Scalar form factor of the pion in the Kroll-Lee-Zumino field theory

The renormalizable Kroll-Lee-Zumino field theory of pions and a neutral rho-meson is used to determine the scalar form factor of the pion in the space-like region at next-to-leading order. Perturbative calculations in this framework are parameter free, as the masses and the rho-pion-pion coupling are known from experiment. Results compare favourably with lattice QCD calculations.

The scalar form factor of the pion [1], and particularly its quadratic radius, plays an important role in chiral perturbation theory (CHPT) [2]. This form factor is defined as the pion matrix element of the QCD scalar current J S = m uū u + m dd d, i.e.
where q 2 = (p 2 − p 1 ) 2 . The associated quadratic scalar radius is given by F S (q 2 ) = F S (0) 1 + 1 6 r 2 π S q 2 + ... , where F S (0) is the pion sigma term The scalar radius fixesl 4 , one of the low energy constants of CHPT, through the relation where F π = 91.9 ± 0.1 MeV is the physical pion decay constant [3]. The low energy constantl 4 , in turn, determines the leading contribution in the chiral expansion of the pion decay constant, i.e.
where F is the pion decay constant in the chiral limit. This scalar form factor is not accessible experimentally, but it has been determined from lattice QCD (LQCD) [4]- [6], or hadronic models [7].
Theoretically, the ideal tool to study this form factor, independently from LQCD, is the Kroll-Lee-Zumino Abelian renormalizable field theory of pions and a neutral ρ-meson [8]. This provides the appropriate field theory platform for the phenomenological Vector Meson Dominance (VMD) model [9], allowing for a systematic calculation of higher order quantum corrections [10]- [11]. Due to the renormalizability of the theory, predictions are parameter free, as the strong ρππ coupling, g ρππ , is known from experiment. In spite of this coupling being a strong interaction quantity, perturbative calculations in the M S scheme make sense because the effective expansion parameter turns out to be (g ρππ /4π) 2 ≃ 0.2. The KLZ theory has been used to compute the next-toleading order (NLO) correction to the tree level (VMD) electromagnetic form factor of the pion in the space-like region with very good results [10]. In fact, it agrees with data up to q 2 ≃ −10 GeV 2 with a chi-squared per degree of freedom χ 2 F = 1.1, as opposed to VMD which gives χ 2 F = 5.0. In addition, the mean-squared radius at NLO is r 2 π = 0.46 fm 2 , compared with the experimental result [3] r 2 π = 0.45 ± 0.01 fm 2 , and the VMD value r 2 π = 0.39 fm 2 . In this note we compute in this framework the scalar form FIG. 2: Next-to-leading order (NLO) contribution to the scalar form factor.
factor of the pion at NLO in the space-like region, and compare with current results from LQCD. The KLZ Lagrangian is given by where ρ µ is a vector field describing the ρ 0 meson (∂ µ ρ µ = 0), φ is a complex pseudo-scalar field describing the π ± mesons, ρ µν is the usual field strength tensor: ρ µν = ∂ µ ρ ν − ∂ ν ρ µ , and J µ π is the π ± current: J µ π = iφ * ← → ∂ µ φ. In spite of the explicit presence of the ρ 0 mass term in the Lagrangian, the theory is renormalizable because the neutral vector meson is coupled to a conserved current [8]. Figures 1 and 2 show, respectively, the LO and the NLO diagrams, where the cross indicates the coupling of the current to the two pions. Notice that while the Lagrangian, Eq.(6), contains a ρρππ quartic coupling, this term only contributes in this application at NNLO and beyond.
Using the Feynman propagator for the ρ-meson, and in d dimensions, the unrenormalized vertex function in Fig.2 in dimensional regularization is given by where ∆(q 2 ) is defined as In the M S scheme, and renormalizing the vertex function at the point q 2 = 0, the NLO contribution in Fig. 2 is with For details on the renormalization procedure for the fields, masses and coupling see [10]. The result of a numerical evaluation of Eq.(9), using g 2 ρππ = 36.0±0.2 from the measured width of the ρ-meson [3], is shown in Fig.3. Regarding the scalar radius, defined in Eq.(2), we confirm the NLO result obtained in [11] r 2 π S = 0.4 fm 2 , with a negligible error due to the strong coupling.
This value is smaller than typical values in the literature [4]- [7]. However, it must be kept in mind that the NLO result is expected to be a lower bound, i.e. with [G(q 2 ) − G(0)] < 0 the NNLO would reduce F S (q 2 ), thus increasing the radius. A rough order of magnitude estimate of the size of the NNLO contribution suggests a correction of some 20% to the NLO term (the NNLO calculation is quite formidable and beyond the scope of this note). This is obtained by estimating a typical two-loop diagram, e.g. the ρ-meson propagator at NNLO and comparing it with the NLO result. The Feynman integrals in the variables x i at NLO and NNLO are of order O(1) in the q 2 range explored here. We find the total contribution from this diagram to be over 20% of the NLO, thus increasing the radius to r 2 π S ≃ 0.5 fm 2 .
A comparison of the KLZ form factor itself at low |q 2 | < 0.5 GeV 2 with LQCD results read from figures in [4] and [6] shows good agreement. It should be mentioned, though, that LQCD results from [4] are for light-quark masses in the range from m s /6 to m s /2, while those from [6] are for m π = 325 MeV. These LQCD determinations find values for the scalar radius higher than in this analysis, Eq.(11), i.e. r 2 π S = 0.6 ± 0.1 fm 2 from [4], and r 2 π S = 0.637 ± 0.023 fm 2 from [6]. These results for the radius are determined from e.g. chiral extrapolations to the physical pion mass. Our results for the form factor are also in agreement within less than 10% with a CHPT calculation [12] in the range −q 2 = 0 − 0.2 GeV 2 .