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/ )

2 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)

3 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

4 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

5 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 = e i P ξ i L i ϕ = 1 2 ( 0 ν + H + corresponding transformation on gauge fields )

6 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

7 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

8 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 = ± 0.029

9 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 (quartic and induced cubic interactions)

10 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

11 No a priori constraint on λ except vacuum stability (λ > 0 0 < M H < ), but t quark loops destabilize vacuum unless M H (Depends on scale Λ at which theory breaks down. supersymmetry) Strong coupling for λ > 1 M H > 1 TeV Triviality: running λ should not diverge below Λ > 130 GeV Doesn t hold in M H < { O(200) GeV, Λ MP = G 1/2 O(750) GeV, Λ 2M H N GeV Experimental bound (LEP 2), e + 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 >

12 Theoretical M H limits, Hambye and Riesselmann, hep-ph/

13 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]

14 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 + e Z ZH) Tevatron, LHC: GG-fusion (GG H via top loop), W W or ZZ fusion (W W H), or associated production ( qq W H, ZH, GG t th) H H t G G W + W

15 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

16 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 nr + (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

17 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

18 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 e, ν µ, ν τ are the weak interaction partners of the e, µ, and τ.) Combinations of A L observable in weak charged current. observable in standard model. A R not

19 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

20 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

21 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 t)γ µ (1 γ 5 )V d s b.

22 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

23 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

24 Effective zero- range 4-fermi interaction (Fermi theory) For Q M W, neglect Q 2 in W propagator L cc eff = G F 2 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) 10 5 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 e, ν µ n µ p, ν µ N µ X

25 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 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 " &

26 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

27 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)

28 Experiment Value of α 1 Difference from α 1 (ae) Deviation from gyromagnetic (52) [ ] ratio, ae = (g 2)/2 for e ac Josephson effect (51) [ ] (0.116 ± 0.051) 10 4 h/mn (mn is the neutron mass) (51) [ ] ( ± 0.051) 10 4 from n beam Hyperfine structure in (83) [ ] (0.064 ± 0.083) 10 4 muonium, µ + e Cesium D 1 line (41) [ ] (0.072 ± 0.041) 10 4 Spectacularly successful: Most precise: e anomalous magnetic moment α Many low energy tests to few 10 8 m γ < ev q γ < e Running α(q 2 ) observed High energy well-measured (PEP, PETRA, TRISTAN, LEP) Muon g 2 sensitive to new physics. Anomaly?

29 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

30 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

31 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

32 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 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)

33 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, M H search 4th generation weak neutral current experiments (atomic parity (Boulder); νe; νn (NuTeV); polarized Møller asymmetry (SLAC)) current future SM Qweak Running ŝ 2 Z in MS scheme sin 2! W APV A PV "-DIS A FB Z-pole Q [GeV]

34 The LEP/SLC Era Z Pole: e + e Z l + l, q q, ν ν LEP (CERN), Z s, unpolarized (ALEPH, DELPHI, L3, OPAL); SLC (SLAC), , P e 75 % (SLD) Z pole observables lineshape: M Z, Γ Z, σ branching ratios e + e, µ + µ, τ + τ q q, c c, b b, s s ν ν N ν = ± if m ν < M Z /2 asymmetries: FB, polarization, P τ, mixed lepton family universality

35 N ν = 3 + N ν = ± N ν = 1 for fourth family ν with m ν < M Z /2 N ν = 1, light ν in supersymmetry 2 N ν = 2, Majoron + scalar in triplet model of m ν with spontaneous L violation rom the ALEPH, DELPHI, L3, and OPAL Collaborations for the cross section in e + e annihilation into of the center-of-mass energy near the Z pole. The curves show the predictions of the Standard Model with t neutrinos. The asymmetry of the curve is produced by initial-state radiation. Note that the error bars have display purposes. References: r. Phys. J. C14, 1 (2000). r. Phys. J. C16, 371 (2000). hys. J. C16, 1 (2000). r. Phys. J. C19, 587 (2001). Collaborations (ALEPH, DELPHI, L3, OPAL)

36 The Z Pole Observables: LEP and SLC (09/05) Quantity Group(s) Value Standard Model pull M Z [GeV] LEP ± ± Γ Z [GeV] LEP ± ± Γ(had) [GeV] LEP ± ± Γ(inv) [MeV] LEP ± ± 0.11 Γ(l + l ) [MeV] LEP ± ± σ had [nb] LEP ± ± R e LEP ± ± R µ LEP ± ± R τ LEP ± ± A F B (e) LEP ± ± A F B (µ) LEP ± A F B (τ ) LEP ±

37 Quantity Group(s) Value Standard Model pull R b LEP/SLD ± ± R c LEP/SLD ± ± A F B (b) LEP ± ± A F B (c) LEP ± ± A F B (s) DELPHI/OPAL ± ± A b SLD ± ± A c SLD ± ± A s SLD ± ± A LR (hadrons) SLD ± ± A LR (leptons) SLD ± A µ SLD ± A τ SLD ± A τ (P τ ) LEP ± A e (P τ ) LEP ± s 2 l (Q F B) LEP ± ± s 2 l (A F B(q)) CDF ±

38 LEP 2 M W, Γ W, B (also hadron colliders) M H limits (hint?) W W production (triple gauge vertex) Quartic vertex SUSY/exotics searches

39 Electroweak Theory Probed at Loop Level Higher order corrections essential for agreement Z(γ) t(b) Z W t W Depend on m t, α s, and M H t(b) b Good t(b) agreement with direct m t t H H Z(γ) and other α s measurements; M Z W W H prediction t(b) b G H H H G H H mixed Typeset by FoilTEX G mixed

40 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]

41 Supersymmetry decoupling limit (M new > GeV): only precision effect is light SMlike Higgs little improvement on SM fit Supersymmetry parameters constrained

42 Gauge Self-Interactions Three and four-point interactions predicted by gauge invariance Indirectly verified by radiative corrections, α s running in QCD, etc. Strong cancellations in high energy amplitudes would be upset by anomalous couplings Tree-level diagrams contributing to e + e W + W

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45 Gauge unification: GUTs, string theories α + ŝ 2 Z α s = ± (MSSM) (non-susy: 0.073(1)) M G GeV Perturbative string: GeV (10% in ln M G ). Exotics: O(1) corrections.! i -1(µ)! i -1(µ) ! 1! 2 SM World Average! 3! S (M Z )=0.117±0.005 sin 2 " =0.2317± MS µ (GeV) 68% CL! 1 MSSM World Average 30! 2 20! U.A. W.d.B H.F µ (GeV) FNAL PiTP 2007 (December (July 17, 13, 2007) 2005) Paul Langacker Paul (Penn/FNAL) Langacker (IAS) 40

46 Implications of Precision Electroweak WNC, Z, W unification are primary predictions and test of electroweak SM correct and unique to first approx. representations) (gauge principle, group, SM correct at loop level (renorm gauge theory; m t, α s, M H ) Watershed: TeV physics severely constrained (unification vs compositeness) unification (decoupling): expect 0.1% TeV compositeness/dynamics: several % unless decoupling Z,W R ; 4-Fermi; exotic fermions/mixings; extended Higgs; Precise gauge couplings (gauge unification)

47 Problems with the Standard Model Lagrangian after symmetry breaking: ( ψ i i m i m ih ν L = L gauge + L Higgs + i g ( ) 2 J µ W 2 W µ + J µ W W + µ ej µ Q A µ ) ψ i g 2 cos θ W J µ Z Z µ Standard model: SU(2) U(1) (extended to include ν masses) + QCD + general relativity Mathematically consistent, renormalizable theory Correct to cm

48 However, too much arbitrariness and fine-tuning: O(27) parameters (+ 2 for Majorana ν) and electric charges Gauge Problem complicated gauge group with 3 couplings charge quantization ( q e = q p ) unexplained Possible solutions: strings; grand unification; magnetic monopoles (partial); anomaly constraints (partial) Fermion problem Fermion masses, mixings, families unexplained Neutrino masses, nature? Probe of Planck/GUT scale? CP violation inadequate to explain baryon asymmetry Possible solutions: strings; brane worlds; family symmetries; compositeness; radiative hierarchies. New sources of CP violation.

49 Higgs/hierarchy problem Expect M 2 H = O(M 2 W ) higher order corrections: δm 2 H /M 2 W 1034 Possible solutions: supersymmetry; dynamical symmetry breaking; large extra dimensions; Little Higgs; anthropically motivated finetuning (split supersymmetry) (landscape) Strong CP problem Can add θ 32π 2 g 2 s F F to QCD (breaks, P, T, CP) d N θ < 10 9, but δθ weak 10 3 Possible solutions: spontaneously broken global U(1) (Peccei- Quinn) axion; unbroken global U(1) (massless u quark); spontaneously broken CP + other symmetries

50 Graviton problem gravity not unified quantum gravity not renormalizable cosmological constant: Λ SSB = 8πG N V > Λ obs ( for GUTs, strings) Possible solutions: supergravity and Kaluza Klein unify strings yield finite gravity Λ? Anthropically motivated fine-tuning (landscape)?

51 Necessary new ingredients Mechanism for small neutrino masses Planck/GUT scale? Small Dirac (intermediate scale)? Mechanism for baryon asymmetry? Electroweak transition (Z or extended Higgs?) Heavy Majorana neutrino decay (seesaw)? Decay of coherent field? CPT violation? What is the dark energy? Cosmological Constant? Quintessence? Related to inflation? Time variation of couplings? What is the dark matter? Lightest supersymmetric particle? Axion? Suppression of flavor changing neutral currents? Proton decay? Electric dipole moments? Automatic in standard model, but not in extensions

52 Conclusions The standard model is spectacularly successful, but is incomplete Promising theoretical ideas at Planck and TeV scale Eagerly anticipate guidance from LHC