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Of the 4 different kinds of forces in the universe, gravity is attractive; electromagnetic force can be attractive or repulsive; the strong force provides a powerful attraction at short distance; the weak force works in very short distance, its action is to change one form of particle into another such as the beta decay of a free neutron in Figure 03k (strong force in stable nucleus inhibits the neutrons from beta decay). Note that the simple picture in Figure 03k already embraced the conservation of mass-energy, charge, and lepton number (n_{l}).
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## Figure 03k Weak Interaction in Picture [view large image] |
The strength of these forces varies according to the separation between particles, but in term of their coupling constants the ratio between the strong, electric, weak, and gravity is about 1 : 10^{-2} : 10^{-13} : 10^{-38}. The understanding of the weak force took many twists and turns. |

Its historical development includes the discovery of neutrino, the introduction of Fermi's theory, the violation of parity, the selection of V - A interaction, its adoption to the gauge theory and finally incorporation into the Standard Model (see next section). All these subjects will be presented in some details below.

- Discovery of neutrino - Beta decay was first observed in 1900. It caused a great deal of anxiety among physicists at
that time because while both the alpha and gamma rays are emitted with discrete spectra following the law of mass-energy conservation, beta rays appeared to exhibit a continuous spectrum. The problem persisted until 1930 when Pauli suggested that an invisible entity called neutrino shares the decay energy with the electron - the two particles together restore the conservation of mass-energy to its rightful place. The neutrino was dully discovered in 1953. It interacts only through the feeble weak interaction and gravity making its detection very difficult. Later on it is realized #### Figure 03l Lifetime of Decays [view large image]

that the elusive entity has to be anti-neutrino to converse lepton number (Figure 03k). It is now understood that all fermions feel the weak force which is characterized by long lifetime as shown in Figure 03l.

- Fermi's Theory - In 1934 Fermi expressed the beta decay mathematically by considering the interaction of the 4 fermions at a single point as shown in Figure 03ma and in the formula below:
M = G _{F}(_{p}u_{n})(_{e}u_{})

where G_{F}is the weak coupling constant, the u's are the Dirac spinors for the proton, neutron, electron, and anti-neutrino; the factor contain the essence of the weak interaction effects, in case it is equated to the gamma matrices, it would prescribe the#### Figure 03m Fermi's Theory [view large image]

current-current interaction (see Table 05 below); M has been labeled as (probability) amplitude, matrix element (of the interaction), (phenomenological) Lagrangian, or (interaction) Hamiltonian similar to the one for QED as shown in Eq.(38b). _{W}^{2}- q^{2}) corresponding to mediation with a massive boson, e.g., the W^{-}(q is the magnitude of 4-momentum for the boson, see Figure 03mb). The problem with non-renormalization remained for a long time until the successful formulation of the Standard Model in 1971. - Parity Violation - Parity is the operation
**P**that reverses the coordinate (x,y,z) to (-x,-y,-z) of a system (e.g., a particle or collection of particles). It is equivalent to a mirror reflection followed by a rotation through 180^{o}. The parity can be +1 or -1 for a system depending on whether the corresponding field remains the same or flips a sign, i.e., in simplemathematical expression: **P**u(**x**,t) = u(-**x**,t). The overall parity is given by the multiplication of its parts, e.g. (+1)x(-1)x(+1) = (-1) etc. It has been long held that parity is conserved before and after all particle interactions. The problem started in the 1950's when it was observed that the and mesons seem to be identical in every respect but their decay products appeared to be very different. While the ends up with three mesons the decays into two. Since the has parity of -1, that means the two end products have different parity, one of them must violate parity. Further study by T. D. Lee and C. N. Yang strongly suggested that is the case.#### Figure 03n Parity Violation [view large image]

Almost immediately C. S. Wu devised and carried a test, which confirmed the idea conclusively (Figure 03n). The result also implied that only the electrons with left-handed spin participate in the

- V - A interaction - It is obvious that the Fermi's Theory is not able to accommodate the experimental fact of parity violation. Theorists were looking for a way to construct a realistic formula from the various types of fermion current as shown in Table 05, which lists the scalar, vector, and tensor under coordinate transformation (where is the Dirac spinor simimlar to the u here). A pseudo or axial is added in front for those involve the
^{5}:

.After a long search for the magic formula, it turns out that parity violation can be expressed in combination of vector and axial vector (V - A) #### Table 05 Types of Fermion Current [view large image]

#### Figure 03o Beta Decay, Modern Version

interaction, e.g., for the electron anti-neutrino portion: _{e}(1 -^{5})u_{}]/2 =_{eL}u_{L}, where the subscript L indicates that only the left-handed elements are involved. There is no right-handed neutrino, and the right-handed electron only participates in the electromagnetic interaction. A similar expression is derived for the d quark (in neutron) and the u quark (in proton) as shown in Figure 03o. The updated version of M for beta decay is now in the form :

where g_{w}is the weak coupling constant and V_{ud}is the u-d element in the CKM matrix. It is the relative probability that the down quark decays into up quark in the weak interaction. - Gauge Theory - The weak interaction in gauge theory (as mentioned in the previous section) belongs to the SU(2) group, which is represented by 2x2 matrices. Thus the Dirac spinor has to be split into one part for the right-handed lepton (electron only) and the other for the left-handed neurtino and electron. This 2-elements representation is known as Weyl spinor. The mediating bosons appear naturally as the result of local gauge invariance.

The concept of gauge invariance can best be visualized with an example, which portrays a boat floating on the water surface with the depth of water at 3 meters (Figure 03p). The gauge (the water level marker) at the left registers a reading of 6 meter on the surface and 3 meter on the bottom, while the measurements from the other one at the right are 9 and 6 meter correspondingly. The differences in the readings are the result of moving the right marker further down, but the depth of water remains the same at 3 meters - the picture has not been changed by moving the gauge around. In theorietical physics it is the governing equation that is unchanged (invariant) under the change of some parameters, which turn out to be the phase angles in an abstract space.

## Figure 03p Gauge Invariance [view large image] |
Local gauge invariance produces mediating bosons, while global gauge invariance is related to conservation law. |

The following mathematical derivation shows the appearance of gauge boson from the requirement of local gauge invariance. The Dirac equation for the free electron has the form :

i

The gauge transformation

i

The invariance is clearly spoiled by the extra term in the middle. The equation can be made invariant only if a boson interaction term -g

i

The mass term would be removed in the Standard Model because the mass for all kinds of particles is generated via the Higgs mechanism. The Dirac spinor would also be replaced by the two-elements Weyl spinor representing the left-handed neutrino and electron in the Standard Model.

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