Nuclear medium effects in Drell-Yan process
We study the nuclear medium effects in Drell-Yan process using quark parton distribution functions calculated in a microscopic nuclear model which takes into account the effects of Fermi motion, nuclear binding and nucleon correlations through a relativistic nucleon spectral function. The contributions of and mesons as well as shadowing effects are also included. The beam energy loss is calculated using a phenomenological approach. The present theoretical results are compared with the experimental results of E772 and E886 experiments. These results are applicable to the forthcoming experimental analysis of E906 Sea Quest experiment at Fermi Lab.
Drell-Yan(DY) production of lepton pairs drellyan () from nucleons and nuclear targets is an important tool to study the quark structure of nucleons and its modification in the nuclear medium. In particular, the proton induced DY production of muon pairs on nucleons and nuclei provides a direct probe to investigate the quark parton distribution functions(PDFs). In a DY process(shown in Fig.1), a quark of beam(target) hadron gets annihilated from the antiquark of target(beam) hadron and gives rise to a photon which in turn gives lepton pairs of opposite charge. The basic process is , where b and t indicate the beam proton and the target nucleon/hadron. A quark(antiquark) in the beam carrying a longitudinal momentum fraction interacts with an antiquark(quark) in the target carrying longitudinal momentum fraction of the target momentum per nucleon to produce a virtual photon.
The cross section per target nucleon in the leading order is given by Amanda ():
where is the fine structure constant, is the charge of quark/antiquark of flavor f, is the photon virtuality and and are the beam(target) quark/antiquark PDFs of flavour f.
This process is directly sensitive to the antiquark parton distribution functions in target nuclei which has also been studied by DIS experiments through the observation of EMC effect. Quantitatively the EMC effect describes the nuclear modification of nucleon structure function for the bound nucleon defined as and gives information about the modification of the sum of quark and antiquark PDFs kenyon (); Geesaman:1995yd () which is dominated by the valence quarks in the high region (). In the low region (), where sea quarks are expected to give dominant contribution, the study of gives information about sea quark and antiquark PDFs. Thus, nuclear modifications are phenomenologically incorporated in and using the experimental data on and are used to analyze the DY yields from nuclear targets. Some authors succeed in giving a satisfactory description of DIS and DY data on nuclear targets using same set of nuclear and eskola09 (), while some others find it difficult to provide a consistent description of DIS and DY data using the same set of nuclear PDFs scheinbein2008 (). On the other hand, there are many theoretical attempts to describe the nuclear modifications of quark and antiquark PDFs to explain DIS which have also been used to understand the DY process on nuclear targets miller ()-Marco:1997xb (). The known nuclear modifications discussed in literature in the case of DIS are (a) modification of nucleon structure inside the nuclear medium, (b) a significantly enhanced contribution of subnucleonic degrees of freedom like pions or quark clusters in nuclei and (c) nuclear shadowing.
However, in the case of DY processes there is an additional nuclear effect due to initial state interaction of beam partons with the target partons which may be present before the hard collisions of these partons giving rise to lepton pairs. As the initial beam traverses the nuclear medium it loses energy due to interaction of beam partons with nuclear constituents of the target. This can be visualized in terms of the interaction of hadrons or its constituents with the constituents of the target nucleus through various inelastic processes leading to energy loss of the interacting beam partons. This has been studied phenomenologically using available parameterization of nuclear PDFs or theoretically in models based on QCD or Glauber approaches taking into account the effect of shadowing which also plays an important role in the low region, however, any consensus in the understanding of physics behind the beam energy loss has been lacking. In this scenario most of the calculations incorporate a phenomenological description of beam energy loss to explain the experimental data on DY yields vasilev (); Duan2005 (); Johnson2002 (); Johnson:2000ph (); Arleo:2002ph (); Garvey2003 (); Brodsky1993 (). In this region of the nuclear modification of sea quark PDF and mesonic contributions also become important. Thus in this process, main nuclear effects are due to nuclear structure, mesonic contributions and shadowing (as in the case of DIS) with additional effect of parton energy loss in the beam parton energy due to the presence of nuclear targets.
In this paper, we present the results of nuclear medium effects on DY production of lepton pairs calculated in a microscopic nuclear model which has been successfully used to describe the DIS of charged leptons and / from various nuclei marco1996 (); sajjadnpa (); prc84 (); prc85 (); Haider:2015vea (); Haider:2016zrk (). The model uses a relativistic nucleon spectral function to describe target nucleon momentum distribution incorporating Fermi motion, binding energy effects and nucleon correlations in a field theoretical model. The model has also been used to include the mesonic contributions from and mesons. The beam energy loss has been calculated using some phenomenological models discussed in the literature Duan2005 ()-vasilev (). The results have been presented for the kinematic region of experiments E772 alde () and E866 e866 (); vasilev () or proton induced DY processes in nuclear targets like , , , and in the region of . The numerical results extended up to , should be useful in analyzing the forthcoming experimental results from the SeaQuest E906 experiment being done at Fermi Lab Seaquest ().
Ii Nuclear effects
When DY process takes place in nuclei, nuclear effects appear which are generally believed to be due to
(a) nuclear structure effects arising from Fermi motion, binding energy and nucleon correlations,
(b) additional contribution due to subnucleonic degrees of freedom like mesons and/or quark clusters in the nuclei,
(c) shadowing effect, and
(d) energy loss of the beam proton as it traverses the nuclear medium before producing lepton pairs.
In the case of proton induced DY processes in nuclei, the target nucleon has a Fermi momentum described by a momentum distribution. The target Bjorken variable is defined for a free nucleon as , where q is the four momentum of pair, and are respectively the beam and target four momenta in the nuclear medium. Moreover, the projectile Bjorken variable expressed covariantly as also changes due to the energy loss of the beam particle caused by the initial state interactions with the nuclear constituents as it travels through the nuclear medium before producing lepton pairs. These nuclear modifications are incorporated while evaluating Eq.(1). Furthermore, there are additional contributions from the pion and rho mesons which are also taken into account.
ii.1 Nuclear Structure
In a nucleus, scattering is assumed to take place from partons inside the individual nucleons which are bound and moving with a momentum within a limit given by the Fermi momentum. The target Bjorken variable becomes Fermi momentum dependent and PDF for quarks and antiquarks in the nucleus i.e. and are calculated as a convolution of the PDFs in bound nucleon and a momentum distribution function of the nucleon inside the nucleus. The parameters of the momentum distribution are adjusted to correctly incorporate nuclear properties like binding energy, Fermi motion and the nucleon correlation effects in the nuclear medium. We use the Lehmann representation of the relativistic Dirac propagator for an interacting Fermi sea in nuclear matter to derive such a momentum distribution and Local Density Approximation to translate these results for a finite nucleus Marco:1997xb (); sajjadnpa (); prc84 (); prc85 (); Haider:2015vea (). The free relativistic propagator for a nucleon of mass is written in terms of positive and negative energy components as
For a noninteracting Fermi sea where only positive energy solutions are considered the relevant propagator is rewritten in terms of occupation number for p while =0 for p:
The nucleon propagator in an interacting Fermi sea is then calculated by making a perturbative expansion of in terms of free nucleon propagator given in Eq. (2) by retaining the positive energy contributions only (the negative energy components are suppressed).
where is the nucleon self energy.
This allows us to write the relativistic nucleon propagator in a nuclear medium in terms of the Spectral functions of hole and particle as FernandezdeCordoba:1991wf ()
where and being the hole and particle spectral functions respectively, which are derived in Ref. FernandezdeCordoba:1991wf (), and is the chemical potential. We use:
The normalization of this spectral function is obtained by imposing the baryon number conservation following the method of Frankfurt and Strikman Frankfurt (). In the present paper, we use local density approximation (LDA) where we do not have a box of constant density, and the reaction takes place at a point , lying inside a volume element with local density and corresponding to the proton and neutron densities at the point . This leads to the spectral functions for the protons and neutrons to be the function of local Fermi momentum given by
and therefore the normalization condition may be imposed as
where the factor of two is to take into account spin degrees of freedom of proton and neutron, and and are the chemical potentials for proton and neutron respectively.
This further leads to the normalization condition given by
The average kinetic and total nucleon energy in a nucleus with the same number of protons and neutrons are given by:
where is the baryon density for the nucleus which is normalized to A and is taken from the electron nucleus scattering experiments. The binding energy per nucleon is given by marco1996 ():
The binding energy per nucleon for each nucleus is correctly reproduced to match with the experimentally observed values. Once the spectral function is normalized to the number of nucleons and we obtain the correct binding energy, there is no free parameter that is left in our model.
Using this, the electromagnetic structure function for a non-symmetric (NZ) nucleus in DIS is obtained as marco1996 (),
For the numerical calculations, we have used CTEQ6.6 cteq () nucleon parton distribution functions(PDFs) for quark() and antiquark() of flavor f.
Following the same procedure as taken for the evaluation of nuclear structure function, we incorporate the nuclear medium effects like Fermi motion, binding energy and nucleon correlations in the evaluation of bound quarks in nucleons of a nucleus. and are expressed in terms of spectral function as Marco:1997xb ():
where is the quark(antiquark) PDFs for flavor f inside a nucleon of kind i and the factor of 2 is because of quark(antiquark) spin degrees of freedom. which is obtained from the covariant expression of with direction.
ii.2 Mesonic contributions
As the nucleons are strongly interacting particles and inside the nucleus continuous exchange of virtual mesons take place, therefore, we have also taken into account the probability of interaction of virtual photons with the meson clouds. In the present work, we have considered and mesons. For this the imaginary part of the meson propagators are introduced instead of spectral function which were derived from the imaginary part of the nucleon propagator. Therefore, in the case of pion, we replace in Eq.(II.1) sajjadnpa ():
where is the pion propagator in the nuclear medium given by
Here, is the form factor, =1GeV, , is the longitudinal part of the spin-isospin interaction and is the irreducible pion self energy that contains the contribution of particle - hole and delta - hole excitations.
Following a similar procedure, as done in the case of nucleon, the contribution of the pions to hadronic tensor in the nuclear medium may be written as marco1996 ()
However, Eq.(19) also contains the contribution of the pionic contents of the nucleon, which are already contained in the sea contribution of nucleon through Eq.(II.1), therefore, the pionic contribution of the nucleon is to be subtracted from Eq.(19), in order to calculate the contribution from the excess pions in the nuclear medium. This is obtained by replacing by marco1996 () as
Using Eq.(19), pion structure function in a nucleus is derived as
This in turn leads to the expression for the pion quark PDF in the nuclear medium. For example, is derived as Marco:1997xb ():
and a similar expression for .
Similarly, the contribution of the -meson cloud to the structure function is taken into account in analogy with the above prescription and the rho structure function is written as marco1996 ()
and the expression for the rho PDF is derived as Marco:1997xb ():
where is now the -meson propagator in the nuclear medium given by:
Here, is the transverse part of the spin-isospin interaction, , is the form factor, =1GeV, , and is the irreducible rho self energy that contains the contribution of particle - hole and delta - hole excitations and . Quark and antiquark PDFs for pions have been taken from the parameterization given by Gluck et al.Gluck:1991ey () and for the rho mesons we have taken the same PDFs as for the pions. It must be pointed out that the choice of and (=1GeV) in NN and NN form factors have been fixed in our earlier workssajjadnpa (); Haider:2015vea (); Haider:2016zrk () while describing nuclear medium effects in electromagnetic structure function to explain the latest data from JLab and other experiments performed using charged lepton beams on several nuclear targets.
We have also taken into account shadowing effect which arises due to coherent multiple scattering interactions of the intermediate states, which is important in DY production at small . Various theoretical calculations have indicated that shadowing in DIS as well as in DY processes has a common origin. For the shadowing effect we have followed the model of Kulagin and Petti kulagin (); Kulagin:2014vsa ().
ii.3 Energy loss of beam partons
The incident proton beam traverses the nuclear medium before the beam parton undergoes a hard collision with the target parton. The incident proton may lose energy due to soft inelastic collisions as it might scatter on its way within the nucleus before producing a lepton pair.
There are many papers in literature Marco:1997xb (); Johnson2002 (); Johnson:2000ph (); Arleo:2002ph (); Garvey2003 (); Duan2005 (); Duan:2006hp (); Duan:2008qt (); Accardi:2009qv (); Gavin (); vasilev (); Kulagin:2014vsa () where the effect of energy loss on DY process is discussed and models are given to incorporate them in the calculation of DY yields. However, there is no model which has the preference over the others. Most of them perform phenomenological fits and the best value of the parameters are those which have been obtained in the independent analysis of the experimental data. The present situation is summarized by Accardi et al.Accardi:2009qv ().
For example Duan et al. Duan2005 (); Duan:2006hp (); Duan:2008qt () have used two different kinds of quark energy loss expression, in which the fractional parton energy is modified to , where in the linear fit are given by
where fm is the average path length of the incident quark in the nucleus A, is the energy of the incident proton. The constants and are varied to get a good fit with the experimental data which were found to be in the range of GeV/fm and Duan2005 (); Duan:2006hp (); Duan:2008qt (); Johnson2002 (); Johnson:2000ph (); vasilev (); Kulagin:2014vsa ().
Gavin and Milana Gavin () have parameterized the energy loss effect as
where . However, in some recent work of Johnson et al. Johnson2002 (), Garvey and Peng Garvey2003 (), and Kulagin and Petti Kulagin:2014vsa (), it has been pointed out that a quantitative estimate of energy loss effect in DY processes depends upon how the shadowing effect is treated. In the presence of shadowing effect, the fitted parameter for energy loss alpha in equation 27 is found to be somewhat smaller in the range of to .
ii.4 Drell-Yan cross sections with nuclear effects
We have taken into account the various nuclear effects discussed above in this section and write the cross section for the DY process as
where is the DY cross section from the nucleons in the nucleus after incorporating the nuclear medium effects like Fermi motion, binding energy, nucleon correlations through the use of spectral function. Furthermore, we have also incorporated shadowing effect following Kulagin and Petti Kulagin:2014vsa () and energy loss effect following the phenomenological model given in Eq. 27 with =1. The expression for is given by:
where is the hole spectral function for the nucleon in the nucleus. and are the nucleon PDFs of flavor f averaged over proton and neutron in the cases of quarks and antiquarks,respectively.
Similarly to include the pionic contribution and the rho contribution , the DY cross sections are respectively written as Marco:1997xb ():
Since in the various experiments the DY cross sections are also obtained in terms of other variables like , , , etc, where, , , , therefore, we have also obtained DY cross sections in terms of some of these variables. For example, using Jacobian transformation Eq.(31) may be written as:
Most of the experimental results for the DY process have been presented in the form of i.e. the ratio of DY cross section in a nuclear target () to the DY cross section in deuteron (). Therefore, to evaluate proton-deuteron DY cross section, we write
and to take into account the deuteron effect, the quark/antiquark distribution function inside the deuteron target have been calculated using the same formula as for the nuclear structure function but performing the convolution with the deuteron wave function squared instead of using the spectral function. The deuteron wave function has been taken from the works of Lacombe et al. Lacombe:1981eg ().
In terms of the deuteron wave function, one may write
where the four momentum of the proton inside the deuteron is described by with as the energy of the off shell proton inside the deuteron and is the deuteron mass. A similar expression has been used for the antiquarks .
Iii Results and Discussion
The results presented here are based on the following calculations:
(1) DY cross section for proton-nucleus scattering i.e.