Pion-nucleon scattering in baryon chiral perturbation theory combined with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msu

This work implements the combined baryon chiral perturbation theory (BChPT) and <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:msub> <a:mi>N</a:mi> <a:mi>c</a:mi> </a:msub> </a:math> expansions for pion-nucleon elastic scattering. The effective theory is based on the baryon sector dynamical spin-flavor <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>S</c:mi> <c:mi>U</c:mi> <c:mo stretchy="false">(</c:mo> <c:mn>4</c:mn> <c:mo stretchy="false">)</c:mo> </c:math> symmetry emergent in the large <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>N</g:mi> <g:mi>c</g:mi> </g:msub> </g:math> limit, whose breaking is controlled by the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mn>1</i:mn> <i:mo>/</i:mo> <i:msub> <i:mi>N</i:mi> <i:mi>c</i:mi> </i:msub> </i:math> expansion. The noncommutativity of the chiral and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mn>1</k:mn> <k:mo>/</k:mo> <k:msub> <k:mi>N</k:mi> <k:mi>c</k:mi> </k:msub> </k:math> expansions in unitarity corrections (loops) requires a linking of both expansions. As it was shown in the case of baryon masses and currents, the natural linking is the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>ξ</m:mi> </m:math> expansion, in which <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi mathvariant="script">O</o:mi> <o:mo stretchy="false">(</o:mo> <o:mi>p</o:mi> <o:mo stretchy="false">)</o:mo> <o:mo>=</o:mo> <o:mi mathvariant="script">O</o:mi> <o:mo stretchy="false">(</o:mo> <o:mrow> <o:mn>1</o:mn> <o:mo>/</o:mo> <o:msub> <o:mi>N</o:mi> <o:mi>c</o:mi> </o:msub> </o:mrow> <o:mo stretchy="false">)</o:mo> <o:mo>=</o:mo> <o:mi mathvariant="script">O</o:mi> <o:mo stretchy="false">(</o:mo> <o:mi>ξ</o:mi> <o:mo stretchy="false">)</o:mo> </o:math> . The spin-flavor symmetry requires that the ground state baryons span an <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>S</z:mi> <z:mi>U</z:mi> <z:mo stretchy="false">(</z:mo> <z:mn>4</z:mn> <z:mo stretchy="false">)</z:mo> </z:math> symmetric irreducible representation which implies that in particular <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mi>N</db:mi> </db:math> and <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mi mathvariant="normal">Δ</fb:mi> </fb:math> are active degrees of freedom in the effective theory. The scattering amplitude is expanded to the next-to-next-to leading order in the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>ξ</ib:mi> </ib:math> expansion, corresponding to the one-loop contributions with the leading-order Lagrangian. The results are given for generic <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:msub> <kb:mi>N</kb:mi> <kb:mi>c</kb:mi> </kb:msub> </kb:math> in order to demonstrate the consistency of the framework. The spin-flavor symmetry plays a central role in maintaining the consistency of the effective theory with respect to the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:mn>1</mb:mn> <mb:mo>/</mb:mo> <mb:msub> <mb:mi>N</mb:mi> <mb:mi>c</mb:mi> </mb:msub> </mb:math> expansion. This consistency manifests itself in an improvement in the convergence of the low energy expansion with respect to the case of the ordinary BChPT without an explicit dynamical <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi mathvariant="normal">Δ</ob:mi> </ob:math> , which is known to be inconsistent with the constraints of <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:msub> <rb:mi>N</rb:mi> <rb:mi>c</rb:mi> </rb:msub> </rb:math> scaling. Fits to the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mi>π</tb:mi> <tb:mi>N</tb:mi> <tb:mo stretchy="false">→</tb:mo> <tb:mi>π</tb:mi> <tb:mi>N</tb:mi> </tb:math> , <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:mi>S</wb:mi> </wb:math> , <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"> <yb:mi>P</yb:mi> </yb:math> , and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"> <ac:mi>D</ac:mi> </ac:math> partial wave amplitudes from the SAID data base are finally used to test the framework and to determine the energy range of its applicability.