Abstract
We perform a systematic study of the modifications to the QCD vacuum energy density ${\ensuremath{\epsilon}}_{\mathrm{vac}}$ in the zero-temperature case ($T=0$) caused by a small, but nonzero, value of the parameter $\ensuremath{\theta}$, using different effective Lagrangian models which include the flavor-singlet meson field and implement the $U(1)$ axial anomaly of the fundamental theory. In particular, we derive the expressions for the topological susceptibility $\ensuremath{\chi}$ and for the second cumulant ${c}_{4}$ starting from the $\ensuremath{\theta}$ dependence of ${\ensuremath{\epsilon}}_{\mathrm{vac}}(\ensuremath{\theta})$ in the various models that we have considered. Moreover, we evaluate numerically our results, so as to compare them with each other, with the predictions of the chiral effective Lagrangian and, finally, also with the available lattice data.
Highlights
The discovery of instantons in the 1970s [1] made it clear that topology was a relevant aspect of the dynamics of the low-energy degrees of freedom (d.o.f.)in QCD [2,3,4], but it raised another important issue: if one introduces in the QCD Lagrangian an additional term Lθ 1⁄4 θQ, where QðxÞ 1⁄4g2 64π2 εμνρσ Faμν ðxÞFaρσ ðxÞ is the so-called topological charge density, despite the fact that Q 1⁄4 ∂μKμ, where Kμ is the so-called Chern-Simons current, its contribution in the quantum theory would be nonzero thanks to the existence of configurations with nontrivial topology
Is the so-called topological charge density, despite the fact that Q 1⁄4 ∂μKμ, where Kμ is the so-called Chern-Simons current, its contribution in the quantum theory would be nonzero thanks to the existence of configurations with nontrivial topology. This term, usually referred to as a topological term or as θ-term, is interesting as it introduces an explicit breaking of the CP symmetry in QCD, absent in the original theory
We notice that, if we take the limit κ → ∞, the expressions for the topological susceptibility and for the second cumulant obtained in the ENLσ model reduce precisely to those found in the previous section using the chiral effective Lagrangian
Summary
The discovery of instantons in the 1970s [1] made it clear that topology was a relevant aspect of the dynamics of the low-energy degrees of freedom (d.o.f.)in QCD [2,3,4], but it raised another important issue: if one introduces in the QCD Lagrangian an additional term Lθ 1⁄4 θQ, where. The strategy of this paper consists in computing the dependence on θ of the vacuum energy density, so as to obtain, exploiting the relations (1.8) and (1.9), the expressions of the topological susceptibility χ and of the second cumulant c4 in terms of the fundamental parameters of the theory, not using directly the fundamental theory For the benefit of the reader, we have decided to report here some details of the calculations of χ and c4 in this case as this will allow us to introduce the basic notations and the main techniques for performing the calculations in the other cases This case is an important frame of reference for the other effective models that we shall discuss in the rest of the paper. VI we shall draw our conclusions, summarizing the analytical results that we have obtained for the topological susceptibility χ and the second cumulant c4 in the four different frameworks mentioned above and evaluate numerically our results, so as to critically compare them with each other and with the available lattice results
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