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 L = ψ¯ (i∂/ − m )ψ − g ψ¯ A/ ψ − 1F F µν − 1 (∂.A )2 0 0 0 0 0 0 0 0 4 0µν 0 2ξ0 0 – µ µ µ with the abbreviations ∂/ = γ ∂µ, A/ = γ Aµ, and (∂.A0)= ∂µA0 – and the fieldstrength Fµν = ∂µAν − ∂νAµ
• 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 ∗ ψ¯ = ψ†γ0 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 µ −1/2 µ • introduces the renormalised fields ψ = Z2 ψ0 and A = Z3 A0 • the renormalised Lagrangian includes gauge-fixing
¯ / ¯ / 1 µν 1 2 L = ψ(i∂ − m)ψ − gψAψ − 4FµνF − 2ξ(∂.A) ¯ ¯ 1 µν +ψ[(Z2 − 1)i∂/ − δm]ψ − (Z1 − 1)gψAψ/ − 4(Z3 − 1)FµνF – 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 a coupling counterterm (Z1 − 1) = δg + g(δZψ + 2δZA) ∗ 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) initial state particle, coming from the past ✲ . p . ∗ u¯α(p,s) final state particle, going into the future ✲ . p . ∗ vα(p,s) final state antiparticle, going into the future, ✲ . p but fermion-arrow enters the diagram . ∗ ¯vα(p,s) initial state antiparticle, coming from the past, ✲ but fermion-arrow leaves the diagram . 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) initial state boson, coming from the past ✲ . k . ∗ ε∗µ(k, λ) final state boson, going into the future ✲ . k ∗ the momentum k points into the future
Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 3 5. Quantum Field Theory (QFT) — QED QED in renormalised perturbation theory • diagrammatic Feynman rules can be obtained from the pathintegral µ i L[ψ,ψ,A¯ ;g]+ψη¯ +¯ηψ+JµA Z[η, η,J¯ ; g]= N × Dψ¯ Dψ DAµ e x µ µ Z R • spinors ψ and ψ¯ are related by ψ¯ = ψ†γ0 – but used as independent variables ∗ in the same way as using the complex numbers z andz ¯ = z∗ instead of real and imaginary parts
• η andη ¯ are anticommuting and spinorvalued source functions ⇒ (ψη¯ ) and (¯ηψ) are commuting Lorentz scalars – the funtional derivative δ is also anticommuting: δη δ δ (ψ¯(y)η(y)) = −ψ¯(y) η(y)= −ψ¯(y)δ(x − y)= −ψ¯(x) δη(x) δη(x) 2 • for QED the Faddeev-Popov determinant ∆g[Aµ] = Det[∂ ] – 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 4 5. Quantum Field Theory (QFT) — QED QED in renormalised perturbation theory • Feynman rules obtained from the pathintegral can be pictured as β β β – [0] β i /pα +mδα fermion propagator SFα (p)= p−m = i 2 2 ./ α p −m +iǫ α β ∗ carries the spinor index ✲ ∗ has a direction: . p the momentum direction is counted by the propagator arrow – [0] i kµkν gauge boson propagator ∆µν (k)= 2 −gµν + (1 − ξ) 2 . k +iǫ k µ ν ∗ carries the vector index ✲ ∗ has no direction . k the momentum direction does not change the propagator µ – [0] ′ k fermion-gauge boson vertex −igΓµ (p, p ; k)= −ig(γµ) ❄ . ∗ connects one vector index with two spinor indices α✏✶ PPq β . ✏ ′ ∗ the momenta of the fermion follow the fermion lines, p p ∗ the momentum of the gauge boson follows from momentum conservation ∗ together with spinors and polarisation vector: ′ ′ [0] ′ (∗)µ 4 4 ′ α ′ ′ β (∗)µ −igu¯(p , s )Γµ (p, p ; k)u(p, s)ε (k, λ)= −ig(2π) δ (p + k − p )¯u (p , s )(γµ)α uβ(p, s)ε (k, λ)
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 . α β – fermion field counterterm i[(Z2 − 1)/p − δm] ✲ ✲ . p p ∗ includes two renormalisation constants, Z2 and δm ∗ has two spinor indices to couple to two fermion propagators ∗ has a direction: the momentum direction is counted by the propagator arrow . µν 2 µ ν µ ν – gauge boson field counterterm i[−g k + k k ](Z3 − 1) ✲ ✲ . k k ∗ includes one renormalisation constant and a projection operator 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 µ µ – vertex counterterm −igγ (Z1 − 1) k ❄ ∗ includes the renormalisation constants, δZψ, δZA, and δg . ∗ has two spinor indices to couple to two fermion propagators α✏✏✶ PPq β . p p′ ∗ and a vector indices to couple to one gauge boson propagators
∗ is related to the fermion field counterterm Z2 by a Ward identity
Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 6 5. Quantum Field Theory (QFT) — QED Renormalisation conditions in QED • somewhat similar like in the ABC-theory • the full fermion propagator at p2 = m2 should be S = i F /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 −ig at q2 = 0 should be ∆ (q)= µν µν 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 ✛k
• fermion self energy . 4 α β [2] 2 d k µ ν ✲ ✲ ✲ −iΣ (p) = (−ig) γ SF (p + k)γ ∆µν(k) p p + k p Z (2π)4 . 4 d k k/ + /p + m i kµkν = (−ig)2 γµi γν −g + (1 − ξ) 4 2 2 2 µν 2 Z (2π) (k + p) − m + iǫ k + iǫ k 2 2 2 2 2 – the first denominator D1 =(k + p) − m = k +2k.p + p − m – the numerator can be simplified using the γ-matrix identities:
−2 −2 2 2 2 2 2 −2 k k/(k/ + /p + m)k/ = k (k k/ + 2(k.p)k/ − /pk + mk ) = [D1 − (p − m )]k k/ − /p + m µ and −γ (k/ + /p + m)γµ = 2(k/ + /p) − 4m
d4k 2(k/ + /p) − 4m k/ (p2 − m2)k/ /p − m −iΣ[2](p) = g2 + (1 − ξ) − − 4 2 2 2 2 2 2 Z (2π) [D1 + iǫ][k + iǫ] [k + iǫ] [D1 + iǫ][k + iǫ] [D1 + iǫ][k + iǫ]
– 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:u ¯(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 . p + q p + k + q 4 ✏✏✶ PPq [2] ′ 3 d q ρ ′ µ ν −igΓ (p, p ) = (−ig) γ S (p + q)γ S (p + q)γ ∆ (q) α . β µ 4 F F ρν ✏✏✶ PPq Z (2π) p p′ ✛ d4q /p ′ + /q + m /p + /q + m q = (−ig)3 γρi γµi γν Z (2π)4 (p′ + q)2 − m2 + iǫ (p + q)2 − m2 + iǫ i qρqν × −g + (1 − ξ) 2 ρν 2 q + iǫ q 4 ′ µ ν d q γν(/p + /q + m)γ (/p + /q + m)γ = −g3 Z (2π)4 [q2 + iǫ][(p′ + q)2 − m2 + iǫ][(p + q)2 − m2 + iǫ] d4q /q(/p ′ + /q + m)γµ(/p + /q + m)/q +(1 − ξ)g3 = ... Z (2π)4 [q2 + 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 : ′ [2] ′ lim gu¯(p )Γµ (p,p )u(p)= gu¯(p)γµu(p) k=p′−p→0 – from ∂ ( )−1 =( )−1 ( )−1 gives the Ward identity Z1 = Z2 −∂pµ /p − m /p − m γµ /p − m
Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 9 5. Quantum Field Theory (QFT) — QED Elementary one-loop diagrams in QED [2] ✛p • gauge boson self energy iΠ (q) µν . – µ ν has an additional (−1) due to the closed fermion loop ✲ ✲ d4p k . k iΠ[2](q) = (−1)(−ig)2 (γ ) αS β(p + q)(γ ) γS δ(p) µν 4 µ δ Fα ν β F γ ✲ Z (2π) p + k 4 d p Tr[γµi(/p + /q + m)γνi(/p + m)] = g2 Z (2π)4 [(p + q)2 − m2 + iǫ][p2 − m2 + iǫ] 2 4 2 ig d p 4[(p + q)µpν + pµ(p + q)ν − gµν(p.(p + q) − m )] = − (4π)2 Z iπ2 [(p + q)2 − m2 + iǫ][p2 − m2 + iǫ] 1 – combining the denominators with a Feynman parameterintegral [AB]−1 = dx[xA +(1 − x)B]−2 0 1 ig2 d4p 4[(p + q) p + p (p + q) − g (p.(p +Rq) − m2)] iΠ[2](q) = − dx µ ν µ ν µν µν (4 )2 2 [ 2 +2 + 2 2 + ]2 π Z0 Z iπ p xp.q xq − m iǫ – replacing the integration variable p → p′ = p + xq and omitting terms odd in p′ 2 2 2 ′2 2 2 ′2 ∗ gives for the denominator p +2xp.q + xq − m = p + x(1 − x)q − m := p − ∆γ := D 2 ∗ and the numerator N =(p + q)µpν + pµ(p + q)ν − gµν (p.(p + q) − m ) ′ ′ ′ ′ ′2 ′ 2 2 N = 2pµpν + (1 − 2x)[qµpν + pµqν] − 2x(1 − x)qµqν − gµν(p + (1 − 2x)(p .q) − x(1 − x)q − m ) ′ ′ ′2 2 ′ = 2pµpν − gµν(p − ∆γ)+2x(1 − x)(gµνq − qµqν )+ terms linear in p
2 1 4 ′ ′ ′ 2 1 4 ′ [2] 4ig d p 2pµpν gµν 8ig 2 d p x(1 − x) ⇒ iΠ (q) = − dx − − [gµνq − qµqν] dx µν (4 )2 2 [ + ]2 [ + ] (4 )2 2 [ + ]2 π Z0 Z iπ D iǫ D iǫ π Z0 Z iπ D iǫ µ [2] – the first part vanishes when regulating the integral, the second part is transverse: q Πµν (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 gµν gives a scalar integral: 4 4 2 d p 2pµpν gµν d p 2p 4 gµν − = − Z (2π)4 [D + iǫ]2 [D + iǫ] Z (2π)4 [D + iǫ]2 [D + iǫ] – doing a Wick rotation p0 → ik4 and ~p → ~k we get ∗ d4p → id4k, p2 =(p0)2 − ~p2 → (ik4)2 − ~k2 = −k2 =: −ℓ2, the Euclidean length 3 4 3 3 ∗ and we can split off the angles into a Euclidean solid angle dΩE: d k = ℓ dℓdΩE ∗ then we can go from our Euclidean four dimensions to D dimensions: d4p 2p2 4 iℓ3dℓdΩ3 −2ℓ2 4 − → E − 4 2 4 2 2 2 Z (2π) [D + iǫ] [D + iǫ] Z (2π) [−ℓ − ∆γ + iǫ] [−ℓ − ∆γ + iǫ] ∞ ∞ dΩ3 −2ℓ2 4 dΩD−1 −2ℓ2 D = i E ℓ3dℓ + → i E ℓD−1dℓ + (2 )4 [ 2 +∆ ]2 [ 2 +∆ ] (2 )D [ 2 +∆ ]2 [ 2 +∆ ] Z π Z0 ℓ γ ℓ γ Z π Z0 ℓ γ ℓ γ dΩD−1 D/2 – E 2π 2 with the integration over the solid angle (2 )D = (2 )DΓ( D ) = (4 )D/2Γ( D ) π π 2 π 2 – ∆γ and a change of variables to y = 2 R ℓ +∆γ
Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 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(∞)= ∆γ = 0 ∞+∆γ – −2ℓdℓ∆γ 2 1−y the measure dy = 2 2 and the inverse function ℓ = ∆γ [ℓ +∆γ ] y ⇒ the contracted scalar integral is
∞ 2 0 2 −2ℓdℓ D[ℓ +∆γ] 2 1 − y D 1 i ℓD 1 − = i (∆ )D/2dy 1 − D 2 2 2 D γ (4π)D/2Γ( ) [ℓ +∆γ] 2ℓ (4π)D/2Γ( ) y 2 1 − y 2 Z0 2 Z1 1 2∆D/2 = −i γ dy(1 − y)D/2y−D/2 − D (1 − y)D/2−1y−D/2 D/2 D 2 (4π) Γ( 2 ) Z0 – recognising the definition of the Beta-function 1 Γ(α)Γ(β) B(α, β)= = dy yα−1(1 − y)β−1 Γ( + ) α β Z0 ⇒ the contracted scalar integral becomes ( using Γ(2)=1!=Γ(1)=0!=1 )
D/2 D D D D D/2 D 2∆ Γ(1 − )Γ(1 + ) D Γ(1 − )Γ( ) 2∆γ Γ(1 − ) D D D −i γ 2 2 − 2 2 = −i 2 Γ(1 + ) − Γ( ) (4π)D/2Γ(D ) Γ(2) 2 Γ(1) (4π)D/2Γ(D ) 2 2 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 8ig2 1 d4p x(1 − x) iΠ[2](q) = − [g q2 − q q ] dx =: i[g q2 − q q ]Π[2](q) µν 2 µν µ ν 2 2 µν µ ν γ (4π) Z0 Z 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 ¯ [2] iΠµν (q)= iΠµν (q) − i[gµνq − qµqν](Z3 − 1) = i[gµνq − qµqν]Πγ (q) – this can be used to resum the gauge boson propagator: [0] [0] ¯[2]ρσ [0] [0] ¯[2]ρσ [0] ¯[2]κλ [0] i∆µν = i∆µν + i∆µρ iΠ i∆σν + i∆µρ iΠ i∆σκiΠ i∆λν + ... µ [2] [0] – since q Π¯ µν (q) = 0, the gauge dependent part of ∆µν does not contribute [0] [2] [2] the product Π¯ [2]ρσ ∆ = [ ρσ 2 ρ σ]Π¯ ( )−igσν =: ρΠ¯ ( ) ∗ i i σν i g q − q q γ q q2 Pν γ q ρ where ρ := [ ρ q qν ] is a projection operator: ρ κ = ρ and ρ = ρ ν = 0 ∗ Pν δν − q2 Pκ Pν Pν qρPν Pν q ρ ρ i qµqρ P q qν ⇒ i∆ = i∆[0](δρ + P ρΠ¯[2](q)+ P ρ[Π¯[2](q)]2 + ... ) = (−g + (1 − ξ) ) ν + µν µρ ν ν γ ν γ 2 µρ 2 [2] 2 q q 1 − Π¯ γ (q) q −iP iξq q −ig iq q 1 = µν − µ ν = µν + µ ν − ξ 2 [2] 4 2 [2] 4 [2] q [1 − Π¯γ (q)] q q [1 − Π¯ γ (q)] q 1 − Π¯ γ (q) – 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 − – for that we have to consider elastic scattering e . p ❈❖ ✄✗ ∗ the particles that are bound should stay the same . . . qe ❈ ✄ qp – and compare the QFT amplitude ✲ ✲ pe ✄✗ k k ❈❖ pp ∗ that we calculate ✄ ❈ e− . p – with the QM amplitude ∗ that we assume: i.e. the potential in the Schr¨odinger equation − 4 4 QFT • the amplitude for elastic e p scattering: i(2π) δ (pe + pp − qe − qp)M 4 d k 4 4 µ 4 4 ν = (−igQe)(2π) δ (pe + k − qe)¯u(qe)γ u(pe)i∆µν(k)(−igQp)(2π) δ (pp − k − qp)¯u(qp)γ u(pp) Z (2π)4 4 4 2 µ −igµν ν = i(2π) δ (pe + pp − qe − qp)g QeQpu¯(qe)γ u(pe) u¯(qp)γ u(pp) 2 [2] k [1 − Π¯γ (k)] – since e− and p are on-shell , their energies do not change µ µ µ ⇒ k = qe − pe = (0, ~qe − ~pe) is space-like ′ ′ QM ~ • the QM amplitude is i2πδ(Ee + Ep − Ee − Ep)M = h~qe, ~qp|V (k; ~pe, ~pp)|~pe, ~ppi – with properly normalized wave functions for e− and p – the potential should not depend on the initial momenta V (~k; ~p , ~p )= V (~k)= igµν ≈ igµν [1 + Π¯[2](~k)] ⇒ e p ~ 2 ¯[2] ~ ~ 2 γ k [1−Πγ (k)] k 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 8ig2 1 d4p x(1 − x) iΠ[2](q) = − dx γ 2 2 2 2 2 2 (4π) Z0 Z iπ [p + x(1 − x)q − m + iǫ] – for small momentum transfer ~q 2 ≪ m2 – with a Wick rotation and dimensional regularisation we get 1 2 Π[2]( ) = 8ig (1 )( 2 + (1 ) 2)D/2−2Γ(2 D ) i γ q (4π)D/2 dx x − x m x − x ~q − 2 Z0 – for taking the limit D → 4 we have to expand Γ( ) 1 + + ∗ ǫ ≈ ǫ γE.M. ... ∗ (m2 + x(1 − x)~q 2)ǫ = eǫ ln[m2+x(1−x)~q 2] ≈ 1+ ǫ ln[m2 + x(1 − x)~q 2]+ ...
8ig2 1 iΠ[2](q) = dx x(1 − x)( 2 + γ + ... )(1 + 4−D ln[m2 + x(1 − x)~q 2]+ ... ) γ D/2 4−D E.M. 2 (4π) Z0 2 1 8ig 2 = dx x(1 − x)( 2 + γ + ln[m2]+ln[1+ x(1 − x) ~q ]+ ... ) D/2 4−D E.M. m2 (4π) Z0 8ig2 1 ~q 2 iα ~q 2 ≈ const + dx x2(1 − x)2 = const + 2 2 2 (4π) Z0 m 15π m
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 q2 → 0 [2] [2] [2] 2 ⇒ Π¯ (q) = Π (q) − Π (0) ≈ iα ~q γ γ γ 15π m2 2 – so V (~q) ≈ 1 [1 + α ~q ]= 1 + α ~q 2 15π m2 ~q 2 15πm2 – which gives a Fourier transformed potential d3q α 4α2 V (r) = ei~q.~rV (~q) ≈− − δ3(r) Z (2π)3 r 15πm2
⇒ gives part of the Lamb shift
• discussing the limit |q2|≫ m2 ⇒ change of the coupling strength with energy ⇒ running coupling constant
Thomas Gajdosik – Concepts of Modern Theoretical Physics 13.09.2012 16