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as you might know, the Standard Model of physics can be seen as a $U(1)\times SU(2)\times SU(3)$ gauge theory where each symmetry group accounts for different force fields.

The behaviour for the force field of a "point" charge in the most simple cases of this model, the electromagnetic interaction expressed as an abelian (elements commute) $U(1)$ theory, is well known and falls of as $r^{-2}$ and is proportional to the total charge of the source. One states that this force is long range as it only falls off polynomial.

Now, the system for the other interactions, the weak and strong ones, is much more complicated since the underlying groups are not abelian which makes the problem intrinsically nonlinear as can be seen from the Yang-Mills equations, $F = DA, DF = 0$ and its dual counterpart. In contrast to electromagnetism, the interaction range falls off rather quickly and different potentials are known to describe different phenomena.

My Question is:

Can one see directly (e.g. from the non-Abelian character of the group) that the decay of the force field must be faster than for electrodynamics?

Thank you in advance.

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I chose to interpret your question more freely as asking about the nature of the interactions of the whole Standard Model and little bit of QFT because it seemed to me that you are not really acquainted with this. As a result, the part of my answer that actually deals with your original question is quite negligible -- more or less because there is nothing to talk about once you know how Standard Model and QFT works. –  Marek Dec 8 '10 at 23:58
    
@Marek: Your answer is well appreciated. As a student I had some interest in differential geometry and its application to physics. I was very unhappy that no lecture on the standard model was given so I educated myself a little. But of course this left me at a handwaving level where any further insight is very helpful. –  Robert Filter Dec 9 '10 at 8:35
    
why did you edit the question to leave out the $ A \wedge A $ term? For Yang-Mills case the field strength is not given by $ F = DA$ but by $ F = dA + A \wedge A $. –  user346 Dec 10 '10 at 2:23
    
@space_cadet: Thank you for your question. I edited it since the former equation was just wrong. I wrote $DF = dF + A\wedge A$ but I think it must have been $DF = dF + A\wedge F$. Since the covariant derivative is given by $D = d + A$, the form above is exactly this equation put in somehow more condensed form :) –  Robert Filter Dec 10 '10 at 8:48
    
@all: Thank you very much for your thoughts so far. I went for the answer of space_cadet since he is directly adressing the point of the question guided by a nice argumentation in terms of explicit equations. –  Robert Filter Dec 12 '10 at 8:01

6 Answers 6

up vote 3 down vote accepted

This is a very nice question. There is indeed a simple way to see that a non-abelian theory will be of shorter range than an abelian one.

The action of a gauge theory, generically contains terms of the form $ Tr[F F] $ or $ Tr[F \star F] $, where $ F $ is the curvature or field strength of the gauge connection. For an abelian connection $A_\mu$, the field strength is of the form:

$$ F_{\mu\nu} = \partial_{[\mu}A_{\nu]} $$

where the $ [ \dots ] $ represents antisymmetrization over the indices within the brackets.

Consequently the $F^2$ type terms in the action are of the form:

$$ F^2 \sim (\partial A) (\partial A) $$

For a non-abelian connection $A_\mu^I$, where $I$ is now an index in the lie algebra of some non-abelian group, we have:

$$ F^I_{\mu\nu} = \partial_{[\mu}A^I_{\nu]} + f^I_{JK}[A^J_\mu,A^K_\nu] $$

where $ f_{IJK} $ are the structure constants of the group in question.

Consequently the $ \mathcal{O}(F^2) $ terms in the action now contain terms of the form:

$$ (\partial A) A^2 \textrm{ and } A^4 $$

These are self-interaction terms which will, in general, endow the connection $A^I_\mu$ with a mass - in a suitable symmetry broken phase of the theory. And a massive gauge particle leads to a short range (and/or confining) interactions.

That's the gist of it. There are likely other ways to approach the problem, but this is the one I'm most familiar with.


In response to some comments I'd like to quote the following line from the Jaffe-Witten paper introducing the Yang-Mills problem as part of the Clay math prize:

" ... One view of the mass gap in Yang–Mills theory suggests that it could arise from the quartic potential $(A \wedge A)^2$ in the action, where $ F = dA + g A \wedge A $, see [11], and may be tied to curvature in the space of connections, see [44].

The reference [11] cited in the line above is a paper by Feynman where he studies SU(2) gauge theory in 2+1 dimensions and concludes the gauge invariance dictates the presence of a mass gap.

One can argue about fixed points and phases and whatnot at different temperatures. But unless you have something that beats Jaffe, Witten and Feynman I guess it is safe to conclude that @robert's intuitive guess that the non-linear nature of non-abelian gauge theory is responsible for its short-range/massive/confining character is right on target.

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@space_cadet: Thank you very much for your well-thought and investigative answer. Your link between the self-interaction terms and its manifestation as mass of the interaction-particles seems like the missing part of the puzzle :) –  Robert Filter Dec 9 '10 at 9:09
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Hm, you completely left out asymptotic freedom. Which shows that what you've written is not really correct. You don't see the behavior of theory just from the fact that it's not abelian and has self-interaction terms. You should know that renormalization running strongly depends on $n$ in $SU(n)$ so if there were different number of colors in nature, the theory could behave completely differently (despite being still non-abelian) and end up free in some regimes. –  Marek Dec 9 '10 at 9:59
    
I am really looking forward to the answer of space_cadet. Discussions like these are always very helpful understanding a subject. –  Robert Filter Dec 9 '10 at 10:10
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@Robert: note that what I've written also has to do with number of quarks (i.e. the true behavior of the QCD depends both on the number of quarks and gluons). If you want to talk just about gluons this is not that relevant. By the way, meanwhile I noticed this was already elaborated by Moshe under his comments: the part about positive and negative contributions to beta functions. In simple terms: this function tells you how strong the coupling is depending on energy of the interacting particles. –  Marek Dec 9 '10 at 10:27
    
@marek: @Robert asked a simple enough question. And yes I left out asymptotic freedom, the large-N approximation, multiverses and the anthropic principle among other topics. @Robert's question boiled down to: "does the non-abelian character of yang-mills theory contribute to the fact that it has a mass-gap at low temperatures" - to which the clear and simple answer is yes. I'm still waiting for someone to give me a situation in which that is not the case. If you have a counter-example you might also want to claim the million dollar clay prize ! –  user346 Dec 9 '10 at 20:07

The picture is a little more complicated. The decay of the weak interaction has to do with whether or not propagators are massive, not with abelianness. The decay of strong force has to do with confinement.

Spontaneous symmetry breaking

The group for electroweak theory is $SU(2) \times U(1)$. There are four gauge bosons here. At high enough energies they are all massless (and therefore provide long-range interactions) and transform among each other. But in nature we can observe weak interactions (which means they have massive propagators that decay quickly). This is because the above group is spontaneously broken at low enough energies by Higgs mechanism.

In the most simple picture, there are four scalar Higgs fields. Three of them couple to the (originally massless) gauge bosons and you obtain massive $W^-, W^+, Z$. These form $SU(2)$. But note that this is a different $SU(2)$ than the original one (i.e. it also contains part of the fourth gauge boson from the original $U(1)$. One of the Higgs fields remains (this is the one people look for at LHC). You also obtain photon, which is massless.

(Note that this model is not a consequence of some particular theory. It was built with all the observations in mind and this is the most natural way to do it).

Now, the weak bosons are very heavy and decay quickly. They are only present in the actual interactions as virtual particles.

Confinement

The other part of the standard model is $SU(3)$. Gluons are massless particles and so they propagate at the speed of light. The difference with respect to $U(1)$ case is that the field theory is not free (except at high energies where the theory possesses an asymptotic freedom) and so the model has complicated dynamics also if no charges are present. But here renormalization comes into play. It tells you that as the relevant energy you are dealing with gets too low, the strong coupling diverges. So you can't separate quarks. If you try, you'll just create jets of hadrons.

Nuclear interactions

So when you are talking about decay of strong force field, you must in fact be talking about the nuclear force between nucleons (protons and neutrons). This is an effective interaction which mediated by pion $\pi$. Pions are massive and described by Yukawa potential which indeed decays exponentially.

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It would appear that you know more about my own research area than I do ;-) +1 –  David Z Dec 9 '10 at 0:07
    
@David: it's easy to provide an answer at this qualitative level for any theoretical physics student, I think. It would be something completely different if you asked me to actually calculate some cross-sections in QCD :-) –  Marek Dec 9 '10 at 0:11
    
@Marek, you're giving a summary of entire the standard model here. Don't you think that is overkill for answering Robert's query? Which is a very nice conceptual question IMHO, and one you don't tackle directly at any point. –  user346 Dec 9 '10 at 2:11
    
@Marek: Thank you for your answer. I have to admit that I do not really have the knowledge to be following you, nevertheless I upvoted it since I think it will be helpfull on the long run. Further I would suggest that you should not assume this kind of qualitative answer is easy for any theoretical (student) physicist. –  Robert Filter Dec 9 '10 at 8:52
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@space_cadet: I don't agree with you. The correct answer is really as short as "interactions by massive particles are short-ranged and it has nothing to do with abelianness" (which I stated). But this is too short, so I decided to also elaborate on related topics. –  Marek Dec 9 '10 at 9:46

The non-linearities and the short range nature if the force are completely independent. You can have Abelian theories which are short ranged (look up the Abelian Higgs model), and non-linear interacting theories that are long ranged (e.g gravity). The same theory can have more than one phase (depending on temperature and other control parameter) in which the forces are long ranged or not, it all depends on the details.

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An Abelian Higgs model needs something more than just the gauge fields. It needs a scalar. As for the phase argument, I mention that caveat in my answer "... endow the connection A with a mass - in a suitable symmetry broken phase of the theory." Given that a non-abelian theory will be effectively deconfined (long-range, massless) at high temperature, it will also typically be confining/massive/short-range below some critical temperatures. This is not true of an abelian theory and I don't know of any non-abelian theory which is deconfining at all energies ! –  user346 Dec 9 '10 at 3:16
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space_cadet: there are many examples of nonabelian theories that are not confining at any energy scale. The easiest to understand are the weakly coupled Banks-Zaks fixed points, where the beta function has a zero at small coupling. An example would be a QCD-like theory with # of flavors = (11/2 - epsilon) times the # of colors (with epsilon small). There are many, many other well-understood examples, with supersymmetry giving the most powerful theoretical tool to study them. –  Matt Reece Dec 9 '10 at 3:48
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In addition to what Matt says, superconductivity is an example of short range Abelian gauge theory, below a certain critical temperature (same is true of the Abelian Higgs model, in which the gauge boson is massive at T=0, but there also there is a phase transition). –  user566 Dec 9 '10 at 4:14
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Simplest to consider the beta function: gluons give it negative contribution (insight which is worth a nobel prize) and matter gives it positive contribution. When you have sufficient matter the beta function will tend to be positive, and then the coupling becomes weaker in the IR. These theories tend to be qualitatively similar to QED. –  user566 Dec 9 '10 at 4:52
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This discussion is diverging, so a couple final points: For pure gauge theory, non-Abelian theory is confining, Abelian theory is free, but I interpreted the question as being more general. Also, confinement and mass gap are not the exactly same thing, they appear together in QCD, but not in every other gauge theory. –  user566 Dec 9 '10 at 5:31

There may be three different r-dependences of interaction: microscopic V(r), effective elastic U(r), and an effective inclusive W(r) derived from an inclusive cross section; see here and in my weblog.

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I well appreciated the lively discussion arising to the question. I am truly interested in grasping an idea of the standard model and understanding it on a level such that I somehow know what is going on in QFT research e.g. at CERN.

I do not know if it is a no-go to answer one's own question but I want to give it a try here to see if I somehow got the ideas of the people contributing, namely Marek, space-cadet and Moshe (thank you!). I might also want to add some additional information.

Following Jaffe & Witten, Quantum Chromodynamics (QCD, the $SU(3)$ "part" of the Standard Model) has to fulfill three properties to be successfully describing the strong force. One of these is the so-called mass gap which means that every excitation of the vacuum must have a truly positive energy $E > 0$.

For QCD, the mass-gap is responsible for massive gauge Bosons and therefore short interaction ranges. So what about the other theories? My thought was that the non-abelian structure of $SU(2)\times U(1)$ and $SU(3)$ somehow directly relates to short interaction ranges. A nice argument was given by space-cadet who points out that the $(A\wedge A)^2$ term in the Langrangian can be interpreted as the term responsible for the mass gap (again from the Jaffe & Witten paper).

And indeed, a recent paper by Frasca shows that for classical Yang-Mills theory

mass gap is simply a dynamical effect arising from the self-interaction term of the equations of motion

So there is evidence that $A\wedge A \neq 0$ might be an explanation of short interaction ranges.

The remaining question is if the non-Abelian character can be the (only) reason for massive gauge particles or if other effects such as asymptotic freedom and confinement are important to understand different ranges of inderaction.

The abelian Higgs model (thanks to Moshe) is an example where symmetry breaking accounts for massive gauge particles. Also, there might be examples of long-range non-Abelian gauge theories.

To sum up, I would state that the situation is indeed complicated as Marek pointed out. But for some theories, the interaction range can be explained by the (non-)Abelian character of the underlying group.

Sincerely

Robert

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To fix ideas, let us talk about mass gap only, not confinement or asymptotic freedom which are related but logically independent. First, this is a highly quantum mechanical effect, staring at the classical action will not tell you much (the paper you quote talk about something else, not what usually people refer to as mass gap). As an indication that mass gap depend on more details than just the non-Abelian self-interaction, there are examples of non-Abelian gauge theories with matter which do not have a mass gap. I don't see a way to distinguish which does or does not without a calculation. –  user566 Dec 11 '10 at 17:50
    
Hi Robert, you say that: "The remaining question is if the non-Abelian character can be the (only) reason for massive gauge particles or if other effects such as asymptotic freedom and confinement are important to understand different ranges of inderaction." As it happens asymptotic freedom and confinement are ALSO direct consequences of the fact that the gauge group is non-abelian. Its a nice little package - mass gap, asymptotic freedom and confinement. I look forward to indignant objections to this statement ;-) There is a very nice book by Quigg which covers these topics very clearly ! –  user346 Dec 11 '10 at 19:20
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Yes, indeed. There are many non-Abelian gauge theories that are also not asymptotically free or confining, in addition to not having mass gap. So, non-Abelian character is neither sufficient nor necessary condition for any of these phenomena, all of which are independent from each other. I am afraid that is all I can contribute to the discussion, I think I am done here. –  user566 Dec 11 '10 at 20:26
    
@space_cadet: Thank you for the literature suggestion. I guess you are referring to "Gauge Theories of the Strong, Weak, and Electromagnetic Interactions" by C. Quigg, if anyone else is also interested in the topic. –  Robert Filter Dec 11 '10 at 22:52
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FYI it's fine to answer your own question if you feel it's necessary. –  David Z Dec 11 '10 at 23:16

@Moshe: As you may know, when people are not able to manage quantum field theories in some limits tend to form a lot of prejudices about how something should come out. This is the situation about mass gap in Yang-Mills theory. You can read this post to have a correct idea about the current situation on this question and, of course, if you are able to find some unsatisfactory points in that papers, published in respectful journals and with an intervention of Terry Tao, I will be happy to hear from you. On the other side, the paper you are discussing is old, unpublished and overcome by 0907.4053 [math-ph] that will appear in the Journal of Nonlinear Mathematical Physics.

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Thank you for your thoughts and welcome to this platform! I hope you don't mind that I was quoting you in my answer. Please let me know if I got you wrong in any way. All the best. –  Robert Filter Dec 11 '10 at 22:48
    
LOL. Hi @jon (marco ?). Good to hear the answer from the horse's mouth - so to speak. –  user346 Dec 12 '10 at 0:25
    
Also, I'm by no means an expert in yang-mills, so as Robert said - if I'm wrong, tell me about it ! On second thoughts you should do that even with the experts ;-) –  user346 Dec 12 '10 at 0:27
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For other helpful comments about nonlinearity and mass gap I have added further comments at marcofrasca.wordpress.com/2010/12/12/what-is-the-right-solution . There is another notable example of a nonlinear theory having a continuous spectrum without mass gap: Liouville field theory. –  Jon Dec 12 '10 at 15:05

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