Rigorous Roy-Steiner equation analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>K</mml:mi></mml:math> scattering at unphysical quark masses

We perform a rigorous analysis of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>π</a:mi><a:mi>K</a:mi></a:math> scattering at an unphysical pion mass 391 MeV using the Roy-Steiner equations, which satisfy unitarity, analyticity, and crossing symmetry, for the first time. Stable solutions of the Roy-Steiner equations with different quantum numbers of isospin and angular momentum are obtained in the elastic energy region, by taking inputs from the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi><c:mi>K</c:mi></c:math> lattice data in the inelastic region, the lattice data from the crossed <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>π</e:mi><e:mi>π</e:mi><e:mo stretchy="false">→</e:mo><e:mi>K</e:mi><e:mover accent="true"><e:mi>K</e:mi><e:mo stretchy="false">¯</e:mo></e:mover></e:math> channels, the masses of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:msub><j:mi>f</j:mi><j:mn>0</j:mn></j:msub><j:mo stretchy="false">(</j:mo><j:mn>500</j:mn><j:mo stretchy="false">)</j:mo></j:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mi>K</n:mi><n:mo>*</n:mo></n:msup></n:math> at the same pion mass from previous study, and the Regge model. Predictions on the elastic <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>π</p:mi><p:mi>K</p:mi></p:math> scattering phase shifts and the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msubsup><r:mi>K</r:mi><r:mn>0</r:mn><r:mo>*</r:mo></r:msubsup><r:mo stretchy="false">(</r:mo><r:mn>700</r:mn><r:mo stretchy="false">)</r:mo></r:math> pole content are made. Contrary to the virtual pole scenario obtained using the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>K</v:mi></v:math>-matrix method in the literature, we find that lightest strange scalar meson <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mrow><x:msubsup><x:mrow><x:mi>K</x:mi></x:mrow><x:mrow><x:mn>0</x:mn></x:mrow><x:mrow><x:mo>*</x:mo></x:mrow></x:msubsup><x:mo stretchy="false">(</x:mo><x:mn>700</x:mn><x:mo stretchy="false">)</x:mo></x:mrow></x:math> remains a broad resonance at <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msub><bb:mi>m</bb:mi><bb:mi>π</bb:mi></bb:msub><bb:mo>=</bb:mo><bb:mn>391</bb:mn><bb:mtext> </bb:mtext><bb:mtext> </bb:mtext><bb:mi>MeV</bb:mi></bb:math>. The cross-channel dynamics is found to play a crucial role in deriving the proper pole position.