Calculation of the pion-nucleon ? term value from the experimental data on pionic atoms

Abstract

The pion-nucleon a term value is calculated using pion-nuclear scattering lengths extracted from the data on pionic atoms with A<24 and isospin I=0. The renormalization of the soft-pion ~rN amplitude in nuclear matter is taken into account. The Adependence of the pion-nuclear scattering lengths is studied. Comparison with other results for a term leads to the conclusion that the most probable interval for it is 30-50 MeV. The dispersive part of the S-wave pion-nuclear potential is discussed. The Sum Rule for the Pion-Nuclear AmplitudesOne of the methods for determination of the pionnucleon a term value (%N) is utilization of the sum rule [1] which connects the pion-nuclear iso-symmetric amplitude on the physical threshold T+ (m~) with the same amplitude in the soft-pion limit (so called Weinberg point) T +(0):Here m= and f~ denote the pion mass and the charged pion decay constant (=0,94 m=) respectively, /) -+ are the axial current divergences, IB> is the nuclear ground state, j+ are the pion source functions; the intermediate state In) runs over all states allowed by the selection rules (actually, the authors of [1] limited the set of states In> to the ground or excited nuclear states without pions or with only one pion; for our approach such a restriction is not necessary); E, and P, are energy and momentum of the intermediate system; c.t. stands for the crossed term q2 ~J+, E,-E B ~-E B-E,). Assuming validity of PCAC the commutator in the right-hand side of (1) is equal to zero. Then, if in the sum (1) we separate the coherent rescattering terms, i.e. In>=lB~>, we obtain (see [13)instead of (1) m2 d3q 2 2). T+(m~)=TB+ q (2=)32mAY~(21T,,q] +lTo~ ] (2)Here m A is the mass of the nucleus, q and (Oq are the intermediate pion momentum and energy; T,,q are iso-symmetric and iso-antisymmetric parts of the coherent scattering amplitudeand fl being isotopic indices, and the state IB(-q) ~(q)> being off-shell. As it is shown in [13 the quantitiy T + includes the soft-pion part of the pionnuclear amplitude and also terms corresponding to the incoherent rescattering (In>=lB'=>) and to the pion absorption (In>=lB'>). It was proved [1] that in the nonrelativistic potential model the quantity TB + is nothing but the Born iso-symmetric amplitude of the pion-nuclear scattering. In the framework of that model the expression for the pion-nuclear scattering length has the same structure as (2) (see [1]), namely

Pion-nuclear optical potential.s Wave scattering

“…For this purpose it is conve-nient to use experimental data on the nuclei with isopin I --0 because for such nuclei the experimentally observed scattering lengths a(x-A) are isosymmetric. It was shown [15,16] that for such nuclei values of Re a + (m~) have well-pronounced linear A-dependence…”

“…It was shown [15,16] that at least for nuclei with 4 < A < 24 it is possible to use the approximation alnc ~acoh (8) which gives (together with (6) and (7)) a + (0) = 2 a~ --a + (ms).…”

“…C(p) being a factor of the renormalization of the softpion zcN amplitude in the nuclear matter calculated in [15,16]. The consistent analysis [17] of different methods for the a~u calculation has led to the most probable interval for the a~N value:…”

“…Then, if in the sum (5) we separate the coherent rescattering terms, i.e. ]n)=JBn), the sum rule (5) can be presented for the isosymmetric pion-nuclear scattering length in a simple form [15,16] : a + (ms) = a~ + a+oh,…”

“…The sum rule, connecting isosymmetric pion-nucleus amplitude T+(m~) at the physical threshold and the same amplitude in the soft-energy limit (Weinberg point), was successfully used [15][16][17] to calculate the pion-nucleon term value. This sum rule reads [18] T + (m~)…”