103. Criteria of confinement. Wilson loop – physical discussion
HEP-TH/PH — By Dmitry Podolsky on November 19, 2008 at 10:00 am
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It is often said that the most physically relevant criterion of confinement is the behaviour of the potential between two fermions: confinement implies the linear growth of potential between two charges with distance. Is it really so? In a gauge theory with fermions in the fundamental representation and the gauge group 
(
corresponds to quantum chromodynamics), the string of chromoelectric field connecting a heavy quark 
and antiquark 
may break down if the distance 
between the fermions becomes sufficiently large. This is due to the production of light quark-antiquark pair in the strong chromoelectric field (I estimate the maximal length of the chromoelectric string in this post.)
As a result, the interquark potential gets screened by light quark-antiquark pairs at large distances in analogues of quantum chromodynamics and does not grow at 
.
By the way, people, who know QCD much better than me, I have a question to you – do you know how to derive 
smarter?
But let me turn back to the original criterion of confinement. When exactly the unbounded growth of the interquark potential is equivalent to the quark confinement? Let us turn the light fermions off. The resulting theory is pure gluodynamics – gauge theory, and its vacuum structure can be tested by infinitely heavy charges (potential between heavy test charges gives the gluon propagator) in the fundamental representation. If the distance between test charges grows large, the string of chromoelectric field cannot break down, since gluons in adjoint representation cannot form an object which is in the fundamental representation of the gauge group. Then, infinite growth of potential between test charges would imply confinement.
Following Wilson, let us consider a closed contour 
and introduce the quantity
, (1)
where 
denotes the -ordered (along the contour ) exponential
(2)
The quantity 
is called the Wilson loop, and its correlation functions can say quite a lot about the vacuum structure of the theory. For example, let us take a rectangular contour of the length 
in the timelike direction and the length in the spacelike direction (let us take it along 
-axis for simplicity). Then, the VEV of the Wilson loop
(3)
carries the information about potential between two test charges.
Exercise. Check that the -ordered exponent (2) is not gauge invariant, while its trace is.
Indeed, we can immediately see from simple physical considerations that the Eq. (3) holds. Let us take a test charge and carry it along the closed contour . The corresponding quantum mechanical amplitude is given by 
. On the other hand, we can interpret this process as the creation of quark-antiquark pair, its propagation during some finite time and subsequent annihilation. It is clear from the form of the evolution operator that the corresponding correlation function can be written as 
, where 
is the interaction energy between quark and antiquark. Indeed, in the limit 
only the ground state of the Hamiltonian contributes into the expression for the amplitude. The energy of the ground state only depends on the distance between the quarks – the latter are heavy and only propagate in time.
If the theory is in the confinement phase, then 
(interquark potential grows linearly!), and the VEV of the Wilson loop is
, (4)
where 
is the minimal area spanned by the contour . Further, since the spectrum of elementary excitations (gluons) in the theory has a mass gap, we conclude that
, (5)
i.e., the area law (4) should hold for arbitrary contours, not just rectangular ones in the confinement phase.
Exercise. What would happen in a theory without mass gap? Is it possible to have confinement in such a theory?
Now, suppose that our theory is not in the confinement phase but in the Coulomb phase instead – that is, the potential between two test charges behaves as 
, where 
is the coupling constant. Using Stokes theorem, we can write for the VEV of the Wilson loop
(6)
For 
this expression can be reduced to
, (7)
i.e., the Wilson loop VEV behaves according to the perimeter law.
To be continued…
Want to know more before I wrote the next post?
Nice brief introduction into Wilson loops can be found in Peskin, Schroeder, “Introduction to quantum field theory”, which is an extremely good text book on QFT anyway. If you want something more advanced, then definitely Polyakov’s book is the best choice (you need the Chapter 5 entirely devoted to confinement).
8 Comments
Just to make sure, in expressions (1) and (2), what are you taking the trace over? The group representation? In that case, you’re defining A = A^a \lambda^a, right? Thanks.
Hi EG
Yes, A is in the adjoint representation, so A=A^a T^a, where T^a are generators of SU(N). For SU(3) T^a is given by Gell-Mann matrices \lambda^a.
Cheers
What kind of explanation is this?
You say that the circulation of the gluon field around a closed path can be interpreted as a quark-antiquark loop just like that? And this is the “physical introduction”?
Also for biologist?
If you really were writing for people without the ph-d in the field this kind of “obvious” reasoning would not appear. Do you really think a standard physics graduate would understand this only from your explanation?
To recommend reading Peskin book we don’t need posts!
Dear Angry One!
Yes!!! I finally see the one who cares!
Let me warmly welcome you. Be angry enough, and you shall be given – that’s how, I think, it works in science.
Normal physical explanation, and what exactly makes you unhappy? Heavy quark propagating back in time is equivalent to antiquark propagating forward in time. Ask more and thou shalt receive.
Yes, it is written at reasonably high level of complexity – because, I think, it is better to take, say, good monograph or recent reviews and learn the topic using them rather than some oversimplistic textbook. This would prepare you for real science career instead of teaching assistant career in some not so good college. Anyway, I would always try playing as high as I could.
Yes, but recommending reading it one more time does not hurt.
Anyway, you see, this is what all this blogging thing for – it is interacting, you can ask the blogger any thing you want, and you are very welcome to do that.
Cheers
First, thanks for your warm answer.
Then, more questions because you promise answers. Or the same but more down to the point.
Why do you interpret a mathematical expresion that displays a gluon field (Amu) as a qqbar loop? Where are the q fields? Why they don’t appear in the Wilson loop but you still interpret they’re there? And where did the gluon go?
About the style and level, if you let me say it, I find that it is what you say it is (reasonably high, not oversimplistic) only apparently. You’re not oversimplifying if you’re able to take people from graduate physics to your post by the same intellectual road that you took to understand the ideas. But your post does, as the large majority of blog expositions, assume that saying things like
“On the other hand, we can interpret this process as the creation of quark-antiquark pair, its propagation during some finite time and subsequent annihilation. It is clear from the form of the evolution operator that the corresponding correlation function can be written as , where is the interaction energy between quark and antiquark. Indeed, in the limit only the ground state of the Hamiltonian contributes into the expression for the amplitude. The energy of the ground state only depends on the distance between the quarks – the latter are heavy and only propagate in time.”
at which you arrive, I’m sure, after some thinking and studying, are understandable to plain physicists or even biologists !?! Not to mention that you talk about a number of things like E(L), evolution operator, correlation function… that you don’t introduce and neither think you need to spend a word to explain their relationship with the Wilson loop integral. Surely this things are in your mind very clear (or maybe not) but you shouldn’t consider them to be equally clear in ours so that, with that fast-paced sentence we can follow the reasoning of such a deep kind of physics.
Frankly speaking, the impression an outsider gets is that you’re repeating standard (obscure) text or paper assumtions that one finds many times stated but never explained. In that way knowledge is only transmited to the initiated, excluding every outsider.
Please, forgive the extension of the comment. Only your kind, fast and warm answer made me so loquacious.
Hi again the Angry One!
I decided to make a separate post to address these questions. If the thing is still unclear, please keep asking!
Do you feel I need to explain these things as well? If you do, then I probably will.
Well, in my defence I can say that (as you might have read in the Welcome page) some of the posts are related to my scientific diary. The posts about confinement will hopefully transform into review I intend to write.
Also, please take into consideration that quite different people read this blog. There is Lubos Motl around for example. He is a former Harvard professor, professional string theorist and a very very good physicist – he could convince me that I am wrong even if I am right
I just want to keep standards high.
And you are again very welcome.
Cheers
First, thanks for your time again. Wilson loop understood.
It looks like the obscurity came out of a so-called “writing vice” typical of people too used to write only for their own peers, a small group of insiders that move fluidly through a maze of “supposs?s” and “understoods”.
Your reference on your own defence to Lubos Motl just confirms that these kind of blogs are only written for insiders and innocent bystanders would only get frustration and inferiority complex out of reading them.
But then, why all of you think your standards are higher by explaining less instead of more? Or by using post-graduate jargon instead of plain english?
Anyway if you judge your standards by the yardstick of “thinkers” like Mr. Motl, I definitely fell on the wrong blog.
Great. You are welcome.
First, if you will be able to overcome your internal pain and decide to read this blog as well as leave comments on it, I would really appreciate if you do not discuss intellectual abilities of other readers.
Second, you did ask some physical questions – it took 3 lines. Other 256 or so lines were mostly devoted to complaining and explanation how innocent you are. Although I did appreciate that something bothers you very much, I should say that I am physicist and I am mostly read by physicists. If you are looking for a personal psychoanalyst, you indeed fell in the wrong blog.
Cheers
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