Generalized parton distributions of the pion^{1}^{1}1Talk presented by WB at SCADRON 70 Workshop on “Scalar Mesons and Related Topics”, Lisbon, 11-16 February 2008
Abstract
Generalized Parton Distributions of the pion are evaluated in chiral quark models with the help of double distributions. As a result the polynomiality conditions are automatically satisfied. In addition, positivity constraints, proper normalization and support, sum rules, and soft pion theorems are fulfilled. We obtain explicit expressions holding at the low-energy quark-model scale, which exhibit no factorization in the -dependence. The crucial QCD evolution of the quark-model distributions is carried out up to experimental or lattice scales. The obtained results for the Parton Distribution Function and the Parton Distribution Amplitude describe the available experimental and lattice data, confirming that the quark-model scale is low, around 320 MeV.
Keywords:
generalized parton distributions, double distributions, light-cone QCD, exclusive processes, chiral quark models:
12.38.Lg, 11.30, 12.38.-taddress=The H. Niewodniczański Institute of Nuclear Physics PAN, PL-31342 Kraków, Poland, and Institute of Physics, Rzeszòw University, PL-35959 Rzeszów, Poland
Generalized Parton Distributions (GPD’s) carry “tomographic” information on the partonic structure of hadrons (for reviews see e.g. Ji (1998); Radyushkin (2000); Goeke et al. (2001); Diehl (2003); Ji (2004); Belitsky and Radyushkin (2005); Feldmann (2007); Boffi and Pasquini (2007)). In this talk we present our recent calculation of the GPD’s of the pion in the framework of chiral quark models Broniowski et al. (2008), which extends the previous calculations of PDF’s Davidson and Ruiz Arriola (1995); Ruiz Arriola (2001); Davidson and Ruiz Arriola (2002), PDA’s Ruiz Arriola and Broniowski (2002); Ruiz Arriola (2002), and GPD in the impact parameter space Broniowski and Ruiz Arriola (2003). Recently, the Transition Distribution Amplitudes (TDA) Pire and Szymanowski (2005a, b) have also been evaluated in the same framework Broniowski and Arriola (2007). Other quark-model calculations of GPD’s and related quantities have been reported in Refs. Polyakov and Weiss (1999a, b); Anikin et al. (2000a, b); Tiburzi and Miller (2003a, b); Theussl et al. (2004); Tiburzi and Miller (2003a); Praszalowicz and Rostworowski (2003); Bzdak and Praszalowicz (2003); Noguera and Vento (2006); Courtoy and Noguera (2007); Kotko and Praszalowicz (2008).
Chiral quark models yield parton distributions at a given low energy scale . The result for a quantity is matched to QCD order by order in the twist expansion, , hence . Then the functions are evolved to higher scales . It turns out that in order to describe the available pion phenomenology the initial scale in the considered quark models must be very low Davidson and Ruiz Arriola (1995, 2002); Ruiz Arriola and Broniowski (2002); Ruiz Arriola (2002); matching the momentum fraction carried by the valence quark at to Sutton et al. (1992); Capitani et al. (2006) yields
(1) |
with MeV and three flavors. At such a low scale , which makes the evolution very fast for the scales close to the initial value.
The kinematics of the process and the assignment of momenta (in the asymmetric notation) is displayed in Fig. 1, representing the large- quark-model evaluation of GPD’s. We adopt the standard notation , , . The leading-twist GPD of the pion is defined as
(2) |
where and are isospin indices for the pion, is the isospin matrix equal for the isoscalar and for the isovector case, is the null vector, and is the light-cone coordinate. In the symmetric notation one introduces and . The following sum rules hold on general grounds:
(3) |
where is the electromagnetic form factor, while and are the gravitational form factors of the pion. Finally, for the equality relates the GPD’s to the the pion’s parton distribution function (PDF). The polynomiality conditions Ji (1998); Radyushkin (2000) and the positivity bound Pobylitsa (2002) are satisfied in our approach.
We work for simplicity in the chiral limit, . Two quark models are considered: the Spectral Quark Model (SQM) Ruiz Arriola and Broniowski (2003) and the NJL model. SQM implements the vector-meson dominance, predicting the form factors
(4) |
The explicit results for the full GPD’s have been provided in Ref. Broniowski et al. (2008). Importantly, their form does not exhibit a factorized -dependence. A sample result for and several values of is shown in Fig. 2. For the NJL model the results are qualitatively the same. For the case of the GPD’s simplify to the well-know Polyakov and Weiss (1999b); Theussl et al. (2004) step-function results
(5) |
Another simple case is in SQM for and any value of Broniowski and Ruiz Arriola (2003)
(6) |
For the QCD evolution we use the leading-order ERBL-DGLAP equations with three flavors. In the left panel of Fig. 3 we confront the result for at the scale GeV with the data at this scale from the E615 Drell-Yan experiment Conway et al. (1989). We note agreement between the model and the data.
In the right panel of Fig 3 we compare our results to the data from lattices Dalley and van de Sande (2003). We take the liberty of moving the scale, as its determination on the lattice is not very precise. As we see, the agreement is qualitatively good if one considers the uncertainties of the data, especially when the lower scale is used.
PDA’s have been intensely studied in the past in several contexts (see Ref. Bakulev et al. (2007) for a brief review). At the quark model scale the PDA of the pion Ruiz Arriola and Broniowski (2002), which can be related to the isovector GPD through the soft pion theorem Polyakov (1999) is Ruiz Arriola and Broniowski (2002). The evolved PDA is shown in Fig. 4, where it is compared to the E791 di-jet measurement Aitala et al. (2001) and to lattice calculations Dalley and van de Sande (2003). Again, good agreement is observed.
For the case of general kinematics, the explicit form of the LO QCD evolution equations for the GPD’s can be found in Mueller et al. (1994); Ji (1997); Radyushkin (1997); Blumlein et al. (1997); Golec-Biernat and Martin (1999); Kivel and Mankiewicz (1999a, b). In this paper we solve them with the numerical method developed in Golec-Biernat and Martin (1999), based on the Chebyshev polynomial expansion.
The results of the LO evolution from the SQM initial condition for at the scale to subsequent values of are shown in Figs. 5. The evolution is fastest at low values of , where the coupling constant is large, and it immediately pulls down the end-point values to zero. Then, the strength gradually drifts from the DGLAP regions to the ERBL region. The approach towards the asymptotic form is very slow, with the tails in the DGLAP region present. The highest displayed in the figure is GeV and the asymptotic form is reached at “cosmologically” large values of , which are never achieved experimentally. The results for the NJL model are very similar to the case of SQM.
In conclusion, we remark that our calculation provides a link between the non-perturbative soft-energy physics in terms of matrix element of operators and the high-energy processes as deduced from perturbative QCD evolution. The overall agreement with the pionic data from experiments and lattices, available for the PDF and PDA, is very reasonable, supporting the presented methodology.
Supported by Polish Ministry of Science and Higher Education grant N202 034 32/0918, Spanish DGI and FEDER funds with grant FIS2005-00810, Junta de Andalucía grant FQM225-05, and EU Integrated Infrastructure Initiative Hadron Physics Project contract RII3-CT-2004-506078.
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