PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 32 Tuesday, March 17, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: Overview of Renormalization in QED. Asymptotic Behavior of QED and the Landau Pole. Overview of Renormalization in QED Let us briefly glance through renormalization of QED. There are finite number of superficially divergent diagrams in QED. Therefore QED is, in principle, renormalizable. We can regularize the Feynman integrals using dimensional regularization. We can also use Pauli-Villars regularization for QED. (Note that Pauli-Villars regularization does not preserve gauge invariance in non-Abelian gauge theories.) From our earlier experiences with the renormalization of φ4 theory, we would expect renormal- ization of the electron mass, electric charge, and the photon and electron fields. Electron Self-energy The fermion self-energy diagram, see Fig. 1, gives the following integral d µ Z d k γµ(p= − k= + m)γ Σ(p) = −ie2µ4−d : (1) (2π)d [(p − k)2 − m2]k2 In d = 4 − dimensions this integral can be evaluated to e2 Σ(p) = (−p= + 4m) + finite: (2) 8π2 Vacuum Polarization The photon self-energy is denoted by Πµν. It is also called vacuum polarization. Unlike electron self-energy, it has no classical counterpart. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 k p p − k p Figure 1: Electron self-energy diagram in QED. It is given as Σ(p) in Eq. (1) In d dimensions we get the expression for the vacuum polarization tensor (see Fig. 2) d Z d p Tr [γµ(p= + m)γν(p= − k= + m)] Π (k) = ie2µ4−d : (3) µν (2π)d (p2 − m2)[(p − k)2 − m2] Evaluating this is d = 4 − dimensions e2 Π (k) = (k k − g k2) + finite: (4) µν 6π2 µ ν µν p k k p − k Figure 2: Photon self-energy or vacuum polarization diagram in QED. It is given as Πµν(k) in Eq. (3) Vertex Correction The vertex correction (see Fig. 3) is given by Z d 0 ν 2− d 0 2−d=2 3 d k γν(p= − k= + m)γµ(p= − k= + m)γ −ieµ 2 Λ (p; q; p ) = −(eµ ) : (5) µ (2π)d k2[(p − k)2 − m2][(p0 − k)2 − m2] The above integral contains convergent and divergent pieces. The part of the numerator quadratic in k is divergent, the rest is convergent. So let us decompose Λµ in the following way (1) (2) Λµ = Λµ + Λµ : (6) 2 / 4 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 q k k − q p p + q 2− d 0 Figure 3: The vertex correction diagram in QED. It is given as −ieµ 2 Λµ(p; q; p ) in Eq. (5). This diagram contains divergent and convergent pieces, Λ(1) and Λ(2), respectively. We have the divergent part e2 Λ(1)(p; q; p0) = γ + finite: (7) µ 8π2 µ The divergent terms may be cancelled by counterterms in the Lagrangian, and subsequent renormalization. The convergent terms have a physical consequence; it gives an anomalous magnetic moment to the electron. Renormalized QED Lagrangian After absorbing all the infinite quantities into the definitions of bare quantities, we can write down the renormalized QED Lagrangian as 1 L = i ¯ γµ − m ¯ − e Aµ ¯ γ − (@ A − @ A )2; (8) B B B B B B B B B µ B 4 µ Bν ν Bµ where p B = Z2 ; (9) µ p µ AB = Z3A ; (10) /2 Z1 /2 −1=2 eB = eµ p = eµ Z3 ; (11) Z2 Z3 e2 Z = Z = 1 − ; (12) 1 2 8π2 e2 Z = 1 − : (13) 3 6π2 The above results are for QED at one-loop. It is also possible to show that QED is renormalizable to all orders in perturbation theory. 3 / 4 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Asymptotic Behavior of QED and the Landau Pole We would like to know how the coupling eB scales with µ. The asymptotic behavior of QED (in the one-loop approximation) may be inferred from e2 e = eµ/2Z−1=2 = eµ/2 1 + : (14) B 3 12π2 Differentiating the above equation gives, to order e3, @e e3 µ = − e + : (15) @µ 2 12π2 Letting ! 0 gives @e e3 µ = β(e) = : (16) @µ 12π2 Thus we see that, like φ4 theory, the beta function of QED is positive, and thus the running coupling constant e increases with scale µ. That is, the coupling gets stronger asymptotically, indicating that perturbation theory in QED is not good at large momenta. The solution to Eq. (16) is 2 2 e (µ0) e (µ) = 2 : (17) e (µ0) µ 1 − 2 ln 6π µ0 We also see that the coupling constant become infinity at a scale 6π2 µ = µ0 exp 2 : (18) e (µ0) This particular value of the scale is called Landau singularity or Landau pole. Since we get to know about the existence of the Landau pole through perturbative one-loop calculations, it is possible that the pole is merely a sign that the perturbative approximation breaks down at strong coupling. References [1] L. H. Ryder, Quantum Field Theory, Cambridge University Press (1996). 4 / 4.