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The presence of numbness in the toes following hip arthroplasty can be attributed to an injury to the peroneal nerve. This nerve, a branch of the sciatic nerve, is responsible for the sensation in the skin of the lower leg and the toes. If the peroneal nerve is compressed or injured during surgery—perhaps due to positioning or direct trauma—it can result in numbness or a loss of sensation in the areas it innervates, including the toes.

While the femoral artery injury could lead to other complications, such as ischemia or bleeding, it would not specifically cause numbness in the toes. Similarly, an injury to the Achilles tendon relates to plantar flexion functions of the ankle and foot and would not typically cause toe numbness. An injury to the sciatic nerve could potentially cause broader issues affecting both lower extremities, but the specific symptom of toe numbness is more closely tied to the peroneal nerve. Thus, the correct explanation for the symptom of numbness in the patient's toes is indeed associated with an injury to the peroneal nerve.