The Weak Currents

The apparently couples to the weak charged current

where the suffix denotes the left-handed projections of fermion fields, and is a unitary matrix of mixing angles which connects the quark fields participating in weak interactions to the mass eigenstates.

(In the sector, is the Cabibbo rotation matrix with ; on including may contain -violating phases.) Approximating for now by the identity matrix, the Lorentz vector part of (l) is related to the isovector part of the electromagnetic current by a weak isospin rotation . Differences between vector current ``decays'' and isovector ``decays'' arise from violations of weak isospin invariance by mass differences between pairs of quarks (and leptons) in the same weak isomultiplet (e.g., ). If quark masses are introduced explicitly into the original Lagrangian, then the rate for at high in the free quark approximation , while the rate for vector current decays is (Ref. [1]). The electromagnetic current is exactly conserved (so that the spin-0 components of are decoupled): this conservation (CVC) for the weak vector current holds only in the weak isospin symmetric limit. The weak vector current in general has a divergence (1) , so that spin-0 may decay through the vector current at a rate . In the chiral symmetry limit where all quark masses are neglected, axial vector current decays should become identical to vector current ones: for finite , the axial current decay rate . The divergence of the axial current (so that spin-0 decay through the axial current at a rate ), which is non-zero whenever finite quark masses are present to break chiral symmetry. The large masses of quarks presumably arise primarily from weak interaction effects (via the Higgs mechanism, etc.), and must be inserted as explicit bare mass terms in the QCD Lagrangian. However, the confinement of quarks into color singlets with radii presumably contributes additional dynamical effective quark masses (which decrease at short distances) independent of flavor. These contributions are important for the divergence of the axial current, but tend to cancel in the divergence of the vector current: the validity of CVC results is thus presumably a consequence of the fact that most hadronic interactions involve , and .

Weak interactions occur between the currents (1) by exchanges. In the low-energy limit, and ignoring QED and QCD corrections (free quark approximation), the interactions are represented by an effective four-fermion vertex As discussed below, QCD corrections to these interactions are sensitive to their short distance structure, so that their softening at must be accounted for. QED corrections are in some cases also sensitive to the possibility of couplings, which must be treated in the complete Weinberg-Salam model. In these notes, I shall not consider higher-order weak processes, as presumably induce , , etc. I shall also not discuss -violating effects.

The mixing matrix appears to be arranged so that , and presumably . I shall assume this ordering below.

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