Collective modes of the Nambu–Jona-Lasinio model with an external U(1) gauge field

The effect of external color fields on the collective modes of the ${\mathrm{SU}}_{\mathit{L}}$(2)×${\mathrm{SU}}_{\mathit{R}}$(2) chiral flavor version of the Nambu–Jona-Lasinio model is studied analytically in a U(1) approximation to the gauge fields. We show that the scalar and pseudoscalar modes respond differently to external chromomagnetic and -electric fields. In the former case, in which chiral asymmetry is enhanced, the modes remain well separated and vary slowly with the field, while in the latter case the scalar mode drops rapidly to become degenerate with the pseudoscalar mode in the chiral limit. In this regime, both modes are weakly coupled to quark matter, and the pseudoscalar pion mode in particular survives as a well-defined excitation as it enters the pair continuum. The Goldberger-Treiman relation, which is shown to hold in the presence of external fields, is responsible for this behavior. Chromoelectric and -magnetic polarizabilities are seen to be equal and opposite with absolute values ${\mathrm{\beta }}_{\mathrm{\sigma }}$=2.0${\mathrm{\alpha }}_{\mathit{s}}$ and ${\mathrm{\beta }}_{\mathrm{\pi }}$=0.03${\mathrm{\alpha }}_{\mathit{s}}$ for the scalar and pseudoscalar modes respectively.