Medium separation scheme effects on the magnetized and cold two-flavor superconducting quark matter

TL;DR

This work tackles regularization artifacts in the NJL description of cold, dense, magnetized two-flavor quark matter by comparing traditional (TRS) regularization with Magnetic Field Independent Regularization (MFIR) and the Medium Separation Scheme (MSS). It demonstrates that MFIR isolates pure magnetic contributions while MSS cleanly separates medium (density) effects, removing unphysical oscillations and producing a monotonic, physically reasonable behavior of the diquark gap $\Delta$ and a consistently positive magnetization $M$ across the explored $\mu$–$eB$ domain. The resulting phase diagrams differ markedly from TRS, restoring a color-superconducting region at high $\mu$ and yielding a softer equation of state $p_N^{\parallel}(\varepsilon_N)$ with a smoother approach to the conformal limit, in line with lattice and RG expectations. These findings bolster the reliability of NJL-based descriptions of magnetized dense matter and have implications for astrophysical applications to magnetars and neutron-star mergers, while pointing to fruitful future work including finite-temperature extensions and lattice cross-checks.

Abstract

We analyze the impact of the Medium Separation Scheme (MSS) on two-flavor color superconducting (2SC) dense quark matter under the influence of a constant external magnetic field. The effects of the proper treatment of the model divergences are examined through a comparison of different approaches, including the combined implementation of the Magnetic Field Independent Regularization (MFIR) and the MSS, as well as the standard use of smooth form factors. Our findings for the Nambu--Jona-Lasinio model emphasize the critical role of properly separating medium effects from vacuum contributions in the model. The combined MFIR-MSS scheme suppresses spurious unphysical oscillations, often misinterpreted in the literature as de Haas--van Alphen oscillations, and ensures the correct high-density behavior of the diquark condensate. Furthermore, within the MSS framework, the magnetization remains positive across the explored parameter space, in sharp contrast with the behavior obtained in the traditional approach.

Figures (8)

• Figure 1: Normalized quark mass $M / M_{0}$ as a function of $eB$ for $G_D/G_S = 0.0$, comparing MFIR and nMFIR.
• Figure 2: Constituent quark mass $M$ (panel a) and diquark condensate $\Delta$ (panel b) as functions of the magnetic field, comparing nMFIR, MFIR and MFIR + MSS.
• Figure 3: Constituent quark mass $M$ and diquark condensate $\Delta$ as functions of the chemical potential, comparing TRS (panel a) and MSS (panel b) at $eB$ = 0.
• Figure 4: Diquark condensate $\Delta$, as a function of $\mu$, comparing MFIR and MFIR + MSS at finite magnetic field.
• Figure 5: Phase diagram in the $eB \times \mu$ plane, comparing the MFIR method in panel (a) and the MFIR + MSS method in panel (b). These diagrams display the chiral symmetry breaking phase ($\chi SB$), the color superconducting phase (CSC), and a phase with restored chiral symmetry (Normal).