1 Fermion masses In class we wrote out the three Yukawa couplings of the Standard Model: L Ye (L¹) + h.c. àa Hª (еR) & ² Hª (dr) (1.1) (1.2) + Ya (Q¹) ama +h.c. + Yu (Q¹) ama (H¹) ab (H¹) (u) + h.c. ¹ªß (1.3) b The numbers ye, yd, and yu are called the Yukawa couplings. They're just numbers that characterize the strength of the interaction. They are analogous to the electric coupling e in the fine structure constant, a = e²/4π = 1/137. The "+ h.c." means Hermitian conjugate. For example, [Ye (L¹) à Hª (Cn) ¡ eªâ] ¹ = y² Lªª (H¹) (ch)²³ cas · (1.4) Q The + h.c. term simply gives the interaction where all particles are replaced by their anti-particles. Observe that all of these terms contain two fermions and a Higgs. That means when we swap the Higgs with the Higgs vev, the terms only contain a pair of fermions and no derivative. The rule in class was that any Lagrangian term with two particles and no derivative is a mass. Let's figure out what these masses look like. فيني
1.1 Mass terms from the Higgs vev Identify which of the left-chiral and right-chiral fermions pair up to become massive and write out the value of the mass. EXAMPLE: when we insert (H) = (0, v/√2) into the electron Yukawa, one finds Ye ezzei (ch) ² (ek) cas + h.c.. √2 (1.5) This means that the left-chiral electron e and the right-chiral electron er pair up and have a mass me = yev/√2. We say that the electron' is a massive particle that is a mixture of both the left-chiral and right-chiral electrons. In other words: the physical electron is a massive fermion with four degrees of freedom. We call this type of mass a Dirac mass, and we call the resulting particle a Dirac fermion. This is in contrast to massless fermions that are chiral and have only two degrees of freedom. 1.2 Numbers Given that m 2 MeV, ma≈ 5 MeV, and e≈ 0.5 MeV: find the values of ye, yd, and y. Which of these three particles has the largest Yukawa interaction with the Higgs?
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