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This is known as the Weinberg-Salam Model. The lepton Lagrangian density

- Meaning of the symbols in the Lagrangina density
- The symbols
_{}and_{}are indices for the space-time components running from 1 to 4. Whenever an index appears in both the subscript and superscript, it signifies a summation over these components. - W
^{a}_{}is related to the three gauge (vector) bosons with the index "a" running from 1 to 3. *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*_{}.- f
^{abc}is the antisymmetric tensor such that f^{123}= +1, f^{213}= -1, f^{113}= 0, ... etc. - 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 -

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). The axial vector interaction is hidden in the last term of Eq.(42d). 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. Note that in Eq.(42c) the right-handed field R does not participate in the weak interaction involving the vector bosons W---------- (42f) ^{a}_{}. _{}are the Pauli matrices as shown in Eq.(10) in the appendix on "Groups" with i running from 1 to 3. The four 4X4 gamma metrices_{}are constructed from the Pauli matrices and the identity matrix,_{}, and_{}is the adjoint spinor of_{}where_{}^{}is the Hermitean conjugate ,_{}.- g and g' are the coupling constants associated with the W
_{}and B_{}boson respectively. G_{e}is the Yukawa coupling constant, which defines the strength of the interaction between the Higgs field and the lepton fields. - The first three terms in Eq.(41c) are responsible for generating the mass of the gauge bosons, while the last term takes care of the fermion mass.
_{}is the Higgs doublet with four real components, three of which will be absorbed by the W^{}, and Z^{0}to become massive. The diagram below shows a simplified Higgs field in the shape of a Mexican hat:

This form of field exhibits a drastic effect called "spontaneous symmetry breaking". When the origin of the field is shifted to v =_{}(-2m^{2}/)^{1/2}(the scalar field at minimum energy) by the transformation ' = - v, the fields*W*^{a}_{}and*B*_{}recombine and reemerge as the physical photon field*A*_{}, a neutral massive vector particle*Z*_{}, and a charged doubled of massive vector particles*W*^{}_{}. In terms of the Weinberg angle (mixing angle) tan_{}= g'/g,

Thus, the transformation from (B_{}, W_{}^{3}) to (A_{}, Z_{}) can be considered as a rotation of the mixing angle.

The masses of the gauge bosons are obtained from the formula:

where we have used the known constants for G_{F}= 1 / v^{2}2^{1/2}= 1.14x10^{-5}Gev^{-2}, e = g sin_{W}= (4)^{1/2}= 0.3028, and from the measurement of sin^{2}_{W}= 0.226_{}0.004 to evalutate the masses for the gauge bosons. The theoretical values are in good agreement with M_{W}= 80.4 Gev, and M_{Z}= 91.19 Gev determined by experiments.- The mass of the electron is generated by the last term in Eq.(41c). Together with shift of the Higgs field v, it emerges as m
_{e}= G_{e}v. While the neutrino remains massless. This formulation is necessary because the usual mass term in the Dirac equation is not gauge invariant when the left-handed component L and the right-handed component R of the fermion wave functions transform differently under SU(2). Thus it is assumed that the fermions start out with no mass, they acquire the mass by interacting with the Higgs field. Each fermion mass is therefore an unknown parameter (determined experientially now) in the Standard model. The attempt to find a theory which can predict these masses, remains an important but unsolved problem.

The above formulism can be carried over to the electro-weak interactions between quarks with the massless neutrino replaced by the up quark u (which has mass), and the electron replaced by the down quark d. In order to get the correct quantum numbers, such as the charge, the covariant derivatives are different from the lepton as shown below:

where the coupling constants g and g' are also different from the case of leptons.

The theory for the strong interaction is called quantum chromodynamics (QCD), which has the Lagrangian density:

---------- (43) |

- Meaning of the symbols in the QCD Lagrangina density :
- F
^{a}_{}is similar to the SU(2) gauge field in Eq.(42b). This is essentially the Yang-Mills field developed back in 1953 by generalizing the gauge invariance used in QED. It represents the eight massless gluons carrying the SU(3) "colour" force with the index "a" running from 1 to 8. - The index "i" is taken over the flavors, which labels the up, down, strange, charm, top, and bottom quarks. The flavor index is not gauged; it represents a global symmetry. However, the quarks also carry the local colour SU(3) indices red, green, and blue (which is suppressed here). In other words, quarks come in six flavors and three colours, but only the colour index participates in the local gauge symmetry. It is the colour force, which binds the quarks together.
- The mass of the quarks m
_{i}is derived from the interaction with the Higgs field in a formula similar to Eq.(41c). - = .

---------- (43a) |

The Standard Model can be written either in mathematical form as summarized in Eq.(43a), or in pictorial form, using the Feynman diagrams as shown in Figure 04. It is convenient to rewrite Eq.(43a) into a shorthand notation in describing the Feynman diagrams:

---------- (44) |

- Explanation of the terms in Eq.(44), and Feynman diagrams in Figure 04:
- The square terms G
^{2}, ... are the energy-momentum tensor of the boson fields, they also include terms that describe interaction among themselves except for the photon, which don't have this kind of interaction (see Figure 04a, b, and g). - The represents the covariant derivative. These terms signify interaction with the gauge bosons.
- The f
_{j}Hf_{k}terms indicate interaction of the fermions with the Higgs. The c_{jk}'s are the coupling constants, which determine the masses of all the quarks and leptons. - In Figure 04c, the vector mesons interacts with the left-handed electrons and right-handed anti-neutrino (not shown) only, while the right-handed electron interacts only with the photon.
- In Figure 04d, the last diagram actually involves a quark from the second generation, the strange quark, as well as a quark from the first generation, the up quark. This is an example of quark mixing, in which quarks of different generations get jumbled up. The amount of mixing is controlled by a parameter known as the Cabibbo angle
_{1}such that the mixed state

d' = d cos_{1}+ s sin_{1}.

A LHCb report in 2015 confirms that only the left-handed version of the quarks participates in the flavor (generation) changing process. The experiment observed the decay of trillions of the_{b}baryons. The bottom quark within these baryons can turn into an up quark during the process. It is from such observations that the weak interaction involves only left-handed fermions is re-affirmed. - Interactions between quark and qluon can be expressed similar to Figure 04d.
- The Higgs boson interacts with just about all elementary particles (Figure 04e, and f) except the neutrino and gluon. These are the interactions that give the vector mesons and fermions their masses after spontaneous symmetry breaking. The neutrinos and gluons remain massless in the Standard Model. The annihilation of the vector mesons in Figure 04e indicates the short lifetime (and thus the short interaction range) of these particles.
- Figure 04g shows the Higgs interacts with itself. These self-interactions generate the Mexican hat potential so crucial for spontaneous symmetry breaking.
- Only the diagrams for the first generation are given in Figure 04; there are similar diagrams for the other fermion families.
- Figure 04 also leaves out diagrams that can be obtained by exchanging all particles with their antiparticles (antiparticles have the same mass, mean lifetime, and spin as their respective particles, but the electric charge and other charges have the opposite sign).

## Figure 04 Standard Model |
The actual numerical values of these coupling constants are not given by the theory and must be measured from experiments. |

where |

c_{ij} = cos_{ij}, s_{ij} = sin_{ij}, and the phase angle tunrs some matrix elements into complex numbers, thereby violates CP invariance (CP invariance demands that V* = V).Experimentally, the mixing angles _{ij} are either smaller than or comparable to the Cabibbo angle _{1} ~ 15^{o}. Thus, the quark mixing is relatively unimportant. A similar mixing matrix exists in neutrino mixing between the flavor states (_{}_{e}, _{}_{}, _{}_{})
and the mass (mixed) states (_{}_{1}, _{}_{2}, _{}_{3}). Neutrino mixing is large in comparison to the quark mixing as shown inFigure 04h. It leads to the detection of neutrino mass. | |

## Figure 04h Flavor Mixing |
The mixing angle must have the same value for every electroweak process. It is observed to have the same value everywhere, to an accuracy of about one percent. Other successful predictions include the existence of the W and Z bosons, the gluon, the charm and the top quarks. Z boson decays have been confimred by LEP in 20 million of such events. |

3 coupling constants + 2 Higgs parameters + 2 x [3 generations x (2 fermions masses) + 4 CKM parameters] + 1 instanton

For massless neutrinos and no leptonic mixing angles, there are still 19 free parameters. With so much arbitrariness, the Standard Model should be considered only as the first approximation to the true theory of subatomic particles, i.e., it is an effective theory to be explained by more fundamental theory.

- Following is a list of subjects that the Standard Model fails to explain. Either it is not in the formulation or it is just plugged into the theory without explanation of its origin.
- The cosmological constant or vacuum energy.
- Dark energy.
- The inflaton in the first fraction of a second of the Big Bang.
- Matter-antimatter asymmetry.
- Cold dark matter.
- The form of the Higgs field.
- Hierarchy problem - huge Higgs boson mass implies huge mass for all elementary particles.
- Gravity.
- Masses of the quarks and leptons.
- Three generations of elementary particles.

In spite of these shortcomings as mentioned above, the Standard Model has been proven to be remarkably resilient under various verifications including the latest measurement for the mass of the Higgs particle. The most recent attempt (in 2013) to break SM is to measure the shape of some nuclei, which would become pear-shaped (Figure 05a) in the presence | |

## Figure 05a Pear-shaped Nucleus [view large image] |
of permanent electric dipole moment (EDM). Since EDM would violate the T symmetry (and thus also introduce CP violation) in SM, the detection of specific radiation patterns (from the pear-shaped nuclei) will indirectly indicate the necessity of new physics. It is |

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