Why (your) mass isn’t
gauge dependent
Des Johnston
Nielsen Identities
Edinburgh, May 2010
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Apologies and some justifications
• It’s been a VERY long time since I’ve done PPT related
stuff..
• The topic is something that has been picked over for many
years
• The Higgs (particle) is back on the agenda
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Plan of Talk
• Symmetry breaking
• Higgs mechanism
• Quantum corrections, gauge fixing, effective potential
• The problem (and the solution)
• Other models - Standard, gravity...
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Form of Potential
Spontaneously broken symmetry
1
λ
V (|φ|2 ) = − µ2 |φ|2 + |φ|4
2
4!
φ = (φ1 + iφ2 )
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Goldstone bosons and all that
• Complex scalar field theory
1
L(x) = ∂µ φ ∂ µ φ∗ − V (|φ|2 )
2
• Minimum: φ0 =
6µ2
λ
1/2
• Take: φ(x) = φ0 + φ1 (x) + iφ2 (x)
• Lagrangian: 21 (∂µ φ1 )2 − µ2 φ21 + 21 (∂µ φ2 )2 + . . .
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Abelian Higgs
• Scalar electrodynamics - Lagrangian
1
1
L(x) = − Fµν F µν + Dµ φi Dµ φi − V (φ2 )
4
2
Fµν = ∂µ Aν − ∂ν Aµ , φ2 = φ21 + φ22
Dµ = ∂µ + ieAµ
• Same symmetry breaking as before
1
1
1
(∂µ φ1 )2 −µ2 φ21 + (∂µ φ2 )2 + e2 φ20 Aµ Aµ +eφ0 Aµ ∂µ φ2 +· · ·
2
2
2
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Abelian Higgs II
• φ2 is surplus to requirements
• Gauge invariance to the rescue
φ(x) = φ(x) exp[iθ(x)]
Aµ (x) = Aµ (x) + ∂µ θ(x)
• Unitary gauge
φ(x) = v + φ1 (x)
• φ2 is “gauged away”
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Quantum Abelian Higgs
• Scalar electrodynamics - Lagrangian in gory detail
1
1
1
− Fµν F µν + ∂µ φi ∂ µ φi −eij (∂µ φi )φj Aµ + e2 Aµ Aµ φ2 +V (φ2 )
4
2
2
Fµν = ∂µ Aν − ∂ν Aµ , φ2 = φ21 + φ22
ij =
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Gauge fixing/regularization
• Add (Rξ ) gauge fixing terms
1
1
1
− Fµν F µν + ∂µ φi ∂ µ φi −eij (∂µ φi )φj Aµ + e2 Aµ Aµ φ2 +V (φ2 )
4
2
2
1
+ ξB 2 + B(∂µ Aµ + eξvi φi ) + ∂µ ψ ∗ ∂ µ ψ − e2 ξψ ∗ ψij vi φj
2
• Use dimensional regularization/MS subtraction
• vi = ij φi0 is ’t Hooft gauge
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The usual setup
• Functional integral
Z
Z[J] =
Z
[DΦ] exp i d4 x [L(x) + JΦ]
• Connected generating functional
W [J] = −i~ ln Z[J]
• One particle irreducible generating functional
Γ[Φ̄] = W [J] − J Φ̄
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Feynman diagrams 0
Some useful definitions before we start...
vi = v ei ,
ei = (0, 1)
φi0 = φ0 ηi ,
ηi = (1, 0)
1
m21 = λφ2 − µ2
2
1
2
m2 = λφ2 − µ2
6
DN = k 4 − k 2 (m22 − 2e2 ξvφ) + e2 φ2 (e2 ξ 2 v 2 + ξm22 )
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Feynman diagrams I
Propagators for the theory
Ghost:
i
k 2 − e2 ij ξvi φj
Scalar:
iηi ηj
i(k 2 − ξe2 φ2 )
(δij − ηi ηj ) + 2
DN
k − m21
Gauge:
kµ kν
kµ kν
−iC gµν − 2
− iD 2
k
k
C=
1
ξ(k 2 − m22 − e2 ξv 2 )
,
D
=
k 2 − e 2 φ2
DN
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Feynman diagrams II
Propagagators in the ’t Hooft gauge
Ghost:
k2
i
− e2 ξφ2
Scalar:
(k 2
−
iηi ηj
i
(δij − ηi ηj ) + 2
2
2
k − m21
− ξe φ )
m22
Gauge:
1
−i 2
k − e 2 φ2
kµ kν
kµ kν
ξ
gµν − 2
−i 2
2
2
k
(k − e ξφ ) k 2
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Feynman diagrams III
Graphically:
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Feynman diagrams IV
Vertices for the theory
Scalar/Scalar/Gauge:
−eij (k1 + k2 )µ
Scalar/Ghost/Ghost:
−ie2 ξvi ij
Scalar/Gauge/Gauge:
2ie2 φi gµν
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Feynman diagrams V
Vertices for the theory
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Effective Potential I
• Quantum equivalent of classical potential
• Should use this to discuss corrections to classical
symmetry breaking picture
• Done perturbatively using loop (i.e. ~) expansion
V =i
X 1
Γi ...i (0, . . . , 0)φi1 · · · φin
n! 1 n
n
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Effective Potential II
V
(1)
ξ
3
= i d k ln(k 2 + e2 ξvφ) − ln(−k 2 + e2 φ2 )
2
1
1
−
ln(k 2 − m21 ) − ln DN
2
2
∂V (1)
∂ξ
Z
4
1
= − ieξφm22
2
Z
d4 k
(2v + φ)k 2 − vξe2 φ2
(k 2 + e2 ξvφ)DN
Ugly, gauge parameter dependent
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And the mass too...
• Masses, 2nd derivatives of (effective) potentials
• Choose λ ∼ O(e4 ) (Coleman-Weinberg)
m2(1) = m21 + Σ(1) (0) + m21
∂Σ(1) (p2 )
∂p2
1
m21 = λφ2 − µ2
2
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Included Diagrams
Work that can’t be avoided
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Excluded Diagrams
Work that can be avoided
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And finally...
Maybe not so ugly, but still gauge parameter dependent:
∂m2(1)
ξ
∂ξ
=
2 2
e2 ξλφ20
e ξφ0
ln
32π 2
M2
One loop correction to effective potential and Higgs mass look
like they depend on ξ
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What’s going on here?
• Effective potential and mass shouldn’t be gauge parameter
dependent
• Nielsen (1975) - don’t forget about implicit dependence
• All would be well if
∂V
∂V
+ C(φ, ξ)
=0
∂ξ
∂φ
∂m2
∂m2
ξ
+ C(φ, ξ)
=0
∂ξ
∂φ
ξ
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What’s going on here?
• Why is this a fix?
• Minimum of effective potential at some φ̄
∂V (φ, ξ)
=0
∂φ
• Vmin stays the same under ξ → ξ + dξ
V (φ̄ + δ φ̄, ξ + dξ) = V (φ̄, ξ) = Vmin
∂V
dφ̄ ∂V
+
=0
∂ξ
dξ ∂ φ̄
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Nielsen Identities
• The man himself N. K. Nielsen - not H. B. Nielsen
• Proved requisite identities exist using BRST symmetry
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(Standard) BRST symmetry I
• “Modern” way of dealing with gauge invariance
• Valid for the gauge-fixed Lagrangian
δ2 = 0
δψ ∗ = B , δψ = 0 , δB = 0 , δAµ = ∂µ ψ
δφi = eij ψφj
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(Standard) BRST symmetry II
• The BRST identities are encapsulated in:
Z
S(Γ) =
4
d x
δΓ δΓ
δΓ δΓ
δΓ δΓ
δΓ
+
+
+B ∗
µ
δρ δAµ δKi δφi δσ δψ
δψ
=0
• ρµ , Ki , σ sources for δAµ , δφi and δψ
∂V
• Nothing that looks like ∂Γ
∂ξ → ∂ξ
• Trick, Piguet and Sibold
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Extended BRST symmetry
• Introduce a BRST transform on the gauge parameter
δξ = χ
• This requires extra terms in the Lagrangian too
1
+ χψ ∗ B + eχψ ∗ vi φi
2
• The BRST identity now reads
S(Γ) + χ
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∂Γ
=0
∂ξ
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Extended BRST symmetry II
• Differentiate w.r.t. χ, substitute for B, set everything but φ
to zero
Z
δΓ(O(x)) δΓ
eξvi φi δΓ(O(x))
∂Γ
4
d x
−
−
=0
∗
δKi δφi
ξ
δψ
∂ξ
• Take constant field limit
ξ
∂V
−
∂ξ
Z
d4 x
δΓ(O(x)) ∂V
=0
δKi
∂φ
• Explicit expression for C(φ, ξ)
Z
−
d4 x
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δΓ(O(x))
δKi
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Some observations
• We have seen that
ξ
∂V
∂V
+ C(φ, ξ)
=0
∂ξ
∂φ
is true
• Also true for other physical quantities with the same C(φ, ξ)
• Applies in non-Abelian theories (e.g. standard model)
• Also in gravity
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Matters of gravity...
• Suppose you are made of scalar, bosonic matter
• And you are happy to be described by 1-loop quantum
gravity
L(x) = −
+
2√
−g(R + Λ) + Lgf
κ2
√
1
−g[(∇µ φ)(∇ν φ)g µν − m2 φ2 ]
2
• Then there is a Nielsen identity
ξ
∂m2
∂m2
+ η µν τ (φ, ξ)
=0
∂ξ
∂hµν
• Interpreted as
ξ → ξ + dξ
η
µν
→ (1 + τ dξ)η µν
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Matters of higher derivative gravity I
• A renormalizable, asymptotically free theory of gravity
exists (Stelle 1977)
• Unfortunately, it is also non-unitary
L(x) =
√
−g
γ
Λ
1
1
R + 4 + βR2 − 2 (Rµν Rµν − R2 )
κ2
κ
α
3
• Spin two propagator
2
1
1
D(q ) = 2
−
α γ q 2 q 2 − (α2 γ/κ2 )
2
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Matters of higher derivative gravity II
• Massive ghost decays
• General recipe
D(q 2 ) = (D0 (q 2 )−1 − Σ(q 2 ))−1
• Large N
2
1
D(q ) = 2
α γ q 2 [1 − AN q 2 ln(q 2 /L2 )]
2
• Poles at q 2 = 0, q = r exp(iθ) with r ∼ γ/N κ2
• Loop corrections ∂M 2 /∂ξ 6= 0
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Summary
• At first sight it looks as if physical quantities depend on
gauge parameters in a loop expansion
• They don’t really, you are saved by:
∂V
∂V
+ C(φ, ξ)
=0
∂ξ
∂φ
∂m2
∂m2
ξ
+ C(φ, ξ)
=0
∂ξ
∂φ
ξ
• C(φ, ξ) explicitly calculable from (extended) BRST identities
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A reference or ten
N.K. Nielsen, Nucl. Phys. B 101 (1975) 173.
R. Fukuda and T. Kugo, Phys. Rev. D 13 (1976) 3469.
I.J.R. Aitchison and C.M. Fraser, Ann. Phys. 156 (1984) 1.
D. Johnston, Nucl. Phys. B 253 (1985) 687; 283 (1987) 317;
297 (1988) 721.
S. Ramaswamy, Nucl. Phys. B 453 (1995) 240.
O. M. Del Cima, D. H.T. Franco and O. Piguet, Nucl.Phys.B 551
(1999) 813.
P. Gambino and P. A. Grassi, Phys. Rev. D 62 (2000) 076002.
L. P. Alexander and A. Pilaftsis, J.Phys.G 36 (2009) 045006.
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THE END :)
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