Scattering length

Before going into a discussion of the present knowledge of the $\pi N$ scattering length, we briefly introduce different notations existing in the literature, which are relevant for the pionic hydrogen problem. With

denoting the isospin of the $\pi N$ system, the scattering lengths

can be represented by linear combinations of the scattering lengths

$\displaystyle a^h_{\pi^-p\to\pi^-p}$	$\textstyle =$	$\displaystyle \frac{2a_{1/2}+a_{3/2}}{3}$
$\displaystyle a^h_{\pi^-p\to\pi^0n}$	$\textstyle =$	$\displaystyle \frac{\sqrt{2}(a_{1/2}-a_{3/2})}{3}$

Instead of $a_{1/2}$ and $a_{3/2}$ sometimes the notations $a_{1}$ and $a_{3}$ are used. The scattering lengths $a_{I}$ are related to the isospin-even and isospin-odd scattering length,

and

, respectively, as follows

$\displaystyle a^+$	$\textstyle =$	$\displaystyle \frac{a_{1/2}+2a_{3/2}}{3}$
$\displaystyle a^-$	$\textstyle =$	$\displaystyle \frac{a_{1/2}-a_{3/2}}{3}.$

The present knowledge of the values for the scattering lengths

, and $a^h_{\pi^-p\to\pi^-p}=(a^{+}+a^{-})$ is shown in Table 1.

Table 1: The $\pi N$ scattering lengths (in units $m_{\pi }^{-1}$ ).

Reference			$(a^{+}+a^{-})$
Experiment Data Analysis
$\pi^-p$ atom [19]	-0.0037(50)	0.0920(42)	0.0883(8)
Koch [21]			0.083(4)
KH83 [25]	-0.0097(17)	0.0913(17)
$\pi^-p\to\gamma n$ [27]		0.0842(21)
$\gamma p\to\pi^0p$ [25]		0.0877(16)
SAID [28]	-0.000(1)	0.088(1)	0.088(1)
Theory
Weinberg [29], Tomozawa [30]	0.	0.079	0.079
Bernard et al. [31]		0.0916(15)
Bernard et al. [32]		0.092(4)
Mojzis[34]	-0.0098(130)	0.0935(140)
Goudsmit et al. [35]	-0.003(4)	0.083(1)

The measurements of the hadronic shift in the $\pi^-p$ atom with the relative accuracy of about 1% give the most precise value of the linear combination $(a^{+}+a^{-})$ and can be used to constrain the phase shift analysis of the scattering data. On the other hand, the current accuracy of the widths measurements of about 10% is not sufficient for extracting a precise value of

. This situation is in striking contrast with the theory where the isospin-odd scattering length

is relatively better determined than the isospin-even scattering length

, because large cancellations occur for the latter, so that the result is sensitive to certain theoretical parameters. A better experimental determination of the scattering length

will be of big importance for reducing this theoretical uncertainty. Another problem that requires precision experimental data is related to the isospin symmetry breaking due to the mass difference of the up and the down quarks ( see [17] and references therein).