Theoretical systematic bias in long-baseline neutrino oscillation parameters derived from analytic approximations

Next-generation long-baseline neutrino experiments aim to determine the charge-parity (CP) violating phase Formula: see text with high precision, requiring theoretical predictions with uncertainties well below statistical limits. We present a systematic assessment of the theoretical bias introduced by widely used analytic approximations to three-flavor neutrino oscillation probabilities in matter, focusing on the standard Cervera et al. expansion. Using an exact Hamiltonian-based treatment of flavor evolution in constant-density matter as a benchmark, we quantify the systematic error budget for appearance probabilities, CP asymmetries, and spectral fits relevant to T2K, NOvA, and DUNE. We map the fractional error in Formula: see text over the Formula: see text plane and derive the induced local phase biases. We further define a global CP-phase bias by fitting the approximate spectrum to an exact three-flavor spectrum over a wide-band energy range. For a DUNE-like baseline of Formula: see text, we find that while the Cervera formula typically reproduces probabilities at the few-percent level, it can induce systematic phase shifts of up to Formula: see text in Formula: see text due to spectral distortions, with larger biases observed for inverted mass ordering. The maximum global phase bias scales with baseline, reaching Formula: see text for Formula: see text. Our results establish a systematic error floor for analyses relying on this class of perturbative approximations and demonstrate the necessity of using exact probabilities or higher-order expansions for precision studies in the DUNE era.