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Fermions of the Electroweak Theory

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Fermions of the Electroweak Theory Fermions of the ElectroWeak Theory The Quarks, The Leptons, and their Masses. This is my second set of notes on the Glashow{Weinberg{Salam theory of weak and electromagnetic interactions. The first set was about the bosonic fields of the theory | the ± gauge fields of the SU(2) × U(1) gauge theory and the Higgs fields that give mass to the Wµ 0 and Zµ vector particles. This set is about the fermionic fields | the quarks and the leptons. From the fermionic point of view, the electroweak gauge symmetry SU(2)W × U(1)Y is chiral | the left-handed and the right-handed fermions form different types of multiplets | and consequently, the weak interactions do not respect the parity or the charge-conjugation symmetries. Specifically, all the left-handed quarks and leptons form doublets of the SU(2)W while the all right-handed quarks and leptons are singlets, so the charged weak currents are purely left-handed, 1 − γ5 Jµ = 1 V µ − Aµ = Ψγµ Ψ = y σ¯µ without a term: (1) ± 2 2 L L R The left-handed and the right-handed fermions also have different U(1) hypercharges, which is needed to give them similar electric charges Q = Y +T 3. For example, the LH up and down 1 quarks | which form an SU(2)W doublet | have Y = + 6 , while the RH quarks are SU(2) 2 1 singlets and have Yu = + 3 and Yd = − 3 . Consequently, their electric charges come out to be 3 1 1 2 9 Q(u; L) = Y (u; L) + T (u; L) = + 6 + 2 = + 3 = same; 3 2 2 Q(u; R) = Y (u; R) + T (u; R) = + 3 + 0 = + 3 ; (2) 3 1 1 1 9 Q(d; L) = Y (d; L) + T (d; L) = + 6 − 2 = − 3 = same: 3 1 1 Q(d; R) = Y (d; R) + T (d; R) = − 3 + 0 = − 3 ; In light of different quantum numbers for the LH and RH quarks, their Lagrangian cannot y y have any mass terms L R or R L. Instead, the physical quark masses arise from the Yukawa couplings of the quarks to the Higgs scalars Hi. In general, the Yukawa couplings of fermions 1 to scalars (or pseudoscalars) have form gφ × ΨΨ or gφ × Ψ(iγ5)Ψ; (3) or in terms of the Weyl fermions, y ∗ ∗ y gφ × L R + g φ × R L : (4) The theories with multiple fermionic and scalar fields may have different Yukawa couplings for different scalar and fermionic species, as long as they are invariant under all the required symmetries. For the electroweak symmetry at hand, the L are SU(2) doublets while the R y y are singlets, so the bi-linears L R and R L are SU(2) doublets, which may couple to the i ∗ SU(2) doublet of scalars such as the Higgs fields H or their conjugates Hi . Taking the U(1) hypercharges into account, the allowed Yukawa terms for the up and down quarks comprise ∗ dy i i y d LYukawa = −gdHi × R L − gdH × L;i R (5) i uy j ij ∗ y u − guijH × R L − gu Hi × L;j R : When the Higgs develops a non-zero Vacuum Expectation Value ! v 0 hHi = p × ; v ≈ 247 GeV; (6) 2 1 the Yukawa couplings of the fermions to this VEV give rise to fermionic mass terms, LYukawa −! Lmass + couplings to the physical Higgs field; (7) Lmass = LYukawa for H ! hHi v dy 2 2y d v uy 1 1y u = −gd p × + − gu p × + (8) 2 R L L R 2 R L L R d d u u ≡ −md × Ψ Ψ − muΨ Ψ ; (9) 2 where the Dirac fermions Ψu and Ψd comprise 1 ! 2 ! Ψu = L ; Ψd = L (10) u d R R and their masses follow from the Higgs VEV and the Yukawa couplings as v v mu = gu × p ; md = gd × p : (11) 2 2 The other 4 quark flavors | charm, strange, top, and bottom | have similar quantum numbers to the up and down quarks. The left-handed quarks form SU(2) doublets (c; s)L and 1 (t; b)L with Y = + 6 while the right-handed quarks are singlets with hypercharges Y (cR) = 2 1 Y (TR) = + 3 and Y (sR) = Y (bR) = − 3 , which lead to non-chiral electric charges 2 Q(cL or R) = Q(tL or R) = Q(uL or R) = + 3 ; (12) 1 Q(sL or R) = Q(bL or R) = Q(dL or R) = − 3 : y Again, the SU(2) × U(1) quantum numbers of these quarks forbid any mass terms L R or y R L in the Lagrangian, but they allow the Yukawa couplings to the Higgs fields similar to (5). The physical masses obtain from those Yukawa couplings when the Higgs scalar develops a non-zero VEV and breaks the SU(2)×U(1) symmetry down to the U(1)EM; similar to eq. (11), v v v v ms = gs × p ; mc = gc × p ; mb = gb × p ; mt = gt × p : (13) 2 2 2 2 2 Note that the charge = + 3 quarks u; c; t have exactly similar electroweak quantum num- bers but very different values of the Yukawa couplings, gu gc gt, and hence very different 1 physical masses, mu mc mt. Likewise, the charge = − 3 quarks d; s; t have exactly similar electroweak quantum numbers but different Yukawa couplings, gd gs gb, and hence different physical masses, md ms mb. Experimentally mu ≈ 2:15 MeV mc ≈ 1:28 GeV mt ≈ 173 GeV; (14) md ≈ 4:7 MeV ms ≈ 94 MeV mb ≈ 4:2 GeV; (15) but we do not have a good explanation of this hierarchical pattern. In the Standard Model, the Yukawa couplings are arbitrary parameters to be determined experimentally. Beyond the 3 Standard Model, there have been all kinds of speculative explanations over the last 40+ years, but none of them can be supported by any experimental evidence whatsoever. Besides the quarks, there are 3 species of charged leptons | the electron e−, the muon − − µ , and the tau τ | and 3 species of neutrinos, νe, νµ, ντ . The left-handed fermions of − − − 1 these 6 species form three SU(2) doublets (νe; e )L,(νµ; µ )L, and (ντ ; τ )L with Y = − 2 , so the bottom halves of these doublets have electric charges − − − 1 Q(eL ) = Q(µL ) = Q(τL ) = Y − 2 = −1 (16) while the top-halves | the neutrinos | are electrically neutral, 1 Q(νeL) = Q(νµL) = Q(ντL) = Y + 2 = 0: (17) The right-handed electron, muon, and tau are SU(2) singlets with Y = −1, so their electric charge Q = Y + 0 = −1 is the same as for the left-handed e; µ, τ. As to the right-handed neutrino fields, there are two theories: In one theory, the neutrino fields are left-handed Weyl spinors L(ν) rather than Dirac spinors, so the R(ν) simply do not exist! In the other theory, the R(ν) do exist, but they are SU(2) singlets with Y = 0 and so do not have any in weak interactions. Since they also do not have strong or EM interactions, this makes the RH neutrinos completely invisible to the experiment | that's why we do not know if they exist or not. For the moment, let me focus on the simplest version without the R(ν); I'll come back to the other theory later in these notes when I discuss the neutrino masses. Similar to the quarks, the SU(2) × U(1) quantum numbers of the leptons do not allow any mass terms in the Lagrangian, but they do allow the Yukawa couplings of leptons to the Higgs fields, ∗ y i i y LYukawa = −geHi × R(e) L(νe; e) − geH × L;i(νe; e) R(e) ∗ y i i y − gµHi × R(µ) L(νµ; µ) − gµH × L;i(νµ; µ) R(µ) (18) ∗ y i i y − gτ Hi × R(τ) L(ντ ; τ) − gτ H × L;i(ντ ; τ) R(τ): When the Higgs field H develop non-zero VEV pv , these Yukawa couplings give rise to the 2 2 4 lepton masses; similar to the quarks, v v v m = g × p ; m = g × p ; m = g × p : (19) e e 2 µ µ 2 τ τ 2 Experimentally, these masses are me = 0:511 MeV mµ = 106 MeV mτ = 1777 MeV: (20) Similar to the quarks, the masses of charged leptons form a hierarchy; we do no not know why. similar to the quark mass hierarchies (14) and (15). Weak Currents Altogether, the fermionic fields of the electroweak theory and their couplings to the bosonic gauge and Higgs fields can be summarized by the Lagrangian X y µ X y µ LF = i Lσ¯ Dµ L + i Rσ Dµ R + LYukawa : (21) LH quarks RH quarks & leptons & leptons In the first section of these notes I was focused on the Yukawa couplings that give rise to the fermion masses when the Higgs field gets its VEV, but now let's turn our attention to the interactions of quarks and leptons with the electroweak SU(2) × U(1) gauge fields. In the Lagrangian (21), the gauge interactions are hidden inside the covariant derivatives Dµ, so let me spell them out in detail: • The left-handed quarks form SU(2) doublets u ! c ! t ! i = or or (22) L d s b L L L 1 of hypercharge Y = + 6 , so for the LH quark fields ig ig D i = @ i + 2 W a(τ a)i j + 1 B i : µ L µ L 2 µ j L 6 µ L 5 • The left-handed leptons also form SU(2) doublets ! ! ! νe νµ ντ i = or or (23) L e− µ− τ − L L L 1 but of hypercharge Y = − 2 , so for the LH lepton fields ig ig D i = @ i + 2 W a(τ a)i j − 1 B i : µ L µ L 2 µ j L 2 µ L 2 1 • The right handed quarks are SU(2) singlets of hypercharges Y = + 3 or Y = − 3 , thus 2ig1 for R = uR or cR or tR ;Dµ R = @µ R + Bµ R ; 3 (24) ig for = d or s or b ;D = @ − 1 B : R R R R µ R µ R 3 µ R • The right-handed charged leptons are SU(2) singlets of hypercharge Y = −1, thus − − − for R = eR or µR or τR ;Dµ R = @µ R − ig1 Bµ R : (25) | Finally, if the right-handed neutrino fields exist at all, they are SU(2) singlets and have zero hypercharge, thus e µ τ for R = νR or νR or νR ;Dµ R = @µ R + 0: (26) Now let's plug these covariant derivatives into the Lagrangian (21), extract the terms containing the gauge fields, and organize the fermionic fields interacting with those gauge fields into the currents according to a µ µ L ⊃ −g2Wµ JT a − g1BµJY ; (27) cf.
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