Improved Wald formalism and First Law of Dyonic Black Strings with Mixed Chern-Simons Terms

We study the first law of thermodynamics of dyonic black strings carrying a linear momentum in type IIA string theory compactified on K3 with leading order $\alpha'$ corrections. The low energy effective action contains mixed Chern-Simons terms of the form $-2B_{(2)}\wedge {\rm tr}(R(\Gamma_\pm)\wedge R(\Gamma_\pm))$ which is equivalent to $2H_{(3)}\wedge \mathrm{CS}_{(3)}(\Gamma_\pm)$ up to a total derivative. We find that the naive application of Wald entropy formula leads to two different answers associated with the two formulations of the mixed Chern-Simons terms. Surprisingly, neither of them satisfies the first law of thermodynamics for other conserved charges computed unambiguously using the standard methods. We resolve this problem by carefully evaluating the full infinitesimal Hamiltonian at both infinity and horizon, including contributions from terms proportional to the Killing vector which turn out to be nonvanishing on the horizon and indispensable to establish the first law. We find that the infinitesimal Hamiltionian associated with $-2B_{(2)}\wedge {\rm tr}(R(\Gamma_\pm)\wedge R(\Gamma_\pm))$ requires an improvement via adding a closed but non-exact term, which vanishes when the string does not carry either the magnetic charge or linear momentum. Consequently, both formulations of the mixed Chern-Simons terms yield the same result of the entropy that however does not agree with the Wald entropy formula. In the case of extremal black strings, we also contrast our result with the one obtained from Sen's approach.