'For a clear discussion of the definition and properties of helicity states see the classic paper by Jacob and Wick [JW 59],

Note that the ratio of this decay channel to the muon decay channel is

Hence the decay of the n~ into p + i>M is 10,000 times more probable, even though the phase space is much smaller. [For a discussion of phase space, refer back to Eq. (9.21).] This surprising result is an example of a general feature of vector (and axial vector) interactions. Generalizing the discussion which led to Eq. (9.150), we can show that the coupling of a vector (or axial vector) current to massless fermions conserves helicity, which in this example means that helicity combinations Xe + Xv which add to zero (the total helicity of the initial state) should be favored over those which add to unity. However, in this case the vector current couples to a spin zero state and angular momentum conservation requires Xe + X,y = 1 (i.e., both particles must be either right- or left-handed). The resulting matrix element will therefore be suppressed by the factor 1— p/(Ep + m) = m/Ev (if Ep » m), which is proportional to the mass of the fermion and a measure of the extent to which helicity conservation can be violated by vector interactions. In our discussion, this suppression factor of the fermion mass appeared automatically when we used the Dirac equation to reduce Eq. (9.147).

Since W,r in Eq. (9.151) has dimension of mass, it can be converted to inverse seconds by dividing by h = 6.6 x 10~22 MeV/sec. Substituting numbers into the formula gives

in good agreement with the measured decay rate of 2.60 x 1CT8 sec.

PROBLEMS

9.1 Suppose the interaction Hamiltonian contains a term like g tU^-} >

where 4> is the 7r+ meson field and 4>k is the field corresponding to a neutral Kq meson of mass 498 MeV. (The mass of the charged pion is 140 MeV, and anti-Ko meson, denoted Kq, is not identical to Kq.) Suppose g = 4 x 10-4 MeV. Compute the lifetime of the K0 meson assuming that its dominant decay mode is into charged n pairs. (The real Kq decays are more complicated than described above.)

9.2 The neutral pion, it0, usually decays into two photons but can also decay into an electron-positron pair. The effective Hamiltonian density for the latter decay is where <j> is the field and ip is the electron field.

(a) Show that Hi is Hermitian.

(b) Calculate the rate for the decay 7r° —► e~ + e+. [Note that m„o = 135 MeV and me = 0.511 MeV, so you may approximate me = 0.]

9.3 [Taken from Sakurai (1967).] Consider the decay of the A0 into a n + 7r° (which happens about 35% of the time). Represent the A0, which is a neutral Dirac particle, by where and are Dirac spinors for the A and bAe destroys a A of momentum k, spin s s destroys a À of momentum k, spin s, and bA\ bA, dA, and dAt satisfy the usual anticommutation relations characteristic of a Dirac field. Represent the interaction describing the decay

Wint --<f>(x) [flAlMzb^AoCz) - 0Atf,Ao(z)7!Vn(aO] = , where g\ is a constant and <p and ipn are the 7r° and neutron fields, respectively. Since 7T° is neutral, </>*(:r) = 4>(x).

(a) Compute the transition rate for the decay A0 —> n + n°. Express your answer in terms of numbers times |<7a|2-

(b) From the experimental lifetime of the A0 and from the fact that 35% of all A°'s decay into n -I-1r°, compute

9.4 Compute the phase space integral p(M;mi,7712) for the most general two-body decay. Be sure to express your results only in terms of numbers and the masses M, mi, and 77129.5 Suppose the two poles in the propagator (9.43) are placed in the upper half plane. Show that the resulting propagator differs from the Feynman propagator only by a homogeneous solution of the KG equation. Discuss the physical difference between this propagator and the Feynman propagator. [Compare with the discussion in Sec. 4.9.]

Fig. 9.9 Diagrams for nn scattering (Prob. 9.6).

6 Suppose the nucleon interacts with a neutral, scalar, meson according to ftint = grp{x)if>(x)<f>(x) , where V is the nucleon field and <p = $ is the neutral scalar meson field.

(a) Compute the scattering amplitude to order g2, and using Eq. (9.72), find the precise form of the NN potential which arises from scalar meson exchange.

(b) Investigate nucleon-antinucleon scattering to order g2 in the scattering amplitude. There are two terms corresponding to the diagrams shown in Fig. 9.9. By explicit calculation of the second order 5-matrix, find the M-matrix for each of these diagrams and extract the Feynman rules for the treatment of antiparticles.

7 Prove the relation (9.137) which is needed in the derivation of the propagator for a massive vector meson.

Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH

CHAPTER 10

We now turn to what is perhaps the most important of all theories — Quantum Electrodynamics (referred to as QED). In addition to being important in its own right, it is also the prototype for Quantum Chromodynamics (QCD), which will be discussed in greater detail in Chapter 15.

Quantum Electrodynamics is the theory which describes how structureless (point-like) charged particles (usually with spin ¿) interact with the EM field. As such, it is the foundation of the subject of Atomic Physics and of fundamental importance to Condensed Matter, Nuclear, and Particle Physics. We already presented a preliminary discussion of some of these topics in Chapters 2 and 3. The quantization of the EM field follows the development given in Chapter 2, and the results obtained there will be carried over without further change. The new aspect of our discussion in this chapter is the treatment of the charged fermions, which will now be described by a fermion quantum field of the type introduced in the preceding three chapters.

We start with two species of charged fermions and the neutral electromagnetic field. For definiteness we take the fermions to be electrons and protons, so that our fields are ipe destroys electrons ipp destroys protons destroys and creates photons.

Quantum Electrodynamics assumes that these fermions only interact through the EM field, which is an excellent approximation for electrons (and muons, which will be considered later in the chapter), but a poor approximation for quarks and protons, which also interact strongly. Nevertheless, in some situations the strong interactions can be taken into account without explicitly calculating their effects. For example, it turns out that a good estimate for the total cross section

for the production of hadronic matter in e+ e~ annihilation at high energy can be calculated from QED alone, and we will discuss this in Sec. 10.4. And even though the proton is a bound state of quarks and gluons with a complex structure and a significant size, it is still very useful to calculate electron-proton scattering by first ignoring these effects, as we do in Sec. 10.2, and then include them by modifying the calculation later. With these applications in mind, we include a point-like hadron (which we call the proton) in our description of QED.

The Lagrangian density for this theory is

where J^ is the current, written

J"

£e and Cp are free Dirac Lagrangians (discussed in Sec. 7.4), and and the electromagnetic Lagrangian with current were encountered before in Sec. 2.2. Normal ordering is understood. We leave it as an exercise to show that this Lagrangian gives the correct Dirac equation with minimal electromagnetic substitution for both the electron and the proton as well as the correct Maxwell equations with JM as current. Note also that the Dirac equation insures that the current is conserved.

Next, we impose the Coulomb gauge, solve for A0, and eliminate it from the Lagrangian just as we did in Sec. 2.2. This gives us

dV p(r,t)-p{r',t) 47T |r — r'l where p = J°, and

Next calculate the corresponding Hamiltonian. The reduction is easier in this case than it was for the nonrelativistic case (Sec. 2.3) because the current does not depend on the generalized velocities. Neglecting the self-energy terms in the Coulomb interaction gives immediately p ■"^¿/^^Mtfir'.t)

H° = J dzrH°e = J d3r ^ (r, t) [-¿a • V + me0] </>e (r, t)

With this Hamiltonian we may treat both relativistic atoms and relativistic scattering problems, but the method is somewhat different in the two cases.

Relativistic Atoms — Here the instantaneous Coulomb term is treated to all orders by solving the Dirac equation with a Coulomb interaction exactly. Then the interaction with the radiation field is treated perturbatively. [In bound state problems, the weak binding potential must always be treated to all orders, since the bound state owes its existence to higher order effects of the potential—see Chapter 12.] Therefore, for problems of this type we separate the Hamiltonian into the free (unperturbed) and interacting parts as follows:

where, as in Chapter 3, we omit the self-energies of the electron and nucleus.

Relativistic Scattering — Here the interaction takes place only over a short time, so that both the instantaneous Coulomb and the radiation interaction are treated perturbatively. Hence,

\Ve shall show later that these two interaction terms combine to give an explicitly covariant result.

Before treating either of these cases, it is helpful to note that the structure of the unperturbed fermion fields depends on the structure of the unperturbed Hamiltonian H0. We will now study this correspondence in general. Suppose the unperturbed Hamiltonian can be written

where O is an operator which operates on the Dirac space, and the unperturbed fields satisfy the usual anticommutation relations:

{<Mr, O.lMr'.i)} = 0 {ipa(r,t),fi>l(r',t)} =6Q063(r-r')

Furthermore, we require that H0 be the generator of time translations:

To find the structure of the field ip implied by these conditions, note first that (10.11) and (10.10) imply

J dV [^(r',t)Va(r,i) + Va(r,t)Vß(r',i)] Ô01^{r',t)

or, in a more compact notation, i—il> = OrJj .

The operator O therefore specifies the wave equation which the unperturbed fields must satisfy.

Therefore, if the field is expanded in terms of annihilation and creation operators,

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