tZ ′ production at hadron colliders

We study the production of a single top quark in association with a heavy extra Z ′ at hadron colliders in new physics models with and without ﬂavor-changing neutral-current (FCNC) couplings. We use QCD soft-gluon resummation and threshold expansions to calculate higher-order corrections for the total cross section and p T spectrum for tZ ′ production. The impact of the uncertainties due to the structure of the proton and scale dependence is also analyzed.


Introduction
The top quark is the heaviest particle in the three quark generations. Its mass, of approximately m t = 172.5 GeV in the pole mass approximation, has been measured with very high accuracy at the Large Hadron Collider (LHC) [1-3], and being close to that of the Higgs boson it makes the top quark one of the best candidates to probe the Electroweak (EW) sector of the Standard Model (SM) and its extensions.
The accumulated data at the LHC have not yet provided us with evidence of deviations from the SM, but Run II of the LHC and its upgrade to a High Luminosity phase (HL-LHC) [4], and especially future center-of-mass energy upgrades are going to record a large number of high-energy collision data which will allow us to probe rare processes that may hint at or provide direct evidence of new physics. In particular, for physics Beyond the Standard Model (BSM) and beyond the LHC, there are several projects going on which provide a synergy of various new-generation facilities like the Future Circular Collider (FCC) [5] and the Super proton proton Collider (SppC) [6]. With a center-of-mass energy of approximately 100 TeV, these new-generation hadron colliders represent the new frontier for discovery at high energies and will be critical to identify particles with mass of O(10) TeV. At these energies, we will be able to investigate properties of the Higgs boson and the top quark, and EW symmetrybreaking phenomena with unprecedented precision and sensitivity. Moreover, the statistics will be enhanced by several orders of magnitude with respect to that of the LHC, and this is going to be ideal to study BSM physics and rare processes. In this respect, a process of interest is the production of a single top quark in association with a new heavy particle.
Regardless of the type of the new heavy particle, many aspects of this reaction are interesting at quantum-field-theoretical level and because of the phenomenological implications on BSM physics. For example, the kinematics of the final state and decay products can be relevant to investigate extensions of the Higgs sector (two-Higgs-doublet model (2HDM), SUSY, etc.), and of the EW sector with enlarged gauge symmetry.
In this work we shall focus on the production of a top quark in association with an extra Z ′ vector boson coming from distinct BSM theories, and we will analyze higher-order QCD corrections to this process due to soft gluon emissions.
Extra vector gauge bosons, generically referred to as extra Z ′ s, are almost ubiquitous in extensions of the EW sector of the SM. Z ′ s are associated with additional abelian U ′ (1) gauge symmetries which were suggested in SM extensions such as left-right symmetric models, Grand Unified Theories (GUTs) and string-inspired constructions (see Refs. [7][8][9][10][11][12] for reviews and references). In the past decade, Z ′ gauge bosons at the TeV scale gathered considerable attention and triggered a vigorous program of experimental searches at the LHC. At high energies, Z ′ s can in principle have different signatures: they can be produced as intermediate resonances in Drell-Yan processes as well as in association with another SM vector or scalar boson, or in association with a jet or single top quark such as in the case of pp → tZ ′ .
The dynamics of this process is non-trivial because of several hard scales entering the cross section. In fact, in high-energy reactions in which the final-state heavy particle has a mass much heavier than the top quark mass, m t , the cross section is affected by large (collinear) logarithmic contributions of the type α n s log n (Q 2 /m 2 t ) (where Q ≈ m Z ′ , the Z ′ -boson mass, and α s is the QCD coupling constant) that can spoil the convergence of the perturbative series in calculations at fixed order [13]. Therefore, there is the necessity of resumming these logarithmic contributions using DGLAP evolution defining a top-quark parton distribution function (PDF) inside the proton. Details of factorization schemes with different number of flavors with consistent treatment of the top quark as a massless degree of freedom at high energies are discussed in Refs. [14,15] and references therein.
In processes with very heavy final states the near-threshold kinematic region becomes particularly important. Soft-gluon corrections typically become large and dominant in such circumstances. In this study, we adopt and extend the soft-gluon resummation formalism used in [16][17][18] for tZ and tγ production (see also applications to top-antitop pair production [19] and single-top production [20], and a review in [21]) to calculate approximate next-to-next-toleading order (aNNLO) cross sections for tZ ′ associated production in two case scenarios: i) the case of Z ′ s with flavor-changing anomalous couplings, ii) the case of Z ′ s originating from low-energy realizations of string models. We explore the impact of the corrections due to multiple emission of soft-gluons as well as the cross section suppression due to Z ′ s of different mass and couplings. Moreover, we analyze the uncertainties in the cross section associated to the PDFs of the initial state protons and to the factorization µ F and renormalization µ R scales. Finally, we generate prospects for the cross section for the case studies mentioned above, at future generation ultra-high energy colliders.
The paper is organized as follows. In Sec. 2 we discuss the BSM effective Lagrangians, couplings, and leading-order cross sections. In Sec. 3 we illustrate the soft-gluon formalism and calculate the higher-order corrections. In Sec. 4 we present results for the total cross sections and p T distributions in tZ ′ production via the processes gu → tZ ′ and gc → tZ ′ with anomalous couplings, and also via the process gt → tZ ′ . We conclude in Sec. 5.  Figure 1: Leading-order diagrams for gu → tZ ′ with anomalous t-u-Z ′ coupling and gc → tZ ′ with anomalous t-c-Z ′ coupling.
A FCNC term in the Lagrangian that includes the anomalous coupling of a t, q pair to a Z ′ boson is given by where κ tqZ ′ is the anomalous t-q-Z ′ coupling, with q an up or charm quark; e is the electron charge; Λ is an effective new physics scale in the few TeV's range; F µν Z ′ is the Z ′ field tensor; and σ µν = (i/2)(γ µ γ ν − γ ν γ µ ) with γ µ the Dirac matrices.
The partonic processes involved are gu → tZ ′ and gc → tZ ′ . Leading-order diagrams for these processes are shown in Fig. 1. Related processes involving Z bosons with anomalous couplings were studied in Refs. [16,17].

2.2
Lagrangian for a stringy Z ′ The Lagrangian for a Z ′ coming from string-inspired models is given below, where we adopt the notation introduced in Refs. [22,23]. Here we report the most basic definitions for completeness. The fermion-fermion-Z ′ interaction is given by where the coefficients z t,L , and z t,R are the charges of the left-and right-handed top quarks respectively. The Z ′ coupling is indicated by g Z ′ .
The mass of the Z gauge boson is parametrized in terms of the vacuum expectation values (vev's) of the Higgs sector v H 1 , v H 2 as follows where the mixing parameter ε is defined perturbatively, z H 1 and z H 2 are the charges of the Higges, g = e/ sin θ W , g Y = e/ cos θ W , and θ W is the Weinberg angle. We consider m Z ′ as a free parameter in the TeV's range. We restrict our attention to the interaction Lagrangian for the top-quark sector only, which is written as where the left-handed (L) and right-handed (R) couplings are where Y t,L/R is the hypercharge and T 3,L is the weak isospin. Based on this Lagrangian, we will study below the process gt → tZ ′ . The leading-order diagrams for this process are shown in Fig. 2.

Hadronic cross section
The hadronic cross section for p(P 1 ) + p(P 2 ) → t(p t ) + Z ′ (p Z ′ ) is expressed in terms of Mandelstam variables We also define T 1 = T − m 2 t and U 1 = U − m 2 t . The factorized differential cross section can be written as where f j/p (x, µ 2 F ) is the parton distribution function representing the probability of finding the parton j in proton p, µ F and µ R are the factorization and renormalization scales respectively, andσ ij→tZ ′ is the hard scattering cross section. Here, Λ QCD is the QCD scale while the scale Λ is of the order of m Z ′ , and power suppressed terms Λ 2 QCD /Λ 2 are neglected. The lower integration limits in the factorization formula are given by The double-differential cross section in Eq. (2.7) can be written in terms of the transverse momentum p T of the top quark and its rapidity y using where the transverse mass m T is defined as m T = p 2 T + m 2 t .

Leading-order cross sections
For the partonic process g(p g ) + q(p q ) → t(p t ) + Z ′ (p Z ′ ), we define the kinematical variables s = (p g + p q ) 2 , t = (p g − p t ) 2 , and u = (p q − p t ) 2 . The leading-order (LO) double-differential partonic cross section for gq → tZ ′ , with q and up or charm quark, via anomalous couplings is with α = e 2 /(4π).
For the partonic process g(p g ) + t(p q ) → t(p t ) + Z ′ (p Z ′ ), we again define the kinematical variables s = (p g + p q ) 2 , t = (p g − p t ) 2 , and u = (p q − p t ) 2 . The LO cross section for gt → tZ ′ is given by (2.12) where the vector and axial coupling of the Z ′ boson to the top quark are where we set sin θ W = s w and cos θ W = c w for brevity.

Soft-gluon corrections
We next describe the formalism and procedure for calculating soft-gluon corrections in the cross section for tZ ′ production. For the processes gq → tZ ′ and gt → tZ ′ , we defined the usual kinematical variables s, t, and u, in the previous section. We can also define a threshold kinematical variable, s 4 = s + t + u − m 2 t − m 2 Z ′ , that measures distance from partonic threshold, and vanishes at partonic threshold where there is no energy available for additional radiation. More specifically, s 4 is the squared invariant mass of additional final-state radiation. We also define The resummation of soft-gluon contributions to the partonic process follows from the factorization of the cross section as a product of functions that describe soft and collinear emission. Taking the Laplace transformσ(N) = (ds 4 /s) e −N s 4 /sσ (s 4 ), we have a factorized expression in 4 − ǫ dimensions, where H gq→tZ ′ is a hard function, S gq→tZ ′ is a soft function for noncollinear soft-gluon emission, and J i are jet functions for soft and collinear emission from the incoming quark and gluon. Our considerations are identical for all three processes to be studied in this paper, i.e. gu → tZ ′ , gc → tZ ′ , and gt → tZ ′ . The dependence of the soft function S gq→tZ on N is resummed via renormalization group evolution [17][18][19][20]24], with S b gq→tZ the unrenormalized quantity and Z S a renormalization constant. The function S gq→tZ obeys the renormalization group equation where g 2 s = 4πα s , β(g s , ǫ) = −g s ǫ/2 + β(g s ) with β(g s ) the QCD beta function, and is the soft anomalous dimension that determines the evolution of S gq→tZ . The soft anomalous dimension Γ S gq→tZ is calculated in dimensional regularization from the coefficients of the ultraviolet poles of the loop diagrams involved in the process [17][18][19][20][21][24][25][26].
The resummed partonic cross section in moment space is then given by Soft-gluon resummation is the exponentiation of logarithms of N. The first exponent in Eq.
(3.5) includes soft and collinear corrections [27,28] from the incoming partons, and can be found explicitly in [20]. We write the perturbative series for the soft anomalous dimension for gq → tZ ′ as Γ To achieve resummation at next-to-leading-logarithm (NLL) accuracy we require the one-loop result which is given, in Feynman gauge, by is the number of colors. Upon expanding the resummed cross section to fixed order and inverting from the transform moment space back to momentum space, the logarithms of N produce "plus" distributions of logarithms of s 4 /m 2 Z ′ . The highest power of these logarithms is 1 at NLO and 3 at NNLO. The NLO soft-gluon corrections for gq → tZ ′ are where β 0 = (11C A − 2n f )/3 is the lowest-order QCD β function, with n f the number of light quark flavors. We set n f = 5 for gu → tZ ′ and gc → tZ ′ , and n f = 6 for gt → tZ ′ . In top-quark production processes, the NLO soft-gluon corrections approximate very well the complete NLO corrections. We denote the sum of the LO cross section and the NLO soft-gluon corrections as approximate NLO (aNLO).
The NNLO soft-gluon corrections for gq → tZ ′ are (3.8) The cross section with the inclusion of the soft-gluon corrections through NNLO is denoted as approximate NNLO (aNNLO).

Phenomenological analysis
In the following sections we present the results of our phenomenological analysis in which we investigate the impact of the QCD corrections due to soft gluon emissions to the production of a single top quark in association with a Z ′ for the case studies previously discussed.
4.1 gu → tZ ′ and gc → tZ ′ We first study tZ ′ production via FCNC interactions with anomalous couplings. The partonic processes involved are gu → tZ ′ and gc → tZ ′ , where the Z ′ anomalously couples to the top quark and the u and c quarks through the flavor-changing coefficients k tuZ ′ /Λ and k tcZ ′ /Λ, respectively. The scale Λ is set equal to ten times the top quark mass m t and the couplings k tuZ ′ and k tcZ ′ are considered as parameters of the theory. As a case study we select k tuZ ′ = k tcZ ′ = 0.1. Thus, in our results below we set k tuZ ′ /Λ = k tcZ ′ /Λ = 0.01/m t . We also set m t = 172.5 GeV. Recent experimental searches for and phenomenological studies of FCNC interactions between the top quark and a Z boson can be found in Refs. [29][30][31][32].
We explore cross sections at 13 and 14 TeV LHC energies for a large range of Z ′ masses, and also explore the cross sections as functions of pp collider energy for future colliders. The theory predictions in this case are obtained by using the CT14NNLO PDFs [33] which lead to the numerical results illustrated in Figs. 3-15.
The initial-state parton combinations g(x 1 )u(x 2 ) + g(x 2 )u(x 1 ) and g(x 1 )c(x 2 ) + g(x 2 )c(x 1 ) are probed in various kinematic regions depending on the collider center-of-mass energy and on the mass of the Z ′ . At  The total cross sections at collider energies of 13 and 14 TeV are illustrated in Figs. 3 and 4, respectively, where we show the theory predictions at LO, aNLO, and aNNLO for the process gu → tZ ′ with anomalous k tuZ ′ coupling, and the process gc → tZ ′ with anomalous k tcZ ′ coupling, as functions of Z ′ mass. The factorization and renormalization scales are equal and set to µ = m Z ′ . We observe a very strong dependence of the cross section on the Z ′ mass. The cross section drops over many orders of magnitude as the Z ′ mass varies from 500 GeV to 8000 GeV. The cross section for gc → tZ ′ is significantly smaller than for gu → tZ ′ . The inset plots show the aNLO/LO and aNNLO/LO K-factors with scale uncertainty bands which are obtained by varying µ in the interval [1/2µ, 2µ]. The K-factors are large and increase with larger Z ′ masses, as expected. The aNLO corrections are large and furthermore the additional aNNLO corrections are very significant. At 14 TeV, the cross sections are of course larger than at 13 TeV, but the dependence on the Z ′ mass and the size of the corrections are very similar.
In Fig. 5 we show the total cross sections at LO, aNLO, and aNNLO for the processes gu → tZ ′ and gc → tZ ′ at 13, 20, 50, and 100 TeV collider energies together with CT14NNLO PDF uncertainties evaluated at the 68% confidence level (C.L.). The PDF uncertainty of both gu and gc channels are larger at lower collider energy and high m Z ′ .
The behavior of the cross section with collider energy is illustrated in Fig. 6, where we show results at LO, aNLO, and aNNLO for the gu and gc channels as functions of the collider energy up to 100 TeV for four choices of Z ′ mass, m Z ′ = 1, 3, 5, and 8 TeV. The cross sections are of course smaller for larger Z ′ masses. The inset plots show the aNLO/LO and aNNLO/LO K-factors. As expected, the K-factors are larger at smaller energies and also for higher Z ′ masses, since we are then closer to threshold.
In the case of tZ ′ production with FCNC couplings, the anomalous couplings entering both channels of the cross section are considered as free parameters. We have therefore performed a two dimensional scan to assess the sensitivity of the cross section. In Fig. 7 we show a case study in which we plot aNNLO total cross sections as functions of the couplings k tuZ ′ /Λ and k tcZ ′ /Λ, at a collider energies of 13 and 100 TeV, for different values of m Z ′ . We notice that if we let both couplings to vary in 0.001 ≤ k/Λ ≤ 0.1 TeV −1 , the cross section spans approximately 5 orders of magnitude. The cross section suppression is larger for larger values of m Z ′ .
It is interesting to study kinematic distributions such as the top-quark p T spectrum, dσ/dp T , and how Z ′ s of different masses affect the p T suppression in various kinematic ranges. We illustrate the top-quark p T distributions in Figs. 8 and 9 where the spectra for the gu → tZ ′ and gc → tZ ′ processes are shown at LO, aNLO, and aNNLO for three choices of the Z ′ mass at a center of mass energy of 13 and 14 TeV respectively. In both cases, the p T distributions decrease quickly as m Z ′ is increased. The aNLO/LO and aNNLO/LO K-factors are shown in the inset plots: they are large and increase for larger Z ′ masses, as expected. The aNLO corrections are large and furthermore the additional aNNLO corrections are important.
Top-quark p T distribution prospects for future-generation pp colliders are presented in Fig. 10 where we display the p T spectrum for four choices of Z ′ mass of 1, 3, 5, and 8 TeV at a collider energy of 100 TeV. The K-factors are shown in the inset plots. The distribu-       tions are significantly larger than at LHC energies and are non-negligible even for large Z ′ masses, indicating that the number of events predicted by these models can be validated at the high-luminosity FCC or SppC colliders.
Next, we explore the extent of correlation between the PDFs and the aNNLO cross section for these processes in pp collisions at √ S=13 and 100 TeV. In particular, in Fig. 11 we show the correlation cosine between the gluon (and the u quark) and the total cross section for the gu → tZ ′ process as a function of the momentum fraction x at the 68% CL at √ S = 13 and 100 TeV. We have chosen the gu channel as it provides the dominant contribution. The definition of the correlation cosine between two quantities determined within the Hessian method is given in the Appendix. At collider energies of 13 TeV, we observe a strong correlation (cos φ ≥ 0.8) between the gluon and the gu → tZ ′ cross section at large x ≥ 0.1 as expected, and the correlation peak shifts towards larger x values for larger m Z ′ . Anti-correlation of approximately 50% in the 10 −4 ≤ x ≤ 10 −2 interval is also observed. The correlation between the u quark and the cross section is much milder and less than 50% at very large x. These patterns change as we move to higher collider energies, where for the gluon the correlation peak for each value of m Z ′ is shifted to lower x-values, while for the u quark correlations are slightly more pronounced.
Besides the correlation with PDFs, important information can also be gathered from the study of simultaneous uncertainty boundaries of the cross section of the gu and gc channels. The allowed regions are represented by correlation ellipses which can be compared to pseudo data in BSM simulations. In Fig. 12 and 13 we show the elliptical confidence regions, at 68% CL, in pp collisions at 13 and 100 TeV, for m Z ′ = 1, 3, 5, and 8 TeV. These can be used to read off PDF uncertainties and correlations for each pair of cross sections. At √ S = 13 TeV, we notice that the two channels are highly correlated and the induced PDF uncertainties on the σ gc channel are very large for this choice of the collider energy. This is reflected by the fact that there is a small portion of the ellipse where the PDF induced errors on the cross sections are larger than the cross section central value itself, allowing for negative values. At √ S = 100 TeV, the gu and gc channels are still highly correlated, but the induced PDF uncertainties on both the cross sections are smaller as in this kinematic domain the PDFs are probed at intermediate x where they are better constrained.
Finally, we study the impact of the scale and PDF uncertainties on the aNNLO/LO Kfactors as functions of the collider energy for large √ S values and different values of m Z ′ . In Fig. 14 and 15, we illustrate the K-factors for the gu and gc channels with PDF and scale uncertainties respectively. Scale variation refers to m Z ′ /2 ≤ µ ≤ 2m Z ′ as before. In Fig. 14 the PDF uncertainties for each m Z ′ value are shown using bands with different hatches and color, and are compared to those relative to scale uncertainty which have no hatches and are shown with black contours with different dashing. At collider energies below 20 TeV PDF uncertainties are large because PDFs are probed in the large-x region. In the gc channel, PDF uncertainties are dominant because the charm-quark PDF is less constrained with respect to the gluon and u-quark. In Fig. 15 the scale dependence in the aNNLO K-factors for the gu and gc channels is illustrated separately.

4.2
String-inspired Z ′ s: the gt → tZ ′ channel In this section we discuss the phenomenological results obtained from the study of tZ ′ production where the Z ′ originates from a low-energy realization of string-inspired models. The interaction Lagrangian in Sec. 2.2, and the choice of the parameters we have examined, are based on the models published in Refs. [22,23]. The leading-order cross section is given by the s-and t-channels of the gt → tZ ′ process and the structure of the couplings is given in Sec.
2.2. The gt → tZ ′ process with m Z ′ in the TeV range requires the top-quark PDF in the initial state. In our phenomenological application, µ = m Z ′ ≫ m t and we consider the top quark as an active flavor inside the proton with very good approximation. Therefore, in the rest of this analysis we work with the 6-flavor scheme and use the NNPDF3.1 PDFs [34] with N f = 6 where N f is the number of active flavors. We set m t = 0 in the initial state lines in the calculation of the LO cross section. We postpone to a future analysis the treatment of this process in a more general context within a general mass variable flavor number (GMVFN) scheme based on an amended version of the S-ACOT-χ scheme, developed for deep inelastic scattering (DIS) in Refs. [35][36][37][38] and extended to NNLO in Ref. [39]. The Z ′ coupling g Z ′ is considered as a free parameter and as a case study we choose g Z ′ = 1 as the default choice. A g Z ′ parameter scan is illustrated in Fig. 16 (left), where the aNNLO cross section is plotted as a function of √ S for different values of m Z ′ which correspond to bands with different dashing. We explore g Z ′ variations in 0.01 ≤ g Z ′ ≤ 1.5 and observe that when g Z ′ varies the cross section is basically rescaled and it spans approximately two order of magnitudes.
In Fig. 16 (right) we show total cross sections at LO, aNLO, and aNNLO at proton-proton colliders with PDF uncertainties at 68% C.L. as functions of Z ′ mass for four different collider energies, 13, 20, 50, and 100 TeV. Prospects at the LHC 13 and 14 TeV collision energies are shown in Fig. 17 where the inset plots show the aNLO/LO and aNNLO/LO K-factors. We note the large effect of the higher-order corrections, which triple the LO result for a 6 TeV Z ′ 10 -4 10 -3 10 -2 10 -1 Figure 11: Correlation cosine at √ S=13 and 100 TeV between σ gu→tZ ′ and the gluon as a function of x gluon (left column) and σ gu→tZ ′ and the up-quark as a function of x up (right column). The four panels show aNNLO results rescaled at the 68% C.L. for different values of the Z ′ mass.

mass.
Total cross section results as functions of the collider energy up to 100 TeV for different values of m Z ′ are given in Fig. 18. Here the scale choice is µ = m Z ′ . The inset plot shows the aNLO/LO and aNNLO/LO K-factors. While the cross sections get smaller with increasing Z ′ mass, the K-factors get larger because this kinematic region is closer to the partonic threshold. In Fig. 19 we illustrate the induced NNPDF3.1 PDF uncertainty on the K-factors as well as the scale uncertainty. The large uncertainty of the top-quark PDF dominates at all collider energies and for every value of m Z ′ . The scale uncertainty due to factorization scale variation in m Z ′ /2 ≤ µ ≤ 2m Z ′ is much smaller.
We continue with the top-quark p T distributions in this process. Fig. 20 shows the topquark p T distributions in the gt → tZ ′ process at LO, aNLO, and aNNLO for different m Z ′ values at LHC energies of 13 and 14 TeV. The K-factors are shown in the inset plots, they are significant, and they are bigger for larger Z ′ masses. The p T distributions prospects for future hadron colliders at higher energy are shown in Fig. 21 where the top-quark p T spectra 12 Figure 13: Same as in Fig. 12, but for pp collisions at √ S = 100 TeV.    Figure 15: Scale uncertainty in the aNNLO K-factors for the gu → tZ ′ and gc → tZ ′ channels. The figures show a scan in the center of mass energy of the collisions √ S for different values of the Z ′ mass. Scale variation refers to m Z ′ /2 ≤ µ ≤ 2m Z ′

Conclusions
We have studied tZ ′ production in various BSM models at hadron colliders. We performed a phenomenological QCD analysis where we scrutinized tZ ′ production in the presence of FCNC and in the case in which the extra Z ′ is generated within a low-energy realization of string theory models. We have calculated theoretical predictions for cross sections and top-quark p T distributions that include higher-order soft-gluon corrections. In particular, theory predictions are obtained at aNLO and aNNLO in QCD by extending the soft-gluon resummation formalism to the case in which a top quark is produced in association with a heavy neutral vector boson in pp collisions at energies that relevant for the LHC and for future new-generation hadron colliders like FCC-hh and SppC. We have found that QCD corrections due to soft-gluon emissions are considerable and need to be included in precision studies. We have investigated the impact of uncertainties due to proton PDFs as well as uncertainties due to scale variation. PDFs uncertainties represent the major source of uncertainty in this analysis. Moreover, we explored the parameter space for the BSM models we scrutinized by performing parameter scans and studying the sensitivity of the cross section to parameter changes. We have found that the total tZ ′ cross section has large sensitivity on the mass of the Z ′ .
These theoretical results will be useful for tZ ′ production searches at the LHC and future hadron colliders.

A Appendix
If A(f ) and B(f ) are two quantities that depend on a generic PDF f , determined within the Hessian method, the extent of correlation between A and B can be assessed by calculating the correlation cosine and the uncertainties on A and B can be obtained by using the symmetric formula The best-fit estimate for A 0 is defined as A(f 0 ) and f ± k represent the n PDF eigenvector sets in the positive and negative direction respectively. When A and B are strongly correlated, then cos φ AB ≈ 1. Anticorrelation corresponds to cos φ AB ≈ −1, and uncorrelation to cos φ AB ≈ 0. The simultaneous uncertainty boundaries on A and B, representing the allowed regions, can be obtained with the Lissajous parametric ellipse, defined as where the parameter θ is in the interval 0 < θ < 2π (see Ref. [40]).