Extensive data on strong interaction effects in pionic atoms
are analyzed with a density-dependent isovector scattering amplitude
suggested recently by Weise to result from a density dependence of the
pion decay constant. Most of the so-called ‘missing -wave repulsion’
is removed when adopting this approach, thus indicating
a partial chiral symmetry restoration in dense matter.
The resulting potentials describe
quite well also elastic scattering of 20 MeV pions on Ca. Further tests
with elastic scattering are desirable.

: 13.75.Gx; 25.80Dj

Keywords: pionic atoms, -wave repulsion, chiral restoration

Corresponding author: E. Friedman,

Tel: +972 2 658 4667,
FAX: +972 2 658 6347,

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July 20, 2021

The interaction of low energy pions with nuclei has been known for years to be described well by a theoretically-motivated phenomenological optical potential [1], particularly at zero energy where strong interaction effects in pionic atoms have been studied extensively both experimentally and theoretically [2]. The traditional method of spectroscopy of pionic X-rays has been supplemented very recently by the observation of ‘deeply bound’ pionic atom states through the He) reaction [3, 4], thus adding a new dimension to the ability to study pion interactions at threshold in the nuclear medium. Whereas the -wave part of the pion-nucleus optical potential, which is effective only near the nuclear surface, is described rather well by the free pion-nucleon amplitudes (plus a two-nucleon absorption term), this is not the case for the -wave part of the potential. This part of the potential, which is effective throughout the nuclear volume, is a natural source of information on possible modifications by the nuclear medium of the pion-nucleon interaction. This is the topic of the present Letter which deals with the strong -wave repulsion of pions in nuclear matter and its possible origins in a density dependence of the pion decay constant which reflects the change of QCD vacuum structure in dense matter.

The interaction between low energy pions and nuclei is traditionally described [2] by an optical potential as follows:

(1) |

with the -wave part given by

(2) | |||||

where and are the neutron and proton density distributions normalized to the number of neutrons and number of protons , respectively, is the pion-nucleus reduced mass and is the mass of the nucleon. The parameter is given in terms of the pion-nucleon (minus) isoscalar and isovector scattering lengths and , respectively,

(3) |

where is the local Fermi momentum. This second order term is included because of the extremely small value of [5] and it will be shown to play a decisive role in what follows. The term with the complex parameter represents absorption on a neutron-proton pair. The term with is referred to as the -wave potential, see Eqs.(20-22) of [2].

Values of the various parameters of the potential are obtained from fits to experimentally determined strong interaction level shifts and widths and ‘upper’ level yields. Modern data sets containing at least 50 data points along the periodic table lead to rather well defined values for the various parameters and to good agreement between calculation and experiment, with typically per point of about 2. Addressing the real part of the -wave potential, it has been found [2] that both and Re are well determined by the data, contrary to earlier conclusions [6] which were based on considerably more restricted data. A somewhat confusing situation arises when values of Re are found to be large and repulsive, i.e. 3 to 5 times larger than the values of Im, whereas expectations are that Re is attractive and of about the same magnitude as the imaginary part. This unexpected phenomenological repulsion has been referred to as a ‘missing repulsion’ [2, 7].

In the present work fits have been made to 60 experimental values of level shifts, widths and upper level yields for targets from O to U, including the very recently determined binding energies and widths for the deeply bound 1s and 2p states in Pb [8, 9]. As the parameters of the linear term of the -wave part of the potential were always found to be very close to the free pion-nucleon values when the Lorentz-Lorenz parameter was close to 1, we have kept subsequently these parameters fixed at the free pion-nucleon values together with =1 and varied only the parameters of the -wave part of the potential and the phenomenological quadratic (absorptive) term in the -wave part. The latter, traditionally denoted by , was found to be independent of variations in the -wave part of the potential and its real part was essentially zero. In order to focus on the various components of the real part of the -wave potential, we show in Fig. 1 results of such fits, as a function of the parameter Re. The upper part shows values of for the 60 data points and the lower part shows the corresponding values of the other parameters of the real part of the -wave potential, namely, and . The values of Im were found to be remarkably constant at 0.056 . Also shown as horizontal bands in the lower part are the free pion-nucleon values of and which have been determined very recently to high precision [5].

Three features are easy to observe from the solid curves labelled ‘C’: (i) the values of all three parameters are well determined, (ii) the parameter Re is repulsive and large (ReIm) and (iii) and are well determined and are significantly different from the corresponding free pion-nucleon values. Note that the parameter is found to be more than 35% larger in absolute value than its free pion-nucleon value and that it contributes significantly to the repulsion, also in nuclei, through the term in Eq.(3). In fact, for Ca it contributes as much as 35% of the real potential. Thus the ‘missing repulsion’ appears as a very repulsive dispersion term Re and an enhanced parameter. This information is lost when one adopts the ‘effective density’ approach of lumping together and Re [6, 10]. As the free pion-nucleon is extremely small and the empirical values of are much smaller than values of , we discuss only the parameter, where the extra repulsion observed may be associated with medium modification of the pion-nucleon interaction.

The in-medium -wave interactions of pions have been discussed very recently by Weise [11] in terms of partial restoration of chiral symmetry in dense matter where the isospin-odd in-medium pion-nucleon amplitude is inversely proportional to the square of the pion decay constant . The square of the latter is given, in leading order, as a linear function of the nuclear density,

(4) |

with the pion-nucleon sigma term. This leads to a density-dependent isovector amplitude such that becomes

(5) |

for =50 MeV and with in units of fm. Note that expanding this expression in powers of the density leads naturally to a repulsive term in the pion-nucleus potential. We have introduced this expression for into the potential, using for the local nuclear density and repeated the fits to the experimental results. The dashed curves labelled ‘W’ in Fig. 1 show the results obtained with this prescription. From the minimum of it is clear that now Re is considerably less repulsive (ReIm) and that agrees with the free pion-nucleon value. The value of is now much closer to the free pion-nucleon value, although still a little more repulsive. It is seen, therefore, that the introduction of the theoretically motivated medium dependence into the repulsive terms containing removes a major fraction of the excessive phenomenological repulsion, as evidenced by the significantly reduced magnitude of both and Re. The best fit parameters for the above potentials are summarized in the first two rows of Table 1. The third row (‘W65’) is for the same prescription but with a larger value for the pion-nucleon sigma term of =65 MeV (not shown in the figure.) It is seen from the table that for this value of the empirical and parameters are consistent with the free pion-nucleon values. Higher order terms have also been considered very recently [12] and found to be small.

Among several previous attempts to account for the missing -wave repulsion we mention a relativistic impulse approximation (RIA) approach [13] which showed, following Birbrair and others [14, 15, 16], that a specific version of the RIA is able to provide a significant part of the missing repulsion through the modification of the nucleon mass in the nuclear medium. We have therefore looked again into this possibility, noting, however, that in Ref.[13] it was shown that there was no unique way of introducing RIA effects into the pion-nucleus interaction [17]. This specific version of the RIA was included using the following parameterization [16]

(6) |

with =2.7 fm, amounting to for the nucleon mass ratio at normal nuclear density. The results are shown in Fig.2 where it is seen that when the RIA correction is applied to the conventional potential (‘CB’, solid curves) no repulsion is required through the term, but the values of and are still not in agreement with the free pion-nucleon values. Also seen from the figure is that when the RIA term is included together with the theoretically motivated density dependence of Eq.(5) (‘WB’ dashed curves), an attractive Re is found whose magnitude is close to the magnitude of the absorptive part and, then, both and agree with the corresponding free pion-nucleon values. The best fit values of the potential parameters for these two versions of the potential (with =50 MeV) are also summarized in Table 1.

The value of the real potential at the center of the Pb nucleus has received some attention recently [18, 19]. All five potentials yield values between 34 and 39 MeV for this quantity. Obviously these values are extrapolated from the better determined values of the potential near the nuclear surface. Taking e.g. the real potential at the 50% density point, then all five potentials yield values between 12.4 and 13.7 MeV for this quantity.

It is interesting to study the above mentioned features at energies just above threshold through the elastic scattering of low energy pions by nuclei, thus testing further the validity of the chirally motivated approach. Indeed it has been shown [6, 20, 21] that pion-nucleus potentials develop smoothly from the bound states regime to the elastic scattering regime. Here we examine only the elastic scattering of 19.5 MeV pions by Ca with the help of the experimental results of Wright et al. [22] which seem to be the only fairly extensive data for and on the same nucleus and from the same experiment at such low energies. Using the parameters of Table 1 we have calculated the differential cross sections for elastic scattering of pions by Ca at 19.5 MeV and found reasonable agreement with the data. The agreement with the data is a little better for the two potentials that include the RIA corrections (last two rows of the table). Fig. 3 shows comparisons between experiment and calculations for these two potentials. Improving the agreement by adjusting the complex parameter , we find that the values of Im hardly change at all but Re has to be made a little more repulsive, typically by 0.02 . Alternatively, if is adjusted, the extra repulsion is compatible with the energy dependence of this parameter. It is therefore concluded that the limited experimental results for elastic scattering of very low energy pions by nuclei support the picture that emerges from the extensive studies of pionic atoms regarding the nature of the missing -wave repulsion. However, the data on elastic scattering are for one nucleus only, whereas we have included 23 nuclei in the study of pionic atoms. Therefore precision data for the elastic scattering of very low energy on some additional nuclei are desirable.

In conclusion, we have shown that most of the ‘missing -wave repulsion’ in the interaction of pions at threshold with nuclei can be removed by adopting a density-dependent isovector amplitude as suggested by Weise [11] to result from a density dependence of the pion decay constant. The underlying picture is that of partial restoration of chiral symmetry in dense matter. When an additional RIA term is included, the best fit pionic atom potential is in full agreement with the chirally motivated model based on the free pion-nucleon amplitudes.

I wish to acknowledge fruitful discussions with H. Clement, A. Gal and G.J. Wagner. This research was partly supported by the trilateral DFG contract GR 243/51-2.

## References

- [1] M. Ericson, T.E.O. Ericson, Ann. Phys. [NY] 36 (1966) 323.
- [2] For a recent review see C.J. Batty, E. Friedman, A. Gal, Phys. Rep. 287 (1997) 385.
- [3] T. Yamazaki et al., Z. Phys. A 355 (1996) 219.
- [4] H. Gilg et al., Phys. Rev. C 62 (2000) 025201; K. Itahashi et al., Phys. Rev. C 62 (2000) 025202.
- [5] H.-Ch. Schröder et al., ETHZ-IPP PR-2001-1 preprint; Euro. Phys. J. in press; H.-Ch. Schröder et al., Phys. Lett. B 469 (1999) 25.
- [6] R. Seki, K. Masutani, Phys. Rev. C 27 (1983) 2799.
- [7] E. Oset, C. García-Recio, J. Nieves, Nucl. Phys. A 584 (1995) 653.
- [8] A. Gillitzer, Proc. Int. Workshop XXIX on Gross Properties of Nuclei and Nuclear Excitations, Hirschegg, Austria, Jan 14-20, 2001, p. 56.
- [9] H. Geissel et al., submitted to Phys. Rev. Lett. (2001).
- [10] L.L. Salcedo, K. Holinde, E. Oset, C. Schütz, Phys. Lett. B 353 (1995) 1.
- [11] W. Weise, Nucl. Phys. A 690 (2001) 98.
- [12] N. Kaiser, W. Weise, Phys. Lett. B 512 (2001) 283.
- [13] A. Gal, B.K. Jennings, E. Friedman, Phys. Lett. B 281 (1992) 11.
- [14] B.L. Birbrair, V.N. Fomenko, A.B. Gridnev, Yu.A. Kalashnikov, J. Phys. G: Nucl. Phys. 9 (1983) 1473; 11 (1985) 471.
- [15] P.F.A. Goudsmit, H.J. Leisi, E. Matsinos, Phys. Lett. B 271 (1991) 290.
- [16] B.L. Birbrair, A.B. Gridnev, Nucl. Phys. A 528 (1991) 647; B.L. Birbrair, A.B. Gridnev, L.P. Lapina, A.A. Petrunin, A.I. Smirnov, Nucl. Phys. A 547 (1992) 645.
- [17] S. Chakravarti, B.K. Jennings, Phys. Lett. B 323 (1994) 253.
- [18] T.Waas, R. Brockmann, W. Weise, Phys. Lett. B 405 (1997) 215.
- [19] E. Friedman, A. Gal, Phys. Lett. B 432 (1998) 235.
- [20] K. Stricker, H. McManus, J.A. Carr, Phys. Rev. C 19 (1979) 929.
- [21] O. Meirav et al., Phys. Rev. C 40 (1989) 843.
- [22] D.H. Wright et al., Phys. Rev. C 37 (1988) 1155.

potential | () | () | Re () | Im () | Im () | |
---|---|---|---|---|---|---|

C | 117.3 | 0.0180.010 | 0.1220.004 | 0.140.04 | 0.0560.002 | 0.0560.004 |

W | 116.3 | 0.0070.009 | 0.1020.004 | 0.060.04 | 0.0560.002 | 0.0560.004 |

W65 | 117.7 | 0.0010.009 | 0.0950.004 | 0.030.04 | 0.0550.002 | 0.0550.004 |

CB | 118.2 | 0.0050.010 | 0.1100.004 | 0.000.04 | 0.0560.002 | 0.0560.004 |

WB | 118.3 | 0.0040.010 | 0.0920.004 | 0.060.04 | 0.0550.002 | 0.0560.004 |