The curvature of the pseudo-critical line in the QCD phase diagram from mesonic lattice correlation functions

In the QCD phase diagram, the dependence of the pseudo-critical temperature, $T_{\rm{pc}}$, on the baryon chemical potential, $μ_B$, is of fundamental interest. The variation of $T_{\rm{pc}}$ with $μ_B$ is normally captured by $κ$, the coefficient of the leading (quadratic) term of the polynomial expansion of $T_{\rm{pc}}$ with $μ_B$. In this work, we present the first calculation of $κ$ using hadronic quantities. Simulating $N_f=2+1$ flavours of Wilson fermions on {\sc Fastsum} ensembles, we calculate the ${\cal O}(μ_B^2)$ correction to mesonic correlation functions. By demanding degeneracy in the vector and axial-vector channels we obtain $T_{\rm{pc}}(μ_B)$ and hence $κ$. While lacking a continuum extrapolation and being away from the physical point, our results are consistent with previous works using thermodynamic observables (renormalised chiral condensate, strange quark number susceptibility) from lattice QCD simulations with staggered fermions.

💡 Research Summary

The paper addresses the curvature of the pseudo‑critical line in the QCD phase diagram, i.e. how the pseudo‑critical temperature Tₚc varies with baryon chemical potential μ_B. Traditionally this curvature is quantified by the coefficient κ in the expansion Tₚc(μ_B)/Tₚc(0)=1−κ(μ_B/Tₚc(0))², and κ has been extracted from thermodynamic observables such as the renormalised chiral condensate or the strange‑quark number susceptibility. The authors introduce a novel, hadron‑based method: they compute mesonic two‑point correlation functions on lattice ensembles, expand them to second order in the light‑quark chemical potential μ_q (with μ_B=3μ_q), and use the degeneracy of vector (V) and axial‑vector (A) channels as a signal of chiral symmetry restoration.

The study uses two sets of anisotropic Wilson‑fermion ensembles generated by the Fastsum collaboration, labelled “Generation 2” (M_π≈391 MeV) and “Generation 2L” (M_π≈239 MeV). Both have N_f=2+1 dynamical flavours, a fixed temporal lattice spacing a_τ, and varying N_τ to change the temperature T=1/(a_τ N_τ). The meson operators J_H=ψ̄Γ_Hψ are projected to zero momentum, and the correlators G(τ;μ_q) are Taylor‑expanded in μ_q/T. Odd derivatives vanish, and the second derivative G″(τ) is evaluated explicitly, including both connected and disconnected contributions; the latter are estimated with ~2000 Gaussian noise vectors.

To probe chiral symmetry, the authors define a ratio R(τ;μ_q)= (Ĝ_V(τ;μ_q)−Ĝ_A(τ;μ_q))/(Ĝ_V(τ;μ_q)+Ĝ_A(τ;μ_q)), where the correlators are normalised at the midpoint τ=N_τ/2. At low temperature the vector meson is lighter than its axial partner, giving a positive curvature of R near the centre; at high temperature the system approaches the free‑quark limit, where Wilson discretisation effects prevent exact degeneracy but the shape of R matches that of non‑interacting lattice quarks. The authors therefore identify the temperature at which the curvature of R at the centre changes sign as the pseudo‑critical temperature Tₚc(μ_q). To suppress short‑distance artefacts they average R over a window τ_min≈0.2 T⁻¹ around the centre, weighting by the inverse variance, yielding a single number R(μ_q,T).

For each μ_q the function R(μ_q,T) is interpolated with a cubic spline; the zero‑crossing gives Tₚc(μ_q). The resulting Tₚc values decrease with increasing μ_q, as expected. Fitting the set {Tₚc(μ_B)} to the quadratic form (1) yields κ=0.0131(23)(23) for Generation 2 and κ=0.034(14) for Generation 2L. These numbers are compatible with previous determinations using staggered‑fermion actions (κ≈0.015–0.020), despite the present study employing a single lattice spacing, unphysical pion masses, and Wilson fermions that explicitly break chiral symmetry.

The paper discusses systematic uncertainties: the lack of a continuum extrapolation, the use of only one lattice spacing, and the fact that the pion masses are heavier than physical. The Generation 2L results carry larger statistical errors because the lighter pion makes the correlators noisier. Nevertheless, the agreement with established results suggests that mesonic correlators can serve as reliable probes of the pseudo‑critical line.

In conclusion, the authors demonstrate that the curvature κ of the QCD pseudo‑critical line can be extracted from hadronic observables alone, without recourse to bulk thermodynamic quantities. This provides an independent cross‑check of existing methods and opens the possibility of studying the phase diagram with a broader set of lattice observables. Future work should include multiple lattice spacings for a continuum limit, simulations at the physical pion mass, and possibly other fermion discretisations (e.g., staggered or domain‑wall) to reduce discretisation‑induced symmetry breaking.