Note that in Eq.(42c) the right-handed field R does not participate in the weak interaction involving the vector bosons W a. The asymmetric forms for the fermion fields in Eq.(42f) is a way to portrait the chiral nature of the objects in weak interaction - the left-handed version is different from the right-handed one. The axial vector interaction is hidden in the last term of Eq.(42d). This curious feature, that the electron is split into two parts is a consequence of the fact that the weak interactions violate parity and are mediated by the V-A interactions, where V stands for vector and A stands for axial vector (also known as pseudovector, which changes sign under a parity transformation). Since there is no right-handed neutrino in nature, the fermionic field consists of a left-handed isodoublet and a right-handed isosinglet as shown below.f abc is the antisymmetric tensor such that f 123 = +1, f 213 = -1, f 113 = 0.F is the anti-symmetric tensor for the electromagnetic field as shown in Eq.(24), where the vector potential A is now denoted by B.W a is related to the three gauge (vector) bosons with the index "a" running from 1 to 3.Whenever an index appears in both the subscript and superscript, it signifies a summation over these components. The symbols and are indices for the space-time components running from 1 to 4.Meaning of the symbols in the Lagrangina density : 1 is the gauge bosons part 2 is the fermionic part and 3 is the scalar Higgs sector, which generates mass for the gauge bosons and the fermions. The lepton Lagrangian density in Eq.(40) consists of three parts. This is known as the Weinberg-Salam Model. Thus instead of writing down the field equations explicitly such as in Eqs.(17) - (20) or Eqs.(13) and (38a), the dynamics of the electro-weak interaction can be expressed in term of the Lagrangian density: The field equations are derived by minimizing the action, which is related to the Lagrangian density. The following is a crude attempt to provide a glance of the subject matter by introducing the Lagrangian density for the Standard Model. In fact, there is no piece of experimental data that violates the Standard Model. Nevertheless, not only is it renormalizable, it can explain a vast number of results from all areas of particle physics. The theory is rather unwieldy and inelegant. It involves 18 unknown parameters, cannot explain the origin of the quark masses or the various coupling constants. It was created by crudely splicing the electroweak theory and the theory of quantum chromodynamics (QCD). However, the Standard Model is certainly not the final theory of particle interactions. It can describe all known fundamental forces except gravity. By imposing the space-time (transformation) and gauge (mixing particles in an internal space) symmetries to its formulation, the Standard Model is one of the great successes in modern physics.
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