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Mass gap in YangMills theories 
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#37
Oct704, 04:13 PM

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Correct, The best model up till now that describes the confinement phenomenon is the dual abelian higgs model. In this model gluons with both colour and NO colour are predicted. So not all gluons undergo confinement since not all gluons contain colour. These last gluons are called the abelian gluons. If somebody wanna know more, please consult the last link i provided in the "elementary particles presented thread" regards marlon 


#38
Oct804, 10:23 AM

P: 4,006

You are right on this statement although the 10/90 comparison in weight is something i have never heard of... Your post is a very good one in my opinion since it insists on making clear the different kinds of mass we need to look at in these subjects. For starters, we have the real physical mass that we measure in experiments. this mass is gained by the Higgsfield in the case of real matterparticles. Now ofcourse massless gaugebosons can also acquire mass (yes, i am primarily referring to GLUONS here via interaction with the Higgsfield) This mass is of a different kind though since it is called effective mass. This is mass generated by the selfinteractions of such particles. Just look at how gluoncondensates are formed out of dynamical massgeneration. When you are talking about the glue, you are basically referring to this kind of mass. I am sure you know these things like effectivemass in solid state physics and the quasiparticles in manybodyproblems. These particles reduce one manybodyproblem that we cannot solve, by many one body problems that we CAN solve by lumping together all the interactions of one particle with many surrounding particles and putting this into the selfenergy of one particle and "forgetting about all the other surrounding particles". This final particle (the quasiparticle) is then considered to be free at first extent.... regards marlon 


#39
Oct804, 10:25 AM

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Do recall that the massive gauge bosons of the weak force have a real physical mass....
marlon 


#40
Oct804, 12:33 PM

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Best regards, Pat 


#41
Oct804, 01:52 PM

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Now I am quite confused. Could you point to me the interaction term between the gluons and the Higgs in the Standard Model? As far as I know, the only interaction between gauge bosons and the Higgs is through the covariant derivative [itex] \vec D_\mu \Phi = (\partial_\mu  {i\over 2} g \vec \tau \cdot \vec A_\mu {i \over 2} g' B_\mu) \Phi [/itex] (see Cheng and Li, page 349). Where the A's and the B are the fields which will become the W^+, W^, Z^0 and the photon after SSB. There is no coupling between the Higgs and the gluons. So what am I missing? Regards Pat 


#42
Oct1004, 04:21 PM

P: 4,006

I never said that, sorry.
Gluonmass is EFFECTIVE mass, that is my point. Your formula is the standard one in QFT but not complete for the needs of QCD. This is a QEDthing. There are models (like the dual Landau Ginzburgmodel) that predict massive gluons via the Higgsfield. The criticism on this model is the fact that this mass value is quite low and the Higgsparticles themselves have low mass. The big question then is ofcourse : if this mass is so low, how come we did not see these Higgsparticles yet. I am sure you will agree this is a very powerful counterstatement. Problem is that this model does the BEST job in describing the nature of meson and baryonconfigurations. Probably (this is just my thought, so it sure ain't no FACT) the idea of magnetic monopoles forming the fluxtube is the correct way to look at confinement (because of elegance and more over SYMMETRY) yet problems arise with how to construct the linear interquarkpotential. regards marlon 


#43
Oct1104, 11:45 AM

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So : here are quotations from "Gauge Fields and strings". Even the negative results are interesting, remember that it is an excellent book to read.



#44
Oct1104, 07:47 PM

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P: 1,687

I'm a little skeptical of that mass gap paper.
Probably b/c im partial to lattice QCD, and its rather apparent that the way the gap appears in that (admittedly numerical) formalism strikes me at odds with some of the claims of the paper. I'll reread it again more thoroughly later. 


#45
Oct2704, 09:01 AM

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Progress towards understanding the massgap in QCD :
Precise Quark Mass Dependence of Instanton Determinant 


#46
Nov904, 05:43 PM

P: 859

W. P. Joyce "Quark state confinement as a consequence of the extension of the BoseFermi recoupling to SU(3) colour" J. Phys. A: Math. Gen. 36 (2003) 12329  12341 This work can now be fit into a much more general framework, either via Joyce's socalled omega algebras (recent work) or equivalently from the perspective of higher categories where these algebraic structures appear naturally. Moreover the mathematics has a close tie to LQG (this is mostly unpublished) and a big motivation for it was instantons, or rather Twistor theory, because the biggest hurdle seemed to be a sufficiently rich nonabelian cohomology. More on all this elsewhere, and at a later date. Cheers Kea 


#47
Nov1204, 09:49 PM

P: 859

Penrose developed Twistor theory from a deep understanding
(I believe) of GR. The correspondence is in terms of sheaf cohomology. This was extended to H2 by Hughston and Hurd in the 80s to study the KleinGordon equation (ie. adding mass). Ross Street, in his classic '87 paper on Oriented Simplices, explains why nonAbelian cohomology in higher dimensions is difficult. This paper lays out the structure of a 'nerve' of a strict ncategory. But for reasons I won't go into here, physics seems to require much more than this: a fully higher categorical cohomology, which is still being developed by Street and others. The question is: what does this have to do with the mass gap issue? Recall that Heisenberg said that he was led to the uncertainty principle by recalling Einstein's words to the effect "the theory always dictates what is observable". In other words, the classical theory is reproduced in a very different (and complicated) way to the idea of taking 'hbar to zero'. For instance, in a topos one must be careful to define what one means by the reals, because the Cauchy reals and Dedekind reals aren't the same. Well, the crazy physical idea...... the classical limit we should be thinking about is something to do with twistors. Now it turns out that Roy Kerr discovered his solution to Einstein's equations by thinking about this sort of maths. Anyway, if there IS NO 'fixed background', which of course there isn't, then the mass gap that we have in the MORE FUNDAMENTAL unified theory goes away because the only 'proper' classical solutions concentrate the mass in things like Kerr black holes. Kea 


#48
Nov1304, 09:50 AM

PF Gold
P: 2,932

A question: in which measure is true that to explain the massgap implies to explain the mass of the proton? It is sort of assuming that a proton and a glueball are almost the same thing, isn't it?



#49
Nov1304, 03:31 PM

P: 859

gravity  and if we understand that, the mass of the proton should follow. 


#50
Nov1604, 11:31 PM

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Sorry that doesn't make much sense. Gravitational effects are completely negligable at that length scale. Even if the mass gap is seen via perturbative effects, gravity will miss it order by order in the series. However if gravity did couple to the theory in some way, it would not only lead to some complicated lagrangian, but presumably incorporate a host of gauge symmetry breaking terms to make it feasible. Moroever, we would have to introduce fine tuning terms many orders of magnitude uglier than the dual abelian higgs model.
As clearly stated in the millenium problem writeup, most people expect the mass gap to appear in the quartic interaction sector of the theory (A ^ A)^2. Not only b/c of duality transitions, but also b/c it would make sense and generalize simpler toy model lagrangians, where existence of mass gaps have been rigorously shown to exist. Finally, the mass gap has been solved by computer and found to be within 12% of the predicted value, via lattice QCD. It means we have the right equation, solving it analytically is what now remains to be done. Adding adhoc speculation about extra non field theoretic interactions is more or less ruled out. 


#51
Nov1704, 03:09 PM

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#52
Nov1804, 10:53 AM

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P: 1,687

There are no gravity terms (either mass terms or interactions) in the theory, why do you think it would need quantum gravity?



#54
Nov1804, 03:29 PM

P: 859

Heisenberg said that particles were not fundamental, because every
particle in some sense contained all others. It appears that the same should be true of Bekenstein's atoms for spacetime. There is no physical difficulty in thinking of spacetime degrees of freedom in a quantum manner. The difficulty arises in coupling matter and spacetime degrees of freedom in a mathematically sensible way. It appears that this is not at all possible unless one addresses some basic issues in quantum logic. Categorical internalisation is an essential element here. There is mounting support for this point of view from studies of, for instance, the Hopf algebra structure of renormalisation (see Connes and Marcolli) and its connection with noncommutative geometry. Twistor theory is one investigation that attempted to respect background independence, and which played an important role in the development of state sum models for quantum gravity. The first interesting step towards a modern category theoretic understanding of mass is perhaps the study of the KleinGordon equation in the Hughston and Hurd paper, in which they combine two solutions to the massless equations for spin s particles thought of as elements of a sheaf cohomology group on a twistor space. The KleinGordon equation solutions then belong to a second cohomology group. Naively at least, therefore, a quantisation of this origin of mass involves a nonAbelian sheaf theoretic second cohomology group. And an understanding of such an object leads one inexorably in the direction of topos cohomology. The first cocycle condition may be thought of as a triangle. Such triangles make sense in any category, so the coefficients for H1 may be generalised, in particular to nonAbelian groups. The difficulty arises in understanding categories deeply enough to develop a sufficiently subtle higher dimensional analogue. The interplay of categories and logic (ie. topos theory) in physics has already been carefully considered by Markopoulou in the context of causal sets, and Isham and others in the context of quantum theory. A topological space is a category of objects the open sets, with inclusions for arrows. For example, the celestial sphere of the twistor correspondence is considered as such a category. Already in two dimensions, YangMills theory involves some beautiful combinatorics (see Witten's work). This uses a generalisation of the Abelian localisation principle from equivariant cohomology. Localisation reaches a pinnacle of abstraction in an adjunction between the inclusion of a topos of sheaves into the presheaf category and the socalled sheafification functor (see Mac Lane and Moerdijk). Sheaves are defined with respect to a topology on the base category. As the String theorists like to tell us, path integrals are heinously complex and unsmooth things. They are now telling us that maybe 4D YangMills is pretty amazing all on its own. And they seem to be saying that twistors are cool too. In other words, we want a higher categorical analogue of the evaluation of path integrals like 2D YangMills. The intended interpretation of pieces of categories is that they are geometric entities. Objects are zero dimensional and arrows are one dimensional etc. I won't go into this now. Objects in a category such as Rep(SU(2)) are representation spaces rather than 'particle states', so to capture the notion of a state properly in category theoretic terms it is necessary to internalise this picture further than is normally considered and to replace the Mac Lane pentagon by at least its tricategorical analogue. The truly fascinating thing is that tensor products in higher dimensional categories are no longer stable dimensionally. For the pentagon this leads to a sort of symmetry breaking. This has already been used to explain confinement RIGOROUSLY (see Joyce). 


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