Interaction ranges in the Standard Model - Electrodynamics vs QCD

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

Answer 6

@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.