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The Higgs Mechanism Electroweak Symmetry Breaking 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Higgs Mechanism False Vacuum Decay References The Higgs Mechanism Electroweak Symmetry Breaking Christopher Dessert November 27, 2017 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Lagrangian Finite Temperature Higgs Mechanism Higgs Potential Standard Model Higgs Mechanism Symmetry Breaking False Vacuum Decay Acquiring a mass References Higgs Mechanism Lagrangian The Lagrangian L is the zero-temperature analogue of the free energy F. 1 2 1 1 1 L = φ(_x) − (rφ(x))2 − µ2φ(x)2 + λφ(x)4 2 2 2 4 1 1 V (φ) = − µ2φ(x)2 + λφ(x)4 2 4 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Lagrangian Finite Temperature Higgs Mechanism Higgs Potential Standard Model Higgs Mechanism Symmetry Breaking False Vacuum Decay Acquiring a mass References Higgs Mechanism Higgs Potential Higgs potential withμ=1,λ=1 1 2 2 1 4 V(ϕ) V (φ) = − µ φ(x) + λφ(x) 1.0 2 4 0.8 Initially the system has the Z2 0.6 symmetry φ $ −φ. 0.4 2 2 d V (φ) 2 2 2 0.2 m = = −µ + 3λφ = −µ φ dφ2 ϕ -2 -1 0 1 2 µ Vmin = v = p λ Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Lagrangian Finite Temperature Higgs Mechanism Higgs Potential Standard Model Higgs Mechanism Symmetry Breaking False Vacuum Decay Acquiring a mass References Higgs Mechanism Symmetry Breaking Higgs potential withμ=1,λ=1 Let φ(x) = v + σ(x). V(ϕ) 1.0 p 2 2 3 1 4 0.8 V (σ) = µ σ(x) + λµσ(x) + λσ(x) 4 0.6 Does not have symmetry σ $ −σ. 0.4 This symmetry is broken sponta- 0.2 neously. ϕ 2 2 -2 -1 0 1 2 mσ = 2µ Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Lagrangian Finite Temperature Higgs Mechanism Higgs Potential Standard Model Higgs Mechanism Symmetry Breaking False Vacuum Decay Acquiring a mass References Higgs Mechanism Acquiring a mass If there is another field in L then it Higgs potential withμ=1,λ=1 V(ϕ) gets a mass even if it is initially mass- 1.0 less. 0.8 2 0.6 V (φ, ) = V (φ) + V ( ) + gφψ 0.4 m2 = 0 0.2 V (σ; ) = V (σ)+V ( )+gv 2+gσ 2 ϕ -2 -1 0 1 2 2 m = 2gv Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Lagrangian Finite Temperature Higgs Mechanism Higgs Potential Standard Model Higgs Mechanism Symmetry Breaking False Vacuum Decay Acquiring a mass References Higgs Mechanism Higgs Mechanism The field φ gives other fields masses when the φ $ −φ symmetry breaks. We call σ the Higgs field. The process by which other fields get masses is called the Higgs mechanism. Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Free Energy F = H − T S 1 2 1 1 1 1 π2 = φ(_x) − (rφ(x))2 − µ2φ(x)2 + λφ(x)4 + m2 T 2 − T 4 2 2 2 4 24 φ 90 Effective potential 1 1 1 π2 V (φ) = − µ2φ(x)2 + λφ(x)4 + m2 T 2 − T 4 2 4 24 φ 90 1 1 1 1 π2 = − µ2φ(x)2 + λφ(x)4 − µ2T 2 + λφ2T 2 − T 4 2 4 24 8 90 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Symmetric State Higgs symmetric potential withμ=1,λ=1 V(ϕ) 1.0 0.8 At high temperatures early in the uni- 0.6 verse (<10s), the Higgs potential was 0.4 symmetric and the Higgs had a posi- tive mass. 0.2 ϕ -2 -1 0 1 2 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Critical State Higgs critical potential withμ=1,λ=1 V(ϕ) 1.0 0.8 At the critical temperature (≈10s), 0.6 the second order phase transition oc- 0.4 curs and the Higgs is massless. 0.2 ϕ -2 -1 0 1 2 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Broken State Higgs broken potential withμ=1,λ=1 V(ϕ) 1.0 After the phase transition, the temper- 0.8 ature continues to fall and the Higgs (σ) has a positive mass. The ground 0.6 state moves away from φ = 0 and any 0.4 particles interacting with the Higgs ac- 0.2 quire a mass. At zero temperature we ϕ reach the final state. -2 -1 0 1 2 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Order Parameter ϕ VEV withμ=1,λ=1 ϕmin 8 1.0 0 T ≥ 2v > <> 0.5 φmin(T ) = r 1 > 2 2 T > v − T T < 2v 1 2 3 4 : 4 -0.5 φmin is an order parameter for this -1.0 system! Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Higgs mass ϕ mass2 withμ=1,λ=1 2 mϕ 3.0 2.5 8 1 >−µ2 + λT 2 T ≥ 2v 2.0 > 4 2 < 1.5 mφ(T ) = 1.0 > 1 :> 2µ2 − λT 2 T < 2v 0.5 2 T 1 2 3 4 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Free Energy Finite Temperature Higgs Mechanism Phase Transition Standard Model Higgs Mechanism Order Parameter False Vacuum Decay Acquiring a mass References Psi mass mψ withμ=1,λ=1,g=1 mψ 2 1.4 V (σ; ) ⊃ gφmin 1.2 1.0 8 0 T ≥ 2v 0.8 > > 0.6 < m2 (T ) = 0.4 r > 1 0.2 > 2 2 :>2g v − T T < 2v T 4 1 2 3 4 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Standard Model Electroweak Lagrangian 3 ! 2 g X g 0 X X L ⊃ −V (φ)+ − W i + B φ − λ qq¯ φ− λ ll¯ φ SM 2 2 q l i=1 quarks leptons Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Vector Bosons When we take φ(x) = v + σ(x), 3 ! 2 g X g 0 X X L ⊃ −V (φ)+ − W i + B φ − λ qq¯ φ− λ ll¯ φ SM 2 2 q l i=1 quarks leptons 2 2 1 1 1 2 1 1 2 2 ! gv W + gv W 2 2 2 2 1 g 2 gg 0 W 3 + v 2 W 3 B 8 gg 0 g 02 B 1 W 1 and W 2 bosons get a mass m = gv, these are the charged W 2 vector bosons W ± that mediate the weak interactions. Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Vector Bosons When we take φ(x) = v + σ(x), 3 ! 2 g X g 0 X X L ⊃ −V (φ)+ − W i + B φ − λ qq¯ φ− λ ll¯ φ SM 2 2 q l i=1 quarks leptons 2 2 1 1 1 2 1 1 2 2 ! gv W + gv W 2 2 2 2 1 g 2 −gg 0 W 3 + v 2 W 3 B 8 −gg 0 g 02 B Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Vector Bosons When we take φ(x) = v + σ(x), 3 ! 2 g X g 0 X X L ⊃ −V (φ)+ − W i + B φ − λ qq¯ φ− λ ll¯ φ SM 2 2 q l i=1 quarks leptons 2 2 1 1 1 2 1 1 2 2 ! gv W + gv W 2 2 2 2 1 0 0 A + v 2 AZ 8 0 g 2 + g 02 Z W 3 and B turn into A, the massless photon field that mediates electromagnetism, and Z, the final massive vector boson that 1 mediates weak interactions, with m2 = v 2(g 2 + g 02). Z 4 Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Fermions When we take φ(x) = v + σ(x), 3 ! 2 g X g 0 X X L ⊃ −V (φ)+ − W i + B φ − λ qq¯ φ − λ ll¯ φ SM 2 2 q l i=1 quarks leptons X X ! − λqvqq¯ − λl vll¯ quarks leptons 2 So the fermions get masses mf = 2λf v. Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Electroweak Lagrangian Standard Model Higgs Mechanism Spontaneous Symmetry Breaking False Vacuum Decay Electromagnetic and Weak Standard Model References Electromagnetic and Weak Standard Model We identify the field σ as the Standard Model Higgs boson that gives the vector bosons and the leptons their masses.
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