6 where In the conventional treatment insisting on the presence of the local gauge-invariant operators only, the sum over n starts with n = 2, while the admission of the dimension-2 condensate includes the n = 1 term as well. In Sect. VI we will analyze this issue further. The results of the previous subsections can be directly translated into the language of the effective Lagrangeans. In particular, we may write the PCDC (Partial Conser- vation of the Dilaton Current) equation, and hence for the on-shell particles we may use the equa- tions of motion, ( − ∂ $^{µ}$∂$_{µ}$ − m 2 $_{n}$) ϕ$_{n}$ = 0, to obtain We now match the QCD result (21) to the large- N$_{c}$ for- mula (19). Firstly, we notice that to generate the log q 2 term we need infinitely many states to contribute. Sec- ondly, approximating the series with an integral, we find via the Euler-Maclaurin formula that For the off-shell particles we need to introduce the im- proved energy momentum tensor [57], which has a better high energy behavior, but the final result is the same. The above properties may be described with a suitable extension of the effective dilatonic Lagrangeans [58–61] of the form For the Regge spectrum in Eq. (2) this specifically means that f 2 n does not asymptotically depend on n . 2 The asymptotic q $^{4}$log q behavior allows one to write down a twice-subtracted dispersion relation, In our case, the spectral density is hence we can write Here, the potential fulfills the conditions that the minimum is at 〈 Φ$_{n}$ 〉 = f$_{n}$ and the mass is V $^{′′}$( f$_{n}$ ) = m 2 $_{n}$. The physical field ϕ$_{n}$ is defined by Φ$_{n}$ = f$_{n}$ + ϕ$_{n}$ . The value at the minimum is We have imposed the condition to have a well behaved object at high energies. With this prescription the low- energy theorem [51, 55, 56], is written as where B$_{v}$ is the energy density of the vacuum. The first derivative with respect to q 2 yields which corresponds to the vacuum energy density (27). The coupling to the pion can be considered by working in the non-linear representation and with the interaction Lagrangean a dimension-2 object which cannot be expressed as an expectation value of a local operator. The second deriva- tive is equal to where U ( x ) = e i⃗ τ · π/f π is the non-linearly transforming pion field and 〈 . 〉 represents the trace in the isospin space. In the vacuum, Φ$_{n}$ = f$_{n}$ , we get a dimensionless object which corresponds to the gravi- ton wave-function renormalization and which may be di- vergent. In general, if we use the fact that f$_{n}$ becomes asymptotically n -independent and m 2 n ∼ n , as suggested by the radial Regge trajectories, the sums (27) and (28) need regularization (see Sect. VI for an application of the ζ -function) and the naively positive combination of Eq. (28) may in fact turn into a vanishing or even nega- tive contribution. which leads to the condition ∑ n g$_{nππ}$f$_{n}$ = 1 . (37) L (0) π = f 2 π 4 ∑ n g$_{nππ}$f$_{n}$ 〈 ∂ $^{µ}$U $^{†}$∂$_{µ}$ U $^{†}$〉 , (36) 1 2 Π ′′ $_{ΘΘ}$(0) = ∑ n f 2 n m 2 n , (29) L π = f 2 π 4 ∑ n g$_{nππ}$ f$_{n}$ Φ 2 n 〈 ∂ $^{µ}$U $^{†}$∂$_{µ}$ U 〉 , (35) Π ′ $_{ΘΘ}$(0) = ∑ n f 2 n ∼ ∫ d $^{4}$xx$^{2}$〈 Θ( x )Θ(0) 〉 , (28) ϵ$_{v}$ = ∑ n V ( f$_{n}$ ) = − 1 16 ∑ n m 2 $_{n}$f 2 $_{n}$, (34) Π$_{ΘΘ}$(0) = ∑ n f 2 $_{n}$m 2 n = − 4 〈 Θ 〉 = − 16 B$_{v}$, (27) Π$_{ΘΘ}$( q ) = ∑ n f 2 $_{n}$m 4 n m 2 n − q $_{2}$. (26) V (Φ$_{n}$) = m 2 n 8 f 2 n Φ 4 n [ log ( Φ$_{n}$ f$_{n}$ ) 2 − 1 2 ] (33) L ( x ) = ∑ n [ 1 2 ∂ $^{µ}$Φ$_{n}$ ∂$_{µ}$ Φ$_{n}$ − V (Φ$_{n}$) ] . (32) 1 π ImΠ( s ) = ∑ n f 2 $_{n}$m 4 $_{n}$δ ( m 2 n − s ) , (25) Π( q $^{2}$) = Π(0) + Π $^{′}$(0) q 2 + q 4 π ∫ dt t 2 ImΠ( t ) t − q 2 − iϵ . (24) f 2 $_{n}$/ ( dm 2 $_{n}$/dn ) → const . (23) Θ( x ) = − ∑ n f$_{n}$m 2 $_{n}$ϕ$_{n}$ ( x ) + O ( ϕ $^{2}$) . (31) D $^{µ}$( x ) = ∑ n f$_{n}$∂ $^{µ}$ϕ$_{n}$ ( x ) + O ( ϕ $^{2}$) , (30) C$_{0}$ = − 1 2 π $_{2}$( N 2 c − 1) ( β ( α ) α ) 2 . (22) C. Effective Lagrangeans