J 2n3J0 f

where q2 = (p — p')2 is the three-momentum transferred by the scattering. Do not forget that the "pole" at k'Q = 0 is to be ignored, so that only the double pole at k'0 = y/fi2 + k2 + q2x( 1 - x)

is evaluated in going from the first to second line of Eq. (12.19).

If q2 = 0 (forward scattering) we see that the ratio of the meson pole contributions to the leading pole contribution (12.5), with p = i6, is

M\ eading

If the meson pole terms are regarded as a correction to the leading contribution, and if the mass of the exchanged meson is very small (or zero), then the correction is of order

— = - , 12.21 m c where v is the typical velocity of the bound constituents. This is a significant relativistic correction, and we conclude that the meson poles terms cannot be ignored, unless they are canceled by some other contribution.

It turns out that the contribution from the crossed ladder diagram, Fig. 12.1B, is of the same size as the meson pole contribution (12.19), and hence the crossed ladder diagram must be included in order to obtain an accurate relativistic description of bound states. In fact, for the 03 example under discussion, and for a large class of other theories, the contribution from the crossed ladder diagram cancels the meson pole contribution. Before we show this, it is instructive to recast (12.19) in a dispersion, or spectral, form.

Since the meson pole contributions to M are real and local (i.e., depend on q2 only), it is appropriate to regard them as a candidate for a new contribution to the meson exchange potential between particles 1 and 2. Recall that the potential

12.2 THE ROLE OF CROSSED LADDERS 383

Aq complex plane

Fig. 12.6 The location of the singularities of the crossed box diagram in the complex kg plane. Compare with Fig. 12.4 and note that the only difference is that ploe 1 is replaced by 1 x and pole 8 is replaced by 8 x .

are negligible, the major difference between the box and crossed box is that pole 1, which dominated the box, has moved from the lower half plane to the upper half plane. These two poles are located at:

Evaluating the crossed box by closing the contour in the lower half plane (as we did before) leads us to the following observations:

• The contribution which dominated the box (pole 1), is no longer present in the lower half plane, and hence this leading contribution is missing from the crossed box. (The curious reader may wonder how this argument would be affected if we were to close the contour in the upper half plane. In this case the two poles 5 and 1 x would cancel, giving a similar result.) We conclude immediately that the crossed box is smaller than the box by the ratio (12.20).

• The meson poles dominate the crossed box, and the only difference between their contribution to the crossed box and the box is the denominator D2.

Introducing k0 = k'0 + E2(p), as we did in our discussion of the meson pole contribution to the box, these two denominators become k2

Fig. 12.11 Diagrammatic representation of the derivation of Hq. (12.36).
Fig. 12.14 Diagrammatic representation of the Born series generated by the integral equation (12.40).

12.4 NORMALIZATION OF BOUND STATES 391

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