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We present the results of calculations of the $p^4$ and $p^6$ orderlow-energy constants for the chiral Lagrangian with tensor sources for both twoand three flavors of pseudoscalar mesons. This is a generalization of ourprevious work on similar calculations without tensor sources in terms of thequark self-energy $\Sigma(p^2)$, based on the first principle derivation of thelow-energy effective Lagrangian and computation of the low-energy constantswith some rough approximations. With the help of partial integration and someepsilon relations, we find that some $p^6$ order operators with tensor sourcesappearing in the literature are related to each other. That leaves 98independent terms for $n$-flavor, 92 terms for three-flavor, and 65 terms fortwo-flavor cases. We also find that the odd-intrinsic-parity chiral Lagrangianwith tensor sources cannot independently exist in any order of low-energyexpansion.

Computation of the $p^6$ order low-energy constants with tensor sources