UDC 539.128.417+539.125

PHYSICS

V. B. BELYAEV, DAO VONG DYK, NGUYEN VAN HIEU

ELECTROMAGNETIC DECAY AND ELECTROPRODUCTION OF A \(3/2^+\)-RESONANCE

(Presented by Academician N. N. Bogolyubov on 12 VI 1965)

The success of the theory of unitary symmetry of elementary particles increases interest in the study of electromagnetic decay and electroproduction of baryon resonances. This is due to the fact that, by studying these processes, one can, with the aid of one or another symmetry scheme, judge the structure of the electromagnetic form factors of baryons.

The aim of the present work is to give a general expression for the probability of electromagnetic decay and for the cross section of electroproduction of a resonance and, in particular, to compare the experimental data for these processes with the theoretical predictions following from the \(U(12)\) symmetry scheme \((^1)\).

Let us write the general expression for the vector current for the transition \(1/2^+ \to 3/2^+\):

\[ V_\mu=\bar{\psi}_\nu(q)\left\{\frac{a_1}{m}(k_\nu\gamma_\mu-\delta_{\mu\nu}\hat{k})+ \frac{a_2}{m^2}k_\nu k_\lambda\sigma_{\mu\nu}+ \frac{a_3}{m^2}(k_\nu k_\mu-\delta_{\mu\nu}k^2)\right\}\gamma_5\psi(p), \tag{1} \]

where \(a_1, a_2, a_3\) are certain form factors depending on \(k^2\); \(m\) is a quantity of the dimension of mass; \(p, q\) are the 4-momenta of the initial and final baryons, respectively; \(k=q-p\).

In the \(U(12)\) symmetry scheme the matrix element of the vector current for the transition between the states of \(1/2^+\) and \(3/2^+\) baryons belonging to the 364-plet has the form

\[ V_\mu=\bar{D}^{abc}\varepsilon_{ade}Q_b^dN_c^e\mu_p \frac{f(k^2)}{m_0^2}\varepsilon_{\mu\nu\sigma\rho}\bar{\psi}_\nu\psi_p\gamma_\sigma k_\rho, \tag{2} \]

where \(D_{abc}\) and \(N_b^a\) are the unitary wave functions of the baryon decuplet and octet; \(Q\) is the charge operator; \(\mu_p\) is the magnetic moment of the proton; \(f(k^2)\) is the magnetic (Sachs) form factor of the proton; \(m_0\) is the mass of the multiplet in the case of exact symmetry (see \((^2,^3)\)).

Let us first consider the electromagnetic decay

\[ \mathrm{B}^*\to \mathrm{B}+\gamma, \]

whose matrix element has the form

\[ M=\frac{1}{\sqrt{2\omega}}\,\bar{\psi} \left\{a_1\left(e_\mu\gamma_5-\frac{i k_\mu e_\nu\gamma_5}{m_{\mathrm B}+m_{\mathrm B^*}}\right) +\frac{a_2}{m^2}\varepsilon_{\nu\mu\sigma\rho}e_\nu p_\sigma k_\rho\right\}\psi_\mu, \tag{3} \]

whence for the decay probability we obtain

\[ W=\frac{e^2m_{\mathrm B^*}}{24\pi} \left(1-\frac{m_{\mathrm B}^2}{m_{\mathrm B^*}^2}\right) \left\{ |a_1|^2\left[ \left(1-\frac{m_{\mathrm B}}{m_{\mathrm B^*}}\right)^2 -\frac{1}{4}\left(1-\frac{m_{\mathrm B}}{m_{\mathrm B^*}}\right)^3 \left(1+\frac{m_{\mathrm B}}{m_{\mathrm B^*}}\right) \right]\right. \]

\[ \left. +\frac{|a_2|^2}{4} \left(\frac{m_{\mathrm B^*}}{m}\right)^4 \left(1-\frac{m_{\mathrm B}^2}{m_{\mathrm B^*}^2}\right)^2 \left(1+\frac{m_{\mathrm B}}{m_{\mathrm B^*}}\right)^2 -\frac{\operatorname{Re}a_1a_2^*}{4} \left(\frac{m_{\mathrm B^*}}{m}\right)^2 \left(1-\frac{m_{\mathrm B}^2}{m_{\mathrm B^*}^2}\right)^2 \left(3+\frac{m_{\mathrm B}}{m_{\mathrm B^*}}\right) \right\}. \tag{4} \]

In the case of \(U(12)\), instead of (4) we obtain

\[ W=\frac{e^2}{96\pi}\,m_{B^*}|g|^2 \left(\frac{m_{B^*}}{m_0}\right)^4 \left(1-\frac{m_B^2}{m_{B^*}^2}\right)^3 \left(1+\frac{m_B}{m_{B^*}}\right)^2, \tag{4′} \]

where \(g=1/\sqrt{3}\,\mu_p\) for the decay \(N^{*+}\to p+\gamma\). For the remaining decays we have the relation

\[ g_{N^{*0},\,n\gamma} =-\frac{2}{\sqrt{3}}g_{Y^{*0},\,\Lambda\gamma} =-g_{Y^{*+},\,\Sigma^+\gamma} \]

\[ =2g_{Y^{*0},\,\Sigma^0\gamma} =-g_{\Xi^{*0},\,\Xi^0\gamma} =g_{N^{*+},\,p\gamma} =-\frac{1}{\sqrt{3}}\mu_p . \]

Using expression (4′) with \(m_0\) equal to the mean mass of the multiplet, we obtain the following estimate for the decay probabilities:

\[ \begin{aligned} \Gamma(N^{*+}\to p+\gamma)&=0.177\ \text{MeV},\\ \Gamma(Y^{*0}\to \Lambda+\gamma)&=0.144\ \text{MeV},\\ \Gamma(Y^{*+}\to \Sigma^+ +\gamma)&=0.083\ \text{MeV},\\ \Gamma(\Xi^{*0}\to \Xi^0+\gamma)&=0.179\ \text{MeV}. \end{aligned} \]

Let us now turn to the consideration of electroproduction of a \(3/2^+\)-baryon on a \(1/2^+\)-baryon:

\[ e^-+B\to B^*+e^- . \]

The diagram of this process is shown in Fig. 1. It corresponds to a matrix element of the form

Fig. 1

\[ M=\frac{1}{k^2}V_\mu \bar u(k_2)\gamma_\mu u(k_1), \tag{5} \]

where \(V_\mu\) is given by expression (1).

Hence, for the differential cross section we obtain the expression

\[ \begin{aligned} d\sigma={}&\frac{2}{3}\left(\frac{e^2}{4\pi}\right)^2 \frac{1}{v_1 k^4 \varepsilon_1 \varepsilon_2 E_B E_{B^*}}\, \delta(E_B+\varepsilon_1-E_{B^*}-\varepsilon_2)\,d\mathbf{k}_2 \\ &\times \Bigg\{ \left|\frac{a_1}{m}-\frac{i a_2}{m^2}(m_{B^*}-m_B)\right|^2 \,2\left(k^2+\frac{(kq)^2}{m_{B^*}^2}\right) \\ &\times \left(\frac{1}{2}m_Bm_{B^*}k^2+k_1p\cdot k_2q+k_1q\cdot k_2p\right) \\ &+\left|\frac{a_2}{m^2}\right|^2 \left(k^2+\frac{(kq)^2}{m_{B^*}^2}\right)(m_Bm_{B^*}+pq) \\ &\times \left[-\frac{k^2}{2}(p+q)^2 -2(pk_1+qk_1)(pk_2+qk_2)\right] \\ &+\left|\frac{i a_1}{m}(m_B+m_{B^*}) +\frac{a_3}{m^2}k^2\right|^2 (m_Bm_{B^*}+pq) \left(-\frac{k^2}{2}-\frac{2qk_1\cdot qk_2}{m_{B^*}^2}\right) \\ &+2\,\operatorname{Im}\left(\frac{a_1}{m} -\frac{i a_2}{m^2}(m_{B^*}-m_B)\right) \frac{a_2^*}{m^2} \left(k^2+\frac{(kq)^2}{m_{B^*}^2}\right) \\ &\times \left[ (m_{B^*}-m_B)(m_Bm_{B^*}+pq)\frac{k^2}{2} +(m_{B^*}-m_B)(qk_1\cdot pk_2+qk_2\cdot pk_1) \right. \\ &\qquad\left. +2m_{B^*}pk_1\cdot pk_2 -2m_B qk_1\cdot qk_2 \right] \\ &+2\,\operatorname{Im}\left(\frac{a_1}{m} -\frac{i a_2}{m^2}(m_{B^*}-m_B)\right) \left(\frac{i a_1^*}{m}(m_B+m_{B^*}) -\frac{a_3^*}{m^2}k^2\right) \\ &\times \left[ \frac{m_{B^*}}{4}k^2(pk-k^2) +\frac{1}{m_{B^*}}\left( \frac{3}{4}pq\cdot kq\cdot k^2 +\frac{1}{2}kq\cdot qk_2\cdot pk_1 \right.\right. \\ &\qquad\left.\left. +\frac{1}{2}kq\cdot qk_1\cdot pk_2 +kp\cdot qk_1\cdot qk_2 \right) +\frac{m_B}{2}k^2\cdot qk -\frac{2m_B}{m_{B^*}}kq\cdot qk_1\cdot qk_2 \right] \\ &+2\,\operatorname{Re}\frac{a_2}{m^2} \left(\frac{i a_1^*}{m}(m_B+m_{B^*}) -\frac{a_3^*}{m^2}k^2\right) \\ &\times (m_Bm_{B^*}+pq)\,qk \left[ \frac{k^2}{2}\left(1-\frac{pq}{m_{B^*}^2}\right) -\frac{1}{m_{B^*}^2} (2qk_1\cdot qk_2+qk_1\cdot pk_2+qk_2\cdot pk_1) \right] \Bigg\}. \end{aligned} \tag{6} \]

In the case of the \(\tilde U\) symmetry (12), formula (6) reduces to the simpler expression

\[ \frac{d\sigma}{d\varepsilon_2\,d\Omega} = \frac{1}{3}\left(\frac{e^2}{4\pi}\right)^2 |g f(k^2)|^2 \frac{m_B}{E_{B^*}} \frac{\varepsilon_2}{\varepsilon_1} \left[1+\frac{(m_B+m_{B^*})^2}{k^2}\right] \times \frac{\varepsilon_1^2+\varepsilon_2^2+k^2/2}{m_0^4} \frac{\delta(\varepsilon_2-\varepsilon_2^0)} {1+(\varepsilon_2^0-\varepsilon_1\cos\theta)/E_{B^*}}, \tag{7} \]

where \(\varepsilon_1, E_B\) and \(\varepsilon_2^0, E_{B^*}\) are the energies of the electron and baryon, respectively, in the initial and final states.

Since \(f(k^2)\) has been measured* in a fairly wide range of transferred momenta, one can compare expression (7) with the available experiments on electroproduction of \(\pi\)-mesons. It is clear that expression (7) must be compared with experiment in the region of resonant produc-

Table 1

Table 2

tion of the \(\pi\)-meson. For convenience, Table 1 gives experimental data taken from Ref. \((^5)\) on electroproduction of \(\pi\)-mesons. Table 2 gives the values of the ratios of cross sections at different energies, as well as

* The values of \(f(k^2)\) are given in Ref. \((^4)\).

corresponding theoretical values. From this table it is seen that the theoretical values for the ratios of cross sections, obtained on the basis of the symmetry $\widetilde U(12)$ using the experimental data on the proton form factor, do not contradict experiment. It should be noted, however, that formula (7) gives the correct absolute values only for the value $m_0 \approx 1900(1 \pm 0.2)$ MeV.

The authors express their deep gratitude to Acad. N. N. Bogolyubov and Prof. Ya. A. Smorodinskii for their interest in the work.

Note added in proof. It should be noted that the electroproduction of an isobar in the $SU_6$ scheme was considered in the recently published work of Geshkenbein (6).

Joint Institute
for Nuclear Research

Received
12 VI 1965

CITED LITERATURE

¹ A. Salam, R. Delbourgo, J. Strathdee, Proc. Roy. Soc., 284, 146 (1965).
² N. N. Bogolyubov, Nguyen Van Hieu, D. Stoyanov, B. V. Struminskii, A. N. Tavkhelidze, V. P. Shelest, Preprint, Joint Inst. Nucl. Res., 1965.
³ P. Winternitz, A. A. Makarov, Nguyen Van Hieu, L. G. Tkachev, M. Uglirzh, Preprint, Joint Inst. Nucl. Res., 1965.
⁴ R. Salin, Nuovo Cimento, 32, 521 (1963).
⁵ G. G. Ohlsen, Phys. Rev., 120, 584 (1960).
⁶ B. V. Geshkenbein, Phys. Lett., 16, No. 3 (1965).