# What are threshold corrections?

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.

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!

• 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.... – user5800 Apr 19 '14 at 18:22
• ...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. – user5800 Apr 19 '14 at 18:29
• @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. – Siva Apr 19 '14 at 18:58

Threshold corrections are the finite renormalization corrections coming from fields that were integrated out in an effective field theory (EFT).

First you have to realize that constants appearing in the Lagrangian are defined just in the given theory even if both would use the same experiments to derive their values. E.g. $\alpha_{em}^{QED}$ is different from $\alpha_{em}^{SM}$.

For example you use an effective theory to describe physics below certain threshold scale, from which effects of heavy fields/particles become relevant. Above it you have to use full theory.

If the EFT is to make sense those two theories have to make predictions that are equal at the specified perturbative level, you might e.g. require that all Green's functions without heavy fields are equal. This imposes a condition on couplings of both theories. They might play the same role e.g. $\alpha_{em}$, but their values would differ. Below threshold of e.g. $M_Z$ you have $\alpha_{em}^{QED}$ and above $\alpha_{em}^{SM}$ that receives additional renormalization corrections arising from weak interactions.

Some reference: http://arxiv.org/abs/hep-th/0701053

• So, are you saying that an example of threshold corrections is simply $\alpha_{em}^{SM} - \alpha_{em}^{QED}$ evaluated at the matching scale ($M_Z$ in this case)? – user5800 May 11 '14 at 6:08