Does touching one's private parts without lust invalidate ablution and necessitate its repetition?

Touching the male organ with the hand nullifies ablution (Wudu') according to many scholars, and this is the most preponderant and religiously safer opinion, whether the touching is with or without desire. So, if you touch the male organ with the palm of your hand, ablution is nullified. Touching with other parts of the hand, or touching the testicles, or anything other than the male organ, does not nullify ablution.

Ghusl (definition) (ritual bath) suffices for Wudu' if it is performed after Janabah (definition) (major ritual impurity), but touching the male organ nullifies Wudu' absolutely. If a person touches his male organ after performing Wudu' without a barrier, his ablution is nullified, and he must repeat it.