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It might be helpful to read this article along with the companion overview of the Standard Model . THE CLASSICAL THEORY A chiral gauge theory This article uses the older Dirac notation instead of the more appropriate Weyl notation. The Weyl notation is more appropriate because there is no natural correspondence between the left handed and right handed fermion fields other than that generated dynamically through the Yukawa couplings after the Higgs field has acquired a VEV . The Helicity projections of a Dirac field ψ are ::left helicity: ψL = ½(1-γ5)ψ and the right helicity: ψR = ½(1+γ5)ψ. are needed, because the SM is a chiral gauge theory, ie, the two helicities are treated differently. Right handed singlets, left handed doublets Under the Weak Isospin SU(2) the left handed and right handed helicities have different Charges . The left handed particles are weak-isospin doublets (2), whereas the right handed are singlets (1). The right handed neutrino does not exist in the standard model. (However, in some ''extensions'' of the standard model they do) The ''up-type quarks'' are charge 2/3 quarks: u, c, t. The charge -1/3 quarks (d, s, b) are called ''down-type quarks''. The theory contains ::the left handed doublet of quarks QL= (uL, dL) and leptons EL= (νL, eL) ::the right handed singlets of quarks uR and dR and the electron eR. There is no right-handed neutrino in the SM. This is essentially by definition. When the Standard Model was written down, there was no evidence for neutrino mass. Now, however, a series of experiments including Super-Kamiokande have indicated that neutrinos indeed have a tiny mass. This fact can be simply accommodated in the Standard Model by adding a right-handed neutrino. This, however, is not strictly necessary. For example, the dimension 5 operator also leads to neutrino oscillations. This pattern is replicated in the next generations. We introduce a generation label i=1,2,3 and write ui to denote the three generations of up-type quarks, and similarly for the down type quarks. The left handed quark doublet also carries a generation index, QiL, as does the lepton doublet, EiL. Why this? What dictates this form of the weak isospin charges? The coupling of a right handed Neutrino to matter in Weak Interaction s was ruled out by experiment long ago. Benjamin Lee and J. Zinn-Justin , and Gerardus 't Hooft and Martinus Veltman in 1972 suggested the inclusion of left and right handed fields into the same Multiplet . This possibility has been ruled out by experiment. This leaves the construction given above. For the Lepton s, the gauge group can then be SU(2)L×U(1)L×U(1)R. The two U(1) factors can be combined into U(1)Y×U(1)l where l is the Lepton Number . Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2)L×U(1)Y. A similar argument in the quark sector also gives the same result for the electroweak theory. This form of the theory developed from a suggestion by Sheldon Glashow in 1961 and extended independently by Steven Weinberg and Abdus Salam in 1967 (and in rudimentary form by Julian Schwinger in 1957). The gauge field part The gauge group has already been described. Now one needs the Field s. The non-Abelian gauge field strength tensor ::Faμν = ∂μAaν - ∂νAaμ + g fabcAbμAcν in terms of the gauge field Aaμ, where the subscript μ runs over spacetime dimensions (0 to 3) and the superscript a over the elements of the Adjoint Representation of the gauge group, and g is the gauge Coupling Constant . The quantity fabc is the Structure Constant of the gauge group, defined by the commutator tb =fabctc. In an Abelian group, since the generators ta all commute with each other, the structure constants vanish, and the field tensor takes its usual Abelian form. We need to introduce three gauge fields corresponding to each of the subgroups SU(3)×SU(2)×U(1) —
The gauge field Lagrangian The gauge part of the electroweak Lagrangian is ::Lg = -¼ (Waμν Wa,μν + BμνBμν) The standard model Lagrangian consists of another similar term constructed using the gluon field tensor. The W, Z and photon The charged W Boson s are the linear combinations ::W±μ = W1μ ± i W2μ. Z Boson s (Zμ) and Photon s (Aμ) are mixtures of W3 and B. The precise mixture is equivalent to deciding on the relation between the Electric Charge , Weak Isospin and Weak Hypercharge — ::Q = (e/g) T3 + (e/g') Y = sinθW T3 + cosθW Y, where we have introduced the Weinberg Angle , θW. In terms of this, one can write ::Zμ=cosθW W3μ - sinθW Bμ, and Aμ=sinθW W3μ + cosθW Bμ. The charged and neutral current couplings The charged currents J±=J1±iJ2 are :: These charged currents are precisely those which entered the Fermi Theory Of Beta Decay . The action contains the charge current piece :: It will be discussed later in this article that the W boson becomes massive, and for energy much less than this mass, the effective theory becomes the current-current interaction of the Fermi theory. However, gauge invariance now requires that the component W3 of the gauge field also be coupled to a current which lies in the triplet of SU(2). However, this mixes with the U(1), and another current in that sector is needed. These must be currents must be uncharged in order to conserve charge. So we require the neutral currents :: :: The neutral current piece in the Lagrangian is then :: There are no mass terms for the fermions. Everything else will come through the scalar (Higgs) sector. Quantum chromodynamics Leptons carry no colour charge; quarks do. Moreover, the quarks have only vector couplings to the gluons, ie, the two helicities are treated on par in this part of the standard model. So the coupling term is given by :: Here Ta stands for the generators of SU(3) colour. The mass term in QCD arises from interactions in the Higgs sector. The Higgs field One requires masses for the W, Z, quarks and leptons. Recent experiments have also shown that the Neutrino has a mass. However, the details of the mechanism that give the neutrinos a mass are not yet clear. So this article deals with the ''classic'' version of the SM (circa 1990s , when Neutrino Mass es could be neglected with impunity). The Yukawa terms Giving a mass to a Dirac field requires a term in the Lagrangian which couples the left and right helicities. A complex scalar doublet (charge 2) Higgs field, (φ+, φ0) is introduced, which couples through the Yukawa Interaction :: where Gu, d are 3×3 matrices of Yukawa couplings, with the ij term giving the coupling of the generations i and j. Symmetry breaking The Higgs part of the Lagrangian is
where λ>0 and μ2<0, so that the mechanism of Spontaneous Symmetry Breaking can be used. In an Unitarity Gauge one can set φ+=0 and make φ0 real. Then <φ0>=v is the non-vanishing Vacuum Expectation Value of the Higgs field. Putting this into LYU, a mass term for the fermions is obtained, with a mass matrix v Gu, d. From LH, quadratic terms in Wμ and Bμ arise, which give masses to the W and Z bosons |
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