#### 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.