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Dirac Operator in Dense QCD

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Book cover Dirac Spectra in Dense QCD

Part of the book series: Springer Theses ((Springer Theses,volume 124))

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Abstract

This and the next chapter constitute the main part of this thesis. In this chapter, we will show that exact analysis of the Dirac operator is possible in the high density BCS phase of QCD for two and three colors, where the symmetries are spontaneously broken by the diquark condensate \(\langle qq \rangle \), rather than by the conventional chiral condensate \(\langle \bar{q}q \rangle \). In Sects. 3.1 and 3.2 we define and analyze the \(\varepsilon \)-regime at high density. We will show a novel link between the spectrum of low-lying Dirac eigenvalues and the BCS pairing gap. In Sects. 3.3 and 3.4 we introduce and solve ChRMT for two-color QCD at nonzero chemical potential. We will analyze in detail the dependence of the microscopic Dirac spectrum on the quark masses and the chemical potential. Finally in Sect. 3.5 we analyze the sign problem based on the exact results in the previous section.

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Notes

  1. 1.

    Precisely speaking, this symmetry is slightly violated by nonzero chiral condensate generated by instanton effects. At asymptotically large \(\mu \) it is negligibly small and one can neglect it in the following.

  2. 2.

    \(\Delta \) diverges as \(\mu \rightarrow \infty \), see Chap. 2 Eq. (2.2).

  3. 3.

    Definition: \(\delta ^2(z)\equiv \delta (\mathrm {Re} z)\delta (\mathrm {Im} z)\).

  4. 4.

    It acquires a nonzero expectation value once the gauge fixing is performed.

  5. 5.

    The Wess-Zumino-Witten term is necessary for completeness of the theory, but it is irrelevant to the ensuing analysis and neglected in the following.

  6. 6.

    For \(N_f=2\), \(UI_2U^T=(\det U)I_2=I_2\) is constant, in accordance with the isomorphism \(\text{ SU}(2)\cong \text{ Sp}(2)\).

  7. 7.

    If the quark masses are non-degenerate, the \(\Pi ^a\) modes acquire masses.

  8. 8.

    We thank Thomas Schäfer for a communication on this point.

  9. 9.

    A pfaffian of an \(n\times n\) antisymmetric matrix \(X\) (for even \(n\)) is defined as

    $$ \mathrm {Pf} (X)=\frac{1}{2^{n/2}(n/2)!}\sum _{\sigma } \,\mathrm {sgn} (\sigma )X_{\sigma (1)\sigma (2)}\dots X_{\sigma (n-1)\sigma (n)}\,, $$

    where the sum runs over all permutations \(\sigma \) of \(1,\,2,\,\dots ,n\). Note that this definition differs by a factor of \(1/2^{n/2}\) from [[38] Eq. (A.8)].

  10. 10.

    Note that the mass term at \({\mathcal O}(M^2)\) is unique, in contrast to the CFL phase (cf. p. 66).

  11. 11.

    We repeatedly used the identities \(I_n^{\prime }(x)=[I_{n-1}(x)+I_{n+1}(x)]/2\) and \(I_n(x)/x=[I_{n-1}(x)-I_{n+1}(x)]/2n\).

  12. 12.

    The model (3.74) is not new. It corresponds to class \(2P\) in Magnea’s mathematical classification of non-Hermitian random matrix ensembles, see ([[45] Table 2]). What is new is the realization that (3.74) describes dense two-color QCD.

  13. 13.

    To avoid confusion: On page 41 the pattern is denoted as \(\text{ SU}(2N_f)\rightarrow \text{ Sp}(2N_f)\) because \(\text{ U}(1)_A\) is not regarded as an exact symmetry there.

  14. 14.

    See p. 39.

  15. 15.

    The large-\(N\) limit with \({\mathcal O}(1)\) non-Hermitian parameter is called the limit of strong non-Hermiticity in the conventional classification (p. 47).

  16. 16.

    For finite \(N\), the massive partition functions were obtained in [66] with the aid of skew-orthogonal polynomials. However we do not consider finite \(N\), since the universal correspondence of ChRMT to QCD emerges only in the large-\(N\) limit.

  17. 17.

    It should not be confused with the BCS gap \(\Delta \).

  18. 18.

    Note that no NG mode appears in \(N_f=1\) two-color QCD. The above \(m_{\pi }\) refers to the mass of the NG modes that appear in \(N_f\ge 2\) two-color QCD.

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  1. Department of Physics, University of Regensburg, Universitätsstraße 31, Regensburg, 93053, Germany

    Takuya Kanazawa

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Kanazawa, T. (2012). Dirac Operator in Dense QCD. In: Dirac Spectra in Dense QCD. Springer Theses, vol 124. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54165-3_3

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