Characteristics of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Z</mml:mi> </mml:math> boson in Chern-Simons matter theory

We consider the Chern-Simons theory coupled to massive fundamental matter in three spacetime dimensions, focusing on the Higgsed phase of the bosonic matter in the large- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> </a:math> limit. To study the characteristics of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>Z</c:mi> </c:math> boson in the Higgsed phase and address an apparent puzzle regarding its fermionic analog, we employ the 3D Bose-Fermi duality to derive the exact large- <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>N</e:mi> </e:math> result for the two-point function of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>Z</g:mi> </g:math> boson. We study the spectral representation of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>Z</i:mi> </i:math> boson and find that, in the large- <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>N</k:mi> </k:math> limit, it behaves as an unstable resonance with a nonperturbatively large lifetime at weak coupling.