Distribution amplitudes of pseudoscalar mesons within the light-front quark model

In this paper, we investigate the distribution amplitudes (DAs) of pseudoscalar mesons within the light-front quark model, and the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>n</a:mi> </a:math> th Gegenbauer moment <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>a</c:mi> <c:mi>n</c:mi> </c:msub> <c:mo stretchy="false">(</c:mo> <c:mi>μ</c:mi> <c:mo stretchy="false">)</c:mo> </c:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>ξ</g:mi> </g:math> moment <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mo stretchy="false">⟨</i:mo> <i:msup> <i:mi>ξ</i:mi> <i:mi>n</i:mi> </i:msup> <i:mo stretchy="false">⟩</i:mo> </i:math> , and transverse moment <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mo stretchy="false">⟨</m:mo> <m:msubsup> <m:mi mathvariant="bold">k</m:mi> <m:mo>⊥</m:mo> <m:mi>n</m:mi> </m:msubsup> <m:mo stretchy="false">⟩</m:mo> </m:math> are also studied. Two parameter sets are adopted by fitting to mesonic decay constants or mass spectra, leading to distinct schemes for numerical analysis. Our results reveal several key insights: (1) Under scheme I, the second Gegenbauer moment for the pion, <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>a</r:mi> <r:mrow> <r:mn>2</r:mn> <r:mo>,</r:mo> <r:mi>π</r:mi> </r:mrow> </r:msub> <r:mo>=</r:mo> <r:mn>0.054</r:mn> </r:math> , is in excellent agreement with predictions from the platykurtic model and the nonlocal chiral quark model. Meanwhile, scheme II yields <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi>a</t:mi> <t:mrow> <t:mn>2</t:mn> <t:mo>,</t:mo> <t:mi>π</t:mi> </t:mrow> </t:msub> <t:mo>=</t:mo> <t:mn>0.126</t:mn> </t:math> , which is consistent with lattice QCD, Dyson-Schwinger equation, and data-driven light-cone sum rules analyses. Furthermore, we observe that <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>ϕ</v:mi> <v:mrow> <v:mn>2</v:mn> <v:mo>,</v:mo> <v:mi>π</v:mi> </v:mrow> </v:msub> <v:mo>=</v:mo> <v:msubsup> <v:mi>ϕ</v:mi> <v:mrow> <v:mn>3</v:mn> <v:mo>,</v:mo> <v:mi>π</v:mi> </v:mrow> <v:mi>P</v:mi> </v:msubsup> </v:math> with the replacement <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>M</x:mi> <x:mo stretchy="false">→</x:mo> <x:msub> <x:mi>M</x:mi> <x:mn>0</x:mn> </x:msub> </x:math> . (2) Flavor symmetry breaking effects are found to be more significant in higher-twist DAs, with a notable relation <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msubsup> <ab:mi>a</ab:mi> <ab:mn>1</ab:mn> <ab:mi>P</ab:mi> </ab:msubsup> <ab:mo>≈</ab:mo> <ab:mn>2</ab:mn> <ab:msub> <ab:mi>a</ab:mi> <ab:mn>1</ab:mn> </ab:msub> </ab:math> emerging in heavy meson systems. (3) Interestingly, numerical analysis indicates that the transverse moment <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mo stretchy="false">⟨</cb:mo> <cb:msubsup> <cb:mi mathvariant="bold">k</cb:mi> <cb:mo>⊥</cb:mo> <cb:mi>n</cb:mi> </cb:msubsup> <cb:mo stretchy="false">⟩</cb:mo> </cb:math> is twist independent with the replacement <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>M</hb:mi> <hb:mo stretchy="false">→</hb:mo> <hb:msub> <hb:mi>M</hb:mi> <hb:mn>0</hb:mn> </hb:msub> </hb:math> . Additionally, we propose an empirical scaling relation <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mfrac> <kb:mrow> <kb:mi mathvariant="normal">Δ</kb:mi> <kb:msup> <kb:mi>m</kb:mi> <kb:mn>0.61</kb:mn> </kb:msup> </kb:mrow> <kb:msup> <kb:mi>β</kb:mi> <kb:mn>1.10</kb:mn> </kb:msup> </kb:mfrac> <kb:msub> <kb:mi>x</kb:mi> <kb:mi>p</kb:mi> </kb:msub> <kb:mo>=</kb:mo> <kb:mn>0.67</kb:mn> </kb:math> to effectively describe the shape of DAs for heavy mesons. Collectively, these results demonstrate that mass asymmetry remarkably influences DAs and kinds of moments; these theoretical predictions will be tested in future experiments.