Novel B -decay signatures of light scalars at high energy facilities

We study the phenomenology of light scalars of masses m 1 and m 2 coupling to heavy flavor-violating vector bosons of mass m V . For m 1 ; 2 ≲ few GeV, this scenario triggers the rare B meson decays B 0 s → 3 μ þ 3 μ − , B 0 → 3 μ þ 3 μ − , B þ → K þ 3 μ þ 3 μ − , and B 0 s → K 0 (cid:2) 3 μ þ 3 μ − ; the last two being the most important ones for m 1 ∼ m 2 . None of these signals have been studied experimentally; therefore, we propose analyses to test these channels at the LHCb. We demonstrate that the reach of this facility extends to branching ratios as small as 6 . 0 × 10 − 9 , 1 . 6 × 10 − 9 , 5 . 9 × 10 − 9 , and 1 . 8 × 10 − 8 for the aforementioned channels, respectively. For m 1 ; 2 ≫ O ð 1 Þ GeV, we show that slightly modified versions of current multilepton and multitau searches at the LHC can probe wide regions of the parameter space of this scenario. Altogether, the potential of the searches we propose outperform other constraints such as those from meson mixing.


I. INTRODUCTION
Searches for new physics in final states often considered as "standard candles," most notably in searches for supersymmetry (SUSY), have not provided any evidence of physics beyond the Standard Model (BSM) so far. This fact does not necessarily disproves low energy SUSY or other popular BSM extensions [1], such as composite Higgs models (CHM) [2,3]. However, it supports the search for new physics in radically new and still unexplored channels.
In this paper, we focus on light singlet scalars a 1;2 that can be produced in rare decays of B mesons mediated by heavy flavor-violating vector bosons V. This scenario is especially motivated, as it arises naturally in nonminimal CHMs [4][5][6][7][8][9]. (V and a 1;2 can be seen as the counterparts of the ρ and the pions in QCD.) Likewise, such vector boson can explain the apparent anomalies observed in tests of lepton flavor universality [10][11][12][13][14][15][16][17]. Moreover, the bounds on such vector boson are weakened when it decays into lighter composite resonances [17], such as the aforementioned scalars. Finally, also supersymmetric models can trigger similar decays, mediated by scalar and pseudoscalar sgoldstino particles [18].
If, similarly to the Higgs boson, the scalars couple stronger to the muon than to the electron, processes such as B 0 s → a 1 a 2 can lead to four muon final states. To the best of our knowledge, the corresponding signal has been studied experimentally only at the LHCb [19], the most stringent limit being BðB 0 s → 2μ þ 2μ − Þ < 2.5 × 10 −9 . However, there are different reasons to consider alternative B meson decay modes. To start with, the partial width for a 2 → a 1 a 1 can very easily dominate over the corresponding leptonic width. In this case, six-muon final states rather than four muon ones are to be studied. And second, the scalars couple to the mediator as a vector current ∼a 1 ∂a 2 . When the latter is conserved, namely for m 1 ∼ m 2 (and in particular in the massless limit), the B meson decay into such scalars vanishes. In other words, ΓðB 0 s → a 1 a 2 Þ ∼ ðm 2 1 − m 2 2 Þ=m B . In this regime, one should rather explore three body decays of B with emitted mesons. In this work, we focus mostly on B þ → K þ 3μ þ 3μ − . (The inclusion of conjugate modes of charged decays is implied throughout the paper. ) We also extend previous works on this topic [18,20,21] by studying the regime of large scalar masses. In such regime, a 1;2 can no longer show up in rare decays of B mesons. However, they can appear in decays of the vector mediator if it is at the TeV scale and therefore be produced in pp collisions at the LHC.
This article is organized as follows. In Sec. II, we provide the Lagrangian relevant for our study and define the region of the parameter space of phenomenological interest. In Sec. III, we focus on the regime m 1;2 ≲ few Gev and provide analyses for the LHCb and estimate the reach for different B decays. We do not circumscribe to any particular value of m 1;2 , but rather scan over different values of these. In Sec. IV, we focus instead on the regime m 1;2 > few GeV and study the corresponding LHC signatures.
Unless otherwise stated, all limits given in this article stand for 95% CL.
We conclude in Sec. V, while we dedicate Appendix to building a complete model that predicts definite values of several of the parameters that we scan over.

II. FRAMEWORK
Let us consider the Lagrangian of the Standard Model (SM) extended with a heavy vector V and two light scalars a 1 , a 2 . The relevant Lagrangian before electroweak symmetry breaking (EWSB) (on the basis in which up quark and lepton Yukawas are diagonal) reads with m V ≫ m 1;2 . The ellipsis stands for terms not relevant for this study. Without loss of generality, we assume m 2 > m 1 . The scalars a 1;2 can be more naturally thought of as the real and imaginary components of a complex field Φ; the Lagrangian being invariant under Φ → exp ðiθÞΦ up to Oð1 − m 2 =m 1 ; m 12 Þ. In Appendix, we match a concrete CHM to the Lagrangian above. Assuming that V interacts mostly with the third generation quarks, after EWSB it couples to b L b L and t L t L as well as b L s L þ H:c: with strengths ∼g qq and respectively. We distinguish two different regimes depending on the masses of the scalars: 1 GeV ≲ m 1;2 ≲ 4 GeV (low-mass regime) and m 1;2 > 4 GeV (high-mass regime). Likewise, we consider two possible scenarios for the couplings of a 1;2 to the fermions. First, we assume that a 1;2 are muonphilic. As a second possibility, we assume that they couple only to the SM leptons and with Higgs-like strength, namely ∼g 1;2 y l a 1;2 l þ l − , with y l the SM Yukawa couplings and g 1;2 free dimensionless parameters and lepton independent.
In the low-mass regime, a 1 decays mostly into muons irrespectively of whether it is muonphilic or just leptophilic. In the high-mass regime, it decays mostly into taus unless it is muonphilic.
Regarding the decay of a 2 , if m 2 > 2m 1 , then a 2 can either decay into a 1 a 1 or into lepton pairs, depending on m 12 =g 2 , In what follows, we assume that m 12 =m 2 ≫ g 2 y l in this regime, so that Bða 2 → a 1 a 1 Þ ≫ Bða 2 → l þ l − Þ. Note that this inequality holds almost trivially, since one expects m 12 ∼ m 2 whereas the Yukawas are tiny. If instead m 2 < 2m 1 , a 2 can either decay into pairs of leptons as before, or into a 1 l þ l − with width This decay mode dominates if g 1 ≳ 100g 2 . We assume this hierarchy hereafter. Thus, for example, for g 1 ¼ 3 and g 2 ¼ 0.01, a 2 decays always into four leptons mediated by a 1 , which can be either on shell or off shell. Also, they both have widths smaller than 10 MeV and lifetime shorter than 10 fs. As a consequence, both a 1;2 would seem to have vanishing experimentally measurable widths and flight distances. Furthermore, note that the Yukawa suppression helps also avoiding bounds from BABAR and even the future Belle-II [22]. At low energies, the vector boson V triggers B meson decays into the light scalars; see Fig. 1. Depending on the relative size between m B and m 1;2 , we distinguish the following two cases: (Other three body decays, e.g., We do not consider any other cases in this paper; see The decay width for B 0 s → a 1 a 2 reads and f B ∼ 0.23 GeV [23]. The amplitude for B 0 s → a 1 a 1 a 1 reads where we have defined the transferred momenta q 2 After integrating over q 2 23 , we obtain which should be evaluated at The final width is obtained integrating over q 2 12 between 4m 2 1 and ðm B − m 1 Þ 2 .
In the limit m 1 , m 2 → 0, the integrated width simplifies to Finally, the amplitude for B þ → K þ a 1 a 2 is given by with and again For convenience, we trade these variables for with Following Ref. [24], we parametrize the form factor as with r 2 ¼ 0.330 and m 2 fit ¼ 37.46 GeV 2 . Finally, in the approximation m 1 , m 2 , m K → 0, and f 0 , f þ ðq 2 Þ → 1, we obtain In Fig. 3, we show the magnitude of three body decays under consideration and their dependence with ðp 1 þ p 2 Þ 2 . In Fig. 4, we show the ratio of ΓðB 0 s → a 1 a 2 Þ to ΓðB þ → K þ a 1 a 2 Þ. It is very worth noting that it vanishes in the limit m 1 → m 2 ; see also Eq. (6). In this regime, searches for B 0 s decaying only to muons are irrelevant; extra mesons have to be tagged instead. There are however no analyses (not even prospects) in this respect, and this is a gap that we try to overcome in this work.
At high energies, V can be produced on shell in pp collisions initiated by bottom quarks and subsequently decay into third generation quarks and into a 1 a 2 with respective widths, with q ¼ t, b. Note that the scalar decay mode dominates already for g 12 ≳ 3g qq .

III. LOW-MASS REGIME AT THE LHCb
In the low-mass regime, the smoking gun signature of the Lagrangian in Eq. (1) is rare decay of B mesons into final states containing six muons (and possibly other lighter mesons). Let us focus first on the channel B 0 s → 3μ þ 3μ − . As we have already commented, there are no searches for this decay mode, and so neither constraints nor any direct way to estimate the potential of the LHCb to test this process. We therefore suggest the first analysis in this respect.
FIG. 3. Differential branching ratios as a function of Due to the different kinematic regions where these decays take place, we have set m 2 ¼ 2.5 GeV and We first require events with at least one muon with p T > 1.7 GeV; this cut ensures that the events pass the same hardware trigger used at ffiffi ffi s p ¼ 8 TeV [19]. We subsequently require exactly six muons, with vanishing total charge. We also require all muon tracks to have p T > 0.5 GeV and 2.5 < η < 5.0. Finally, we require all muons tracks to have total momentum larger than 2.5 GeV to simulate the threshold for muon identification based on the penetration power through absorption plates in the detector. Due to the six muons in the final state, the SM backgrounds are negligible to very good approximation. They arise mostly from resonant production of J=Ψ and φ with subsequent decays into muons; we completely remove them by enforcing that no zero charge muon pair has an invariant mass in the range ½0.95; 1.09 ∪ ½3.0; 3.2 GeV. (We lose sensitivity to signal events with m 1 in that region, though.) Even searches for four muons are background free [19,21], so it is guaranteed that any observed event in the six lepton final state is due to the signal.
We generate signal B meson events using PYTHIA v8 [25] and MadGraph v5 [26] with Feynrules v2 [27] for the decays. (We have cross-checked our event distributions using EvtGen [28].) Following Ref. [21], we compare the (mass dependent) efficiencies for selecting events in the channel B 0 The former is shown in Table I, while we estimate the latter to be ε 2μ þ 2μ − ∼ 0.14. The explanation for the smaller efficiencies for the six-muon process is twofold. First, due to the larger number of final state tracks, there are more events with no single muon with p T > 1.7 GeV, which therefore do not pass the trigger; see Fig. 5. And second, there are more muons with at least one track with p T < 0.5 GeV which is therefore not detected; see Fig. 6.
Given the absence of background, we can estimate the upper limit on the branching ratio of the new processes at ffiffi ffi s p ¼ 14 TeV and luminosity L 0 as where B 2μ þ 2μ − max is the upper limit on BðB 0 s → 2μ þ 2μ − Þ ¼ 2.5 × 10 −9 , obtained in Ref. [19] with L ¼ 3 fb  Table I.
We also consider the channel B þ → K þ 3μ þ 3μ − . In this case, on top of the selection criteria proposed before, we require the presence of a charged kaon which is also required to have p T > 0.5 GeV and 2.5 < η < 5.0. The corresponding efficiencies are shown in Table I. The limit on the branching ratio can be again obtained as The upper limits (×10 −9 ) on the corresponding branching ratios for 3 fb −1 of data are also shown. We vary m 1;2 in the colored region of Fig. 2, with m 2 < 10 GeV and m 1 ≥ 1.1 GeV. (For smaller values of m 1 , the efficiency is negligible.) where the factor 3.7 stands for the larger B þ production cross section [29]. The bounds obtained this way are also shown in Table I. It is worth noting that the prospective limits on this channel are comparable or even more stringent than that on the decay mode without the extra meson (due mostly to the larger cross section, that compensates the smaller efficiency). This fact, together with the observation that theoretically this decay mode dominates for m 2 ∼ m 1 , strongly motivates searches for For illustration, we translate the expected limits in Table I to the plane ðg sb ; m V Þ in Fig. 7 for definite values of g 12 , m 1 , m 2 , and m 12 (when relevant). Prospects for the Upgrade II, defined by L 0 ¼ 300 fb −1 , are also shown. It is interesting to see that with our proposed analyses we can easily test masses larger than 15 TeV, thereby outperforming constraints obtained from ΔM s and completely probing the region in which the anomalies in lepton flavor universality can be explained.
Likewise, we also translate the aforementioned bounds to the plane ðm 1 ; m 2 Þ in Fig. 8, fixing g sb ¼ 0.04 as well as m V ¼ 4 TeV. Such values are not yet excluded by measurements of ΔM s ; see Ref. [30]. In both figures, only the weakest limits of Table I are used. We also note that, if a signal is observed in these sixmuon channels, the mass of the scalar particles involved could be reconstructed due to the outstanding detector resolution of the LHCb. To this aim, we provide two different algorithms, depending on whether m 2 > 2m 1 (in which case a 2 → a 1 a 1 ) or rather m 2 < 2m 1 (and therefore a 2 → a 1 μ þ μ − ).
For the first case, we minimize the difference jm rec 11 − m rec 12 j þ jm rec 12 − m rec 13 j, where m rec i is the invariant mass of each combination of opposite-sign muons. The two a 1 s that reconstruct the heavier scalar are those with the minimum ΔR among themselves; see Fig. 9 for an example.
Concerning the second case, the muon pairs reconstructing the two a 1 s are selected as those minimizing the difference jm rec 11 − m rec 12 j among the three pairs of muons. Then, a 2 is reconstructed from the two muons not assigned to any a 1 and the a 1 that minimizes ΔRðp 1 ; p μμ Þ (with p 1 FIG. 7. Maximum value of m V that can be tested in the searches for B 0 s → 3μ þ 3μ − and B þ → K þ 3μ þ 3μ − at the current run of the LHCb (solid lines) and for Upgrade II (dashed lines). The red dotted line delimits the area excluded by measurements of ΔM s . In the dash-dotted line, the anomalies in R K and R K Ã can be explained at the 1σ level assuming g Vll ∼ 1 [30]. We have fixed g 12 ¼ 0.5 as well as m 1 ¼ 1.2 GeV. We have set m 2 ¼ 2.0 GeV for both B s 0 → a 1 a 2 and B þ → K þ a 1 a 2 . For B s 0 → a 1 a 1 a 1 , we have fixed instead m 2 ¼ m 12 ¼ 5 GeV.

IV. HIGH-MASS REGIME AT THE LHC
In the high-mass regime, a 1;2 can no longer be produced in the decay of B mesons. However, if V is light enough (m V ≲ few TeV), it can be produced on shell at colliders, giving rise to a 1;2 pair production upon decay. The tree level signal cross section for g qq ¼ 0.5 and g 12 ¼ 1 ranges between ∼0.04 pb and ∼10 −5 pb for m V between 1 and 5 TeV.
There are multilepton searches at the LHC which are very sensitive to this scenario. Most of them rely on substantial missing energy, being therefore not relevant for our model. In this work, we consider the signal region dubbed SR0A in the analysis of Ref. [31]. The main selection cuts of that study are (i) at least four isolated leptons, (ii) no hadronic taus, (iii) no pair of opposite-sign leptons with invariant mass in the range [81.2, 101.2] GeV, and (iv) m eff > 600 GeV, where m eff stands for the scalar sum of the p T of all leptons, jets with p j T > 40 GeV and missing energy.
Only hadronic tau candidates with p τ T > 20 GeV are considered in (ii); jets are reconstructed using the anti-k t algorithm with R ¼ 0.4. The experimental analysis reports the observation of 13 events, while 10.2 AE 2.1 are predicted in the background-only hypothesis. Using these numbers including the systematic uncertainty on the SM prediction, we obtain that the maximum number of allowed signal events is 12. Scaling the expected number of background events with the larger luminosity, and assuming the same uncertainty, the expected maximum number of signal events at the HL-LHC is 300.
We recast this analysis using homemade routines based on ROOT v5 [32], HepMC v2 [33], and FasJet v3 [34]. We define hadronic taus as jets with angular separation smaller than 0.2 from a true hadronic decayed tau lepton.
We establish a flat tau-tagging efficiency of 0.5. We consider light leptons to be isolated if the hadronic activity around ΔR ¼ 0.2 of the corresponding lepton is smaller than 10% of its transverse momentum. On top of the cuts above, we require that the angular separation between any pair of muons is larger than 0.05, to simulate their correct reconstruction at detectors.
We generate signal events for pp → V → a 1 a 2 with the corresponding scalar decays with MadGraph v5 [26] with no parton level cuts. For the PDFs, we use the NNPDF23LO set [35]. Signal events are subsequently passed through PYTHIA v8 [25] to account for initial and final state radiation, fragmentation, and hadronization effects.
If the light scalars couple mostly to the tau lepton (second scenario introduced in Sec. II), the aforementioned signal region has no sensitivity. We can rely instead on the signal region SR2 defined in the same experimental paper of Ref. [31], which requires (i) exactly two light leptons with invariant mass not in the range [81.2, 101.2] GeV; (ii) at least two hadronic taus with p τ T > 30 GeV; m eff > 650 GeV. The experimental collaboration reports the observation of two events, the SM prediction being 2.3 AE 0.8. Using again the CL s method, we obtain 6 (121) events as the current (future) maximum allowed signal.
We scan over 20 values of m 1 and m 2 in logarithmic scale in the range [1,500] GeV, with special attention to low masses as well as masses close to the Z pole.
In Fig. 11, we depict the region in the ðm 1 ; m 2 Þ plane for g qq ¼ 0.5 and g 12 ¼ 1 that is already excluded in the muonphilic case and also in the case with couplings to taus. The exclusion prospects for the HL-LHC, defined by 3 ab −1 , are also shown. The tau analysis is much less constraining (mainly due to the small branching ratio to leptons), and thus we only show results for m V ¼ 1 TeV. FIG. 11. Region in the plane ðm 1 ; m 2 Þ that is excluded by multilepton searches (solid red) and lepton-tau searches (solid green) [31]. The dashed lines represent the corresponding prospects at the HL-LHC. We have fixed g qq ¼ 0.5, g 12 ¼ 1. The low sensitivity in the small m 1 region is due to muons being very collimated. (Decays into taus are furthermore forbidden for m 1 ≲ 4 GeV.) If it were possible to resolve muons with angular separations as small as 0.001, then almost the whole small mass range could be tested in the muonphilic case.
Likewise, the nonexcluded region around m 1 ∼ 100 GeV results from the Z veto of the analysis. This region could be covered if the veto on the Z pole is removed and, instead of m eff , the invariant mass of all final state observable objects (which in our signal, and contrary to the SUSY targets of the analysis, presents a narrow peak) is used. Such improvement would also extend the reach to smaller masses. It is therefore desirable that future updates of the experimental work consider different versions of the cut on m eff .
In the same vein, in Fig. 12, we plot the minimum value of g qq that can be tested for different values of m V and for fixed values of m 1;2 . We have also fixed g 12 to the value for which BðV → a 1 a 2 Þ ∼ 0.25.

V. CONCLUSIONS
We have studied the phenomenology of light leptophilic scalars a 1;2 that couple to a heavy flavor violating (mostly b − s like) spin-1 resonance V. We have shown that, under very mild conditions, a 2 decays mostly into a 1 , which subsequently decays into pairs of leptons. Thus, for scalar masses ≲ few GeV, this scenario produces new B meson decays into six muons, namely B 0 s → 3μ þ 3μ − and B þ → K þ 3μ þ 3μ − . Interestingly, the later dominates over the second when m 1 ∼ m 2 . None of them has been explored experimentally; we have therefore proposed dedicated analyses to explore these signals at the LHCb. We have found that branching ratios as small as 6.0 × 10 −9 (5.9 × 10 −9 ) for the first (second) process can be already tested with the current luminosity. Branching ratios hundred times smaller could be probed at the Upgrade-II of the LHCb.
For larger scalar masses, a 1=2 arise rather in the decay of V, which can be produced on shell at pp collisions at the LHC. Current multilepton searches in final states with muons (taus) constrain most of the parameter space for m 1 ≳ 10 GeV provided that σðpp → V → a 1 a 2 Þ ≳ 0.001ð0.01Þ pb. Smaller masses give rise to very collimated leptons (or jets) that are difficult to disentangle at detectors. However, at the HL-LHC, the reach can be extended to m 1 ≲ 5. And even further if the current analyses cut on the invariant mass of all visible objects.
Finally, let us comment how these results would get modified if different flavor assumptions are made. To start with, if a 1;2 are not leptophilic but rather they couple to all SM fermions with Yukawa-like couplings, the branching ratio of a 1 into leptons would get reduced by 1-2 orders of magnitude. In turn, LHCb would be only sensitive to exotic branching ratios thousand times larger. (Note that such branching ratios are not excluded by any current measurement, though.) However, LHC searches in multilepton final states would lose almost all sensitivity in this case.
On the other hand, V might also induce b − d transitions. In that case, we expect new rare decays such as B 0 → 3μ þ 3μ − . The production cross section for B 0 is ∼3.7 larger than for B 0 s [29], from where we estimate that BðB 0 → 3μ þ 3μ − Þ ≳ 1.6 × 10 −9 (∼10 −11 ) can be probed currently (in the Upgrade-II of the LHCb).
On the theory side, this channel vanishes also at tree level when m 1 ∼ m 2 . In this regime, we propose searching for B 0 s → K Ã0 3μ þ 3μ − , with K Ã0 → K þ π − , whose branching ratio is around 2=3 [36]. Upon performing an equivalent analysis to that described in Sec. III, we obtain efficiencies of about 2 times smaller, in comparison to the B 0 s → 3μ þ 3μ − channel. Consequently, we estimate the LHCb reach to be BðB 0 s → K Ã0 3μ þ 3μ − Þ ≳ 1.8 × 10 −8 currently and again about hundred times stronger in the Upgrade-II.
At high scalar masses, the prospects are only slightly better than for b − s transitions, because the production cross section for V at the LHC grows only by a very small factor. Both low and high energy searches are also more constraining than bounds on ΔM d [37] on a wide region of the parameter space.
Overall, our study motivates new searches for B 0 s → ðK 0Ã Þ3μ þ 3μ − and B þ → K þ 3μ þ 3μ − at the LHCb as well as small modifications of current multilepton and multitau analyses at CMS and ATLAS.

ACKNOWLEDGMENTS
We are grateful to Ulrik Egede for previous collaboration that opened this new line of research. M. C. is supported by the Royal Society under the Newton International Fellowship programme. M. R. is supported by Fundação   FIG. 12. Minimum value of g qq that it is excluded by multilepton searches (solid red) and lepton-tau searches (solid green) [31]. The dashed lines represent the corresponding prospects at the HL-LHC. We have assumed BðV → a 1 a 2 Þ ∼ 0.25. para a Ciência e Tecnologia (FCT) under the Grant No. PD/ BD/142773/2018 and also acknowledges financing from LIP (FCT, COMPETE2020-Portugal2020, FEDER, POCI-01-0145-FEDER-007334). M. R. would like to thank the IPPP Durham for hospitality where the main part of this work was carried out. M. S. acknowledges the hospitality of the University of Tuebingen and support of the Humboldt Society during the completion of parts of this work.

APPENDIX: CONCRETE COMPOSITE HIGGS MODEL
Nonminimal CHMs is the context where heavy vector bosons and new light scalars, separated by a large mass gap, arise more naturally. The reason is that the latter are pseudo-Nambu-Golstone bosons (pNGBs) from the spontaneous breaking of G=H, at a scale f ∼ TeV.
The smallest coset for which the scalar sector consists of the Higgs degrees of freedom as well as two SM singlets is SOð7Þ=SOð6Þ [7][8][9]. The corresponding 15 unbroken and 6 broken generators, T and X, respectively, can be written as Without loss of generality, the pNGB matrix can be written as with Σ 2 ¼ 1 − ðh 2 þ a 2 1 þ a 2 2 Þ=f 2 . Following the partial compositeness paradigm [38], the couplings of a 1;2 to the SM fermions, as well as the scalar potential, depend on the quantum numbers of the composite operators that the SM fermions mix with breaking the global symmetry. Or equivalently, they depend on how the SM fermions are embedded in representations of SOð7Þ.
We assume that q L þ u R ∼ 7 þ 21. Likewise, we assume that l L þ e R ∼ 27 þ 1. Explicitly, L L ≡ ν L Λ e þ e L Λ ν , where the vectors read v 1 ¼ ðe L ; −ie L ; ν L ; iν L Þ and v 2 ¼ ðie L ; e L ; iν L ; −ν L Þ and θ and γ are real parameters. (Note that the different embeddings for quarks and leptons are primarily justified by the fact that the lepton and quark masses and mixings are completely different.) The scalar potential can be written as Vðh; a 1;2 Þ ¼ V q ðh; a 1;2 Þ þ V l ðh; a 1;2 Þ, where the first and second contributions of the rhs come from loops of quarks and leptons, respectively. It can be also shown that the quark sector respects a symmetry a 1;2 → −a 1=2 , as well as the shift symmetry of the singlets. Consequently, V q ðh; a 1;2 Þ ¼ V q ðhÞ. It is completely fixed by the measurements of the Higgs mass and its vacuum expectation value.
The only model dependence come from V l ðh; a 1;2 Þ, which to leading order in the expansion in the global symmetry breaking parameters reads where the dressed spurion reads Λ α D ≡ U T Λ α U with α ¼ e, ν. (The indices 1 and 6 indicate the projection into the singlet and the sextuplet in the decomposition 27 ¼ 1 þ 6 þ 20 from SOð7Þ to SOð6Þ.) The constants c 1 and c 2 are free parameters encoding the (unknown) details on the strongly coupled UV. Writing explicitly the one-loop induced potential, we find We further expand this expression in powers of 1=f, and keep only terms up to dimension four, V l ∼ 4f 3 c 2 γa 2 þ 2f 2 c 2 ½ðγ 2 − 1Þa 2 2 þ ðθ 2 − 1Þa 2 1 þ 2fγ½ðc 1 − 3c 2 Þa 2 h 2 − c 2 ða 2 1 þ a 2 2 Þa 2 þ ðc 1 − 2c 2 Þ½ðθ 2 − 1Þa 2 1 þ ðγ 2 − 1Þa 2 2 h 2 þ …; ðA5Þ where the three dots encode terms involving the Higgs boson solely. The requirements c 1 ∼ 3c 2 and γ ∼ 1 make the interactions between a 2 and the Higgs (in particular mixings) very small. In order to avoid bounds from Higgs searches, we restrict to this case hereafter. The tadpole can then be removed by the field redefinition a 2 → a 2 þ ffiffiffiffiffiffiffi ffi 2=3 p f. Let us also fix f ¼ 1 TeV, as well as c 2 ∼ g 2 Ã y 2 l = ð4πÞ 2 ∼ 10 −6 . The latter is the value expected from strongly-interacting light Higgs power counting [39] for g Ã ∼ 3, with g Ã the typical strong coupling between composite resonances. This choice fixes both m 2 and m 12 to ∼3.1 and ∼0.002 GeV, respectively; while m 1 depends solely on θ. We compute numerically this dependence and it is depicted in Fig. 13.
On another front, the Yukawa Lagrangian to dimension five reads L Y ¼ yfl L α ðΛ α D Þ † 77 e R þ H:c: The vector resonance associated to the generator T 56 is the only one that couples to a 1;2 . We identify it with V. The interaction between V and the pNGBs is entirely determined by the Callan-Coleman-Wess-Zumino formalism [40][41][42], reading where e μ is the trace of the Maurer-Cartan form ω along the unbroken generators We expect m V ∼ g Ã f. Therefore, the interaction between the vector resonance and the light scalars reads Finally, the vector resonance cannot couple directly to the left-handed quarks. The coupling g qq is therefore suppressed by v 2 =f 2 ≲ 0.1. Altogether, this model matches into the parametrization in Eq. (1). For example, let us take θ ¼ 1.2. We obtain m 1 ∼ 1.3 GeV, m 2 ∼ 3.1 GeV, m 12 ∼ 0.002 GeV, g qq ∼ 0.1, g 12 ∼ 2, g 2 ∼ 0.17, g 1 ∼ 0.22.
These numbers are also obtained if the leptons are embedded in 7 þ 7.