Electroweak Theory: 3

Electroweak Theory: 3 Introduction QED The Fermi theory The standard model Precision tests CP violation; K and B systems Higgs physics Prospectus STIAS January, 2011 Paul Langacker IAS 55

References Slides at http://www.sns.ias.edu/ pgl/talks/ subject to revision P. Langacker, The Standard Model and Beyond, CRC Press, 2010 especially chapters 6, 7 P. Langacker, Introduction to the Standard Model and Electroweak Physics, [arxiv:0901.0241 [hep-ph]] TASI 2008 E.D. Commins and P.H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge, 1983 P. Renton, Electroweak Interactions, Cambridge, 1990 Precision Tests of the Standard Electroweak Model, ed. P. Langacker World, 1995 especially Fetscher and Gerber; Herczog; Deutsch and Quin K. Nakamura et al. [Particle Data Group], J. Phys. G 37, 075021 2010electroweak and other reviews STIAS January, 2011 Paul Langacker IAS 56

Previous Lecture The Weak Interactions of Hadrons Semi-Leptonic Processes in the Fermi Theory π and K Decays, and the Strong Interactions β Decay and Related Processes Charm c Quark and Third Family CKM Universality STIAS January, 2011 Paul Langacker IAS 57

This Lecture The Standard Electroweak Model Beyond the Fermi Theory Gauge Theories and the Standard Model The Standard Electroweak Model Spontaneous Symmetry Breaking and the Higgs Mechanism The Lagrangian after Symmetry Breaking STIAS January, 2011 Paul Langacker IAS 58

Beyond the Fermi Theory Fermi theory successfully describes large number of WCC processes However, not renormalizable Radiative corrections divergent but needed Unitarity violated at high energy E CM 1 TeV Loop processes such as K 0 K 0 mixing meaningless/incorrect More complete theory needed e e ν e ν e e e ν e ν e ν e ν e e e s d J µ J µ J µ J µ J µ J µ J µ J e ν e µ e ν e u u ν e ν e e e ν e ν e e e d s e e ν e ν e e e ν e ν e STIAS January, 2011 Paul Langacker IAS 59

Gauge Theories Standard Model is remarkably successful gauge theory of the microscopic interactions Gauge symmetry apparently massless spin-1 vector, gauge bosons Interactions group, representations, gauge coupling Like QED U1, but gauge self interactions for non-abelian Application to strong short range confinement Application to weak short range spontaneous symmetry breaking Higgs or dynamical Unique renormalizable field theory for spin-1 STIAS January, 2011 Paul Langacker IAS 60

Non-Abelian n non-interacting fermions of same mass m: iγ µ x m ψ µ a = 0, a = 1 n, invariant under global SUn group, ψ 1. ψ n expi N β i L i ψ 1. ψ n i=1. L i are n n generator matrices N = n 2 1; β i are real parameters [L i, L j ] = ic ijk L k c ijk are structure constants STIAS January, 2011 Paul Langacker IAS 61

Gauge local transformation: β i β i x iγ µ N x µδ ab g γ µ A i µ Li ab mδ ab ψ b = 0 i=1 Invariant under Φ ψ 1. ψ n Φ UΦ A µ L A µ L U A µ LU 1 + i g µu U 1 U e i β L 1 STIAS January, 2011 Paul Langacker IAS 62

Gauge invariance implies: N apparently massless gauge bosons A i µ Specified interactions up to gauge coupling g, group, representations, including self interactions A i µ ψ a igl i ab γµ ψ b g g 2 Generalize to other groups, representations, chiral L R Chiral Projections: ψ LR 1 Typeset by FoilTEX 2 1 γ 5ψ independent fields Chirality = helicity up to Om/E STIAS January, 2011 Paul Langacker IAS 63

The Standard Model Gauge group SU3 SU2 U1; gauge couplings g s, g, g u d L u d L u d L νe e u R u R u R ν er? d R d R d R e R L = left-handed, R = right-handed L SU3: u u u, d d d 8 gluons SU2: u L d L, ν el e L W ± ; phases W 0 U1: phases B Heavy families c, s, ν µ, µ, t, b, ν τ, τ STIAS January, 2011 Paul Langacker IAS 64

The Standard Electroweak Model Gauge part Field strength tensors L SU2 U1 = L gauge + L φ + L f + L Yukawa L gauge = 1 4 F i µν F µνi 1 4 B µνb µν B µν = µ B ν ν B µ F i µν = µ W i ν νw i µ gɛ ijkw j µ W k ν, i = 1 3 gg is SU2 U1 gauge coupling; ɛ ijk is totally antisymmetric symbol Three and four-point self-interactions for the W i g g 2 B and W 3 mix to form γ, Z STIAS January, 2011 Paul Langacker IAS 65

U1: Φ j expig y j βφ j, y j = q j t 3 j = weak hypercharge Scalar part where φ = φ + φ 0 Gauge covariant derivative: L φ = D µ φ D µ φ V φ is the complex Higgs doublet with y φ = 1/2 D µ φ = µ + ig τ i 2 W i µ + ig 2 B µ φ where τ i are the Pauli matrices Three and four-point interactions between gauge and scalar fields g g 2 STIAS January, 2011 Paul Langacker IAS 66

Higgs potential V φ = +µ 2 φ φ + λφ φ 2 φ φ = φ }{{} φ + φ + + φ 0 φ 0 Allowed by renormalizability and gauge invariance Spontaneous symmetry breaking for µ 2 < 0 Vacuum stability: λ > 0 Quartic self-interactions φ 0 φ + λ φ 0 φ STIAS January, 2011 Paul Langacker IAS 67

Fermion part L f = F m=1 q 0 ml i Dq0 ml + l 0 ml i Dl0 ml + ū 0 mr i Du0 mr + d 0 mr i Dd0 mr + ē0 mr i De0 mr + ν0 mr i Dν0 mr L-doublets R-singlets u q 0 ml = 0 m d 0 m L l 0 ml = ν 0 m e 0 m u 0 mr, d0 mr, e 0 mr, ν0 mr? L F 3 families; m = 1 F = family index; 0 = weak eigenstates definite SU2 rep., mixtures of mass eigenstates flavors; quark color indices α = r, g, b suppressed e.g., u 0 mαl. Can add gauge singlet ν 0 mr for Dirac neutrino mass term STIAS January, 2011 Paul Langacker IAS 68

Different chiral L and R representations lead to parity and charge conjugation violation maximal for SU2 Fermion mass terms forbidden by chiral symmetry Triangle anomalies absent for chosen hypercharges and 3 colors includes quark-lepton cancellations exercize j i k STIAS January, 2011 Paul Langacker IAS 69

Gauge covariant derivatives D µ q 0 ml = µ + ig 2 τ i W i µ + ig 6 B µ q 0 ml D µ l 0 ml = µ + ig 2 τ i W i µ ig 2 B µ D µ u 0 mr = µ + i 2 3 g B µ u 0 mr D µ d 0 mr = µ i g 3 B µ d 0 mr D µ e 0 mr = µ ig B µ e 0 mr D µ ν 0 mr = µ ν 0 mr l 0 ml Read off W and B couplings to fermions W i µ i g 2 τ i 1 γ5 γ µ 2 B µ ig 1 γ5 yγ µ 2 STIAS January, 2011 Paul Langacker IAS 70

Spontaneous Symmetry Breaking Higgs 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 Higgs field v = 0 φ 0 = constant, φ = φ + φ 0 µ v 0 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 STIAS January, 2011 Paul Langacker IAS 71

SU2 U1: introduce Hermitian basis φ = φ + φ 0 = 2 1 φ 1 + iφ 2 1 2 φ 3 + iφ 4 where φ i = φ i. 2 V φ = 1 2 µ2 4 i=1 φ 2 i + 1 4 λ 4 i=1 φ 2 i is O4 invariant. w.l.o.g. choose 0 φ i 0 = 0, i = 1, 2, 4 and 0 φ 3 0 = ν V φ V v = 1 2 µ2 ν 2 + 1 4 λν4 STIAS January, 2011 Paul Langacker IAS 72

For µ 2 < 0, minimum at V ν = νµ 2 + λν 2 = 0 ν = µ 2 /λ 1/2 SSB for µ 2 = 0 also; must consider loop corrections φ 1 2 0 ν v the generators L 1, L 2, and L 3 Y spontaneously broken, L 1 v 0, etc L i = τ i 2, Y = 1 2 I 1 0 Qv = L 3 + Y v = 0 0 SU2 U1 Y U1 Q v = 0 U1 Q unbroken STIAS January, 2011 Paul Langacker IAS 73

Rewrite Lagrangian in New Vacuum Physical Higgs scalar oscillations around minimum: M H = 2λν Higgs covariant kinetic energy terms: D µ φ D µ φ = 1 2 0 ν [ g 2 τ i W i µ + g 2 B µ]2 0 ν + H terms M 2 W W +µ W µ + M 2 Z 2 Zµ Z µ + H kinetic energy and gauge interaction terms W e L W e L ν ν g 2 M W = gν h ν e g 2 2 h e ν W ν e R m e = h eν W 2 ν e R STIAS January, 2011 Paul Langacker IAS 74

Mass eigenstate bosons: W, Z, and A photon W ± = 1 2 W 1 iw 2 Z = sin θ W B + cos θ W W 3 A = cos θ W B + sin θ W W 3 Weak angle: tan θ W g /g Masses: M W = gν 2, M Z = g 2 + g 2 ν 2 = M W cos θ W, M A = 0 Goldstone scalars eaten longitudinal components of W ±, Z STIAS January, 2011 Paul Langacker IAS 75

Will show: Fermi constant G F / 2 g 2 /8M 2 W G F = 1.1663645 10 5 GeV 2 from muon lifetime Electroweak scale: ν = 2M W /g 2G F 1/2 246 GeV Will show: g = e/ sin θ W α = e 2 /4π 1/137.036 M W = M Z cos θ W = gν 2 πα/ 2G F 1/2 sin θ W Weak neutral current: sin 2 θ W 0.23 M W 78 GeV, and M Z 89 GeV increased by 2 GeV by loop corrections Discovered at CERN: UA1 and UA2, 1983 STIAS January, 2011 Paul Langacker IAS 76

Fermion Masses and Mixings Yukawa Higgs-fermion interaction + analogous u, e, ν terms L Y uk = = F Γ d mn q 0 ml φd0 nr + h.c. m,n=1 F m,n=1 Γ d mn [ū0 ml φ+ d 0 nr + d 0 ml φ0 dnr] 0 + h.c. u 0 ml d 0 ml φ + iγ d mn φ 0 iγ d mn d 0 nr d 0 nr STIAS January, 2011 Paul Langacker IAS 77

For φ 1 2 L Y uk = i 0 ν + H F M d d 0 mn ml d0 nr m,n=1 unitary gauge m i dil d ir 1 + H ν 1 + H ν + h.c. + h.c. = i m i di d i 1 + H ν with M d Γ d ν/ 2 d i = d il + d ir are mass eigenstates of mass m i For F = 3 A d L M d A d R = m d 0 0 0 m s 0 0 0 m b, d s b L,R = A d L,R }{{} unitary d0 1 d 0 2 d 0 3 L,R Higgs H couplings to fermions are diagonal in flavor and mass STIAS January, 2011 Paul Langacker IAS 78

The Weak Charged Current Fermi Theory incorporated in SM and made renormalizable W -fermion interaction L = g 2 2 J µ W W µ + J µ W W + µ Charge-raising current ignoring ν masses J µ W = F m=1 0 [ ν m γµ 1 γ 5 e 0 m + ] ū0 m γµ 1 γ 5 d 0 m = ν e ν µ ν τ γ µ 1 γ 5 e µ τ + ū c tγ µ 1 γ 5 V d s b STIAS January, 2011 Paul Langacker IAS 79

Ignore ν masses for now Pure V A maximal P and C violation; CP conserved except for phases in V V = A u L Ad L is F F unitary Cabibbo-Kobayashi-Maskawa CKM matrix from mismatch between weak and Yukawa interactions Cabibbo matrix for F = 2 V = cos θc sin θ c sin θ c cos θ c sin θ c 0.23 Cabibbo angle Good zeroth-order description since third family almost decouples General unitary 2 2: 1 angle and 3 unobservable q L phases STIAS January, 2011 Paul Langacker IAS 80

CKM matrix for F = 3 involves 3 angles and 1 CP -violating phase after removing unobservable q L phases 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 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 STIAS January, 2011 Paul Langacker IAS 81

Effective zero- range 4-fermi interaction Fermi theory For Q M W, neglect Q 2 in W propagator L cc eff = g 2 2 2 J µ W gµν Q 2 M 2 W J ν W g2 8M 2 W J µ W J W µ Fermi constant: G F 2 g2 8M 2 W = 1 2ν 2 Muon lifetime: τ 1 = G2 F m5 µ 192π 3 G F = 1.17 10 5 GeV 2 Weak scale: ν = 2 0 φ 0 0 246 GeV Excellent description of β, K, hyperon, heavy quark, µ, and τ decays, ν µ e µ ν e, ν µ n µ p, ν µ N µ X STIAS January, 2011 Paul Langacker IAS 82

Full theory probed: e ± p ν e X at high energy HERA Electroweak radiative corrections loop level Very important. Only calculable in full theory. M KS M KL, kaon CP violation, B B mixing loop level 1.5 excluded area has CL > 0.95 excluded at CL > 0.95 " $m d 1 0.5 sin 2! $m s & $m d CKMFITTER group: http://ckmfitter.in2p3.fr/ ' 0 % K V ub /V cb " #! # -0.5-1 # sol. w/ cos 2! < 0 excl. at CL > 0.95 % K C K M f i t t e r EPS 2005 " -1.5-1 -0.5 0 0.5 1 1.5 2 & STIAS January, 2011 Paul Langacker IAS 83

Quantum Electrodynamics QED Incorporated into standard model Lagrangian: L = gg µ g2 + g 2J Q cos θ W B µ + sin θ W W 3 µ 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 STIAS January, 2011 Paul Langacker IAS 84

Electromagnetic current: F J µ Q = [ 2 m=1 F = m=1 3ū0 m γµ u 0 m 1 3 [ 2 µ u m 1 3ūmγ 3 d 0 m γµ d 0 m ē0 m γµ e 0 m ] d m γ µ d m ē m γ µ e m ] Electric charge: Q = T 3 + Y, where Y = weak hypercharge 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 STIAS January, 2011 Paul Langacker IAS 85

Prediction of SU2 U1 The Weak Neutral Current g2 + g 2 L = J µ Z sin θ W B µ + cos θ W W 3 µ 2 g = J µ Z 2 cos θ Z µ W Neutral current process and effective 4-fermi interaction for Q M Z STIAS January, 2011 Paul Langacker IAS 86

Neutral current: J µ Z = [ū0 ml γµ u 0 ml d 0 ml γµ d 0 ml + ν0 ml γµ ν 0 ml ] ē0 ml γµ e 0 ml m = m 2 sin 2 θ W J µ Q [ūml γ µ u ml d ml γ µ d ml + ν ml γ µ ν ml ē ml γ µ ] e ml 2 sin 2 θ W J µ Q Flavor diagonal: Same form in weak and mass bases because fields which mix have same charge 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 Parity and charge conjugation violated but not maximally: first term is pure V A, second is V STIAS January, 2011 Paul Langacker IAS 87

Effective 4-fermi interaction for Q 2 M 2 Z : L NC eff = G F 2 J µ Z J Zµ Coefficient same as WCC because G F 2 = g2 8M 2 W = g2 + g 2 8M 2 Z STIAS January, 2011 Paul Langacker IAS 88

Cross-section pb 10 5 10 4 10 3 Z e + e hadrons 10 2 CESR DORIS PEP W + W - 10 KEKB PEP-II PETRA TRISTAN SLC LEP I LEP II 0 20 40 60 80 100 120 140 160 180 200 220 Centre-of-mass energy GeV 10. Electroweak model and constraints on new physics 15!"$!"$'%!"$'!"$''!"$'$ " WW pb 30 20 LEP PRELIMINARY 17/02/2005 *+, $ θ ' µ!"$'!"$&%!"$&!"$&'!"$&$!"$&!"$$% -. /0123 -. /43 CD556E -F4<G 4@AB8 ν*+,- 5617# 859 :4;<=>?,!"!!!#!"!!#!"!#!"# # #! #!! #!!! #!!!! µ!"#$%& 10 0 YFSWW/RacoonWW no ZWW vertex Gentle only # e exchange Gentle 160 180 200!s GeV STIAS January, 2011 Paul Langacker IAS 89

The Z, the W, and the Weak Neutral Current Primary prediction and test of electroweak unification WNC discovered 1973 Gargamelle at CERN, HPW at FNAL 70 s, 80 s: weak neutral current experiments few % Pure weak: νn, νe scattering Weak-elm interference in ed, e + e, atomic parity violation Model independent analyses νe, νq, eq SU2 U1 group/representations; t and ν τ exist; m t limit; hint for SUSY unification; limits on TeV scale physics W, Z discovered directly 1983 UA1, UA2 STIAS January, 2011 Paul Langacker IAS 90

90 s: Z pole LEP, SLD, 0.1%; lineshape, modes, asymmetries LEP 2: M W, Higgs search, gauge self-interactions Tevatron: m t, M W, Higgs search 4th generation weak neutral current experiments atomic parity Boulder; νe; νn NuTeV; polarized Møller asymmetry SLAC STIAS January, 2011 Paul Langacker IAS 91

SM correct and unique to zeroth approx. gauge principle, group, representations SM correct at loop level renorm gauge theory; m t, α s, M H TeV physics severely constrained unification vs compositeness Consistent with light elementary Higgs Precise gauge couplings SUSY gauge unification Measurement Fit O meas O fit /σ meas 0 1 2 3 α 5 had m Z 0.02758 ± 0.00035 0.02768 m Z [GeV] 91.1875 ± 0.0021 91.1874 Γ Z [GeV] 2.4952 ± 0.0023 2.4959 σ 0 had [nb] 41.540 ± 0.037 41.478 R l 20.767 ± 0.025 20.742 A 0,l fb 0.01714 ± 0.00095 0.01645 A l P τ 0.1465 ± 0.0032 0.1481 R b 0.21629 ± 0.00066 0.21579 R c 0.1721 ± 0.0030 0.1723 A 0,b fb 0.0992 ± 0.0016 0.1038 A 0,c fb 0.0707 ± 0.0035 0.0742 A b 0.923 ± 0.020 0.935 A c 0.670 ± 0.027 0.668 A l SLD 0.1513 ± 0.0021 0.1481 sin 2 θ lept eff Q fb 0.2324 ± 0.0012 0.2314 m W [GeV] 80.399 ± 0.023 80.379 Γ W [GeV] 2.098 ± 0.048 2.092 m t [GeV] 173.1 ± 1.3 173.2 August 2009 0 1 2 3 STIAS January, 2011 Paul Langacker IAS 92