Meson-baryon scattering lengths without annihilation diagrams to order p4 in heavy baryon chiral perturbation theory

We calculate the threshold <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math> matrices of the meson and baryon processes that have no annihilation diagrams: <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>π</c:mi><c:mo>+</c:mo></c:msup><c:msup><c:mi mathvariant="normal">Σ</c:mi><c:mo>+</c:mo></c:msup></c:math>, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msup><f:mi>π</f:mi><f:mo>+</f:mo></f:msup><f:msup><f:mi mathvariant="normal">Ξ</f:mi><f:mn>0</f:mn></f:msup></f:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>K</i:mi><i:mo>+</i:mo></i:msup><i:mi>p</i:mi></i:math>, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>K</k:mi><k:mo>+</k:mo></k:msup><k:mi>n</k:mi></k:math>, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mover accent="true"><m:mi>K</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mn>0</m:mn></m:msup><m:msup><m:mi mathvariant="normal">Ξ</m:mi><m:mn>0</m:mn></m:msup></m:math> to the fourth order in heavy baryon chiral perturbation theory. By performing least-squares and Bayesian fits to the nonphysical lattice QCD data, we determine the low-energy constants through both perturbative and nonperturbative iterative methods. By using these low-energy constants, we obtain the physical scattering lengths in these fits. The convergence behavior is not good across all channels in the perturbative method. The scattering lengths for the five channels, obtained by taking the median values from four different fitting approaches, are <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi>a</r:mi><r:mrow><r:msup><r:mi>π</r:mi><r:mo>+</r:mo></r:msup><r:msup><r:mi mathvariant="normal">Σ</r:mi><r:mo>+</r:mo></r:msup></r:mrow></r:msub><r:mo>=</r:mo><r:mo>−</r:mo><r:mn>0.16</r:mn><r:mo>±</r:mo><r:mn>0.07</r:mn><r:mtext> </r:mtext><r:mtext> </r:mtext><r:mi>fm</r:mi></r:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>a</u:mi><u:mrow><u:msup><u:mi>π</u:mi><u:mo>+</u:mo></u:msup><u:msup><u:mi mathvariant="normal">Ξ</u:mi><u:mn>0</u:mn></u:msup></u:mrow></u:msub><u:mo>=</u:mo><u:mo>−</u:mo><u:mn>0.04</u:mn><u:mo>±</u:mo><u:mspace linebreak="goodbreak"/><u:mn>0.04</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>fm</u:mi></u:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>a</y:mi><y:mrow><y:msup><y:mi>K</y:mi><y:mo>+</y:mo></y:msup><y:mi>p</y:mi></y:mrow></y:msub><y:mo>=</y:mo><y:mo>−</y:mo><y:mn>0.41</y:mn><y:mo>±</y:mo><y:mn>0.11</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:msub><ab:mi>a</ab:mi><ab:mrow><ab:msup><ab:mi>K</ab:mi><ab:mo>+</ab:mo></ab:msup><ab:mi>n</ab:mi></ab:mrow></ab:msub><ab:mo>=</ab:mo><ab:mo>−</ab:mo><ab:mn>0.19</ab:mn><ab:mo>±</ab:mo><ab:mn>0.10</ab:mn><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>fm</ab:mi></ab:math>, and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>a</cb:mi><cb:mrow><cb:msup><cb:mover accent="true"><cb:mi>K</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover><cb:mn>0</cb:mn></cb:msup><cb:msup><cb:mi mathvariant="normal">Ξ</cb:mi><cb:mn>0</cb:mn></cb:msup></cb:mrow></cb:msub><cb:mo>=</cb:mo><cb:mo>−</cb:mo><cb:mn>0.30</cb:mn><cb:mo>±</cb:mo><cb:mn>0.07</cb:mn><cb:mtext> </cb:mtext><cb:mtext> </cb:mtext><cb:mi>fm</cb:mi></cb:math>, where the uncertainties are conservatively estimated by taking the maximum deviation between the median and extreme values of the statistical errors.