Volume 72B, number 1 PHYSICS LETTERS 5 December 1977 BOREL SUMMATION OF PERTURBATION SERIES IN QUANTUM MECHANICS AND FIELD THEORY V.S. POPOV, V.L. ELETSKY and A.V. TURBINER hTstitute of Theoretical and Experimental Physics, Moscow, USSR Received 8 September 1977 The Borel summation of factorial divergent perturbation series is considered. The relation between the asymptotics of the coefficients in the series and the character of the exact function singularity is established. The applicability lim- its for the improved perturbation theory are obtained. The results are tested on a number of physical problems for which the exact solutions are available. In the last few years some effective methods were suggested to compute the high orders of perturbation series (PS) in quantum mechanics [1,2] (in particular, the anharmonic oscillator was especially detailed [2]). Lipatov developed [3, 4] a quasi-classical method of . computation of the functional integral in the quantum field theory and obtained the asymptotic values of the PS coefficients of the Gell-Mann-Low function (GLF) in the scalar theory with interaction Hint. = gf~o n dDx/ n !, D = 2n/(n - 2). This approach is now being inten- sively developed [5-10]. Lipatov's method was ap- plied to treat the following problems: the n-compo- nent scalar field [5], the fermion theory with the Yukawa interaction [6], scalar electrodynamics [8], Next in turn is the computation of PS asymptotics for realistic field theories (quantum electrodynamics, Yang-Mills theory, etc.). In this connection a question arises: what addition- al information about the exact solution can be ex- tracted from the asymptotics of the PS coefficients in comparison with the first few PS terms? The purpose of this note is to clarify this question. Let f(z) be a function represented by divergent power series ¢o f(z)= Z) ak(-z)k, (1) k=ko which is asymptotic at z ~ 0. For most of the theories investigated [3-8] the coefficients are of the form: a k = (ks)! akk~(c 0 + Cl/k + c2/k 2 + ...), (2) where z! -= P(z + 1); s, 13, a, c i are computable con- stants (s, a ~> 0). Another and more convenient repre- sentation of the asymptotic values at k -+ oo is ~o a k = ((ka)!/k!) a k ~ C m ( k + 13 - m)!. (3) m=0 Coefficients c and C are connected by a linear trans- formation c i = ~ o SijC/" We have obtained explicit formulae for the elements of the matrices S and S -1 . Substituting eq. (3) into eq. (1) and using the Borel summation method (B', a) [11] we get (see also [10]): f(z) = (az)- (~+ l) exp(1/~!z) X ~ Cm(13- rn)!(az)ml(az;a,m - 13), (4) m=0 where oo i(x;s,13)=x- f dte-t/x[(t- I) a + 1] ~-1. 0 When x ~ 0 we have I(x; a, 13) ec x 1 -~ exp(1/X ) and when x -~ oo, I ~ x (a-1)(/J-1) for 13 ) 1 - s -1 , Ic~x-~ X lnxfor13= 1-s -1 andlccx-~for/3<l-s -1. In the typical case s = 1 this integral reduces to the in- complete gamma-function oo f(z)=(az) -(~+1) exp(1/az) ~ Cm(13-m)! m =0 (5) × (az) m F(m - ~, 1/az). Summing up the series (l) with coefficients (3) one gets the same result using the Sommerfeld-Watson in- tegral transform as suggested in [3]. The possibility of obtaining explicit expressions (4), (5) for the sum 99