full slope a = 1 . 31 GeV is equivalent to taking every sec- ond state in the scalar-isoscalar trajectory, hence we are implicitly assuming that every second state on the tra- jectory is a glueball, alternating with q ¯ q scalar-isoscalar mesons. This alternation is suggested by the two parallel trajectories, Eq. (5). For the case of the dimension-2 condensate the ζ - function regularization [84] (see Appendix A) applied on the spectrum M 2 n = an + M 2 yields The condition for a vanishing dimension-2 condensate is thus equivalent to requesting that the mass of lowest state be given by the equality which for a = 1 . 31(12) GeV from Eq. (9) means M = 810(40) MeV. The value is not far from the mass of f$_{0}$ (980), it is also in the PDG range for m$_{σ}$ (cf. Fig. 9). 6 Thus, on the basis of a simplest Regge model considered here it is not possible to sort out which state is a glue- ball, and which one is a q ¯ q meson, if the dimension-2 condensate were indeed to vanish. The condition for the vanishing C$_{2}$ is identical to that for the vector or axial flavor channels with C$_{2}$$_{,V}$ = ∑ $_{n}$f 2 n,V and C$_{2}$$_{,A}$ = ∑ $_{n}$f 2 $_{n,A}$, respectively [82]. Note that in that case the first Weinberg sum rule requires a non-vanishing dimension-2 object f 2 π = C$_{2}$$_{,A}$ − C$_{2}$$_{,V}$ . This shows that both dimension-2 operators cannot van- ish simultaneously, despite the fact that they cannot be represented by local and gauge invariant operators. One may extend the Regge model by modifying the parameters of the lowest states on the trajectory, for in- stance the f constants. It is known from similar studies that such modifications cause strong effects [82]. Also, the 1 /N$_{c}$ effects may play a role. Nevertheless, we see that the fine-tuning needed to cancel the dimension-2 object leads to a quite natural condition for the lowest mass of the Regge family. For the dimension-4 condensate the matching of Eq. (96) to Eq. (95) leads to the large- N$_{c}$ identity (see Appendix A) The condition for the gluon condensate to be positive is therefore a > $_{(3+}$√$_{3)}$$_{M}$ 2 or a < (3 − √$_{3)}$$_{M}$ $^{2}$. Combining We have presented a wide-scope analysis of the σ -sta t e in the large- N$_{c}$ Regge framework. Our main points are the following: This quantity is plotted against the mass M in Fig. 1 3 . We note that we have two windows for the values of M , above 1.3 GeV and below 0.45 GeV, which match the predictions of the considered Regge model to the QCD estimate of Ref. [74] giving 〈 α $_{π}$G $^{2}$〉 = 0 . 009(7) GeV $^{4}$. A l- ternative determinations yield about twice larger values, 〈 α $_{π}$G $^{2}$〉 = 0 . 022(3) GeV 4 [107], or even negative values , α $_{π}$G = − 0 . 005(1) GeV 4 [108]. In our model the gluo n condensate is related to the lighest glueball mass M a c - corging to Eq. (103). Modifications of the Regge model along the lines of Ref. [82, 84] may modify or lift these constraints, making it possible for the approach to be fully consistent with the requirements of the Operator Product Expansion. QCD sum rules Regge 0.4 0.6 0.8 1.0 1.2 GLYPH<Minus> 0.005 0.000 0.005 0.010 0.015 0.020 M GLYPH<LBracket1> GeV GLYPH<RBracket1> GLYPH<Less>Α GLYPH<Slash1> Π G 2 GLYPH<Greater> GLYPH<LBracket1> GeV 4 GLYPH<RBracket1> FIG. 13: (Color online) The gluon condensate, 〈 α $_{π}$G $^{2}$〉 , in t h e one-family Regge model plotted as a function of the mass of the lowest state in the glueball trajectory, M . We u s e a = 1 . 31(12) GeV and α = 0 . 2(1). The band indicate s the uncertainty in these quantities. The horizontal dashe d line and the band indicate the estimate of Ref. [74], equal to 0 . 009(7) GeV $^{4}$. Eq. (103) and (100) yields C$_{2}$ = ∑ n f 2 n = f 2 ( 1 2 − M 2 a ) . (101) M 2 = 1 2 a, (102) 〈 α π G $^{2}$〉 = αN 2 c ( a 2 − 6 aM 2 + 6 M $^{4}$) 24 π 3 . (10 4 ) 15 121 N 2 $_{c}$α 144 π 〈 α π G $^{2}$〉 = − 1 2 ∑ n f 2 $_{n}$M 2 n = f 2 2 ( M 4 a + a 6 − M 2 ) . (103) VII. CONCLUSIONS 1. The σ state fits very well to the scalar-isoscalar Regge trajectory including all the known state s from the PDG review. From the viewpoint of the spectrum, there seems to be no particular distinc- tion among the scalar-isoscalar states. The glueb a ll or meson nature has to be sought based on the N$_{c}$ scaling of the couplings of the 0 ++ states to t h e vacuum or to hadrons. 6 As described in [32], the shift from the pole resonance to the $^{Breit-Wigner resonance is}$O( N − 2 c ) suppressed, suggesting that the leading N$_{c}$ values lies in between.