2.1.
Leading order and fragmentation dilepton production
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The conventional large transverse momentum $ (P_{\rm T}) $ dilepton can be directly produced by the hard parton scattering in the relativistic heavy ion collisions. The leading order subprocesses are the quark anti-quark annihilation ($ q\bar{q}\rightarrow g l^{+}l^{-} $) and Compton processes ($ qg\rightarrow q l^{+}l^{-} $). According to the perturbative QCD factorization approach, the cross section of the leading order dileton production $A+B\rightarrow $$ l^{+}l^{-}+X $ (Fig. 1) can be written as[3]
where $ {\rm d}\hat{\sigma}/{\rm d}M^{2}{\rm d}\hat{t} $ are the cross section of the subprocesses of parton collisions, and
where $ \alpha $ is the QED coupling constant, $ M $ is the invariant mass of the dilepton pair and $ m_{\rm l} $ is the lepton mass. These subprocesses are given by the leading order QCD calculations of the interactions $ q\bar{q}\rightarrow g \gamma^{*} $ and $ qg\rightarrow q \gamma^{*} $[7].
The Mandelstam variables of these subprocesses are $ \hat{s} = x_{a}x_{b}s $, $ \hat{u} = M^{2}-sx_{b}x_{1} $ and $ \hat{t} = M^{2}-sx_{a}x_{2} $. The momentum fractions of the parton $ a $ and $ b $ of the nucleons $ A $ and $ B $ are defined as $ x_{a}^{\min} = (x_{1}-\tau)/ $$(1-x_{2}) $ and $ x_{b} = (x_{a}x_{2}-\tau)/(x_{a}-x_{1}) $, where $ \tau = M^{2}/ $s,$ x_{1} = (P_{\rm T}^{2}+M^{2})^{1/2}{\rm e}^{y}/\sqrt{s} $ and $ x_{2} = (P_{\rm T}^{2}+M^{2})^{1/2}{\rm e}^{-y}/ $$\sqrt{s} $. $ \sqrt{s} $ is the center-of-mass energy of the colliding nucleons.
The parton distribution function $ f_{a/A}(x,Q^{2}) $ including the iso-spin effect of the nucleon is in the form
where $ R(x,Q^{2},A) $ is the nuclear modification factor[8], $ Z $ is the proton number of the nucleus and $ A $ is the nucleon number. $ p(x,Q^{2}) $ is the proton's parton distribution function, and $ n(x,Q^{2}) $ is the neutron's parton distribution function[9]. The scale of the transverse momentum is $ P_{\rm T}^{2} = Q^{2} $.
In the hard parton scattering, the final states $ c $ of hard parton collisions $ ab\rightarrow cd $ can smash into dileptons by using the Born approximation of the virtual photon bremsstrahlung $ c\rightarrow c \gamma^{*} $($ \gamma^{*}\rightarrow l^{+}l^{-} $)(Fig. 1). The cross section of the fragmentation dileptons is
where the momentum fractions are $ x_{a}^{\min} = x_{1}/(1-x_{2}) $, $ x_{b}^{\min} = x_{a}x_{2}/(x_{a}-x_{1}) $ and $ z_{c} = (x_{a}x_{2}+x_{b}x_{1})/x_{a}x_{b} $. The cross sections $ {\rm d}\hat{\sigma}/{\rm d}\hat{t}(ab\rightarrow cd,\hat{s},\hat{u},\hat{t}) $ of the subprocesses of the hard parton collisions ($ qq'\rightarrow qq' $, $ q\bar{q}'\rightarrow q\bar{q}' $, $ qq\rightarrow qq $, $ q\bar{q}\rightarrow q'\bar{q}' $, $ q\bar{q}\rightarrow q\bar{q} $, $ gg\rightarrow q\bar{q} $, $ qg\rightarrow qg $, $ q\bar{q}\rightarrow gg $ and $ gg\rightarrow gg $) can be found in Ref. [7]. Here the Mandelstam variables are $ \hat{s} = x_{a}x_{b}s $, $ \hat{u} = -sx_{b}x_{1}/z_{c} $ and $ \hat{t} = -sx_{a}x_{2}/z_{c} $. The full QCD evolution of the dilepton fragmentation function $ D_{l^{+}l^{-}/c}(z_{c},Q^{2}) $ of quarks and gluons is studied in Ref. [10].
2.2.
Dileptons from the photon-nucleon collisions
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The electron-proton deep inelastic scattering at HERA reveals the parton structure of the photon and the nucleon. Since the leading order $ \gamma n\rightarrow X $ is the QED processes, the photon-nucleon collisions play an important role at HERA[11-12]. However, leading dilepton production from the central heavy ion collisions is in the order of $ \alpha\alpha_{s} $, the photoproduction processes is in the order of $ \alpha^{2}\alpha_{s} $. The QED coupling constant reduces the contribution of the photon-nucleon collisions in the central heavy ion collisions[13-14].
In the peripheral heavy ion collisions, the average number of binary nucleon-nucleon collisions $ (\langle N_{\mathrm{coll}}\rangle $) of the leading QCD processes is decreased. However, because of the electromagnetic interaction is a kind of the long range interaction, all of the nucleons in the ion can participate in the photon-nucleon collisions. In the overlap area of the two colliding ions, high energy photons are emitted from the charged partons in the nucleon due to the inelastic scattering of nucleons. In the non-overlap area the photons emitted from the protons of the ion.
2.2.1.
Photon flux from proton
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In the Weizsäcker-Williams approximation (WWA), the electromagnetic field emitted from the high energy charged nucleon A may be treated as a flux of nearly-real photons[15-16]. The cross section of dileptons produced from the interaction of high energy photon $ \gamma $ and the nucleon $ B $ (Fig. 2) is given by the following
where $ F_{\gamma/A} $ is the photon distribution function from the proton, and we have,
where $ z $ denotes the fraction of the proton energy taken by the photon. The kinematic limits are $ P^{2}_{\max} = z^{2}s/4 $ and $ P^{2}_{\min} = m^{2}_{\rm p}z^{2}/(1-z) $, $ m_{\rm p} $ is the mass of proton. The momentum fractions are defined as $ z_{\min} = (x_{1}-\tau)/(1-x_{2}) $ and $ x_{b} = (zx_{2}-\tau)/(z-x_{1}) $.
In the subprocesses of the photon-nucleon collisions, the real photon emitted from the charged nucleon $ A $ interacts with the parton $ b $ of nucleon $ B $ by the QED Compton process $ q\gamma\rightarrow q\gamma^{*} $. The Mandelstam variables of the subprocesses $ {\rm d}\hat{\sigma}/{\rm d}\hat{t} $$(\gamma b\!\rightarrow\!\! \gamma^{*}d,\hat{s},\hat{u},\hat{t}) $ are $ \hat{s} = zx_{b}s $, $ \hat{u} = M^{2}-sx_{b}x_{1} $ and $ \hat{t} = M^{2}-szx_{2} $. One can find the cross sections of subprocess of $ q\gamma\rightarrow q\gamma^{*} $ in the Ref. [7].
The Heisenberg's uncertainty principle allows the flux of the high energy photons to fluctuate into a quark-antiquark pairs. In such interactions, the photons emitted from the charged nucleons can be regarded as an extended object consisting of quarks and also gluons. These photons are the so-called resolved photons[11-12]. In the factorization approach, the structure of the resolved photon can be defined by the photon parton distribution function. In stead of the QED subprocess($ q\gamma\rightarrow q\gamma^{*} $) of the real photon-nucleon collisions, the subprocesses of the resolved photon-nucleon collisions are the hard parton scattering $( q\bar{q}\rightarrow g \gamma^{*} $ and $ qg\rightarrow q \gamma^{*} $).
The cross section of dileptons from the resolved photon-nucleon collisions (Fig. 2) can be written in the form
where $ f_{a'/\gamma}(z_{a'},Q^{2}) $ is the parton distribution function of the resolved photon[17]. The momentum fractions are $ z_{\min} = (x_{1}-\tau)/(1-x_{2}) $ and $ x_{a'}^{\min} = (x_{1}-\tau)/ $$z(1-x_{2}) $.
The Mandelstam variables of the cross sections $ {\rm d}\hat{\sigma}/{\rm d}\hat{t}(a'b\rightarrow \!\!\gamma^{*}d,\hat{s},\hat{u},\hat{t}) $ are $ \hat{s} = zx_{a'}x_{b}s $, $\hat{u} = M^{2}- $$ sx_{b}x_{1} $ and $ \hat{t} = M^{2}-szx_{a'}x_{2} $. In the subprocesses $ a'b\rightarrow \!\!\gamma^{*}d $ of the resolved photon-nucleon collisions, the parton $ a'( = q,g) $ is from the resolved photon. The hard scattering of parton $ a' $ and $ b $ are the quark anti-quark annihilation ($ q\bar{q}\rightarrow g \gamma^{*} $) and QCD Compton $( qg\rightarrow q \gamma^{*} )$ processes.
2.2.2.
Photon flux from charged parton
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In the inelastic nucleon-nucleon collisions, high energy photons can be emitted from the charged partons. The cross section of dileptons produced from the interaction of high energy photon $ \gamma $ and the nucleon $ B $ (Fig. 3) is given by the following
where the photon distribution function from the charged parton in the nucleon is
where $ e_{a} $ is the charge of the charged parton, here the kinematic limits are $ P^{2}_{\max} = (x_{a}z_{a})^{2}s/4 $ and $ P^{2}_{\min} = m^{2}_{a}z_{a}^{2}/(1-z_{a}) $, $ m_{a} $ is the mass of the quark. The momentum fractions are defined as $ x_{a}^{\min} = (x_{1}-\tau)/(1-x_{2}) $, $ x_{b}^{\min} = (x_{a}x_{2}-\tau)/(x_{a}-x_{1}) $ and $ z_{a} = (x_{b}x_{1}-\tau)/(x_{a}x_{b}-x_{a}x_{2}) $. The Mandelstam variables of the subprocesses $ {\rm d}\hat{\sigma}/{\rm d}\hat{t}(\gamma b\!\rightarrow\!\! \gamma^{*}d,\hat{s},\hat{u},\hat{t}) $ are $ \hat{s} \!=\! x_{a}x_{b}z_{a}s $, $ \hat{u} \!=\! M^{2}\!-\!sx_{b}x_{1} $ and $ \hat{t}\! =\! M^{2}\!-\!sx_{a}x_{2}z_{a} $.
The cross section of dileptons from the resolved photon-nucleon collisions (Fig. 3) can be written in the form
where the momentum fractions are $ x_{a}^{\min} = (x_{1}-\tau)/ $$(1-x_{2}) $ and $ x_{b}^{\min} = (x_{a}x_{2}-\tau)/(x_{a}-x_{1}) $, $ z_{a'}^{\min} = $$ (x_{b}x_{1}-\tau)/(x_{a}x_{b}-x_{a}x_{2}) $ and $ z_{a} = (x_{b}x_{1}-\tau)/(x_{a}x_{b}z_{a'}- $$x_{a}z_{a'}x_{2}) $. The Mandelstam variables of the cross sections $ {\rm{d}}\hat{\sigma}/{\rm{d}}\hat{t}(a'b\rightarrow \!\!\gamma^{*}d,\hat{s},\hat{u},\hat{t}) $ are $ \hat{s} = x_{a}x_{b}z_{a}z_{a'}s $, $ \hat{u} = M^{2}-sx_{b}x_{1} $ and $ \hat{t} = M^{2}-sx_{a}x_{2}z_{a}z_{a'} $.