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With the experimental observable O exp i and the calculated value O cal i , by minimizing the square deviation 

the ensemble of parameters a can be obtained. Furthermore, in order to balance the influence of different observables, the weight ω$_{i}$ is introduced for binding energies, charge radii, and empirical pairing gaps respectively. The corresponding weight ω$_{i}$ is roughly determined by the desired accuracy. Here the weights ω$_{i}$ are respectively 1 . 00 MeV for binding energies, 0 . 02 fm for charge radii, and 0 . 05 MeV for empirical pairing gaps. A new parameter set PC-PK1, which contains the nine coupling constants in Eq. (2) and the pairing strength in Eq. (20), is obtained and listed in Table I. 

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By scaling the coupling constants in accordance with the QCD-based Lagrangian, the naturalness in effective theories can be investigated [12, 13]. According to the QCD-based Lagrangian [13], 

with ψ the nucleon field, f$_{π}$ = 92 . 5 MeV the pion decay constant, and Λ = 770 MeV a generic QCD large-mass scale respectively, by taking into account the role of chiral symmetry in weakening N -body forces by Δ = l + n − 2 ⩾ 0 [26, 27], it has been found that six of the nine coupling constants in PC-LA and all of them in PC-F1 are natural, i.e., the QCD-scaled coupling constants c$_{ln}$ are of order unity [11, 12]. 

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Similarly, the nine coupling constants of PC-PK1 are also tested for the naturalness and all the dimensionless coefficients c$_{ln}$ are of order 1, as shown in the last column of Table I, which indicates that all the coupling constants in PC-PK1 are natural. 

Tables II and III list respectively the binding energies and charge radii for nuclei selected in the determination of PC-PK1, PC-F1, PC-LA, and NL3* [28] effective interactions. The corresponding root mean square (rms) deviation Δ together with the root of relative square (rrs) deviation δ for the binding energy and charge radius are given in the last two rows of Table II and III, respectively. Compared with the other effective interactions, the newly obtained PC-PK1 provides a much better description for the experimental binding energies and the same good description for the charge radii. 

