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 galaxy density field is constructed entirely from Galileon-type operators, which also characterize the intrinsic nonlinear evolution of dark matter. This formulation ensures that the bias expansion is well defined at the field level, automatically satisfies the statistical conditions of vanishing ensemble and volume averages, and eliminates the need for any ad hoc renormalization procedures. We derive analytic expressions for the one-loop galaxy-galaxy and galaxy-matter power spectra and implement an efficient numerical algorithm using fftlog and fast-pt, enabling rapid and accurate evaluation of the full power spectrum. The resulting model requires only a minimal set of bias parameters, comprising three for correlation functions and four for power spectra. To assess its predictive accuracy, we perform joint fits to the halo-halo auto- and halo-matter cross-power spectra obtained from the dark emulator, considering nine combinations of redshift and halo mass, with 100 cosmological models sampled for each combination. We find that a single set of bias parameters successfully and simultaneously reproduces both spectra with better than $\ensuremath{\sim}1%$ accuracy up to $k\ensuremath{\simeq}0.3h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$ for typical linear bias values in the range ${b}_{1}\ensuremath{\sim}0.8$ to 2. For more strongly biased tracers with ${b}_{1}\ensuremath{\sim}3$, the agreement remains within $\ensuremath{\sim}1%$ up to $k\ensuremath{\simeq}0.2h\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. We further confirm that the same bias parameters consistently describe the two-point correlation functions in configuration space down to $r\ensuremath{\simeq}15{h}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{Mpc}$ with comparable accuracy. Moreover, ULPT predicts the theoretical relation ${b}_{{K}^{2}}^{\mathrm{E}}=\ensuremath{-}\frac{3}{4}{b}_{2}^{\mathrm{E}}$ between second-order Eulerian local and tidal bias parameters, which is validated through comparison with empirical fitting formulas calibrated on $N$-body simulations. These findings demonstrate that the ULPT framework offers a physically interpretable, statistically consistent, and computationally efficient model for nonlinear galaxy bias, with promising applicability to other observables such as redshift-space distortions, bispectra, and density-field reconstruction. The numerical implementation developed in this work is publicly released as the open-source python package ulptkit (https://github.com/naonori/ulptkit).