Exact results for scaling dimensions of neutral operators in scalar conformal field theories

We determine the scaling dimension <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="normal">Δ</a:mi><a:mi>n</a:mi></a:msub></a:math> for the class of composite operators <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi>ϕ</d:mi><d:mi>n</d:mi></d:msup></d:math> in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>λ</f:mi><f:msup><f:mi>ϕ</f:mi><f:mn>4</f:mn></f:msup></f:math> theory in <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>d</h:mi><h:mo>=</h:mo><h:mn>4</h:mn><h:mo>−</h:mo><h:mi>ε</h:mi></h:math> taking the double scaling limit <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>n</j:mi><j:mo stretchy="false">→</j:mo><j:mi>∞</j:mi></j:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>λ</m:mi><m:mo stretchy="false">→</m:mo><m:mn>0</m:mn></m:math> with fixed <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>λ</p:mi><p:mi>n</p:mi></p:math> via a semiclassical approach. Our results resum the leading power of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>n</r:mi></r:math> at any loop order. In the small <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>λ</t:mi><t:mi>n</t:mi></t:math> regime we reproduce the known diagrammatic results and predict the infinite series of higher-order terms. For intermediate values of <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>λ</v:mi><v:mi>n</v:mi></v:math> we find that <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mi mathvariant="normal">Δ</x:mi><x:mi>n</x:mi></x:msub><x:mo>/</x:mo><x:mi>n</x:mi></x:math> increases monotonically approaching a <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mo stretchy="false">(</ab:mo><ab:mi>λ</ab:mi><ab:mi>n</ab:mi><ab:msup><ab:mo stretchy="false">)</ab:mo><ab:mrow><ab:mn>1</ab:mn><ab:mo>/</ab:mo><ab:mn>3</ab:mn></ab:mrow></ab:msup></ab:math> behavior in the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>λ</eb:mi><eb:mi>n</eb:mi><eb:mo stretchy="false">→</eb:mo><eb:mi>∞</eb:mi></eb:math> limit. We further generalize our results to neutral operators in the <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msup><hb:mi>ϕ</hb:mi><hb:mn>4</hb:mn></hb:msup></hb:math> in <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:mi>d</jb:mi><jb:mo>=</jb:mo><jb:mn>4</jb:mn><jb:mo>−</jb:mo><jb:mi>ε</jb:mi></jb:math>, <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:msup><lb:mi>ϕ</lb:mi><lb:mn>3</lb:mn></lb:msup></lb:math> in <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi>d</nb:mi><nb:mo>=</nb:mo><nb:mn>6</nb:mn><nb:mo>−</nb:mo><nb:mi>ε</nb:mi></nb:math>, and <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:msup><pb:mi>ϕ</pb:mi><pb:mn>6</pb:mn></pb:msup></pb:math> in <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:mi>d</rb:mi><rb:mo>=</rb:mo><rb:mn>3</rb:mn><rb:mo>−</rb:mo><rb:mi>ε</rb:mi></rb:math> theories with <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi>O</tb:mi><tb:mo stretchy="false">(</tb:mo><tb:mi>N</tb:mi><tb:mo stretchy="false">)</tb:mo></tb:math> symmetry.