Interacting Field Theories

With this chapter we begin a systematic study of interacting field theories in which all particles are described by relativistic quantum fields. This means that all particles are handled in a similar way and that the annihilation or creation of particles or particle pairs can be treated in a way consistent with the description of their scattering. The particles are isolated from their surroundings and interact only with each other, so that momentum (as well as energy) is conserved, and the recoil of the target in a collision process can be properly described. One of the great successes of this treatment is that it leads naturally to a description of particle forces.

We often want a simple interacting field theory to use as an example when we begin the discussion of a new subject. In this book we will use 4>3 theory for this purpose. While this theory has very few applications, it has the virtue of being one of the more simple theories which can be constructed and is also rich enough to illustrate the general techniques used to treat any interacting theory.

In its most complete form, our illustrative 03 theory will include three kinds of scalar particles: two charged scalar particles with masses rrii, i = 1,2, and a neutral scalar particle with mass fi. As discussed in Sec. 7.5, each particle will be described by a separate quantum field, which is for particles 1 and 2 and <j> for the neutral particle. The total Lagrangian is the sum of four terms,

The free Lagrangians for the charged fields were discussed in Sec. 7.3 and are

9.1 <t>3 THEORY: AN EXAMPLE 239

where i is not summed over, and for the neutral scalar field, where = <f>, the Lagrangian is as discussed in Prob. 7.2. All three of these Lagrangians describe spin 0 KleinGordon particles, and hence their corresponding fields satisfy commutation relations. The interaction term will have a <f>3 structure and consist of three possible terms,

Ant = -Ai:$i(®)$i(i)0(a;):-A2:$$(®)$a(a:)0(®):-^:03(a:): , (9.4)

where the coupling constants of the theory are Ai, A2, and A, all of which are real because (9.4) must be Hermitian. In some references, the term "4>3 theory" is reserved exclusively for a purely neutral self-interaction term like the last term given in (9.4), but we will refer to any of the interaction terms in (9.4) as a "4>3 theory." The reason for dividing the neutral <f>3 term by 3! will be discussed later. In some applications we will take some of these constants to be zero, giving simpler theories. According to the discussion in Sec. 7.5, this interaction Lagrangian describes elementary processes in which three particles interact at a point.

The fields €>i(x) and <p(x) have the following structure:

d>(x) = £ |afc0i+)(x) + 4^(x)} , k where the subscripts on the annihilation and creation operators, alp, label both the particle type, i, and the momentum, p, and the corresponding operators for the neutral particle will be distinguished by having only one subscript, the momentum k. The subscripts on the wave functions $ label both the particle type and the momentum.

The interaction Hamiltonian corresponding to the <£3 theory proposed in Eq. (9.2) is

Wtot = -£int = Ai: blix^ixWx): +A2: Sj(®)*2(i)0(a;): ¿3(x): .

The perturbation theory for the time translation operator, worked out in Sec. 3.1, can also be applied to a perturbative treatment of interacting second quantized theories, and we will carry it over without further discussion. Expressed in terms of Hamiltonian densities, the time translation operator to second order is

U! = l-iJ d4xnint(x) + ^-J dixldix2T(Hint{xi)'Hint{x2)) + ... ,


where the T operation is the time-ordered product of the Hamiltonian densities, with the later times on the left.

In the next few sections, we will discuss particle decay (which happens in first order in <t>3 theory) and particle scattering (which happens in second order). Our discussion here parallels the discussion in Sec. 3.2 but is more general because all particles are treated relativistically, with the possibility of particle production and annihilation, and there is no longer any fixed center of force. The latter allows us to conserve momentum as well as energy.


The Hamiltonian (9.6) will permit the neutral particle to decay into a particle and antiparticle of type-1 if p. > 2mi. In this section we will calculate this decay rate and the corresponding lifetime of the neutral particle. While the details of this calculation are given for this example, nearly all of our results apply to any decay process and thus are very general. We will extract the general features of the calculation as we go along. The S-matrix for decay is

where k is the momentum of the neutral, heavy particle and p and p' are the momenta of particles and antiparticles of type-1. The bar over a momentum variable will be used to denote an antiparticle. To lowest order in the interaction, this matrix element is

where, if A2 ^ 0, the term with $2 $2 must be zero because the interaction term is normal ordered and there are no annihilation or creation operators for particles of type-2 in either the initial or the final state to prevent these operators from acting directly on the vacuum state and giving zero. Now, recalling the expansions for and <j> given in Eq. (9.5), and the commutation relations satisfied by these operators, we see that the only non-zero term can come from the a terms in (p, the a\ terms in ${, and the c\ terms in Furthermore, using the result

the momentum sums in these three fields must collapse to a single term, giving simply


where the calculation was carried out in the rest system of the neutral particle, so u)(k) = ¡j. and E\(p) = Ei(p'), and the relative momenta of the two decay products is obtained from the equation so the final result can be written

where r is the lifetime.


Decays exhibit the following features:

• The decay rate is proportional to A2, and hence, if Ai is small so that the decay is well described by the lowest order perturbative result, the magnitude of the coupling constant Ai can be determined from the decay.

• The decay will not take place unless /i > 2mi.

• The decay rate is proportional to the phase space, which for this simple S wave two-body decay is p(p.;mumi)

If /i is very close to 2mi, the rate is low (the lifetime long), even if Ai is large.

The phase space of an n-particle decay is defined to be d3px ■■ ■ d3pn p(p\m i,

where k is the four-momentum of the decaying particle and k2 = p2. The phase space p is an integral operator, but if it acts on a constant yVf-matrix, all the integrals can be carried out and it can be reduced to a known function of the masses. Evaluation the phase space integral is a useful way to estimate many body decay rates and cross sections. For two-body decays it is easy to work out, and the task of obtaining the general result is left to the reader (Prob. 9.4).


To describe scattering, we need to go to second order in the </>3 interaction Hamil-tonian (a (fi4, term would be needed for scattering to occur in lowest order). As our first example, consider the case of a particle of type-1 scattering from a particle of type-2. The process is represented diagrammatically in Fig. 9.1, where the

(O |T (4>(R + ir) d>(R - ir))| 0) = (0|T (0(r)0(O)) |0) (9.27)

around the pole in the complex plane, the e2 term in the last denominator can be ignored, and it' — 2itujk is completely equivalent to it (because u>k > 0), and in all subsequent calculations we will assume this equivalence by taking e' = e without comment. Putting this all together gives

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