# On scheme dependence in QFT renormalization

I searched for the answer to my question quite a while and it seems nobody ever asked similar questions or it is written explicitly in any textbooks. The question is,

1. If physical parameters of any theory change with energy scale at which the theory tried to describe, then why are those physical parameters listed on the cover page of any physics textbook (say, on the cover of many modern physics textbooks the electric charge $$e = -1.602 \times 10^{-19}\;\mathrm{C}$$, electron mass = ..., and so on are listed) are not associated with any energy scale at which the experiments are conducted to measure them? Is it because these parameters are quite insensitive to the energy scale under which the parameters are measured? but then how insensitive?

Below is what I know and my basic understanding of my problem. I just learned that parameters in a QFT change with the energy scale at which the theory tried to describe. I have also seen some path integrals associated with some kinds of Feynman diagrams diverge, that is, the embarassing perturbative expansion where each term diverges.

The initial renormalization idea is that parameters are not physical. They are just parameters to describe the physics of interest. So, there can be a gap between physical parameters like physical mass and electric charge and bare parameters. (Well, I think this idea is really...lame honestly speaking, how can you just change your face and turn around to say those parameters are not parameters you thought just because your way to calculating things does not give you a consistent answer?)

Well, so one simply cheats by separating the bare parameters into a physical, finite piece which is what we want and a divergent piece which is unphysical. For example, m_b = m_phy + m_div, where m_b is bare mass and m_phy is physical mass which is finite and m_div is infinity. This is sort of absorbing path integral infinity into the counter term part m_div.

This comes to my second question (though I think this one will be answered if I keep learning renormalization of QFT):

1. How do we deal with the arbitrariness of writing (infinity) = (finite) + (infinity)?

I have read the regularization part of QFT textbook and OK with the calculation to separate the divergent part of a path integral, i.e. introducing some cutoff or arbitrary energy scale and to write the divergent path integral as some pole (which is the divergent part) plus the finite part which usually depends on the arbitrary energy scale if using dimensional regularization. However, as expected, the finite part depends on some arbitrary new parameters you introduced to "regularize" the divergent path integral. Then, how can we start to do prediction? how do we get rid of this arbitrariness? What comes to my mind is to conduct a few experiments and fit the finite part (which contains arbitrary energy scale) with the experimental values you measured. After this is done, one can start to predict in other experiment.

A classical explanation of the changing electric charge under different scale is the screening effect caused by vacuum polarization. I think this argument is OK as I accept vacuum fluctuations. However, if this is true, why did no one ever told us that?

"Be careful, the values of the physical parameters showing on the cover page of your modern physics textbook actually change if you conducted the experiment at a different energy scale!"

If you look at the PDG, equation ($$10.7d$$), you'll see that they define the electromagnetic constant $$\alpha$$ at a precise scale (the mass of the muon in this case). Later on they also give the value at different scales and it is indeed different. When quoting a value, if it is unclear from the context, one must always specify the scale.

They also explicitly say what is the energy scale of the "textbook" value of the electric charge

[...], with $$\alpha^{-1} \sim 137$$ appropriate at very low energy, i.e. close to the Thomson limit

If you keep scrolling to, for example, Table $$\mathbf{10.2}$$ you'll see some names under the voice "Scheme." These are shorthands for different prescriptions to separate the "bare" contribution into "renormalized" + "$$\infty$$". Again, unless clear from the context one must always specify what scheme has been used when quoting a Lagrangian parameter.

For instance the $$\overline{\mathrm{MS}}$$ stands for "modified minimal subtraction scheme." In there divergences appear as poles in a regulator $$\varepsilon$$ as $$\varepsilon\to0$$ and they are subtracted together with a constant $$\gamma_E - \log (4\pi)$$ chosen simply to make the expressions nicer in the end.

• Many thanks! That's spot on. Feel much more comfortable and motivated to keep learning renormalizatoin, to keep being "renormalized"! – chichi Aug 9 '19 at 2:56
• Just want to be confirmed by an expert before I go on. May I make sure if my understanding of my 2nd question correct? i.e. we need to "fit" the finite part with arbitrary "unknown parameter" with an experiment before starting prediction. I think this is answered by the scheme session of your post. However, I am wondering if different schemes gives same physical value at same energy scale? Physics should not depend on whether you absorb Euler constant or not, am I right? – chichi Aug 9 '19 at 3:20
• Correct, the physics doesn't care. Within any given scheme you can compute observables (like cross sections or life times) and use those to fix your constants in that scheme. Once the Lagrangian is fixed all other predictions that you can make with the theory are unambiguous. – MannyC Aug 9 '19 at 3:28