The formation of low-mass helium white dwarfs orbiting pulsars Evolution of low-mass X-ray binaries below the bifurcation period

Context. Millisecond pulsars (MSPs) are generally believed to be old neutron stars (NSs) that have been spun up to high rotation rates via accretion of matter from a companion star in a low-mass X-ray binary (LMXB). This scenario has been strongly supported by various pieces of observational evidence. However, many details of this recycling scenario remain to be understood. Aims. Here we investigate binary evolution in close LMXBs to study the formation of radio MSPs with low-mass helium white dwarf companions (He WDs) in tight binaries with orbital periods P-orb similar or equal to 2-9 h. In particular, we examine i) if the observed systems can be reproduced by theoretical modelling using standard prescriptions of orbital angular momentum losses (i.e. with respect to the nature and the strength of magnetic braking), ii) if our computations of the Roche-lobe detachments can match the observed orbital periods, and iii) if the correlation between WD mass and orbital period (M-WD, P-orb) is valid for systems with P-orb < 2 days. Methods. Numerical calculations with a detailed stellar evolution code were used to trace the mass-transfer phase in similar to 400 close LMXB systems with different initial values of donor star mass, NS mass, orbital period, and the so-called gamma-index of magnetic braking. Subsequently, we followed the orbital and the interior evolution of the detached low-mass (proto) He WDs, including stages with residual shell hydrogen burning. Results. We find that severe fine-tuning is necessary to reproduce the observed MSPs in tight binaries with He WD companions of mass < 0.20 M-circle dot, which suggests that something needs to be modified or is missing in the standard input physics of LMXB modelling. Results from previous independent studies support this conclusion. We demonstrate that the theoretically calculated (M-WD, P-orb)-relation is in general also valid for systems with P-orb < 2 days, although with a large scatter in HeWD masses between 0.15-0.20 M-circle dot. The results of the thermal evolution of the (proto) He WDs are reported in a follow-up paper (Paper II).