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Millisecond pulsars (MSPs) are generally believed to be old neutron stars(NSs) which have been spun up to high rotation rates via accretion of matterfrom a companion star in a low-mass X-ray binary (LMXB). However, many detailsof this recycling scenario remain to be understood. Here we investigate binaryevolution in close LMXBs to study the formation of radio MSPs with low-masshelium white dwarf companions (He WDs) in tight binaries with orbital periodsP_orb = 2-9 hr. In particular, we examine: i) if such observed systems can bereproduced from theoretical modelling using standard prescriptions of orbitalangular momentum losses (i.e. with respect to the nature and the strength ofmagnetic braking), ii) if our computations of the Roche-lobe detachments canmatch the observed orbital periods, and iii) if the correlation between WD massand orbital period (M_WD, P_orb) is valid for systems with P_orb < 2 days.Numerical calculations with a detailed stellar evolution code were used totrace the mass-transfer phase in ~ 400 close LMXB systems with differentinitial values of donor star mass, NS mass, orbital period and the so-calledgamma-index of magnetic braking. Subsequently, we followed the orbital and theinterior evolution of the detached low-mass (proto) He WDs, including stageswith residual shell hydrogen burning. We find that a severe fine-tuning isnecessary to reproduce the observed MSPs in tight binaries with He WDcompanions of mass < 0.20 M_sun, which suggests that something needs to bemodified or is missing in the standard input physics of LMXB modelling. Wedemonstrate that the theoretically calculated (M_WD, P_orb)-relation is ingeneral also valid for systems with P_orb < 2 days, although with a largescatter in He WD masses between 0.15-0.20 M_sun. The results of the thermalevolution of the (proto) He WDs are reported in a follow-up paper (Paper II).

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