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Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients Missouri University of Science and Technology Scholars' Mine Physics Faculty Research & Creative Works Physics 01 Sep 2000 Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients Ulrich D. Jentschura Missouri University of Science and Technology, [email protected] Jens Becher Ernst Joachim Weniger Gerhard Soff Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork Part of the Physics Commons Recommended Citation U. D. Jentschura et al., "Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients," Physical Review Letters, vol. 85, no. 12, pp. 2446-2449, American Physical Society (APS), Sep 2000. The definitive version is available at https://doi.org/10.1103/PhysRevLett.85.2446 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. VOLUME 85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER 2000 Resummation of QED Perturbation Series by Sequence Transformations and the Prediction of Perturbative Coefficients U. D. Jentschura,1,* J. Becher,1 E. J. Weniger,2,3 and G. Soff 1 1Institut für Theoretische Physik, TU Dresden, D-01062 Dresden, Germany 2Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38-40, 01187 Dresden, Germany 3Institut für Physikalische und Theoretische Chemie, Universität Regensburg, D-93040 Regensburg, Germany (Received 10 November 1999) We propose a method for the resummation of divergent perturbative expansions in quantum electro- dynamics and related field theories. The method is based on a nonlinear sequence transformation and uses as input data only the numerical values of a finite number of perturbative coefficients. The results obtained in this way are for alternating series superior to those obtained using Padé approximants. The nonlinear sequence transformation fulfills an accuracy-through-order relation and can be used to predict perturbative coefficients. In many cases, these predictions are closer to available analytic results than predictions obtained using the Padé method. PACS numbers: 12.20.Ds Perturbation theory leads to the expansion of a physical The polynomials Pl͑g͒ and Qm͑g͒ are constructed so that quantity P ͑g͒ in powers of the coupling g, the Taylor expansion of the Padé approximation agrees X` with the original input series Eq. (1) up to terms of order n P ͑g͒ϳ cng . (1) l 1 m in g, n෇0 P ͑g͒ 2 ͓l͞m͔ ͑g͒ ෇ O͑gl1m11͒, g ! 0. (3) The natural question arises as to how the power series P on the right-hand side is related to the (necessarily finite) For the recursive computation of Padé approximants we quantity on the left. It was pointed out in [1] that perturba- use Wynn’s epsilon algorithm [9], which in the case of the tion theory is unlikely to converge in any Lagrangian field power series (1) produces Padé approximants according to theory. Generically, the asymptotic behavior of the pertur- ͑n͒ e2k ෇ ͓n 1 k͞k͔P ͑g͒. Further details can be found in bative coefficients is assumed to be of the form [2] Chap. 4 of [10]. n! In this Letter, we advocate a different resummation ϳ g ! cn Kn n , n ` , (2) scheme.P For an infinite series whose partial sums are S ෇ n sn j෇0 aj, the nonlinear (Weniger) sequence transfor- where K, g, and S are constants. S is related to the first mation with initial element s0 is defined as [see Eq. (8.4-4) coefficient of the b function of the underlying theory. of [10] ] P ͑ ͒ In view of the probable divergence of perturbation ex- n j n b1j n21 sj ෇ ͑21͒ ͑ ͒ ͑ ͒ pansions in higher order, a number of prescriptions have ͑0͒͑ ͒ ෇ j 0 j b1n n21 aj11 dn b, s0 P ͑ ͒ , (4) n ͑ ͒j͑ n ͒ b1j n21 1 been proposed both for the resummation of divergent per- j෇0 21 j ͑b1n͒ a turbation series and for the prediction of higher-order per- n21 j11 ͑ ͒ ෇ ͑ ͒͞ ͑ ͒ turbative coefficients. A very important method is the where a m G a 1 m G a is a Pochhammer symbol. Borel summation procedure whose application to QED The shift parameter b is usually chosen as b ෇ 1, and this perturbation series is discussed in [3,4]. The Borel method, choice will be exclusively used here (see also [10]). The while being useful for the resummation of divergent series, power of the d transformation and related transformations cannot be used for the prediction of higher-order perturba- [e.g., the Levin transformation, Eq. (7.3-9) of [10] ] is due tive coefficients in an obvious way. to the fact that explicit estimates for the truncation error of In recent years, Padé approximants have become the the series are incorporated into the convergence accelera- standard tool to overcome problems with slowly conver- tion or resummation process (see Chap. 8 of [10]). Note gent and divergent power series [5]. Padé approximants that the d transformation (4) has led to numerically stable have also been used for the prediction of unknown per- and remarkably accurate results [11,12] in the resumma- turbative coefficients in quantum field theory [6–8]. The tion of the perturbative series of the quartic, sextic, and ͓l͞m͔ Padé approximant to the quantity P ͑g͒ represented octic anharmonic oscillator whose coefficients display a by the power series (1) is the ratio of two polynomials similar factorial pattern of divergence as the quantum field Pl͑g͒ and Qm͑g͒ of degree l and m, respectively, theoretic coefficients indicated in Eq. (2). We consider as a model problem the QED effective ac- ͑ ͒ l ͓ ͞ ͔ ͑ ͒ ෇ Pl g ෇ p0 1 p1g 1 ··· 1 plg tion in the presence of a constant background magnetic l m P g m . Qm͑g͒ 1 1 q1g 1 ··· 1 qmg field for which the exact nonperturbative result can be 2446 0031-9007͞00͞85(12)͞2446(4)$15.00 © 2000 The American Physical Society VOLUME 85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER 2000 expressed as a proper-time integral: ͑21͒n114njB j Z Ω æ µ ∂ c ෇ 2n14 , (7) 2 2 ` m2 n ͑ ͒͑ ͒͑ ͒ ෇ e B ds 1 s e 2n 1 4 2n 1 3 2n 1 2 SB 2 2 2 coths 2 2 exp 2 s . 8p 0 s s 3 eB where B2n14 is a Bernoulli number, display an alternating (5) sign pattern and grow factorially in absolute magnitude, Here, B is the magnetic field strength, and e is the elemen- ͑ ͒n11 ͑ ͒ 21 G 2n 1 2 2͑2n14͒ E B cn ϳ ͓1 1 O͑2 ͔͒ (8) tary charge. The general result for arbitrary and field 8 p2n14 can be found in Eq. (3.49) in [13] and in Eq. (4-123) in [14]. The nonperturbative result for SB can be expanded as n ! `. The series differs from “usual” perturbation ෇ 2 2͞ 4 in powers of the effective coupling gB e B me, which series in quantum field theory by the distinctive property results in the divergent asymptotic series that all perturbation theory coefficients are known. X` 2e2B2 The numerical results in the fifth column of Table I show ϳ n ! that the application of the d transformation (4) to the partial SB 2 2 gB cngB , gB 0. (6) p ෇ n 0 sums sn͑gB͒ of the perturbation series (6) produces conver- The expansion coefficients gent results even for a coupling constant as large as gB ෇ 10. In the third column of Table I, we display the sequence ͓0͞0͔, ͓1͞0͔, ͓1͞1͔,...,͓n͞n͔, ͓n11͞n͔, ͓n11͞n11͔,... of Padé approximants, which were computed using Wynn’s epsilon algorithm [9]. With the help of the The d transformation (4), when applied to the partial notation ͓͓ x͔͔ for the integral part of x, the elements of this sums Pn͑g͒ of the power series (1), fulfills the accuracy- sequence of Padé approximants can be written compactly through-order relation [11]: ͓͓͓͑ ͒͞ ͔͔͓͓͞ ͞ ͔͔ ͔ as n 1 1 2 n 2 . Obviously, Padé approximants P ͑g͒ 2d͑0͒͑1, P ͑g͒͒͒ ෇ O͑gn12͒, g ! 0. (9) converge too slowly to the exact result to be numerically n 0 useful. The Levin d transformation defined in Eq. (7.3-9) Upon reexpansion of the d transform a prediction for the in [10], which is included because it is closely related next higher-order term in the perturbation series may there- to the d transformation (4), fails to accomplish a resum- fore be obtained. mation of the perturbation series, as shown in the fourth In Table II we compare predictions for the coefficients column of Table I. cn of the perturbation series (6) obtained by reexpand- So far, predictions for unknown perturbative coefficients ing the Padé approximants ͓͓͓n͞2͔͔͓͓͑͞n 2 1͒͞2͔͔ ͔ and the ͑0͒ were usually obtained using Padé approximants. The transforms dn22͑1, s0͑gB͒͒͒, which were computed from accuracy-through-order relation (3) implies that the Taylor the partial sums s0͑gB͒, s1͑gB͒,...,sn21͑gB͒. For higher expansion of a Padé approximant reproduces all terms orders of perturbation theory in particular, the Weniger used for its construction. The next coefficient obtained transformation yields clearly the best results, whereas for in this way is usually interpreted as the prediction for low orders the improvement over Padé predictions is only the first unknown series coefficient (see, e.g., [6–8]). gradual. For example, let us assume that for a particular TABLE I.
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