Fermions of the ElectroWeak Theory

Fermions of the ElectroWeak Theory The Quarks, The eptons, 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 U1 gauge theory and the Higgs fields that give mass to the W µ ± and Zµ 0 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 W U1 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 W while the all right-handed quarks and leptons are singlets, so the charged weak currents are purely left-handed, J µ ± = 1 V µ A µ = Ψγ µ 1 γ5 Ψ = σµ without a R term. 1 The left-handed and the right-handed fermions also have different U1 hypercharges, which is needed to give them similar electric charges Q = Y +T 3. For example, the H up and down quarks which form an SU W doublet have Y = + 1 6, while the RH quarks are SU singlets and have Y u = + 3 and Y d = 3 1. Consequently, their electric charges come out to be Qu, = Y u, + T 3 u, = + 6 1 + 1 = + 3 Qu, R = Y u, R + T 3 u, R = + 3 + 0 = + 3 Qd, = Y d, + T 3 d, = + 6 1 1 = 1 3 Qd, R = Y d, R + T 3 d, R = 3 1 + 0 = 1 3 same, same. In light of different quantum numbers for the H and RH quarks, their agrangian cannot have any mass terms R or R. Instead, the physical quark masses arise from the Yukawa couplings of the quarks to the Higgs scalars H i. In general, the Yukawa couplings of fermions 1

to scalars or pseudoscalars have form gφ ΨΨ or gφ Ψiγ 5 Ψ, 3 or in terms of the Weyl fermions, gφ R + g φ R. 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 are SU doublets while the R are singlets, so the bi-linears R and R are SU doublets, which may couple to the SU doublet of scalars such as the Higgs fields H i or their conjugates Hi. Taking the U1 hypercharges into account, the allowed Yukawa terms for the up and down quarks comprise Yukawa = g d Hi d R i g dh i,i d R g u ɛ ij H i u R j g uɛ ij Hi,j u R. 5 When the Higgs develops a non-zero Vacuum Expectation Value H = v 0, v 47 GeV, 6 1 the Yukawa couplings of the fermions to this VEV give rise to fermionic mass terms, Yukawa mass + couplings to the physical Higgs field, 7 mass = Yukawa for H H v = g d d R + d R g u v u R 1 + 1 u R 8 m d Ψ d Ψ d m u Ψ u Ψ u, 9

where the Dirac fermions Ψ u and Ψ d comprise 1 Ψ u =, Ψ d = u R d R 10 and their masses follow from the Higgs VEV and the Yukawa couplings as m u = g u v, m d = g d v. 11 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 doublets c, s and t, b with Y = + 1 6 while the right-handed quarks are singlets with hypercharges Y c R = Y T R = + 3 and Y s R = Y b R = 1 3, which lead to non-chiral electric charges Qc or R = Qt or R = Qu or R = + 3, Qs or R = Qb or R = Qd or R = 1 3. 1 Again, the SU U1 quantum numbers of these quarks forbid any mass terms R or R in the agrangian, 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 U1 symmetry down to the U1 EM ; similar to eq. 11, m s = g s v, m c = g c v, m b = g b v, m t = g t v. 13 Note that the charge = + 3 quarks u, c, t have exactly similar electroweak quantum numbers but very different values of the Yukawa couplings, g u g c g t, and hence very different physical masses, m u m c m t. ikewise, the charge = 1 3 quarks d, s, t have exactly similar electroweak quantum numbers but different Yukawa couplings, g d g s g b, and hence different physical masses, m d m s m b. Experimentally m u.15 MeV m c 1.8 GeV m t 173 GeV, 14 m d 4.7 MeV m s 94 MeV m b 4. 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, ν e, ν µ, ν τ. The left-handed fermions of these 6 species form three SU doublets ν e, e, ν µ, µ, and ν τ, τ with Y = 1, so the bottom halves of these doublets have electric charges Qe = Qµ = Qτ = Y 1 = 1 16 while the top-halves the neutrinos are electrically neutral, Qν e = Qν µ = Qν τ = Y + 1 = 0. 17 The right-handed electron, muon, and tau are SU 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 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 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 U1 quantum numbers of the leptons do not allow any mass terms in the agrangian, but they do allow the Yukawa couplings of leptons to the Higgs fields, Yukawa = g e H i R ei ν e, e g e H i,i ν e, e R e g µ H i R µi ν µ, µ g µ H i,i ν µ, µ R µ g τ H i R τi ν τ, τ g τ H i,i ν τ, τ R τ. 18 When the Higgs field H develop non-zero VEV v, these Yukawa couplings give rise to the 4

lepton masses; similar to the quarks, m e = g e v, m µ = g µ v, m τ = g τ v. 19 Experimentally, these masses are m e = 0.511 MeV m µ = 106 MeV m τ = 1777 MeV. 0 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 agrangian F = H quarks & leptons i σµ D µ + RH quarks & leptons i R σµ D µ R + Yukawa. 1 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 U1 gauge fields. In the agrangian 1, the gauge interactions are hidden inside the covariant derivatives D µ, so let me spell them out in detail: The left-handed quarks form SU doublets c t i = u d or s or b of hypercharge Y = + 1 6, so for the H quark fields D µ i = µ i + ig W a µτ a i j j + ig 1 6 B µ i. 5

The left-handed leptons also form SU doublets i νe = e or νµ µ or ντ τ 3 but of hypercharge Y = 1, so for the H lepton fields D µ i = µ i + ig W a µτ a i j j ig 1 B µ i. The right handed quarks are SU singlets of hypercharges Y = + 3 or Y = 1 3, thus for R = u R or c R or t R, D µ R = µ R + ig 1 3 B µ R, for R = d R or s R or b R, D µ R = µ R ig 1 3 B µ R. 4 The right-handed charged leptons are SU singlets of hypercharge Y = 1, thus for R = e R or µ R or τ R, D µ R = µ R ig 1 B µ R. 5 Finally, if the right-handed neutrino fields exist at all, they are SU singlets and have zero hypercharge, thus for R = ν e R or νµ R or ντ R, D µ R = µ R + 0. 6 Now let s plug these covariant derivatives into the agrangian 1, extract the terms containing the gauge fields, and organize the fermionic fields interacting with those gauge fields into the currents according to g W a µj µ T a g 1B µ J µ Y, 7 cf. eq. 1 from the first set of my notes on the electroweak theory. Since the right-handed 6

quarks and leptons are SU singlets, the SU currents turn out to be purely left-handed, H quarks J µ T a = u,d,c,s,t,b,i τ a i j H leptons σ µ j + ν e,e,ν µ,µ,ν τ,τ,i τ a i j σ µ j. 8 However, the U1 current has both left-handed and right handed contributions, J µ Y = u,c,t quarks + 16 σµ + 3 R σµ R e,µ,τ leptons + 1 σµ R σµ R d,s,b quarks + neutrinos 16 σµ 1 3 R σµ R 1 σµ + 0. 9 In the first set of notes I had re-organized these 4 gauge currents into currents which couple to the specific electroweak gauge field, namely the electric current J µ EM = Jµ T 3 + Jµ Y 30 which couples to the EM field A µ, the charged weak currents J +µ = J µ T 1 ijµ T and J µ = J µ T 1 + ijµ T 31 which couple to the charges W ± µ massive vector fields, and the neutral weak current J µ Z = Jµ T 3 sin θj µ EM 3 which couples to the neutral massive vector field Z 0 µ. Now let s spell out all these currents in terms of the fermionic fields. For the charged currents, reorganizing the weak isospin currents 8 into J ±µ amounts to combining the 7

isospin Pauli matrices τ a in the same way as the currents 31, 0 0 τ + τ 1 iτ = 0, τ τ 1 + iτ = 0 0 0. 33 Consequently, in eqs. 8 we have,i and therefore τ + i j τ σ µ j =, σµ 1,,i i j σ µ j =,1 σµ, 34 J +µ J µ = d σµ u + s σµ c + b σµ t + e σµ ν e + µ σµ ν µ + τ σµ ν τ, = u σµ d + c σµ s + t σµ b + ν e σ µ e + ν µ σ µ µ + ν τ σ µ τ. 35 In terms of Dirac fermions for the quarks and leptons, σµ = Ψγ µ 1 γ5 Ψ, 36 hence J +µ J µ = Ψ d γ µ 1 γ5 Ψ u + Ψ e γ µ 1 γ5 Ψ νe = Ψ u γ µ 1 γ5 Ψ d + Ψ νe γ µ 1 γ5 Ψ e + Ψ s γ µ 1 γ5 Ψ c + Ψ µ γ µ 1 γ5 Ψ νµ + Ψ c γ µ 1 γ5 Ψ s + Ψ νµ γ µ 1 γ5 Ψ µ + Ψ b γ µ 1 γ5 Ψ t + Ψ τ γ µ 1 γ5 Ψ ντ, + Ψ t γ µ 1 γ5 Ψ b + Ψ ντ γ µ 1 γ5 Ψ τ. As promised, these charged weak currents are purely left-handed, so they completely violate the parity and the charge-conjugation symmetries. But please note that this left-handedness is in terms of chirality of the fermionic fields rather than helicities of the fermionic particles. In terms of helicities, the quarks and the leptons participating in charged-current weak interactions are polarized left, but the antiquarks and the antileptons are polarized right; the degree of polarization = β = v/c, which approaches 100% for the ultrarelativistic particles. 37 8

On the other hand, the electric current is left-right symmetric, J µ EM = 3 quarks q=u,c,t Ψ q γ µ Ψ q 1 3 quarks q=d,s,b Ψ q γ µ Ψ q leptons l=e,µτ Ψ l γ µ Ψ l. 38 Finally, the neutral weak current has both left-handed and right-handed components but it is not left-right symmetric. In terms of Dirac spinor fields, J µ Z = Jµ T 3 [left-handed] sin θ J µ EM [left-right symmetric] = quarks q=u,c,t + Ψ q γ µ + 1 γ5 4 leptons l=e,µ,τ Ψ l γ µ 1 γ5 4 3 sin θ Ψ q + + sin θ Ψ l + quarks q=d,s,b Ψ q γ µ 1 γ5 4 neutrinos Ψ ν γ + µ 1 γ5 4 ν=ν e,ν µ,ν τ + 1 3 sin θ Ψ q 0 Ψ ν. 39 Flavor Mixing and the Cabibbo Kobayashi Maskawa Matrix Actually, the charged weak currents are more complicated then I wrote down in eq. 37. Since we have 3 quark flavors of each charge + 3 or 1 3, we need to be careful as to how they form 3 SU doublets. Normally, one defines the specific flavors of quarks as eigenstates of the quark mass matrix, but this definition does not respect the doublet structure: the SU partner of say the u quark is not the d quark but rather some linear combination of the d, s, b quarks, and likewise for the partners of the c and t quarks. Thus, the SU doublets are u, d c, s t, for b d s b = V d s 40 b where V is a unitary 3 3 matrix called the Cabibbo Kobayashi Maskawa matrix CKM. In this section, I shall first explain where this matrix comes from, and then I ll tell you its physical consequence for the weak interactions. In the un-broken SU U1 theory the quarks are massless and we cannot tell which quark is u, which is c, etc., etc.; we cannot even tell which left-handed Weyl field pairs up 9

with which right-handed Weyl field into a Dirac spinor. We can use the SU symmetry to form doublets, but we are free to choose any basis we like for the 3 doublets let s call them Q α for α = 1,, 3 and we are free to change this basis by a unitary field re-definition, i Q α i Q α = β U Q α,β i Q β, 41 where U Q is a unitary 3 3 matrix. Similarly, we may use any basis D α for the 3 right-handed quarks of charge 1 3, any basis U α for the 3 right-handed quarks of charge + 3, and we are free to change these two bases by unitary transforms R U α R U α = β U U α,β R U β, R D α R D α = U D α,β R D β, where U U and U D are two independent unitary 3 3 matrices. However, we cannot mix the U α with the D α because of their different U1 hypercharges. ikewise, we are free to use any basis α for the 3 doublets of left-handed leptons, any basis E α for the 3 right-handed charged leptons, and we are free to changes all these bases by unitary transforms, β 4 i α i α = β I ll take care of the neutrinos in a later section. β 43 U α,β i β, R E α R E α = U E α,β R E β. The Yukawa couplings involve one Higgs field H i or Hi and two fermion fields, one left-handed, one right-handed and for each choice of their SU U1 quantum numbers, there three fields and three R fields. Consequently, there is a big lot of the Yukawa terms in the agrangian Yukawa Y U αβ R U α i Q β ɛ ij H j Y D αβ R D α i Q β H i = α,β α,β Y E αβ R E α i β H i α,β + Hermitian conjugates, 44 which involve three 3 3 complex matrices Yα,β U, Y α,β D, and Y α,β E of the Yukawa coupling constants. And when the Higgs develops symmetry-breaking VEV, these matrices of Yukawa 10

couplings give rise to the complex 3 3 mass matrices M U α,β = v Y U α,β, M U α,β = v Y D α,β, M E α,β = v Y E α,β, 45 mass = α,β M U αβ R U α 1 Q β α,β M D αβ R U α Q β 46 α,β M E αβ R α β + Hermitian conjugates. 47 To get the physical masses of quarks and leptons, we need to diagonalize these mass matrices via suitable unitary transforms 41 43. In matrix notations, these transforms lead to Y U = U U Y U U Q, Y D and consequently = U D Y D U Q, Y E = U E Y E U, 48 M U = U U M U U Q, M D = U D M D U Q, M E = U E M E U. 49 Now, any complex matrix M can be written as a product M = W 1 DW where W 1 and W are unitary matrices while D is diagonal, real, and non-negative. Consequently, using appropriate unitary matrices U E and U Q we can make the charged lepton s mass matrix diagonal and real M E M E = U E M E m e 0 0 U = 0 m µ 0 0 0 m τ. 50 Note that it is the transformed bases where the M E is diagonal that the H and RH To prove, start with a polar decomposition M = UH where U is unitary and H = M M is hermitian and positive semi-definite. Then diagonalize the hermitian matrix H, i.e., write it as H = W DW for some unitary matrix W. Consequently, M = UW DW = W 1 DW for W = W and W 1 = UW. 11

Weyl fields combine into Dirac fields of the physical electron, muon, and the tau, Ψ e 1 = 1β β R E 1 = U 1β E R E, β Ψ µ = β β R E = U β E R E, β Ψ τ 3 = 3β β R E 3 = U 3β E R E. β 51 ikewise, using the U U and the U Q unitary matrices we may diagonalize the mass matrix for the charge + 3 quarks, M U M U = U U M U m u 0 0 U Q = 0 m c 0, Ψ u = Ψ c = Ψ t = 1 Q 1 = U Q 1β 1 Q β R U 1 = U 1β U R U, β 1 Q = U Q β 1 Q β R U = U β U R U, β 1 Q 3 = U Q 3β 1 Q β R U 3 = U 3β U R U, β 0 0 m t 5 and similarly for the charge 1 3 quarks, M D M D = U D M D m d 0 0 Ũ Q = 0 m s 0, Ψ d = Ψ s = Ψ b = Q 1 = Ũ Q 1β Q β R U 1 = U 1β D R D, β Q = Ũ Q β 1 Q β R U = U β D R D, β Q 3 = Ũ Q 3β 1 Q β R U 3 = U 3β D R D, β 0 0 m b 53 1

However, it takes different unitary matrices U Q Ũ Q to diagonalize the up-type and downtype quark mass matrices, and that s what messes up the SU doublet structure! Indeed, in terms of the upper components 1 Q α of the original doublets, the left-handed u, c, t quarks of definite mass are linear combinations u c t = U Q 1 Q 1 1 Q, 54 1 Q 3 so their SU partners are similar linear combinations of the lower components Q α of the original doublets, d s b = U Q Q 1 Q, 55 Q 3 for the same U Q matrix as the up-type quarks. On the other hand, the d, s, b quarks defined as mass eigenstates obtain from different linear combinations d s b = Ũ Q Q 1 Q. 56 Q 3 Comparing the sets of down-type quark fields, we immediately see that d s b = U Q Ũ Q d s b, 57 which gives us the Cabibbo Kobayashi Maskawa matrix V CKM = U Q Ũ Q. 58 Now let s go back to the charged weak currents J ±µ. Since they are gauge current of the SU W, they connect a fermion in some SU doublet into the other fermion in exactly same doublet! Thus, the J + current would turn the u quark into its partner d, or the c quark 13

into its partner s, etc., and vice verse for the J current. In terms of the Dirac spinor fields, this means J µ quarks = Ψ u γ µ 1 γ5 = α=u,c,t β=d,s,b J +µ quarks = Ψ d γ µ 1 γ5 = α=u,c,t β=d,s,b Ψ d + Ψ c γ µ 1 γ5 Ψ s V α,β Ψ α γ µ 1 γ5 Ψ β, Ψ u + Ψ s γ µ 1 γ5 Ψ c Vα,β Ψβ γ µ 1 γ5 Ψ α, + Ψ t γ µ 1 γ5 Ψ b + Ψ b γ µ 1 γ5 Ψ t 59 where V α,β is the CKM matrix. The CKM matrix is very important for the physics of weak interactions. For example, without this matrix the strange particles like the K-mesons or Λ-baryons would be stable because the s quark would not be able to decay. Indeed, the SU partner of the s quark is the c quark, so without the CKM matrix the only flavor-changing processes involving the s quark would be s c. However, the c quark is much heavier than s, so the decay can only go from c to s but not from s to c. But thanks to the CKM matrix specifically, to the non-zero matrix element V u,s the s quark may also decay to the u quark which is lighter than s, albeit with a reduced amplitude V u,s 0.. There are may other interesting flavor-changing weak processes involving the charged currents and the CKM matrix. I wish I could spend a few weeks talking about them, but alas I do not have the time for this in my QFT class. I hope professor Can Kiliç will explain the subject in some detail in his Phenomenology class in Fall 017. But in these notes, I have to move on to the next subject. Eq. 59 give the charged weak currents of the quarks, but what about the leptons? Again, we need to pick the bases for the 3 charged leptons and for the neutrinos, and if the two bases disagree with the SU doublet structure, we would get a CKM-like matrix for the leptons. For the charged leptons, the mass is important, so people always use the basis of mass eigenstates e, µ, τ as in eqs. 50 and 51. But the neutrino masses are so small, they only matter in long-baseline interferometry experiments, so for all other purposes people use 14

the interaction basis ν e, ν µ, ν τ of species defined as the SU partners of the corresponding charged leptons e, µ, τ. In this basis, there are no CKM-like matrices and J +µ leptons = Ψ e γ µ 1 γ5 Ψ νe J µ leptons = Ψ νe γ µ 1 γ5 Ψ e + Ψ µ γ µ 1 γ5 Ψ νµ + Ψ νµ γ µ 1 γ5 Ψ µ + Ψ τ γ µ 1 γ5 Ψ ντ, + Ψ ντ γ µ 1 γ5 Ψ τ. 60 On the other hand, in this basis the neutrino mass matrix is non-diagonal, which makes neutrinos slowly oscillate from one species to another. I ll come back to this issue later in these notes. Now consider the neutral weak current J µ Z. The unitary transforms that diagonalize the fermion s mass matrices can only mix up fields with similar chiralities and R and electric charges. In the Standard Model, this limits the mixing to fermions that have both similar weak isospins T 3 and similar hypercharges Y, which makes for similar contributions to the neutral weak current J µ Z H species α T 3 sin θq el α σµ α or RH species α T 3 sin θq el R ασµ R α. The sums here are invariant under all unitary field redefinitions that mix only fields with similar T 3 sin Q el, so regardless of such redefinitions the neutral weak current remains diagonal. Specifically, 61 J µ Z = Jµ T 3 [left-handed] sin θ J µ EM [left-right symmetric] = quarks q=u,c,t + Ψ q γ µ + 1 γ5 4 leptons l=e,µ,τ Ψ l γ µ 1 γ5 4 3 sin θ Ψ q + + sin θ Ψ l + quarks q=d,s,b Ψ q γ µ 1 γ5 4 neutrinos Ψ ν γ + µ 1 γ5 4 ν=ν e,ν µ,ν τ and there are no flavor-changing neutral weak currents in the Standard Model. + 1 3 sin θ Ψ q 0 Ψ ν, Note that this is a peculiar property of the Standard Model where all fermions of the same electric charge and chirality also have the same T 3. Historically, before the Standard 39 15

Model was fully developed and confirmed experimentally, people used to consider models with different quantum numbers for different quarks. In particular, back in the 1960 s when only 3 quark flavors u, d, s were known, people assumed the left-handed s quark was un-paired SU singlet with Y = 3 1 to give it the right electric charge. The mass matrix somehow mixed the two charge 1 3 quarks d and s, so the SU doublet was u, d while the singlet was s, where d = d cos θ c + s sin θ c, s = s cos θ c d sin θ c, θ c 13. 6 In such a model, the s and the d have different T 3 sin θq el, so their mixing makes for off-diagonal terms in the J µ Z. In other words, there would be the s d flavor changing neutral current, which would lead to processes like the K 0 µ + µ decay. But experimentally, there are no such decays, nor any other signatures of the flavor-changing neutral currents. This made Glashow, Illiopoulos, and Maiani conjecture in 1970 that the s quark or rather the s should be a member of a doublet just like the d quark which would give them the same T 3 and hence keep the neutral weak current flavor-diagonal and consequently there must be a fourth quark flavor c to form the c, s doublet. And in 1974 this fourth flavor called the charm was experimentally discovered at SAC and BN. ater, when the fifth flavor b was discovered in 1977, most physicists expected it to also be a part of the doublet, so everybody was looking for the sixth flavor t. This expectation turned out to be correct, and the t quark was duly discovered in 1995. The delay was due to the very large mass of the top quark, m t 173 GeV, much heavier that the other 5 flavors. CP violation ike any chiral gauge theory, the weak interactions do not have the parity symmetry P or the charge conjugation symmetry C. In particular, the charged currents involve only the left-chirality Weyl spinors, which in particle terms mean left-helicity quarks and leptons but right-helicity anti-quarks or anti-leptons. However, the chirality is perfectly consistent with the combined CP symmetry, which does not mix the and the R fields; instead it acts as CP : x, t ±σ x, t, R x, t ±σ R x, t. 63 16

By the CPT theorem, the CP symmetry is equivalent to the time-reversal or rather motionreversal symmetry T, so it would be nice to have it as an exact symmetry of Nature. But in 1964, Cronin and Fitch have discovered that weak decays of the neutral K-mesons are only approximately CP-symmetric, but sometimes a CP-odd state of the kaon decays into a CPeven pair of pions. ater experiments found CP violations in weak other processes involving mesons containing b quarks or c quarks. All the experimentally measured CP-violating effects can be explained by the imaginary part IV α,β of the CKM matrix. The relation between those effects and the CKM matrix is rather complicated and involves interference between different orders of perturbations theory; at the tree level, there is no CP violation. I am not going to work out such complicated issues in these notes; instead, let me simply show that complex CKM matrix violates the CP symmetry of the electroweak agrangian. Since the neutral weak current does not care about the CKM matrix, let me focus on the charged currents. Under CP, the charged vector fields W µ ± x transform as CP : W ± 0 x, t W 0 x, +t, W ± i x, t +Wi x, +t, 64 where the exchange W + W is due to charge conjugation while different signs for 3-scalar and 3-vector components are due to reflection x x of the space coordinates. Consequently, in a CP symmetric theory we would need a similar relation for the charged currents, CP : J 0± x, t J 0± x, +t, J i± x, t +J i x, +t, 65 In terms of fermions, the charged weak currents are sums of left-handed currents terms of general form j µ = 1 σµ = Ψ1 γ µ 1 γ5 Ψ, 66 so let s work out how such terms transform under CP. Assuming the Weyl fermions 1 and 17

have the similar intrinsic CP signs as members of the same SU doublet, we have CP : 1 σµ = +1 σ σ µ σ = + 1 σ σ µ σ = σ µ = σµ 1 = σµ 1 { +1 for µ = 1,, 3, 1 for µ = 0. 67 The µ dependence of the overall sign here which comes from comparing σ µ to + σ µ is in perfect agreement with eq. 65. In Dirac notations, eq 67 amounts to CP : Ψ 1 γ µ 1 γ5 Ψ Ψ γ µ 1 { γ5 +1 for µ = 1,, 3, Ψ 1 1 for µ = 0. 68 Besides the µ dependent sign, the CP exchanges the two fermionic species Ψ 1 Ψ involved in the current j µ. For the leptonic charged weak currents 60, this exchange leads to J+µ J µ, exactly as in eq. 65; indeed, J +µ Ψ e γ µ 1 γ5 Ψ νe while J µ Ψ νe γ µ 1 γ5 Ψ e, etc., etc. 69 Consequently, the interactions = g W µ + J µ leptonic + W µ J µ+ leptonic 70 of the leptons with the vector fields W ± µ are invariant under CP. But for the charged currents of the quarks, we have J µ quarks = J +µ quarks = α=u,c,t α=u,c,t β=d,s,b β=d,s,b V α,β Ψ α γ µ 1 γ5 Ψ β, Vα,β Ψβ γ µ 1 γ5 Ψ α, 59 18

which transform into CP : J µ quarks ±µ β=d,s,b V α,β Ψ β γ µ 1 γ5 Ψ α, which is almost like ± µ J +µ quarks, except for V α,β instead of V α,β ; CP : J +µ quarks ±µ β=d,s,b Vα,β Ψα γ µ 1 γ5 Ψ β, which is almost like ± µ J µ quarks, except for V α,β instead of V α,β ; 71 Consequently, the net effect of CP on the interactions = g W µ + J µ quark + W µ J µ+ quark 7 of the W ± µ with the quarks is equivalent to complex conjugating the CKM matrix, CP : V α,β V α,β. 73 Thus, the weak interactions of quarks and hence hadrons are CP symmetric if and only if the CKM matrix is real. For three families of quarks and leptons, the CKM matrix V is a unitary 3 3 matrix. To parametrize such a matrix we need 3 real angles as for a real orthogonal O3 matrix and 6 phases, for the total of 6 + 3 = 3 parameters. However, some of the 6 phases can be eliminated by the unitary field redefinitions which commute with the mass matrices, namely the abelian symmetries Ψ u e iθ u Ψ u, Ψ c e iθ c Ψ c, Ψ t e iθ t Ψ t }, Ψ d e iθ d Ψ d, Ψ s e iθ s Ψ s, Ψ b e iθ b Ψ b, = V α,β e iθα iθ β V α,β. Note that the common phase change for all the quarks does not affect the CKM matrix, but the differences between phases for different quarks do make a difference. Thus, 5 out of 6 phases in V can be eliminated, but we are stuck with one remaining phase. It is this one phase that s responsible for all the CP violations by the weak interaction! 19

BTW, if we had only two families of quarks and leptons the u, s, c, s quarks but not the b and t, and the e, ν e, µ, ν µ leptons but not the τ, ν τ, the CKM matrix would have 1 real angle the Cabibbo angle θ c 13 and 3 phases, but all these phases could be eliminated by the remaining abelian symmetries of the quarks. Consequently, there would be no CP violation! Back in 1973, only two families were known in fact, even the charm quark was predicted but not yet discovered experimentally and the origin of the weak CP violation was a complete mystery although there were many far-out speculations. At that time, Kobayashi and Maskawa speculated that maybe there is a third family similar to the first two; in this case, there Cabibbo mixing matrix would be 3 3 instead of, so one of its complex phases could not be eliminated by field redefinition, and that would be a source of CP violation. Their speculation turned out to be correct, and in 008 Kobayashi and Maskawa got a Nobel prize. Neutrino masses Back when the Glashow Weinberg Salam theory was formulated, the neutrinos were though to be exactly massless. But the later discovery of neutrino oscillations between the ν e, ν µ, and ν τ species calls for tiny but non-zero neutrino masses, m ν < 1 ev. Or rather, the oscullations call for the neutrino mass matrix M ν α,β that is non-diagonal in the weak-interactions basis ν e, µ µ, ν τ. Indeed, consider the effective Hamiltonian for a single ultra-relativistic neutrino particle; in the momentum-species basis, Ĥ = p + M p + M p. 74 While a free neutrino flies from the point where it is produced to the point where it is detected, the second term here causes its species state to oscillate, p, α β i exp p M α,β p, β up to an overall phase. 75 To illustrate how this works, let me spell out the oscillation matrix for neutrino species, say 0

ν e and ν µ. In this case, the mass matrix can be written as cos θ sin θ m M 1 0 cos θ sin θ = sin θ cos θ 0 m sin θ cos θ 76 where m 1 and m are the eigenvalues and θ is the mixing angle between the mass eigenbasis ν 1, ν and the weak-interaction basis ν e, µ µ. Consequently, the oscillation matrix in eq. 75 becomes up to an overall phase i m exp p M = cos 1 m 1 0 4p 0 1 m + i sin 1 m cosθ sinθ. 4p sinθ cosθ For the three neutrino species, the 3 3 oscillation matrix is more complicated, so I am not writing it here. et me simply say that it depends on the differences m 1 m and m m3 between the mass eigenvalues and the CKM-like mixing angles between the weak-interactions basis ν e, µ µ, ν τ and the mass eigenbasis ν 1, ν, ν 3. Experimentally, the neutrino mixing angles are rather large θ 1 33, θ 3 40, θ 13 9 much larger than the CKM angles for the quarks while the m differences are very small, m 1 76 10 6 ev and m 3.3 10 3 ev. Theoretically, there are two ways to add the neutrino masses to the Glashow Weinberg Salam theory. The first possibility is to make the neutrino fields Dirac spinors and give them masses via Yukawa couplings to the Higgs doublet, just like the other fermions of the theory. In terms of the original theory with massless neutrinos, this means adding 3 SU singlet, Y = 0 right-handed Weyl fields R N α and give them Yukawa couplings 77 Yukawa α,β Y N α,β R N α i β ɛ ij H j + Hermitian conjugates. 78 When the Higgs gets its VEV, these Yukawa couplings give rise to the Dirac mass terms for the neutrinos mass α,β M N α,β R N α i β + H. c., M N α,β = v Y N α,β. 79 The only problem with this setup is that it does not explain why the neutrinos are so light 1

compared to the other fermions of the Standard Model million times lighter than even the electron, never mind the heavier leptons or quarks. Al we can say is that somehow, the Yukawa couplings for the neutrinos are extremely weak Y N 10 1, but why?!? we do not have a clue. The other possibility is the Majorana neutrinos. In Dirac-spinor notations, a Majorana fermion is a neutral field Ψx = γ Ψ x. In terms of the Weyl spinor fields, Majorana Ψx = x σ x for the same x, 80 there is no independent R x. Thus, a majorana fermion is equivalent to a single Weyl fermion x together with its conjugate x. The agrangian for the free Majorana field is = 1 Ψi mψ = i σµ µ + m σ + m σ. 81 In a general theory of multiple fermions, mass terms like in this formula are called the Majorana masses. To give the neutrinos Majorana masses we do not need the independent right-handed neutrino fields R N α. All we need are the left-handed neutrino fields 1 α and their conjugates, plus some interactions that would give rise to the Majorana mass terms mass 1 Mα,β ν 1 α σ 1 β + 1 Mα,β ν 1 α σ 1 β, 8 α,β Note that the mass matrix in this formula may be complex rather than real, but it should be symmetric M ν β,α = M ν α,β because α,β 1 β σ 1 α = + 1 α σ 1 β 83 the σ matrix is antisymmetric, but the fields are anticommuting fermions. The neutrino mass terms 8 break the SU U1 gauge symmetry so we cannot put them directly into the agrangian of the high-energy theory. Instead, they obtain from the gauge-invariant couplings of the leptons and Higgs fields, which give rise to the mass terms

after the Higgs gets its vacuum expectation value. The simplest couplings that will do this job are the Yukawa-like couplings involving two left-handed lepton fields and two Higgs scalars, HH = 1 C α,β Hi i α σ H j j β + H. c. 84 α,β Note that the product H i i of the Higgs doublet and the left-handed epton doublet is a gauge-invariant Weyl spinor, so we can combine two such products into a gauge-invariant, orentz-invariant agrangian term. When the Higgs VEV breaks the electroweak gauge symmetry, it also makes neutrino mass terms from the couplings 84. Indeed, substituting Higgs VEV H i into the interaction terms 84, we obtain the Majorana mass terms for the neutrinos that look exactly like in eq. 8 for M ν α,β = v C α,β. 85 Unlike the dimensionless gauge and Yukawa couplings, the C α,β couplings have dimensionality energy 1. We shall see later in class that such couplings make trouble for perturbation theory at high energies, so they are not allowed in UV-complete quantum field theories. However, if the Standard Model is only an effective theory that s valid up to some maximal energy E max but at higher energies must be superseded by a more complete theory, then it s OK for the SM to have small negative-dimensionality couplings C < 1/E max. The key word here is small it explains why the neutrinos are so much lighter than the other fermions: If C < 1/E max, then m ν < v E max v. 86 In particular, for E max 10 15 GeV this limit tells us m ν < 0.1 ev, which is in the right ballpark for the neutrino masses inferred from the neutrino oscillations. 3