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Phenomenological Issues in Beyond the Standard Model

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Phenomenological Issues in Beyond the Standard Model Phenomenological Issues in Beyond the Standard Model • The Structure of the Standard Model • The MSSM • Testing the Standard Model • Neutrino Physics • Beyond the MSSM TASI (June 2, 2003) Paul Langacker (Penn) The Structure of the Standard Model Remarkably successful gauge theory of the microscopic interactions. 1. The Standard Model Lagrangian 2. Spontaneous Symmetry Breaking 3. The Gauge Interactions (a) The Charged Current (b) QED (c) The Neutral Current (d) Gauge Self-interactions 4. Problems With the Standard Model (See “Structure Of The Standard Model,” hep-ph/0304186) TASI (June 2, 2003) Paul Langacker (Penn) Gauge Transformations Φ → Φ0 ≡ UΦ i 0 −1 −1 A~µ · L~ → A~ µ · L~ ≡ UA~µ · LU~ + (∂µU) U g U = eiβ~·L~ where Φ is an n component representation vector for fermions or spin-0, Li is the n×n dimensional representation matrix for the ith generator (i = 1, ··· ,N), βi (i = 1, ··· ,N) is an arbitrary differentiable real function of space and time, and Aµ are N Hermitian gauge fields. TASI (June 2, 2003) Paul Langacker (Penn) The Standard Model Lagrangian Group: SU(3)×SU(2)×U(1) 0 Gauge couplings: gs (QCD); g, g (electroweak) Generators: SU(3) (QCD): Li, i = 1, ··· , 8 SU(2): T i, i = 1, 2, 3 U(1): Y (weak hypercharge) Gauge bosons: i SU(3) (QCD): Gµ, i = 1, ··· , 8 i SU(2): Wµ, i = 1, 2, 3 U(1): Bµ TASI (June 2, 2003) Paul Langacker (Penn) Quarks/leptons: 1 Chiral Projections: ψL(R) ≡ 2(1 ∓ γ5)ψ (Chirality = helicity up to O(m/E)) L-doublets: u0 ν0 q0 = m l0 = m mL d0 mL e−0 m L m L 0 0 −0 0 R-singlets: umR, dmR, emR, (νmR) (F ≥ 3 families; m = 1 ··· F = family index; 0 = weak eigenstates (definite SU(2) rep.), mixtures of mass eigenstates (flavors); quark color indices 0 α = r, g, b suppressed (e.g., umαL). ) ϕ+ Higgs: Complex scalar doublet ϕ = ϕ0 TASI (June 2, 2003) Paul Langacker (Penn) 3 U(1)Q: Electric charge generator Q = T + Y 1 1 1 Yq = ,Yl = − ,Yψ = qψ,Yϕ = + L 6 L 2 R 2 Lagrangian: L = LSU(3) + LSU(2)×U(1) TASI (June 2, 2003) Paul Langacker (Penn) Quantum Chromodynamics (QCD) 1 i iµν X α β LSU(3) = − FµνF + q¯rαi D6 β qr 4 r F 2 term leads to three and four-point gluon self-interactions. i i i j k Fµν = ∂µGν − ∂νGµ − gsfijk Gµ Gν i is field strength tensor for the gluon fields Gµ, i = 1, ··· , 8., gs = QCD gauge coupling constant. No gluon masses. Structure constants fijk (i, j, k = 1, ··· , 8), defined by i j k [λ , λ ] = 2ifijkλ where λi are the Gell-Mann matrices. TASI (June 2, 2003) Paul Langacker (Penn) τ i 0 λi = , i = 1, 2, 3 0 0 0 0 1 0 0 −i λ4 = 0 0 0 λ5 = 0 0 0 1 0 0 i 0 0 0 0 0 0 0 0 λ6 = 0 0 1 λ7 = 0 0 −i 0 1 0 0 i 0 1 0 0 λ8 = √1 0 1 0 3 0 0 −2 The SU3 (Gell-Mann) matrices. TASI (June 2, 2003) Paul Langacker (Penn) α β Quark interactions given by q¯rαi D6 β qr th qr = r quark flavor; α, β = 1, 2, 3 are color indices; Gauge covariant derivative α i i Dµβ = (Dµ)αβ = ∂µδαβ + igs Gµ Lαβ, for triplet representation matrices Li = λi/2. TASI (June 2, 2003) Paul Langacker (Penn) Quark color interactions: Diagonal in flavor uβ gs i Off diagonal in color −i λ γµ @ 2 αβ §¤§¤i§¤ ¦¥¦¥¦¥ @ Gµ I@ @ Purely vector (parity conserving) uα @ Bare quark mass allowed by QCD, but forbidden by the chiral symmetry of LSU(2)×U(1) (generated by spontaneous symmetry breaking) Additional ghost and gauge-fixing terms 2 θgs i ˜iµν ˜iµν Can add (unwanted) CP-violating term Lθ = 32π2 FµνF , F ≡ 1 µναβ i 2 Fαβ TASI (June 2, 2003) Paul Langacker (Penn) QCD now very well established • Short distance behavior (asymptotic freedom) • Confinement, light hadron spectrum (lattice) • Approximate global SU(3)L×SU(3)R symmetry and breaking (π, K, η are pseudogoldstone bosons) • Unique field theory of strong interactions TASI (June 2, 2003) Paul Langacker (Penn) The Electroweak Sector L = L + L + L + L SU2×U1 gauge ϕ f Yukawa Gauge part 1 i µνi 1 µν Lgauge = − F F − BµνB 4 µν 4 Field strength tensors Bµν = ∂µBν − ∂νBµ i i j k Fµν = ∂µWν − ∂νWµ − gijkWµWν 0 g(g ) is the SU2 (U1) gauge coupling; ijk is the totally antisymmetric symbol Three and four-point self-interactions for the Wi B and W3 will mix to form γ, Z TASI (June 2, 2003) Paul Langacker (Penn) Scalar part µ † Lϕ = (D ϕ) Dµϕ − V (ϕ) ϕ+ where ϕ = . Gauge covariant derivative: ϕ0 i 0 ! τ i ig Dµϕ = ∂µ + ig W + Bµ ϕ 2 µ 2 where τ i are the Pauli matrices Three and four-point interactions between the gauge and scalar fields TASI (June 2, 2003) Paul Langacker (Penn) Higgs potential V (ϕ) = +µ2ϕ†ϕ + λ(ϕ†ϕ)2 Allowed by renormalizability and gauge invariance Spontaneous symmetry breaking for µ2 < 0 Vacuum stability: λ > 0. Quartic self-interactions TASI (June 2, 2003) Paul Langacker (Penn) Fermion part (F families) F X 0 0 ¯0 0 LF = q¯mLi Dq6 mL + lmLi Dl6 mL m=1 0 0 ¯0 0 0 0 +u ¯mRi Du6 mR + dmRi Dd6 mR +e ¯mRi De6 mR L-doublets u0 ν0 q0 = m l0 = m mL d0 mL e−0 m L m L R-singlets 0 0 −0 umR, dmR, emR Different (chiral) L and R representations lead to parity violation (maximal for SU(2)) Fermion mass terms forbidden by chiral symmetry 0 Can add gauge singlet νmR for Dirac neutrino mass term TASI (June 2, 2003) Paul Langacker (Penn) Gauge covariant derivatives 0 0 ig i i g 0 Dµq = ∂µ + τ W + i Bµ q mL 2 µ 6 mL 0 0 ig i i g 0 Dµl = ∂µ + τ W − i Bµ l mL 2 µ 2 mL 0 2 0 0 Dµu = ∂µ + i g Bµ u mR 3 mR 0 0 g 0 Dµd = ∂µ − i Bµ d mR 3 mR 0 0 0 DµemR = (∂µ − ig Bµ) emR Read off W and B couplings to fermions g i 1−γ5 0 1∓γ5 −i τ γµ −ig yγµ @ 2 2 @ 2 §¤§¤§i¤ §¤§¤§¤ ¦¥¦¥¦¥ @ ¦¥¦¥¦¥ @ Wµ I@ Bµ I@ @ @ @ @ TASI (June 2, 2003) Paul Langacker (Penn) Yukawa couplings (couple L to R F X u 0 0 d 0 0 −LYukawa = Γmnq¯mLϕu˜ mR + Γmnq¯mLϕdnR m,n=1 e ¯0 0 ν ¯0 0 + ΓmnlmnϕenR (+ΓmnlmLϕν˜ mR) + H.C. Γmn are completely arbitrary Yukawa matrices, which determine fermion masses and mixings d, e terms require doublet ϕ = + mL ϕ 0 with Yϕ = 1/2 ϕ Γmn @ @ ϕ I@ @ u (and ν) terms require doublet nR @ Φ0 Φ = with Y = −1/2 Φ− Φ TASI (June 2, 2003) Paul Langacker (Penn) 0† In SU(2) the 2 and 2∗ are similar ⇒ ϕ˜ ≡ iτ 2ϕ† = ϕ −ϕ− 1 transforms as a 2 with Yϕ˜ = −2 ⇒ only one doublet needed. Does not generalize to SU(3), most extra U(1)0, supersymmetry, etc ⇒ need two doublets. (Does generalize to SU(2)L×SU(2)R×U(1) ) TASI (June 2, 2003) Paul Langacker (Penn) 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 = h0|ϕ|0i = constant ∂µv 6= 0 increases energy, but important for monopoles, strings, domain walls TASI (June 2, 2003) Paul Langacker (Penn) Minimize V (v) to find v and quantize ϕ0 = ϕ − v SU(2)×U(1) : introduce Hermitian basis 1 ! ϕ+ √ (ϕ1 − iϕ2) ϕ = = 2 , ϕ0 √1 (ϕ − iϕ 2 3 4 † where ϕi = ϕi . 4 ! 4 !2 1 X 1 X V (ϕ) = µ2 ϕ2 + λ ϕ2 2 i 4 i i=1 i=1 is O4 invariant. w.l.o.g. choose h0|ϕi|0i = 0, i = 1, 2, 4 and h0|ϕ3|0i = ν 1 1 V (ϕ)→V (v) = µ2ν2 + λν4 2 4 TASI (June 2, 2003) Paul Langacker (Penn) For µ2 < 0, minimum at V 0(ν) = ν(µ2 + λν2) = 0 1/2 ⇒ ν = −µ2/λ SSB for µ2 = 0 also; must consider loop corrections 0 ϕ→√1 ≡ v ⇒ the generators L1, L2, and L3 − Y 2 ν 1 i τ i 1 spontaneously broken, L v 6= 0, etc (L = 2 ,Y = 2I) 1 0 Qv = (L3 + Y )v = v = 0 ⇒ U(1) unbroken ⇒ 0 0 Q SU(2)×U(1)Y →U(1)Q TASI (June 2, 2003) Paul Langacker (Penn) Quantize around classical vacuum • Kibble transformation: introduce new variables ξi for rolling modes 1 P i i 0 ϕ = √ ei ξ L 2 ν + H • H = H† is the Higgs scalar • No potential for ξi ⇒ massless Goldstone bosons for global symmetry • Disappear from spectrum for gauge theory (“eaten”) • Display particle content in unitary gauge P i i 1 0 ϕ→ϕ0 = e−i ξ L ϕ = √ 2 ν + H + corresponding transformation on gauge fields TASI (June 2, 2003) Paul Langacker (Penn) Rewrite Lagrangian in New Vacuum Higgs covariant kinetic energy terms 0 2 † µ 1 g i i g 0 (Dµϕ) D ϕ = (0 ν) τ W + Bµ + H terms 2 2 µ 2 ν 2 2 +µ − MZ µ → M W W + Z Zµ W µ 2 + H kinetic energy and gauge interaction terms Mass eigenstate bosons: W, Z, and A (photon) 1 W ± = √ (W 1 ∓ iW 2) 2 3 Z = − sin θW B + cos θW W 3 A = cos θW B + sin θW W TASI (June 2, 2003) Paul Langacker (Penn) Masses gν ν M p 2 02 W MW = ,MZ = g + g = ,MA = 0 2 2 cos θW (Goldstone scalar transformed into longitudinal components of W ±,Z) 0 Weak angle: tan θW ≡ g /g √ 2 2 Will show: Fermi constant GF / 2 ∼ g /8MW , where GF = 1.16639(2)×10−5 GeV −2 from muon lifetime Electroweak scale √ −1/2 ν = 2MW /g ' ( 2GF ) ' 246 GeV TASI (June 2, 2003) Paul Langacker (Penn) 2 Will show: g = e/ sin θW , where α = e /4π ∼ 1/137.036 ⇒ √ 1/2 (πα/ 2GF ) MW = MZ cos θW ∼ sin θW 2 Weak neutral current: sin θW ∼ 0.23 ⇒ MW ∼ 78 GeV , and MZ ∼ 89 GeV (increased by ∼ 2 GeV by loop corrections) Discovered at CERN: UA1 and UA2, 1983 Current: MZ = 91.1876 ± 0.0021 MW = 80.449 ± 0.034 TASI (June 2, 2003) Paul Langacker (Penn) The Higgs Scalar H Gauge interactions: ZZH, ZZH2,W +W −H, W +W −H2 1 0 ϕ→√ 2 ν + H µ † Lϕ = (D ϕ) Dµϕ − V (ϕ) 2 1 2 2 µ+ − H = (∂µH) + M W W 1 + 2 W µ ν 2 1 2 µ H + M Z Zµ 1 + − V (ϕ) 2 Z ν TASI (June 2, 2003) Paul Langacker (Penn) Higgs potential: V (ϕ) = +µ2ϕ†ϕ + λ(ϕ†ϕ)2 µ4 λ → − − µ2H2 + λνH3 + H4 4λ 4 Fourth term: Quartic self-interaction Third: Induced cubic self-interaction Second: (Tree level) H mass-squared, √ p 2 MH = −2µ = 2λν TASI (June 2, 2003) Paul Langacker (Penn) No a priori constraint on λ except vacuum stability (λ > 0 ⇒ 0 < MH < ∞), but > t quark loops destabilize vacuum unless MH ∼ 115 GeV > > Strong coupling for λ ∼ 1⇒MH ∼ 1 TeV Triviality: running λ should not
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