Introduction

In the study of electron/positron-proton ($ep$) collisions, a precise estimation of the dilepton¹ production cross-sections in the electroweak (EW) interaction is important since it could become a significant background for various physics analyses such as, for example, exclusive $J/\psi$ or $\Upsilon$ production and new physics searches. So far only the generator LPAIR [#!LPAIR!#] has been used in experimental analyses to estimate the dilepton background [#!Dirk!#]. The calculation of LPAIR is based on the diagrams of the photon-photon collision process [#!GAMGAM!#], so-called two-photon Bethe-Heitler (2-$\gamma$ BH), corresponding to the diagrams of Fig.1-(a) or Fig.2-(a) with the photon contribution only. This process is dominant in most of the phase space. It is, however, expected that in the region of low invariant masses of the dilepton system, QED-Compton type (CO) diagrams ( i.e. photon internal conversion process) as seen in Fig.1-(b) and Fig.2-(b) become dominant. In the high mass region, there is an additional interesting process, i.e. Z$^{0}$ production, which is implemented in the MC event generator EPVEC [#!EPVEC!#]. EPVEC, however, does not include 2-$\gamma$ BH nor CO diagrams. In the di-$e$ channel, interference effects of the final state $e^-e^-$ or $e^+e^+$ should also be taken into account, which are included neither in LPAIR nor in EPVEC.

In this paper, a new MC event generator GRAPE-Dilepton for the dilepton production in $ep$ collisions is presented. The FORTRAN code to calculate the Feynman amplitudes is generated by GRACE [#!GRACE!#] which is an automatic calculation system. GRACE has been used mainly for $e^+e^-$ interactions so far. This is the first time for GRACE to be applied to the case where there is a composite particle ( i.e. proton) in the initial state. GRAPE stands for a GRAce-based generator for Proton-Electron collisions.

This generator has the following features.

• The cross-section calculation is based on the exact matrix elements in the electroweak theory at tree level. Not only 2-$\gamma$ BH but also the dilepton productions via $\gamma$Z$^0$ and Z$^0$Z$^0$ collisions are taken into account. CO and Z$^0$ on/off-shell production are also included. Interference effects of the final state $e^{\pm}e^{\pm}$ are taken into account in the di-$e$ channel. It is possible to select any sub-set of diagrams in the calculation.
  
• All fermion masses are kept non-zero both in the matrix elements and in the kinematics, which makes it possible to use this program with arbitrary small scattering angles of $e^{\pm}$ and/or small invariant masses of dilepton down to the kinematical limits.
  
• The calculation of the proton vertex covers the whole kinematical region by dividing it into 3 categories of elastic, quasi-elastic and DIS (Deep Inelastic $eq$ Scattering) processes, as described in the next section in detail.
  
• Both of Initial State Radiation (ISR) and Final State Radiation (FSR) can be included.