Renormalization-Free Galaxy Bias in Unified Lagrangian Perturbation Theory

We present a renormalization-free framework for modeling galaxy bias based on Unified Lagrangian Perturbation Theory (ULPT). In this approach, the biased density fluctuation is built solely from Galileon-type operators associated with the intrinsic nonlinear growth of dark matter. This ensures the bias expansion is well defined at the field level, automatically satisfies statistical conditions of vanishing ensemble and volume averages, and removes the need for ad hoc renormalization. We derive analytic one-loop expressions for the galaxy-galaxy and galaxy-matter power spectra and implement an efficient numerical algorithm using \texttt{FFTLog} and \texttt{FAST-PT}, enabling rapid and accurate evaluation. The model requires only a minimal set of bias parameters: three parameters are sufficient to describe correlation functions in configuration space, while four parameters are needed for power spectra in Fourier space. To test accuracy, we jointly fit halo auto- and cross-spectra from the \textit{Dark Emulator}, covering nine redshift-mass combinations with 100 cosmologies each. A single set of bias parameters reproduces both spectra within $\sim1\%$ up to $k \simeq 0.3\,h\,\mathrm{Mpc}^{-1}$ for typical linear bias $b_1 \sim 0.8$-2, and to $k \simeq 0.2\,h\,\mathrm{Mpc}^{-1}$ for $b_1 \sim 3$. The same parameters also match two-point correlation functions down to $r \simeq 15\,h^{-1}\mathrm{Mpc}$. Moreover, ULPT predicts the relation $b_{K^2}^{\mathrm{E}} = -\tfrac{3}{4} b_2^{\mathrm{E}}$, validated against $N$-body results. These results demonstrate that ULPT provides a physically consistent and efficient model for nonlinear galaxy bias, with applications to redshift-space distortions, bispectra, and reconstruction. The numerical implementation is released as the open-source Python package https://github.com/naonori/ulptkit.