Normally just casting another spell that doesn't require concentration doesn't break the first spell. But you must pay attention to the details (as always). If you ready the second spell for example (in order to let the spell effects happen on a specific trigger) this DOES require concentration and breaks the first spell, even though the second spell in itself doesn't require concentration. It is the Ready action that does, though.
It says so on pg. 193 of the PHB under "Ready", last paragraph:
To be readied, a spell must have a casting time of 1 action, and holding onto the spell's magic requires concentration (explained in chapter 10).