As we all know, the idea of confinement consists in forbidding free propagation of free color-chagres, instead keeping them inside color-neutral states -- mesons and baryons. This phenomenological property of quarks is true only at low energy transfers, while at high energy scales (>few 100MeV?) quarks acquire (asimtotic) freedom and well described by perturbative methods.
The very interesting and challenging question is: "How we describe quark interaction at low E regime?"
In quantum mechanical (i.e. non-relativistic) approach one suggests some phenomenological quark-quark potantials (like V(r)~ar+br3, etc.) and solve some bound state problems, etc.
Another (powerful but rather sophisticated) approach is the lattice QCD. It reached some obviuos successes, but has (IMHO) essentially computing destination.
Is it possible in some effective theory to obtain quark-gluon relativistic potentials for low-energy regime?
I was looking for the answer or some attemts to that, but found quite nothing.
The one idea could be to perform a kind of unitary transformation from bare partons to "clothed" ones (different from usual renarmalization), which will account for confinement properties.
The idea was inspired by several works of Weinberg, Greenberg, Schweber, Kazes, Shirokov, Shebeko (http://arxiv.org/abs/nucl-th/0102037), Stefanovich (http://arxiv.org/abs/physics/0504062) and some others. What they suggest is an alternative approach to QFT, where renormalization procedure is achieved by special unitary transformation of particle operators. This transformation require that only interaction with non-empty energy shell are allowed between physical particles. This idea is not trivial, but describing it will take too much space...
In case of quark-gluon system, the idea of unitary transformation suggests we require only those interactions that transfrom color-neutral states to color-neutral states. New one-quark states will be effectively confined in color-neutral multy-quark states.
One presumes, that is such approach the strong coupling will be effectivelyrenormalized and weaken down, while quark-quark (and multy-quark) potantials will have property of confinement (i.e. diverge as positive power of r) at low energy transfer and assymtotic freedom at high.
I recited the core of these ideas here: http://arxiv.org/abs/1404.4383
Some considerations there may not be completed... But in case anyone finds it interesting or has any comments, I'll be extremely grateful.
If someone consider these ideas essentially empty or incorrect, please, let me know as well!
Thanks a lot!
Ivan
Hi Ivan! Nice to meet you here.
Would this paper be of interest to you?
http://arxiv.org/abs/nucl-th/9901025
Misha
You may see the following book.
QCD as a Theory of Hadrons: From Partons to Confinement, By Stephan Narison, PAGES: 37-39
Thank you for the answers!
The first aproach (mentioned by Mikhail) is familiar to me and indeed suggests a nice way to obtain the confinement potentials. The main idea there is to use the expansion in q/m, which means the method is valid only when q
Instead of solving the problem using approximation methods such as expansion in coupling, why don't you use numerical methods ? Do you get some extra physical insight by using analytical methods ?
Well, there are several reasons for not going for numeric methods.
First of all, for me it is interesting from academic point of view... It is probably interesting, what are properties of new (after clothing transformation) one-quark states; how look interactions between them, especially when moving from low energies to higher; how quark masses and coupling renormalized via new method, strong coupling effects are presumed to be 'screened' or, say, removed into confinement potentials... thus giving new description of low-energy regime...
Secondly, numeric methods usually do not have enough predictive force... From analytic theory one can probably hope to get predictions of new effects or better understanding and better description of known ones. That concerns, for example, some new relativistic effects or some new multi-quark forces that we can obtain from analytic theory.
Pure QCD will not lead to quark confinement potentials. Instead, one needs to introduce dual fields, which can be viewed as dual gluons, by the principle of interaction dynamics. Then one can derive the strong interaction potentials, clearly demonstrating quark confinement. See Tian Ma & Shouhong Wang, Duality of Strong Interaction, EJTP, 11:31(2014), 101-124.
Dear Shouhong. Thank you very much for your comment!
QCD in it's conventional field-theoretical formulation indeed do not naturally produce confinement property.
What I try to suggest is the representation of QCD in terms of new quark-gluon operators (which I call "color-clothed" quarks and gluons, I again refer to http://arxiv.org/abs/1404.4383). These oparators are related the the original "bare" operators via unitary transformations (unitarity guaranties that observables, like time-evolution operator and S-operator stays the same).
These unitary transformations are chosen in the particular way: we forbid interactions between non-color-singlet states (e.g. q->gq, qq->qq, etc.).
Thus, we get an effective theory, where, being expressed in terms of new operators, the Hamiltonian and S-operator contain all possible multi-quark-gluon interaction operators between color-singlet states (e.g. qqq->qqq, qqbar->qqbar, qqbar->qqqbarqbar, gg->qqbar, etc), that (I assume so) naturally produce the property of confinement and asymtotic freedom. As I mentioned, during this "clothing" procedure, quark masses and coupling get renormalized.
All mentioned above approaches (if I am not mistaking) are non-relativistic and, for example, can not suggest multi-quark potentials (say, 3-quark potential).
I'm not trying here to diminish the utility and importance of the different approaches suggested.
In fact, in my consideration, it is quite cumbersome (mathematically) to obtain milty-quark confinement potentials and calculate higher-order corrections to quark masses and couplings. I cannot now decidedly say, how will they look... That is why I was asking for help of some QCD experts, who (maybe) can assess the perspectives of the method suggested.
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