Distribution functions of a radially excited pion

Abstract A nonperturbatively-improved, symmetry-preserving approximation to the quantum field equations relevant in calculations of meson masses and interactions is used to deliver predictions for all distribution functions (DFs) of the ground state pion, $$\pi _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , and its first radial excitation, $$\pi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , viz. valence, glue, and sea. Regarding Mellin moments of the valence DFs, the $$m=0,1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> moments in both states are identical; but for each $$m\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , that in the $$\pi _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is greater than its partner in the $$\pi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Working with such information, pointwise reconstructions of the hadron-scale $$\pi _{0,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> valence DFs are developed. The predicted $$\pi _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> valence DF is consistent with extant results. The $$\pi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> valence DF is novel: it possesses three-peaks, with the central maximum partnered by secondary peaks on either side, each separated from the centre by a zero: the zeroes lie at $$x\approx 0.2,0.8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0.2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.8</mml:mn> </mml:mrow> </mml:math> and the secondary peaks at $$x\approx 0.1,0.9$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0.1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.9</mml:mn> </mml:mrow> </mml:math> . Evolution to $$\zeta =3.2\,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3.2</mml:mn> <mml:mspace/> </mml:mrow> </mml:math> GeV, a typical scale for nonperturbative calculations, is accomplished using an evolution scheme for parton DFs that is all-orders exact. At this higher scale, differences between the $$\pi _{0,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>π</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> valence DFs remain significant, but analogous differences between glue and sea DFs are far smaller. This analysis confirms that, owing to constraints imposed by chiral symmetry and the pattern by which it is broken in Nature, there are noticeable differences between the structural properties of the pion ground state and its radial excitations.