# independence of the bare parameters on μ for beta function

So I know re-normalization has bean "beaten to death". I want to understand something a bit specific which might seem trivial. Independence of the bare parameters on $\mu$ and relevance to the beta function derivation. This seems to be an important assumption or rather fact that is built upon in making further renormalization and renormalization group arguments. In fact, even explicit statements about locality trace to this assumption. Again there is a chance this is an extremely lousy question, but can someone give me a precise mathematical argument describing this independence.

I am also looking to put together some reading lists comprising of the original papers treating the subject at a pedagogic level($\phi ^{\text{something}}$ theories). Not those directly addressing the QED problems yet still assuming ignorance of the reasons behind the divergences. Papers making explicit and hopefully proven statements accounting for some of the assumptions taken.

I want to know the deal with independence of the bare parameters.

• If you're confused about renormalization (as it seems to be the case, but that's normal), you should simply read Collins - Renormalization (Cambridge). Old QFT papers are not good to start with. – Vibert Jan 14 '14 at 12:26
• I just downloaded the book. I am going to see what they say about $\mu$ independence. I've been looking for a non physically motivated explanation for about 6 months, may be there is something I am missing. – user37343 Jan 14 '14 at 17:08
• Can you help make manifest mathematically the "independence of the bare parameters on μ " – user37343 Jan 14 '14 at 17:17

The bare parameters in your original Lagrangian are just some unknown constants with little physical meaning. The original Lagrangian has nothing to do with renormalization; it doesn't "know" it will get renormalized.

When we renormalize the Lagrangian we rewrite the theory in terms of physical parameters that we can go out and measure. However, when doing this we have to choose a scale to be able to define our physical couplings at. Thus these physical values can depend on the renormalization scale, $\mu$.

To illustrate this point consider instead renormalization of a ball moving in a fluid. If you wanted to write down Newton's law in a vacuum (where density $=0$) then you write:

$F=m_0 a$

where $m_0$ is the mass of the ball in a vacuum, some parameter. However if you want to track a ball's motion in a water, which has a given density $\rho$ (i.e. at a different "scale"), then your expression will not work. It doesn't work because now the water applies a force on the ball.

Instead you need to renormalize the value of your bare parameter. How much the mass needs to be renormalized is dependent on the density of the water. In other words in the fluid you can write,

$F = m_{ren} (\rho) a$

where now $m_{ren}$ refers to the mass of the ball in the water. The renormalized parameter depends on the scale but the bare parameter $m_0$ is just the mass in a vacuum; it isn't dependent on the renormalization density $\rho$!

Coming back to high energy physics. In this same vein bare parameters in the Lagrangian are just the original values, not the ones that match reality. Only when your renormalized parameters can depend on your renormalization conditions and hence the scale, $\mu$

• your physical motivation is beautiful. Can you help illuminate the mathematics of the statement.Can you help make manifest mathematically the "independence of the bare parameters on μ " – user37343 Jan 14 '14 at 17:16
• @kevinTahN. I added an analogy. I hope this helps. – JeffDror Jan 14 '14 at 18:45