Revisiting Roy-Steiner-equation analysis of pion-kaon scattering from lattice QCD data

A comprehensive analysis of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>π</a:mi><a:mi>K</a:mi><a:mo stretchy="false">→</a:mo><a:mi>π</a:mi><a:mi>K</a:mi></a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>π</d:mi><d:mi>π</d:mi><d:mo stretchy="false">→</d:mo><d:mi>K</d:mi><d:mover accent="true"><d:mi>K</d:mi><d:mo stretchy="false">¯</d:mo></d:mover></d:math> amplitudes at large unphysical pion mass for all important partial waves is presented. A set of crossing-symmetric partial-wave hyperbolic dispersion relations is used to describe lattice quantum chromodynamics (QCD) data at <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>m</i:mi><i:mi>π</i:mi></i:msub><i:mo>=</i:mo><i:mn>391</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>MeV</i:mi></i:math>. In the present analysis, the amplitudes for the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi></k:math>- and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>P</m:mi></m:math>-waves are formulated by combining the constraints of analyticity, unitarity, and crossing symmetry, fulfilling Roy-Steiner-type equations. We use these results to investigate the low-lying strange-meson resonances and resolve the instability problem tied to analytic continuation in prior lattice QCD studies based on the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>K</o:mi></o:math>-matrix formalism. At <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>m</q:mi><q:mi>π</q:mi></q:msub><q:mo>=</q:mo><q:mn>391</q:mn><q:mtext> </q:mtext><q:mtext> </q:mtext><q:mi>MeV</q:mi></q:math>, the rigorous Roy-Steiner-type equation approach allows us to determine the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>S</s:mi></s:math>-wave scattering lengths, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>m</u:mi><u:mi>π</u:mi></u:msub><u:msubsup><u:mi>a</u:mi><u:mn>0</u:mn><u:mrow><u:mn>1</u:mn><u:mo>/</u:mo><u:mn>2</u:mn></u:mrow></u:msubsup><u:mo>=</u:mo><u:mrow><u:mo stretchy="false">(</u:mo><u:mn>0.9</u:mn><u:msubsup><u:mn>2</u:mn><u:mrow><u:mo>−</u:mo><u:mn>0.28</u:mn></u:mrow><u:mrow><u:mo>+</u:mo><u:mn>0.06</u:mn></u:mrow></u:msubsup><u:mo stretchy="false">)</u:mo></u:mrow></u:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>m</y:mi><y:mi>π</y:mi></y:msub><y:msubsup><y:mi>a</y:mi><y:mn>0</y:mn><y:mrow><y:mn>3</y:mn><y:mo>/</y:mo><y:mn>2</y:mn></y:mrow></y:msubsup><y:mo>=</y:mo><y:mo>−</y:mo><y:mrow><y:mo stretchy="false">(</y:mo><y:mn>0.3</y:mn><y:msubsup><y:mn>2</y:mn><y:mrow><y:mo>−</y:mo><y:mn>0.02</y:mn></y:mrow><y:mrow><y:mo>+</y:mo><y:mn>0.05</y:mn></y:mrow></y:msubsup><y:mo stretchy="false">)</y:mo></y:mrow></y:math>, and the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>κ</cb:mi></cb:math> [also known as <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msubsup><eb:mi>K</eb:mi><eb:mn>0</eb:mn><eb:mo>*</eb:mo></eb:msubsup><eb:mo stretchy="false">(</eb:mo><eb:mn>700</eb:mn><eb:mo stretchy="false">)</eb:mo></eb:math>] pole position, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msqrt><ib:msub><ib:mi>s</ib:mi><ib:mi>κ</ib:mi></ib:msub></ib:msqrt><ib:mo>=</ib:mo><ib:mrow><ib:mo stretchy="false">(</ib:mo><ib:mn>96</ib:mn><ib:msubsup><ib:mn>6</ib:mn><ib:mrow><ib:mo>−</ib:mo><ib:mn>24</ib:mn></ib:mrow><ib:mrow><ib:mo>+</ib:mo><ib:mn>41</ib:mn></ib:mrow></ib:msubsup><ib:mo>−</ib:mo><ib:mi>i</ib:mi><ib:mn>19</ib:mn><ib:msubsup><ib:mn>8</ib:mn><ib:mrow><ib:mo>−</ib:mo><ib:mn>17</ib:mn></ib:mrow><ib:mrow><ib:mo>+</ib:mo><ib:mn>38</ib:mn></ib:mrow></ib:msubsup><ib:mo stretchy="false">)</ib:mo></ib:mrow><ib:mtext> </ib:mtext><ib:mtext> </ib:mtext><ib:mi>MeV</ib:mi></ib:math>. We also provide a detailed analysis of the complex validity domain of the Roy-Steiner-type equations.