Abstract
We study the η-η′ mixing up to next-to-next-to-leading-order in U(3) chiral perturbation theory in the light of recent lattice simulations and phenomenological inputs. A general treatment for the η-η′ mixing at higher orders, with the higher-derivative, kinematic and mass mixing terms, is addressed. The connections between the four mixing parameters in the two-mixing-angle scheme and the low energy constants in the U(3) chiral effective theory are provided both for the singlet-octet and the quark-flavor bases. The axial-vector decay constants of pion and kaon are studied in the same order and confronted with the lattice simulation data as well. The quark-mass dependences of m η , m η ′ and m K are found to be well described at next-to-leading order. Nonetheless, in order to simultaneously describe the lattice data and phenomenological determinations for the properties of light pseudoscalars π, K, η and η′, the next-to-next-to-leading order study is essential. Furthermore, the lattice and phenomenological inputs are well reproduced for reasonable values of low the energy constants, compatible with previous bibliography.
Highlights
In this article we show that this large–NC χPT framework yields an excellent description of the η and η′ masses from lattice simulations at different light-quark masses [15,16,17,18,19]
In order to show the results step by step, we present the discussions split in three parts: we consider fits performed at leading order, next-to-leading order and next-to-next-to-leading order
We explicitly present the fit results with F in the theoretical expressions and use the alternative fits expressed in terms of Fπ to estimate the systematic errors, due to working up to next-to-next-to-leading order (NNLO) in δ and neglecting higher orders
Summary
At leading order in the δ expansion, i.e. O(δ0), the U(3) χPT Lagrangian consists of three operators. Comparing with the NLO results in eq (15) from our previous paper [45], we have generalized the expression to the NNLO case in the present eq (2.15) Another way to treat the mixing of pseudoscalar mesons in χPT was previously studied in ref. Since the correlation function is the second derivative with respect to the axial-vector external source aμ, and aμ always appears in the Lagrangian together with the partial derivative ∂μ as shown in eq (2.2), the absence of the mixing for the ηq and ηs decay constants in eq (2.25) implies that there are no kinematic mixing terms for the quark-flavor states ηq and ηs in the FKS formalism. No kinematic mixing terms for the ηq and ηs fields result from these chiral operators Since general terms up to NNLO in δ expansion are kept in our discussion, unlike in the previous works [14, 42,43,44,45, 49, 50] where different assumptions, such as the preference of the higher order p2 and 1/NC effects, are made, it is important and interesting for us to justify these assumptions in later discussions
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