Scalar diquarks in the QCD vacuum

Technical Summary: Scalar Diquarks in the QCD Vacuum

Problem Statement
The phase structure of dense, strongly interacting matter, particularly in the cores of neutron stars, remains an open problem in modern physics. While phenomenological models are essential for interpreting astrophysical observations, they rely on low-energy constants (LECs)—such as particle masses and coupling strengths—that are often poorly constrained. This is especially true for scalar diquarks, which are crucial for modeling the two-flavor color-superconducting (2SC) phase but have never been measured directly. Existing theoretical estimates often conflate "curvature masses" (accessible in Euclidean space calculations) with "pole masses" (physical observables defined by peaks in scattering processes). The authors note that while these masses coincide for light pions, their relationship for heavier states like diquarks is unexplored. Consequently, there is a need for a first-principles determination of diquark properties and other LECs from Quantum Chromodynamics (QCD) to provide rigorous input for effective models of dense quark matter.

Methodology
The authors employ the Functional Renormalization Group (FRG) approach to two-flavor QCD in the vacuum. The framework is built on the Wetterich flow equation, which describes the scale-dependent effective action $\Gamma_k$ as quantum fluctuations are integrated from an ultraviolet (UV) scale $\Lambda$ down to the infrared (IR).

Key technical features of the methodology include:

1. Truncation Scheme: The effective action is decomposed into glue, quark-gluon, and composite sectors. The composite sector explicitly includes scalar and pseudoscalar mesons ($\sigma, \vec{\pi}$) and scalar diquarks ($\Delta$).
2. Dynamical Hadronization: To handle the emergence of bound states without introducing arbitrary bosonization scales, the authors use a dynamical hadronization technique. This maps four-quark interactions onto composite bosonic fields continuously along the RG flow, ensuring that composite fields are auxiliary in the UV and become relevant degrees of freedom in the IR.
3. Parameter Fixing: The only free parameters are the strong coupling $\alpha_s$ and current quark masses, fixed at a high UV scale ($\Lambda = 20$ GeV). The strong coupling is tuned to reproduce a constituent quark mass of $m_q = 350$ MeV, and the explicit chiral symmetry breaking term is fixed to the pion mass $m_\pi = 137$ MeV. The gluon mass gap parameter is fine-tuned to achieve the confining "scaling solution" for the gluon and ghost propagators.
4. Real-Time Analysis: To distinguish between curvature and pole masses, the authors compute momentum-dependent two-point functions. They employ a three-dimensional regulator for the spatial momentum to preserve the pole structure in the complex frequency plane and perform an analytic continuation from Euclidean to Minkowski space to extract spectral functions and pole masses.

Key Contributions

• First-Principles LECs: The paper provides a self-consistent determination of vacuum LECs for the 2SC phase, including diquark masses, quark-diquark couplings, and diquark self-interactions, without phenomenological input beyond QCD parameters.
• Distinction of Mass Schemes: The work explicitly calculates and compares both curvature and pole masses for the scalar diquark, pion, and $\sigma$-meson, addressing a gap in previous literature where these were often assumed to be identical.
• Validation of Framework: The authors demonstrate the independence of IR observables from UV initial conditions for composite fields, validating the predictive power of the dynamical hadronization approach.

Results

1. Gauge Sector: The gluon propagator exhibits a mass gap, and the ghost and gluon dressing functions follow the expected scaling behavior in the deep IR ($\bar{Z}_A \propto (p^2)^{-2\kappa}$, $\bar{Z}_c \propto (p^2)^\kappa$ with $\kappa \approx 0.64$). The strong coupling avatars diverge or vanish according to scaling predictions.
2. Mesons: The pion is confirmed as a stable single-particle excitation with pole and curvature masses nearly identical ($m_{\pi, \text{pole}} \approx 120-123$ MeV vs. $m_{\pi, \text{curv}} = 137$ MeV). The $\sigma$-meson appears as a broad resonance rather than a stable particle.
3. Scalar Diquark:
   • Curvature Mass: The diquark curvature mass is found to be heavy, $m_{\Delta, \text{curv}} = 1030-1187$ MeV.
   • Pole Mass: In contrast, the pole mass is significantly lower, $m_{\Delta, \text{pole}} = 580-619$ MeV.
   • Interpretation: The pole mass is consistent with the quark-diquark picture of the nucleon, where the diquark mass is approximately twice the constituent quark mass ($2m_q \approx 700$ MeV) minus binding energy. The large discrepancy between curvature and pole masses indicates strong frequency dependence in the diquark self-energy.
4. Couplings: The quark-diquark Yukawa coupling $g_{\Delta qq}$ is found to be stronger than the quark-meson coupling $g_{\phi \bar{q}q}$ in the IR. Diquark-diquark quartic couplings are larger than meson-meson couplings, suggesting a stiffer effective potential in the diquark direction.

Significance
The authors claim that their results provide crucial microscopic input for constraining effective models of dense quark matter, particularly those describing the 2SC phase in neutron stars. By providing genuine first-principles predictions for diquark properties, the work reduces the number of free parameters in phenomenological models. Specifically, the predicted diquark pole mass implies a condensation threshold at a baryon chemical potential $\mu_B = 870-929$ MeV, which is consistent with the onset of nuclear matter. This supports the physical viability of the 2SC phase emerging after the dissociation of nucleons at high densities. The paper positions this framework as a bridge between fundamental QCD and low-energy effective models, enhancing the predictive power of theoretical studies on neutron star interiors.