Friday, December 14, 2007

Asymptotic Freedom and the Coupling Constant of QCD

[Source: Siegfried Bethke, Experimental Tests of Asymptotic Freedom, arXiv:hep-ex/0606035, Fig. 17.]

Life without interaction is boring. If quarks would not interact, there would be no protons, no neutrons, no atoms, no readers of blogs. In quantum chromodynamics (QCD), the theory that in principle explains how quarks bind to protons, this interaction between quarks is described by the exchange of gluons - particles that glue together the quarks. The very naive idea is the following: two quarks exchange a gluon with momentum Q, and depending on the colour charge of the quarks, this exchange results in an attraction or a repulsion between both quarks.

The strength of the interaction depends of a factor which is called the "coupling constant", which for quarks and gluons is usually denoted as αs. Here, the index "s" stands for "strong", since the interaction between quarks and gluons had been called the "strong interaction" for reasons that show up nicely in the plot above, as we will see in a second. The exchange of one gluon is proportional to a factor g² = 4παs - in the diagram, each on of the two vertices where the gluon and the quark get in touch contributes a factor of g, the square root of 4παs.

This is completely analogous to quantum electrodynamics, where the exchange of a photon between two electrons is proportional to e², the product of the electrical charges of the two interacting particles - the very same factor that has been known since a long time from Coulomb's law for the force between charges. In electrodynamics, the constant α = e²/4π is called the fine structure constant - it's a pure number, without dimensions of length or mass, and has the value α ≈ 1/137. Moreover, it has the nice property to be more or less independent of the momentum Q of the photon that is exchanged. The smallness and the constancy of α in QED allow all kinds of calculations that are in pretty good agreement with experiment.

In QCD, alas, things are more complicated, and the main reason for this is encoded in the plot above. It shows a compilation of the values for αs, derived from many different experiments, and for different momenta Q of the exchanged gluons. Gluon momentum is measured in GeV/c (and c, the speed of light, is set to 1), and a logarithmic scale has been used to allow to show a bigger range of values.

There are two features of the curve which correspond to two main characteristics of quantum chromodynamics:

The "coupling constant" αs is not a constant at all - it decreases with increasing momentum. Moreover, it lies in the range 0.1 - 0.3 at values of Q that can be probed in experiment, which means that it's about 50 times larger than the fine structure constant of electrodynamics - that's why the "strong interactions" are strong -, and the factor g² = 4παs is on the order of 1, and bigger than 1 for small momenta.

This second feature is called "asymptotic freedom", and it means that quarks are nearly free, or non-interacting, when the exchange momentum is very big. As a result, the computational tools which are so successful for electrons and photons can be applied to quarks and gluons at very high energies.

The other side of the coin, however, is that phenomena at lower energies are much harder to calculate. And, for example, in the regime where quarks bind together to protons or other hadrons, αs is too big to use the recipes of quantum electrodynamics. So far, there are only numerical methods available to solve the full equations of QCD for hadrons, and many different analytic approximation schemes.

Which makes, on the other hand, the question of how quarks interact to build a proton as challenging as interesting.

More than you ever wanted to know about the running coupling of QCD can you find, e.g., in the paper by Siegfried Bethke: Experimental Tests of Asymptotic Freedom, arXiv:hep-ex/0606035, and Progress in Particle and Nuclear Physics 58 (2007) 351-386, and in the review Quantum Chromodynamics and its coupling by I. Hinchliffe for the Particle Data Group (PDF file).

On asymptotic freedom, and QCD in general, you can check out QCD Made Simple (Physics Today) and Asymptotic Freedom: From Paradox to Paradigm (arXiv: hep-ph/0502113) by Frank Wilczek, who together with David Gross and David Politzer was awarded the Nobel Prize in Physics 2004 for the discovery of asymptotic freedom in QCD.

This post is part of our 2007 advent calendar A Plottl A Day.


  1. Sorry about all the flubs.
    Here's my question, and BTW that "repaired link" (heh, it's is good for middle-brow-sers to get good scoop.

    Does “color charge” have physical units the way electric charge does? From f = q1q2/r^2, we have Q = M^0.5 L^3 T^-1. I heard color force follows a different exponent but isn’t really consistent or simple over an extended range anyway, so I wonder if an equivalent in MLT can be found?

  2. In what sense can QCD be simulated by a quantum computer that depends purely on the physics of QED.

    And then how is the SU(3)'s role subsumed into physics based on SU(2).

  3. The coupling constant runs in QED as well, but it's a much weaker effect and opposite in sign to QCD.

  4. Bee and Stefan,
    thanks for the Adventskalender.....
    I found you blog after reading Smolin, which is one of the good things I got from that book. Have not yet managed to read through all the entries about it here, but am enjoying the plots.

    Wanted to complain about a link, but this is fixed already. So will just send you
    greetings from Munich

  5. Hi Neil',

    Does “color charge” have physical units the way electric charge does?

    Thanks for asking! It may sound a bit odd, but all the charges of particles in the gauge theories of the standard model are actually dimensionless.

    For electricity, this is usually hidden by the choice of one of the many systems of units that are used, so that, for example, electrical charge has the unit 1 Coulomb = 1 Ampère × Second, but then, the Ampère is defined via the forces between currents in wires, and in the end, this would cancel out.

    So, electrical charge and colour charge, or weak charge, are all dimensionless. In your example, you could conclude (I think there was a type in the exponent of length)

    [Q] = M^(0.5) L^(1.5) T^(-1),

    and now using the "c = 1" convention, L and T have the same dimension, which is that of 1/M, so all dimensions cancel.

    There is a potential which works very well for the interaction among heavy quarks such as charm and bottom which starts just with the Coulomb 1/r term and than has an additional linear term, proportional to r - I'm planning to comment in more detail on that in one the next posts.

    Best, Stefan

  6. Hi Thud,

    In what sense can QCD be simulated by a quantum computer that depends purely on the physics of QED.

    Sorry, maybe I was a bit brief... What I wanted to say is that in QED, calculating things by summing up ever more complicated Feynman diagrams works pretty well and gives amazingly precise results when comparing with experiment. This so-called "perturbative approach" (you think of free charges as moving around with the interaction by photon exchange etc. acting as a small perturbation to free motion) is so successful because the α of QED is about 1/137, much smaller than 1 - what you actually do in perturbation theory is to sum up a series like α + α² + α³ + ..., and this series has only a chance to work somehow if α is small enough.

    In QCD, however, this perturbative approach does only work at very high energies, just because the α is then only sufficiently small to allow perturbations theory to work. At low energies, relevant for the binding of quarks to protons, this approach just makes no sense, because α is too big.

    So you need other methods, and one is to "discretise" space and time in a model, and formulate the equations of QCD on this space-time lattice, and solve the resulting equations on a computer. That's what is called "Lattice QCD". If you want to read more about it with much more technical details, you may find useful in the blogosphere this post and the blog "Life on the Lattice", and on the web e.g. "Lattice QCD for novices" by G. Peter Lepage (PDF file).

    Hi Stephen,

    The coupling constant runs in QED as well, but it's a much weaker effect and opposite in sign to QCD.

    thanks for mentioning that - I remember that I as a bit surprised when I first heard about the QED running coupling, since you usually hear about α = 1/137... as "the fine structure constant", while in fact that actual value is the low-energy limit, and at the Z-pole is has increased to 1/128.9... already...

    Hi Liesel,

    thanks for the kind words :-)...

    We are currently experiencing some problems with the server where the plots are stored, but I hope this has mostly been fixed by now.

    Best, Stefan

  7. Hi thud,

    And then how is the SU(3)'s role subsumed into physics based on SU(2).

    The SU(3) of QCD which is "responsible" for the interaction among quarks and gluons because of the colour charge is the SU(3) gauge symmetry of colour. "Gauge" here means in physics terms that there a particles, bosons, being exchanged between quarks, in this case the gluons.

    This SU(3) of colour is completely independent of the SU(3) of flavour, which was used by Gell-Mnnn and Ye'eman to classify hadrons including strangeness. Flavour-SU(3) has as a subgroup the SU(2) of isospin (meaning that the nuclear force is independent of electrical charge, and that, in a sense, proton and neutron are just different states of the "nucleon"), but there is not such a subgroup with a physical meaning for colour-SU(3).

    However, for the weak interactions, there is again a gauged SU(2) symmetry, which is called "weak isospin", and where the W and Z bosons are the gauge particles exchanged between fermions. This SU(2) of the weak interactions is, again, completely independent of the SU(3) of colour.

    I hope that somehow helps to answer your question?

    Best, Stefan

  8. Yes, I noticed the links to hyperphysics because it is a tool I have to use as well often

    Now what use any understanding if one didn't point to Quark Confinement?

    Fig. 1. In quantum chromodynamics, a confining flux tube forms between distant static charges. This leads to quark confinement - the potential energy between (in this case) a quark and an antiquark increases linearly with the distance between them.

    see picture here.

    If you did not give the space a "dimensional attribute" what use any theoretics like Garrett lisi's if you cannot speak to the "value of that distance?"

    So "gravity" had to be included to make sense to understand that "new physics" :)

  9. Hi Plato,

    yes, quark confinement is the other big thing about QCD... We did not mention it in the post, but then, I am not so sure about this, but in my understanding confinement is idependent of the running of the coupling. Essentially, it comes about from the self-interaction of the gluon, and I wonder if there could be a theory that has a running of the coupling similar to QCD, but no confinement.

    Best, Stefan

  10. Quark confinement happens when alpha_s is approximately equal to unity, which you can see from the plot happens around 1 GeV. Note that hadrons and mesons all of masses around this scale.

  11. Hi Eric,

    thanks for you comment. Let me ask the question in another way:

    Would QED confine electrical charges if the coupling constant was of the order of 1 instead of 0.01?

  12. Thanks Stefan for the clarification about units. Since you relate L and T to M, I suppose you need to use G to fit it all together, which I wouldn't have expected (or do you use the Compton wavelength somehow?) But what if there was no gravity to calibrate the magnitude of mass? I get the impression that the deep theoretical implications would be more than just no attraction between masses.

    In any case, another thing I wonder: with three colors, you can't use simple plus and minus to represent color charge - how does the math work? It would have to be relative I'm sure, showing no inherent preference of one color over another (the apparent "preference" of positive over negative in electric charge of course is an illusion, since if Ben Franklin had picked otherwise, we'd call electrons the "positive" charge etc.

  13. Stefan,
    Yes, QED would become confining if the coupling costant were of order one at some energy and if it had a negative beta function, so that the coupling is of order one for all values below this energy. However, I think this can only happen for nonAbelian gauge symmetries. Essentially, any nonAbelian gauge group SU(N) with a negative beta function will become confining at some energy.

  14. Hi Neil',

    no, Newton's gravitational constant G is not involved here, but, as you suggest, the Compton wavelength. Maybe you see better what is going on when focussing on the potential energy U between two electrical charges Q and q at distance r, which is (in SI or MKS units)

    U = qQ/4πεr .

    Now, you measure U in units of the rest energy of the electron, E = mc², r in units of the electron Compton wavelength, λ = ħ/mc, and the charges Q and q in units of the electron charge. You can put this in the formula by expanding to

    U/E = (e²/(4πεEλ))·(q/e)(Q/e)·(1/(r/λ)).

    Now, the first fraction on the right hand side is just the fine structure constant α - the electron mass cancels, since Eλ = ħc - and everything is expressed in a nice dimensionless way. You have to keep in mind that the units of electrical charge such as the Coulomb are defined via the force between currents, and that this definition is hidden in the constant 4πε of the Coulomb law.

    As for the representation of the three colour charges, you can plot them as arrows in plane, starting at the origin, having equal length and pointing at the corners of an equilateral triangle - that's what is called the weight diagram of SU(3). Anticolour is represented by an arrow pointing in the opposite direction as the colour. Addition of charges is represented by adding up the arrows, and for colourless hadrons, they have to sum up to the zero vector.

    Best, Stefan

  15. Hi Eric,

    thanks again - I am not so familiar with the beta function, will have to check that some time...

    But then, the point remains, is the non-abelian nature of QCD, in contrast to abelian QED, relevant for confinement?

    OK, if in QED the coupling constant is increased to 1, there is the argument that the vacuum gets instable and that real electron-positron pairs are popping up. That's similar to the picture of the string-breakup invoked to explain why quarks cannot be separated, but I'm not sure if it's really the same.

    I mean, in QCD string breakup occurs because the colour-electric field is squeezed into flux tubes, by the dual superconductor mechanism, and this is caused by the self-interaction of the gluon field, hence the non-abelian nature of QCD. This would happen even if the coupling constant was much less than one, I guess?

    Now, is the dual superconductor equivalent to confinement? I think so. But then, would QED with α=1 be a dual superconductor? I am a bit confused... Does the Gribov theory of confinement answer these questions?

    Best, Stefan

  16. hi - i've only recently started reading Backreaction and I really enjoy your surveys of important historical topics, especially the links to modern survey articles.

    Since we're on the topic of quark confinement, let me ask something I've wondered about since the 80's, which I hope some of you will find entertaining: does quark confinement require quantum field theory, or does is show up in the "classical" gauge theory? Let's make that question more precise. First some background:

    As many of you know, gauge field theory (such as QCD and QED) can be expressed on the "classical" (that is, non quantum-field) level as the evolution of the particle fields (electron, quark, photon, etc.) according to some partial differential equations (here I'm specifically thinking the Dirac and Yang-Mills equations). It turns out the equations for the intermediate vector bosons such as the photon and gluon, and their interaction with fermion particle fields, are the same equations that arise in a purely differential geometric description of a curved space containing the particle fields attached to every point of spacetime. This way of looking at things is called the "fiber bundle" formalism, described in many math-physics books, and some QFT books use it as the pre-quantization starting point. In this formalism QED and QCD take on a flavor that looks and feels much like general relativity, where particle motion is described as geodesics on a curved space (not the same as spacetime).

    The most comprehensive book I know on this point if view is David Bleeker's "Gauge Theory and Variational Principles", which has recently been reissued by Dover. In this book Bleeker takes this approach all the way, developing a description of geodesics in the curved "bundle space". These geodesics can be projected down to spacetime to give the paths of particles observed by us spacetime observers. When this formalism is applied to the case of QED, using the gauge group U(1), the projected geodesic equation turns out to be exactly the Lorentz force law found in classical electrodynamics! Now things are looking a whole lot like general relativity - though it's a little unclear what to think of the "bundle space" where the fermion field lives.

    You can play exactly the same game with the gauge group SU(3) and get a different projected geodesic equation, which presumably gives the classical motion of quarks under the strong nuclear force. Problem is, this one is nonlinear and very difficult to solve so it's hard to connect it with observation. The non-abelian nature of SU(3) produces non-linear terms in the differential equations for the curvature (which happen to vanish for abelian groups such as U(1)), making the problem analytically untractable. But I have to wonder: if we were able to solve the PDEs that determine the SU(3) bundle curvature and look at projected geodesics, would we find that an isolated pair of quarks is confined? I've long dreamed of extending computational electrodynamics to this problem, but that's a very difficult numerical problem. Perhaps in my retirement... my home computer should be fast enough by then.

    Anyway this discussion focuses the question: is confinement due to the non-abelian nature of SU(3) (if confinement shows up at the classical level as described above, the answer is certainly "yes"), or does confinement only arise after quantization, though presumably still only for non-abeliean gauge groups? At the moment our only insight is after quantization, via renormalization group methods, where you find the beta function. It would be nice to see another kind of analysis succeed. As an aside this suggests that renormalization group techniques in QCD may turn out to be useful in analyzing solutions of classical non-linear partial differential equations.

    Does anyone know of anyone who's looked at these issues?

  17. Steve my guess is no. You can get confining theories even for abelian models in lower dimensions.

    Also if you stick things on a lattice, confinement/nonconfinement is really seen as a sort of phase transition.

  18. Hi Steve:

    That's an interesting question, one I've asked myself, and I still have no really satisfactory answer for this. Sure, you can write down classical SU(3) Yang-Mills theory, and I think there are general ways to find solutions to this kind of theory. The formalism for this is the frame bundle over the base-manifold how you describe.

    You need in addition however the currents. I.e. the frame bundle gives you the gauge fields (via the connection), but not the fermions. You need to add these to get the full theory. Please note, the Lorentz force law is for the motion of the current (electrons), yet classical ED doesn't have the Dirac equation for their motion, so is actually incomplete.

    Now what is the question you actually want to address with the confinement. It's the question what happens if you pull apart two (or three) quarks that are together color neutral. The standard picture in QCD is the strong interaction grows stronger with the distance, forming a flux tube that will eventually, when stretched sufficiently, break and fall apart into pieces that are however again color neutral. I don't know how one could address this question classically since it relies on particle-antiparticle creation which is an explicitly quantum feature. What I've been wondering though is whether one can derive classically the observed potential with the 1/r term and the linear growth at larger distances (@Stefan: I keep forgetting the name of that potential, do you know what plot I'm talking about?).

    Also, I'd like to mention that the sign of the beta coefficient depends not only on the N in SU(N) but also on the number of flavors in the theory. I think if there were more quark flavors (possibly heavier than observed), the beta function would eventually change sign, so the question of whether abelian or non-abelian can't be the full story.



  19. Hi Bee - First off I'll agree: there's a really neat question here involving an awful lot of physics, and one way to address it is to try to separate out some simpler parts and see if the interesting stuff is still there.

    I want to clarify a couple things: The picture I'm referring to is not on the frame bundle, it's instead a principle bundle with the appropriate gauge group. The fermion fields are then sections of this bundle, and the gauge fields (e.g. gluons) are connections on this bundle. The geometry (curvature etc) in this bundle is independent of the curvature of the base manifold. The frame bundle is intimately related to the tangent space of the base manifold and contains the geometry of the base manifold. Frame bundle geometry is where you find standard general relativity. In gauge theory the bundle space is this very abstract thing whose ontological status begs all kinds of questions regarding, for example, the interpretation of quantum theory.

    But the mathematics works remarkably well, which is the point of the Lorentz force law example. In other words the only point of mentioning the Lorentz force law is that this non-geometrical equation (in the old way of thinking) can be derived from a geometrical picture in exactly the same way that the motion of a particle under gravity can be explained by geometry in General Relativity.

    Hmm... I'm having a hard time finding a nice website on the geometry of fiber bundles... Wikipedia is lacking in this regard. Perhaps that's a call for action!

    Now what is the question you actually want to address with the confinement. It's the question what happens if you pull apart two (or three) quarks that are together color neutral.

    Exactly! But then you talk about flux tubes breaking, which lands us squarely in the realm of quantum field theory. I don't have any issues with the standard QFT picture of confinement, I just want to tease out the parts. Clearly flux tube breaking cannot happen at the "classical" non-quantized field level. But yes, a great result on the classical side would be to discover that the behavior of the projected geodesics is well described by a 1/r -> constant potential. But I'd be happy just showing the classical energy of the interaction diverged with increasing distance.

    Re the beta function and its dependence on the various N's, again that's in the context of full quantum theory. Here the situation may well be much richer than the simpler classical theory. The conventional wisdom in QCD is that confinement is somehow deeply related to the non-abelian nature of the gauge group, but is it necessary? sufficient? I'm not sure if these things are known.

    For Anonymous, yes, different things happen in low dimensions, but there's no gravity in 2+1 dimensions either: empty space is not curved in 2+1 dimensional General Relativity. So I don't know what conclusions to draw from abelian confinement in low dimensions. Your comment about what happens on the lattice is certainly correct, but that's in QFT-land, evaluating Feynman integrals. I'm simply looking at another way to view this problem.

    This is a great blog with great comments!

  20. Hi Steve:

    Sorry, the 'frame bundle' was a typo, I meant to say 'fibre bundle', apologies for the confusion.

    In other words the only point of mentioning the Lorentz force law is that this non-geometrical equation (in the old way of thinking) can be derived from a geometrical picture in exactly the same way that the motion of a particle under gravity can be explained by geometry in General Relativity.

    Yes, that's the idea of KK theory. You can do the same thing with every gauge group (that's been done some time in the early 70ies I believe). Just put the appropriate bundle over the manifold with the right symmetry etc. That however doesn't give you the fermions as I mentioned earlier. You derive the Geodesic motion of the particle, but what is the particle? See, the Geodesic equation isn't actually an independent postulate of GR, it's the curve you obtain for a pointlike source without backreaction (essentially it's a consequence of Energy conservation). If you know the symmetry group is U(1), you can however as well get the Lorentz law by making a minimal coupling of the current to the gauge field in the usual way, and you don't even need to know anything about fibre bundles for that. In principle you can do the same thing for any gauge group, just replace \partial with \partial + A. And yes, you can do that classically - the question is then what is the relation to the quantum theory?



  21. Hi Bee - Great discussion! I can see that to some extent we're talking past each other, and I hope resolving this will be interesting to anyone listening.

    The bundle formalism I'm describing is not Kaluza-Klein, though the superficial resemblance has led many authors to confuse them. KK posits structure beyond the standard model (whatever that is at the time) in the hopes of explaining the features of the standard model. Specifically KK posits new compact spatial dimensions that are just like normal space (or time) but are topologically closed and presumably very small. Thus, as you explain in your excellent blog entry, the spacetime metric is extended with new terms. Of course there are no particle fields in this theory and I've always found it a little sterile.

    The bundle formalism I'm describing does nothing more and nothing less than provide (as opposed to "posit") a mathematical framework that is designed to contain the elements of QED or QCD. In that sense it is nothing more than "just mathematics". The bundle space is completely different from our spatial dimensions and is carefully designed so that its sections are the spinor fields required by QED or QCD. In that sense it is exactly an expression of quantum mechanics.

    If the bundle formalism were built by just putting in by hand whatever is needed to express gauge field theory then it would be an utterly trivial exercise. But if you take the bundle space to allow curvature, that is to have a connection described by classical differential geometry as developed by Cartan and that school, several things fall in your lap:
    - The connection is (sort of) on the tangent space of the fiber in the bundle space (gotta be careful here because the connection is not a tensor, but this captures the idea), so if your particles are spin 1/2 the connection must be spin 1. You've discovered that your force-carrying particles are Bosons! The degrees of freedom of these Bosons is the dimension of the tangent space: 8 for SU(3) (8 gluons!), 1 for SU(1) (1 photon!).
    - The formula for the curvature in terms of your connection is exactly the same as the formula for the "force tensor" = Yang-Mills tensor.
    - All derivatives in your formulas must include the connection A through \partial -> \partial + A.

    What impresses me about all this is the above three items are, so far as I know, relatively ad hoc without the bundle formalism. In the standard model:
    - Force carrying Bosons are posited independently of the fermion fields and their having spin 1 is (so far as I understand) ad hoc to match observation.
    - The formula for the Yang-Mills tensor is completely ad hoc without being motivated by geometry (being, as it is, exactly the formula for a curvature in terms of a connection).
    - Minimal coupling (\partial -> \partial + A) is entirely ad hoc when not motivated by geometry. Even the name "minimal coupling" makes this point, implying that the coupling could have been otherwise.

    So even though the bundle formalism is "just mathematics", and I've had no less than Bruno Zumino tell me that it is useless, I can't ignore what falls out from it. It's true that it has not helped physicists do calculations, but I can't help but point out that the work in gauge theory instantons was done by people deep in this bundle formalism (Atiyah, Singer, Taubes, etc.). Returning to my original comment, what's particularly inspiring to me about it is that it provides a context to explore the classical structure of the theory. I suppose I could have just taken the equations without their geometrical context and solved them anyway, which is what I suppose Zumino was telling me, but the bundle formalism is a mature body of mathematics. We should use it when appropriate.

    Another way to make the contrast between the bundle formalism and KK: the bundle formalism may be useless and a trivial exercise (not my opinion), but it is not wrong so long as the standard model is not wrong. KK can be wrong (which I think it is) even if the standard model is right.

    I've been concentrating on QED and QCD. What about the full standard model, in other words including spontaneously broken SU(2)? Much of the same bundle formalism works (dimension of SU(2) = 3 implies three vector Bosons matching the W+, W- and Z, same minimal coupling, same Yang-Mills formula), but the spontaneous symmetry breaking is added in an ad-hoc way just like in the standard model. It fits, just not in the satisfying way that QED and QCD fit.

    It's been fun, Best to you too

  22. Oops - when I said ... The connection is (sort of) on the tangent space of the fiber in the bundle space... I should have just said ... The connection is (sort of) on the tangent space of the bundle space...


  23. Theres nothing useless about the bundle formalism, thats really the fundamental mathematics of Yang Mills theory. Be sure most HEP theorists are intimately familiar with it.

    Huge progress was made in the eighties using this geometrical picture, particularly with respect to topological field theories.

    Not that it really told us something new, just that it made the physical picture easier and allowed/hinted us to use some useful cohomology machinery.

    Anyhoo. I can't see how you would derive the 1/r potential classically in a correct fashion (say for the process u + dbar --> u + dbar) you sort of need to use the feynman rules to derive the potential (and its a trivial two line calculation). Of course this is perturbation theory, and you will miss the confining potential so its hard to see how one can even expect to recover it classically.

  24. Hi Steve:

    Thanks for your explanation. I guess I share Anonymous' confusion above. I certainly don't say the fibre bundle formalism is useless. I as many others find it attractive because of the similarity to GR. Yes, if you construct your gauge bundle with a Lie-Group then the tangential space of the group is the Lie-Algebra. Fields over the manifolds are cuts in the bundle. The connection is gauge invariant under the symmetry by construction. And yes, the gauge fields are vectors over the manifolds. For the fermions things get somewhat more nasty.

    The bundle space is completely different from our spatial dimensions and is carefully designed so that its sections are the spinor fields required by QED or QCD. In that sense it is exactly an expression of quantum mechanics.

    Where does the 'Q' come from? Nothing of what you've said referred to quantization. Best,


  25. Classical QCD (with massless fermions) is a scale invariant theory, so it cannot have a string tension or a mass gap. Scale invariance is broken by quantum effects. The QCD scale comes from the running coupling and dimensional transmutation.



  26. Good morning -

    For the most recent Anonymous - I would be very surprised if we could find a 1/r or any other specific potential with a classical analysis. The equations are very non-linear and it's hard for me to imagine anything other than a numerical study making headway. As I said before, I would be content with purely qualitative results, like "the energy diverges as the quarks separate" or "the initially separating geodesics always re-converge". Even that is a really hard numerical problem, likely technically harder than black hole merger in numerical general relativity.

    I don't know the observational status of the 1/r potential - does that have observational support? Can our measurements tell the difference between that and a similar but not so simple confining potential?

    For Bee - Yes, you've very succinctly summarized the bundle formalism in many fewer words than I. The "Q" in this case (aside from wanting to avoid defining new terms) rests on the particle fields in the bundle formalism being solutions to the equations of quantum mechanics, e.g. the Dirac equation. Them being such solutions is not part of the geometry, by the way, it's imposed from the outside. Such fields are continuous objects that obey partial differential equations which, in my world, we call "classical". In other words the bundle formalism is in the world of first quantization, prior to second quantization.

    Thomas' point is interesting and goes beyond my competence. Is the classical theory really scale invariant? Isn't there an implied coupling constant in the minimal coupling? I imagine I've learned about this scale invariance in my education, but that was a long time ago and I missed the implication Thomas is making. I'll check my library, in the meantime are there any good descriptions of QCD scale invariance you can point to?

    I don't think anyone here implied that the bundle formalism is useless - that memory of Zumino's comment sticks in my head for obvious reasons. He may not have said it as strongly as I remember, and it was about 1985. Perhaps the results since then that Anonymous alludes to have changed his mind.

    I've looked at a few books since last night and see that some authors (including Bleeker who I reference in my first comment) define Kaluza-Klein as any account based on the product of spacetime with another space. I think this obscures the very important difference I described in my last comment, between Kaluza-Klein as originally intended and as described in Bee's post and general fiber bundles. I think it best to reserve the label "Kaluza-Klein" for taking spacetime to be a product of the normal 3+1 manifold of GR with some other spatial thing, excluding exotic spinor-valued associated bundles found in the standard model bundle formalism. What do you think?


  27. Thomas' point is interesting and goes beyond my competence. Is the classical theory really scale invariant?

    If it's massless.

  28. From Bee: If it's massless.

    Interesting point here: pure QCD in the standard model is massless, but is my proposed (and probably too-difficult-to-do) classical study constrained to be massless? I think it would be easier with massive quarks. The formalism doesn't care.


  29. Well, I don't know. As I said earlier, since the confinement comes with particle-antiparticle production I can't quite see how one could address it classically. I mean, one can investigate the classical solutions but I wouldn't know really what to conclude from it. I think there are some solutions known to classical YM theory, let me see if I find a reference, I had one at hand recently...

    Here is one:

    PRL 40, 22 p. 1411 (1978)
    Sikivie and Weiss
    "Screening solutions to Classical Yang Mills Theory"


    and here is another one

    Comm Math Phys 159, 593 (1994)
    Sniatycki and Schwarz
    "The Existence and Uniqueness of Solutions of Yang Mills Equations with Bag Boundary Conditions"

    Then I have a couple of reference about classical SU(2), in case you're interested.



  30. Dear all,

    thanks for these interesting contributions!

    As for confinement, I am not sure how one would measure it in a purely classical theory - would the Wilson Loop criterion work?

    I guess as a starter, it would be "natural" - as coming from electrodynamics - to look at the theory in 3+1D and with massive charges - after all, electrons have mass as well? It would probably be very interesting if one could deduce something like the Cornell potential (the -α/r + κr potential) from classical Yang-Mills, but I have never heard about something like that. Cornell is not a solution to the classical equations, as far as I know.

    On the other hand, if I am not completely confused, the dual superconductor mechanism is classical, isn't it? It uses monopoles. which appear as classical solutions - so this may work? However, I am not sure about the role of temperature?

    As I've said before, spontaneous particle creation may not be equivalent to confinement, as it is expected to happen also in supercritical QED, and I am not sure if this then is confined...

    BTW, since this has not been mentioned, the equation for classical particles coupled to Yang-Mills, as analogous to Dirac-Maxwell, is the the Wong equation (Nuovo Cimento 65A (21 Feb 1970) 689-94) ... that may be a useful keyword for further literature search...

    Best, Stefan

  31. Thanks, Bee, for the papers. They are interesting and it's nice to know the Cauchy problem has a solution in YM. Not sure these papers directly address confinement... But I will enjoy reading them.

    Thanks Stefan for the comments. I've only seen Wilson loops in use in a lattice QCD context, but they are classical objects that apparently were designed to address the confinement problem. According to the links I found, though, so far there has been no success. Perhaps someone out there knows more.

    My whole point about projecting classical geodesics of the curved QCD bundle is this provides an alternative classical way to explore confinement. It's rather prosaic: somehow study what happens to the (classical) particle motion as, say, a u and ubar are separated. I think the only hope is numerical and I'm sure that's very difficult. All I'm pointing out is it's another avenue of approach. No particle creation, certainly not quantitatively physically correct, just a toy model that may or may not exhibit confinement.

    I had not heard of the Wong equation - Thanks for pointing to that. The modern references I find give it in a form that (I'm pretty sure) is exactly what Bleeker derives as the projection of geodesics in the curved bundle space! So yes, my scheme about projected geodesics can be thought of as just using Wong's equations to explore classical confinement. Bleeker apparently didn't know about it either since he does not reference Wong. I'm curious how Wong derived it, but I won't be near an appropriate library until January and so far as I can tell Wong's paper is not online. I look forward to seeing it.

    Best to all

  32. Don't mind me.

    I've been criticized on my thinking of circles many times:)

    The KK tower is equally interesting, although this had been dismissed from the conversation early on.

    The familiar extended dimensions, therefore, may very well also be in the shape of circles and hence subject to the R and 1/R physical identification of string theory. To put some rough numbers in, if the familiar dimensions are circular then their radii must be about as large as 15 billion light-years, which is about ten trillion trillion trillion trillion trillion (R= 1061) times the Planck length, and growing as the universe expands. If string theory is right, this is physically identical to the familiar dimensions being circular with incredibly tiny radii of about 1/R=1/1061=10-61(should allow html sup tags) times the Planck length! There are our well-known familiar dimensions in an alternate description provided by string theory. [Greene's emphasis]. In fact, in the reciprocal language, these tiny circles are getting ever smaller as time goes by, since as R grows, 1/R shrinks. Now we seem to have really gone off the deep end. How can this possibly be true? How can a six-foot tall human being 'fit' inside such an unbelievably microscopic universe? How can a speck of a universe be physically identical to the great expanse we view in the heavens above? (Greene, The Elegant Universe, pages 248-249)

    I had also been criticized on this as well. It can all really be confusing.

    A circle of radius r has a curvature of size 1/r. Therefore, small circles have large curvature and large circles have small curvature. The curvature of a line is 0. In general, an object with zero curvature is "flat.

    Hmmm....speed of light is flat?

  33. Hi Steve:

    Yeah, I just thought the refs could be useful to see what others have done, techniques etc. Good luck, and let us know from time to time how its going! Best,



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