Lattice QCD determination of the normalization of the leading-twist photon distribution amplitude and susceptibility of the quark condensate

The normalization of the leading-twist photon distribution amplitude (DA), <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msubsup><a:mrow><a:mi>f</a:mi></a:mrow><a:mrow><a:mi>γ</a:mi></a:mrow><a:mrow><a:mo>⊥</a:mo></a:mrow></a:msubsup></a:mrow></a:math>, is an important ingredient in the study of exclusive processes involving the photon emission by means of QCD sum-rules. In this paper we determine the up-, down- and strange-quark contribution to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msubsup><c:mi>f</c:mi><c:mi>γ</c:mi><c:mo>⊥</c:mo></c:msubsup></c:math> by exploiting its relation to the zero-momentum two-point correlation function of the electromagnetic current <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msubsup><e:mi>J</e:mi><e:mrow><e:mi>em</e:mi></e:mrow><e:mi>μ</e:mi></e:msubsup></e:math> and the electric component of the tensor current <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>T</g:mi><g:mrow><g:mi>μ</g:mi><g:mi>ν</g:mi></g:mrow></g:msup></g:math>. To that end we employ the gauge ensembles obtained by using <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>N</i:mi><i:mi>f</i:mi></i:msub><i:mo>=</i:mo><i:mn>2</i:mn><i:mo>+</i:mo><i:mn>1</i:mn><i:mo>+</i:mo><i:mn>1</i:mn></i:math> Wilson-Clover twisted-mass quark flavors, generated by the Extended Twisted Mass (ETM) Collaboration, and after adding all sources of systematic uncertainties, we obtain a total error of 1.5% and 3.5%, respectively, for the light- (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>u</k:mi></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>d</m:mi></m:math>) and strange-quark contribution to <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msubsup><o:mi>f</o:mi><o:mi>γ</o:mi><o:mo>⊥</o:mo></o:msubsup><o:mo stretchy="false">(</o:mo><o:mn>2</o:mn><o:mtext> </o:mtext><o:mtext> </o:mtext><o:mi>GeV</o:mi><o:mo stretchy="false">)</o:mo></o:math> in the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mover accent="true"><s:mi>MS</s:mi><s:mo stretchy="true">¯</s:mo></s:mover></s:math> scheme, thus improving their accuracy by a factor of 2.3 and 2.8, respectively. For the strange-quark contribution <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msubsup><w:mi>f</w:mi><w:mrow><w:mi>γ</w:mi><w:mo>,</w:mo><w:mi>s</w:mi></w:mrow><w:mo>⊥</w:mo></w:msubsup><w:mo stretchy="false">(</w:mo><w:mn>2</w:mn><w:mtext> </w:mtext><w:mtext> </w:mtext><w:mi>GeV</w:mi><w:mo stretchy="false">)</w:mo></w:math>, we observe a discrepancy with respect to previous lattice calculations. By combining our result with the world average lattice value of the chiral condensate, we obtain for the susceptibility of the quark condensate <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mrow><ab:msubsup><ab:mrow><ab:mi>χ</ab:mi></ab:mrow><ab:mrow><ab:mi>d</ab:mi></ab:mrow><ab:mrow><ab:mover accent="true"><ab:mi>MS</ab:mi><ab:mrow><ab:mo accent="true" stretchy="true">¯</ab:mo></ab:mrow></ab:mover></ab:mrow></ab:msubsup><ab:mo stretchy="false">(</ab:mo><ab:mn>2</ab:mn><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>GeV</ab:mi><ab:mo stretchy="false">)</ab:mo><ab:mo>≃</ab:mo><ab:msubsup><ab:mrow><ab:mi>χ</ab:mi></ab:mrow><ab:mrow><ab:mi>u</ab:mi></ab:mrow><ab:mrow><ab:mover accent="true"><ab:mi>MS</ab:mi><ab:mrow><ab:mo accent="true" stretchy="true">¯</ab:mo></ab:mrow></ab:mover></ab:mrow></ab:msubsup><ab:mo stretchy="false">(</ab:mo><ab:mn>2</ab:mn><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>GeV</ab:mi><ab:mo stretchy="false">)</ab:mo><ab:mo>=</ab:mo><ab:mspace linebreak="goodbreak"/><ab:mn>2.17</ab:mn><ab:mo stretchy="false">(</ab:mo><ab:mn>12</ab:mn><ab:mo stretchy="false">)</ab:mo><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:msup><ab:mi>GeV</ab:mi><ab:mrow><ab:mo>−</ab:mo><ab:mn>2</ab:mn></ab:mrow></ab:msup></ab:mrow></ab:math>. Published by the American Physical Society 2024