Can want retain the old issues(defects) which links to Jira in each new test cycle?

Hi scoleman 

Zephyr Enterprise can link multiple defects to a given test cycle(In ZE you can re-execute  test cycles and still obtain all of the execution history which is great also!) 

If a new test cycle is created  you can still report this via traceability in ZE. 

Given this is available in ZE I suspect you are using a plugin in Jira. 

If this is case please post in the relevant community as this community is  Zephyr  Enterprise, a standalone fully fledge Test Management solution that has a bi directional real-time integration with Jira. 

Zephyr Scale Community 

https://community.smartbear.com/t5/Zephyr-Scale/ct-p/zephyr-scale-community

Zephyr Squad Community 

https://community.smartbear.com/t5/Zephyr-Squad/ct-p/zephyr-squad-community

Hope this helps 

Vinnie