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Consider the transmission of a polar code of block length and rate over a binary memoryless symmetric channel and let be the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters , , , and the quality of the channel quantified by its capacity and its Bhattacharyya parameter . In previous work, two main regimes were studied. In the error exponent regime, the channel and the rate are fixed, and it was proved that the error probability scales roughly as . In the scaling exponent approach, the channel and the error probability are fixed and it was proved that the gap to capacity scales as . Here, is called scaling exponent and this scaling exponent depends on the channel . A heuristic computation for the binary erasure channel (BEC) gives and it was shown that, for any channel  …