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rpp2010-rev-guts - 15 Grand Unified Theories 1 15 GRAND...

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15. Grand Unified Theories 1 15. GRAND UNIFIED THEORIES Revised October 2005 by S. Raby (Ohio State University). 15.1. Grand Unification 15.1.1. Standard Model: An Introduction : In spite of all the successes of the Standard Model [SM], it is unlikely to be the final theory. It leaves many unanswered questions. Why the local gauge interactions SU(3) C × SU(2) L × U(1) Y , and why 3 families of quarks and leptons? Moreover, why does one family consist of the states [ Q, u c , d c ; L, e c ] transforming as [(3 , 2 , 1 / 3) , ( ¯ 3 , 1 , 4 / 3) , ( ¯ 3 , 1 , 2 / 3); (1 , 2 , 1) , (1 , 1 , 2)], where Q = ( u, d ), and L = ( ν, e ) are SU(2) L doublets, and u c , d c , e c are charge conjugate SU(2) L singlet fields with the U(1) Y quantum numbers given? [We use the convention that electric charge Q EM = T 3 L + Y/ 2 and all fields are left-handed.] Note the SM gauge interactions of quarks and leptons are completely fixed by their gauge charges. Thus, if we understood the origin of this charge quantization, we would also understand why there are no fractionally charged hadrons. Finally, what is the origin of quark and lepton masses, or the apparent hierarchy of family masses and quark mixing angles? Perhaps if we understood this, we would also know the origin of CP violation, the solution to the strong CP problem, the origin of the cosmological matter-antimatter asymmetry, or the nature of dark matter. The SM has 19 arbitrary parameters; their values are chosen to fit the data. Three arbitrary gauge couplings: g 3 , g, g (where g , g are the SU(2) L , U(1) Y couplings, respectively) or equivalently, α s = ( g 2 3 / 4 π ) , α EM = ( e 2 / 4 π ) ( e = g sin θ W ), and sin 2 θ W = ( g ) 2 / ( g 2 + ( g ) 2 ). In addition, there are 13 parameters associated with the 9 charged fermion masses and the four mixing angles in the CKM matrix. The remaining 3 parameters are v, λ [the Higgs VEV (vacuum expectation value) and quartic coupling] (or equivalently, M Z , m 0 h ), and the QCD θ parameter. In addition, data from neutrino oscillation experiments provide convincing evidence for neutrino masses. With 3 light Majorana neutrinos, there are at least 9 additional parameters in the neutrino sector; 3 masses, 3 mixing angles and 3 phases. In summary, the SM has too many arbitrary parameters, and leaves open too many unresolved questions to be considered complete. These are the problems which grand unified theories hope to address. 15.1.2. Charge Quantization : In the Standard Model, quarks and leptons are on an equal footing; both fundamental particles without substructure. It is now clear that they may be two faces of the same coin; unified, for example, by extending QCD (or SU(3) C ) to include leptons as the fourth color, SU(4) C [1]. The complete Pati-Salam gauge group is SU(4) C × SU(2) L × SU(2) R , with the states of one family [( Q, L ) , ( Q c , L c )] transforming as [(4 , 2 , 1) , ( ¯ 4 , 1 , ¯ 2)], where Q c = ( d c , u c ) , L c = ( e c , ν c ) are doublets under SU(2) R . Electric charge is now given by the relation Q EM = T 3 L + T 3 R + 1 / 2( B L ), and SU(4) C contains the subgroup SU(3) C × ( B L ) where B ( L ) is baryon (lepton) number. Note ν c has no SM quantum numbers and is thus completely “sterile.” It is introduced to complete the SU(2)
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