content_image/377964.jpg content_image/377965.jpg content_image/377966.jpg content_image/377963.jpg content_image/377957.jpg content_image/377959.jpg content_image/377961.jpg content_image/377958.jpg content_image/377960.jpg content_image/377962.jpg
2. The mass of the lowest 0 ++ state (the σ ) can also be extracted from the analysis of gravitational form factors of the pion and nucleon, available from the lattice calculations. Extracted masses are com- patible with the benchmark determinations within the expected accuracy of the large- N$_{c}$ framework, showing the consistency of the approach and the proximity of the large- N$_{c}$ limit to the real world in the scalar-isoscalar sector. 

3. The constraints of chiral symmetry and the high- energy behavior of the energy-momentum correla- tor imposes a set of sum rules involving various cou- pling constants. Saturation of these sum rules with just two states suggest that f$_{0}$ (600) is a meson and f$_{0}$ (980) is a glueball. Such an assignment, specific to the simple two-state model, results in natural N$_{c}$ -scaling of the ratios of the coupling constants of these states. 

4. Correlation functions of the energy-momentum tensor are saturated by (infinitely many) scalar- isoscalar states. The analysis of the Operator Product Expansion shows that there appears the dimension-2 condensate, whose vanishing requires a fine tuning of parameters. Then, in the simplest Regge model the resulting mass of the lowest glue- ball state is ∼ 800 MeV. 

We thank J. Prades for clarifications and S. Nari- son for remarks. This work is partially supported by the Polish Ministry of Science and Higher Education (grants N202 034 32/0918 and N202 249235), Span- ish DGI and FEDER funds (grant FIS2008-01143/FIS), Junta de Andaluc´ıa (grant FQM225-05), the EU Inte- grated Infrastructure Initiative Hadron Physics Project (contract RII3-CT-2004-506078). 

## Appendix A: The ζ -regularization 

In the large N$_{c}$ limit only tree-level diagrams survive, however, with infinitely many resonances. Thus, the infi- nite sums may need to be regularized. In the dimensional regularization the coupling of the resonance to the cur- rent acquires an additional dimension f$_{n}$ → f$_{n}$µ ϵ with ϵ = d − 4. By choosing as a natural scale µ = m$_{n}$ , one gets f 2 n → f 2 $_{n}$m 2 ϵ n , which means 2 2 k 2 2 s 

This is completely equivalent to formally expanding the two-point correlator at large Q 2 and reinterpreting the coefficients within the zeta-function regularization [84]. In our case the mass formula is m 2 n = an + m 2 $_{0}$, hence one has to consider the sums of the form 

<img src='content_image/377964.jpg'>

which is the generalized Riemann zeta function. Then, we arrive at the explicit formulas 

<img src='content_image/377965.jpg'>

<img src='content_image/377966.jpg'>

<img src='content_image/377963.jpg'>

## Appendix B: The linear Lagrangean for one scalar 

For only one scalar-isoscalar state, Lagrangean (35) with condition (37) reads 

<img src='content_image/377957.jpg'>

Introducing the Cartesian decomposition 

one gets an extension of the linear sigma model with the so-called A -term [109], allowed in the linear realization of the chiral symmetry: 

<img src='content_image/377959.jpg'>

where in the present case 

Note that σ and ⃗ π are chiral partners. 

Imposing the condition f$_{σ}$ = f$_{π}$ removes the A term. Note, however, that the identity f$_{π}$ = f$_{σ}$ is not a conse- quence of any symmetry or physical requirement. With this condition the Lagrangean reads 

<img src='content_image/377961.jpg'>

Note the resemblance to the bosonic part of the dilated chiral quark model of Ref. [110]. The main difference between potential (B5) and the standard Mexican hat potential, 

is the scale behaviour of the latter, a different vacuum energy density (factor of two difference), and modified couplings containing more than one scalar. The σππ ver- tex is the same on shell , thus yields the same decay width of the σ into two pions. 

<img src='content_image/377958.jpg'>

<img src='content_image/377960.jpg'>

<img src='content_image/377962.jpg'>

