Ratio for comparison between groups and correlation analyses

I'm trying to compare two groups on their ratio between two scores and then investigate the correlation of this ratio for each participant with scores obtained on other tests. What would be an appropriate way of calculating this ratio and what statistical test would I then use to see if there is a significant difference between the groups?
NB: some of the scores involved = 0

personID c_world_right c_world_left group
<int> <int> <int> <chr>
1 3489 3945 control
2 3754 3952 patient
3 5154 2386 control
4 4173 4183 patient
6 2919 2617 control
7 4109 1270 patient