NLO production of $W^{\pm }$ and $Z^0$ vector bosons via hadron collisions in the frameworks of Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution functions

In a series of papers, we have investigated the compatibility of the Kimber-Martin-Ryskin (KMR) and Martin-Ryskin-Watt (MRW) unintegrated parton distribution functions (UPDFs) as well as the description of the experimental data on the proton structure functions. The present work is a sequel to that survey, via calculation of the transverse-momentum distribution of the electroweak gauge vector bosons in the $k_t$-factorization scheme, by the means of the KMR, the leading-order (LO) MRW, and the next-to-leading-order (NLO) MRW UPDF, in the NLO. To this end, we have calculated and aggregated the invariant amplitudes of the corresponding involved diagrams in the NLO and counted the individual contributions in different frameworks. The preparation process for the UPDF utilizes the parton distribution functions of Martin et al., MSTW2008-LO, MSTW2008-NLO, MMHT2014-LO, and MMHT2014-NLO, as the inputs. Afterward, the results have been analyzed against each other as well as the existing experimental data, i.e., D0, CDF, ATLAS, and CMS collaborations. Our calculations show excellent agreement with the experiment data. It is, however, interesting to point out that the calculation using the KMR framework illustrates a stronger agreement with the experimental data, despite the fact that the LO MRW and the NLO MRW formalisms employ a better theoretical description of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation. This is of course due to the use of the different implementation of the angular ordering constraint in the KMR approach, which automatically includes the resummation of $\ln (1/x)$, Balitski-Fadin-Kuraev-Lipatov logarithms, in the LO Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation.

Figure 1

The NLO ladder-type diagrams for the production of $W^{\pm }$ and $Z^0$ in the $k_t$-factorization framework. The $f_g(x,k_t^2,{\mu }^2)$ and $f_q(x,k_t^2,{\mu }^2)$ represent the corresponding UPDF in the KMR, the LO-MRW, or the NLO-MRW framework, i.e., Eqs. (6), (7), (9), and (11).

Figure 2

The individual contributions into the matrix elements of the partonic scattering. The diagrams in the panel (a) correspond to the $q+{\overline{q}}^{\prime }\to W^{\pm }/Z^0+g+g$ subprocess, panel (b) corresponds to the $q+g\to W^{\pm }/Z^0+q+g$ subprocess, and panel (c) corresponds to the $g+g\to W^{\pm }/Z^0+q+{\overline{q}}^{\prime }$ subprocess. It should be pointed out that one may find additional non-ladder-type diagrams which contribute to these matrix elements. We have eliminated these undesirable contributions using our choice of the gluon gauge, Eq. (21).

Figure 3

The differential cross section of the productions of $W^{\pm }$ bosons in a DIS at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$, against the transverse-momentum distribution of the produced particle. The panels (a) and (b) illustrate the up-down and charm-strange contributions, respectively. The contribution of each partonic subprocess is singled out: the green-dashed histogram is for $g+g\to W^{\pm }+q+{\overline{q}}^{\prime }$, the red-dotted histogram is for $q+g\to W^{\pm }+q^{\prime }+g$, and the blue-dashed-dotted histogram is for $q+{\overline{q}}^{\prime }\to W^{\pm }+g+g$. The black full histogram is the total contribution of the given quark pairs. The histograms are produced using the KMR UPDF with the PDF of MSTW2008.

Figure 4

The differential cross section of the productions of $Z^0$ boson in a DIS at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$, against the transverse-momentum distribution of the produced particle. The contributions of the up and the down quarks [the panels (a) and (b), respectively] and the lightest sea quarks [the panel (c) for the charm quark and the panel (d) for the strange quark]. The green-dashed histogram is for $g+g\to Z^0+q+\overline{q}$, the red-dotted histogram is for $q+g\to Z^0+q+g$, and the blue-dashed-dotted histogram is for $q+\overline{q}\to Z^0+g+g$. The black full histogram is the total contribution of the given quarks. The data are produced using the KMR UPDF, with the PDF of MSTW2008.

Figure 5

The contributions of the individual partonic subprocesses to the differential cross section of the productions of $W^{\pm }$ bosons in a DIS at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$ vs the transverse-momentum distribution of the produced particle. The panels (a), (b), and (c) correspond to the $g+g\to W^{\pm }+q+{\overline{q}}^{\prime }$, $q+{\overline{q}}^{\prime }\to W^{\pm }+g+g$ and $g+q\to W^{\pm }+g+q^{\prime }$ sub-processes, respectively. The data have been obtained using the UPDF of KMR, with the PDF of MSTW2008.

Figure 6

The contributions of the individual partonic subprocesses into the differential cross section of the productions of $Z^0$ bosons in a DIS at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$ vs the transverse-momentum distribution of the produced particle. The notions of the diagrams are the same as in Fig. 5.

Figure 7

The comparison of the differential cross section of the $W^{\pm }$ production in the NLO in the KMR [panel (a)], LO MRW [panel (b)], and NLO MRW [panel (c)] frameworks. Panel (d) illustrates this comparison with the help of experimental data of the D0 Collaboration, Ref. [37]. Panels (e), (f), and (g) are the same values, but this time they divided by the total cross sections in their respective frameworks and compared to an older set of data points from the D0 Collaboration, [34]. Again, an overall comparison with the experiment is presented in panel (h). To perform these calculations, we have utilized the PDF of MSTW2008.

Figure 8

The comparison of the differential cross section of the $Z^0$ production in the NLO in the KMR [panel (a)], LO MRW [panel (b)], and NLO MRW [panel (c)] frameworks. Panel (d) illustrates this comparison with the help of the experimental data of D0 and CDF collaborations, Refs. [32, 34].

Figure 9

The production rate of the $W^{\pm }$ boson in $E_{\mathrm{CM}}=1.8\mathrm{TeV}$. The labels (a), (b), and (c) compare the contributions of the individual subprocesses in their respective frameworks. The total values of differential cross section in these frameworks are subjected to a comparison with the data of the D0 Collaboration [37] separately, in label (d). This very same notion is also presented in labels (e) through (f), where the $1/\sigma d\sigma /d{k}_t$ histograms are compared with each other and with the data from Ref. [34].

Figure 10

The production rate of the $Z^0$ boson in $E_{\mathrm{CM}}=1.8\mathrm{TeV}$. The notions of the diagrams are the same as in Fig. 9.

Figure 11

Comparison of the differential cross section of the $W^{\pm }$ production, using the UPDF of KMR, prepared with the PDF of MSTW2008 [label (a)] and MMHT2014 [label (b)]. Label (c) shows their difference relative to the experimental data of the D0 Collaboration, Ref. [37].

Figure 12

Comparison of the differential cross section of the $Z^0$ production, using the UPDF of KMR, prepared with the PDF of MSTW2008 [label (a)] and MMHT2014 [label (b)]. Label (c) shows their difference relative to the experimental data of the D0 and CDF collaborations, Refs. [32, 34].

Figure 13

The differential cross section of the production of the $W^{\pm }$, calculated in different frameworks, against the transverse momentum of the produced gauge boson at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$. The notions of the histograms are as follows: the continuous black histogram represents the calculation in using the KMR UPDF, the dotted green histogram is prepared in the LO MRW framework, and the short-dotted red line is prepared in the NLO MRW. To perform these calculations, we have utilized the PDF of MSTW2008. The brown dot-dot-dashed histogram is produced using the CCFM TMD PDF (Ref. [40]). The yellow dotted-dashed histogram is calculated, utilizing the doubly DUPDF in the framework of $(k_t-z)$ factorization [9]. The purple short-dashed histogram is calculated in the collinear framework.

Figure 14

The differential cross section of the production of the $Z^0$, calculated in different frameworks, against the transverse momentum of the produced gauge boson at $E_{\mathrm{CM}}=1.8\mathrm{TeV}$. The notions of the histograms are the same as in Fig. 13.

Figure 15

The cross section of the production of the $W^{\pm }$ bosons as a function of the center-of-mass energy, $E_{\mathrm{CM}}$. The experimental data are acquired from the UA1, UA2, D0, and CDF collaborations, Refs. [29, 30, 31, 32, 33, 34, 35, 36, 37]. The calculations are performed using the KMR and the LO MRW UPDF. We have omitted the NLO UPDF results here, to save computation data.

Figure 16

The cross section of the production of the $Z^0$ bosons as a function of the center-of-mass energy, $E_{\mathrm{CM}}$. The notation of the diagram is the same as in Fig. 15.

Figure 17a

(continued).

Figure 17b

Production of the $Z^0$ boson in $E_{\mathrm{CM}}=8\mathrm{TeV}$, using the KMR approach. The individual contributions from the partonic subprocesses are presented, and the total values of (single and double) differential cross sections are subjected to comparison with the data of the ATLAS (black circles) and CMS (white circles) collaborations [38, 39]. The labels (a) through (f) illustrate the results of our calculations for a single differential cross section of the production of $Z^0$, in the given rapidity regions. The results for double differential cross section are presented in the this figure with labels (g) through (h).