Understanding of renormalization

[nrqed]

Allow me to ask a naive or somehow stupid question. Actually, I still haven't really completely understood the renormalization group and renormalization theory.

Can I realize the running coupling constant as follows: The running of coupling constants tells us how the renormalization procedure adjusts by itself such that the unrenormalized n-point function is independent of the arbitrary scale M, since, however, the arbitrary scale M is only introduced when we perform renormalization.

I just wrote a long reply and then it got erased.

I might get some time to rewrite it later.

The short answer is that it is the renormalized n-point function which is independent of the scale M. Since it should not make any difference if one has chosen to fix their coupling constant at M or, say , 2M. So the RG equation essentially says that

\frac{d G_{phys}}{dM} = 0

There are two dependence on M which cancels out (I am keeping things simple here, no worrying about wavefunction or mass renormalization). On one hand, the one-loop (say) renormalized coupling constant depends on M through the one-loop calculation that was used to fix it at energy M. But the actual physical value measured at M also depends on M. These two dependence cancel out, leaving the result for G(M') independent of M.

 

[jdstokes]

Uhh, I think you might have made a typo here: it is the unremormalized n-point function which is independent of M, not the renormalized.

Quoting Ryder (p. 324)

``The renormalized 1PI function \Gamma^{(n)}_{\mathrm{r}} depends on \mu [the renormalization scale], as shown in equation (9.48), through the dependence of Z_\phi on \mu. In other words, the unrenormalized function \Gamma^{(n)} given by (9.49) is independent of \mu...''

 

[jdstokes]

What I was saying is that the final result for the process you calculate (after regularization and renormalization) will contain a factor g^2 log(E/E') times a bunch of stuff (constants, angular dependence, etc which I assume all give a result of order 1). I was talking about the whole, final expression.

I see. So given that increasing orders in perturbation theory are not giving terms of decreasing magnitude, we deduce that perturbation theory is failing. Interesting, I never throught about it that way for some reason.

It is quite neat that the one-loop result can be used to include all those higher order contributions. This is what is special and powerful about the renormalization group.

That is an amazing claim. What's even more amazing is that I can't find any mention of this in the usual textbooks on the subject.

What I would really like to see is a full perturbative QCD calculation, starting with analysis at one energy scale, and then using the running couplings to analyse the same process at a different energy scale. Is there somewhere I can find this in the literature?

 

[nrqed]

Uhh, I think you might have made a typo here: it is the unremormalized n-point function which is independent of M, not the renormalized.

Quoting Ryder (p. 324)

``The renormalized 1PI function \Gamma^{(n)}_{\mathrm{r}} depends on \mu [the renormalization scale], as shown in equation (9.48), through the dependence of Z_\phi on \mu. In other words, the unrenormalized function \Gamma^{(n)} given by (9.49) is independent of \mu...''

Ok, I was not careful enough with the exact wording. 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.

I will have a look at Ryder when I get home. Thanks for pointing this out.

 

[nrqed]

I see. So given that increasing orders in perturbation theory are not giving terms of decreasing magnitude, we deduce that perturbation theory is failing. Interesting, I never throught about it that way for some reason.

exactly.

That is an amazing claim. What's even more amazing is that I can't find any mention of this in the usual textbooks on the subject.

What I would really like to see is a full perturbative QCD calculation, starting with analysis at one energy scale, and then using the running couplings to analyse the same process at a different energy scale. Is there somewhere I can find this in the literature?

Just to make sure what I wrote is clear: solving the RG equation using a one-loop result only sums up the *leading log* corrections to all orders. Now, if one uses he coupling constant fund by solving the RG differential equation in a calculation, say to one loop, the mu dependence then does not cancel out because one is using a coupling constant that contains corrections from a higher number of loops. In that case, one has to estimate a sensible value of mu. This is the case in many QCD calculations, including calculations on a lattice.

I don't have my books with me but I would say that Greiner's book on QCD is the most likely place to see a thorough discussion. Maybe Aitchison and Hey do it well too.

 

[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, \alpha \approx 1/137 [/tex]. After renormalization, the final answer for your process will be of the form<br /> <br /> f \bigl( \alpha = 1/137, \ln(E&amp;#039;/m_e) \bigr)<br /> <br /> 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&#039; :<br /> <br /> f \bigl( \alpha = 1/128, \ln(E&amp;#039;/m_Z) \bigr)<br /> <br /> 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.<br /> <br /> <br /> 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. <br /> <br /> Let&#039;s say that the scatering process you are interested in is at, say, E&amp;#039; = 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.<br /> 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) .<br /> <br /> But let&#039;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 <br /> \ln(5 M_z/m_e) multiplying a coupling constant that is not very small and you cannot trust your result anymore.<br /> <br /> 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 <br /> <br /> \alpha(M_z ) \approx \alpha (m_e) ( 1 + \alpha(m_e) \ln(M_z/m_e) )<br /> <br /> which just brings back the problem of a large log. So we are stuck.<br /> <br /> <br /> It is at this point that the renormalization group enters. It allows to sum to all orders the large logs in order to &quot;run&quot; 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 <br /> <br /> \alpha(M_Z) = \frac{ \alpha(m_e) }{ 1 + \alpha(m_e) \ln(M_z / m_e) }<br /> <br /> which is now fine! (i.e. the large log does not cause any problem now)<br /> <br /> <br /> So this way, we can use the experiment at scale m_e and use the RG to &#039;&#039;run&#039;&#039; the coupling constant to a scale near the energy at which we are actually interested in. <br /> <br /> 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