The Higgs Mechanism Electroweak Symmetry Breaking

0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Higgs Mechanism False Vacuum Decay References

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) − (∇φ(x))2 − µ2φ(x)2 + λφ(x)4 2 2 2 4 1 1 V (φ) = − µ2φ(x)2 + λφ(x)4 2 4

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 = √ λ

Higgs potential withμ=1,λ=1 Let φ(x) = v + σ(x). V(ϕ) 1.0 √ 1 0.8 V (σ) = µ2σ(x)2+ λµσ(x)3+ λσ(x)4 4 0.6

0.4 Does not have symmetry σ ↔ −σ. This symmetry is broken sponta- 0.2 neously. ϕ 2 2 -2 -1 0 1 2 mσ = 2µ

If there is another ﬁeld ψ 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, ψ) = V (φ) + V (ψ) + gφψ

0.4 2 mψ = 0 0.2 V (σ, ψ) = V (σ)+V (ψ)+gvψ2+gσψ2 ϕ -2 -1 0 1 2 2 mψ = 2gv

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) − (∇φ(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

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 positive mass. 0.2

ϕ -2 -1 0 1 2

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

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

ϕ VEV withμ=1,λ=1 ϕmin  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!

ϕ mass2 withμ=1,λ=1 2 mϕ

3.0

2.5  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

mψ withμ=1,λ=1,g=1 mψ 2 1.4 V (σ, ψ) ⊃ gφminψ

1.2

1.0  0 T ≥ 2v 0.8   0.6  m2 (T ) = 0.4 ψ 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

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.

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

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.

We identify the ﬁeld σ as the Standard Model Higgs boson that gives the vector bosons and the leptons their masses. The electroweak interaction has broken into separate Lagrangians for the electromagnetic interaction and the weak interaction. X X LEM ⊃ − mqqq¯ − ml ll¯ quarks leptons

2 2 1 1 + 2 1 1 − 2 1 2 2 02 2 LWeak ⊃ gv W + gv W + v (g +g ) |Z| 2 2 2 2 8

Christopher Dessert Higgs Mechanism 0K Higgs Mechanism True Vacuum Finite Temperature Higgs Mechanism Critical Temperature Standard Model Higgs Mechanism Metastability False Vacuum Decay Current Bounds References Broken State

False Vacuum Decay Possibility V(ϕ)

2 This is, to our best understanding, the Higgs potential today. ϕ -2 -1 1 2

-2

False Vacuum Decay Possibility V(ϕ) It could be that the Higgs has a diﬀer- ent potential than we think, allowing 4 for the possibility of a ﬁrst order tran-

2 sition. After the electroweak symmetry breaking, the universe continued ϕ -2 -1 1 2 to cool and more minima may have appeared. In this plot, however, the -2 Higgs is still in a global minimum.

False Vacuum Decay Possibility V(ϕ)

4 As the universe cools more, the extra

2 minima keep falling. At the critical temperature, both minima have the ϕ -2 -1 1 2 same value.

-2

Now the extra minima are the global minima! The current Higgs state False Vacuum Decay Possibility V(ϕ) is only metastable and becomes the "false vacuum." It could quantum 4 tunnel into the true vacuum at any

2 time. When this happens the transition will release energy, and the walls ϕ -2 -1 1 2 will expand at the speed of light, leav- ing behind the true vacuum state. -2 Most of the particle masses will be changed and the universe as we know it will cease to exist.

Christopher Dessert Higgs Mechanism 0K Higgs Mechanism Finite Temperature Higgs Mechanism Standard Model Higgs Mechanism False Vacuum Decay References References

Papers Broken Symmetries and the Masses of Gauge Bosons (P. Higgs, Phys. Rev. Lett. 13 (1964) 508) A Model of Leptons (S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264) Electroweak phase transition in the early universe and Baryogenesis – Arka Banerjee Quantum Field Theory An Introduction to QFT – Peskin and Schroeder QFT for the Gifted Amateur – Lancaster and Blundell Lecture Notes Higgs Mechanism – Vadim Kaplunovsky Cosmology II – Hannu Kurki-Suonio

Christopher Dessert Higgs Mechanism