Three-Flavor Partially Quenched Chiral Perturbation Theory at NNLO for Meson Masses and Decay Constants

TL;DR

The authors extend PQχPT to NNLO for the mesonic sector with three nondegenerate sea quarks, delivering complete analytic formulas for charged pseudoscalar masses and, for the first time, the decay constants at NNLO. They introduce a compact, symmetry-aware notation to manage the otherwise enormous NNLO expressions arising from the neutral-sector propagators and multiple quark-mass configurations, and provide extensive checks and numerical explorations. The results confirm sizable NNLO corrections and emphasize their dependence on LECs L_i^r and K_i^r, highlighting the need for lattice data to determine these constants. The work lays a rigorous foundation for using PQχPT at NNLO to analyze lattice QCD results and to extract physical low-energy constants with improved precision.

Abstract

We discuss Partially Quenched Chiral Perturbation Theory (PQ$χ$PT) and possible fitting strategies to Lattice QCD data at next-to-next-to-leading order (NNLO) in the mesonic sector. We also present a complete calculation of the masses of the charged pseudoscalar mesons, in the supersymmetric formulation of PQ$χ$PT. Explicit analytical results are given for up to three nondegenerate sea quark flavors, along with the previously unpublished expression for the pseudoscalar meson decay constant for three nondegenerate sea quark flavors. The numerical analysis in this paper demonstrates that the corrections at NNLO are sizable, as expected from earlier work.

Figures (8)

• Figure 1: Feynman diagrams up to ${\mathcal{O}}(p^6)$ or two loops, for the self-energy $\Sigma(M_{\mathrm{phys}}^2,\chi_i)$. Filled circles denote vertices of the ${\mathcal{L}}_2$ Lagrangian, whereas open squares and diamonds denote vertices of the ${\mathcal{L}}_4$ and ${\mathcal{L}}_6$ Lagrangians, respectively. In the top row, the first diagram from the left is of ${\mathcal{O}}(p^2)$, whereas the other two diagrams are of ${\mathcal{O}}(p^4)$. The diagrams of ${\mathcal{O}}(p^6)$, which give the NNLO correction to the meson mass, are shown in the bottom row. Of those, the third diagram from the left is called the "sunset" diagram in the text.
• Figure 2: Feynman diagrams up to ${\mathcal{O}}(p^6)$ or two loops, for the matrix element $F(M_{\mathrm{phys}}^2,\chi_i)$ of the axial current operator $A_a^\mu(0)$. Filled circles denote vertices of the ${\mathcal{L}}_2$ Lagrangian, whereas open squares and diamonds denote vertices of the ${\mathcal{L}}_4$ and ${\mathcal{L}}_6$ Lagrangians, respectively. In the top row, the first diagram from the left is of ${\mathcal{O}}(p^2)$, whereas the other two diagrams are of ${\mathcal{O}}(p^4)$. The diagrams of ${\mathcal{O}}(p^6)$, which give the NNLO correction to the decay constant, are shown in the bottom row.
• Figure 3: The relative shifts of the charged pseudoscalar meson mass $\Delta_M$ and decay constant $\Delta_F$ to NNLO for $d_{\mathrm{val}} = 1$ and $d_{\mathrm{sea}} = 1$, as a function of the valence and sea-quark masses $\chi_1$ and $\chi_4$. The quantity plotted represents the sum of the NLO and NNLO shifts, and the difference between two successive contour lines in the plots is $0.10$. The values chosen for the LEC:s correspond to "fit 10" as discussed in the text.
• Figure 4: Comparison of the NLO, NNLO and total NLO+NNLO shifts of the charged meson mass for $d_{\mathrm{val}} = 1$ and $d_{\mathrm{sea}} = 1$. The left-hand plot shows the NLO and NLO+NNLO results for the set of LEC:s labeled 'fit 10', which has been used for all the other figures in this paper. The right-hand plot shows the result when "fit 10" is augmented by $L_0^r = -0.2 \times 10^{-2}$, $L_4^r = 0.1 \times 10^{-3}$, and $L_6^r = -0.1 \times 10^{-3}$. The effects of introducing a nonzero value for the $\mathcal{O}(p^6)$ LEC $K_{25}^r$ is also demonstrated for the NNLO and NLO+NNLO results.
• Figure 5: The combined NLO and NNLO shifts $\Delta_M$ of the charged pseudoscalar meson mass, plotted for $d_{\mathrm{val}} = 2$ and $d_{\mathrm{sea}} = 1$, for $\theta=60^\circ$ in the $\chi_1 - \chi_4$ plane, and for the proportionality factors $x=\{0.25, 1.1, 2.0, 5.0, 10.0\}$ between the valence quark masses $\chi_3$ and $\chi_1$.