Confinement-Higgs and deconfinement-Higgs transitions in four-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo

We reexamine by numerical simulation the phase structure of the ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>3</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> )-dimensional <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>S</c:mi> <c:mi>U</c:mi> <c:mo stretchy="false">(</c:mo> <c:mn>2</c:mn> <c:mo stretchy="false">)</c:mo> </c:math> lattice gauge theory with gauge fields coupled to Higgs fields at finite temperature. Concretely, we explore two different order parameters, which are able to distinguish the three phases of the theory: (i) the Fredenhagen-Marcu operator used to discriminate between deconfinement and confinement/Higgs phases and (ii) the Greensite-Matsuyama overlap operator proposed recently to distinguish confinement and Higgs phases.