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j1221

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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.