Download 5. Quantum Field Theory (QFT) — QED Quantum Electrodynamics

5. Quantum Field Theory (QFT)
—
QED
Quantum Electrodynamics (QED)
• the bare Lagrangian including gauge-fixing
– bare means: one writes the Lagrangian from the theorists viewpoint,
no connection to observation yet
µν
1 (∂.A )2
/ 0 ψ0 − 1
−
F
F
L0 = ψ̄0(i∂/ − m0)ψ0 − g0ψ̄0A
0
0µν
0
4
2ξ0
/ = γ µAµ, and (∂.A0) = ∂µAµ
– with the abbreviations ∂/ = γ µ∂µ, A
0
– and the fieldstrength Fµν = ∂µAν − ∂ν Aµ
• QED includes one charged particle (ψ) and the photon (Aµ)
– the charged particle ψ (i.e. the electron) has the bare mass m0
∗ ψ is understood as a 4-component Dirac spinor
∗ fulfilling the Dirac equation (i∂/ − m0)ψ = 0
∗ ψ̄ = ψ †γ 0 is the adjoint spinor
– and couples to the photon with the bare interactionstrength g0
– ξ0 is the bare (unrenormalised) gauge-fixing parameter
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
1
5. Quantum Field Theory (QFT)
—
QED
QED in renormalised perturbation theory
−1/2
• introduces the renormalised fields ψ = Z2
−1/2 µ
A0
ψ0 and Aµ = Z3
• the renormalised Lagrangian includes gauge-fixing
µν
1 (∂.A)2
/ −1
F
F
−
L = ψ̄(i∂/ − m)ψ − g ψ̄ Aψ
µν
4
2ξ
1 (Z − 1)F F µν
/ −4
+ψ̄[(Z2 − 1)i∂/ − δm]ψ − (Z1 − 1)g ψ̄ Aψ
3
µν
– with field counterterms δZψ = Z2 − 1 and δZA = Z3 − 1
∗ since ψ is a spinor, δZψ will in principle be matrix-valued, treating different helicities differently
∗ but in QED the matrix is just a number times the unit matrix in spin space
– a mass counterterm δm = Z2m0 − m
1 δZ )
– a coupling counterterm (Z1 − 1) = δg + g(δZψ + 2
A
∗ the change of the coupling, δg, has to combined with the changes in the fields
– and the redefined gauge-fixing parameter ξ = ξ0/Z3
∗ in the full treatment ξ should also receive a counterterm δξ
∗ the value of δξ is then determined by a (new) renormalisation condition
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
2
5. Quantum Field Theory (QFT)
—
QED
QED in renormalised perturbation theory
• Feynman rules describing the incoming and outgoing states
– fermion spinors distinguish in- or outgoing particle or antiparticle
∗ fermion lines carry an arrow, indicating the fermion flow
∗ uα(p, s)
∗
ūα(p, s)
∗ vα(p, s)
∗
v̄ α(p, s)
initial state particle, coming from the past
✲
.
✲
.
✲
.
p
final state particle, going into the future
.
final state antiparticle, going into the future,
but fermion-arrow enters the diagram
.
initial state antiparticle, coming from the past,
but fermion-arrow leaves the diagram
✲
.
p
p
.
p
.
∗ the momentum p points into the future, s describes the helicity state
– gauge boson polarisation vectors distinguish in- or outgoing bosons
∗ in QED the gauge boson lines carry no arrow
since there is no conserved charge connected to the photon
∗ in QED the gauge boson is its own antiparticle
∗ εµ(k, λ)
∗
ε∗µ(k, λ)
✲
initial state boson, coming from the past
final state boson, going into the future
∗ the momentum k points into the future
Thomas Gajdosik – Concepts of Modern Theoretical Physics
k
.
✲
.
.
k
.
13.09.2012
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5. Quantum Field Theory (QFT)
—
QED
QED in renormalised perturbation theory
• diagrammatic Feynman rules can be obtained from the pathintegral
µ
Z[η, η̄, J ; g] = N ×
Z
i x L[ψ̄,ψ,Aµ;g]+ψ̄η+η̄ψ+J µAµ
D ψ̄ Dψ DAµ e
R
• spinors ψ and ψ̄ are related by ψ̄ = ψ †γ 0
– but used as independent variables
∗ in the same way as using the complex numbers z and z̄ = z ∗ instead of real and imaginary parts
• η and η̄ are anticommuting and spinorvalued source functions
⇒ (ψ̄η) and (η̄ψ) are commuting Lorentz scalars
– the funtional derivative
δ
δη
is also anticommuting:
δ
δ
(ψ̄(y)η(y)) = −ψ̄(y)
η(y) = −ψ̄(y)δ(x − y) = −ψ̄(x)
δη(x)
δη(x)
• for QED the Faddeev-Popov determinant ∆g [Aµ] = Det[∂ 2]
– which is constant and absorbed in the normalisation constant N
– Det[∂ 2] means summing over the spectrum of the differential operator ∂ 2
∗ this can be written as a pathintegral over the introduced ghosts
∗ but it is completely independent from the physical fields in a U (1)-gauge theory
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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5. Quantum Field Theory (QFT)
—
QED
QED in renormalised perturbation theory
• Feynman rules obtained from the pathintegral can be pictured as
– fermion propagator
β
β
β
p
[0] β
/ +mδ α
i
= i p2α−m2+iǫ
SF α (p) = p−m
/
α
∗ carries the spinor index
∗ has a direction:
α
✲
β
.
p
.
the momentum direction is counted by the propagator arrow
[0]
∗ has no direction
k k
i
∆µν (k) = k2+iǫ
−gµν + (1 − ξ) kµ2ν
– gauge boson propagator
∗ carries the vector index
µ
✲
ν
.
k
.
the momentum direction does not change the propagator
µ
[0]
– fermion-gauge boson vertex −igΓµ (p, p′; k) = −ig(γµ)
∗ connects one vector index with two spinor indices
∗ the momenta of the fermion follow the fermion lines,
k
❄ .
✶
✏✏
α
.
p
PP
q
′
β
p
∗ the momentum of the gauge boson follows from momentum conservation
∗ together with spinors and polarisation vector:
′
(∗)µ (k, λ) = −ig(2π)4 δ 4 (p + k − p′ )ūα (p′ , s′ )(γ ) β u (p, s)ε(∗)µ (k, λ)
−igū(p′ , s′ )Γ[0]
µ α β
µ (p, p ; k)u(p, s)ε
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
5
5. Quantum Field Theory (QFT)
—
QED
QED in renormalised perturbation theory
• Feynman rules for countertems can be pictured as
– fermion field counterterm i[(Z2 − 1)p
/ − δm]
α
.β
✲
✲
p
∗ includes two renormalisation constants, Z2 and δm
p
.
∗ has two spinor indices to couple to two fermion propagators
∗ has a direction:
the momentum direction is counted by the propagator arrow
– gauge boson field counterterm
µ
i[−g µν k2 + kµ kν ](Z3 − 1)
∗ includes one renormalisation constant and a projection operator
.ν
✲
✲
k
k
.
that guarantees that the photon only couples with transverse polarisations
∗ has two vector indices to couple to two gauge boson propagators
∗ has no direction:
changing of the momentum direction does not affect the counterterm
– vertex counterterm −igγ µ(Z1 − 1)
µ
k
❄ .
∗ includes the renormalisation constants, δZψ , δZA, and δg
∗ has two spinor indices to couple to two fermion propagators
✶
✏✏
α
.
∗ and a vector indices to couple to one gauge boson propagators
p
PP
q
′
β
p
∗ is related to the fermion field counterterm Z2 by a Ward identity
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
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5. Quantum Field Theory (QFT)
—
QED
Renormalisation conditions in QED
• somewhat similar like in the ABC-theory
i
• the full fermion propagator at p2 = m2 should be SF = p−m
/
– this fixes the mass counterterm δm
– and the fermion field counterterm Z2
∗ by considering the fermion self energy diagram
• the gauge independent part of the full gauge boson propagator
−igµν
at q 2 = 0 should be ∆µν (q) = q2+iǫ
– this fixes the gauge boson field counterterm Z3
∗ by considering the gauge boson self energy diagram
• the fermion-gauge boson vertex should give the classical scattering
of photons on electrons at low energies: Thomson scattering
– this gives a condition for nearly real particles
∗ the decay e → e + γ is kinematically not allowed
– it also enforces the Ward identity, which gives Z1 = Z2
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
7
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
• are the lowest order diagrams that include loops
• can be calculated from the given Feynman rules
• fermion self energy Z
[2]
−iΣ (p)
=
=
(−ig)
2
(−ig)2
Z
✛k
α
d4 k µ
γ SF (p + k)γ ν ∆µν (k)
4
(2π)
✲
✲
p
p+k
.
✲
β
p
.
/+p
k
d4 k µ
i
/+m
ν
γ
i
γ
(2π)4
(k + p)2 − m2 + iǫ k2 + iǫ
−gµν
kµ kν
+ (1 − ξ) 2
k
– the first denominator D1 = (k + p)2 − m2 = k2 + 2k.p + p2 − m2
– the numerator can be simplified using the γ-matrix identities:
2
2
2
2
−2
/ (k
/+p
/ = k−2 (k2 k
/ + 2(k.p)k
/−p
/−p
k−2 k
/ + m)k
/k + mk ) = [D1 − (p − m )]k k
/+m
and
−iΣ[2] (p)
=
/+/
/+/
−γ µ (k
p + m)γµ = 2(k
p) − 4m
g2
Z
p
/
/
(p2 − m2 )k
k
/−m
−
−
+
(1
−
ξ)
(2π)4 [D1 + iǫ][k2 + iǫ]
[k2 + iǫ]2
[D1 + iǫ][k2 + iǫ]2
[D1 + iǫ][k2 + iǫ]
d4 k
/+p
2(k
/) − 4m
– the gauge dependent part vanishes for external lines:
∗ the first term vanishes with the integration over d4k
∗ the second term is zero, as for external particles p2 = m2
∗ the third term vanishes when acting on a spinor: ū(p)(/
p − m) = (/
p − m)u(p) = 0
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
8
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
µ
k
❄
• fermion-gauge boson vertex correction
′
−igΓ[2]
µ (p, p )
=
=
=
(−ig)3
Z
d4 q ρ
γ SF (p′ + q)γ µ SF (p + q)γ ν ∆ρν (q)
4
(2π)
p + q✏
✶
α
p′ + /
q+m
p+/
q+m
d4 q ρ
/
/
µ
γ
i
γ
i
γν
(−ig)
4
′
2
2
2
2
(2π)
(p + q) − m + iǫ
(p + q) − m + iǫ
qρ qν
i
−gρν + (1 − ξ) 2
× 2
q + iǫ
q
3
−g
3
Z
p+k+q
P
q
P
✏
Z
✶
✏✏
p
.
PP
q
′
.
✛
β
p
q
γν (/
p′ + /
q + m)γ µ(/
p+/
q + m)γ ν
d4 q
(2π)4 [q 2 + iǫ][(p′ + q)2 − m2 + iǫ][(p + q)2 − m2 + iǫ]
+(1 − ξ)g
3
Z
q (p
q + m)γ µ (p
q + m)/
q
/
d4 q
/′ + /
/+/
= ...
(2π)4 [q 2 + iǫ]2 [(p′ + q)2 − m2 + iǫ][(p + q)2 − m2 + iǫ]
– has three denominators
– without additions it cannot describe an allowed process
∗ as four momentum conservation forces the momentum of the photon to vanish
∗ Thomson limit: very low energy scattering of photons on electrons Eγ ≪ m
∗ this limit is used to define the renormalisation condition :
lim
′
k=p −p→0
′
gū(p′ )Γ[2]
µ (p, p )u(p) = gū(p)γµ u(p)
– gives the Ward identity Z1 = Z2 from − ∂p∂ µ (/p − m)−1 = (/p − m)−1γµ(/p − m)−1
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
9
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
✛p
[2]
• gauge boson self energy iΠµν (q)
– has an additional (−1) due
Z to the closed fermion loop
iΠ[2]
µν (q)
=
d4 p
(−1)(−ig)2
Z
(2π)
µ
γ
β
α
SF γ δ (p)
(p
+
q)(γ
)
S
(γ
)
ν
µ
δ
F
α
β
4
.
g2
=
ig 2
−
(4π)2
−
✲
✲
ν
k
p+k
Tr[γµ i(/
p+/
q + m)γν i(/
p + m)]
d4 p
(2π)4 [(p + q)2 − m2 + iǫ][p2 − m2 + iǫ]
=
=
✲
k
Z
d4 p 4[(p + q)µ pν + pµ (p + q)ν − gµν (p.(p + q) − m2 )]
iπ 2
[(p + q)2 − m2 + iǫ][p2 − m2 + iǫ]
– combining the denominators with a Feynman parameterintegral [AB]−1 =
iΠ[2]
µν (q)
.
ig 2
(4π)2
Z
0
1
dx
Z
R1
0
dx[xA + (1 − x)B]−2
d4 p 4[(p + q)µ pν + pµ (p + q)ν − gµν (p.(p + q) − m2 )]
iπ 2
[p2 + 2xp.q + xq 2 − m2 + iǫ]2
– replacing the integration variable p → p′ = p + xq and omitting terms odd in p′
∗ gives for the denominator p2 + 2xp.q + xq 2 − m2 = p′2 + x(1 − x)q 2 − m2 := p′2 − ∆γ := D
∗ and the numerator N = (p + q)µ pν + pµ (p + q)ν − gµν (p.(p + q) − m2 )
N
⇒
iΠ[2]
µν (q)
=
2p′µ p′ν + (1 − 2x)[qµ p′ν + p′µ qν ] − 2x(1 − x)qµ qν − gµν (p′2 + (1 − 2x)(p′.q) − x(1 − x)q 2 − m2)
=
2p′µ p′ν − gµν (p′2 − ∆γ ) + 2x(1 − x)(gµν q 2 − qµ qν ) + terms linear in p′
=
4ig 2
−
(4π)2
Z
1
dx
0
Z
gµν
d4 p′ 2p′µ p′ν
8ig 2
−
[gµν q 2 − qµ qν ]
−
2
2
2
iπ [D + iǫ]
[D + iǫ] (4π)
Z
0
1
dx
Z
d4 p′ x(1 − x)
iπ 2 [D + iǫ]2
– the first part vanishes when regulating the integral, the second part is transverse: qµΠ[2]
µν (q) = 0
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
10
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
[2]
• regulating the gauge boson self energy iΠµν (q)
– the momentum integration d4p averages over all possible directions
⇒ the result has to be independent from the directions of p
∗ the only term possible is gµν
– contraction with g µν gives a scalar integral:
Z
Z
4p
d4p
2p
p
2p2
g
4
d
µ ν
µν
µν
=
−
−
g
(2π)4 [D + iǫ]2 [D + iǫ]
(2π)4 [D + iǫ]2 [D + iǫ]
– doing a Wick rotation p0 → ik4 and p
~ → ~k we get
∗ d4p → id4k, p2 = (p0)2 − p
~2 → (ik4)2 − ~k2 = −k2 =: −ℓ2, the Euclidean length
∗ and we can split off the angles into a Euclidean solid angle dΩ3E : d4 k = ℓ3dℓdΩ3E
∗ then we can go from our Euclidean four dimensions to D dimensions:
Z
Z 3
3
4
2
2
d p
2p
4
−
(2π)4 [D + iǫ]2
[D + iǫ]
=i
Z
dΩ3E
(2π)4
Z
∞
0
4
−2ℓ2
+
ℓ3dℓ 2
[ℓ + ∆γ ]2
[ℓ2 + ∆γ ]
iℓ dℓdΩE
→
→
(2π)4
i
– with the integration over the solid angle
– and a change of variables to y =
∆γ
ℓ2 +∆γ
D−1
dΩE
Z
(2π)D
R
Thomas Gajdosik – Concepts of Modern Theoretical Physics
D−1
E
D
dΩ
(2π)
−2ℓ
4
−
[−ℓ2 − ∆γ + iǫ]2
[−ℓ2 − ∆γ + iǫ]
Z
0
=
∞
ℓD−1 dℓ
D
−2ℓ2
+
[ℓ2 + ∆γ ]2
[ℓ2 + ∆γ ]
2π D/2
(2π)D Γ( D2 )
13.09.2012
=
2
(4π)D/2 Γ( D2 )
11
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
[2]
• regulating the gauge boson self energy iΠµν (q)
– we get the boundaries y(0) = 1 and y(∞) =
−2ℓdℓ∆γ
[ℓ2 +∆γ ]2
– the measure dy =
i
(4π)D/2 Γ( D
)
2
=
−i
0
−2ℓdℓ
ℓD 2
[ℓ + ∆γ ]2
D/2
2∆γ
(4π)D/2Γ( D
)
2
Z
=0
and the inverse function ℓ2 = ∆γ 1−y
y
⇒ the contracted scalar integral is
Z ∞
2
2
∆γ
∞+∆γ
D[ℓ + ∆γ ]
1−
2ℓ2
=i
2
(4π)D/2 Γ( D
)
2
Z
1
0
D 1
1 − y D/2
)
dy 1 −
(∆γ
y
2 1−y
1
dy(1 − y)D/2 y −D/2 −
D
(1
2
− y)D/2−1 y −D/2
0
– recognising the definition of the Beta-function
Z 1
B(α, β) =
Γ(α)Γ(β)
=
Γ(α + β)
dy y α−1 (1 − y)β−1
0
⇒ the contracted scalar integral becomes ( using Γ(2) = 1! = Γ(1) = 0! = 1 )
D/2
D/2
D
D
D
D
D −i
2∆γ
(4π)D/2 Γ( D
)
2
Γ(1 −
2
)Γ(1 +
Γ(2)
2
)
D Γ(1 − 2 )Γ( 2 )
−
2
Γ(1)
= −i
2∆γ
Γ(1 −
2
(4π)D/2 Γ( D
)
2
)
D
D D
Γ(1 + ) − Γ( )
2
2
2
– since Γ(z + 1) = zΓ(z), the contracted scalar integral vanishes identically
• this was dimensional regularisation
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
12
5. Quantum Field Theory (QFT)
—
QED
Elementary one-loop diagrams in QED
• the regulated gauge boson self energy
8ig 2
[2]
iΠµν (q) = −
[gµν q 2 − qµ qν ]
2
(4π)
Z
1
dx
0
Z
d4 p x(1 − x)
=: i[gµν q 2 − qµ qν ]Π[2]
γ (q)
2
2
iπ [D + iǫ]
has the same structure as the gauge boson field counterterm
– together they form the renormalised one-loop selfenergy
2
2
[2]
[2]
iΠ̄[2]
µν (q) = iΠµν (q) − i[gµν q − qµ qν ](Z3 − 1) = i[gµν q − qµ qν ]Π̄γ (q)
– this can be used to resum the gauge boson propagator:
i∆µν
=
[2]κλ
[2]ρσ
[0]
[2]ρσ
[0]
i∆[0]
i∆[0]
i∆[0]
i∆[0]
σκ iΠ̄
σν + i∆µρ iΠ̄
µν + i∆µρ iΠ̄
λν + . . .
[0]
– since qµ Π̄[2]
µν (q) = 0, the gauge dependent part of ∆µν does not contribute
[2]
ρ [2]
−igσν
ρσ 2
ρ σ
∗ the product iΠ̄[2]ρσ i∆[0]
σν = i[g q − q q ]Π̄γ (q) q 2 =: Pν Π̄γ (q)
∗ where Pνρ := [δνρ −
⇒
i∆µν
=
=
ρ
i∆[0]
µρ (δν
+
Pνρ Π̄[2]
γ (q)
−iPµν
q 2 [1
−
q ρ qν
]
q2
Π̄[2]
γ (q)]
−
is a projection operator: PκρPνκ = Pνρ and qρPνρ = Pνρq ν = 0
ρ
+
2
Pνρ [Π̄[2]
γ (q)]
i
qµ qρ
+ . . . ) = 2 (−gµρ + (1 − ξ) 2 )
q
q
−igµν
iξqµ qν
iqµ qν
=
+
q4
q4
q 2 [1 − Π̄[2]
γ (q)]
1
1−
Π̄[2]
γ (q)
−ξ
q ρ qν
+ 2
q
1 − Π̄[2]
(q)
γ
Pν
– when this propagator attaches to a physical fermion line, the last term vanishes
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
13
5. Quantum Field Theory (QFT)
—
QED
Physics of the renormalised gauge boson propagator
• since the photon propagator describes the interaction of charges
– one should be able to obtain the potential of a bound state
e−
– for that we have to consider elastic scattering
. p
✗✄q
qe ❖❈❈
∗ the particles that are bound should stay the same . . .
✄ p
✲
✲
– and compare the QFT amplitude
❖❈ pp
pe ✗✄
k
k
❈
✄
∗ that we calculate
− .
p
e
– with the QM amplitude
∗ that we assume: i.e. the potential in the Schrödinger equation
• the amplitude for elastic e− p scattering: i(2π)4δ 4(pe + pp − qe − qp)MQFT
Z
4
=
=
d k
(−igQe)(2π)4δ 4 (pe + k − qe )ū(qe )γ µ u(pe )i∆µν (k)(−igQp )(2π)4δ 4 (pp − k − qp )ū(qp )γ ν u(pp )
4
(2π)
−igµν
i(2π)4δ 4 (pe + pp − qe − qp )g 2 Qe Qp ū(qe )γ µu(pe )
ū(qp )γ ν u(pp )
[2]
k2 [1 − Π̄γ (k)]
– since e− and p are on-shell , their energies do not change
⇒ kµ = qeµ − pµe = (0, ~
qe − p
~e) is space-like
• the QM amplitude is i2πδ(Ee + Ep − Ee′ − Ep′ )MQM = h~qe, q~p|V (~k; p~e, p~p)|~pe, p~pi
– with properly normalized wave functions for e− and p
– the potential should not depend on the initial momenta
~
⇒ V (~k; p~e , p~p ) = V (~k) = ~k2 [1−igΠ̄µν[2] (~k)] ≈ ig~kµν2 [1 + Π̄[2]
γ (k)]
γ
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
14
5. Quantum Field Theory (QFT)
—
QED
Physics of the renormalised gauge boson propagator
• evaluating the regularised gauge boson self energy
8ig 2
[2]
iΠγ (q) = −
(4π)2
Z
1
dx
0
Z
d4 p
x(1 − x)
iπ 2 [p2 + x(1 − x)q 2 − m2 + iǫ]2
– for small momentum transfer ~
q 2 ≪ m2
– with a Wick rotation and dimensional regularisation we get
Z 1
2
8ig
iΠ[2]
dx x(1 − x)(m2 + x(1 − x)~
q 2 )D/2−2Γ(2 −
γ (q) = (4π)D/2
0
D
)
2
– for taking the limit D → 4 we have to expand
∗ Γ(ǫ) ≈ 1ǫ + γE.M. + . . .
2
2
∗ (m2 + x(1 − x)~
q 2)ǫ = eǫ ln[m +x(1−x)~q ] ≈ 1 + ǫ ln[m2 + x(1 − x)~
q 2] + . . .
Z 1
2
8ig
2
4−D
2
2
iΠ[2]
dx
x(1
−
x)(
+
γ
+
.
.
.
)(1
+
ln[m
+
x(1
−
x)~
q
] + ...)
E.M.
γ (q) =
4−D
2
(4π)D/2 0
Z 1
8ig 2
q2
~
2
2
=
dx
x(1
−
x)(
+
γ
+
ln[m
]
+
ln[1
+
x(1
−
x)
] + ...)
E.M.
4−D
m2
(4π)D/2 0
Z 1
8ig 2
iα q~ 2
~2
2
2q
≈ const +
dx x (1 − x) 2 = const +
(4π)2 0
m
15π m2
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
15
5. Quantum Field Theory (QFT)
—
QED
Physics of the renormalised gauge boson propagator
• Fourier transforming the potential V (~k)
– the renormalised gauge boson self energy has to vanish for q 2 → 0
[2]
[2]
[2]
q~ 2
iα
⇒ Π̄γ (q) = Πγ (q) − Πγ (0) ≈ 15π m2
q2
~
1
α
α
– so V (~
q ) ≈ ~q 2 [1 + 15π m2 ] = q~12 + 15πm
2
– which gives a Fourier transformed potential
V (r) =
Z
4α2 3
α
d3q i~q.~r
e V (~
q) ≈ − −
δ (r)
3
2
(2π)
r
15πm
⇒ gives part of the Lamb shift
• discussing the limit |q 2| ≫ m2
⇒ change of the coupling strength with energy
⇒ running coupling constant
Thomas Gajdosik – Concepts of Modern Theoretical Physics
13.09.2012
16