Anomalous dimensions from conformal field theory: Generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mm

We investigate <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>ϕ</a:mi><a:mrow><a:mn>2</a:mn><a:mi>n</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:msup></a:math> deformations of the generalized free theory in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ε</c:mi></c:math> expansion, where the canonical kinetic term is generalized to a higher-derivative version. For <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi><e:mo>=</e:mo><e:mn>1</e:mn></e:math>, we use the conformal multiplet recombination method to determine the leading anomalous dimensions of the fundamental scalar operator <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ϕ</g:mi></g:math> and the bilinear composite operators <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="script">J</i:mi></i:math>. Then, we extend the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>n</l:mi><l:mo>=</l:mo><l:mn>1</l:mn></l:math> analysis to the Potts model with <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>S</n:mi><n:mrow><n:mi>N</n:mi><n:mo>+</n:mo><n:mn>1</n:mn></n:mrow></n:msub></n:math> symmetry and its higher-derivative generalization, in which <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>ϕ</p:mi></p:math> is promoted to an <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>N</r:mi></r:math>-component field. We further examine the Chew-Frautschi plots and their <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>N</t:mi></t:math> dependence. However, for each integer <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>n</v:mi><v:mo>&gt;</v:mo><v:mn>1</v:mn></v:math>, the leading anomalous dimensions of <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>ϕ</x:mi></x:math> and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="script">J</z:mi></z:math> are not fully determined and contain one unconstrained constant, which in the canonical cases can be fixed by the results from the diagrammatic method. In all cases, we verify that the multiplet-recombination results are consistent with crossing symmetry using the analytic bootstrap methods.