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Renormalization in QED Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED Renormalization in QED Divergences, regularisation and renormalization As we will see, in calculating loop Feynman diagrams in QED one finds integrals which diverge. Such apparent pathologies are dealt with by the process of renormalization. This procedure can be viewed in several ways. From one perspective it is a formal manipulation, part of the definition of the quantum field theory, which allows one to calculate finite, testable expectation values and scattering amplitudes. From a more physical perspective, one starts by noting that the key divergences come from the high energy limits of the momentum integration. However, experimentally, we do not now what physics describes high energy processes. For instance, there maybe be new very heavy particles which contribute to the loop diagrams (as virtual particles in the loop) and would change the divergence of the integral. From this perspective, renormalization is a procedure which allows us to sensibly calculate the effects of the low-energy physics, independent of how it is corrected at high energies. Concretely renormalization proceeds as follows. As an example, suppose we are considering the electron self-energy. We define the full electron propagator GF (p) as including order-by-order in the perturbation theory the corrections to the Feynman Green function. Writing e0 and m0 for the “bare” charge and mass parameters which enter the QED action we have 4 GF (p) = ¡ + ¢ + O(e0) i (1) 4 = + GF (p) + O(e0): p= − m0 + i (1) The term GF (p) is divergent. The three steps in renormalization are as follows. (1) (1) regularisation: Introduce a new finite integral GF (p; Λ) which depends on a parameter Λ, sometimes known as the “cut-off scale”. This has the property (1) Λ!1 (1) GF (p; Λ) −! GF (p) We can then split the new function into divergent and finite pieces Adiv and Ac respectively (1) GF (p; Λ) = Adiv(p; Λ) + Ac(p; Λ) The finite part Ac(Λ) leads to physically measurable effects and is known as a radiative correc- tion. (2) renormalization: If the theory is renormalizable the divergent part Ac(p; Λ) can be combined with the tree-level propagator SF (p), so that i 4 GF (p; Λ) = + Adiv(p; Λ) + Ac(p; Λ) + O(e0) p= − m0 + i iZ (Λ) = 2 + A (p; Λ) + O(e4) p= − m(Λ) + i c 0 111 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED Thus the divergent terms can be incorporated as a modification of the tree-level propagator by a rescaling or “wavefunction renormalization” Z2(Λ) and a renormalised mass parameter m(Λ). (3) removing Λ-dependence: Finally we consider taking the limit Λ ! 1. In doing we let the original bare parameters m0 and e0 become singular so that the new physical parameters m and e are finite. The perturbation expansion is then really a series in e rather than e0. While this procedure may seem a sleight of hand, it is well-defined and gives definite (and testable) physical results independent of the particular choice of regularisation. In general we say a theory is renormalizable if all divergences can be removed by renormalization of a finite number of couplings in the Lagrangian. In what follows, we will discuss the structure of the renormalization procedure for the photon and electron self-energy graphs as well as the vertex correction. Photon self-energy We define the full photon propagator with contributions from tree- and loop-level terms 0 4 4 0 F hΩj TAµ(k)Aν(k ) jΩi ≡ (2π) δ (k + k )Gµν(k) p + k k k k µ µ 4 = £ ν + ¤β να + O(e0) p 4 4 0 F 4 4 0 F (1) 4 = (2π) δ (k + k )Dµν(k) + (2π) δ (k + k )Gµν (k) + O(e0) 4 where O(e0) denotes the contribution from higher-order Feynman diagrams. Evaluating the second Feynman diagram we have Z d4p h i GF (1)(k) = DF (k) · (−) ie γα Sbc(p + k) ie γβ Sda(p) · DF (k) µν µα (2π)4 0 ab F 0 cd F βν F 2 αβ F ≡ Dµα(k) · ie0Π (k) · Dβν(k) Substituting for the SF Feynman Green functions we have Z 4 αβ d p α 1 β 1 Π (k) = i 4 Tr γ γ (2π) p= + k= − m0 + i p= − m0 + i This integral is quadratically divergent. To see this write p = E(1; v). Then d4p ∼ E3dEd3v, extracting the overall dependence on E we have 3 large E Z E dE Z Π ∼ ∼ EdE E2 222 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED which diverges as E2. We say this is a ultra-violet divergence since it diverges due to the high-energy behaviour of the integral. To remove the divergence we cold introduce a cut-off at some energy scale Λ to simulate the effect of unknown physics. However this would break Lorentz invariance and is not very elegant. A simple alternative is to remove the divergence by regulating the integral. This is done by introducing a new function dependent on a parameter Λ, such that Παβ(k; Λ) ! Παβ(k) as Λ ! 1. For instance one can write Z 4 4 αβ d p α 1 β 1 Λ Π (k; Λ) = i 4 Tr γ γ 2 2 2 (2π) p= + k= − m0 + i p= − m0 + i (Λ − p ) At finite Λ, as p ! Λ this is now convergent since Π ∼ R dE=E3 at large E1. As Λ ! 1 the new function Παβ(k; Λ) tends to the original integral Παβ(k). It is easy to show that the leading Λ behaviour in an expansion of Παβ(k; Λ) is Λ2, reflecting the divergence of Παβ(k). Thus analysing the Λ dependence of Παβ(k; Λ) we can separate out the divergent parts from the finite contributions which will correspond to those of the full theory as we approach the limit Λ ! 1 . Note that this type of regulator breaks gauge invariance but more complicated regulators can preserve it. One can explicitly calculate the value of this integral, taking care to deal correctly with the i defining the poles in the fermion Green functions. However, here we will simply discuss the structure of the result of the integration. First we note that by Lorentz invariance Παβ must have the form Παβ(k; Λ) = −gαβA(k2; Λ) + kαkβB(k2; Λ) Concentrating on A(k2; Λ) and expanding in k2 gives 2 2 2 2 A(k ; Λ) = A0(Λ) + k A1(Λ) + k Πc(k ; Λ) 2 2 2 αβ where Πc(k ; Λ) ∼ k for small k . From the original expression for Π (k) we have, keeping the k dependence, Π ∼ R EdEf(k=E). Thus expanding f(k=E), we see that in the k2 expansion each successive term introduces an extra power of E−2. Thus the leading term diverges as R EdE ∼ E2, the next as R dE=E ∼ log E and the third R dE=E3 is finite. This implies 2 2 A0(Λ) ∼ Λ A1(Λ) ∼ log Λ Πc(k ; Λ) is finite Given this expansion the full propagator has the form ig 1 1 GF (k; Λ) = − µν + ig e2A(k2; Λ) + O(e4) µν k2 + i µν k2 + i 0 k2 + i 0 ig e2A (Λ) ie2g = − µν 1 − 0 0 + 0 µν A (Λ) + Π (k2; Λ) + O(e4) k2 + i k2 k2 + i 1 c 0 2 2 2 2 −1 4 2 Writing 1 − e0A0(Λ)=k = [1 + e0A0(Λ)=k ] + O(e0) we have, keeping only terms of order e0, 2 2 F igµν 1 − e0A1(Λ) ie0gµν 2 4 Gµν(k; Λ) = − 2 2 + 2 Πc(k ; Λ) + O(e0) k + e0A0(Λ) + i k + i 2 (0.1) iZ3(Λ)gµν ie0gµν 2 4 ≡ − 2 2 + 2 Πc(k ; Λ) + O(e0) k − mγ(Λ) + i k + i 1To see this expand the regulator in the limit Λ2p2 1 and show by power counting. 333 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED where 2 4 Z3(Λ) = 1 − e0A1(Λ) + O(e0) 2 2 4 mγ(Λ) = −e0A0(Λ) + O(e0) Comparing with the general Kallen–Lehmann¨ expression for the two-point function in an inter- acting scalar theory Z 1 2 F iZ dM 2 i G (k) = 2 2 + ρ(M ) 2 2 k − m + i ∼4m2 2π p − M − i we see that Z3(Λ) corresponds to a wavefunction renormalization Z3(Λ), while we also introduce a 2 new renormalised mass term mγ(Λ). Physically, we measure that the photon is massless. Thus we expect 2 mγ(Λ) = 0 One finds that this is indeed the case provided the regularization preserves the gauge symmetry. (Note that the regularization mentioned above, does not preserve the gauge symmetry. Such regularizations can be dealt with, but the analysis is a bit more complicated.) By fixing the gauge one can also ensure that the B(Λ) function vanishes. We can incorporate the effect of the wavefunction renormalization by defining a rescaled field ph −1=2 wavefunction renormalization: Aµ ! Aµ = Z3(Λ) Aµ: such that −ig hΩj TAph(k)Aph(k0) jΩi = (2π)4δ4(k + k0) µν + ::: µ ν k2 − i ph 1=2 Physically this means that the actual measured field Aµ is Z3 times the field which appears in the ph “bare” Lagrangian. Taking the Λ ! 1 limit, we hold the rescaled physical field Aµ fixed, while the bare field Aµ becomes singular. Electron self-energy This calculation is analogous to the photon self-energy just discussed.
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