Orateur
Description
The resummation of perturbative expansions of Quantum Field Theories has been studied extensively in the case when there are only a few coupling constants, with no results yet for functional field theories (FFT), where all powers of interaction remain relevant under the renormalization group flow.
We address this gap by examining the large-order behaviour (LOB) of a functional field theory describing disordered elastic manifolds in equilibrium.
This FFT involves two fields $\phi$ and $\tilde{\phi}$ and a control parameter $w$ which sets the expectation value $<\phi>=w$. To understand how we can gain non-perturbative access to the functional fixed point, we considered the zero-dimensional limit ($d \to 0$). This “toy” model reveals a surprisingly rich complexity: the standard saddle-point method fails to yield the correct asymptotics due to the non-analyticity of the Borel sum.
To resolve this problem, we used the fact that each term of the perturbative expansion can be rewritten as a contour integral circling the origin. This allows us to resum the series and reach the strong coupling limit. However, this approach only works in $d=0$ dimensions. For $d>0$ we could use a saddle-point method to obtain the LOB. This method has problems even in d = 0: the action at the saddle point corresponding to the closest singularity in the Borel plane develops a branch cut for positive values of $w > 0.31$. As a result, the inverse Borel transform acquires a complex ambiguity. The resolution requires resurgence theory, where this complex contribution is cancelled by non-perturbative corrections obtained from the discontinuity of the Borel sum (the Stokes jump).
