Introduction to the Standard Model Lecture 12 Classical Goldstone Theorem and Higgs Mechanism The Classical Goldstone Theorem: To each broken generator corresponds a massless field (Goldstone boson). Proof: 2 ∂ V Mij = ∂φi∂φj φ~= φ~ h i V (φ~)=V (φ~ + iεaT aφ~) ∂V =V (φ~)+ iεaT aφ~ + ( ε 2) ∂φj O | | ∂ ∂V a Tjlφl =0 ⇒∂φk ∂φj 2 ∂ ∂V a ∂ V a ∂V a Til φl = Tjlφl + Tik ∂φk ∂φi ∂φi∂φj ∂φi φ~= φ~ h i 0= M T a φ +0 ki il h li =~0i | 6 {z } If T a is a broken generator one has T a φ~ = ~0 h i 6 ⇒ Mik has a null eigenvector null eigenvalues massless particle since the eigenvalues of the mass matrix are the particle⇒ masses. ⇒ We now combine the concept of a spontaneously broken symmetry to a gauge theory. The Higgs Mechanism for U(1) gauge theory Consider µ 2 2 1 µν = D φ∗ D φ µ φ∗φ λ φ∗φ F F L µ − − − 4 µν µν µ ν ν µ with Dµ = ∂µ + iQAµ and F = ∂ A ∂ A . Gauge symmetry here means invariance under Aµ Aµ ∂µΛ. − → − 2 2 2 case a) unbroken case, µ > 0 : V (φ)= µ φ∗φ + λ φ∗φ with a minimum at φ = 0. The ground state or vaccuum is U(1) symmetric. The corresponding theory is known as Scalar Electrodynamics of a massive spin-0 boson with mass µ and charge Q. 1 case b) nontrivial vaccuum case 2 µ 2 v iα V (φ) has a minimum for 2 φ∗φ = − = v which gives φ = e . λ √2 We may choose α =0 as α is arbitrary but fixed. h i eiΛQ φ = φ SSB h i 6 h i → We may parameterise the field in polar coordinates 1 1 φ = Reiθ = v + H + i π √2 √2 The kinetic term is 1 iθ Dµφ = ∂µ + iQAµ Re √2 1 iθ = ∂µR + iR∂µθ + iQAµR e √2 1 iθ Dµφ∗ = ∂µR iR∂µθ iQAµR e− √2 − − The potential term is 1 2 2 λ 4 V (φ∗φ)= µ R + R 2 4 Note: θ is in the sense of Hamiltonian dynamics a cyclic field, the Hamiltonian does not depend on it. 1 1 1 1 = ∂µR∂ R + R2∂µθ∂ θ Q2R2AµA + QR2A ∂µθ V (R2) F µνF L 2 µ 2 µ − 2 µ µ − − 4 µν 1 1 1 1 2 = ∂µR∂ R V (R2) F µνF + Q2R2 A + ∂ θ 2 µ − − 4 µν 2 µ Q µ 1 Now we note that the term Aµ + Q ∂µθ looks like a gauge field transformation. In fact, it 1 1 can be gauged away : A A′ = A + ∂ θ. Now, we see that R = v as R = v + H. µ → µ µ Q µ h i 1 µ 2 1 µν 1 2 2 µ = ∂ H∂ H v (v + H) F ′ F ′ Q (v + H) A′ A′ L 2 µ − − 4 µν − 2 µ 1 µ 2 1 µν 1 2 2 µ 1 2 2 µ = ∂ H∂ H v (v + H) F ′ F ′ Q v A′ A′ Q (2vH + H )A′ A′ 2 µ − − 4 µν − 2 µ − 2 µ At the start we had Aµ (the massless gauge boson) with 2 d.o.f’s and H, π R, θ with 2 d.o.f’s giving 4 d.o.f’s. ↔ 1This was first noted in condensed matter physics by Anderson. P. W. Higgs applied it to theoretical particle physics. 2 SSB: θ is a cyclic, massless Goldstone boson that can be gauged away; and Aµ is massive with 3 d.o.f’s: 2 transverse + 1 longitudinal. We say that “θ is eaten by the gauge field.” H is massive: 1 degree of freedom. The number of d.o.f’s (=4) is unchanged. The Goldstone boson is absorbed by the gauge boson leading to a longitudinal degree of freedom from the gauge field. This is called the Higgs mechanism after P. W. Higgs (1964). 3.