Walking Technicolor in the light of $Z^{\prime}$ searches at the LHC

We investigate the potential of the Large Hadron Collider (LHC) to probe one of the most compelling Beyond the Standard Model (BSM) frameworks --- Walking Technicolor (WTC), involving strong dynamics and having a slowly running (walking) new strong coupling. For this purpose we use recent LHC Run2 data to explore the full parameter space of the minimal WTC model using dilepton signatures from heavy neutral $Z^{\prime}$ and $Z^{\prime\prime}$ resonances predicted by the model. This signature is the most promising one for discovery of WTC at the LHC for the low-intermediate values of the $\tilde g$ coupling -- one of the principle parameters of WTC. We have demonstrated complementarity of the dilepton signals from both resonances, have established the most up-to-date limit on the WTC parameter space, and provided projections for the the LHC potential to probe the WTC parameter space at higher future luminosities and upgraded energy. We have explored the whole four-dimensional parameter space of the model and have found the most conservative limit on the WTC scale $M_A$ above 3 TeV for the low values of $\tilde g$ which is significantly higher than previous limits established by the LHC collaborations.


Introduction
With the discovery of a Higgs boson at the LHC [1,2] it has become not only possible, but imperative to discover the true origin of mass in the Universe. The traditional Standard Model (SM) Higgs mechanism of mass generation via spontaneous electroweak symmetry breaking (SEWSB) leads to the hierarchy problem, associated with the large fine tuning between the EWSB scale and the Planck mass. Several classes of Beyond the Standard Model (BSM) theories have been proposed to address the shortcomings of the SM, and one of them is Technicolor which is based on new strong dynamics [3,4]. In Technicolor, EWSB is generated dynamically by the formation of a chiral condensate under the new strong dynamics, providing a natural scale for mass generation without fine tuning. Experimental bounds from Electroweak Precision Data (EWPD) disfavour TC models with QCD-like dynamics [5], so modern Technicolor models must have a modified strong coupling. Walking Technicolor (WTC) [6][7][8][9][10][11] and its recent developments [12][13][14][15][16][17][18] is a very compelling BSM candidate for the underlying theory of Nature. It has a strong coupling α T C with a very slowly running ("walking") regime between the TC energy scale and high energy Extended-TC scale. The lightest scalar resonance of WTC can be identified as the experimentally consistent Higgs boson, whose mass scale is naturally generated thus does not incur a hierarchy problem [19,20]. WTC also provides a rich phenomenology of composite spin-0 and multiple triplets of composite spin-1 resonances, making this a prime candidate for experimental particle physics searches. Using LHC Run 1 dilepton data, the ATLAS Collaboration have interpreted experimental limits on a new heavy neutral resonance in the context of the WTC parameter space in Ref. [21] and Ref. [22] using dilepton and HV searches respectively. These WTC interpretations have been following the phenomenological exploration of WTC parameter space performed in [23] for a 2-dimensional(2D) bench mark from the whole 4-dimensional(4D) parameter space of the model.
This study makes the next step in exploration of the LHC potential to test WTC. First of all, we perform analysis in the full 4D parameter space of the model. Secondly, we study the complementarity of the dilepton signals from both heavy neutral vector mesons of WTC and demonstrate its importance. In this work we focus exclusively on Drell-Yan (DY) processes, and provide justification for the single peak analysis of current LHC constraints in the context of this model. Finally, we are establishing here the most up-to-date limit on WTC parameter space from LHC dilepton searches for the whole 4D parameter space of the model, and give projections for the the LHC potential to probe WTC parameter space at higher integrated luminosity in the future.
In Section 2 we discuss the WTC model together with the constraints on its parameter space. In Section 3 we explore the phenomenology of WTC and the LHC potential to probe the model. Finally in Section 4 we summarise the results of this work, and comment on the future prospects for WTC exploration at the LHC.

Minimal Walking Technicolor Model
Throughout this paper we focus on the global symmetry breaking pattern SU (2) L × SU (2) R → SU (2) V . This pattern is realized by the Next to Minimal Walking Technicolor model (NMWT) [12,14,23], which features two Dirac fermions transforming in the 2-index symmetric representation of the technicolor gauge group SU (3). However, any technicolor model must feature SU (2) L × SU (2) R → SU (2) V as a subgroup breaking pattern of the full symmetry breaking pattern G → H. This is required to ensure mass generation for the W and Z bosons and to preserve an SU (2) V custodial symmetry in the new strong dynamics sector like that in the SM higgs sector.
More generally, theories of composite dynamics with a Technicolor limit will feature this as a global symmetry breaking subpattern. Examples are composite Higgs and partially composite Higgs models [24,25] with an underlying 4 dimensional realization, e.g. [26][27][28][29][30]. Another example is bosonic technicolor [31][32][33][34]. In both partially composite Higgs models and in bosonic technicolor, the Higgs particle is a mixture of an elementary and composite scalar. Some aspects of how the spin-1 resonance phenomenology is affected by aligning the theory away from the Technicolor vacuum in composite Higgs and partially composite Higgs models are given in [28,35]. In general the mass scale set by the Goldstone boson decay constant of the strong interactions, F π , is larger in composite Higgs models than in ordinary technicolor while it is smaller in the bosonic technicolor models. In partially composite Higgs models, the scale depends on the relative size of the elementary doublet vacuum expectation value v and the vacuum alignment angle θ. In general F π in these different composite models is determined through a constraint of the form where N D is the number of electroweak doublet fermion families, θ = π/2 corresponds to the technicolor vacuum and 0 < θ < π/2 to the composite Higgs vacuum. In this study we restrict ourselves to the technicolor limit, which provides dynamical electroweak symmetry breaking, and a composite Higgs resonance, but requires a further extension to provide SM fermion masses. The composite Higgs resonance has been argued to be heavy in the Technicolor limit with respect to the electroweak scale, by analogy with scalar resonances in QCD which are heavy compared to the QCD pion decay constant. However for composite sectors which are not a copy of QCD it is a non-perturbative problem to determine the lightest scalar mass. Both model computations [14,36] and lattice simulations [37,38] of models like the NMWT model, which appear to be near the conformal window, have indicated the presence of a scalar 0 ++ resonance that is much lighter than expected from simply scaling up scalar masses in QCD. The physics behind the origin of the fermion masses can also play a role in reducing this TC Higgs mass to the observed value at LHC and at the same time provide SM Higgs like couplings [19,20] to the SM particles. It is possible to probe the origin of the fermion masses, whether they are due to extended technicolor, fermion partial compositeness or a new elementary scalar, via the pseudo scalar sector of the theory, the analogues of the QCD η and η resonances as discussed in [39,40].
We follow the same prescription for constructing an effective theory of the underlying composite dynamics as in [23,41] by introducing composite spin-1 resonances transforming under the SU (2) L × SU (2) R global symmetry: Two new triplets of heavy spin-1 resonances are introduced at interaction eigenstate level A L/R as gauge fields under SU (2) L/R respectively. The SU (2) L is gauged as SU (2) W such that the A L fields form a weak triplet analogous to the triplet in W models while the A R fields are SU (2) W singlets.
Together with the Standard Model electroweak fields in the gauge eigenbasis,W µ andB µ , we define chiral fields where g, g are the usual Standard Model EW coupling constants, andg is the coupling constant of the NMWT gauge interactions. These fields transform homogeneously when the A L/R fields are introduced formally as gauge fields. The scalar composite Higgs resonance H and the triplet of pions π a absorbed by the W and Z bosons are introduced as a bi-doublet field under the SU (2) L/R symmetries described via the 2 × 2 matrix M, where a = 1, 2, 3, v = µ/ √ λ is the vacuum expectation value associated with the breaking of the chiral symmetry, and T a are the generators of the SU (2) groups, related to the Pauli Matrices by With these definitions we write the low-energy effective Lagrangian of the model, up to dimension 4 operators as in [23]: whereW µν andB µν are the SM electroweak field strength tensors, and F L/Rµν are the field strength tensors corresponding to the vector meson fields. The global symmetry breaking pattern SU is triggered by the vev of M and provides the 3 Goldstone degrees of freedom for the massive W and Z bosons. The heavy vector resonances, here introduced via the A L/R triplets, can equivalently be treated as Higgs'ed gauge fields of a 'hidden local symmetry' copy of the above global symmetry group [42], as discussed in [18]. The physical spectrum of the model then consist of the 2 triplets of spin-1 mesons which in the absence of electroweak interactions form a vector triplet V under SU (2) V and the axial-vector partner triplet A, analogous to the ρ and a 1 vector mesons in QCD. In this study we focus on the two neutral resonance mass eigenstates which, in the presence of SM electroweak interactions, we for convenience refer to as Z and Z although these are distinct from sequential Z resonances.
The spin-1 sector of the Lagrangian in equation 2.5 contains five parameters, m,g, r 2 , r 3 and s. The masses and decay constants of the vector and axial-vector resonances, in the limit of zero electroweak couplings, are given in terms of these parameters as The techni-pion decay constant F π may be expressed in terms of F V and F A as and with F π = 246 √ N D GeV in technicolor models with N D families of technifermions and no elementary doublet scalars. Here we assume N D = 1 as in the NMWT model. We can now make use of the Weinberg Sum Rules (WSRs) [43] to constrain the number of parameters in the effective model and connect them to the underlying fermionic dynamics.
The assumed asymptotic freedom of the effective theory implies the 1st and 2nd WSRs respectively where Π LR (s) is the Lorentz invariant part of the LR correlation function: Assuming that only the lowest spin-1 resonances A, V saturate the WSRs, the vector and axial vector spectral densities are given in terms of the spin-1 masses and decay constants as The 1st WSR therefore implies that ω = 0 and In terms of the Lagranian parameter this gives the relation The second WSR is less dominated by the infrared dynamics than the first WSR as seen from Eq. 2.10. We therefore allow for a modification of the second WSR encoded by the dimensionless parameter a following [44] a  where d(R) is the dimension of the gauge group representation of the underlying technifermions. The parameter a measures the contribution of the underlying dynamics to the integral in Eq. 2.10 from intermediate energies, above the confinement scale. Finally the electroweak Peskin-Takeuchi S parameter is related to a zeroth Weinberg sum rule, Combining the first and second WSRs it follows that the a parameter gives a negative contribution to axial-vector mass difference M 2 A − M 2 V and a negative contribution to the S parameter. We therefore expect a to be positive in a near conformal theory yielding a smaller S-parameter and a more degenerate axial-vector mass spectum than in QCD. This is in line with e.g. model computations [36,45] based on Schwinger-Dyson analysis, but here we take it as an assumption.
In QCD we expect a 0 while in near-conformal theories, where the coupling constant is assumed to be approximately constant in a region above the confinement scale, we expect a > 0 [44] . This allows us to trade one of the Lagrangian parameters for the S parameter via the relation We are left with a four dimensional parameter space that describes the model: M A ,g, S, s. The Lagrangian constant s parametrises the interactions of the Technicolor spin-1 mesons with the Higgs sector (see 2.5). Since we do not consider the composite Higgs phenomenology the only relevant effect of the s parameter is on the branching ratio of the Z and Z states into dileptons. The branching ratios into dileptons are maximal for s = 0 so we therefore restrict to this throughout. This leaves 3 relevant parameters We show the value of the a parameter in the M A ,g plane for different values of S in Figure 1. Restricting to positive values of a we get an upper limit on the mass parameter M A which compliments the experimental limits we derive from dilepton searches.
3 Phenomenology and LHC potential to probe WTC parameter space In our analysis of heavy neutral spin-one resonances in the NMWT parameter space, we conduct a 3-dimensional scan over M A ,g and S. The results in this section are presented in the M A ,g parameter space for discrete values of S such as S = −0.1, 0.0, . . . , 0.3. The largest value, S = 0.3 for the range we choose, is already disfavoured by EWPD [46], however we include it in this work for direct comparison to results of the previous work [23]. The remaining limits of the scan over S ensure that the tension with EWPD is minimised (for the zero T -paramter). In this section we present results at the benchmark S = 0.1; fixed values of S = 0.1 are given in Appendix A.4.
There is an upper bound ong:g which follows from Eq. 2.17 and ensures that all physical quantities are real as we will see below. For S = 0.3, the biggest value of S we consider here, the upper limit isg = 9.15. Therefore we present all results in the M A ,g space withg ≤ 9 to avoid unphysical parameter space. The phenomenology of the NMWT model is explored using the CalcHEP package [47] which allows to perform simple and robust analysis of tree-level collider events. The Lagrangian for NMWT was implemented using LanHEP [48], from which all interaction vertices are generated for use in CalcHEP. We focus on neutral heavy spin-one resonances in the Drell-Yan channel, with di-leptons signature. The mass spectra of the Z /Z are presented in section 3.1.1, the coupling strength of Z /Z vertices in section 3.1.2, followed by a discussion of the total widths and dilepton branching ratios in section 3.2, production and total cross sections for DY processes of Z /Z are given in section 3.3, section 3.4 explores the interference between the neutral resonances and discusses the validity of reinterpreting LHC constraints for the NMWT model, and finally section 3.5 explored the LHC potential to probe the WTC parameter space.

Mass spectra
Besides numerical analysis it is informative also to perform analytical one as we do for some masses and couplings to understand the qualitative properties of the model and the limits of the parameter space. Diagonalising the neutral mixing matrix (see details in Appendix A.1), we find the Z /Z masses to 2nd order ing −1 take the form where from equation 2.18 we express χ as The mass spectrum of the Z is shown in Figure 2a and numerically presented in Table 1 where we also present the Z mass for the 3D grid in (M A ,g, S) space. One can see that forg 2, M Z M 2 A as follows from Eq. 3.2. In Figure 2b we present the spectrum for the relative mass difference, ∆M/M Z , where ∆M = M Z − M Z . One can see that M Z behaviour is less trivial which reflects the 'competition' ofg and F π /M A ratios in Eq. 3.3. For large M A one can observe that Z starts to mildly depend ong. This change in behaviour is due to a change of state of the Z (Z ) from mostly axial(vector) to mostly vector(axial) [23]. Figure 2b clearly reflects this mass inversion forg > 1 at a fixed M inv = M A which to 2nd order ing −1 takes the form Using the benchmark S = 0.1 the mass inversion occurs at M A = 2760GeV, we clearly observe this behaviour in Figure 2b. The mass splitting is large at low M A , highg, opening new decay channels such as Z → W + W − . This is discussed further in section 3.2.

Couplings
Here we explore analytic form of Z and Z couplings to fermions. These are composed of elements of the neutral diagonalisation matrix N ij [18], details of the mixing matrix calculation are included in Appendix A.1.
For the vertices with fermions, the coupling strengths can be decomposed into left and right handed parts and to 2nd order in 1/g, g Z ff and g Z ff couplings take the form: where I 3 = ±1/2 is usual 3rd componet of the weak Isospin for up and down-femions respectively, Y = q f − I 3 is their hypercharge, and q f is the charge of the fermions.
The parameter dependence of the Z and Z dilepton couplings are given as a ratio to the SM g Zl + l − in Figures 3.6 and 3.7 respectively. Both L and R components of the Z dilepton coupling increase asg → 1, however as the coupling is diluted through the mixing effects between the gauge fields, g Z l + l − ≥ g Zl + l − is never realised.
Similarly, the L component of the Z dilepton coupling grows asg → 1, however this is not the case for the R component. The R component is suppressed in comparison to the Z as the mixing with the photon is smaller for γ − Z than γ − Z ; such mixing effects are discussed further in 3.2.
Again we see that the axial(vector) composition of the Z (Z ) affects both L and R coupling strengths, suppressing the coupling as the Z (Z ) becomes mostly vector(axial).

Widths and branching ratios
The width-to-mass ratio Γ/M for Z and Z is shown in Fig. 5.
One can see that Z is generically narrow in the whole parameter space -the Γ/M is always below 10%. One should also note that for large values ofg and M A < M inv the main contribution to the width is coming from Z → ZH decay as one can see from Fig. 6(a,b) where we present Br(Z )(a,b) and Br(Z )(c,d) for all decay channels as a function of M A at the fixed values of (a,c)g = 3, (b,d)g = 8, at benchmark values of S = 0.1 and s = 0. This happens because of the following asymptotic of g Z ZH coupling at largeg, which makes Γ(Z ) increase with the increase ofg. One can see the numerical results confirming this effect in Table 2, where we present Γ(Z ) and Γ(Z ) for the 3D grid in (M A ,g, S) space. For M A > M inv , Z "switches" its properties from pseudo-vector to vector, and its width is enhanced then by the Z → W + W − decay for largeg with the respective g Z W W coupling proportional tog.  Figure 3.
Coupling of Z to charged lepton pairs as a ratio to its SM equivalent separated into left and right handed components,  Coupling of Z to charged lepton pairs as a ratio to its SM equivalent separated into left and right handed components, (a) | g L . In this region Z does not contribute to the dilepton signature at the LHC and therefore this region can be safely explored and interpreted using Z dilepton signature at the LHC.
Let us take a closer look at dilepton signature and the respective Z and Z branching ratios in 2D (M A ,g) parameter space, presented in Fig. 7 for S=0.1 and Table 3 presenting numerical values for Br(Z → e + e − ) and Br(Z → e + e − ) for 3D grid in (M A ,g, S) space. Besides an expected 1/g suppression, in Fig. 7 one can observe that for low values ofg both Br(Z → + − ) and Br(Z → + − ) are enhanced above the 3% value corresponding to Br(Z → + − ) in SM. One can see from Table 3   1 one can check numerically that photon-Z mixing is enhanced, while Z − Z is suppressed, which leads to a relative suppression of Br(Z → νν) and Br(Z → q dqd ) with respect to Br(Z → + − ) and Br(Z → q uqu ).
Talking about all other decay channels, which actually define Br(Z → e + e − ) and Br(Z → e + e − ), there are four more decays: qq, νν, V V and V h channels as one can see from see Branching ratios of Z vs M A for S=0.1,g=8 500 1000 1500 2000 2500 3000 3500 4000 Branching ratios of Z vs M A for S=0.1,g=3 Branching ratios of Z vs M A for S=0.1,g=8 terms from the Lagrangian defined by Eq. 2.5. One should note that in case of such cancellation and absence of ZH  Table 3.
Di-electron branching fraction of Z , Z in the MA,g, S parameter space, displayed in the format Br(Z → e + e − )(Br(Z → e + e − )) in %.
signal, which has been explored by the ATLAS collaboration to probe WTC parameter space [22], the role of dilepton searches in probing WTC parameter space becomes especially appealing as a crucial complementary channel.

Cross sections
Both Z and Z can be resonantly produced in a DY process, giving rise to dilepton signatures. The cross section rates are directly related to Z and Z coupling to fermions and dilepton branching ratios discussed earlier.  Table 4 as a numerical results for 3D (M A ,g, S) grid. Cross sections are calculated using CalcHEP [47] via the High Energy Physics Model Database HEPMDB [49], linked to the IRIDIS4 supercomputer. The PDF set used is NNPDF23 LO as_0130_QED [50], and the QCD scale Q is set to be the dilepton invariant mass, Q = M (e + e − ). The cross section has been evaluated in the narrow width approximation (NWA) to be consistent with the latest CMS limit [51] which we use for the interpretation of our signal as we discuss below. In the experimental CMS paper the cross section for Z models was calculated in a mass window of ±5% √ s at the resonance mass, following the prescription of Ref. [52] where it was checked that for this cut the cross section is close to the one from the NWA to within 10%. To account for NNLO QCD effects in our analysis below, the LO cross sections are multiplied by a mass-dependent K-factor which was found using WZPROD program [53][54][55] which we have modified to evaluate the cross sections for Z and W resonances and linked to LHAPDF6 library [56] as described in Ref. [57]. The resulting NNLO K-factors are presented in Table 5.
From Fig. 8 one can observe for Z and Z DY cross sections an expected 1/g suppression discussed above as well as eventual PDF suppression with the increase of the mass of the resonances. Also, one should make an important remark that in the large mass region for low-intermediate values ofg the signal from the Z is higher than the one from the Z . This highlights the complementarity between the two resonances, indicating that the Z and Z DY processes will exclude different areas of the parameter space. This motivates our study of both resonances in conjunction, as we will exclude a greater portion of the parameter space with combined searches.    Table 5. K-factors for NNLO QCD corrections to Drell-Yan cross sections at √ s = 13TeV evaluated with the help of the modified ZWPROD program as described in the text, using NNPDF23 LO as_0130_QED and NNPDF23 NNLO as_0119_QED [58] PDFs for LO and NNLO cross sections respectively.

Z /Z interference and validity of the re-interpretation of the LHC limits
Following our results in the previous section, we explore the interference between the Z and Z boson which gives rise to the di-lepton signature. This is an important point for our study since we aim to re-interpret the LHC limits based on a single resonance search in the di-lepton channel. Besides interference, the validity of such an interpretation also depends on how well these resonances are separated, their relative contribution to the signal and their width-to-mass ratio.  Figure 9. Left(a): the contour levels for pp → Z → e + e − production cross section at the LHC@13TeV as well as relative ratio of di-lepton rates for Z vs Z production for S=0.1. Right(b): the interference between Z and Z contributing to di-lepton signature from pp → Z /Z → + − process.
In Fig. 9(a) the contour levels for pp → Z → e + e − production cross section at the LHC@13TeV as well as relative ratio of di-lepton rates for Z vs Z production for S=0.1. As in the recent experimental CMS paper, the cross section for Z and Z was evaluated using finite width and mass window of ±5% √ s at the resonance mass to correctly estimate the size of the Z /Z interference. Qualitatively the picture is similar for other values of S-parameter. First of all one can notice that with a luminosity of roughly 40 fb −1 for which the limits on di-lepton resonances are publicly available, one can expect a di-lepton cross section of the order of 0.1 fb, which translates to M A of about 3 TeV for low g values. As we will see in the following section this rough estimation agrees with an accurate limit we establish later in our paper. Second, one can clearly see that the role of Z becomes important and even dominant for M A above 1.5 TeV andg below about 4. Fig. 9(b) presents the interference between Z and Z contributing to di-lepton signature.
One can see that the interference is at the percent level and can be safely neglected. This is an important condition for interpretation of the LHC limits on single resonance search. Taking this into account and the fact that the Z contribution to di-lepton signature is dominant, in the region of small M A < 1 TeV we conclude that one can use LHC limits for di-lepton single resonance searches. Using similar logic, one can see that in the region of intermediate and large M A > 1.5 TeV where M Z contribution to di-lepton signature is dominant one can use LHC limits for single resonance di-lepton searches in the case of Z . Finally, in the intermediate region of M A between 1 and 1.5 TeV when di-lepton signals from Z and Z are comparable, well separated in mass (above 10%) recalling Z − Z mass difference from Fig. 2 and their width-to-mass ratio is small (few percent) (Fig. 5), the LHC limits can be applied separately to Z or Z signatures. Therefore in the whole parameter space of interest (with σ(pp → Z /Z → + − ) 0.1 fb) one can use the signal either from Z or Z to best probe the model parameter space. This procedure sets the strategy which we use in the following section. The statistical combination of signatures from both resonances is outside of the scope of this paper since it requires also the change the procedure in setting the limit at the experimental level.

The setup for the LHC limits
The CMS Z dielectron 13TeV limits [51] which we use for the interpretation of the WTC parameter space are expressed as R σ = σ(pp → Z → e + e − )/σ(pp → Z → e + e − ), which is the ratio of the cross section for dielectron production through a Z boson to the cross section for dielectron production through a Z boson. The limits are expressed as a ratio in order to remove the dependency on the theoretical prediction of the Z boson cross section and correlated experimental uncertainties.
To reproduce these limits, a simulated dataset of the CMS mass distribution is generated using a background probability density function: m κ e α+βm+γm 2 +δm 3 + m 4 (3.9) where κ, α, β, γ, δ and are function parameters. This probability density function was used by to describe the dielectron mass background distribution, where the background is predominantly Drell-Yan dielectron events. A simulated CMS dataset is obtained by normalising the Z boson region (60 < m ee < 120 GeV) in simulation to data. The total number of data events corresponding to a given integrated luminosity is N Lumi . Using the above probability density function we generate hundreds of datasets, each with a total number of events which is a Poisson fluctuation on N Lumi . For each dataset we step through mass values and set a 95% confidence level (CL) limit on R σ . The limits are set using a Bayesian method with an unbinned extended likelihood function. Using both the signal and background probability density functions, the likelihood distribution is calculated as a function of the number of signal events for a given mass. The 95% CL upper limit on the number of signal events N 95 for a given mass is taken to be the value such that integrating the likelihood from 0 to N 95 is 0.95 of the total likelihood integral. This number N 95 is converted to a limit on the ratio of cross sections by dividing by the total number of acceptance and efficiency corrected Z bosons, the signal acceptance and efficiency. At each mass point, a limit is calculated for each of the hundreds of simulated datasets. Using the limits computed from each simulated dataset, the median 95% CL limit and the one and two sigma standard deviations on the 95% CL limit for each mass point can be calculated. The signal probability distribution used in the likelihood is a convolution of a Breit-Wigner function and a Gaussian function with exponential tails to either side. The limits are calculated in a mass window of ± 6 times the signal width, with this window being symmetrically enlarged until there is a minimum of 100 events in it.
To generate 14TeV dataset limits, the above procedure is repeated but the background probability density function is multiplied by an NNPDF scale factor to convert the 13TeV background distribution into a 14TeV distribution. In this work the PDF set NNPDF LO as_0130_QED is applied.

LHC potential to probe Walking Technicolor Parameter Space
With the set up described above we have evaluated limits on the NMWT parameter space according to Run 2 at CMS. We use the 95% CL observed limit on σ(pp → Z → e + e − )/σ(pp → Z → e + e − ) at √ s = 13TeV based on a dataset of integrated luminosity 36fb −1 [51].  The SM DY cross section at NNLO is given to be σ(pp → Z/γ * → e + e − ) = 1.928nb, which we use to convert the ratio of cross sections to a limit on σ(pp → Z → e + e − ). This limit is then projected onto the (M A ,g) plane and compared to the signal cross sections for Z and Z which we have evaluated at NNLO level. Figure 10a presents the NMWT parameter space in the (M A ,g) plane for S=0.1 which is already excluded with the recent CMS results. One can observe an important complementarity of Z and Z ; as was expected from the plots with cross sections, Z extends the coverage of the LHC in largeg and M A region. Analogous exclusion plots for different values of S are presented in Fig. 26a, Fig. 27a, Fig. 28a and Fig. 29a for S = −0.1, 0.0, 0.2 and 0.3 respectively.

Exclusion on M
We have also found the projected LHC limit for higher integrated luminosities. To do this we have simulated the SM DY background and have obtain an expected limit for 36f b −1 , confirming to within a few % the CMS expected limits using the method described in the previous section for the sake of its validation. Then we have obtained analogous expected limits for 100fb −1 at √ s = 13TeV as well as for 300fb −1 and 3000fb −1 at √ s = 14TeV. We follow the CMS limit setting procedure except for mass points with less than 10 events where we set limits using Poisson statistics. The excluded regions of the M A ,g parameter space are shown in Figure 10b, c, d respectively. Analogous exclusion plots for different values of S are presented in Fig. 26, Fig. 27, Fig. 28 and Fig. 29 for S = −0.1, 0.0, 0.2 and 0.3 respectively.
Already at 100fb −1 the excluded region visibly increases in M A andg for both Z and Z resonances. For example for small values ofg it increases for M A from 3.5 TeV to about 3.8 TeV. Figure 10 also shows the theoretical upper limit on M A imposed by the a parameter (see section 2). Requiring a > 0 and combining it with the current or projected experimental limits one gets the full picture of the surviving parameter space.
With the beam energy increase to √ s = 14TeV and total integrated luminosity 300 fb −1 or more the entire range of M A that we explore is excluded in the region ofg < 2, and the predictions for the final high-luminosity run of the LHC (Figure 10d) increase the exclusions in both the M A andg directions ruling out the whole parameter space for g < 3. To see the picture of the LHC sensitivity to the whole NMWT parameter space we have performed a scan of the full 4D (M A ,g, S, s) parameter space with ∼ 1x10 7 random points. In Fig. 11 we present the projection of this scan into (M A ,g) plane, with S and s range (−0.1, 0.3) and (−1, 1) respectively for LHC@13TeV and 36 fb −1 integrated luminosity. In Fig. 11a we overlayed the excluded points from Z or Z signals on top of the allowed points to show the (M A ,g) parameter space which is allowed for all values of S and s parameters, while in Fig. 11b we overlayed the allowed points on top of the excluded points to show the (M A ,g) parameter space which is excluded for all values of S and s parameters. The excluded points from the Z cross section (dark grey) are layered on top of those excluded by the Z cross section (light grey). It is important to stress that the most conservative limit on M A (parameter space which is excluded for all values of S and s parameters) is about 3.1 TeV for low values ofg and that this limit is significantly higher (by about 1 TeV) than previous limits established by the ATLAS collaboration in Refs. [21,22] for S = 0.3, s = 0 benchmark in (M A ,g, S, s) plane which actually gives one of the most optimistic limits for NMWT.

Conclusions
Walking Technicolor remains one of the most appealing BSM theories involving strong dynamics. In this study we have fully explored the 4D parameter space of WTC using dilepton signatures from Z /Z production and decay at the LHC. This signature is the most promising one for discovery of WTC at the LHC for the low-intermediate values of theg parameter.
We have studied the complementarity of the dilepton signals from both heavy neutral vector resonances and have demonstrated its importance. As a result, we have established the most up-to-date limit on the WTC parameter space and provided projections for the the LHC potential to probe WTC parameter space at higher future luminosity and upgraded energy.
Our results on the LHC potential to probe WTC parameter space are presented in Fig. 26,27,10,28 and 29 for the (M A ,g) plane for S = −0.1, 0.0, 0.1, 0.2 and 0.3 respectively, which gives a clear idea how the properties of the model and the respective LHC reach depend on the value of the S parameter. This extends the results found previously for just S = 0.3 which is not quite motivated in light of present EWPD. Moreover, as another new element of the exploration of WTC, we have provided an analytic description for features such as the Z /Z masses, the mass inversion M inv point as well as some couplings in our paper. We have also presented all these properties in the form of figures in the (M A ,g) plane and 3D (M A ,g, S) tables for clear insight into the model behaviour, and for direct comparison with prior works. We have discussed the theoretical upper limit on M A from the requirement of "walking" dynamics, and in combination with the exclusions from experiment we have found the strongest constraints on WTC to date. The predicted exclusions indicate that within the scope of the LHC, the lowg regions of the WTC parameter space can be closed completely.
We have explored the effect of the S and s parameters on the WTC exclusions using a very detailed scan of the 4D parameter space and establishing the current LHC limit in this 4D space which we present in Fig. 11. The results we have found reflect the most conservative limit on M A around 3.1 TeV, which for low values ofg is significantly higher (by about 1 TeV) than previous limits established by the ATLAS collaboration in Refs. [21,22] for the most optimistic benchmark with S = 0.3. The complete 4D scan also indicates the important influence of the value of the S-parameter on the dilepton signal rate, while the s parameter has little effect on the rate of the dilepton signal but could be important for the complementary V V and V H signatures.
Besides Z and Z complementarity for the exploration of the dilepton signal in the low-intermediateg region, it is important to note the complementarity for the V V and V H signatures which would allow us to probe the large values ofg. This is the subject of the upcoming study [59].

A.1 Mass Matrices in NMWT
We calculate N ij by diagonalising the bosonic mixing matrices A.3, perturbatively calculating the eigenvalues and eigenvectors of the matrices that diagonalise M 2 C and M 2 N order by order in 1/g. Details of the calculation are presented in here, with the results for C ij and N ij to 2nd order in 1/g 2 . At 0th order, the eigenvalues for the γ, Z are degenerate and m 2 γ , m 2 Z = 0, so the eigenvectors cannot be uniquely defined at this stage. To resolve this degeneracy we introduce a generic parameter x which is fixed at 2nd order to be x = g 2 /g 1 .
From the covariant derivative terms of the effective bosonic Lagrangian 2.5, we construct the mixing matrices that diagonalise to give physical masses for the vector bosons. The Lagrangian of the vector bosons in the mass eigenbasis is where these mass matrices for the charged and neutral bosons are In order to perform the analytic diagonalisation of these matrices, we perform an expansion in 1/g and calculate the eigenvectors and eigenvalues of the matrix order by order. Rephrasing the χ and M 2 V parameters such that we can rewrite these matrices in terms of the parameters of the model that we have used in this paper. Further to this, from the WSRs [43] we set ω = 0 and fix F π = 246GeV, so the mass matrices are written entirely from the free parameters, M A ,g and S.
Consider the diagonalisation of the neutral matrix M 2 N A.3. From the logic above we see that this can be written as To expand in powers of 1/g we can rewrite the independent M A , S and F π parameters in terms ofg and dependent parameters of the model. As stated above, in the regime of largeg, M 2 A is dominated by the r 2 term of equation 2.6, however it is not obvious to see that in the case of smallg, the m 2 term dominates. We can determine the scaling of m 2 from the 1st WSR and the definition of the pion decay constant in NMWT. From equation 2.8 we see that F 2 π can be written in terms of M 2 A /g 2 . In the lowg regime this would lead to F 2 π ∝ m 2 /g 2 , so one would naïvely expect F π ∝ 1/g. However, F π is fixed to avoid deviations from the 1st WSR, so m 2 must itself scale withg 2 . Finally, from equation 2.17 we see that S can be written in terms ofg −2 .
At leading order in 1/g, the mass squared terms for the neutral bosons are As there are two degenerate eigenvalues= 0, we must define the eigenvectors at 0th order with a generic term x which is fixed only at 2nd order in the 1/g expansion. The 0th order eigenvectors are then We can now construct the higher order corrections order by order. To calculate the 1st order corrections, we consider the eigenvalue equation where M = M 0 + M 1 + M 2 + . . . is the mixing matrix,v =v 0 +v 1 +v 2 + . . . are the eigenvectors of M , and λ = λ 0 + λ 1 + λ 2 + . . . are the eigenvalues of M . At first order we have where we have used the 0th order eigenvalue equation M 0v0 = λ 0v0 to remove 0th order terms, and have discarded terms of order > 1.
We can immediately see that the 1st order eigenvalues are λ i 1 = 0 for all i = 1, . . . , 4, as M 2 N does not have any diagonal components at order 1/g. We do not expect to see corrections to the squared masses of the vector bosons at odd order in 1/g as then we would find mass terms dependent on fractional powers in the coupling. The eigenvectors will contribute to the 2nd order mass corrections, and in terms of model parameters and the unknown x we find To find the 2nd order eigenvalues, we follow the same procedure as above, and keeping only 2nd order terms we find where we use the fact that λ 1 = 0 to reduce this to At this order we can now fix x, which turns out to be x = g 2 /g 1 , and we arrive at the 2nd order corrections to the neutral vector boson masses; Finally, the rotation matrices C and N can be constructed from the transpose of the sum of 0th, 1st and 2nd order eigenvectors; where N and C diagonalise the neutral and charged mass matrices respectively. It is the elements of these rotation matrices that comprise the vector boson couplings in NMWT, as discussed in section 3.1.2.
The we resolve v by subtracting equation A.17 from equation A.15, and substituting the definitions of χ and M V from equations 2.18 and A.15 respectively: Then we substitute this into equation A.17 to find r 2 , Now we find m in terms of the NMWT parameters from equation 2.6, Then we relate the Fermi constant G F to the model parameters by finding the form of F π . We combine the definitions of F V and F A with the 1st WSR, and substitute the definition of χ from equation 2.7, Finally, we arrive at the expression for F π ,

A.3 Solving for EW couplings
The other important quantities to derive analytic formulae for are the EW equivalent couplings g 1 and g 2 in terms of the independent parameters. These couplings can be derived as roots of the characteristic equation for the Z boson eigenvalue, i.e we can solve the equation det[M 2 N − M 2 Z ] = 0. Taking the absolute values of the roots, we find two solutions to this equation which correspond to the couplings g 2 and g 1 respectively, , and we have not replaced M V and χ, as they are purely functions of the independent parameters and not of either g 1 or g 2 .

A.4 Effect of S on Z /Z properties
Here we provide the additional figures and information relevant to the phenomenological study presented in this paper. Throughout the paper we have chosen S = 0.1 and s = 0 as the benchmark parameter space values, the effect of varying S is discussed here. As s is the Lagrangian parameter that quantifies Higgs interactions with the WTC gauge bosons, we continue to assume s = 0 throughout.

Couplings
In Figures 14-15 and Figures 16-17 we present the L-R components of the dilepton couplings for the Z and Z , respectively, for different values of S. These are analogous to the couplings presented in section 3.1.2, where the analytic form for the coupling components are also presented. The S dependence of these couplings is implicit in χ, g 1 , and g 2 , and the effect on the parameter space dependence for varying S is presented here. The width to mass ratio for Z and Z for different S are shown in Figures 18 and 19. The widths largely show similar behaviour to those at the benchmark value of S = 0.1 ( Figure 5), with the exception of S = 0. At S = 0, the Z width to mass ratio is very small (less than % level), so the Z resonance is always narrow at this S. The Z also has a narrower width for much of the parameter space at S = 0, however the region of Γ Z ≥ M Z nevertheless appears in the region with low M A and highg. The branching ratio spectra for the Z withg = 3, 8 is presented in Figures 20,21), and for the Z withg = 3, 8 -in Figures 22, 23 for various values S. The features of the branching ratio spectra such as the dips in the V V /V h channels are discussed in section 3.2, and again we note that the Z → W + W − channel is opened at low M A , high g at all values of S. Also note that for the Z , at S = 0 where the resonance is very narrow, the dilepton and diquark branching ratios are boosted and are the dominant decay channels across the whole (M A ,g) parameter space.
Again, the mass inversion point can also be identified as the point at which the W + W − and Zh branching ratios have a crossing point, hence the lack of crossing point at S = −0.1, 0. The DY production cross sections at LO for pp → Z → e + e − pp → Z → e + e − processes are presented in Fig. 24 and Fig. 25 respectively as contour levels of the cross section in (M A ,g) space for different S. As noted in section 4, the S parameter could be of great importance in determining the excluded region of WTC parameter space. As such, we present a set of figures for each discrete S in which we show the current and future limits on the WTC parameter space for fixed S. This is for direct comparison to the exclusions quoted and discussed in section 3.5.2. Figures 26, 27, 28, 29 show the excluded regions of M A ,g for S = −0.1, 0, 0.2, 0.3 respectively. The projected limits depend strongly on the S parameter, and for large S , the limit from dilepton searches at the LHC covers less of the parameter space, while the theoretical limit requiring a > 0 excludes a large portion of the M A parameter space from above.