Nonanalytic structure of effective potential at finite temperature on compactified space

We thoroughly investigate nonanalytic terms in the finite-temperature effective potential in one-loop approximation on a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>D</a:mi> </a:math> -dimensional spacetime, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi>S</c:mi> </c:mrow> <c:mrow> <c:mi>τ</c:mi> </c:mrow> </c:msub> <c:mo>×</c:mo> <c:msup> <c:mrow> <c:mi>R</c:mi> </c:mrow> <c:mrow> <c:mi>D</c:mi> <c:mo>−</c:mo> <c:mo stretchy="false">(</c:mo> <c:mi>p</c:mi> <c:mo>+</c:mo> <c:mn>1</c:mn> <c:mo stretchy="false">)</c:mo> </c:mrow> </c:msup> <c:mo>×</c:mo> <c:msubsup> <c:mrow> <c:mo>∏</c:mo> </c:mrow> <c:mrow> <c:mi>i</c:mi> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:mrow> <c:mrow> <c:mi>p</c:mi> </c:mrow> </c:msubsup> <c:mtext> </c:mtext> <c:msubsup> <c:mrow> <c:mi>S</c:mi> </c:mrow> <c:mrow> <c:mi>i</c:mi> </c:mrow> <c:mrow> <c:mn>1</c:mn> </c:mrow> </c:msubsup> </c:mrow> </c:math> , using a mode recombination formula. Such nonanalytic terms cannot be expressed as positive powers of field-dependent mass squared. The formula provides a clear separation of the effective potential into a part that contains the nonanalytic terms and a part that is purely analytic, and clarifies the origin of the nonanalytic terms. We obtain all the nonanalytic terms and show that only two types of nonanalytic terms arise from the modes with zero Matsubara frequency. For a real scalar field with periodic boundary conditions, if the number of noncompacted spatial dimensions is odd (even), there are odd powers of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>M</g:mi> </g:math> ( <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>log</i:mi> <i:mi>M</i:mi> </i:math> terms) but no <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>log</k:mi> <k:mi>M</k:mi> </k:math> terms (no odd powers of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>M</m:mi> </m:math> ). For fermions with general boundary conditions, we find that neither of the two types appears. These results clarify the nonanalytic structure of the finite-temperature effective potential on the spacetime with compactified spatial dimensions.