# Mass gap in Yang-Mills theories

1. Aug 10, 2004

### 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/ [Broken]

Thank you for help !

Last edited by a moderator: May 1, 2017
2. Aug 10, 2004

### humanino

The original description of the problem by Jaffe and Witten :

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

Last edited by a moderator: May 1, 2017
3. Aug 10, 2004

### marlon

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

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4. Aug 14, 2004

### humanino

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. :uhh:

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

A selection from google : (not easy)
http://solid13.tphys.physik.uni-tuebingen.de/reinhardt/langfeld/qcd/node6.html [Broken]
http://www.idsia.ch/~marcus/physics/pdise.htm [Broken]

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

Last edited by a moderator: May 1, 2017
5. Aug 14, 2004

Staff Emeritus
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.

6. Aug 15, 2004

### marlon

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

Cristal clear explanation...

regards
marlon

7. Aug 15, 2004

### Olias

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?

8. Aug 15, 2004

### sol2

Last edited: Aug 16, 2004
9. Aug 16, 2004

### 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.

10. Aug 16, 2004

### humanino

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 ?

Last edited: Aug 17, 2004
11. Aug 19, 2004

### humanino

I knew I should not have written that :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 ) 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.

12. Aug 19, 2004

### 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?

13. Aug 22, 2004

### marlon

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

14. Aug 22, 2004

### 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?

Last edited: Aug 22, 2004
15. Aug 24, 2004

### marlon

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

16. Aug 24, 2004

### 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?

17. Aug 24, 2004

### marlon

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

18. Aug 25, 2004

### kurious

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

19. Aug 25, 2004

### marlon

right on, brother

20. Aug 25, 2004

### 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.