The arrow for the electron antineutrino is pointing down which is against time, why is that?
The math gives us the right answers in some problems (and this is one of them) if we treat an anti-particle as a regular particle moving backwards in time. That doesn't mean that an anti-particle is "really" a regular particle moving back in time.
Be aware that Feynman diagrams such as this one are easily misunderstood - they are not picctures of what the particles are really doing, they're a way of organizing the calculations to find the probability of various interactions.
Purely a convention.
so why does the math permit the arrow pointing downwards? which equation implies that? so do all anti-particles by convention point downwards? Thanks!
Its simply a logical consequence of the math. Logical consequences may have a deeper meaning, or not. Experience in this case has shown its helps sometimes, but can lead to issues if pushed too far. For example one can view an electron as a positron going backwards in time. So which is it? Is it the electron going backwards in time or is it the positron? The theory is silent about that.
Before asking questions like that you need to understand the underlying theory, including the math. And the math of QFT is no walk in the park by a long shot.
Nothing is going backwards in time. The arrows are a convention for writing down the mathematics. Nothing more, nothing less.
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