Program for Multipole Analysis
Abstract Generated abstract
This paper describes a computational program for extracting multipole amplitudes in pion photoproduction by least-squares fitting experimental angular distributions to theoretical expressions for the transverse cross section and the function H. The program uses linearization minimization, represents photoproduction amplitudes through electric and magnetic multipoles with higher partial waves treated in the Born approximation, and incorporates isospin decompositions and Fermi-Watson phase relations to connect real and imaginary parts. It is formulated for both charged and neutral pion photoproduction on the proton, although the reported calculations are limited to the charged channel because of insufficient neutral-channel data.
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PROGRAM FOR MULTIPOLE ANALYSIS
G. M. Radutskii, N. I. Sablin, B. M. Shumilov
(Presented by the seminar of the High-Energy Laboratory, Research Institute of Nuclear Physics, Electronics, and Automation)
The problem consists in determining the multipole amplitudes in the isotopic parts in \(\pi\)-photoproduction, which is accomplished by fitting the experimental data to a certain theoretical function, into which 8 unknown amplitudes enter as variable parameters:
\[ M=\sum_{i=1}^{N}\left(\frac{\sigma_{\perp}^{s}(\theta_i)-\sigma_{\perp}(\theta_i)}{\Delta\sigma_{\perp}(\theta_i)}\right)^2 +\sum_{i=2}^{N-1}\left(\frac{H^{s}(\theta_i)-H(\theta_i)}{\Delta H(\theta_i)}\right)^2, \]
where \(i=\overline{1,N}\) denotes a set of angles from \(0^\circ\) to \(180^\circ\).
\[ \sigma_{\perp}(\theta_i)=\frac{d\sigma_{\perp}(\theta_i)H(\theta_i)}{a\Omega} \]
are functions of the variable parameters.
Minimization of the functional \(M\) is performed by means of the linearization method [1], implemented at JINR in the form of the standard program SP-123 [2].
The initial values of the parameters were taken from the calculation of BDE, which gives the real parts of 8 multipole amplitudes; the imaginary parts were determined by multiplying the real parts by \(\operatorname{tg}\) of the corresponding phases; the phases were taken by us from the calculation [3].
The program is written for the analysis of both \(\pi^+\)-photoproduction and \(\pi^0\)-photoproduction. However, calculations were carried out only for the \(\pi^+\)-photoproduction reaction, since at present there is still insufficient experimental information on \(\pi^0\)-photoproduction, which does not make it possible, by calculation, to extract from this information the required parameters with the necessary reliability.
In the arithmetic part of the program, the algorithmic part of which is described below, the functions \(\sigma_{\perp}(\Theta)\), \(H(\Theta)\), and their derivatives with respect to the multipole amplitudes are calculated.
In Sec. 1 the dependences of \(\sigma_{\perp}\) and \(H\) on the photoformation amplitudes \(F_i\) are presented. In Sec. 2 the expansion of \(F_i\) in multipole amplitudes up to the state with \(I^P=(3/2)^+\) is given, including the contribution of higher angular states in the Born approximation. In Secs. 3 and 4 all the necessary relations are given for calculating the Born amplitudes and the multipole amplitudes in the Born approximation \(M_{l\pm}^{B}\) and \(E_{l\pm}^{B}\). In Secs. 5 and 6 expansions of the amplitudes \(F_i\) and \(M_{l\pm}\) are given for the reactions \(\gamma p \to \pi N\) as functions of amplitudes with a definite isospin state and relations expressing the connection between the real and imaginary part—
…amplitudes. In this case the multipole amplitudes according to F. Berends et al. [3] are used. A block diagram of the arithmetic program is presented in Fig. 1.
- From the general expression for the photoproduction amplitude [4]
\[ F=iF_1\vec{\sigma}\vec{\varepsilon} +F_2\frac{(\vec{\sigma}\cdot\vec{q})(\vec{\sigma}\cdot\vec{k}\cdot\vec{\varepsilon})}{qk} +iF_3\frac{(\vec{\sigma}\cdot\vec{k})(\vec{q}\cdot\vec{\varepsilon})}{qk} +iF_4\frac{(\vec{\sigma}\vec{q})(\vec{q}\cdot\vec{\varepsilon})}{q^2} \]
the cross section is extracted,
\[ \frac{k}{q}\frac{d\sigma_\perp}{d\Omega}(\theta,E_\gamma) = |F_1|^2+|F_2|^2-2\,\operatorname{Re}F_1^*F_2\cos\theta \tag{1.1} \]
and the function \(H(\xi)\) [5]:
\[ H(\xi)=\left[|F_3|^2+|F_4|^2 +2\operatorname{Re}\left(F_2^*F_3+F_1^*F_4+F_3^*F_4\cos\theta\right)\right]\xi - 2\left(\frac{M}{W}\right)^2\beta^2 e^2 f^2 \left[1+\left(\frac{q}{k}\right)^2-2\frac{E}{k}\pi\right]\frac{1}{\xi}. \]
\[ \xi=1-\beta\cos\Theta \]
- Each of the amplitudes \(F_i\) is expanded in multipole amplitudes—magnetic \(M_{l\pm}\) and electric \(E_{l\pm}\). The contribution of multipole amplitudes with \(l\ge 2\) is given in the Born approximation. Then \(F_i\) can be written as:
\[ \begin{aligned} F_1&=E_0+3(E_{1+}+M_{1+})\cos\theta+\varphi_1;\\ F_2&=2M_{1+}+M_{1-}+\varphi_2;\\ F_3&=3(E_{1+}-M_{1+})+\varphi_3;\\ F_4&=\varphi_4, \end{aligned} \tag{2.1} \]
where
\[ \begin{aligned} \varphi_1&=F_1^B-\left[E_0^B+3(E_{1+}^B+M_{1+}^B)\cos\theta\right];\\ \varphi_2&=F_2^B-\left[2M_{1+}^B+M_{1-}^B\right];\\ \varphi_3&=F_3^B-\left[3(E_{1+}^B-M_{1+}^B)\right];\\ \varphi_4&=F_4^B \end{aligned} \tag{2.2} \]
is the total contribution of amplitudes with \(l\ge 2\) in the Born approximation, \(M_{l\pm}^B\) and \(E_{l\pm}^B\) are multipole amplitudes which are found as projections of the Born amplitudes \(F_i^B\) onto a state of definite angular momentum.
\[ F^B_3,(t)=[C(s)]^{-1}[B(st)]A^B(s,t), \tag{3.1} \]
where \([C]\) and \([B]\) are taken from the DDB;
\(F^B\) is a column with elements:
\[ F_1^B,\;F_2^B,\;F_3^B,\;F_4^B; \]
\[ A^B(s,t)= \left[ \frac{1}{s-m_N^2} + \frac{[\xi]}{u-m_N^2} \right]\tau, \tag{3.2} \]
Fig. 1. Block diagram of the arithmetic program:
- What reaction? \(J_1; J_2\)
- \(m_i=m_{J+}\)
- \(m_\pi=m_{\pi^0}\)
- Reading the energies \(E_k\) and \(x=\cos\theta\)
- \(E_{ki}=E_{ki-1},\; J_{1i}=J_{1i-1}\); yes / no; \(x_i=x_{i-1}\); yes / no
- Calculation of kinematic variables independent of the angle
- Calculation of Legendre polynomials
- Calculation of total multipoles and Born multipoles
- Calculation of kinematic variables dependent on the angle
- Calculation of the Born amplitudes with a specified \(\beta\)-parity
- Calculation of the total photoproduction amplitudes for the given reaction
- Which type of information? \(J_2=1,2\)
- 1: Calculation of cross sections and derivatives
- 2: Calculation of \(H(\xi)\) and derivatives
and
\[ |\xi|=\xi \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&1 \end{pmatrix}; \]
\[
\xi=+1 \text{ for } (0) \text{ and } (+) \text{ isospin amplitudes;}
\]
\[
\xi=-1 \text{ for } (-1) \text{ isospin amplitudes;}
\]
\(\tau\) is a column with elements
\[ \tau_1^{(0,\pm)}=\frac{eg}{2}, \qquad \tau_2^{(0,\pm)}=\frac{-eg}{t^2-m_\pi^2}, \tag{3.3} \]
\[ \tau_3^{(\pm)}=\tau_4^{(\pm)} =\frac{-eg}{4m_N}\left(\mu_p'-\mu_n'\right), \qquad \tau_3^{(0)}=\tau_4^{(0)} =\frac{-eg}{4m_N}\left(\mu_p'+\mu_n'\right), \]
\[ \frac{e^2}{4\pi}=\frac{1}{137}; \qquad \mu_p'=1.793; \qquad \mu_n'=-1.913; \qquad \frac{g^2}{4\pi}=14.4 \text{—constants.} \]
From the kinematics of photoproduction processes it follows that:
\[ s=W^2; \qquad t=-2kq(1-\beta\cos\theta_1)+m_\pi^2; \]
\[ u=-2kE_2\left(1-\frac{q}{E^2}\cos\theta_2\right)+m_N^2; \qquad \beta=\frac{q}{q_0}. \]
\[ E_2=\frac{W^2+m_N^2-m_\pi^2}{2W}; \qquad E_\pi=q_0=\frac{W^2-m_N+m_\pi^2}{2W}; \]
\[ E_1=\frac{W^2+m_N^2}{2W}, \]
where \(W=E_\gamma+m_N\) is the total energy in the c.m.s. (center-of-mass system).
4.
\[
E_{0+}^{B}=\frac{Z}{\mu}
\left\{
\frac{\xi-1}{2}\left(\mu-\frac{m_N}{W}\right)
-\frac{\xi+1}{2}\cdot\frac{m_N}{W}(\mu+1)
-\xi(\mu+1)T_{0+}^{N}
+\right.
\]
\[
\left.
+(\xi-1)\left[-\frac{2m_N}{W-m_N}\cdot\frac{q}{m_N+E_2}R_1^\pi\right]
-\xi\left(\mu-\frac{2m_N}{W-m_N}\right)\frac{q}{m_N+E_2}R_1^N
\right\},
\tag{4.1}
\]
\[
E_{1+}^{B}=\frac{1}{2}\frac{Z}{\mu}
\left\{
(\xi-1)\left[
\frac{2m_N}{W+m_N}R_1^\pi
-\frac{2m_N}{W-m_N}\cdot\frac{2q}{m_N+E_2}R_2^\pi
\right]
\right.
\]
\[
\left.
-\xi(\mu+1)T_{1+}^{N}
-\xi\left(\mu+\frac{2m_N}{W+m_N}\right)R_1^N
-\xi\left(\mu-\frac{2m_N}{W-m_N}\right)\frac{2q}{m_N+E_2}R_2^N
\right\},
\]
\[ M_{1+}^{B}=\frac{1}{2}\frac{Z}{\mu} \left\{ (1-\xi)\frac{2m_N}{W+m_N}R_1^\pi -\xi(1+\mu)T_{1+}^{N} +\xi\left(\mu+\frac{2m_N}{W+m_N}\right)R_1^N \right\}, \]
\[
M_{1-}^{B}=\frac{Z}{\mu}
\left\{
\frac{q}{E_2+m_N}
\left[
\frac{\xi-1}{2}\left(\mu+\frac{m_N}{W}\right)
-\frac{\xi+1}{2}\cdot\frac{m_N}{W}(\mu+1)
\right]
+\right.
\]
\[
\left.
+\xi(\mu+1)T_{1-}^{N}
+(\xi-1)\frac{2m_N}{W+m_N}R_1^\pi
-\xi\left(\mu+\frac{2m_N}{W+m_N}\right)R_1^N
\right\},
\]
where
\[ R_1^\pi=\frac{1}{3}\left[Q_2\left(\frac{q_0}{q}\right)-Q_0\left(\frac{q_0}{q}\right)\right]; \]
\[ R_1^N=-\frac{1}{3}\left[Q_2\left(\frac{E_2}{q}\right)-Q_0\left(\frac{E_2}{q}\right)\right]; \]
\[ R_2^\pi=\frac{1}{5}\left[Q_3\left(\frac{q_0}{q}\right)-Q_1\left(\frac{q_0}{q}\right)\right]; \tag{4.2} \]
\[ R_2^N=\frac{1}{5}\left[Q_3\left(\frac{E_2}{q}\right)-Q_1\left(\frac{E_2}{q}\right)\right]; \]
\[ T_{1+}^N=\left[-\frac{m_N}{q}Q_1\left(\frac{E_2}{q}\right)+ \frac{m_N}{E_2+m_N}Q_2\left(\frac{E_2}{q}\right)\right]; \]
\[ T_{0+}^N=\left[\frac{m_N}{q}Q_0\left(\frac{E_2}{q}\right)- \frac{m_N}{E_2+m_N}Q_1\left(\frac{E_2}{q}\right)\right]; \]
\[ T_{1-}^N=\left[-\frac{m_N}{q}Q_1\left(\frac{E_2}{q}\right)+ \frac{m_N}{E_2+m_N}Q_0\left(\frac{E_2}{q}\right)\right]; \]
\[ Z=\frac{\mu}{4m_N}\sqrt{\frac{E_2+m_N}{2W}}\frac{eq}{4\pi}. \]
Here
\[ \mu^\pm=\mu'_p-\mu'_n,\qquad \mu^0=\mu'_p+\mu'_n, \]
\(Q_l(x)\) are Legendre functions of the second kind.
- All expressions for each of the following processes have been introduced into the program with the index
\[ \gamma+p\to\pi^++n,\quad J_1=1, \tag{5.1} \]
\[ \gamma+p\to\pi^0+p,\quad J_1=2. \]
\(F_i(J_1)\) or \(M_{l\pm}(J_1)\) are expanded in terms of the isoscalar amplitudes \(F_i^{1/2}, M_{l\pm}^{1/2}\) and the isovector \(F_i^{3/2}, M_{l\pm}^{3/2}\), or in terms of amplitudes with definite \(G\)-parity \(F_i^0, F_i^\pm(M_{l\pm}^0, M_{l\pm}^\pm)\). For the multipole amplitudes we have:
\[ M_{l\pm}(1)=\sqrt{2}\{M_{l\pm}^0+M_{l\pm}^-\} =\frac{\sqrt{2}}{3}\left[M_{l\pm}^{1/2}-M_{l\pm}^{3/2}\right]; \]
\[ M_{l\pm}(2)=\sqrt{2}\{M_{l\pm}^0-M_{l\pm}^-\} =\frac{1}{3}\left[M_{l\pm}^{1/2}+2M_{l\pm}^{3/2}\right]. \tag{5.2} \]
- The multipole amplitudes are complex quantities. The relation between the real and imaginary parts of each multipole amplitude is given by the Fermi–Watson theorem [6].
\[ \operatorname{Im} M_{l\pm}^{2T}(W)=\operatorname{Re} M_{l\pm}^{2T}(W)\tan \delta_{2I,2T}(W), \]
where \(\delta_{2I,2T}(W)\) is the phase of the \(\pi N\to\pi N\) scattering process,
\(W\) is the c.m. energy, the same for photoproduction and scattering.
All expressions for derivatives with respect to the parameters of the functions \(\dfrac{d\sigma_\perp}{d\Omega}\) and \(H\) have been introduced into the program with the index:
\[ \frac{d\sigma_\perp}{d\Omega}(\cos\theta,E_\gamma) \;-\; (J_2=1) \quad \text{and} \quad H(\xi)-J_2=2. \]
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