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# Scalar-isoscalar states in the large- N$_{c}$ Regge approach 

$^{3}$Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland 

Enrique Ruiz Arriola 1, ∗ and Wojciech Broniowski 2,3, † 

$^{1}$Departamento de F´ısica At´ omica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain. 

$^{2}$The H. Niewodnicza´ nski Institute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Krak´ ow, Poland 

(Dated: November 13, 2018) 

Scalar-isoscalar states ( J P C = 0 $^{++}$) are investigated within the large- N$_{c}$ Regge approach. We elaborate on the consequences of including the lightest f 0 (600) scalar-isoscalar state into such an analysis, where the position of f 0 (600) fits very well into the pattern of the radial Regge trajectory. Furthermore, we point out that the pion and nucleon spin-0 gravitational form factors, recently measured on the lattice, provide valuable information on the low-mass spectrum of the scalar- isoscalar states on the basis of the scalar-meson dominance in the spin-0 channel. Through the fits to these data we find m$_{σ}$ = 450 − 600 MeV. We compare the predictions of various fits and methods. An analysis of the QCD condensates in the two-point correlators provides further constraints on the parameters of the scalar-isoscalar sector. We find that a simple two-state model suggests a meson nature of f 0 (600), and a glueball nature of f 0 (980), which naturally explains the ratios of various coupling constants. Finally, we note that the fine-tuned condition of the vanishing dimension-2 condensate in the Regge approach with infinitely many scalar-isoscalar states yields a reasonable value for the mass of the lighest glueball state. 

PACS numbers: 12.38.Lg, 11.30, 12.38.-t Keywords: σ meson, scalar-isoscalar states, large- N$_{c}$ Regge models, meson dominance, pion and nucleon gravitational form factors, dimension-2 condensate 

The history and status of the σ -meson has been quite vacillating (for reviews see e.g. [1–3] and references therein). A scalar-isoscalar state with a mass of ∼ 500 MeV was originally proposed in the fifties [4] as an ingredient of the nucleon-nucleon force providing satura- tion and binding in nuclei. Along the years, there has always been some arbitrariness in the “effective” or “fic- titious” σ meson mass and the coupling constant to the nucleon, partly due to the lack of other sources of infor- mation. For instance, in the very successful Charge De- pendent (CD) Bonn NN-potential [5], any partial wave 2 S $^{+1}$L$_{J}$ -channel is fitted with a different scalar-isoscalar meson mass and coupling. The σ -meson was also intro- duced as the chiral partner of the pion to account for spontaneous breaking of the chiral symmetry [6]. The lack of confidence in its existence motivated taking its mass to infinity, yielding the non-linear sigma model [7], which is the modern starting point for the Chiral Pertur- bation Theory [8]. 

During the last decade, the situation has steadily changed, and the σ -meson has been finally resur- rected [9], culminating with the inclusion of the 0 ++ res- onance in the Particle Data Group review (PDG) [10] as the f$_{0}$ (600) state, seen as a ππ resonance. It has wide- spread values for the mass, 400 − 1200 MeV, and for the 

## I. INTRODUCTION 

width, 600 − 1200 MeV [11]. A rigorous definition of the σ as a ππ resonance requires that it be a pole of the ππ scattering amplitude in the ( J, T ) = (0 , 0) channel in the second Riemann sheet in the Mandelstam variable s . Within such a framework, the uncertainties have recently been narrowly sharpened with a benchmark determina- tion based on the Roy equations with constraints from the chiral symmetry [12], yielding the value 1 

The analysis of Ref. [13] yields a result with somewhat higher m$_{σ}$ , 473 ± 6 ± 11 − i 257 ± 5 ± 2 MeV (with the errors statistical and systematic, respectively), while the unitarized Chiral Perturbation Theory ( χ PT) gives a bit lower value of the mass, 401 +12 − 16 − i 277 +23 − 26 MeV [14]. Nev- ertheless, various determinations of the pole mass agree with the values (1) within the uncertainties. These ac- curate determinations make somewhat tricky the orig- inal question on what σ mass should be used a priori within a meson-exchange picture, due to the very large width of the resonance. Moreover, the determinations mentioned above do not imply necessarily the standard assignment of the linear sigma-model where one takes ( σ, ⃗ π ) as chiral partners in the ( 1 $_{2}$, 1 $_{2}$) representation of the chiral SU (2)$_{R}$ ⊗ SU (2)$_{L}$ group. A priori, the σ state 

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