Understanding of renormalization

[Avodyne]

By "renormalized" function, I meant the whole, final, renormalized amplitude, including the coupling constant. Clearly this cannot depend on the value of mu since mu is completely arbitrary.

A technical point: in general, an amplitude is not independent of $\mu$; only infrared safe quantities are observable, and hence independent of $\mu$.

 

[nrqed]

A technical point: in general, an amplitude is not independent of $\mu$; only infrared safe quantities are observable, and hence independent of $\mu$.

You are correct if $\mu$ is the $\mu$ of dimensional regularization which regularizes both infrared and ultraviolet divergences. I meant the ultraviolet regulator only. This is one thing that makes dim reg confusing when learning about renormalization, in my opinion.

Thanks for pointing that out.

 

[jdstokes]

What renormalization group allows is, using only the one-loop result , to sum up all the leding log cntributions into a new, "running" coupling constant.

I haven't been able to find a discussion of this anywhere. Would it be possible for you to elaborate a bit more on this or suggest a reference?

Another thing which is bothering me is the physical interpretation of the running couplings. As you say, they are arbitrary mathematical definitions, so what is the physical sense in saying that, e.g., the gauge couplings unify at the GUT scale?

 

[nrqed]

I haven't been able to find a discussion of this anywhere. Would it be possible for you to elaborate a bit more on this or suggest a reference?

The best introductory level discussion I know is the book by Aitchison and Hey.


A more detailed discussion is probably found in the QCD book by Greiner who is alway svery thorough and detailed. Unfortunately, I don't have access to my book right now so I can't check.

Another thing which is bothering me is the physical interpretation of the running couplings. As you say, they are arbitrary mathematical definitions, so what is the physical sense in saying that, e.g., the gauge couplings unify at the GUT scale?

I did not mean to imply that they are mathematical definitions.
After renormalizing the coupling constant (using a process at some energy "E" let's say) and one uses this renormalized coupling to calculate a process at some energy E', the final result is of the form

$$f(\alpha(E), log (E/E') )$$

This appears to be a disaster because it seems to depend on E . But actually the E dependence of the coupling constant cancels the E dependence of the log.

but the value alpha *does* vary with the energy. And that's a physical effect.

 

[nrqed]

The best introductory level discussion I know is the book by Aitchison and Hey.


A more detailed discussion is probably found in the QCD book by Greiner who is alway svery thorough and detailed. Unfortunately, I don't have access to my book right now so I can't check.


I did not mean to imply that they are mathematical definitions.
After renormalizing the coupling constant (using a process at some energy "E" let's say) and one uses this renormalized coupling to calculate a process at some energy E', the final result is of the form

$$f(\alpha(E), log (E/E') )$$

This appears to be a disaster because it seems to depend on E . But actually the E dependence of the coupling constant cancels the E dependence of the log.

but the value alpha *does* vary with the energy. And that's a physical effect.

Just to add what I worte earlier:


Letès say you rae calculating a process at some energy E' in QED. And let's say you use low energy scattering at E of the order $m_e$ to fix the fine structure constant. At the energy, [itex] \alpha \approx 1/137 [/tex]. After renormalization, the final answer for your process will be of the form

$$f \bigl( \alpha = 1/137, \ln(E'/m_e) \bigr)$$

If instead, you use scattering at $E = m_Z$ to fix the coupling constant, you find, from the experiment, that $\alpha$ is now about 1/128. This is a physical effect. Now, you will get for your process at energy E' :

$$f \bigl( \alpha = 1/128, \ln(E'/m_Z) \bigr)$$

The two results will give the same answer, of course. The change of value of the coupling constant compensates for the different mass scale appearing in the log.


If the coupling constant is small, this can be the end of the story. Now consider a coupling constant which is not small compared to 1. The calculation contains the product of the coupling constant with a log. So if the log is large, we run into trouble because perturbation theory is no longer reliable.

Let's say that the scatering process you are interested in is at, say, $E' = 5 M_Z$. Then, you would much prefer to renormalize using an experiment at $E = M_Z$ than an experiment at $E = m_e$ to avoid a large log in your final result.
For example, the alpha at m_e may be, say, 0.1 whereas the alpha at M_z may be 0.3. You prefer to have a factor of 0.3 ln(5) than 0.1 ln(M_z/m_e) .

But let's say that the experimental result at $E = M_Z$ is not available. You only have the one at $E = m_e$ . Then you will encounter large logs of the form
$\ln(5 M_z/m_e)$ multiplying a coupling constant that is not very small and you cannot trust your result anymore.

What you would like to do is to use the experimental result at $E = m_e$ and predict theoretically what the coupling constant would have been if it had been measured at, say, $E = M_z$ or event at $E = 5 M_z$. Doing this the most naive way will simply give something of the form

$$\alpha(M_z ) \approx \alpha (m_e) ( 1 + \alpha(m_e) \ln(M_z/m_e) )$$

which just brings back the problem of a large log. So we are stuck.


It is at this point that the renormalization group enters. It allows to sum to all orders the large logs in order to "run" a coupling constant between two scales. One finds that the alpha at M_Z is related to the alpha at m_e by a relation of the form

$$\alpha(M_Z) = \frac{ \alpha(m_e) }{ 1 + \alpha(m_e) \ln(M_z / m_e) }$$

which is now fine! (i.e. the large log does not cause any problem now)


So this way, we can use the experiment at scale m_e and use the RG to ''run'' the coupling constant to a scale near the energy at which we are actually interested in.

Hope this helps.

 

[jdstokes]

Thanks for your detailed reply nrqed.

I understand your explanation in principle. It's hard to believe that even though the calculation at the different energy scales gives the same result, one method is more reliable at higher orders in perturbation theory. I guess this intuition comes from actually doing the calculation.

It seems to me that there is an important distinction to be made between the running of alpha with scale M^2, and the running of the effective alpha with q^2. Given the definition of alpha, that it runs with M^2 is almost tautology. On the other hand, the effective alpha is defined by an equation like

$\Gamma^{(n)}(tq,\alpha,M) = f(t)\Gamma^{(n)}(q,\alpha(t),M)$.

These two different methods of understanding the running of alpha are almost never distinguished in the literature making it extremely difficult to understand what is going on.

 

[jdstokes]

By the way, one of the best explanations of all of this stuff I've seen is John Gunion's lecture notes on renormalization,

http://higgs.ucdavis.edu/gunion/home.html#courses