PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 19

Wednesday, February 12, 2020

Topics: Radiative Corrections, The Vertex Function.

Radiative Corrections

We have already looked at some tree-level processes - that is, with diagrams that contain no loops. All such processes receive higher-order contributions, known as radiative corrections, from diagrams that contain loops. Another source of radiative corrections in QED is - the emission of extra final-state during a reaction. Let us look at the simplest possible context in order to illustrate these ideas. Let us consider the process of electron scattering from another, very heavy, particle. At the next order in perturbation theory, we encounter the diagrams given in Fig. 1.

Figure 1: The process of electron scattering from another, very heavy, particle, at one-loop order.

The order-α correction to the cross section comes from the interference term between these diagrams and the tree-level diagram. There are six additional one-loop diagrams involving the heavy particle in the loop. But they can be neglected in the limit where that particle is much heavier than the electron, since the mass appears in the denominator of the . The PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 physical meaning is that the heavy particle accelerates less, and therefore radiates less, during the collision. Of the four diagrams given in Fig. 1, the first, known as the vertex correction, is the most intricate and gives the largest variety of new effects. It gives rise to anomalous magnetic moment for the electron, for example. The next two diagrams are external leg corrections. We will neglect them, for now, as they are not amputated, as required by our formula for S-matrix elements. The final diagram is called . The study of these corrections will be complicated by the fact that they are ill-defined. Each diagram in Fig. 1 involves an integration over the undetermined loop momentum. In each case, the integral is divergent in the k → ∞ or ultraviolet region. Fortunately, the infinite parts of these integrals will always cancel out of expressions for observable quantities such as cross sections. The first three diagrams in Fig. 1 also contain infrared divergences - infinities coming from k → 0 end of the loop-momentum integrals. It is possible to show that these divergences are canceled when we also include the bremsstrahlung diagrams given in Fig. 2

Figure 2: The the bremsstrahlung diagrams that help in cancelling the infrared divergences in the first three diagrams of Fig. 1.

These diagrams are divergent in the limit where the energy of the radiated tends to zero. In this limit, the photon (called soft photon) cannot be observed by any physical detector. So it makes sense to add the cross section for producing these low-energy photons to the cross section for scattering without radiation.

The Electron Vertex Function

Let us look the the correction to electron scattering that comes from the presence of an additional virtual photon. See Fig. 3. Before embarking on computing the , let us see what form we expect this correction to take and how to interpret its various possible terms. We will see that the basic

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Figure 3: Vertex correction to electron scattering in the presence of an additional virtual photon. requirements of Lorentz invariance, the discrete symmetries of QED, and the Ward identity strongly constrain the form of the vertex. Let us consider the effective vertex diagram given on the left hand side of Fig. 4.

Figure 4: The effective vertex in the process of electron scattering in the presence of another heavy particle.

The gray filled circle in Fig. 4 indicates the sum of the lowest-order electron-photon vertex and all amputated loop corrections. Let us call this sum of vertex diagrams

−ieΓµ(p0, p). (1)

Then the amplitude for electron scattering from a heavy target is

h i 1 h i iM = ie2 u¯(p0)Γµ(p0, p)u(p) u¯(k0)γ u(k) . (2) q2 µ

We can use general arguments to restrict the form of Γµ(p0, p). To lowest order we have

Γµ = γµ. (3)

In general, Γµ is some expression that involves p, p0, γµ and constants such as m, e, and pure numbers. Fortunately, this list is exhaustive. The only other object that could appear in nay theory is µνρσ (or equivalently γ5). But this is forbidden in any -conserving theory.

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Let us further narrow down the form of Γµ by appealing to Lorentz invariance. Since Γµ transforms as a vector, in the same sense that γµ does), it must be a linear combination of the vectors from the list above: γµ, pµ, p0µ. Using the combinations p0 + p and p0 − p for convenience, we have Γµ = γµ · A + (p0µ + pµ) · B + (p0µ − pµ) · C. (4)

The coefficients A, B, and C could involve Dirac matrices dotted into vectors, that is, p/ or p/0. We have

pu/ (p) = m · u(p), (5) u¯(p0)p/0 =u ¯(p0) · m. (6)

Thus we can write the coefficients in terms of ordinary numbers without loss of generality. The only non-trivial scalar available is q2 = −2p0 · p + 2m2. (7)

So A, B and C must be functions of q2 and of constants such as m. The list of allowed vectors can be further shortened by applying the Ward identity

µ qµΓ = 0. (8)

Dotting qµ into Eq. (4) we get

µ µ 0µ µ 0µ µ qµΓ = qµγ · A + qµ(p + p ) · B + qµ(p − p ) · C. (9)

The first term vanishes when sandwiched between u¯(p0) and u(p). We have, for on-shell (the external spinors are on-shell)

pu/ (p) = mu(p), (10) u¯(p0)p/0 = mu¯(p0). (11)

This implies

u¯(p0)/qu(p) =u ¯(p0)(p/0 − p/)u(p) =u ¯(p0)p/0u(p) − u¯(p0)pu/ (p) = mu¯(p0)u(p) − u¯(p0)mu(p) = 0. (12)

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The second term also vanishes since the external particles (electrons) are on-shell. We have

0µ µ 0 0µ µ qµ(p + p ) · B = (pµ − pµ)(p + p ) · B = (p02 − p2) · B 2 2 = (me − me) · B = 0. (13)

The third term does not automatically vanish, so C must be zero. We can make no further simplifications of Eq. (4) on general principles. Thus we have

Γµ = γµ · A + (p0µ + pµ) · B. (14)

It is conventional to write Eq. (4) by means of the Gordon identity, for on-shell spinors

u¯(p0)(pµ + p0µ)u(p) = 2mu¯(p0)γµu(p) + iu¯(p0)σµν(pµ − p0µ)u(p), (15) giving p0µ + pµ iσµνq  u¯(p0)γµu(p) =u ¯(p0) + ν u(p). (16) 2m 2m 0 µν This identity allows us to swap the (p + p) term for one involving σ qν. We write the final result as

Γµ(p0, p) = γµ · A + (p0µ + pµ) · B (p0µ + pµ) = γµ · A + · 2mB 2m  iσµνq  = γµ · A + γµ − ν · 2mB 2m iσµνq = γµ · (A + 2mB) + ν · (−2mB). (17) 2m

Let us denote

2 F1(q ) = (A + 2mB), 2 F2(q ) = (−2mB). (18)

Thus the final result is

iσµνq Γµ(q, q0) = γµF (q2) + ν F (q2), (19) 1 2m 2

2 where F1 and F2 are unknown functions of q called form factors.

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To lowest order, we have

F1 = 1, (20)

F2 = 0. (21)

In principle, the form factors can be computed to any order in perturbation theory. The form factors F1 and F2 contain complete information about the influence of an electromagnetic field on the electron. They should, in particular, contain the electron’s gross electric and magnetic couplings.

References

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

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