mass gap in Yang-Mills theories

[humanino]

Could someone rephrase in a short and casual manner the famous Millenium Problem of the Clay Math institute ?

http://www.claymath.org/millennium/Yang-Mills_Theory/

Thank you for help !

[humanino]

The original description of the problem by Jaffe and Witten :

http://www.claymath.org/millennium/Yang-Mills_Theory/Official_Problem_Description.pdf

[marlon]

The original description of the problem by Jaffe and Witten :

http://www.claymath.org/millennium/Yang-Mills_Theory/Official_Problem_Description.pdf


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

[humanino]

Hey Marlon, what's up old dude ! :wink: :biggrin: :cool:

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. :cry: :uhh:

You are certainly more likely to undestand that than I am :approve: : the instantons indeed spontaneaously break chiral invariance by giving the quark condensate a non vanishing value :eek: :uhh: :surprise: :rolleyes: :biggrin: :tongue2: :bugeye:

A selection from google : (not easy)
http://solid13.tphys.physik.uni-tuebingen.de/reinhardt/langfeld/qcd/node6.html
http://www.idsia.ch/~marcus/physics/pdise.htm
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1999hep.ph....4353R

On the one hand, I read a few of Diakonov's papers on instantons and the way they break chiral invariance, and it seems to the most promising way to generate the mass gap. On the other hand, I don't know where the "10%Higgs/ 90% glue" comes from.

I would appreciate I if people could elaborate :confused:

[selfAdjoint]

The up and down quarks are light, and there are only three of them in a proton or neutron. You add up those three masses and you don't get anywhere near the mass of a proton or neutron.

Enter the "sea of gluons". Just hordes of these little massless bosons being exchanged not only by the quarks, but by each other. Yes the gluons, unlike photons, can and do interact with each other. This is all in aid of holding the proton or neutron together, and it generates a lot of potential energy; the energy keeping the quarks from wandering off. This potential energy is added to the quark mass (also a form of potential energy according to Einstein) to make up the mass of the nucleon.

[marlon]

selfAdjoint is completely right, humanino. This is how it is, point final.

Cristal clear explanation...

regards
marlon

[Olias]

The up and down quarks are light, and there are only three of them in a proton or neutron. You add up those three masses and you don't get anywhere near the mass of a proton or neutron.

Enter the "sea of gluons". Just hordes of these little massless bosons being exchanged not only by the quarks, but by each other. Yes the gluons, unlike photons, can and do interact with each other. This is all in aid of holding the proton or neutron together, and it generates a lot of potential energy; the energy keeping the quarks from wandering off. This potential energy is added to the quark mass (also a form of potential energy according to Einstein) to make up the mass of the nucleon.

So Quark Confinement is akin to that of a micro-Quantum well, locked within every single quark?..it may be that then, the energy for a Quark to free itself from its internal>>surrounding space is equivilent to that of the energy of a Single photon(or equivilent particle-sea) overcoming the restrianing factor of a Micro-Blackhole Event Horizon/Barrier?

Solving the Mass/Energy/Gap is like producing a single Quark out of a single portion of Quantum/well/space?..and we all know that the said Geometry is not yet Available?

[sol2]

and we all know that the said Geometry is not yet Available?


At one dimension, we start out with U(1) :smile: The planck epoch presides this. And then, there was light? (http://www.damtp.cam.ac.uk/user/gr/public/images/bb_history.gif)

If Photons are held to the brane, they are never to far away from home?

After all these years, how could we be considered bots? :confused:

Pure arrogance I guess, or they do not have the slightest clue about what quantum geometry means? Imagine a physicist?:smile:

[Haelfix]

The post above is completely meaningless and goobledygook, a bunch of physics jargon assembled together by a bot would probably have a better chance of making sense.

[humanino]

selfAdjoint is completely right, humanino. This is how it is, point final.

Cristal clear explanation...

regards
marlon
Of course, since selfAdjoint is a superMentor, and one of the best available in this forum. :tongue2: Thanks for answer sA. I am not flattering. :shy:

Let me elaborate a little. Instantons are selfAdjoint (too !) classical solutions of the pure YM dynamics, and thus minimize the action as :
S\geq \frac{8\pi^2}{g\2}Q_T
with the topological charge (Pontryagin index) related to Chern-Simmons number as Q_T = N_{CS}(+\infty)-N_{CS}(-\infty) = \int d^4x \partial_\mu K_\mu
where \frac{1}{32\pi^2}F \tilde F = \partial_\mu K_\mu is the topological term that can be added to the usual QCD lagrangian.

The energy of the field is a periodic function of the topological charge Q_T, and oscillator-like in the other directions. This leads to the interpretation of instantons as tunneling effect between different vacua (similar to solitons).

The tunneling amplitude is given by
{\cal A} \sim e^{-S} = e^{-\frac{8\pi^2}{g^2} }= e^{-\frac{2\pi}{\alpha_s} }
which makes it clear that instantons are non-perturbative in the coupling constant.

Let me come to the point : chiral symetry breaking by instantons.
It is obvious from the well-known fact that the quark condensate acquire a nonzero value in the presence of instantons :
\langle \bar{q}_i q_i\rangle \approx -(250 MeV )^3

To see this, one has to calculate the partition function of QCD, separate the pure-gluon contribution, and in the remaining part, interpret the fermionic functional integral as a determinant :

{\cal Z} = \int DA_{\mu} D\Psi D\Psi^{\dagger}
\exp[-\frac{1}{4g^2}\int F^2 + \sum_f\int \Psi_f^{\dagger}(\imath \gamma_\mu \nabla^\mu+\imath m_f) \Psi_f]
=\int DA_{\mu}
\exp[-\frac{1}{4g^2}\int F^2] \prod_f \det(\imath \gamma_\mu \nabla^\mu+\imath m_f)
= \overline{det(\imath \gamma_\mu \nabla^\mu+\imath m_f) }

with the average taken over the instanton gas. I am beginning to think that only those already knowing the Banks-Casher relation are following :uhh:

The classical problem with this determinant is that it is formally not hermitean because of the \imath m term. Here, by acting on a solution with \gamma_5, one obtains another eigenvector of the Dirac operator, with opposite eigenvalue (classic trick in chiral stuff) :
det(\imath \gamma_\mu \nabla^\mu+\imath m_f) = \sqrt{ \prod_n (\lambda_n^2 + m^2)}
=\exp[\frac{1}{2}\sum_n (\lambda_n^2 + m^2)]
=\exp[\frac{1}{2}\int_{-\infty}^{+\infty} d\lambda \overline{\nu(\lambda)} \ln(\lambda_n^2 + m^2)]

with the spectral density of the Dirac operator \nu(\lambda)} averaged over the instanton ensemble.
A few more manipulations lead to the celebrated Banks-Casher relation :
\langle \bar{q}_i q_i\rangle = -\frac{1}{V}
\frac{\partial}{\partial m}
\left[ \frac{1}{2}\int_{-\infty}^{+\infty} d\lambda \overline{\nu(\lambda)} \ln(\lambda^2 + m^2) \right] _{m \rightarrow 0}
= -\frac{1}{V}
\left[ \int_{-\infty}^{+\infty}
d\lambda \overline{\nu(\lambda)} \frac{m}{\lambda^2 + m^2}\right] _{m \rightarrow 0}


And finally :
\langle \bar{q}_i q_i\rangle = -\frac{1}{V} sign(m)\pi \overline{\nu(0)}

I hope that was not too long, or at least will motivate those not already familiar who could be interested. I made it technical because I am not able to sum up with concepts in a clear manner those tools I recently discovered in the literature.

The Banks-Casher relation relates the quark condensate to the spectral density of the Dirac operator at the origin.
I would like to know if other people think it is (as I am convincing myself) an appealing direction to compute the mass gap ?

[humanino]

I knew I should not have written that :cry: :uhh:

The problem with my previous post : people who actually understand it, know it is the very beggining of the instantons story. They know the story goes far beyond, and what is written here is somehow naiv or even trivial.

The people not already knowing it, won't try to undestand. Too many equations.

I thought it was worth describing those basic steps, because instantons seem to me a very promising path for solving the mass gap problem. Then of course, I just discovered them, I did not go too far yet, and I don't know where/when I am going to meet serious difficulties preventing to solve the mass gap with instantons, and I read in Diakonov's paper, as well as in Polyakov's, that there are serious evidence disproving the validity of instantons to solve the mass gap but I can't find those evidences, neither in the litterature nor (obviously :wink: ) by myslelf . Does someone know where/why the instantons become useless in this context ?

I want to add that instantons must have some relevance to describe the strong glue field. There are many numerical results indicating this. We can even say : the few spots where instantons are unable to fournish good approximations are actually the interesting issues one should address in QCD. Instantons tell us "If I can't deal with it, that means you should look closer, because somethin funny must happen there". For instance, glueball people always try to identify/predict in their glueballs spectra funny hadronic states according to instantons.

Well, please do NOT hesitate to post any comment/argument/advise/equation/good joke :tongue2:

[kurious]

Marlon:
The mass-gap clearly states that there is to be a certain minimum mass for those carriers

Kurious:

The pdf file by Jaffe and Witten says a minimum energy.
And doesn't the creation of mass just require the creation
of a particle with one type of chirality?

[marlon]

Marlon:
The mass-gap clearly states that there is to be a certain minimum mass for those carriers

Kurious:

The pdf file by Jaffe and Witten says a minimum energy.
And doesn't the creation of mass just require the creation
of a particle with one type of chirality?


No, no, no certainly not. It is well eshtablished in QFT that a mass-term mixes up the two different chiralities. Because chirality is a basic FUNDAMENTAL property of particles, all elementary particles must be massless in QFT. The massterm generated through interaction with the Higgs-field then mixes them chiralities up, which offcourse corresponds to symmetry-loss of the groundstate. I am referring to the spontanuous breaking of symmetry, ok ??

regards
marlon

[kurious]

What causes the interaction with the Higgs field?
When you say "mass mixes up chiralities" do
you mean that mass allows left and right handed particles to exist?
The Lagrangian of the Higgs bosons is not invariant
under a gauge transformation. Does this remove asymmetry
from the electric field and the other fields?

[marlon]

When you say "mass mixes up chiralities" do
you mean that mass allows left and right handed particles to exist?
The Lagrangian of the Higgs bosons is not invariant
under a gauge transformation. Does this remove asymmetry
from the electric field and the other fields?

No, i am just saying that when mass "comes into play" then the two chiralities will be mixed up into the mass-term. You can write down the formula for the mass in terms of products of the left and right-handed chirality.

The Higgs-lagrangian not invariant under a gauge transform ??? How do you know that ? I never heard of such a thing, besides what gauge-transformation are you referring to.

What do you mean by assymmetry between the electric field and other fields ???

The Higgs-interaction is "caused" by the spontanuous breakdown of symmetry of the groundstate in QCD (for example). This idea was "stolen" by the theoretical physicists, from solid-state-physics where it was incorporated in models explaining superconductivity...

regards
marlon

[kurious]

I read on the web that the Lagrangian of the Higgs boson
is not invariant under a gauge transformation.Wasn't the Higgs theory
put into the standard model to remove assymetry?

[marlon]

I read on the web that the Lagrangian of the Higgs boson
is not invariant under a gauge transformation.Wasn't the Higgs theory
put into the standard model to remove assymetry?

Well, i know what you mean and you are right. yet it is better to look at it like this : in order to be sure that spontanuous breakdown of symmetry of the groundstate can occur, an extra field is added to the filedtheory. So that when one looks to the groundstate, you can be sure it is degenerate before the breakdown at the "moment" that all particles are massless. This extra field is the Higgsfield and this is the reason why it was put into the Yang-Mills-gauge-theories. It is indeed some sort of a trick or manipulation that we need in order to gain massive gauge-bosons so that we are able to explain short-range-interactions...

Questions remains offcourse (and you are gonna like this, because i have heard you are no fan of the Higgs-mechanism, just like Hawking isn't) WHAT the F*** is this Higgs-field

regards
marlon and pardon the emotional language

[kurious]

Your last point is one I have heard other people make!

[marlon]

Your last point is one I have heard other people make!

right on, brother

[humanino]

Hey guys !

Remeber the enormous succes of Higgs' mechanism in condensed matter (not to mention electroweak model). So basically, this mechanism is useful.

I am trying here to get information on another generating mass mechanism, chiral symmetry breaking by glue. I would especially like to undestand why the instanton approach has failed.

[marlon]

Hey guys !

Remeber the enormous succes of Higgs' mechanism in condensed matter (not to mention electroweak model). So basically, this mechanism is useful.

I am trying here to get information on another generating mass mechanism, chiral symmetry breaking by glue. I would especially like to undestand why the instanton approach has failed.


I know that instantons are used in order to implement the QFT-varaint of the QM-tunneling through a potential barrier. This process lowers the groundstate energy and this matters cannot be described using perturbation theory.

I don't really know how the instanton approach has failed ???

When a particle interacts with an instanton, it's chirality will change "direction"

regards
marlon

[selfAdjoint]

Instantons I believe are defined in Euclidean 4-space, with \tau = it . I don't think they work in Minkowski spacetime. As such, though, they are good for discovering effects which can later try to derive in relativistic spacetime by other means. One of several tools the standard model physicists have for probing the non-perturbative sector. Lattice is another.

[humanino]

sA, I think instantons are real. When used to compute hadronic masses, they are so efficient ! The hadronic spectrum contains hundreds of particles, and instanton-based computations are able to reach the 10 percent accuracy level. Indeed, the few parts of the spectra where those computations failed are often already known to be "funny", that is typically where glueballs people try to find candidates for pure glueballs or composite particles.

So, I don`t understand why you say they don`t work in Minkowski spacetime. Instantons occur in the real world, or at least, are the main part of the glue field.

[marlon]

Humanino,

Why has the instanton-approach failed???

regards
marlon

[humanino]

Yes Marlon ! That's the question. That's THE question I am asking here for a while now. If nobody is able to give an answer, I will eventually display the poor informations I have. :frown:

[marlon]

Yes Marlon ! That's the question. That's THE question I am asking here for a while now. If nobody is able to give an answer, I will eventually display the poor informations I have. :frown:


I won't be able to answer your question since i don't really know what you mean. Give me your poor informations please...


regards
marlon

[humanino]

Part of this information is in the "Gauge fields and strings" by Polyakov. We already discussed about this masterpiece did not we :wink:

The remaining is easily available, I found it in Diakonov's paper. I have no time to sum up right now, as I told you I might do it later.

The question is simple, I keep restating it :
Why is the instanton approach to the mass-gap problem not well-suited ?

[kurious]

What are instantons?
A quick search on the web says they are a configuration of a gauge field at a particular region of space at a particular moment in time.Does this mean a configuration of gluons at a specific time? And why is an idea like that useful to calculate hadron masses? Also, the phrase " pure gauge at infinity" : does this mean that their is no colour field at infinity because the gluons have a limited range as force carriers?

[humanino]

kurious : Let me try to help you a little here. Solitons are localized configurations in (euclidian) space connecting different vacua at infinity. Instantons are solitons in euclidian spacetime. The different vacua here have different Pontryagin index (see previous posts). So, meeting an instantons is like suddenly switching from one vacuum to another through a tunnel effect.

Instantons are classical solution to the pure glue (YM) equations. Those equations are non-linear, therefore one should not be allowed to superpose several instantons and consider this is another solution. But the average size (0.3 f) compared to the average separation (1 f) makes it very unlikely that one meets two instantons simultaneously (remember d=4). This motivates a model of the glue field as a gas of instantons. In this model, one is able to compute partition functions as in the previous post, by averaging over the instanton ensemble.

A model for hadrons requires to include other ingredients, such as symmetry constraints, or renormalization (dressed propagators etc). But thanks to the computability of the generating functionnal (=partition function) one can evaluate hadron masses. The good agreement indicates that indeed, the glue field is almost only instantons.

Now I'll answer your last question about "pure gauge at infinity". Remember, under the gauge transformation U, the field transforms as A_{\mu} \rightarrow U^{\dagger} A_{\mu} U + \imath U^{\dagger} \partial_{\mu}U. Well, simply drop the first term. You get that A_{\mu} \rightarrow_{\infty} \imath U^{\dagger} \partial_{\mu}U with U an arbitrary element of the gauge group.

References : (I already gave those in another thread on instantons : http://www.physicsforums.com/showthread.php?t=34636)

A "final exam" at NYU by Marko Kolanovic (thanx Marko !) on "Instantons and Vacuum tunneling" :
eprints.fizika.org:2101/archive/00000027/01/seminar.ps

Please read Diakonov whenever you want to learn about QCD and instantons :
nac21.uv.es/pdf/9602375
or arXiv:hep-ph/9602375 v1 23 Feb 1996 (this is the same)

A very good review on instantons by Diakonov :
arXiv:hep-ph/0212026

[kurious]

Thanks for the information and references Humanino.

[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 :smile:
The origin of the mass gap in QCD (http://www.arxiv.org/abs/hep-th/0410043)
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 wanna 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 ! :wink: :biggrin: :cool:

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. :cry: :uhh:



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.

Best regards,

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' 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 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.

[humanino]

Progress towards understanding the mass-gap in QCD :
Precise Quark Mass Dependence of Instanton Determinant (http://lanl.arxiv.org/abs/hep-th/0410190)
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]

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]

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



:wink:

[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]

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. :smile:

[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]

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]

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]

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 dont 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]

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.

[Chronos]

Is there actually any missing mass? I don't think so. Yang-Mills suggests a slight CPT violation, but no missing energy [mass] that I can see. No matter how much you twist and turn space time around, mass does not go away. The observational evidence for its existence is fairly solid.

[ohwilleke]

Is there actually any missing mass? I don't think so. Yang-Mills suggests a slight CPT violation, but no missing energy [mass] that I can see. No matter how much you twist and turn space time around, mass does not go away. The observational evidence for its existence is fairly solid.

The mass isn't missing from the observational universe. The mass is missing from the theory. The gap is between the presence of mass in the observational world, and the absence of mass in the theory, not the other way around. The issue is how to get mass into the theory (with a Higgs like mechanism a possible solution) so that it can fit what we observe, because expect for the mass problem Y-M does a good job of giving us to QCD that we observe.

[Lester]

You can find a simple definition of mass gap on Wikipedia (http://en.wikipedia.org/wiki/Mass_gap). A Yang-Mills theory, in the limit of the coupling gauge going to infinity, displays at the classical level a mass gap. This because there is a theorem proved in

http://arxiv.org/abs/0709.2042 (appeared in Physics Letters B)

http://arxiv.org/abs/0903.2357 (appeared in Modern Physics Letters A)

that maps classical solutions of a massless quartic scalar field on the Yang-Mills field. These solutions appear to describe free massive fields notwithstanding we started from massless theories. One can use these solutions to build a quantum field theory and obtain an identical situation once is proved that quantum corrections do not modify it.