Abstract
Full Text
PHYSICS
A. R. FRENKIN
GREEN FUNCTIONS IN THE STRONG-COUPLING THEORY
(Presented by Academician N. N. Bogolyubov on 3 V 1962)
1. The strong interaction of an extended nucleon with a charged meson field has been studied on models in a number of works ($^1$, $^2$), all the authors attempting in one way or another to diagonalize the Hamiltonian of the system and thereby determine both the ground-state energy and the renormalized meson charge. In doing so, attempts to pass to a point nucleon (removal of the cutoff) led to a logarithmic divergence of the renormalized coupling constant ($^2$). There is reason to suppose that this is connected with the presence in the system of degeneracy with respect to rotations in isotopic space. Therefore, in studying the interaction of a point nucleon with a charged meson field we shall remove this degeneracy, correspondingly changing the Hamiltonian of the system. In contrast to works ($^1$, $^2$), we shall use a more convenient method connected with the application of Green functions ($^3$, $^4$).
2. Let us take the Hamiltonian of a system of charged mesons interacting with an infinitely heavy nucleon in the form:
\[ \mathcal{H}=\sum_{(k)} \omega_k\left(b_{k+}^{+}b_{k+}+b_{k-}^{+}b_{k-}\right)-g\left(Q\tau+\tau^{+}Q^{+}\right)+v\left(Q-Q^{+}\right)^2, \tag{1} \]
where
\[ Q=\sum_{(k)}\frac{\lambda_k}{\sqrt{2\omega_k}}\left(b_{k+}+b_{k-}^{+}\right);\qquad Q^{+}=\sum_{(k)}\frac{\lambda_k}{\sqrt{2\omega_k}}\left(b_{k-}+b_{k+}^{+}\right); \tag{2} \]
$\tau,\tau^{+}$ are the creation and annihilation operators of the nucleon charge, possessing the property $\tau\tau^{+}+\tau^{+}\tau=1$; $b_{k+}^{+}(b_{k+})$ and $b_{k-}^{+}(b_{k-})$ are the creation (annihilation) operators of positive and negative mesons with momentum $k$; $\lambda_k=\lambda_k^{*}=\lambda_{(k^2)}$ is a form factor equal to 1 for a point nucleon, and $v\ne 0$ is a real number.
Owing to the addition to the ordinary Hamiltonian of the term $v(Q-Q^{+})^2$ ($^3$), the total charge of the system
\[ q=\tau\tau^{+}+\sum_{(k)} b_{k+}^{+}b_{k+}-\sum_{(k)} b_{k-}^{+}b_{k-} \tag{3} \]
ceases to be an integral of motion, which leads to the removal of the rotational degeneracy in isotopic space. Indeed, if $v\ne0$, Hamiltonian (1) is not invariant with respect to the simultaneous replacement of the operators:
\[ b_{k+}\to e^{i\varphi}b_{k+};\qquad b_{k-}\to e^{-i\varphi}b_{k-};\qquad \tau\to \tau e^{-i\varphi}; \tag{4} \]
\[ b_{k+}^{+}\to e^{-i\varphi}b_{k+}^{+};\qquad b_{k-}^{+}\to e^{i\varphi}b_{k-}^{+};\qquad \tau^{+}\to \tau^{+}e^{i\varphi}, \]
where $\varphi$ is a real phase. The removal of charge degeneracy makes it possible to obtain a finite renormalization of the meson charge not only for an extended, but also for a point nucleon.
- Following the work \((^4)\), let us construct equations for the retarded Green functions of commutator type. Taking into account the equations of motion for the Heisenberg operators \(\tau\), \(\tau^+\) and \(\sigma=\tau\tau^+-\tau^+\tau\), we obtain, in the energy representation:
\[ \begin{aligned} E\langle\langle \tau \mid \tau^+ \rangle\rangle &= \frac{\langle \sigma\rangle}{2\pi} - g\langle\langle \sigma Q^+ \mid \tau^+ \rangle\rangle;\\ E\langle\langle \tau^+ \mid \tau^+ \rangle\rangle &= g\langle\langle \sigma Q \mid \tau^+ \rangle\rangle;\\ E\langle\langle \sigma \mid \tau^+ \rangle\rangle &= -\frac{\langle \tau^+\rangle}{\pi} -2g\langle\langle \tau Q \mid \tau^+ \rangle\rangle +2g\langle\langle \tau^+ Q^+ \mid \tau^+ \rangle\rangle . \end{aligned} \tag{5} \]
Here the averaging is carried out over the ground state of the Hamiltonian (1).
To solve this system of equations, by analogy with Wick’s theorem, we decouple the higher Green functions, assuming that in the principal approximation \(\langle \tau\rangle=\langle \tau^+\rangle\), \(\langle \sigma\rangle=0\).
Then we obtain:
\[ \begin{aligned} E\langle\langle \tau \mid \tau^+ \rangle\rangle &= -g\langle Q^+\rangle\langle\langle \sigma \mid \tau^+ \rangle\rangle;\\ E\langle\langle \tau^+ \mid \tau^+ \rangle\rangle &= g\langle Q\rangle\langle\langle \sigma \mid \tau^+ \rangle\rangle;\\ E\langle\langle \sigma \mid \tau^+ \rangle\rangle &= -\frac{\langle \tau\rangle}{\pi} -2g\langle \tau\rangle\langle\langle Q \mid \tau^+ \rangle\rangle -2g\langle Q\rangle\langle\langle \tau \mid \tau^+ \rangle\rangle \\ &\quad +2g\langle \tau\rangle\langle\langle Q^+ \mid \tau^+ \rangle\rangle +2g\langle Q^+\rangle\langle\langle \tau^+ \mid \tau^+ \rangle\rangle . \end{aligned} \tag{6} \]
The mean values \(\langle Q\rangle\) and \(\langle Q^+\rangle\) are determined from the equations of motion for the meson operators:
\[ \langle Q\rangle=\langle Q^+\rangle=gI\langle \tau\rangle, \tag{7} \]
where
\[ I=\sum_{(\mathbf{k})}\frac{\lambda_k^2}{\omega_k^2}, \tag{8} \]
and the Green functions of mixed type are expressed through the nucleon Green functions in the following way:
\[ \langle\langle Q \mid \tau^+ \rangle\rangle = \frac{gJ(E)}{1-4\nu J(E)} \left\{ (1-2\nu J(E))\langle\langle \tau^+ \mid \tau^+ \rangle\rangle -2\nu J(E)\langle\langle \tau \mid \tau^+ \rangle\rangle \right\}; \tag{9} \]
\[ \langle\langle Q^+ \mid \tau^+ \rangle\rangle = \frac{gJ(E)}{1-4\nu J(E)} \left\{ -2\nu J(E)\langle\langle \tau^+ \mid \tau^+ \rangle\rangle +(1-2\nu J(E))\langle\langle \tau \mid \tau^+ \rangle\rangle \right\}, \]
where
\[ J(E)=\sum_{(\mathbf{k})}\frac{\lambda_k^2}{\omega_k^2-E^2}. \tag{10} \]
Then, from the system of equations (6), we obtain:
\[ \begin{aligned} \langle\langle \tau \mid \tau^+ \rangle\rangle &= \frac{1}{\pi} \frac{g^2 I\langle \tau\rangle^2} {E^2-\dfrac{4g^4\langle \tau\rangle^2 I^2}{1-4\nu J(E)}\,\Delta(E)};\\ \langle\langle \tau^+ \mid \tau^+ \rangle\rangle &= -\langle\langle \tau \mid \tau^+ \rangle\rangle;\\ \langle\langle \sigma \mid \tau^+ \rangle\rangle &= -\frac{E}{\pi} \frac{\langle \tau\rangle} {E^2-\dfrac{4g^4\langle \tau\rangle^2 I^2}{1-4\nu J(E)}\,\Delta(E)}, \end{aligned} \tag{11} \]
where
\[ \Delta(E)=1-4\left(\nu+\frac{1}{4I}\right)J(E). \tag{12} \]
- A similar result can also be obtained by means of perturbation theory in inverse powers of the interaction constant \(g\). Indeed, representing the solution of the Schrödinger equation
\[ \mathscr{H}\Psi=E\Psi \tag{13} \]
in the form of an expansion in the eigenfunctions of the nucleon charge operators \(\tau\) and \(\tau^{+}\):
\[ \Psi=\Phi_p\delta(n_p-1)\delta(n_n)+\Phi_n\delta(n_n-1)\delta(n_p), \tag{14} \]
where \(\Phi_p\) and \(\Phi_n\) depend only on the meson operators, we obtain, in the zeroth approximation in powers of \(g^{-1}\), the following equation for the ground state \(A_0=\Phi_p+\Phi_n\):
\[ (\mathscr{H}^{0}-E^{0})A_0=0. \tag{15} \]
The Hamiltonian
\[ \mathscr{H}^{0}=\sum_{(\mathbf{k})}\omega k\,(b_{k+}^+b_{k+}+b_{k-}^+b_{k-})+ \left(v+\frac{1}{4I}\right)(Q-Q^+)^2 \tag{16} \]
can be diagonalized by means of the \(u,v\)-transformation \((^5)\), and its eigenvalues \(E_\mu^0\) are determined as the roots of the secular equation
\[ \Delta(E^0)=0. \tag{17} \]
It is easy to see that all \(E_\mu^0\ne 0\).
Since \(\Delta(E)\) also appears in the denominators of the nucleon Green functions, all three Green functions (11) have no poles at \(E=0\), which indicates the absence of any kinds of degeneracies in the system.
- The Green functions written out make it possible to determine the mean values of the operators for a point nucleon.
In the principal approximation we obtain:
\[ \langle\tau\rangle=\langle\tau^+\rangle=\frac{1}{2}. \tag{18} \]
In contrast to the degenerate case \((^{1,2})\), all mean values are finite. Defining the renormalized coupling constant as \(g_r=g\langle\tau\rangle\), we have:
\[ g_r=\frac{1}{2}g. \tag{19} \]
The same result can also be obtained by means of perturbation theory in powers of \(g^{-1}\).
It is interesting to note that, in weak coupling as well, removal of the degeneracy leads to a finite renormalization of the charge \(g\) for a point nucleon.
The author expresses his deep gratitude to Acad. N. N. Bogolyubov for constant attention and valuable advice, and to V. D. Kukin for useful discussions.
Moscow State University
named after M. V. Lomonosov
Received
28 IV 1962
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