Novel effects of the W-boson mass shift in the 3-3-1 model

The recent precision measurement of the W-boson mass reveals an exciting hint for the new physics as of the 3-3-1 model. We indicate that the 3-3-1 model contains distinct sources by itself that may cause the W-mass deviation, as measured, such as the tree-level Z-Z′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z'$$\end{document} mixing, the tree-level W-Y and Z-Z′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z'$$\end{document}-X mixings, as well as the non-degenerate gauge vector (X, Y) and new Higgs doublets. We point out that the gauge vector doublet negligibly contributes to this mass shift, whereas the rest of the effects with tree-level mixings governed by Z-Z′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z'$$\end{document} and new Higgs doublets are significant. A discussion of scalar sextet contributions is also given.


I. INTRODUCTION
The CDF collaboration has recently announced a new result of W -boson mass m W | CDF = 80.4335 ± 0.0094 GeV [1] which deviates from the standard model prediction m W | SM = 80.357 ± 0.006 GeV [2] at 7σ.Such a high precision measurement of W mass may be a significant indication for the new physics beyond the standard model.
On theoretical grounds, the CDF W -mass anomaly possibly originates from (i) a nonminimal Higgs sector that contains the standard model Higgs field and directly contributes to this mass deviation via relevant Higgs mechanism, (ii) tree level mixings of the standard model Z and even W bosons with new particles that cause the W mass as shifted, and/or (iii) loop-level quantum corrections due to the presence of new particles to gauge boson selfenergies that modify the Peskin-Takeuchi parameters S, T, U [3][4][5].Several efforts have been made in the literature to provide possible solutions to this puzzle, see Refs.  for an incomplete list.In this work, we show that the model based upon SU (3) C ⊗SU (3) L ⊗U (1) X (called 3-3-1) gauge symmetry [54][55][56][57][58][59] manifestly accommodates the CDF W -mass anomaly.
The reason for this model choice is that all the effects dedicated above of the new physics are actually dictated by the 3-3-1 gauge principle, and thus they are very predictive.
The 3-3-1 model can be classified, based upon the embedding of electric charge operator in the new gauge symmetry, say Q = T 3 + βT 8 + X, through the β parameter, where T j (j = 1, 2, 3, • • • , 8) and X are SU (3) L and U (1) X charges, respectively.Generally, the 3-3-1 model possesses a lepton triplet of form (ν L , e L , E q L ) where E is some field with electric charge q, related to the β parameter as β = −(1 + 2q)/ √ 3. Notice that switching representations with conjugated representations of SU (3) L , e.g.(ν L , e L , E q L ) → (e L , −ν L , E q L ), changes β → −β and leads to a version with rather similar phenomenology, including the W mass, thus structures cause the W -mass shift at tree level.Additionally, the new non-Hermitian gauge vector doublet (X, Y ) and inert Higgs multiplets presented in 3-3-1 models for neutrino mass and/or dark matter also contribute to this mass shift at loop level.
The rest of this work is organized as follows.In Sec.II we set up a generic 3-3-1 model in which relevant Higgs mechanism important for the W -mass shift is determined and classified by electric charge conservation and B − L behavior.In Sec.III, we investigate various novel contributions of the model to the W -mass shift.A remark of scalar sextet contribution to the W -mass shift is given in Sec.IV.The extra important constraints for 3-3-1 model are discussed in Sec.V. We make a conclusion in Sec.VI.

II. DESCRIPTION OF THE MODEL
We first present the necessary features of the 3-3-1 model with arbitrary β embedding (or q charge) parameter.We then determine distinct gauge-boson mass spectra according to profiles of Higgs vacuum structures, which affect differently to the W -mass anomaly.

A. Particle content
The 3-3-1 gauge symmetry is given by where the first factor is the usual color group, while the last two are directly extended from the electroweak group, as mentioned.The decomposition scheme of the extended gauge sector into the usual gauge groups takes the form, where T j (j = 1, 2, 3, • • • , 8) and X stand for SU (3) L and U (1) X charges, respectively.Additionally, the hypercharge Y and the electric charge Q are embedded, respectively, as In other words, when the 3-3-1 symmetry is broken down to the standard model symmetry, the hypercharge is composed of the two new diagonal charges, T 8 and X, as broken.When the electroweak symmetry is broken down to electromagnetic symmetry, the electric charge is composed of the third weak-isospin component and the hypercharge, as usual.The coefficient β is related to a basic charge parameter q that is the electric charge of the third component (E) of a lepton triplet, such as It is noted that the 3-3-1 model can possess variants that differ by corresponding β (or q) values, as imposed in Table I.
Despite of the same q, the relevant models have alternative phenomenologies.Specially for q = 0, N a may gain a charge B −L = 0 different from that of ν aR , revealing a theory for dark matter [99,107].For q = −1, we obtain the 3-3-1 model with heavy charged leptons (E − a ) [122,123].Interestingly, this version also implies dark matter stability if E − a have B − L = 0 different from that of the usual charged leptons (see, e.g., that in [109]).Along the line for q = −1, the flipped 3-3-1 model puts all quark families in antitriplets, while two lepton families in triplets and the remaining lepton family in sextet, which differs from the above arrangement [124,125].This kind of the model has an extra chiral fermion triplet resided in the sextet, but it is highly degenerate in mass, negligibly contributing to W mass.The other sources that affect W mass are identical to the unflipped version with q = −1.That said, it is able to collect all the viable lepton sectors (family and left-handed indices omitted, right-handed counterparts if viable are in singlet) in Table I, while the corresponding quark sectors are not listed, since they have a common form differing only in electric charge for exotic quarks.
Three scalar triplets are generally introduced as for which χ breaks the 3-3-1 symmetry down to the standard model, giving mass for new particles, while ρ, η break the standard model symmetry down to SU (3) C ⊗U (1) Q , providing mass for ordinary particles.It is noted that one of the triplets ρ, η may be excluded [105,118].
However, the following investigation does not depend on such changes of scalar multiplet number; instead, it results from the vacuum structure of scalar fields.As shown in [126], it is sufficient to consider three triplets (given above) and one sextet, where in this work the sextet will be separately treated, without loss of generality.This vacuum structure depends on q as well as B − L behavior, studied below in order, along with implied gauge boson masses.
−n − 1, while the rest of fields takes usual value.Since T 8 is gauged, B − L and N must be gauged.We impose ν aR for cancelling U (1) N anomalies and a superheavy scalar singlet φ B−L with B − L = 2 that couples to ν R ν R and breaks U (1) N .After symmetry breaking, the neutrinos gain a small mass via canonical seesaw, while there exists a residual matter parity P M = (−1) 3(B−L)+2s not commuted with SU (3) L [107,109].The 3-3-1 model with n = 0, as mentioned, possesses a nontrivial matter parity for new fields, E a , J a , η 3 , ρ 3 , χ 1,2 , X, and Y , such as Hence, in spite of q = 0 (q = −1), the relevant scalars η 3 , χ 1 (ρ 3 , χ 2 ) cannot develop a vacuum expectation value (VEV) due to the matter parity conservation.Alternatively, the 3-3-1 model with n = 1, including the minimal 3-3-1 model, the 3-3-1 model with righthanded neutrinos, and even the 3-3-1 model with heavy charged leptons, transform trivially under P M , i.e.P M = 1 for every field.In this case, the scalars η 3 , χ 1 (ρ 3 , χ 2 ) if electrically neutral can develop a VEV, not protected by P M , actually governed by B − L (or P M ) violating interactions.With the aid of P M , a summary of possible vacuum structures for 3-3-1 variants is given in Table I.
If the B − L symmetry is approximate, we avoid a U (1) N extension as given above.In this case, the interactions violating B − L enter, such as for the 3-3-1 model with right-handed neutrinos (q = 0, n = 1), besides the normal Yukawa couplings (h's) and scalar self-couplings (µ's and λ's)-as in the usual theory-which conserve B −L.It is easily verified that all these violating couplings violate L by 2 units, except for λ1 by 4 units, while preserve B and a lepton parity (−1) L .Thus, the proton stability is protected by B and (−1) L .Furthermore, the violating couplings must be small, e.g.μ µ, λ λ, and s h, since by contrast L-conservation sets μ, λ, s = 0 but µ, λ, h = 0.In this case, the potential minimization gives L-violating VEVs, η 3 , χ 1 ∼ μ2 /w, as suppressed, where w is the 3-3-1 breaking scale (cf.[127]).If discarding the unwanted coupling ψ L ψ L ρ by a symmetry ρ → −ρ, the neutrinos gain a naturally small mass via L-violating effective interaction, s w ψ L ψ L ηη, to be m ν ∼ s u 2 /w, doubly suppressed by s h and u w, where u is a weak scale.The presence of small η 3 , χ 1 and s's lead to a small mixing between usual quarks and exotic quarks causing flavor changing Z-currents, as studied below (Sec.V).
B. Gauge spectrum for q = 0, −1 or B − L conservation For q = 0 and q = −1, the scalar components that are electrically neutral can develop VEVs, such as Since w breaks the 3-3-1 symmetry, while u, v break the standard model symmetry, we impose w v, u for consistency.This standard vacuum alignment has been extensively studied in the literature, even applying for the 3-3-1 model with arbitrary q (see Table I).
However, notice that for the model with q = 0 (q = −1), an extra symmetry such as B − L and its residual matter parity is needed to prevent the other neutral scalars η 3 , χ from developing a VEV, ensuring the standard vacuum structure, as mentioned [107,109].
Particularly for q = 0, this section applies for the 3-3-1 model with neutral (heavy) fermions, not for the 3-3-1 model with right-handed neutrinos.
The mass spectrum of the gauge bosons is given by where Φ runs over the scalar triplets, and the covariant derivative is Here (g s , g, g X ) and (G j , A j , B) are the gauge couplings and gauge bosons of 3-3-1 subgroups, respectively, and t j is SU (3) C charges.
Define non-Hermitian gauge bosons, which are coupled to the weight-raising and -lowering operators, respectively.W ± , X ∓q , and Y ∓(1+q) are physical fields by themselves with masses respectively.W is identical to that of the standard model, while (X, Y ) form a new, heavy gauge vector doublet with a mass splitting Consider neutral gauge bosons.The photon field A that is coupled to the electric charge Q with coupling e is given by A/e = (A 3 + βA 8 )/g + B/g X , which is determined from Q = T 3 + βT 8 + X by substituting each generator with corresponding gauge field over coupling.This is a direct result of electric charge conservation, neither depending on VEVs nor necessarily diagonalizing the relevant mass matrix, as proved in [126].The normalization of photon field implies s W ≡ e/g = g X / g 2 + (1 + β 2 )g 2 X , which matches the sine of the Weinberg angle in the standard model.Hence, the photon field A can be rewritten as where the expression in the parentheses is just the hypercharge field defined by ( 3).The standard model Z field is given orthogonally to A, as usual, while a new Z field is obtained orthogonally to the hypercharge field, The photon A is massless and decoupled, as a physical field, while Z and Z mix via a symmetric mass matrix with elements, given by Diagonalizing the Z-Z mass matrix, we obtain two physical fields, where the Z-Z mixing angle (ϕ) and Z 1,2 masses are The mixing angle ϕ is small, suppressed by (v, u) 2 /w 2 .Additionally, Z 1 has a mass approximating that of Z, called the standard model Z-like boson, while Z 2 is a new, heavy gauge boson with mass proportional to w.
Because the physics obtained is the same independent of q = 0 or −1 for phenomena interested in this work, we consider only q = 0, thus β = −1/ √ 3.In this case, although both η and χ transform the same under the gauge symmetry, they differ in B − L numbers, possibly obtaining VEVs at the first and third components.That said, the vacuum alignment under consideration is generically given by In contrast to u, v, and w that conserve B −L, the remaining VEVs u , w break this number, suppressed by relevant violation interactions.To be consistent with the standard model, we impose u u and w w, in addition to u, v w [76].
Substituting the VEVs in (33) to the Lagrangian, we get where 2 defined as before mix, while the real and imaginary parts of X 0,0 * = (A 4 ∓ iA 5 )/ √ 2 behave differently, i.e.A 4 mixes with A 3,8 and B, whereas A 5 does not.That said, A 5 is decoupled, as physical field, with mass, The mass matrices of the charged and neutral gauge bosons are given, respectively, by where t X = g X /g.It is noteworthy that both the mixing of W and Y and the mixing of (A 3 , A 8 , B) and A 4 are caused by u , w .Diagonalizing M 2 c , we obtain two physical fields where the W -Y mixing angle (θ) is given by which implies that θ u /w + (u/w)(w /w) is very small, because of u u and u, w w.
The W 1 , Y 1 masses are W 1 is the standard model W -like boson, whereas Y 1 is a new, heavy charged gauge boson.
To diagonalize M2 0 , we define the photon A, the usual Z, and the new Z as in the previous model, but for β = −1/ √ 3, such as where s W = √ 3g X / 3g 2 + 4g 2 X .In the new basis (A, Z, Z , A 4 ), the photon A is massless and decoupled, as usual, while Z, Z , and A 4 mix by themselves via the mass matrix, It is easily verified that M 2 0 (thus M 2 0 ) contains an exact eigenvalue, with a corresponding exact eigenstate, where s θ = s 2θ /2c W is very small as θ is, hence A 4 A 4 .It is noteworthy that A 4 and A 5 always have equal masses. 2 Hence, we identify to be a physical non-Hermitian field, with the common mass, To diagonalize M 2 0 , we choose two gauge vectors orthogonally to A 4 , such as where Z Z and Z Z similar to In the new basis (Z, Z , A 4 ), the field A 4 is decoupled, while Z and Z mix by themselves via a symmetric mass matrix with elements, given by Diagonalizing this mass matrix, we obtain physical fields, where the Z-Z mixing angle (ϕ) and Z 1 , Z 2 masses are The ϕ angle is small, suppressed by (u, v) 2 /w 2 .Additionally, the field Z 1 is the standard model Z-like boson, while Z 2 is a new, heavy gauge boson with mass at w.

III. SOURCES OF THE W -MASS SHIFT
The W -mass shift in the 3-3-1 model arises from various new physics sources, Z-Z mixing, X-Y and Z-Z -X mixings, a gauge vector doublet as well as a new Higgs doublet that are not degenerate in mass.They are newly recognized, depending on relevant 3-3-1 model.
Let us remind the reader that as already done in the literature of electroweak precision fit [6,10,11,13,14,18,20,22,31], a positive and dominant contribution of the Peskin- Takeuchi T -parameter can generate an enhancement of the W -boson mass consistent with the recent CDF measurement.
A. Z-Z mixing in the model with q = 0, −1 or B − L conservation The 3-3-1 model under consideration reveals a tree-level mixing of Z with Z , while W is retained.Because of the Z-Z mixing, the observed Z 1 -boson mass (m Z 1 ) is reduced in comparison with the standard model Z-boson mass (m Z ).This gives rise to a positive contribution to the T parameter at tree level, where note that m W = m Z c W [128]. Since m Z 1 is precisely measured and fixed, this enhances the mass of W boson proportionally to αT , such as [3-5] We input the parameters as α 1/127.955,s 2 W 0.231, and Additionally, we consider the three 3-3-1 models according to β = 1/ √ 3, β = −1/ √ 3, and , where the last two are identical to the 3-3-1 model with neutral (heavy) fermions and the minimal 3-3-1 model, respectively.We make a contour of ∆m 2 W = 80.4335 2 −80.357 2 GeV 2 taking central values [1,2] as function of v and w for the mentioned models as in Fig. 1.
Here, the black line is for the central value.Additionally, the 1σ, 2σ, and 3σ ranges are also shown (in cyans).Besides, the excluded region (light red) by the FCNCs and the Landau pole limit if it applies (taken as 5 TeV) have appropriately been included to plot (cf.Sec.V).
From Fig. 1, we obtain the viable ranges for the new physics and weak scales, namely, a phenomenon occurs similarly to the economical 3-3-1 model [120].New observation is that the reduction of Z mass is bigger than that of W mass. Correspondingly, this produces a ρ-parameter bigger than 1, causing the CDF W -mass shift, as measured.
That said, from ( 61) and ( 62) we obtain a positive contribution to the T parameter at tree level, such as It is noteworthy that the three terms that do change the W, Z masses in ( 61) and (62) are at the same order, because of u /u ∼ w /w ∼ (u, v)/w.Additionally, the first two of these terms caused by u , w come from W -Y and (Z, Z )-X mixings, while the last term suppressed by (u, v) 2 /w 2 arises from the Z-Z mixing.Although both kinds of the mixings reduce the W, Z masses, only the Z-Z mixing governs the ρ-parameter deviation which subsequently affects the W mass as shifted, similar to the previous model.

D. Oblique contributions of a non-degenerate scalar doublet
Under the standard model symmetry, the three scalar triplets as given correspondingly contain three SU (2) L scalar doublets, since each triplet is decomposed as 3 = 2 ⊕ 1.One of the doublets is eaten by W, Z leaving only the usual physical Higgs field, other one of the doublets is completely eaten by the X, Y gauge vector doublet.The remaining scalar doublet is really a new physical Higgs doublet, which potentially contributes to the W -boson mass via S, T, U parameters, partly noted in [132].Such a physical Higgs doublet also exits in the 3-3-1 models for dark matter as the first and second entries of inert scalar triplets [104,105], or in 3-3-1 models with flavor symmetries as contained in scalar flavon triplets [77][78][79].Even if one includes a scalar sextet [56,[76][77][78][79], a new Higgs doublet correspondingly arises, since 6 = 3 ⊕ 2 ⊕ 1 under SU (2) L .In this case, the SU (2) L scalar triplet also contributes to S, T, U parameters, but this contribution is neglected because of its significant tree-level contribution as discussed below.
That said, a new physical Higgs doublet in addition to the usual Higgs doublet is popularly presented in the 3-3-1 model.Without loss of generality, we consider a generic scalar triplet, which contains the new SU (2) L Higgs doublet, i.e. (φ 1 , φ 2 ), as desirable.Here φ can have an arbitrary X-charge, making a significant contribution to the oblique parameters independent of electric charge.It is easily shown that the contributions of φ to S, U parameters are more smaller than that to T , if the scalar fields are radically heavier than W, Z masses.Indeed, in the 3-3-1 model, both φ 1,2 are typically heavy at TeV scale, while the mass-squared splitting of φ 1,2 is only proportional to the weak scale [104,105].In this case, the W -boson mass shift is governed by the T parameter, such as where m 1,2 are the masses of φ 1,2 , respectively.
Taking the central values of W mass from the CDF experiment [1] and the standard model prediction [2], respectively, in Fig. 4 we contour ∆m 2 W as function of δm = m 2 − m 1 and m 1 by the black line.We obtain the viable value, δm 98 GeV, as appropriate.Further, the contributions to ∆m 2 W depend only on the mass splitting δm, nearly insensitive to m 1 .For clarity, the 1σ, 2σ, and 3σ ranges of the measured W mass are also shown (in cyans).Last, but not least, it is clear that m 1 ∼ λ φ−H v 2 /δm where λ φ−H relates to the coupling of φ to the usual Higgs field.Hence, φ obtains a mass at TeV, i.e. m 1 ∼ TeV, only if λ φ−H is at the perturbative limit [52].

IV. TREE-LEVEL CONTRIBUTION OF A SEXTET
A scalar sextet of type, often studied in the 3-3-1 models [56,57,72,76,133], can develop a vacuum value such as Other fields if electrically neutral can also have a VEV.But they belong to a SU (2) L doublet or singlet, giving a contribution similar to the above cases of scalar triplets, and are not interested.Here, the nontrivial vacuum structure is associated with the SU (2) L scalar triplet, (S 11 , S 12 , S 22 ), contained in the sextet, unlike those in the cases of scalar triplets.
Unfortunately, this sextet gives a negative contribution to the ρ-parameter, as shown in [76], incompatible with the experiment.This case is similar to a scalar triplet in the type II seesaw mechanism added to the standard model.
However, if we consider an alternative scalar sextet of type, studied in the 3-3-1 model for dark matter [105], the SU (2) L scalar triplet (σ 11 , σ 12 , σ 22 ) contained in the sextet develops a VEV in different way, such as It gives a positive contribution to the ρ-parameter, such as Comparing to the W -mass shift, κ is about 4.5 GeV, similarly in size to u bounded above.
This case is similar to a scalar triplet with Y = 0 added to the standard model [134].
It is stressed that the mentioned scalar sextets contribute to the gauge boson mass spectra differently from the scalar triplets in previous sections.However, the results obtained according to the scalar triplets are easily generalized for the sextets with the aid of [126].

V. EXISTING BOUNDS
We now present the running coupling and Landau pole limit which place a constraint on β, q parameters.We also discuss FCNCs which reveal important information on B − Lviolating VEVs and new physics scale.Last, but not least, we examine collider bounds, validating the previous constraints, as well as we summarize the relevant bounds used for model classification and updated with W -mass measurement.

A. Running coupling and Landau pole
A gauge coupling commonly denoted as g = {g s , g, g X } changes with renormalization scale µ through the RG equation, where the 1-loop beta function is given by where V, L/R, S indicate vector, left/right fermion, and scalar field representations under the relevant gauge group, respectively.For SU (3), C V = 3 and C L = C R = C S = 1/2 for (anti)triplets, while for U (1) X , C V = 0 and C L,R,S = X 2 L,R,S for L, R, S fields, respectively.We obtain b gs = 5 > 0 and b g = 13/2 > 0. Thus, g s , g decrease when µ increases.There is no Landau pole associated with g s , g.However, because of C V = 0 and C L,R,S = X 2 L,R,S > 0, we always have b g X < 0. Hence, g X increases when µ increases.
In contrast to the standard model and grand unification, a finite Landau pole associated with g X potentially arises because of U (1) X along with SU (3) L embedded in U (1) Q .Indeed, as given before, the electric charge operator defines the photon field that couples to it and that the normalization of the photon field implies where note that g is always finitely nonzero.When the energy scale increases, g X /g increases as long as s 2 W approaches the r.h.s of the inequality.The model encounters a Landau pole µ at which s 2 W (µ) = 1 1+β 2 , or equivalently g X (µ) = ∞, where the scale of µ depends heavily on β value.It is stressed that the model is valid only if the Landau pole µ is larger than the new physics scale, i.e. µ > w, thus than the weak scale u, v. Correspondingly, we have . This yields |β| < cot W (u, v) 1.824 for s 2 W (u, v) 0.231, which translates to −2.08 < q < 1.08, due to β = −(1 + 2q)/ √ 3.
For q = −2, 1, we have |β| = √ 3 and s 2 W (µ) = 1/4 just above that at the weak scale.In this case, the model presents a low Landau pole at µ = 4-5 TeV, as shown in [135,136] for the minimal 3-3-1 model.However, if q = −1, 0, which include the 3-3-1 model with heavy charged leptons and the 3-3-1 model with right-handed neutrinos, we have |β| = 1/ √ 3 and s 2 W (µ) = 3/4 that is much beyond the typical value s 2 W = 3/8 at the grand unification scale, close to the Planck regime.In this case, the model still possesses a Landau pole, but this pole is beyond the Planck scale.

B. FCNCs
There are two sources for FCNCs in the 3-3-1 model that come from the nonuniversality of quark families under the gauge group and the possible mixing of ordinary quarks and exotic quarks, respectively.The former occurs for every q associated with Z current, while the latter arises only if q = 0 or −1 concerning Z current.Particularly for q = 0, it happens with the 3-3-1 model with right-handed neutrinos but not with the 3-3-1 model with neutral (heavy) fermions.These FCNCs are actually induced at tree-level and indeed dangerous.
The model with q = −1 happening similarly to the case q = 0, as well as the FCNCs that are possibly arisen/coupled to the Higgs and new Higgs fields, will not be discussed.
In the 3-3-1 model with q = 0, the ordinary and exotic quarks of up-type (u a , J 3 ) and of down-type (d a , J α ) each mix by themselves due to the lepton-number violating VEVs of η 3 , χ 1 as well as the lepton-number violating Yukawa couplings (called s's) that like ordinary Yukawa couplings (called h's) but appropriately interchange the right-handed components of ordinary and exotic quarks, as supplied above [76].We define mixing matrices, , such that the 4 × 4 mass matrix of (u a , J 3 ) and the 5 × 5 mass matrix of (d a , J α ) are diagonalized.
Because the ordinary and exotic quarks have different T 3 weak isospin, the tree-level FCNCs of Z boson arise, such as i.e. the standard model CKM mechanism does not work [108].Here, we denote q as either u for up-type or d for down-type quarks, which should not be confused with the q charge parameter used throughout, i, j = 1, 2, 3 label ordinary physical quarks, and I = J 3 and plus sign applies for V u , while I = J α and minus sign applies for V d .Integrating Z out as well as using m 2 Z (g 2 /4c 2 W )(u 2 + v 2 ), we obtain effective interactions, where ∆C ds ≡ [(V * dL ) I1 (V dL ) I2 ] 2 /(u 2 + v 2 ), and "• • • " indicates its conjugate as well as other four-quark systems.The strongest bound comes from the K 0 -K0 mixing that requires the relevant coupling, ∆C ds , to be smaller than (10 4 TeV) −2 [2], implying The mixing of ordinary and exotic quarks is much smaller than the smallest mixing element of CKM matrix (around 5 × 10 −3 ), which may be understood in the 3-3-1-1 model [107].
For the present model, we safely assume ( It is noted that the model with q = 0, −1 has only FCNCs associated with nonuniversal Z couplings, because the ordinary and exotic quarks do not mix.The following computation would apply for all cases of q-charge, since for q = 0 (or −1) the ordinary and exotic quark mixing negligibly contributes to this kind of the FCNCs, as shown in [108].Because the third family of quarks transforms under SU (3) L ⊗ U (1) X differently from the first two, there must be tree-level FCNCs.Indeed, using X = Q − T 3 − βT 8 , the interaction of the neutral , where F runs over fermion multiplets.It is clear that the terms of T 3 and Q, as well as all terms of ν a , e a , E a , J α , and J 3 , do not flavor change.The relevant part includes only T 8 with ordinary quarks, , where q denotes either up-type or down-type quarks and T 8q = 1 2 √ 3 diag(−1, −1, 1) combines their T 8 charge.Changing to the mass basis, q aL,R = (V qL,R ) ai q iL,R , we have which implies FCNCs for i = j.Integrating Z out and using m 2 where ∆C sb ≡ [(V * dL ) 32 (V dL ) 33 ] 2 /w 2 , and "• • • " stands for its conjugate and other four-quark systems.The strongest bound comes from the B 0 s − B0 s system, given by [2] [ Assuming V uL = 1, the CKM factor is |(V * dL ) 32 (V dL ) 33 | 3.9 × 10 −2 , which translates to w > 3.9 TeV.This bound is independent of β, i.e. applying for every 3-3-1 model.Last, notice that the K 0 -K0 mixing gives a bound, w > 3.6 TeV, slightly smaller than the given one, which has not been signified.

C. Collider searches
The LEPII studied the process e + e − → f f , where f is an ordinary fermion, through exchange of a new heavy gauge boson Z , described by effective interactions, where as usual.Considering f = µ, τ , the charged leptons have equal couplings, and we rewrite where the successive terms differ from the first one only in chiral structure, and The LEPII supplied bounds for such chiral couplings, typically [137] g which are 5.7 TeV, 3.9 TeV, 2.1 TeV, and 0.3 TeV, for q = −2, −1, 0, and 1, respectively.
The first case is ruled out by the Landau pole, while the last case means that Z negligibly contributes to the process, since w would be at TeV due to the FCNCs above.That said, the 3-3-1 model with q = 0 and ±1 would be viable, as taken into account.
The LHC searched for dilepton signals through the process pp → f f c that is contributed by Z , supplying a bound m Z ∼ 4 TeV for Z couplings identical to those of the usual Z boson [138].It is noticed that Z in our model couples similarly to Z, governed by the common g coupling but having a small difference due to β.Therefore, the bound as given applies to our model with some extent since the couplings are not identical.The Z mass limit thus converts to a w bound comparable to the LEPII.

D. Summary of the existing bounds with updates for model building and W mass
The viable range of β, q and the Landau pole limit determined in Sec.V A have appropriately been supplied to Table I for classifying 3-3-1 versions.Additionally, all the relevant, existing bounds of this section, i.e.Secs.V A, V B, and V C, have appropriately been combined with the W -mass measurement in the previous sections.
For convenience in reading, we give a summary of all the bounds obtained in this section as well as their application to the W -mass deviation in the previous sections, as in Table II.
That said, the constraints previously given concerning the W mass, such as (w, v) that results from Figs.  Used appropriately to Sec.II C (u , w , s) as well as Eqs.( 61) and (62) TABLE II.A summary of existing bounds and their updates with W -mass result.

VI. CONCLUSION
We have examined a variety of variants of the 3-3-1 model, characterized by a basic charge parameter q and the behavior of B − L symmetry through its residual matter parity P M .
The 3-3-1 model generally has a standard vacuum structure, except for the 3-3-1 variant with right-handed neutrinos or with heavy charged leptons which can develop an abnormal vacuum structure, not protected by P M .We have diagonalized the gauge spectra according to the two kinds of vacuum in detail.
There are two kinds of constraints for this model, which are jointly considered and consistently updated: New experiment: We have investigated the various contributions of the 3-3-1 model to the CDF W -mass anomaly, and we conclude that the Z-Z mixing due to the 3-3-1 breakdown by the normal VEVs u, v, w and a non-degenerate physical Higgs doublet popularly existed in the model can explain this anomaly, separately.Additionally, a scalar sextet that contains a SU (2) L Higgs triplet with Y = 0 can also solve this puzzle.Furthermore, considering the special 3-3-1 versions with q = 0, ±1, the viable regimes of u, v, w have been derived, obeying the FCNC, collider, and relevant Landau-pole limits.For the case of the heavy non-degenerate Higgs doublet, its mass splitting should be at 98 GeV, whereas for the case of the scalar sextet, the tiny VEV should be about 4.5 GeV.
[Verify Sec.V for the β range and Landau pole.]A star " * " shows a vacuum expectation value viably for the corresponding scalar component.For q = 0, −1, the standard vacuum applies for N, E − versions with [B − L](N, E − ) = 0, protected by P M , whereas the abnormal vacuum happens for ν c , E − versions with [B − L](ν c , E − ) = 1, not protected by P M .
J a in fermion multiplets generically have B − L charge differently from that of the usual leptons and quarks, respectively.Let [B − L](E a ) = n.We obtain B − L = diag(−1, −1, n) for lepton triplets, which neither commutes nor closes algebraically with SU (3) L .If B − L is conserved, an extra U (1) N group is required by symmetry principles such that B − L = β T 8 + N , where β
result in the previous sector for β = −1/ √ 3 applies to this model without change, however.Although the phenomenologies of the two mentioned models are distinct, characterized by B − L conservation or violation, the Z-Z mixing is crucial to set the CDF W -mass anomaly.The mixing effects caused by u , w effectively not contributing to the W -mass shift as observed are probably due to the fact that u , w break lepton number, associated with Majoron fields that are eaten by the corresponding non-Hermitian gauge bosons (X, Y ).Hence, u , w only affect the X, Y observables.C.Oblique contributions of the non-degenerate vector doubletAt one-loop level, the two new non-Hermitian gauge bosons predicted by the 3-3-1 model X ±q and Y ±(1+q) , which form an SU (2) L doublet according to the decomposition of SU (3) L gauge adjoint 8 = 3 ⊕ 2 ⊕ 2 * ⊕ 1, contribute to the oblique parameters S, T , and U , through transverse self-energies.The W -boson mass shift induced by these oblique corrections can be expressed as follows[3][4][5]