We study the phase structure of the two-flavor Nambu–Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [M. Sadzikowski, Phys. Lett. B 553, 45 (2003); M. Sadzikowski, Phys. Lett. B 642, 238 (2006)]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in the presence of a homogeneous 2SC condensate, if the dispersion relations in the absence of color superconductivity are known. In our calculations, we employ two different Ansätze for the inhomogeneous chiral condensate, the CDW as well as the real-kink crystal (RKC). Depending on the value of the diquark coupling we find a region of the phase diagram where the inhomogeneous chiral and the 2SC condensates coexist, confirming results of M. Sadzikowski [Phys. Lett. B 553, 45 (2003); 642, 238 (2006)]. Decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase. On the other hand, increasing the diquark coupling leads to a larger 2SC phase, while the inhomogeneous chiral and the coexistence phases become smaller. In agreement with previous studies the RKC Ansatz is energetically preferred over the CDW Ansatz. Both Ansätze lead to a qualitatively similar phase diagram, however, the coexistence phase is smaller for the RKC Ansatz.

Germany

Phillip Lakaschus

Michael Buballa

Dirk H. Rischke

Competition of inhomogeneous chiral phases and two-flavor color superconductivity in the NJL model

We consider the two-flavor version of the extended linear sigma model (eLSM), which contains (pseudo)scalar and (axial-)vector quark-antiquark mesons, a scalar glueball [predominantly corresponding to <math><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mrow><mo>(</mo><mn>1710</mn><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math>, as well as the nucleon and its chiral partner. We extend this model by the additional light scalar meson <math><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>500</mn><mo>)</mo></mrow></mrow></math>, predominantly a putative four-quark state. We investigate various interaction terms of the four-quark and glueball states with the other particles, some of which preserve and some of which explicitly break the <math><mrow><mtext>U</mtext><msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mi>A</mi></msub></mrow></math> symmetry. We test our model by performing a global fit to masses and decay widths of the scalar resonances and pion-pion scattering lengths. We also discuss the influence of the scalar four-quark state and the glueball on the baryon sector by evaluating pion-nucleon scattering parameters. We find that the inclusion of <math><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>500</mn><mo>)</mo></mrow></mrow></math> improves the description of pion-pion and pion-nucleon scattering lengths.

Justin L.P. Mauldin

Poland

Francesco Giacosa

China

Role of a four-quark and a glueball state in pion-pion and pion-nucleon scattering