Shedding light on the intrinsic transversal momentum distributions of pions and kaons

We propose to introduce the intrinsic transversal momentum distribution functions (iTMDs), in conjunction with the light-cone distribution amplitudes, to elucidate the probability amplitude of encountering a meson state wherein the partons swiftly traverse along the longitudinal axis while gently oscillating in the transversal plane. The primary motivation stems from the oversight of soft transverse dynamics within the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>k</a:mi><a:mi>T</a:mi></a:msub></a:math> factorization formalism of an exclusive QCD process, which confines perturbative QCD (pQCD) predictions to scenarios involving large-momentum transfers. We meticulously investigate the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>K</e:mi></e:math> electromagnetic form factors using the iTMD-improved pQCD calculation at next-to-leading order. By analyzing data in the timelike physical regions, we obtain the transversal size parameters <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msubsup><g:mrow><g:mi>β</g:mi></g:mrow><g:mrow><g:mi>π</g:mi></g:mrow><g:mrow><g:mn>2</g:mn></g:mrow></g:msubsup><g:mo>=</g:mo><g:mn>0.51</g:mn><g:mo>±</g:mo><g:mn>0.04</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:msup><g:mrow><g:mi>GeV</g:mi></g:mrow><g:mrow><g:mo>−</g:mo><g:mn>2</g:mn></g:mrow></g:msup></g:mrow></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msubsup><i:mrow><i:mi>β</i:mi></i:mrow><i:mrow><i:mi>K</i:mi></i:mrow><i:mrow><i:mn>2</i:mn></i:mrow></i:msubsup><i:mo>=</i:mo><i:mn>0.30</i:mn><i:mo>±</i:mo><i:mn>0.05</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:msup><i:mrow><i:mi>GeV</i:mi></i:mrow><i:mrow><i:mn>2</i:mn></i:mrow></i:msup></i:mrow></i:math>. We then extract the chiral mass of pions to be <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:msubsup><k:mrow><k:mi>m</k:mi></k:mrow><k:mrow><k:mn>0</k:mn></k:mrow><k:mrow><k:mi>π</k:mi></k:mrow></k:msubsup><k:mo stretchy="false">(</k:mo><k:mn>1</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>GeV</k:mi><k:mo stretchy="false">)</k:mo><k:mo>=</k:mo><k:mn>1.84</k:mn><k:mo>±</k:mo><k:mn>0.07</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>GeV</k:mi></k:mrow></k:math> and explain the precise measurements of kaon form factors in the perturbative timelike region. As a remarkable byproduct, we found that the incorporation of iTMDs improves the pQCD predictions for electromagnetic form factors, extending the applicable range to a few <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:msup><o:mrow><o:mi>GeV</o:mi></o:mrow><o:mrow><o:mn>2</o:mn></o:mrow></o:msup></o:mrow></o:math>. This improvement allows for direct comparison with existing measurements and lattice QCD evaluations.