This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent μ of polar codes for a memoryless channel q_Y|X with capacity I(q_Y|X) characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed ε∈ (0, 1), the gap between the capacity I(q_Y|X) and the maximum non-asymptotic rate R_n^* achieved by a length-n polar code with average error probability ε scales as n^-1/μ, i.e., I(q_Y|X)-R_n^*

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... = Θ(n^-1/μ). It is well known that the scaling exponent μ for any binary-input memoryless channel (BMC) with I(q_Y|X)∈(0,1) is bounded above by 4.714, which was shown by an explicit construction of polar codes. Our main result shows that 4.714 remains to be a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O(n^-1/μ√( n)) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O(n^-1/μ n) by using a superposition of n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.