Feynman Rules for Quantum Electrodynamics

Home , Photon polarization, Quantum electrodynamics, Yukawa interaction

PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 13

Monday, January 27, 2020

Topics: Feynman Rules for Quantum Electrodynamics. The Coulomb Potential from QFT.

Feynman Rules for Quantum Electrodynamics

From Yukawa theory, let us now move towards Quantum Electrodynamics (QED).

QED Interaction Lagrangian

We replace the scalar particle φ with a vector particle Aµ. Here the 4-vector Aµ represents a real vector ﬁeld representing the photon. The Yukawa interaction Hamiltonian then takes the form Z 3 µ Hint = d x eψγ ψAµ. (1)

We have already looked at the rules for fermions. The additional Feynman rules for QED are given in Fig. 1. These rules are easy to guess though diﬃcult to prove.

We use wavy lines to denote photons. The symbol µ(p) stands for the polarization vector of the initial- or ﬁnal-state photon.

Equation of Motion for Photon

We have the Maxwell’s equation

µν ∂µF = 0, µ ν ν µ ∂µ(∂ A − ∂ A ) = 0, 2 ν ν µ ∂ A − ∂ (∂µA ) = 0. (2)

In Lorentz gauge µ ∂µA = 0, (3) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Figure 1: Feynman rules for QED.

we get the ﬁeld equation for Aµ 2 ∂ Aµ = 0. (4)

This tells us that each component of A separately obeys the Klein-Gordon equation, with m = 0. Our experience with the WKlein-Gordon ﬁeld tells us that we can expect a plane-wave expansion for the photon ﬁeld in the form

3 Z d3p 1 X A (x) = ar r (p)e−ipx + ar†r∗(p)eipx , (5) µ (2π)3 p p µ p µ 2Ep r=0

2 where r = 0, 1, 2, 3 labels the basis of polarization vectors, with p = 0 and thus Ep = |p|.

Photon Polarization

The solutions to Eq. 2 in the Lorentz gauge are plane waves given in Eq. (5). To make sure that we are still in the Lorentz gauge we require that

µ ∂ Aµ = 0, (6) and this implies µ r p µ = 0, (7) by uniqueness of Fourier transform. In order to make our calculations simpler, let us consider a photon moving in the z direction:

pµ = (E, 0, 0,E), (8)

2 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 with E = |p|. The constraint when dotted with p gives zero, gives the following basis that obeys the condition

1 1 = √ (0, 1, +i, 0), (9) 2 1 2 = √ (0, 1, −i, 0), (10) 2 where 1 and 2 correspond to left and right handed circular polarized waves (or helicity states). Gauge invariance in QED ensures that there are only two independent polarizations for the photon. We can have the polarization states     1 1     0  0  3 0 =   , =   , (11)  0  0     −1 1 representing the scalar and longitudinal polarization. Within Lorentz gauge there is still gauge freedom.

Aµ → Aµ + ∂µχ (12) with µ ∂µA = 0. (13)

We can still take any function χ(x) satisfying ∂2χ = 0. If we take r r µ → µ + βpµ (14)

2 µ for an arbitrary β, we see that nothing will change since p = pµp = 0 in

r r µ pµµ → pµµ + β pµp . (15)

We have

µ 0 p µ = (|p|, 0, 0, |p|) · (1, 0, 0, 1) = 2|p| 6= 0. (16)

µ r Since p µ 6= 0 for r = 0, it does not respect the gauge we are in, and thus we must reject the 0 polarization vector µ.

3 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

We also have

µ 3 p µ = (|p|, 0, 0, |p|) · (1, 0, 0, −1) = |p| − |p| = 0, (17) which is a good thing. However, now let us look at the transformation

r r µ → µ + β pµ (18) with 1 β = − . (19) |p|

3 For µ we have 3 3 µ → µ + β pµ. (20)

That is         1 1 |p| 0         0 0 1  0  0   →   −   =   (21) 0 0 |p|  0  0         1 1 |p| 0

This gives   0   3 0 →   . (22) µ 0   0 That is, the original vector is now changed to a zero vector. In summary, the Lorentz gauge condition implies that the timelike and longitudinal photons does not contribute to physical processes. It is possible for them to contribute to virtual processes.

QED Vertex

In a QED amplitude, the γ matrix will sit between spinors or other γ matrices, with the Dirac indices contracted along the fermion line. In general, for a Dirac particle with electric charge Q|e|, we have the vertex given in Fig. 2. An electron has Q = −1, and up quark has Q = +2/3, and a down quark has Q = −1/3.

4 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Figure 2: A general vertex with charge Q|e|.

Photon Progagator

It is not easy to derive the form of the photon propagator. We saw that the electromagnetic ﬁeld in Lorentz gauge obeys the massless Klein-Gordon equation. The photon propagator is nearly identical to the massless Klein-Gordon propagator. Thus we write down the photon propagator in Lorentz gauge as

Lorentz invariance of the theory (QED here) dictates that the photon propagator be an isotropic second-rank tensor that can dot together the γµ and γν from the vertices at each end. The simplest candidate is gµν.

Later we will see that the three states created by Ai, with i = 1, 2, 3, have positive norm. These states include all real (non-virtual) photons, which always have spacelike polarizations.

As a result of gµν not being positive deﬁnite, the states created by A0 will have negative norm. This is a serious problem for any theory with vector particles. Fortunately, for QED, the negative- norm states created by A0 are never produced in physical processes.

The Coulomb Potential

Let us repeat the non-relativistic scattering calculation but now for the case of QED interaction. The leading-order contribution is given in Fig. 3. In the limit |p| → 0 we have

! √ ξs us = 2m (24) 0 √ u¯s = 2m ξs† 0 (25)

In the representation ! ! 0 I 0 σ γ0 = , γ = , (26) I 0 −σ 0

5 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Figure 3: Computing the Coulomb potential in QED. we have  ss0 0 2mδ if µ = 0 u¯sγµus = (27) 0 if µ 6= 0

This gives, in the non-relativistic limit

0 u¯(p0)γ0u(p) = u†(p0)u(p) ≈ +2mξ †ξ. (28)

We also have p = (m.p), k = (m, k), p0 = (m, p0), k0 = (m, k0) (29) leading to (p0 − p)2 = −|p0 − p|2. (30)

The matrix element takes the form

−ig iM ≈ (−ie)2u¯(p0)γ0u(p) 00 u¯(k0)γ0u(k) −|p0 − p|2 2 +ie 0 0 = (2mξ †ξ) (2mξ †ξ) · g −|p0 − p|2 p k 00 2 −ie 0 0 = (2m)2(ξ †ξ) (ξ †ξ) . (31) |p0 − p|2 p k

Comparing this to the Yukawa case,

2 ig 2 ss0 rr0 iMYukawa = 0 2 2 (2m) δ δ . (32) |p − p| + mφ we see that there is an extra factor of −1. Thus the potential is a repulsive Yukawa potential with m = 0: that is, a repulsive Coulomb potential

e2 α V (r) = = , (33) 4πr r

6 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where e2 1 α = ≈ , (34) 4π 137 is the ﬁne-structure constant. For the particle-anti-particle scattering, we need to deal with the term v¯(k)γµv(k0). We note that v¯(k)γ0v(k0) = v†(k)v(k0) ≈ +2mξ†ξ0, (35) and the rest is zero. The non-relativistic scattering amplitude is given in Fig. 4.

Figure 4: Particle anti-particle scattering in QED.

The (−1) in Fig 4 is the same fermion minus sign we saw in the Yukawa case. This is an attractive potential. Similarly we can show that for antifermion-antifermion scattering the potential is repulsive. Thus, in QFT, when a vector particle is exchanged, like charges repel while unlike charges attract. We note that the repulsion in fermion-fermion scattering came entirely from the extra factor

−g00 = −1 in the vector boson propagator. A tensor boson, such as the graviton, would have a propagator, as given in Fig. 5.

Figure 5: The graviton propagator.

2 In non-relativistic collisions this propagator gives a factor −(g00) = +1. This will result in a universally attractive potential.

7 / 8 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020

Exchanged particle ff and f¯f¯ ff¯ scalar (Yukawa) attractive attractive vector (electricity) repulsive attractive tensor (gravity) attractive attractive

Table 1: Exchanged particle and the nature of the force.

References

[1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).

8 / 8