Modelling vector-like quarks in partial compositeness framework

Composite Higgs models, together with partial compositeness, predict the existence of new scalars and vector-like quarks (partners) at and above the TeV scale. Generically, the presence of these additional scalars opens up new decay topologies for the partners. In this paper we show how to systematically construct the general low energy Lagrangian to capture this feature. We emphasize the specific pattern in the top-partner spectrum arising in this class of models. We then present a concrete realization in the context of the SU(5)/SO(5) coset. We show that the top-partners in this model can have significant branching ratios to the additional scalars and a third generation quark, compared to the usual Standard Model channels. Amongst the most promising signatures at the LHC are final states containing a diphoton resonance along with a top quark.


Introduction
The composite Higgs model with partial compositeness (PC) is amongst the few mechanisms available for a symmetry-based explanation of the Electroweak (EW) hierarchy problem in the Standard Model (SM). It realizes the Higgs boson as a pseudo-Nambu-Goldstone boson (pNGB) arising from the breaking of a global symmetry [1] and uses linear couplings between some SM fermions and composite fermions to generate their masses [2] and to give the Higgs boson a vacuum expectation value (vev) [3][4][5].
We first present the structure of the IR lagrangian describing the interactions of the third family of quarks and the SM gauge bosons with the fermionic partners and pNGBs of the strongly coupled sector. This is needed since the spectra and the couplings of these theories are highly non-generic, and thus simplified models may miss interesting signatures for searches or falsely point to some that cannot be realized in the full theory. Also, tools are being developed to automatize the simulation of general models of PC, with future hadron colliders in mind. Extending the field content without guidance from symmetry principles leads to many undetermined parameters. However, once the symmetries of the underlying theory are employed, the number of these parameters is greatly reduced.
We use the PC paradigm to construct the low energy Lagrangian of the composite sector and present it for the relevant cosets. We further chalk out the steps to extract the interactions of the elementary fields with the composite ones, and provide details in the appendix. The mass spectrum of the fermionic partners in the PC framework has some specific patterns, independent of the choice of coset. In particular, a group of nearly degenerate fermionic bound states is expected in this class of models.
We then consider a specific realization of the real (SU(5)/SO(5)) case and survey its experimental signatures. The choice of focusing on this coset is partly motivated by the recent interest in searching for exotic signatures of top-partner decays t → t (S → γ γ) in the notation of [55], (see also [56]), which are more easily realized in this coset. It follows that the branching ratios to the SM final states such as th, tZ and bW are reduced compared to those into pNGBs and third generation quarks. In particular, we analyze a specific scenario where the pair production of top-partners, with one of them decaying into a tγγ final state, has a cross section of the order of a few femtobarn.
Our focus is on the production and decay of composite fermions and EW pNGBs, but it should be noticed that the bound states arising from underlying theories of this kind include additional types of composite particles. Some of these additional particles have been studied elsewhere: colored pNGBs [57,58], fermions in non-triplet irreps of color [59], vector resonances [60][61][62][63][64], and axion-like particles [65,66]. One can also envisage a variety of additional decay channels of the fermionic partners as discussed in [67][68][69], see also [70][71][72][73].
The paper is structured as follows. Sections 2 and 3 together with appendix A present the general construction of the models while section 4 together with appendices B and C deals with the specific SU(5)/SO(5) coset and its diphoton signal. We offer our conclusions in section 5.

IR Lagrangian
The purpose of this section is to present the various components of the IR Lagrangian describing the interactions of the composite sector with the SM vector bosons and the quarks of the third generation. At this stage we keep the presentation general, including all the minimal cosets arising in these underlying models, while the main interest of the later phenomenological section (section 4 and appendix B) is in the SU(5)/SO(5) coset.
We split the Lagrangian into several parts for ease of discussion The elementary Lagrangian L elem for the third generation quarks and vector bosons is identical to the SM Lagrangian In what follows we discuss each part of the composite Lagrangian L comp . More details are given in appendix A. In this work, for simplicity we assume CP invariance and all couplings to be real.
The explicit pNGB content are given in appendix A.1 for the three cosets under consideration. Throughout this paper we assume that the vacuum misalignment, leading to EW symmetry breaking (EWSB), is caused by a nonzero vev of the Higgs doublet alone, while other pNGB scalars do not receive any vevs. The true vacuum after EWSB can be obtained by exponentiating the Higgs vev and is denoted by a matrix Ω(θ) as in [10] where the angle θ parameterizes the misalignment of the vacuum.
For the real and pseudoreal cases one can also construct a matrix U ≡ Σ Σ T which transforms as U → gU g T . Here represents a symmetric (antisymmetric) H invariant tensor h h T = for the real (pseudoreal) case respectively. For the complex case it is more convenient to split the elements of SU(4) l ×SU(4) r as g = (g l , g r ), Our explicit formulas will be mostly based on the real/pseudoreal case with the modification for the complex case left understood. 1 While Σ is required to define the interactions of the vector-like quarks with the SM fermions through PC, the self interactions and gauge interactions of the pNGBs can be expressed more conveniently using U . At O(p 2 ), the kinetic term of the pNGBs can be JHEP03(2022)200 written using the matrix U as We introduce β in order to simultaneously canonically normalize the kinetic terms of the pNGBs and ensure that the masses of W and Z bosons can be written uniformly as Here c W ≡ cos θ W denotes the cosine of the weak mixing angle, s θ ≡ sin θ and the definition of electroweak scale is fixed as v = f s θ = 246 GeV. Note that the tree level relation ρ = 1 is preserved since we assume that the vevs of all other non-standard pNGB scalars are zero. The interactions of the weak gauge bosons with the 125 GeV Higgs boson can thus be written as (2.5) The hV V and hhV V couplings (V = W, Z) are modified with respect to the SM by a universal factor c θ and c 2θ respectively for any compact coset [76]. The Lagrangian eq. (2.3) also describes the interactions of one or two EW gauge bosons with pairs of additional (non-Higgs) pNGBs, generically denoted as π, via the covariant derivative [10]. Note that πV V terms are absent because of the absence of a vev for π.

The anomalous terms
In the absence of a vev for π, the interactions of a single pNGB with the gauge bosons are given by the hyperquark anomaly, described by a Wess-Zumino-Witten term (WZW). The WZW terms involving one pNGB and two gauge bosons can be written in a coset independent way in terms of differential forms as [77,78] where dim(ψ) is the dimension of the hypercolor irrep of the hyperfermion ψ giving rise to the EW coset, and A denotes the Lie algebra valued one form A ≡ (gW a µ T a L + g B µ T 3 R )dx µ . For the SU(4)/Sp(4) and SU(5)/SO(5) [10] cosets A = −A T while for the SU(4) l × SU(4) r /SU(4) d [15] case A = A. After integrating by parts and expanding to leading order in the pNGBs the anomaly Lagrangian L anom can be written in terms of the physical gauge fields as Table 1. Coefficients of the anomaly terms in eq. (2.7) uniformly normalized by a factor e 2 dim(ψ) 48π 2 f . Here η always denotes a SM singlet while the remaining coefficients are expressed in the custodial basis but otherwise agree with those in [10]. The pNGBs not appearing in the table do not couple to the anomaly terms.
Here π 0 i , π ± j and π ±± k represent any pNGBs for a generic coset with electric charge Q = 0, ±1 and ±2, respectively. In table 1 we list the coefficients of the L anom for different pNGBS in the custodial basis for the three minimal cosets (see appendix A.1 for notations). Note that for the SU(4) l × SU(4) r /SU(4) d coset only η couples to the anomaly term and for both SU(4)/Sp(4) as well as SU(4) l × SU(4) r /SU(4) d cosets K i γγ = 0, as already pointed out in [15]. Also, for the real case χ 0 3 does not appear in L anom .

The Lagrangian for vector-like quarks
Vector-like fermionic partners (Ψ) are built out of G HC -invariant trilinears involving two types of hyperfermions ψ and χ. The SU(3) c quantum number of the partner is carried by the χ-type hyperfermions, which are however not charged under G. On the other hand, ψ transforms as a fundamental (F ) of G. 2 The trilinear composite operators can thus be divided into two major categories of the type χψχ (one-index irrep of G) and ψχψ (two-index irrep of G) respectively. Below the G → H symmetry breaking scale the irreps of G should be decomposed under the unbroken global H. This implies that χψχ-type partners transform as the irreps in the decomposition of F of G on restriction to H, while the ψχψ-type partners belong to the irreps in the decomposition of F × F . Thus, in matrix notation, the top-partners transform

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as Ψ F → hΨ F (for fundamental of H), Ψ S,A → hΨ S,A h T (for symmetric or anti-symmetric of H) and Ψ D → hΨ D h † (for adjoint of H in the complex case).
Taking into account these informations from the UV, we consider a single irrep (up to two-index) of H for the fermionic partner as the relevant dynamic degree of freedom. For any coset, it is convenient to decompose Ψ in terms of SU(2) L × SU(2) R ⊂ H. Explicit matrices for Ψ in different irreps are given in the appendix A.2 for the three minimal cosets. The Lagrangian for the fermionic partners is then given by 3 where the covariant derivative is In the second term of the above equation v µ acts on Ψ in the appropriate representation. The third term in eq. (2.9) corresponds to the interactions along the additional factor of U(1) X that is the minimal additional gauge degree of freedom necessary to reproduce the correct hypercharge (given by Y = T 3 R + X) of the SM quarks. 4 The matrix-valued d µ and v µ symbols can be calculated using the CCWZ formalism [74,75] and are given by following expressions, whereT i (T a ) denotes the broken (unbroken) generators of G and V µ is given by The term proportional to κ in eq. (2.8) leads to the derivative interactions of the partners with the pNGB fields and belongs exclusively to the strong sector.

The partial compositeness Lagrangian
The PC mechanism relies on the linear mixing between the top quark and the top-partner which explicitly breaks the global symmetry of the strong sector. In order to parameterize this explicit breaking by coupling the SM third generation quarks to the top-partners and the pNGBs, we use spurionic embeddings of the SM fermions into the irreps of G.
For the real and pseudoreal cases we consider spurions which transform as N, F, A, S, and D of G (for the explanation of the notation, see section 2.3). For the complex case the spurions are classified according to the representations (ρ l , ρ r ) where ρ l,r = N, F . . . D, with the additional possibility of a Bifundamental B, but the idea behind the construction is the same. (See [10,45] for more details.) Table 2. Formally G invariant operators at leading order giving rise to PC of the SM quarks. The × symbol implies no possible invariant can be constructed while the zeros denote that the operators vanish identically due to symmetry properties of the spurions and the irreps of the fermionic partners. For the real (pseudoreal) case the expression tr A † Σ Σ T , (tr S † Σ Σ T ) vanishes, since is symmetric (antisymmetric).

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The transformation properties of the spurions under a global g ∈ G are given, in matrix notation, by (2.12) The left-handed quark doublet (q L ) should be embedded into a (2, 2) 2/3 of SU(2) L ×SU(2) R × U(1) X , while for the right-handed top (t R ) either a (1, 1) 2/3 , or the T 3 R = 0 component of a (1, 3) 2/3 can be used. The embedding of q L in (2, 2) ensures that the corrections to the Z → b LbL decay width is under control due to the custodial protection [79]. The explicit embedding matrices for the pseudoreal, real and complex cases are given in appendix A.3.
The structure of the formally G invariant PC Lagrangian is given as Hereq L andt R denote the embedding of the elementary quarks into the incomplete G multiplets above. For example, if both q L and t R are embedded in the adjoint irrep of G, we can writeq To construct the invariant operators O L,R , the partner Ψ is dressed with appropriate insertions of the pNGB matrix Σ. In table 2 we present the leading order invariant terms of eq. (2.13) for the real and pseudoreal case. The construction is the same in the real and pseudoreal case if one uses the appropriate invariant tensor (symmetric or anti-symmetric).
For the complex case one simply needs to keep track of the difference between the left and right factors of G. For instance, one can have the following two invariants with antisymmetric tensors: and similarly for the adjoint and symmetric. In the complex case one can also use the bifundamental B → g l Bg T r and similar for various combinations of fundamentals and anti-fundamentals e.g. B → g l B g † r .
Here the invariants are simply tr B † Σ l Ψ A Σ T r or tr B † Σ l Ψ S Σ T r in the first case and tr B † Σ l Ψ N Σ † r or tr B † Σ l Ψ D Σ † r in the second case.

The scalar potential
We have now come to the last term in eq. (2.1), namely the scalar potential. This term is the most model dependent and prone to computational difficulties since it is fully generated by the terms explicitly breaking the global symmetry H. The contributions of the symmetry breaking interactions to the potential can be split as where V m denotes the contribution from the bare hyperfermion mass term which can be written as The coefficients B m , B m are dimensionless low energy parameter encoding the strong dynamics. We have included a B m contribution, in the same spirit as [80][81][82]. Although f should be thought of as an inverse coupling and inserted according to the rules of naive dimensional analysis, we chose to use it as the only dimensionful quantity for convenience, hence the unusual f dependence. Similarly one-loop contributions from the gauge bosons V g are given by for the real and pseudoreal cases, while for the complex case both T a * L and T 3 * R should be replaced by −T a L and −T 3 R , respectively. The top quark contribution to the potential V t depends on the specific spurionic representations in which q L and t R are embedded. Generically, the lowest order invariants are formed using the spurions in the various representations and are given by terms like for the real and pseudoreal case, and similarly for the complex case. Note that the antisymmetric spurion does not contribute to the potential in the real case and the symmetric one does not contribute in the pseudoreal case. Also, none of the irreps (1, ρ) or (ρ, 1) contribute in the complex case.
Writing the full potential as in eq. (2.14) one proceeds first by selecting those spurions that guarantee the absence of tadpoles for the non-Higgs pNGBs. For these selected representations one then fixes two linear combinations of the low energy coefficients B by imposing the correct Higgs mass and vev. The remaining coefficients are then varied in the stability region of the potential to read off the spectrum of pNGBs. Detailed discussions about the scalar potential for the various cosets can be found in [10,11,15,37].

Spectrum of the fermionic partners
Having constructed the full Lagrangian in section 2, we can now study the generic properties of the fermionic spectrum arising in these models. Here we discuss the classical mass matrices arising from combining the Dirac mass in eq. (2.8) and the contribution of eq. (2.13) after EWSB. The only partners involved in the quadratic part of eq. (2.13) are those with the same quantum number as the top or bottom quarks; the other exotic partners are unaffected by PC and remain degenerate with tree level mass set by M in eq. (2.8). We thus focus our attention on these two sectors. The generic structure of their mass matrix is that of a n × n matrix In some cases, such as when both |ω t L (θ)| and |ω t R (θ)| ∼ O(θ), the top quark mass is O(θ 2 ) and must be discarded as it is too small. In the left panel of figure 1 we show the contours satisfying m t = 173 GeV in the y R − y L plane for different values of M/f . The solid (dashed) contours in the left panel of figure 1 represent models of Type I (II). Note that fairly large values for y L and y R are required to reproduce the correct top mass.
We now discuss the spectrum of the composite fermions with electric charge Q = 2/3. The generic spectrum at tree level is shown in the right panel of figure 1. The presence of the (n − 1) × (n − 1) diagonal block ensures that M 2/3 has n − 3 exactly degenerate states with mass M in its singular value decomposition. This can be seen by noticing that the ω t L,R can be brought by a field redefinition to have only at most the first two components non-zero. The masses of the remaining two top-partners (recall that n × n mass matrix is composed of top quark and n − 1 top-partners) are given by M 2 + y 2 L f 2 and M 2 + y 2 R f 2 , for models of Type I. In case of Type II models one of the top-partners has a mass slightly heavier than M : M 2 + y 2 L,R v 2 (note that this leads to a very small mass splitting ≈ y 2 L,R v 2 /2M ), while the other is heavier with a mass of M 2 + y 2 R,L f 2 . In the presence of a partner for the bottom quark leading to a non-zero ω b R (θ) the singular value decomposition of M −1/3 proceeds exactly as in the previous case. In the cases where there is no b R partner we assume that the mass of the bottom quark can be generated by some bilinear operator. Incorporating this assumption, we get a modified M −1/3 as

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where µ b denotes the contribution from the bilinear operator of the typeq L Ob R . The generic expression for the bottom quark mass to O(θ) can also be of two types as follows 3) now implies that there are n − 2 degenerate states with mass equal to M . The remaining state can have a mass M 2 + y 2 L f 2 for Type I, and M 2 + y 2 L v 2 for Type II models.

One loop self energy
In this subsection we discuss the one loop self energy of the top-partners and its effect on their masses and widths. Due to the presence of explicit breaking of G through gauge interactions and PC, we expect that the one loop corrections to the fermionic self energy JHEP03(2022)200 lift the degeneracy in the top-partner spectrum. The relevant self energy diagrams are shown in figure 2 where T i,j represent the degenerate top-partners while Ψ k denotes either a partner or a SM quark running in the loop. The contributions from the two diagrams to the self-energy are The γ 5 terms can be rotated away by a field redefinition and are ignored. The real part Re[Σ( / p)] is logarithmically divergent and contributes to the mass correction. We regularize this divergence using a UV cut-off around the compositeness scale Λ ∼ 4πf , where the non-perturbative dynamics from the strongly interacting hypercolor sector kicks in. We denote the corrected central value of the mass of the semi-degenerate multiplet by M T and the mass splitting by δM ij as where N T denotes the number of degenerate states (typically N T = 2 or 3 in these models). Just as the mass splitting, the total decay width (Γ T ) of these states can be expressed as a hermitian matrix with non-zero off-diagonal elements using the optical theorem as The Γ T is the sum of the matrix valued partial decay widths for the appropriate channels as given below π and Γ ± V are given as and, λ(x, y, z) ≡ x 4 + y 4 + z 4 − 2x 2 y 2 − 2y 2 z 2 − 2x 2 z 2 is the Källén function.

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Thus, the outcome of the analysis above is that in case of nearly degenerate states with |Γ T | ∼ |δM | M T we obtain a matrix propagator (3.11) When considering top-partners pair production and their subsequent decays one must take into account the interference between the channels involving the nearly degenerate states using the above matrix propagator, leading to an interesting computational challenge. We will show a way to handle this puzzle for the specific example discussed in the next section.

Phenomenology of top-partners: an explicit example
Having presented the general construction of these models in the previous sections we now move to describe a specific example of such models displaying various unusual phenomenological features. We choose to employ the SU(5)/SO(5) coset since it has a rich pNGB sector with anomalous couplings to dibosons leading to a wide variety of decay channels for the top-partners. This coset leads to 14 pNGBs whose decomposition under SU(2) L × SU(2) R ⊃ SU(2) cust is given by 6 (also see appendix A.1) After chosing the coset, the remaining discrete choices to be made are the irreps for the fermionic partner and the spurions of the third quark family. We choose the following irreps for the construction of our model (see appendix A.3 for details) The spurions are involved in the construction of both the scalar potential and the Yukawa couplings, while the choice of the partner only affects the latter. The relevant parts of the Lagrangian are given in appendix B. We choose Ψ A ∈ 10 for simplicity, since it leads to a more restricted number of partners while retaining the main interesting features. Chosing the symmetric irrep for Ψ would give rise to additional fermions with exotic charges 8/3 and −4/3. There are only a few possible choices for the spurions [11] satisfying the necessary requirements (such as no vevs for the triplets) and eq. (4.2) complies with them.
To set the notation we present the decomposition of the partners' irrep 10 2/3 under . 6 The superscripts denote the electric charges and G denotes the Goldstones eaten by the SM gauge bosons.  Table 3. Possible decay channels of the vector-like fermionic partners for the model discussed in this section, with the choice given in eq. (4.2). The interesting novel feature of this model is having top partners with much reduced branching ratios to the usual SM channels t h, t Z, and b W and instead a large branching ratio to beyond the SM (BSM) mediated channels such as t(η → γ γ) as we now proceed to discuss.
We consider only pair production of top-partners and their subsequent decays. All the possible two body decay channels for this model are listed in table 3. Recall that pair production of the vector-like quarks is model independent and is a function of their mass only. The pair production cross-section σ(pp → ΨΨ) for any color triplet is calculated at the NNLO+NNLL accuracy with Top++ [83] using the sets NNPDF4.0 parton densities and is shown in figure 3 (see [84] for corrections arising from the finite size of the partners).
We now need to consider the various decay modes. In this paper we are only concerned with the total cross section for each channel without any detailed study of the reach attainable at colliders. This is a first step needed to justify further studies and it already involves an interesting challenge, since we must deal with a nearly degenerate spectrum and want to take into account the interference between channels. This problem has been encountered and studied before in similar contexts [85][86][87], but here we present a different angle on it, showing how to treat off-diagonal contributions to masses and widths in the particular limit of interest for this paper. In what follows, we will be interested in the phenomenology of the lightest two toppartners T 2 3 and X 2 3 with tree level mass M . The benchmark parameters used for this study are displayed in the table 4. The pNGB masses are obtained by minimizing the potential given in eq. (2.14). The benchmark shown in the table 4 has mass eigenstates of the pNGBs approximately aligned to the custodial direction. Here m 3 , m 5 and m 1 denote the masses of the custodial triplet χ 3 , quintet χ 5 and the singlet χ 0 1 , while m η denotes the mass of the pure singlet η.

Decay of nearly degenerate top-partners
We consider the processes shown in the figure 4 pp → T T → AB, where T ≡ T 2 3 , X 2 3 collectively denotes the two nearly degenerate top-partners and A, B ≡ (bW + ), (tZ) . . . (bχ + 5 ) (B just being the charge conjugate of B). Thus A and B run over the possible SM and BSM decay modes shown in the first row of table 3. Since the pair production mode is universal we can factor out the production cross section σ(pp → T 2 figure 3. However, note that even at one-loop their mass splittings, computed using the eqs. (3.5) and (3.6), are smaller than their individual decay widths. Therefore we need to consider the interference between their identical decay channels and the fact that the propagator in eq. (3.11) is not diagonal.
We have been able to show that in the narrow width approximation (NWA), even in the presence of an off-diagonal matrix propagator one can factorize the cross-section as This is the only factorization allowed in eq. (4.4) if one wants to keep the interference between the channels. The details of the calculation will be reported in an upcoming paper, however, the main steps for calculating BR 2 (T T → AB) are given in appendix C. Notice that it is not possible to write BR 2 as a product of two branching ratios, as one would do if there was no degeneracy. It is however possible to define an effective branching ratio of T → A as

Decay of pNGBs
To extract the observable signal we need to combine σ(pp → T T → AB) with the branching ratio of the pNGBs into the appropriate SM channels. While still promptly decaying, the pNGBs have a very small width so that in this case we can employ the usual NWA and ignore any further interference. The specific combination of cross-sections and branching ratios that needs to be employed depend on the final states one wants to target and how inclusive one wants to be. As mentioned in the Introduction, we now focus on the diphoton channel and calculate the branching ratios of the neutral pseudoscalar pNGBs using the expressions given in [58]. In the specific scenario we are considering, among the BSM pNGBs only η couples to the tt pair with a strength (at O(θ)) proportional to while the other pseudoscalars χ 0 5 and χ 0 1 do not couple to tt at O(θ). As a result, η couples to a pair of gluons through a top quark loop which has a strength comparable to the WZW terms.
In figure 6 we present the branching ratios of η, χ 0 5 and χ 0 1 , respectively using the benchmark values of M , f , y L,R and κ given in table 4. Note that η decays dominantly to JHEP03(2022)200 tt pair if m η > 2m t , while for χ 0 5 and χ 0 1 the branching ratio to diphoton dominates over a large range of masses. For our benchmark masses, the three body decays χ 0 1 → χ 0 3 ff and χ 0 1 → χ ± 3 ff (where f, f denote leptons or light quarks) via off-shell W or Z are orders of magnitude smaller compared to the two body decays (BR(χ 0 1 → χ 0 3 ff ) = 3.3 × 10 −4 and BR(χ 0 1 → χ ± 3 ff ) = 1.2 × 10 −4 ). Having the branching ratios of the neutral pNGBs we can compute the cross sections for various signals of interest. Focusing on the diphoton, we could consider the inclusive process with a top quark and a diphoton resonance in the final state: where, for ease of notation we now omit all the intermediate states in the expression of the cross section. For the specific scenario discussed above and for our choice of benchmark parameters we find σ (pp → (tγγ) . . . ) ∼ 1.31 fb. The cross section given in eq. (4.8) is relevant for the leptonically decaying top quark where the distinction between a top and an anti-top is possible. However, for a hadronically

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decaying top, a more inclusive cross section as given below is of relevance: (4.9) We subtracted the processes with four photons in the final state to avoid double counting. For our specific model, the choice given in table 4 yields σ pp → (t/t γγ) . . . ∼ 2.43 fb.
Note that more exclusive processes can be obtained by restricting the sum overB in eq. (4.8). On the other hand, more inclusive quantities, such as the fully inclusive diphoton signal σ(pp → γγ . . . ) would require the cross section of the processes involving additional intermediate fermionic partners with nearly equal masses, as well as the branching ratios of the charged pNGBs, which is beyond the scope of this paper.

Conclusions
In this paper we constructed the low energy Lagrangian of a class of models with vector-like quarks and pNGB scalars, addressing the electroweak hierarchy problem within the partial compositeness framework. We presented the Lagrangian for three minimal cosets arising from strongly coupled gauge theories with fermionic matter in the UV. Our approach, based on symmetries motivated from specific UV realizations, greatly reduces the number of free parameters compared to the simplified models with the same field content.
In these models, the structure of the fermion mass matrix and spectrum follows a specific pattern, and in particular, we highlighted the presence of nearly degenerate fermionic partners. We further estimated the one loop mass splitting in the degenerate sector.
We then focused on a concrete example based on the SU(5)/SO(5) coset to investigate the possible signatures of top-partners at colliders. We considered a minimalistic choice of both the irreps of the fermionic partners as well as the spurionic embedding of the SM third family quarks. We calculated the cross sections for production of a pair of lightest and nearly degenerate top-partners followed by their decays into SM and pNGB final states. The pNGBs were allowed to decay either into dibosons or a pair of third family quarks. For the nearly degenerate states we incorporated the effects of the off-diagonal self energy and the full quantum interference in the resonant production cross section and branching ratios. We showed that for the specific model in question the lightest top-partners decay

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dominantly into a top quark and a pseudoscalar pNGB, which in turn can decay to diphoton, primarily through anomalous interactions. This leads to a promising channel to search for top-partners at the LHC with an inclusive cross section of the order of a few femtobarns.

Acknowledgments
This work is supported by the Knut and Alice Wallenberg foundation under the grant KAW 2017.0100 (SHIFT project). We would like to thank Luca Panizzi and Venugopal Ellajosyula for discussions and Thomas Flacke, Manuel Kunkel, and Leonard Schwarze for comments on the draft.

A Building blocks for the IR theory
In this section we provide more technical details on the basic building blocks needed to write the IR Lagrangian for a generic G/H coset. The explicit expression for the generators of SU(2) L × SU(2) R (T i L , T i R ), the pNGB matrix Π, and the vacuum matrices Ω(θ) are the same as in [10] and will not be repeated.

A.1 pNGBs
We present the decompositions of the pNGBs under the various subgroups of H, paying attention to the transformation to the custodial basis, to set the notation. This is the same field content of the model [37]. In our convention the Sp(4) invariant tensor is the symplectic matrix 0 ≡ iσ 3 ⊗ σ 2 so that U = Σ 0 Σ T . Because the additional pNGB η is a singlet of SU(2) L × SU(2) R nothing needs to be done to go to the custodial basis, other than, of course the usual decomposition already present in the SM (2, 2) → 1(h)+3(G ± , G 0 ), where G ±,0 are the would be Goldstone bosons eaten by the W ± and Z.

II. SU(5)/SO(5) coset. The 14 pNGBs in this coset transforming as 14 of SO(5) can be decomposed on restriction to
Therefore the pNGBs comprise of the usual Higgs doublet and a real singlet η. For this coset the symmetric invariant matrix is taken to be the identity matrix so that U = ΣΣ T .

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It is also relevant to show the pNGB content in the custodial basis, since this is a good approximate symmetry of the potental. Under SU(2) L × SU(2) R → SU(2) cust the pNGBs decompose as Since we assume that only the Higgs doublet receives a vev, the custodial triplet originating from the (2, 2) serves as the would be Goldsotne boson to be eaten up by the W ± and Z bosons, as before. The physical pNGB spectrum contains one custodial quintet , and three singlets (h, χ 0 1 , η), out of which the singlet h associated with the (2, 2) is identified with the 125 GeV Higgs boson. The relation between the triplets Φ 0 and Φ ± with the custodial basis (χ 0 1 , χ 3 , χ 5 ) is given by Note that the phase convention for the Clebsch-Gordan coefficients in (A.4) is sligthly different from the conventional one for consistency with the choice of generators. The convention used here is the same as the one in [11] apart from the sign of φ + 0 . The H is the same Higgs doublet as before, while H is an additional doublet. ∆ = (∆ ± , ∆ 0 ) is a triplet of SU(2) L and N = (N ± , N 0 ) a triplet of SU(2) R decomposed according to its hypercharge, which in this case coincides with the electric charge. ∆ and N are already custodial triplets.

A.2 Fermionic partners
The fermionic partners transform under an irrep of H. In table 5, we present the decomposition of the relevant irreps of SO(5) × U(1) X (same as those of Sp(4) × U(1) X ) and of Below we show the explicit notations for the pseudo real, real and the complex cosets, respectively.
where the submultiplets are given by The adjoint irrep is equivalent to the symmetric, but the explicit form in terms of the fields is Ψ D = Ψ S 0 since the transformations under a generic generator T a of Sp(4) are respectively T a Ψ S + Ψ S T aT and [T a , Ψ D ], where T a 0 = − 0 T aT .

III. SU(4) l × SU(4) r /SU(4) d coset.
The field content of the complex coset can be inferred directly from that of the pseudoreal one, since the generators of SU(2) L × SU(2) R are the same. Specifically, the antisymmetric of SU(4) d is the same as that of Sp(4) in (A.6) augmented by an additional Sp(4) singlet 1 2 T 2 3 0 , the symmetric is exactly the same, and the adjoint can be taken as Ψ A 0 + Ψ S 0 for two generic symmetric and antisymmetric matrices as in (A.6).

A.3 Spurions
Finally we list the explicit expressions for the spurion matrices realizing the embedding of the elementary fields t L , b L and t R . The ones relevant for section 4 are the adjoint spurions D 1 t L , D 1 b L , and D 2 t R of SU(5)/SO(5) coset as discussed below. Notice that all spurions must carry X = 2/3 since we need to form a Yukawa coupling with the fermionic partners by PC. This restricts the possible embeddings quite significantly. In particular, looking at the decomposition of the various irreps under SU(2) L × SU(2) R , the only allowed embedding for the LH doublet q L is in the (2, 2) 2 3 , while t R can be embedded into the singlet (1, 1)

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The antisymmetric matrices A 1,2 t R and the adjoint D 1 t R are in the (1, 1). (A 1 t R is a full Sp(4) singlet proportional to the invariant tensor 0 .) The symmetric matrix S t R and the adjoint D 2 t R are T 3 R = 0 components of (1, 3). The matrices for embedding t L and b L are and for the adjoint
With the same notation as in the previous case, we have, for t R 14) The matrices in the adjoint are given by As far as the spurions for the LH fields are concerned we use (A. 15) with the matrices in the adjoint given by

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III. SU(4) l × SU(4) r /SU(4) d coset. Lastly we consider the complex case. Here, in principle, we need to specify the irrep of both SU(4) l,r but, again, the explicit matrices are independent on these details and we can lump, e.g. (1, 10), (1,10), (10, 1), (10, 1) into a S of SU (4). Having reduced the problem to a single SU(4) the spurions N, A, S, and D in the complex case are given by the same numerical expressions as in the pseudoreal case, (A.12) and (A.13), and need not be repeated. In addition, we must also consider the Bifundamental matrices B. (4, 4), (4,4) are given by the same expression as the symmetric and the antisymmetric given in the pseudoreal case. Similarly, (4,4), (4,4) are the same as the adjoint and an additional singlet only viable for the RH top (B t R = 1 2 1 4 ).

B The Lagrangian of the model in section 4
The interaction Lagrangian for the model presented in section 4 involving pNGB scalars in the custodial basis and the fermions in the gauge basis can be written as where π 0 i ≡ h, χ 0 3 , χ 0 5 , χ 0 1 , η and π + j ≡ χ + 3 , χ + 5 and In contrast to the section 4, in this appendix we denote the column vector containing all fermions with Q = 2/3 as T . The interactions between the weak gauge bosons and the fermionic partners are similarly given by the follwoing Lagrangian.
Note that both the Lagrangians above are obtained from eqs. (2.2), (2.8) and (2.13) together with the choice given in eq. (4.2). We only keep terms with one pNGB (or gauge boson) and at least one fermion with Q = 2/3 in eqs. (B.1) and (B.3). The mass matrix M 2/3 is given by

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Similarly for B the mass matrix M −1/3 is given by while the biunitary rotation matrices, defined by B L,R →Ũ L,R B L,R , are given at the O(θ) as (B.7)

C Branching ratios involving degenerate states
As mentioned in the section 4, in the presence of N T nearly degenerate top-partners the cross section for the process pp → T T → AB shown in figure 4 can be factorized as where σ(pp → T T ) is the pair production cross section, which is the same for each partner, and BR 2 (T T → AB) denotes the joint branching ratio of T → A and T →B. The full cross section (C.1) arises from squaring the amplitude in figure 4 and contains a term tr ∆(p 2 1 ) † Γ A (p 2 1 )∆(p 2 1 )∆(p 2 2 )ΓB(p 2 2 )∆(p 2 2 ) † , where the trace is over the N T states and p 1,2 are the momenta of the two fermionic partners to be integrated over. The ∆s are defined in eq. (3.11) with a total matrix-valued width Γ T = A Γ A (M 2 T ) ≡ B ΓB(M 2 T ), where Γ A (M 2 T ) and ΓB(M 2 T ) are the on-shell matrix-valued partial decay widths.
If it wasn't for the matrix-valued nature of the propagators and widths, one could further simplify eq. (C.2) by commuting the ∆s and use the NWA to write the expression as the product of two ordinary branching ratios after integrating over p 2 1 and p 2 2 . Here instead we need to treat each of the two pieces under the trace, namely ∆(p 2 1 ) † Γ A (p 2 1 )∆(p 2 1 ) and ∆(p 2 2 )ΓB(p 2 2 )∆(p 2 2 ) † as matrix-valued distributions in p 2 1 and p 2 2 respectively. Let us consider the p 2 1 term for definiteness and define Z = Γ T + 2iδM . One can show that in the limit Z ∼ 0 where all components of Z become small compared to M T dp 2 1 ∆(p 2 1 ) † Γ A (p

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and the coefficients c ± n only depend on Z and can be extracted by tracing the expression (C.3) over four linearly independent matrices, e.g. 1 2 , σ a . 7 Applying the above result to our problem we can now write the joint branching ratio as (C.5) where the matrix operatorŌB ± m is defined similarly as Note that in (C.5) we extract a factor 1/N T to normalize BR 2 to one. Solving for c ± n for our benchmark parameters in section 4, we find c + 1 = 1.30, c + 2 = −0.296, c − 1 = 0.192, and c − 2 = −0.072.
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