Questions about chiral symmetry breaking

[j1221]

Hello everyone,

I was learning about the topic "chiral symmetry breaking" recently and got couple questions. I try to describe my understandings below, then list the questions:



From the QCD Lagrangian level (quark level), I can understand the exact chiral symmetry exists when we take the quark mass to be zero.
If we have a mechanism to induce a term like m \bar{q}q, then we say the chiral symmetry is dynamical broken.
However, when I read the review papers, they always said if there is a quark condensate(chiral order parameter) <\bar{q}q> =\ 0, it means the chiral symmetry breakdown.
Since the <\bar{q}q> is invariant under SU(2)_V, not SU(2)_A. (Suppose we consider only 2 flavors here.)

The question is:

1. the condensate <\bar{q}q> is a "expectation value", it should be a number. why we can discuss its transformation property?
2. Is the "condensate exists" a necessary condition of "chiral symmetry breaks" or it's a sufficient condition?
3. Similar questions as 2. Is the "condensate exists" a necessary condition of "nonzero quark mass" or it's a sufficient condition? since I do not see the direct connection between the "quark condensate" and the mass of quarks. Could we really derive a relation between them?

My final question is:
Do we really understand the mechanism of chiral symmetry breaking?
Or do physicists reach a consensus about how chiral symmetry breaking (in quark level, not effective meson level, such as NJL model)?
Is this topic still under investigation?

Several questions, please help. Thanks in advance.

 

[tom.stoer]

Regarding 1) afaik the condensate is a two-dim order parameter $$<\bar{q}q>, <\bar{q}\gamma_5q>$$ which explains why it has an orientation.

2) a necessary condition; the condenstae is the order parameter of the phase transition; in the symmetric phase it's zero, in the broken phase it has some non-zero value.

3) the quark condensate has a much stronger effect on chiral symetry breaking than the tiny u- and d-quark masses. The condensate breaks the symmetry spontaneously, whereas the quark mass term breaks the symmetry explicitly. Think about magnetism; the magnetic field of the Earth can be compared to the quark mass term, but the effect due to intrinsic magnetization in a ferromagnet is much stronger than the tiny magnetic field from the earth.

I guess the mechanism is well-understood in lattice QCD.

 

[daschaich]

I believe the mechanism was well-understood long before lattice QCD came along. I would imagine that QCD itself gained a lot of early acceptance thanks to its clear explanation for the dynamics behind chiral symmetry breaking, which was already familiar from the meson spectrum and current algebra. The paper I'm always told to read, but haven't yet, is Dashen (1971), "Some Features of Chiral Symmetry Breaking".

While it's easy to measure $$\langle\overline\psi\psi\rangle$$ on the lattice, the signal is dominated by the explicit chiral symmetry breaking that results from the unphysically large quark masses we need to use (to avoid critical slowing down $$\propto m_q^{-4.5}$$), as well as the explicit chiral symmetry breaking introduced by some common discretizations of the quark fields. Check out Fig. 4 in this paper: the chiral condensate relevant to spontaneous symmetry breaking is the intercept $$\lim_{m \to 0}\langle\overline\psi\psi\rangle(m)$$, which is an order of magnitude smaller than any data point. (There's nothing special about that paper, I just knew this plot was in there.)

It's just as easy to measure $$\langle\overline\psi\gamma_5\psi\rangle$$, but I think the only new information it gives you is some insight into the topology. Or at least, that's all I've ever used it for.

The main chiral-symmetry-related question I've seen under consideration in hep-lat is the restoration of chiral symmetry at high temperatures, for instance in the formation of quark-gluon plasma. In this region of the phase diagram, there is both a chiral transition as well as a deconfinement transition, and the question is whether these two phenomena are related, and if so how. There looks to be some background in these recent lecture notes, but I don't see too much on this issue itself. I don't know enough about it to say any more.

 

[tom.stoer]

I believe the mechanism was well-understood long before lattice QCD came along. I would imagine that QCD itself gained a lot of early acceptance thanks to its clear explanation for the dynamics behind chiral symmetry breaking, which was already familiar from the meson spectrum and current algebra.

I don't think that the dynamics was well understood. If you look at affective low-energy theory, chiral perturbation theory etc. they use mesons (Goldston e bosons) but they do not explain how the symmetry breaking occures (in terms of fundamental degrees of freedom). That's why I guessed that lattice QCD will do the job.

 

[daschaich]

I think QCD should suffice; I don't see what the lattice is supposed to provide.

 

[tom.stoer]

I think QCD should suffice; I don't see what the lattice is supposed to provide.

You should be able to calculate the value of the quark condensate, find the order of the phase transition, calculate the eqiation of state etc. I don't know whether appropriate calculational tools are available in QCD w/o any computer support.

Do you know references where they do this in QCD?