Two- and three-meson scattering amplitudes with physical quark masses from lattice QCD

We study systems of two and three mesons composed of pions and kaons at maximal isospin using four CLS ensembles with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>a</a:mi> <a:mo>≈</a:mo> <a:mn>0.063</a:mn> <a:mtext> </a:mtext> <a:mtext> </a:mtext> <a:mi>fm</a:mi> </a:math> , including one with approximately physical quark masses. Using the stochastic Laplacian-Heaviside method, we determine the energy spectrum of these systems including many levels in different momentum frames and irreducible representations. Using the relativistic two- and three-body finite-volume formalism, we constrain the two- and three-meson K matrices, including not only the leading <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>s</c:mi> </c:math> wave, but also <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>p</e:mi> </e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>d</g:mi> </g:math> waves. By solving the three-body integral equations, we determine, for the first time, the physical-point scattering amplitudes for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mn>3</i:mn> <i:msup> <i:mi>π</i:mi> <i:mo>+</i:mo> </i:msup> </i:math> , <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mn>3</k:mn> <k:msup> <k:mi>K</k:mi> <k:mo>+</k:mo> </k:msup> </k:math> , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msup> <m:mi>π</m:mi> <m:mo>+</m:mo> </m:msup> <m:msup> <m:mi>π</m:mi> <m:mo>+</m:mo> </m:msup> <m:msup> <m:mi>K</m:mi> <m:mo>+</m:mo> </m:msup> </m:math> , and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>K</o:mi> <o:mo>+</o:mo> </o:msup> <o:msup> <o:mi>K</o:mi> <o:mo>+</o:mo> </o:msup> <o:msup> <o:mi>π</o:mi> <o:mo>+</o:mo> </o:msup> </o:math> systems. These are determined for total angular momentum <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msup> <q:mi>J</q:mi> <q:mi>P</q:mi> </q:msup> <q:mo>=</q:mo> <q:msup> <q:mn>0</q:mn> <q:mo>−</q:mo> </q:msup> </q:math> , <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mn>1</s:mn> <s:mo>+</s:mo> </s:msup> </s:math> , and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mn>2</u:mn> <u:mo>−</u:mo> </u:msup> </u:math> . We also obtain accurate results for <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mn>2</w:mn> <w:msup> <w:mi>π</w:mi> <w:mo>+</w:mo> </w:msup> </w:math> , <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mi>π</y:mi> <y:mo>+</y:mo> </y:msup> <y:msup> <y:mi>K</y:mi> <y:mo>+</y:mo> </y:msup> </y:math> , and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mn>2</ab:mn> <ab:msup> <ab:mi>K</ab:mi> <ab:mo>+</ab:mo> </ab:msup> </ab:math> phase shifts. We compare our results to chiral perturbation theory and to phenomenological fits.