Abelian charge inspired by family number

Quark has an electric charge either -1/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1/3$$\end{document} or 2/3 and a baryon number 1/3, where the divisions 3’s match the color number. Although the electric charge and the baryon number have a nature distinct from the color charge, the matching is necessary for the standard model or a relevant B-L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B-L$$\end{document} extension consistent at quantum level, since the relevant anomaly [SU(2)L]2U(1)A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[SU(2)_L]^2U(1)_A$$\end{document} for A=Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=Y$$\end{document} or B-L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B-L$$\end{document} must vanish. If elementary particles have a new U(1) charge differently from A, such anomaly is not cancelled for each family. However, if we demand that the anomaly is cancelled over all families, this relates the color number to the family number instead of the electric charge and baryon number, and interestingly the family number guides us to a novel U(1) theory. We will discuss the implication of this theory for neutrino mass, recent W-boson mass anomaly, FCNC, and particle colliders.


Motivation
The weak isospin symmetry SU (2) L , which forms isodoublets l L = (ν L , e L ) and q L = (u L , d L ) while the righthanded partners e R , u R , and d R are isosinglets, is not completed. The electric charge Q, which is diag(0, −1) for l L and diag(2/3, −1/3) for q L , is neither commuted nor closed algebraically with SU (2) L , because of [Q, T 1 ± i T 2 ] = ±(T 1 ± i T 2 ) = 0 and Tr(Q) = 0. Since [T 3 , T 1 ±i T 2 ] = ±(T 1 ±i T 2 ) is similar to Q, a new abelian charge Y ≡ Q − T 3 arises, such that [Y, T 1 ± i T 2 ] = 0 = [Y, T 3 ]. This defines a complete group, SU (2) L ⊗U (1) Y , which establishes the standard electroweak model [1][2][3]. The model successfully works at quantum level because every anomaly vanishes for which the most important one is [SU (2) L ] 2 U (1) Y ∼ 3Y q L + Y l L = a e-mail: dong.phungvan@phenikaa-uni.edu.vn (corresponding author) b e-mail: hung.tranngoc@phenikaa-uni.edu.vn c e-mail: loi.duongvan@phenikaa-uni.edu.vn 3 × 1 6 − 1 2 = 0. This cancellation occurring for each family is due to the fact that the color number 3 relates the electric charge of quark to that of lepton. Besides the hypercharge Y , the model contains an accident gauge symmetry of baryon number minus lepton number B − L, if a right-handed neutrino ν R is simply imposed for each family. In this case, every anomaly is cancelled too for which the most important one is 1 3 −1 = 0. This cancellation is due to the fact that the color number 3 relates the B − L number of quark to that of lepton, analogous to the case of electric charge. Who arranges such matchings, since the electric charge and the B − L number have a nature distinct from the color number?
This work does not directly answer such question. We guess that at high energy the partial interactions are unified in a simple group, well-known as grand unification. The unification group would contain the color charge and weak isospin, besides neutral generators that determine the hypercharge and even B − L. This work does not search for a specific unification group. Instead, we investigate possible neutral charges that lead to the hypercharge. The hypothetical symmetry of B − L although mentioned will not be addressed. Obviously, an individual neutral charge, called X , does not necessarily have the above property as of the hypercharge, i.e. 3X q L + X l L = 0 for a single family. The anomaly [SU (2) L ] 2 U (1) X is cancelled, needed to a correlative contribution with exotic particles or between families. The latter case would make a constraint on the number of families. The minimal grand unifications such as SU (5) [4], SO (10) [5,6], and Pati-Salam model [7] do not contain X , since they leave only the hypercharge and/or B − L, T 3R . The trinification [8] and E 6 [9] as well as their variants [10][11][12][13][14] would provide such a X .
In what follows, we search for such neutral charges and propose the model of interest. We discuss the new physics insights that the new proposal reveals. Interestingly, this approach provides a potential explanation to neutrino mass, W -boson mass deviation, compelling phenomena for flavor physics, as well as hints for particle colliders.

Proposal of the model
Instead of Y, B − L charges and even their combination, we look for an entirely alternative charge X such that the relation 3X q L + X l L = 0 does not hold for each family. That said, the new electroweak group takes the form, where 3X q L + X l L = 0 for each family, and the charge N is necessarily included such that The N charge obeys 3N q L + N l L = −(3X q L + X l L ) = 0 as X does, for each family. Thus X, N have a nature different from Y (and B − L too), but the breaking of X, N explains Y and the matching. Given a grand unification, X and N would be unified with SU (2) L in a higher isospin group. Hence, the magnitudes of X, N can be fixed as the neutral charges of the large group. Obviously, X (or N ) is not originated from a charge, such as T 3R or B − L, as in minimal left-right symmetry, Pati-Salam group, and SO (10). Looking for trinification, we find that T 8L ,R have a property of X , namely X ∼ T 8L + T 8R , while N is followed by N = Y − X . Note that fermion doublets may be arranged in trinification triplets/antitriplets. Hence, we restrict ourselves by assigning X l L = x for all lepton doublets, X q L = x for a number of quark doublets, while X q L = −x for the rest of quark doublets. Here the normalization of x is left arbitrarily. The condition 3X q L + X l L = 0 for each family simply requires x = 0. Summing over N f families, the anomaly [SU (2) , where m and n = N f − m denote numbers of q L doublets that possess x and −x, respectively. It is clear that the similar anomaly for N , i.e. [SU (2) L ] 2 U (1) N , vanishes as a result. Hence, the family number N f = 3(n − m) is an integer multiple of the color number, 3, for which fixing N f = 3 leads to n = 2 and m = 1. We define two kinds of family indices, like a, b, ... run over 1,2,3 according to N f = 3 and α, β, ... run over 1,2 according to n = 2. We then assign (X, N ) charges to fermion representations as in Table 1, where the color and isospin quantum numbers are also included for completeness.
That said, the third quark doublet has a X -charge as of lepton doublets, while the first and second quark doublets have an opposite X -charge. Hence, 3X q L + X l L = 4x for the third family, while 3X q L + X l L = −2x for the first and second families. The anomaly [SU (2) L ] 2 U (1) X ∼ 3X q L + X l L vanishes when summing over three families. It is verified that all the other anomalies, say [SU 3 , and [U (1) N ] 3 , vanish too. The anomaly cancel- lation is due to the fact that the family number matches the color number. Last, but not least, the right-handed neutrinos are presented, required for anomaly cancellation. Additionally, for symmetry breaking and mass generation, we have introduced the scalar fields: (i) H is identical to that of the standard model, (ii) couples the two kinds of quarksq α and q 3 necessary for recovering CKM matrix, and (iii) χ couples to right-handed neutrinos ν R ν R responsible for neutrino mass generation. Up to the gauge fixing and ghost terms, the total Lagrangian takes the form, The kinetic term is where F and S run over fermion and scalar multiplets, respectively. The covariant derivative and field strength tensors are given by where (g s , g, g X , g N ), (t p , T j , X, N ), and (G pμ , A jμ , B μ , C μ ) denote the coupling constants, the generators, and ) groups, respectively. A kinetic mixing term between the two U (1) gauge fields is suppressed, due to its small effect. The Yukawa Lagrangian and the scalar potential are given by Above, f ν , h's, and λ's are dimensionless, while μ 1,2,3,4 have a mass dimension. Additionally, we defineH = iσ 2 H * and˜ = iσ 2 * . The last two terms in (4) before the H.c. are radiatively induced by one-loop diagrams despited in Fig. 1  The gauge symmetry is broken as follows Here the vacuum expectation values (VEVs) are given by and we require v 1,2 and v 2 1 + v 2 2 = (246 GeV) 2 for consistency with the standard model. As stated, the combined field gets a value matching that of , i.e.

Fermion mass
Substituting the VEVs to the Yukawa Lagrangian, we obtain fermion masses. First, charged leptons gain a mass via h e coupling to be Diagonalizing this mass matrix, we get the masses of electron, muon, and tau, as usual.
The neutrinos ν L ,R gain a Dirac mass via h ν coupling, while ν R obtains a Majorana mass via f ν coupling, The total mass matrix of neutrinos takes the form, Because of v 1 , i.e. m D m M , the active neutrinos ∼ ν L receive a small mass via the canonical seesaw, Taking v 1 ∼ 100 GeV and ∼ 5-10 TeV, the observed values m ν ∼ 0.1 eV require h ν ∼ 10 −5 , given that f ν ∼ 1. The heavy neutrinos ∼ ν R obtain a mass m M at scale.
Concerning quarks, their masses are given by for down-type quarks, while for up-type quarks. Hence, the quark masses are governed by two weak scales v 1,2 as well as two kinds of couplings h and h . The small mixing between the third quark family and the first two quark families can be understood by h < h and v 2 < v 1 . Of course, diagonalizing the quark mass matrices, we get the masses of u, d, c, s, t, b and CKM matrix, as expected.

Gauge boson mass
Substituting the VEVs to the scalar kinetic term, i.e.
S (D μ S) † (D μ S), we get gauge boson masses. First notice that the gluons are massless since SU (3) C is conserved, unbroken. Concerning the electroweak sector, the charged 2 is a physical field by itself with mass, Whereas, the neutral gauge bosons A 3 , B, and C mix via a mass matrix, where The mass matrix of neutral gauge bosons has an exact, zero eigenvalue (i.e. photon mass) corresponding to the exact mass eigenstate (i.e. photon field), such as which is normalized by The photon field properly couples to the electric charge Q = T 3 +Y = T 3 +X +N , since the above solution can be obtained by substituting each generator by corresponding gauge field over coupling, say Q by A/e, T 3 by A 3 /g, X by B/g X , and N by C/g N . The electromagnetic coupling is identified as e = gs W through the sine of the Weinberg's angle, while the tan of this angle is given by We choose two other fields: the usual Z field, which is given orthogonally to A, as usual, and the new field, which is given orthogonally to the hypercharge field in parentheses.
In the new basis (A, Z , Z ), the photon A is decoupled as a physical field, while the remainders Z , Z mix through a mass matrix, where Diagonalizing the Z , Z mass matrix, we obtain two physical fields, where the Z -Z mixing angle (ϕ) obeys and the Z 1,2 masses are given by The Z -Z mixing, i.e. ϕ, is small as suppressed by (v 1 , v 2 ) 2 / 2 . The field Z 1 has a mass approximating that of Z , called the standard model Z -like boson, whereas the field Z 2 is a new heavy gauge boson with mass proportional to .

W -boson mass deviation
The model under consideration yields a tree-level mixing of Z with Z . Because of this mixing, the observed Z 1 mass is reduced, compared with the standard model Z mass. Additionally, this reduction gives rise to a positive contribution to T -parameter, with the aid of m W = m Z c W and m Z 1 approximated above. Since the Z 1 mass is fixed as precisely measured, the positive value of αT dominantly enhances the W mass (cf. [15,16]), Use the recent measurement m W | CDF = 80.4335±0.0094 GeV [17], which deviates from the standard model prediction, m W | SM = 80.357 ± 0.006 GeV, at 7σ . We make a contour of m 2 W = m 2 W | CDF − m 2 W | SM as function of v 1 and for a choice of x = ± 1 6 and ± 1 2 , as in Fig. 2 (solid, color curves). Here we have used v 2 = (246 GeV) 2 − v 2 1 , fixed t θ = g N /g X = 1, and taken s 2 W = 0.231 and α = 1/128. It is clear from the figure that concerning the W -mass deviation, the new physics scale depends on x for each v 1 . For each |x|-value, is larger for positive x, while is smaller for negative x. Additionally, is appropriate to the seesaw setup as well as the FCNC and collider bounds (discussed below) for interested values of x with an appropriate choice of v 1 . However, the scale is being constrained by Z -fermion couplings, specified in the following section.

Interactions
Substituting the gauge fields in terms of the physical fields obtained above, the covariant derivative takes the form, where Substituting this covariant derivative into the fermion kinetic term, FF iγ μ D μ F, the interactions of gauge bosons with fermions are derived.
It is clear that gluons, photon, and W ± interact with fermions as in the standard model, where λ p for p = 1, 2, . . . , 8 denotes the Gell-Mann matrices. Additionally, q and f indicate every quark and fermion in mass eigenstate basis, while ν, e, u, and d stand for neutrinos, charged leptons, up quarks, and down quarks in gauge basis, respectively. The interactions of Z 1,2 with fermions have the form, where I = 1, 2, and The V, A couplings of Z 1,2 with fermions are collected in Tables 2 and 3, respectively. Notice that |ϕ| 1 as substantially suppressed by (v 1 , v 2 ) 2 / 2 . In the limit ϕ = 0, the field Z 1 becomes the standard model Z boson, and the V, A couplings of Z 1 become those of Z in the standard model, i.e. g Z 1 V In other words, the current theory is consistent with the standard model at low energy. When 0 = |ϕ| 1, the well-measured couplings of Z with fermions are modified by an amount proportional to ϕ,  Table 3 Couplings of Z 2 with fermions where note that θ is always finite. With the aid of the electroweak precision data [18], the new physics contribution is safe, given that |ϕ| 10 −3 which applies for most Z models including ours. Indeed, the electroweak precision data include low-energy weak neutral current experiments, precision Z -pole physics, and others. The low-energy weak neutral current experiments are modified by Z exchange, mainly sensitive to its mass. The precision Z -pole physics at the LEP and SLC are mainly sensitive to Z -Z mixing, which lowers the Z mass and shifts the Z -fermion couplings, by contrast. Notice that the Z -fermion couplings will determine the various Z -pole observables, such as asymmetries and partial widths. The precision data strongly constrain the Z -Z mixing angle, ϕ, which is extracted by comparing the very precise measurements of the coupling constants at the Z pole with the standard model predictions including radiative corrections (cf. [19]). They also provide lower limits on the Z mass, but these limits are generally weaker than the LEPII and LHC, obtained later (these high-energy experiments that are above the Z -pole will supply strong constraints on the Z mass, but generally not sensitive to small Z -Z mixing). The precision data constraint can be presented for two cases of ρ 0 -paramter, ρ 0 free for arbitrary Higgs structure and ρ 0 = 1 for Higgs doublets and singlets like ours, but there is only a slight difference between the limits given. Alternatively, performing a global fit to the full electroweak data set necessarily gives comparable bounds as |ϕ| ∼ 10 −3 , see [20] for instance.
Last, but not least, in contrast to the standard model, both Z 1,2 flavor-change when interacting with quarks, while conserving lepton flavors. This is due to the fact that quark families transform differently under X, N , while lepton families transform the same under such charges. Additionally, quark families transform the same under T 3 and Y that define Z . Hence, the flavor changing effect associated with Z 1 results from a Z -Z mixing to be small. Whereas, the flavor change dominantly occurs associated with Z 2 , even for ϕ = 0. We will investigate the FCNCs coupled to Z 2 , while those with Z 1 are safe, given that |ϕ| 10 −3 .

FCNCs
The couplings of Z 1,2 with fermions arise from Changing to the basis of (A, Z , Z ), i.e.
and by using Q = T 3 + Y , Y = X + N , e = gs W , and g X s θ = g N c θ = gt W , we obtain Because Q, T 3 , and Y are universal for every repeated fermion flavors, such as neutrinos, charged leptons, up-type quarks, down-type quarks, the flavor change is only associated with X -charge. Additionally, lepton flavors are universal under X . Hence, this flavor change occurs only for quarks.

The relevant Lagrangian is
where q a denotes quarks of either up-types (u α , u 3 ) or down- types (d α , d 3 ), and notice also that q a is vector-like under X . Changing to the mass basis, where q = u, d, and i = 1, 2, 3 labels mass eigenstates, u i = (u, c, t) and d i = (d, s, b), so that the quark mass matrices m u,d are diagonalized, Here, note that the CKM matrix is V CKM = V † uL V d L . Additionally, the quark current in (49) couples only the same quark chirality, i.e. qγ μ q = q L γ μ q L + q R γ μ q R . Using the unitarity condition, where i, j are summed. The first term conserves quark flavors, whereas the last two terms give rise to FCNCs for i = j, where we have defined, Substituting Z = c ϕ Z 2 −s ϕ Z 1 into the FCNC Lagrangian and integrating Z 1,2 out, we get effective interactions, where the last two terms differ from the first one only in chiral structures. We have neglected the contribution of Z 1 , because it is more suppressed than that of Z 2 , i.e.
due to t ϕ ∼ m 2 Z 1 /m 2 Z 2 ; additionally, we have utilized c ϕ 1. The effective couplings in (55) are further approximated as and similar for the others, which are all independent of θ, x, The effective interactions in (55) contribute to the neutralmeson mixing amplitudes, extensively studied in the literature. Assuming the dominant new physics effects come from the first term, L 2 i j , the existing data imply according to K 0 -K 0 mixing, B 0 d -B 0 d mixing, and B 0 s -B 0 s mixing, respectively [21,22]. Aligning the quark mixing to the down quark sector, without loss of generality, we have Hence, the constraints in (58) yield corresponding to the mentioned meson mixing systems, compatible to the previous bounds. Alternatively, the effective interactions (L R) and (R R) also contribute to the neutral-meson mixing amplitudes through switching on the right-handed quark mixing matrix, V q R . Since V q R is left arbitrarily as in the standard model (where only the left-handed quark mixing is constrained by the CKM matrix V CKM = V † uL V d L ), an issue arises that this V q R contribution leads to dangerous FCNCs? However, as given from outset, our theory originates from a trinification which conserves a left-right symmetry. We expect that the left-right symmetry is still good at the energy scale of the current theory, i.e. V q R ∼ V q L . This would suppress the dangerous (L R) and (R R) contributions to neutral-meson mixings as the CKM factor does so for the (L L) coupling. To be concrete, we first consider the contribution of all effective interactions in (55) to the mass difference of K 0 -K 0 mixing system, such as where K 0 is constructed from two quarks (q i , q j ) = (d, s). Further, we have with the aid of in agreement with [23]. Here, the hadronic matrix elements have been determined by the vacuum insertion approximation using PCAC [24]. Similarly for B 0 d,s -B 0 d,s mixings, i.e. (q i , q j ) = (d, b) and (s, b), respectively, we have The contribution terms of new physics in (62), (65), and (66) are equally suppressed by an approximate left-right symmetry, V d R ∼ V d L , which implies R i j ∼ L i j . Assuming the contributions (L R) and (R R) equivalent to (L L), using the approximation (57), the bounds from the mass differences (cf. [23]) are compatible to the ones determined in (60).

Collider bounds
Since Z and Z slightly mix, we will omit their mixing effect in this section, without loss of generality. That said, we set ϕ → 0 and Z 2 → Z as a physical field. Because Z interacts with both usual quarks and leptons, it definitely presents interesting signals at the current hadron and lepton colliders. The LEPII experiment [25] has studied the process e + e − → ff that produces a pair of ordinary fermions ( ff ) through exchange of Z . This process is best described by the following effective interactions, where the chiral gauge couplings a Z L , , as usual. Considering dilepton signals, f = μ, τ , all the leptons possess universal couplings, such that where the explicit couplings are given, using Table 3, The LEPII experiment has searched for such chiral interactions, giving several constraints on respective couplings, commonly achieved at order of a few TeV. Considering the bound derived for a new U (1) gauge boson like ours, it typically obeys [26] Using the approximation, m 2 for t θ = 1 and |x| = 1/2, as before, in agreement with the previous bounds. At the LHC experiment, both new physics signals of dijet [27] and dilepton [28] have been investigated. However, since the current bound for dijet is less sensitive than dilepton, we will only examine the latter. The cross-section producing a final state of dilepton can be estimated, using narrow width approximation, such as where the luminosity d L qq /dm 2 Z can be extracted from [29] for LHC √ s = 13 TeV. The partonic cross-section and branching Br(Z → ff ) = (Z → ff )/ Z take the form, Fig. 3 Dilepton production cross-section as function of Z mass, where dotted lines denote observed limit for different widths extracted at dilepton invariant-mass resonance, using 36.1 fb −1 of pp collision at √ s = 13 TeV by ATLAS [28]. The black and gray dotted lines are theoretical prediction according to x = 1/2 and −1/2, respectively where we exclude the decay Z → ν RνR , and note also that Z decay to other fields negligibly contributes to Z width. In Fig. 3, we plot the dilepton production cross-section for f = e, μ, τ , commonly labelled l, which have the same Z couplings, thus cross-section. The ATLAS [28] yields a negative result for new dilepton event of high mass, thus this translates to a lower limit for Z mass m Z > 4.6 TeV and m Z > 2.8 TeV for the model with x = 1/2 and x = −1/2, respectively, where t θ = 1 is taken as above. Hence, using m Z (4|x|gt W /s 2θ ) leads to > 6.4 TeV and > 3.9 TeV according to the Z mass bounds. These LHC bounds are in agreement with the previous constraints.

Conclusion
A guide from potential grand unification is that fermion families at low energy might be not universal, unlike the conventional standard model. This work has shown that such abelian remnants are presented, making a match of family number to color number. Additionally, the hypercharge is a result of their breaking, making an electrically-neutral world whose proton and electron charges are cancelled out.
Interestingly, the Higgs fields experience the new abelian charges, causing a mixing of the usual Z boson with new Z boson, explaining the recently-measured W -boson mass anomaly. The electroweak precision data on Z -Z mixing combined with the W mass measurement implies the new physics scale = 4.5-8.5 TeV and a weak scale v 1 = 0-185 GeV for the charge parameter x = 1/2, while ∼ 5 TeV and v 1 = 0-55 GeV for x = −1/2, where for all cases, the relative new gauge couplings set as t θ = g N /g X = 1. Additionally, the values of x = ±1/6 are obviously ruled out.
This theory contains the right-handed neutrinos as fundamental constituents, and the symmetry breaking leads to naturally-small neutrino masses, suppressed by the new physics scale that sets Majorana right-handed neutrino masses also at TeV scale.
The family nonuniversality occurs only for quarks, and this leads to FCNCs of quarks coupled to Z . An analysis for the contribution of left currents (L L) implies a bound for new physics scale > 3.6 TeV, 4 TeV, and 4.4 TeV according to the mixings, K 0 -K 0 , B 0 s -B 0 s , and B 0 d -B 0 d , respectively. The contribution of right currents such as (L R) and (R R) to the mixings may be large, but safely suppressed by assuming an approximate left-right symmetry. This kind of constraint is independent of x, θ.
Considering Z contribution to dilepton signals at LEPII gives a bound > 4.5 TeV for |x| = 1/2 and t θ = 1, while the dilepton production cross-section at LHC induced by Z implies > 6.4 TeV and 3.9 TeV with respect to x = 1/2 and x = −1/2, respectively, while keeping t θ = 1. Hence, the collider, FCNC, electroweak precision data, and W -mass measurement on the new physics scale are all compatible.
If the current theory is embedded in a trinification [8] or E 6 [9], this implies x = ±1/6, as well as the family nonuniversality is lost. An analysis shows that our proposal can be properly embedded in a flipped trinification [13,14] for arbitrary x as the smallest choice of a unification symmetry.

Data Availability Statement
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