UDC 529.121.72

PHYSICS

V. P. SHELEST

ON THE POSSIBILITY OF TAKING THREE-PARTICLE FORCES INTO ACCOUNT IN THE RELATIVISTIC THREE-BODY SCATTERING PROBLEM

(Presented by Academician N. N. Bogolyubov, June 29, 1965)

1°. At present the convenience of using L. D. Faddeev’s integral equations \((^{1})\) for studying the nonrelativistic three-body scattering problem is generally recognized. The merits of these equations are also preserved in the case of applying the relativistic generalization obtained by D. Stoyanov and A. N. Tavkhelidze \((^{2,3})\). In \((^{4})\), with the aid of this relativistic generalization, a method was indicated for obtaining matrix elements of the amplitudes of three-particle processes; namely, it was shown that these matrix elements have the form

\[ T_{ij}=\chi_{(i)}^{0+} M_{ij}\chi_{(j)}^{0}, \tag{1} \]

where the transition operators \(M_{ij}\) satisfy the equations

\[ M_{ij}=\sum_{\alpha\ne i} K_\alpha+\sum_{\beta\ne j} M_{i\beta}g_\beta K_\beta . \tag{2} \]

Here \(g_i\) is the two-particle Green’s function

\[ g_i=g_0+g_0K_i g_i, \tag{3} \]

\(K_\alpha\) are the two-body Bethe–Salpeter kernels, \(g_0\) is the free Green’s function, and \(\chi_{(i)}^{0}\) are solutions of the two-particle Bethe–Salpeter equations for the \(i\)-th two-particle subsystem.

The transition operators \(M_{ij}\) can also be defined by the relations \((^{4})\)

\[ g_iM_{ij}g_j=g_i(K_\Pi-K_i)g_\Pi = g_\Pi(K_\Pi-K_i)g_i, \tag{4} \]

where \(g_\Pi\) is the complete three-particle Green’s function (for the case of pair interaction)

\[ g_\Pi=g_i+g_i(K_\Pi-K_i)g_\Pi, \tag{5} \]

\[ K_\Pi=\sum_\alpha K_\alpha . \tag{6} \]

From relation (4) it is seen that

\[ g_iM_{ij}g_j=g_iM_{ik}g_k . \tag{7} \]

Using (7), we have, instead of (2),

\[ M_{ij}=(K_\Pi-K_i)+M_{ij}g_j(K_\Pi-K_j). \tag{8} \]

2°. We shall distinguish two types of possible states of a two-particle system—scattering states and bound states. In an arbitrary coordinate system the total energy \(E_i\) of scattering states is equal to \(\sqrt{\mathbf p_j^2+m_j^2}+\sqrt{\mathbf p_k^2+m_k^2}\), whereas the energy of a bound state is \(E_i^n=\sqrt{(\mathbf p_j+\mathbf p_k)^2+\mu_n^{i\,2}}\), where \(\mu_n^i\) is the mass of the bound state. Since

for which the inequality \(\mu_n^i < m_j + m_k\) is satisfied, it is obvious that, for identical momenta entering the expressions for \(E_i\) and \(E_i^n\), we have \(E_i^n < E_i\). Making use of this circumstance, one can obtain the following expression for the two-particle Green function \((5)^*\) (in Jacobi coordinates)

\[ \hat g(P_i\bar x_i\bar y_i) = P(E_i^n)\int f_{P_i\bar p_i}(\bar x_i)\hat g_0(P_i\bar p_i)\varphi_{P_i\bar p_i}^{+}(\bar y_i)\,d\bar p_i + \sum_n \varphi_{P_i}^{n}(\bar x_i)\omega_{P_i}^{n+}(\bar y_i) \delta(P_i^2-\mu_n^{i2})\theta(P_{i0}). \tag{9} \]

Here \(P(E_i^n)\) means that the integral is taken in the sense of the principal value at all points corresponding to the energies of the bound states; \(\omega^n\) satisfies the Bethe—Salpeter equation for the wave function of the bound state

\[ \omega_{P_i}^{n}(\bar x_i) = \frac{1}{(2\pi)^8} \int \hat g_0(P_i\bar x_i\bar u_i)\hat K_i(P_i\bar u_i\bar v_i) \omega_{P_i}^{n}(\bar v_i)\,d\bar u_i d\bar v_i; \tag{10} \]

\(\omega^{+n}\) satisfies the corresponding adjoint equation; \(f\) is the Bethe—Salpeter wave function describing the scattering of two particles, and satisfies the equation

\[ f_{P_i\bar p_i}(\bar x_i) = \varphi_{P_i\bar p_i}^{-}(\bar x_i) + \frac{1}{(2\pi)^8} \int \hat g_0(P_i\bar x_i\bar u_i)\hat K_i(P_i\bar u_i\bar v_i) f_{P_i\bar p_i}^{-}(\bar v_i)\,d\bar u_i d\bar v_i, \tag{11} \]

\[ \varphi_{P_i\bar p_i}^{-}(\bar x_i)=\exp[-iP_i\bar x_i]. \tag{12} \]

Introducing the notation

\[ \chi_{\bar p\,\bar p_i}^{000}(X\tilde x_i\bar x_i) = \exp[-iPX-i\tilde p_i\tilde x_i]\, \varphi_{\frac{M_i}{M}P+\tilde p_i,\,p_i}(\bar x_i); \tag{13} \]

\[ \chi_{\tilde p_i\bar p_i}^{00}(X\tilde x_i\bar x_i) = \exp[-iPX-i\tilde p_i\tilde x_i]\, f_{\frac{M_i}{M}P+\tilde p_i,\,p_i}^{-}(\bar x_i); \tag{14} \]

\[ \chi_{P\tilde p_i}^{0n}(X\tilde x_i\bar x_i) = \exp[-iPX-i\tilde p_i\tilde x_i]\, \omega_{\frac{M_i}{M}P+\tilde p_i}^{n}(\bar x_i); \tag{15} \]

\[ M_i=m_j+m_k,\qquad M=m_i+m_j+m_k, \tag{16} \]

multiplying (8) on the right and on the left by the corresponding \(\chi\)’s and integrating over the intermediate variables, we obtain, in accordance with (1) and (9), the following systems of equations:

\[ \begin{aligned} T_{ii}^{nm}(P\tilde p_i\tilde p_i) &= \langle K_{\Pi}-K_i\rangle_{ii}^{nm}(P\tilde p_i\tilde p_i) + \frac{P}{(2\pi)^{12}i} \int T_{i0}^{n0}(P\tilde p_i\tilde p_i^{\prime\prime}\bar p_i^{\prime\prime}) \\ &\quad\times S_i\!\left(\frac{m_i}{M}P-\tilde p_i^{\prime\prime}\right) \hat g_0^{\,i}\!\left(\frac{M_i}{M}P+\tilde p_i^{\prime\prime},\bar p_i^{\prime\prime}\right) \langle K_{\Pi}-K_i\rangle_{ii}^{00,m} (P\tilde p_i^{\prime\prime}\bar p_i^{\prime\prime}\tilde p_i) \,d\tilde p_i^{\prime\prime}d\bar p_i^{\prime\prime} \\ &\quad+ \frac{\sum_{n'}}{(2\pi)^{12}i} \int T_{ii}^{nn'}(P\tilde p_i\tilde p_i^{\prime\prime}) S_i\!\left(\frac{m_i}{M}P-\tilde p_i^{\prime\prime}\right) \delta\!\left[\left(\frac{M_i}{M}P+\tilde p_i^{\prime\prime}\right)^2-\mu_{n'}^{i2}\right] \\ &\quad\times \theta\!\left(\frac{M_i}{M}P_0+\tilde p_{i0}^{\prime\prime}\right) \langle K_{\Pi}-K_i\rangle_{ii}^{n'm}(P\tilde p_i^{\prime\prime}\tilde p_i)\,d\tilde p_i^{\prime\prime}; \tag{17a} \end{aligned} \]

\[ \begin{aligned} T_{i0}^{n0}(P\tilde p_i\tilde p_i\bar p_i) &= \langle K_{\Pi}-{}^{1}K_i\rangle_{ii}^{n0}(P\tilde p_i\tilde p_i\bar p_i) + \frac{1}{(2\pi)^{12}i} P\int T_{i0}^{n0}(P\tilde p_i\tilde p_i^{\prime\prime}\bar p_i^{\prime\prime}) \\ &\quad\times S_i\!\left(\frac{m_i}{M}P-\tilde p_i^{\prime\prime}\right) \hat g_0\!\left(\frac{M_i}{M}P+\tilde p_i^{\prime\prime},\bar p_i^{\prime\prime}\right) \langle K_n-K_i\rangle_{ii}^{00,0} (P\tilde p_i^{\prime\prime}\bar p_i^{\prime\prime}\tilde p_i\bar p_i) \,d\tilde p_i^{\prime\prime}d\bar p_i^{\prime\prime} \\ &\quad+ \frac{\sum_{n'}}{(2\pi)^{12}i} \int T_{ii}^{nn'}(P\tilde p_i\tilde p_i^{\prime\prime}) S_i\!\left(\frac{m_i}{M}P-\tilde p_i^{\prime\prime}\right) \delta\!\left[\left(\frac{M_i}{M}P+\tilde p_i^{\prime\prime}\right)^2-\mu_{n'}^{i2}\right] \theta\!\left(\frac{M_i}{M}P_0+\tilde p_{i0}^{\prime\prime}\right) \\ &\quad\times \langle K_{\Pi}-K_i\rangle_{ii}^{n'0}(P\tilde p_i^{\prime\prime}\tilde p_i\bar p_i) \,d\tilde p_i^{\prime\prime}. \tag{17б} \end{aligned} \]

\[ \text{* The caret } \hat{\ } \text{ indicates that the quantity is two-particle.} \]

\[ \begin{aligned} T_{ii}^{nm}(P\widetilde{p}_i\widetilde{p}_i')={}& \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{nm}(P\widetilde{p}_i\widetilde{p}_i')+\\ &+\frac{P}{(2\pi)^{12}i}\int \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{n,00}(P\widetilde{p}_i\widetilde{p}_i''\widetilde{p}_i''')\, S_i\!\left(\frac{m_i}{M}P-\widetilde{p}_i''\right)\times\\ &\times \hat g_0\!\left(\frac{M_i}{M}P+\widetilde{p}_i'',\widetilde{p}_i'''\right) T_{ii}^{0m}(P\widetilde{p}_i''\widetilde{p}_i''')\,d\widetilde{p}_i''\,d\widetilde{p}_i'''\\ &+\frac{\sum_{n'}}{(2\pi)^{12}i}\int \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{nn'}(P\widetilde{p}_i\widetilde{p}_i'')\times\\ &\times S_i\!\left(\frac{m_i}{M}P-\widetilde{p}_i''\right) \delta\!\left[\left(\frac{M_i}{M}P+\widetilde{p}_i''\right)^2-\mu_{n'}^{i\,2}\right] \theta\!\left(\frac{M_i}{M}P_0+\widetilde{p}_{i0}''\right) T_{ii}^{n'm}(P\widetilde{p}_i''\widetilde{p}_i')\,d\widetilde{p}_i'' , \end{aligned} \tag{18a} \]

\[ \begin{aligned} T_{0i}^{0m}(P\widetilde{p}_i\widetilde{p}_i')={}& \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{0m}(P\widetilde{p}_i\widetilde{p}_i')+\\ &+\frac{P}{(2\pi)^{12}i}\int \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{0,00}(P\widetilde{p}_i\widetilde{p}_i''\widetilde{p}_i''')\, S_i\!\left(\frac{m_i}{M}P-\widetilde{p}_i''\right)\times\\ &\times \hat g_0\!\left(\frac{M_i}{M}P+\widetilde{p}_i'',\widetilde{p}_i'''\right) T_{0i}^{0m}(P\widetilde{p}_i''\widetilde{p}_i''')\,d\widetilde{p}_i''\,d\widetilde{p}_i'''\\ &+\frac{\sum_{n'}}{(2\pi)^{12}i}\int \langle K_{\mathrm{p}}-K_i\rangle_{ii}^{0n'}(P\widetilde{p}_i\widetilde{p}_i'')\times\\ &\times S_i\!\left(\frac{m_i}{M}P-\widetilde{p}_i''\right) \delta\!\left[\left(\frac{M_i}{M}P+\widetilde{p}_i''\right)^2-\mu_{n'}^{i\,2}\right] \theta\!\left(\frac{M_i}{M}P_0+\widetilde{p}_{i0}''\right) T_{ii}^{n'm}(P\widetilde{p}_i''\widetilde{p}_i')\,d\widetilde{p}_i'' . \end{aligned} \tag{18b} \]

In formulas (17)—(18) the function \(S_i\) is the one-particle Green’s function, and \(\langle K_{\mathrm{p}}-K_i\rangle_{ii}^{0m}(P\hat p_i\hat p_i'\bar p_i')\), etc., denote \(\langle K_{\mathrm{p}}-K_i\rangle\) integrated with the corresponding functions \(\chi\) (5).

Thus, we have obtained two systems of equations: system (17), which relates the scattering amplitudes on a bound state (without its decay, but with a possible transition to another energy state within the same two-particle subsystem) \(T_{ii}^{nm}\) and the scattering amplitudes of three unbound particles with the formation of two-particle bound states \(T_{i0}^{n0}\), and system (18), which relates the same scattering amplitudes on a bound state \(T_{ii}^{nm}\) and the scattering amplitudes on a bound state with its decay \(T_{0i}^{0m}\).

3°. All of the preceding discussion referred to the case of pair interaction. At the same time, the inclusion of three-particle forces may prove essential for certain problems involving three particles \((^6)\). Therefore we shall include three-particle forces, assuming that the kernel of the three-particle Bethe—Salpeter equation now has the form

\[ K=\sum_{\alpha}K_\alpha+K_{\mathrm{T}}=K_{\mathrm{p}}+K_{\mathrm{T}}. \tag{19} \]

Let us introduce new transition operators \(W_{ik}\), defining them by the equations

\[ W_{ik}=(K_{\mathrm{p}}-K_i+K_{\mathrm{T}})+W_{ij}g_j(K_{\mathrm{p}}-K_k+K_{\mathrm{T}}). \tag{20} \]

Representing \(W_{ik}\) in the form

\[ W_{ik}=M_{ik}+A_{ik}, \tag{21} \]

we obtain

\[ A_{ik}=K_{\mathrm{T}}+M_{ij}g_jK_{\mathrm{T}}+A_{ij}g_j(K_{\mathrm{p}}-K_k+K_{\mathrm{T}}). \tag{22} \]

Hence, using (4) and (5), we have:

\[ g_iA_{ik}g_k=g_{\mathrm{p}}K_{\mathrm{T}}g_{\mathrm{p}}+g_iA_{ik}g_kK_{\mathrm{T}}g_{\mathrm{p}}. \tag{23} \]

Putting \(g_iA_{ik}g_k=g_{\mathrm{p}}Tg_{\mathrm{p}}\), we find the equation for \(T\)

\[ T=K_{\mathrm{T}}+Tg_{\mathrm{p}}K_{\mathrm{T}}=K_{\mathrm{T}}+Tg_iK_{\mathrm{T}}+Tg_iM_{ik}g_kK_{\mathrm{T}}. \tag{24} \]

Again using equations (4) and (5), we obtain from (21)

\[ W_{ik}=M_{ik}+T+Tg_kM_{kk}+M_{ij}g_jT+M_{ij}g_jTg_kM_{kk}. \tag{25} \]

4°. Now, applying a technique analogous to that described in 2°, one can obtain expressions for the matrix elements of the transition operators with allowance for three-particle forces, assuming that the matrix elements of the transition operators without allowance for three-particle forces \(M_{ik}\) are known (which can be obtained from equations (17)—(18)) and the matrix elements of the amplitude \(T\) (which should be taken from (24)). Because of lack of space these relations are not presented here.

However, one can use the approximation of strong binding inside the two-particle bound state (7). Then instead of (25) and (24) we obtain, respectively,

\[ \begin{aligned} \mathcal{J}_{ii}^{nm}(P\widetilde p_i\widetilde p_i) &=\langle K_T\rangle_{ii}^{nm}(P\widetilde p_i\widetilde p_i) +\frac{\sum_{n'}}{(2\pi)^{13}} \int \frac{ d\widetilde p_i''\, \mathcal{T}_{ii}^{nn'}(P\widetilde p_i\widetilde p_i'')\, S_i\!\left(\frac{m_i}{M}P-\widetilde p_i''\right) }{ \frac{M_i}{M}P_0+\widetilde p_{i0}''- \sqrt{\left(\frac{M_i}{M}\mathbf P+\widetilde{\mathbf p}_i''\right)^2+\mu_{n'}^{2}+i\varepsilon} } \times \\ &\quad\times \left\{ \langle K_T\rangle_{ii}^{n'm}(P\widetilde p_i''\widetilde p_i) +\sum_{n''}\int \frac{ d\widetilde p_i'''\, T_{ii}^{n'n''}(P\widetilde p_i''\widetilde p_i''')\, S_i\!\left(\frac{m_i}{M}P-\widetilde p_i'''\right) \langle K_T\rangle_{ii}^{n''m}(P\widetilde p_i'''\widetilde p_i) }{ \frac{M_i}{M}P_0+\widetilde p_{i0}'''- \sqrt{\left(\frac{M_i}{M}\mathbf P+\widetilde{\mathbf p}_i'''\right)^2+\mu_{n''}^{2}+i\varepsilon} } \right\}; \end{aligned} \tag{26} \]

\[ \begin{aligned} \overset{*}{T}_{ii}^{nm}(P\widetilde p_i\widetilde p_i') &=T_{ii}^{nm}(P\widetilde p_i\widetilde p_i') +\mathcal{J}_{ii}^{nm}(P\widetilde p_i\widetilde p_i') \\ &\quad +\sum_{n'}\int \frac{ d\widetilde p_i''\, S_i\!\left(\frac{m_i}{M}P-\widetilde p_i''\right) }{ \frac{M_i}{M}P_0+\widetilde p_{i0}''- \sqrt{\left(\frac{M_i}{M}\mathbf P+\widetilde{\mathbf p}_i''\right)^2+\mu_{n'}^{2}+i\varepsilon} } \times \\ &\quad\times \left\{ \mathcal{J}_{ii}^{nn'}(P\widetilde p_i\widetilde p_i'')\, T_{ii}^{n'm}(P\widetilde p_i''\widetilde p_i') + T_{ii}^{nn'}(P\widetilde p_i\widetilde p_i'')\, \mathcal{J}_{ii}^{n'm}(P\widetilde p_i''\widetilde p_i') \right. \\ &\qquad\left. +\sum_{n''}\int \frac{ T_{ii}^{nn'}(P\widetilde p_i\widetilde p_i'')\, \mathcal{J}_{ii}^{n'n''}(P\widetilde p_i''\widetilde p_i''')\, S_i\!\left(\frac{M_i}{M}P-\widetilde p_i'''\right)\, T_{ii}^{n''m}(P\widetilde p_i'''\widetilde p_i')\, d\widetilde p_i''' }{ \frac{M_i}{M}P_0+\widetilde p_{i0}'''- \sqrt{\left(\frac{M_i}{M}\mathbf P+\widetilde{\mathbf p}_i'''\right)^2+\mu_{n''}^{2}+i\varepsilon} } \right\}; \end{aligned} \tag{27} \]

where

\[ \mathcal{J}_{ii}^{nm}(P\widetilde p_i\widetilde p_i') = \int \chi_{Pp_i}^{0n}(X\widetilde x_i\overline{x}_i)\, T(X-Y,\widetilde x_i\overline{x}_i\widetilde y_i\overline{y}_i)\, \chi_{Pp_i'}^{+\,0m}(Y\widetilde y_i\overline{y}_i)\, dX\,dY\,d\overline{x}_i\,d\overline{y}_i\,d\widetilde x_i\,d\widetilde y_i; \tag{28} \]

\[ \overset{*}{T}_{ii}^{nm}(P\widetilde p_i\widetilde p_i') = \int \chi_{Pp_i}^{0n}(X\widetilde x_i\overline{x}_i)\, W_{ii}(X-Y,\widetilde x_i\overline{x}_i\widetilde y_i\overline{y}_i)\, \chi_{Pp_i'}^{+\,0m}(Y\widetilde y_i\overline{y}_i)\, dX\,dY\,d\overline{x}_i\,d\overline{y}_i\,d\widetilde x_i\,d\widetilde y_i. \tag{29} \]

In conclusion, the author expresses gratitude to Academician N. N. Bogolyubov, D. Stoyanov, and A. N. Tavkhelidze for valuable discussions.

Joint Institute for Nuclear Research

Received
29 VI 1965

CITED LITERATURE

1. L. D. Faddeev, ZhETF, 39, 1459 (1960).
2. D. Stoyanov, Preprint of the Joint Institute for Nuclear Research, Dubna, R-1777, 1964.
3. D. Stoyanov, A. N. Tavkhelidze, Phys. Lett., 13, 76 (1964).
4. V. P. Shelest, D. Stoyanov, Phys. Lett., 13, 253 (1964).
5. D. Stoyanov, V. P. Shelest, Preprint of the Joint Institute for Nuclear Research, Dubna, R-2066, 1965.
6. R. Sawyer, Preprint, Cambridge, Mass., 1965.
7. D. Stoyanov, V. P. Shelest, Preprint, Dubna, E-2108, 1965.