Advances in High Energy Physics

Advances in High Energy Physics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 9548737 |

Xu-Hong Zhang, Fu-Hu Liu, "Statistical Behavior of Lepton Pair Spectrum in the Drell-Yan Process and Signal from Quark-Gluon Plasma in High-Energy Collisions", Advances in High Energy Physics, vol. 2021, Article ID 9548737, 21 pages, 2021.

Statistical Behavior of Lepton Pair Spectrum in the Drell-Yan Process and Signal from Quark-Gluon Plasma in High-Energy Collisions

Academic Editor: Theocharis Kosmas
Received20 Aug 2020
Revised29 Mar 2021
Accepted11 May 2021
Published01 Jun 2021


We analyze the transverse momentum () spectra of lepton pairs () generated in the Drell-Yan process, as detected in proton-nucleus (pion-nucleus) and proton-(anti)proton collisions by ten collaborations over a center-of-mass energy or if in a simplified form) range from  GeV to above 10 TeV. Three types of probability density functions (the convolution of two Lévy-Tsallis functions, the two-component Erlang distribution, and the convolution of two Hagedorn functions) are utilized to fit and analyze the spectra. The fit results are approximately in agreement with the collected experimental data. Consecutively, we obtained the variation law of related parameters as a function of and invariant mass . In the fit procedure, a given Lévy-Tsallis (or Hagedorn) function can be regarded as the probability density function of transverse momenta contributed by a single quark () or anti-quark (). The Drell-Yan process is then described by the statistical method.

1. Introduction

There are more than one processes that can generate a pair of charged leptons () in experiments of high-energy collisions. In 1970, Sidney Drell and Tung-Mow Yan firstly proposed production in a high-energy hadron scattering in a process we now call the “Drell-Yan” process [1]. In this process, a quark () in one hadron and an antiquark () in another hadron are annihilated, and a virtual photon () or boson is generated, which then decays into . This process is expressed as , where and are collision hadrons and denotes other particles produced in the collisions. The Drell-Yan process has been extensively studied experimentally, theoretically, and phenomenologically.

The literature about the theoretical description of the Drell-Yan process within quantum chromodynamics (QCD) is well known and settled [29]. The framework for the description of the transverse momentum dynamics [sometimes, indicated as Collins-Soper-Sterman (CSS) formalism] is summarized in the well-known book by John Collins [2]. Some recent reviews on the subject of transverse momentum distributions in the Drell-Yan process can be found in refs. [46] in which many works were cited. More theoretical works at both small and large are available in literature [79]. At the same time, lots of phenomenological works were published in the past [1015], recent [1621], and very recent years [2225].

Several phenomenological interpretations of experimental Drell-Yan data collected in the previous many years have been released by various groups, particularly in recent years where first extractions of quark transverse momentum distributions are becoming available from highly accurate theoretical descriptions of QCD perturbative ingredients. The factorization theorem for the Drell-Yan process allows to write the transverse momentum differential cross-section as a convolution of two transverse momentum-dependent (TMD) parton distribution functions (PDFs), which are, under certain conditions, very complicated. This complicated factorization involves soft factors that resum soft gluon radiation regularizing a certain class of divergences that arise in the theoretical formulae. The soft gluon resummation is especially important in the description of the quark-gluon plasma (QGP), where can be produced in a process similar to that of Drell-Yan but with different origins of quarks.

QGP is a new form of matter which is created in the central region of high-energy nucleus-nucleus collisions, where extreme density and high-temperature environment are developed. It has become one of the important areas of research in the field of nuclear and particle physics. The gradual maturity of QCD and gauge field theory provides a powerful explanation for this novel matter and phenomenon. In fact, QGP is particularly short-lived. In QGP, a quark and anti-quark can soon be annihilated into a virtual photon or boson, which then decays to a pair of leptons . This happens in the QGP degeneration process in which most particles are produced. The yield, invariant mass, rapidity (), and transverse momentum () distribution of depend on the momentum distribution of and gluons in QGP in the collision region. Therefore, the information of can be used to judge whether QGP is generated and further to study its thermodynamic status making the production one of the most important signals generated by QGP. Consequently, the study on becomes particularly critical.

From the above, it is clear that can be produced in high-energy hadronic/nucleonic collisions in two main ways: the Drell-Yan process and QGP degeneration process. To study the properties of QGP, we should remove the influence of the Drell-Yan process and vice-versa. Generally, we may use the same methodology to describe the two processes. At present, one mainly uses the statistical method to study the properties of QGP. Correspondingly, we may also use the statistical method to study the production of in the Drell-Yan process, especially because the factorization theorem is very hard to model. In short, the statistical description for the Drell-Yan process is necessary to better understand the properties of QGP.

The measurement of lepton-pair physical quantities (including energy, , and ) in experiments studying the Drell-Yan process provides lots of valuable information about the dynamic properties and evolution process of the produced particles. In particular, is Lorentz invariant in the beam direction and can be used to describe the particles’ motion and system’s evolution. There are different functions that can be used to describe the spectra in statistics. For example, we can use the Lévy-Tsallis function [2630], the (two-component) Erlang distribution [3133], and the Hagedorn function [34, 35] to fit the experimental data to obtain the analytical parameters of the spectrum. Since the Drell-Yan process is the result of the interactions of and , we can use the convolution of two functions to describe the spectra. The idea of convolution is concordant to the factorization theorem for the Drell-Yan process.

In this paper, we use three functions to fit and analyze the Drell-Yan spectra obtained by ten collaborations from the experiments of high-energy proton-nucleus (pion-nucleus) and proton-(anti)proton collisions. These experimental studies provide a great resource for us to better understand the collision mechanism and dynamic characteristics of the mentioned process.

2. Formalism and Method

Naturally, the spectra of Drell-Yan depend on collision energy. For that reason, we should use different probability functions to study these spectra at different energies. Here, we briefly describe the three functions which will be used in this study. In the following, and are the transverse momenta of the two quarks, and is the transverse momentum of the two quark system, which equals the transverse momentum of the dilepton system at leading order.

2.1. The Lévy-Tsallis Function

The Boltzmann distribution is the most important probability density function in thermodynamic and statistical physics. We present the probability density function of as a simple Boltzmann distribution [3638]: where is the number of identical particles of mass produced in the collisions, is the normalization constant, and is the effective temperature of the collision system.

The Boltzmann distribution is a special form of the Tsallis distribution, and the latter has a few alternative forms [2630]. As one of the Tsallis distribution and its alternative forms, the Lévy-Tsallis function of the spectrum of hadrons [2630] is used in this work. We have the following form to describe the transverse momentum () distribution of (anti-)quark: where is the normalization constant, and are the fitted parameters, and is the mass of or taking part in the reaction. In general, we use in the Drell-Yan process because or is from the participant hadrons. The same or is for sea quarks and those in baryons, where the sea quarks of higher mass are not considered in this work. In QGP, and because the quarks are approximately bare [39]. It has been verified that the Tsallis distribution is just a special case of the Lévy distribution, but not the opposite [30].

2.2. The (Two-Component) Erlang Distribution

The Erlang distribution [3133] is proposed to fit the spectra in the multisource thermal model [40]. Generally, a two-component Erlang distribution [3133] is used to describe both the soft and hard processes. The contribution fractions of the two components are determined by fitting the experimental data. The numbers of parton sources participating in the soft and hard processes are represented by and , respectively. The contribution () of each parton source to of final-state particle is assumed to obey an exponential function: where represents the average contributed by the -th source. Because is the same for different sources, the index in is omitted.

The distribution contributed by () sources is the convolution of () exponential functions, which gives the Erlang distribution. Let denote the contribution fraction of the first component (soft process). The two-component Erlang distribution is

Fitting the data with the two-component Erlang distribution, we can get the changes of parameters , , and .

We should discuss the values of and further. If , the participant partons are expected to be and gluons in the soft or nonviolent annihilation process. Considering that the probability of multiparton participating together in the process is low, we have usually or 3 in this work. Generally, the larger the , the sharper the distribution peak. In many cases, means that and a gluon participate in the soft process. For all cases, (always true in this work) means that only participates in the violent annihilation in the hard process.

2.3. The Hagedorn Function

The Hagedorn function is an inverse power law [34, 35] which is an empirical formula derived from perturbative QCD. Generally, this function can only describe the spectra at large , but not the entire interval. In the case of using the Hagedorn function in a wide range of , the probability density function of can be expressed as where is the normalization constant, and and are the fitted parameters. The final-state particles with high momenta are mainly produced by the hard scattering process during the collisions. However, both the soft and hard processes contribute to the spectra. In some cases, the soft excitation process in the low range can also be described by the Hagedorn function. We try to use the Hagedorn function to describe the transverse momentum distribution of (anti-)quarks. That is, we may use instead of in Eq. (5) to obtain the transverse momentum distribution of (anti-)quarks, which is a new form of Eq. (5) and will be used in the following section.

We have tested the Hagedorn function with different revisions in which in Eq. (5) is replaced by , is replaced by , or is replaced by , and is replaced by , where and vary in different revisions. The uses of the revised Hagedorn functions result in some overestimations in low (or high) region comparing to the Hagedorn function. Contrarily, these revisions result in some underestimations in high (or low) region due to the normalization. The revisions of the Hagedorn function are beyond the focus of the present work, and we shall not discuss them further.

2.4. The Convolution of Functions

The convolution of functions is an important operation process in functional analysis that can be used to describe the weighted superposition of input and system response (that is, two subfunctions). The Drell-Yan process is the result of the interactions of and in high-energy collisions, which means that we need the convolution of two functions to describe this process. Indeed, the above Eq. (2) or (5) can be used to describe the transverse momentum distribution, , of a single (anti-)quark’s contribution. The second (anti-)quark’s contribution is , where is still the transverse momentum of the system. So, the convolution of two probability density functions should be used to describe the spectrum of in the Drell-Yan process. We have the convolution of two Eq. (2) or (5) to be expressed as where [] is shown as Eq. (2) if we use the Lévy-Tsallis function or Eq. (5) if we use the Hagedorn function.

It should be noted that the total transverse momentum before (of the two quarks system) and after (of the two leptons system) are equal. The assumption that the total transverse momentum is equal to the sum of the scalar transverse momenta of the two partons that is for the particular case in which the vectors and are parallel. Our recent works [41, 42] show that this assumption is in agreement with many data. Naturally, we do not rule out other assumptions such as the particular case in which and are perpendicular and the general case which shows any azimuth for and . The particular case used in this work is more easier than other particular or general case in the fit to data. We are inclined to use the parallel case.

As a legitimate treatment, the convolution formula Eq. (6) can be used to fit the spectrum of in the Drell-Yan process, where and are from empirical guess which is simpler than the factorization theorem based on perturbative QCD. On one hand, Eq. (6) can reflect the weighted contribution of the transverse momentum of each (anti-)quark to the spectrum in the process. On the other hand, Eq. (6) can also reflect the system in which two main participants take part in the interactions. Using the convolution to fit the data is a good choice for us, which allows us to more accurately understand the interaction process and mechanism between interacting partons, more completely describe the energy dependence and interdependence of the function parameters, and further better analyze the spectrum.

3. Results and Discussion

3.1. Comparison with Data

Figure 1 shows the spectra of [with different invariant masses () or Feynman variables ()] produced by the Drell-Yan process in different collisions at different energies (with different integral luminosities if available in literature), where the concrete type of is also given in each panel. The symbols and on the vertical axis denote the energy of and the cross-section of events, respectively. Among them, the data points presented in Figures 1(a)1(c) are quoted from the proton-copper (-Cu) collision experiments performed by the E288 Collaboration [43], and the collision energy per nucleon pair or if in a simplified form) is 19.4, 23.8, and 27.4 GeV, respectively. The data points shown in Figure 1(d) are the results of the -Cu collision experiment performed by the E605 Collaboration [44] at a collision energy of 38.8 GeV. For the E288 Collaboration, the invariant mass ranges from 4 to , while the corresponding invariant mass ranges of the E605 Collaboration are from 7 to . The experimental data points in Figures 1(e) and 1(f) are from negative pions () induced wolfram (W) (-W) collisions at 21.7 GeV performed by the FNAL-615 Collaboration [45]. The different symbols in Figure 1(e) represent invariant mass in the range of 4.05– with different scalings, where the units GeV/ are not shown in the panel due to crowding space. The Feynman variables range from 0 to 1 with a step of 0.1, as shown in Figure 1(f). Different collaborations have different intervals of and , while the detailed binning information is marked in the panels. In some cases, the range of rapidity is not available due to other selection conditions such as the complex polar coverages and sensitivities of detector components or Feynman variable being used [45]. The total experimental uncertainties are cited from refs. [4345] which include both the statistical and systematic uncertainties if both are available. The solid, dashed, and dotted curves in all panels are the results of our fittings with the convolution of two Lévy-Tsallis functions, the two-component Erlang distribution, and the convolution of two Hagedorn functions, respectively. The histograms in this and following figure correspond to QCD calculations which will be discussed later. We use the minimum- to evaluate the goodness of the fits, where and are for the -th data. We list the results of the fits (parameters), the and the number of degrees of freedom (ndof) in Table 1. The numbers and which are not listed in the table to avoid trivialness. One can see that the three functions can approximately describe the spectra of produced by the Drell-Yan process in high-energy -Cu and -W collisions. The two-component Erlang distribution describes better than the other two functions.

Figure (GeV) (GeV/) (GeV) (GeV/) (GeV/) (GeV/)ndof

Figure 1(a)19.44-501225216410

Figure 1(b)23.84-51551315711

Figure 1(c)27.45-61033515911

Figure 1(d)38.87-8196427

Figure 1(e)21.74.05-4.5041210756313