UDC 539.17.01

PHYSICS

M. A. MESTVIRISHVILI

INTEGRAL REPRESENTATION FOR THE TWO-PARTICLE AMPLITUDE IN THE NONRELATIVISTIC CASE

(Presented by Academician N. N. Bogolyubov, January 4, 1965)

1. In the work of L. D. Faddeev (¹) a system of coupled equations was proposed that is suitable for describing the three-body problem when the particles interact with one another by pairwise forces. A remarkable feature of these equations is, first, that in solving them the three-body scattering amplitude \(T\) is expressed in terms of the two-particle amplitudes \(T_{ij}\), and, second, that one obtains comparatively easily an expansion for \(T\) in a perturbation-theory series and, as was shown in (²), one can formulate the corresponding diagram technique. Diagrams of this kind are completely determined by specifying vertices, which are two-particle amplitudes off the energy surface; therefore, in order to study the analytic properties of these diagrams it is essential to know the analytic properties of \(T_{ij}\).

Fig. 1

1. The purpose of the present work is to give an integral representation for the two-particle amplitude off the energy surface in the nonrelativistic case. Let us write the matrix element of the operator \(t\) in the form:

\[ \delta(\mathbf{p}_i+\mathbf{k}_i-\mathbf{p}_f-\mathbf{k}_f) \left\langle (\mathbf{p}_f-\mathbf{k}_f)/2 \left| t\left(z-\frac{(\mathbf{p}_f+\mathbf{k}_f)^2}{2}\right) \right| (\mathbf{p}_i-\mathbf{k}_i)/2 \right\rangle . \tag{1} \]

The operator \(t\) satisfies the Lippmann—Schwinger equation

\[ t=V-VG_0t, \tag{2} \]

where \(V\) is the interaction potential depending on the difference of coordinates (for convenience in what follows a Yukawa potential is chosen), and \(G_0\) is the Green function, having the form (in a system of units in which \(\hbar^2=2m_1=2m_2=1\))

\[ G_0=\frac{1}{\mathbf{k}^2-\left[z-(\mathbf{p}_f+\mathbf{k}_f)^2/2\right]} . \]

The independence of the combinations \((\mathbf{p}+\mathbf{k})/2\) and \((\mathbf{p}-\mathbf{k})/2\) makes it possible to fix the expression \(E=z-(\mathbf{p}_f+\mathbf{k}_f)^2/2\), and then the matrix element (1) will depend on three scalar squares: \(((\mathbf{p}_f-\mathbf{k}_f)/2)^2\), \(((\mathbf{p}_i-\mathbf{k}_i)/2)^2\), \((\mathbf{k}_f-\mathbf{k}_i)^2\). Iterating equation (2), we obtain a perturbation-theory series whose corresponding diagrams are depicted in Fig. 1.* The contribution from each diagram in the \(\alpha\)-representation (up to constant factors) can be written in the form of the integral (⁴):

\[ F_n(\mathbf{q}_i)\sim \int_0^\infty \frac{ \prod_{i=1}^{n-1} d\alpha_i\, \prod_{j=0}^{n-1} d\beta_{j,j+1} }{ D^{3/2}(\alpha,\beta) } \exp\left[ \frac{R(\alpha,\beta,\mathbf{q}_1,\mathbf{q}_2)}{D(\alpha,\beta)} -\left(\sum_{i=1}^{n-1}\alpha_i\right)E +\left(\sum_{j=0}^{n-1}\beta_{j,j+1}\right)\mu^2 \right], \tag{3} \]

where, for convenience, the notation \(\mathbf{q}_1=(\mathbf{p}_f-\mathbf{k}_f)/2\) and \(\mathbf{q}_2=(\mathbf{p}_i-\mathbf{k}_i)/2\) has been introduced.

* Nonrelativistic diagrams are usually drawn in a different form, namely: the thin lines do not emerge from a single point \(a\), as in Fig. 1, but from separate points. We shall agree that at the vertex \(a\) we shall write the interaction constant to the power whose value is equal to the number of incoming lines. Thus, the first diagram in Fig. 1 will be of second order, the second of fourth order, etc.

In (3), \(D(\alpha,\beta)\) is the determinant of the matrix

\[ d= \left( \begin{array}{cccc} \beta_{01}+\alpha_1+\beta_{12} & -\beta_{21} & 0 & \ldots\\ -\beta_{12} & \beta_{12}+\alpha_2+\beta_{23} & -\beta_{23} & \ldots\\ 0 & -\beta_{23} & \beta_{23}+\alpha_3+\beta_{34} & \ldots\\ \ldots & \ldots & \ldots & \ldots \end{array} \right). \tag{4} \]

\(R(\alpha,\beta,\mathbf q_1,\mathbf q_2)\) is the bordered determinant of the determinant \(D(\alpha,\beta)\):

\[ R(\alpha,\beta,\mathbf q_1,\mathbf q_2)= \left| \begin{array}{cc} 0 & B_i\\ B_i & d \end{array} \right|, \quad \text{where}\quad B_1=-\beta_{01}\mathbf q_1,\quad B_{n-1}=-\beta_{n-1,n}\mathbf q_2,\quad B_i=0, \]

\(i=2,3,\ldots,n-2\). It is easy to verify that (4)

\[ \min_{\mathbf k_\nu}\left[ \sum_{i=1}^{n-1}\alpha_i(\mathbf k_i^2-E) + \sum_{j=0}^{n-1}\beta_{j,j+1}(\mathbf k_j^2+\mu^2) \right] = \frac{R(\alpha,\beta,\mathbf q_1,\mathbf q_2)}{D(\alpha,\beta)} - \]

\[ - \left(\sum_{i=1}^{n-1}\alpha_i\right)E + \left(\sum_{j=0}^{n-1}\beta_{j,j+1}\right)\mu^2 = A_1(\alpha,\beta)\mathbf q_1^2 + A_2(\alpha,\beta)\mathbf q_2^2 + \]

\[ + A_3(\alpha,\beta)(\mathbf q_1-\mathbf q_2)^2 - \left(\sum_{i=1}^{n-1}\alpha_i\right)E + \left(\sum_{j=0}^{n-1}\beta_{j,j+1}\right)\mu^2 = Q(\alpha,\beta,\mathbf q_i,\mu^2,E). \tag{5} \]

Here the vectors \(\mathbf q_{1,2}\); \(\mathbf k_i\), \(\mathbf k_j\) satisfy the conservation law at each vertex. The minimization is performed over the independent momenta that remain after integration over the \(\delta\)-functions;

\[ A_1(\alpha,\beta) = \beta_{01} + \frac{1}{D(\alpha,\beta)} \left| \begin{array}{cc} 0 & \beta_{01}\;0\ldots 0\\ \beta_{01} & \\ \vdots & d\\ \beta_{n-1,n} & \end{array} \right|, \]

\[ A_2(\alpha,\beta) = \beta_{n-1,n} + \frac{1}{D(\alpha,\beta)} \left| \begin{array}{cc} 0 & 0\ldots \beta_{n-1,n}\\ \beta_{01} & \\ \vdots & d\\ \beta_{n-1,n} & \end{array} \right|, \tag{6} \]

\[ A_3(\alpha,\beta) = - \frac{1}{D(\alpha,\beta)} \left| \begin{array}{cc} 0 & 0\ldots \beta_{n-1,n}\\ \beta_{01} & \\ \vdots & d\\ 0 & \end{array} \right|. \]

We note that (3) \(D(\alpha,\beta)\ge 0\); \(A_i(\alpha,\beta)\ge 0\), \(i=1,2,3\).

3. Theorem. The analyticity domain of any diagram of order \(2n>4\) (\(n\) is the number of thin lines in the diagram) includes in itself (or coincides with) the analyticity domain of the fourth-order diagram, provided only that \(E=z-(\mathbf p_f+\mathbf k_f)^2/2<0\) \((-E=m^2)\).

Proof. Let, for the functions \(Q_{2n}\) and \(Q_{2n'}\) corresponding to diagrams of orders \(2n\) and \(2n'\) (where \(n<n'\)), the inequality

\[ Q_{2n'}\ge Q_{2n}; \tag{7} \]

hold; then we shall say that the diagram of order \(2n\) majorizes the diagram of order \(2n'\), and these words have the following meaning: the analyticity domain \(G_{n'}\) of the diagram of order \(2n'\) contains in itself (or coincides with) the analyticity domain \(G_n\) of the diagram of order \(2n\), i.e. \(G_n\subseteq G_{n'}\).

Let the diagram of order \(2n'\) contain the figure shown in Fig. \(2a\), and let the diagram of order \(2n\) be obtained from it by replacing figure \(2a\) by figure \(2b\). According to (5), in order to establish inequality (1) under such a replacement it is sufficient to show that the minimum with respect to \(\mathbf k\) of \(f(\alpha,\beta,\mathbf k,\mathbf q_i)\) is greater than \(\varphi(\gamma,\mathbf q_i)\), with an appropriate choice of \(\gamma\), where

\[ f(\alpha,\beta,\mathbf k,\mathbf q_i) = \beta_1(\mathbf q_1^2+m^2) + \beta_2(\mathbf q_2^2+m^2) + \alpha_1(\mathbf k^2+m^2) + \alpha_2\bigl[(\mathbf q_1-\mathbf k)^2+\mu^2\bigr] + \alpha_3\bigl[(\mathbf q_2-\mathbf k)^2+\mu^2\bigr], \]

\[ \varphi(\gamma,\mathbf q_i) = \gamma_1(\mathbf q_1^2+m^2) + \gamma_2(\mathbf q_2^2+m^2) + \gamma_3\bigl[(\mathbf q_1-\mathbf q_2)^2+\mu^2\bigr]. \]

Indeed:

\[ \min_{\mathbf{k}} f(\alpha,\beta,\mathbf{k},\mathbf{q}_i) = \left(\beta_1+\frac{\alpha_1\alpha_2}{\sum_{i=1}^{3}\alpha_i}\right)(\mathbf{q}_1^{\,2}+m^2) + \left(\beta_2+\frac{\alpha_1\alpha_3}{\sum_{i=1}^{3}\alpha_i}\right)(\mathbf{q}_2^{\,2}+m^2) + \frac{\alpha_2\alpha_3}{\sum_{i=1}^{3}\alpha_i}\bigl[(\mathbf{q}_1-\mathbf{q}_2)^2+\mu^2\bigr] + \frac{\alpha_1^2}{\sum_{i=1}^{3}\alpha_i}m^2 + \frac{\alpha_2(\alpha_1+\alpha_2)+\alpha_3\left(\sum_{i=1}^{3}\alpha_i\right)} {\sum_{i=1}^{3}\alpha_i}\,\mu^2, \]
and if we define
\[ \gamma_1=\beta_1+\alpha_1\alpha_2 \bigg/ \sum_{i=1}^{3}\alpha_i;\qquad \gamma_2=\beta_2+\alpha_1\alpha_3 \bigg/ \sum_{i=1}^{3}\alpha_i;\qquad \gamma_3= \]
\[ =\alpha_2\alpha_3 \bigg/ \sum_{i=1}^{3}\alpha_i, \]
then
\[ \min_{\mathbf{k}} f(\alpha,\beta,\mathbf{k},\mathbf{q}_i)\geqslant \varphi(\gamma,\mathbf{q}_i). \tag{8} \]

When one of the two \(\mathbf{q}_i\) (say, \(\mathbf{q}_1\)) corresponds to an external line, the inequalities (8) are satisfied if \(\mathbf{q}_1^{\,2}>-(m+\mu)^2\).

Applying the indicated substitution to the diagram of Fig. 1, we arrive at the proof of the theorem. Thus, in order to find the minimal real domain of holomorphy of the amplitude, it suffices to restrict ourselves to the study of the analytic properties of the fourth-order diagram.

Fig. 2

Fig. 3

4. The analytic properties of the triangular diagram in field theory are well known \((^{4-5})\). Omitting the calculations, we present the results obtained. The domain in which \(Q>0\) is determined by the inequalities
\[ \xi_i>0,\ i=1,2,3;\quad \xi_i+\xi_j<0;\quad \xi_k>\xi_i\xi_j-\sqrt{(1-\xi_i^2)(1-\xi_j^2)}, \tag{9} \]
where the \(\xi_i\) are given by the relations
\[ \xi_1=(m^2+\mu^2+z_1)/2m\mu,\quad \xi_2=(m^2+\mu^2+z_2)/2m\mu,\quad \xi_3=(2\mu^2+z_3)/2\mu^2 \]
(Fig. 3).

To obtain the integral representation, we shall carry out a study analogous to the works \((^{3,6})\). Denote by \(G_0\) the domain (9). Following By (7), one can find a certain complex domain of analyticity \(G\) over the prescribed real domain of analyticity \(G_0\). Indeed, let \(G\) denote the set of complex points \(z_i=x_i+iy_i\) \((i=1,2,3)\), for which there exists a real \(\lambda\) such that \(x_i+\lambda y_i\in G_0\). Then, from the linearity of \(Q\) with respect to \(z_i\), it follows that \(Q\ne0\), which proves our assertion. The domain \(G\) is a natural domain of holomorphy in the space \(z_i\) \((i=1,2,3)\) \((^{6,7})\).

In what follows it is convenient to rewrite (3) in a somewhat different (but identical) form:
\[ F_n(z_i)\sim \int_0^1\cdots\int_0^1 \frac{ \delta\left(1-\sum_{i=1}^{n-1}\alpha_i'-\sum_{j=0}^{n-1}\beta'_{j,j+1}\right) \prod_{i=1}^{n-1}d\alpha_i'\prod_{j=0}^{n-1}d\beta'_{j,j+1} }{ D^{3/2}(\alpha',\beta')\,[Q(\alpha',\beta',z_i,m^2,\mu^2)-i0]^{(n+1)/2} }. \tag{10} \]
where \((n+1)\) is the number of vertices in the diagram. It is expedient to consider separately the diagrams for even \(n\) and odd \(n\).

We denote the corresponding functions by \(F_{2k}(z_i)\) and \(F_{2k+1}(z_i)\). Let

\[ \eta_i=\frac{A_i(\alpha',\beta')}{\sum_{i=1}^{3} A_i(\alpha',\beta')},\quad i=1,2,3; \qquad \rho=\frac{\left(\sum_{i=1}^{n-1}\alpha'_i\right)m^2+\left(\sum_{j=0}^{n-1}\beta'_{j,j+1}\right)\mu^2}{\sum_{i=1}^{3} A_i(\alpha',\beta')}. \]

Substituting into (10) the identity

\[ \begin{aligned} 1={}&\int_0^1 d\eta_1\int_0^1 d\eta_2\int_0^1 d\eta_3\int_{-\infty}^{\infty}d\rho\, \delta\!\left(\eta_1-\frac{A_1(\alpha',\beta')}{\sum_{i=1}^{3}A_i(\alpha',\beta')}\right) \delta\!\left(\eta_2-\frac{A_2(\alpha',\beta')}{\sum_{i=1}^{3}A_i(\alpha',\beta')}\right) \\ &\times \delta\!\left(\eta_3-\frac{A_3(\alpha',\beta')}{\sum_{i=1}^{3}A_i(\alpha',\beta')}\right) \delta\!\left(\rho-\frac{\left(\sum_{i=1}^{n-1}\alpha'_i\right)m^2+\left(\sum_{i=0}^{n-1}\beta'_{i,i+1}\right)\mu^2}{\sum_{i=1}^{3}A_i(\alpha',\beta')}\right) \end{aligned} \]

and integrating with respect to \(\rho\) \(k\) times by parts, we obtain the desired integral representations

\[ F_{2k}(z_i)= \int_0^1\!\!\int_0^1\!\!\int_0^1 d\eta_1 d\eta_2 d\eta_3 \int_{\rho_0(\eta_i)}^{\infty} d\rho\, \frac{f_{2k}(\eta_i,\rho)\,\delta\!\left(1-\sum_{i=1}^{3}\eta_i\right)} {\left[\eta_1z_1+\eta_2z_2+\eta_3z_3+\rho-i0\right]^{1/2}}, \tag{11a} \]

\[ F_{2k+1}(z_i)= \int_0^1\!\!\int_0^1\!\!\int_0^1 d\eta_1 d\eta_2 d\eta_3 \int_{\rho_0(\eta_i)}^{\infty} d\rho\, \frac{f_{2k+1}(\eta_i,\rho)\,\delta\!\left(1-\sum_{i=1}^{3}\eta_i\right)} {\left[\eta_1z_1+\eta_2z_2+\eta_3z_3+\rho-i0\right]}, \tag{11b} \]

\[ \begin{aligned} \rho_0(\eta_i) &=\max_{z_i\in G_0}[-\eta_1z_1-\eta_2z_2-\eta_3z_3] \\ &=\eta_1(m^2+\mu^2)+\eta_2(m^2+\mu^2)+2\eta_3\mu^2 -2\min_{\zeta_i\in G_0}(m\mu\eta_1\zeta_1 \\ &\qquad\qquad\qquad\qquad\qquad +m\mu\eta_2\zeta_2+\mu^2\eta_3\zeta_3). \end{aligned} \tag{12} \]

The minimum with respect to \(\zeta_i\) in (12) is attained in the region \(G_0\), where the \(\zeta_i\) may be specified in the form \(\zeta_i=\cos\theta_i\), and gives the following values for \(\rho_0(\eta_i)\) \({}^{(6)}\):

\[ \rho_0(\eta_i)=\Phi(\eta_i)\left[\mu^2\left(\frac{1}{\eta_1}+\frac{1}{\eta_2}\right)+\frac{m^2}{\eta_3}\right], \quad \text{if }\; \mu\left|\frac{1}{\eta_1}-\frac{1}{\eta_2}\right|<\frac{m}{\eta_3}<\mu\left(\frac{1}{\eta_1}+\frac{1}{\eta_2}\right), \]

\[ \rho_0(\eta_i)=\eta_1(m+\mu)^2+\eta_2(m-\mu)^2+4\mu^2\eta_3, \quad \text{if }\; \frac{\mu}{\eta_2}>\frac{\mu}{\eta_1}+\frac{m}{\eta_3}, \]

\[ \rho_0(\eta_i)=\eta_1(m+\mu)^2+\eta_2(m+\mu)^2, \quad \text{if }\; \frac{m}{\eta_3}>\mu\left(\frac{1}{\eta_1}+\frac{1}{\eta_2}\right), \]

\[ \rho_0(\eta_i)=\eta_1(m-\mu)^2+\eta_2(m+\mu)^2+4\mu^2\eta_3, \quad \text{if }\; \frac{\mu}{\eta_1}>\frac{\mu}{\eta_2}+\frac{m}{\eta_3}; \]

\[ \Phi(\eta_i)=\eta_1\eta_2+\eta_2\eta_3+\eta_1\eta_3. \]

In conclusion, the author expresses his deep gratitude to A. A. Logunov and A. N. Tavkhelidze for their interest in the work and for discussion.

United Institute
for Nuclear Research

Received
12 XII 1964

References

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