# What are threshold corrections?

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As the title goes, what are threshold corrections in quantum field theory?

In particular, I would be glad if a good reference is provided. Standard QFT books such as Peskin, Weinberg, etc seem to have nothing to say about them.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user whistles

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user QuantumDot

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Threshold corrections is a term that appears when you discuss effective field theories (EFTs). An EFT is an approximation of a full theory which is valid at low energies, ie below some threshold.

Let $A_{\mbox{eff}}$ be any amplitude as calculated in the EFT and $A_{\mbox{full}}$ the amplitude for the same process calculated in the full theory.

The threshold correction is defined as $$A_{\mbox{full}}-A_{\mbox{eff}}$$ and it describes all the information that has been "integrated out" in the EFT.

It needs to be calculated/approximated somehow when the accuracy of the result of the EFT is not sufficient.

The threshold (and the region of validity) of the EFT are not fixed a priori. There are two different ways to think about this:

i)We can either set the threshold/scale at a value we desire and then keep as many terms in the expansion as required (truncating the rest), so that the theory is valid to the desired accuracy below that scale.

or

ii) Decide to keep a certain number of terms and then calculate the threshold to which this calculation gives acceptably accurate results.

If you are thinking more along the lines of i), then it is often the case that the threshold you have set initially for yourself is no longer good enough. For example you might have upgraded your accelerator to higher energies. If that's the case, then you will have to include corrections (extra terms) with regards to the old threshold, that will make your theory valid up to the new desired threshold. An example would be a new particle whose rest mass is accessible to the upgraded accelerator but not the old one. You would then have to include some further operators in your lagrangian to take the new particle into account.

Hope this helps!

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user Heterotic
answered Apr 19, 2014 by (525 points)
It's a good start. Thanks! However it's not the complete answer I'm looking for, yet. In principle, any high energy (HE) theory can be exactly described at the EFT level if one takes into account the infinite set of operators that appear hen one integrates out the heavy fields. Of course doing precisely that is impossible, one must always truncate the series somewhere and that introduces an error in the EFT calculation. However, I understand this is NOT what threshold corrections are about. Specifically, they appear when one crosses a mass scale (the "threshold") while the error....

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user whistles
...I mentioned above is always there, no matter if there are some other relevant scales between the one you are interested in and the scale at which the full theory takes over. So, where are these corrections coming from exactly? I would also like to know how to calculate/estimate them. Do you know a reference where this is even loosely explained. I can fill details myself.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user whistles
@whistles: I think it might be exactly related to your infinite set of higher-dimensional operators. Each of these operators is suppressed by a suitable power of the mass scale where you've integrated out something. As you approach that mass scale (from below), you cannot truncate your calculation... but need to "resum" the full set of effective higher-dim operators. I think some such "non-perturbative" correction that happens close to a threshold scale is what the phrase refers to. A "threshold" typically refers to an energy scale where you start producing/sensing new degrees off freedom.

This post imported from StackExchange Physics at 2014-04-21 16:24 (UCT), posted by SE-user Siva

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