  The (classical) zero bias is a transformation of centered, finite variance
probability distributions, often expressed in terms of random variables
$X\mapsto X^\ast$, characterized by the functional equation \[ E[Xf(X)] =
\sigma^2 E[f'(X^\ast)] \quad \text{for all }\; f\in C_c^\infty(\mathbb{R}) \]
where $\sigma^2$ is the variance of $X$. The zero bias distribution is always
absolutely continuous, and supported on the convex hull of the support of $X$.
It is related to Stein's method, and has many applications throughout
probability and statistics, most recently finding new characterization of
infinite divisiblity and relation to the L\'evy--Khintchine formula.
  In this paper, we define a free probability analogue, the {\em free zero
bias} transform $X\mapsto X^\circ$ acting on centered $L^2$ random variables,
characterized by the functional equation \[ E[Xf(X)] = \sigma^2 E[\partial
f(X^\circ)] \quad \text{for all }\; f\in C_c^\infty(\mathbb{R}) \] where
$\partial f$ denotes the free difference quotient. We prove existence of this
transform, and show that it has comparable smoothness and support properties to
the classical zero bias. Our main theorem is an analogous characterization of
free infinite divisibility in terms of the free zero bias, and a new
understanding of free L\'evy measures. To achieve this, we develop several new
operations on Cauchy transforms of probability distributions that are of
independent interest.
