Today in arxiv appeared a relevant paper by Osaka and Berlin groups (see here). This is a really important paper as these two groups merged their data for the lattice computation of the gluon and ghost propagators for SU(3) in the Coulomb gauge. As usual I give here a picture summing up their results about gluon propagator

As you can see one has again the propagator reaching a finite, non-null value in the infrared limit. About the ghost propagator they obtain again a result very near to the case of a free particle. In this case the agreement was perfect for SU(2). For SU(3) there is a tiny disagreement.

I would like to emphasize a couple of points that should be discussed with these results at hand. There is a paper, published on Physical Review Letters, that was claiming that the gluon propagator in the Coulomb gauge should take the Gribov form going to zero at lower momenta. You can find this paper here and here. I think that authors should reconsider their computations as the disagreement with lattice is really serious. All the research lines aimed at a proof of confinement scenarios heavily relying on Gribov ideas seem to have reached a failure point. There could be a lot of reasons for this but it seems to me that, as lattice computations improve, we are left with the only option that the starting points of all these studies are to be reconsidered.

A second point to be made is the completely missing link between people working on the computation of propagators and those working on the spectrum of QCD. I think this is the moment to try to connet these two relevant areas as times are mature to try a consistency check between them. After the failure in view of some functional methods do we have to believe yet that Kaellen-Lehman formula does not apply in the infrared limit?

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11 Responses to Osaka and Berlin merge their data!

It is a beautiful result and I would issue some comments about it that worries me.
As far as I understand, at least in Landau Gauge (but Coulomb should be more particular), one still can save confinement with just positive-violations of gluon propagators… which is known to happen in both scalling and decoupling solutions.
However, it is the “mechanism” responsible for confinement that is at check, since the BRST-invariat Kugo-Ogima and Gribov-Zwanziger scenarios seems unable to describe current lattice data.
Functional methods on their side allow both aforementioned solutions, being a question of setting “correct boundary condictions” (BC) to get the “correct solution”.
The problem (as I see) is that by chosing BC that imply the decoupling solution (which fits well lattice data), one just gets too loose constraints on propagators, which can not (for example) determine the gluonic dynamical-mass ab initio.
Another point would be that decoupling solution seems to imply no RG fixed point at IR at all, so, there is no well-defined alpha_s at IR.
If decoupling is confirmed as correct (let’s wait for data from the Australian group and their new lattice-BRST scheme), IR QCD will also bring a really bad time for AdS/QCD theorists! Since the allegedly QCD almost-conformal invariance used in their models will have to be droped as well.
Summing up: It seems that nowadays “mainstream” does not present a faithful first-principle analytical method able to deal with QCD from UV to IR. If the role of BC had been better understood in Dyson-Schwinger approach maybe it would provide some insight on gluon mass… anyway, it still would be a problem for them to explain why single massiveg-gluons doesn’t “click” on detectors…
Thus, it is the appropriate time for investing intelectual resources in non-usual approaches!
Cheers,

Thank you for the very useful comment. My view on this matter is quite unconventional and not yet accepted by the community being anyhow published.

The present problems arise from considering Gribov ideas essential for understanding confinement. These ideas are proved wrong by lattice computations. BRST invariance does not seem to apply as Yang-Mills theory in the infrared does not appear to behave as a bona fide gauge theory. This situation is really shocking and is the ground for a new, revolutionary understanding of quantum field theory. This is a point I am emphasizing since the start of my blog pointing out that here a breakthrough is in the verge. This is a typical situation where cherished physical principles do not appear to work properly.

My view about gluon mass is that, in the infrared, true excitations of the theory are not gluons anymore but some kind of glue anyhow colorless. There are evidence of this from the experimental side as sigma resonance, proton spin and so on.

Functional methods are good but they should be properly applied due to truncation problems. Ambiguities arising from them are taken us to the present messy situation.

RG fixed point simply does not exist. This is a prejudice formed in the course of time without any sound reason. In order to have confinement a system must be charge neutral. There are some papers on arxiv by Nishijima proving this. Nishijima gives a correct scenario for Yang-Mills even if his papers went unnoticed until now.

I am not really 100% sure, but since decoupling and scalling solutions presents positivity-violation of gluon propagator, both solutions don’t have gluons as physical particles (they are sweept out of the YM spectrum by destructive interference).
On the other hand, colorful massive decoupling solutions would present no real-masses as (gauge-invariant glueballs).
A stuning point is that Elitzur’s theorem does not allow spontaneous local symmetry breaking, however, it can of course be globally broken, producing (screening) massive colorfull gluons… when temperature is turned on.
Thus, gauge symmetry requires gluons to be massless, notwithstanding by breaking BRST this constraint seems to be avoided somehow. What some theorists seems to argue is that the lattice formulation breaks BRST explicitly, so, one should not thrust completely this LQCD results.
Anyway, Adelaide’s group claim to have a well-posed BRST formulation on the lattice and their future results will (probably) settle down this “Functional-Methods vs. Lattice” arguments…
Cheers,

For these solutions a change of gauge boils down to a change of coordinates but the theory is completely decoupled to a product of U(1) components. I think you can check this by yourself.

What is really stunning is that lattice computations seem to recover them. So, I would like to understand if this is just an artifact of lattice or something real.

Dear Marco,
but (if I understood your paper properly) these are gauge-invariant solutions of classical YM, so, most-likely you have reproduced the (colorless) glue-ball spectrum of QCD…
On the other hand, the gluonic masses I was refering to (from decoupling-DSE) are the ones that colorful gluons, in contradistinction to their colorless bounded states, should not have (because they are forbiden by local gauge symmetry).
However, you are wright, LQCD is observing this puzzling effects of massive colorful gluons!

Sorry for moderation but I have a little problem today as you may know.

I just would say that the situation is quite similar to condensed matter physics. In the UV the right particles are colorful gluons while for IR you should find the right ones. My idea is that such classical solutions describe that particles but I would like to be sure that LQCD hits the right solution ( I believe this but belief does not make a truth).

You have said that on using Smilga’s choice one sees that “theory is completely decoupled to a product of U(1) components”. Since it remembers me the gluon propagator on “Maximally Abelian Gauges” I would like to ask you:
Do you precisely mean that SU(2) gets restricted to an U(1) subgroup and, for SU(3), this would be the U(1)xU(1) subgroup?
Cheers,

Let me explain in a simple way what I mean. Classical Yang-Mills theory admits chaotic and integrable solutions. I do a first hypothesis that chaotic solution cannot be used to build a quantum field theory (this is not proved yet but a conjecture if you like).

Now one has to identify a set of integrable solutions that can give you a starting point for a quantum field theory. Smilga, in his book, showed that, for Y-M Hamiltonian equations, one can choose a set with all equal components that has very interesting properties. You have to choose your components is such a way that some are zero and other not. So let us take SU(2). You will have

and your theory reduces to a massless quartic scalar theory. This theory admits classical solutions given by

being

and are two arbitrary constants. This classical solution is massive notwithstanding we started from a massless theory.

This is the starting point. The theory retains its scale invariance as should be.

What does it happen with the gauge term in the Lagrangian? Just a redefinition of coordinates, after the mapping, as you get corrections to the kinematic term of the scalar theory. The ghost field decouples instead leaving a free particle.

So you should consider your symmetries on the vector

Thus, it seems me that by taking a representations of A on SU(2) as: A = a0*I + a_i*sigma_i, with a0²+a_i²=1, you have selected the “Maximall Abelian Generators” of this group, just to say U(1). If I am correct, your choice in:

is composed of two copies of U(1), which means a product of U(1)xU(1), again, it is the Maximal Abelian Group of SU(3).
It raises a interesting point. MAG is a very interesting gauge in many aspects, see for instance:

were authors fit, and compare, gluon-propagators on MAG by Yukawa and Gribov-like forms. They find a “gluon mass” of about 700 Mev.
A very interesting issue would be if one could also fit that LQCD behavior by using your solutions, since, they seem very similar… and, in an affirmative case, which mass could come from that!
Regards,

Thank you very much for your clarifications that give an understanding of such solutions. I think I will use this view in my future papers.

Of course, I have a deep estimation for the work of Cucchieri and Mendes. Their papers are cornerstone on the path toward understanding Yang-Mills theory in the infrared. If my work fits well in their findings I can be only very satisfied.

[…] classical solutions of Yang-Mills equations Following the discussion with Rafael Frigori (see here) and returning to sane questions, I discuss here a class of exact classical solutions that must be […]

Dear Marco,

It is a beautiful result and I would issue some comments about it that worries me.

As far as I understand, at least in Landau Gauge (but Coulomb should be more particular), one still can save confinement with just positive-violations of gluon propagators… which is known to happen in both scalling and decoupling solutions.

However, it is the “mechanism” responsible for confinement that is at check, since the BRST-invariat Kugo-Ogima and Gribov-Zwanziger scenarios seems unable to describe current lattice data.

Functional methods on their side allow both aforementioned solutions, being a question of setting “correct boundary condictions” (BC) to get the “correct solution”.

The problem (as I see) is that by chosing BC that imply the decoupling solution (which fits well lattice data), one just gets too loose constraints on propagators, which can not (for example) determine the gluonic dynamical-mass ab initio.

Another point would be that decoupling solution seems to imply no RG fixed point at IR at all, so, there is no well-defined alpha_s at IR.

If decoupling is confirmed as correct (let’s wait for data from the Australian group and their new lattice-BRST scheme), IR QCD will also bring a really bad time for AdS/QCD theorists! Since the allegedly QCD almost-conformal invariance used in their models will have to be droped as well.

Summing up: It seems that nowadays “mainstream” does not present a faithful first-principle analytical method able to deal with QCD from UV to IR. If the role of BC had been better understood in Dyson-Schwinger approach maybe it would provide some insight on gluon mass… anyway, it still would be a problem for them to explain why single massiveg-gluons doesn’t “click” on detectors…

Thus, it is the appropriate time for investing intelectual resources in non-usual approaches!

Cheers,

Rafael

Dear Rafael,

Thank you for the very useful comment. My view on this matter is quite unconventional and not yet accepted by the community being anyhow published.

The present problems arise from considering Gribov ideas essential for understanding confinement. These ideas are proved wrong by lattice computations. BRST invariance does not seem to apply as Yang-Mills theory in the infrared does not appear to behave as a bona fide gauge theory. This situation is really shocking and is the ground for a new, revolutionary understanding of quantum field theory. This is a point I am emphasizing since the start of my blog pointing out that here a breakthrough is in the verge. This is a typical situation where cherished physical principles do not appear to work properly.

My view about gluon mass is that, in the infrared, true excitations of the theory are not gluons anymore but some kind of glue anyhow colorless. There are evidence of this from the experimental side as sigma resonance, proton spin and so on.

Functional methods are good but they should be properly applied due to truncation problems. Ambiguities arising from them are taken us to the present messy situation.

RG fixed point simply does not exist. This is a prejudice formed in the course of time without any sound reason. In order to have confinement a system must be charge neutral. There are some papers on arxiv by Nishijima proving this. Nishijima gives a correct scenario for Yang-Mills even if his papers went unnoticed until now.

Stay tuned!

Cheers,

Marco

Dear Marco,

I am not really 100% sure, but since decoupling and scalling solutions presents positivity-violation of gluon propagator, both solutions don’t have gluons as physical particles (they are sweept out of the YM spectrum by destructive interference).

On the other hand, colorful massive decoupling solutions would present no real-masses as (gauge-invariant glueballs).

A stuning point is that Elitzur’s theorem does not allow spontaneous local symmetry breaking, however, it can of course be globally broken, producing (screening) massive colorfull gluons… when temperature is turned on.

Thus, gauge symmetry requires gluons to be massless, notwithstanding by breaking BRST this constraint seems to be avoided somehow. What some theorists seems to argue is that the lattice formulation breaks BRST explicitly, so, one should not thrust completely this LQCD results.

Anyway, Adelaide’s group claim to have a well-posed BRST formulation on the lattice and their future results will (probably) settle down this “Functional-Methods vs. Lattice” arguments…

Cheers,

Rafael.

Dear Rafael,

The point is that integrable exact solutions of classical Yang-Mills equations exist that have a massive dispersion relation. See here

http://arxiv.org/abs/0807.2179

For these solutions a change of gauge boils down to a change of coordinates but the theory is completely decoupled to a product of U(1) components. I think you can check this by yourself.

What is really stunning is that lattice computations seem to recover them. So, I would like to understand if this is just an artifact of lattice or something real.

Marco

Dear Marco,

but (if I understood your paper properly) these are gauge-invariant solutions of classical YM, so, most-likely you have reproduced the (colorless) glue-ball spectrum of QCD…

On the other hand, the gluonic masses I was refering to (from decoupling-DSE) are the ones that colorful gluons, in contradistinction to their colorless bounded states, should not have (because they are forbiden by local gauge symmetry).

However, you are wright, LQCD is observing this puzzling effects of massive colorful gluons!

Rafael.

Dear Rafael,

Sorry for moderation but I have a little problem today as you may know.

I just would say that the situation is quite similar to condensed matter physics. In the UV the right particles are colorful gluons while for IR you should find the right ones. My idea is that such classical solutions describe that particles but I would like to be sure that LQCD hits the right solution ( I believe this but belief does not make a truth).

Marco

Dear Marco,

You have said that on using Smilga’s choice one sees that “theory is completely decoupled to a product of U(1) components”. Since it remembers me the gluon propagator on “Maximally Abelian Gauges” I would like to ask you:

Do you precisely mean that SU(2) gets restricted to an U(1) subgroup and, for SU(3), this would be the U(1)xU(1) subgroup?

Cheers,

Rafael.

Dear Rafael,

Let me explain in a simple way what I mean. Classical Yang-Mills theory admits chaotic and integrable solutions. I do a first hypothesis that chaotic solution cannot be used to build a quantum field theory (this is not proved yet but a conjecture if you like).

Now one has to identify a set of integrable solutions that can give you a starting point for a quantum field theory. Smilga, in his book, showed that, for Y-M Hamiltonian equations, one can choose a set with all equal components that has very interesting properties. You have to choose your components is such a way that some are zero and other not. So let us take SU(2). You will have

and your theory reduces to a massless quartic scalar theory. This theory admits classical solutions given by

being

and are two arbitrary constants. This classical solution is massive notwithstanding we started from a massless theory.

This is the starting point. The theory retains its scale invariance as should be.

What does it happen with the gauge term in the Lagrangian? Just a redefinition of coordinates, after the mapping, as you get corrections to the kinematic term of the scalar theory. The ghost field decouples instead leaving a free particle.

So you should consider your symmetries on the vector

Marco

Dear Marco,

Thus, it seems me that by taking a representations of A on SU(2) as: A = a0*I + a_i*sigma_i, with a0²+a_i²=1, you have selected the “Maximall Abelian Generators” of this group, just to say U(1). If I am correct, your choice in:

https://marcofrasca.wordpress.com/2008/10/25/smilgas-choice-and-the-mapping-theorem/

is composed of two copies of U(1), which means a product of U(1)xU(1), again, it is the Maximal Abelian Group of SU(3).

It raises a interesting point. MAG is a very interesting gauge in many aspects, see for instance:

http://en.wikipedia.org/wiki/Gauge_fixing about MAG,

as it is said, SU(N) components are fixed just partialy, so, YM excitations would be colorless.

If you are interested, there is a beautiful paper

http://xxx.lanl.gov/abs/hep-lat/0611002

were authors fit, and compare, gluon-propagators on MAG by Yukawa and Gribov-like forms. They find a “gluon mass” of about 700 Mev.

A very interesting issue would be if one could also fit that LQCD behavior by using your solutions, since, they seem very similar… and, in an affirmative case, which mass could come from that!

Regards,

Rafael.

Dear Rafael,

Thank you very much for your clarifications that give an understanding of such solutions. I think I will use this view in my future papers.

Of course, I have a deep estimation for the work of Cucchieri and Mendes. Their papers are cornerstone on the path toward understanding Yang-Mills theory in the infrared. If my work fits well in their findings I can be only very satisfied.

Marco

[…] classical solutions of Yang-Mills equations Following the discussion with Rafael Frigori (see here) and returning to sane questions, I discuss here a class of exact classical solutions that must be […]