Introduction to the Standard Model

Transcription

1 Introduction to the Standard Model Origins of the Electroweak Theory Gauge Theories The Standard Model References: 2008 TASI lectures: arxiv: [hep-ph] and The Standard Model and Beyond, CRC Press

2 The Weak Interactions Radioactivity (Becquerel, 1896) β decay appeared to violate energy (Meitner, Hahn; 1911) Neutrino hypothesis (Pauli, 1930) (Reines, Cowan; 1953) ν µ (Lederman, Schwartz, Steinberger; 1962) ν τ (DONUT, 2000) (τ, 1975)

3 Fermi theory (1933) Loosely like QED, but zero range (non-renormalizable) and nondiagonal (charged current) p H G F J µ J µ J µ pγ µn+ γ µ [n p, ] J µ J µ J µ J µ J µ nγ µ p+ēγ µ [p n, ( )] n G F GeV 2 [Fermi constant] e p g g g W W + g

4 Fermi theory modified to include parity violation (V A) (Lee, Yang; Wu; Feynman-Gell-Mann) µ, τ decay strangeness (Cabibbo) quark model heavy quarks and CP violation (CKM) ν mass and mixing Fermi theory correctly describes (at tree level) Nuclear/neutron β decay/inverse (n p ; p n) µ, τ decays (µ ν µ ; τ µ ν µ ν τ, ν τ π, ) π, K decays (π + µ + ν µ, π 0 e + ; K + µ + ν µ, π 0 e +, π + π 0 ) hyperon decays (Λ pπ ; Σ nπ ; Σ + Λe + ) heavy quark decays (c se + ; b cµ ν µ, cπ ) ν scattering (ν µ µ ; ν µ n µ p }{{} X elastic deep inelastic ; ν µ N µ }{{ ) }

5 J µ Fermi theory violates unitarity at high energy (non-renormalizable) (s E 2 J µ J CM µ ν ) e σ( ) G2 F s π pure S-wave unitarity: σ < 16π s g W + g Fermi theory: divergent g integrals iltex n 1 e fails for E CM 2 π G F Born not unitary; often restored by H.O.T. p ( ) ( ) k k d 4 k J µ J µ k 2 k GeV J µ J µ e p

6 J µ J µ J µ J µ Intermediate vector boson theory (Yukawa, 1935; Schwinger, 1957) n p e J µ J µ G F g g g W 2 W + g2 8Mg W 2 for M W Q n e Typeset by FoilTEX 1 g W + g no longer pure S-wave better behaved 1

7 W W + W but, e W + + d W + W s violates g K 0 unitarity for s 500 GeV g g W 0 polarization (non-renormalizable) g e + e + d introduce W 0 to cancel fixes W 0 W + W and e + W 0 vertices requires [ J, J ] J 0 (like SU(2)) e + ɛ µ k µ /M W for longitudinal Typeset by FoilTEX 2 not realistic W W + K 0 g g W 0 Z W W + g s g e + d d K 0 K 0 Z

8 Glashow model (1961) (W ±, Z, γ, but no mechanism for M W,Z ) Weinberg-Salam (1967): Higgs mechanism M W,Z Renormalizable (1971) ( t Hooft, ) Flavor changing neutral currents (FCNC) very large K 0 K 0 mixing s K 0 d s K 0 d GIM mechanism (c quark) (1970) Z u W W u c discovered (1974) d K 0 s d K 0 s

9 Weak neutral current (1973) QCD (1970 s) W, Z (1983) Precision tests ( ) CKM unitarity ( 1995-) t quark (1995) ν mass ( ) Measurement Fit O meas O fit /σ meas α (5) had (m Z ) ± m Z [GeV] ± Γ Z [GeV] ± σ 0 had [nb] ± R l ± A 0,l fb ± A l (P τ ) ± R b ± R c ± A 0,b fb ± A 0,c fb ± A b ± A c ± A l (SLD) ± sin 2 θ lept eff (Q fb ) ± m W [GeV] ± Γ W [GeV] ± m t [GeV] ±

10 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 (U(1)), 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

11 QED Free electron equation, ( iγ µ ) x m ψ = 0, µ is invariant under U(1) (phase) transformations, ( iγ µ ) x m ψ = 0, where ψ iβ ψ µ Not invariant under local (gauge) transf., ψ ψ iβ(x) ψ, x ( x, t)

12 Introduce vector field A µ ( A, φ): ( iγ µ ) A x µ+eγµ µ m ψ = 0, (e > 0 is gauge coupling) is invariant under ψ iβ(x) ψ, A µ A µ 1 β e x µ Quantization of A µ massless gauge boson Gauge invariance γ, long range force, prescribed (up to e) amplitude for emission/absorption e γ e p p

13 Non-Abelian n non-interacting fermions of same mass m: ( iγ µ ) x m ψ µ a = 0, a = 1 n, invariant under (global) SU(n) group, ψ 1. ψ n exp(i 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)

14 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)

15 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 L(R) 1 2 (1 γ 5)ψ (Chirality = helicity up to O(m/E)) Typeset by FoilTEX

16 The Standard Model Gauge group SU(3) SU(2) U(1); gauge couplings g s, g, g ( u d ) L ( u d ) L ( u d ) L ( νe ) u R u R u R R (?) d R d R d R R ( L = left-handed, R = right-handed) L SU(3): u u u, d d d (8 gluons) SU(2): u L d L, L L (W ± ); phases (W 0 ) U(1): phases (B) Heavy families (c, s, ν µ, µ ), (t, b, ν τ, τ )

17 Quantum Chromodynamics (QCD) L SU(3) = 1 4 F i µν F iµν + r q α r i Dβ α q rβ F 2 term leads to three and four-point gluon self-interactions. F i µν = µg i ν νg i µ g sf ijk G j µ Gk ν is field strength tensor for the gluon fields G i µ, i = 1,, 8. g s = QCD gauge coupling constant. No gluon masses. Structure constants f ijk (i, j, k = 1,, 8), defined by [λ i, λ j ] = 2if ijk λ k where λ i are the Gell-Mann matrices.

18 λ i = ( τ i ), i = 1, 2, 3 λ 4 = λ 6 = λ 8 = λ 5 = λ 7 = 0 0 i i i 0 i 0 The SU(3) (Gell-Mann) matrices.

19 Quark interactions given by q α r i Dβ α q rβ q r = r th quark flavor; α, β = 1, 2, 3 are color indices Gauge covariant derivative D µβ α = (Dµ ) αβ = µ δ αβ + ig s G iµ L i αβ, for triplet representation matrices L i = λ i /2.

20 Quark color interactions: u α Diagonal in flavor Off diagonal in color Purely vector (parity conserving) G i µ i g s 2 λi αβ γµ u β Bare quark mass allowed by QCD, but forbidden by chiral symmetry of L SU(2) U(1) (generated by spontaneous symmetry breaking) Additional ghost and gauge-fixing terms Can add (unwanted) CP-violating term L θ = θg2 s F i F 32π 2 µν iµν, F iµν 1 2 ɛµναβ Fαβ i Typeset by FoilTEX

21 QCD now very well established Short distance behavior (asymptotic freedom) Confinement, light hadron spectrum (lattice) g s = O(1) (α s (M Z ) = gs 2 /4π 0.12) Strength + gluon self-interactions confinement Yukawa model dipole-dipole Approximate global SU(3) L SU(3) R (π, K, η are pseudo-goldstone bosons) symmetry and breaking Unique field theory of strong interactions

22 Quantum Chromodynamics (QCD) Modern theory of the strong interactions Average Hadronic Jets e + e - rates Photo-production Fragmentation Z width ep event shapes Polarized DIS Deep Inelastic Scattering (DIS)! decays Spectroscopy (Lattice) " decay # s (M Z )! s (µ) µ GeV 22nd Henry Primakoff Lecture Paul Langacker (3/1/2006)

23 The Electroweak Sector Gauge part Field strength tensors L SU(2) U(1) = 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 g(g ) is SU(2) (U(1)) gauge coupling; ɛ ijk is totally antisymmetric symbol Three and four-point selfinteractions for the W i g g 2 B and W 3 will mix to form γ, Z

24 U(1): Φ j exp(ig y j β)φ 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

25 Higgs potential V (ϕ) = +µ 2 ϕ ϕ + λ(ϕ ϕ) 2 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 Typeset by FoilTEX

26 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 SU(2) 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

27 Different (chiral) L and R representations lead to parity and charge conjugation violation (maximal for SU(2)) Fermion mass terms forbidden by chiral symmetry Triangle anomalies absent for chosen hypercharges (quark-lepton cancellations)

28 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 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

29 Spontaneous Symmetry Breaking (Higgs mechanism) 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 W e L e L ν M W = gν 2 ν g 2 ν g 2 h e m e = h eν 2 ν ν h e ν W W e R e R

30 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 )

31 Will show: Fermi constant G F / 2 g 2 /8M 2 W (G F = (5) 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/ ) 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

32 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 µ ν τ )γ µ (1 γ 5 ) e µ τ + (ū c t)γ µ (1 γ 5 )V d s b.

33 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 0.22 Cabibbo angle Good zeroth-order description since third family almost decouples

34 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

35 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 = GeV 2 Weak scale: ν = 2 0 ϕ GeV Excellent description of β, K, hyperon, heavy quark, µ, and τ decays, ν µ e µ, ν µ n µ p, ν µ N µ X

36 Full theory probed: e ± p ( ) 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 sin 2! $m s & $m d (CKMFITTER group: ' 0 % K V ub /V cb " #! # # sol. w/ cos 2! < 0 (excl. at CL > 0.95) % K C K M f i t t e r EPS 2005 " &

37 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

38 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)

39 Experiment Value of α 1 Precision e ae = (g 2)/ (94) [ ] h/m (Rb, Cs) (69) [ ] 0.33 ± 0.69 Quantum Hall (25) [ ] 3.3 ± 2.5 h/m (neutron) (28) [ ] 8.0 ± 2.8 γ p, 3 He (J. J.) (43) [ ] 12.2 ± 4.3 µ + hyperfine (80) [ ] 2.0 ± 8.0 Spectacularly successful: Most precise: e anomalous magnetic moment α Many low energy tests to few 10 8 m γ < ev, Running α(q 2 ) observed q γ < e Muon g 2 sensitive to new physics. Anomaly?

40 Prediction of SU(2) U(1) 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

41 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

42 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

43 Cross-section (pb) Z e + hadrons CESR DORIS PEP W + W - 10 KEKB PEP-II PETRA TRISTAN SLC LEP I LEP II Centre-of-mass energy (GeV) Tevatron " WW (pb) LEP PRELIMINARY 17/02/2005 sin 2! W (µ) Q W (APV) Q W (e) MOLLER "-DIS Qweak e-dis LEP 1 SLC YFSWW/RacoonWW no ZWW vertex (Gentle) only # e exchange (Gentle) µ [GeV] !s (GeV)

44 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 +, atomic parity violation Model independent analyses (νe, νq, eq) SU(2) U(1) 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)

45 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))

46 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 α (5) had (m Z ) ± m Z [GeV] ± Γ Z [GeV] ± σ 0 had [nb] ± R l ± A 0,l fb ± A l (P τ ) ± R b ± R c ± A 0,b fb ± A 0,c fb ± A b ± A c ± A l (SLD) ± sin 2 θ lept eff (Q fb ) ± m W [GeV] ± Γ W [GeV] ± m t [GeV] ± August