The pion-nucleon $\sigma$ term from pionic atoms

TL;DR

This paper determines the pion-nucleon sigma term from pionic atom data, finding a value of 57±7 MeV, which aligns with phenomenological studies but conflicts with recent lattice QCD results.

Contribution

The study provides a robust extraction of the sigma term from pionic atom data, highlighting its consistency with phenomenology and discrepancy with lattice QCD.

Findings

Sigma term value: 57±7 MeV

Agreement with phenomenological studies

Disagreement with lattice QCD results

Abstract

Earlier work suggested that the in-medium $\pi N$ threshold isovector amplitude $b_1(\rho)$ gets renormalized in pionic atoms by about 30% away from its $\rho=0$ free-space value, relating such renormalization to the leading low-density decrease of the in-medium quark condensate $<\bar q q>$ and the pion decay constant $f_{\pi}$ in terms of the pion-nucleon $\sigma$ term $\sigma_{\pi N}$. Accepting the validity of this approach, we extracted $\sigma_{\pi N}$ from a large-scale fit of pionic-atom level shift and width data across the periodic table. Our fitted value $\sigma_{\pi N}=57\pm 7$ MeV is robust with respect to variation of $\pi N$ interaction terms other than the isovector $s$-wave term with which $\sigma_{\pi N}$ was associated. Higher order corrections to the leading order in density involve some cancellations, suggesting thereby only a few percent overall systematic uncertainty. The value of $\sigma_{\pi N}$ derived here agrees with values obtained in several recent studies based on near-threshold $\pi N$ phenomenology, but sharply disagrees with values obtained in recent direct lattice QCD calculations.

Equations

$${\sigma }_{\pi N}=\frac{{\bar{m}}_q}{2m_N}\sum _{u,d}⟨N|\bar{q}q|N⟩,{\bar{m}}_q=\frac{1}{2}(m_u+m_d),$$

$$b_0^{\mathrm{free}}=0.0076\pm 0.0031m_{\pi }^{-1},b_1^{\mathrm{free}}=-0.0861\pm 0.0009m_{\pi }^{-1},$$

$$b_0^{\mathrm{TW}}=0,b_1^{\mathrm{TW}}=-\frac{{\mu }_{\pi N}}{8\pi f_{\pi }^2}=-0.079m_{\pi }^{-1},$$

$$\frac{f_{\pi }^2(\rho )}{f_{\pi }^2}=\frac{<\bar{q}q{>}_{\rho }}{<\bar{q}q>}\simeq 1-\frac{{\sigma }_{\pi N}}{m_{\pi }^2{f}_{\pi }^2}\rho ,$$

$$b_1=b_1^{\mathrm{free}}{\left(1-\frac{{\sigma }_{\pi N}}{m_{\pi }^2{f}_{\pi }^2}\rho \right)}^{-1}.$$

$$E^2-{\vec{p}}^2-m_{\pi }^2-\Pi (E,\vec{p},\rho )=0,$$

$$\left[{\nabla }^2-2\mu (B+V_{\mathrm{opt}}+V_c)+(V_c+B{)}^2\right]\psi =0,$$

$$2\mu V_{\mathrm{opt}}(r)=q(r)+\vec{\nabla }\cdot \left(\frac{{\alpha }_1(r)}{1+\frac{1}{3}\xi {\alpha }_1(r)}+{\alpha }_2(r)\right)\vec{\nabla },$$

$$q(r)$$

$${\alpha }_1(r)=4\pi (1+\frac{\mu }{m_N}{)}^{-1}\{c_0[{\rho }_n(r)+{\rho }_p(r)]+c_1[{\rho }_n(r)-{\rho }_p(r)]\}$$

$${\alpha }_2(r)=4\pi (1+\frac{\mu }{2m_N}{)}^{-1}4C_0{\rho }_n(r){\rho }_p(r),$$

$$b_0\to b_0-\frac{3}{2\pi }(b_0^2+2b_1^2)p_F,$$

$$r_n-r_p=\gamma \frac{N-Z}{A}+\delta ,$$

$$\frac{<\bar{q}q{>}_{\rho }}{<\bar{q}q>}=1-\frac{\rho }{f_{\pi }^2}\left[\frac{{\sigma }_{\pi N}}{m_{\pi }^2}\left(1-\frac{3p_F^2}{10m_N^2}+\frac{9p_F^4}{56m_N^4}\right)+\frac{\partial E(\rho )/A}{\partial m_{\pi }^2}\right],$$

$$\frac{f_{\pi }^2(\rho )}{f_{\pi }^2}=\frac{m_{\pi }^2}{m_{\pi }^2(\rho )}\frac{<\bar{q}q{>}_{\rho }}{<\bar{q}q>}\simeq \frac{m_{\pi }^2}{m_{\pi }^2(\rho )}\left(1-\frac{{\sigma }_{\pi N}}{m_{\pi }^2{f}_{\pi }^2}\rho \right),$$

$$b_1=b_1^{\mathrm{free}}\frac{m_{\pi }^2(\rho )}{m_{\pi }^2}{\left(1-\frac{{\sigma }_{\pi N}}{m_{\pi }^2{f}_{\pi }^2}\rho \right)}^{-1}.$$

$$\delta m_{\pi }^2\approx -4\pi \left(1+\frac{m_{\pi }}{m_N}\right){\overline{b}}_0{\rho }_{\mathrm{eff}}=0.045(8)m_{\pi }^2,$$

$$\delta m_{\pi }^2\approx -\frac{4\pi (1+\frac{m_{\pi }}{m_N}{)}^{-1}{c}_0{\rho }_{\mathrm{eff}}}{1+\frac{1}{3}4\pi (1+\frac{m_{\pi }}{m_N}{)}^{-1}{c}_0{\rho }_{\mathrm{eff}}}{p}_{\mathrm{eff}}^2=-0.025m_{\pi }^2,$$

$${\sigma }_{\pi N}\approx (59\pm 3)\mathrm{MeV}+1.116\Delta b_0^{\mathrm{free}}+0.390\Delta b_1^{\mathrm{free}},$$

$$\mathrm{BJ}:(-0.9,-85.3)\cdot 10^{-3}{m}_{\pi }^{-1},(+7.9,-85.4)\cdot 10^{-3}{m}_{\pi }^{-1},$$

Full text

††journal: Physics Letters B

1 Introduction

The $\pi N$ $\sigma$ term

[TABLE]

records the contribution of explicit chiral symmetry breaking to the nucleon mass $m_{N}$ arising from the non-zero value of the $u$ and $d$ quark masses in QCD. A wide spectrum of evaluated $\sigma_{\pi N}$ values, from about 20 to 80 MeV, was compiled by Sainio back in 2002 [1]. Recent evaluations roughly fall into two classes: (i) pion-nucleon low-energy phenomenology, using $\pi N$ $s$-wave scattering lengths derived precisely from pionic hydrogen and deuterium, results in calculated values of $\sigma_{\pi N}\sim(50-60)$ MeV [2, 3, 4, 5, 6], the most recent of which is 58$\pm$5 MeV, whereas (ii) recent lattice QCD calculations reach values of $\sigma_{\pi N}\sim(30-50)$ MeV [7, 8, 9, 10, 11, 12], the most recent of which is 26$\pm$7 MeV. However, when augmented by chiral perturbation expansions such lattice calculations may lead also to values of about 50 MeV, see e.g. Refs. [13, 14, 15, 16]. This spread of calculated $\sigma_{\pi N}$ values is discussed further in the concluding section.111Some works denote the entity defined by the r.h.s. of (1) as the nucleon $\sigma$ term $\sigma_{N}$. The notation adopted in the present work, $\sigma_{\pi N}$, follows that of recent works rooted in the low-energy non-strange sector of hadronic physics, e.g. [4].

Here we show that the wealth of data on pionic atoms across the periodic table provides a precise determination of $\sigma_{\pi N}$. The experimental database for pionic atoms is the most extensive of all hadronic atoms [17, 18], offering a useful test-ground for studying in-medium effects. On the theory side, the near-threshold pion-nucleus optical potential $V_{\rm opt}$ is given by single-nucleon $\pi N$ interaction terms approximated by their free-space values, with relatively small contributions from absorption on two nucleons. Our recent analysis of pionic atoms [19] demonstrated robustness in the quality of fitting the data against details of the applied analysis methodology.

The starting point in discussing in-medium renormalization in pionic atoms is that the free-space isoscalar and isovector $\pi N$ scattering lengths derived in a chiral perturbation calculation [20] from pionic hydrogen and pionic deuterium precise X-ray measurements [21, 22],

[TABLE]

are well approximated by the Tomozawa-Weinberg (TW) leading-order chiral limit [23]

[TABLE]

where $\mu_{\pi N}$ is the pion-nucleon reduced mass and $f_{\pi}=92.2$ MeV is the free-space pion decay constant. This expression for the isovector amplitude $b_{1}$ suggests that its in-medium renormalization is directly connected to that of the pion decay constant $f_{\pi}$, given to first order in the nuclear density $\rho$ by the Gell-Mann - Oakes - Renner (GOR) expression [24]

[TABLE]

where $<\bar{q}q>_{\rho}$ stands for the in-medium quark condensate and $\sigma_{\pi N}$ is the pion-nucleon $\sigma$ term. The decrease of $<\bar{q}q>_{\rho}$ with density in Eq. (4) marks the leading low-density behavior of the order parameter of the spontaneously broken chiral symmetry. Recalling the $f_{\pi}$ dependence of $b_{1}^{\rm TW}$ in Eq. (3), Eq. (4) suggests the following density dependence for the in-medium $b_{1}$:

[TABLE]

In this model, introduced by Weise [25, 26], the explicitly density-dependent $b_{1}(\rho)$ of Eq. (5) figures directly in the pion-nucleus $s$-wave near-threshold potential. Studies of pionic atoms [27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and low-energy pion-nucleus scattering [37, 38] confirmed that the $\pi N$ isovector $s$-wave interaction term is indeed renormalized in agreement with Eq. (5). It is this in-medium renormalization that brings in $\sigma_{\pi N}$ to the interpretation of pionic-atom data. However, the value of $\sigma_{\pi N}$ was held fixed around 50 MeV in these studies, with no attempt to determine its optimal value.

In the present work, we kept to the $\pi N$ isovector $s$-wave amplitude $b_{1}$ renormalization given by Eq. (5), but adopted a reversed approach of fitting $\sigma_{\pi N}$ to a comprehensive set of pionic atoms data across the periodic table. Other real $\pi N$ interaction parameters fitted simultaneously with $\sigma_{\pi N}$ converged at expected free-space values. Holding these parameters fixed at the converged values, except for the tiny isoscalar $s$-wave amplitude $b_{0}$ which is renormalized primarily by a double-scattering term (see below), we get a best-fit value of $\sigma_{\pi N}=57\pm 7$ MeV.

The paper is organized as follows. In Sect. 2 we outline the methodology applied to fitting pionic atoms data. Results are given in Sect. 3, followed by discussion in Sect. 4 of estimated deviations from the linear-density expression (4) and their impact on the value derived for $\sigma_{\pi N}$.

2 Methodology

Here we briefly review the methodology applied in our recent work [19] to dealing with pionic atoms data, using energy-dependent optical potentials within a suitably constructed subthreshold model. For a recent review focusing on $K^{-}$ and $\eta$ nuclear near-threshold physics, see Ref. [39]. The pion self-energy operator $\Pi(E,\vec{p},\rho)$ in nuclear matter of density $\rho$ enters the in-medium pion dispersion relation [18]

[TABLE]

where ${\vec{p}}$ and $E$ are the pion momentum and energy, respectively, in nuclear matter of density $\rho$. The resulting pion-nuclear optical potential $V_{\rm opt}$, defined by $\Pi(E,\vec{p},\rho)=2EV_{\rm opt}$, enters the wave equation for the pion at or near threshold:

[TABLE]

where $\hbar=c=1$ was implicitly assumed in these equations. In this expression, $\mu$ is the pion-nucleus reduced mass, $B$ is the complex binding energy, $V_{c}$ is the finite-size Coulomb interaction of the pion with the nucleus, including vacuum-polarization terms, all added according to the minimal substitution principle $E\to E-V_{c}$. Interaction terms negligible with respect to $2{\mu}V_{\rm opt}$, i.e. $2V_{c}V_{\rm opt}$ and $2BV_{\rm opt}$, were omitted. We use the Ericson-Ericson form [40]

[TABLE]

with its $s$-wave part $q(r)$ and $p$-wave part, $\alpha_{1}(r)$ and $\alpha_{2}(r)$, given by [18]

[TABLE]

[TABLE]

[TABLE]

where $\rho_{n}$ and $\rho_{p}$ are the neutron and proton density distributions normalized to the number of neutrons $N$ and number of protons $Z$, respectively. The coefficients $b_{0}$ and $b_{1}$ in Eq. (9) are effective, density-dependent pion-nucleon isoscalar and isovector $s$-wave scattering amplitudes, respectively, evolving from the free-space scattering lengths $b_{0}^{\rm free}$ and $b_{1}^{\rm free}$ of Eq. (2), and are essentially real near threshold. Similarly, the coefficients $c_{0}$ and $c_{1}$ in Eq. (10) are effective $p$-wave scattering amplitudes which, since the $p$-wave part of $V_{\rm opt}$ acts mostly near the nuclear surface, are close to the free-space scattering volumes $c_{0}^{\rm free}$ and $c_{1}^{\rm free}$ provided $\xi=1$ is applied in the Lorentz-Lorenz renormalization of $\alpha_{1}$ in Eq. (8). The parameters $B_{0}$ and $C_{0}$ represent multi-nucleon absorption and therefore have an imaginary part. Their real parts stand for dispersive contributions which often are absorbed into the respective single-nucleon amplitudes. Below we focus on the $s$-wave part $q(r)$ of $V_{\rm opt}$.

Regarding the isoscalar amplitude $b_{0}$, since the free-space value $b_{0}^{\rm free}$ in Eq. (2) is exceptionally small, it is customary in the analysis of pionic atoms to supplement it by double-scattering contributions induced by Pauli correlations which give rise to explicit density dependence of the form [40, 41]

[TABLE]

where $p_{F}$ is the local Fermi momentum corresponding to the local nuclear density $\rho=2p_{F}^{3}/(3\pi^{2})$.222Note added in proof: the double-scattering term in (12) is missing a kinematical factor ($1+m_{\pi}/m_{N}$) [42] which, when included, hardly affects our results. The role of double-scattering contributions in general will be discussed by us in a forthcoming paper.

Regarding the isovector amplitude $b_{1}$, it is given by the r.h.s. of Eq. (5) in terms of a free-space $b_{1}^{\rm free}$ and $\sigma_{\pi N}$. It affects primarily level shifts in pionic atoms with $N-Z\neq 0$. However, it affects also $N=Z$ pionic atoms through the dominant quadratic $b_{1}$ contribution to the r.h.s. of Eq. (12). This dominance follows already at the level of the free-space $b_{1}^{\rm free}$ from a systematic expansion of the pion self-energy up to $O(p^{4})$ in nucleon and pion momenta within chiral perturbation theory [43]. To understand why the in-medium $b_{1}$ of Eq. (5) enters the Pauli-correlation double-scattering contribution to Eq. (12), we recall how it was introduced in Ref. [31]. The energy dependence of the pion self-energy operator $\Pi(E,\vec{p},\rho)$ in a uniform medium of density $\rho$ was traded there for an equivalent energy independent optical potential, with $b_{1}^{\rm free}$ promoted to a density dependent in-medium $b_{1}$, Eq. (5). Once done, it is this $b_{1}$, not $b_{1}^{\rm free}$, that enters as input the formal derivation [44] of the Pauli-correlation double-scattering term. This approach has been practised in numerous global fits to pionic atoms by us [18, 19] as well as by other groups, e.g., Geissel et al. [29].

An important ingredient in the analysis of pionic atoms are the nuclear densities that enter the potential, Eqs. (9)–(11). With proton densities determined from nuclear charge densities, we vary the neutron densities searching for a best agreement with the pionic atoms data. A linear dependence of $r_{n}-r_{p}$, the difference between the root-mean-square (rms) radii, on the neutron excess ratio $(N-Z)/A$ has been recognized as a useful and relevant representation, parameterized across the periodic table as

[TABLE]

with $\gamma$ close to 1.0 fm and $\delta$ close to zero. Two-parameter Fermi distributions were used for $\rho_{p}$ and $\rho_{n}$ with the same diffuseness parameter for protons and neutrons, the so-called ‘skin’ shape [18, 45] which was found to yield lower values of $\chi^{2}$ than other shapes do for pions. Here we used $\delta=-$0.035 fm and varied the parameter $\gamma$. With $\gamma$=1 fm, for example, the ‘neutron skin’ of 208Pb is $r_{n}-r_{p}=0.177$ fm which agrees well with recent values derived specifically for 208Pb from several sources.333For example, 0.16$\pm$0.02$\pm$0.04 fm from $\bar{p}$ atoms [46], 0.156${}^{+0.025}_{-0.021}$ fm from $E1$ polarizability studies [47], 0.15$\pm$0.08 fm from $\pi^{-}$ atoms [48], 0.11$\pm$0.06 fm from $\pi^{+}$ total reaction cross sections [48], and 0.15$\pm$0.03${}^{+0.01}_{-0.03}$ fm from coherent pion photoproduction measurements at MAMI [49]. In what follows, rather than show results as a function of the neutron-excess parameter $\gamma$ of Eq. (13), we present results as a function of the implied value of $r_{n}-r_{p}$ for 208Pb, as this quantity has been discussed extensively in recent years, e.g. Refs. [50, 51], particularly in the context of neutron stars.

3 Results

In line with our previous studies of pionic atoms [18, 19] we performed global fits to strong interaction level shifts and widths across the periodic table, from Ne to U, including ‘deeply bound’ states in Sn isotopes and in 205Pb. This approach provides an average behavior of the $\pi N$ interaction parameters within an optical potential model, Eqs. (8)–(11). Fits were made over a wide range of values for the neutron-excess radius parameter $\gamma$. The lowest values of $\chi^{2}$ were obtained, as expected, when varying all eight parameters of the optical potential. Whereas the imaginary parts Im$\,B_{0}$ and Im$\,C_{0}$ were well-determined and hardly varied over a wide range of values tested for $\gamma$, their real parts Re$\,B_{0}$ and Re$\,C_{0}$ were poorly determined and varied over a broad range. Moreover, they displayed correlations with the two resulting scattering volumes $c_{1}$, $c_{0}$, respectively. With Re$\,B_{0}$ and Re$\,C_{0}$ kept zero, all the other parameters turned out to be well determined. Consequently most of the resulting $\pi N$ interaction terms, but not the $\pi N$ $\sigma$ term $\sigma_{\pi N}$, turned out to be independent of the neutron-excess radius parameter $\gamma$. We note that when $b_{1}^{\rm free}$ was used in the double scattering term (12) instead of the in-medium $b_{1}$ form (5), the lowest $\chi^{2}$ increased by about 5 units. With an achieved $\chi^{2}$ per degree of freedom of 1.7 this increase means three standard deviations.

Extensive fits essentially displayed correlations between rms radii of the neutron density distribution and the resulting $\sigma_{\pi N}$, as shown in Fig. 1. The figure shows fits with six adjusted parameters, namely $b_{0}$, $\sigma_{\pi N}$, Im$\,B_{0}$, $c_{0}$, $c_{1}$ and Im$\,C_{0}$. As in earlier work [18] a finite range (FR) folding of rms radius of 0.9 fm was applied to the $\pi N$ $p$-wave interaction terms. The bottom part of the figure shows the derived $\sigma_{\pi N}$ values with their uncertainties. An interesting by-product of these fits is the value 0.15$\pm$0.03 fm of the implied neutron skin of 208Pb, taken from the minimum of the $\chi^{2}$ curve in the top part of the figure, in agreement with the values cited in a footnote to the text at the end of Sect. 2 above.

In the fits shown in Fig. 1, the single-nucleon isoscalar $c_{0}$ and isovector $c_{1}$ parameters of the $\pi N$ $p$-wave potential $\alpha_{1}(r)$ turned out to agree with the corresponding values of the free-space $\pi N$ scattering volumes. This is shown in Fig. 2.

With $c_{0}$ and $c_{1}$ hardly dependent on the neutron densities, one could keep these fixed during fits to reduce the uncertainties of the resulting values of $\sigma_{\pi N}$. Figure 3 shows two such fits with fixed values, both analogous to Fig. 1, one with $p$-wave finite-range folding (FR, solid lines, black), and one without folding (ZR, dashed lines, red). In both parts of Fig. 3 the red curves are shifted to the right of the corresponding black curves, but for the best fit values of $\sigma_{\pi N}$, at the minima of $\chi^{2}$, there is hardly any difference between the FR and ZR models, regardless of the $\sim$0.06 fm difference between the best implied values of the 208Pb skin in these models. With fixed $c_{0}$ and $c_{1}$, the fitting errors are indeed smaller than those in Fig. 1. The average value for $\sigma_{\pi N}$ from Fig. 3 is $\sigma_{\pi N}=57\pm 7$ MeV. For $p$-wave FR folding with rms radius smaller than 0.9 fm the resulting curves in Fig. 3 are located between the 0.9 fm FR curve and the ZR curve. We note that a size of order 1 fm appears naturally for $\pi N$ $p$-wave form factors fitted to the $\Delta$ resonance $P_{33}$ phase shifts [53].

4 Discussion and summary

The pionic atoms fits and the value of the $\pi N$ $\sigma$ term $\sigma_{\pi N}$ extracted in the present work are based on the in-medium renormalization of the near-threshold $\pi N$ isovector scattering amplitude $b_{1}$ as given by Eq. (5), derived from Eq. (4) for the leading order in-medium decrease of the quark condensate $<\bar{q}q>$. Higher order corrections to this simple form have been proposed in the literature and are discussed briefly below to see how much they affect our fitted value of $\sigma_{\pi N}$. Generally, one does not expect appreciable corrections simply because typical nuclear densities probed in pionic atoms are only about 0.6 [30] or even 0.5 [54] of nuclear matter density. A representative effective density of $\rho_{\rm eff}=0.1$ fm*-3* is used for the two types of corrections discussed below.

Kaiser et al. [55] extended Eq. (4) to

[TABLE]

accounting for kinetic energy contributions up to order $m_{N}^{-3}$ in the Fermi gas model plus $NN$ correlation contributions from one- and two-pion interaction terms. At density $\rho_{\rm eff}=0.1$ fm*-3* and for $\sigma_{\pi N}=60$ MeV the r.h.s. of Eq. (14) is about 0.75, larger than the purely linear density expression by about 0.03. Most of this increase is owing to the $NN$ correlation contributions. If we wish to absorb at $\rho_{\rm eff}$ this departure from linearity in $\rho$ into an effective linear density form, Eq. (4), we need to increase our fitted $\sigma_{\pi N}$ value by about 7 MeV. A smaller increase results by following a chiral approach up to next-to-leading order [56].

Jido and collaborators [57] extended the GOR expression Eq. (4) by including the increase of the in-medium pion mass $m_{\pi}(\rho)$ in symmetric nuclear matter444In asymmetric nuclear matter the charged pion masses split, with $m_{\pi^{-}}$ increasing further owing to the repulsive $b_{1}$ isovector $s$-wave $\pi N$ interaction term [43]. This effect is disregarded in the estimate given below. from its free-space value $m_{\pi}$:

[TABLE]

and also by adding corrections of order $\rho^{4/3}$ [58, 59] which at $\rho_{\rm eff}$ are negligible. The pion mass dependence in Eq. (15) leads to the following modification of Eq. (5) for the near-threshold $\pi N$ isovector amplitude:

[TABLE]

The pion mass in $N=Z$ isospin-zero symmetric nuclear matter increases from its free-space value $m_{\pi}$ to $m_{\pi}(\rho)$ owing to the weakly repulsive $b_{0}$ isoscalar $s$-wave $\pi N$ interaction term. Identifying the in-medium pion mass with $E(\vec{p}=0)$ in the dispersion equation (6) and using Eq. (9) with an appropriate subthreshold value [19] $\overline{b}_{0}=-0.011(2)\,m_{\pi}^{-1}$ corresponding to our best fit threshold value $b_{0}=-0.022(2)\,m_{\pi}^{-1}$, we obtain for the difference $\delta m_{\pi}^{2}=m_{\pi}^{2}(\rho)-m_{\pi}^{2}$ at $\rho_{\rm eff}=0.1$ fm*-3*:

[TABLE]

or equivalently $m_{\pi}^{2}(\rho_{\rm eff})/m_{\pi}^{2}=1.045(8)$. Interestingly, this isoscalar contribution to $m_{\pi}^{2}(\rho)$ agrees with a rescattering contribution to the pion in-medium self energy, derived in Ref. [60] with the purpose of providing additional renormalization of $b_{1}^{\rm free}$ beyond the leading $\sigma_{\pi N}$ contribution given by Eq. (5).555Ref. [60] deals with several other in-medium modifications that require separate discussion. In particular, their rescattering contribution, which as pointed out here is equivalent to in-medium pion mass renormalization, was considered earlier by Delorme et al. [61]. For this contribution, adding a factor $\pi$ missing in the denominator of Eq. (28) in Ref. [60], one obtains $(\delta b_{1}/b_{1}^{\rm free})\approx(3p_{F}\rho_{\rm eff})/(16\pi^{2}f_{\pi}^{4})=0.0454$, in agreement with Eq. (17) above. With this increased in-medium pion mass, our best-fit central value of $\sigma_{\pi N}$=57 MeV decreases, by just (7$\pm$1) MeV, to (50$\pm$1) MeV. Perhaps fortuitously, the two higher-order effects considered here upon deriving $\sigma_{\pi N}$ from pionic atoms, Eqs. (14) and (16), cancel each other.

It is worth recalling that the attractive isoscalar $p$-wave $\pi N$ interaction term was disregarded in this uniform nuclear matter estimate where the pion momentum vanishes. In finite-size nuclei, however, multiplying $p_{F}(\rho_{\rm eff})$ by $m_{\pi}/(m_{N}+m_{\pi})$ a representative pion effective momentum of $p_{\rm eff}=29.1$ MeV is obtained. This leads to the following $p$-wave contribution:

[TABLE]

using $c_{0}^{\rm free}=0.230\,m_{\pi}^{-3}$. Adding up these $s$-wave and $p$-wave contributions, we get $m_{\pi}^{2}(\rho_{\rm eff})/m_{\pi}^{2}=1.020(8)$, leading to a decrease of our best-fit $\sigma_{\pi N}$ central value of 57 MeV, by only (3$\pm$1) MeV, to (54$\pm$1) MeV.

To conclude the discussion, we note that unlike most determinations of $\sigma_{\pi N}$ that rely heavily on the vanishingly small and highly model dependent value of the free-space $\pi N$ isoscalar scattering length $b_{0}^{\rm free}$, the present work is based on the considerably larger and nearly model independent value of the free-space $\pi N$ isovector scattering length $b_{1}^{\rm free}$. The dependence of $\sigma_{\pi N}$ on the input free-space $\pi N$ scattering lengths, within any specific hadronic model calculation, is given according to the Bonn-Jülich (BJ) group [62] by

[TABLE]

where (59$\pm$3) MeV is the BJ calculated $\sigma_{\pi N}$ value [4] and $\Delta b_{j}^{\rm free}$, $j=0,1$, is the difference between the values of $b_{j}^{\rm free}$ (in units of $10^{-3}\,m_{\pi}^{-1}$) used in that specific model and in the BJ calculation. Two sets were suggested by BJ for $(b_{0}^{\rm free},\,b_{1}^{\rm free})$,

[TABLE]

depending on how charge dependence is considered. These two sets differ mostly in the $b_{0}^{\rm free}$ values. To demonstrate the use of Eq. (19) we refer to the evaluation of the $\pi N$ $\sigma$ term in Ref. [63] from $\pi^{\pm}p$ scattering data taken by the CHAOS group at TRIUMF [64]. Extrapolating from the lowest pion kinetic energy of 19.9 MeV reached in the experiment, the value used in Ref. [63] was $b_{0}^{\rm free}=(-9.7\pm 0.9)\cdot 10^{-3}\,m_{\pi}^{-1}$. Eq. (19) ‘predicts’ then $\sigma_{\pi N}$=49$\pm$3 or 39$\pm$3 MeV, depending on the choice made for $b_{0}^{\rm free}$ in Eq. (20), in rough agreement with the value $\sigma_{\pi N}$=(44$\pm$12) MeV derived in Ref. [63]. Similarly, the increase of $\sigma_{\pi N}$ from the older value (45$\pm$8) MeV derived by Gasser, Leutwyler and Sainio [65] to the very recent (59$\pm$3) MeV [4] is related, according to Eq. (19), to the use by the BJ group of the more precise $\pi N$ scattering lengths as extracted recently from $\pi^{-}$H and $\pi^{-}$D atoms.

In conclusion, we have derived in this work a value of $\sigma_{\pi N}=57\pm 7$ MeV from a large scale fit to pionic atoms observables, in agreement with the relatively high values reported in recent studies based on modern hadronic $\pi N$ phenomenology [6], but in disagreement with the low $\sigma_{\pi N}$ values reached in the modern lattice QCD calculations, e.g. [12]. Our derivation is based on the model introduced by Weise and collaborators [25, 26, 31] for the in-medium renormalization of the $\pi N$ near-threshold isovector scattering amplitude, using its leading density dependence Eq. (5), and was found robust in fitting the wealth of pionic atoms data against variation of other $\pi N$ interaction parameters that enter the low-energy pion self-energy operator. The two types of model corrections beyond the leading density dependence considered here were found to be relatively small, a few MeV each, and partly canceling each other. Further model studies are necessary to confirm this conclusion.

Acknowledgments

We thank Wolfram Weise for a useful communication [66] on in-medium and partial restoration of chiral symmetry effects on $\sigma_{\pi N}$.