Step 1: Localise the modality first. Sharp pain and thermal sensation travel in the lateral spinothalamic pathway, while the dorsal columns ($fasciculus\ gracilis$ and $fasciculus\ cuneatus$) handle vibration, fine touch and joint position sense.
Step 2: The key feature of the spinothalamic system is early crossing. The axons decussate in the anterior white commissure of the cord within a couple of segments of entering, so the ascending tract on one side already represents the opposite half of the body.
Step 3: Therefore a lesion of the lateral spinothalamic tract on one side removes pain and temperature from the contralateral body below the lesion. A dorsal column lesion would instead give ipsilateral loss of proprioception, and the anterior spinothalamic tract relates to crude touch, so neither fits a contralateral pain-temperature loss.
Step 4: The single tract whose injury explains the finding is the lateral spinothalamic tract.\[\boxed{Lateral\ spinothalamic\ tract}\]