PHYSICS

V. S. BARASHENKOV and XIANG DING-CHANG

THE REAL PART OF THE AMPLITUDE OF ELASTIC SCATTERING AT HIGH ENERGIES

(Presented by Academician N. N. Bogolyubov, 21 IV 1960)

1. In paper (¹) the “sum rule” was obtained

[
\Delta(\mu)-\Delta(\infty)=\frac{4f^2}{\mu^2}+\frac{1}{2\pi^2}\int_\mu^\infty \frac{dE}{k}\,[\sigma_-(E)-\sigma_+(E)],
\tag{1}
]

where

[
\Delta(E)=[D_-(E)-D_+(E)]/E;
]

(D_\pm(E)) is the real part of the elastic-scattering amplitude; (\sigma_\pm(E)) are the cross sections for the interaction of (\pi^+)- and (\pi^-)-mesons with the proton. If the value of the constant (\Delta(\infty)) is known, relation (1) proves very useful for applications of dispersion relations.

We shall show that (\Delta(\infty)=0), and shall consider some consequences of this relation.

Experimental data indicate that the interaction cross sections of particles at high energies become constant or, at any rate, do not increase (²). In accordance with these data we shall assume that at energies (E \ge a) the interaction cross sections have the form

[
\sigma_\pm(E)=\sigma_\pm+\alpha_\pm/E+\beta_\pm/E^2+f_\pm(E),
\tag{2}
]

where (\sigma_\pm) are the limiting values of the cross sections; (\alpha_\pm) and (\beta_\pm) are constant coefficients; (f_\pm(E)) are small functions tending to zero as (E) increases.

Within the accuracy of present-day experiments one may put (f_\pm(E)=0) at high energies. As will be seen below, the presence of the functions (f_\pm(E)) does not fundamentally change our arguments.

Taking expression (2) into account, by means of dispersion relations the real part of the elastic-scattering amplitude (D_\pm(E)) can be represented in the form of a series

[
D(E)=A_1E\ln E+B_1E+A_0\ln E+B_0+A_{-1}\ln E/E+O(1/E)
\tag{3}
]

with known coefficients (A) and (B). Some of these coefficients must be equal to zero. To show this, let us represent the elastic-scattering cross section in the form

[
\sigma_{el}=\pi\lambda^2\sum_{l=0}^{\infty}(2l+1)\left|1-\beta_l e^{2i\delta_l}\right|^2 =
]

[
=\pi\lambda^2\sum_{l=0}^{\infty}(2l+1)(1-\beta_l)^2
+4\pi\lambda^2\sum_{l=0}^{\infty}(2l+1)\beta_l\sin^2\delta_l
=\Sigma_1+\Sigma_2.
\tag{4}
]

Here the first term is entirely due to inelastic processes ((\beta_l\ne 1)) and represents the cross section of purely diffractive scattering. The second term is associated with nondiffractive scattering.

For (l \gg 1),

[
\Sigma_2 \simeq 8\pi \lambda^2 \int_0^\infty \beta_l \delta_l^2\, dl
\simeq 8\pi \int_0^R \rho \beta(\rho E)\delta^2(\rho E)\, d\rho
= \beta(E)\delta^2(E)\,\mathrm{const},
\tag{5}
]

where (\rho=\lambda l=l/E) is the impact parameter, and (\beta(E)) and (\delta(E)) are the mean values of the functions (\beta_l) and (\delta_l) in the region of action of the nuclear forces. (Clearly, (\beta_l \simeq 0) for (l>R/\lambda), where (R) is the effective radius of the nuclear forces.)

Since at high energies all scattering becomes purely diffractive ((^3)), (\beta(E)\to \mathrm{const}), (\delta(E)\to 0). It follows from this that the real part of the amplitude

[
D(E)=2\lambda \sum_{l=0}^{\infty}(2l+1)\beta_l \sin 2\delta_l
\simeq E\delta(E)\cdot \mathrm{const}
\tag{6}
]

in any case grows more slowly than (E), and the coefficients (A_1) and (B_1) in expansion (3) must be equal to zero:

[
A_1=\frac{1}{4\pi^2}(\sigma_+-\sigma_-)=0;
\tag{7}
]

[
B_1 \equiv \frac{1}{2\mu}(D_-^0-D_+^0)-\frac{2f^2}{\mu^2}
+\frac{1}{4\pi^2}\int_\mu \frac{dE}{k}\,[\sigma_+(E)-\sigma_-(E)]+
]

[
+\frac{1}{4\pi^2 a}\left[\alpha_+-\alpha_-+\frac{1}{2a}(\beta_+-\beta_-)\right]=0.
\tag{8}
]

The first of these relations was previously obtained by I. Ya. Pomeranchuk ((^4))*, and from comparison of the second with (1) for (a\to\infty) it follows that (\Delta(\infty)=0).

1. At present there is no theory that would determine how rapidly the nondiffractive part of the scattering (\Sigma_2) decreases with increasing energy. The optical model often used for interpreting experimental data gives one or another energy behavior of this cross section depending on various assumptions about the structure of the interacting particles. However, the inverse formulation of the problem is possible: on the basis of experimental data, to obtain information about the dependence (\Sigma_2(E)).

Analysis of the data presently known on the interaction of (\pi)-mesons with nucleons leads to the conclusion that the cross section (\Sigma_2) decreases with increasing energy, in any case, no faster than (1/E^2). Indeed, by virtue of relations (5) and (6), a faster decrease of (\Sigma_2) is possible only under the condition that the coefficients (A_0) and (B_0) in expansion (3) vanish:

[
A_0 \equiv -\frac{1}{4\pi^2}(\alpha_+ + \alpha_-)=0;
\tag{9}
]

[
B_0=\frac{1}{2}(D_-^0+D_+^0)+\frac{f^2}{M}
-\frac{1}{4\pi^2}\int_\mu^a \frac{E\,dE}{k}\,[\sigma_+(E)+\sigma_-(E)]+
]

[
+\frac{a}{4\pi^2}\left[2\sigma-\frac{1}{a^2}(\beta_+ + \beta_-)\right]=0,
\tag{10}
]

where (\sigma_+=\sigma_-=\sigma).

* This conclusion also follows from the work of S. Z. Belen’kii ((^6)), where it is shown that for (E\to\infty) the real part of the elastic-scattering amplitude is much smaller than its imaginary part: (D(E)\ll A(E)\simeq \sigma_t E/4\pi). However, from this it still cannot be concluded that (D(E)) grows more slowly than (E).

We give the values of (B_0), calculated from the experimental values of the cross sections (\sigma_{\pm}(E)) (see (2)), for (D_-^0=(0.121\pm0.018)\cdot10^{-13}) cm; (D_+^0=-(0.170\pm0.016)\cdot10^{-13}) cm; (f^2=0.08\pm0.01)*.

As can be seen, these values differ noticeably from zero. One can obtain the values (B_0=0) if one assumes that in the experimental values of the cross sections (\sigma_{\pm}(E)) at energies (Ea\gg(3\div4)) BeV, while in the interval (4) BeV (