The economical 3-3-1 model revisited

We show that the economical 3-3-1 model poses a very high new physics scale of the order of 1000~TeV due to the constraint on the flavor-changing neutral current. The implications of the model for neutrino masses, inflation, leptogenesis, and superheavy dark matter are newly recognized. Alternatively, we modify the model by rearranging the third quark generation differently from the first two quark generations, as well as changing the scalar sector. The resultant model now predicts a consistent new physics at TeV scale unlike the previous case and may be fully probed at the current colliders. Particularly, due to the minimal particle contents, the models under consideration manifestly accommodate dark matter candidates and neutrino masses, with novel and distinct production mechanisms. The large flavor-changing neutral currents that come from the ordinary and exotic quark mixings can be avoided due to the approximate $B-L$ symmetry.


INTRODUCTION
There have been up to now certain experimental evidences for the physics beyond the standard model prediction [1]. The most important issues of which must include neutrino oscillations, baryon-number asymmetry, dark matter, and inflation. The traditional proposals such as supersymmetry, extradimension, and grand unification solve only some of the questions separately, and obviously they obey several theoretical and experimental issues [1]. In this work, we will show that the model based on the gauge symmetry SU (3) C ⊗ SU (3) L ⊗ U (1) X (3-3-1) [2,3] may be an intriguing choice for the new physics due to its ability to solve the underlined problems integrally.
The weak isospin group SU (3) L that is directly extended from SU (2) L of the standard model is well-established as it is able to determine the number of generations to match that of colors as observed by the [SU (3) L ] 3 anomaly cancelation. However, the electric charge Q neither commutes nor closes algebraically with SU (3) L . Hence, a new Abelian group is deduced as a result to close those symmetries by the gauge group SU (3) C ⊗ SU (3) L ⊗ U (1) X , where Q = T 3 + βT 8 + X with T n (n = 1, 2, 3, ..., 8) and X indicating to the SU (3) L and U (1) X charges, respectively (cf. [4]).
For this aim, we first consider the 3-3-1 model with arbitrary β and extract the bound for the 3-3-1 breaking scale from FCNCs, which depends only on the arrangement of quark representations, as presented in Sec. II. Turning to the model under investigation, two folds for it, i.e. the economical 3-3-1 model, are derived, and the corresponding consequences are discussed, as given in Sec. III.
Note that these economical 3-3-1 models are not limited by a Landau pole since this pole is actually higher than the Planck scale [28]. We finally conclude this work in Sec. IV.

II. FCNCS
The 3-3-1 model with arbitrary β is given by the electric charge operator as mentioned Supposing that the first quark generation transforms differently from the last two under SU (3) L , the fermion content which is anomaly free is achieved as e aR ∼ (1, 1, −1), k aR ∼ (1, 1, q), ν aR ∼ (1, 1, 0), where a = 1, 2, 3 and α = 2, 3 are generation indices, and the electric charges of the new particles are related to the basic electric charge, q = −(1 + √ 3β)/2, by Q(k a ) = q, Q(j 1 ) = q + 2/3, and Q(j α ) = −q − 1/3, as aforementioned. The numbers in parentheses denote representations with respect to the 3-3-1 groups. ν aR are sterile which may be imposed or not. This similarly applies for k aR if q = 0. Two minimal 3-3-1 versions have been studied, where the singlets ν aR and k aR are omitted, while k aL are replaced by (e aR ) c or (N aR ) c , respectively [2,3]. Here N aR are some neutral fermions like ν aR . The above ingredient does not apply for quarks since SU (3) C , SU (3) L , and spacetime symmetry commute. By contrast, if the second or third quark generation is arranged differently from the two others, α takes values, α = 1, 3 or α = 1, 2, respectively.
The quark generations are not universal under the SU (3) L ⊗ U (1) X symmetry, therefore there could be FCNCs. The neutral current takes the form where F runs over all fermion multiplets, and note that the covariant derivative is D µ = ∂ µ + ig s t n G nµ + igT n A nµ + ig X XB µ , with the last three terms containing gauge coupling, generators, and gauge bosons for the 3-3-1 groups, respectively. We have also used X = Q − T 3 − βT 8 and where we denote either q = (u 1 , u 2 , u 3 ) for up quarks or q = (d 1 , d 2 , d 3 ) for down quarks, and , and q 3L , respectively. In the mass basis, we have q L,R = V qL,qR q ′ L,R , where either q ′ = (u, c, t) or q ′ = (d, s, b), and V qL,qR are quark mixing matrices that diagonalize the corresponding quark mass matrices such which causes FCNCs for i = j, where i, j = 1, 2, 3 indicate to respective physical quark states. Z ′ might mix with Z and V = A 4 or V = A 6 for q = 0 or q = −1, respectively. The contribution of V to the FCNCs is negligible. We write Z ′ = −s ϕ Z 1 + c ϕ Z 2 , where Z 1,2 are two physical neutral gauge bosons with masses and the Z-Z ′ mixing angle is Substituting Z ′ into the above FCNCs and integrating Z 1,2 out, we obtain the effective Lagrangian describing meson mixings, The Z 1 contribution is negligible too, since which is suppressed due to v w ≪ w, where we have used |β| < 1/t W derived from the photon field normalization and gauge coupling matching, s W = e/g = t X / 1 + (1 + β 2 )t 2 X , as partly mentioned, and v 2 w = u 2 + v 2 = (246 GeV) 2 identified from the W boson mass. It is easily proved that the ρ-parameter deviation from the standard model value due to Z-Z ′ mixing is , which again implies the nonsignificant contribution of Z 1 due to ∆ρ < 0.0006 from the global fit [1]. Therefore, only Z 2 governs the FCNCs, leading to which is independent of β and the Landau pole if presented for large |β|, which is a new observation of this work and in agreement with a partial conclusion in [25].
In both economical 3-3-1 models discussed below, the ordinary (u a , d a ) and exotic (U, D α ) quarks correspondingly representing in the same triplet/antitriplet with the same electric charge might mix. Hence, the mixing matrices are now redefined as ( such that the 4 × 4 mass matrix of up-type quarks (u a , U ) and the 5 × 5 mass matrix of down-type quarks (d a , D α ) are diagonalized [34]. The FCNC Lagrangian as coupled to Z ′ is now changed to − g for down-type quarks. The corresponding effective Lagrangian is achieved as As mentioned in the above footnote, the ordinary and exotic quark mixings also lead to the FCNCs associated with Z, obtained by the Lagrangian, where " + " and I = 4 are applied for V u , whereas " − " and I = 4, 5 are applied for V d . Integrating Z out, the corresponding effective Lagrangian is which would spoil the standard model prediction for the neutral meson mass differences if the mixing of the ordinary and exotic quarks was compatible to the ordinary quark mixing. For which is much smaller than the smallest CKM matrix element. To avoid the large FCNCs, we assume so that (22) is insignificant, and (20) is thus reduced to (18). The above inequality is also valid when 1's are replaced by α = 2, 3, due to the unitarity condition, (V † qL V qL ) ij = 0. Furthermore, the B −L conservation demands that the exotic and ordinary quark mixings vanish [29,35]. Hence, the suppressions like (23) are naturally preserved by an approximate B − L symmetry, as interpreted in [4,25,36]. Lastly, there may exist tree-level FCNCs induced by new non-Hermitian gauge bosons X 0,0 * that couple u 1 with U , and d α with D α , given by the Lagrangian, where I = 2 + α. This yields the effective Lagrangian, where we have used m 2 X = g 2 4 (u 2 + w 2 ) ≃ g 2 w 2 /4. The X contributions to FCNCs as (25) are radically smaller than those of Z ′ in (18) due to the conditions (23). In summary, for any 3-3-1 model the FCNCs due to Z ′ in (18) would dominate, which will be taken into account.
Without loss of generality, by alignment in the up quark sector, i.e. V uL = 1, the CKM matrix The CKM factor is |(V * dL ) 11 (V dL ) 12 | ≃ 0.22 [1], which implies This bound applies for the considering model with nonuniversal first quark generation. If one arranges the second quark generation differently from the others, the CKM factor is similarly , which presents the same bound for w as the previous case. Furthermore, putting the third quark generation differently from the first two, the CKM factor is now smaller, Let us stress again that the bounds achieved in (27) and (28) are independent of β, applying for every 3-3-1 model with appropriate fermion content, i.e. quark arrangement, which is a new investigation of this work, in agreement with the special cases in [4,24].
We can similarly study for the B 0 [1,37] for nonuniversal first quark generation, and so forth for other cases. With the aid of the CKM factors in [1], if the second or third quark generation is different from the two others, it gives a bound w > 4 TeV. Otherwise, when the first quark generation is different, it gives a negligible contribution to the B meson mixing. We see that the B mixing effect does not discriminate the second and third quark generations, unlike the case of the kaon mixing. The B mixing gives the bound in agreement with the K mixing when the third generation is different.
However, it gives a negligible contribution to the B mixing when the kaon mixing bound is applied to the model with nonuniversal first or second quark generation.
Let us remind the reader that the detailed outcomes of the FCNCs (18) using the neutral meson mass differences are worth studying, but the overall bounds as obtained above would be expected (see, for instance, [31,32]). In other words, it is sufficient for the purpose of this work as to classify and interpret the new directions of the economical 3-3-1 models, to be discussed below.

III. TWO SCENARIOS FOR THE ECONOMICAL 3-3-1 MODEL
An economical 3-3-1 model is defined as to work with the minimal fermion and scalar content that includes ν aR in lepton triplets and contains only two scalar triplets, either (χ, ρ) or (χ, η).
Such theory has electric charge operator Q = T 3 − 1 √ 3 T 8 + X. As a result of the above analysis, there are two distinct economical 3-3-1 models. The first one has particle content as the original economical 3-3-1 model (i.e., possessing nonuniversal first quark generation and χ, ρ), but the 3-3-1 breaking scale is beyond 1000 TeV order, called type-I economical 3-3-1 model. By contrast, the second one has nonuniversal third quark generation and works with χ, η, which implies a TeV 3-3-1 breaking scale, called type-II economical 3-3-1 model.

A. Type-I economical 3-3-1 model
The fermion and scalar content is given as [8,10] Recall that α = 2, 3, and U, D have ordinary electric charges like u, d, respectively.
The 3-3-1 breaking scale is bounded by w >2200 TeV. Since χ 0 1 has L = 2 = 0, its VEV, u ′ , which breaks this charge should be much smaller than the weak scale, u ′ ≪ v. Because of u ′ = 0 there mix in the gauge boson sectors, the charged W -Y and the neutral Z-Z ′ -A 4 , in addition to the ordinary Z-Z ′ mixing. Diagonalizing these sectors we get physical eigenstates and masses similarly to [8]. Consequently, from the W boson mass, m 2 W = g 2 v 2 /4, we determine the weak scale v ≃ 246 GeV, as usual. The mixings in both gauge boson sectors shift the tree-level ρ-parameter from the standard model prediction by ∆ρ = ρ − 1 = to the global fit ∆ρ < 0.0006 [1]. Therefore, the mixings between exotic and ordinary quarks are proportional to u ′ /w ∼ 10 −6 which does not affect the FCNCs due to the Z exchange as well as the non-unitarity of ordinary quark mixing matrices as remarked before [29].
Note that all the new particles including Higgs bosons H 1,2 , gauge bosons Z ′ , Y, X, and exotic quarks U, D gain the masses proportional to the w scale, which are quite heavy as expected. The ordinary particles get consistent masses after the electroweak symmetry breaking, expect for the followings. Because of the minimal scalar content, there are 3 light quarks possessing vanishing tree-level masses. However, they can obtain appropriate masses induced by radiative corrections or effective interactions since the Peccei-Quinn symmetry is completely broken [11,17].
At the tree-level, the neutrinos have Dirac masses, one zero and two degenerate, which are unacceptable. But up to the five dimensional interactions, the relevant Yukawa Lagrangian is where Λ is a cut-off scale which can be taken as Λ ∼ w. Therefore, the observed neutrinos (∼ ν L ) gain Majorana masses via a seesaw mechanism, evaluated to be which naturally fits the data since w is large as 2200 TeV, e.g. taking m ν ∼ 0.1 eV and h ′ν ∼ 1 yields h ν ∼ 10 −4 which looks like the charged lepton Yukawa couplings. It is to be noted that the above neutrino mass generation scheme may be radiatively induced [12].
The scalar field that breaks SU , which provides the masses for the new particles as well as setting the seesaw scale, as mentioned.
Further, the imaginary part is the Goldstone boson of Z ′ , while the real part includes a new neutral Higgs boson living in the w scale. In the early universe, the real field Φ = √ 2ℜ(χ 0 3 ) can be interpreted as an inflaton field involving (in time) toward the potential minimum Φ min = w, driving the cosmic inflation. Let us consider its potential when the inflation scale is either not too high but significantly larger than w or close to the Planck scale.
For the first case, the inflation potential is radiatively contributed by the gauge bosons, fermions, and scalars which takes the form (up to the leading-log approximation) [38], where the renormalization scale has been fixed at w, and Here, g X = gt W / 1 − t 2 W /3 is used; h U,Dα denote the Yukawa couplings of inflaton with exotic quarks U, D α ; and λ, λ ′ correspond to the self-inflaton and Higgs-inflaton quartic couplings, respectively. This potential yields an appropriate local minimum given that a/λ > −63.165. Additionally, since w is radically smaller than the Planck scale, the inflation potential is governed by the quartic and log terms. The number of e-folds will be chosen N > ∼ 40 so that the inflation scale is correspondingly higher than the expected 2200 TeV value. The CMB measurements yield a constraint on the curvature perturbation which leads to λ < ∼ 10 −12 [1]. Further, the spectral index n s , the tensor-to-scalar ratio r, and the running index α can be evaluated as functions of a ′ ≡ a/λ and fitting to the data [1]. Then we obtain a ′ ∼ −10, and thus g ∼ h U,Dα ∼ √ λ, λ ′ < ∼ 10 −2.75 , which contradicts the electroweak data g ∼ 0.5. Conversely, this regime of the potential is not flat to reproduce a suitable inflation scenario.
For the second case, the interaction of inflaton to gravity via a non-minimal coupling ξ may be important, where R is the scalar curvature and m P = (8πG N ) −1/2 ≃ 2.4 × 10 18 GeV is the reduced Planck mass. We assume ξ > ∼ 1 and the action can be rewritten in the Einstein frame as [39] where the inflation potential is related to the canonically-normalized inflaton field φ as That said, the inflation potential is flat due to the large field values, φ ≫ m P or Φ ≫ m P / √ ξ, and it successfully fits the data if ξ ∼ 10 4 √ λ, in agreement to [39]. In this case, the number of e-folds set is about 60. Since λ = m 2 H 1 /(2w 2 ) can be small for a H 1 mass of few TeVs, the unitarity condition ξ < ∼ O(10) is recognized, and the inflation begins from the Planck regime Φ ∼ M P . The reheating happens when the inflaton decays into the exotic quarks or new gauge bosons. Considering the first case, it yields T R ∼ h U,Dα (w/1000 TeV) 1/2 × 10 11 GeV ∼ 10 11 GeV.
Since the right-handed neutrinos do not directly couple to the inflaton, they could only be produced from the thermal bath of radiations. The CP asymmetry decays of these right-handed neutrinos into a heavy charged Higgs boson and a charged lepton ν R → H ± 2 e ∓ due to the Yukawa couplings h e abψ aL ρe bR + H.c. can generate the expected baryon asymmetry via a leptogenesis mechanism similarly to the standard technique provided that m ν R > ∼ m H 2 [40]. However, it differs from the standard prediction due to the fact that the channels ν R → G ± W e ∓ via the couplings h ν abψ c aL ψ bL ρ + H.c. are negligible as suppressed by h ν ≪ h τ and m W ≪ m H 2 . Additionally, like the neutral field H 1 , the finding of the charged field H 2 with some mass in the TeV regime can mark the existence of this baryon-asymmetry production scheme.
Let us emphasis that the economical 3-3-1 model has a natural room for dark matter as basic scalars filling up the model [25,36]. As studied in [36], the dark matter candidate might be resided in an inert scalar triplet η, as a replication of χ and odd under a Z 2 symmetry. We may have another inert scalar triplet for dark matter as a replication of ρ. However, in this model, the candidate has a mass proportional to the 3-3-1 scale of 1000 TeV order. Therefore, if its mass is in or beyond this scale, it cannot be generated as thermal relics as of [36], since otherwise it overcloses the universe due to the unitarity constraint [41]. Interestingly, this superheavy dark matter can be generated in the early universe by some mechanisms such as gravitational and thermal productions, associated with the discussed inflation and reheating, analogous to [42]. By contrast, the thermal generations may be interpreted as in [36] if the inert field masses are at TeV scale.
Hence, by the realization of a high 3-3-1 breaking scale, the 3-3-1 model might integrally explain the neutrino masses and the cosmological issues, comparable to the other theories [4,24,29,35,[42][43][44][45]. Note that the usual 3-3-1 models do not reveal the inflation and associated superheavy dark matter. A detailed investigation of all the issues for this kind of the model is out of the scope of the present work, which should be published elsewhere [46].

B. Type-II economical 3-3-1 model
A low bound for the 3-3-1 breaking scale is only if the third quark generation is discriminative.
For this case, the scalar that breaks the electroweak symmetry should be η instead of ρ, in order to generate the top quark mass consistently (by contrast, with the scalar content as the previous model retained, the top quark has vanishing tree-level mass and it is impossible to be induced by radiative corrections or effective interactions, cf. [25], for instance). Thus, the fermion and scalar content is appropriately derived as where note that α = 1, 2 and t ξ = u ′ /w ′ as well as the physical scalar spectrum as explicitly displayed can be obtained from the following scalar potential. Recall also the FCNC bounds: w > 3.5 TeV for the K mixing and w > 4 TeV for the B s mixing.
The total Lagrangian is L = L kinetic + L Yukawa − V scalar , where where F, S run over fermion and scalar multiplets, respectively. G nµν , A nµν , and B µν are field strength tensors corresponding to the 3-3-1 groups, and D µ is covariant derivative as previously supplied. The Yukawa Lagrangian and scalar potential are By the criteria in [4,35] with L(ν R ) = 1 and B(ν R ) = 0, the baryon minus lepton number It is easily checked that the leptons and three ordinary quarks have vanishing tree-level masses.
Furthermore, the Lagrangian automatically contains the Peccei-Quinn like symmetries, similarly to the original economical 3-3-1 model [17]. Such massless particles can get appropriate masses when the Peccei-Quinn like symmetries are completely broken via radiative corrections or effective interactions [17]. Let us impose the latter, which is up to five dimensions, given by where as usual the unprimed couplings conserve B − L, while the primed couplings stand for the violating ones. The cutoff scale Λ can be taken in the same order as w. Specially, f ′ν ab and g ′ν ab are symmetric, while h ′ν ab is general, in flavor indices. The fermion mass matrices can be derived when substituting the VEVs of the scalars into the Lagrangians (47) and (49). Using the conditions that the violating parameters are radically smaller than the corresponding conserved ones, the charged leptons obtain masses, [M e ] ab ≃ h e ab u w 2Λ , where w ∼ Λ and u = 246 GeV as imposed from the W boson mass, which can fit the data as the standard model. Since u ≪ w, we have seesaw mechanism for the neutrino masses. Indeed, the right-handed The new observation is that the neutrinos get masses when both the Peccei-Quinn and B − L symmetries are broken. The strength of the symmetry breakings is set by the primed couplings of the effective interactions, commonly called h ′ , thus M ν ∼ u 2 Λ h ′ . Note that for the 3-3-1 model, if B − L is conserved, it must be a gauged charge, and thus the effective interactions must absent themselves [4,24,29,35,[42][43][44]. Therefore, h ′ measures the approximate B − L symmetry as well as the nonunitarity of the 3-3-1 model, as imprinted from the 3-3-1-1 model. The h ′ strength can be obtained by integrating U (1) N gauge boson out in the 3-3-1-1 model, which matches h ′ /Λ = g N /Λ N . Further, we have h ′ ∼ Λ/Λ N ∼ 10 −11 , where Λ N ∼ 10 14 GeV is just inflation scale and g N ∼ 1 [42,43]. This implies M ν ∼ 0.1 eV as desirable. Alternatively, comparing M ν /M e ∼ u w h ′ h with u/w ∼ 0.1 and M ν /M e ∼ 10 −6 , it yields h ′ /h ∼ 10 −5 , and thus the breaking strength h ′ is suitably smaller than the electron Yukawa coupling, in agreement to [25].
The mixings of the exotic and ordinary quarks are proportional to u ′ /u, w ′ /w, and h ′ /h-the ratios of the B − L violating parameters over the corresponding normal ones [34]. Again, the VEVs u ′ , w ′ and couplings h ′ should be small, u ′ ≪ u, w ′ ≪ w, h ′ ≪ h, in order to suppress the dangerous FCNCs coming from Z boson exchange due to the ordinary and exotic quark mixings.
Generalizing the result in [29], we obtain where (V dL ) Ii is the element that connects the corresponding exotic and ordinary quarks in the mixing matrix. It yields u ′ < ∼ 0.77 GeV due to u = 246 GeV, and w ′ < ∼ 3.16, 15.8, and 31.6 GeV for w = 1, 5, and 10 TeV, respectively, as well as h ′ is more suppressed, similarly to the ones for the neutrino masses. In practice, the VEVs u ′ , w ′ break B − L (i.e., the lepton number), and they are suppressed to be small by the corresponding lepton-number violating scalar-potential. From the potential, we have roundly u ′ ∼ λ ′ 7 u and w ′ ∼ λ ′ 7 w. Thus, u ′ and w ′ should be small since its absence, i.e. λ ′ 7 = 0, enhances the 3-3-1-1 gauge symmetry. Following the approach in [25,36], the model can provide realistic dark matter. The inert triplet ρ, which is odd under a Z 2 symmetry and analogous to the one in the 3-3-1 model with right-handed neutrinos, if introduced cannot be dark matter since the candidate ρ 0 2 = 1 √ 2 (H + iA) yields degenerate masses for H and A, which implies a large direct detection cross-section via Z exchange, already ruled out by the experiment [47]. However, an inert triplet as replication of η or χ, called ζ = (ζ 0 1 , ζ − 2 , ζ 0 3 ), provides a consistent candidate as the combination of either real or imaginary parts of ζ 0 1,3 . The inert scalar sextet responsible for dark matter can be also interpreted, similarly to the simple 3-3-1 model in [25].
In summary, the 3-3-1 model with right-handed neutrinos has a nontrivial vacuum for u ′ = 0 and w ′ = 0, and this yields the appropriate new-physics consequences as obtained. Interestingly, the type II economical 3-3-1 model is a minimal realization of this vacuum, while explicitly indicating to dark matter. See [48] for other interpretations. Note that the previous studies [3] only consider the vacuum with u ′ = w ′ = 0, and thus the above consequences were not recognized, although they include more than two scalar triplets.

IV. CONCLUSION
As a fundamental element, the 3-3-1 model presents the FCNCs via Z ′ boson due to nonuniversal fermion generations under the gauge symmetry. We have proved that the FCNCs describing meson mixings are independent of the embedding of electric charge operator as well as the potential Landau pole. Applying for the K and B s mixings, we obtain the new physics scale w > 2200 TeV if the first or second fermion generation is discriminative, and w > 3.5 TeV for the K system or w > 4 TeV for the B s system if the third fermion generation is discriminative.
Due to the above constraint, the original economical 3-3-1 model works in a large energy regime, yielding the seesaw mechanism, inflation, leptogenesis, and superheavy dark matter integrally. The 3-3-1 breaking field, χ 0 3 , is important to set the seesaw scale originating from inflation scale and define inflaton. Dark matter is a hidden/inert field, a replication of χ, called η, or of ρ, called ρ ′ , which might be created in the early universe by nonthermal processes/mechanisms associated with the inflation and reheating. The imprints at TeV scale of the inflation and leptogenesis mechanisms are the new Higgs fields H 1,2 which may be verified at the LHC.
Alternatively, we have introduced a new economical 3-3-1 model, where the third fermion generation is rearranged differently from the first two and the scalar content includes η, χ. This model works naturally at the TeV scale. The lepton number breaking/violating parameters are suppressed, u ′ ≪ u, w ′ ≪ w, h ′ ≪ h, and λ ′ ≪ λ. The strength of the lepton number breaking might have a source from the 3-3-1-1 breaking to be actually small, responsible for the neutrino masses.
It is shown that a hidden scalar field ζ as a replication of η or χ can provide WIMPs as thermal relics. However, ρ if included as an inert scalar cannot be dark matter.
Let us stress that the discrimination of fermion generations as recognized at 1000 TeV order is surprisingly close to the WIMP mass bound. Although the 3-3-1 model does not directly solve this coincidence, it provides both scenarios for dark matter as nonthermal and thermal relics. Therefore, these two economical 3-3-1 models would predict and connect the particle physics to the cosmological issues with rich phenomenologies, attracting much attention [46].