  We consider Rayleigh-B\'enard convection as modeled by the Boussinesq
equations, in case of infinite Prandtl number. There is a broad interest in
bounds of the upwards heat flux, as given by the Nusselt number ${\rm Nu}$, in
terms of the forcing via the imposed temperature difference, as given by the
Rayleigh number in the turbulent regime ${\rm Ra}\gg 1$. In several works, the
background field method applied to the temperature field has been used to
provide upper bounds on ${\rm Nu}$ in terms of ${\rm Ra}$. In these
applications, the background field method comes in form of a variational
problem where one optimizes a stratified temperature profile subject to a
certain stability condition; the method is believed to capture marginal
stability of the boundary layer. The best available upper bound via this method
is ${\rm Nu}$ $\lesssim {\rm Ra}^\frac{1}{3}(\ln {\rm Ra})^\frac{1}{15}$; it
proceeds via the construction of a stable temperature background profile that
increases logarithmically in the bulk. In this paper, we show that the
background temperature field method cannot provide a tighter upper bound in
terms of the power of the logarithm. However, by another method one does obtain
the tighter upper bound ${\rm Nu}\lesssim {\rm Ra}^\frac{1}{3}(\ln\ln {\rm
Ra})^\frac{1}{3}$, so that the result of this paper implies that the background
temperature field method is unphysical in the sense that it cannot provide the
optimal bound.
