The paired t test analyzes the differences between pairs. For each pair, it calculates the difference. Then it calculates the average difference, the 95% CI of that difference, and a P value testing the null hypothesis that the mean difference is really zero.
The paired t test makes sense when the difference is consistent. The control values might bounce around, but the difference between treated and control is a consistent measure of what happened.
With some kinds of data, the difference between control and treated is not a consistent measure of effect. Instead, the differences are larger when the control values are larger. In this case, the ratio (treated/control) may be a much more consistent way to quantify the effect of the treatment.
Analyzing ratios can lead to problems because ratios are intrinsically asymmetric – all decreases are expressed as ratios between zero and one; all increases are expressed as ratios greater than 1.0. Instead it makes more sense to look at the logarithm of ratios. Then no change is zero (the logarithm of 1.0), increases are positive and decreases are negative.
A ratio t test averages the logarithm of the ratio of treated/control and then tests the null hypothesis that the population mean of that set of logarithms is really zero.
Because the ratio t test works with logarithms, it cannot be computed if any value is zero or negative. If all the values are negative, and you really want to use a ratio t test, you could transform all the values by taking their absolute values, and doing the ratio t test on the results. If some values are negative and some are positive, it makes no sense really to think that a ratio would be a consistent way to quantify effect.