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We take advantage of the fact that in λφ 4 problems, a large field cutoff φmax makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of φmax. For perturbative series terminated at even order, it is in principle possible to adjust φmax in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift m 2 → m 2 (1+η) in order to obtain the exact result. We discuss weak and strong coupling methods to determine φmax and η. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear δ-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Padé and Padé-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours.

We revisit the question of the convergence of lattice perturbation theory for a pure SU (3) lattice gauge theory in 4 dimensions. Using a series for the average plaquette up to order 10 in the weak coupling parameter β −1 , we show that the analysis of the extrapolated ratio and the extrapolated slope suggests the possibility of a non-analytical power behavior of the form (1/β − 1/5.7(1)) 1.0(1) , in agreement with another analysis based on the same asumption. This would imply that the third derivative of the free energy density diverges near β = 5.7. We show that the peak in the third derivative of the free energy present on 4 4 lattices disappears if the size of the lattice is increased isotropically up to a 10 4 lattice. On the other hand, on 4 × L 3 lattices, a jump in the third derivative persists when L increases, and follows closely the known values of βc for the first order finite temperature transition. We show that the apparent contradiction at zero temperature can be resolved by moving the singularity in the complex 1/β plane. If the imaginary part of the location of the singularity Γ is within the range 0.001 < Γ < 0.01, it is possible to limit the second derivative of P within an acceptable range without affecting drastically the behavior of the perturbative coefficients. We discuss the possibility of checking the existence of these complex singularities by using the strong coupling expansion or calculating the zeroes of the partition function.

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We discuss the weak coupling expansion of a one plaquette SU (2) lattice gauge theory. We show that the conventional perturbative series for the partition function has a zero radius of convergence and is asymptotic. The average plaquette is discontinuous at g 2 = 0. However, the fact that SU (2) is compact provides a perturbative sum that converges toward the correct answer for positive g 2 . This alternate method amounts to introducing a specific coupling dependent field cut, that turns the coefficients into g-dependent quantities. Generalizing to an arbitrary field cut, we obtain a regular power series with a finite radius of convergence. At any order in the modified perturbative procedure, and for a given coupling, it is possible to find at least one (and sometimes two) values of the field cut that provide the exact answer. This optimal field cut can be determined approximately using the strong coupling expansion. This allows us to interpolate accurately between the weak and strong coupling regions. We discuss the extension of the method to lattice gauge theory on a D-dimensional cubic lattice.