Effective quark operator models ofUA(1)symmetry breaking

We undertake a systematic investigation of ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry-breaking, $C$-, $P$-, $T$-, and ${\mathrm{SU}}_L{(N}_f)\times {\mathrm{SU}}_R{(N}_f)$-invariant effective fermion operators and their consequences for pseudoscalar and scalar mesons. We construct four types of such operators that exist for any number of flavors $N_f>~2$, two of which can be identified with ’t Hooft’s interaction and the quark self-interaction leading to the Veneziano-Witten meson-interaction term. We isolate the ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry-breaking effect from the quark mass- and electromagnetic interaction-induced chiral symmetry-breaking effects and quantify it as the deviation from zero of $f_0^2{m}_{\mathrm{U}\left(1\right)}^2{=f}_{{\eta }^{\prime }}^2{m}_{{\eta }^{\prime }}^2{+f}_{\eta }^2{m}_{\eta }^2-f_K^2{(m}_{K^{+}}^2{+m}_{K^0}^2{)+f}_{\pi }^2{(m}_{{\pi }^{+}}^2-m_{{\pi }^0}^2)$, where $m_{\phi }{,f}_{\phi }$ are the pseudoscalar $\phi$ meson mass and weak decay constant, respectively. Then we use Dashen’s general formula to evaluate the masses and the mixing angle of isoscalar pseudoscalar mesons in the presence of the current quark masses and each one of these four types of ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry-breaking interactions. We find that both the ’t Hooft and the Veneziano-Witten interaction push the sum of the ${\eta }^{\prime }$ and $\eta$ masses squared upward and the mixing angle to negative values, in accord with empirical evidence. The other two types of ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry-breaking operators do not influence the pseudoscalar meson spectrum to leading order in $N_C,$ so long as no new higher-order quark condensates are assumed. In an attempt to determine which linear combination of the ’t Hooft and the Veneziano-Witten operators is responsible for the observed ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry breaking, we calculate the scalar meson masses in the three-flavor Nambu–Jona-Lasino model in the presence of either of these two interactions. Presently available data do not allow a definitive answer to that question, though they can be interpreted as favoring the ’t Hooft interaction.