Electro-Weak Interactions

Home , Electroweak interaction, Hypercharge, Isospin, Weak hypercharge, Weak isospin, X (charge)

Marcello Fanti

Physics Dept. — University of Milan

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 1 / 36 The ElectroWeak model

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 2 / 36 Electromagnetic vs weak interaction

Electromagnetic interactions mediated by a photon, treat left/right fermions in the same way

g M = [¯u (eγµ)u ] − µν [¯u (eγν)u ] 3 1 q2 4 2

1 − γ5 Weak charged interactions only apply to left-handed component: ψ = ψ L 2 Fermi theory (eﬀective low-energy theory):

GF µ 5 ν 5 M = √ u¯3γ (1 − γ )u1 gµν u¯4γ (1 − γ )u2 2

Complete theory with a vector boson W mediator: g 1 − γ5 g g 1 − γ5 √ µ µν √ ν M = u¯3 γ u1 − 2 2 u¯4 γ u2 2 2 q − MW 2 2 2 g µ 5 ν 5 −−−→ u¯3γ (1 − γ )u1 gµν u¯4γ (1 − γ )u2 2 2 low q 8 MW √ 2 2 g −5 −2 ⇒ GF = — and from weak decays GF = (1.1663787 ± 0.0000006) · 10 GeV 8 MW

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 3 / 36 Experimental facts

e e Electromagnetic interactions γ Conserves charge along fermion lines ¡ Perfectly left/right symmetric e e Long-range interaction electromagnetic µ ⇒ neutral mass-less mediator ﬁeld A (the photon, γ) currents

eL νL Weak charged current interactions Produces charge variation in the fermions, ∆Q = ±1 W ± Acts only on left-handed component, ψ !! ¡ L u Short-range interaction L dL ⇒ charged massive mediator ﬁeld (W ±)µ weak charged − − − currents E.g. weak decays, n → pe ν¯e , µ → e ν¯eνµ

Weak neutral current interactions νµ νµ Conserves fermion charge, ∆Q = 0 Acts on both left-handed and right-handed components ¡Z (but with diﬀerent strengths!) e e Short-range interaction weak neutral µ ⇒ neutral massive mediator ﬁeld Z currents − − E.g. neutrino scattering, νµe → νµe

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 4 / 36 Seeking a symmetry group

4 gauge ﬁelds: 2 neutral (γ, Z) and 2 charged (W ±) ⇒ symmetry group must have 4 parameters

Cannot be just one group (e.g. U(2)), because electromagnetic interaction couples to uL, uR, vL, vR with same strength, while weak charged current only couples to uL, vR

g0 P3 k Easiest way is a composition of two groups, SU(2) ⊗ U(1) : ψ → ei 2 θ0Y · eig k=1 θk T ψ (note diﬀerent couplings g, g 0 for groups SU(2), U(1)) Naively one could think at U(1) as the group for electromagnetism and at a SU(2) for weak (charged and neutral) interactions, but this again does not work, because weak charged currents couples only to uL, vR, while weak neutral currents also couple with uR, vL — despite with diﬀerent strength ⇒ The gauge boson of the U(1) group cannot be the photon!

U(1) group:

g0 i θ0Y µ e 2 ⇒ one parameter θ0 ⇒ one gauge ﬁeld B Y is the weak hypercharge operator — each particle is an eigenstate of Y SU(2) group:

P3 k ig k=1 θk T µ e ⇒ 3 parameters θk ⇒ 3 gauge ﬁelds Wk (k = 1, 2, 3) 2 2 2 2 2 T k are the weak isospin operators — each particle is an eigenstate of T and T 3 (T ≡ T 1 + T 2 + T 3)

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 5 / 36 Weak isospin and weak hypercharge for fermions

T = weak isospin, Y = weak hypercharge : must commute with each other

Weak charged currents couple to ψL, not to ψR ⇒ treat ψL, ψR differently. uL ψL are isospin doublets: ψL ≡ with same hypercharge YL dL τ k ⇒ T k ≡ ; L 2 0 1 0 −i 1 0 recall Pauli’s matrices: τ = ; τ = ; τ = 1 1 0 2 i 0 3 0 −1 k ψR are isospin singlets: uR , dR ⇒ T R = 0, no coupling to W -fields — and different hypercharge eigenvalues, Yu,R, Yd,R T 3 QY Here uL, dL stay for generic “up” or “down” states in isospin space, ν +1/2 0 u νe e can be e.g. up-/down-quarks, or leptons: ≡ − −1 d e− e −1/2 −1 L L L

On the opposite, uR , dR are their chiral counterparts, but they (νe)R 0 0 0 don’t belong to doublets. However, uL, uR have same electric (e)R 0 −1 −2 charge, and likewise for dL, dR

T3 eigenvalues are quantized (as from SU(2) algebra) u +1/2 +2/3 1/3 d −1/2 −1/3 Y eigenvalues are free ⇒ chosen such to fulﬁll L

Y = 2(Q − T 3) (u)R 0 +2/3 4/3 (d)R 0 −1/3 −2/3

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 6 / 36 Weak hypercharge and weak isospin for gauge bosons

3 Extend the relation Y = 2(Q − T ) to any particle — i.e. also to vector bosons. U(1) and SU(2) are separate groups The B-boson has T = 0 and Y = 0 ⇒ has charge Q = 0 ⇒ no B-Wk interactions in L ⇒ TB = 0 and YW = 0 (if B had Y 6= 0, it’d couple to itself: not foreseen since U(1)Y is abelian) The 3 W-bosons

are assigned Y = 0( ⇒ have charge Q = T3)  1   0   0  are in the adjoint representation of SU(2) W1 =  0  ; W2 =  1  ; W3 =  0  ⇒ dimension 3 0 0 1  0 0 0  T = −i 0 0 1 δW µ = −g θ W µ 1   a abc b c 0 −1 0  0 0 −1  µ µ ⇒ [Tb] = −ibac T = −i 0 0 0 δWa = igθb [TbW ]a ac 2   µ 1 0 0 = igθb [Tb]ac Wc  0 1 0  T3 = −i  −1 0 0  ⇒ T3 W3 = 0 ; T3 W1 = iW2 ; T3 W2 = −iW1 0 0 0

T 3 QY B 0 0 0 ± 1 Deﬁne W = √ (W1 ± iW2) W 0 0 0 2 3 ± ± W + +1 +1 0 ⇒ T3 W3 = 0 and T3 W = ± W W − −1 −1 0

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 7 / 36 Interaction Lagrangian

0 0 g P3 k g X k µ ψ → ei 2 θ0Y · eig k=1 θk T ψ ⇒L EW = − ψγ¯ Y ψ Bµ − g ψγ¯ T ψ W k int 2 µ µ | {zY } k | {zk } Jµ Jµ

Recall: k uL k τ L-fermions: ψL ≡ , with T L ≡ and one common YL dL 2 0 1 0 −i 1 0 recall Pauli’s matrices: τ = ; τ = ; τ = 1 1 0 2 i 0 3 0 −1 k R-fermions: separate uR, dR , with T R ≡ 0 and diﬀerent Yu,R, Yd,R

g 0 g g 0 g 0 g LEW = − ψγ¯ BµY ψ − ψ¯ γ W k,µτ kψ = − ψ¯ Y B/ψ − ψ¯ Y B/ψ − ψ¯ τ kW/ kψ int 2 µ 2 L µ L 2 R R 2 L L 2 L L ! g 0 g 0 1 g 0Y B/ + gW/ 3 g(W/ 1 − iW/ 2) u = − u¯ Y B/ u − d¯ Y B/ d − u¯ d¯ L L R u,R R R d,R R L L 1 2 0 3 2 2 2 g(W/ + iW/ ) g Y B/ − gW/ dL | {z } L R-couplings | {z } L-couplings

k def ¯ k Y def ¯ fermionic currents: Jµ = ψγµT ψ and Jµ = ψγµY ψ Y 1 Q = + T 3 ⇒ electromagnetic current: Jem def= ψγ¯ Qψ = JY + J3 2 µ µ 2 µ µ

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 8 / 36 Physical electroweak ﬁelds

g 0 Recall: LEW = − JY · B − g J1 · W 1 + J2 · W 2 + J3 · W 3 int 2 Weak charged currents: ± 1 2 + − ¯ ± 1 1 2 deﬁne J = J ± iJ (explicitely, J =u ¯LγµdL and J = dLγµuL) and W = √ W ± iW µ µ 2 ! / − 1 1 2 2 1 + − − + 1 ¯ 0 W uL ⇒ J · W + J · W = √ W · J + W · J = √ u¯L dL + 2 2 W/ 0 dL NOTE: W/ ∓ change isospin by ∆T 3 = ±1 and keep Y unchanged ⇒ change fermion charge by ∆Q = ±1 Y 1 Electromagnetic currents: (recall: Q = + T 3 ⇒ Jem = JY + J3 ) 2 2 1 1 we need a term −e Jem · A ≡ −e JY + J3 · A , can we extract it from − g 0JY · B + gJ3 · W 3 ? 2 2 Yes, if A is a “mixture” of B, W 3 ⇒ introduce a rotation (B, W 3) → (A, Z)

B = A cos θW − Z sin θW 3 W = A sin θW + Z cos θW θW is the “Weinberg” electroweak mixing angle

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 9 / 36 Physical electroweak ﬁelds

1 g 0 g 0JY · B + gJ3 · W 3 = (A cos θ − Z sin θ ) JY + g (A sin θ + Z cos θ ) J3 2 2 W W W W g 0 g 0 = JY cos θ + gJ3 sin θ A + gJ3 cos θ − JY sin θ Z 2 W W W 2 W

 Jem  z }| { 0 1 Y 3 g cos θW = e ⇒ can recover the e  J + J  · A term if: 2  g sin θW = e

e e Useful relations: B = 0 A − Z = A cos θW − Z sin θW 0 g g e = g sin θW = g cos θW e e 0 2 2 W 3 = A + Z = A sin θ + Z cos θ gg e e g g 0 W W e = — to enforce + = 1 pg 2 + g 02 g 0 g Then the part coupled to Z becomes: 0 3 g Y e cos θW 3 e sin θW em 3 e 3 2 em gJ cos θW − J sin θW = J − J − J = J − sin θW J 2 sin θW cos θW sin θW cos θW

Electroweak interactions:  weak charged weak neutral  z }| { e.m. z }| { EW g + − − + z }| em{ e 3 2 em ⇒L int = − √ W · J + W · J + e A · J + Z · J − sin θW J   2 sin θW cos θW 

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 10 / 36 Electroweak interactions

EW g + − − + em e 3 2 em Lint = − √ W · J + W · J − e A · J − Z · J − sin θW J 2 sin θW cos θW | {z } | {z } | {z } weak charged e.m. weak neutral

1 − γ5 † 1 − γ5 1 − γ5 1 − γ5 Recall L-projector: (ψ¯ )γµ(ψ ) = ψ γ0γµ ψ = ψ¯ γ0 γ0γµ ψ 1,L 2,L 2 1 2 2 1 2 2 2 5 1 − γ 1 + γ5 1 − γ5 1 − γ5 1 − γ5 ψL = ψ = ψ¯ γ0γ0 γµ ψ = ψ¯ γµ ψ 2 1 2 2 2 1 2 2 2 1 − γ5 = ψ¯ γµ ψ (only 1 L-projector) 1 2 2 Charged weak interactions:

g + − − + g h − ¯ + i −√ W · J + W · J = −√ u¯LW/ dL + dLW/ uL 2 2 g 1 − γ5 1 − γ5 = −√ u¯W/ − d + d¯W/ + u 2 2 2 Neutral weak interactions:

5 e 3 2 em e ¯ 3 1 − γ 2 ¯ − Z · J − sin θW J = − ψ Z/T ψ − sin θW ψ Z/Q ψ sin θW cos θW sin θW cos θW 2   e ¯ 3 2 3 5 = − ψZ/ (TL − 2Q sin θW ) − TL γ  ψ 2 sin θW cos θW | {z } |{z} CV CA

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 11 / 36 Particles’ classiﬁcation wrt EW interactions

fermion TT3 YQ 1 νeL , νµL , ντ L 1 +2 0 Summary: 2 1 −1 eL , µL , τL −2 −1 Left-handed fermions occur in isospin doublets νe , νµ , ντ 0 0 leptons R R R (T = 1/2, T 3 = ±1/2) with same hypercharge 0 0 eR , µR , τR −2 −1 3 Y = 2(Q − T ). u0 , c0 , t0 +1 +2 L L L 1 2 −1 3 Right-handed fermions occur in isospin singlets 0 0 0 2 1 3 1 dL , sL , bL −2 −3 (T = T 3 = 0), with hypercharge Y = 2Q. 0 0 0 4 2 quarks uR , cR , tR +3 +3 0 0 0 0 0 2 1 dR , sR , bR −3 −3 Interactions

interaction lagrangian comments

¯ µ electromagnetic −eQ Aµ ψγ ψ L/R-symmetric, pure V structure, don’t mix flavours g − √ W − u¯γµ(1 − γ5)d charged weak 2 2 µ V − A structure, only L-fermions interact, mix flavours g − √ W + d¯γµ(1 − γ5)u 2 2 µ e ¯ µ 5 neutral weak − Zµ ψγ (CV − CAγ )ψ mixed V /A structure, don’t mix flavours 2 sin θW cos θW

3 2 3 Neutral weak V /A couplings: CV = TL − 2Q sin θW ; CA = TL

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 12 / 36 Particles’ masses

We observe a wide mass hierarchy in the know fundamental particles

But masses are not allowed for gauge bosons.

Also for fermions, we are in troubles, due to the chiral structure of the electroweak interaction. In princi- ¯ µ ple Lfermions = ψ(iγ Dµ − m)ψ . . . but the “mass matrix” m does not commute with the electroweak gener- ators T a, Y — otherwise fermions in iso-doublets (e.g. e, νe) would have the same mass.

Fermion masses spoil the SU(2)L ⊗ U(1)Y gauge invariance — troublesome, isn’t it?? We can only allow a mass-less lagrangian for fermions: ¯ µ Lfermions = ψ(iγ Dµ)ψ ¯ µ uL = u¯L dL (iγ Dµ) dL µ ¯ µ +u ¯R(iγ Dµ)uR + dR(iγ Dµ)dR

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 13 / 36 Massive particles : the Higgs mechanism

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 14 / 36 Simple example: QED with “heavy photon” (1)

QED lagrangian: 1 L = ψ¯ (iγµD − m) ψ − F µνF ( D ≡ ∂ + ieA ) QED µ 4 µν µ µ µ Introduce a complex scalar fieldΦ, transforming asΦ → eieα(x)Φ, with lagrangian µ ∗ ∗ 2 2 ∗ LHiggs = (D Φ) (DµΦ) − λ(Φ Φ) − µ Φ Φ | {z } U(Φ) µ2 1 such that U(Φ) has a minimum for any Φ such that |Φ |2 = def= υ2 0 0 2λ 2 ⇒ The vacuum state (aka status of minimal energy) is degenerate, and characterized by a uniform, non-vanishing scalar field. All particles are excitations from the vacuum state ⇒ rewrite fields as variations from the vacuum: def υ υ + φ + iχ chooseΦ 0 = √ andΦ= √ (φ, χ real fields) — the vacuum choice (Φ0) breaks the gauge invariance 2 2 υ2 Little displacements in χ would keep Φ∗Φ = ⇒ stay in the set of vacuum states ⇒ χ is not a physical field 2 1 φ is the physical field that matters!:Φ= √ (υ + φ) 2 ∗ 2 µ µ υ + φ υ + φ µ 2 λ 4 LHiggs = (∂ + ieA ) √ (∂µ + ieAµ) √ + (υ + φ) − (υ + φ) 2 2 2 4 1 e2 µ2υ2 µ2 µ2 = ∂µφ∂ φ + AµA (υ + φ)2 − + µ2φ2 + φ3 + φ4 2 µ 2 µ 4 υ 4υ2

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 15 / 36 Simple example: QED with “heavy photon” (2)

Putting all together and rearranging: 2 ¯ µ ¯ µ 1 µν (eυ) µ L = LQED + LHiggs = ψ(iγ ∂µ − m)ψ −eAµψγ ψ − F Fµν + A Aµ | {z } | {z } 4 2 fermion propagator fermion-gauge interaction | {z } gauge boson propagator 1 µ2 µ2 + ∂µφ∂ φ − 2µ2φ2 − φ3 + φ4 2 µ υ 4υ2 | {z } | {z } Higgs ﬁeld propagator Higgs ﬁeld self-interaction e2 +(e2υ)AµA φ + φ2AµA µ 2 µ | {z } Higgs-gauge interaction

Summary We started with a simple gauge theory, with mass-less gauge field A. Then we introduced a scalar field Φ, interacting with A and with non-zero vacuum expectation value (VEV) υ. This brought us to: a choice of vacuum that is not gauge-invariant a mass term for the gauge field A, such that m = eυ — measuring m would fix the VEV υ A √ √ A an observable real scalar field φ, with mass mH = 2 µ = υ 2λ — mH is a free parameter as long as λ is not known 1 m 2 Higgs-gauge interactions proportional to m2 : A 2υφAµA + φ2AµA A 2 υ µ µ This is the Higgs mechanism for “spontaneous symmetry breaking”: the gauge field has acquired mass dynamically, thanks to the interaction with a “pervasive” scalar field, without spoiling the gauge symmetry. CAVEAT this was just an “exercise”, QED with massive photon is not implemented in nature. . .

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 16 / 36 Higgs mechanism in EW theory (1)

W ±, Z fields are massive, A is massless ⇒ break the gauge symmetry for W ±, Z, not for A Choose a set of complex scalar fields Φ with non-vanishing vacuum expectation value (VEV): 0 µ † h † 2 2 † i g k k LΦ = (D Φ) (DµΦ) − λ Φ Φ − µ Φ Φ Dµ ≡ ∂µ + i BµY + igWµ T | {z } 2 kinetic, K(Φ) | {z } potential, U(Φ) µ2 1 Vacuum defined such that U(Φ) be minimal ⇒ Φ†Φ = def= υ2. 0 0 2λ 2 Φ must couple to W ±-fields to give them mass ⇒ must be an isospin multiplet (T =6 0) 1 1 Φ+ ⇒ minimal model: choose isospin doublet of scalar complex fields i.e. T = and T 3 = ± ⇒ Φ ≡ 2 2 Φ0 Y Y + τ 1 Y + 1 0 Φ must not couple to A-field ⇒ must have Q = 0: recalling Q def= + T ≡ 3 ≡ 2 3 2 2 0 Y − 1 1 0 0 the eigenvalue equation QΦ = 0 implies Y = ±1 ⇒ choose Y = 1 ⇒ Q ≡ ⇒ Φ ≡ 0 0 Φ0 1 0 1 0 ⇒ choose the vacuumΦ 0 ≡ √ ⇒ defineΦ ≡ √ 2 υ 2 υ + φ φ is the physical field i.e. the displacement from vacuum state (1)

χ1 + iχ2 1 √1 † 1 2 other little displacements from Φ0, such as Φ = , would keep Φ Φ = 2υ to 1st order, i.e. would only move to other 2 υ + iχ3 vacuum states ⇒ these are not physical ﬁelds

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 17 / 36 Higgs mechanism in EW theory (2)

Recall:

0 g g i 1 0 1 D Φ = ∂ Φ + i B Φ + i τ k W k Φ = ∂ (Φ − Φ ) + g 0B Φ + gτ k W k Φ τ = µ µ 2 µ 2 µ µ 0 2 µ µ 1 0 1 0 i (g 0B + gW 3) g(W 1 − iW 2) 0 0 −i √ √ µ µ µ µ τ 2 = ; Y = 1 = ∂µ + 1 2 0 3 i 0 2 φ 2 2 g(Wµ + iWµ )(g Bµ − gWµ ) υ + φ 1 0 τ 3 = 0 −1 0 2 2 2 g g k k The kinetic term K(Φ) ≡ |DΦ| = |∂Φ| + i BΦ + i τ W Φ contains a term: 2 2

√ 2 0 2 1 2 2 2 g 2W − ! g g k k i(υ + φ) g(W − iW ) (υ + φ) i BΦ + i τ W Φ = √ = g 2 2 g 0B − gW 3 8 − Z 2 2 cos θ W W 1 ± iW 2 gυ2 1 gυ 2 W ± = √ = W −W + + Z 2 2 2 2 2 cos θW B = A cos θW − Z sin θW g 2υ 1 g 2υ 3 + φW −W + + φZ 2 W = A sin θW + Z cos θW 2 0 2 4 cos θW g sin θW = g cos θW g 2 1 g 2 + φ2W −W + + φ2Z 2 2 2 2 cos θW

gυ2 1 gυ 2 We can identify therefore two mass terms for the gauge bosons: W −W + and Z 2 2 2 2 cos θW gυ gυ mW ⇒ mW = ; mZ = = 2 2 cos θW cos θW

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 18 / 36 Higgs mechanism in EW theory (3)

. . . The rest describes the Higgs-gauge interactions, proportional to gauge masses squared: m 2 1 m 2 W 2υφW +W − + φ2W +W − + Z 2υφZZ + φ2ZZ υ 2 υ W ± φ W ± Z φ Z φ ¡ ¡ φ ¡ ¡ W ± φ W ± Z φ Z

The Higgs kinetic term |∂Φ|2 and its potential U(Φ) can be expressed as 1 λ µ2 |∂Φ|2 − U(Φ) = (∂µφ)(∂ φ) − (υ + φ)4 − (υ + φ)2 2 µ 4 2 1 φ4 λυ4 = (∂µφ)(∂ φ) − µ2φ2 − λ υφ3 + + 2 µ 4 4 | {z } | {z } Higgs propagator Higgs self-interaction √ √ hence describing a Higgs mass mH = 2µ = 2λ υ and Higgs triple and quadruple self-interactions: ¡ ¡

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 19 / 36 Higgs mechanism in EW theory (4)

Fermion masses 0 0 γ is the parity operator ⇒ ﬂips L/R: γ ψL,R = ψR,L ¯ † 0 ψL † † ⇒ mψψ = m (ψL ψR) γ = m ψLψR + ψRψL ψR ⇒ Dirac fermion mass terms mix Left/Right components: uL But in EW model ψL ≡ is a doublet, while uR, dR are singlets. How to combine? We need to build a dL gauge-invariant piece of lagrangian ⇒ must be an isospin singlet! 1 0 Use Higgs doubletΦ= √ and its charge-conjugate 2 υ + φ def ∗ 1 0 1 0 1 υ + φ Φc = −iτ 2Φ = √ = √ 2 −1 0 υ + φ 2 0 Introduce the Yukawa terms: f

h † † † † i LYukawa = − gd uL dL ΦdR + gu uL dL ΦcuR + hermitian conjugate φ 1 h i ¡ † † = −√ gd (υ + φ)d dR + (h.c.) + gu(υ + φ)u uR + (h.c.) 2 L L f

1 m ¯ So we identify mass terms mu,d = √ υ · gu,d and Higgs-fermion couplings φψLψR + (h.c.) 2 υ

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 20 / 36 Higgs mechanism in EW theory : Summary

We have 4 interactions: electromagnetic, weak neutral, and two weak charged ⇒ we need 4 mediator fields A, Z, W ± Three are massive: Z and W ± — the photon is massless ⇒ need a scalar field with non-vanishing VEV υ to give mass to W ±, must belong to an isospin multiplet to keep the photon massless, must be electrically neutral ⇒ the minimal choice is an isospin doublet of complex scalar fields ⇒ 4 degrees of freedom Three are “eaten” by the gauge bosons Z, W ± when they acquire mass (massless vector fields have only 2 transverse polarization states ±1, massive gauge fields have a longitudinal polarization state too). The remaining degree of freedom should manifest itself as a free particle. . . the Higgs boson, with spin=0, charge=0, unknown mass What we got: gυ The W -boson mass is predicted: m = W 2 mW The ratio of W , Z masses is fixed: = cos θW — this is distinctive of the Higgs iso-doublet structure, other mZ choices would give different results. Independent measurements of cos θW , mW , mZ confirm this relation, thus the doublet model. m2 The Higgs boson couples with gauge fields with strength proportional to the square of the field mass: g = 2 V V υ Fermion masses mf are also generated, but are free parameters — they depend on the Yukawa couplings that are chosen to produce exactly the observed mass values m The Higgs boson couples with fermions with strength proportional to their masses g = f f υ

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 21 / 36 Higgs mechanism in EW theory : Quantities

The parameters of the Higgs sector are µ, λ. These are connected to measurable quantities, mH, υ: m 1 m 2 µ = √H ; λ = H 2 2 υ

√ s 2 g 2 gυ 1 The VEV is known: from GF = and mW = we get υ = √ 8 mW 2 2 GF − − −5 −2 GF is known with high precision from µ → e νµν¯e decay: GF = (1.1663787 ± 0.0000006) · 10 GeV ⇒ υ = (246.21965 ± 0.00006) GeV

The Higgs boson mass mH is a free parameter: recent observation at ATLAS and CMS give mH ' 125 GeV (≈ 1% precision) ⇒ the Higgs self-coupling would result λ ' 0.129 (great! quite perturbative)

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 22 / 36 Feynman rules for EW physics

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 23 / 36 What follows are screenshots from Appendix B of the book:

Yorikiyo Nagashima : “Elementary Particle Physics: Foundations of the Standard Model, Volume 2”

http://onlinelibrary.wiley.com/doi/10.1002/9783527648887.app2/pdf

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 24 / 36 Fermion - gauge boson interactions

From the fermion-gauge EW lagrangian:

EW em Lint = − e A · J

| e.m.{z } g − √ W + · J− + W − · J+ 2 | {z } weak charged e 3 2 em − Z · J − sin θW J sin θW cos θW | {z } weak neutral

(note: for Wqq vertices, the CKM matrix is used)

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 25 / 36 Triple and Quartic Gauge Couplings (TGC and QGC)

Trilinear and 4-linear gauge boson terms in the lagrangian:

1 L = g W µW ν (∂ W − ∂ W ) TGC 2 abc b c µ aν ν aµ 1 L = g 2 W µW νW W QGC 4 abc ars b c rµ sν 1 = g 2(δ δ − δ δ )W µW νW W 4 br cs bs br b c rµ sν produce gauge bosons self-interaction. Using (W 1, W 2) → (W +, W −) and (W 3, B) → (Z, A) transforms, we get: W +W −γ and W +W −Z vertices (TGC) (note: no ZZZ, ZZγ, Zγγ, γγγ)

W +W −γγ, W +W −Zγ, W +W −ZZ, W +W −W +W − (QGC)

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 26 / 36 Higgs-fermion and Higgs-gauge interactions

Some help to go back to our notations: e e gW ≡ = ; gZ = sin θW sin θW cos θW

gυ m = W 2

Therefore

1 gW mf mf ≡ 2 mW υ m2 g m ≡ 2 W W W υ m2 g m ≡ 2 Z Z Z υ

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 27 / 36 The Standard Model

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 28 / 36 Particles’ zoo in Standard Model

Particles can be classiﬁed according to their interactions ⇒ according to their representations of the symmetry groups SU(3)C ⊗ SU(2)L ⊗ U(1)Y SU(3) SU(2) U(1) particle C L Y Q spin remarks (dimension) TT3 Y 1 νeL , νµL , ντ L 1 1 +2 0 2 1 −1 eL , µL , τL 1 −2 −1 νe , νµ , ντ 1 0 0 (do νR’s exist?) leptons R R R 0 0 eR , µR , τR 1 −2 −1 1 0 0 0 1 2 uL , cL , tL 3 1 +2 1 +3 2 0 0 0 2 1 +3 1 dL , sL , bL 3 −2 −3 0 0 0 4 2 mass eigenstates mix fermion generations

quarks uR , cR , tR 3 +3 +3 0 0 0 0 0 2 1 dR , sR , bR 3 −3 −3 g (gluons) 8 0 0 0 0 γ (photon) 1 not def. 0 0 0 mixings of B, W Z 1 not def. 0 0 0 1 3 W + 1 +1 +1 1 0 gauge bosons W − 1 −1 −1 1 1 H (Higgs) 1 2 +2 +1 0 0

Q: are L-fermion and R-fermion (e.g. eL, eR) the same particle? A: according to this classiﬁcation, no — but they have same charge, same mass — quite peculiar, isn’t it? (they are connected together via the interaction with the Higgs ﬁeld)

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 29 / 36 Assignment of weak hypercharge (1)

Anomalies are quantum processes that would spoil a symmetry holding in non-quantum ﬁelds. W g B B ¡ B ¡ B ¡ W g B (a) BWW anomaly (b) Bgg anomaly (c) BBB anomaly

Zeroing the anomalies: [recall: quarks are 3× due to color states ; L-fermions are 2× due to isospin doublet] [using L-chirality everywhere ⇒ use fermion doublets and anti-fermion singlets (with opposite Y )]

short notations: ν L ≡ e L (a) only L-fermions enter (due to W -couplings):2 · (YL + 3 YQ) = 0 ν ≡ νR

e ≡ eR u (b) only quarks enter (due to g-couplings):3 · (2 YQ − Yu − Yd ) = 0 Q ≡ d L u ≡ uR 3 3 3 3 3 3 (c) all fermions enter:2 YL − Ye − Yν + 3 · (2 YQ − Yu − Yd ) = 0 d ≡ dR

NOTE: SU(2) singlets (νR , eR , uR , dR ) enter as L-antifermions (¯νL, e¯L, u¯L, d¯L) to have same chirality in the vertices ⇒ opposite sign in Y

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 30 / 36 Assignment of weak hypercharge (2)

Consider Yukawa couplings in Lagrangian: every term must have overall Y = 0 (due to U(1)Y invariance)

† U(1)Y i(−Y +Y +Y )θ † † U(1)Y i(−Y +Y −Y )θ † [e.g. the term L φe −−−−−→ e L e H 0 L φe while the term L φC ν −−−−−→ e L ν H 0 L φC ν ]

† (d) L φe ⇒− YL + Ye + YH = 0 Note: adding eqs (f) and (g) we get eq (b) † (e) L φC ν ⇒− YL + Yν − YH = 0 Also, from eqs (a), (d), (e), (f), (g) and some † (f) Q φd ⇒− YQ + Yd + YH = 0 algebra, (c) becomes an identity! † (g) Q φC u ⇒− YQ + Yu − YH = 0 (see next slide)

⇒ 7 unknowns: YL, Ye, Yν, YQ, Yd , Yu, YH Y ⇒ 5 independent homogeneous equations (a), (d), (e), (f), (g) ⇒ solutions will be in terms of ratios k Yj Assign electric charge Q = −1 to eL, eR [recall: Y = 2(Q − T3)] ⇒ YL = −1 and Ye = −2

Then express all other hypercharges as functions of YL, Ye, using eqs (a), (d), (e), (f), (g): (details in next slide)

1 1 Now the hypercherges are not anymore completely arbi- YQ = − YL = + 3 3 trary. Yν = 2 YL − Ye = 0 2 4 Yu = −Ye + 3 YL = +3 Triangular anomalies and Yukawa couplings settle the 4 2 Yd = Ye − 3 YL = −3 spectrum of hypercharges — and therefore of electric 1 YH = 2 (YL − Ye) = +1 charges — once the electron charge is set.

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 31 / 36 Assignment of weak hypercharge (computations)

From Yukawa couplings:

−Y + Y + Y = 0  L e H ⇒ 2 Y = Y + y ; Y − Y = 2 Y  −Y + Y − Y = 0 L e ν ν e H  L ν H  ⇒ Yν − Ye = Yu − Yd −YQ + Yd + YH = 0  ⇒ 2 YQ = Yd + yu ; Yu − Yd = 2 YH  −YQ + Yu − YH = 0 

Working on BBB anomaly:

3 3 3 3 2 2 2 2 2 2YL − (Ye + Yν ) = 2YL − (Ye + Yν)(Ye − YeYν + Yν ) = 2YL YL − (Ye − YeYν + Yν ) " # Y + Y 2 −3Y 2 − 3Y 2 + 6Y Y = 2Y e ν − (Y 2 − Y Y + Y 2) = 2Y e ν e ν L 2 e e ν ν L 4 3 = − Y (Y − Y )2 2 L e ν 3 (in the same way) 2Y 3 − (Y 3 + Y 3) = − Y (Y − Y )2 Q d u 2 Q d u 3 h i 2Y 3 − (Y 3 + Y 3) + 3 · 2Y 3 − (Y 3 + Y 3) = − Y (Y − Y )2 + 3Y (Y − Y )2 L e ν Q d u 2 L e ν Q d u 3 (using Y − Y = Y − Y ) = − (Y + 3Y )(Y − Y )2 ν e u d 2 L Q e ν (using YL + 3YQ = 0) = 0 i.e. eqn (c) is not independent!

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 32 / 36 Quarks’ ﬂavour mixing: the CKM matrix

u c t Quark mass eigenstates are — these can propagate as free particles : e∓i(Et−p~·~x) with E = +p(p~)2 + m2 d s b u0 c0 t0 Electroweak (Y , T ) eigenstates are not mass eigenstates! d 0 s0 b0  u0   u   d 0   d  0 0 ⇒ need mixing matrices:  c  = VU  c  and  s  = VD  s  t0 t b0 b 1 1 V and V are 3 × 3 unitary matrices, that mix respectively T = + and T = − states. U D 3 2 3 2 Charged currents  d 0   d  L L 0 0 ¯0 − 0 ¯ − † (u ¯L c¯L tL ) W/  sL  = (u ¯L c¯L tL ) W/ VUVD  sL  0 bL | {z } bL  W ±γµ  VCKM µ / ± ± µ  0    W ≡  Wµ γ  uL uL ± µ Wµ γ ¯0 0 ¯0 + 0 ¯ ¯ + † dL s¯L bL W/  cL  = dL s¯L bL W/ VDVU  cL  t0 t L | {z† } L VCKM   Vud Vus Vub VCKM ≡  Vcd Vcs Vcb  is the Cabibbo-Kobayashi-Maskawa matrix Vtd Vts Vtb

† † Neutral currents introduce combinations VDVD ≡ 1 or VUVU ≡ 1 ⇒ no mixing

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 33 / 36 Flavour-Changing Neutral Currents (FCNC)

At tree-level, neutral currents have no ﬂavour mixing ⇒ the only Zqq¯ vertices are: Zdd¯ , Zuu¯ , Zss¯ , Zcc¯ , Zbb¯ , Ztt¯ What if we consider 1-loop diagrams like these: d¯ d¯

u; c; t W Z ¡u; c; t W Z ¡W u; c; t s s

Z → sd¯ Z → sd¯ . . . and analogous for all other possible final states: bd¯, bs¯, cu¯ and their charge-conjugates. They do exist, but must be summed over all the possible quark flavours in the loop. In the example of the figure, the Wqq0 vertices imply CKM coefficients involved in a sum like ∗ ∗ ∗ Vud Vus + Vcd Vcs + Vtd Vts, which is null due to the unitarity of VCKM Actually, the cancellation is not perfect, due to the different quark masses in the loops ⇒ the V ∗V terms are weighted differently by quarks propagators. 0 0 0 0 + − However, a large suppression still holds. Therefore FCNC meson decays like e.g. K , D , B , BS → µ µ are highly suppressed.

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 34 / 36 The complete Standard Model lagrangian

µ µ µ 0 µ def P µ W3 W1 − iW2 νL def g g νL W = k Wk τ k ≡ µ µ µ Dµ = ∂µ − i Bµ + i W µ W1 + iW2 −W3 eL 2 2 eL µ def 1 P µ g 0 g G = G λk (3 × 3 matrices) uL def uL 2 k k Dµ = ∂µ + i Bµ + i W µ + igG µ def d d Bµν = ∂µBν − ∂νBµ L 6 2 L def ig D ν = ∂ ν W µν def= ∂µW ν − ∂νW µ + (W µW ν − W νW µ) µ R µ R def 0 2 DµeR = [∂µ − ig Bµ] eR µν def µ ν ν µ µ ν ν µ 0 G = ∂ G − ∂ G + igs (G G − G G ) 2g D u def= ∂ + i B + igG u µ R µ 3 µ µ R 0 0 def g g 1 0 def g DµΦ = ∂µ + i Bµ + i W µ Φ with Φ ≡ √ D d = ∂ − i B + igG d 2 2 2 υ + φ µ R µ 3 µ µ R

Higgs dynamics and Higgs-gauge interactions gauge bosons’ dynamics z }| { z 1 1 }| 1 { m2 υ2 2 L = − BµνB − Tr [W µνW ] − Tr [G µνG ] + (DµΦ)† (D Φ) − H Φ†Φ − 4 µν 8 µν 2 µν µ 2υ2 2  leptons’ dynamics and lepton-gauge interactions quarks’ dynamics and quark-gauge interactions  3 z }| { z }| { X  νk uk  + ν¯k e¯k iγµD L +ν ¯k iγµD νk +e ¯k iγµD ek + u¯k d¯k iγµD L +u ¯k iγµD uk + d¯k iγµD d k L L µ ek R µ R R µ R L L µ d k R µ R R µ R k=1  L L    j, k are fermion family indexes:   √ leptons’ masses and Yukawa couplings e.g. ν1 ≡ ν , ν2 ≡ ν , ν3 ≡ ν 2 X z }| { e µ τ − ν¯k e¯k ΦMjk ej + h.c. + −e¯k ν¯k Φ∗Mjk νj + h.c. e.g. e1 ≡ e , e2 ≡ µ , e3 ≡ τ υ L L e R L L ν R j,k | {z } | {z } 1 2 3  charged leptons neutrinos  e.g. u ≡ u , u ≡ c , u ≡ t e.g. d 1 ≡ d , d 2 ≡ s , d 3 ≡ b   √ quarks’ masses and Yukawa couplings Oﬀ-diagonal terms in M-matrices 2 X z }| { − u¯k d¯k ΦMjk d j + h.c. + −d¯k u¯k Φ∗Mjk uj + h.c. describe ﬂavour mixing: υ L L d R L L u R j,k | {z } | {z }  down-type quarks up-type quarks  EW eigenstates are not mass eigenstates

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 35 / 36 Free parameters in the Standard Model

Couplings: 0 gs, g, g for the 3 gauge groups SU(3)C ⊗ SU(2)L ⊗ U(1)Y ⇒ 3 parameters e 0 e Measurable quantities θW , e are related to them by g = and g = ; sin θW cos θW Higgs sector: µ, λ — or alternatively mH, υ — or GF instead of υ ⇒ 2 parameters gυ mW Weak boson masses are predicted: mW = and mZ = 2 cos θW Fermion masses: 6 quarks, 3 leptons ⇒ 9 parameters

If we had nf fermion families, we’d have

2nf − 1 relative phases CKM matrix for quark ﬂavour mixing: 2 nf real parameters in the CKM (unitary) matrix it’s a U(3) matrix ⇒ 9 real parameters; but 5 are related 2 2 ⇒ nf − (2nf − 1) = (nf − 1) physical parameters. to relative phases among quarks ⇒ unphysical 2 1 An ortogonal matrix (rotation) has nf − (nf + 2nf (nf − 1)) = ⇒ 4 physical parameters 1 2nf (nf − 1) parameters (3 rotation angles and 1 complex phase) 1 ⇒ Number of complex phases: 2(nf − 1)(nf − 2) [A complex phase in VCKM makes the EW Hamiltonian non- ⇒ nf = 3 is the minimal amount of fermion families to achieve hermitian ⇒ CP violation] CP violation

⇒ Overall, 18 parameters [here we did not count neutrinos’ masses and mixings]

M. Fanti (Physics Dep., UniMi) Fundamental Interactions 36 / 36