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I found some references, for instance here and here, but I still don't really understand the concept of why it happens, or why it's interesting.

In particular, I'd like to know:

  1. What is hadron condensation? (e.g. pions, kaons, ..)
  2. Why is it interesting?
  3. What is the difference between $s$-wave and $p$-wave condensation? (is there also $d$-wave, etc.?)

edit: I do have some random bits of knowledge in my head and would like to connect it to a more thorough explanation. I'd appreciate it if the answer would include terms like "negative mass", "chemical potential", "sigma term", and "Bose-Einstein condensate".

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    $\begingroup$ Just a quick question as a clarification: when you ask "why is it interesting", what exactly are you looking for? I started writing an extended comment rather than a full answer, and then I looked at the papers you have posted and there is a nice, clear explanation as to why this phenomenon would be interesting. There's some clear, concise discussion regarding high density QCD and the phase diagram. $\endgroup$
    – user172341
    Commented May 13, 2021 at 8:12
  • $\begingroup$ @DiSp0sablE_H3r0 I guess if you would point me to the relevant sections or even summarize the main point, I‘d be very grateful! $\endgroup$
    – ersbygre1
    Commented May 13, 2021 at 12:33
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    $\begingroup$ perhaps I did not explain myself well enough. In the most recent work you have cited, it is explicitly mentioned that pion condensates might be a new state in the high density QCD regime. this relates to the QCD phase diagram. I meant to say, would you like some clarifications on that, or are you looking for some lattice studies -which are first principle computations- experimental results, etc? $\endgroup$
    – user172341
    Commented May 14, 2021 at 12:50
  • $\begingroup$ @DiSp0sablE_H3r0 Thanks for your comment; connecting all this to the QCD phase diagram would certainly help! $\endgroup$
    – ersbygre1
    Commented May 14, 2021 at 21:36

1 Answer 1

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The original pion condensation was a liquid crystal like phase in neutron matter conjectured to occur at high densities. The main neutron coupling to pions is given by a term like $\vec \sigma \cdot \vec \nabla \Pi$, where $\vec \sigma$ is the neutron spin operator and the $\Pi$ represents the pion quantum field. This interaction gives an attraction when the relative angular momentum of the nucleon and pion is 1, or a p-wave attraction. If it results in a pion condensate it is sometimes called a p-wave condensate. Similarly for the s-state with angular momentum 0.

The conjectured liquid crystal phase has a neutron liquid where the spin of the neutrons tend to be aligned in layers, with opposite spin in opposite layers as if there were an interaction $\vec \sigma \cdot [\hat z\cos(qz)]$. That is the pion field with wave vector $q$ has a component that can be viewed classically, or alternatively that field mode has a macroscopic occupation number. This is then called a pion condensate.

If this drives a phase transition (rather than a cross over), then the compressibility of the neutron matter will change dramatically at the transition. This change in state will change things like the mass-radius relation of neutron stars. Now that LIGO observations on neutron star mergers etc. can give more information about the equation, exotic states like these may be either observed or ruled out.

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  • $\begingroup$ If the πN coupling involves a pion derivative (and therefore a pion momentum), isn't it always a p-wave interaction? $\endgroup$
    – ersbygre1
    Commented May 14, 2021 at 1:02
  • $\begingroup$ Yes the usual interaction is p-wave. $\endgroup$
    – user200143
    Commented May 14, 2021 at 19:08
  • $\begingroup$ I've thought about your answer for a bit, and I'd like to ask some more follow-up questions: 1) is the attractiveness of the πN interaction the necessary condition for the pions to get massless (or zero chemical potential)? 2) Could you please elaborate on the difference between phase transition and cross over? Why should the compressibility of neutron stars change? 3) What would an s-wave interaction look like? Is it also attractive? $\endgroup$
    – ersbygre1
    Commented May 17, 2021 at 9:52
  • $\begingroup$ In neutron stars the temperature is low compared to the fermi energy, so new equilibrium phases at high density need to lower the energy. This requires something attractive. A phase transition has discontinuous derivatives of the free energy. A first order transition has the entropy discontinuous and a latent heat across the transition. A crossover is like going from a liquid to a gas above the critical point where there is no place that the derivatives are discontinuous. An example of an s-wave interaction would be one proportional to the pion field or density. $\endgroup$
    – user200143
    Commented May 18, 2021 at 1:52

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