Generalized Gauge U(1) Family Symmetry for Quarks and Leptons

If the standard model of quarks and leptons is extended to include three singlet right-handed neutrinos, then the resulting fermion structure admits an infinite number of anomaly-free solutions with just one simple constraint. Well-known examples satisfying this constraint are $B-L$, $L_{\mu}-L_{\tau}$ , $B-3L_{\tau}$ , etc. We derive this simple constraint, and discuss two new examples which offer some insights to the structure of mixing among quark and lepton families, together with their possible verification at the Large Hadron Collider.


Introduction :
In the standard model of particle interactions, there are three families of quarks and leptons.
Under its SU(3) C × SU(2) L × U(1) Y gauge symmetry, singlet right-handed neutrinos ν R do not transform. They were thus not included in the minimal standard model which only has three massless left-handed neutrinos. Since neutrinos are now known to be massive, ν R should be considered as additions to the standard model. In that case, the model admits a possible new family gauge symmetry U(1) F , with charges n 1,2,3 for the quarks and n ′ 1,2,3 for the leptons as shown in Table 1.
Ref. [12] would also satisfy Eq. (5). This may then be considered [13] as the separate gauging of B and L.
In this paper, we discuss two new examples which offer some insights to the structure of mixing among quarks and lepton families. Both have nontrivial connections between quarks and leptons. Their structures are shown in Table 3. In both cases, with only one Higgs doublet with zero charge under U(1) F , quark and lepton mass matrices are diagonal except for the first two quark families. This allows for mixing among them, but not with the third family. It is a good approximation to the 3×3 quark mixing matrix, to the extent that mixing with the third family is known to be suppressed. In the lepton sector, mixing also comes from the Majorana mass matrix of ν R which depends on the choice of singlets with vacuum expectation values which break U(1) F . Adding a second Higgs doublet with nonzero U(1) F charge will allow mixing of the first two families of quarks with the third in both cases. As for the leptons, this will not affect Model A, but will cause mixing in the charged-lepton and Dirac neutrino mass matrices in Model B. Flavor-changing neutral currents are predicted, with interesting phenomenological consequences.
Basic structure of Model A : Consider first the structure of the 3 × 3 quark mass matrix M d linking (d L ,s L ,b L ) to (d R , s R , b R ). Using with φ 0 1 = v 1 , it is clear that M d is block diagonal with a 2 × 2 submatrix which may be rotated on the left to become where s L = sin θ L and c L = cos θ L . We now add a second Higgs doublet is obtained. At the same time, M u is of the form where it has been rotated on the right. Because of the physical mass hierarchy m u << m c << m t , the diagonalization of Eq. (10) will have very small deviations from unity on the left. Hence the unitary matrix diagonalizing Eq. (9) on the left will be essentially the experimentally observed quark mixing matrix V CKM which has three angles and one phase.
Now M d of Eq. (9) has exactly seven parameters, the three diagonal masses m ′ d , m ′ s , m ′ b , the angle θ L , the off-diagonal mass m ′ sb which can be chosen real, and the off-diagonal mass m ′ db which is complex. With the input of the three quark mass eigenvalues m d , m s , m b and V CKM , these seven parameters can be determined.
Consider the diagonalization of the real mass matrix where s 1,2 << 1 and a << b << c have been assumed. We obtain and Hence where α is the phase transferred from m ′ db .
Comparing the above with the known values of V CKM [14], we obtain to a very good approximation.

Scalar sector of Model A :
In addition to Φ 1,2 , we add a scalar singlet then the Higgs potential containing Φ 1,2 and σ is given by Let φ 0 1,2 = v 1,2 and σ = u, then the minimum of V is determined by For m 2 2 large and positive, a solution exists with v 2 Hence the scalar particle spectrum of Model A consists of a Higgs boson h very much like that of the SM with m 2 h ≃ 2λ 1 v 2 1 , a heavy Higgs boson which breaks U(1) F with m 2 σ ≃ 2λ 3 u 2 , and a heavy scalar doublet very much like Φ 2 with m 2 (φ + 2 , φ 0 2 ) ≃ m 2 2 + λ 23 u 2 .

Gauge sector of Model A :
With the scalar structure already considered, the Z − Z F mass-squared matrix is given by .
The Z − Z F mixing is then (g Z /2g F )(v 2 2 /u 2 ). For v 2 ∼ 10 GeV and u ∼ 1 TeV, this is about 10 −4 , well within the experimentally allowed range.
Since Z F couples to quarks and leptons according to n 1,2,3 and n ′ 1,2,3 , its branching fractions to e − e + and µ − µ + are given by 2n ′

Flavor-changing interactions :
Whereas the SM Z boson does not mediate any flavor-changing interactions, the heavy Z F does because it distinguishes families. For quarks, Using Eqs. (12) and (13) to express the above in terms of mass eigenstates for the d sector, and keeping only the leading flavor-changing terms, we find From the experimental values of the B 0 −B 0 , B 0 S −B 0 S , and K L −K S mass differences, severe constraints on g 2 F /m 2 Z F are obtained, coming from the operators respectively. Using typical values of quark masses and hadronic decay and bag parameters [17], we estimate the various Wilson coefficients to find their contributions as follows: Using Eq. (15) and assuming that the above contributions are no more than 10% of their experimental values [14], we find the lower limits on m Z F /g F to be 10.2, 9.5, 0.84 TeV respectively. This is easily satisfied for m Z F > 4.0 TeV with g F = 0.13 from the LHC bound discussed in the previous section.
In the scalar sector, since Φ 1,2 both contribute to M d , the neutral scalar field orthogonal to the SM Higgs field will also mediate flavor-changing interactions. The Yukawa interactions are Extracting again the leading flavor-changing terms, we obtain where the physical scalar Assuming negligible mixing between H or A with the SM h (identified as the 125 GeV particle observed at the LHC), we consider the following effective operators [18]: The upper bounds on (1/v 2 respectively, whereas those on (1/v 2 For v 2 = 10 GeV, these are easily satisfied with for example m H = 500 GeV and m A = 520 GeV. Lepton sector of Model A : With the chosen U(1) F charges (0, −2, −4) of Table 3 Call that σ 1 and add σ 2,4 ∼ 2, 4, with vacuum expectation values u 1,2,4 respectively. Then where M 0 is an allowed invariant mass term, M 1 comes from u 2 , and M 2,3 from u 4 . The seesaw neutrino mass matrix is then where the two texture zeros appear because of the form of M R and M D being diagonal [19].
This form is known to be suitable for a best fit [20] to current neutrino-oscillation data with normal ordering of neutrino masses.
In the gauge sector, again Z F → e − e + is zero, and the branching fraction Z F → µ − µ + is now 2/51. The c u,d coefficients are then For the same choice of g F = 0.13 for Model A, the present experimental lower bound from LHC data is reduced from 4.0 TeV to 3.7 TeV. For quarks, Using Eqs. (12) and (13) to express the above in terms of mass eigenstates for the d sector, and keeping only the leading flavor-changing terms, we find This differs from Eq. (25) only by an overall factor of −2. As for the scalar sector, Eqs. (30) and (31) remain the same.
Lepton sector of Model B : With the chosen U(1) F charges (0, −1, −2) of Table 3, the charged-lepton and Dirac neutrino mass matrices are given by Using the scalar singlets σ 1 ∼ 1 as well σ 2 , the ν R Majorana mass matrix is again given by Eq. (37). Now even though M D is not diagonal, Eq. (38) is still obtained, thereby guaranteeing a best fit to current neutrino-oscillation data. The difference from Model A is the presence of τ − e transitions from the nondiagonal M l . However, for m ′ eτ /m ′ τ < 0.1, the branching fraction of τ → eµ − µ + is less than 2 × 10 −11 , far below the current bound of 4.1 × 10 −8 .
Application to LHC anomalies : Whereas Z F also mediates b → sµ − µ + , its effect is too small in Models A and B to explain the tentative LHC observations of B → K * µ − µ + and the ratio of B + → K + µ − µ + to B + → K + e − e + [21]. The reason is the stringent bound on m Z F from LHC data as a function of g F through the parameters c u,d of Eqs. (23) and (40).