The 21 cm-Lyman-$α$ forest cross-power spectrum with a minimal effective field theory

Intensity maps of the 21 cm line and the Lyman-$α$ (Ly$α$) forest trace the same cosmic volume at 2 z 4. Their cross-power spectrum is insensitive to foregrounds that do not correlate between the two probes and, together with the Ly$α$ auto-spectrum, separates the mean 21 cm brightness temperature from the neutral hydrogen (HI) bias. Effective field theory models of each tracer exist, but at one loop they carry tens of nuisance parameters. We test whether a minimal effective field theory (mEFT), consisting of the tree-level Kaiser model with leading-order counterterms and stochastic terms, previously validated for the Ly$α$ auto-spectrum, describes the 21 cm auto-, Ly$α$ auto- and cross-power spectra jointly. We use the Sherwood 80 h⁻¹Mpc hydrodynamical simulation with HI painted onto halos at $z=2.4$-$4.2$, and reduce sample variance with Zel'dovich control variates. Each spectrum is validated separately and then in a joint fit with shared parameters. With scale cuts kₘₐₓ=0.5, $1.2$ and 2.0\,h\, Mpc⁻¹ for the 21 cm auto-, cross- and Ly$α$ auto-spectra, the mEFT fits all three spectra at all five redshifts. In the joint fit the mEFT determines the HI bias to 2.0-2.5% and the Ly$α$ density bias to 2.1-2.3%. The scale cuts follow the HI velocity dispersion and the Ly$α$ pressure-smoothing scale. When the brightness temperature is left free, the fits that include the cross-spectrum prefer values 20-30% above the true one, at 1-2$σ$, while the two auto-spectra together do not constrain it. A prior on the HI bias recovers the true value of the brightness temperature.