content_image/383121.jpg content_image/383116.jpg content_image/383114.jpg content_image/383125.jpg content_image/383122.jpg content_image/383120.jpg content_image/383117.jpg content_image/383112.jpg content_image/383124.jpg content_image/383123.jpg content_image/383119.jpg content_image/383118.jpg content_image/383115.jpg content_image/383113.jpg content_image/383111.jpg content_image/383110.jpg
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. 

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 Regge spectrum in Eq. (2) this specifically means that f 2 n does not asymptotically depend on n . The asymptotic q $^{4}$log q 2 behavior allows one to write 

down a twice-subtracted dispersion relation, 

<img src='content_image/383121.jpg'>

In our case, the spectral density is 

hence we can write 

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 

<img src='content_image/383116.jpg'>

where B$_{v}$ is the energy density of the vacuum. The first derivative with respect to q 2 yields 

<img src='content_image/383114.jpg'>

a dimension-2 object which cannot be expressed as an expectation value of a local operator. The second deriva- tive is equal to 

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. 

<img src='content_image/383125.jpg'>

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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 

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 

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 

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 

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 

which leads to the condition 

<img src='content_image/383124.jpg'>

<img src='content_image/383123.jpg'>

<img src='content_image/383119.jpg'>

<img src='content_image/383118.jpg'>

<img src='content_image/383115.jpg'>

<img src='content_image/383113.jpg'>

<img src='content_image/383111.jpg'>

## C. Effective Lagrangeans 

<img src='content_image/383110.jpg'>

