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I Determinant problem in an article about QCD phase diagram

  1. Jul 17, 2017 #1
    Hi.
    I'm reading an article about QCD phase diagram. https://arxiv.org/abs/1005.4814.
    I want to derive eq(20), but I don't know how.
    Does anyone know how to derive this?
     
    Last edited by a moderator: Jul 20, 2017
  2. jcsd
  3. Jul 18, 2017 #2

    vanhees71

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    It just uses that the determinant of a matrix is given by the product of its eigenvalues. The eigenvalues of ##D(0)## come in pairs, ##\gamma_i## and ##\gamma_i^*##. The Mass matrix is proportional to the unit matrix and thus the eigenvalues are ##\gamma_i+m_q## and ##\gamma_i^*+m_q##. Do you get
    $$\det (D(0)+m_q)=\prod_i (\gamma_i+m_q)(\gamma_i^*+m_q)$$
    which is Eq. (20) in the paper.
     
  4. Jul 18, 2017 #3
    Thanks.
    If there is no mass, (##m_q=0##), then will it be like this?:
    $$
    \det D(0)=\prod_i \gamma_i \gamma_i^*.
    $$
    I'm not familiar with 'a pair (##\gamma_i, \gamma_i^*##)' part. Why do we have to think about pair of eigenvalues?
     
  5. Jul 18, 2017 #4

    vanhees71

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    The point is to show that without baryo-chemical potential you have always pairs of conjugate omplex eigenvalues and that's why in this case the fermion determinant is positive. For finite ##\mu_{\text{B}}## it's not longer real (except for imaginary chemical potential). That's why you cannot use Lattice QCD so easily to evaluate the QCD phase diagram at finite ##\mu_{\text{B}}##. Ways out of this trouble is subject of vigorous ungoing research in the nuclear-physics/finite-temperature-lattice community.
     
  6. Jul 20, 2017 #5
    Thanks!
    Problem solved.
     
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