Subatomic Physics

184 Structure of Subatomic Particles
Figure 6.32: Slope parameters, b, corresponding to the cross sections shown in Fig. 6.30 [From
M.M. Block and F. Halzen, Phys. Rev. D 63, 114004 (2001).] It is seen from the figure that b(¯pp)
approaches b(pp) asymptotically.
can be interpreted in terms of a Gaussian profile function, as given in Eq. (6.99).
Identification of Eqs. (6.100) and (6.103) leads to the relation
ρo = �(2b) 1/2 . (6.104)
ρo characterizes the width of the Gaussian profile function describing the scattering
of two extended protons by hadronic forces. It is therefore not legitimate to compare
ρ 2 o, or a corresponding mean-square radius, directly with the mean-square radius of
the proton as determined with electromagnetic probes. Nevertheless, it is reassuring
that the two measures of the proton size are comparable: The electromagnetic radius
is given by Eq. (6.46) as 〈r 2 〉≈0.7 fm, whereas a value of b = 10( GeV/c) −2 ,taken
from Fig. 6.32, leads to ρ0 ≈ 0.9 fm.
The “size” of the proton and slope parameter b(s) are related through
Eq. (6.104); a constant ρ0 implies a constant b(s). Fig. (6.32) shows, however,
that at the highest energies b(s) increases logarithmically with the square of the
c.m. energy, s. Sinceb(s) describes the width of the diffraction peak, an increase of
b(s) means a shrinking diffraction peak, and it suggests an increase in the size, ρ0,
of the interaction region. This behavior can be understood with a geometric picture
in which the area of the interaction region is related to the total cross section. (72)
We saw in Fig. (6.30) that the total cross section increases with s or laboratory
momenta at very high energies. Indeed, the ratio b/σtot ≈ constant, (73) as can be
noted from a comparison of Figs. (6.32) and (6.30).
72 M. Kamran, Phys. Rep. 108, 275 (1984); K. Goulianos, Phys. Rep. 101, 169 (1983).
73 M.M. Block and F. Halzen, Phys. Rev. D 63, 114004 (2001).