Cross section of the ?N interaction in the region of the S11N* (1535) resonance
Abstract Generated abstract
This paper estimates low energy eta nucleon interaction cross sections in the region of the S11 N*(1535) resonance, where direct eta beam measurements are not feasible. The authors model elastic eta N scattering and eta N to pi N conversion through s and u channel N*(1535) pole amplitudes, derive differential and total cross section formulas using isospin relations, and determine the eta NN* and pi NN* couplings from assumed resonance branching fractions. Numerical results at E* about 1575 MeV are tabulated for several resonance widths, and comparison with pion induced eta production data and existing eta N total cross section estimates favors a resonance width near 160 MeV, giving a total eta N cross section of about 68 mb.
Full Text
CROSS SECTION OF THE $\eta N$ INTERACTION IN THE REGION OF THE RESONANCE $S_{11}N^*(1535)$
G. N. Dudkin, V. N. Eponeshnikov, Yu. F. Krechetov, V. A. Tryasuchev
(Presented by the scientific-technical seminar of the High-Energy Laboratory of the Research Institute of Nuclear Physics)
As is known, the lifetime of the $\eta$ meson does not permit experiments with an $\eta$-meson beam. However, the cross section of the $\eta N$ interaction can be determined by measuring the dependence of the photoproduction cross section of $\eta$ mesons on nuclei on the atomic weight [1]. In this connection, estimates of the $\eta N$ interaction cross section are of interest. We assumed that, in the interaction of low-energy $\eta$ mesons (50–200 MeV) with a nucleon, the basic processes are
\[ \eta + N \to \eta + N, \tag{1} \]
\[ \eta + N \to \pi + N, \tag{2} \]
proceeding through the resonance $S_{11}N^*(1535)$. The pole parts of the amplitudes, including the $s$- and $u$-channels of the unstable intermediate particle—the resonance $S_{11}N^*(1535)$—for processes (1) and (2) were written in the form [2]
\[ M = |g^2_{\eta NN^*}| \left\{ \left( \frac{1}{m^2 - s - im\Gamma} + \frac{1}{m^2 - u - im\Gamma} \right)(m + m_N)\times \right. \]
\[ \left. \times \bar{u}(P_2)u(P_1) + \frac{1}{2} \left( \frac{1}{m^2 - s - im\Gamma} - \frac{1}{m^2 - u - im\Gamma} \right) \times \right. \]
\[ \left. \times \bar{u}(P_2)\hat{Q}u(P_1) \right\}, \tag{1} \]
\[ M = g_{\pi NN^*}g^*_{\eta NN^*} \left\{ \left( \frac{1}{m^2 - s - im\Gamma} + \frac{1}{m^2 - u - im\Gamma} \right)(m + m_N)\times \right. \]
\[ \left. \bar{u}(P_2)u(P_1) + \frac{1}{2} \left( \frac{1}{m^2 - s - im\Gamma} - \frac{1}{m^2 - u - im\Gamma} \right) \times \right. \]
\[ \left. \times \bar{u}(P_2)\hat{Q}u(P_1) \right\}, \tag{2} \]
where $g_{\eta NN^*}$, $g_{\pi NN^*}$ are the constants of the $\eta NN^*$ and $\pi NN^*$ interactions; $m$, $m_N$ are the masses of the resonance and the nucleon; $\hat{Q}=\gamma_\mu q_1^\mu+\gamma_\mu q_2^\mu$, $P_1 q_1$; $P_2 q_2$ are the momenta of the nucleon and meson, respectively, in the initial and final states; $\Gamma$ is the width of the resonance $S_{11}N^*(1535)$.
Isotopic invariance of processes (1) and (2) leads to the following relations:
\[ g_{\eta p N^*}=g_{\eta n N^*}, \tag{3} \]
\[ g_{\pi^0 p N^*}=-g_{\pi^0 n N^*} =\frac{1}{\sqrt{2}}\,g_{\pi^- p N^*} =\frac{1}{\sqrt{2}}\,g_{\pi^+ p N^*}. \tag{4} \]
Within the framework of \(SU(2)\) symmetry we have:
\[ \sigma(\eta p\to \eta p)=\sigma(\eta n\to \eta n), \tag{5} \]
\[ \sigma(\eta p\to \pi^0 p)=\sigma(\eta n\to \pi^0 n) =\frac{1}{2}\sigma(\eta p\to \pi^+ n) =\frac{1}{2}\sigma(\eta n\to \pi^- p). \tag{6} \]
The differential cross section for process (1) in the center-of-mass system (c.m.s.) is written in the form
\[ \frac{d\sigma}{dt} = -\frac{g_{\eta NN^*}^{4}\,m_N^2}{32\pi\cdot s\cdot P_i^2} \left[ \frac{K_1+M_1t}{(m^2-s)^2+m^2\Gamma^2} +\right. \]
\[ \left. +\frac{ 2\left[(m^2-s)(m^2-h+s+t)+m^2\Gamma^2\right](K_2+M_2t) }{ \left[(m^2-s)^2+m^2\Gamma^2\right] \left[(m^2-h+s+t)^2+m^2\Gamma^2\right] } +\right. \]
\[ \left. +\frac{K_3+M_3t}{(m^2-h+s+t)^2+m^2\Gamma^2} \right]. \tag{7} \]
Here \(t=-2P_i^2(1-\cos\Theta)\); \(\bar P_i,\Theta\) are the three-dimensional momentum and the emission angle of the \(\eta\)-meson in the c.m.s.
\[ K_1=\frac{1}{2}\left(2m+2m_N+\frac{2s-h}{2m_N}\right)^2, \]
\[ K_2=2(m+m_N)^2-\frac{(2s-h)^2}{8m_N^2}, \]
\[ K_3=\frac{1}{2}\left(2m+2m_N-\frac{2s-h}{2m_N}\right)^2, \]
\[ M_1=\frac{s-m^2}{2m_N^2}, \tag{8} \]
\[ M_2=-\frac{s+m^2+2mm_N}{2m_N^2}, \]
\[ M_3=\frac{s-(m+2m_N)^2}{2m_N^2}, \]
\[ h=s+t+u=2(m_N^2+\mu_\eta^2). \]
\(\mu_\eta\) is the mass of the \(\eta\)-meson. The total cross section of elastic scattering will have the form
\[ \sigma= \frac{g_{\eta NN^*}^{4}\,m_N^2}{32\pi\cdot s\cdot P_i^2} \left\{ \frac{ 4K_1P_i^2-8M_1P_i^4+8M_2P_i^2(m^2-s) }{ (m^2-s)^2+m^2\Gamma^2 } +\right. \]
\[ \left. +\left[ \frac{ m\Gamma(2K_2+2M_2h-4m^2M_2) }{ (m^2-s)^2+m^2\Gamma^2 } + \frac{ K_3-M_3(m^2-h+s) }{ m\Gamma } \right]\times \right. \]
\[ \left. \times \arctg \frac{ 4P_i^2m\Gamma }{ m^2\Gamma^2+(m^2-h+s)(m^2-h+s-4P_i^2) } +\right. \]
\[ \left. +\left[ \frac{ K_2(m^2-s)+M_2m^2\Gamma^2-M_2(m^2-h+s)(m^2-s) }{ (m^2-s)^2+m^2\Gamma^2 } + \frac{1}{2}M_3 \right]\cdot \ln \frac{ (m^2-h+s)^2+m^2\Gamma^2 }{ (m^2-h+s-4P_i^2)^2+m^2\Gamma^2 } \right\}. \tag{9} \]
For process (2) the differential cross section in the c.m. system is written in the form
\[ \frac{d\sigma}{dt}= -\frac{g_{\pi NN^{*}}^{2}\,g_{\eta NN^{*}}^{2}\,m_N^{2}}{32\pi\cdot s\cdot P_i^{2}} \left[ \frac{N_1+R_1t}{(m^2-s)^2+m^2\Gamma^2} +\right. \]
\[ \left. +\frac{2\left[(m^2-s)(m^2-h+s+t)+m^2\Gamma^2\right](N_2+R_2t)} {\left[(m^2-s)^2+m^2\Gamma^2\right]\left[(m^2-h+s+t)^2+m^2\Gamma^2\right]} +\frac{N_3+R_3t}{(m^2-h+s+t)^2+m^2\Gamma^2} \right], \tag{10} \]
\[ N_1=K_1-\frac{(\mu_\eta^2-\mu_\pi^2)^2}{8m_N^2}, \]
\[ N_2=K_2+\frac{(\mu_\eta^2-\mu_\pi^2)^2}{8m_N^2}, \tag{11} \]
\[ N_3=K_3-\frac{(\mu_\eta^2-\mu_\pi^2)^2}{8m_N^2}, \]
\[ R_1=M_1,\qquad R_2=M_2,\qquad R_3=M_3, \]
\[ h=2m_N^2+\mu_\eta^2+\mu_\pi^2 . \]
\[ t=(P_1-P_2)^2=(E_1-E_2)^2-\left[P_i^2+P_f^2+2P_iP_f\cos(\pi-\theta)\right], \]
\[ t(\theta=0)=t_1=(E_1-E_2)^2-(P_i-P_f)^2, \tag{12} \]
\[ t(\theta=\pi)=t_2=(E_1-E_2)^2-(P_i+P_f)^2 . \]
\(\mu_\pi\) is the pion mass.
\(E_1,\mathbf P_i;\ E_2,\mathbf P_f\) are the energy and momentum of the initial and final nucleon in the c.m. system. The total cross section of process (2) will be equal to
\[ \sigma= \frac{m_N^2 g_{\pi NN^{*}}^{2}g_{\eta NN^{*}}^{2}} {32\pi\cdot s\cdot P_i^{2}} \left\{ \frac{ N_1(t_1-t_2)+\frac12 R_1(t_1^2-t_2^2)+2R_2(m^2-s)(t_1-t_2)} {(m^2-s)^2+m^2\Gamma^2} +\right. \]
\[ +\left[ \frac{m\Gamma(2N_2+2R_2h-4R_2m^2)} {(m^2-s)^2+m^2\Gamma^2} + \frac{N_3+R_3(m^2-h+s)}{m\Gamma} \right]\times \]
\[ \times \operatorname{arctg} \frac{m\Gamma(t_1-t_2)} {m^2\Gamma^2+(m^2-h+s+t_2)(m^2-h+s+t_1)} + \]
\[ +\left. \left[ \frac{N_2(m^2-s)+R_2m^2\Gamma^2-R_2(m^2-h+s)(m^2-s)} {(m^2-s)^2+m^2\Gamma^2} +\frac{R_3}{2} \right] \ln \frac{(m^2-h+s+t_1)^2+m^2\Gamma^2} {(m^2-h+s+t_2)^2+m^2\Gamma^2} \right\}. \]
The constants \(g_{\eta NN^{*}}\), \(g_{\pi NN^{*}}\) can be determined from the decay of the resonance \(S_{11}N^{*}(1535)\) as functions of \(\Gamma\). The matrix elements of the processes
\[ N^{*}(1535)\to N+\eta, \]
\[ N^{*}(1535)\to N+\pi \tag{14} \]
were written, according to [4], as
\[ g_{\eta NN^{*}}\bar u(P_N)u(P_{N^{*}})\bar N^\alpha N^{*}_{\alpha}, \]
\[ g_{\pi NN^{*}}\bar u(P_N)u(P_{N^{*}})\bar N^\alpha(\tau_i)_{\alpha}^{\beta}N^{*}_{\beta}\pi_i, \tag{15} \]
where \(N, P_N;\ N^{*}, P_{N^{*}}\) are the isospin and four-momentum, respectively, of the nucleon and resonance. In the rest system of the resonance \(S_{11}N^{*}(1535)\), the probabilities of the processes (14) have the form
\[ W_\eta=\Gamma_\eta=\frac{\lvert g_{\eta NN^*}\rvert^2\bigl((m+m_N)^2-\mu_\eta^2\bigr)}{8\pi m^2}\lvert \bar P_f\rvert, \]
\[ W_\pi=\Gamma_\pi=3\frac{g_{\pi NN^*}^{2}}{8\pi}\frac{(m+m_N)^2-\mu_\pi^2}{m^2}\lvert P'_f\rvert, \tag{16} \]
where \(P_f, P'_f\) are the momenta in the c.m. system. According to [3], it was assumed that
\[ \frac{\Gamma_\eta}{\Gamma}=0.66,\qquad \frac{\Gamma_\pi}{\Gamma}=0.34. \tag{17} \]
The cross sections of processes (1) and (2), with the contribution of the resonance \(S_{11}N^*(1535)\) to the amplitude, are given in Table 1 as functions of the resonance width \(\Gamma\). The coupling constants \(g_{\pi NN^*}^{2}\) and \(g_{\eta NN^*}^{2}\) are also given there. The cross sections were calculated for the total energy \(E^*=1575\) MeV in the \(\eta\)—\(N\) center-of-mass system, which corresponds to a nucleon at rest and an \(\eta\)-meson with kinetic energy \(T_\eta \simeq 140\) MeV in the laboratory coordinate system.
Table 1
| \(\Gamma\) (MeV) | 50 | 100 | 160 |
|---|---|---|---|
| \(\lvert g_{\eta NN^*}\rvert^2\) | 1.84 | 3.69 | 5.91 |
| \(\lvert g_{\pi NN^*}\rvert^2\) | 0.118 | 0.236 | 0.377 |
| \(\sigma(\eta p\to \eta p)=\sigma(\eta n\to \eta n)\) (mb) | 16.6 | 36.5 | 48.4 |
| \(\sigma(\eta p\to \pi^0 p)=\sigma(\eta n\to \pi^0 n)\) (mb) | 2.59 | 4.81 | 6.40 |
| \(\sigma(\eta p\to \pi^+ n)=\sigma(\eta n\to \pi^- p)\) (mb) | 5.18 | 9.62 | 12.8 |
| \(\sigma_t(\eta p)=\sigma_t(\eta n)\) (mb) | 24.37 | 50.93 | 67.6 |
At present there are a number of works [5—8] on the determination of the total and differential cross sections of the process
\[ \pi^-+p\to\eta+n. \tag{18} \]
The cross section of this process, starting from threshold, has a form characteristic of a resonance and is close to isotropic [5—8]. Therefore there is reason to suppose that at the total energy \(E^*=1575\) MeV in the \(\pi\)—\(N\) center-of-mass system (which corresponds to the kinetic energy of the pion \(T_\pi\simeq 700\) MeV in the laboratory coordinate system) the resonance \(S_{11}N^*(1535)\) dominates in the cross section of process (18). The detailed-balance method, taking isospin conservation into account, gives
\[ \sigma(\eta n\to \pi^-p)=\frac{3}{2}\sigma(\pi^-p\to \eta n)\cdot\left(\frac{\bar P^*_\pi}{\bar P^*_\eta}\right)^2. \tag{19} \]
We take the value \(\sigma(\pi^-p\to \eta n)\) for \(E^*=1576\) MeV in the \(\pi\)—\(N\) center-of-mass system from [5]. Then, from formula (19), we have
\[ \sigma=(\eta n\to\pi^-p)\simeq 14.5\ \text{mb}. \tag{20} \]
Comparing this result with the tabulated one (Table 1), we see that the best agreement is achieved at \(\Gamma=160\) MeV. The authors of [9] arrived at the same value of the width of the resonance \(S_{11}N^*(1535)\). They studied the process
\[ \gamma+p\to\eta+p. \tag{21} \]
The value of the total cross section of the \(\eta N\) interaction at \(T_\eta=145\) MeV, \(\sigma_{\text{total}}\simeq 65\) mb, which was obtained in Frascati [1], also indicates the width \(\Gamma=160\) MeV.
It follows from Ref. [10] that \(g_{\eta NN}^{2}/4\pi = 0.002\). Therefore, the nucleon contribution to the cross sections of processes (1) and (2) does not exceed 5% for \(\Gamma = 160\) MeV.
At present, preparations are under way at the Nuclear Physics Research Institute of Yerevan State University to measure the total cross section of the \(\eta N\) interaction in the energy region \(E^{*} = 1575\) MeV.
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