Loop Heisenberg-Virasoro Lie Conformal algebra

Let $HV$ be the loop Heisenberg-Virasoro Lie algebra over $\text{C}$ with basis $\{L_{\text{a},i},H_{\text{b},j}\mathrm{\mid }\text{a},\text{b},i,j\in \mathbb{Z}\}$ and brackets $[L_{\a,i},L_{\b,j}]=(\a-\b)L_{\a+\b,i+j}, [L_{\a,i},H_{\b,j}]=-\b H_{\a+\b,i+j},[H_{\a,i},H_{\b,j}]=0$. In this paper, a formal distribution Lie algebra of $HV$ is constructed. Then the associated conformal algebra $CHV$ is studied, where $CHV$ has a $\text{C}[\mathrm{\partial }]$-basis $\{L_i,H_i\mathrm{\mid }i\in \mathbb{Z}\}$ with $\lambda$-brackets $[L_i, {}\lambda , L_j]=(\partial+2\lambda) L{i+j}, [L_i, {}\lambda , H_j]=(\partial+\lambda) H{i+j}, [H_i, {}\lambda , L_j]=\lambda L{i+j}$and$[H_i, {}_\lambda , H_j]=0$. In particular, the conformal derivations of $CHV$ are determined. Finally, rank one conformal modules and $\mathbb{Z}$-graded free intermediate series modules over $CHV$ are classified.