QCD sum rules 0.4 0.6 0.8 1.0 1.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 m Σ GLYPH<LBracket1> GeV GLYPH<RBracket1> GLYPH<LParen1> GLYPH<Minus>Ε v GLYPH<RParen1> 1 GLYPH<Slash1> 4 GLYPH<LBracket1> GeV GLYPH<RBracket1> FIG. 12: (Color online) The fourth root of the minus energy density of the vacuum, ( − ϵ$_{v}$ ) 1 / 4 in the two-state model as a function of m$_{σ}$ . The horizontal line and the band indicate the estimate with uncertainties of Eq. (94) [74]. Very recently, the presence of the dimension-2 objects has been linked to the truncation of the full perturbative series [105], indicating duality between the resummed perturbation theory and the presence of the lowest power corrections. This is supported by the correlation between perturbative and non-perturbative contributions found in Ref. [106]. As we have already discussed in Sect. III B, in the Θ-Θ correlator the dimension-2 object, C$_{2}$ = ∑ $_{n}$f 2 $_{n}$, appears naturally. Clearly, if there is a finite num- ber of states, the sum is necessarily positive and hence non-vanishing. This is in an open contradiction to the Operator-Product-Expansion philosophy, where only dimension-4 and higher local gauge-invariant operators are allowed. Therefore, the only scenario where the dimension-2 object can possibly vanish is by assuming an infinite number of resonances, such that the sum diverges requiring regularization. In this case the asymptotic con- dition, Eq. (23), holds. As usual in the Regge models, we assume f$_{n}$ to be independent on n for all values of n , not only asymptotically, as requested by Eq. (2). The basic object of our study is the Regge sum (in this section M denotes the lowest glueball mass) case. For instance, as discussed in Sect. III B, proper matching of the two-point functions to QCD necessar- ily requires infinitely many intermediate states present in the correlator. Only that way the leading logarithmic behavior (22) can be reproduced. Similar studies have been carried out in the past for other correlators in the Regge approach [75–85]. Here we focus on the Π$_{ΘΘ}$ cor- relator defined in Eq. (16). In QCD, keeping the leading terms in α , this object has the explicit twist expansion of the form [50–54] which can be evaluated using the formula (see Ap- pendix A) (note a factor of 16 difference between our definition (16) and Ref. [54], carrying over to the definition of C$_{2}$ ). The dimension-2 gluon condensate, originally pro- posed by Celenza and Shakin [86], appears as an elu- sive gauge-invariant non-perturbative and non-local op- erator which generates the lowest 1 /Q 2 power correc- tions in the twist expansion. Its dynamical origin re- mains unclear [87, 88], despite certain evidence provided by the instanton model [89], phenomenological QCD sum rules reanalyses [90], phenomenological studies of the τ decay data [91], further theoretical considerations [92– 95], quark-model calculations [96, 97], or lattice simu- lations [98, 99] and their relevance for the confinement- deconfinement phase transition [100]. Phenomenologi- cally, upper bounds for the dimension-2 operators coming from the e $^{+}$e − data were analyzed in [101–103], yielding 16 C$_{2}$ ≤ 0 . 1 GeV 2 ∼ (0 . 3 GeV) $^{2}$. The possible appear- ance of the dimension-2 condensates in Regge-like models can be traced in Ref. [104]; it was also briefly discussed in Ref. [81]. A more quantitative analysis was carried out by us in Refs. [82, 84]. which for α = 0 . 2, a = 1 . 31GeV $^{2}$, and N$_{c}$ = 3 gives f ∼ 135 MeV, a reasonable value. Note that taking the We read off that f ∼ N$_{c}$ , in accordance with the fact that the glueballs provide the saturation at the leading- N$_{c}$ level. Explicitly, We note that the infinite Regge sum generates properly the Q $^{4}$log Q 2 term. We assume that we have one family of states contributing to the Regge sum at the leading- N$_{c}$ order. The matching with the leading term of the OPE expansion gives immediately is the polygamma function. where Π$_{ΘΘ}$ = ∑ n Q $^{4}$f2 n m 2 n + Q 2 = ∞ ∑ n =0 Q $^{4}$f2 n M 2 + an + Q $_{2}$, (96) ∞ ∑ n =0 [ 1 M 2 + an + Q 2 − 1 M 2 + an ] = 1 a [ Ψ ( M 2 a ) − Ψ ( M 2 + Q 2 a )] , (97) Ψ( z ) = Γ $^{′}$( z ) / Γ( z ) = log z − 1 2 z − 1 12 z 2 + 1 120 z 4 + O ( z − $^{6}$) , (98) Π$_{ΘΘ}$ = − α $^{2}$( N 2 c − 1) β 2 0 32 π 4 Q $^{4}$log Q 2 µ 2 + C$_{2}$Q 2 + αβ 2 0 16 π 〈 α π G $^{2}$〉 + . . . (95) C$_{0}$ = − f 2 a = − α $^{2}$( N 2 c − 1) β 2 0 32 π 4 . (99) f = $_{11}$√$_{aαN}$ 2 c $_{12}$√$_{2}$$_{π}$ 2 , (100) 14