# Z Boson Production via p-p and Pb-Pb Collisions at spp=5.02TeV

Authors: Leonard S. Kisslinger; Debasish Das
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

We estimate the production ofZa bosons, with a as the component of a Z vector boson, via p-p collisions using previous work on J/Ψ,Ψ2S production in p-p collisions, with the new aspect being the creation of Za bosons via quark interactions. We then estimate the production of Za bosons via Pb-Pb collisions using modification factors from previous publications.


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## Body

## 1. Introduction

This is an extension of our recent work on heavy quark state production via Xe-Xe collisions atspp=5.44TeV [1] and heavy quark state production in Pb-Pb collisions at spp=5.02TeV [2]. More than three decades ago, W and Z bosons were observed at CERN via proton-antiproton experiments at spp=540GeV [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, aZ boson is a quantum of weak interactions with no electric charge. The Z boson with mass MZ≃91GeV is a vector boson with quantum spin 1 [5]. Therefore, a Z boson has three components, Za, with a=−1, 0, and 1.Our present estimate of the production ofZa bosons, via Pb-Pb collisions, is motivated by the fact that since Z bosons have only a weak interaction [5], they have little interaction with the nuclear medium and by ALICE experiments that measured Z boson production in Pb-Pb collisions [6] and in p-Pb collisions [7, 8] at sNN=5.02TeV. The estimate of Z boson production via Pb-Pb collisions makes use of Ref [9], which was based estimates of heavy quark state production in p-p collisions [10]. Note that when the final calculation and results are presented in Sections 3 and 4, the momentum pa→pZ, the momentum of the Z boson produced by Pb-Pb collisions at spp=5.02TeV, and Za→Z, a Z boson.Our present work is also related to an estimate ofΨ2S to J/Ψ decay to π mesons [11] except the quarks have a vertex with Z bosons rather than pions and there is no gluon-Z vertex. Also, it was shown [12] that the Ψ2S state is approximately a 50%-50% mixture of a standard charmonium and hybrid charmonium state:
(1)Ψ2S>≃−0.7cc¯2S>+1−0.5cc¯g2S>,while J/Ψ is essentially a standard qq¯ state J/Ψ1S>≃cc¯1S>, which we use in our estimate of Za boson production via Ψ2S→J/Ψ1S+Za. Having a hybrid component, cc¯g, is important for Z boson production from Ψ2S decay as the active gluon component of Ψ2S produces a Z boson, as shown in Figure 1 (Section 2).Figure 1
Ψ2S, a hybrid component, to J/psi1S+Za.


## 2.J/Ψ+Z Production in p-p Collisions with spp=5.02TeV

The cross section forpp→Ψ2S→J/Ψ1S+Za in terms of fq[10, 13], the quark distribution function, is
(2)σpp→Ψ2S→JΨ1S+Za=fqx¯y,2mfqax¯y,2mσΨ2S→JΨ1S+Za,where y is the rapidity, a=4m2/s=3.6×10−7, and x¯y=1.058xy. We take y=0 in the present work, so x¯0≃6.4×10−4.Fors=5.02TeV, the quark distribution functions fq [10, 13] are
(3)fqx¯0,2m≃82.37−63582.36x0≃41.6,fqax¯0,2m≃82.37−ax¯0≃82.4.Therefore, from equations (30), (2), and (3),
(4)σPbPb→Ψ2S→JΨ1S+Za≃4.46×105σpp→Ψ2S→JΨ1S+Za.We use the following notation:(5)σpp→Ψ2S→JΨ1S+Za≡σHHZp=gμν∏HZaμνp+∏HHZaμνp,with ΠHZaμνp and ΠHHZaμνp defined below.The normal∣cc¯2S> component of Ψ2S decaying to ∣cc¯1S> with Za production via quark-Z coupling is shown in Figure 2.Figure 2
Ψ2S, a standard component, to J/psi1S+Za.In Figure1, Za production with the hybrid ∣cc¯g2S> component of Ψ2S is shown.Figure3 shows the coupling processes needed for Figures 1 and 2.Figure 3
(a) Gluon-quark coupling and (b)c-Z coupling.
(a)(b)In Figure3(a), the operator giving the gluon sigma coupling is
(6)gluon‐quarkcoupling=14SκδGkGκδ0,SκδGk=σκδ,Sk+=σκδSk+Skσκδ,where σκδ=iγκγδ−gκδ and Gκδ is the gluon field.In Figure3(b), with Za with a as the component of the vector Z boson and defining gc≡gcV≃0.25 [5], the ccZα coupling is [14]
(7)SZa=γagc−gcAγ5.As shown in Section3.2, the γaγ5 term does not contribute to σHHZp, so we define gcA=gc.


## 3.Ψ2S Decay to J/Ψ+Za

In this section, we estimate the decay of∣Ψ2S> to ∣J/Ψ1S>+Za for both the standard and hybrid components of ∣Ψ2S> as shown in Figures 1 and 2.


### 3.1.Ψ2S Decay to J/Ψ+Za via the Standard Component of Ψ2S

As in Ref [11], the correlator corresponding to Figure 2 is
(8)∏HZaμνp=∑abg2∫d4k2π4TrSkγμSZaSp−kγν,where the quark propagator Sk=k+M/k2−M2, M is the mass of a charm quark (Mc), k=∑μkμγμ, and g2=4παs≃1.49 [5]. Since TrSkγμSZakSp−kγν is independent of color, ∑ab=3.Thus, the correlator forΨ2Snormal decay to J/Ψ1S+Z is
(9)∏HZaμνp=3g2gc∫d4k2π4TrSkγμγa1−γ5Sp−kγν.The trace in equation (9) is
(10)TrSkγμγa1−γ5Sp−kγν=Trk+Mγμγa1−γ5p−k+Mγνk2−M2k−p2−M2.Using the fact that the trace of an odd number ofγs vanishes and Trγ5γαγβγδγλ=−4iεαβδλ,
(11)Trk+Mγμγa1−γ5p−k+Mγν=4Mpνgaμ+pagμν−pμgaν+iεμaαν+2kμgaν−2kagμν−2ikαεαμaν.Therefore,(12)∏HZaμνp=12g2gcM∫d4k4π4pνgaμ+pagμν−pμgaν+ipαεμaαν+2kμgaν−2kagμν−2ikαεαμaνk2−M2k−p2−M2.Using(13)∫d4k4π41k2−M2k−p2−M2=2M2−p2/24π2I0p,∫d4k4π4kμk2−M2k−p2−M2=pμ2M2−p2/24π2I1p,one finds
(14)∏HZaμνp=AM2M2−p2/24π2pνgaμ−pμgaν+ipαεμaανI0p+2pμgaν−2pagμν−2ipαεμaανI1pA=12g2gc,with
(15)I0p=∫01dα1p2α−α2−M2,I1p=∫01dααp2α−α2−M2.


### 3.2.Ψ2S Decay to J/Ψ+Za via the Hybrid Component of Ψ2S

The two-point correlator for the hybridΨ2S-J/Ψ, corresponding to Figure 1, without the gluon-Z or quark-Z coupling is [12] (see equation (18))
(16)∏HHμνp=3g24∫d4k2π4TrσκδSk+γλSp−kγμTrGνλ0Gκδ0.The correlatorΠHHqq¯Zaμν, obtained from Figure 1, is
(17)∏HHZaμνp=3g24∫d4k2π4TrσκδSk+γλSZaSp−kγμTrGνλ0Gκδ0.Note that [15] (with G2=0.476GeV2)
(18)TrGνλ0Gκδ0=2π41296G2gνκgλδ−gνδgκλ,(19)σκδSk+=i−2gκδk+M+2Mγκγδ+kαγκγδγα+γαγκγδk2−M2.Therefore,(20)∏HHZaμνp=B∫d4k2π4ik2−M2p−k2−M2gνκgλδ−gνδgλκTrA1,M2B=3g2gc42π41296G2≃25.87GeV2,with
(21)TrA1≡Trkαγκγδγα+γαγκγδγa1−γ5γλp−k+Mγμ−2gκδk+M+2Mγκγδγλγa1−γ5p−k+Mγμ.Note thatgνκgλδ−gνδgλκgκδ=0.0, so the gκδ term in equation (21) vanishes. Therefore, from equation (21),
(22)gνκgλδ−gνδgλκTrA1=2Trkαγνγλγα+γαγνγλ−2gλνγαγa1−γ5γλ+4Mγν−γλγa1−γ5p−k+Mγμ.SinceTroddnumberofγs=0,
(23)gνκgλδ−gνδgλκTrA1=2MTrkαγνγλγα+γαγνγλ−2gλνγαγa1−γ5γλγμ+4pβ−kβγν−γλγa1−γ5γβγμ.As in equation (11), using εαβλλ=0, one obtains for the 4−γ terms
(24)2MTr−2kαgλνγαγa1−γ5γλγμ+4pβ−kβγν−γλγa1−γ5γβγμ=16M−kagμν+kμgaν−kνgaμ+ikαεαaνμ+2pμ−kμgνa+pa−kagνμ−pν−kνgμa−pμ−kμgνa−pa−kagνμ+pν−kνgμa+2ipβ−kβενaβμ−ενaβμ.For the6−γ terms in equation (22),
(25)Trγαγβγδγλγμγν=4gαβTrγδγλγμγν+gαδTrγβγλγμγν+gαλTrγβγδγμγν+gαμTrγβγδγλγν+gανTrγβγδγλγμ,(26)Trγαγβγδγλγμγνγ5=−16igαβεδλμν+gαδεβλμν+gαλεβδμν+⋯.From equations (22), (25), and (26) and using εββμν=0, the 6−γ terms in equation (22) are
(27)2MTrkαγνγλγα+γαγνγλγa1−γ5γβγμ=4M3kagνμ+kμgaν+kνgaμ.DefiningΠZμνp=ΠHZaμνp+ΠHHZaμνp, with pZ as the Z boson momentum, from equations (14), (22), (23), (24), (27), and (13),
(28)∏Zμνp=M2M2−p2/24π2Apνgaμ−pμgaν+ipαεμaαν+B3pμgνa+pZgνμ−pνgμa−pμgνa−pZgνμ+pλgμa+pλgμaI0p+A2pμgaν−2pZgμν−2ipαεμaαν−32BpZgνμ+pμgaν−pνgaμ−pZgμν−pμgaν+pμgaν+ipαεαaνμ−2pμgνa+pZgνμ−pνgμa+pμgνa−pZgνμ+pνgμaI1p.From equations (5) and (20), taking the μ sum with gμν,
(29)σHHZp=4.46×105M2M2−p2/24π2BpZI0p+I1p=7.4×104GeV−2×pZ2M2−p2/2MI0p+I1p,with the Z boson momentum pZ≃1−3MeV, so pz≪p as p≃2−3GeV in our calculation.


## 3.1.Ψ2S Decay to J/Ψ+Za via the Standard Component of Ψ2S

As in Ref [11], the correlator corresponding to Figure 2 is
(8)∏HZaμνp=∑abg2∫d4k2π4TrSkγμSZaSp−kγν,where the quark propagator Sk=k+M/k2−M2, M is the mass of a charm quark (Mc), k=∑μkμγμ, and g2=4παs≃1.49 [5]. Since TrSkγμSZakSp−kγν is independent of color, ∑ab=3.Thus, the correlator forΨ2Snormal decay to J/Ψ1S+Z is
(9)∏HZaμνp=3g2gc∫d4k2π4TrSkγμγa1−γ5Sp−kγν.The trace in equation (9) is
(10)TrSkγμγa1−γ5Sp−kγν=Trk+Mγμγa1−γ5p−k+Mγνk2−M2k−p2−M2.Using the fact that the trace of an odd number ofγs vanishes and Trγ5γαγβγδγλ=−4iεαβδλ,
(11)Trk+Mγμγa1−γ5p−k+Mγν=4Mpνgaμ+pagμν−pμgaν+iεμaαν+2kμgaν−2kagμν−2ikαεαμaν.Therefore,(12)∏HZaμνp=12g2gcM∫d4k4π4pνgaμ+pagμν−pμgaν+ipαεμaαν+2kμgaν−2kagμν−2ikαεαμaνk2−M2k−p2−M2.Using(13)∫d4k4π41k2−M2k−p2−M2=2M2−p2/24π2I0p,∫d4k4π4kμk2−M2k−p2−M2=pμ2M2−p2/24π2I1p,one finds
(14)∏HZaμνp=AM2M2−p2/24π2pνgaμ−pμgaν+ipαεμaανI0p+2pμgaν−2pagμν−2ipαεμaανI1pA=12g2gc,with
(15)I0p=∫01dα1p2α−α2−M2,I1p=∫01dααp2α−α2−M2.


## 3.2.Ψ2S Decay to J/Ψ+Za via the Hybrid Component of Ψ2S

The two-point correlator for the hybridΨ2S-J/Ψ, corresponding to Figure 1, without the gluon-Z or quark-Z coupling is [12] (see equation (18))
(16)∏HHμνp=3g24∫d4k2π4TrσκδSk+γλSp−kγμTrGνλ0Gκδ0.The correlatorΠHHqq¯Zaμν, obtained from Figure 1, is
(17)∏HHZaμνp=3g24∫d4k2π4TrσκδSk+γλSZaSp−kγμTrGνλ0Gκδ0.Note that [15] (with G2=0.476GeV2)
(18)TrGνλ0Gκδ0=2π41296G2gνκgλδ−gνδgκλ,(19)σκδSk+=i−2gκδk+M+2Mγκγδ+kαγκγδγα+γαγκγδk2−M2.Therefore,(20)∏HHZaμνp=B∫d4k2π4ik2−M2p−k2−M2gνκgλδ−gνδgλκTrA1,M2B=3g2gc42π41296G2≃25.87GeV2,with
(21)TrA1≡Trkαγκγδγα+γαγκγδγa1−γ5γλp−k+Mγμ−2gκδk+M+2Mγκγδγλγa1−γ5p−k+Mγμ.Note thatgνκgλδ−gνδgλκgκδ=0.0, so the gκδ term in equation (21) vanishes. Therefore, from equation (21),
(22)gνκgλδ−gνδgλκTrA1=2Trkαγνγλγα+γαγνγλ−2gλνγαγa1−γ5γλ+4Mγν−γλγa1−γ5p−k+Mγμ.SinceTroddnumberofγs=0,
(23)gνκgλδ−gνδgλκTrA1=2MTrkαγνγλγα+γαγνγλ−2gλνγαγa1−γ5γλγμ+4pβ−kβγν−γλγa1−γ5γβγμ.As in equation (11), using εαβλλ=0, one obtains for the 4−γ terms
(24)2MTr−2kαgλνγαγa1−γ5γλγμ+4pβ−kβγν−γλγa1−γ5γβγμ=16M−kagμν+kμgaν−kνgaμ+ikαεαaνμ+2pμ−kμgνa+pa−kagνμ−pν−kνgμa−pμ−kμgνa−pa−kagνμ+pν−kνgμa+2ipβ−kβενaβμ−ενaβμ.For the6−γ terms in equation (22),
(25)Trγαγβγδγλγμγν=4gαβTrγδγλγμγν+gαδTrγβγλγμγν+gαλTrγβγδγμγν+gαμTrγβγδγλγν+gανTrγβγδγλγμ,(26)Trγαγβγδγλγμγνγ5=−16igαβεδλμν+gαδεβλμν+gαλεβδμν+⋯.From equations (22), (25), and (26) and using εββμν=0, the 6−γ terms in equation (22) are
(27)2MTrkαγνγλγα+γαγνγλγa1−γ5γβγμ=4M3kagνμ+kμgaν+kνgaμ.DefiningΠZμνp=ΠHZaμνp+ΠHHZaμνp, with pZ as the Z boson momentum, from equations (14), (22), (23), (24), (27), and (13),
(28)∏Zμνp=M2M2−p2/24π2Apνgaμ−pμgaν+ipαεμaαν+B3pμgνa+pZgνμ−pνgμa−pμgνa−pZgνμ+pλgμa+pλgμaI0p+A2pμgaν−2pZgμν−2ipαεμaαν−32BpZgνμ+pμgaν−pνgaμ−pZgμν−pμgaν+pμgaν+ipαεαaνμ−2pμgνa+pZgνμ−pνgμa+pμgνa−pZgνμ+pνgμaI1p.From equations (5) and (20), taking the μ sum with gμν,
(29)σHHZp=4.46×105M2M2−p2/24π2BpZI0p+I1p=7.4×104GeV−2×pZ2M2−p2/2MI0p+I1p,with the Z boson momentum pZ≃1−3MeV, so pz≪p as p≃2−3GeV in our calculation.


## 4. Calculation ofσpp→Ψ2S→Ψ1S+Za via Calculation of σHHZp for p≃Mc≃1.27GeV

Carrying out the integrals forI0p,I1p shown in equation (15), one obtains from equation (29) the values of σHHZp, with pZ=1,2,3MeV=0.001,0.002,0.003GeV, which from equation (5) is the cross section σpp→Ψ2S→J/Ψ1S+Za with the proton-proton energy=5.02TeV, shown in Figure 4.Figure 4
σpp→Ψ2S→J/Ψ1S+Za with pZ=a1,b2,andc3MeV.Note that [5] the units for a cross section are nb∝hcGeV−2. As it is customary, we take h=c=1.


## 5.J/Ψ+Z Production in Pb-Pb Collisions with spp=5.02TeV

The cross section for the production of a heavy quark stateΦ with helicity λ=0 (for unpolarized collisions [10]) in the color octet model in Pb-Pb collisions is given by [9]
(30)σPbPb→Φ=RPbPbENbinPbPbσpp→Φ,where NbinPbPb is the number of binary collisions, RPbPbE is the nuclear modification factor, and E is the total energy in Pb-Pb collisions.From [2], RPbPbENbinPbPb≃130. Therefore,
(31)σPb‐Pb→Ψ2S→JΨ1S+Za≃130×σpp→Ψ2S→JΨ1S+Za,or σPb‐Pb→Ψ2S→J/Ψ1S+Za is approximately 130 times the results shown in Figure 4.


## 6. Conclusions

Using the relationship between the cross sectionsσPbPb→Ψ2S→J/Ψ1S+Z and σpp→Ψ2S→J/Ψ1S+Z shown in equation (30) and Ψ2S decay to J/Ψ+Z for both the standard and hybrid components of Ψ2S, the cross section σHHZp≡σPbPb→Ψ2S→J/Ψ1S+Z was estimated for spp=5.02TeV and the Z boson momentum pZ=1,2,and3MeV, as shown in the figure. This should be useful for the experimental measurement of Z boson production via Pb-Pb collisions at spp=5.02TeV. For simplicity, we assumed that the rapidity=y=0, where y≡lnE+pL/E−pL with pL as the longitudinal momentum. Current experiments [6] measure Z boson production via Pb-Pb collisions at spp=5.02TeV at large rapidities.



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