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fine-tuning condition on the σ mass is needed. Finally, in Section VII we come to our conclusions. 

In this section the mass of the σ -meson is determined by extrapolating the upper part of the J P C = 0 ++ spec- trum in the large- N$_{c}$ Regge approach. In Ref. [37] the analysis of the radial and rotational Regge trajectories was carried out. In Ref. [38] the scalar sector was stud- ied in a greater detail, including the modeling of the me- son widths. Two parallel radial trajectories could then be identified, including three states per trajectory, which has resulted in rather poor statistics. In Fig. 1 we show the squared mass of all nine J P C = 0 ++ isoscalar states listed in the PDG tables [36], given for completeness in Table I. Note that the four heaviest 0 ++ states are omit- ted from the PDG summary table and in principle need confirmation. As we can see, an approximate straight line behavior looks quite appealing. To improve the statistics, we propose to fit with half the standard slope, thus in- cluding all the PDG scalar-isoscalar 0 ++ states together in the mass formula 

where a = 2 πσ , and σ is the string tension. The same strategy was recently applied in Ref. [39]. 

In the first method, called Fit I, we minimize the χ 2 

## II. SCALAR-ISOSCALAR REGGE TRAJECTORIES 

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variable 

We get, after excluding the lowest n = 0 state and using the 8 remaining PDG states, the central parameter values a = 1 . 35 GeV 2 and m$_{σ}$ = 530 MeV, with χ $^{2}$/ DOF = χ $^{2}$/ (8 − 2) = 9 . 9. A non-trivial issue, not addressed in the analyses of 

Refs. [37, 38], is the determination of errors of the pa- rameters in the Regge fits. Actually, the large value of χ $^{2}$/ DOF ∼ 10 ≫ 1 found above prevents a reliable er- ror determination of the model parameters a and m$_{σ}$ and calls for corrections from different sources. A com- mon way to overcome this difficulty is to rescale the weights in the χ 2 variable. In our case we simply re- place Δ M$_{f,n}$ → 3Δ M$_{f,n}$ , such that χ $^{2}$/ DOF ∼ 1. In this case we get 

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In the usual approaches (see, e.g., Refs. [38, 40]) a dif- ference between the bare and physical pole masses, based on the K -matrix or the Breit-Wigner formula, is made. This generally introduces some model dependence and thus there may be systematic uncertainties in the masses related to the precise definition of the resonance param- eters. 

As we can see from Eq. (4), the fit is quite accept- able and implies the value of the string tension √$_{σ}$ = 463(9) MeV. We note that the phenomenological value is √$_{σ}$ = 420 MeV from a global fit to the excited meson spectrum [37]. 

Formula (2) is equivalent to two parallel radial Regge trajectories with the standard slope 

From here we get the mass-splitting formula 

and, in particular, 

which works extremely well for the value m$_{f}$$_{0}$ = 980 MeV and m$_{σ}$ = 530 MeV, yielding 1 . 36 GeV 2 for the l.h.s. vs. 1 . 35 GeV 2 for the r.h.s. 

Given the fact that all states fit quite naturally into the pattern, we see no obvious way how glueball states could be singled out solely on the basis of belonging to one of the trajectories (5). Also note that a glueball corresponds to a bound state in gluodynamics, or equiv- alently QCD for infinitely heavy quarks (zero active fla- vors), while the states obtained here correspond to the limit of light quarks (two or three active flavors). 

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