The $$\eta ^{(\prime )}$$ -mesons in the quark-flavor basis are mixtures of two mesonic states $$|\eta _{q}\rangle =|{\bar{u}} u+{\bar{d}} d\rangle /\sqrt{2}$$ and $$|\eta _{s}\rangle =|{\bar{s}} s\rangle .$$ In previous work, we have made a detailed study on the $$\eta _{s}$$ leading-twist distribution amplitude by using the $$D^+_s$$ meson semileptonic decays. As a sequential work, in the present paper, we fix the $$\eta _q$$ leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of $$\eta _q$$ leading-twist distribution amplitude $$\phi _{2;\eta _q}$$ at the initial scale $$\mu _0\sim 1$$ GeV are fixed by using those moments. The QCD sum rules for the $$0_{\textrm{th}}$$ -order moment can also be used to fix the magnitude of $$\eta _q$$ decay constant, giving $$f_{\eta _q}=0.141\pm 0.005$$ GeV. As an application of $$\phi _{2;\eta _q},$$ we calculate the transition form factors $$B(D)^+ \rightarrow \eta ^{(\prime )}$$ by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the leading-twist part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays $$B(D)^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell .$$
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"title": "Properties of the $$\\eta _q$$ <math> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </math> leading-twist distribution amplitude and its effects to the $$B/D^+ \\rightarrow \\eta ^{(\\prime )}\\ell ^+ \\nu _\\ell $$ <math> <mrow> <mi>B</mi> <mo>/</mo> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03b7</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> <msup> <mi>\u2113</mi> <mo>+</mo> </msup> <msub> <mi>\u03bd</mi> <mi>\u2113</mi> </msub> </mrow> </math> decays"
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"value": "The $$\\eta ^{(\\prime )}$$ <math> <msup> <mi>\u03b7</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> </math> -mesons in the quark-flavor basis are mixtures of two mesonic states $$|\\eta _{q}\\rangle =|{\\bar{u}} u+{\\bar{d}} d\\rangle /\\sqrt{2}$$ <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> <mrow> <mo>\u27e9</mo> <mo>=</mo> <mo>|</mo> <mover> <mrow> <mi>u</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>u</mi> <mo>+</mo> <mover> <mrow> <mi>d</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>d</mi> <mo>\u27e9</mo> <mo>/</mo> </mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> </math> and $$|\\eta _{s}\\rangle =|{\\bar{s}} s\\rangle .$$ <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>\u03b7</mi> <mi>s</mi> </msub> <mrow> <mo>\u27e9</mo> <mo>=</mo> <mo>|</mo> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>\u00af</mo> </mrow> </mover> <mi>s</mi> <mo>\u27e9</mo> <mo>.</mo> </mrow> </mrow> </math> In previous work, we have made a detailed study on the $$\\eta _{s}$$ <math> <msub> <mi>\u03b7</mi> <mi>s</mi> </msub> </math> leading-twist distribution amplitude by using the $$D^+_s$$ <math> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> </math> meson semileptonic decays. As a sequential work, in the present paper, we fix the $$\\eta _q$$ <math> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </math> leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of $$\\eta _q$$ <math> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </math> leading-twist distribution amplitude $$\\phi _{2;\\eta _q}$$ <math> <msub> <mi>\u03d5</mi> <mrow> <mn>2</mn> <mo>\u037e</mo> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </mrow> </msub> </math> at the initial scale $$\\mu _0\\sim 1$$ <math> <mrow> <msub> <mi>\u03bc</mi> <mn>0</mn> </msub> <mo>\u223c</mo> <mn>1</mn> </mrow> </math> GeV are fixed by using those moments. The QCD sum rules for the $$0_{\\textrm{th}}$$ <math> <msub> <mn>0</mn> <mtext>th</mtext> </msub> </math> -order moment can also be used to fix the magnitude of $$\\eta _q$$ <math> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </math> decay constant, giving $$f_{\\eta _q}=0.141\\pm 0.005$$ <math> <mrow> <msub> <mi>f</mi> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </msub> <mo>=</mo> <mn>0.141</mn> <mo>\u00b1</mo> <mn>0.005</mn> </mrow> </math> GeV. As an application of $$\\phi _{2;\\eta _q},$$ <math> <mrow> <msub> <mi>\u03d5</mi> <mrow> <mn>2</mn> <mo>\u037e</mo> <msub> <mi>\u03b7</mi> <mi>q</mi> </msub> </mrow> </msub> <mo>,</mo> </mrow> </math> we calculate the transition form factors $$B(D)^+ \\rightarrow \\eta ^{(\\prime )}$$ <math> <mrow> <mi>B</mi> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> <mo>+</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03b7</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> </mrow> </math> by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the leading-twist part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays $$B(D)^+ \\rightarrow \\eta ^{(\\prime )}\\ell ^+ \\nu _\\ell .$$ <math> <mrow> <mi>B</mi> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> <mo>+</mo> </msup> <mo>\u2192</mo> <msup> <mi>\u03b7</mi> <mrow> <mo>(</mo> <mo>\u2032</mo> <mo>)</mo> </mrow> </msup> <msup> <mi>\u2113</mi> <mo>+</mo> </msup> <msub> <mi>\u03bd</mi> <mi>\u2113</mi> </msub> <mo>.</mo> </mrow> </math>"
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