Hello and welcome to just answer.
I am sorry to hear about your son's pain and discomfort.
Sensitivity to cold indicates an inflamed pulp. The inflammation is considered REVERSIBLE if the pain from the cold stimulus disappears within a few seconds of removing the cold stimulus i.e 15 seconds.
Reversible pulpitis may be caused by any agent that is capable of injuring the pulp such as:
1)It can happen while preparing the cavity with a dull bur or keeping the bur(used for cutting the cavity)in contact with the tooth for too long 2)Excessive dehydration of the cavity with chlorform or alcohol. 3)Bacteria,as from caries.
In your son's case, there is a possibility of injuring the nerve during the filling if the decay ( the cavity)he drilled lied close to the pulp. This trauma to the nerve can cause the nerve to die or necrosed. When a pulp dies it can become infected and can give rise to an abscess/infection.
If this is the case then the tooth will have to go for baby root canal /pulpotomy to remove the necrosed pulp.
I would recommend you to take your son to a pedodontist (Childs specialist) and get his teeth evaluated by getting an x-ray done before considering extraction.
Extracting the baby teeth should not be considered as the first option if the pulp is injured/ nerve is dead These teeth (baby teeth) must be saved till the time permanent ones are scheduled to erupt as they are important in maintaining the physiologic space of the permanent teeth. These teeth (baby teeth) also prevent malocclusion and bite problems which can arise if they are lost early.
I hope this helps!
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