Zài (在) and zhèng zài (正在) both could be used before verbs to mark the progressive aspect, indicating that an action is ongoing or in progress.
lǎo bǎn zài /zhèng zài kāi huì.
The boss is having a meeting.
Zài (在) could replace zhèng zài (正在) in most cases. Nevertheless, when emphasizing the moment in which an action is being carried out in a sentence, zhèng zài (正在) is preferable.
zuó tiān wǒ jìn mén de shí hòu, tā zhèng zài chàng gē.
He was singing when I came in yesterday.
Compared with zhèng zài (正在), the semantic meaning of zài (在) focuses more on the ongoing status of an action, therefore zài (在) is compatible with durative processes which contain an internal interval (Liu, 2015) and cannot be replaced by zhèng zài (正在) in these cases:
(1) “Zài (在) VP” could occur with temporal adverbs conveying the semantic meaning of continuance, but zhèng zài (正在) cannot.
wǒ yì zhí zài xiě zuò yè.
I've been doing my homework all the time.
wǒ yì zhí zhèng zài xiě zuò yè
(2) The ongoing status or the continuance of the action introduced by zài (在) can be repeatable, which could be characterized by using together with temporal adverbs like měi tiān每天(every day), cháng cháng 常常(often) and so on.
tā men měi tiān dōu zài tán lùn nà chǎng jiāo tōng shì gù.
They keep talking about that traffic accident every day.
(3) Zhèng zài (正在) cannot replace zài (在) when the action is already taken, signaling by the appearance of temporal adverbs like yǐ jīng已经(already) in a sentence, which also implies the continuance of an action.30183617
jǐng fāng yǐ jīng zài diào chá zhè gè àn zi le.
The police are already investigating the case.
Hái zài (还在) is also used before verbs, which signifies that some phenomenon continues to exist or the actions is still ongoing. Hái zài (还在) generally translates to “still” in English.
wài miàn hái zài xià yǔ.
It’s still raining outside.
yè shēn le, tā hái zài xiě zuò yè.
It was late at night and he was still working away at his homework.
Reference：Meichun Liu, Tense and Aspect in Mandarin Chinese, The Oxford Handbook of CHINESE LINGUISTICS p284