The first batch of Constitution Light Cruisers built by the Federation used tried and true laser weaponry as phasers remained in their infancy. Likewise, photon torpedo technology had not yet been successfully reverse engineered. Federation ships instead used the long range accelerator cannon to meet their needs in this area.
After the development of phaser technology, the Constitution Light Cruisers received upgrades to this new weapons technology. This was a priority for Starfleet and was done relatively quickly.