Bootstrapping gauge theories

We consider asymptotically free gauge theories with gauge group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:msub><a:mi>N</a:mi><a:mi>c</a:mi></a:msub><a:mo stretchy="false">)</a:mo></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>N</e:mi><e:mi>f</e:mi></e:msub></e:math> quarks with mass <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>m</g:mi><g:mi>q</g:mi></g:msub><g:mo>≪</g:mo><g:msub><g:mi mathvariant="normal">Λ</g:mi><g:mi mathsize="small">QCD</g:mi></g:msub></g:math> that undergo chiral symmetry breaking and confinement. We propose a bootstrap method to compute the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>S</k:mi></k:math> matrix of the pseudo-Goldstone bosons (pions) that dominate the low-energy physics. For the important case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>N</m:mi><m:mi>c</m:mi></m:msub><m:mo>=</m:mo><m:mn>3</m:mn></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>N</o:mi><o:mi>f</o:mi></o:msub><o:mo>=</o:mo><o:mn>2</o:mn></o:math>, a numerical implementation of the method gives the phase shifts of the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>S</q:mi><q:mn>0</q:mn></q:math>, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>P</s:mi><s:mn>1</s:mn></s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>S</u:mi><u:mn>2</u:mn></u:math> waves in good agreement with experimental results. The method incorporates gauge theory information (<w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>N</w:mi><w:mi>c</w:mi></w:msub></w:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>N</y:mi><y:mi>f</y:mi></y:msub></y:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>m</ab:mi><ab:mi>q</ab:mi></ab:msub></ab:math>, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi mathvariant="normal">Λ</cb:mi><cb:mi mathsize="small">QCD</cb:mi></cb:msub></cb:math>) by using the form-factor bootstrap recently proposed by Karateev, Kuhn and Penedones together with a finite energy version of the Shifman-Vainshtein-Zakharov (SVZ) sum rules. At low energy we impose constraints from chiral symmetry breaking. The only low-energy numerical inputs are the pion mass <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>m</gb:mi><gb:mi>π</gb:mi></gb:msub></gb:math> and the quark and gluon condensates. Published by the American Physical Society 2024