Exotic Vector-Like Quark Phenomenology in the Minimal Linear σ Model

Extensions of the Standard Model that include vector-like quarks commonly also include additional particles that may mediate new production or decay modes. Using as example the minimal linear σ model, that reduces to the minimal SO(5)/SO(4) composite Higgs model in a specific limit, we consider the phenomenology of vector-like quarks when a scalar singlet σ is present. This new particle may be produced in the decays T → tσ, B → bσ, where T and B are vector-like quarks of charges 2/3 and −1/3, respectively, with subsequent decay σ → W+W−, ZZ, hh. By scanning over the allowed parameter space we find that these decays may be dominant. In addition, we find that the presence of several new particles allows for single T production cross sections larger than those expected in minimal models. We discuss the observability of these new signatures in existing searches.


I. INTRODUCTION
The electroweak (EW) hierarchy problem remains one of the weaknesses of Standard Model (SM) of particle physics. Indeed, a potentially strong fine tuning in the scalar sector is required once the need for heavy new physics is considered, for example to explain nonvanishing neutrino masses and the presence of dark matter. Among the best known scenarios to solve this problem, the composite Higgs (CH) framework [1-3] has received a renewed interest [4][5][6][7]. The traditional CH model is based on a global symmetry G, broken spontaneously to a subgroup H, such that the coset G/H is symmetric. A certain number of Nambu-Goldstone bosons (GBs) arise from this breaking: in particular, the SM GBs, which are the would-be longitudinal components of the EW gauge bosons, and the Higgs itself. This technically solves the hierarchy problem, providing a symmetry protection for the Higgs mass.
The idea is that an undetermined strong dynamics, acting at a high-scale, generates exotic fermionic bound states that spontaneously break the group G. Moreover, the gauging of the SM symmetries, together with nonuniversal fermion masses, provide an explicit breaking of G such that a small mass is generated for the Higgs. SM fermion masses are generated by means of the fermion partial compositeness mechanism [8][9][10], which consists in introducing in the spectrum a series of exotic (typically vector-like) fermions which act as partners of the SM fermions, with a role similar to that of right-handed (RH) Majorana neutrinos in the Type I Seesaw mechanism [11][12][13] for neutrino masses. In the mass basis, this results in light SM fermions and heavier exotic counterparts. The top quark is the heaviest among the light fermions and it turns out to be largely composed of exotic states. On the other hand, the lightest exotic fermions are the counterparts of the top and bottom quarks. These are labelled, depending on their electric charge, as top-partner T and bottom-partner B, respectively. The search for heavy partners of the SM quarks is a very effervescent field from the experimental side [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. The discovery of such particles would not only represent a fundamental step toward the understanding of the theory beyond the SM, but also a window to study the electroweak symmetry breaking (EWSB) sector.
In this paper the focus is in the so-called Minimal Linear σ Model (MLσM) [31], that attempts to improve the limitations of the traditional CH models. In the latter, the indetermination of the strong dynamics leads to describe the model Lagrangian with an effective approach, which however leads to a limited range of application. The MLσM, instead, is a renormalisable model based on the global spontaneous symmetry breaking SO(5) → SO(4) that matches the minimal CH model [4,[32][33][34] in a specific limit. The main idea is the introduction of a scalar quintuplet of SO(5) that contains as degrees of freedom the three SM GBs, the Higgs and an additional EW singlet σ. Moreover, the spectrum is enlarged by the introduction of several exotic fermions in the trivial and fundamental representations of SO(5) that will give rise to the fermion partial compositeness mechanism.
Several studies have been conducted on this model to analyse its features at low-energies [35], projecting to the so-called Higgs effective field theory Lagrangian [36][37][38][39][40][41][42][43][44][45][46][47][48][49], to access its ability to solve the strong CP problem [50][51][52]. The focus of this paper, instead, is to study the phenomenology associated to the exotic vector-like quarks (VLQs). Indeed, being a renormalisable model, all the parameters describing the low-energy theory are fixed in terms of the original Lagrangian parameters. In particular, the whole spectrum of the exotic VLQs and their interactions are fixed in terms of the initial parameters: this is a novelty with respect to previous studies where no direct link was present between the masses of the different VLQs, neither with their interactions, which were taken to be independent parameters.
The new VLQ states can be produced in pairs via QCD interactions in hadron collisions, with cross sections that only depend on the VLQ mass. In addition, it is also possible to singly produce the VLQs at colliders via their mixing with SM quarks. In both cases, these cross sections quickly decrease as the VLQ mass increases, and therefore for collider phenomenology the most interesting signals are generically those of the lightest VLQ. 1 The electric charge of the lightest VLQ (and therefore its decay modes) depends on the model parameters. In case it has exotic charge 5/3 or −4/3, its phenomenology is quite the same as in minimal models [53,54], because it does not couple to the scalar σ. On the other hand, if it has charge 2/3 (T ) or charge −1/3 (B), these VLQ states inherit new decay modes beyond the standard decays due to the presence of the singlet scalar σ. These new decays open a plethora of new possible signatures for VLQs, both in the pair and the single production channels. Current searches, while targeting the standard decays in Eq.
(1), are sensitive to the new decays to a varying degree. Non-standard VLQ top and bottom partner decays into a singlet scalar field like those in Eq. (2) have already been explored in the literature [55][56][57][58][59] (see also [60,61]). These studies mainly consider the singlet scalar σ being invisible at colliders [55,56,62], or decaying into Zγ, γγ final states [58], yet not focusing on the singlet scalar decays from Eq. (3), which generically are the main decay modes for a singlet scalar σ mixing with the SM Higgs boson and with mass m σ > 200 GeV. Moreover, the singlet scalar σ has previously been considered to be a GB of the coset G/H (together with the Higgs boson), while in the MLσM the nature of the σ field is qualitatively very different from that of the Higgs (e.g. σ is naively expected to be significantly heavier that the Higgs boson).
The paper is organised as follows. Section II describes the relevant aspects of the model, writing down the interaction Lagrangians that determine the phenomenology of the VLQs. The constraints on the parameter space of the model arising either from precision electroweak data, Higgs boson measurements, and direct searches, are discussed in Sec. III. In Sec. IV we outline the procedure used for the exploration of the MLσM parameter space. The results of our parameter scan are analyzed in Sec. V, focusing on the phenomenology of the lightest VLQ, the presence of new decay modes and opportunities for single VLQ production at the LHC. We also include a discussion on how these new decay modes can be experimentally searched for. We finally conclude in Sec. VI.

II. THE MINIMAL LINEAR σ MODEL
The MLσM is based on the group SO(5) × U (1) X , where the last factor ensures the correct hypercharge assignments for the SM fields. The spectrum contains the four SM gauge bosons associated to the SM gauge symmetry, a real scalar field φ in the fundamental representation of SO(5), the elementary fermions with the same quantum numbers as in the SM and finally exotic vectorlike quarks in the trivial and fundamental representations of SO (5).
The scalar field φ includes the three would-belongitudinal components of the SM gauge bosons π i , i = 1, 2, 3, the Higgs field h and the additional scalar field s, singlet under the SM group: where the last expression holds in the unitary gauge (but in the unbroken phase). The associated scalar potential is responsible for the spontaneous SO(5) → SO(4) breaking. As discussed in the following, explicit breaking terms are also contained in the scalar potential and are responsible for the SO(4) and EW breaking. The elementary fermions do not directly couple to the scalar φ, and therefore neither to the SM Higgs once SO(5) is broken. These couplings only arise through the mediation of the exotic fermions that do have tree level interactions with φ. The proto-Yukawas are couplings between the SO(5) quintuplet VLQs ψ and the singlets χ: these fields have U (1) X charge equal to 2/3 (−1/3) and are associated to the up-type (down-type) sector. Table I reports the fields in the spectrum together with their transformation properties under SO(5) × U (1) X . While in full generality one set of VLQs can be introduced per SM generation, in this paper only the third generation SM fermions and their VLQ siblings are considered, accordingly to the description of the original publication [31]. This does not represent a restriction in the present analysis, because the top-and bottom-partners are indeed the lightest VLQs and therefore they are the first exotic states that may show up in experiments, as discussed before. Moreover, their contributions are the only numerically relevant ones and for this reason the tree-level couplings of the first two generation quarks are taken to be purely SM like (deviations from the SM can however be induced at loop-level and will be discussed in Sec. III).
In the remainder of this section, the scalar and fermionic sectors will be discussed following Ref. [31], fixing the notation and the conventions that will be used in the rest of the paper.

A. The Scalar Sector
The part of the Lagrangian describing the interactions of the scalar fields and the symmetry breaking is where the covariant derivative containing the SU (2) L × U (1) Y gauge bosons is defined by where Σ a L and Σ a R denote the generators of SU (2) L × SU (2) R , isomorphic of SO(4) , subgroup of SO(5), and rotated with respect to the SO(4) residual group after the spontaneous breaking of SO (5). The scalar potential V (φ) contains terms responsible for the spontaneous breaking of SO(5) down to SO(4), plus additional terms that break explicitly the residual SO(4), where f is the scale at which the SO(5) breaking takes place. The last two terms are just a subset of all the soft breaking terms, but are the ones necessary to absorbe divergencies once the one-loop Coleman-Weinberg contributions are considered (see Ref. [51] for a different treatment). For values λ 1, the non-linear model would be recovered, corresponding to a decoupling of the σ field.
The SO(4) and EW breaking then require h and s to develop a vacuum expectation value (VEV) where the normalisation has been chosen to match Eq. (4), note in particular that v h = 246 GeV. For α = 0 = β, the VEVs turn out to be undergoing the condition Requiring the SO(5) breaking (f 2 > 0) and that the Higgs arises as a GB (|v h | < |v s |) imply that which in the strongly interacting limit, α, β λ, reduces to Then, in order to get v 2 h f 2 from Eq. (9), α/2β ∼ 1 is needed.
Once the scalar fields develop their VEVs, a non diagonal 2 × 2 mass matrix results from Eq. (7). Diagonalising this mass matrix, the mass eigenstates h, σ turn out to be where the mixing angle is given by and the masses of these two states, in the limit of α, β λ and positive β, read As a concluding remark, the possibility m σ < m h is viable but extremely fine-tuned and therefore will not be considered. As a consequence, the angle γ will always be taken in the interval [−π/4, π/4].
In the interaction basis, only the SM GBs and h couple to the EW gauge bosons. Due to the rotation in Eq. (13), σ inherits couplings with W ± and Z weighted by sin γ, while the h couplings acquire a suppression with cos γ, The modification in the h couplings with respect to SM predictions constrain the possible values of γ, as it will be seen in Section III.

B. The Fermionic Sector
We now turn to discuss the fermionic sector of the MLσM. In the basis of canonical kinetic terms for the gauge fields, the renormalisable fermionic Lagrangian can be written as follows [31] where the U (1) X charge of the exotic fermion fields ψ and χ has been made explicit throughout. The first line contains the canonical kinetic terms for the elementary quarks, q L for the left-handed (LH) SU (2) L -doublet, t R and b R for the RH SU (2) L singlets. The second line describes the kinetic and mass terms for the exotic quarks. The proto-Yukawa interactions among the exotic fermions and the scalar quintuplet are written in the third line. Finally the last two lines show the SO(5) breaking interactions between the exotic and the elementary quarks: the terms ∆ 2×5 and ∆ 5×1 denote spurion fields [63][64][65][66][67] that connect the exotic and the elementary sector and are the responsible of the light fermion masses. According to the fermion partial compositeness paradigm, no direct elementary fermion couplings to φ are allowed. All the parameters in the previous Lagrangian are taken to be real: this is equivalent to assuming CP conservation in these interactions; as only the third generation of light fermions are considered and therefore the CKM matrix cannot be described, this hypothesis is viable. In terms of the SU (2) L quantum numbers, the scalar quintuplet φ and the VLQs can be decomposed as follows where H is the SM SU (2) L doublet, with H ≡ iσ 2 H * , and K ( ) and Q ( ) are SU (2) L doublets and T 1,5 and B 1,5 are singlets. The rest of the charge assignments can be seen in Table II, where the hypercharge follows from the relation with X the U (1) X charge and Σ R the third component of the global SU (2) R , which is part of the residual SO(4) group after the breaking of SO (5).
In terms of the SU (2) L components, the fermionic Lagrangian acquires the following form where the scalar fields H and s still denote the unshifted and unrotated fields defined in Eq. (4). In order to match with the notation adopted in the previous section, notice that in the unitary gauge Light fermion masses arise via a series of interactions a la Seesaw and once the Higgs doublets develop a VEV. The diagram in Fig. 1 exemplifies the top-quark case. A first-order approximation for the values of the top and bottom quark masses is given by A more precise result can be obtained by diagonalising the whole fermion mass matrix that includes elementary and exotic quarks. It is useful to group all the fermions within a single vector, where the ordering of the components is based on their electric charges, +5/3, +2/3, −1/3, and −4/3, respectively. Moreover, T and B list together all the states with same electric charge, +2/3 and −1/3, respectively, The whole fermion mass term, still in the interaction basis, can then be written as where the mass matrix M(v h , v s ) is a 14 × 14 block diagonal matrix, with and similarly for M B (v h , v s ), replacing the unprimed parameters with the primed ones and vice-versa. The matrix in Eq. (26) can be diagonalised through a bi-unitary transformation, where Ψ L,R stand for the mass eigenstates and M for the diagonal matrix. The two unitary matrices can be written as block-diagonal structures where U T L,R and U B L,R diagonalise M T (v h , v s ) and M B (v h , v s ), respectively. Finally, the diagonalised mass matrix is given by and the mass eigenstate fermion fields are defined as with both charge 2/3 and charge −1/3 mass eigenstates ordered by increasing masses, so that the lightest states correspond to the top and bottom quarks, respectively. The exotic charge states do not mix with the other fields and then K u ≡ K u and K d ≡ K d . The full expression for the tree level top quark mass is given by and similarly for the bottom quark mass, replacing the unprimed parameters with the primed ones. This expression matches the one in Eq. (22) once the contributions proportional to y 2 and Λ 2 are neglected.

Interactions with the EW gauge bosons
A large part of the phenomenological discussion in the next sections will follow from the fermion interactions with the EW gauge bosons, which are modified with respect to the SM case due to the mixing between the elementary and the exotic quarks. This is the case for the Zbb-coupling, the electroweak precision observables (EWPOs) and the single production of the VLQs and their decays.
Adopting a compact notation, the Lagrangian terms describing these interactions with the W ± and Z gauge bosons read where Ψ are the mass eigenstates in Eq. (31), while the matrices V and C are obtained by means of the unitary matrices U L,R in Eq. (29) and the matrices containing the fermion interactions in the interaction basis. Notice that V L,R are block off-diagonal and C L,R block diagonal because of charge conservation. P R,L = (1±γ 5 )/2 are the usual chirality projectors, g is the SU (2) L gauge coupling and c W ≡ cos θ W , with θ W the weak mixing angle. For L W , the coupling matrices in this basis read being V L,R the W -coupling matrices in the interaction basis, whose form can be deduced from Table II, where and V T B R is identical to V T B L , except for its (1, 1) entry that is vanishing, corresponding to the fact that right-handed weak eigenstates are SU (2) L singlets. For L Z , the coupling matrices for mass eigenstates are given by with s W ≡ sin θ W , Q the electromagnetic charge matrix and C L,R the isospin-dependent coupling matrices in the interaction basis The normalisation adopted is such that the diagonal entries equal two times the isospin of the weak eigenstates.

Interactions with the scalars
The Lagrangian terms involving the fermion couplings to the scalar fields is where Y h and Y σ are the respective Yukawa coupling matrices of the scalar mass eigenstates, given by with with c γ and s γ being the cosine and sine of the mixing angle γ, respectively.

III. PARAMETER SPACE CONSTRAINTS
As described in the previous section, the presence of the additional scalar singlet σ and of the exotic fermions modifies the interactions of the light fermions with respect to the SM. This section is dedicated to summarise the theoretical and experimental constraints on the parameter space of the MLσM. A few theoretical constraints in the parameter space have already been listed in Section II A. First of all, f 2 > 0 in order to guarantee that SO(5) → SO(4) breaking takes place. Moreover, α and β should satisfy the condition in Eq. (11). Finally, by the diagonalisation of the scalar mass matrix, β > 0 allows a positive mass for the physical h as shown in Eq. (15).

A. Bounds from LHC Higgs searches
The measurements of the 125 GeV Higgs signal strengths at the LHC by ATLAS and CMS constrain the Higgs-singlet mixing sin γ, which universally suppresses the couplings of h to SM particles with respect to their SM values. Very recently, ATLAS has performed a √ s = 13 TeV analysis of Higgs signal strengths with 80 fb −1 of integrated luminosity [68], from which we derive the bound sin 2 γ 0.11 (44) at 95% C.L. using a χ 2 fit to the ATLAS data by assuming a universal suppression of Higgs couplings. This bound is shown in Fig. 2 as an excluded shaded red region in the (m σ , sin 2 γ) plane. In addition, Fig. 2 highlights the impact of theoretical constraints on the MLσM parameter space: the area under the yellow curve is ruled out when requiring a feasible spontaneous symmetry breaking of the MLσM SO(5) group. Note that realisations where v 2 h /f 2 ≡ ξ = 1 (with ξ the non-linearity parameter typically introduced in CH models), depicted by the orange line, and even regions where v 2 h > f 2 are not excluded by experimental bounds. Finally, the bound from Eq. (44) is consistent with existing bounds on ξ in the non-linear limit m σ m h , for which ξ sin 2 γ [31]. For completeness, Fig. 2 shows other lines where the SO(5) preserving parameter λ takes particular values and where it is equal to the remaining parameters of the scalar potential of Eq. (7).
Additional constraints on the MLσM parameter space arise from ATLAS and CMS searches for heavy scalars decaying into SM gauge boson pairs, W W and ZZ. In order to derive the corresponding constraints from these searches, it is useful to introduce an effective Higgssinglet mixing angle sin 2 γ eff as the ratio between the cross sections for gluon fusion production of σ in the MLσM (taking into account the VLQ loop contributions to gg → σ, whose expressions can be found in Ref. [31]) and for gluon fusion production of a SM-like scalar h mσ , with mass m σ In the absence of VLQ loop contributions, one simply has sin 2 γ eff = sin 2 γ. Considering the latest √ s = 13 TeV ATLAS search for scalar resonances in di-boson final states with 36 fb −1 of integrated luminosity [69], Fig. 2 shows the the 95% C.L. limits in sin 2 γ eff as an excluded blue area, in the absence of VLQ loop contributions. These latter contributions of VLQs to gg → σ are instead included when performing the MLσM viable parameter space scan in Sec. IV. For the case y 2 = y 2 = 0, these corrections turn out to be negligible, and sin 2 γ eff sin 2 γ. On the other hand, for y 2 ∈ [3.0, 6.0] (see below) the VLQ loop contributions are important and may interfere constructively with the SM top quark contribution, significantly enhancing the gg → σ production cross section. Numerically the excluded region on the (m σ , sin 2 γ eff ) plane corresponds to the blue region in Fig. 2, replacing γ by γ eff , up to small difference due to the variation of the branching ratios of σ → tt.

B. Bounds From Zbb Coupling
The couplings of the Z boson to the c and b quarks have been precisely measured at LEP. The tree-level mixing of the bottom quark with the VLQs induces deviations from the SM prediction of these couplings (at tree-level for the bottom quark and at loop-level for the charm quark). The effective Zbb vertex can be written as [54] The coefficients can be written as the sum of the SM prediction and its deviation, where the SM values are given by Comparing Eqs. (33) and (46), the deviations δc L,R read where k L = 1 and k R = 0. Effects of deviations from the SM values can be seen in several observables, such as the ratios of partial widths R b ≡ Γ(bb)/Γ(hadrons) and R c ≡ Γ(cc)/Γ(hadrons), the forward-backward (FB) charge asymmetry A b FB , and the coupling parameter A b from the LR FB asymmetry [54,70]: The SM values read while the experimental results that will be used to constrain the parameter space are [71] R exp b = 0.21629 ± 0.00066 , and, with same ordering as in Eq. (52), the correlation matrix is given by

C. Bounds From EWPOs
The EWPOs also set constraints on the parameter space. In our analysis we restrict ourselves to the T and S oblique parameters, which are the most relevant ones.
Due to the mixing in the scalar sector, the physical σ acquires couplings with the EW gauge bosons, weighted by sin γ, and therefore it contributes to the oblique parameters. On the other side, the h couplings get suppressed by cos γ with respect to the SM and this also affects the contributions to T and S. All in all, the T and S contributions can be straightforwardly derived from the SM one: so that The general expression for the T, S parameters considering contributions from VLQs with arbitrary couplings can be found in Refs. [72,73], and are given in appendix A for completeness. In the SM scenario, that is considering only the SM top and bottom quarks with their EW couplings, the contribution to the T parameter is It is then possible to define the deviation from the SM contribution as the difference where T f MLσM corresponds to Eq. (A5) considering the whole fermionic content of the MLσM. The SM contribution to the S parameter can be obtained considering only the top and bottom contributions in Eq. (A7), and it turns out to be The deviation from the SM can be expressed in terms of the difference, where S f MLσM is obtained from Eq. (A7) considering the whole fermionic content of the MLσM.
The sum of scalar and fermionic contributions to both T and S must agree with experimental data, ∆T ≡ T − T SM = 0.06 ± 0.06 ∆S ≡ S − S SM = 0.02 ± 0.07 (60) with a correlation of 0.92.

IV. MLσM PARAMETER SPACE SCAN
Here we discuss the details of our parameter scan of the MLσM. For the mass and mixing of the new scalar, two benchmark points are chosen within the allowed region in Fig. 2, In the fermionic sector, two different choices of parameters are considered: 1. Both y 2 and y 2 are set to zero, since it is sufficient to have y 1 and y 1 different from zero to reproduce the measured values of top and bottom quark masses at tree level. With y 2 = y 2 = 0, y 1 and y 1 can be determined from other parameters and the quark masses in Eq. (32) which, up to second order, take the form and similarly for the bottom sector, replacing the unprimed parameters with the primed ones and vice-versa.
2. A Yukawa coupling y 2 different from zero and relatively large (y 2 ∈ [3.0, 6.0]) is considered. The motivation for this choice is that, with this range of y 2 , the decay T → tσ has quite sizeable branching ratio as can be seen below.
For the remaining MLσM parameters a numerical scan is performed. After numerically diagonalising the mass matrices M T and M B , the constraints discussed in Secs. III A-III C are enforced, such that all the model predictions for these observables agree with the experimental measurements at 95% C.L. In addition to the above, the top quark mass resulting from the numerical diagonalisation is required to be in the interval [172,174] GeV, and the bottom quark mass in the interval [4.4, 4.8] GeV. At the same time, all Yukawa couplings are required to be far from the non-perturbativity limit (y 1 , y 1 < 6). The combination of all constraints can be satisfied for M i , M i , Λ i around a few TeV and Λ i around a few hundred GeV. In the next section, the results of this scan are presented with a focus on the salient phenomenology of the model. In particular, we explore the relevant phenomenological features of the lightest exotic quarks, which generally dominate the phenomenology of CH scenarios as discussed previously in this work.

V. LIGHTEST VLQ PHENOMENOLOGY
As it is well-known, VLQs can be produced at colliders like the LHC either in pairs or via single production. Examples of Feynman diagrams representing these two possibilities are shown in Fig. 3. In this work, the focus is on scenarios where the lightest VLQ has electric charge 2/3 or −1/3, since it is in this case where the phenomenological features of the MLσM may be qualitatively different to those of minimal CH models. Once produced, such lightest VLQ state may decay into a third generation SM quark and a scalar or gauge boson [74].
The Lagrangian terms involved in the production and decay of the lightest exotic quarks of charge 2/3 (T for the top-partner) and −1/3 (B for the bottom-partner) read L σQq = t y σ,L tT P L + y σ,R tT P R T + + b y σ,L bB P L + y σ,R bB P R B σ + h.c. , where λ a are the Gell-Mann matrices, a is a colour index, G a µ are the gluon fields and g s is the SU (3) c gauge cou-pling. In terms of the V matrices introduced in Eq. (34), being U T ,B L,R the rotation matrices given at Eq. (29) and V T B L,R the matrices defined at Eqs. (36) and (37). In terms of the C matrices given at Eq. (38), where C T ,B L,R are defined at Eqs. (40). Finally, the Yukawa couplings of Eq. (63) are given by with s being either h or σ, Y s given in Eq. (42) and Y T ,B s in Eq. (43). In the following we consider separately the scenarios where the lightest VLQ has charge −1/3 (B) or charge 2/3 (T ).

A. Bottom partner (B) phenomenology
For scenarios with the B state as lightest VLQ the most salient phenomenological feature is the large branching ratio BR(B → σb) that is possible in wide regions of the parameter space. This non-standard VLQ decay may even be the dominant one. In Fig. 4, the results of our MLσM scan for m σ = 700 GeV and y 2 = y 2 = 0 are shown for BR(B → σb) as a function of the bottom partner mass m B (the varying density of points has no physical meaning but is rather an artifice of the parameterisation used in the scan), showing that values BR(B → σb) > 0.5 may easily be obtained. The corresponding results for m σ = 1 TeV are found to be quantitatively very similar. A specific example of model parameters that result in the lightest VLQ being a bottom partner state B predominantly decaying into σb is given in the first line of Tab. III.
The dominant contribution to the single B production cross section is given by the processes 2 pp → Bbq, pp → 2 Other single B production channels like pp → Btq and pp →Btq, Bbq (through the Feynman diagram in Fig. 3 (right), with V being the Z boson), which are controlled by the strength of the BbZ flavour-changing neutral coupling defined in terms of the couplings from Eq. (63) as Fig. 5 shows the size of X bB as a function of m B for our   MLσM scan parameter points with m σ = 700 GeV and y 2 = y 2 = 0. The values for X bB barely exceed 0.02 as seen in Fig. 5, resulting in single production cross sections for B much smaller than the pair production cross section σ(pp → BB) at √ s = 13 TeV, except for very large mass values m B 3.5 TeV. This can be seen explicitly in Fig. 6, which shows the B pair production cross section (independent of X bB ) and the single B production for X bB = 0.01, both obtained with Madgraph aMC@NLO [75], using a Feynrules [76] implementation of the model in Eq. (63).

B. Top partner (T ) phenomenology
For scenarios with the T state as lightest VLQ, the branching ratio for the non-standard decay T → σt is generally small for y 2 = y 2 = 0 but can be quite large for sizable y 2 , as shown in Fig. 7 respectively for m σ = 700 GeV (upper panel) and m σ = 1 TeV (lower panel). The increase in m σ generically leads however to a decrease in the maximum possible size for BR(T → σt), as seen by comparing the upper and lower panels of Fig. 7. The second and third lines of Tab. III respectively give an example of model parameters that result in the lightest VLQ being a T quark with a large branching ratio to σt for m σ = 700 GeV and m σ = 1 TeV.
The mixing of the T state with the SM top and bottom quarks can be sizeable in these scenarios, leading to large values of the single T production cross section, but this is anti-correlated with the presence of a large BR(T → σt): for large mixing the branching ratio to σt is smaller, and vice versa. Fig. 8 shows the allowed values for the coupling V T b , given in terms of the couplings from Eq. (63) as as a function of the T mass m T . The different colours of the scan points correspond to different ranges of BR(T → tσ). As Fig. 8 highlights, sizeable mixings V T b 0.1 are possible, but only for a small branching ratio for T → tσ.
Notice that there appears to be a minimum value of V T b as a function of m T : V T b can indeed be below such apparent minima in Fig. 8, but in such case a charge −1/3 B state is lighter than T . The coupling V T b controls the dominant contributions to single T production via the processes pp → Tbq, pp →T bq through the Feynman diagram in Fig. 3 (right), with V being the W boson. Figure 9 shows together the T pair production (independent of V T b ) and the single T production cross sections for two benchmark values V T b = 0.15, 0.05, computed with Madgraph aMC@NLO [75]. It becomes clear that for O(0.1) mixings V L T b , the single T production cross section becomes more important than the pair production one above m T 1 TeV.
Finally, let us comment that a general feature of models with more than just one VLQ state or multiplet is that the mixing of charge-2/3 exotic quarks T with the third generation SM quarks can be larger than in minimal models [54]. As shown, this also happens in the MLσM due to the interplay of different contributions to EWPO from the several new fermions and the new scalar. From the experimental point of view, the possibility of large mixing is very interesting, as it leads to much larger single T production cross sections, which can in fact dominate the overall production of VLQ states above m Q 1 TeV. not very boosted (as a result of m σ and m T /B not being very far apart, which is something to be expected in the MLσM), its decay products are well separated and the signals resulting from single or pair production of VLQs followed by the exotic decays are similar to the ones already searched for at the LHC (i.e. the standard decays), but with higher SM gauge/Higgs boson multiplicities. It is then pertinent to ask ourselves to which extent the existing ATLAS and CMS searches are sensitive to the B → σb and T → σt decay signatures. The answer de- . This analysis primarily targets the process with two same-sign or three leptons produced from the decays of the W bosons. The events are required to have a large amount of missing transverse energy E T , as well as large transverse energy H T (defined as the scalar sum of all transverse momenta of jets and leptons). This search is also sensitive to the processes pp → TT → σt σt and pp → BB → σb σb, with σ → W + W − . In the former case, the final state contains two extra W bosons compared to Eq. (69), and hence the combinatorial factor to get two same-sign or three leptons is larger. The latter process yields the same final state as Eq. (69). For illustration, Fig. 10 presents the normalised kinematical distributions of the missing transverse energy E T , transverse energy H T and leading lepton transverse momentum (p T ), for the process in Eq. (69) with a standard VLQ decay mode (B → W t) and for the process pp → BB → σb σb with σ → W + W − , respectively for m σ = 700 GeV and m σ = 1 TeV, as obtained at parton level using Madgraph aMC@NLO [75] and MadAnalysis 5 [77]. The similarity of the kinematical distributions make it apparent that, despite being designed for a specific decay mode, the ATLAS search [22] is flexible enough to provide limits on other non-standard decay modes of VLQ. In addition, we note that this search is also sensitive to single T production followed by the exotic decay T → σt (σ → W + W − ): since two same-sign or three leptons can be produced from the decay of a single T state. Therefore, the same inclusive search can be used to probe both the mass and the coupling of new T quarks with exotic decays.
2. ATLAS search for opposite-sign dileptons or three leptons plus b-tagged jets [21]. This analysis focuses on the decays of vector-like quarks that involve a Z boson, with X being additional particles, that is, another heavy VLQ in pair production or light SM quarks in single production. The event selection requires one same-flavour opposite-sign lepton pair with an invariant mass consistent with M Z , as well as b-tagged jets and large H T (now defined with jets only). The analysis defines several signal regions according to the number of large-radius jets present with large mass. This search is sensitive to single and pair production of T or B quarks followed by the exotic decays in Eq. (2) and σ → ZZ, with the advantage of the combinatorial factor stemming from the two Z bosons that can decay leptonically.
3. ATLAS search for multi-jets with several b-tagged jets together with one charged lepton or large E T [18]. This analysis focuses on pair production that is, when either of the heavy quarks decays into H (−) t and the other one decays via any of the three standard VLQ decay modes in Eq. (1). Besides identifying top and Higgs candidates from the reconstructed objects, a large effective mass m eff = H T + E T is required. This event selection is rather sensitive to pair production of T states followed by the decay T → σt with σ → hh, where there are two Higgs bosons instead of only one as in the standard VLQ decay T → ht.
With these examples one can easily understand that event selections that require high H T or m eff (characteristic of the production of heavy particles) and leptons or b quarks are sensitive to -but not optimised for -the new VLQ decay modes discussed in this paper. The results of these searches can be reinterpreted by relaxing the conditions and introducing the additional decays in Eq. (2). On the other hand, there are searches that attempt to reconstruct the heavy quark masses by assuming a particular decay chain and kinematics [15,20]. Their sensitivity to the new modes in which either there are additional particles (e.g. in T decays) or the kinematical configuration is different (in B decays) is degraded.

VI. CONCLUSIONS
Extensions of the SM that include new VLQs usually include extra new particles that may mediate new production or decay mechanisms for the VLQs. In this paper, the focus is the study of the phenomenology of the Minimal Linear σ Model, that is a renormalisable extension of the SM based on the global spontaneous symmetry breaking SO(5) → SO(4) that improves on several limitations of traditional Composite Higgs models. The model includes several VLQs in the spectrum, as well as a new scalar σ that can mediate novel decay channels for the VLQs.
A scan over the allowed parameter space shows that the new decay modes T → σt, B → σb can have large branching ratios, and even dominate over the 'standard' decays in Eqs. (1). Moreover, the single production of T quarks can be sizable and dominate over pair production, as a consequence of the interplay among the new VLQ states and the third generation SM quarks. This is a salient feature of realistic models (as opposed to simplified VLQ models), not generally possible with just one VLQ state or multiplet.
The pair and single productions of VLQs lead final states with multiple bosons, e.g.
These new decay modes can in principle be captured by the current searches at the LHC. In this sense, the observability of such new decay modes requires that VLQ searches must be rather generic, i.e. requiring the presence of multiple b quarks, W/Z/h bosons, high transverse energy, etc, and a few examples of ATLAS and CMS searches, which are sensitive to the new decay modes, have been discussed in this work, and can be straightforwardly interpreted to set limits on those decays.
To conclude, the possibility of new VLQ decays is open and demands more generic experimental analyses, with a wide sensitivity beyond the 'standard' decays, and providing limits on the new modes. Dedicated searches for these new modes would benefit from the conspicuous multi-boson signatures produced in these decays, and would also be welcome. The explicit expression for ∆T (h and σ) is given by 3 On the other side, for the computation of ∆S (h and σ) , the explicit expression of S h SM can be found in Refs. [78,79] 4 : where x ≡ m 2 /m 2 Z , and with the F (x) and F (x) functions given by for x < 4, whereas for x > 4 5 The generalized contribution to the T parameter from VLQs with arbitrary couplings was presented at Ref. [72]: where N c is the number of colours, x i ≡ m 2 i /m 2 Z with m i the VLQ masses, and V L,R and C L,R are the matrices given in Eqs. (34) and (38). The θ ± functions are defined as where ∆ is a divergent quantity arising in the dimensional regularisation. This contribution disappears in the sum of all the contributions. Similarly for the S parameter, the generic expression can be found in Ref. [73] and reads where here X L,R are the isospin-dependent coupling matrices in the mass basis (that is, the U L,R C L,R U † L,R matrices present at Eq. (38)), the Y L,R are the hypercharge matrices of the fermions and the χ ± functions are defined as being µ an arbitrary scale that disappears as long as Tr X † L Y L + X † R Y R = 0. In the MLσM, the hypercharge matrices are defined by The explicit computation of the decay rates leads to the following results: for the process involving a scalar, Γ ψj →ψis = 1 32π instead, for a process with a gauge boson, with k V = 1 for V = W ± and k V = 1/ 2c 2 W for V = Z and the couplings given again at Tab. IV.