CURRENT ALGEBRA AND LOW-ENERGY $\pi N$ SCATTERING

G. M. Radutskii, A. N. Tabachenko

(Presented by the Scientific-Technical Seminar of the High-Energy Laboratory of the Research Institute of Nuclear Physics)

The method of current algebra is based on the properties of the vector and axial-vector currents. The basic hypotheses are conservation of the vector current (CVC), partial conservation of the axial-vector current (PCAC), and equal-time commutation relations, which determine the algebra of currents (CCR). These assumptions concerning the properties of the vector and axial-vector currents first made it possible to approach, from a unified point of view, the problem of weak, electromagnetic, and strong interactions.

The hypotheses of current algebra in the soft-pion approximation (i.e., pions whose momentum and mass are zero or close to zero) lead to low-energy theorems for the amplitudes of various pion processes. In comparing the results of these theorems with experiment, an essential role is played by the assumption that the matrix elements change slowly in the transition to the four-momenta of physical particles. One may therefore expect that the soft-pion model is applicable only for describing threshold phenomena. In addition, the approximation of physical amplitudes by matrix elements of reactions with soft pions in a number of cases leads to an ambiguous result.

Recently, various methods have been proposed for calculating reaction amplitudes without using extrapolation in the pion momentum, in which one deals from the very beginning with physical, or hard, pions [1–7]. An important step in this direction was first taken by Weinberg [1], who considered sum rules for the spectral functions of the vector and axial-vector currents in the approximation of single-meson dominance. Subsequently, using the idea of [1], Schnitzer and Weinberg [2] formulated a method of current algebra with hard pions, based on Ward identities. In this method, in addition to the hypotheses of current algebra CVC, PCAC, and CCR, it is assumed that: a) sums over intermediate states in the vacuum expectation values of $T$-products of vector and axial-vector currents are saturated by the lowest-lying single-meson states; b) the vertex functions of particles can be approximated by polynomials of finite degree in the momenta.

Arnowitt, Friedman, Nath, and Suitor [3] showed that these dynamical assumptions, together with the requirements of current locality in calculating the vacuum expectation values of $T$-products of currents, are equivalent to adopting current-field identities and using an effective meson-meson Lagrangian in the approximation of diagrams of the “tree” and “cha-

…es.” The use of the lowest order of perturbation theory is not an approximation of weak coupling, but only an algorithm that provides single-particle saturation.

In works [8, 9] the current-algebra model proposed by Arnowitt et al. [3], based on effective Lagrangians describing the interaction of \(\pi\), \(\rho\), \(A_1\), and \(\sigma\)-mesons, is generalized to a system of particles that also includes nucleons and the \(\Delta(1236)\) resonance. For this system of particles, effective interaction Lagrangians were constructed that make it possible to calculate the amplitudes of various meson-baryon reactions. In particular, in works [10, 11], with the aid of effective Lagrangians constructed in this way, low-energy \(\pi N\)-scattering was considered.

Low-energy \(\pi N\)-scattering is one of the most thoroughly studied processes experimentally. The application of current algebra with soft pions to \(\pi N\)-scattering makes it possible to obtain a number of interesting experimentally verifiable results. These include, first of all, Weinberg’s calculation, and that of a number of other authors [12], of the \(S\)-wave lengths of \(\pi N\)-scattering. A special feature of the result of Weinberg et al. is that it does not depend on any model assumptions concerning the amplitude of axial-vector scattering and is a consequence only of current algebra and the PCAC hypothesis.

As was shown by Schnitzer [13] and Raman [14], in the case of the \(P\)-wave lengths of \(\pi N\)-scattering and the \(S\)-wave effective radii, the amplitude of axial-vector scattering becomes important. Approximating its contribution by the nucleon and the \(\Delta(1236)\) resonance leads to a satisfactory description of the isospin-odd \(P\)-wave scattering lengths. However, inclusion of the \(\Delta(1236)\) resonance spoils the agreement with experiment for the isospin-symmetric \(S\)-wave scattering length \(a_{0+}^{(+)}\) obtained by Weinberg, unless the scalar term is taken into account. Taking it into account again gives the required result for \(a_{0+}^{(+)}\) and at the same time improves the agreement with experiment for the isospin-even \(P\)-wave scattering lengths. Thus a consistent method arises for calculating the \(S\)- and \(P\)-wave lengths and the \(S\)-wave effective radius of \(\pi N\)-scattering.

In the approaches to \(\pi N\)-scattering considered above, what is common is the use of one or another method of extrapolating from the amplitude with momenta \(k=q=0\) to the amplitude with physical values of the pion four-momenta. As we have already seen, the extrapolation hypothesis is unsatisfactory for a number of reasons. The fact that, when the parameters of low-energy \(\pi N\)-scattering are calculated using different methods of extrapolation to the physical amplitudes, the results differ only slightly simply indicates that, in the region of threshold phenomena, this difference is of order

\[ \frac{m_\pi^2}{m_N^2}. \]

The latter, however, may not be the case in the energy region above threshold. Moreover, the ambiguity is unsatisfactory in itself, since it is unclear what is actually being tested by experiment—the extrapolation procedure or the consequences of current algebra.

Low-energy \(\pi N\)-scattering with amplitudes on the mass shell was considered in a number of works [15—17]. Ray and Osypowski [17], in order to describe \(\pi N\)-scattering, used current algebra within the framework of the hypothesis of single-particle dominance. In work [16], the Ward-identity method, developed by Weinberg and Schnitzer, was applied for the first time to the calculation of the parameters of low-energy \(\pi N\)-scattering. In doing so, the form of the commutators of the vector and axial-vector currents with the nucleon fields was additionally assumed. The results obtained for the isospin-odd \(S\)- and \(P\)-wave scattering lengths and the \(S\)-wave effective radius agree satisfactorily with

by experiment. The values of the parameters connected with the isospin-even amplitude turned out somewhat worse. Although Ray [16] had two free parameters at his disposal, he was unable to obtain a satisfactory result simultaneously for all four isospin-even parameters.

Osipovskii [17], relying on the Ward-identity method together with an additional assumption about the form of the commutator of the vector and axial-vector currents with the nucleon fields, obtained the \(S\)- and \(P\)-wave scattering lengths and the \(S\)-wave effective radii, as well as the values of the partial amplitudes at a pion kinetic energy of 58 MeV. In contrast to Ray, Osipovskii’s work took into account the contribution of the \(\sigma\)-meson by means of the \(\sigma\)-commutator, and the \(\pi N\)-scattering amplitude was considered up to second order in the momenta. This leads to the appearance of 5 unknown parameters in the amplitude, which describes eight quantities: the \(S\)-wave lengths \(a^{(\pm)}_{0+}\), the \(P\)-wave lengths \(a^{(\pm)}_{1\pm}\), and the \(S\)-wave effective radii \(R^{(\pm)}\). As a result he succeeded in obtaining values of the \(\pi N\)-scattering parameters, as well as amplitudes at a pion kinetic energy of 58 MeV, which agree well with experiment.

In the current-algebra method, based on the use of effective Lagrangians [8, 9], low-energy \(\pi N\)-scattering is described by the following interaction Lagrangian:

\[ L_I = L_{(3)} + L_{(4)}, \tag{1} \]

where

\[ \begin{aligned} L_{(3)}={}& \varepsilon_{abc}\left\{ g_{\pi\rho\rho}\, v_{\mu b}\,\varphi_c^{\nu}\varphi_a +\frac{1}{2}\lambda_{\pi\rho}\,\varphi_{\mu b}\,G_c^{\lambda\mu}\varphi_{\mu a} \right\} +\frac{1}{2}g_{\pi\sigma}\,\varphi_a\varphi_a\sigma \\ &+\frac{1}{2}\lambda_{\pi\pi\sigma}\,\varphi_a^\mu\varphi_{\mu a}\sigma +i f_{NN\pi}\,\bar{\psi}\gamma^5\tau_a\psi\varphi_a +i\lambda_{NN\pi}\,\bar{\psi}\gamma^\nu\gamma^5\tau_a\psi\varphi_{a\mu} \\ &+d_{\Delta N\pi}\left\{ \bar{\psi}(3g^{\nu\mu}+\sigma^{\nu\mu})T_a^{+}\psi_\nu +\bar{\psi}_\nu(3g^{\nu\mu}-\sigma^{\nu\mu}T_a)\psi \right\}\varphi_{a\mu} \\ &+i g_{NN\rho}\,\bar{\psi}\gamma^\mu\tau_a\psi\,v_{\mu a} +\frac{i}{2}\lambda_{NN\rho}\,\bar{\psi}\sigma^{\mu\nu}\tau_a\psi\,G_{\mu\nu}^{a}, \\[0.8em] L_{(4)}={}& \frac{1}{2F_\pi}\left\{ f_{NN\pi}-\frac{b_{NN\sigma}}{\lambda_1} \right\}\bar{\psi}\psi\,\varphi_a\varphi_a +\left( \frac{\lambda_{NN\pi}}{F_\pi} -\frac{m_\rho^2}{2g_\rho^2} \right) \\ &\times \bar{\psi}\gamma^\mu \frac{1}{2}[\tau_b\tau_a]\psi\,\varphi_{\mu b}\varphi_a . \end{aligned} \]

Here \(\psi\), \(\psi_\nu\), \(v_{\mu a}\), \(\varphi_a\), \(\sigma\) are phenomenological Heisenberg interpolating fields of the isobar \(\Delta(1236)\), the nucleon \(N\), and the \(\rho\)-, \(\pi\)-, and \(\sigma\)-mesons; the parameter \(\lambda_1\) describes the interaction of the \(\sigma\)-meson with the \(\pi\)- and \(A_1\)-mesons.

The structure of the \(\Delta N\pi\) vertex is written taking into account the requirement that the \(\pi N\) system interact only with the \(3/2\) part of the \(\Delta\) field both on the mass shell and off the mass shell.

The part of the interaction Lagrangian \(L_{(3)}\) gives contributions to low-energy \(\pi N\)-scattering through exchange of \(\sigma\), \(\rho\), \(N\), \(\Delta\) particles in the direct and crossed channels, while \(L_{(4)}\) represents a contact interaction. The form of the Lagrangian \(L_I\) follows from the basic assumptions of the model [8, 9], in which, along with the \(SU(2)\times SU(2)\) algebra of the vector \(V_a^\mu\) and axial-vector \(A_a^\mu\) currents, the commutators of the baryon field \(\psi\) with the currents

\[ \delta(x^0-y^0)[V_a^0(x),\psi(y)] = -\delta^{(4)}(x-y)\frac{\tau_a}{2}\psi(x)+c-\mathrm{No.S.T}, \]

\[ \delta(x^0-y^0)[A_a^0(x),\psi(y)] = -\delta^{(4)}(x-y)\frac{\tau_a}{2}\gamma^5\psi(x)+c-\mathrm{No.S.T} \]

under the hypotheses of CVC and PCAC, two assumptions of a dynamical character are adopted: a) the sums over intermediate states in the vacuum averages of the T-products of currents and baryon fields are saturated by one-particle states of the \(\pi\), \(\rho\), \(A_1\), \(\sigma\)-mesons and the \(\Delta\), \(N\)-baryons; b) the particle vertex functions are approximated by polynomials of finite degree in the momenta. The algebra of currents, the CVC and PCAC hypotheses impose a number of restrictions on the coupling constants entering the Lagrangian \(L_1\). The meson-meson constants are expressed in terms of three free parameters: the anomalous magnetic moment of the \(A_1\) meson and the parameters describing the interaction of the \(\sigma\)-meson with the \(\pi\) and \(A_1\) mesons. Two of them were determined by Arnowitt et al. [3], proceeding from the width of the decay \(\rho \to 2\pi\) and an analysis of \(\pi\pi\)-scattering. The meson-baryon constants are expressed in terms of the pion-nucleon coupling constant \(g = 13.5\) and the magnetic form factor of the nucleon [8]. Finally, the constant \(d_{\Delta N\pi}\) can be found from the experimental value of the decay width of the \(\Delta\) (1236).

Using the interaction Lagrangian (1), one can obtain the amplitudes of \((A^{(\pm)}, B^{(\pm)})\) \(\pi N\)-scattering, which are defined in the following way:

\[ \langle Np_2;\,\pi q_2 b\,\mathrm{out}\mid Np_1;\,\pi q_1 a\,\mathrm{in}\rangle = -i(2\pi)^4 \delta(p_2+q_2-p_1-q_1)\frac{1}{(2\pi)^6}\times \]

\[ \times \frac{m_N}{\sqrt{4q_1^0,p_1^0,q_2^0p_2^0}}\, \bar u(p_2) \left\{ -A^{(+)}(s,t,u)\delta_{ba} -A^{(-)}(s,t,u)\frac{1}{2}[\tau_b\tau_a] +\right. \]

\[ \left. +\,i\,\frac{\hat q_1+\hat q_2}{2} \left( B^{(+)}(s,t,u)\delta_{ba} +B^{(-)}(s,t,u)\frac{1}{2}[\tau_b\tau_a] \right) \right\}u(p_1). \]

The expressions for the different contributions to the amplitudes due to \(\rho\)-, \(\sigma\)-mesons, the nucleon, the isobar \(\Delta\) (1236), and the contact-interaction Lagrangian \(L_{(4)}\) have the form:

from \(\rho\)-exchange

\[ {}_{\rho}A^{(+)} = {}_{\rho}B^{(+)} = 0, \]

\[ {}_{\rho}A^{(-)} = \frac{s-u}{2}\lambda_{NN\rho} \frac{2g_{\pi\pi\rho}-m_\rho^2\lambda_{\pi\pi\rho}}{m_\rho^2-t}, \]

\[ {}_{\rho}B^{(-)} = \left(g_{NN\rho}-2m_N\lambda_{NN\rho}\right) \frac{2g_{\pi\pi\rho}-m_\rho^2\lambda_{\pi\pi\rho}}{m_\rho^2-t} + g_{NN\rho}\lambda_{\pi\pi\rho}; \]

from \(\sigma\)-exchange

\[ {}_{\sigma}A^{(+)} = b_{NN\sigma}\, \frac{ g_{\pi\pi\sigma} +\dfrac{t-2m_\pi^2}{2}\lambda_{\pi\pi\sigma} }{ m_\sigma^2-t }, \]

\[ {}_{\sigma}A^{(-)} = {}_{\sigma}B^{(\pm)} = 0; \]

from \(N\)-exchange

\[ {}_{N}A^{(+)} = -4f_{NN\pi}\lambda_{NN\pi} + 4m_N\lambda_{NN\pi}^{2}, \]

\[ {}_{N}A^{(-)}=0, \]

\[ {}_{N}B^{(+)} = g^2 \left( \frac{1}{m_N^2-s} - \frac{1}{m_N^2-u} \right), \]

\[ {}_{N}B^{(-)} = g^2 \left( \frac{1}{m_N^2-s} + \frac{1}{m_N^2-u} \right) - 2\lambda_{NN\pi}^{2}, \]

where \(g=f_{NN\pi}-2m_N\lambda_{NN\pi}\) is the ordinary constant of the \(\pi N\)-interaction, \(g^2/4\pi=14.4\) [8];

from \(\Delta\)-exchange

\[ {}_{\Delta}A^{(+)}=-\frac{2}{3}\,d_{\Delta N\pi}^{2} \left[ \frac{f(s,t)}{m_{\Delta}^{2}-s}+\frac{f(u,t)}{m_{\Delta}^{2}-u} \right], \]

\[ {}_{\Delta}A^{(-)}=\frac{1}{3}\,d_{\Delta N\pi}^{2} \left[ \frac{f(s,t)}{m_{\Delta}^{2}-s}-\frac{f(u,t)}{m_{\Delta}^{2}-u} \right], \]

\[ {}_{\Delta}B^{(+)}=\frac{2}{3}\,d_{\Delta N\pi}^{2} \left[ \frac{g(s,t)}{m_{\Delta}^{2}-s}-\frac{g(u,t)}{m_{\Delta}^{2}-u} \right], \]

\[ {}_{\Delta}B^{(-)}=-\frac{1}{3}\,d_{\Delta N\pi}^{2} \left[ \frac{g(s,t)}{m_{\Delta}^{2}-s}+\frac{g(u,t)}{m_{\Delta}^{2}-u} \right]. \]

Here

\[ \begin{aligned} f(s,t)={}&-8(m_{\Delta}+m_N)(t-2m_\pi^2)+(m_N^2-s)(4m_{\Delta}+6m_N)+{}\\ &+\frac{4}{3m_{\Delta}}\left[-s^2+s(m_N^2-4m_\pi^2)+4m_\pi^2(m_N^2-m_\pi^2)\right]+{}\\ &+\frac{2}{3}\frac{m_N}{m_{\Delta}^{2}}\left[-3s^2+s(7m_N^2-8m_\pi^2)-4(m_N^2-m_\pi^2)^2\right]. \end{aligned} \]

\[ \begin{aligned} g(s,t)={}&8(t-m_\pi^2)-2(m_N^2-s)-m_N(8m_{\Delta}+12m_N)+{}\\ &+\frac{8m_N}{3m_{\Delta}}(s-2m_N^2+2m_\pi^2)+{}\\ &+\frac{2}{3m_{\Delta}^{2}}\left[s^2-s(3m_N^2-4m_\pi^2)+4(m_N^2-m_\pi^2)^2\right]. \end{aligned} \]

Finally, the contribution from the contact-interaction Lagrangian is equal to:

\[ {}_{\kappa}A^{(+)}=\frac{f_{NN\pi}}{F_\pi}-\frac{b_{NN\sigma}}{F_\pi\lambda_1}, \]

\[ {}_{\kappa}A^{(-)}={}_{\kappa}B^{(+)}=0, \]

\[ {}_{\kappa}B^{(-)}=\frac{2\lambda_{NN\pi}}{F_\pi}-\frac{m_\rho^2}{g_\rho^2}. \]

The contributions to the amplitudes from exchange of the \(\rho\), \(N\), and \(\Delta(1236)\) particles do not depend on free parameters, while the contributions to the isospin-odd amplitudes from the \(\sigma\)-meson and the contact term are expressed through two free parameters \(b_{NN\sigma}\) and \(\lambda_1\). The amplitudes written above make it possible to find the \(S\)- and \(P\)-wave scattering lengths, as well as the energy dependence of the \(S\)- and \(P\)-phases of \(\pi N\)-scattering on the pion energy.

Table 1

Table 1 presents numerical values of the various contributions to the S- and P-wave lengths of \(\pi N\) scattering, the S-wave effective radii, obtained in [10, 11], as well as the corresponding experimental data. The value of \(a^{(-)}_{0+}\) is determined mainly by the \(\rho\)-meson contribution, whose magnitude is proportional to \(G_A^{-2}\), which agrees well with experiment and coincides with the result of the current-algebra method with soft pions for the isospin-odd S-wave scattering length. The second term in \(a^{(-)}_{0+}\) arises from the sum of the nucleon \({}_N B^{(-)}_0\) and contact \({}_K B^{(-)}_0\) amplitudes. A feature of this quantity is that it is very small in comparison with the \(\rho\)-meson term, although separately the nucleon and contact contributions exceed the \(\rho\)-meson term by an order of magnitude. Another feature of the result for the scattering length \(a^{(-)}_{0+}\) is the compensation between the contributions of the \(\Delta(1236)\)-resonance in \(A^{(-)}_0\) and \(B^{(-)}_0\), each of which separately exceeds the \(\rho\)-meson term by two orders of magnitude.

The scattering length \(a^{(-)}_{1+}\) is determined mainly by the contribution of the nucleon and of the \(\Delta(1236)\) isobar. This quantity agrees rather well with experiment. There is also reasonable agreement with experiment for the scattering length \(a^{(-)}_{1-}\).

Similarly, in the isospin-even S-wave scattering length \(a^{(+)}_{0+}\), the nucleon and contact terms almost completely cancel one another, although separately they are large. In contrast to the scattering length \(a^{(-)}_{0+}\), in \(a^{(+)}_{0+}\) there is no compensation of the \(\Delta(1236)\)-resonance contributions in \(A^{(+)}_0\) and \(B^{(+)}_0\).

Because of the excessively large \(\Delta(1236)\)-resonance contribution and the cancellation of the nucleon and contact terms, \(a^{(+)}_{0+}\), without taking into account the term associated with the \(\sigma\)-meson parameter, agrees poorly with experiment. Owing to the uncertainty of the \(\sigma\)-meson parameters, the combination of these constants can be found from the condition that the theoretical value of \(a^{(+)}_{0+}\) coincide with its experimental value.

For the P-wave scattering length \(a^{(+)}_{1+}\), without taking the \(\sigma\)-meson into account, one obtains a result that is too small compared with experiment. Again, one can find a combination of the \(\sigma\)-meson parameters from the condition that the experimental and theoretical values coincide.

Then the P-wave scattering length \(a^{(+)}_{1-}\) is completely determined through known quantities. In this case inclusion of the \(\sigma\)-meson contribution improves the prediction for \(a^{(+)}_{1-}\), bringing it closer to the experimental value.

Determination of the \(\sigma\)-meson parameters from \(a^{(+)}_{0+}\) and \(a^{(+)}_{1+}\) [10, 11] leads to invariant amplitudes written in terms of known quantities. This makes it possible to calculate the S-wave effective radii and to find partial-wave amplitudes as functions of the kinetic energy of the \(\pi\)-meson.

Calculation of the S-wave effective radii \(R^{(\pm)}\) with the obtained values of the \(\sigma\)-meson interaction parameters leads to \(R^{(\pm)}\) which, as is seen from Table 1, do not contradict the data of Hamilton and Woolcock [18].

To find the energy dependence of the phases, in [10, 11] the \(K\)-matrix approach is used to unitarize the amplitude [3, 22]

\[ \operatorname{R} f_{1\pm} = \frac{1}{q}\tan \delta_{1\pm}. \]

Figures 1–6 present the experimental data [23] (curve I) and the results of [10, 11] (curve II) for the dependence of the S- and P-phases of \(\pi N\) scattering on the kinetic energy of the \(\pi\)-meson in the c.m. system.

The obtained energy dependence of the S and P partial amplitudes up to the energy of the first \(\Delta(1236)\) resonance is in satisfactory agreement with the experimental data.

Fig. 1. Phase shifts for \(\pi N\)-scattering in the \(S_{11}\) channel.

Fig. 2. Phase shifts for \(\pi N\)-scattering in the \(S_{31}\) channel.

Fig. 3. Phase shifts for \(\pi N\)-scattering in the \(P_{11}\) channel.

Fig. 4. Phase shifts for \(\pi N\)-scattering in the \(P_{31}\) channel.

Fig. 5. Phase shifts for \(\pi N\)-scattering in the \(P_{13}\) channel.

Fig. 6. Phase shifts for \(\pi N\)-scattering in the \(P_{33}\) channel.

Thus, the algebra of currents in the formulation of effective Lagrangians, as applied to \(\pi N\)-scattering, satisfactorily describes simultaneously the entire set of parameters, which includes the \(S\)- and \(P\)-wave scattering lengths and the \(S\)-wave effective radii.

The results for the isospin-odd scattering lengths \(a^{(-)}_{0+}\) and \(a^{(-)}_{1+}\) \([10, 11]\) are close to the values of these parameters in the hard-pion model used by Reay [16] and by Osypowski [17].

The value \(a^{(-)}_{1-}\) differs from that obtained in Osypowski’s work; however, both of these values lie within the existing experimental estimates of \(a^{(-)}_{1-}\). We note that, in contrast to Reay’s work, where two free parameters were used, and to Osypowski’s work, in which there is one parameter, the isospin-odd \(S\)- and \(P\)-wave scattering lengths and the effective radius obtained in \([10, 11]\) contain no free parameters.

In the case of the isospin-even \(S\)- and \(P\)-wave scattering lengths and the \(S\)-wave effective radius, the results of \([10, 11]\) are close to the corresponding quantities of Osypowski [17]. In [17] the isospin-even quantities depend on four free parameters, two of which are connected with the contribution of the \(\sigma\)-meson. Agreement with experiment implies that the best choice of free parameters is one in which the fit to the experimental data \(a^{(+)}_{0+}\), \(a^{(+)}_{1\pm}\), \(R^{(+)}\) is carried out by means of parameters characterizing the contribution of the \(\sigma\)-meson, while the other two are set equal to zero.

From the results obtained in \([10, 11]\) for the \(\pi N\)-scattering phases it follows that the method of effective Lagrangians also makes it possible to describe \(\pi N\)-scattering considerably above threshold, where methods of current algebra with soft pions are not applicable.

Thus, the algebra of currents with hard pions, based on the use of effective Lagrangians, when applied to low-energy \(\pi N\)-scattering successfully describes the entire set of experi-

experimental data, as are models based on the use of Ward identities. At the same time, the effective-Lagrangian model is distinguished by the simplicity and transparency of its interaction mechanism.

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