Electroweak Theory, SSB, and the Higgs: Lecture 2

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1 1 Electroweak Theory, SSB, and the iggs: Lecture Spontaneous symmetry breaking (iggs mechanism) - Gauge invariance implies massless gauge bosons and fermions - Weak interactions short ranged spontaneous symmetry breaking for mass; also for fermions - Allow classical (ground state) expectation value for iggs field ( ) φ + v = φ = constant, φ = φ - µ v increases energy, but important for monopoles, strings, domain walls, phase transitions (e.g., EWPT, baryogenesis) - Minimize V (v) to find v and quantize φ = φ v - SU() U(1): introduce ermitian basis ( ) ( φ + 1 ) (φ 1 + iφ ) φ = φ = 1, φ (φ 3 + iφ 4 ) i = φ i ) ) V (φ) = µ φ φ + λ(φ φ) 1 µ ( 4 i=1 φ i λ ( 4 i=1 φ i - V (φ) is O(4) invariant (higher symmetry than L gauge + L f + L Y uk ) - w.l.o.g., choose φ i =, i = 1,, 4 and φ 3 = V (φ) V (v) = 1 µ λ4

2 - For µ >, minimum at = (no SSB) - For µ <, minimum at V () = (µ + λ ) = = ( µ /λ ) 1/ φ 1 ( ) v - Generators L 1, L, and L 3 Y = ( 1 ) spontaneously broken, e.g., L 1 v (L i = τ i, Y = 1 I) ( ) 1 - Qv = (L 3 + Y )v = v = U(1) Q unbroken SU() U(1) Y U(1) Q - SSB for µ = also (need loop corrections) Quantize around classical vacuum - Define new quantum field φ with φ = : ( ) w φ v + φ + ++iz - Kibble transformation: introduce new variables ξ i for rolling modes φ = 1 e i ( ) ( ) ξ i L i w iz

3 3 - = is the iggs scalar - φ φ = ( + ) / no potential for ξ i 3 massless Goldstone bosons for global symmetry - Disappear from spectrum for gauge theory ( eaten ) - Display particle content in unitary gauge φ e i ξ i L i φ = 1 ( ) + = 1 ( (+ corresponding transformation on gauge fields) Rewrite Lagrangian in new vacuum - Physical iggs scalar (oscillations around minimum) ( will show M = λ) ) [ 1 + ] - iggs covariant kinetic energy terms: (D µ φ) D µ φ = 1 ( µ) + 1 [ ] g ( ( ) τ i Wµ i + g B µ ) [ 1 + ] M W W +µ W µ + M Z Zµ Z µ + kinetic energy and gauge interaction terms W W e L M W = g e L g g m e e = h e h e W W e R e R - Mass eigenstate bosons: W, Z, and A (photon) W ± = 1 (W 1 iw ) Z = sin θ W B + cos θ W W 3 A = cos θ W B + sin θ W W 3 Typeset by FoilTEX 1 Typeset by FoilTEX 1

4 4 - Weak angle: tan θ W g /g - Masses: M W = g, M Z = g + g = M W cos θ W, M A = (Goldstone scalars eaten longitudinal components of W ±, Z ) - Weak charged current (W ± ) and QED (A) known before SU() U(1). Weak neutral current (Z) predicted - Will show: Fermi constant G F / g /8M W (G F = GeV from muon lifetime) - Electroweak scale: = M W /g ( G F ) 1/ 46 GeV - Will show: g = e/ sin θ W (α = e /4π 1/137.36) M W = M Z cos θ W = g (πα/ G F ) 1/ sin θ W - Weak neutral current (Z): sin θ W.3 M W 78 GeV, M Z 89 GeV (increased by GeV by loop corrections) - Weak neutral current discovered 1973 (Gargamelle at CERN, PW at FNAL) - W ± and Z discovered in 1983 (UA1 and UA at CERN) - M Z now measured to 1 ppm!

5 5 The iggs Scalar - Gauge interactions: ZZ, ZZ, W + W, W + W φ 1 ( + ) (unitary gauge) L φ = (D µ φ) D µ φ V (φ) = 1 ( ( µ) + MW W µ+ Wµ 1 + ) + 1 ( M ZZ µ Z µ 1 + ) V (φ) (quartic and induced cubic interactions, ()M / ) W + µ W µ f igµ M W igµ M W i m f W W + f Zµ Zµ igµ M Z igµ M Z Z Z 3i M 3i M Typeset by FoilTEX

6 6 - iggs potential: V (φ) = +µ φ φ + λ(φ φ) µ4 4λ µ + λ 3 + λ 4 4 Fourth term: quartic self-interaction Third: induced cubic self-interaction Second: (tree level) mass-squared, M = µ = λ First: constant (irrelevant until gravity added cosmological constant) λ λ Fermion Masses and Mixings - Yukawa (iggs-fermion) interaction (+ analogous u, e, terms) L d Y uk = = 3 m,n=1 3 m,n=1 Γ d mn q mlφd nr + h.c. Γ d [ū mn ml φ + d nr + d mlφ d ] nr + h.c. Typeset by FoilTEX u ml d ml φ + iγ d mn φ iγ d mn d nr d nr

7 7 ( - For φ 1 L d Y uk + 3 m,n=1 = d LM d d R = i ) M d mn d mld nr (unitary gauge) ( 1 + ) m i dil d ir ( 1 + ( 1 + ) + h.c. + h.c. ) + h.c. = i ( m i di d i 1 + ) with M d Γ d / - d i = d il + d ir are mass eigenstates of mass m i ( bare QCD masses) - Mixing matrices diagonalize M d (A L = A R up to phases for ermitian M d ): d LM d d R = d L A d LA d L M d A d RA d R d R = d ( ) L A d L M d A d R d R A d L M d A d R = m d m s m b, d s b L,R = A d L,R }{{} unitary d 1 d d 3 L,R - couplings to fermions are diagonal in flavor and mass (stringent constraint from off-diagonal in some extended iggs models) - Mass eigenvalues: m u MeV, m c 1.3 GeV, m t 173 GeV m d 5 MeV, m s 95 MeV, m b 4. GeV m e.5 MeV, m µ 15 MeV, m τ 1.8 GeV m u,d,s break global (flavor) chiral SU(), SU(3) symmetries Consitutent (dynamical QCD) masses 3 MeV m p /3 m.1 ev (may be additional dynamics, Majorana or Dirac)

8 8 The Weak Charged Current - Fermi theory incorporated in SM and made renormalizable - W -fermion interaction L f = q L i DqL + l L i DlL + ū R i Du R + d R i Dd R + ē R i De R+ R i DR q g L SU() τ i Wµq i L l g L τ i Wµl i L τ i W i = τ 3 W 3 + τ + W + + τ W W ± = W 1 iw, τ ± = τ 1 ± iτ L = g ( J µ W W µ + J µ W W + µ ) - Charge-raising current J µ W = 3 m=1 [ m γ µ (1 γ 5 )e m + ū mγ µ (1 γ 5 )d m] = ( 1 3 )γ µ (1 γ 5 )V l e µ τ + (ū c t)γ µ (1 γ 5 )V d s b = L γ µ V l e L + ū L γ µ V d L (V l = A L Ae L V P MNS, V = Au L Ad L = V CKM ) - For negligible masses define: ( 1 3 )V l ( e µ τ ) - Pure V A maximal P and C violation; CP conserved except for phases in V - V = A u L Ad L is (unitary) Cabibbo-Kobayashi-Maskawa (CKM) matrix from mismatch between weak and Yukawa interactions

9 9 - Cabibbo matrix if third family ignored ( cos θc sin θ V = c sin θ c cos θ c ) sin θ c.3 Cabibbo angle Good zeroth-order description since third family almost decouples General unitary : 1 angle and 3 phases; remove 3 (unobservable) external q L phases ( = 1 angle) ( ) (ū L c e iα u ( L) e γ µ cos θc sin θ ) ( ) ( ) c e iα d d L iαc sin θ c cos θ c e iαs s L }{{}}{{} (ū L c L ) d L s L - CKM matrix for 3 families involves 3 angles and 1 CP -violating phase (after removing 5 unobservable q L phases, = 3 + 1) (new interations involving q R could make observable) V = V ud V us V ub V cd V cs V cb V td V td V td 1 λ λ 3 λ 1 λ λ 3 λ 1 (λ sin θ c.3) V ij Third family nearly decoupled - Only V (and V l ) observable unless new interactions and individual A L s not observable) (A u,d,e, R - Extensive studies, especially in B decays, to test unitarity of V as probe of new physics and test origin of CP violation - Need additional source of CP breaking for baryogenesis

10 1 Effective zero- range 4-fermi interaction (Fermi theory) - For Q M W, neglect Q in W propagator L cc eff = ( g ) ( ) J µ gµ W Q MW J W g 8M W J µ W J W µ - Fermi constant: G F g 8MW - Muon lifetime: τ 1 = G F m5 µ 19π 3 (µ e µ e ) = 1 G F = GeV - Weak scale: = φ 46 GeV - Excellent description of β, K, hyperon, heavy quark, µ, and τ decays, µ e µ e, µ n µ p, µ N µ X - Full theory probed: e ± p ( ) e X at high energy (ERA) Electroweak radiative corrections (loop level) (Very important. Only calculable in full theory.) M KS M KL ; kaon CP violation; D D and B B mixing (loop level)

11 1.5 excluded area has CL >.95 excluded at CL > " $m d 1 sin! $m s & $m d.5 ' % K V ub /V cb " #! # # sol. w/ cos! < (excl. at CL >.95) % K C K M f i t t e r EPS 5 " (CKMFITTER group: & Quantum Electrodynamics (QED) - Incredibly successful (but a µ, muonic Lamb shift?) - Incorporated into standard model - Lagrangian: gg L = J µ g + g }{{ Q (cos θ W B µ + sin θ W Wµ 3 ) }{{}} A µ g sin θ W - Photon field: A µ = cos θ W B µ + sin θ W W 3 µ - Positron electric charge: e = g sin θ W, where tan θ W g /g

12 1 - Electromagnetic current: 3 [ J µ Q = 3ū mγ µ u m 1 ] 3 d mγ µ d m ē mγ µ e m m=1 3 [ = µ 3ūmγ u m 1 ] 3 d m γ µ d m ē m γ µ e m m=1 = r q r ψr γ µ ψ r (q r = electric charge in units of e ) - Electric charge: Q = T 3 + Y, where Y = weak hypercharge (y = coefficient of ig B µ in covariant derivatives) - Flavor diagonal: same form in weak and mass bases because fields which mix have same charge - Purely vector (parity conserving): L and R fields have same charge (q i = t 3 i + y i is the same for L and R fields, even though t 3 i and y i are not) The Weak Neutral Current - Prediction of SU() U(1) g L = + g J µ ( Z sin θw B µ + cos θ W W 3 µ) g = J µ Z cos θ Z µ W - Neutral current process and effective 4-fermi interaction for Q M Z

13 13 - Neutral current: J µ Z = m [ū ml γ µ u ml d mlγ µ d ml + mlγ µ ml ē mlγ µ e ml] = m sin θ W J µ Q [ūml γ µ u ml d ml γ µ d ml + ml γ µ ml ē ml γ µ ] e ml sin θ W J µ Q = r t 3 rl ψ r γ µ (1 γ 5 )ψ r sin θ W J µ Q (t 3 rl = + 1 [u, ], 1 [d, e] = q r y rl ) - Flavor diagonal: same form in weak and mass bases because fields which mix have same charges - GIM mechanism: c quark predicted so that s L could be in doublet to avoid unwanted flavor changing neutral currents (FCNC) at tree and loop level (c mass successfully predicted from M KS M KL ; c observed 1974) - Parity and charge conjugation violated but not maximally: first term is pure V A, second is V Effective 4-fermi interaction for Q M Z : L NC eff = G F J µ Z J Zµ - Coefficient same as WCC because G F = g 8M W = g + g 8M Z

14 14 Cross-section (pb) Z e + e hadrons CESR DORIS PEP W + W - 1 KEKB PEP-II PETRA TRISTAN SLC LEP I LEP II Centre-of-mass energy (GeV) 1. Electroweak model and constraints on new physics 15!"$)!"$'%!"$'(!"$''!"$'$ " WW (pb) 3 LEP PRELIMINARY 17//5 *+, $ θ ' (µ)!"$'!"$&%!"$&( -. /13 -. /43 *+,-!"$&'!"$&$!"$&!"$$% CD556E -F4<G 4@AB8 5617# 859 :4;<=>?,!"!!!#!"!!#!"!#!"# # #! #!! #!!! #!!!! µ!"#$%& 1 YFSWW/RacoonWW no ZWW vertex (Gentle) only # e exchange (Gentle) 16 18!s (GeV)

15 15 The Z, the W, and the Weak Neutral Current - Primary prediction and test of electroweak unification - WNC discovered 1973 (Gargamelle at CERN, PW at FNAL) - 7 s, 8 s: weak neutral current experiments (few %) Pure weak: N, e scattering Weak-elm interference in ed, e + e, atomic parity violation Model independent analyses (e, q, eq) SU() U(1) group/representations; t and τ exist; m t limit; hint for SUSY unification; limits on TeV scale physics - W, Z discovered directly 1983 (UA1, UA) - 9 s: Z pole (LEP, SLD),.1%; lineshape, modes, asymmetries; m t prediction - LEP : M W, iggs search, gauge self-interactions - Tevatron: m t, M W, iggs search; M prediction - 4th generation weak neutral current experiments (atomic parity (Boulder); e; N (NuTeV); polarized Møller asymmetry (SLAC))

16 16 SM correct and unique to zeroth approx. (gauge principle, group, representations) SM correct at loop level (renorm gauge theory; m t, α s, M ) TeV physics severely constrained (unification vs compositeness) Consistent with light elementary iggs Precise gauge couplings (SUSY gauge unification) Measurement Fit O meas O fit /σ meas 1 3 α (5) had (m Z ).758 ± m Z [GeV] ± Γ Z [GeV].495 ± σ had [nb] ± R l.767 ±.5.74 A,l fb.1714 ± A l (P τ ).1465 ± R b.169 ± R c.171 ± A,b fb.99 ± A,c fb.77 ± A b.93 ±..935 A c.67 ± A l (SLD).1513 ± sin θ lept eff (Q fb ).34 ± m W [GeV] ± Γ W [GeV].98 ±.48.9 m t [GeV] ± August LC (1): boson consistent with SM iggs at 15 GeV