12, 2017

### Urs Schreiber

Thanks! But now help me: You seem to be saying that even the 4th order contribution is not much smaller than the experimental precision and uncertaintly. This makes me ask again how you know that adding the 5th, 6th, 7th etc. contribution will necessarily further improve the match to experiment?

Or if you feel my questions are not going in the right direction, could you lay out again from scratch the argument by which you conclude that all the first 430 loop orders should keep improving the match between theory and experiment (in the given example). Because that's what I am still missing. Feel free to tell me that I am missing the obvious, but please do state the obvious then. Thanks!

14. Sep 12, 2017

### Staff: Mentor

The 4th order contribution is large (~50 times the experimental uncertainty). Its uncertainty is small (~1/10 times the experimental uncertainty).

Without the 4th order the theoretical prediction wouldn't match at all, with it but without the 5th order it would still have a notable tension, only with the 5th order we get good agreement. The 6th order is again a factor ~500 smaller, so it won't play a role for quite some time.

15. Sep 12, 2017

### Urs Schreiber

Thanks for your replies, I'll stop insisting and thank you for your patience. In closing, I'll just point out once more that the smallness alone of the contribution at some loop order does not address the question which I tried to raise in #3: The contributions beyond some order may be tiny, and still push the theoretical result away from the physical value, instead of towards it. But probably it is just not known for available theories at which order this happens.

16. Sep 12, 2017

### Staff: Mentor

We know the value goes away from the physical value at some point, but it is not expected that this happens before the point where the contributions stop getting smaller, and no instance where this would happen early has been observed so far.

17. Sep 12, 2017

### Urs Schreiber

Thanks. What is the basis of this expectation?

18. Sep 12, 2017

### Staff: Mentor

General experience. See above, I'm not aware of mathematical proofs of it.

19. Sep 12, 2017

### Urs Schreiber

That's fine, I would be content with general experience. But which experiences are you referring to, could you give me pointers? Or do you rather mean "general feeling" than "general experience"?

20. Sep 12, 2017

### Staff: Mentor

"That's what the theorists working on these calculations say".
I'm not a theorist.