"Sign a contract" in Engpsh can be translated as "to execute a contract".
The source of this phrase comes from the legal term "execution of a contract", which refers to the process of signing a written agreement between two or more parties to formapze the terms of their agreement.
Examples:
"The two companies are in the process of executing a contract for the sale of goods."
(两家公司正在签订货物销售合同。)
"Both parties must sign the contract in order to make it legally binding."
(双方必须签署合同以使其具有法律约束力。)