Quark Mass Matrices from a Softly Broken U(1) Symmetry

Assigning U(1) charges to the quarks of the standard model, and allowing one extra scalar doublet with m^2>0, the correct pattern of the up and down quark mass matrices is obtained, together with their charged-current mixing matrix.

In the standard model of particle interactions, quark masses and the charged-current mixing matrix, V CKM , which links the (d, s, b) L quarks to the (u, c, t) L quarks, are known to exhibit a hierarchical pattern [1].
where the magnitude range of each matrix element is denoted.
With the one Higgs doublet of the standard model, this pattern (or any other) is certainly allowed, but then Yukawa couplings spanning 5 decades of magnitude are needed. On the other hand, if two Higgs doublets exist with v 1 = 174 GeV, but v 2 /v 1 ∼ 10 −3 ∼ 10 2 MeV, then Yukawa couplings spanning only 2 decades of magnitude are sufficient. In other words, m c,b,t are proportional to v 1 , but m u,d,s are proportional to v 2 . Of course, the hierarchical structure of the 2 vacuum expectation values (VEVs) is yet to be explained. As shown below, this may be attributed to the soft breaking of an assumed U(1) symmetry and is easily implemented if Φ 2 has m 2 > 0 while Φ 1 has m 2 < 0 as in the standard model.
The puzzle of quark masses and the charged-current mixing matrix, usually denoted by V ij , with i = u, c, t and j = d, s, b, has received a great deal of continuing attention.
One approach is to restrict the number of independent parameters necessary for a general description of all masses and mixing angles, so that a relationship among them may be derived, such as [2] This is usually postulated without recourse to a well-defined symmetry of the Lagrangian nor the extra particle content required to sustain it [3]. Another shortcoming of this approach is that the mass hierarchy of Eq. (1) remains largely unexplained.
The present approach is different. It looks for a way to understand why m u,d,s << v = 174 GeV, i.e. the scale of electroweak symmetry breaking, as well as the pattern of Eq. (2).
However, no precise prediction such as Eq.
(3) will be made. This approach was used in a radiative scheme some years ago [4], but the model itself is rather complicated. In contrast, the model to be described below is much simpler, requiring only one extra Higgs doublet together with a softly broken global U(1) symmetry.
The U(1) assignments of the 3 generations of quarks and the 2 Higgs doublets are given as follows.
As a result, the up quark mass matrix linking (u, c, t) L to (u, c, t) R is given by where v i = φ 0 i , and the freedom to rotate among (u, d) L and (c, s) L has been used to set theū L c R element to zero; whereas the down quark mass matrix linking (d, s, b) L to (d, s, b) R is given by where the freedom to rotate among the (d, s, b) R states has been used to set the 3 lower off-diagonal entries to zero.
As for V CKM , the contribution from M u is negligible because they are of order (m u /m 2 Hence Comparing the above with Eq. (2), it is also clear that the Yukawa coupling ratios f ds /f d , f db /f d , and f sb /f s may all be of order unity. Thus the correct pattern of quark masses and mixing angles is obtained. Obviously, the charged-lepton masses may be treated in the same way, i.e.
What remains to be shown is how v 2 << v 1 can arise naturally.
The most general scalar potential of the 2 assumed scalar doublets is given by where the µ 2 12 term breaks the U(1) symmetry softly. The equations of constraint for the VEVs are then Let m 2 1 < 0, m 2 2 > 0, and |µ 2 12 | << m 2 2 , then Since the µ 2 12 term breaks the U(1) symmetry, it is natural [5] for it to be small compared to is obtained. where but since V u = 1 to a very good approximation, V CKM ≃ V d , and the (d, s, b) L states have to be rotated by V d to become mass eigenstates. For example, b L in Eq. (9) becomes In the up quark sector, the roles of V and U are reversed, i.e. and which is the identity matrix to a very good approximation, as mentioned earlier.
in the mass-eigenstate basis. The most severe constraint on m 2 comes from the b → sµ + µ − rate through φ 0 2 exchange, i.e.
The K L − K S mass difference ∆m K gets its main contribution from (d L s R )(d R s L ) in this model through φ 0 2 exchange. Thus Similarly, the ∆m B 0 and ∆m B 0 s contributions are and Using f B = 170 MeV, B B = 1.0, |V tb | = 1, and the other parameter values as before, these contributions are respectively less than 3.7 × 10 −15 and 8.5 × 10 −12 , to be compared against the experimental value of 5.9 × 10 −14 for the former and the experimental lower bound of 1.3 × 10 −12 for the latter.
In the case of D 0 − D 0 mixing, the main contribution comes from (c L u R )(c R u L ), i.e. Other FCNC processes are also suppressed. For example, Using the previously chosen values for all the parameters, this contribution is less than There is also a contribution from Φ 2 to the muon anomalous magnetic moment [6]. It is easily calculated to be which is of the order 10 −11 or less, and thus negligible. However, the present model can be extended to allow for neutrino masses using a leptonic Higgs doublet [7], then the possible observed discrepancy in ∆a µ may be explained [8], but a nearly degenerate neutrino mass matrix is required. The extra contributions from Φ 2 to the oblique parameters S, T, U in precision electroweak measurements are all suppressed by λ 4 v 2 1 /m 2 2 and do not upset the excellent fit of the standard model.
In summary, a new realization of the generation structure of quarks and leptons has been presented in this paper, as given by Eqs. (4) to (7). The one extra scalar doublet is heavy with m 2 > 0. Typical values are m 2 ∼ few TeV with v 2 ∼ fraction of a GeV, whereas Φ 1 has m 2 < 0, resulting in v 1 = 174 GeV and m H = 115 GeV or greater. This is accomplished by a softly broken U(1) symmetry with |µ 12 | 2 /m 2 2 ∼ 10 −3 . The pattern of the observed quark masses (with m u,d,s from v 2 and m c,b,t from v 1 ) and the corresponding charged-current mixing matrix (V CKM ) is realized without severely hierarchical Yukawa couplings. All