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 } \simeq 2 - 9 { hr } .
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 \sim 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 _ { \odot } , 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 He WD masses between 0.15 - 0.20 M _ { \odot } .
The results of the thermal evolution of the ( proto ) He WDs are reported in a follow-up paper ( Paper II ) .

Conclusions :