Spontaneous Symmetry Breaking and Goldstone`s Theorem

Spontaneous Symmetry Breaking and Goldstone`s Theorem

Spontaneous Symmetry Breaking and Goldstone’s Theorem
i
• Exact symmetry: T , L = 0 (equations of motion); T i|0i = 0 (vacuum)
i
i
• Explicit breaking: T , L 6= 0, e.g., L = L0 + L1 with T , L1 6= 0
• Spontaneous breaking: T i|0i =
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|0i = 0, or
– SSB: T i|0i =
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):
1
2
µ2φ2
λφ4
V (φ) =
+
L = (∂µφ) − V (φ) ,
2
2
4
2
2
∂
∂V
2
2
~
= − µ + λφ φ
−∇ φ=−
2
∂t
∂φ
• Lowest energy solution for classical field: φclass ≡ h0|φ|0i ≡ hφi
(vacuum expectation value [VEV])
– h0|φ|0i = constant in x, minimizes V
2
∂ V ∂V = 0,
>0
2
∂φ hφi
∂φ hφi
– Require λ > 0 (V bounded below); µ2 arbitrary
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φ(x)
ν
V (φ)
−ν
-d
ν
φ
d
x
-ν
– µ2 > 0: minimum at h0|φ|0i = 0, symmetry unbroken
2
p
– µ < 0: h0|φ|0i = 0 is unstable; minima at ±ν ≡ ± −µ2/λ
(φ → −φ symmetry spontaneously broken)
– Define φ = ν + φ0; φ0 is ordinary quantum field (h0|φ0|0i = 0)
0
L (φ) = L (ν + φ ) =
0
V (φ ) =
−µ4
|4λ
{z }
cosm. const.
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1
2
2
(∂µφ0) − V (φ0)
02
03
λ
04
−µ
φ
+
λνφ
+
φ
| {z }
| {z }
4
2
2
induced cubic
2
µ 0 =−2µ >0
φ
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λ
λ
ν
• Can add explicit Z2-breaking terms (φ or φ3), e.g.,
V (φ) =
µ2φ2
2
− aφ +
λφ4
4
,
a>0
– ⇒ h0|φ|0i =
6 0, even for µ2 > 0 (Coleman’s theorem)
– For µ2 > 0 and a small: ν = hφi = a/µ2 + O(a3)
– Typeset by FoilTEX –
1
0
V (φ ) = −
a2
2µ2
+
µ2
2
02
φ
+ λνφ
03
+
– For µ2 < 0: global (true) minimum at ν = ν0 +
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λ
4
a
2
2ν0
φ0 4
+ O ( a2 )
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A Complex Scalar
• Complex scalar (λ > 0):
†
µ
L0 = (∂µφ) ∂ φ − V (φ) ,
2
V (φ) = µ φ φ + λ φ φ
2
†
†
– continuous global U (1) symmetry, φ → eiβ φ
√
• Hermitian basis: φ = (φ1 + iφ2)/ 2 ⇒ O(2)symmetry
i
1h
2
2
(∂µφ1) + (∂µφ2) −V (φ1, φ2) ,
L0 =
2
λ
2
µ2 2
2
2
2
φ1 + φ2 +
φ1 + φ2
V =
2
4
cos β − sin β
φ1
→
φ2
sin β
cos β
(rotation; U (1) and SO (2) equivalent)
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φ1
φ2
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V (φ)
V (φ)
φ2
ν
!
!
!
!
!
φ1
!
!
φ2
!
!
!
!
φ1
!
!
!
!
!
φ2
φ1
• 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 − φ22
2
m21 = µ2,
m22 = µ2 + P529
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• µ2 < 0 and = 0 (Nambu-Goldstone realization): degenerate minima of
Mexican hat potential along
φ21 + φ22 = ν 2 ≡
−µ2
λ
– Choose axes so that φ1 = ν + φ01,
L=
V =
1
0 2
∂µφ1
2
−µ4
4λ
2
−µ
+
1
2
φ012
>0
φ2 = φ02:
0 2
∂µφ2
−V
φ01, φ02
λνφ01
φ012
φ022
+
+
+
λ
4
φ012
+
02 2
φ2
– m21 = −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 −φ22/2 ⇒ unique vacuum (up to
sign), with
m21 = −2µ2,
m22 = m21
• 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:
†
†
µ
L = ψ̄Li 6 ∂ψL + ψ̄Ri 6 ∂ψR − hψ̄LψRφ − hψ̄RψLφ + (∂µφ) ∂ φ − V (φ)
2
2 †
†
with V (φ) = µ φ φ + λ φ φ
– Chiral symmetry:
φ → eiβ φ,
ψL → ψL,
– For µ2 < 0 (and λ > 0): φ1 = ν + φ01,
LY uk
0
φ1
hν
= − √ |{z}
ψ̄ψ 1 +
ν
2 scalar
ψR → e−iβ ψR
φ2 = φ02
h
5
0
− √ i ψ̄γ
ψ
φ
2
| {z }
2 pseudoscalar
(ψ̄LψR + ψ̄RψL = ψ̄ψ and ψ̄LψR − ψ̄RψL = ψ̄γ 5ψ )
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ψL
h
√
2
ν
ψR
– Massless Goldstone boson φ02
– Effective ψ mass: mψ =
hν
√
2
– Scalar (pseudoscalar) couplings of φ1(φ2), strength
√
h/ 2 = mψ /ν
– Typeset by FoilTEX –
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Possibilities for Continuous Symmetry
Exact Lagrangian Symmetry ([UG, L] = 0)
UG|0i = |0i
U G |0 i =
6 |0i
exact symmetry
spontaneous symmetry breaking
(Wigner-Weyl)
(Nambu-Goldstone)
degenerate multiplets
conserved charges
relations between interactions
chiral: massless fermions
gauge: massless gauge bosons
Explicit breaking
multiplet splitting, etc.
chiral: fermions acquire mass
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chiral: fermions acquire mass
global: Goldstone bosons
gauge: gauge bosons acquire mass
by Higgs or dynamical mechanism
([UG, L] 6= 0) (global only)
multiplet splitting, etc.
Goldstone bosons acquire mass
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