<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>π</mml:mi><mml:mi>

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>I</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi><c:mi>π</c:mi></c:math> elastic <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>s</e:mi></e:math>-wave scattering phase shift is measured by lattice quantum chromodynamics with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>N</g:mi><g:mi>f</g:mi></g:msub><g:mo>=</g:mo><g:mn>3</g:mn></g:math> flavors of the Asqtad-improved staggered fermions. The lattice-calculated energy-eigenvalues of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>π</i:mi><i:mi>π</i:mi></i:math> systems at one center of mass frame and some moving frames using the moving wall source technique are utilized to secure phase shifts by Lüscher’s formula. Our computations are fine enough to obtain threshold parameters: scattering length <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>a</k:mi></k:math>, effective range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>r</m:mi></m:math>, and shape parameter <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>P</o:mi></o:math>, which can be extrapolated at the physical point by next-to-leading order in chiral perturbation theory, and our relevant next-to-next-to-leading order predictions from expanding nuclear physics with lattice quantum chromo dynamics collaboration’s works are novelly considered as the systematic uncertainties. Our outcomes are consistent with Roy equation determinations, newer experimental data, and lattice estimations. Numerical computations are performed with a coarse (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>a</q:mi><q:mo>≈</q:mo><q:mn>0.12</q:mn><q:mtext> </q:mtext><q:mtext> </q:mtext><q:mi>fm</q:mi></q:math>, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>L</s:mi><s:mn>3</s:mn></s:msup><s:mi>T</s:mi><s:mo>=</s:mo><s:mn>3</s:mn><s:msup><s:mn>2</s:mn><s:mn>3</s:mn></s:msup><s:mn>64</s:mn></s:math>), two fine (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>a</u:mi><u:mo>≈</u:mo><u:mn>0.09</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>fm</u:mi></u:math>, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msup><w:mi>L</w:mi><w:mn>3</w:mn></w:msup><w:mi>T</w:mi><w:mo>=</w:mo><w:mn>4</w:mn><w:msup><w:mn>0</w:mn><w:mn>3</w:mn></w:msup><w:mn>96</w:mn></w:math>) and a superfine (<y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>a</y:mi><y:mo>≈</y:mo><y:mn>0.06</y:mn><y:mtext> </y:mtext><y:mtext> </y:mtext><y:mi>fm</y:mi></y:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msup><ab:mi>L</ab:mi><ab:mn>3</ab:mn></ab:msup><ab:mi>T</ab:mi><ab:mo>=</ab:mo><ab:mn>4</ab:mn><ab:msup><ab:mn>8</ab:mn><ab:mn>3</ab:mn></ab:msup><ab:mn>144</ab:mn></ab:math>) lattice ensembles at four pion masses of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>m</cb:mi><cb:mi>π</cb:mi></cb:msub><cb:mo>∼</cb:mo><cb:mn>247</cb:mn><cb:mtext> </cb:mtext><cb:mtext> </cb:mtext><cb:mi>MeV</cb:mi></cb:math>, 249 MeV, 275 MeV, and 384 MeV, respectively. Published by the American Physical Society 2024