Goldstone Theorem, Higgs Mechanism

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).