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Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients

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Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients 1, 1 2 1 U. D. Jentschura ∗,J.Becher,E.J.Weniger, and G. Soff 1Institut f¨ur Theoretische Physik, TU Dresden, D-01062 Dresden, Germany 2Institut f¨ur Physikalische und Theoretische Chemie, Universit¨at Regensburg, D-93040 Regensburg, Germany (November 9, 1999) have also been used for the prediction of unknown per- We propose a method for the resummation of divergent turbative coefficients in quantum field theory [6–8]. The perturbative expansions in quantum electrodynamics and re- [l/m]Pad´e approximant to the quantity (g) represented lated field theories. The method is based on a nonlinear se- by the power series (1) is the ratio ofP two polynomials quence transformation and uses as input data only the numer- P (g)andQ (g) of degree l and m, respectively, ical values of a finite number of perturbative coefficients. The l m results obtained in this way are for alternating series superior P(g) p+pg+...+p gl to those obtained using Pad´e approximants. The nonlinear [l/m] (g)= l = 0 1 l . m sequence transformation fulfills an accuracy-through-order re- P Qm(g) 1+q1g+...+qm g lation and can be used to predict perturbative coefficients. In many cases, these predictions are closer to available analytic The polynomials Pl(g)andQm(g) are constructed so that results than predictions obtained using the Pad´e method. the Taylor expansion of the Pad´e approximation agrees with the original input series Eq. (1) up to terms of order PACS numbers: 12.20.Ds, 02.70.-c, 02.60.-x l + m in g, (g) [l/m] (g)=O(gl+m+1) ,g0.(3) P − P → I. INTRODUCTION For the recursive computation of Pad´e approximants we use Wynn’s epsilon algorithm [9], which in the case of the Perturbation theory leads to the expansion of a phys- power series (1) produces Pad´e approximants according (n) ical quantity (g) in powers of the coupling g, to 2k =[n+k/k] (g). Further details can be found in P Ch. 4 of [10]. P ∞ In this Letter, we advocate a different resummation (g) c gn . (1) P ∼ n scheme. For an infinite series whose partial sums are sn = n=0 n X j=0 aj, the nonlinear (Weniger) sequence transforma- tion with initial element s is defined as [see Eq. (8.4-4) The natural question arises as to how the power series P 0 on the right-hand side is related to the (necessarily finite) of [10]]: quantity on the left. It was pointed out in [1] that pertur- n bation theory is unlikely to converge in any Lagrangian j n (β + j)n 1 sj ( 1) − field theory. Generically, the asymptotic behavior of the − j (β + n) a j=0 n 1 j+1 perturbative coefficients is assumed to be of the form [2] δ(0)(β,s )=X − . (4) n 0 k j n (β + j)n 1 1 ( 1) − γ n! − j (β + n) a cn Kn ,n , (2) j=0 n 1 j+1 ∼ Sn →∞ X − where K, γ and S are constants. S is related to the first The shift parameter β is usually chosen as β =1,andthis coefficient of the β function of the underlying theory. choice will be exclusively used here (see also [10]). The In view of the probable divergence of perturbation power of the δ transformation and related transforma- expansions in higher order, a number of prescriptions tions [e.g., the Levin transformation, Eq. (7.3-9) of [10]] is have been proposed both for the resummation of diver- due to the fact that explicit estimates for the truncation gent perturbation series and for the prediction of higher- error of the series are incorporated into the convergence order perturbative coefficients. A very important method acceleration or resummation process (see Ch. 8 of [10]). is the Borel summation procedure whose application to Note that the δ transformation (4) has lead to numer- QED perturbation series is discussed in [3,4]. The Borel ically stable and remarkably accurate results [11,12] in method, while being useful for the resummation of diver- the resummation of the perturbative series of the quartic, gent series, cannot be used for the prediction of higher- sextic and octic anharmonic oscillator whose coefficients order perturbative coefficients in an obvious way. display a similar factorial pattern of divergence as the In recent years, Pad´e approximants have become the quantum field theoretic coefficients indicated in Eq. (2). standard tool to overcome problems with slowly conver- We consider as a model problem the QED effective gent and divergent power series [5]. Pad´e approximants action in the presence of a constant background magnetic 1 field for which the exact nonperturbative result can be for the first unknown series coefficient (see, e.g., [6–8]). expressed as a proper-time integral: Note that the δ transformation (4), when applied to the partial sums n(g) of the power series (1), fulfills an 2 2 ∞ 2 accuracy-through-orderP relation [11] analogous to that of e B ds 1 s me SB = coth s exp s . (5) Pad´e approximants: − 8π2 s2 − s − 3 −eB Z0 (0) n+2 (g) δn 1, 0(g) =O(g ) ,g0.(9) Here, B is the magnetic field strength, and e is the el- P − P → ementary charge. The general result for arbitrary E Upon re-expansion of the δ transform a prediction for and B field can be found in Eq. (3.49) in [13] and the next higher-order term in the perturbation series may in Eq. (4-123) in [14]. The nonperturbative result for therefore be obtained. The accuracy-through-order rela- SB can be expanded in powers of the effective coupling tion (9) has far-reaching consequences. For example, it 2 2 4 gB = e B /me, which results in the divergent asymptotic ensures that invariance properties of the S matrix which series can be formulated order-by-order in perturbation theory (notably unitarity) are conserved under the Weniger re- 2e2B2 ∞ S g c gn ,g0. (6) summation process. B ∼− π2 B n B B→ In Table II we compare predictions for the coefficients n=0 X cn of the perturbation series (6) obtained by re-expanding The expansion coefficients the Pad´e approximants [[ n/2]]/[[ ( n 1)/2]] and the − transforms δ(0) 1,s (g ) , which were computed from n+1 n n 2 0 B ( 1) 4 2n+4 the partial sums− s (g ), s (g ),...,s (g ). For higher cn = − |B | , (7) 0 B 1 B n 1 B (2n + 4)(2n + 3)(2n +2) orders of perturbation theory in particular,− the Weniger transformation yields clearly the best results, whereas where is a Bernoulli number, display an alternat- B2n+4 for low orders the improvement over Pad´e predictions ing sign pattern and grow factorially in absolute magni- is only gradual. For example, let us assume that for tude, a particular problem only three coefficients c0, c1 and ( 1)n+1 Γ(2n +2) c2 are available and c3 should be estimated by a ra- c 1+O(2 (2n+4)) (8) tional approximant. Because of the accidental equal- n − 2n+4 − ∼ 8 π (0) ity [1/1] (g)=δ1 1, 0(g) , the predictions for c3 ob- as n . The series differs from “usual” perturbation tained usingP the Pad´eschemeandtheP δtransformation, series→∞ in quantum field theory by the distinctive property are equal. Differences between the Pad´e predictions and that all perturbation theory coefficients are known. those obtained using the δ transformation start to accu- The numerical results in the fifth column of Table I mulate in higher order. show that the application of the δ transformation (4) to We now turn to the case of the uniform background the partial sums sn(gB) of perturbation series (6) pro- electric field, for which the effective action reads [13] duces convergent results even for a coupling constant as large as gB = 10. In the third column of Table I, we 2 2 ∞ 2 e E ds 1 s me display the sequence SE = coth s exp i s+i . 8π2 s2 − s − 3 eE Z0 [0/0], [1/0], [1/1],...,[ν/ν],[ν +1/ν], [ν +1/ν +1],... This result can be derived from (5) by the replacements of Pad´e approximants, which were computed using B i E and the inclusion of the converging factor. With Wynn’s epsilon algorithm [9]. With the help of the no- → 2 2 4 the convention gE = e E /me the divergent asymptotic tation [[x]] for the integral part of x, the elements of this series sequence of Pad´e approximants can be written compactly 2 2 ∞ as [[ ( n +1)/2]]/[[ n/2]] . Obviously, Pad´e approximants 2e E n S g c0 g ,g0, (10) converge too slowly to the exact result to be numerically E ∼ π2 E n E E→ n=0 useful. The Levin d transformation defined in Eq. (7.3-9) X in [10], which is included because it is closely related to is obtained. The expansion coefficients the δ transformation (4), fails to accomplish a resumma- n tion of the perturbation series, as shown in the fourth 4 2n+4 c0 = |B | (11) column of Table I. n (2n + 4)(2n + 3)(2n +2) So far, predictions for unknown perturbative coeffi- cients were usually obtained using Pad´e approximants.
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