The scientific, Latin name for paan ki jad (पान की जड़) is Alpinia galanga. Other common name is kulanjan, as referenced in for paan ki jad in Royal Hyderabadi Cooking by Sanjeev Kapoor in the section for making a Potli ka Masala. The common English name is "Greater Galangal" or "Galangal" and available as fresh and dried forms in Asian supermarkets.
The cover image credit goes to Munnalal Dawasaz at https://munnalaldawasaz.in. Harsh from their Product Team verified and confirmed:
"Pan ki jad is also called as 'kulanjan' and 'Greater Galangal'. It is used for cold, cough and throat infection."
It's a common misnomer that paan ki jad is listed in the ingredients as "Betel Leaf Root" or "Betel Root." Because, the betel leaf or the paan leaf is from the Betel plant known in scientific name as "Piper betel." If you have seen the actual roots/vines of the Betel plant, it does **NOT** resemble anywhere close to the rhizome roots that is paan ki jad. It's my guess that some chefs/cooks and culinary journalists are directly and literally translating paan = betel and jad = root to falsely and incorrectly. I found both fresh and dried products of paan ki jad at a local Asian supermarket called "County Square Market" here in Vacaville, CA. I was calling almost all the major Indian groceries in my local areas of Vacaville, Fairfield, Fremont, Yuba City, and Union City asking for "Betel Leaf Root" and/or "Betel Roots." No one, I repeat—no one—knows or stock them.
With the corrected name in common English, of course they stock/sell them. Some of the larger specialty Indian grocers with Ayurvedic sections. You don't have to make your trip to an Indian grocer with Ayurvedic section, simply look for galangal at an Asian supermarket near you. You'll have 100% luck finding it in Thai grocers, as galangal is one of the culinary ingredient staples in Thai cooking.