f Σ f$_{f}$ f Π 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 m Σ GLYPH<LBracket1> GeV GLYPH<RBracket1> decay constant GLYPH<LBracket1> GeV GLYPH<RBracket1> g$_{f}$ $_{ΠΠ}$GLYPH<Slash1> g ΣΠΠ g$_{fNN}$ GLYPH<Slash1> g Σ NN f $_{Σ}$GLYPH<Slash1> f$_{f}$ 0.4 0.6 0.8 1.0 1.2 GLYPH<Minus> 0.6 GLYPH<Minus> 0.4 GLYPH<Minus> 0.2 0.0 0.2 0.4 0.6 m Σ GLYPH<LBracket1> GeV GLYPH<RBracket1> ratio FIG. 11: (Color online) The meson constants f$_{n}$ as functions of the σ mass. FIG. 10: (Color online) Ratios of coupling constants of the σ and f 0 (980) in the two-state model. All ratios are of the order of 1 $_{/}$√$_{N}$$_{c}$$_{, indicating the interpretation of the the}$ σ as a q ¯ q meson and f 0 (980) as a glueball, cf. Table II. In addition, the dimension-2 object of Eq. (21,28) is that σ is a q ¯ q meson, while f$_{0}$ (980) is a glueball . With this assignment the results of Fig. 10 emerge naturally. Certainly, the conclusion relies on the assumption of hav- ing just two states dominating the sum rules. Yet, it is appealing in its simplicity. In the following section we will see that if infinitely many states are considered this con- clusion would not necessarily follow when a further sum rule (the vanishing of a dimension-2 condensate) is im- posed. Note that in the two-state model the dimension-2 object, C$_{2}$ , is necessarily positive, see Eq. (93). Figure 11 shows the dependence of the constants f$_{σ}$ and f$_{f}$ on m$_{σ}$ . For comparison, we also plot the pion de- cay constant f$_{π}$ = 93 MeV. We note that f$_{σ}$ is of the or- der f$_{π}$ near m$_{σ}$ ∼ 500 MeV. As discussed in Appendix B, f$_{π}$ and f$_{σ}$ need not be equal even if we postulate the scalar to be the chiral partner of the pion. Finally, we examine the last sum rule (92), concerning the energy density of the vacuum due to the gluon con- densate. According to Ref. [74], we have (for three active flavors) It is clear that the satisfaction of the second sum rule (87) with a finite number of states requires the presence of at least two scalar-isoscalar states. The presently-available lattice data do not allow to extend the analysis of Sect. IV to account for two or more states. Nevertheless, one can perform a qualitative analysis based solely on Eq. (87). When we saturate the sum rules (87) with two states, we get Let us identify σ with f$_{0}$ (600), the state f with f$_{0}$ (980), and treat m$_{σ}$ as a variable. From the decay widths Γ$_{σ}$$_{→}$$_{ππ}$ = 540(20) MeV [12] and Γ$_{f}$$_{0}$$_{→}$$_{ππ}$ = 70(30) MeV we infer (up to the sign) with the help of Eq. (39) the m$_{σ}$ -dependent values of the coupling constants g$_{σππ}$ and g$_{fππ}$ . Then, equations (89) are used to find the constants f$_{σ}$ and f$_{f}$ . We choose the convention where both of these constants are positive. Next, assuming g$_{σNN}$ = 9 . 5(5), we obtain g$_{fNN}$ from Eq. (91). Finally, from Eq. (92) we compute the vacuum energy density. The results are displayed in Figs. 10, 11, and 12. The most important qualitative feature can be in- ferred from the analysis of the ratios of coupling con- stants. Figure 10 shows the ratios of the constants f$_{σ}$/f$_{f}$ , g$_{fππ}$/g$_{σππ}$ , and g$_{fNN}$/g$_{σNN}$ . All these ratios are of the order 1 $_{/}$√$_{N}$$_{c}$ ∼ 0 . 6 for the whole PDG range of m$_{σ}$ . Ac- cording to the scaling of Table II, this supports the view In the previous section we have explored phenomeno- logical consequences of saturating the sum rules in the large- N$_{c}$ limit with a finite number of states (two states). Such an approach is justified for observables where the coupling constants to higher-mass states are sharply sup- pressed, and we are essentially left with the dominance of the low-mass states. This is, however, not always the We note that the two-state model favors the lower values of m$_{σ}$ , up to 550 MeV, where the sum rule (92) is satisfied within the values (94). VI. DIMENSION TWO CONDENSATES IN THE SCALAR SECTOR ∑ n f 2 n = C$_{2}$. (88) 1 = g$_{σππ}$f$_{σ}$ + g$_{fππ}$f$_{f}$, (89) C$_{2}$ = f σ + f $_{f}$. (93) ϵ$_{V}$ = − 1 16 ( f 2 $_{σ}$m 2 σ + f 2 $_{f}$m 2 f ) , (92) = f 2 + f 2 $_{f}$. (93) 0 = g$_{σππ}$f$_{σ}$m 2 σ + g$_{fππ}$f$_{f}$m 2 $_{f}$, (90) M$_{N}$ = g$_{σNN}$f$_{σ}$ + g$_{fNN}$f$_{f}$, (91) ϵ$_{v}$ = − 9 32 〈 α π G $^{2}$〉 = − (224 +35 − 70 MeV) $^{4}$. (94) 13