Exploring nonperturbative behavior of moments and cumulants in quantum theories

The dynamics of quantum fields become nonperturbative when their interactions are probed by a large number of particles. To explore this regime, we study correlation functions that involve a large number of fields, focusing on massive scalar theories that feature arbitrary self-interactions, <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>p</a:mi> </a:mrow> </a:msup> </a:math> . Treating quantum fields as operator-valued distributions, we investigate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>n</c:mi> </c:math> -point correlation functions at ultrashort distances and compute moments and cumulants of fields, using a semiclassical saddle point approximation in the double scaling limit of weak coupling, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>λ</e:mi> <e:mo stretchy="false">→</e:mo> <e:mn>0</e:mn> </e:math> , large quantum number, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>n</h:mi> <h:mo stretchy="false">→</h:mo> <h:mi>∞</h:mi> </h:math> , while keeping <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>λ</k:mi> <k:mi>n</k:mi> </k:math> constant. Addressing the nonperturbative regime, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>λ</m:mi> <m:mi>n</m:mi> <m:mo>≳</m:mo> <m:mn>1</m:mn> </m:math> , requires a resummation of the effective saddle point to all orders in <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>λ</o:mi> <o:mi>n</o:mi> </o:math> . We perform this resummation in zero and one dimensions and show that the moments, corresponding to correlation functions including disconnected contributions, grow exponentially with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>n</q:mi> </q:math> . This growth is significantly reduced for higher-order self-interactions, i.e., for larger <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>p</s:mi> </s:math> . On the other hand, we argue that the cumulants, which represent connected correlation functions, grow even more rapidly and are mostly independent of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>p</u:mi> </u:math> .