On the one-loop curvature function in the sl(2) sector of N=4 SYM

We consider twist J operators with spin S in the sl(2) sector of N = 4 SYM. The small spin expansion of their anomalous dimension defines the so-called slope functions. Much is known about the linear term, but the study of the quadratic correction, the curvature function, started only very recently. At any fixed J , the curvature function can be extracted at all loops from the Pmu-system formulation of the Thermodynamical Bethe Ansatz. Here, we work at the one-loop level and follow a different approach. We present a systematic double expansion of the Bethe Ansatz equations at large J and small winding number. We succeed in fully resumming this expansion and obtain a closed explicit simple formula for the one-loop curvature function. The formula is parametric in J and can be evaluated with minor effort for any fixed J . The result is an explicit series in odd-index z values. Our approach provides a complete reconciliation between the Pmu-system predictions and the large J approach.