Can you confirm that the local index files are present in the /AppData/CMSModules/SmartSearch folder of the live site application?
If they are not there, please check that your web farms are configured properly and tasks are being processed: https://docs.xperience.io/configuring-xperience/setting-up-web-farms/troubleshooting-web-farms. If everything looks okay, can you please try rebuilding the index in the CMS admin UI? This should generate a web farm task to generate the index files on the live site application as well.