Formation of \(\pi^+\)-Mesons on Hydrogen by Linearly Polarized Photons

V. N. Zabaev, V. M. Kuznetsov, B. N. Kalinin, A. P. Potylitsyn,
E. V. Repenko, O. I. Stukov, N. M. Filipenko

(Submitted by the scientific-technical seminar of the High-Energy Laboratory, Research Institute of Nuclear Physics)

Introduction

Experimental studies of pion photoproduction on nucleons consist in the study of four elementary processes:

\[ \gamma + p \to \pi^+ + n \tag{I} \]

\[ \gamma + p \to \pi^0 + p \tag{II} \]

\[ (\gamma + n \to \pi^- + p) \tag{III} \]

\[ (\gamma + n \to \pi^0 + n). \tag{IV} \]

To create a consistent theory of photoproduction, it is necessary to have experimental data for these reactions with all possible combinations of photon and nucleon states in the initial and final states. In all, 64 different combinations can be found; of these, for constructing the amplitude of the process \(\gamma + N \to \pi + N\), which contains four unknown complex functions, it is sufficient to carry out eight independent experiments for all photon energies and pion emission angles. Such a minimal set of experiments is called a “complete experiment.”

At present, only the first two processes (I) and (II) have been studied comparatively well, on unpolarized photon beams and with unpolarized nucleons. The study of reactions (IV) and (III) is difficult, since there are no “free” neutrons in nature.

Experiments with polarized photon beams and targets involve great technical difficulties. However, at present there are possibilities for carrying out these experiments as well. Polarized photon beams have been obtained at Stanford, Frascati, Kharkov, and Tomsk and are being successfully used to study meson photoproduction processes.

We note that analysis of only the differential cross sections for \(\pi\)-meson production on hydrogen by unpolarized photons cannot serve as a criterion for the validity of existing theoretical models [1—3]. Therefore there is a need for experimental data with polarized photons and nucleons. It is possible that only they can serve as a sufficient criterion.

In the present work, the asymmetry coefficients of the process of \(\pi^+\)-meson production by linearly polarized photons were measured in the region of large angles, where there are almost no experimental data. For the measurements, the emission angles of the \(\pi^+\)-mesons and the photon energies were chosen

such for which the differential cross sections are known with high accuracy.

1. Amplitude of the process of \(\pi^+\)-meson production by linearly polarized photons and experimentally studied parameters

From consideration of the amplitude of pion photoproduction on nucleons one can obtain the following expression for the differential transverse cross section:

\[ \frac{d\sigma}{d\Omega} = \frac{d\sigma}{d\Omega}_{\text{unpol}} - \alpha(0)\sin^2\theta\cos 2\varphi, \tag{1} \]

where \(\alpha(\Theta)\) is a function of the multipole amplitudes,

\[ \frac{d\sigma}{d\Omega}_{\text{unpol}} \]

is the production cross section for unpolarized photons.

In \(\pi^+\)-meson photoproduction, in addition to the principal mechanism of isobar production, mechanisms not associated with isobars also occur. One of them is the interaction of the photon with the meson current, as a result of which a charged meson is produced. Quantitatively this interaction is determined by the photon polarization vector \(\varepsilon\) and the meson-current vector \(eq\) as \(e\varepsilon q\). This interaction, obviously, is absent if the polarization vector is perpendicular to the reaction plane, defined by the direction of the photon momentum \(k\) and the momentum of the \(\pi^+\)-meson \(q\). If \(\varepsilon\) lies in the reaction plane, then \(e\varepsilon q\) contains the coefficient \(\sin\Theta\), and consequently the contribution of the direct process to the transverse cross section, summed over all directions of polarization, is proportional to \(e^2\sin^2\Theta q^2\). This leads to the appearance in the expression for the amplitude of an additional term, which has been called the photoelectric (or “retarded”) term:

\[ \frac{1}{2kE_\pi}\times \frac{1}{1 - V_\pi\cos\theta}, \tag{2} \]

where \(E_\pi, q\) are the energy and momentum of the \(\pi\)-meson;
\(V_\pi = q/E_\pi\) is the velocity of the \(\pi\)-meson.

It is clear from expression (2) that its contribution becomes significant at small angles and when \(V_\pi\) tends to the speed of light. Further, the photoelectric term will not contribute to the differential cross section

\[ \frac{d\sigma_\perp}{d\Omega} \]

for photons with an electric vector perpendicular to the production plane, and at \(\pi^+\)-meson emission angles of \(0\) and \(180^\circ\).

If the angular distribution of the differential cross sections is represented in the form

\[ \frac{d\sigma}{d\Omega} = \sum_{i=0,1,\ldots}^{n} A_i\cos^i\theta_\pi, \tag{3} \]

where \(A_i\) are functions of the imaginary and real parts of the multipole amplitudes, then one can determine the coefficients \(A_i\) and the states in which the \(\pi^+\)-mesons are produced. If the differential cross sections are measured with unpolarized photons, there is a contribution from the photoelectric term, which introduces into \(\pi^+\)-photoproduction partial waves with \(l>1\) even near threshold, which does not allow the principal production mechanism to be isolated in pure form.

The cross sections

\[ \frac{d\sigma_\perp}{d\Omega}, \]

measured on a linearly polarized photon beam, ...

tons, are free from the contribution of the photoelectric term. Further, in these measurements one can obtain the parameter [4], linear in \((1-\beta\cos\theta_\pi)\):

\[ [\Delta]^M=\alpha(1-\beta\cos\theta_\pi)+\gamma, \tag{4} \]

where the coefficient \(\gamma\) is determined by the interference of the photoelectric amplitude with the photoproduction amplitude. Thus, measurements with linearly polarized photons make it possible to obtain qualitatively new coefficients \(A_i\) and new additional parameters \(\alpha\) and \(\gamma\), needed for carrying out a multipole analysis.

2. Experimental studies of the production of \(\pi^+\)-mesons by linearly polarized photons

In the region of the first resonance \((E_\gamma \sim 300\ \text{MeV})\) such an experiment was first carried out at Stanford [5]. To obtain a polarized photon beam, the results of the work of May [6] were used; he found an analytic expression for calculating the number of polarized photons produced when electrons are decelerated on a thin amorphous target. The magnitude of the polarization and the direction of the electric vector were found as a function of the region used in the photon-beam spectrum. Registration of the process \(\gamma+p\to\pi^+ + n\) was based on detection of the \(\pi^+\)-meson through the decay \(\pi\to\mu\to e\) with the aid of a magnetic analyzer and a telescope of counters. The asymmetry measured was:

\[ \Sigma(\theta_\pi,E_\gamma)= \frac{ \frac{d\sigma_\perp}{d\Omega}(\theta_\pi,E_\gamma)- \frac{d\sigma_\parallel}{d\Omega}(\theta_\pi,E_\gamma) }{ \frac{d\sigma_\perp}{d\Omega}(\theta_\pi,E_\gamma)+ \frac{d\sigma_\parallel}{d\Omega}(\theta_\pi,E_\gamma) } \tag{5} \]

for angles \(\theta_\pi\) in the c.m.s. of \(45^\circ\), \(90^\circ\), and \(135^\circ\), and for photon energies 227, 234, 342, 360, 373 MeV. From an analysis of the results it followed that it is necessary to introduce partial waves with \(l>1\). It was found that the contribution of the photoelectric term is significant. Agreement of the results with the predictions of the theory of MacKinley [7], based on dispersion-relation theory, was not observed even when an attempt was made to combine a phase shift and a constant \(\gamma\pi\rho\)-interaction.

Subsequently, measurements of the asymmetry of photoproduction of \(\pi^+\)-mesons were begun at Frascati [8—10]. To obtain linearly polarized photons, the process of coherent bremsstrahlung of 1 GeV electrons on a crystalline diamond target was used [11]. Registration of the process \(\gamma+p\to\pi^+ + n\) was carried out by detecting the \(\pi^+\)-meson with the aid of a magnetic analyzer and a telescope of scintillation counters. The advantage of this experiment was the high polarization of the \(\gamma\)-beam, amounting on average to 0.35, and the drawback was the background from meson-pair-production processes, reaching 50%.

At Frascati, 62 values of the asymmetry were measured in the energy interval \(E_\gamma\) from 200 to 450 MeV and at angles \(\Theta_\pi=30, 45, 71, 90, 120, 133\), and \(144^\circ\); of these, 54 values were measured in the region of angles of the forward hemisphere, and only 8 values in the backward hemisphere. On the basis of an analysis of the experimental data [12], the supposition was made that in the region up to 300 MeV \(\pi^+\)-mesons are produced in \(s\)-, \(p\)-, and \(d\)-states, contrary to the existing hypothesis of production in \(s\)- and \(p\)-states. An estimate was given of the coefficients \(A_i\) and of the parameters \(\alpha\) and \(\gamma\). It was found that the contribution of the photoelectric term is significant at small angles. Experiment-

experimental data are in agreement with a number of predictions [1, 3, 13, 14] based on the theory of dispersion relations.

3. Calculation of the Beam Polarization

Our measurements of the asymmetry of $\pi^+$-meson production on the proton were carried out at the “Sirius” electron synchrotron. A linearly polarized photon beam was used, and a method was developed for measuring the asymmetry of the process $\gamma+p\to\pi^+ + n$ using a hydrogen-containing polyethylene target. The photon beam was collimated by lead collimators I and II, 20 mm in diameter and 350 mm long. The $\gamma$ beam was then cleaned of charged particles by a magnetic field of $\sim 3000$ oersteds, passed through the target, and was monitored by a Gaussian quantameter [15]; the diameter of the photon beam at the target was 25 mm. Registration of the process $\gamma+p\to\pi^+ + n$ was performed by means of a charged-particle telescope ($\pi^+$ channel) and a neutron detector ($n$ channel), connected in fast coincidence (Fig. 1).

Fig. 1. Experimental setup.

The polarized beam was produced by coherent bremsstrahlung of electrons on a diamond target [16]. The diamond target, of size $10\times 6\times 2$ mm, was oriented by means of a goniometer (by rotation about perpendicular axes, also orthogonal to the incident electron beam). The radiation spectrum was measured with a paired magnetic $\gamma$ spectrometer [17].

Figure 2 shows the spectrum from a tantalum target, and Fig. 3 that from a diamond target.

Coherent bremsstrahlung of electrons on crystalline targets has many features in common with the well-known diffraction of X rays by crystals. In both cases the radiation has sharp maxima due to the ordered arrangement of the centers of radiation. When ultrarelativistic electrons pass through a crystalline target, bremsstrahlung occurs in the Coulomb field of nuclei screened by the electron shell. This radiation is polarized, and its intensity and degree of polarization depend on the parameters of the crystal and on its orientation relative to the incident beam.

The theory of coherent bremsstrahlung on single-crystal targets was developed in [18, 19, 20]. We note that the form of the spectrum and the magnitude of the polarization depend very strongly on the orientation of the target. Their calculations are carried out on the basis of the Bethe–Heitler differential cross section [18, 19].

Figure 4 presents the results of the calculation for the spectrum and degree of polarization as functions of the energy of the $\gamma$ quanta, obtained for crystal orientation angles (relative to the incident electron beam) $\Theta=1^\circ 59'$, $\alpha=76^\circ 50'$. Here $\Theta$ is the angle between the crystal axis $[110]$ and the momentum of the incident electron $p_0$, $\alpha$ is the angle between the cry-

Fig. 2. Spectrum of bremsstrahlung from a tantalum target.

Fig. 3. Measured spectrum of coherent bremsstrahlung from a diamond target.

Fig. 4. Spectrum and polarization of coherent bremsstrahlung for ideal conditions.

Fig. 5. Spectrum and polarization of real coherent bremsstrahlung.

crystallographic plane (001) and a plane drawn through the axis \([110]\) and \(\mathbf{p}_0\).

The orientation of the crystal was chosen on the basis of the following considerations. The experimental error in measuring the asymmetry will decrease with increasing statistics (the number of recorded \(\pi^+\)-mesons) and degree of polarization. Since the statistics are proportional to the radiation intensity, the problem of optimization arises—choosing such a crystal orientation for which the intensity and the degree of polarization will both be sufficiently large.

Mathematically, the problem is formulated as the search for the orientation angles \(\Theta\) and \(\alpha\) at which the time required to accumulate statistics for a prescribed error in the asymmetry, at a given energy of the \(\gamma\)-quanta, is minimized.

The problem is complicated by the following restrictions: the angles \(\Theta\) and \(\alpha\) must lie within the region of angles permitted by the goniometer; the degree of linear polarization is calculated for the plane \(\beta = 0\) (the median plane of the accelerator).

To find the optimal angles \(\Theta\) and \(\alpha\), a program was written for the BESM-4 computer; as a result of the calculations the angles given above were found.

The experimentally measured intensity spectrum for this orientation is shown in Fig. 3. A noticeable discrepancy with the theoretically calculated spectrum is observed (Fig. 4). It may also be expected that the actual value of the degree of polarization will differ from the calculated one.

It should be noted that the theory of Überall, according to which the calculations [18, 19] were carried out, is valid for an infinitely thin ideal crystalline target and for a monoenergetic and monodirectional electron beam. The conditions of the real experiment are, naturally, far from ideal. This explains the difference between the theoretically calculated and the measured intensity spectra.

As already noted, even a small change in the angles at which the electron enters the crystal (i.e., a change in orientation) can lead to a significant change in the spectrum. The principal factors leading to a change in the entrance angles are the following:

a) a crystalline target of finite thickness is used;
b) lattice defects are present;
c) the synchrotron electron beam has a certain angular divergence.

Using the experimental spectrum and the theoretical calculation of the spectrum and polarization, one can determine a value of the degree of polarization closer to reality. By introducing a function of the effective angular distribution of the incident electrons, which takes into account the influence of all these factors, it is possible to calculate more accurately the spectrum and the degree of polarization:

\[ \overline{I}(\theta_0, x) = \int I(\theta', x) K(\theta', \theta_0)\, d\theta', \tag{6} \]

where \(\overline{I}(\Theta_0, x)\) is the radiation intensity averaged over the effective angular distribution of the electron beam;

\(\Theta_0\) is the orientation angle;

\(x = E_\gamma / E_{\max}\) is the energy of the \(\gamma\)-quanta;

\(I(\Theta', x)\) is the radiation intensity of \(\gamma\)-quanta with energy \(x\) at the entrance angle of the primary electron \(\Theta'\) (relative to the axis \([110]\));

\(K(\Theta', \Theta_0)\) is the function of the effective angular distribution of electrons relative to the initial direction.

(In principle, \(K(\Theta', \Theta_0)\) will be a function of the orientation angles \(\Theta'\) and \(\alpha\). However, as calculations showed, the dependence on \(\alpha\) in our case was negligibly small. The averaged degree of polarization \(P\) is calculated analogously.

It is practically impossible to choose the function \(K(\Theta', \Theta_0)\) from theoretical considerations, but it can be obtained from the experimentally measured intensity spectrum. To find the function \(K(\Theta', \Theta_0)\), the ill-posed inverse problem was solved:

\[ I_{\text{exp}}(\theta_0, x_i) \pm \delta_i = \int dx' \int d\theta' I(\theta', x')K(\theta', \theta_0)\xi(x_i, x') = \tilde I(\theta_0, x_i), \]

where \(I_{\text{exp}}(\Theta_0, x_i)\) is the experimental value of the intensity for energies of \(\gamma\)-quanta \(x_i\);

\[ \begin{aligned} \Theta_0 &\text{ is the orientation angle;}\\ \delta_i &\text{ is the experimental error;}\\ \xi(x_i, x') &\text{ is the energy-resolution function of the spectrometer,}\\ &\text{ i.e., the probability of registering a }\gamma\text{-quantum}\\ &\text{ with energy }x'\text{ as having energy }x_i; \end{aligned} \]

\(\xi(x_i, x')\) was approximated by a Gaussian:

\[ \xi(x_i, x') = \frac{1}{\sqrt{2\pi}\,D(x_i)} \exp\left\{ -\frac{1}{2} \left[ \frac{x_i-x'}{D(x_i)} \right]^2 \right\}, \tag{8} \]

where the dispersion \(D(x_i)\) varies from \(3.6\%\) for small \(x_i\) to \(1.9\%\) for \(x_i=1\) (\(E_\gamma=800\) MeV) [17]. Finding an exact solution is very difficult, and a simpler method may be used.

By specifying, instead of \(K(\Theta_0,\Theta')\), some simple function with several fitting parameters, one can choose these parameters so that the relation [7] is satisfied. In our case, after several trials we settled on the following function:

\[ K(\theta_0,\theta') = A_1 \exp\left\{ -\frac{(\theta_0-\theta')^2}{2D_1^2} \right\} + A_2 \exp\left\{ -\frac{(\theta_0-\theta')^2}{2D_2^2} \right\}. \tag{9} \]

\(A_1, A_2, D_1, D_2\) are fitting parameters (the 4 parameters are related by the normalization condition: \(\int K(\Theta_0,\Theta')\,d\Theta' = 1\)). The fitting parameters were found by the maximum-likelihood method. Under the assumption that the experimental errors \(\delta_i\) for different \(i\) are independent and normally distributed with zero mean, the likelihood function is written in the form

\[ P = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi}\,\delta_i} \exp\left\{ -\frac{1}{2\delta_i^2} \left[ I_{\text{exp}}(\theta_0,x_i) - \tilde I(\theta_0,x_i) \right]^2 \right\}. \tag{10} \]

To find the fitting parameters entering into \(I\) and maximizing this functional, and to compute the spectrum and polarization with the found function \(K(\Theta_0,\Theta')\), a program was written for the BESM-4 computer. The results of the calculations are shown in Fig. 5.

4. Target

The choice of target was made taking into account the following requirements: a) a high percentage content of hydrogen nuclei (the 1st component), b) minimum content of other elements and impurities, c) minimum atomic number of the element that is the 2nd component of the target, d) sufficiently good homogeneity, e) ease of manufacture, f) the 2nd component of the target must be an element that can be used as a separate simple target (for the difference method). Polyethylene \(\mathrm{CH}_2\) satisfied these requirements well. Existing technology makes it possible to manufacture a polyethylene target of the required dimensions.

Fig. 6. Data on the measurement of differential cross sections.

without impurities and with good homogeneity. The target had dimensions of Ø 50 mm, length 80 mm, density \(0.931 \pm 0.003\ \mathrm{g/cm^3}\). The carbon target was a polyethylene capsule filled with spectrally pure carbon with density \(1.242 \pm 0.008\ \mathrm{g/cm^3}\). The polyethylene capsule was a hollow cylinder 80 mm long, with an outer diameter of 46 mm and an inner diameter of 30 mm. The inlet and outlet openings of the capsule were closed with Plexiglas plates 0.5 mm thick.

5. Charged-particle telescope

The charged-particle telescope (Fig. 1) consisted of scintillation counters \(S_1\) \((100 \times 100 \times 10\ \mathrm{mm})\), \(S_2\) \((150 \times 150 \times 10\ \mathrm{mm})\), \(S_3\) \((170 \times 170 \times 10\ \mathrm{mm})\), \(S_4\) \((240 \times 240 \times 10\ \mathrm{mm})\), and two copper absorbers. The characteristics of the counters and of the telescope were studied in detail with 50 MeV positrons [20] from one of the channels of a pair magnetic \(\gamma\)-spectrometer. The average efficiency of the counters was 0.98. Subsequently, with the charged-particle telescope, differential cross sections for photoproduction of \(\pi^+\)-mesons on hydrogen in the region of the first resonance [21] were measured for the angle \(\Theta_\pi = 90^\circ\) in the c.m.s. as a function of photon energy. Fig. 6 gives the data of our measurements and those of Bonn [22].

6. Neutron detector

As the neutron detector a scintillation counter was used, constructed on the basis of a plastic scintillator Ø \(100 \times 200\) mm (at present a crystal Ø \(300 \times 300\) mm is used). The counter was enclosed in a lead casing with walls 50 mm thick. The efficiency of the neutron detector was measured in the energy range 30–60 MeV and was on average 18% [21].

7. Electronic apparatus

A block diagram of the electronic apparatus is shown in Fig. 7. The shaped signals from counters \(S_1\), \(S_2\), \(S_3\) were fed to fast-coincidence circuits \(CC_1\), \(CC_2\), and \(CC_3\) with a resolving time \(t \simeq 2\) ns. Next, the signals from \(CC_2\) and \(CC_3\) were fed to an anticoincidence circuit \(CA\). The veto signal was the signal from the coincidence circuit \(CC_2\) \((t \simeq 2\ \mathrm{ns})\). To circuit \(CC_2\) were fed the shaped signals of counter \(S_3\) and counter \(S_4\), which selected electrons and high-energy \(\pi^+\)-mesons. Thus ob—

At the same time, the charged-particle telescope selected the charged particles stopped in counter \(S_3\) and in the absorber installed behind counter \(S_3\). The shaped signals from the n-detector were fed to a delay line, by means of which \((\pi^+ n)\)-coincidences were set up. Signals from the charged-particle telescope and the neutron detector were simultaneously fed to differential coincidence circuits CC \((\tau = 1.5\ \text{ns})\), which made it possible to measure simultaneously both the sum of true and random coincidences and only random coincidences.

8. Selection of the process \(\gamma + p \to \pi^+ + n\). Control experiments

The use of a complex target leads to a large background from competing reactions on nuclei if one uses a measurement technique with a single recording channel. Recording two particles simultane-

Fig. 7. Block diagram of the electronics.

Fig. 8. Dependence of the counting rate of \((\pi^+ n)\)-coincidences on setting the n-detector at different angles \(\Theta_n\).

but it is possible to reduce substantially the background from reactions whose kinematics differs from the kinematics of the process under study. Indeed, if the reaction products \(\gamma+p\to\pi^+ + n\) fly apart only in a plane, then in the processes \(\gamma+A\to\pi^+ + n + B' + \ldots\) they fly apart without satisfying this condition. For the process \(\gamma+A\to\pi^+ + n + B+\ldots\) the reaction products interact with nuclear matter. Furthermore, in the photoproduction of mesons on nuclei, additional energy is required for the recoil nucleon in order to break the energy bonds holding it in the nucleus. If a \(\pi^+\)-meson and a neutron are registered, the contribution of a number of competing processes is excluded, since the emission angles and the energies of the neutron and the \(\pi^+\)-meson for the process \(\gamma+p\to\pi^+ + n\) are strictly determined, whereas for the processes \(\gamma+A\to\pi^+ + n + B+\ldots\) they take arbitrary values (within the conservation laws).

Thus, by registering in coincidence \(\pi^+\)-mesons and recoil neutrons at angles corresponding to the kinematics of single production of \(\pi^+\)-mesons on hydrogen, one may expect a pronounced peak in the counting rate of \((\pi^+ n)\)-coincidences against the background of competing processes occurring on complex nuclei of a hydrogen-containing target.

In the present work, using a charged-particle telescope, the emission angle and range of the \(\pi^+\)-meson were registered and determined. A neutron detector placed 1 meter from the target registered, in coincidence with the telescope, neutrons at a definite emission angle. A series of preliminary control experiments was carried out.

a. Difference method. The counting rate of registered \(\pi^+\)-mesons produced on hydrogen was determined according to

\[ V_{\mathrm{H}_2}=V_{\mathrm{CH}_2}-\xi V_c, \tag{11} \]

where \(C_{\mathrm{CH}_2}\), \(V_c\) are the counting rates of \((\pi n)\)-coincidences from the polyethylene and carbon targets, respectively,

\[ \xi \]

is a coefficient taking into account the relative weight of the number of carbon nuclei of the hydrogen-containing target with respect to the carbon target.

This formula is valid if, in the geometry of the polyethylene and carbon targets, corrections for the different energy losses by particles have been taken into account.

From measurements of the counting rates \(V_{\mathrm{CH}_2}\) and \(V_c\) for various kinematic situations it was found that the quantity \(V_c/V_{\mathrm{CH}_2}\) varies from 5 to 20%.

For a quantitative estimate of the background, the neutron detector was moved out of the plane of the \(\gamma\)-beam and the direction of emission of the \(\pi^+\)-meson by an angle \(\varphi\). The measured counting rate of \((\pi^+ n)\)-coincidences \(V^{\varphi}_{\mathrm{CH}_2}\) for such an experimental geometry is determined by processes on carbon, by multiple production, i.e., by all competing processes except \(\gamma+p\to\pi^+ + n\). The counting rate on hydrogen \(V_{\mathrm{H}_2}\) for single production of \(\pi^+\)-mesons is determined by the formula

\[ V_{\mathrm{H}_2}=V_{\mathrm{CH}_2}-aV^{\varphi}_{\mathrm{CH}_2}, \tag{12} \]

where \(a=V_c/V_c^{\varphi}\) is a coefficient determined by the ratio of counting rates on carbon when the \(n\)-detector lies in the plane and outside it at the angle \(\varphi\).

It was found that the quantities \(\xi V_c\) and \(aV^{\varphi}_{\mathrm{CH}_2}\) are equal within the 10% error. We note that the quantity \(\xi V_c\) does not take into account the background of pion pair production on hydrogen, whereas \(aV^{\varphi}_{\mathrm{CH}_2}\) should take this background into account. From the equality of these quantities it followed that this background is absent. Independently, the same result for higher energies was obtained in Ref. [23].

b. Violation of the kinematics of the process $\gamma+p\to\pi^{+}+n$.

The counting rates of $(\pi^{+}+n)$ coincidences were measured for various values of the angle $\Theta_n$ at a fixed value of $\Theta_\pi$. The measurement results for $E_\gamma=230$ MeV and $\Theta_\pi=90^\circ$ in the c.m. system are shown in Fig. 8. It is seen from the figure that the maximum counting rate occurs at the angle $\Theta_n=31^\circ$, corresponding to the kinematics of the process $\gamma+p\to\pi^{+}+n$. For other values of the angle the counting rate falls sharply.

c. Measurement of the time of flight of the recoil neutron.

The time-of-flight measurements were carried out with the aid of a time-to-amplitude converter (TAC). Signals from the $\pi$-meson channel served as the start pulse. Figure 9 shows, for $E_\gamma=230$ and $\Theta_\pi=100^\circ$, histogram I (the neutron counter is installed in the charged-particle telescope next to counter $S_3$) and histogram II (the neutron counter in

Fig. 9. Dependence of the counting rate of $(\pi^{+}n)$ coincidences on the time of flight. Histogram II (neutron counter in the working position in the n-channel).

the working position—in the n-channel). The displacement of the peak of histogram II corresponds to the calculated time, determined from the simple formula:

\[ \Delta t_n = t_n - t_\pi , \tag{13} \]

where

where $t_n$, $t_\pi$ are the times of flight of the neutron and the $\pi^{+}$ meson from the target to the counters.

This displacement $\Delta t$ was calculated by the Monte Carlo method and is equal to the mean value 7.57 ns.

As a result of control experiments it was established that the process $\gamma+p\to\pi^{+}+n$ is indeed registered. It was found that the background from carbon is of a magnitude comparable with the background from the container of the liquid-hydrogen target, while the background of multiple pion production is absent.

9. Method for measuring the asymmetry

In the experiment the counting rates of $\pi^{+}$ mesons VI (VII) are measured in the plane perpendicular (parallel) to the polarization plane of the photon beam. Expressing these quantities through differential se-

values \(d\sigma_{\perp}/d\Omega\), \(d\sigma_{\parallel}/d\Omega\) and the number of photons \(N_{\perp}\), \((N_{\parallel})\) with the electric vector perpendicular (parallel) to the reaction plane [21], we obtain an expression for the asymmetry:

\[ \Sigma(E_{\gamma},\theta_{\pi})= P^{-1} \frac{ \left[V^{\perp}(E_{\gamma},\theta_{\pi})-\xi V_{c}^{\perp}(E_{\gamma},\theta_{\pi})\right] - \left[V^{\parallel}(E_{\gamma},\theta_{\pi})-\xi V_{c}^{\parallel}(E_{\gamma},\theta_{\pi})\right] }{ \left[V^{\perp}(E_{\gamma},\theta_{\pi})-\xi V_{c}^{\perp}(E_{\gamma},\theta_{\pi})\right] + \left[V^{\parallel}(E_{\gamma},\theta_{\pi})-\xi V_{c}^{\parallel}(E_{\gamma},\theta_{\pi})\right] } \tag{14} \]

where \(\xi V_{c}^{\perp}(E_{\gamma},\theta_{\pi})\) \([\xi V_{c}^{\parallel}(E_{\gamma},\theta_{\pi})]\) is the background counting rate for the \(\perp\)-th (\(\parallel\)) reaction planes relative to the polarization plane of the \(\gamma\)-beam.

Fig. 10. Photograph of the setup.

Whereas in Frascati [8—10], in order to measure the direction of the polarization vector of the \(\gamma\)-beam, the orientation of the crystalline target was measured and then two spectra were fitted, in our experiment [24] the orientation of the recording device was changed at a fixed position of the crystal. This eliminates a number of errors associated with reorientation of the diamond target. Figure 10 shows a setup that makes it possible to change the orientation of the apparatus within 1—2 minutes. Relatively frequent changes of orientation make it possible to reduce a number of errors associated with drift of the apparatus characteristics.

10. Experimental results

The asymmetry \(\Sigma(E_{\gamma},\theta_{\pi})\), which is a function of 5 variables — \(V_{\mathrm{CH}_2}^{\parallel}\), \(V_{\mathrm{CH}_2}^{\perp}\), \(V_{c}^{\parallel}\), \(V_{c}^{\perp}\), and the polarization \(P\) — can be represented in the form

\[ \Sigma(E_{\gamma},\theta_{\pi})= P^{-1} \frac{ R_{\mathrm{CH}_2}(E_{\gamma},\theta_{\pi}) \left[ \frac{1-V_{c}^{\perp}/V_{\mathrm{CH}_2}^{\perp}} {1-V_{c}^{\parallel}/V_{\mathrm{CH}_2}^{\parallel}} \right] -1 }{ R_{\mathrm{CH}_2}(E_{\gamma},\theta_{\pi}) \left[ \frac{1-V_{c}^{\perp}/V_{\mathrm{CH}_2}^{\perp}} {1-V_{c}^{\parallel}/V_{\mathrm{CH}_2}^{\parallel}} \right] +1 } \tag{15} \]

\[ R_{\mathrm{CH}_2}(E_{\gamma},\theta_{\pi})= V_{\mathrm{CH}_2}^{\perp}/V_{\mathrm{CH}_2}^{\parallel} \]

is a directly measured quantity.

Let us consider the statistical error in measuring the quantity \(R_{\mathrm{CH}_2}\), which ultimately determines the error in determining \(\Sigma(E_{\gamma},\)

\(\theta_\pi)\). The causes of the error in the measurement of \(R_{\mathrm{CH}_2}\) are the following: displacement of the central axis of the target from the axis of the \(\gamma\)-beam, and noncoincidence of the central angles of the counter telescopes and the neutron counter. Analysis of these causes [21] showed that they lead to a mismatch of the operating regions in photon energy when registration is carried out in two orthogonal planes, and the error in \(R_{\mathrm{CH}_2}\) is expressed as

\[ \frac{\Delta R_{\mathrm{CH}_2}}{R_{\mathrm{CH}_2}} = \frac{1}{R_{\mathrm{CH}_2}}\, \Delta E_\gamma/E_\gamma . \tag{16} \]

Uncertainty in the alignment of the \(\gamma\)-beam up to \(1.5\) mm gives an uncertainty in \(\Delta E_\gamma/\theta_\gamma\) of less than \(0.3\%\), and an error in setting the angles
\(\Delta\theta_\pi = |\theta_\pi^\perp-\theta_\pi^\parallel|\) up to \(2^\circ\) gives no more than \(1\%\) in the error \(\Delta E_\gamma = |E_\gamma^\perp-E_\gamma^\parallel|\). An estimate of this error for our setup showed that it is negligibly small and amounts to less than \(0.25\%\) [21].

The most important source of error is the drift of the apparatus characteristics, since measurements of the asymmetry even for one kinematic point are carried out over several days. To eliminate this error the orientation of the apparatus was changed frequently. It is difficult to give an exact estimate of such an error, since all the factors affecting the registering apparatus are not clear. Therefore the method of measuring the quantity \(R_{\mathrm{CH}_2}\) with unpolarized photons was used for each kinematic point. Thus, for example, for \(E_\gamma=230\) MeV and \(\Theta_\pi=90^\circ\) in the c.m.s., this quantity was \(0.99\pm0.026\).

A systematic error may be introduced by uncertainty in setting the energy \(E_0\). In this connection the energy of the accelerated electrons \(E_0\) was carefully monitored. We note that an uncertainty in \(E_0\) of up to \(1\%\) introduces a negligibly small error into the value of the polarization of the \(\gamma\)-beam and, thus, does not affect the measured value of the asymmetry. The error due to the noncoincidence of the calculated polarization plane with the reaction plane was small and was not taken into account in the calculation of the asymmetry. The most significant errors in the experiment were those of a random nature, i.e., statistical.

The total error was calculated by the formula

\[ \Delta\Sigma = \frac{2R_{\mathrm{H}_2}}{P\left(R_{\mathrm{H}_2}+1\right)^2} \sqrt{ \left( \frac{\Delta N_{\mathrm{CH}_2}^{\perp}} {N_{\mathrm{CH}_2}^{\perp}-N_c^{\perp}} \right)^2 + \left( \frac{\Delta N_{\mathrm{CH}_2}^{\parallel}} {N_{\mathrm{CH}_2}^{\parallel}-N_c^{\parallel}} \right)^2 + h^2\left(\frac{\Delta N_c}{N_c}\right)^2 + \left(\frac{R_n^2-1}{2R}\right)^2 \frac{\Delta P^2}{P^2} }, \]

where

\[ R_{\mathrm{H}_2} = \left(V_{\mathrm{CH}_2}^{\perp}-V_c^{\perp}\right)/ \left(V_{\mathrm{CH}_2}^{\parallel}-V_c^{\parallel}\right), \]

\[ h = V_c^{\perp}\left(R_{\mathrm{H}_2}-1\right)/ \left(V_{\mathrm{CH}_2}^{\perp}-V_c^{\perp}\right), \]

\(N_{\mathrm{CH}_2}(N_c)\) is the number of events for the corresponding counting rate \(V_{\mathrm{CH}_2}(V_c)\).

It follows from this that the errors are practically determined only by statistics. Indeed, the asymmetry is the difference of two close, almost equal quantities. Small deviations in the measured counting rates lead to large deviations in the value of the calculated asymmetry. Therefore, for each block of measurements a statistical analysis of the selected events was carried out. The analysis consisted in testing the hypothesis of a Poisson distribution of the random variables \(V^\perp(V^\parallel)\). For each value \(V_i^{\perp(\parallel)}\) the quantity [25]

\[ \sum_i \frac{(r_i-m_i)^2}{m_i}, \tag{18} \]

was calculated.

Table 1

where \(r_i\) is the actual number of events for the \(i\)-th counting rate, taken from the experiment,

\(m_i\) is the expected number of events according to the Poisson distribution for the \(i\)-th counting rate, and the quantity \(\chi^2_\nu\) was calculated.

Next, for the entire series of measurements the significance level was calculated, which served as the selection criterion for each block of measurements.

The experimental data are given in Table 1. The energy and angular intervals in which the asymmetry was measured are not included in the errors \(\Delta\Sigma\), but determine the size of the kinematic region \(E_\gamma,\ \theta_\pi\). They were calculated by the Monte Carlo method. Figures 11 and 12 show the event distributions as functions of \(E_\gamma\) and \(\theta_\pi\) for the measured asymmetry at \(E_\gamma = 230\ \mathrm{MeV}\) and \(\theta_\pi = 100^\circ\).

Conclusion

Despite the difference in the methods used for registering the process \(\gamma + p \to \pi^+ + n\), the methods for measuring \(\Sigma\), and the methods for obtaining a linearly polarized \(\gamma\)-beam, the results obtained are, on the whole, in agreement with the Stanford [5] and Frascati [10] data (Fig. 11). This makes it possible to combine these data into one group for analysis. From Fig. 13 it is seen that our data do not contradict the theory of dispersion relations [1, 3, 13]. However, in the angular distribution at \(E_\gamma = 230\ \mathrm{MeV}\), for the backward hemisphere in the angular region from \(90\) to \(120^\circ\), there is a tendency toward larger asymmetry values than predicted by theory [1–3].

Fig. 11. Histogram of energy resolution.

Fig. 12. Histogram of angular resolution.

Figure 13 labels:
\(\Sigma = \dfrac{d\sigma_\perp - d\sigma_\parallel}{d\sigma_\perp + d\sigma_\parallel}\);
\(\gamma + p \to \pi^+ + n\), \(E_\gamma = 230\ \mathrm{MeV}\).
Legend: our data; Frascati [10]; Stanford [5]; theory [2]; theory [1]; theory [3].

Fig. 13. Angular distribution of the asymmetry.

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