By James R. Munkres
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Extra info for Elementary Differential Topology
Thus the result is proportional to mass differences. For K 0 − K¯ 0 mixing the contribution of virtual u quarks is negligible due to the small value of mu and the contribution of the t quark is also small due to the mixing factors Vts∗Vtd ∼ o(A2 λ 5 ). 85 is a QCD correction factor and O∆s=2 = d¯L γµ sL s¯L γµ dL is the 4-quark dimension 6 relevant operator. 77) 3 where fK ∼ 113MeV is the kaon pseudoscalar constant. 76) non perturbative additional contributions must be added, some of them of o(m2K /m2c ), because the smallness of mc makes a completely partonic dominance inadequate.
This is the see-saw mechanism for neutrino masses . Note that if no νiR exist a Majorana mass term could still be built out of ν jL . But ν jL have weak isospin 1/2, being part of the left handed lepton doublet l. Thus, the left handed Majorana mass term has total weak isospin equal to 1 and needs 2 Higgs fields to make a gauge invariant term. 83) with M a large scale (apriori comparable to the scale of MRR ) and λ a dimensionless coupling generically of o(1), is a non renormalizable operator of dimension 5.
In fact the SM lagrangian should contain all terms allowed by gauge symmetry and renormalizability. The most general renormalizable lagrangian, built from the SM fields, compatible with the SM gauge symmetry, in absence of νiR , is automatically B and L conserving. 80) c = Cν T is the charge conjugated neutrino field and C is the charge conjugation matrix in Dirac ¯ iR where νiR spinor space. The Majorana mass term is an operator of dimension d = 3 with ∆L = 2. Since the νiR are gauge singlets the Majorana mass MRR is fully allowed by the gauge symmetry and a coupling with the Higgs is not needed to generate this type of mass.