Physics aspects

This generator simulates the $ep$ interaction: $e^{\pm}_{(in)}\,p_{_{(in)}} \hspace{-0.1cm} \rightarrow e^{\pm} l^+ l^- X$ where $e^{\pm}_{(in)}$ and $p_{_{(in)}}$ indicate the electron/positron and the proton in the initial state respectively, $e^{\pm}$ and $l^+ l^-$ 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($Q_p^2$) and the invariant mass of the hadronic system($M_{had}$);

$$\hspace*{-0.85cm} Q_{p}^2 \stackrel{\rm def}{\equiv} - \le... ...)}} - (p_{_{e^{\pm}}} + p_{_{l^+}} + p_{_{l^-}}) \right\}^2,$$ (1)

$$M_{had}^2 \stackrel{\rm def}{\equiv} \left\{ (p_{_{e^{\pm}... ...}}) - (p_{_{e^{\pm}}} + p_{_{l^+}} + p_{_{l^-}}) \right\}^2,$$ (2)

where $p_{_{e^{\pm}(in)}}$ and $p_{_{p(in)}}$ are the 4-momenta of the incoming lepton and the proton after ISR, respectively. $p_{_{e^{\pm}}}$ and $p_{_{l^{\pm}}}$ are those of the scattered lepton and the produced leptons before FSR, respectively. The 3 categories are

• $M_{had}=M_p$( elastic),
  
• $Q_{p}^2<Q_{min}^2$ OR $M_p+M_{\pi^{0}}<M_{had}<M_{cut}$( quasi-elastic),
  
• $Q_{p}^2>Q_{min}^2$ AND $M_{had}>M_{cut}$( DIS),
  

where $M_p$ and $M_{\pi^{0}}$ are the masses of the proton and the neutral pion, respectively. $Q_{min}$ is set to around 1GeV depending on the Parton Density Function (PDF) used in the DIS process. The recommended value for $M_{cut}$ 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 ( $\Gamma^{\mu}_{pp\gamma}$) with the on-shell proton. The general form of the elastic proton vertex can be written as

$$\Gamma^{\mu}_{pp\gamma} {\small =e_p\left( F_1(Q_{p}^2)\ga... ...kappa_p}{2M_p}F_2(Q_{p}^2) \,i\sigma^{\mu\nu}q_{\nu}\right) }$$ (3)

where $e_p$ indicates the electric charge of the proton, $q$ is the 4-momentum transfer at the proton vertex ($q^2=-Q_{p}^2$), $F_1(Q_{p}^2)$ and $F_2(Q_{p}^2)$ are the 2 independent form factors, and $\kappa_p$ is the anomalous magnetic moment of the proton (see, for example, [#!Q_and_L!#].). The electric and magnetic form factors $G_E^p(Q_{p}^2)$ and $G_M^p(Q_{p}^2)$, respectively are defined as follows,

$$\left( \begin{array}{@{\,}c@{\,}} G_E^p(Q_{p}^2) \\ G_M... ...}^2)&+&\,\,\,\kappa_p\,\,\, F_2(Q_{p}^2) \end{array} \right).$$ (4)

Using the Gordon decomposition and the scaling law of the form factor,

$$G_E^p(Q_{p}^2) = G_M^p(Q_{p}^2) / \vert\mu_p\vert,$$ (5)

the following formula which is used in this program is obtained,

$$\Gamma^{\mu}_{pp\gamma} =e_p\left(\mu_p G_E^p(Q_{p}^2)\gam... ...rac{\kappa_p}{1+\frac{Q_{p}^2}{4M_p^2}} G_E^p(Q_{p}^2) \right)$$ (6)

where $\mu_p = (1 +\kappa_p)\mu_B$, $\mu_B$ is the Bohr magneton, and $p_{p(out)}$ indicates the 4-momentum of the scattered proton. $G_E^p(Q_{p}^2)$ is calculated according to the formula of the dipole fit,

$$G_E^p(Q_{p}^2) = \left(1+\frac{Q_{p}^2}{0.71\,{\rm GeV}^2}\right)^{-2}.$$ (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!#].),

$$W^{\mu\nu} = W_1\left( -g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^2} \right)$$

$$+ W_2\frac{1}{M_p^2} \left( p_{_{p(in)}}^{\mu}-\frac{p_{_{p(in)}}\c... ...ight) \left( p_{_{p(in)}}^{\nu}-\frac{p_{_{p(in)}}\cdot q}{q^2}q^{\nu} \right).$$ (8)

$W_1(Q_p^2,M_{had})$ and $W_2(Q_p^2,M_{had})$ are the electromagnetic proton structure functions. The hadron tensor is contracted with the lepton tensor $L^{\mu\nu}$ numerically to obtain the cross section,

$$d\sigma \sim L_{\mu\nu} W^{\mu\nu} .$$ (9)

In this version, $W_1$ and $W_2$ are parameterized with Brasse et al.[#!BRASSE!#] for $M_{had}<$2GeV (the proton resonance region), and with ALLM97 [#!ALLM97!#] for $M_{had}>$2GeV. These two parameterizations are based on fits to the experimental data on the measurement of the total $\gamma^*p$ 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 $Q_p^2$ 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

$$u \stackrel{\rm def}{\equiv} \bigl\vert\bigl\{ p_{_{q(in)}... ...-}}) \bigr\}^2\bigr\vert \,\,\,\gtrsim\,\,\, 25\,{\rm GeV}^2,$$ (10)

where $p_{_{q(in)}}$ is the 4-momentum of the incoming quark. The value of $u$ corresponds to the virtuality of the $u$-channel quark in the diagrams in Fig.2-(b),(c). When it is nearly or smaller than 25 GeV$^2$, 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: $u>25$GeV$^2$ 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. $\bigl\{ p_{_{e^{\pm}(in)}} - p_{_{e^{\pm}}}\bigr\}^2$ 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.

Fig. 1: Feynman diagrams included in the (quasi-)elastic process. $e$= $\left\{ e^+,e^- \right\}$, l$^{\pm}$= $\left\{ e^{\pm}, \mu^{\pm}, \tau^{\pm} \right\}$. N means a (dissociated) proton or a nucleon resonance.

Fig. 2: Feynman diagrams included in the DIS process. $e$= $\left\{ e^+,e^- \right\}$, l$^{\pm}$= $\left\{ e^{\pm}, \mu^{\pm}, \tau^{\pm} \right\}$ and $q$= $\left\{\right.$$u$ $^{^{(}}$ $^{-}$ $^{^{)}}$,$d$ $^{^{^{(}}}$ $^{^-}$ $^{^{^{)}}}$,$s$ $^{^{(}}$ $^{-}$ $^{^{)}}$,$c$ $^{^{(}}$ $^{-}$ $^{^{)}}$,$b$ $^{^{^{(}}}$ $^{^-}$ $^{^{^{)}}}$,$t$ $^{^{^{(}}}$ $^{^-}$ $^{^{^{)}}}$ $\left.\right\}$.