Mass gap in Yang-Mills theories

In summary, the Clay Math Institute is investigating the possibility that the mass gap in YM-theory is due to the existence of a Higgs field. This field has not yet been observed, but is only estimated to have an energy of around 125 GeV.
  • #1
humanino
2,527
8
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 !
 
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  • #2
The original description of the problem by Jaffe and Witten :

http://www.claymath.org/millennium/Yang-Mills_Theory/Official_Problem_Description.pdf [Broken]
 
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  • #3
humanino said:
The original description of the problem by Jaffe and Witten :

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


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
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 [Broken]
http://www.idsia.ch/~marcus/physics/pdise.htm [Broken]
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:
 
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  • #5
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
selfAdjoint is completely right, humanino. This is how it is, point final.

Cristal clear explanation...

regards
marlon
 
  • #7
selfAdjoint said:
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?
 
  • #9
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
marlon said:
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 :
[tex]S\geq \frac{8\pi^2}{g\2}Q_T[/tex]
with the topological charge (Pontryagin index) related to Chern-Simmons number as [tex]Q_T = N_{CS}(+\infty)-N_{CS}(-\infty) = \int d^4x \partial_\mu K_\mu[/tex]
where [tex] \frac{1}{32\pi^2}F \tilde F = \partial_\mu K_\mu[/tex] 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 [tex]Q_T[/tex], 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
[tex]{\cal A} \sim e^{-S} = e^{-\frac{8\pi^2}{g^2} }= e^{-\frac{2\pi}{\alpha_s} }[/tex]
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 :
[tex]\langle \bar{q}_i q_i\rangle \approx -(250[/tex] MeV [tex])^3[/tex]

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 :
[tex]
{\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) }
[/tex]
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 [tex]\imath m[/tex] term. Here, by acting on a solution with [tex]\gamma_5[/tex], one obtains another eigenvector of the Dirac operator, with opposite eigenvalue (classic trick in chiral stuff) :
[tex] 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)]
[/tex]
with the spectral density of the Dirac operator [tex]\nu(\lambda)}[/tex] averaged over the instanton ensemble.
A few more manipulations lead to the celebrated Banks-Casher relation :
[tex] \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}
[/tex]

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

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 ?
 
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  • #11
nobody cares about instantons

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:
 
  • #12
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
kurious said:
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
 
  • #14
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?
 
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  • #15
kurious said:
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
 
  • #16
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
kurious said:
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 going to 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
Your last point is one I have heard other people make!
 
  • #19
kurious said:
Your last point is one I have heard other people make!

right on, brother
 
  • #20
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.
 
  • #21
humanino said:
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
 
  • #22
Instantons I believe are defined in Euclidean 4-space, with [tex]\tau = it [/tex]. 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.
 
  • #23
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.
 
  • #24
Humanino,

Why has the instanton-approach failed?

regards
marlon
 
  • #25
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:
 
  • #26
humanino said:
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
 
  • #27
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 ?
 
  • #28
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?
 
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  • #29
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 [tex]U[/tex], the field transforms as [tex]A_{\mu} \rightarrow U^{\dagger} A_{\mu} U + \imath U^{\dagger} \partial_{\mu}U[/tex]. Well, simply drop the first term. You get that [tex]A_{\mu} \rightarrow_{\infty} \imath U^{\dagger} \partial_{\mu}U[/tex] with [tex]U[/tex] an arbitrary element of the gauge group.

References : (I already gave those in another thread on instantons : https://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
 
  • #30
Thanks for the information and references Humanino.
 
  • #31
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
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.
 
  • #32
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!
 
  • #33
Yes sA, I am aware this is nothing close to the prize of the Clay Institute.
 
  • #34
marlon said:
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
 
  • #35
nrqed said:
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
 

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