Fig. 12.27 The complex u plane showing motion of uq, the end point of the contour C'2, as Mg increases. The contour Cj begins at the fixed point (mi + £i)2 and m be deformed in order to avoid the moving singularity.

This displays the imaginary part as a dispersion integral with singularities along the real axis from a to uq. The upper limit, uq, will turn out to be the same «0 which appears in (12.105), but the lower limit (which, in this application, is actually three numbers describing two disconnected line segments) will play no role in the subsequent discussion. The locus of singularities, and hence the value of uo, can be found from the zeros of the denominator, which is a function of both u and Mg rrc,

where 2 is the cosine of the scattering angle, Eg and Ex are the energies of the bound state and the on-shell intermediate particle 1 in the rest system of the final virtual particle 1, and p and k are the magnitudes of the respective three-momenta. Expressing these momenta and energies in terms of the energy y/u, one can find uo. which is the largest value of u at which the denominator (12.114) is zero (which occurrs at the end point z = 1 of the angular integration). (This is a straightforward but tedious calculation; see Prob. 12.3.)

Now examine (Prob. 12.3) the behavior of this upper limit u0 as a function of the bound state mass Mg. Observe that u0 < (mi + fi)2 for small Mg, but that as Mg increases, u0 increases to a maximum value of (mi + /i)2 and then decreases. The critical value of Mg at which u0 is equal to this maximum is easily found by differentiating u0 with respect to Mg and is

Furthermore, if we give Mg a small imaginary part (in order to define the singular denominators) we can show that u0 moves in such a way that it circles above the point (mi + /i)2. Specifically, when Mg = Mc2rit ± it,

The significant fact here is that ReuC[lt > (mj + fi)2, even if only by an infinitesimal amount. Therefore, the moving upper limit of the integral (12.113)

anomalous region exchanged particle on-shell

Fig. 12.28 The discontinuity in the anomalous region has the exchanged particle on-mass-shell, as in the spectator equation.

anomalous region exchanged particle on-shell

Fig. 12.28 The discontinuity in the anomalous region has the exchanged particle on-mass-shell, as in the spectator equation.

follows the path C2 shown in Fig. 12.27. If Ci is the cut from (mi + p.)2 to oo which defines the original dispersion integral (12.105), and if the overall dispersion integral (12.105) is to be a single analytic function for all values of Mg, then this cut must be deformed into the complex plane in order to avoid the moving singularity at the end of the contour C2 as Mg increases beyond Mc2rit. The contour C\ then surrounds the path of integration C2. This deformation is illustrated in Fig. 12.27. As the bound state mass increases, the protruding branch cut continues to move to the left toward smaller values of uo, moving the anomalous threshold further and further toward m2, as suggested in the figure. This is the mathematical origin of the anomalous threshold; the physical origin has already been discussed.

Finally, observe that the integrand of the new dispersion integral obtained from Eq. (12.105) in the anomalous region is the discontinuity (or imaginary part) of the dispersion integral for the exchanged particle pole, Eq. (12.104). But this integral is only singular when the exchanged denominator is singular, which means that the exchanged particle is on-shell. We see that the contribution in the anomalous region, which is closest to the physical region when u ~ m2, arises from the condition that the internal particle 2 be on-shell. In this way we recover the spectator equation, as illustrated in Fig. 12.28. Another way to describe the spectator equation is to observe that it sums up the anomalous contributions exactly.

We now turn to the study of gauge symmetries and gauge field theories, which will occupy our attention for the remainder of this book.


12.1 Write down the spectator equation for a Dirac particle of mass m and a scalar particle of mass m2, exchanging a scalar meson of mass /¿, and show that it



Relativistic Quantum Mechanics and Field Theory

FRANZ GROSS Copyright© 2004 WILEY-VCH Verlag GmbH


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