# A Meaning of phase transitions in lattice gauge theories

#### diegzumillo

Hi all

Decided to post in 'beyond standard model' because lattice gauge theories are usually used to explore these models. Hope that's all right.

So in lattice simulations, phase transitions in beta mass plane seem to have an important meaning. I understand phase transitions from a mathematical stand point and its simple interpretations in Ising model, as ferromagnetic and paramagnetic phases. But whenever lattice is used to study QCD or QCD-like theories, this phase transition seems to acquire some importance that just isn't obvious to me. From context I can tell it's related to the model being confining or not, which I can more or less anticipate, seeing how beta is inversely proportional to the coupling constant, but that's as far as I go. Any help would be appreciated.

#### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

#### ohwilleke

Gold Member
Hi all

Decided to post in 'beyond standard model' because lattice gauge theories are usually used to explore these models. Hope that's all right.

So in lattice simulations, phase transitions in beta mass plane seem to have an important meaning. I understand phase transitions from a mathematical stand point and its simple interpretations in Ising model, as ferromagnetic and paramagnetic phases. But whenever lattice is used to study QCD or QCD-like theories, this phase transition seems to acquire some importance that just isn't obvious to me. From context I can tell it's related to the model being confining or not, which I can more or less anticipate, seeing how beta is inversely proportional to the coupling constant, but that's as far as I go. Any help would be appreciated.
It is a little hard to know what you are asking, which I don't fault you for because it sounds like the main problem is, in part, knowing how to pose the question you really want to ask. I'll mention a couple of concepts that seem related to see if this is what you are interested in or if I'm not getting what you are asking at all.

When QCD undergoes a transition from being confining to being not confining, that non-confining phase is called a quark-gluon plasma. The linked Wikipedia article on this topic explains its significance:

QCD is one part of the modern theory of particle physics called the Standard Model. Other parts of this theory deal with electroweak interactions and neutrinos. The theory of electrodynamics has been tested and found correct to a few parts in a billion. The theory of weak interactions has been tested and found correct to a few parts in a thousand. Perturbative forms of QCD have been tested to a few percent. Perturbative models assume relatively small changes from the ground state, i.e. relatively low temperatures and densities, which simplifies calculations at the cost of generality. In contrast, non-perturbative forms of QCD have barely been tested. The study of the QGP, which has both a high temperature and density, is part of this effort to consolidate the grand theory of particle physics.

The study of the QGP is also a testing ground for finite temperature field theory, a branch of theoretical physics which seeks to understand particle physics under conditions of high temperature. Such studies are important to understand the early evolution of our universe: the first hundred microseconds or so. It is crucial to the physics goals of a new generation of observations of the universe (WMAP and its successors). It is also of relevance to Grand Unification Theories which seek to unify the three fundamental forces of nature (excluding gravity).
As I said, I'm not sure if this answer your question or not, because it isn't clear what you question is.

On the other side of the UV (i.e. high energy) phenomena of QGP in the infrared (low energy) of QCD is asymptotic freedom (which is also in a sense non-confining at the immediate micro-level), because the QCD coupling constant gets very small at both low and high energies, while it peaks at approximately the energy scale of a nucleon like a proton or neutron. A nice historical review of the discovery of asymptotic freedom can be found in this article.

Asymptotic freedom could be viewed as a phase, although I'm not sure that this would be a conventional terminology for asymptotic freedom in the IR. In infrared QCD one of the big open questions in QCD is whether the QCD coupling constant goes to zero at negligible energy scales or to a low but finite and non-zero fixed point. This his important implications that go to the mathematical rigor and structure and consistency of QCD as a theory.

A phase diagram for QCD might also be useful to you in making sense of the questions you are asking (I tried but can't figure out how to get it into the body of the post).

Other less commonly known but hypothesized phase is a color-flavor-lock phase at high energy scales but low temperatures, and non-CFL quark liquids at similarly low temperatures but energy scales intermediate between CFL and hadronic matter, both of which might be found within neutron stars.

FWIW, lattice QCD is not really beyond the Standard Model, even when non-physical values of physical constants like the pion mass and non-physical numbers of color charges or quark flavors are used to asymptotically estimate the physical QCD behavior. Lattice QCD is pretty much ordinary SM QCD physics.

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#### diegzumillo

I know it can be hard to answer vague questions. Thanks for the effort! it actually did help. There is a constant struggle between relating lattice models with finite volume with the phases of continuum limit (QCD) and I didn't have the full picture of what the actual QCD phase space looked like (or can look like).

One more thing I learned is that the quark condensate is one of the things changing in this transition, and that helps make sense of it. If the vacuum expectation value goes from almost zero to a larger-than-almost-zero value it could mean some symmetry got broken in the continuum limit, and if we're talking about a lattice that tries to imitate QCD I guess one could infer that transition pinpoints loss of chiral symmetry. Maybe. Depending on how that behaves as volume increases.

Lots of "I guess", "I think", "maybe", "if" here but things are clearing up.

#### diegzumillo

Hey again. Was about to make a new topic but the title\topic of this one is still appropriate so I'll just revive it. Let me know if creating a new one is more appropriate.

So as I'm reading through a book and some papers about lattice one thing often mentioned is the continuum limit. Seems pretty simple to me; you take lattice spacing to zero. However a strong importance is given to second order phase transitions and it's never made abundantly obvious what this critical point means to the continuum limit, except that it's important.

My current understanding of it is that since correlation length* diverges at this critical point, we know that lattice effects (small scale phenomena) will be irrelevant, so whatever effective theory we want to approach with lattice is probably being described by the parameters near this critical point.

But, again, this is never made obvious anywhere I read so I'm pretty uncertain about this. Surely the continuum limit at other points are also relevant and the continuum theory we want to reach must exist at other points away from critical point. Does that mean the continuum theory is plagued by small distance effects at other points in parameter space?

* Correlation length $\xi$ is the parameter describing the rate the correlation between two points decrease with distance $\propto e^{-|x-y|/\xi}$

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