Analysis of Experimental Data on the Photoproduction of ?+-Mesons at a Photon Energy of 230 MeV
Abstract Generated abstract
This paper analyzes experimental data on positive pion photoproduction on nucleons at a photon energy of 230 MeV in order to extract quantities related to multipole amplitudes, which individual observables alone do not adequately test. Using measured asymmetries and unpolarized differential cross sections from the authors and other laboratories, the authors compute the perpendicular photon cross section and fit its angular dependence by least squares to obtain coefficients associated with assumed dominant s and p wave contributions. Two data sets give consistent fitted coefficients with acceptable chi square probabilities, while comparison with a Frascati fit shows discrepancies in two coefficients, possibly due to different energy ranges used. The study also applies the Nigro and Schiavuta formulation to estimate additional multipole related parameters, yielding five independent quantities proposed as useful criteria for evaluating theoretical models.
Full Text
ANALYSIS OF EXPERIMENTAL DATA ON PHOTOPRODUCTION OF $\pi^+$ MESONS AT A PHOTON ENERGY OF 230 MeV
V. N. ZABAEV, V. M. KUZNETSOV, O. I. STUKOV
(Presented by the scientific-technical seminar of the High-Energy Laboratory of the Nuclear Physics Research Institute)
Comparisons of experimental data on photoproduction of $\pi^+$ mesons on nucleons with the predictions of a number of theoretical models [1—3] have now shown that individual measured quantities—namely, differential cross sections $\dfrac{d\sigma}{d\Omega}$ (unpol.), asymmetry coefficients $\Sigma$, etc.—are not criteria for the validity of any given model. This is explained by the good agreement of the measurements [4—7] with the existing models. However, the multipole amplitudes calculated on the basis of these theories differ. In this connection it is desirable to carry out the comparison directly through the multipole amplitudes, which can be estimated only from a joint analysis of various measured quantities.
The aim of the present work was to calculate independent numerical values—functions of the multipole amplitudes—on the basis of our experimental data [5] and the data of other laboratories [5, 6, 7] for a photon energy of 230 MeV.
Knowing the asymmetry $\Sigma(E_\gamma,\theta_\pi)$ and the photoproduction cross section of $\pi^+$ mesons by unpolarized photons $\dfrac{d\sigma}{d\Omega}$ (unpol.), one can calculate the quantity
\[ \frac{d\sigma_{\perp}}{d\Omega} = \frac{d\sigma}{d\Omega}(\text{unpol.})\,[1+\Sigma(\theta_\pi,E_\gamma)], \tag{1} \]
where $\dfrac{d\sigma_{\perp}}{d\Omega}$ is the differential cross section for photons with an electric vector perpendicular to the reaction plane. This cross section is of special interest, since it is free from the influence of the photoelectric term.
There exists a hypothesis that only $s$ and $p$ waves contribute to $\dfrac{d\sigma_{\perp}}{d\Omega}$; that is, it can be written as
\[ \frac{d\sigma_{\perp}}{d\Omega} = A_1 + A_2 \cos\theta_\pi + A_3 \cos^2\theta_\pi . \tag{2} \]
Thus, by confirming formula (2), one can obtain the coefficients $A_1$, $A_2$, $A_3$, which are functions of the multipoles. The procedure for find-
the coefficients \(A_i\) is as follows: a) from the experimental data \(\Sigma(E_\gamma,\theta_\pi)\) one calculates \(\dfrac{d\sigma_\perp}{d\Omega}E_\gamma,\theta_\pi\), b) the obtained values \(\dfrac{d\sigma_\perp}{d\Omega}(E_\gamma,\theta_\pi)\) are approximated by the method of least squares [8]. The curve is drawn through the points in such a way that the condition
\[ M=\sum_{r=1}^{n}\frac{(Y_{\mathrm{exp}}-Y_{\mathrm{theor}})^2}{\sigma_r^2}=\min, \tag{3} \]
is satisfied, where \(Y_{\mathrm{exp}}(Y_{\mathrm{theor}})\) are the experimental (approximated) values of \(\dfrac{d\sigma_\perp}{d\Omega}\),
\[ \sigma_r^2 \]
is the variance of the corresponding measurement.
The equations for \(A_i\) will be
\[ A_i=\sum_{k=1}^{p}\sum_{r=1}^{n} \frac{\left(\dfrac{d\sigma_\perp}{d\Omega}\right)_r f_k(x_r)}{\sigma_r^2} [D(A)]_{ki}, \tag{4} \]
where \(n\) is the number of selected measurements;
\(p\) is the number of unknown coefficients, with
\[ f_1(x_r)=1;\quad f_2(x_r)=\cos\theta_\pi;\quad f_3(x_r)=\cos^2\theta_\pi; \]
\[ \sigma_i^2 \]
is the variance of each value of \(\dfrac{d\sigma_\perp}{d\Omega}\);
\[ C=[D^{-1}(A)]_{ki} \]
is the error matrix, where
\[ [D^{-1}(A)]_{ki}=\sum_{r=1}^{n} \frac{f_k(x_r)\cdot f_i(x_r)}{\sigma_r^2}. \]
By this procedure the values of the coefficients were calculated for two sets of experimental data.
- In the first set only our values of the asymmetry and \(\dfrac{d\sigma}{d\Omega}\) (unpol.) measured at FIAN [7] and coinciding with them in \(E_\gamma\) and \(\theta_\pi\) were taken. At angles \(\theta_\pi=0^\circ\) and \(180^\circ\) the expected values \(\Sigma(E_\gamma,\theta_\pi)=0\) were used.
As a result of the fit to the first set, the following curve was obtained (Fig. 1):
\[ \frac{d\sigma_\perp}{d\Omega} = (14.84-3.10\cos\theta_\pi-4.45\cos^2\theta)\cdot 10^{-30} \frac{\mathrm{cm}^2}{\mathrm{sr}} \tag{5} \]
The error matrix for this case is:
\[ D= \begin{pmatrix} 0.689 & -0.174 & -1.022\\ -0.174 & 0.840 & 0.984\\ -1.022 & 0.984 & 2.431 \end{pmatrix} \tag{6} \]
Since the quantity \(M\) obeys a \(\chi^2\) distribution with the number of degrees of freedom equal to \((n-p)\), the \(\chi^2\) goodness-of-fit criterion may be used to assess the quality of the fit. For the first set we have \(p(\chi^2)=0.96\). Such fairly good agreement makes it possible to conclude, within our statistics, that the interaction proceeds mainly through \(s\)- and \(p\)-states for \(E_\gamma=230\) MeV [10]. The set considered has the advantage that, for the calculation of \(\dfrac{d\sigma_\perp}{d\Omega}\), there is no need—
to use the interpolated values of \(\dfrac{d\sigma}{d\Omega}\) (unpol.). This eliminates a number of uncertainties in the cross sections, although a smaller amount of data is used, which leads to a certain increase in the errors of the coefficients.
- To reduce these errors, the second set included the asymmetry data of Frascati [6] and Stanford [5]. For them it was necessary to use the values of \(\dfrac{d\sigma}{d\Omega}\) (unpol.) [7]. For the second set the following curve was obtained:
\[ \frac{d\sigma_{\perp}}{d\Omega} = \left(14.55 - 3.69\cos\theta_{\pi} - 4.20\cos^{2}\theta_{\pi}\right) \cdot 10^{-30}\ \frac{\mathrm{cm}^{2}}{\mathrm{sr}} . \tag{7} \]
The error matrix for this case is:
\[ D = \begin{matrix} 0.091 & -0.018 & -0.013\\ -0.018 & 0.796 & 0.784\\ -0.013 & 0.784 & 1.207 \end{matrix} \tag{8} \]
Fig. 1. Angular distribution of differential cross sections for photon energies of 230 MeV.
In Table 1 the data of both fits are presented, where \(\nu\) is the number of degrees of freedom.
Table 1
| Set | \(A_{1}\) | \(A_{2}\) | \(A_{3}\) | \(\chi^{2}\) | \(\nu\) | \(P(\chi^{2})\) |
|---|---|---|---|---|---|---|
| I | \(14.84 \pm 0.83\) | \(-3.10 \pm 0.915\) | \(-4.45 \pm 1.56\) | 0.625 | 4 | 0.96 |
| II | \(14.55 \pm 0.3\) | \(-3.67 \pm 0.89\) | \(-4.20 \pm 1.12\) | 1.05 | 10 | 0.73 |
As expected, in the second set \(\chi^{2}\) became worse, but the error magnitudes decreased. Table 2 presents our data in comparison with the results of the Frascati fit.
Table 2
| Set | \(A_1\) | \(A_2\) | \(A_3\) | \(\chi^2\) | \(\nu\) | \(P(\chi^2)\) |
|---|---|---|---|---|---|---|
| II Tomsk | \(14.55 \pm 0.3\) | \(-3.67 \pm 0.89\) | \(-4.10 \pm 1.12\) | 7.05 | 10 | 0.73 |
| III Frascati | \(15.4 \pm 0.4\) | \(-5.16 \pm 0.12\) | \(-5.83 \pm 0.26\) | — | — | 0.71 |
Comparison shows that the coefficients \(A_2\) and \(A_3\) in these fits differ. One possible reason for the discrepancy is that in the Frascati fit values of \(\Sigma(\theta_\pi,E_\gamma)\) were used that lie in a wider range of energies \(E_\gamma\) than in our case.
- Nigro and Skjævuta [11] showed that in photoproduction of \(\pi^+\)-mesons one can obtain from experiment the quantity \([\Delta]^M\), linear in \((1-\beta\cos\theta_\pi)\), if the contribution of the photoelectric term to this process is significant:
\[ [\Delta]^M = \alpha(1-\beta\cos\theta_\pi)+\gamma, \tag{9} \]
where \(\alpha\) and \(\gamma\) are functions of the multipoles.
The presence of a contribution from the photoelectric term can be confirmed by carrying out the expansion:
\[ -\frac{k}{q}\cdot 1( - \beta \cos\theta_\pi)^2 \frac{d\sigma}{d\Omega}_{\text{unpol}} \frac{\Sigma(E_\gamma,\sigma_\pi)}{\sin^2\theta_\pi} = \alpha'(1-\beta\cos\theta_\pi)^2+ \gamma'(1-\beta\cos\theta_\pi)+\delta'. \tag{10} \]
In this case the coefficient \(\delta'\) determines the contribution of the photoelectric term. Using the least-squares method we obtained the value \(\delta'=3.8\pm5.3\). Having confirmed the premise of the hypothesis of Nigro and Skjævuta, we used the same procedure to find the coefficients \(\alpha\) and \(\gamma\). In doing so, the approximated values \([\Delta]^M\) were calculated by the formula:
\[ [\Delta]^M = \frac{ -\dfrac{k}{q}\left[ 2\dfrac{d\sigma}{d\Omega}_{\text{unpol}}\Sigma\cdot \dfrac{(1-\beta\cos)\theta_\pi)^2}{\sin^2\theta_\pi} \right]-\delta }{ (1-\beta\cos\theta_\pi) } \tag{11} \]
where \(k\) and \(q\) are the momenta of the photon and the \(\pi^+\)-meson.
The theoretical value of \(\delta\), according to [9], was chosen equal to \(4.3\ \mu\text{b}/\text{sr}\). To determine the coefficients \(\alpha\) and \(\gamma\), data [5–6, 7, 9] were used. The results are given in Table 3.
Table 3
| \(\gamma\) | \(\alpha\) | \(\chi^2\) | \(\nu\) | \(P(\chi^2)\) |
|---|---|---|---|---|
| \(4.15 \pm 2.8\) | \(-6.45 \pm 3.0\) | 3.55 | 9 | 0.9 |
Thus, we have succeeded in determining 5 independent quantities, \(A_1\), \(A_2\), \(A_3\), \(\alpha\), \(\gamma\), necessary for calculating the multipole amplitudes, which can serve as a criterion for the validity of existing theoretical models.
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