Interaction between productivity growth and interest rates
Starting at the end, the simple answer is that intreset rates are decided as a counter measure for slow growth, which may derive from productivity slump. Meaning interest policy is a result, not a cause

On a wider level, the interntion of Central bankers is that there is a productivity rate, internally set in the economy set it's pace of growth. Imagine the added value of internet applications, as opposed to the world before that, as an example.

Their goal, is to set interest rate to match the proper appetite for risk, to encourage that productivity growth: take loans to pay for internet services, as per the example

The main parameter lacking in your equation, is that monitary policy cannot be the main economic policy to encourage productivity. It's the government in general: investinh in education, lowering barries for good healthy markets, provide a supporting legal system for business disputes etc.

Monitary policy can only be effective to a certain extent, in a way, like a wide (regressive) tax deduction