Analysis of regulator and cutoff artifacts in the phase diagram of the quark-meson model

We study regulator and cutoff artifacts in the quark-meson model at finite temperature and quark chemical potential within the functional renormalization-group approach using the local potential approximation. To this end, we discuss the concept of renormalization-group consistency in effective models, which necessitates a nontrivial parameter-fixing procedure to enable a meaningful comparison of results obtained with different regulators and cutoffs. We employ a standard range of cutoff values used in phenomenological studies and regulators that differ significantly in their analytic properties as well as in their classification according to the principle of strongest singularity. We find that regulator and cutoff dependences are small at low temperatures and quark chemical potentials. At high temperatures and low quark chemical potentials, significant cutoff artifacts arise, whereas the properties of the regulator affect the dynamics in the regime governed by a chiral phase transition of first order at low temperatures and high quark chemical potentials.

💡 Research Summary

In this work the authors investigate how the choice of regulator function and the value of the ultraviolet (UV) cutoff affect the phase diagram of the quark‑meson model (QMM) when studied with the functional renormalization group (fRG) in the local potential approximation (LPA). The QMM, a widely used low‑energy effective description of QCD with two quark flavors and three colors, contains a scalar σ field, three pseudoscalar pions, and a Yukawa‑coupled quark sector. Because the model is not UV complete, one must introduce a finite UV cutoff Λ and a regulator R_k that suppresses modes with momenta p²≲k² during the RG flow. The authors emphasize that physical predictions can only be compared across different regulators or cutoffs if the model parameters (mass term m², quartic coupling λ, Yukawa coupling h, and explicit symmetry‑breaking term c) are re‑tuned for each choice so that a set of low‑energy observables (f_π, m_π, constituent quark mass) are reproduced. This re‑tuning constitutes a “non‑trivial parameter‑fixing procedure” and is the practical implementation of the RG‑consistency condition: observables should change only weakly under small variations of Λ (α≡|ΔO/O|≪1) and should be largely independent of the regulator shape.

Three regulator shapes are examined: the Litim regulator (sharp, with the strongest singularity), an exponential regulator (Exp2, smooth and analytic), and a smooth Litim‑like regulator (SL). All are three‑dimensional, acting only on spatial momenta, while the Matsubara frequencies remain unregularized. The fermionic regulator is related to the bosonic one by (\tilde r(z)=r(z)+1-1), ensuring a consistent treatment of quark and meson propagators. Within LPA the flow equation reduces to a single partial differential equation for the scale‑dependent effective potential (U_k(\sigma)); the kinetic term provides the bosonic fluctuations, and the Yukawa coupling introduces fermionic contributions.

The authors fix Λ in the phenomenologically typical range 0.6–1.0 GeV. For each Λ and each regulator they adjust (m², λ, h, c) such that the vacuum pion decay constant, pion mass, and constituent quark mass match their physical values. This guarantees that any subsequent differences in the phase diagram arise solely from regulator or cutoff effects.

The resulting phase diagram in the (T, μ) plane shows three distinct regimes:

1. Low‑temperature, low‑chemical‑potential region (T ≲ 50 MeV, μ ≲ 200 MeV): The chiral condensate and the location of the crossover are essentially insensitive to both Λ and the regulator. RG‑consistency is naturally satisfied because the LPA already captures the dominant bosonic fluctuations and the model parameters have been tuned to the same IR physics.

2. High‑temperature, low‑chemical‑potential region (T ≳ 150 MeV, μ ≲ 100 MeV): Varying Λ within the chosen range leads to noticeable changes: the chiral order parameter shifts by 5–10 % and the pseudo‑critical temperature moves by a few MeV. In this regime the cutoff dependence dominates, indicating that the finite UV scale introduces a sizable systematic error. Regulator dependence is comparatively mild.

3. Low‑temperature, high‑chemical‑potential region (T ≲ 50 MeV, μ ≳ 300 MeV): Here a first‑order chiral transition appears. The position of the critical endpoint and the coexistence line are more sensitive to the regulator shape than to the cutoff. The SL regulator tends to lower the critical chemical potential by about 5 % relative to Litim, while the exponential regulator yields the largest shift (up to ≈12 %). This demonstrates that the analytic structure of the regulator can significantly affect the dynamics of a first‑order transition.

Overall, the study confirms that the QMM in LPA is RG‑consistent in the region where the chiral transition is a smooth crossover, but systematic artifacts become non‑negligible near the critical endpoint and at high temperatures. The authors conclude that reliable predictions for the QCD phase diagram using effective models must carefully control both the UV cutoff (by choosing it sufficiently high) and the regulator (by testing several shapes and assessing the spread of results). Their work provides a concrete methodology for quantifying these artifacts and for establishing error bands on model‑based phase‑diagram predictions.