Modified $\pi \pi$ amplitude with $\sigma$ pole

A set of well-known once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multichannel $S$ ($\pi \pi$, $K\overline{K}$, and $\eta \eta$) and $P$ ($\pi \pi$, $\rho 2\pi$, and $\rho \sigma$) wave amplitudes mostly below 1 GeV. Before the modifications, these amplitudes significantly did not satisfy the crossing symmetry condition and did not describe the $\pi \pi$ threshold region. Moreover, the pole of the $S$ wave amplitude related with the $f_0(500)$ meson (former $f_0(600)$ or $\sigma$) had much smaller imaginary part and bigger real one in comparison with those in the newest Particle Data Group Tables. Here, these amplitudes are supplemented by near threshold expansion polynomials and refitted to the experimental data in the effective two pion mass from the threshold to 1.8 GeV and to the dispersion relations up to 1.1 GeV. In result the self consistent, i.e., unitary and fulfilling the crossing symmetry condition, $S$ and $P$ wave amplitudes are formed and the $\sigma$ pole becomes much narrower and lighter. To eliminate doubts about the uniqueness of the so obtained sigma pole position short and purely mathematical proof of the uniqueness of the results is also presented. This analysis is addressed to a wide group of physicists and aims at providing a very effective and easy method of modification of, many presently used, $\pi \pi$ amplitudes with a heavy and broad $\sigma$ meson without changing of their original mathematical structure.

Figure 1

Phase shifts for the $S0$ wave (a) and $P1$ wave (b) amplitudes as a function of the effective two pion mass $m_{\pi \pi }$ for the original (solid line), extended (dashed line) and refitted (dash-dotted line) amplitudes considered in the text. Data are taken from [14].

Figure 2

Shift of the $\sigma$ pole on sheet II after fitting to the GKPY equations. The stars in the lower part of figure denote the pole positions for the “Old” (original) and “New” amplitudes while the stars indicated by the arrows their positions after refitting. The big (PDG2010) and small (PDG2012) rectangles show area allowed by the Particle Data Tables in 2010 and 2012, respectively. Black points are positions of the poles listed in the Particle Data Tables published in 2010.

Figure 3

Input (dashed line) and output (solid line) real parts of the $S0$ wave (upper figure) and $P1$ wave (lower figure) extended amplitudes, i.e., before fitting.

Figure 4

Input (dashed line) and output (solid line) real parts of the $S0$ (upper figure) and the $P1$ (lower figure) amplitudes after fitting.

Figure 5

Effective two-pion mass dependence of the phase shifts corresponding to the extended (dashed line) and refitted (solid line) “Old” amplitudes.

Figure 6

Dependence of the real and imaginary part of the amplitude on the phase shifts.

Figure 7

Gradient of the real and imaginary part of the amplitude as a function of the phase shifts.

Figure 8

Effective two pion mass dependence of the $K{T}_{00}^{00}$ term.