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# Physics aspects

This generator simulates the interaction: where and indicate the electron/positron and the proton in the initial state respectively, and are the scattered electron/positron and the produced dilepton respectively. The relevant processes are classified into 3 categories using the negative momentum transfer squared at the proton vertex() and the invariant mass of the hadronic system();
 (1)

 (2)

where and are the 4-momenta of the incoming lepton and the proton after ISR, respectively. and are those of the scattered lepton and the produced leptons before FSR, respectively. The 3 categories are
• ( elastic),
• OR ( quasi-elastic),
• AND ( DIS),
where and are the masses of the proton and the neutral pion, respectively. is set to around 1GeV depending on the Parton Density Function (PDF) used in the DIS process. The recommended value for is 5GeV.

For the elastic process, the diagrams in Fig.1 are calculated with the following dipole form factor for the proton-proton-photon vertex ( ) with the on-shell proton. The general form of the elastic proton vertex can be written as

 (3)

where indicates the electric charge of the proton, is the 4-momentum transfer at the proton vertex (), and are the 2 independent form factors, and is the anomalous magnetic moment of the proton (see, for example, [#!Q_and_L!#].). The electric and magnetic form factors and , respectively are defined as follows,
 (4)

Using the Gordon decomposition and the scaling law of the form factor,
 (5)

the following formula which is used in this program is obtained,
 (6)

where , is the Bohr magneton, and indicates the 4-momentum of the scattered proton. is calculated according to the formula of the dipole fit,
 (7)

The only difference between the elastic and the quasi-elastic processes is the treatment of the proton vertex and the simulation of the hadronic final state. The quasi-elastic proton vertex can be described using the hadron tensor in the following form assuming parity and current conservation (for example, see [#!Q_and_L!#].),

 (8)

and are the electromagnetic proton structure functions. The hadron tensor is contracted with the lepton tensor numerically to obtain the cross section,
 (9)

In this version, and are parameterized with Brasse et al.[#!BRASSE!#] for 2GeV (the proton resonance region), and with ALLM97 [#!ALLM97!#] for 2GeV. These two parameterizations are based on fits to the experimental data on the measurement of the total cross-sections. The exclusive hadronic final state is generated using the MC event generator SOPHIA [#!SOPHIA!#] in the event generation step.

For the DIS process with the Quark Parton Model, the diagrams in Fig.2 are calculated. PDFLIB [#!PDFLIB!#] is linked to obtain parton densities with as a QCD scale. The simulation of the proton remnant and the hadronization are performed by PYTHIA [#!PYTHIA!#]. It should be noted that the lowest order calculation in this process is valid only for the region of

 (10)

where is the 4-momentum of the incoming quark. The value of corresponds to the virtuality of the -channel quark in the diagrams in Fig.2-(b),(c). When it is nearly or smaller than 25 GeV, the lowest order calculation is not correct as explained in [#!EPVEC!#] since QCD corrections become large. In this case, the dilepton production should be treated as Drell-Yan process between the proton and the resolved photon from the beam lepton, which is not implemented in this program. The cut: GeV is explicitly applied in this program if the diagrams other than BH are included.

The effect of ISR is included in the cross-section calculation using the structure function method described in [#!ISR_SF!#], where the momentum transfer squared on the beam lepton, i.e. is used as a QED scale. When ISR turns on, the correction for the photon self energy, i.e. the vacuum polarization, is included according to the parameterization in [#!QEDVAC!#] by modifying photon propagators. FSR is performed by PYTHIA using the parton shower method when the event is generated.

Next: Program structure Up: paper Previous: Introduction
Tetsuo Abe
2001-07-12