Introduction to the Standard Model

Transcription

1 Introduction to the Standard Model Origins of the Electroweak Theory Gauge Theories The Standard Model Lagrangian Spontaneous Symmetry Breaking The Gauge Interactions Problems With the Standard Model ( Structure Of The Standard Model, hep-ph/ ) Lecture 1 1

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) Lecture 1 2

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 J µ J µ pγ µn+ γ µ [n p, ] J µ J µ n p J µ nγ µ p+ēγ µ [p n, ( )] G F GeV 2 e [Fermi constant] g g g W g W + Lecture 1 3

4 Fermi theory modified to include µ, τ decay strangeness (Cabibbo) quark model heavy quarks (CKM) ν mass and mixing parity violation (V A) (Lee, Yang; Wu; Feynman-Gell-Mann) Fermi theory correctly describes (at tree level) Nuclear/neutron β decay (n p ) 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 {z } X elastic deep inelastic ; ν µ N µ {z ) } Lecture 1 4

5 Fermi theory violates unitarity at high energy (non-renormalizable) σ( ) G2 F s π, s E2 CM J µ J µ J µ pure S-wave unitarity: σ < 16π s fails for E CM 2 π G F e 500 GeV g Born not unitary; often restored by H.O.T. g p W + k 2 e m 2 e k 2 n iltex 1 g Fermi theory: divergent integrals J µ J µ d 4 k k + m e k J µ J µ e p Lecture 1 5

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 Lecture 1 6

7 W W + W but, W e + W d + 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) U(1)) 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 Lecture 1 7

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) W W + very large K 0 K 0 mixing W W + g d K 0 s GIM mechanism (c quark) (1970) g g W 0 Z c discovered (1974) e + g e + d K 0 s Lecture 1 8

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 /" meas #$ (5) had (m Z ) ± m Z [GeV] ± % Z [GeV] ± " 0 had [nb] ± R l ± A 0,l fb ± A l (P & ) ± R b ± R c ± A 0,b fb ± A 0,c fb ± A b ± A c ± A l (SLD) ± sin 2 ' lept eff (Q fb ) ± m W [GeV] ± % W [GeV] ± m t [GeV] ± Lecture 1 9

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 Lecture 1 10

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) Lecture 1 11

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 Lecture 1 12

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) Lecture 1 13

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) Lecture 1 14

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 Lecture 1 15

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 ) Lecture 1 16

17 Quantum Chromodynamics (QCD) L SU(3) = 1 4 F i µν F iµν + r q rα i D α β qβ r 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 where λ i are the Gell-Mann matrices. [λ i, λ j ] = 2if ijk λ k Lecture 1 17

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

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

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 Lecture 1 20 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 Lecture 1 21

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) Lecture 1 22

23 The Electroweak Sector Gauge part Field strength tensors L = L SU(2) U(1) 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 Lecture 1 23

24 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. 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 Lecture 1 24

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 Lecture 1 25 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 Lecture 1 26

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 Lecture 1 27

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 Lecture 1 28

29 Yukawa couplings (couple L to R) L Yukawa = F m,n=1 [ Γ u mn q0 ml ϕu0 mr + Γd mn q0 ml ϕd0 nr + Γ e mn l 0 mn ϕe0 nr (+Γν mn l 0 ml ϕν0 mr )] + H.C. Γ mn are completely arbitrary Yukawa matrices, which determine fermion masses and mixings d, ( e terms ) require doublet ϕ = ϕ + ϕ 0 with Y ϕ = 1/2 u (and( ν) terms ) require doublet Φ 0 Φ = Φ with Y Φ = 1/2 φ Γ mn ψ ml ψ nr Lecture 1 29

30 ( In SU(2) the 2 and 2 are similar ϕ iτ 2 ϕ = ϕ 0 ϕ transforms as a 2 with Y ϕ = 1 only one doublet needed. 2 ) Does not generalize to SU(3), most extra U(1), supersymmetry, SO(10) etc need two doublets. (Does generalize to SU(2) L SU(2) R U(1) and SU(5)) Lecture 1 30

31 Spontaneous Symmetry Breaking Gauge invariance implies massless gauge bosons and fermions Weak interactions short ranged spontaneous symmetry breaking for mass; also for fermions Color confinement for QCD gluons remain massless Allow classical (ground state) expectation value for Higgs field v = 0 ϕ 0 = constant µ v 0 increases energy, but important for monopoles, strings, domain walls, phase transitions (e.g., EWPT, baryogenesis) Lecture 1 31

32 Minimize V (v) to find v and quantize ϕ = ϕ v SU(2) U(1): 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 λ ( 4 i=1 ϕ 2 i is O(4) 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 λν4 Lecture 1 32

33 For µ 2 < 0, minimum at V (ν) = ν(µ 2 + λν 2 ) = 0 ν = ( µ 2 /λ ) 1/2 SSB for µ 2 = 0 also; must consider loop corrections ( ) ϕ v the generators L ν 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 = v = 0 U(1) 0 0 Q unbroken SU(2) U(1) Y U(1) Q Lecture 1 33

34 Quantize around classical vacuum Kibble transformation: introduce new variables ξ i for rolling modes ϕ = 1 ( ) e i P ξ i L i 0 2 ν + H H = H is the Higgs scalar No potential for ξ i symmetry massless Goldstone bosons for global Disappear from spectrum for gauge theory ( eaten ) Display particle content in unitary gauge ϕ ϕ = i P ξ i L i ϕ = 1 2 ( 0 ν + H + corresponding transformation on gauge fields ) Lecture 1 34

35 Rewrite Lagrangian in New Vacuum 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 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 Lecture 1 35

36 Masses M W = gν 2, M Z = g 2 + g 2ν 2 = M W cos θ W, M A = 0 (Goldstone scalar transformed into longitudinal components of W ±, Z) Weak angle: tan θ W g /g Will show: Fermi constant G F / 2 g 2 /8M 2 W, where G F = (1) 10 5 GeV 2 from muon lifetime Electroweak scale ν = 2M W /g ( 2G F ) 1/2 246 GeV Lecture 1 36

37 Will show: g = e/ sin θ W, where α = e 2 /4π 1/ M W = M Z cos θ W (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 Current: M Z = ± M W = ± Lecture 1 37

38 The Higgs Scalar H Gauge interactions: ZZH, ZZH 2, W + W H, W + W H 2 ϕ 1 2 ( 0 ν + H ) L ϕ = (D µ ϕ) D µ ϕ V (ϕ) ( = 1 2 ( µh) 2 + M 2 W W µ+ W µ 1 + H ν ( M 2 Z Zµ Z µ 1 + H ) 2 V (ϕ) ν ) 2 Lecture 1 38

39 Higgs potential: V (ϕ) = +µ 2 ϕ ϕ + λ(ϕ ϕ) 2 µ4 4λ µ2 H 2 + λνh 3 + λ 4 H4 Fourth term: Quartic self-interaction Third: Induced cubic self-interaction Second: (Tree level) H mass-squared, M H = 2µ 2 = 2λν λ λ ν Lecture 1 39

40 No a priori constraint on λ except vacuum stability (λ > 0 0 < M H < ), but t quark loops destabilize vacuum unless M > H 130 GeV (Depends on Λ. Doesn t hold in supersymmetry) Strong coupling for λ > 1 M H > 1 TeV Triviality: running λ should not diverge below scale Λ at which theory breaks down M H < { O(200) GeV, Λ MP = G 1/2 O(750) GeV, Λ 2M H N GeV Experimental bound (LEP 2), e + Z ZH M H GeV at 95% cl (can evade with singlet or in MSSM) Hint of signal at 115 GeV Indirect (precision tests): M H < 189 GeV, 95% cl MSSM: much of parameter space has standard-like Higgs with M H < 130 GeV > Lecture 1 40

41 Theoretical M H limits, Hambye and Riesselmann, hep-ph/ Lecture 1 41

42 1000 all data (90% CL) M H [GeV] Γ Ζ, σ had, R l, R q asymmetries low-energy M W m t excluded m t [GeV] Lecture 1 42

43 Decays: H bb dominates for M < H 2M W (H W + W, ZZ dominate when allowed because of larger gauge coupling) Production: LEP: Higgstrahlung (e + Z ZH) Tevatron, LHC: GG-fusion (GG H via top loop), WW fusion (W W H), or associated production ( qq W H, ZH) Lecture 1 43

44 First term in V : vacuum energy 0 V 0 = µ 4 /4λ No effect on microscopic interactions, contribution to cosmological constant but gives negative Λ SSB = 8πG N 0 V Λ obs Require fine-tuned cancellation Λ cosm = Λ bare + Λ SSB Also, QCD contribution from SSB of global chiral symmetry Lecture 1 44

45 Yukawa Interactions L Yukawa F m,n=1 ū 0 ml Γu mn = ū 0 L (M u + h u H) u 0 R ( ν + H 2 ) u 0 mr + (d, e) terms + H.C. + (d, e, ν) terms + H.C. u 0 L = ( u 0 1L u0 2L u0 F L) T is F -component column vector M u is F F fermion mass matrix M u mn = Γu mn ν/ 2 (need not be Hermitian, diagonal, symmetric, or even square) h u = M u /ν = gm u /2M W is the Yukawa coupling matrix Lecture 1 45

46 Diagonalize M by separate unitary transformations A L ((A L = A R ) for Hermitian M) A u L M u A u R = M u D = m u m c m t is diagonal matrix of physical masses of the charge 2 3 Similarly and A R quarks. (may also be Majorana masses for ν R ) A d L M d A d R = M d D A e L M e A e R = M e D (A ν L M ν A ν R = M ν D ) Find A L and A R by diagonalizing Hermitian matrices MM and M M, e.g., A L MM A L = M 2 D Lecture 1 46

47 Mass eigenstate fields u L = A u L u0 L = (u L c L t L ) T u R = A u R u0 R = (u R c R t R ) T d L,R = A d L,R d0 L,R = (d L,R s L,R b L,R ) T e L,R = A e L,R e0 L,R = (e L,R µ L,R τ L,R ) T ν L,R = A ν L,R ν0 L,R = (ν 1L,R ν 2L,R ν 3L,R ) T (For m ν = 0 or negligible, define ν L = A e L ν0 L, so that ν i, ν µ, ν τ are the weak interaction partners of the e, µ, and τ.) Lecture 1 47

48 Typical estimates: m u = MeV, m d = 4 8 MeV, m s = MeV, m c 1.3 GeV, m b 4.2 GeV, m t = ± 1.8 GeV Implications for global SU(3) L SU(3) R of QCD These are current quark masses. M i = m i + M dyn, M dyn Λ MS 300 MeV from chiral condensate 0 qq 0 = 0 m t is pole mass; others, running masses at m or at 2 GeV 2 Lecture 1 48

49 Yukawa couplings of Higgs to fermions L Yukawa = i ψ i ( m i gm ) i H ψ i 2M W Coupling gm i /2M W is flavor diagonal and small except t quark H bb dominates for M H < 2M W (H W + W, ZZ dominate when allowed because of larger gauge coupling) Flavor diagonal because only one doublet couples to fermions fermion mass and Yukawa matrices proportional Often flavor changing Higgs couplings in extended models with two doublets coupling to same kind of fermion (not MSSM) Stringent limits, e.g., tree-level Higgs contribution to K L K S mixing (loop in standard model) h ds/m H < 10 6 GeV 1 Lecture 1 49