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Double parton distributions of the pion in the NJL model

We evaluate the valence double parton distribution (dPDF) of the pion in the Nambu--Jona-Lasinio model. At the low-energy quark-model scale and in the chiral limit a particularly simple factorized form $D(x_1,x_2, \vec{q}) = δ(1-x_1-x_2) F(\vec{q})$ follows, where $x_{1,2}$ denote the longitudinal momentum fractions of the valence quark and antiquark, and $\vec{q}$ is their relative transverse momentum. For $\vec{q}=\vec{0}$ our result complies to the Gaunt-Sterling sum rules. We carry out the necessary dDGLAP evolution to higher scales via the Mellin moments and explore its impact on the correlation defined as the ratio of dPDF to the product of single parton distributions, $D(x_1,x_2, \vec{q}=\vec{0})/D(x_1)D(x_2)$. Since the ratios of the valence Mellin moments $\langle x_1^n x_2^m \rangle / \langle x_1^n \rangle \langle x_2^m \rangle $ are invariants of the dDGLAP evolution, they may serve as robust measures of these correlations. Model predictions, which can be tested in the upcoming lattice simulations, are provided. We also discuss the transverse form factor related to the dPDF of the pion.