spontaneous symmetry breaking and goldstone’s...
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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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V (!)
x
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d-d
!(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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– 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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!2!
!!
!!
!!
V (!)
!1
!2!
!!
!!
!!
!!
V (!)
!1
!2
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• 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
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