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Register a new userUnlocking the Standard Model. III. 2 Generations of quarks : calculating the Cabibbo angle
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Bruno Machet
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Maximally extending the Higgs sector of the Glashow-Salam-Weinberg model by including all scalar and pseudoscalar J=0 states expected for 2 generations of quarks, I demonstrate that the Cabibbo angle is given by tan^2(theta_c) = (1/m_K^2-1/m_D^2)/(1/m_pi^2-1/m_{D_s}^2) approx (m_pi^2 /m_K^2)(1-m_K^2/m_D^2 + m_pi^2/m_{D_s}^2).
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The Glashow-Salam-Weinberg model for N=2 generations is extended to 8 composite Higgs multiplets by using a one-to-one correspondence between its complex Higgs doublet and very specific quadruplets of bilinear quark operators. This is the minimal number required to suitably account, simultaneously, for the pseudoscalar mesons that can be built with 4 quarks and for the masses of the $W$ gauge bosons. They are used as input, together with elementary low energy considerations, from which all other parameters, masses and couplings can be calculated. We focus in this work on the spectrum of the 8 Higgs bosons, on the mixing angles, and on the set of horizontal and vertical entangled symmetries that, within the chiral $U(4)_L times U(4)_R$ group, strongly frame this extension of the Standard Model. In particular, the $u-c$ ($theta_u$) and $d-s$ ($theta_d$) mixing angles satisfy the robust relation $tan(theta_d+theta_u)tan(theta_d-theta_u) = Big(frac{1}{m_{K^pm}^2}-frac{1}{m_{D^pm}^2}Big) big/ Big(frac{1}{m_{pi^pm}^2}-frac{1}{m_{D_s^pm}^2}Big)$. Light scalars (below $90 MeV$) arise and the mass of (at least) one of the Higgs bosons grows like that of the heaviest $bar qgamma_5 q$ bound state. $theta_u$ cannot be safely tuned to zero and several parameters have no reliable expansion in terms of small parameters like $m_pi$ or the mixing angles. This study does not call for extra species of fermions. The effective couplings of scalars, which depend on the non-trivial normalization of their kinetic terms, can be extremely weak. For the sake of (relative) brevity, their rich content of non-standard physics (including astrophysics), the inclusion of the 3rd generation and the taming of quantum corrections are left for a subsequent work.
A very specific two-Higgs-doublet extension of the Glashow-Salam-Weinberg model for one generation of quarks is advocated for, in which the two doublets are parity transformed of each other and both isomorphic to the Higgs doublet of the Standard Model. The chiral group U(2)_L X U(2)_R gets broken down to U(1) X U(1)_{em}. In there, the first diagonal U(1) is directly connected to parity through the U(1)_LX U(1)_R algebra. Both chiral and weak symmetry breaking can be accounted for, together with their relevant degrees of freedom. The two Higgs doublets are demonstrated to be in one-to-one correspondence with bilinear quark operators.
We continue investigating the Standard Model for one generation of fermions and two parity-transformed Higgs doublets K and H advocated for in a previous work, using the one-to-one correspondence, demonstrated there, between their components and bilinear quark operators. We show that all masses and couplings, in particular those of the two Higgs bosons $varsigma$ and $xi$, are determined by low energy considerations. The mass of the quasi-standard Higgs boson, $xi$, is $m_xi approx m_pi$; it is coupled to u and d quarks with identical strengths. The mass of the lightest one, $varsigma$, is $m_varsigma approx m_pi frac{f_pi}{2sqrt{2}m_W/g} approx 34,KeV$; it is very weakly coupled to matter except hadronic matter. The ratio of the two Higgs masses is that of the two scales involved in the problem, the weak scale $sigma=frac{2sqrt{2}m_W}{g}$ and the chiral scale $v=f_pi$, which are also the respective vacuum expectation values of the two Higgs bosons. They can freely coexist and be accounted for. The dependence of $m_varsigma$ and $m_xi$ on $m_pi$, that is, on quark masses, suggests their evolution when more generations are added. Fermions get their masses from both Higgs multiplets. The theory definitely stays in the perturbative regime.
We review the recent experimental and theoretical progress in the determination of |V_{ud}| and |V_{us}|, and the status of the most stringent test of CKM unitarity. Future prospects on |V_{cd}| and |V_{cs}| are also briefly discussed.
The successful description of current data provided by the Standard Model includes fundamental fermions that are color-singlets and electroweak-nonsinglets, but no fermions that are electroweak-singlets and color-nonsinglets. In an effort to understand the absence of such fermions, we construct and study {it gedanken} models that do contain electroweak-singlet chiral quark fields. These models exhibit several distinctive properties, including the absence of any neutral lepton and the fact that both the $(uud)$ and $(ddu)$ nucleons are electrically charged. We also explore how such models could arise as low-energy limits of grand unified theories and, in this more restrictive context, we show that they exhibit further exotic properties.
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