Flavour-changing top quark decays in the alternative left-right model

We examine flavour-changing neutral-current decays of the top quark, $t\to q \gamma$, $t \to qZ$, $t \to q H$, and $ t\to q g$ (with $q=u, c$), in the Alternative Left-Right Model, a left right-symmetric model featuring exotic quarks and light bosons. These decays have a very small probability of occurring within the Standard Model, but they can be enhanced in this model through the presence of the exotic states. While associated signals may be detected directly at the LHC, rare decays have the advantage of offering means to probe new particles indirectly, through loop-contributions. We perform a comprehensive analysis of the model's parameter space to demonstrate the possible existence of enhancements in the corresponding branching ratios, of $10^6$ for the branching ratios $\mathcal{B}(t\to uZ)$ and $\mathcal{B}(t \to uH)$, and in the range of $10^{2} - 10^{4}$ for the other decays, relative to the Standard Model. We subsequently determine the preferred parameter space regions of the model in terms of potential of being reached in the near future.


Introduction
The LHC continues its testing of the Standard Model (SM) while actively searching for physics Beyond the Standard Model (BSM).These searches unfold on varied complementary fronts.On the one hand, they directly look for collider signals of additional particles and interactions, including particularly those indicative of an extended gauge structure.On the other hand, they also probe BSM effects indirectly by scrutinising physical observables associated with rare phenomena in the SM, which may occur more frequently in various BSM scenarios.A prime example of this is the exploration of new top quark interactions via flavour-changing neutral-current (FCNC) processes.The top quark, the heaviest particle in the SM, possesses a mass close to the electroweak scale.Consequently, studying its nature and properties is believed to shed light on the dynamics behind the origin of elementary particle masses and to reveal potentially new characteristics of electroweak and strong interactions.However, since its discovery in 1995 [1,2], the properties of the top quark have consistently aligned with SM predictions, as demonstrated notably at the LHC where top quarks are abundantly produced.
Top FCNC couplings have been studied experimentally both through the production of a top-antitop pair followed by a rare decay, and through FCNC single top production followed by a standard decay.To date, no evidence of new effects has been observed, and constraints have consequently been derived from searches in various experiments, including electron-positron collisions at LEP [33][34][35], deep inelastic scattering processes at HERA [36,37], and  p collisions at the Tevatron [38][39][40].At the LHC, the ATLAS [41][42][43][44][45][46][47] and CMS [48-53] collaborations have significantly improved these bounds by nearly one order of magnitude.The most up to date limits for decays into a  boson via lefthanded (LH) and right-handed (RH)  couplings are: In addition, the best limits on decays into the SM Higgs boson  are determined from the analysis of potential signals in multi-leptonic and di-photon final states given, for the electron/muon, hadronic tau and di-photon channels, by Finally, bounds on decays into massless bosons are: Given the prominent role of top quark physics in future collider projects proposed within the community, significant improvements in the aforementioned bounds could be anticipated in case one of such colliders is built [54][55][56][57][58][59].
In the present work, we analyse for the first time BSM effects on rare top decays in the framework of Alternative Left-Right-symmetric Models (ALRM) [60,61].Such models emerge from the breaking of an  6 grand-unified symmetry group into its  (3) ×  (3) ×  (3) subgroup, which is next broken into the  (3)  ×  (2)  ×  (2)  ×  (1)  group that embeds the SM gauge symmetry.While in usual Left-Right Symmetric Models (LRSM) [62][63][64][65],  (2)  is identified with  (2)  and  (1)  with  (1) − , in the ALRM we associate  (2)  with an  (2)  ′ group differently embedding the SM fermions into doublets [66][67][68][69].In this way, the  (2)  ′ partner of the right-handed upquark   is not its down-type counterpart   but an exotic down-type quark  ′  , and similarly the  (2)  ′ partner of the right-handed charged lepton   is a new neutral lepton, the scotino   , rather than a more traditional right-handed neutrino   .The   and   states hence remain singlets under both the  (2)  and  (2)  ′ symmetries, like the LH partners of the new states,  ′  and   .ALRM scenarios have several advantages over LRSM ones.In the LRSM, the properties of the Higgs sector could imply non-acceptable tree-level flavour-violating interactions, conflicting with the observed properties of kaon and -meson systems.As a consequence, the  (2)  × (1) − symmetry has to be broken at a very high energy scale, leading to extremely massive extra Higgs and gauge bosons unlikely to be detected at the LHC.In the ALRM, the new scalars always couple simultaneously to ordinary and exotic fermions so that such bounds can be evaded even in the presence of light additional Higgs bosons.In addition, unlike in the LRSM [70], the ALRM could predict a viable DM candidate as the scotino   [71].
Previous studies on the ALRM have established conditions for the stability of its vacuum state [72], shown that the model can have a significant impact on neutrinoless double beta decay processes and leptogenesis [73], and that it could embed a consistent dark matter phenomenology [74].In this work, we focus on the indirect effects of ALRM exotic fermions and Higgs bosons through their loop-induced contributions to rare top decays.We demonstrate the possibility of significant enhancements over the SM predictions in specific scenarios, with obtained values close to current limits.ALRM scenarios, therefore, have the potential to be tested indirectly in the near future, as bounds on rare top decays are expected to strengthen.In the following, we briefly describe in Sec. 2 the ALRM before exploring its implications for rare top decays in Sec. 3. We conclude in Sec. 4.

Alternative Left-Right Models
In this section we briefly review the ALRM, and refer to [71] for a comprehensive introduction.The ALRM is based on the  (3)  ×  (2)  ×  (2)  ′ ×  (1) − gauge group, to which we supplement an additional  (1)  global symmetry.The spontaneous breaking of  (2)  ′ ×  (1) − ×  (1)  to the hypercharge group is implemented such that the generalised lepton number  =  +  3 (with  3 being the third generator of  (2)  ′ and  the  (1)  charge) remains unbroken.This is achieved by means of an  (2)  ′ doublet of scalar fields   charged under  (1)  , which we pair with an  (2)  counterpart   to maintain left-right symmetry.This extra  (2)  field is, however, blind to  (1)  .The electroweak symmetry is further broken to electromagnetism by means of a bi-doublet of Higgs fields charged under both  (2)  and  (2)  ′ , but with no  − quantum numbers.We collect all scalar and fermion fields of the model in Table 1, along with the representation of the fields under the model's gauge group and the associated  (1)  quantum numbers.Associated electric charges are derived through a generalised Gell-Mann-Nishijima relation  =  3 +  3 +  − , where  3 is the third generator of  (2)  and  − the  (1) − charge.
In the ALRM, FCNC interactions in the fermionic sector are generated from the Yukawa Lagrangian in which we omitted all flavour indices for simplicity.The Yukawa couplings Ŷ are thus 3 × 3 matrices in the flavour space.Moreover, we introduced the dual fields φ =  2  2 and χ, =  2  , with  2 being the second Pauli matrix.The scalar potential dictating Higgs spectrum and mixings includes bilinear (), trilinear () and quartic (, ) terms, ( Imposing consistent vacuum stability conditions and nontachyonic states, phenomenologically acceptable scenarios always feature  ≡  1 ≃  2 =  3 and  2 = 0. The breaking of the left-right symmetry down to electromagnetism yields a vacuum configuration in which all neutral components of the scalar fields acquire non-vanishing vacuum expectation values (vevs), with the exception of  0 1 , which is protected by the conservation of the generalised lepton number  (which also forbids mixing between the SM  and exotic  ′ quarks), Subsequently, while the (pseudo)scalar components of  0 1 is disallowed from further mixing with any other (pseudo)scalar degree of freedom, the rest of the neutral Higgs bosons do.We refer to [71] for expressions of the corresponding 3 × 3 mixing matrices that are entirely determined by the other parameters of the model (including those introduced below).In the charged sector, the physical massive charged Higgs bosons  ± 1 and  ± 2 , and the two massless Goldstone bosons  ± 1 and  ± 2 to be absorbed by the  ≡   and   charged gauge bosons, are defined by Correspondingly, the squared charged scalar masses, that are relevant for loop-induced FCNC top quark decays, read As state is generally heavy, while the  ± 2 one lies in the sub-TeV to TeV range for a large part of the parameter space.
The breaking of the left-right symmetry generates masses for the gauge bosons and induces their mixing.Due to the conservation of the generalised lepton number (preventing  0  1 from acquiring a vev), the charged  and   bosons do not mix.In contrast, the neutral bosons undergo mixing, which results in a massless photon and two massive  and  ′ states.The masses of the extra gauge bosons are These expressions depend on the sine (  ) and cosine (  ) of the two mixing angles, that are determined from where   and  denote the hypercharge and electromagnetic coupling constant respectively.The mass ratio of the two  (2)  ′ (-like) gauge bosons satisfies    ∕  ′ ∼   .Thus for values of   ≠   , this ratio can be substantially large or small.This is the same as in the LRSM, however there are independent collider limits on    from decays into quarks or lepton-neutrino which do not apply here.Thus the   boson can be much lighter than the  ′ .

Rare top decays in the ALRM
In the SM, FCNC couplings arise at one loop from exchanges of  bosons and down-type quarks.The same mechanism applies to ALRM scenarios, with additional contributions from exchanges of charged Higgs bosons  ± 1,2 , charged  (2)  ′ bosons   , and  ′ -type quarks.Among these contributions, loops featuring exotic quarks ( ′ ,  ′ and  ′ ) and   and  ± 2 bosons dominate over loops involving ordinary quarks (,  and ) and  and  ± 1 states.This dominance originates from the fact that right-handed (lefthanded) top quarks directly couple to exotic  ′  quarks and not to ordinary   -type quarks through interactions with the   ( ± 2 ) boson, from larger relevant Yukawa couplings, and from the absence of any GIM suppression in the BSM sector.
In order to explore the model's parameter space, we consider the independent parameters relevant for rare top decays listed in  Maximum branching ratios obtained in the scan of the ALRM parameter space defined in Table 2. SM predictions for each channel are given for comparison [80].
To compute the branching ratios for all possible top FCNC decays, we rely on FeynArts [75] and FormCalc [76] for the generation of Feynman diagrams, the derivation of associated matrix elements, and their conversion into a Fortran package for numerical evaluation.The ALRM model implementation in FeynArts has been performed with FeynRules [77,78] using the model file developed in [71].As a cross-check, we verified that SM predictions were retrieved after decoupling the BSM sector.Our numerical analysis was conducted by integrating the Fortran output of FormCalc into a custom scan module that explores the parameter space defined in Table 2, within the specified ranges.
Our selection criteria for the parameter space regions to be scanned aims to ensure that at least one of the charged Higgs bosons, the extra gauge bosons, and the exotic quarks all fall within the sub-TeV to TeV range.Additionally, we allow the mixing angles  ′  , which parameterise exotic quark mixings, to vary freely from 0 (no-mixing) to ∕4 (maximum mixing).Given the similarity in masses of exotic quarks, our results are, however, not expected to be significantly dependent on  ′  .The  (2)  ′ coupling   is allowed to vary within the range [0.37, 0.768], where the lower bound ensures that   ∕  exceeds tan  [79] and the upper bound (approximately √ 4) maintains   within the perturbative regime. 1  In addition, the SM parameters are fixed according to the Particle Data Group Review [80], except for the total decay width of the top quark, which is set to Γ  = 1.32 ± 0.5 GeV as in [81].In the scan, we randomly sample 10 7 scenarios, ensuring that   ′ ≥ 4.5 TeV to comply with LHC constraints [82,83], and that   ± 2 ≥ 90 GeV to satisfy LEP limits [84].Furthermore, scenarios with   ± 2 > 100 GeV are subjected to an additional condition, requiring that the associated production cross section at the LHC is less than 0.1 pb, aligning with existing limits.
The objective of our scan is to identify regions in the parameter space that maximise individual branching ratios 1 The minimum value of   is easily obtained from Eq. ( 7), that yields . Requiring sin  ≤ 1 gives   ∕  ≥ tan , thus the minimum value of   = 0.37 [73].associated with the rare top decays  → , ,  and  (with  = , ).The largest obtained branching ratios are collected in Table 3, along with the corresponding SM predictions.We observe that branching ratios can reach up to 10 −11 − 10 −10 for decays into spin-1 bosons and approach 10 −8 for decays into a Higgs boson.These results are also visually presented in Fig. 1, where we include the current experimental limits [48-52].These experimental bounds can be further compared with the expected improvement of about one order of magnitude from the high-luminosity run of the LHC, and the additional order of magnitude anticipated from a future 100 TeV hadronic collider [85,86].
These results unveil that the decays  →  and  →  for  = ,  exhibit a pronounced dependence on the final quark flavour, closely intertwined with the mass values of the additional states.In the case of decays into up quarks, that is illustrated with the plots in the bottom row of Fig. 2  Predictions for top decays into  and  final states exhibit very similar properties, as illustrated for  →  in the plots shown in the top row of Figure 2. The   mass needs to be either within 1.2 -2 TeV or around 4 TeV.In the former case, the charged Higgs boson must have a mass greater than 500 GeV, and tan  ≲ 5, while in the latter case,   ± 2 has to be between 90 and 115 GeV, with tan  remaining unconstrained.For decays into a first (second) generation uptype quark, the right-handed CKM mixing angle  ′ 13 ( ′ 23 ) is required to be in the 25 • − 45 • range, with the other two angles allowed to vary freely, up to unitarity constraints.
The decay processes  →  and  →  have properties quite similar to those involving a final state with a  boson.The main difference is that the   boson has to be light, with a mass below 2 TeV, to guarantee an enhanced branching ratios ( → ), and the charged Higgs boson  ± 2 must be heavy with a mass of a few hundred GeV.Furthermore, tan  is unrestricted, although it has to be smaller than 10 when the charged Higgs mass is larger than 500 GeV.
Finally, it is important to note that, in all cases, the mass of the  ′ boson must be consistent with experimental data.Therefore, it satisfies   ′ ≈ 4.5 − 5 TeV for all classes of decays studied.
The relative masses of the charged Higgs boson  ± 2 and of the gauge boson   determine the significance of the different diagrams contributing to the studied rare top decays.When   ± 2 is in the 100 GeV range, diagrams involving a charged-Higgs exchange dominate.In this case, the distinction between decays into first-generation and secondgeneration quarks is linked to the SM quark mass factor that multiplies specific terms in the corresponding amplitudes.Consequently, charged Higgs-boson exchanges play a more significant role in decays involving charm quarks as   ≫   .While other terms in these amplitudes are multiplied by a factor of the mass of the exotic quarks exchanged, all these masses are in the same ballpark, thus introducing no substantial difference between decays into first and secondgeneration quarks.Scenarios yielding an enhanced branching ratio for one of the rare top decays considered indeed always feature one lower exotic quark mass of about 1 TeV and two larger masses of about 10 TeV.

Conclusions
Rare top decays involving FCNCs are highly suppressed in the SM due to the GIM mechanism, with branching ratios of  →  (with  = ,  and  = , , , ) of the order of 10 −12 − 10 −16 .A significant enhancement in these rates would, therefore, be a signal of new physics.In this work, we examined rare FCNC top quark decays in the ALRM.In addition to extra  (2)  ′ gauge bosons and exotic quarks, this model also contains new charged Higgs bosons.Unlike in the usual left-right model, these charged Higgs states do not lead to FCNCs at tree level and are thus allowed to be light.We analysed the effect of all model's parameters on the corresponding branching ratios, and we explored the consequences of large deviations on the parameter space.We found that an enhancement of 10 2 − 10 6 compared to the SM is achievable while imposing consistency with current theoretical and experimental constraints.Notably, we observed an enhancement of 10 6 for the branching ratios ( → ) and ( → ), while for other decay processes, the enhancement lies in the range of 10 2 − 10 4 .Two significant classes of scenarios lead to such a large enhancement.The first features a charged Higgs boson  ± 2 with a mass greater than 500 GeV together with tan  ≲ 5.The second exhibits a light charged Higgs boson with a mass of about 100 GeV, which is possible if tan  > 5. Related to these findings, the charged gauge boson   can possibly be light, with a mass in the 1.2 -2 TeV regime, or heavy, with a mass of about 4 TeV.Additionally, one of the right-handed CKM mixing angles has to be large ( ′   23   for decays into charm quarks and  ′  13 for decays into up quarks), with the other angles generally small as required by the unitarity of the right-handed CKM matrix.While these enhancements are significant, they fall below observability prospects of a future hadronic collider expected to operate at a centre-of-mass energy of 100 TeV.
Studying rare top decays therefore provides an interesting avenue to explore the ALRM.It serves as indirect probes for the possibility of a light charged Higgs boson or moderately heavy   boson (both currently allowed by data), as well as TeV-scale exotic quarks.It will also allow for an unambiguous distinction of the ALRM from usual LRSM setups in which such a mass spectrum is excluded.

Figure 1 :
Figure 1: Maximum branching ratios for the considered rare top decays obtained from our ALRM scan (blue dots) and in the SM [80] (red dots).For comparison, we include current bounds from LHC data (red dots with arrows) [48-52].
, significant branching ratios favour   -boson masses in the 1.2 -1.5 TeV range, correlating with a charged Higgs boson of mass ranging from a few hundred GeV to 1.5 TeV.Such a light   boson additionally influences the right-handed CKM mixing angle  ′ 13 , requiring it to be at least 25 • and below 45 • (which corresponds to a maximally mixed situation), and pushes exotic quark masses deep into the multi-TeV regime.Furthermore, a TeV-scale charged Higgs boson constrains the possible values of tan  as driven by Eq. (5).Notably, charged Higgs boson masses beyond 1 TeV imply tan  ≲ 5, while a lighter charged Higgs boson below 1 TeV allows for substantial values for ( → ) and ( → ) only with a large tan  value.On the other hand, large enhancements of ( → ) and ( → ) favour   -boson masses in the 3.5 -4 TeV regime, which is compensated by a charged Higgs boson mass between 90 and 115 GeV.Due to the large   mass, these enhancements are independent of the right-handed CKM mixing angles and exotic quark masses, while a light charged Higgs boson remains uncorrelated with any constraint on tan .

Table 2 ,
and in terms of which all other Lagrangian parameters and physical masses or mixings can be expressed.These parameters include the  (2)  ′ gauge coupling   and CKM mixing matrix that we express in terms of inter-generational mixing angles ′  12,  ′ 13 and  ′ 23(ignoring any possible  violating phase as irrelevant for rare top decays); the vev  ′ and mixing angle tan ; the scalar potential parameters  3 ,  and ; and the masses of the exotic quarks   ′ ,   ′ and   ′ .