I Determinant problem in an article about QCD phase diagram

[Ken Gallock]

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?

 

[vanhees71]

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.

 

[Ken Gallock]

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.

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?

 

[vanhees71]

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.

 

[Ken Gallock]

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.

Thanks!
Problem solved.