I'll quote doctor of philosophy and father of seven Taylor Marshall on the matter.
Objection 1. It seems that spanking children is not lawful because the Apostle states, “Fathers, do not provoke your children, lest they become discouraged” (Col 3:21) and again, “Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord” (Eph 6:4). Now physical injury is a provocation. Hence spanking children is not lawful.
Objection 2. Besides, the Psalmist sings, “thy rod and thy staff, they comfort me” (Ps 23:4). Now striking a child with the rod is not comforting. Hence, it is not lawful to spank a child.
Objection 3. Furthermore, God prevented Moses from entering the Promised Land because he angrily struck the rock of salvation with a rod (cf. Num 20:11). By this signification we learn that the rod denotes the species of anger that forfeits the divine promises on account of malice. Hence, it is not lawful to use the rod on children about which Christ spoke: “For to such belongs the kingdom of Heaven” (Mt 19:14) for in these words Christ speaks of a promise.
On the contrary, It is written: “He who spares the rod hates his son, but he who loves him is diligent to discipline him” (Prov 12:24) and again, “Folly is bound up in the heart of a child, but the rod of discipline drives it far from him” (Prov 22:15). Now the dutiful parent disciplines his child and drives away folly. Therefore, it is lawful to spank a child with the rod.
I answer that, striking a child may be accomplished in two ways. A parent may spank a child in wrath so as to injure the child, or a parent may spank a child in a spirit of charity and patience so as to fulfill the words: “he who loves him is diligent to discipline him.” Consequently, spankings administered in charity are virtuous and meritorious.
Reply to the First Objection. Here the Apostle speaks of wrathful provocation whereby the child is wounded. Now the correction of vice in a child is not a wound, but a cure. Hence, the Apostle does not denounce corporeal punishments.
Reply to the Second Objection. The rod comforts the Psalmist spiritually. As stated above, the removal of vice is comforting. Hence, corporeal punishment that seeks to mitigate vice is licit since it leads to spiritual comfort.
Reply to the Third Objection. Moses forfeited his right to the Promise Land by angrily bearing the rod of the Lord. Previously, the Lord also said to Moses: “And you shall take in your hand this rod, with which you shall do the signs” (Ex 4:17). Now rods can be used in two ways. Relatively speaking, rods are neutral in that they can be used for a good or a bad purpose. The rod of discipline, simply speaking, drives away folly, as stated above, and for this reason the Lord says, “with which you shall do the signs.” Hence the term “rod” need not denote wrathful indignation on the part of the one who bears the rod.
tl;dr Beat your child not from wrath but from love. Do not injure him though.