Abstract
We estimate the production of bosons, with as the component of a vector boson, via - collisions using previous work on production in - collisions, with the new aspect being the creation of bosons via quark interactions. We then estimate the production of bosons via Pb-Pb collisions using modification factors from previous publications.
1. Introduction
This is an extension of our recent work on heavy quark state production via Xe-Xe collisions at [1] and heavy quark state production in Pb-Pb collisions at [2]. More than three decades ago, and bosons were observed at CERN via proton-antiproton experiments at [3]. CMS experiments on electroweak boson production via relativistic heavy ion collisions (RHIC) are related to our present research [4].
As the photon is the quantum of electromagnetic interactions, a boson is a quantum of weak interactions with no electric charge. The boson with mass is a vector boson with quantum spin 1 [5]. Therefore, a boson has three components, , with , 0, and 1.
Our present estimate of the production of bosons, via Pb-Pb collisions, is motivated by the fact that since bosons have only a weak interaction [5], they have little interaction with the nuclear medium and by ALICE experiments that measured boson production in Pb-Pb collisions [6] and in -Pb collisions [7, 8] at . The estimate of boson production via Pb-Pb collisions makes use of Ref [9], which was based estimates of heavy quark state production in - collisions [10]. Note that when the final calculation and results are presented in Sections 3 and 4, the momentum , the momentum of the boson produced by Pb-Pb collisions at , and , a boson.
Our present work is also related to an estimate of to decay to mesons [11] except the quarks have a vertex with bosons rather than pions and there is no gluon- vertex. Also, it was shown [12] that the state is approximately a 50%-50% mixture of a standard charmonium and hybrid charmonium state: while is essentially a standard state , which we use in our estimate of boson production via . Having a hybrid component, , is important for boson production from decay as the active gluon component of produces a boson, as shown in Figure 1 (Section 2).
2. Production in - Collisions with
The cross section for in terms of [10, 13], the quark distribution function, is where is the rapidity, , and . We take in the present work, so .
For , the quark distribution functions [10, 13] are
Therefore, from equations (30), (2), and (3),
We use the following notation: with and defined below.
The normal component of decaying to with production via quark- coupling is shown in Figure 2.
In Figure 1, production with the hybrid component of is shown.
Figure 3 shows the coupling processes needed for Figures 1 and 2.
(a)
(b)
In Figure 3(a), the operator giving the gluon sigma coupling is where and is the gluon field.
In Figure 3(b), with with as the component of the vector boson and defining [5], the coupling is [14]
As shown in Section 3.2, the term does not contribute to , so we define .
3. Decay to
In this section, we estimate the decay of to for both the standard and hybrid components of as shown in Figures 1 and 2.
3.1. Decay to via the Standard Component of
As in Ref [11], the correlator corresponding to Figure 2 is where the quark propagator , is the mass of a charm quark (), , and [5]. Since is independent of color, .
Thus, the correlator for decay to is
The trace in equation (9) is
Using the fact that the trace of an odd number of s vanishes and ,
Therefore,
Using one finds with
3.2. Decay to via the Hybrid Component of
The two-point correlator for the hybrid -, corresponding to Figure 1, without the gluon- or quark- coupling is [12] (see equation (18))
The correlator , obtained from Figure 1, is
Note that [15] (with )
Therefore, with
Note that , so the term in equation (21) vanishes. Therefore, from equation (21),
Since ,
As in equation (11), using , one obtains for the terms
For the terms in equation (22),
From equations (22), (25), and (26) and using , the terms in equation (22) are
Defining , with as the boson momentum, from equations (14), (22), (23), (24), (27), and (13),
From equations (5) and (20), taking the sum with , with the boson momentum , so as in our calculation.
4. Calculation of via Calculation of for
Carrying out the integrals for shown in equation (15), one obtains from equation (29) the values of , with , which from equation (5) is the cross section with the proton-proton , shown in Figure 4.
Note that [5] the units for a cross section are . As it is customary, we take .
5. Production in Pb-Pb Collisions with
The cross section for the production of a heavy quark state with helicity (for unpolarized collisions [10]) in the color octet model in Pb-Pb collisions is given by [9] where is the number of binary collisions, is the nuclear modification factor, and is the total energy in Pb-Pb collisions.
From [2], . Therefore, or is approximately 130 times the results shown in Figure 4.
6. Conclusions
Using the relationship between the cross sections and shown in equation (30) and decay to for both the standard and hybrid components of , the cross section was estimated for and the boson momentum , as shown in the figure. This should be useful for the experimental measurement of boson production via Pb-Pb collisions at . For simplicity, we assumed that the , where with as the longitudinal momentum. Current experiments [6] measure boson production via Pb-Pb collisions at at large rapidities.
Data Availability
All data for our article can be found in the references, especially Refs [4–6, 12, 15], as is stated in our article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Author D. Das. acknowledges the facilities of Saha Institute of Nuclear Physics, Kolkata, India. Author L.S. Kisslinger acknowledges support in part as a visitor at the Los Alamos National Laboratory, Group P25. The authors thank Bijit Singha for helpful suggestions.