UDC 539.12

PHYSICS

Academician of the Academy of Sciences of the Ukrainian SSR A. I. AKHIEZER, M. P. REKALO

PHOTOPRODUCTION OF MESONS ON NUCLEONS AND \(SU(6)\) SYMMETRY

In the present note, within the framework of \(SU(6)\) symmetry, we investigate relations between the amplitudes for the production of mesons in a \(p\)-state in the interaction of magnetic-dipole \(\gamma\)-quanta with nucleons. In the case under consideration the photoproduction process is described by two partial amplitudes: one amplitude corresponds to the total angular momentum \(I=\frac12\) \((M_{11})\), the other to the total momentum \(I=\frac32\) \((M_{13})\). The spin structure of the amplitude for photoproduction of pseudoscalar mesons in the center-of-mass system, in terms of \(M_{11}\) and \(M_{13}\), has the form \((^{1})\)

\[ F_M=i(\vec{\sigma}\cdot \vec{e}\,\hat{\mathbf{k}}\cdot\hat{\mathbf{q}} -\vec{\sigma}\cdot\hat{\mathbf{k}}\,\vec{e}\cdot\hat{\mathbf{q}})(M_{13}-M_{11}) -\vec{e}\times\hat{\mathbf{k}}\cdot\hat{\mathbf{q}}(2M_{13}+M_{11}), \tag{1} \]

where \(\vec{e}\) is the polarization vector of the \(\gamma\)-quantum; \(\hat{\mathbf{k}}, \hat{\mathbf{q}}\) are unit vectors in the directions of the momenta of the \(\gamma\)-quantum and of the produced meson.

In \(SU(6)\) symmetry a magnetic-dipole \(\gamma\)-quantum is described by a tensor of second rank \((^{2})\)

\[ Q^{A}_{A'} \equiv Q^{ia}_{i'a'}=(\vec{\sigma}\cdot\vec{e}\times\hat{\mathbf{k}})^{i}_{i'}\,Q^{a}_{a'}, \qquad Q^{a}_{a'}= \begin{pmatrix} 2 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}. \tag{2} \]

Since the mesons are produced in a \(p\)-state, it is necessary to take into account the angular spurion \((^{3})\)

\[ L^{A}_{A'}\equiv L^{ia}_{i'a'}=(\vec{\sigma}\cdot\mathbf{q})^{i}_{i'}\,\delta^{a}_{a'}, \tag{3} \]

where the Latin indices take the values \(1,2\), and the Greek indices \(1,2,3\).

Then the number of independent parameters describing the process of meson photoproduction on nucleons upon absorption of a magnetic-dipole \(\gamma\)-quantum is determined by the number of common multiplets in the products \(56\times56\) and \(35\times35\times35\). Since

\[ 56\times56=1+35+405+2695, \]

from the product \(35\times35\times35\) it is necessary to retain multiplets of dimensions \(1, 35, 405, 2695\). For this purpose we note that

\[ 35\times35\times35=(1+35_1+35_2+189+280+280^{*}+405)\times35. \]

Each of the products \(189\times35\), \(280\times35\), \(280^{*}\times35\), \(405\times35\) contains the multiplet \(35\) once. Further, the products \(280\times35\), \(280^{*}\times35\) contain the multiplet \(405\) once, and the product \(405\times35\) contains the multiplet \(405\) twice. Thus, in the product \(35\times35\times35\) there are contained 2 singlet multiplets, 9 multiplets \(35\), 6 multiplets \(405\), and one multiplet \(2695\). Therefore the photoproduction amplitude under consideration is determined by 18 parameters:

\[ \begin{aligned} F_M={}&\bar{\psi}^{A'B'C'}\{\alpha\delta^{A}_{A'}\delta^{B}_{B'}(LQ)^{D}_{D}M^{C}_{C'} +\beta M^{A}_{A'}Q^{B}_{B'}L^{C}_{C'} \\ &+a_1\delta^{A}_{A'}\delta^{B}_{B'}\delta^{C}_{C'}(LQ_d)^{D}_{D'}M^{D'}_{D} +a_2\delta^{B}_{B'}\delta^{C}_{C'}[(LQ_d)^{A}_{D}M^{D}_{A'}+(LQ_d)^{D}_{A'}M^{A}_{D}] \\ &+a_3\delta^{B}_{B'}\delta^{C}_{C'}[(LQ_d)^{A}_{D}M^{D}_{A'}-(LQ_d)^{D}_{A'}M^{A}_{D}] +a_4\delta^{C}_{C'}(LQ_d)^{A}_{A'}M^{B}_{B'}+ \end{aligned} \]

\[ \begin{aligned} &+ b_1 \delta^A_{A'} \delta^B_{B'} \delta^C_{C'} (LQ_f)^D_{D'} M^{D'}_D + b_2 \delta^B_{B'} \delta^C_{C'} \left[(LQ_f)^A_D M^D_{A'} + (LQ_f)^D_{A'} M^A_D\right] + \\ &+ b_3 \delta^B_{B'} \delta^C_{C'} \left[(LQ_f)^A_D M^D_{A'} - (LQ_f)^D_{A'} M^A_D\right] + b_4 \delta^C_{C'} (LQ_f)^A_{A'} M^B_{B'} + \\ &+ \delta^B_{B'} \delta^C_{C'} \left(c_1 R^{[AD]}_{[A'D']} M^{D'}_D + c_2 R^{AD}_{[A'D']} M^{D'}_D + c_3 R^{[AD]}_{A'D'} M^{D'}_D + c_4 R^{AD}_{A'D'} M^{D'}_D\right) + \\ &+ d_1 \delta^C_{C'} \left(R^{AB}_{[A'D]} M^D_{B'} + R^{AB}_{[B'D]} M^D_{A'}\right) + d_2 \delta^C_{C'} \left(R^{[AD]}_{A'B'} M^B_D + R^{[BD]}_{A'B'} M^A_D\right) + \\ &+ d_3 \delta^C_{C'} \left(R^{AB}_{A'D} M^D_{B'} + R^{AB}_{B'D} M^D_{A'}\right) + d_4 \delta^C_{C'} \left(R^{AD}_{A'B'} M^B_D + R^{BD}_{A'B'} M^A_D\right) \Big\}\psi_{ABC}, \end{aligned} \tag{4} \]

where \(\psi_{ABC}\) is the tensor describing baryons, \(M^A_{A'}\) is the tensor describing mesons,

\[ (LQ_d)^A_{A'} = L^A_B Q^B_{A'} + L^B_{A'} Q^A_B,\qquad (LQ)^A_A = L^A_B Q^B_A, \]

\[ (LQ_f)^A_{A'} = L^A_B Q^B_{A'} - L^B_{A'} Q^A_B, \]

\[ R^{[AB]}_{[A'B']} = L^A_{A'}Q^B_{B'} + L^B_{B'}Q^A_{A'} - L^B_{A'}Q^A_{B'} - L^A_{B'}Q^B_{A'}, \]

\[ R^{[AB]}_{A'B'} = L^A_{A'}Q^B_{B'} - L^B_{B'}Q^A_{A'} - L^B_{A'}Q^A_{B'} + L^A_{B'}Q^B_{A'}, \]

\[ R^{AB}_{[A'B']} = L^A_{A'}Q^B_{B'} - L^B_{B'}Q^A_{A'} + L^B_{A'}Q^A_{B'} - L^A_{B'}Q^B_{A'}, \]

\[ R^{AB}_{A'B'} = L^A_{A'}Q^B_{B'} + L^B_{B'}Q^A_{A'} + L^B_{A'}Q^A_{B'} + L^A_{B'}Q^B_{A'}. \]

In (4) the invariant proportional to \(\alpha\) arises from a term of the type \(1(LQ)\times 35(M)\); the invariant proportional to \(\beta\) corresponds to the extraction of the multiplet \(2695\) from the product of three multiplets \(35\). Further, the invariants \(a_i\) and \(b_i\) \((i=1,2,3,4)\) arise as a result of multiplying \(35_1(LQ)\times 35(M)\) and \(35_2(LQ)\times 35(M)\); the invariants \(c_i\) correspond to the extraction of the multiplet \(35\) from the products

\[ 189(LQ)\times 35,\qquad 280(LQ)\times 35,\qquad 280^*(LQ)\times 35,\qquad 405(LQ)\times 35. \]

and, finally, \(d_i\) correspond to the extraction of the multiplet \(405\) from the products

\[ 280(LQ)\times 35,\qquad 280^*(LQ)\times 35,\qquad 405(LQ)\times 35. \]

The symbol \((LQ)\) after the dimension of a multiplet means that the multiplet under consideration arises in the multiplication of the corresponding multiplets \(35\).

We note that from the explicit form of the tensors \(L\) and \(Q\) it follows that

\[ (LQ)^A_A = 0. \]

In addition, the invariant in (4) proportional to \(b_1\) gives no contribution to the photoproduction of pseudoscalar mesons. Thus, the amplitude for meson production in a \(p\)-state upon absorption of a magnetic-dipole \(\gamma\)-quantum is determined by 16 parameters, with 12 of them, as follows from (4), contributing to the combination of amplitudes \(f_1=M_{13}-M_{11}\), and 13 parameters contributing to the combination \(f_2=2M_{13}+M_{11}\). After examination of the invariants (4), it turns out that in fact the amplitudes \(f_1\) are determined by only 5 parameters. Thus, for the photoproduction amplitudes on the proton we have:

\[ \begin{aligned} f_1(\gamma p \to p\eta) &= 4\sqrt{6}\,x_1,\\ f_1(\gamma p \to p\pi^0) &= 10\sqrt{2}\,x_1 \qquad\quad + 2\sqrt{2}\,x_3,\\ f_1(\gamma p \to n\pi^+) &= 5x_1 \qquad\quad + 5x_2 \qquad\quad + 4x_3,\\ f_1(\gamma p \to \Lambda K^+) &= -\frac{9}{\sqrt{6}}x_1 - \frac{9}{\sqrt{6}}x_2 - \sqrt{6}\,x_3 + \sqrt{6}\,x_5,\\ f_1(\gamma p \to \Sigma^0 K^+) &= \frac{1}{\sqrt{2}}x_1 + \frac{1}{\sqrt{2}}x_2 + \sqrt{2}\,x_3 + 3\sqrt{2}\,x_5,\\ f_1(\gamma p \to \Sigma^+ K^0) &= -2x_1 \qquad\quad + 2x_3; \end{aligned} \tag{5a} \]

for the photoproduction amplitudes on the neutron we have:
\[ \begin{aligned} f_1(\gamma n\to n\eta)&=-\sqrt{6}\,x_1+2\sqrt{6}\,x_3,\\ f_1(\gamma n\to n\pi^0)&=5\sqrt{2}\,x_1-2\sqrt{2}\,x_3,\\ f_1(\gamma n\to p\pi^-)&=5x_1-5x_2+4x_3,\\ f_1(\gamma n\to \Lambda K^0)&=3\sqrt{6}\,x_1-\frac{15}{\sqrt{6}}x_3-\frac{1}{\sqrt{6}}x_4+\sqrt{6}\,x_5,\\ f_1(\gamma n\to \Sigma^0K^0)&=\sqrt{2}\,x_1+\frac{1}{\sqrt{2}}x_3-\frac{1}{\sqrt{2}}x_4+3\sqrt{2}\,x_5,\\ f_1(\gamma n\to \Sigma^-K^+)&=x_1+x_2+x_3+x_4, \end{aligned} \tag{5b} \]
where
\[ x_1=18(b_1+c_2-c_3),\qquad x_2=6(b_2+c_1-c_4),\qquad x_3=b_4, \]
\[ x_4=\frac{1}{9}a,\qquad x_5=\frac{1}{3}(d_1+d_2-d_3-d_4). \]

From (5a) and (5b) the following relations between photoproduction amplitudes follow:
\[ f_1(\gamma p\to n\pi^+)+f_1(\gamma n\to p\pi^-) =\sqrt{2}\,[f_1(\gamma p\to p\pi^0)-f_1(\gamma n\to n\pi^0)], \tag{6a} \]
\[ f_1(\gamma p\to \Sigma^+K^0)+f_1(\gamma n\to \Sigma^-K^+) =\sqrt{2}\,[f_1(\gamma p\to \Sigma^0K^+)-f_1(\gamma n\to \Sigma^0K^0)], \]
\[ \sqrt{2}\,f_1\!\left(\gamma p\to p\,\frac{\pi^0-\sqrt{3}\eta}{2}\right) =f_1(\gamma p\to \Sigma^+K^0), \]
\[ \sqrt{2}\,f_1\!\left(\gamma p\to \frac{\Sigma^0-\sqrt{3}\Lambda}{2}\,K^+\right) =f_1(\gamma p\to n\pi^+), \tag{6b} \]
\[ f_1\!\left(\gamma n\to n\,\frac{\pi^0-\sqrt{3}\eta}{2}\right) =-f_1\!\left(\gamma n\to \frac{\Sigma^0-\sqrt{3}\Lambda}{2}\,K^0\right), \]
\[ f_1(\gamma p\to p\eta)-f_1(\gamma n\to n\eta) =\sqrt{3}\,f_1(\gamma n\to n\pi^0), \tag{6c} \]
\[ f_1(\gamma p\to p\pi^0)+f_1(\gamma n\to n\pi^0) =\frac{\sqrt{75}}{4}\,f_1(\gamma p\to p\eta). \]

Let us note that the relations (6a) are a consequence of the known transformation properties of the Hamiltonian of electromagnetic interactions with respect to isotopic rotations; the relations (6b) are satisfied in \(SU(3)\)-symmetry \((^4)\); and, finally, the relations (6c) are valid only in \(SU(6)\)-symmetry.

For an experimental test, the last relation in (6c) is of special interest. In the impulse approximation, the expression \(f_1(\gamma p\to p\pi^0)+f_1(\gamma n\to n\pi^0)\) is the amplitude for photoproduction of \(\pi^0\)-mesons on the deuteron. Therefore this relation between amplitudes is equivalent to a relation between the cross sections of \(\pi^0\)-photoproduction on the deuteron and \(\eta\)-meson production on the proton, if absorption of magnetic dipole \(\gamma\)-quanta takes place. Such a situation occurs in the region of the first resonance of pion photoproduction on nucleons.

As for the combination of amplitudes \(f_2=2M_{13}-M_{11}\), the relations (6a) and (6b) are satisfied for them. No new relations, valid only in \(SU(6)\)-symmetry, arise.

Physical-Technical Institute
Academy of Sciences of the Ukrainian SSR

Received
6 IX 1965

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