Fate of the first-order chiral phase transition in QCD: Implications for dark QCD studied via a Nambu-Jona-Lasinio model

The first-order nature of the chiral phase transition in QCD-like theories can play crucial roles to address a dark side of the Universe, where the created out-of equilibrium is essential to serve as cosmological and astrophysical probes such as gravitational wave productions, which have extensively been explored. This interdisciplinary physics is built based on a widely-accepted conjecture that the thermal chiral phase transition in QCD-like theories with massless (light) three flavors is of first order. We find that such a first order feature may not hold, when ordinary or dark quarks are externally coupled to a weak enough background field of photon or dark photon (which we collectively call a magnetic" field). We assume that a weak magnetic" background field could be originated from some magnetogenesis" in the early Universe. We work on a Nambu-Jona-Lasinio model which can describe the chiral phase transition in a wide class of QCD-like theories. We show that in the case with massless (light) three flavors, the first-order feature goes away when $2 f_π^2 \lesssim eB ( \ll (4 πf_π)^2)$, where $eB$ is the magnetic" field strength and $f_{\pi }$ the pion decay constant at the vacuum. This disappearance is the generic consequence of the presence of the magnetically" induced scale anomaly and the magnetic" catalysis for the chiral symmetry breaking, and would impact or constrain modeling dark QCD coupled to an external ``magnetic" field.