Bound States And Unitarity

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In the previous chapters it has been implicitly assumed that perturbation theory is adequate and that we can obtain a reasonable estimate of the scattering amplitude by calculating a few Feynman diagrams of lowest order. However, there are many problems for which the calculation of a few Feynman diagrams is inadequate. The study of bound states is one of these problems. A bound state produces a pole in the scattering matrix in the channel in which it appears. If the bound state is truly composite, no such pole exists in any Feynman diagram (or any finite sum); a pole can only be generated by an infinite sum. The same observations apply to the description of low energy elastic scattering; an exact treatment of unitary requires an infinite number of diagrams.

Ideally, we would like to sum all Feynman diagrams which describe the reaction; if we could do this, we assume we would have the correct answer. However, this is not possible in general, and we must settle for an infinite sum of a particular class of diagrams we believe to be particularly important physically. This is done by finding an integral equation, the solution of which can be interpreted as the sum of the class of diagrams under consideration. The equation used depends on the physics of the problem.

The only problems which will be discussed in this chapter are those in which long range peripheral interactions are expected to be important. We consider systems of two heavy particles interacting through the exchange of light mesons, and further assume that self-energy diagrams and vertex corrections can be ignored or treated phenomenologically. Systems which may be approximated in such a way include atomic, nuclear, and heavy quark bound states and low energy elastic scattering (below particle production thresholds).

We will first consider the ladder and crossed ladder sums of Feynman diagrams, which leads to a discussion of the Bethe-Salpeter equation [SB 51], and then to other relativistic two-body equations. All of these equations can be shown to produce bound states and to satisfy elastic unitary. We conclude this chapter with a brief discussion of the application of dispersion theory to bound states, which requires an understanding of anomalous thresholds.

Fig. 12.2 The ladder diagrams to sixth order.

Here we assume the external particles are on-shell, so that if the diagram is evaluated in the center of mass frame, W = E\{p) + E2{p) — E\{p') + E2{p').

kg conpl ex plane

8 •

7 6 5 • • •

1 2 3


Fig. 12.4 The location of the singularities of the box diagram in the complex kg plane, when |£| is small. As |Jfc| increases, the singularities in the lower half plane move to the right and those in the upper half plane move to the left.

Fig. 12.4 The location of the singularities of the box diagram in the complex kg plane, when |£| is small. As |Jfc| increases, the singularities in the lower half plane move to the right and those in the upper half plane move to the left.

To estimate this diagram near threshold (where p and p' are small) it is helpful to examine its singularity structure in the complex ko plane, as we did in Sec. 11.7 when discussing the structure of the vertex function. There are eight poles, as shown in Fig. 12.4. These are found from the zeros of the denominators in (12.2) which can be factored into eight factors,

8 1 D0 = (w - E2(p) + k0- ie) (u + E2{p) - ko - ie)

6 2 D'0 = (u/ - E2(p') + fco - ie) (a/ + E2(p') -kg- ie) , 7 3

where the numbering of the factors in (12.3) corresponds to the numbering of the corresponding poles in Fig. 12.4. If we evaluate the box diagram by closing the contour in the lower half ko complex plane, we see that the pole at E2 will dominate, because it is very close to the singularity at ko = W — Ex in the upper half plane. Keeping this term only, the box diagram reduces to d3k 1

P - (E2 - E2(P))2] [u/2 - (E2 - E2(P'))2] A j A^ f d^fc

Conjecture: Even if the effective <p3 coupling constant glR is much less than unity, so that the use of perturbation theory would normally be justified, there may still exist a bound state. This can occur if the exchange meson mass /i and the wave number <5 are small enough to guarantee that all of the diagrams in the ladder sum are of comparable magnitude, so that the sum of an infinite number of ladder diagrams will diverge, reflecting the appearance of a pole in the scattering matrix M. A sufficient condition for this to occur is that

9eff m

Our argument is not sufficiently polished or complete to constitute a "proof" of the above conjecture; in particular, we have not demonstrated that (12.9) is sufficient to insure that the sixth and higher order ladder diagrams are of comparable size to the fourth order diagram we just estimated [Gr 69], But we will see below that relativistic bound state equations have solutions when condition (12.9) is satisfied, and our main purpose here is to provide a physical understanding of why this is so.

Note that (12.9) tells us that a potential with a finite range (/i / 0) will have a bound state (6 > 0) only when qk. H>i

It also tells us that a potential with an infinite range (/i = 0, as in the Coulomb potential) will always have a bound state. In this case, (12.9) tells us that the ground state energy, which we can estimate from -62/2m, will be of the order of


Recalling that the ground state of a Coulomb potential has a binding energy of E0 — -ma2/2, we see that this is consistent with (12.11).

The condition (12.9) for a finite range potential can also be understood non-relativistically. Consider a particle of mass m bound by a Hulthen potential (introduced in Sec. 3.5)

Then the (exact) solution of the 5-state Schrodinger equation has the form ip(r) = N (1

Anticipating the fact that the value of k'0 fixed by the double pole in (12.18) is much larger than the (p2 - k2)/2mi terms in (12.17), we may approximate D ~ fcg2 and complete the square in (12.18) by shifting k —► k +px +p'{ 1 - x), obtaining

M\>o meson poles

16 miTO2 327r2mim2

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