UDC 539.12.01

MATHEMATICAL PHYSICS

V. B. GOSTEV, A. R. FRENKIN

SCATTERING OF MESONS IN A MODEL WITH A FIXED NUCLEON

(Presented by Academician N. N. Bogolyubov, 11 I 1966)

In papers (1–3) a model of a meson–nucleon system with three possible states of a fixed nucleon was considered. In the present note this model is used to describe the scattering of mesons by a fixed source.

We write the Hamiltonian of the meson–nucleon system in the form

\[ H=m_{A0}A^{+}A+m_{B0}B^{+}B+m_C C^{+}C+\int dk^3 \omega a^{+}(k)a(k)+ \]
\[ +\lambda_{01}\int dk^3 u(\omega)[A^{+}Ba(k)+\text{h. c.}] +\lambda_{02}\int dk^3 [B^{+}Ca(k)+\text{h. c.}], \tag{1} \]

where \(\omega=\omega_k=\sqrt{k^2+\mu^2}\); \(u(\omega)\) is a real cutoff function (for a point source \(u(\omega)=1/\sqrt{2\omega}\)); \(A^{+}(A)\), \(B^{+}(B)\), \(C^{+}(C)\), and \(a^{+}(k)(a(k))\) are the creation (annihilation) operators of fermions \(A\), \(B\), \(C\) and of the boson \(\theta\). All quantities with the subscript 0 are unrenormalized.

In paper (3) the Schrödinger equation was solved exactly for one-nucleon states belonging to the discrete energy spectrum. Now the Schrödinger equation

\[ H|X\rangle=E|X\rangle \tag{2} \]

is solved for states of the continuous spectrum \(|X\rangle\), containing one \(A\)-particle—the states of meson scattering on nucleons.

We seek the state \(|X\rangle\) in the form:

\[ |X\rangle=\{\psi(E)A^{+}+\int dk^3\psi_1(E,\omega_k)B^{+}a^{+}(k)+ \]
\[ +\frac{1}{\sqrt{2!}}\int dk^3 dl^3\psi_2(E,\omega_k,\omega_l)C^{+}a^{+}(k)a^{+}(l)\}|0\rangle, \tag{3} \]

where \(|0\rangle\) is the vacuum state. Equation (2) is equivalent to the following relations between the wave functions:

\[ (E-m_{A0})\psi(E)=\lambda_{01}\int dk^3\psi_1(E,\omega_k)u(\omega_k), \tag{4} \]

\[ (E-m_{B0}-\omega_k)\psi_1(E,\omega_k)= \]
\[ =\lambda_{01}u(\omega_k)\psi(E)+\sqrt{2}\lambda_{02}\int dl^3 u(\omega_l)\psi_2(E,\omega_k,\omega_l), \tag{5} \]

\[ (E-m_C-\omega_k-\omega_l)\psi_2(E,\omega_k,\omega_l)= \]
\[ =\frac{1}{\sqrt{2}}\lambda_{02}[u(\omega_k)\psi_1(E,\omega_l)+u(\omega_l)\psi_1(E,\omega_k)]. \tag{6} \]

Among the states \(|X\rangle\) we choose the states corresponding to an incident plane wave \(B^{+}a^{+}(k_0)|0\rangle\) and an outgoing scattered wave. These states \(|B\theta k_0\rangle_+\) describe the scattering of mesons by \(B\)-particles. For them

\[ \psi_1(E,\omega_k)=\psi_1(\omega_0,\omega_k)=\delta^3(k_0-k)+\varphi_{1+}(\omega_k,\omega_0), \tag{7} \]

where \(\omega_0=E-m_B\) is the energy of the scattered meson; \(m_B\) is the renormalized mass of the \(B\)-particle. Equations (5) and (6) reduce to an integralոփ

equation

\[ h(\omega_0+b-\omega)\varphi_{1+}(\omega,\alpha_0)=\lambda_{01}Z_Bu(\omega)\psi(\omega_0)- \]
\[ -\gamma u(\omega)u(\omega_0)\frac{1}{\omega-b} -\gamma\int \frac{\varphi_{1+}(\omega_l)u(\omega_l)\,dl^3}{\omega+\omega_l-\omega_0-b-i\varepsilon}, \tag{8} \]

in which

\[ h(z)=(z-b)\left[1+4\pi(z-b)\gamma\int_\mu^\infty \frac{\sqrt{\omega^2-\mu^2}\,\omega u^2(\omega)\,d\omega}{(\omega-b)^2(\omega-z-i\varepsilon)}\right], \tag{9} \]

\(\gamma=\lambda_{02}Z_B=\lambda_2^2\) is the square of the renormalized coupling constant of the \(BC\theta\)-interaction; \(Z_B\) is the renormalization constant of the \(B\)-particle, coinciding with the renormalization constant of the \(V\)-particle in the Lee model \((^4)\). We shall restrict ourselves to the case \(0<Z_B\leq 1\) and regard the \(B\)-particle as stable: \(b=m_B-m_C<\mu\). By the substitution

\[ \varphi_{1+}(\omega,\omega_0)=\gamma\frac{u(\omega)u(\omega_0)\varphi(\omega,\omega_0)}{h(\omega_0+b-\omega)} \tag{10} \]

equation (9) is transformed into the form

\[ \varphi(\omega,\omega_0)=-\frac{1}{\omega-b}+K-\frac{1}{\pi}\int_\mu^\infty \frac{\operatorname{Im}h(\omega')\varphi(\omega',\omega_0)\,d\omega'} {h(b+\omega_0-\omega)(\omega'+\omega-\omega_0-b-i\varepsilon)}, \tag{11} \]

\[ K=\frac{\lambda_{01}Z_B}{\gamma}\frac{\psi(\omega_0)}{u(\omega_0)}. \]

An integral equation with such a kernel for the inhomogeneous term \(-1/(\omega-b)\) in the case \(b=0\) was solved in works \((^5,^6)\), and for a constant inhomogeneous term in work \((^3)\). Generalizing the solution \((^5)\) to the case \(b\ne0\) by the method used in \((^3)\), we find

\[ \varphi(\omega,\omega_0)= \frac{h(\omega_0+b-\omega)}{\omega_0-\omega+i\varepsilon} \left[f(\omega,\omega_0)+KZ_B^{-1}j(\omega,\omega_0)\right], \tag{12} \]

where

\[ f(\omega,\omega_0)=- \left[\frac{\omega-b}{h(\omega)}\frac{1}{\omega-b} +\frac{2A(\omega)}{1-h(\omega)A(\omega_0)}\right], \tag{13} \]

\[ j(\omega,\omega_0)= \frac{1+h(\omega_0)[A(\omega)-A(\omega_0)]}{1-h(\omega)A(\omega_0)}, \tag{14} \]

\[ A(\omega)=\frac{1}{\pi}\int_\mu^\infty \frac{(\omega'-b)\,d\omega'} {(\omega'+\omega-\omega_0-b-i\varepsilon)h(\omega_0+b-\omega')} \operatorname{Im}\frac{1}{h(\omega')}. \tag{15} \]

Returning to \(\varphi_{1+}(\omega,\omega_0)\), we obtain

\[ \varphi_{1+}(\omega,\omega_0)= \frac{u(\omega)}{\omega_0-\omega+i\varepsilon} \left[\gamma u(\omega_0)f(\omega,\omega_0)+\lambda_{01}\psi(\omega_0)j(\omega,\omega_0)\right]. \tag{16} \]

Substituting \(\varphi_{1+}(\omega,\omega_0)\) into equation (4), we determine \(\psi(\omega_0)\)

\[ \psi(\omega_0)=\lambda_{01}\frac{u(\omega_0)[1+L(\omega_0)]}{g(\omega_0)}, \tag{17} \]

where

\[ L(\omega_0)=\frac{1}{\pi}\int_\mu^\infty \frac{d\omega\,\operatorname{Im}h(\omega)f(\omega,\omega_0)} {\omega_0-\omega+i\varepsilon}, \tag{18} \]

\[ g(\omega_0)=m_B+\omega_0+i\varepsilon-m_{A_0} -\frac{\lambda_{01}^2}{\gamma}M(\omega_0), \tag{19} \]

\[ M(\omega_0)=\frac{1}{\pi}\int_\mu^\infty \frac{d\omega\,\operatorname{Im}h(\omega)j(\omega,\omega_0)} {\omega_0-\omega+i\varepsilon}. \tag{20} \]

The zeros of the function \(g(\omega_0)\) determine the values of the renormalized mass of the \(A\)-particle \({}^{(3)}\).

Thus, the state \(|B\theta k_0\rangle_+\) is completely determined (\(\psi_2(\omega_0,\omega_k,\omega_l)\) is immediately found from equation (6)). Taking into account the normalization, the \(|B\theta k_0\mathrm{in}\rangle\)-state has the form

\[ |B\theta k_0\mathrm{in}\rangle=Z_B^{1/2}|B\theta k_0\rangle_+ . \tag{21} \]

The last independent solution of the Schrödinger equation (2), corresponding to the scattering of two mesons by a \(C\)-particle \(|C2\theta k_0l_0\rangle_+\), contains the plane wave
\[ \frac{1}{\sqrt{2}}\,C^+a^+(k_0)a^+(l_0)|0\rangle \]
and the outgoing scattered wave
\(\varphi_{2+}(\omega_{k_0},\omega_{l_0},\omega_k,\omega_l)\) (the notation \(E=m_C+\omega_{k_0}+\omega_{l_0}\) is adopted). The main technical difficulty in determining this state consists in solving the integral equation

\[ \eta(\omega)=-\frac{1}{\omega-\omega_{k_0}}+ \frac{1}{\pi}\int_\mu^\infty \frac{\operatorname{Im}h(\omega')\eta(\omega')\,d\omega'} {(\omega'+\omega-\omega_{k_0}-\omega_{l_0}-i\varepsilon)\, h(\omega_{k_0}+\omega_{l_0}-\omega')}, \tag{22} \]

in which the pole of the inhomogeneous term does not coincide with the zero of the function \(h(z)\). Such an equation for \(b=0\) was solved in Refs. \({}^{(2,7)}\). Using the method of Ref. \({}^{(3)}\), it is easy to generalize the solution to the case \(b\ne0\). Because of the cumbersome formulas, the explicit expression for the state \(|C2\theta k_0l_0\rangle_+\) is not given. The state \(|C2\theta k_0l_0\rangle_+\) is normalized to a plane wave and will be denoted by \(|C2\theta k_0l_0\mathrm{in}\rangle\).

The scattering states make it possible to determine the amplitudes for scattering of a \(\theta\)-particle by a \(B\)-particle, \(T(\omega)\), and for production of two \(\theta\)-particles in \(B-\theta\) collisions, \(T(\omega_p,\omega_q)\), from the formulas

\[ (\mathrm{out}\ B\theta k'|B\theta k\ \mathrm{in}) =\delta^3(k-k')-2\pi i\delta(\omega-\omega')T(\omega), \tag{23} \]

\[ (\mathrm{out}\ C2\theta pq|B\theta k\ \mathrm{in}) =-2\pi i\delta(\omega_p+\omega_q-\omega_k-b)T(\omega_p,\omega_q). \tag{24} \]

Calculating the scalar products, we obtain

\[ T(\omega)=u^2(\omega)\left\{\gamma f(\omega,\omega) +\lambda_{01}^2 j(\omega,\omega)g^{-1}(\omega)[1+L(\omega)]\right\}, \tag{25} \]

\[ \begin{aligned} T(\omega_p,\omega_q) &=\frac{1}{\sqrt{2}}\lambda_2 u(\omega_p)u(\omega_q)u(\omega_1) \Bigg\{ \gamma\left[\frac{f(\omega_q,\omega_1)}{\omega_p-b} +\frac{f(\omega_p,\omega_1)}{\omega_q-b}\right] \\ &\qquad\qquad +\lambda_{01}^2\frac{[1+L(\omega_1)]}{g(\omega_1)} \left[ \frac{j(\omega_q,\omega_1)}{\omega_p-b} +\frac{j(\omega_p,\omega_1)}{\omega_q-b} \right] \Bigg\}, \end{aligned} \tag{26} \]

\[ \omega_1=\omega_p+\omega_q-b . \]

With the aid of equalities (13), (14), the \(B-\theta\) scattering amplitude is transformed to the form

\[ T(\omega)=u^2(\omega)\frac{1}{1-h(\omega)A_0(\omega)} \left\{ \lambda_{01}^2\frac{[1+L(\omega)]}{g(\omega)} -\gamma\,\frac{1+h(\omega)A_0(\omega)}{h(\omega)} \right\}, \]

\[ A_0(\omega)=\frac{1}{\pi}\int_\mu^\infty \frac{d\omega'}{h(\omega+b-\omega')} \operatorname{Im}\frac{1}{h(\omega')}. \tag{27} \]

The amplitude \(T(\omega)\), as expected, for \(\lambda_{02}=0\) becomes the \(N\theta\)-scattering amplitude of Lee’s model \({}^{(4)}\), and for \(\lambda_{01}=0\) the \(V\theta\)-scattering amplitude of the same model \({}^{(8)}\).

It should be noted that at incident-meson energies \(\omega\le 2\mu-b\), scattering of \(\theta\)-particles by a \(B\)-particle is purely elastic, whereas at energies \(\omega>2\mu-b\) the competing process of production of two mesons with probability amplitude (26) is switched on, and the scattering becomes inelastic.

The cross section for elastic scattering of \(\theta\)-particles is proportional to \(|T(\omega)|^2\). It may have two maxima associated with resonance denomina-

by \(1-h(\omega)A_0(\omega)\) and \(g(\omega)\). The first denominator leads to a maximum of the cross section when the interaction constant \(\gamma\) is such that, with the \(AB\theta\)-interaction switched off, no bound \(B\theta\)-states are formed (\(V\theta\)-bound states in the Lee model), i.e., for \(Z_B>1/2\) \({}^{(9)}\). The second denominator gives a resonance when the constant \(\gamma\) is so small that no second one-particle state of the \(A\)-particle is formed \({}^{(3)}\). As shown in \({}^{(3)}\), the resonance energy corresponding to the minimum of \(|g(\omega)|\), for fixed coupling constants \(\lambda_{01}, \lambda_2\), is greater than the resonance energy of the first denominator. Therefore, as \(\gamma\) decreases from

\[ \gamma_{\mathrm{crit}}=\left[\int \frac{dk^3 u^2(\omega)}{(b-\omega)^2}\right]^{-1} \]

to 0, first the second resonance appears, then the first; the resonance energies increase, while the distance between the maxima of the cross section decreases—they merge into a single asymmetric maximum. The amplitude for the production of two mesons, \(T(\omega_p,\omega_q)\), has the same features.

The authors express their deep gratitude to Academician N. N. Bogolyubov and V. I. Grigor’ev for their attention and useful discussions.

Moscow State University
named after M. V. Lomonosov

Received
28 XII 1965

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