Theoretical bounds on dark Higgs mass in a self-interacting dark matter model with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:msup><m

Motivated by the null results of current dark matter searches and the small-scale problems, we study a dark sector charged by a spontaneous broken gauge <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:msup><a:mo stretchy="false">)</a:mo><a:mo>′</a:mo></a:msup></a:math>. To explore the parameter space of this model, in addition to the consideration of the small-scale data, we also consider the theoretical bounds on the dark Higgs mass, with the upper bound coming from the tree-level perturbative unitarity and the lower bound from the one-loop Linde-Weinberg bound. We deeply examine the dependence of the Linde-Weinberg bound on gauge choice and energy scale, and present a Linde-Weinberg bound that is gauge and scale independent. Combining the theoretical and observational constraints, we obtain the following ranges for the parameter space: the dark matter mass is 10–500 GeV, the mediator (dark photon) mass is 0.5–5 MeV, the dark Higgs mass is 0.05–50 MeV, and the dark fine-structure constant is 0.001–0.4. We conclude that the dark Higgs in this model cannot be ignored in the phenomenological study of the dark sector.