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) 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 → ∞. 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 hΩ| TAµ(k)Aν(k ) |Ωi ≡ (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 → ∞. 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 → ∞ 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 → ∞ . 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 ∞ 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 hΩ| TAph(k)Aph(k0) |Ωi = (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 → ∞ 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. We deﬁne the full electron propagator with contributions from tree- and loop-level terms

¯ 0 4 4 0 hΩ| T ψ(p)ψ(p ) |Ωi ≡ (2π) δ (p + p )GF (p) ← k

p pp 4 = ¥ + + O(e0) ¦α p − k β 4 4 0 4 4 0 (1) 4 = (2π) δ (p + p )SF (p) + (2π) δ (p + p )GF (p) + O(e0)

444 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED

Evaluating the second Feynman diagram we have

Z d4k h i G(1)(p) = S (p) · ie γα iS (p − k) ie γβiDF (k) · S (p) F F (2π)4 0 F 0 αβ F 2 ≡ SF (p) · ie0Σ(p) · SF (p)

F Substituting for the SF and Dαβ Feynman Green functions we have

Z 4 d k α 1 1 Σ(p) = i 4 γ γα 2 (2π) p/ − k/ − m0 + i k + i

Again the integral is ultraviolet divergent. Writing p = E(1, v) we have

3 large E Z E dE Z Σ ∼ ∼ dE E3 which diverges as E. (Note that the integral also diverges at small E. This is an infrared divergence. It is actually cancelled by a related tree-level divergence and we will not consider it further.) Again there are various ways to regulate the integral. For instance, we cam replace the photon Feynman Green function by a Λ dependent function 1 1 D (k) → D (k, Λ) = − F F k2 + i k2 − Λ2 + i This gives a ﬁnite function Σ(p, Λ) which diverges linearly as Λ → ∞. Recall that Σ(p, Λ) is combination of gamma matrices. The only Lorentz invariant combinations of gamma matrices we can form are the identity 1 and powers of p/. Since p/p/ = p21 we thus can choose to expand

2 Σ(p, Λ) = Σ0(Λ) + Σ1(Λ)(p/ − m0) + Σc(p , Λ)(p/ − m0)

2 2 2 where Σ0, Σ1 and Σc are scalars and Σc(p , Λ) → 0 as p → m0. Note that this expansion implies that, acting on the free-particle spinor u(p), we have Σ(p, Λ)u(p) = Σ0(Λ)u(p). Expanding the integral deﬁning Σ(p) in terms of p/E gives

2 Σ0(Λ) ∼ ΛΣ1(Λ) ∼ log Λ Σc(p , Λ) is ﬁnite

The full propagator can then be written as

i i 2 1 GF (p, Λ) = − e0Σ0(Λ) p/ − m0 + i p/ − m0 + i p/ − m0 + i i 2 i 2 2 4 − e0Σ1(Λ) − e0Σc(p , Λ) + O(e0) p/ − m0 + i p/ − m0 + i For matrices A and B it is easy show that

(A − B)−1 = A−1 + A−1BA−1 + O(B2)

555 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED

2 4 Taking A = p/ − m0 + i and B = e0Σ0(Λ)1 we thus can rewrite, working to order O(e0),

i i 2 i 2 2 4 GF (p, Λ) = 2 − e0Σ1(Λ) − e0Σc(p , Λ) + O(e0) p/ − m0 + e0Σ0(Λ) + i p/ − m0 + i p/ − m0 + i 2 i[1 − e0Σ1(Λ)] i 2 2 4 = 2 − e0Σc(p , Λ) + O(e0) p/ − m0 + e0Σ0(Λ) + i p/ − m0 + i iZ2(Λ) i 2 2 4 ≡ − e0Σc(p , Λ) + O(e0) p/ − m(Λ) + i p/ − m0 + i (0.2) where we have deﬁned

2 4 Z2(Λ) = 1 − e0Σ1(Λ) + O(e0) 2 4 m = m0 − e0Σ0(Λ) + O(e0)

We see that Z2 corresponds to a wavefunction renormalization and we also have a renormalization

of the mass m0,

ph −1/2 wavefunction renormalization: ψ → ψ = Z2(Λ) ψ

mass renormalization: m0 → m = m0 + δm(Λ)

2 4 where δm(Λ) = −e0Σ0(Λ) + O(e0). Thus i hΩ| T ψph(p)ψ¯ph(p0) |Ωi = (2π)4δ4(p + p0) + ... p/ − m + i

Vertex modiﬁcation

Finally we consider renormalization of the vertex. We deﬁne the full vertex with contributions from tree- and loop-level terms

hΩ|TAµ(p3)ψ(p2)ψ¯(−p1) |Ωi 5 = + + + + + O(e0) § ¨ ©

p2

← p3 = p1

4 4 F α = (2π) δ (−p1 + p2 + p3)Gµα(p3)GF (p2) · V (p1, p2) · GF (p1) (0.3)

666 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED

This case is rather more complicated than the previous cases since more diagrams contribute. Note that the ﬁrst three diagrams in brackets correspond to self-energy corrections to the electron and pho- F ton propagators GF (p1), GF (p2) and Gµα(p3), while the last diagram is a correction to the vertex α V (p1, p2). In the following line these different corrections are represented by the solid blobs. Ex- 2 panding the propagators to order e0 then gives the diagrams in the previous line provided the vertex is given by, with p3 = p1 − p2,

p2

p2 p2 − k α ← p3 ← p3 µ 5 V (p1, p2) = µ + µ ↑ k + O(e0) β p1 p1 − k

p1 µ 3 µ 5 = ie0γ + ie0Λ (p1, p2) + O(e0). Evaluating the second Feynman diagram we have Z 4 µ d k α 1 µ 1 1 Λ (p1, p2) = −i 4 γ γ γα 2 (2π) p/2 − k/ − m0 + i p/1 − k/ − m0 + i k + i Again the integral is ultra-violet divergent. Writing p = E(1, v) we have

3 large E Z E dE Z dE Λ ∼ ∼ E4 E which diverges as log E. Again we can regulate the integral introducing a cut-off scale Λ and then expand in terms of momenta p1 and p2. We write

µ µ µ Λ (p1, p2, Λ) = Λ0 (Λ) + Λc (p1, p2, Λ) where (compare with expansion of Σ(p, Λ)) we deﬁne the expansion by requiring µ µ µ u¯(p2)Λc (p1, p2, Λ)u(p1) = 0. This is equivalent to Λc (p1, p2, Λ) = A (p1, p2, Λ)(p/1 − µ µ µ m0) + (p/2 − m0)B (p1, p2, Λ) for some matrices A and B . We then have

µ µ Λ0 (Λ) ∼ log Λ Λc (p1, p2, Λ) is ﬁnite

By Lorentz covariance we have µ µ Λ0 (Λ) = L(Λ)γ and hence µ µ 3 µ 3 µ 5 V (p1, p2, Λ) = ie0γ + ie0L(Λ)γ + ie0Λc (p1, p2, Λ) + O(e0) (0.4) −1 µ 3 µ 5 ≡ iZ1(Λ) e0γ + ie0Λc (p1, p2, Λ) + O(e0) −1 2 4 where we have deﬁned Z1(Λ) = 1 + e0L(Λ) + O(e0) so

2 4 Z1(Λ) = 1 − e0L(Λ) + O(e0)

777 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED

This leads to a renormalization of the electric charge e. To see exactly how we must be careful about how we deﬁne e. In the free theory we have

¯ h0| TAµ(p3)ψ(−p1)ψ(p2) |0i =

4 4 i i i = (2π) δ (−p1 + p2 + p3) 2 · ie0γµ · p3 + i p/2 + m0 + i p/1 + m0 + i From this perspective, the free electric charge is given by the coefﬁcient of the correlation function as 2 2 2 2 p1 → m0, p2 → m0. Similarly the measured physical charge e should be given by the residue of the physical three point function. That is we deﬁne e by the interacting three-point correlation function

ph ph ¯ph hΩ| TAµ (p3)ψ (p2)ψ (−p1) |Ωi

4 4 i i i = (2π) δ (−p1 + p2 + p3) 2 · ieγµ · + ... p3 + i p/2 + m + i p/1 + m + i

2 2 where m is now the physical mass and the dots represent terms which do not diverge as p1 → m and 2 2 p2 → m . Given deﬁnitions of the physical ﬁelds, the relation in the last line of (0.3), the deﬁnition (0.4) of F Z1 and the expressions (0.1) and (0.2) for Gµν(p) and GF (p) we see that

1/2 Z2Z3 charge renormalization: e0 → e = e0. Z1

Ward identity

We have seen that there are three contributions to the renormalization of the charge e: from the photon and fermion wavefunctions and from the vertex modiﬁcation. Thus far we considered only one type of fermion. In fact in QED we have three types: electrons, muons and taus. Each have different mass and so a priori each lead to different Z1 and Z2 renormalizations. Thus, labelling the type of particle by the index (i) we have

Z(i)Z1/2 e → e(i) = 2 3 e (i) = e, µ, τ 0 (i) 0 for Z1 Note that the photon wavefunction renormalization is the same for each particle, though it now has three contributions arising from integrating over a loop of electrons, muons or taus. Generically we see that each particle gets a different renormalized charge. But this is contrary to observation (and the principle of minimal coupling) that all particles couple to the same basic unit of electric charge. Equivalently, it would imply that the quantum loop corrections violate the original gauge invariance of the bare action.

888 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED

In fact, there is an relation, known as the Ward identity which ensures that all the e(i) are equal. It states simply that for any given fermion

Z1 = Z2 Ward identity which ensures that the renormalized theory is still gauge invariant. It is easy to see that this relation holds for our (formally divergent) one-loop unregulated expres- sions. Recall that Z d4k Σ(p) = −i γαS (p − k)γ D (k) (2π)4 F α F Taking a derivative with respect to pµ gives

∂Σ(p) Z d4k ∂S (p − k) = −i γα F γ D (k) ∂pµ (2π)4 ∂pµ α F

−1 −1 Now by deﬁnition SF (q) = i (q/ − m0 + i), so since SF SF = 1, we have

∂S (q) ∂S−1(q) F = −S (q) F S (q) = −iS (q)γµS (q) ∂qµ F ∂qµ F F F

µ This gives, comparing with Λ (p1, p2),

∂Σ(p) Z d4k = γαS (p − k)γµS (p − k)γ D (k) ∂pµ (2π)4 F F α F = Λµ(p, p)

Assuming these relations still hold after regularization, we have, by deﬁnition, that the divergent pieces of each expression are given by

∂Σ(p, Λ) = Σ (Λ)γµ + ... ∂pµ 1 Λµ(p, p, Λ) = L(Λ)γµ + ... where both Σ1 and L diverge as log Λ. Thus we have that Σ1(Λ) = L(Λ) and hence

Z1(Λ) = Z2(Λ)

proving the Ward identity to this order. In fact it is possible to show that the identity holds at all orders in perturbation theory. It is necessary for the quantum theory to be gauge invariant.

Renormalizability of QED

Finally let us brieﬂy sketch how one shows that all potential divergences in QED, to all orders in perturbation theory can be absorbed by renormalization, that is, that the theory is renormalizable. First one has to classify the divergent diagrams. The initial step is to identify “proper” diagrams

999 Quantum Electrodynamics (QED), Winter 2015/16 Renormalization in QED which cannot be reduced to two diagrams simply by cutting an internal line. The ﬁrst diagram below is not proper; the second is.

One then removes all the self-energy and vertex corrections from graphs, since these we know are renormalizable. These give the “skeleton graphs”.

−→

For the remaining (inﬁnite set) of graphs one can then count the expected order of divergence from the powers of E in the propagators and integrals. One ﬁnds that there are only ﬁve possible divergent graphs Z3 Z2, m Z1 zero ﬁnite The ﬁrst three are the familiar diagrams we have already considered. Of the last two, one vanishes and one is ﬁnite essentially as a result of charge conjugation symmetry and gauge invariance. Thus we see the theory is indeed renormalizable.

101010