Collider phenomenology of a unified leptoquark model

We demonstrate that in a recently proposed unified leptoquark model based on the gauge group $SU(4)_C\times SU(2)_L \times U(1)_R$ one can explain $R_{K^{(*)}}$ without the need of extra heavy fermions. Low energy data, in particular lepton flavour violating $\mu$ decays and $K_L\to e \mu$, severely constrain the available parameter space. We show that in the allowed part of the parameter space (i) some of the lepton-flavour-violating tau decay branching ratios are predicted to be close to their current experimental limits. (ii) The underlying scalar leptoquarks can be probed at the LHC via their dominant decay modes into tau-leptons and electrons and the third generation quarks. (iii) The constraints from meson oscillations imply that the masses of scalar gluons, another pair of coloured multiplets around, have to be bigger than around 15 TeV and, thus, they can be probed only at a future 100 TeV collider. In both neutral and charged variants, these scalars decay predominantly into third generation quarks, with up to $O$(10%) branching ratios into family-mixed final states. Moreover, we comment on the phenomenology of the scalar gluons in the current scenarios in case that the $B$-decay anomalies eventually disappear.

Assuming that these anomalies are not a result of experimental systematics, they can be accounted for by leptoquarks (LQs) of various kinds [9][10][11][12][13][14][15][16][17][18][19][20][21]. However, building viable ultraviolet (UV) complete models involving those particles is challenging, especially in light of very stringent constraints on lepton flavor violation (LFV) from various experimental searches, see e.g. [22,23]. Several attempts to build UV completions exist already in the literature [19,[24][25][26][27][28][29][30][31][32][33][34]. Most of them aim for getting the vector leptoquark U 1 , which has quantum numbers (3, 1, 2/3) under the SM gauge group G SM = SU (3) C × SU (2) L × U (1) Y , sufficiently light as it is an excellent candidate to explain the anomalies. It emerges naturally from the breaking of SU (4) C to SU (3) C which fixes the properties of U 1 up to effects from generation mixing of the fermions to which it couples. In these attempts the precise way in which the group is broken is ignored as well as the pattern of the couplings in the scalar sector and its mass spectrum.
In [35] we have presented a detailed analysis of a model based on SU (4) C × SU (2) L × U (1) R , proposed in [36,37], putting particular attention to the scalar sector. Keeping the minimal fermionic particle content minimal one cannot reduce the mass of U 1 well below 1000 TeV due to the constraints stemming from K L decays. Nevertheless, this setup in principle allows for an explanation of R K ( * ) and predicts R D ( * ) to be close to the SM expectation if the scalar leptoquarks are taken into account. In [35] we have used the SO (10) inspired assumption that all Yukawa couplings are symmetric in the flavor indices. As a consequence we found that one cannot explain R K without violating the experimental bounds from K L → eµ. This demonstrates that the scalar sector of such models must not be ignored.
In this paper we show that when releasing the symmetry conditions on the Yukawa matrices while still staying within the well-known minimal model, we can accommodate the R K ( * ) anomalies without violating any other experimental bound. The corresponding parameter space is quite restricted which implies that the properties of the additional scalars are fixed to a high degree. Consequently, this leads to rather specific predictions for LHC searches.
The paper is organized as follows: in Sec. II we summarize the main features of the model with a particular focus on aspects relevant for the B-physics anomalies. In Sec. III we discuss various constraints stemming from low energy data and their consequences for the properties of the new scalars. This is followed by discussion on the resulting collider phenomenology in Sec. IV. A brief summary is given in Sec. V.
For our investigation the SARAH package [38][39][40][41][42] needed to be extended considerably. We present this extension in the appendix. For our numerical calculations we used the generated model files to produce a spectrum generator based on SPheno [43,44]. For the calculation of cross section at hadron colliders we have used the SARAH-generated interface to MadGraph aMC@NLO [45,46].

II. MODEL ASPECTS
We will briefly summarize here the main features of the model that are important for the subsequent discussion. For further details we refer to refs. [35][36][37]. The Model is based on the gauge group G = SU (4) C ⊗ SU (2) L ⊗ U (1) R where the SM SU (3) C emerges as part of the SU (4) C breaking. In this class of models, the leptons (including the right-handed neutrino) are unified with the quarks in SU (4) C representations as summarized in Tab. I. The sub-eV neutrino masses and the observed leptonic mixing pattern are accommodated via an inverse seesaw mechanism [47] by adding extra 3 generations of a gauge-singlet fermion N to the original model of ref. [36] as proposed in [37]. Even though the SU (4) C breaking implies potentially also B and/or L breaking, it turns out that only the lepton number gets eventually broken while B remains a good symmetry to all orders in perturbation theory [35].

A. Symmetry breaking and scalar sector
The scalar sector consists of three irreducible representations of G as given in Tab. I. At the level of G SM , the colourless part of the scalar sector consists of a complex singlet χ 0 and two Higgs doublets H and H 2 . The gauge symmetry is broken by their vacuum expectation values (VEVs) in the two successive steps Fermions Scalars We parametrize the VEVs as where the square brackets denote the SU (2) L structure, v ew 246 GeV, and v χ ≈ 1000 TeV. The later is chosen in such a way that the vector leptoquark mass is consistent with the stringent bound set by the non-observation of K L → eµ. 1 As in the usual two-Higgs-doublet models (2HDM) it is convenient to rotate the SU (2) doublets via which takes one to the basis where h accommodates the entire electroweak VEV and contains also the would-be Nambu-Goldstone-bosons to be eaten by W ± and Z, whereasĤ is a second Higgs doublet which does not participate in the electroweak symmetry breaking. One can follow the analogy with the 2HDMs one step further. In particular, the physical component of the h field defined by transformation Eq. (3) corresponds almost exactly to the SM Higgs because the current setting may be viewed as the 2HDM in the decoupling regime asĤ is expected to be pushed up to the SU (4) C breaking scale v χ . Furthermore, the admixture of χ 0 in the physical Higgs is also suppressed by v ew /v χ . All this can be readily verified by the analysis of the most general renormalizable scalar potential where and j denoting the SU (2) L indices; the matrix notation is used to capture the SU (4) C structure and the traces run only over SU (4) C indices.
The coloured scalar degrees of freedom are theS 1 field originating from χ which dominates the Goldstone mode associated with the vector leptoquark, an SU (2) L doublet G of charged and neutral scalar gluons and two other leptoquark doublets R 2 andR 2 , all of which stem from Φ.
Although we have chosen v χ so large that the effects of the extra vector bosons (the Z and the vector leptoquark U 1 ) are completely negligible the model allows for cases where a certain part of the scalar spectrum is much lighter. This can be easily seen by neglecting for the moment the effects of the SU (2) L breaking VEVs in the masses of the different components of the Φ-field: 2 where µ 2 Φ has been eliminated using the minimization conditions for the potential. This yields the approximate tree-level sum rule [35] It is well known that, unlikeR 2 , the R 2 leptoquark has the potential to simultaneously accommodate R K and R K * . From Eq. (9) one can see that R 2 can be in the TeV range even in case of a rather large v χ if there is an appropriate interplay between λ 4 , λ 6 and λ 14 .
Assuming for the moment being that λ 4 is at least of the order 10 −2 , one sees from Eq. (9) that relatively light scalar gluons are possible in scenarios where R 2 is light andR 2 heavy. We will thus investigate such scenarios. In principle also λ 4 could be smaller yielding somewhat lighterĤ andR 2 states. However, the contribution ofR 2 to lepton flavour violating observables imply that the masses should be in the multi-TeV range. For completeness, we note that the large number of parameters allows to obtain easily a SM-like Higgs boson with a mass m h = 125 GeV.

B. Fermionic sector
The fermion masses are generated by the following Lagrangian: where Y i are Yukawa couplings and µ is a Majorana mass matrix. Without loss of generality, we work in a basis where the lepton mass matrix is flavor-diagonal. The up-and down-type quarks in the mass basis are obtained viâ q L = V q q L andq R = U q q R for q = u, d, with the four arbitrary unitary matrices in the flavour space being constrained by V CKM = V u V d † . Two of the Yukawa couplings are strongly related to the masses of down-type quarks and charged leptons, whereM u,d,e are diagonal matrices of the corresponding fermion masses. The Yukawa interactions of the LQs and scalar gluons are encoded solely in Y 2 and Y 4 . Eq. (11) and Eq. (12) determine Y 4 up to the two rotation matrices. On the other hand, due to the extended neutrino sector, the other important matrix Y 2 , as well as Y 5 , can be chosen essentially arbitrarily. Indeed, the measured up-type quark masses satisfying can be always achieved by a suitable choice of Y 1 . The light Majorana neutrino mass matrix, from which the neutrino masses and PMNS matrix follow, can be then obtained via a proper choice of the Majorana mass matrix µ. While both Y 2 and Y 4 generally contribute to various lepton-flavour violating processes, only the interactions arising from Y 4 are sufficient for a tree-level explanation of the R K ( * ) anomalies. Hence, for simplicity we have assumed small entries in Y 2 in order to fulfill the bounds on LFV violating muon decays [35].

III. CONSTRAINTS FROM RARE LEPTON AND MESON DECAYS
The interactions of R 2 following from the term proportional to Y 4 in Eq. (10) read with the relevant Yukawa matrix parametrized aŝ As we have stated in the previous section, we neglect the other pair of the interaction terms arising from Y 2 . Without referring to the specific pattern of the above matrix imposed by the extended symmetry of the model, a few simple but important observation are to be made. First, the interactions in Eq. (14) involve the right-handed leptons. In view of R K , this implies that the corresponding tree-level contributions to C 9 and C 10 (entering at the scale where the leptoquarks are integrated out) have not only the same magnitude but also the same sign. Thus, there is only a very small interference between the NP and SM contributions in the b → sl + l − amplitudes. Notice that there are ways to circumvent this feature by making the loop contributions dominant, see [49,50]; however, we do not opt here for this scenario.
Second, interaction in Eq. (14) generally induces new sources of LFUV whenever two columns of Eq. (15) differ. In this respect, R K < 1 can be achieved if and only if the LQs couple more to the electrons than to the muons [51], i.e., when |y se y be | > |y sµ y bµ | .
The third point is that the interactions in Eq. (14) mediate lepton flavour violating processes (LFV) whenever there are nonzero entries ofŶ de 4 in two different columns. For example, a very stringent experimental bound arises from the limits on BR(K L → e ± µ ∓ ) ∝ |y se y dµ | 2 + |y de y sµ | 2 or from µ → eγ whose amplitudes are given by linear combinations of y qe y * q µ . It is clear that all the muon number violating process mediated by R 2 will be suppressed if approximately holds. As indicated earlier,Ŷ de 4 cannot be chosen arbitrarily in our model as it is a subject of the extended symmetry constraints. In particular, applying the flavour rotations in Eq. (10) and using the relations from Eq. (11) and Eq. (12) one obtains the following pattern: with U d and V d being arbitrary unitary matrices. The question is now whether this pattern is compatible with significant deviation of R K < 1 and supressed LFV. In Ref. [35], this model was studied under an extra SO(10) inspired assumption V d = U * d and neglecting possible phases in a second step. In such a case, the interaction matrix in Eq. (17) simplifies tô where V ij denotes the elements of the V d mixing matrix. Clearly, the requirements like Eq. (16) are in contradiction with the unitarity of V d and thus LFV is principally unavoidable. In [35] it was found by scanning over the considered parameter space that the experimental bound BR(K L → µe) < 4.7 × 10 −12 [52] inevitably leads to R K ≥ 1, in contradiction with measurements. Consequently, this implies that the assumption V d = U * d is inconsistent with requirement of simultaneously explaining R K and respecting the bound from the K L → µe decay. However, such a model assumption is only fully justified at the scale where one still has the left-right symmetry which, however, is broken well above the SU (4) C -breaking scale, see e.g. [53] and refs. therein for explicit constructions. Renormalization group effects will lead to a breaking of We also note that this model might not emerge from SO(10) but from another framework. In the general case of V d = U * d we have the freedom to choose 6 angles (apart from the phases). We impose the following constraints: GeV mA, mR 2 2 · 10 5 GeV, 900 GeV tan β 50 1. To suppress the muon number violating processes we require the conditions from Eq. (16) to be satisfied.
2. In order to maximize the LQ effect on R K ( * ) , we need the product y se × y be to become as large as possible; due to the smallness of electron mass, this condition affects only U d .
To achieve this, we take as starting point Neglecting in Eq. (17) the mass of the first generation fermions and also the second generation if they appear together with one of the 3rd generation, we arrive at In practice we see that this is not yet sufficient and that we also need y bµ closer to zero. Using now the freedom of the additional mixing angles of O(m µ /m b ) we can achieve the form Note that all other choices of V d and U d satisfying the imposed conditions are, within this approximation, related to the presented ones up to the phases. In the numerical calculations presented below we have fulfilled Eq. (16) exactly, finding four separate closed curves in the parameter space when restricting to real U d and V d . For definiteness, we set U u = U d V † CKM in the following. This construction allows to obtain the experimentally preferred values for R K and R K * for m R2 cos β 18 GeV.
Hence, cos β 1 must apply in order to obey the bounds from direct leptoquark searches. As this and Eq. (21) define a rather special part of the parameter space, the question arises in which other observables such a setting can be tested. There are essentially two broad classes: low energy observables and LHC signals. We will focus here on the low energy part and discuss the collider aspects in the next section.
From the construction it is clear that there will be no additional constraints from any muon number violating decays such as µ → eγ. In fact, we can achieve any value of BR(µ → eγ) between zero and the experimental bound by small deviations from the current limiting scenario Eq. (16) essentially without changing the findings below.
In contrast, we do expect sizable effects in the τ sector. Some of the relevant experimental bounds are BR(τ → eγ) ≤ 3.3 × 10 −8 , BR(τ → 3e) ≤ 2.7 × 10 −8 and BR(τ → eπ) ≤ 8 × 10 −8 [52]. The LQ contributes to the first two processes are loop level whereas to the last one also at the tree level. We have found that the bound on τ → eπ indeed starts to constrain the parameter space. This can be seen from Fig. 1 where we show BR(τ → eπ) versus BR(τ → 3e) scanning over the four lines in the allowed parameter space as described above. In addition we have found that also BR(τ → eγ) is close to its experimental bound varying in the narrow range (2.2-2.7) ×10 −8 , providing another test of the current scenario in upcoming experiments like Belle II. We note for completeness, that in the allowed parameter flavour violating τ decays into muons are strongly suppressed and, thus, an observation of τ → 3µ would rule out our scenario.
We have also checked that the prediction for meson decays like b → sγ are fully consistent with the current experimental data. In the context of leptoquarks a potentially constraining observable is the ratio BR(K + → e + ν)/BR(K + → µ + ν). However, due the required smallness of Y 2 , all leptoquark effects on observables with neutrinos in the final state are suppressed and, thus, also this is consistent with data.
Staying in this part of the parameter space we have also checked whether the low energy data can constrain the masses of the other components of Φ. Our construction implies that the scalar gluons, both the charged one and the neutral one, have flavour mixing couplings to quarks. This means in particular that the neutral one contributes at tree level to K 0 -K 0 and B q -B q (q = d, s) mixing. We find that within the experimental and theoretical uncertainties B s -B s requires m G 0 > ∼ 10 TeV whereas in case of the K 0 -K 0 mixing the bound is m G 0 > ∼ 15 TeV. It might be surprising that K 0 -K 0 mixing is only slightly more stringent than the B-meson mixing but this is a consequence of the specific parameter space considered here. We note for completness, that in other parts of the parameter space this bound increases to m G 0 > ∼ 120 TeV.

A. Collider phenomenology in the presence of flavour anomalies
In the previous section we have found a restricted region in parameter space where R K ( * ) can be explained while being consistent with the constraints from other flavor observables. Here we discuss collider signatures testing this part of the parameter space. Eq. (9) allows for the cases where, apart from R 2 , also the scalar gluons G, or even the whole scalar sector arising from Φ, can be light enough to be tested either at the LHC or a prospective 100 TeV pp-collider.
In the slice of the parameter space, where one explains R K ( * ) while simultaneously respecting other low energy constraints such as µ → eγ and K L → eµ, the leptoquarks have rather special properties. The pattern of their Yukawa couplings Eq. (21) is reflected in their decays. For the charge 2/3 particle one finds, regardless on which point in the allowed regions is chosen, 1.17 is calculated at the scale m R2 . All other channels are negligible. Due to the hierarchical structure of the CKM matrix, the similar pattern appears for the charge 5/3 particle, where the non-negligible decay channels satisfy These particles are searched for by the ATLAS [54] and CMS [55] experiments. Assuming branching ratios of 100 % into a specific channel such as τ b bounds up to 1.1 TeV have been set. However, as we have various combinations of different decay channels involving also τ leptons which are experimentally more difficult to measure, the actual bounds are expected to be somewhat weaker. However, recasting these analyses is beyond the scope of this article and will be left for future work. We now turn to the next component of Φ which can be potentially light, namely the doublet of charged and neutral scalar gluons. In the following we will consider the complex field G 0 even though it is split into its scalar and pseudoscalar component. However, this splitting is at most of O(GeV) and thus can be neglected for the discussion here. The scalar gluon interactions arising from Y 4 generally read where the relevant Yukawa matrix satisfieŝ Note that the interactions of the scalar gluons with right-handed up-type quarks origin from Y 2 which, as mentioned earlier, is suppressed in our model. For this reason our findings differ significantly from the ones of refs. [56][57][58]. Due to the m b enhancement in Eq. (26), the neutral scalar gluons are generally predicted to prefer decays to the b-quarks.
In the slice discussed so far, combining Eq. (19) and Eq. (26) leads tô and one finds the following ranges for the various branching ratios pp -> G 0 bb -/ tt -, G ± bt -, G ± bb -, G + tt at √s=100 TeV The neutral states have also loop induced couplings to the gluons [59]. Denoting the scalar (pseudoscalar) component of G 0 by σ 0 (φ 0 ) we find BR(σ 0 → gg) 0.05 and BR(φ 0 → gg) 0.01. It has been noted already in ref. [59] that the scalar contributions in the loop induced couplings are negligible even for λ i = 1 and, thus, the parametric uncertainties due to the unknown λ i are tiny. The remaining decays are into two quarks of the first two generations. We found in the previous section that the mass of the scalar gluon should be above ∼ 15 TeV due to the constraints on the K 0 -K 0 mixing. This is clearly too heavy for the LHC and, thus, one needs a 100 TeV pp-collider [60,61] to probe these particles. In Fig. 2 we give some of the dominant Feynman diagrams for the processes pp → G 0 qq (q = b, t) and pp → G + bt. The cross sections for a 100 TeV collider are shown in Fig. 3 where we have included all QCD contributions as well as all couplings of scalar gluons to quarks. The relevant Yukawa coupling Y 4 is choosen to be in the slice consistent with low energy data discussed in Sec. III. The cross sections include also the contributions from the production of a scalar gluon pair with the subsequent decay of one of the scalar gluons into the corresponding quark final state. For large scalar masses the production cross sections get a significant contribution from the quark initial states or are even dominated by those. Due to this, for example σ(pp → G + bb) varies by about 20 per-cent because of its dependence on Y 4 in the considered regions of parameter space. Note that the cross sections shown here are calculated at tree-level and we expect sizable QCD corrections. Combining the cross sections with branching ratio information, we have found that the dominant signals will be in the 4 b-jet and 2t+2b-jets channels which are experimentally challenging.

B. Scalar gluons at colliders without flavour anomalies
Since the measurements of the B anomalies still admit the case of being pure statistical fluctuations, in the following we focus on the situation when both leptoquarks are too heavy to contribute significantly to the low energy observables and when the lightest BSM fields are the scalar gluons.
These particles are interesting by their own, and, thus we study here the limiting case, where all flavour violating  implying that BR(G + → tb) is close to one and that the neutral states decay dominantly into bb. The latter can also decay into two gluons. In this model the neutral scalar gluons have suppressed couplings to the top-quark compared to the models discussed for example in refs. [36,59,63] and, thus also the loop induced G 0 gg coupling is significantly smaller. Firstly this implies, that the decays into two gluons have at most a branching ratio of 5 per-cent. Secondly, this also implies that the bounds from processes like obtained by the CMS experiment [64,65] do not constrain our model even when taking QCD corrections via a K-factor of 1.7 [66] into account. Here j can be either a quark or a gluon jet. Instead we have found that the strongest constraints come from an ATLAS search for the H +t b production. [62]. We can see from Fig. 4 that this excludes scenarios with m G + 1 TeV. This is actually a conservative bound in the sense as we assume here that BR(G + → tb) = 1 which maximizes the power of the experimental analysis. We want to stress here, that we have also included here the pair production pp → G + G − combined with the subsequent decay G − →tb. Due the steep decrease of the cross sections with the mass this plot indicates that the reach of the LHC will not be above 1.5 TeV. We therefore show in Fig. 3 the various cross sections at a 100 TeV collider starting from masses in the TeV range which clearly shows that the cross sections in the low mass range is so large that these particles should be found within the first data sets.

V. CONCLUSIONS
In this paper we have studied a model based on the extended SU (4) C × SU (2) L × U (1) R gauge symmetry which is arguably the most minimal UV-complete gauge framework including vector and scalar leptoquark fields. It has been shown recently [35] that, among other features, this setup has the potential to accommodate the measured values of the R K and R K * observables in semileptonic B-decays. It is well known that, in this context, the strongest constraints stem from the non-observation of K L → eµ and µ → eγ. In order for these to be satisfied along with R K and R K * a rather specific flavour pattern of leptoquark interactions with matter is required; for instance, all couplings of R 2 to muons need to be strongly suppressed. We have shown that there exists a narrow region in the parameter space where a fully consistent picture can be achieved. This, in turn, leads to a very predictive scenario in which several other phenomenological conclusions can be drawn.
First, there is a sharp prediction for the branching ratios of τ → eγ and τ → 3e which turn out to be close to their current experimental limits and, thus, should be observable in the next round of experiments such as Belle II.
Second, the charge-2/3 and 5/3 scalar leptoquarks, whose masses should not be much above 1 TeV in order to address the B-anomalies, turn out to have rather specific decay properties which can be tested either at LHC or at a future 100 TeV pp collider. In particular, we find that BR(R . As such, a clear indication, if not a discovery, should be expected in the next LHC run (with the possible exception of scenarios with tan β 75).
Third, there is enough room in the allowed parameter space for relatively light scalar gluons (with electric charges 0 and 1) whose masses are constrained from meson mixing to be above some 15 TeV. Again, the branching ratios of their decays (including those into flavour violating channels) are fixed within narrow ranges which would facilitate their searches at future colliders.
Remarkably enough, the phenomenology of such relatively light scalar gluons in the model under consideration is interesting even if the B-anomalies eventually disappear. It turns out that in such a case the stringent limits from the meson mixing can be alleviated and the bounds on their masses can be lowered into the LHC domain.
In this scenario the most stringent limits stems from the process pp → G +t b where we get a bound m G 1 TeV recasting an ATLAS search for H + . The usual bounds on G 0 do not apply in this model. In that situation the branching ratios into the third generation quarks, namely, BR(G + → tb) and BR(G 0 → bb), amount to almost 100 %.  2. In order to define that SU (4) C get broken to an unbroken group SU (3), the following three steps are necessary: (a) The name of the group which shall be broken as well as the name of the unbroken subgroups are defined via UnbrokenSubgroups Unbr okenSu bgroup s ={ pati -> color }; Here, the first part of the rule must correspond to an entry in Gauge. (b) The features of the unbroken gauge groups in the new array AuxGauge are defined. This is completely analogue to the definition of a group in Gauge.
This relation is defined in the model file using the new array RepGaugeBosons. For each unbroken subgroup a list must be given which consists of pairs of the name of a gauge boson and its dimension. Note, the names for the gauge bosons must always start with V. From this definition, also the mapping of the ghost is derived. The names of the ghost fields are those of the vector boson with V replaced by g.

3.
After the definition of the gauge groups, the matter fields are defined. This is done for non-supersymmetric fields using the arrays FermionField and ScalarField. For fields, which transform non-trivially under the broken gauge groups, the tensor notation is used. Thus, the fundamental representation is a vector of dimension N . If the unbroken subgroup has dimension n, the relation between the components of the fields are The number of fields with a prime is N − n. For the adjoint representation, an N × N matrix is used. This matrix is then decomposed as Here, Ψ is in the adjoint representation of the unbroken subgroup and all primed fields Ψ andΨ are vectors under the unbroken subgroup. The fields α to ω are singlets under the unbroken group.
(a) In the given model, the fermion fields are either singlets or transform in the (anti-) fundamental representation. This is defined via Here, we have introduced the abbreviation dAB only for better readability. Note, that for the tensor representation the name of the second colour index is extended by b (i.e. colorb to prevent any ambiguity. There is one additional subtlety: in SARAH and other codes like MadGraph, CalcHep or WHIZARD the higher dimensional representations of unbroken gauge groups, i.e. the colour group, are not written as tensors but vectors. Therefore, it is necessary to re-write the neutral and charged octets. The necessary definitions are given in the list TensorRepToVector which reads in our case: with a function f to rename the indices. • Finally, one needs to define also the reverse operation, i.e. the relation to re-write the vector into the tensor representation. This is needed to derive the ghost interactions. For all terms but λ 9 and λ 10 the index contraction is unique. For those terms one needs to define the contraction explicitly using Kronecker deltas. The remaining terms coming with λ 11 -λ 19  Although the colour octet doesn't receive a VEV, it's CP even and odd component has a different mass. Therefore, it is also decomposed in real fields.