  The statistical correlations of two copies of a d-dimensional elastic
manifold embedded in slightly different frozen disorder are studied using the
Functional Renormalization Group to one-loop accuracy, order O(eps = 4-d).
Determining the initial (short scale) growth of mutual correlations, i.e. chaos
exponents, requires control of a system of coupled differential (FRG) equations
(for the renormalized mutual and self disorder correlators) in a very delicate
boundary layer regime. Some progress is achieved at non-zero temperature, where
linear analysis can be used. A growth exponent a is defined from center of mass
fluctuations in a quadratic potential. In the case where temperature is
marginal, e.g. a periodic manifold in d=2, we demonstrate analytically and
numerically that a = eps (1/3 - 1/(2 log(1/T)) with interesting and unexpected
logarithmic corrections at low T. For short range (random bond) disorder our
analysis indicates that a = 0.083346(6) eps, with large finite size
corrections.
