QCD-like Theories at Finite Baryon and Isospin Density

TL;DR

The paper investigates QCD-like theories at finite baryon and isospin densities using two-color QCD as a tractable proxy. It combines a low-density chiral Lagrangian for the Goldstone sector with a high-density weak-coupling analysis to map the μ_B–μ_I phase diagram, condensates, and low-energy spectra, including an FFLO-like inhomogeneous phase that occurs only for N_f=2 with β_D=1. Positivity of the Euclidean measure crucially determines the presence or absence of FFLO: positive-measure theories do not support FFLO, while the sign-insensitive two-color theory can. The results yield concrete predictions and benchmarks for lattice simulations and illuminate how isospin and baryon chemical potentials interplay to shape the ground state and excitations in QCD-like systems.

Abstract

We use 2-color QCD as a model to study the effects of simultaneous presence of chemical potentials for isospin charge, $μ_I$, and for baryon number, $μ_B$. We determine the phase diagrams for 2 and 4 flavor theories using the method of effective chiral Lagrangians at low densities and weak coupling perturbation theory at high densities. We determine the values of various condensates and densities as well as the spectrum of excitations as functions of $μ_I$ and $μ_B$. A similar analysis of QCD with quarks in the adjoint representation is also presented. Our results can be of relevance for lattice simulations of these theories. We predict a phase of inhomogeneous condensation (Fulde-Ferrel-Larkin-Ovchinnikov phase) in the 2 colour 2 flavor theory, while we do not expect it the 4 flavor case or in other realizations of QCD with a positive measure.

Figures (7)

• Figure 1: $N_c=2$, $N_f=2$ ($\beta_{\rm D}=1$): The figure on the left shows a schematic version of the phase diagram. To the right: The ratio of the chiral condensate to its value at $\alpha=0$ in the first quadrant of the $(\mu_B,\mu_I)$-plane.
• Figure 2: $\beta_{\rm D}=1$, $N_f=2$: The leftmost plot displays the isospin charge density in the $(\mu_B,\mu_I)$-plane in units of $2 F^2 m_\pi$. On the right hand side is shown the ratio of the pion condensate to the chiral condensate at $\alpha=0$. The first order phase transition is apparent.
• Figure 3: $\beta_{\rm D}=1$, $N_f=2$: The ratio masses of the pion modes to $m_\pi$ as a function of $\mu_B$ and $\mu_I$. From the left $m_{\pi^+}$, $m_{\pi^0}$, and $m_{\pi^-}$. Note that the mass of the $\pi^-$ excitation vanishes at $\mu_B=\mu_I>m_\pi$. The masses of the $\tilde{q}^*$ and the $\tilde{q}$ excitations are the mirrors in the $\mu_B=\mu_I$-plane of $m_\pi^+$ and $m_\pi^-$ respectively.
• Figure 4: Phase diagram of the $N_f=4$ 2-color QCD, determined by a qualitative argument of section \ref{['sec:nf=4']}. Solid lines are (second order) phase transitions where certain diquark/antidiquark condensates, indicated on the appropriate side of each line, appear. Subscripts refer to flavour. Dashed lines show the direction of $\mu_u$ and $\mu_d$ axes.
• Figure 5: Two colour QCD at small and large chemical potentials: Upper figure shows $N_f=2$ and lower figure displays $N_f=4$. Solid lines are phase transitions and gray areas illustrate regions of phase space which remain undetermined. The lines above the clouds are drawn out of scale. Dashed lines show the direction of $\mu_u$ and $\mu_d$ axes.