Abstract
Full Text
Physics
A. A. LOGUNOV, V. A. MESHCHERYAKOV, and A. N. TAVKHELIDZE
ON THE APPROXIMATE $\gamma_5$-INVARIANCE OF THE THEORY OF STRONG INTERACTIONS
(Presented by Academician N. N. Bogolyubov, 14 VIII 1961)
Recently, questions concerning the appearance of symmetry properties at high energies have been intensively discussed. Thus, for example, in Gell-Mann’s work ($^1$) symmetry properties are considered that follow from the invariance of the interaction with respect to a three-dimensional unimodular group. On the other hand, a number of authors (($^2$, $^3$) and others) have raised the question of the existence of vector bosons that carry out strong interactions in the form
$$ \mathcal{L}_{\mathrm{int}}=\sum_{\alpha=1}^{4} I_\alpha(x) B_\alpha(x), \tag{1} $$
where $B_\alpha$ is a vector boson field; $I_\alpha$ is the vector current of the strong interaction, equal to $I_\alpha(x)=\sum g_i^2 \bar{\psi}_i \gamma_\alpha \psi_i$. It should be noted that the interaction Lagrangians for weak, electromagnetic, and strong processes in the form (1) possess one common property: they are invariant with respect to $\gamma_5$-transformations of spinor particles:
$$ \psi_i \to \gamma_5 \psi_i, \qquad \bar{\psi}_i \to \bar{\psi}_i \gamma_5, \qquad \gamma_5^2=-1. \tag{2} $$
In the present note we discuss the hypothesis that at high energies and momentum transfers $s,t \gg m^2$ the matrix elements of all physical processes are invariant with respect to $\gamma_5$-transformations of spinor particles.
The more precise meaning of $\gamma_5$-invariance will be illustrated below for a number of physical processes. Let us note that in the case of a $\gamma_5$-invariant interaction, $\gamma_5$-noninvariant terms in the scattering amplitude arise because of the presence of mass terms of spinor particles in the free Lagrangian.*
On the basis of an analysis of the lowest terms of perturbation theory, one may assume that at high energies and momentum transfers the mass terms play no role and, consequently, a $\gamma_5$-invariant interaction will lead to $\gamma_5$-invariant matrix elements.
Let us turn to concrete examples.
Starting from Lorentz invariance and the gradient transformation, the most general expression for the electromagnetic vertex of the nucleon is found in the form
$$ F_\mu(q^2)=F_1(q^2)\gamma_\mu+i\sigma_{\mu\nu}q_\nu F_2(q^2). \tag{3} $$
From the requirement of $\gamma_5$-invariance it follows that
$$ \lim_{q^2\to\infty} |q|\,F_2(q^2)=0, \tag{4} $$
i.e., the magnetic form factor of the nucleon decreases at sufficiently large momentum transfers.
* Here we deliberately assume the absence of renormalization in the theory, since the latter, as was shown in ($^4$), may lead to the appearance of $\gamma_5$-invariant terms.
Let us clarify the consequences to which the requirement of $\gamma_5$-invariance of the amplitude leads for scattering processes of the type
\[ 0+\frac{1}{2}\to 0+\frac{1}{2}; \tag{5} \]
$0$, $\frac{1}{2}$ are the spins of the particles participating in the reaction.
The scattering amplitude of any of these processes, if the relative parity of the particles at the beginning and at the end of the reaction does not change, may be written in the form
\[ M=\bar u_2(p_2)\left[A(s,t)+\frac{\hat q_1+\hat q_2}{2}\,B(s,t)\right]u_1(p_1), \tag{6} \]
where $q_1$ and $q_2$ are the momenta of the bosons, $p_1$ and $p_2$ are the momenta of the fermions,
\[ s=(p_1+q_1)^2,\qquad t=(p_1-p_2)^2. \]
From the requirement of $\gamma_5$-invariance it follows that
\[ \lim_{s,t\to\infty} A(s,t)=0. \tag{7} \]
Hence it is easy to establish that, if the initial fermion was longitudinally polarized, then the final fermion will also be longitudinally polarized. In the case where the initial state is unpolarized, the final state will also be unpolarized.
It follows from this, for example, that the polarization of $\Lambda$- and $\Sigma^{-}$-particles in the scattering processes
\[ \pi^-+p\to\Lambda+K^0,\qquad \pi^-+p\to\Sigma^-+K^+ \]
at high energies and momentum transfers must decrease.
For nucleon–nucleon scattering the amplitude may be written in the form
\[ \begin{aligned} M(s,t)=&\,\bar u(p_2)\bar u(k_2)\,[G_1-G_2(\gamma^{(1)}P+\gamma^{(2)}K) \\ &+G_3(\gamma^{(1)}P)(\gamma^{(2)}K)-G_4\gamma_5^{(1)}\gamma^{(1)}P\gamma_5^{(2)}\gamma^{(2)}K -G_5\gamma_5^{(1)}\gamma_5^{(2)}]\,u(p_1)u(k_1), \end{aligned} \tag{8} \]
where
\[ K=\frac{k_1+k_2}{2},\qquad P=\frac{p_1+p_2}{2},\qquad Q=k_1-k_2. \tag{9} \]
The requirement of $\gamma_5$-invariance leads to the result that, in the region of large $s$ and $t$, the expression for the amplitude is substantially simplified and has the form
\[ M(s,t)=\bar u(p_2)\bar u(k_2)\,[G_3\gamma^{(1)}P\gamma^{(2)}K -G_4\gamma_5^{(1)}\gamma^{(1)}P\gamma_5^{(2)}\gamma^{(2)}K]\,u(p_1)u(k_1). \tag{10} \]
Hence it is easy to establish that, if the initial state is unpolarized, then the final state will also remain unpolarized.
It is not difficult to see that for other scattering processes as well, $\gamma_5$-invariance substantially simplifies the expression for the scattering amplitude and makes it possible to draw the conclusion that the helicity of the particles is conserved.
In conclusion, we express our gratitude to N. N. Bogoliubov, S. M. Bilenky, S. S. Gershtein, M. G. Meshcheryakov, A. M. Baldin, R. M. Ryndin, and Ya. S. Smorodinsky for valuable discussions and advice.
Joint Institute for Nuclear Research
Received
20 VII 1961
CITED LITERATURE
- M. Gell-Mann, Preprint, 1961.
- Y. Fujii, Progr. Theor. Phys., 21, 232 (1959).
- I. I. Sakurai, Ann. of Phys., 11, 1 (1960).
- N. N. Bogoliubov, Physica, 26, Suppl. 1 (1960); Y. Nambu, G. Ionn-Lasinio, Phys. Rev., 122, No. 1, 345 (1961); B. A. Arbuzov, A. N. Tavkhelidze, R. N. Faustov, DAN, 139, No. 2 (1961).