Pion structure function in the Nambu and Jona-Lasinio model

Abstract

: The pion structure function is studied in the Nambu and Jona-Lasinio (NJL) model. We calculate the forward scattering amplitude of a virtual photon from a pion target in the Bjorken limit, and obtain valence quark distributions of the pion at the low energy hadronic scale, where the NJL model is supposed to work. The calculated distribution functions are evolved to the experimental momentum scale using the Altarelli-Parisi equation. The resulting distributions are in a reasonable agreement with experiment. We calculate also the kaon structure function and compare the ratio of kaon to pion valence u-quark distributions with experiment. * e-mail address : shige@atlas.phys.metro-u.ac.jp † Also at RIKEN, Wako, Saitama, 351, Japan Recently, the nucleon structure functions are discussed in terms of the MIT bag mode [4,5] as well as other models [6]. Those results are based on the assumption that the structure functions at low energy hadronic scale Q 2 = Q 2 0 (unknown) are obtained by calculating the twist 2 matrix elements within the effective models, since the twist 2 operators are dominant in the Bjorken limit and higher twist terms vanish at high enough. Once one obtains them at the hadronic scale, these parton distributions are evolved to the experimental scale by using the perturbative QCD, and the comparison with experiment can be made.In this letter, we concentrate on the pion structure function. The pion is believed to be the Goldstone boson due to the spontaneous breakdown of the chiral symmetry (SBχS).Thus, the information of the pion structure function gives us deeper understandings of SBχS. The experimental data of the pion structure function are extracted from πN scattering using the Drell-Yan process [7]. The valence, sea quark, and gluon distribution functions are obtained from a recent re-analysis of these data [8].We use the Nambu and Jona-Lasinio (NJL) model [9], where the chiral invariance is 2 the main ingredient. The NJL model describes the pion as a collective qq excitation of the non-perturbative QCD vacuum [10]. Recently, the generalized SU(3) f NJL model with the U(1) A anomalous term is demonstrated to reproduce the meson properties successfully, in spite of the lack of confinement [10,11,12]. This model is also applied to the chiral phase transition at finite temperature and density [13]. All these results indicate that the NJL model possesses the essential features of QCD dynamics.The SU(3) f NJL lagrangian is given by [10,11,12]where ψ denotes the quark field with current mass m, λ i are SU(3) flavor matrices, and [10,11,12]. Using this lagrangian, we will calculate the parton distribution of pion with no free parameter.Unfortunately, the NJL model is a non-renormalizable theory, and this model requires the finite momentum cutoff Λ ∼ 1 GeV, which is identified with a scale of SBχS. One may understand the physical meaning of the cutoff as an approximate realization of "asymptotic freedom" in the NJL model. This means that the interaction between two quarks with the relative mom...

Light-Cone Quantization: Foundations and Applications

“…If one compares with other NJL calculations of the structure function (Shigetani et al 1993;Bentz et al 1999) and with the empirical parton distributions in the literature (Glück et al 1999), one finds reasonable qualitative agreement.…”

Nucleon and pion distribution functions in the valence region

“…and solid-curve from (Shigetani et al, 1993), an NJL-model result evolved from to Q 0 = 0.5 GeV we observe that the entire data set is fitted by (see Fig. VI.42)…”

“…On the other hand, NJL models yield a = 1 with a hard cutoff (Shigetani et al, 1993), as do duality arguments (Melnitchouk, 2003). Relativistic constituent quark models (Frederico and Miller, 1994;Szczepaniak et al, 1994) give (1 − x) a with 0 < a < 2 depending on the form of model wave function; and instanton-based models produce (1 − x) a with 1 < a < 2 (Dorokhov and Tomio, 2000).…”

“…Then, the u v quark distribution in the kaon should be weighted to lower values in x than that in the pion. A Nambu-Jona-Lasinio model calculation (Shigetani et al, 1993) Experimentally, a Drell-Yan measurement (Badier et al, 1980) of the ratio of K − to π − is consistent with unity over most of the x region, with a suggestion that the ratio is dropping at high x. However, the data are not of sufficient quality to test and verify our understanding of pion and kaon structure.…”

“…In this case both the interaction and the regularization procedure are momentum-independent, and hence neither can act that is regularized by using a hard cutoff (Shigetani et al, 1993). Under one-loop evolution, a resolving scale of Q 0 = 0.5 GeV optimizes agreement with data parametrizations.…”