High\(p_T\) signatures in vector–leptoquark models
Abstract
We present a detailed analysis of the collider signatures of TeVscale massive vector bosons motivated by the hints of lepton flavour nonuniversality observed in Bmeson decays. We analyse three representations that necessarily appear together in a large class of ultravioletcomplete models: a coloursinglet (\(Z'\)), a colourtriplet (the \(U_1\) leptoquark), and a colour octet (\(G'\)). Under general assumptions for the interactions of these exotic states with Standard Model fields, including in particular possible righthanded and flavour offdiagonal couplings for the \(U_1\), we derive a series of stringent bounds on masses and couplings that constrain a wide range of explicit newphysics models.
1 Introduction
The hints of Lepton Flavour Universality (LFU) violation in semileptonic B decays, namely the deviations from \(\tau /\mu \) (and \(\tau /e\)) universality in \(b\rightarrow c \ell {\bar{\nu }}\) decays [1, 2, 3, 4] and the deviations from \(\mu /e\) universality in \(b\rightarrow s \ell \bar{\ell }\) decays [5, 6], are among the most interesting departures from the Standard Model (SM) reported by experiments in the last few years. The attempt to find a single beyondtheSM (BSM) explanation for the combined set of anomalies has triggered intense theoretical activity, whose interest goes beyond the initial phenomenological motivation. In fact, it has shed light on new classes of SM extensions that turn out to be very interesting per se and that have not been investigated in great detail so far, pointing to nontrivial dynamics at the TeV scale possibly linked to a solution of the SM flavour puzzle.
The initial efforts to address both sets of anomalies have been focused on Effective Field Theory (EFT) approaches via fourfermion effective operators (see [7, 8, 9, 10] for the early attempts). However, the importance of complementing EFT approaches with appropriate simplified models with new heavy mediators was soon realised [9, 11]. Given the relatively low scale of new physics hinted by the chargedcurrent anomalies, the impact of considering a full model rather than an EFT on high\(p_T\) constraints are significant [12, 13, 14]. More recently, a further advancement has been achieved with the development of more complete (and more complex) models with a consistent ultraviolet (UV) behaviour (see in particular [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]).
In early EFT attempts, it was realised that a particularly good mediator accounting for both sets of anomalies is a TeVscale \(U_1\sim (\mathbf {3},\mathbf {1},2/3)\) vector leptoquark, coupled mainly to thirdgeneration fermions [8, 11]. The effectiveness of this state as a single mediator accounting for all available lowenergy data has been clearly established in [28]. However, this state can not be the only TeVscale vector particle in a realistic extension of the SM. Since it is a massive vector, the \(U_1\) can be either a massive gauge boson of a spontaneously broken gauge symmetry \(G_{\mathrm{NP}} \supset G_{\mathrm{SM}}\), as in the attempts proposed in [15, 16, 17, 18], or a vector resonance of some new strongly interacting dynamics, as e.g. in [19, 21]. As we show, in both cases the consistency of the theory requires additional vector states with similar masses. The purpose of this paper is to provide a comprehensive analysis of the high\(p_T\) constraints on the vector leptoquark \(U_1\) and what can be considered its minimal set of vector companions, namely a colour octet \(G^\prime \sim (\mathbf {8},\mathbf {1},0)\), which we will refer to as the coloron, and a colour singlet \(Z^\prime \sim (\mathbf {1},\mathbf {1},0)\).
In our analysis we consider the most general chiral structure for the \(U_1\) couplings to SM fermions. This is in contrast with many recent studies which considered only lefthanded (LH) couplings. While this hypothesis is motivated by the absence of clear indications of righthanded (RH) currents in the present data and by the sake of minimality, it does not have a strong theoretical justification. Indeed, the quantum numbers of the \(U_1\) allow for RH couplings, and in motivated UV completions such couplings naturally appear [18, 26]. We also analyse the impact of a nonvanishing mixing between the second and third family in high\(p_T\) searches, including in particular constraints from \(pp\rightarrow \tau \mu \) and \(pp\rightarrow \tau \nu \). As we show, the inclusion of righthanded couplings and/or a sizeable 2–3 family mixing yields significant modifications to the results found in the existing literature.
The structure of this paper is as follows: In Sect. 2 we motivate our choice of TeVscale vectors and in Sect. 3 we introduce the phenomenological Lagrangian adopted to describe their high\(p_T\) signatures. We then present the results of the searches in Sect. 4 and conclude with Sect. 5.
2 The spectrum of vector states at the TeV scale
 i.
Gauge models. Here \(U_1\) is the massive gauge boson of a spontaneously broken gauge symmetry \(G_{\mathrm{NP}} \supset G_{\mathrm{SM}}\). The need for extra massive vectors follows from the size of the cosetspace of \(G_{\mathrm{NP}}/G_{\mathrm{SM}}\), that necessarily requires additional generators besides the six associated to \(U_1\).
 ii.
Strongly interacting models. Here \(U_1\) appears as a massive resonance for a new strongly interacting sector. In this case the need of additional massive vectors is a consequence of the additional resonances formed by the same set of constituents leading to \(U_1\).
2.1 Gauge models: the need for a \(Z^\prime \)
In gauge models, the presence of an extra massive vector \(Z^\prime \sim (\mathbf {1},\mathbf {1},0)\) associated with the breaking \(U(1)_{BL} \times U(1)_{T^3_R} \rightarrow U(1)_Y\) is thus unavoidable. Since the breaking of \(U(1)_{BL}\) necessarily implies a breaking of SU(4), the breaking terms which lead to a nonvanishing \(Z^\prime \) mass necessarily induce a mass term for the \(U_1\) as well. Hence, the \(Z^\prime \) state cannot be decoupled. The opposite is not true: since the \(U_1\) generators are associated to the \(SU(4)/SU(3)_c\times U(1)_{BL}\) coset, mass terms for the \(U_1\) do not necessarily contribute to the \(Z^\prime \) mass.
2.2 Gauge models: the need for a \(G^\prime \)
While the minimal group in Eq. 5 allows us to build a consistent model for a massive \(U_1\sim (\mathbf {3},\mathbf {1},2/3)\), it does not leave us enough freedom to adjust \(U_1\) and \(Z^\prime \) couplings in order to comply with low and highenergy data.
Under \(G_{\mathrm{NP}}^{\mathrm{min}}\) the interaction strengths of both \(U_1\) and \(Z^\prime \) are unambiguously related to the QCD coupling (\(g_s\)) and to hypercharge, given that they all originate from the same SU(4) group. In particular \(g_U =g_s (M_{U_1})\), in a normalisation where \(\beta _{L,R}^{ij} \le 1\). Moreover, the couplings of the \(Z^\prime \) to SM fermions are necessarily flavour universal.^{1} A flavouruniversal \(Z^\prime \) is constrained by LHC dilepton searches to have \(M_{Z^\prime } \gtrsim 5\) TeV [30, 31]. Within \(G_{\mathrm{NP}}^{\mathrm{min}}\), the \(U_1\) should be necessarily close in mass [22] which, together with the low value of \(g_U\), results in a negligible impact on \(b\rightarrow c \ell \nu \) decays.
The enlargement of the coset space to \((G_{\mathrm{NP}}^{\mathrm{min}})^\prime /G_{\mathrm{SM}}\) directly requires a massive colouroctet vector (the “coloron” \(G^\prime \)) associated to the breaking \(SU(3)_{[4]} \times SU(3)^\prime \), where \(SU(3)_{[4]}\) is the “coloured” subgroup of SU(4). Similarly to the case of the \(Z'\), breaking terms leading to a nonvanishing \(G^\prime \) mass necessarily induces a mass term also for the \(U_1\), while the opposite is not necessarily true.
2.3 Vector spectrum in strongly interacting models
In strongly interacting models, the leptoquark \(U_1\) is a composite state with two elementary fermions charged under the new confining group \(G_{\mathrm{strong}}\) as constituents. These fermions are necessarily charged under \(SU(3)_c\) in order to generate a colourtriplet state.
3 Phenomenological Lagrangian
Having motivated the minimal set \(\{G',Z',U_1\}\) of massive vectors for a meaningful description of TeV scale dynamics, we proceed to set up a versatile framework for analysing the high\(p_T\) signatures of these states in a general way. In our analysis we restrict our attention to the interactions of these vectors with SM fermions and gauge bosons. We neglect possible Higgs couplings to the \(Z^\prime \) since they are severely constrained by electroweak precision data (see e.g. [28]) and are typically very small in the model realisations we are interested in. We also ignore any possible interactions of the extra vectors among themselves and to any other particles related to the UV completion of the model (either scalars or fermions). While some of the high\(p_T\) signatures related to these interactions can be quite interesting [22], they are highly dependent on the details of the UV completion. Here we only consider their possible indirect effects on the widths of the vectors, which we treat as an additional free parameter.^{2}
Summary of the relevant experimental constraints. All searches have a centre of mass energy of 13 TeV
Constrained BSM amplitude  Final state  Data set  Section  Reference 

\(U_1\) pair prod.  \(b\bar{b}\tau ^\tau ^+\)  CMS, 35.9 fb\(^{1}\)  [41]  
\(U_1\) pair prod.  \(t\bar{t}\nu _\tau \bar{\nu _\tau }\)  CMS, 35.9 fb\(^{1}\)  [42]  
\(G'\) pair prod.  \(2b2\bar{b}\)  CMS, 35.9 fb\(^{1}\)  [43]  
\(U_1\) [t chan.] & \(Z'\) [s chan.]  \(\tau _h^+ \tau _h^\)  ATLAS, 36.1 fb\(^{1}\)  [44]  
\(U_1\) [t chan.]  \(\tau _h \nu \)  CMS, 35.9 fb\(^{1}\)  [45]  
\(U_1\) [t chan.] & \(Z'\) [s chan.]  \(\tau _h \mu \)  ATLAS, 36.1 fb\(^{1}\)  [46]  
\(G'\) [s chan.] & \(Z'\) [s chan.]  \(t\bar{t}\)  ATLAS, 36.1 fb\(^{1}\)  [47] 
4 Results
We consider a variety of high\(p_T\) searches at the LHC which place limits on the model discussed above. The most constraining ones, which we discuss in detail below, are shown in Table 1. In some cases the searches are optimised for the BSM processes we are interested in, allowing a simple translation of the reported limits in terms model parameters. In most cases however, a reinterpretation of the reported limits is necessary.
Summary of the experimental constraints on pair produced leptoquarks in the \(b\bar{b}\tau ^\tau ^+\) [41] and \(t\bar{t}\nu _\tau \bar{\nu _\tau }\) [42] final states, assuming the leptoquarks decay solely into third generation SM particles. When \(\kappa _U = 1\), QCD production processes become less important and lepton exchange (which depends on \(g_U\)) is relevant. We thus show how the limit varies in the range \(g_U \in [0,4]\)
Parameters  \(b\bar{b}\tau ^\tau ^+\) final state  \(t\bar{t}\nu _\tau \bar{\nu _\tau }\) final state  

\(\kappa _U\)  \(\beta _R^{33}\)  \(\text {BR}(U_1\rightarrow b \tau ^+)\)  Limit [TeV]  \(\text {BR}(U_1\rightarrow t \bar{\nu _\tau })\)  Limit [TeV] 
0  0  0.51  1.4  0.50  1.6 
0  1  0.67  1.5  0.33  1.3 
1  0  0.51  1.1–1.3  0.49  1.1–1.2 
1  1  0.67  1.2–1.4  0.32  1.0–1.2 
The case of the \(\tau ^+\tau ^\) final state, which constrains both the \(Z^\prime \) (s channel production) as well as the \(U_1\) (t channel exchange), is significantly more involved. Here we reinterpret the limits on resonances decaying into taulepton pairs, with hadronically decaying taus, reported by ATLAS [44]^{3} (bounds from leptonic tau decays turn out to be significantly weaker at large ditau invariant masses). We first consider the bounds placed on the \(U_1\) and on the \(Z^\prime \) in isolation, for various choices of couplings and widths, and then in combination. As we emphasise below, it is essential to include all relevant experimental information when deriving limits in this case.
We extract further bounds on \(U_1\) by recasting CMS searches for \(pp\rightarrow \tau \nu \) [45] and limits on both \(Z^\prime \) and \(U_1\) from the \(pp\rightarrow \tau \mu \) search by ATLAS [46]. In both cases the 23 family mixing of the leptoquark plays a key role. As far as other dilepton final states are concerned, we explicitly checked that constraints from \(pp\rightarrow \mu \mu \) (see e.g. [31]) do not significantly constrain the parameter space relevant to our model.
The leading bound on the \(G^\prime \) is extracted by the unfolded \(t\bar{t}\) invariant mass spectrum provided by ATLAS [47]. In principle, the \(U_1\) and the \(Z'\) could be constrained by dijet searches. However, in our setup resonances tend to be very wide, with a widthovermass \(\sim 25\%\). As a result, the limits reported in the literature on narrow dijet peaks over a data driven background spectrum [50, 51, 52] are not directly applicable. Furthermore, dijet signatures are mostly produced for light quarks and gluons, which couple only weakly to \(Z'\) and \(G^\prime \) in our setup.^{4} Indeed, dedicated recasts of dijet searches performed in a setup similar to ours have shown that these constraints are less significant than those from the \(t\bar{t}\) final state [22]. Although one can envision scenarios where current dijet searches are more constraining than \(t{\bar{t}}\) searches, such as when thirdgeneration couplings are suppressed or when lightgeneration couplings are large, these limits are less relevant for the class of models which fit the flavour anomalies and so we do not consider dijet searches.
To perform recasts of these searches we implement the model described in Sect. 3 in FeynRules 2.3.32 [55] and generate the corresponding UFO model file. The FeynRules model files as well as the corresponding UFO model are available at https://feynrules.irmp.ucl.ac.be/wiki/LeptoQuark. In our Feynrules implementation and in all our results throughout this paper, we include only treelevel effects. While some NLO QCD corrections are available for the vector leptoquark case [56], in specific models these are expected to be supplemented by additional NLO contributions that can be of similar (or even larger) size. Hence we opt not to include them and we add a systematic error in our signal to (partially) account for them. Other Feynrules implementations for the vector leptoquark (but with interactions to thirdgeneration lefthanded fields only) are available [57]. We have crosschecked our leptoquark implementation (with \(\beta _R^{33}=0\)) against the one in [57], finding a perfect agreement between the two.
4.1 Limits from resonance pair production
We first briefly discuss limits on the leptoquark coming from their pair production. For a large fraction of the parameter space, the dominant production modes are governed by QCD and the relevant couplings are the strong gauge coupling and \(\kappa _U\), see Eq. 9. However, for \(\kappa _U\sim 1\) the QCDinduced production crosssection is smaller and pairproduction via lepton exchange becomes relevant for large values of \(g_U\). The most constraining searches in our scenario are those for the \(b\bar{b}\tau ^\tau ^+\) [41] and \(t\bar{t}\nu _\tau \bar{\nu _\tau }\) [42] final states.
In Table 2 we report the limits for various values of \(\kappa _U\) and \(\beta _R^{33}\), which determines the branching ratios (the branching ratios deviate slightly from the expected 1/2, 1/3, 2/3 due to phase space effects). We assume that the leptoquark decays only into third generation SM particles and find that the limits range from 1 TeV to 1.6 TeV. Similar limits have also been obtained in the literature, see e.g. [33, 42, 58], although using lower luminosity in the \(b\bar{b}\tau ^\tau ^+\) channel. Whenever it is possible to compare, we find good agreement between our results and those in the aforementioned references. With \(\kappa _U=0\) there is an extra coupling to the gluon field strength tensor boosting the production crosssection and strengthening the limit. As \(\beta _R^{33}\) increases, the branching ratio to \( b \tau ^+\) increases while the branching ratio to \(t \bar{\nu }_\tau \) decreases, which is reflected in a strengthening and weakening of the limits, respectively. For illustration, we include the strongest bound from pairproduction, i.e the limit \(M_U>1.6\) TeV, in Figs. 1, 6 and 7.
In a similar fashion, bounds on the coloron mass can be extracted from a search for pairproduced resonances decaying to quark pairs, performed by the CMS collaboration [43]. The search excludes a coloron in the whole mass range considered, from 80 GeV to 1500 GeV, so provides an upper bound of \(M_{G'}>1.5\) TeV. However, a stronger upper bound can be estimated by extrapolating the production crosssection and exclusion limit to higher energies, where bounds of 1.7 TeV and 2.1 TeV for \({\tilde{\kappa }}_{G'}=0\) and \({\tilde{\kappa }}_{G'}=1\) are obtained. The stronger bound in the latter case can be understood from the fact that the corresponding operator in Eq. 11 adds significantly to the \(gg\rightarrow G' G'\) amplitude. The estimated limits are practically independent of the choices of the couplings to quarks, because the production cross section is dominated by the gluoninitiated processes. In setting these limits, we fix the coloron gauge coupling to \(g_{G'}=3\), \(\kappa _{G'}=0\) and \(\kappa _{q,u,d}^{33}=1\).
4.2 \(pp\rightarrow \tau \tau \) search
The ATLAS collaboration has performed a search of heavy resonances in the ditau final state using \(36.1~\text {fb}^{1}\) of 13 TeV data [44]. In this section we recast this search to set limits on the \(U_1\) and \(Z^\prime \) masses for different choices of the couplings. In Sect. 4.2.3 and Sect. 4.2.2 we consider separate limits for the \(Z^\prime \) and \(U_1\) assuming that one of the two has fully decoupled. The interplay of the two resonances in this search is considered at the end, in Sect. 4.2.4.
4.2.1 Search strategy
We focus on the analysis with \(\tau _h\tau _h\) since this channel presents the highest sensitivity to highmass resonances. The contributions to the \(pp\rightarrow \tau ^+\tau ^\) process from new heavy resonances, including the interference with the SM, are computed using Madgraph5_aMC@NLO v2.6.3.2 [59], with the NNPDF23_lo_as_0119_qed PDF set [60]. Hadronization of the \(\tau \) final states is performed with Pythia 8.2 [61] with the A14 set of tuned parameters [62]. Detector simulation is done using Delphes 3.4.1 [63]. The ATLAS Delphes card has been modified to satisfy the object reconstruction and identification requirements. In particular we include the \(\tau \)tagging efficiencies quoted in the experimental search [44]. After showering and detector simulation, we apply selection cuts using MadAnalysis 5 v1.6.33 [64] (see Table 3 for details on the applied cuts). We have validated our results by generating the SM DrellYan \(pp\rightarrow \tau \tau \) background and comparing our results with the one quoted by ATLAS. A good agreement is found between the two samples (we find a discrepancy with the quoted central values of less than 20%, well within the given \(1\sigma \) region).
Summary of the experimental cuts for the ATLAS \(\tau _h\tau _h\) search [44]. For the leading \(\tau _h\) we use the \(p_T\) cut \(p_T^{\tau _{h1}}>130\) GeV as quoted in the HEPData entry for Ref. [44]. Note that the corresponding cut was \(p_T^{\tau _{h1}}>85\) GeV for \(10\%\) of the data
Particle selection  At least two \(\tau _h\)’s and no electrons or muons 
Charge  \(\tau _{h1}\) \(\tau _{h2}\) should be of opposite charge 
\(\tau _h\) \(p_T\)  \(p_T^{\tau _{h1}}>130\) GeV, \(p_T^{\tau _{h2}}>65\) GeV 
\(\eta \)  \(\eta _{\tau _h}<2.5\) excluding \(1.37<\eta _{\tau _h}<1.52\) 
\(\phi \)  \(\Delta \phi (\tau _{h1},\tau _{h2})>2.7\) rad 
4.2.2 Limits on the \(U_1\) leptoquark
In this section we decouple the \(Z^\prime \) and concentrate on the limits arising exclusively from the leptoquark exchange. In our search we take maximal values for \(\beta _L^{33}\) (i.e. \(\beta _L^{33}=1\)) and consider three benchmarks for the righthanded coupling: \(\beta _R^{33}=\{0.0,\,0.5,\,1.0\}\). Note that the search is not sensitive to the relative sign choice between \(\beta _R^{33}\) and \(\beta _L^{33}\) but only to their magnitudes. The reason for this is that the New Physics (NP) amplitudes of different chiralities do not interfere with each other and the amplitude proportional to \(\beta _R^{33}\, \beta _L^{33}\) does not interfere with the SM ones. We further fix the leptoquark width to its natural value. The leptoquark width only mildly affects the results of this search, contrary to the \(Z^\prime \) case discussed in the next section, since the NP contribution is generated via a t channel exchange.
Exclusion limits in the \((g_U, M_U)\) plane, setting \(\beta ^{23}_L=0\), are shown in Fig. 1 (left). Similar recasts for the case with \(\beta _R^{33}=0\) can be found in the literature [58, 68]. We obtain slightly stronger limits than those in the previous references. As we show in Fig. 5, this difference can be understood from the fact that we consider the full \(m_T^{\mathrm{tot}}\) distribution and not only the highest bin. The lower bins are important since a t channel exchange gives rise to a broad tail in the spectrum. Exclusion limits for the scenario where \(\beta _R^{33}\ne 0\) have not been discussed in the literature. We find that the additional chirality significantly enhances the cross section, yielding limits that are about \(70\%\) stronger than in the case when \(\beta _R^{33}=0\).
Finally, we also study the limits on \(M_U\) for nonzero values of \(\beta _L^{23}\), Fig. 1 (right). Here we fix \(g_U=3\) and \(\beta _L^{33}=1\) and plot the corresponding exclusion limits for the three benchmark values of \(\beta _R^{33}\) discussed above. As can be seen, only a mild increase of the limits is found for \(\beta _L^{23}\lesssim 0.4\). For larger values of \(\beta _L^{23}\), the PDF enhancement is enough to make \(s{\bar{s}}\rightarrow \tau ^+\tau ^\) the dominant partonic channel and the limits start growing linearly with \(\beta _L^{23}\).
4.2.3 Limits on the \(Z^\prime \) resonance
We now proceed to the limits set on the \(Z'\), decoupling the leptoquark. Throughout this section we fix \(\zeta _{q,u,d}^{33} = \zeta _{\ell ,e}^{33} =1\) and focus on the impact of varying the overall \(Z'\) coupling \(g_{Z'}\), varying the coupling to lefthanded light quarks \(\zeta _q^{ll}\), and varying the width of the \(Z'\).
In the left panel of Fig. 2 we set \(\zeta _q^{ll} = 0\) and show the exclusion in the \((g_{Z'},M_{Z'})\) plane. For small couplings, \(g_{Z'} < 0.5\), the \(Z'\) is not excluded above 1 TeV as the production cross section is too small. In the range \(0.5< g_{Z'} < 1.0\) the limit increases from 1 TeV to 2 TeV and it approaches a regime where it increases linearly with the coupling. This can be understood by the fact that, having set \(\zeta _q^{ll}=0\), the \(Z'\) is dominantly produced from bquarks, which carry only low momentum fractions of the protons. As a result, even for relatively low masses the effective crosssection scales like a contact interaction \(\sigma _Z' \sim g_Z'^4/M_Z'^4\).
Finally, we also show the impact of varying the width. As can be noted, doubling the width (dashed line in Fig. 2) has a relatively minor impact. This is consistent with the observation that the limits does not come from the onshell production of the \(Z^\prime \), but rather from its tail (that scales like a contact interaction).
In Fig. 2 (right) we fix \(g_{Z'} = 3\) and vary the couplings to lefthanded light quarks \(\zeta _q^{ll}\). Since the light quarks have less PDF suppression than the thirdgeneration quarks, the limit increases rapidly. For \(\zeta _q^{ll} \ll 1\), the width is not affected by increasing \(\zeta _q^{ll}\), while for larger values of \(\zeta _q^{ll}\) the width starts to be affected leading to a change of slope.
We again show that doubling the natural width decreases the limit by around 10 %. We also show the impact of changing the relative sign between the light quark couplings and the thirdgeneration coupling. With opposite signs the interference term contributes constructively, strengthening the limit, whereas when the signs are the same the interference term contributes destructively, weakening the limit.
In summary, the \(Z'\) mass limit of the ditau search depends weakly on the universal coupling \(g_{Z'}\), is very sensitive to the lightquark couplings (it is excluded below 5 TeV for \(\zeta _q^{ll} \approx 1\)), and is only weakly relaxed by an increase of the total width of the \(Z'\).
4.2.4 Combined limits for the \(Z^\prime \) and the \(U_1\) leptoquark
We now consider the limits when both the \(Z'\) and the leptoquark are present. For the \(Z'\) we set \(\zeta _{q,u,d}^{33} = \zeta _{\ell ,e}^{33} =1\) and \(\zeta _q^{ll} = 0\). For the leptoquark we set \(\beta _L^{33}=\beta _R^{33}=1\) and \(\beta _L^{23}=0\). In both cases we assume natural widths.
In Fig. 4 we show the exclusion limit on the \((M_U,~M_{Z'})\) plane for a variety of overall coupling strengths, \(g_U = g_{Z'} \in \{2.5,\, 3.0,\, 3.5\}\). The increase of the limits with growing coupling in each step is relatively small for the \(Z'\) (\(\sim 200 {\,\mathrm GeV}\)), while it is larger for the leptoquark (\(\sim 600 {\,\mathrm GeV}\)). We see that the decoupling regimes considered in the previous two sections hold when the \(Z'\) is heavier than (roughly) 3 TeV, and when the leptoquark is heavier than \(56\) TeV.
We now highlight the importance of including more than just the highest bin in \(m_\text {T}^\text {tot}\) in setting the mass limit. In Fig. 5 (left) we plot the \(m_\text {T}^\text {tot}\) distribution of the data and background from [44], along with our simulated leptoquark and \(Z'\) contributions. We show the distributions for \(g_{U} = 3\) and \(g_{Z'} = 3\), for masses at the 95% C.L. limit. After a peak, the background steadily falls with increasing \(m_\text {T}^\text {tot}\). The final bin has a larger number of events than the preceding bin as this bin is wider and as it includes overflow events. As such, the final three bins each contain a similar number of background events. Since tau pair production via a \(Z'\) proceeds through an s channel, it is more peaked in \(m_\text {T}^\text {tot}\) and the events from a multiTeV \(Z'\) cluster in the highest energy bin. However, tau pair production via a leptoquark proceeds through t channel process, so there is no clear peak in the invariant mass distribution. This leads the distribution in \(m_\text {T}^\text {tot}\) to extend to lower values. We see in Fig. 5 (right) the impact of including only the N highest bins in the CLs calculation. For the \(Z'\), the limit obtained with only the highest bin is almost \(200{\,\mathrm GeV}\) lower than the limit including all bins. For the leptoquark, when only the highest bin is included, the 95% C.L. limit is around \(400{\,\mathrm GeV}\) weaker than when all bins are included. When the highest two bins are included the difference reduces to around \(100{\,\mathrm GeV}\), and slowly improves as more bins are added. We see that it is crucial to include more than the highest bin in \(m_\text {T}^\text {tot}\) to produce an accurate estimate of the leptoquark exclusion limit. However, it should be noted that we have not been able to account for possible correlations between the bin errors, which could impact the derived exclusion limits.
4.3 \(pp\rightarrow \tau \nu \) search
The ATLAS and CMS collaborations have performed searches for heavy resonances decaying to \(\tau \nu \) (with hadronically decaying \(\tau \)) using \(36.1~\text {fb}^{1}\) [69] and \(35.9~\text {fb}^{1}\) [45] of 13 TeV data, respectively. In this section we reinterpret this search in the context of the model in Sect. 3 to set limits on the vector leptoquark mass as a function of \(\beta _{23}^L\). In our limits we use the CMS data. Since ATLAS data presents a (small) upper fluctuation with respect to the SM background, a combination of ATLAS and CMS data yields slightly weaker limits than CMS data alone (see e.g. [14]).
4.3.1 Search strategy
We compute the NP contribution to the \(pp\rightarrow \tau _h\nu \) process, including the interference with the SM, using Madgraph5_aMC@NLO v2.6.3.2 [59], with the NNPDF23_lo_as_0119_qed PDF set [60]. Hadronization of the \(\tau \) final state is done with Pythia 8.2 [61] using the CUETP8M1 set of tuned parameters [70]. The detector response is simulated using Delphes 3.4.1 [63]. The CMS Delphes card has been modified to satisfy the object reconstruction and identification requirements, in particular we include the \(\tau \)tagging efficiencies quoted in the experimental search [45].
Summary of the experimental cuts for the CMS \(\tau _h\,\nu \) search [45]
Particle selection 1  No events with an electron (\(p_T^e>20\) GeV, \(\eta _e<2.4\)) 
Particle selection 2  No events with a muon (\(p_T^\mu >20\) GeV, \(\eta _\mu <2.5\)) 
\(\tau _h\) \(p_T\)  \(p_T^{\tau _{h}}>80\) GeV 
Missing energy  \(E_T^{\,\mathrm miss}>200\) 
\(p_T\) vs missing energy  \(0.7<p_T^{\tau _{h}}/E_T^{\,\mathrm miss}<1.3\) 
\(\phi \)  \(\Delta \phi (p_T^{\tau _{h}},p_T^{\,\mathrm miss})>2.4\) rad 
4.3.2 Limits on the \(U_1\) leptoquark
For this search, we fix \(\beta _L^{33}=1\) and consider two different benchmarks for the righthanded coupling, \(\beta _R^{33}=0,\,1\). In this case the relative sign between \(\beta _R^{33}\) and \(\beta _L^{33}\) is not observable in this channel. Since the leptoquark width plays a marginal role, we fix it to its natural value. We furthermore set \(g_U=3\).
We compute exclusion limits for the vector leptoquark in the \((\beta _L^{23},M_U)\) plane, see Fig. 6. For comparison, we overlay the corresponding limits from \(pp\rightarrow \tau \tau \) [see Fig. 1 (right)] and pairproduction limits. As can be seen, these limits give complementary information to those presented in Sect. 4.2.2, offering more stringent limits only when the \(\beta _L^{23}\) coupling becomes large. Analogous limits for the case \(\beta _R^{33}=0\) have already been derived in the past literature [14]; we find good agreement between these limits and the ones quoted here. Interestingly, and as happens in the \(pp\rightarrow \tau \tau \) search, the exclusion bounds get significantly affected by nonzero values of \(\beta _R^{33}\). The different shapes in the exclusion bands can be understood from the fact that, for \(\beta _R^{33}=1\), the dominant partonic process is \(bc\rightarrow \tau \nu \), whose cross section scales as \(\sigma _{bc\rightarrow \tau \nu }\sim \beta _L^{23}^2/M_U^4\) in the EFT limit. On the contrary, for \(\beta _R^{33}=0\), the relative contribution from sc production, for which \(\sigma _{sc\rightarrow \tau \nu }\sim (\beta _L^{23}/M_U)^4\), is important and even becomes dominant for mediumsize values of \(\beta _L^{23}\).
4.4 \(pp\rightarrow \tau \mu \) search
The ATLAS collaboration has published a search for heavy particles decaying into differentflavour dilepton pairs using \(36.1~\text {fb}^{1}\) [46] of 13 TeV data. In this section we recast the ATLAS data and reinterpret the collider bounds in terms of the model in Sect. 3 to set limits on \(\beta _{32}^L\) and \(\zeta ^{23}_\ell \), as a function of the leptoquark and \(Z^\prime \) masses, respectively.
4.4.1 Search strategy
We use Madgraph5_aMC@NLO v2.6.3.2 [59] with the NNPDF23_lo_as_0119_qed PDF set [60] to compute the NP contribution to the \(pp\rightarrow \tau \mu \) process. The output is passed to Pythia 8.2 [61] for tau hadronization and the detector effects are simulated with Delphes 3.4.1 [63]. The ATLAS Delphes card has been adjusted to satisfy the object reconstruction and identification criteria in the search. In particular we have modified the muon efficiency and momentum resolution to match the High\( p_T \) muon operating point, and adjusted the missing energy reconstruction to account for muon effects. We have further included the \(\tau \)tagging efficiencies quoted in the experimental search [46].
After showering and detector simulation, we apply the selection cuts specified in Table 5 using MadAnalysis 5 v1.6.33 [64]. The resulting events are binned according to their dilepton invariant mass. Following the approach described by ATLAS [46], the tau momentum is reconstructed from the magnitude of the missing energy and the momentum direction of the visible tau decay products. This approach relies on the fact that the momentum of the visible tau decay products and the neutrino momentum are nearly collinear.
In order to validate our procedure, we have simulated the \(Z^\prime \) signal quoted in the experimental search [46], finding good agreement between our signal and the one by ATLAS.
Summary of the experimental cuts for the ATLAS \(\tau _h\,\mu \) search [46]
Particle selection  One single \(\tau \) and \(\mu \), no electrons 
\(p_T\)  \(p_T^{\tau _h}>65\) GeV, \(p_T^{\mu }>65\) GeV 
\(\eta \)  \(\eta _{\tau _h}<2.5\) excluding \(1.37<\eta _{\tau _h}<1.52\); \(\eta _\mu <2.4\) 
\(\phi \)  \(\Delta \phi (\tau _h,\mu )>2.7\) rad 
\(\Delta R\)  \(\Delta R(\tau _h,\mu )>0.4\) 
4.4.2 Limits on the \(U_1\) leptoquark
Following a similar strategy as for the other channels, we fix \(g_U=3\) and \(\beta _L^{33}=1\), and take two benchmark values for the righthanded coupling \(\beta _R^{33}=0,1\) (different sign choices for this parameter do not have an impact on the high\(p_T\) signal). Varying the leptoquark width only yields a subleading effect so we keep it fixed to its natural value.
We decouple the \(Z^\prime \) and compute the exclusion limits for the vector leptoquark mass as a function of \(\beta _L^{32}\), see Fig. 7 (left). As in previously analysed channels, the exclusion limits vary significantly for different values of \(\beta _R^{33}\). We additionally overlay the corresponding exclusion limit obtained from the \(pp\rightarrow \tau \tau \) search and searches for pairproduction. The limits from \(pp\rightarrow \tau \mu \) become stronger than those obtained from \(pp\rightarrow \tau \tau \) only for large values of the \(\beta _L^{32}\) parameter, especially in the case when \(\beta _R^{33}=1\).
4.4.3 Limits on the \(Z^\prime \)
4.5 \(pp\rightarrow t{\bar{t}}\) search
We finally turn our attention to searches in the ditop final state, which is subject to NP effects from s channel colorons and \(Z'\) bosons. We focus our analysis on the coloron since the bounds from this channel on the \(Z^\prime \) are weaker than the ones reported in Sect. 4.2.3.
4.5.1 Search strategy
We perform a recast of the ATLAS study [47], using 36 fb\(^{1}\) of collected data. Since the data was unfolded in this work, we can compute partonlevel predictions and directly compare them to the unfolded distributions provided in the reference study.
We choose to derive the constraints from the normalised partonlevel differential crosssections as a function of the \(t{\bar{t}}\)invariant mass, shown in Fig. 14(b) of [47]. As in the other searches, we do not include possible error correlations between the bins in the invariantmass distribution since they are not provided. Our signal predictions are derived by integrating the leadingorder SM QCD partonic crosssections \(q{\bar{q}}\rightarrow t{\bar{t}}\) and \(gg\rightarrow t{\bar{t}}\) and the NP contributions from coloron and \(Z'\) over the parton distribution functions, employing the NNPDF30_nlo_as_0119 PDF set [60] and fixing the factorisation and renormalization scale to the center of the corresponding \(t{\bar{t}}\)invariant mass bin. We use the running strong coupling constant as provided by the PDF set. The only cut applied is on transverse momentum of either top quark: \(p_T^{t}>500~\mathrm {GeV}\). Note that our reference study places the cuts as \(p_T^{t,1}>500~\mathrm {GeV}\) on the leading top, and \(p_T^{t,2}>350~\mathrm {GeV}\) on the subleading one. For a fully exclusive, partonic \(t{\bar{t}}\) final state, \(p_T^{t,1}=p_T^{t,2}\) and hence the second cut does not influence our calculation. However, the unfolded distributions are derived from data which employ this slightly milder cut, leading to slight deviations in bins of lower invariant mass. We therefore drop the bins \(m_{t{\bar{t}}}<1.2~\mathrm {TeV}\) and then find excellent agreement with the SM predictions presented in the ATLAS study. While the analysis also provides unfolded spectra differential in \(p_T\) and various other kinematic observables, we find the invariant mass spectrum to be the most constraining distribution. We therefore focus solely on the invariant mass spectrum and do not consider searches in the angular spectra.
4.5.2 Limits on the coloron
We are now ready to present the constraints on the various parameters related to the coloron. Throughout this section we set \(\kappa _q^{33}=1\) and \(\kappa _q^{ll}=\kappa _u^{ll}=\kappa _d^{ll}=(g_s/g_{G'})^2\), and we fix \(g_{G'}=3\), unless otherwise stated.
In Fig. 9, we show exclusion regions for the coloron with its natural width and with a width enhanced by a factor of two. In the left panel, the exclusion limits in the \((g_{G'},M_{G'})\) plane are shown for the natural width and twice this value. An interesting feature of these exclusion regions is that the boundaries bend towards smaller masses for larger couplings. This can be understood by the fact that while the cross section grows with the coupling, so does the width. For the reasons discussed above, the search then loses sensitivity to the resulting signal.
Finally, Fig. 10 shows exclusion limits with varying widths of the coloron. The different curves (solid, dashed, dotted) show various different choices of relations between the couplings to left and righthanded light quarks. As expected, limits get weaker with increasing width of the resonance. When the sign of \(\kappa _q^{ll}\) is chosen to be opposite of \(\kappa _{u,d}^{ll}\), the bounds also become weaker for the same reason as discussed above. The grey bands denote the regions in which the floating width parameter is below the partial width to quarks. Note that for \(\kappa _{G'}\ne 0\) the coloron can decay to two gluons, in which case the actual width would become significantly larger than the partial width to quarks alone.^{8}
5 Conclusions
The high\(p_T\) phenomenology of models predicting a TeVscale \(SU(2)_L\) singlet vector leptoquark which is able to account for the hints of LFU violations observed in Bmeson decays is quite rich. This is both because this exotic mediator can manifest itself in different final states accessible at the LHC, and also because this state cannot be the only TeVscale exotic vector. As we have shown, the minimal consistent set of massive vectors comprising a \(U_1\) also includes a coloron and a \(Z^\prime \). In this paper we have presented a comprehensive analysis of the high\(p_T\) signatures of this set of exotic TeV states, deriving a series of bounds on their masses and couplings.

In most of the relevant parameter space the most stringent bound on the leptoquark is obtained by the \(pp\rightarrow \tau \tau \) process. In this channel a possible O(1) righthanded coupling (\(\beta _R^{33}\)) has a very large impact, as shown in Fig. 1.

A nonvanishing offdiagonal coupling of the \(U_1\) to quarks has a modest impact in \(pp\rightarrow \tau \tau \), provided \(\beta _L^{23} \lesssim 0.2\) (as expected from a natural flavour structure), but a significantly larger impact in \(pp\rightarrow \tau \nu \). However, the latter search remains subleading compared to \(pp\rightarrow \tau \tau \) up to \(\beta _L^{23}  \lesssim 0.8\) for for \(\beta _R^{33}=1\) (or up to \(\beta _L^{23}  \lesssim 0.6\) for \(\beta _R^{33}=0\)).

For large nonvanishing offdiagonal coupling to leptons, a potentially interesting channel is \(pp\rightarrow \tau \mu \). In the pure lefthanded case, the bound from this channel is stronger than the one from \(pp\rightarrow \tau \tau \) if \(\beta _L^{32} \ge 0.5\) (see Fig. 7).

Taking \(g_U=g_{Z^\prime }\) and assuming dominant thirdgeneration coupling to fermions and small couplings to the light families, the constraints on the \(Z^\prime \) mass are significantly weaker than those on the \(U_1\) (see Fig. 4). The combination of \(U_1\) and \(Z^\prime \) signals in \(pp\rightarrow \tau \tau \) leads to a modest increase on the corresponding bounds, confined to a relatively narrow region of the parameter space.

The bound on the coloron from \(pp\rightarrow t{\bar{t}}\) is quite sensitive to the width of this state, and to the possible coupling to light quarks. Due to the increase of the width, the bounds become weaker at large couplings (see Fig. 10 (left)).
Footnotes
 1.
This statement follows from the fact that the mixing of SM fermions among themselves (in flavour space) and with possible exotic representations necessarily involve states with the same \(BL\) charges. As a result, the mixing acts as a unitary rotation on the \(Z^\prime \) couplings that remains proportional to the identity matrix in flavour space.
 2.
We will consistently assume that righthanded neutrinos, if present, are heavy enough so that they effectively decouple and do not play any relevant role. Models with light \(\nu _R\) in connection to the Banomalies can be found in [34, 35, 36, 37], and in connection to the vector leptoquark in [38].
 3.
We do not consider the corresponding analysis by CMS [49], which focuses on heavy Higgs bosons.
 4.
 5.
The power of each of the bins in excluding a signal is shown in Fig. 5, where we plot the 95% CL exclusion limit in the leptoquark mass, as a function of the number of the bins included in the statistical analysis.
 6.
Current limits from loopmediated transitions, such as \(\tau \rightarrow \mu \gamma \), offer stronger bounds in certain UV completions [26]. However, these bounds are more sensitive to the details of the UV completion and are therefore less robust.
 7.
A very wide resonance also leads to a suppression in the overall signal crosssection, further decreasing the constraining power of the search.
 8.
In this case, the production cross section of the coloron would also be drastically increased, leading to much stronger bounds on its mass from this search.
Notes
Acknowledgements
We thank Claudia Cornella and Admir Greljo for useful comments on the manuscript. JFM is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support while this work was being finalised. We would like to thank Hubert Spiesberger for allowing us to use the THEP Cluster in Mainz for parts of this study. This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200021159720.
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