Fully solvable mathematical scheme in finding out the right mass and width values of f0(500) and ρ0(770) mesons

Abstract. Starting from the phase representations with one subtraction of the pion scalar-isoscalar and vector-isovector charged pion electromagnetic form factor and exploiting the most accurate information on the S-wave isoscalar and the P-wave isovector ππ scattering phase shifts, respectively, to be obtained from the existing inaccurate experimental data by means of the GarciaKamiński-Peláez-Yndurain Roy-like equations with an imposed crossing symmetry condition, in the framework of the so-called "fully solvable mathematical scheme" the most reliable values of the f0(500) and ρ0(770) meson mass and width are found.


Introduction
If there is a function F(t) to be analytic in the whole complex t-plane besides the cuts on the positive real axis from the lowest square root branch point t = 4 to +∞, fulfills the elastic unitarity condition ImF(t) = F(t)e −iδ sin δ in elastic region with δ to be some of the ππ scattering phase shifts, the asymptotic behavior F(t) |t|→∞ ∼ 1/t, the reality condition F * (t) = F(t * ) and it is normalized at t = 0 to one, then one can write down using the Cauchy formula a dispersion relation with one subtraction at the point t = 0, which together with the unitarity condition through the Omnés-Muskelishvili integral equation leads to the phase representation of F(t) to be the starting point for our further investigations. Under the "fully solvable mathematical scheme" [1] it is understood a procedure leading to a very simple form of F(t) in the variable q = [(t − 4)/4] 1/2 (2) by means of an explicit calculation of the integral in the phase representation (1). The function F(t) on the positive real axis for t > 4 is complex with the phase δ F to be defined by the relation which, however, due to the elastic unitarity condition, phenomenologically verified to be valid approximately up to 1 GeV, is identical with the ππ scattering phase shift δ.
Since the transformation (2) is in fact a conformal mapping of the two-sheeted Riemann surface in t variable into one q-plane, elastic cut, generated by the square root branch point t = 4, disappears.
Noticing the conformal mapping (2) in more detail, the first Riemann sheet in t variable, containing only branch points corresponding to opening various thresholds and zeros of F(t), is mapped on the upper half q-plane, whereby position of the branch point t = 4 and the normalization point t = 0 are mapped into q = 0 and q = +i, respectively, and the real axis from −∞ up to t = 4, on which F(t) is a real function, is mapped on the whole positive imaginary axis of the q-plane.
The second Riemann sheet in t variable, containing branch points, again some zeros and also complex conjugate pairs of poles, which control the shape of F(t), is mapped on the lower half q-plane.
If we restrict ourselves only to the elastic region and neglect contributions to F(t) of all inelastic branch points beyond 1 GeV, then there are only zeros in the upper and lower half q-plane to be accounted as roots of a polynomial in numerator and conjugate pairs of poles according to the imaginary axis exclusively in the lower half q-plane to be accounted as roots of a polynomial in denominator of F(t).
As a result one can represent F(t) in the form of the following rational function Multiplying the numerator and denominator by the complex conjugate denominator, Eq. (4) is changed to the form The reality condition F * (t) = F(t * ) results in the reality of (5) on the positive imaginary axis of the q-plane. Then it is straightforward to see that the expression actually fulfils the latter claim and through the definition of the phase of F(t) leads to the following parametrization of the ππ scattering phase shift δ δ = arctan with all coefficients to be real. This parametrization is derived directly from the basic principles like analyticity, unitarity and reality condition.

Pion scalar-isoscalar form factor and f 0 (500) meson parameters
The pion scalar-isoscalar form factor (FF) Γ π (t) is defined by the matrix element of the scalar quark density , 0 (201 E Web of Conferences https://doi.org/10.1051/e onf /201920401020 PJ pjc 9) 204 10 Baldin ISHEPP XXIV 20 with t = (p 2 − p 1 ) 2 andm = 1 2 (m u + m d ), and fulfils all properties of the function F(t) defined in Sect. 1. Even the normalization of physically nonmeasurable pion scalar FF Γ π (t) is equal to one as it is seen from the calculated pion sigma term value with error in the framework of the χPT [2], if the pion mass m π is taken to be one.
Then the phase representation of the pion scalar-isoscalar FF is where δ 0 0 (t) is now the S-wave isoscalar ππ scattering phase shift to be exactly equal to the parametrization (7) with the parameter A 1 to be identified with the S-wave isoscalar ππ scattering length a 0 0 . The number of nonzero parameters in (7) and their numerical values are found in a fitting procedure of the most precise data on δ 0 0 (t) with theoretical errors in Fig. 1 to be generated by Garcia-Kamiński-Peláez-Yndurain (GKPY) equations [3] for the S-wave isoscalar ππ scattering amplitude. Figure 1. The data on δ 0 0 (t) with theoretical errors to be generated by GKPY equations for the Swave isoscalar ππ scattering amplitude. Solid line represents our optimal fit of data with 5 nonzero coefficients A i in (7) The data in Fig. 1 have been analyzed by the relation (7) up to the moment, when the minimum of χ 2 /ndf was achieved. The latter has been found with the first 5 nonzero coefficients A i of the values and the final form of the S-wave isoscalar ππ scattering phase shift δ 0 0 (t) takes the form , 0 (201 E Web of Conferences https://doi.org/10.1051/e onf /201920401020 PJ pjc 9) 204 10 Baldin ISHEPP XXIV 20 Substitution of (11) into (10), however, leads to the expression which does not allow one to calculate the corresponding integral explicitly.
In order to carry out it practically, it is convenient to decompose the integral (14) into a sum of two integrals according to singularities to be placed in the upper or lower half q-plane, as it is sketched in Fig. 2.
Then the explicit form of is obtained in the straightforward way, if in the case of the first integral the contour of integration is closed in the upper half q-plane (see Fig. 2). In a such way one avoids complicated calculations of the cut-contributions to be automatically generated by branch points under logarithms, if the contours are drawn as it is demonstrated in Fig. 2.
The substitution of (17) into (13) leads to the explicit form of the pion scalar-isoscalar FF where P n (t) is any polynomial normalized at t = 0 to one, however, it has not violate the asymptotic behavior of the pion scalar-isoscalar FF. The pole q = q * 3 on the second Riemann sheet in t-variable corresponds to the f 0 (500) meson resonance, now with the mass and the width, m σ = (459 ± 29) MeV and Γ σ = (517 ± 77) MeV [7], respectively, which are compatible with the parameters obtained in [4,5].

Vector pion electromagnetic form factor and ρ 0 (770) meson parameters
The vector isovector charged pion electromagnetic (EM) form factor F EM,I=1 π (t) is defined by the matrix element of the pion EM current J EM µ as follows with e to be the electric charge and t = (p 2 − p 1 ) 2 the momentum transfer squared. The F EM,I=1 π (t) also fulfils all properties of the function F(t) defined in Sect. 1, including the normalization F EM,I=1 π (0) = 1, if the electric charge is put to be one. Then F EM,I=1 π (t) can be represented by the phase representation where δ 1 1 (t) is now the P-wave isovector ππ scattering phase shift and P n (t) is polynomial normalized at t = 0 to one, however, it has not violate the asymptotic behavior of the charged pion EM FF.
In this case the model independent parametrization (7), taking into account a threshold behavior of the δ 1 1 (t) |q|→0 ∼ a 1 1 q 3 , has to be adapted to the form where the parameter A 3 is identified with the P-wave isovector ππ scattering length a 1 1 . The number of nonzero parameters in (23) and their numerical values are again found in a fitting procedure of the most precise data on δ 1 1 (t) with theoretical errors in Fig. 4 to be generated by Garcia-Kamiński-Peláez-Yndurain equations [3] for the P-wave isovector ππ scattering amplitude.
The minimum of χ 2 /ndf was achieved by the first 4 nonzero values of coefficients in (23)   Figure 4. Optimal description of the GKPY δ 1 1 (t) data. Solid line represents our optimal fit of data with 4 nonzero coefficients A i in (23) and the roots of the corresponding polynomials in the numerator and denominator of the equivalent to (23) logarithmic relation are q 1 = −2.5480 ± 0.0020 +i0.2752 ± 0.0016, q * 1 = −q 2 , q 2 = 2.5480 ± 0.0020 +i0.2752 ± 0.0016, q * 2 = −q 1 , A substitution of (25) with (26) into (22) leads to and the integral in (27) is calculated in the same way as it was done in the case of the scalarisoscalar pion FF. The pole q = q * 1 on the second Riemann sheet in t-variable corresponds to the ρ 0 (770) meson resonance, with the mass and the width, m ρ = (763.56±0.51) MeV and Γ ρ = (143.09± 0.82) MeV [7], respectively.

Conclusions
Taking into account phase representations with one subtraction of the pion scalar-isoscalar form factor and vector-isovector charged pion electromagnetic form factor and exploiting the , 0 (201 E Web of Conferences https://doi.org/10.1051/e onf /201920401020 PJ pjc 9) 204 10 Baldin ISHEPP XXIV 20 most accurate up to now information on the S-wave isoscalar and the P-wave isovector ππ scattering phase shifts, respectively, to be obtained from the existing inaccurate experimental data by means of the Garcia-Kamiński-Peláez-Yndurain Roy-like equations with an imposed crossing symmetry condition, in the framework of the so-called "fully solvable mathematical scheme" the most reliable values of the f 0 (500) and ρ 0 (770) meson mass and width are determined.
The work was supported by VEGA grant No.2/0153/17.