RG studies of scalar-field models of long-range interactions

In this work, we study the long-range interactions in nongravitational field theories and their behavior in the deep infrared. To model such effects, we consider a nonlocal scalar theory obtained by adding a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>ϕ</a:mi> <a:msup> <a:mo>□</a:mo> <a:mrow> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msup> <a:mi>ϕ</a:mi> </a:math> term to the local action. Using the functional renormalization group, we analyze its infrared fixed-point structure. Within the local potential approximation (LPA), we show that nonlocality modifies phase-transition patterns and can induce symmetry breaking. Extending the LPA beyond polynomial truncations, we examine the convexity property of the effective potential as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>k</c:mi> <c:mo stretchy="false">→</c:mo> <c:mn>0</c:mn> </c:math> and find that the flow becomes singular for <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msup> <f:mi>λ</f:mi> <f:mn>2</f:mn> </f:msup> <f:mo>&gt;</f:mo> <f:mn>0</f:mn> </f:math> before reaching the deep infrared. Including the scale-dependent wave function renormalization, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi>Z</h:mi> <h:mi>k</h:mi> </h:msub> </h:math> ( <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msup> <j:mi>LPA</j:mi> <j:mo>′</j:mo> </j:msup> </j:math> ), we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalize the model to <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>ϕ</l:mi> <l:msup> <l:mo>□</l:mo> <l:mrow> <l:mi>σ</l:mi> <l:mo>/</l:mo> <l:mn>2</l:mn> </l:mrow> </l:msup> <l:mi>ϕ</l:mi> </l:math> and analyze how the infrared properties depend on <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>σ</n:mi> </n:math> . With appropriate scaling choices, we show that the infrared behavior remains unchanged up to <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>σ</p:mi> <p:mo>=</p:mo> <p:mi>d</p:mi> <p:mo>/</p:mo> <p:mn>2</p:mn> </p:math> and follows Sak’s prediction up to <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>σ</r:mi> <r:mo>=</r:mo> <r:mn>2</r:mn> </r:math> . Finally, we study higher-derivative cases within the LPA, focusing on <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>σ</t:mi> <t:mo>=</t:mo> <t:mn>4</t:mn> </t:math> , which corresponds to isotropic Lifshitz criticality, and obtain results consistent with earlier work.