Two-Higgs-doublet model and quark-lepton unification

We study the Two-Higgs-Doublet Model predicted in the minimal theory for quark-lepton unification that can describe physics at the low scale. We discuss the relations among the different decay widths of the new Higgs bosons and study their phenomenology at the Large Hadron Collider. As a result of matter unification, this theory predicts a correlation between the decay widths of the heavy Higgs bosons into tau leptons and bottom quarks. We point out how to probe this theory using these relations and discuss the relevant flavor constraints.


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and the SM matter fields are unified in the following representations: ∼ (4, 2, 0), (2.1) In this context the leptons can be understood as the fourth color of the fermions. The Lagrangian of this theory can be written as is the strength tensor for the SU(4) C gauge fields, A µ ∼ (15, 1, 0).
The small neutrino masses can be generated while allowing the SU(4) C symmetry to be broken at the low scale using the inverse seesaw mechanism. (3.9) where the mass parameter M L is given by M 2 L = v 2 λ 1 c 4 β + λ 2 s 4 β + 2λ 345 s 2 β c 2 β + 2λ 6 c 2 β s 2β + 2λ 7 s 2 β s 2β , (3.10) where v 2 = v 2 1 + v 2 2 . From eq. (3.9) we can see that the alignment limit can be achieved whenever there is a cancellation between M L and M h or in the decoupling limit when M H M L , M h .
In this limit, the physical Higgs masses are given by

Higgs bosons decays
In this section, we discuss the decay properties of the new Higgs bosons and relations among the decay widths that arise from quark-lepton unification. For simplicity we assume the Yukawa interactions to be flavor-diagonal [13], this will be justified in section 6 where we discuss constraints from flavor violation. In the limit when h is the SM-Higgs, H does not interact with SM gauge bosons, so the following decays channels vanish at tree-level Consequently, the total decay width of the heavy Higgs H corresponds to where the repeated index implies a sum over the different flavors. The trilinear coupling between H and two SM Higgs bosons can be written as (4.4) we refer the reader to appendix A for a complete list of the Feynman rules. In the limit with flavor-diagonal couplings this theory gives clean predictions for the coupling of H and A to down-type quarks and charged leptons. Namely, both couplings depend on the physical masses and the value of tan β as given in appendix A.
In the top panel in figure 1 we present our results for the branching ratio of H as a function of tan β. In this case we fix M H = 300 GeV so the decay H →tt is kinematically closed. The blue (green) line shows the branching ratio for the decay channel H →bb (H →τ τ ). The orange line corresponds to the channel H → hh which depends on the value of λ eff . As can be seen, the branching ratio for the H →bb channel nearly vanishes for tan β ≈ 0.3, this is because there is a cancellation between the two terms in the coupling to down-type quarks.
For small values of tan β, quark-lepton unification predicts the following relation from the top-left panel in figure 1 we can see that this relation is already satisfied for tan β 0.05. For large values of tan β we have that from the plot we can see that for tan β 3 this relation is already satisfied.  In the lower panel in figure 1 we present our results for the branching ratio of A as a function of tan β. In contrast to H, the pseudoscalar A has no trilinear term with hh, this implies that the decay channel A → hh vanishes at tree-level. Quark-lepton unification gives the following relation for small values of tan β and for large values of tan β from this plot we can see that eq. (4.7) is already satisfied for tan β 0.05 while eq. (4.8) is already satisfied for tan β 3. Unfortunately, the theory does not predict the coupling to up-type quarks. However, we can parametrize this coupling by introducing the parameter κ, where C Lud corresponds to the coupling with the charged scalar. Since there is freedom in the U T C M D T ν U term, it can be fixed at each point in order to remove the dependence on the parameter tan β. Since the branching ratios shown in figure 1 are independent of the parameter κ, the LHC bounds can be avoided by choosing a small value for this parameter.
In figure 2 we show the branching ratios for M H,A = 500 GeV when the decay H, A →tt is kinematically open. In the bottom-right panel we show the branching ratios for the decay channels of the charged Higgs as a function of tan β. The decay width for H + →bt depends on the parameter κ given in eq. (4.9), which we fix as before to κ = 0.1 in both plots; however this decay width changes as we vary κ. Since the right-handed neutrinos acquire their mass from the SU(4) C symmetry breaking scale they are expected to be heavy, and hence, the decay channel H + →ē i N j is kinematically closed.
As we discussed in ref. [7], the idea of quark-lepton unification also predicts relations among the decay widths of the scalar leptoquarks present in the theory. Assuming flavordiagonal Yukawa interactions, we obtain the following relations between the decay widths (4.10) Consequently, if scalar leptoquarks are discovered in the near future, then these relations can be used to test whether the underlying theory comes from quark-lepton unification. A detailed study of the collider phenomenology for the scalar leptoquarks is beyond the scope of this paper.

Production at the LHC
The new neutral Higgs bosons can be produced at the LHC through gluon fusion with the top and the bottom quarks running in the loop as shown in the Feynman diagram in figure 3. The collider phenomenology in the 2HDM has been studied before in different contexts, see e.g. [14][15][16][17][18][19][20][21][22][23][24]. The contribution from the bottom quark is relevant only for large and small values of tan β, and hence, the cross-section mostly depends on the κ parameter used to parametrize the coupling to the top-quark. The effective coupling between the neutral Higgs bosons and the gluons is given by where the dual field strength tensor is given by G µν = ε µναβ G αβ /2 and where the sum is over the quarks in the SM, although the dominant contribution comes from the top quark, and the f (τ ) loop function is given by   where We are interested in the regions with large or small values of tan β, since in these regions by measuring pp → H, A →τ τ then the cross-section for pp → H, A →bb can be predicted by using eqs. (4.7) and (4.8). We implement the model using FeynRules 2.0 [25] and calculate the cross-sections using MadGraph5_aMC@NLO [26] which were cross-checked in Mathematica with use of the MSTW2008 [27] set of parton distribution functions.
In figure 4 we present our predictions for the cross-section pp → S →τ τ as a function of the mass of the scalar M S with center-of-mass energy of √ s = 13 TeV. We implement the following cuts on the transverse momentum and the rapidity of the tau leptons, p T > 30 GeV and |η| < 2.  S = H, A and assumes M H = M A , since for large masses their mass splitting cannot be large due the perturbativity of the scalar couplings and the constraints coming from electroweak precision observables [29]. The region shaded in red corresponds to the exclusion limit from searches by the ATLAS [28] collaboration for a heavy scalar decaying into a two tau leptons with integrated luminosity of 139 fb −1 (see also ref. [30]). We fix the trilinear coupling to λ eff = 0.1 which affects only the process involving H.
The two upper panels in figure 4 correspond to κ = 1. For the plot on the left we set tan β = 1/20 and the LHC bound require the mass of H and A to be above 1.2 TeV; for the plot on the right we set tan β = 10 which requires the masses to be above 1 TeV. These bounds can be avoided by choosing a smaller value for κ. In the lower panels we set κ = 0.1 and then the heavy scalars can be around the electroweak scale.

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In figure 5 we present the cross-sections for the process pp → S →bb with center-ofmass energy of √ s = 13 TeV for κ = 1 (κ = 0.1) in the upper (lower) panels. In this case we impose p T > 50 GeV for the transverse momentum and |η| < 2 for the rapidity of the bottom quarks. These cross-sections have a similar magnitude as the ones withτ τ in the final state; however, the search forbb is more challenging experimentally and the current bounds are much weaker than the values predicted [31,32]. Nonetheless, these predictions will be relevant for future searches of a heavy scalar decaying into two bottom quarks. We checked that for large and small values of tan β the relations from quark-lepton unification, eqs. (4.7) and (4.8), are satisfied.
For the charged Higgs the most relevant bound comes from the experimental measurement of the transition b → sγ which requires M H ± > 790 GeV [33,34]; however, this observable depends on the coupling between the charged Higgs and the top quark which is not predicted in this theory and we parametrize in eq. (4.9) using the parameter κ. That bound corresponds to setting κ 1 and it becomes weaker for smaller values of this parameter. Regarding production at the LHC, the charged Higgs can be pair-produced through a Z boson or a photon; however, this cross-section is smaller than the one we have consider for single production of the neutral Higgs bosons, and hence, we do not discuss it any further. The high luminosity stage at the LHC is expected to reach an integrated luminosity of 3000 fb −1 and will probe masses for H and A around the TeV scale. Consequently, the relations between the decay widths predicted from quark-lepton unification can be tested in the near future.
The interactions between H and the SM down-type quarks and charged leptons are defined by where V = D † E and V c = D † c E c have the information about the unknown mixings between the quarks and leptons. Notice that in the above equations, the first term is flavor-diagonal while the second term generically violates flavor but their values are bounded by the quarks or lepton masses. The framework of quark-lepton unification implies that the flavor violating couplings in the quark sector are proportional to the lepton masses and vice versa.
From eqs. (6.1) and (6.2) we see that the effects of flavor violation are proportional to the fermion masses, and hence, the largest effect will involve the third generation. Here we discuss a simple scenario where all flavor violating processes are suppressed by the masses -12 -JHEP08(2022)293 of the quarks and leptons from the first and second generations. In this scenario V and V c are given by Therefore, the elements (C H dd ) a3 = (C H dd ) 3a = (C H ee ) a3 = (C H ee ) 3a = 0 and (C H dd ) 33 = (C H ee ) 33 = 1 with a = 1, 2. The interactions of the CP-odd Higgs, A, also can be written in a similar way, The relations in eqs. (6.1)-(6.5) can be seen as an ansatz for the Yukawa matrices which is motivated by quark-lepton unification. When we use eq. (6.3) one can easily find that Therefore, all flavor-violating couplings will be suppressed by the light quark and lepton masses. Now, we will discuss the predictions for the most relevant lepton violating processes and meson decays to show that the experimental bounds can be satisfied.
The flavor-violating couplings in the quark sector will contribute to the measured mass splitting for the K mesons (see diagram in figure 8). For studies of the general 2HDM and meson mixing see e.g. refs. [43,48]. We require this contribution to -13 -JHEP08(2022)293   sin θ = sin θ c = 1/ √ 2. The blue line corresponds to tan β = 1/20 and the experimental measurement of ∆m K requires M H 18 TeV; while for tan β = 1 (tan β = 10) we find that M H 1.8 TeV (M H 9 TeV) is allowed. This bound becomes weaker as the mixing angles are reduced, this can be seen in the left panel in figure 6, where we present our results for sin θ = sin θ c = 0.1.
In figure 7 we present our results as a contour plot in the tan β vs M H plane. The region shaded in red is ruled out since it gives a larger contribution than the measured value of ∆m EXP K . In the left panel we take maximal mixing angles which requires M H 1.8 TeV but for large and small values of tan β to be allowed it requires M H 10 TeV. Therefore, for the Higgs bosons to be around the TeV scale this bound requires the Yukawa couplings to be very close to flavor-diagonal. On the right panel we take small mixing angles of sin θ = sin θ c = 0.1 which require M H 250 GeV.
The couplings will also induce the decay K L → µ ± e ∓ which for couplings of O(1) excludes masses of 10 3 TeV [52]. However, in this case the four-fermion effective interaction is suppressed by Y ds Y µe ≈ m µ m s /(16v 2 ) ≈ 10 −8 , and hence, this bound is much weaker than the one from ∆m K .
In the leptonic sector, the new neutral scalars can give rise to µ → eγ at one-loop (see diagram in figure 10). Since the coupling to the top-quark is not predicted by the theory and the κ parameter can be small, the two-loop Barr-Zee contribution (with the top quark running in the loop) is subleading. This process has been constrained experimentally by the MEG collaboration to be Br(µ → eγ) ≤ 4.2 × 10 −13 [53]. For the calculation of this branching ratio we adapt the results in ref. [54], where In the context of the general 2HDM, the experimental bounds coming from µ → eee and µ − e conversion have been shown to be subleading to the bound from µ → eγ [35,36].
In summary, the bound that comes from the measurement of ∆m K is much stronger than the constraint from µ → eγ on the mixing angles θ and θ c . For the Higgs bosons to be around the electroweak scale this requires the Yukawa interactions to be very close to flavor-diagonal which justifies the approach taken in section 4.

Summary
Quark-lepton unification remains one of the best-motivated ideas for physics beyond the Standard Model. In this article, we studied the phenomenology of the 2HDM in the minimal theory of quark-lepton unification that can live at the low scale. In the limit with no flavor-violating couplings we gave concrete predictions for the branching ratios of the heavy neutral and charged Higgs bosons. Moreover, we derived relations between the decay widths of the heavy Higgs bosons into quarks and leptons that arise from quarklepton unification. Namely, for small tan β the decay width into bottom quarks should be three times the decay width into tau leptons, while for large tan β it is the opposite.

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We also studied the production cross-sections of the new scalars at the LHC. The current experimental bounds by the ATLAS collaboration already exclude some regions in the parameter space and the future high-luminosity stage will be able to probe this scenario in the TeV regime. In this theory, the cross-section for the processes pp → H, A →τ τ is related to pp → H, A →bb for small and large values of tan β and can be used in the future to probe the idea of quark-lepton unification.
Furthermore, we studied the experimental constraints on the flavor-violating couplings in the quark and leptonic sectors induced by the Higgs bosons. We demonstrated that the experimental measurement of meson mixing gives strong constraints on the off-diagonal entries and for Higgs bosons around the TeV scale this requires the interacting matrices to be very close to flavor-diagonal. If new scalars beyond the Standard Model Higgs are discovered in the near future, this study provides a path to infer whether the underlying theory arises from quark-lepton unification.

A Feynman rules
The Feynman for the physical scalars in the two Higgs doublets: In the alignment limit sin(β − α) 1, the Higgs interactions take the following form The interaction matrices in terms of the physical fermion masses are given by , 2v .

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The interaction matrices of H in the Standard Model limit are given by The interaction matrices are given by: K 1 and K 3 are diagonal matrices containing three phases, K 2 is a diagonal matrix with two phases. As can be seen from the Feynman rules, due to quark-lepton unification the off-diagonal entries for the quark interactions will depend on the lepton masses and vice versa. For the approximation given above the mixing matrices take the form: the above matrices enter in the couplings C dd and C ee . Therefore, eqs. (A.2) and (A.3) can be seen as an ansatz for the Yukawa matrices that arises from quark-lepton unification. This is different from the commonly used Cheng-Sher ansatz [55]. For a general Higgs decay that couples to massive fermions in the following way the decay width corresponds to where N C corresponds to the color factor for the fermions.