Lecture 11

# Introductory Nuclear Physics

• Notes
• davidvictor
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PHY431 Lecture 11 26 5/1/2000 14. Symmetries of the Standard Model: U (1), SU (2), SU (3) We’ve now constructed a powerful toolset: we know how to describe free elementary particles: bosons using the Klein-Gordon Lagrangian; and fermions using the Dirac Lagrangian. We introduce interac- tions by the use of the gauge principle: requiring invariance of the Lagrangian under local phase/gauge transformations of the particle fields, we are led to introduce ‘compensating gauge fields’ of the proper form. The gauge fields have to be massless in order to preserve gauge invariance, which is clearly a problem when considering weak interactions where the gauge bosons are massive. However, we can overcome this problem by invoking the Higgs mechanism: postulating the existence of a boson field with an odd shape, i.e. a non-zero expectation value, and by re-expressing the Higgs field with respect to a true minimum (vacuum), we break the manifest symmetry of the Lagrangian, but reap great bene- fits as well: the gauge boson acquires mass, just what we need for the weak gauge bosons. A final benefit from the use of a theory with local gauge invariance is that such theories are inherently self- consistent when higher-order diagrams are considered. Very elegant cancellations between diagrams occur, which cure divergencies that would otherwise make the theory meaningless. We label the gauge symmetry by their group structure. The simplest gauge transformation is the multi- plication of the field by a phase factor function: exp{ i α ( x )} 1 + i α + ( i α ) 2 /2! + ( i α ) 3 /3! + ... , which is the (infinite) group of complex functions that have modulus 1. The group operation is multiplication (addition of the phase angles), and the unit element is exp{0} = 1. The group’s elements are clearly unitary: [exp{ i α ( x )}] = exp{ i α ( x )} = [exp{ i α ( x )}] 1 . The official name of the group is “unitary group of dimension one”: U (1) for short. More complicated gauge groups are formed by bringing in matrices: e.g. the 2 × 2 Pauli matrices exp{ i ½ σ⋅α ( x )} 1 2 + i ½ σ⋅α + ( i ½ σ⋅α ) 2 /2! + ( i ½ σ⋅α ) 3 /3! + ... . This group is again an infinite group of complex operator functions that have determinant 1 and are unitary. They are unitary because the Pauli matrices are Hermitian (and so is i ½ σ⋅α ); because the Pauli matrices are traceless we also find that det[exp{ i ½ σ⋅α }] = exp{Tr[ i ½ σ⋅α ]} = exp{0} = 1. This group’s name is special unitary group of di- mension 2, SU (2). Its generators are the Pauli Matrices, because any member of the group can be formed by appropriate combinations of the three Pauli matrices. 14.1 The SU (2) L of Weak Isospin The basic particle for the weak interactions is the doublet consisting of the (electron) neutrino and the electron. The weak interaction processes that we have seen – muon decay, or neutron decay – proceed by coupling of the W vector boson to the left-handed neutrino-electron or up-down quark doublets: * * * * * * * * , , , , e e e e W W e n p W W e d u W W e W W e µ µ ν ν ν ν π ν +

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