Spontaneous Symmetry Breaking and Goldstone’s Theorem

• Exact symmetry:[T i,L] = 0 (equations of motion); T i|0〉 = 0 (vacuum)

• Explicit breaking:[T i,L] 6= 0, e.g., L = L0 + L1 with

[T i,L1

] 6= 0

• Spontaneous breaking: T i|0〉 6= 0

• Coleman’s theorem: explicit breaking induces spontaneous

• The Goldstone alternative:[T i,L] = 0 allows either

– Symmetry unbroken (Wigner-Weyl realization): T i|0〉 = 0, or

– SSB: T i|0〉 6= 0⇒ massless Nambu-Goldstone boson (or Higgs

mechanism for gauge symmetry)

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Single Hermitian Field

• No continuous symmetries; can impose discrete Z2 (φ→ −φ):

L =1

2(∂µφ)

2 − V (φ) , V (φ) =µ2φ2

2+λφ4

4(∂2

∂t2− ~∇2

)φ = −∂V

∂φ= − [µ2 + λφ2

]φ

• Lowest energy solution for classical field: φclass ≡ 〈0|φ|0〉 ≡ 〈φ〉(vacuum expectation value [VEV])

– 〈0|φ|0〉 = constant in x, minimizes V

∂V

∂φ

∣∣∣∣〈φ〉

= 0,∂2V

∂φ2

∣∣∣∣〈φ〉

> 0

– Require λ > 0 (V bounded below); µ2 arbitrary

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φ

ν−ν

V (φ)

x

ν

-ν

d-d

φ(x)

– µ2 > 0: minimum at 〈0|φ|0〉 = 0, symmetry unbroken

– µ2 < 0: 〈0|φ|0〉 = 0 is unstable; minima at ±ν ≡ ±√−µ2/λ

(φ→ −φ symmetry spontaneously broken)

– Define φ = ν + φ′; φ′ is ordinary quantum field (〈0|φ′|0〉 = 0)

L (φ) = L (ν + φ′) =1

2(∂µφ

′)2 − V (φ′)

V (φ′) =−µ4

4λ︸ ︷︷ ︸cosm. const.

−µ2φ′ 2︸ ︷︷ ︸µ2φ′=−2µ2>0

+ λνφ′ 3︸ ︷︷ ︸induced cubic

+λ

4φ′ 4

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λ

ν

λ

– Typeset by FoilTEX – 1

• Can add explicit Z2-breaking terms (φ or φ3), e.g.,

V (φ) =µ2φ2

2− aφ+

λφ4

4, a > 0

– ⇒ 〈0|φ|0〉 6= 0, even for µ2 > 0 (Coleman’s theorem)

– For µ2 > 0 and a small: ν = 〈φ〉 = a/µ2 +O(a3)

V (φ′) = − a2

2µ2+µ2

2φ′ 2 + λνφ′ 3 +

λ

4φ′ 4

– For µ2 < 0: global (true) minimum at ν = ν0 + a

2ν20

+O(a2)

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A Complex Scalar

• Complex scalar (λ > 0):

L0 = (∂µφ)†∂µφ− V (φ) , V (φ) = µ2φ†φ+ λ

(φ†φ

)2– continuous global U(1) symmetry, φ→ eiβφ

• Hermitian basis: φ = (φ1 + iφ2)/√

2⇒ O(2)symmetry

L0 =1

2

[(∂µφ1)

2+ (∂µφ2)

2]−V (φ1, φ2) , V =

µ2

2

(φ

21 + φ

22

)+λ

4

(φ

21 + φ

22

)2

(φ1

φ2

)→(

cosβ − sinβ

sinβ cosβ

)(φ1

φ2

)(rotation; U(1) and SO(2) equivalent)

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φ1

φ2!

!!

!!

!!

V (φ)

φ1

φ2!

!!

!!

!!

!!

V (φ)

φ1

φ2

ν

• µ2 > 0: minimum at ν1 = ν2 = 0 (Wigner-Weyl realization);degenerate φ1,2 (or φ, φ†), conserved charge, quartics related

– Can add explicit breaking

L = L0 −ε

2φ2

2

m21 = µ2, m2

2 = µ2 + ε

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• µ2 < 0 and ε = 0 (Nambu-Goldstone realization): degenerate minima ofMexican hat potential along

φ21 + φ2

2 = ν2 ≡ −µ2

λ> 0

– Choose axes so that φ1 = ν + φ′1, φ2 = φ′2:

L =1

2

(∂µφ

′1

)2+

1

2

(∂µφ

′2

)2 − V (φ′1, φ′2)V =

−µ4

4λ− µ2φ′ 21 + λνφ′1

(φ′ 21 + φ′ 22

)+λ

4

(φ′ 21 + φ′ 22

)2– m2

1 = −2µ2 > 0 and m22 = 0 (Nambu-Goldstone boson)

– Can prove for any SSB of continuous global symmetry

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• Add small explicit breaking −εφ22/2⇒ unique vacuum (up to

sign), withm2

1 = −2µ2, m22 = ε� m2

1

• φ2 is pseudo-Goldstone boson (e.g., pions in QCD)

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Spontaneously Broken Chiral Symmetry

• Chiral fermion ψ = ψL + ψR (no mass term) and complex scalar φ:

L = ψ̄Li 6∂ψL + ψ̄Ri 6∂ψR − hψ̄LψRφ− hψ̄RψLφ† + (∂µφ)†∂µφ− V (φ)

with V (φ) = µ2φ†φ + λ

(φ†φ)2

– Chiral symmetry:

φ→ eiβφ, ψL→ ψL, ψR→ e−iβψR

– For µ2 < 0 (and λ > 0): φ1 = ν + φ′1, φ2 = φ′2

LY uk = −hν√2ψ̄ψ︸︷︷︸scalar

(1 +

φ′1ν

)− h√

2i ψ̄γ5ψ︸ ︷︷ ︸

pseudoscalar

φ′2

(ψ̄LψR + ψ̄RψL = ψ̄ψ and ψ̄LψR − ψ̄RψL = ψ̄γ5ψ)

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ψR

ψL

νh√2

– Typeset by FoilTEX – 1

– Massless Goldstone boson φ′2– Effective ψ mass: mψ = hν√

2

– Scalar (pseudoscalar) couplings of φ1(φ2), strengthh/√

2 = mψ/ν

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Possibilities for Continuous Symmetry

Exact Lagrangian Symmetry ([UG, L] = 0)

UG|0〉 = |0〉exact symmetry(Wigner-Weyl)

UG|0〉 6= |0〉spontaneous symmetry breaking(Nambu-Goldstone)

degenerate multipletsconserved chargesrelations between interactionschiral: massless fermionsgauge: massless gauge bosons

chiral: fermions acquire massglobal: Goldstone bosonsgauge: gauge bosons acquire mass

by Higgs or dynamical mechanism

Explicit breaking ([UG, L] 6= 0) (global only)

multiplet splitting, etc.chiral: fermions acquire mass

multiplet splitting, etc.Goldstone bosons acquire mass

P529 Spring, 2013 11