Mass gap in Yang-Mills theories

[humanino]

I am back in Paris, I now have access to my collected notes, I will soon try to post some more informations on the instanton approach.

Yet after the nice "Asymptotic freedom" Royal Swedish Academy of Sciences award we have another good piece of news today
The origin of the mass gap in QCD
V. Gogohia

We have unambiguously established the dynamical source of the mass scale parameter (the mass gap) responsible for the large scale structure of the true QCD vacuum. At the microscopic, Lagrangian level it is the nonlinear fundamental four-gluon interaction. At the level of the corresponding equation of motion for the gluon propagator, it is the two-loop skeleton contribution into the gluon self-energy, which contains the four-gluon vertices only. The mass gap and corresponding infrared singularities are "hidden" in this term, and they show explicitly up when the gluon momentum $q$ goes to zero. The full gluon propagator thus unavoidably becomes severely (i.e., more singular than $1/q^2$) singular in the infrared than its free counter-part. Taking into account the distribution nature of severe infrared singularities, the gluon confinement criterion is formulated in manifestly gauge-invariant way.

 

[selfAdjoint]

Hah! So maybe,contrary to the Clay problem, mass gap isn't a necessary feature of generic Yang-Mills, but holds in phenomenology due to demonstrable physics!

 

[humanino]

Yes sA, I am aware this is nothing close to the prize of the Clay Institute.

 

[nrqed]

This is indeed one very interseting issue. The mass-gap must be present because in YM-gauge-theory the elementary particles are always massless. The reason for this is that mass <mixes the two different chiralities. BUT the strong force has a short range. In order to have this property the messenger-particles are to be massive. The mass-gap clearly states that there is to be a certain minimum mass for those carriers. How this mass is generated is the BIG question of the Higgs-field. The Higgs-particles are the mass giving particles once the symmetry of the groundstate or vacuumstate is spontaniously broken. This Higgs-field has not yet been observed though, only estimates on it's energy are made at the Fermilab

regards
marlon

I am not sure I understand what you mean. If I follow your reasoning, you seem to be saying that

short range of strong force -> massive carriers -> mass generated by SSB

But that's not the explanation at all for the short range of the strong force. The gluons are massless, they don't have a mass generated by the SSB. The short range is explained by the nonlinear nature of QCD.

Pat

 

[marlon]

I am not sure I understand what you mean. If I follow your reasoning, you seem to be saying that

short range of strong force -> massive carriers -> mass generated by SSB

But that's not the explanation at all for the short range of the strong force. The gluons are massless, they don't have a mass generated by the SSB. The short range is explained by the nonlinear nature of QCD.

Pat

First of all it is true that gluons are generally considered to be massless, altough mass-values of a few MeV can also be possible !

After spontaneous breakdown of symmetry gluons DO acquire mass. The process responsible for this is dynamical mass generation. The best example (of a massive-gluon-state...) are the glueball-condensates (constructed solely out of gluons) which give rise to an effective-gluon-mass without breaking gauge-invariance.

Ofcourse, i have to be honest and say that the gluons themselves are massless and we are talking about an EFFECTIVE mass here. That was my point.

I agree on the non lineair nature though...

regards
marlon

 

[selfAdjoint]

Gluon confinement without having mass is still not fully explained in QCD, although a lot of progress on the answer has been made. It is BELIEVED that a reverse effect like the asymptotic freedom, which has just won three physicists the Nobel proze, is responsible.

 

[marlon]

Gluon confinement without having mass is still not fully explained in QCD, although a lot of progress on the answer has been made. It is BELIEVED that a reverse effect like the asymptotic freedom, which has just won three physicists the Nobel proze, is responsible.

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 want to know more, please consult the last link i provided in the "elementary particles presented thread"

regards
marlon

 

[marlon]

Hey Marlon, what's up old dude !

I am not sure that the mass gap is accountable by Higgs field. I heard stuff like "10% of the mass of the proton is due to the higgs field. The 90% remaining is the weight of the glue". But I don't undestand it.

Hi, Humanino...

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 matter-particles. Now ofcourse massless gauge-bosons 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 self-interactions of such particles. Just look at how gluon-condensates are formed out of dynamical mass-generation. When you are talking about the glue, you are basically referring to this kind of mass. I am sure you know these things like effective-mass in solid state physics and the quasi-particles in many-body-problems. These particles reduce one many-body-problem 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 self-energy of one particle and "forgetting about all the other surrounding particles". This final particle (the quasi-particle) is then considered to be free at first extent...

regards
marlon

 

[marlon]

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

marlon

 

[nrqed]

I am not sure I understand what you mean. If I follow your reasoning, you seem to be saying that

short range of strong force -> massive carriers -> mass generated by SSB

But that's not the explanation at all for the short range of the strong force. The gluons are massless, they don't have a mass generated by the SSB. The short range is explained by the nonlinear nature of QCD.

Pat

First of all it is true that gluons are generally considered to be massless, altough mass-values of a few MeV can also be possible !

After spontaneous breakdown of symmetry gluons DO acquire mass. The process responsible for this is dynamical mass generation. The best example (of a massive-gluon-state...) are the glueball-condensates (constructed solely out of gluons) which give rise to an effective-gluon-mass without breaking gauge-invariance.

Ofcourse, i have to be honest and say that the gluons themselves are massless and we are talking about an EFFECTIVE mass here. That was my point.

I agree on the non lineair nature though...

regards
marlon

Hi Marlon, I agree that it is dynamical mass generation. My point was that this has nothing to do with the Higgs mechanism (which seemed to be what you were hinting at in your previous post). That was the only point I wanted to make.



Pat

 

[nrqed]

Hi, Humanino...

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 matter-particles. Now ofcourse massless gauge-bosons can also acquire mass (yes, i am primarily referring to GLUONS here via interaction with the Higgsfield) ...

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

\vec D_\mu \Phi = (\partial_\mu - {i\over 2} g \vec \tau \cdot \vec A_\mu -{i \over 2} g&#039; B_\mu) \Phi (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

 

[marlon]

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 QED-thing. There are models (like the dual Landau Ginzburg-model) that predict massive gluons via the Higgs-field. The criticism on this model is the fact that this mass value is quite low and the Higgs-particles themselves have low mass. The big question then is ofcourse : if this mass is so low, how come we did not see these Higgs-particles yet. I am sure you will agree this is a very powerful counter-statement. Problem is that this model does the BEST job in describing the nature of meson and baryon-configurations. Probably (this is just my thought, so it sure ain't no FACT) the idea of magnetic monopoles forming the flux-tube is the correct way to look at confinement (because of elegance and more over SYMMETRY) yet problems arise with how to construct the linear interquark-potential.

regards
marlon

 

[humanino]

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

(from intro §6)
We have seen in the previous chapters that in Abelian systems the problem of charge confinement is solved by instantons.
In Non-Abelian theories instantons are also present. However, due to the large perturbative fluctuations, dicussed in Chapter 2, it is difficult to judge whether they play a decisive role in forming a mass gap and a confining regime. In such theories we had a kind of instanton liquid which is difficult to treat. It is possible that due to some hidden symmetries, present in these systems, instantons may form a useful set of variables for an exact description of the system, but this has not yet been shown.
At the same time, due to the fact that instantons carry non-trivial topology (they describe configurations of the fields which cannot be "disentangled"), some manifestations of instantons cannot be mixed up with perturbative fluctuations.

[...]
(from end of §6.2)
As happened in the case of n-fields, the instanton contribution has an infrared divergence. This implies that in the multi-instanton picture, individual instantons tend to grow and to overlap. The vaive dilute gas approximation is certainly inapplicable then, and we should expect somethig like dissociation of dipole-like instantons to their elementary constituents, as happened in the case of n-fields. However, even one loop computations on the multi-instanton background have not yet been performed, and nothing similar to the Coulomb plasma of the previous section has been discovered. This is connected partly with the fact that multi-instanton solutions have not yet been explicitely parameterized up to now. I expect many interesting surprises await us, even on the one loop level, in this hard problem.

[...]
(from end of §6.2)
So, our conclusion is that on the present level of understanding of instanton dynamics, we cannot obtain any exact dynamical statements concerning Non-Abelian gauge theory. In the case of n-fields the situation is slightly better, since we were able to demonstrate the apearance of the mass-gap on a qualitative level. Even in this case one would like to have much deeper understanding of the situation. There are reasons to believe that some considerable progress will be achieved in the near future. In the case of gauge fields we have to pray for luck.

At the same time, the existence of fields with topological charge has a deep qualitative influence on the dynamical structure of the theory.

[...]
(from end of §6.3)
(...) exchange of a massless fermion pair leads to long-range forces between instantons and anti-instantons. The result of this may have several alternative consequences. The first one is that since (6.87) implies quenching of large fluctuations in the presence of massless fermions, the system looses the confining property and we would end up with massless gauge fields together with fermions. This option seems highly improbable to me on the basis of some analogies and some model considerations. However, I am not aware of any strict statements permitting us to reject it.
The second possibility, which in my opinion is realized in the theory, is the following. Due to the strong binding force between fermions the chiral symmetry gets spontaneously broken and as a result the fermions acquire mass. After that has happened, the long range force between instantons and anti-instantons disappears, being screened by the fermionic mass term in the effective lagrangian. The only remaining effect of anomalous non-conservation will consist of giving a mass to the corresponding Goldstone boson.
There is also another improbable option, namely that instantons get confined but some type of large fluctuations, not suppressed by fermions, disorder the system.

(follows a short but excellent account on the \theta-term and the (failure of the) search for axion particle)

 

[Haelfix]

I'm a little skeptical of that mass gap paper.

Probably b/c I am 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.

 

[humanino]

Progress towards understanding the mass-gap in QCD :
Precise Quark Mass Dependence of Instanton Determinant

The fermion determinant in an instanton background for a quark field of arbitrary mass is determined exactly using an efficient numerical method to evaluate the determinant of a partial wave radial differential operator. The bare sum over partial waves is divergent but can be renormalized in the minimal subtraction scheme using the result of WKB analysis of the large partial wave contribution. Previously, only a few leading terms in the extreme small and large mass limits were known for the corresponding effective action. Our approach works for any quark mass and interpolates smoothly between the analytically known small and large mass expansions.

Gerald V. Dunne, Jin Hur, Choonkyu Lee, Hyunsoo Min (hep-th/0410190)

 

[Kea]

Confinement

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.

regards
marlon

I believe the best explanation now for confinement begins with
W. P. Joyce "Quark state confinement as a consequence of the
extension of the Bose-Fermi 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 so-called 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 non-abelian
cohomology.

More on all this elsewhere, and at a later date.
Cheers
Kea

 

[Kea]

crazy idea

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 Klein-Gordon equation (ie. adding mass).

Ross Street, in his classic '87 paper on Oriented Simplices, explains
why non-Abelian cohomology in higher dimensions is difficult.
This paper lays out the structure of a 'nerve' of a strict n-category.
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

 

[arivero]

A question: in which measure is true that to explain the mass-gap 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?

 

[Kea]

proton mass

A question: in which measure is true that to explain the mass-gap 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?

I'm saying we can't explain the mass gap without quantum
gravity - and if we understand that, the mass of the proton
should follow.

 

[Haelfix]

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 1-2% 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.

 

[Kea]

It means we have the right equation, solving it analytically is what now remains to be done.

Solving it analytically might require quantum gravity.

 

[Haelfix]

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

 

[marlon]

Solving it analytically might require quantum gravity.

Why is that ? Never heard of this, though...
marlon

 

[Kea]

Hi

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 non-commutative
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 Klein-Gordon
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
Klein-Gordon equation solutions then belong to a second cohomology
group.

Naively at least, therefore, a quantisation of this origin of mass
involves a non-Abelian 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 non-Abelian 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, Yang-Mills 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 so-called
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 Yang-Mills 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 Yang-Mills.

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).

 

[selfAdjoint]

Kea, the trouble with all your references is that nobody who doesn't have a graduate university library can see them. Isam's stuff is online at the arxive; is any of the rest of it. Anything at the level of Joyce?

 

[Kea]

references

Unfortunately, no.

On-line books on topos theory include Barr and Wells
and Goldblatt. John Baez's website is a good source
of references.

Sorry everybody.

 

[Kea]

ref

Here

http://www.cwru.edu/artsci/math/wells/pub/ttt.html

 

[marcus]

I believe the best explanation now for confinement begins with
W. P. Joyce "Quark state confinement as a consequence of the
extension of the Bose-Fermi 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 so-called 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) ...

Hello everybody, selfAdjoint asked for arxiv references for Joyce work because they are accessible to those of us who don't have a university library close by. Here are some:

http://arxiv.org/find/hep-th/1/au:+Joyce_W/0/1/0/all/0/1

hep-th/0307047
Quark confinement without a confining force
P.S. Isaac, W.P. Joyce, J. Links
5 pages

hep-th/0307046
An algebraic origin for quark confinement
P.S. Isaac, W.P. Joyce, J. Links
23 pages

hep-th/0306256
Quark State Confinement as a Consequence of the Extension of the Bose--Fermi Recoupling to SU(3) Colour
W. P. Joyce
15 pages, 4 figures

this list gives not only the one which Kea cited but also two others which are more recent---not yet in hardcopy
I have not examined these papers personally, but only helping as
an assistant librarian (I think I would not understand the papers in any case, or grasp their applicability)

regards to all

*

 

[Kea]

also

http://www.arxiv.org/find/math/1/au:+Joyce_W/0/1/0/all/0/1

 

[selfAdjoint]

Thank you both Kea and Marcus. I will start looking at these tomorrow. I also just found out about Google Scholar (www.scholar.google.com). I typed in topos quantization and got some very interesting results. There's hope for the old presheaf guy yet.