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

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1, 1 2 1 U. D. Jentschura ∗,J.Becher,E.J.Weniger, and G. Soff 1Institut für Theoretische Physik, TU Dresden, D-01062 Dresden, Germany 2Institut für Physikalische und Theoretische Chemie, Universität 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]Padé 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 Padé 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 Padé method. the Taylor expansion of the Padé 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 Padé 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 Padé 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 transformation 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, Padé 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]. Padé 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) Padé 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 Padé 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 Padé 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 PadéschemeandtheP δtransformation, series→∞ in quantum field theory by the distinctive property are equal. Differences between the Padé 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 Padé 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 Padé approximants can be written compactly 2 2 ∞ as [[ ( n +1)/2]]/[[ n/2]] . Obviously, Padé 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 coefficients were usually obtained using Padé approximants. display a nonalternating sign pattern, but are equal in The accuracy-through-order relation (3) implies that the magnitude to the magnetic field case [cf. Eq. (7)]. For Taylor expansion of a Padé approximant reproduces all physical values of gE, i.e., for gE > 0, there is a cut in terms used for its construction. The next coefficient ob- the complex plane, and the nonvanishing imaginary part tained in this way is usually interpreted as the prediction for SE gives the pair-production rate. As is well known,

2 4 (0) resummation procedures for (nonalternating) divergent 1.70 10 is obtained using δ3 whereas the [2/2] Padé series usually fail when the coupling g assumes values on approximant− × yields 1.63 104. the cut in the complex plane [10]. The Borel method We have shown that− the×δ transformation (4) can be fails because of the poles on the integration contour in used to accomplish a resummation of alternating diver- the Borel integral [4]. The δ transformation and Padé gent perturbation series whose coefficients diverge facto- approximations fail for reasons discussed in [10] and [15], rially. In many cases, the δ transforms converge faster respectively. to the nonperturbative result than Padé approximants. We now come to an important observation which to the The δ transformation uses as input data only the numer- best of our knowledge has not yet been addressed in the ical values of a finite number of perturbative coefficients. literature: the prediction of perturbative coefficients by We stress here that the factorial divergence is expected of nonlinear sequence transformations may even work if the general perturbative expansions in quantum field theory resummation of the divergent series fails, i.e. if the cou- [see Eq. (2)]. The Weniger δ transformation can be used pling g lies on the cut. A general divergent series whose for the prediction of higher-order coefficients of alternat- coefficients are nonalternating in sign, evaluated for pos- ing and nonalternating factorially divergent perturbation itive coupling, corresponds to a series with alternating series. Both in model problems and in more realistic ap- coefficients, evaluated for negative coupling. Alternat- plications, the δ transformation yields improved predic- ing series can be resummed with the δ transformation in tions (compared to Padé approximants). It appears that many cases, and predictions for higher-order coefficients the potential of sequence transformations, notably the δ should therefore be possible for both the alternating and transformation, has not yet been widely noticed in the the nonalternating case. For example, the perturbative field of large-order perturbation theory. coefficients in Eqs. (7) and (11) differ only in the sign U.D.J. acknowledges helpful conversations with M. pattern, not in their magnitude. As shown in Table III, Meyer-Hermann, P. J. Mohr and K. Pachucki. E.J.W. rational approximants to the series (6) and (10) produce, acknowledges support from the Fonds der Chemischen after the re-expansion in the coupling, the same predic- Industrie. G.S. acknowledges continued support from tions up to the different sign pattern. BMBF, GSI and DFG. We would like to stress here that the resummation procedure and the prediction scheme presented in this ∗ Electronic address: [email protected]. Letter also work for higher-order terms in the derivative [1] B. Simon, Phys. Rev. Lett. 28, 1145 (1972). expansion of the QED effective action [16]. The resum- [2] A. I. Vainshtein and V. I. Zakharov, Phys. Rev. Lett. 73, mation also works for the partition function for the zero- 1207 (1994). dimensional φ4 theory which is discussed in [14] (p. 464) [3] V. I. Ogievetsky, Proc. Acad. Sci. USSR 109, 919 (1956), and is used in [17] as a paradigmatic example for the [in Russian]. divergence of perturbative expansions in quantum field [4] G. V. Dunne and T. M. Hall, Phys. Rev. D 60, 065002 theory. Results will be presented in detail elsewhere [16]. (1999). An interesting and more “realistic” application is given [5] G. A. Baker and P. Graves-Morris, Padé approximants, by the β function of the Higgs boson coupling in the stan- 2nd ed. (Cambridge University Press, Cambridge, 1996). dard electroweak model [18]. In the MS renormalization [6] M. A. Samuel, G. Li, and E. Steinfelds, Phys. Rev. D 48, scheme, five coefficients of this β function are known. 869 (1993). [7] M. A. Samuel, G. Li, and E. Steinfelds, Phys. Rev. E , Using the first four coefficients, the “prediction” for the 51 3911 (1995). fifth coefficient (which is known) may be obtained and [8] M. A. Samuel, J. Ellis, and M. Karliner, Phys. Rev. Lett. compared to the analytic result. Using the transforma- 74, 4380 (1995). (0) 7 tion δ2 a prediction of β4 4.404 10 is obtained [9] P. Wynn, Math. Tables Aids Comput. 10, 91 (1956). ≈ × 7 which is closer to the analytic result of β4 4.913 10 [10] E. J. Weniger, Comput. Phys. Rep. 10, 189 (1989). ≈ × than the predictions obtained using the [2/1] and [1/2] [11] E. J. Weniger, J. C´ˇıˇzek, and F. Vinette, J. Math. Phys. 7 Padé approximants (these yield β4 3.969 10 and 34, 571 (1993). 7 ≈ × β4 4.188 10 , respectively). The prediction for [12] E. J. Weniger, Phys. Rev. Lett. 77, 2859 (1996). ≈ × (0) [13] J. Schwinger, Phys. Rev. 82, 664 (1951). the unknown coefficient β5 obtained using δ3 is β5 9 9 ≈ [14] C. Itzykson and J. B. Zuber, Quantum Field Theory 3.938 10 as compared to β5 3.756 10 from the −[2/2] Pad´×e approximant. ≈− × (McGraw-Hill, New York, NY, 1980). For the β function of the scalar φ4 theory the situation [15] M. Pindor, Los Alamos preprint hep-th/9903151. is similar to the Higgs boson case. Five coefficients are [16] U. D. Jentschura, J. Becher, M. Meyer-Hermann, P. J. Mohr, E. J. Weniger, and G. Soff, in preparation (1999). known analytically [19]. Again, the prediction for the (0) [17] J. Zinn-Justin, Quantum Field Theory and Critical Phe- fifth coefficient obtained using the transformation δ2 nomena, 3rd ed. (Clarendon Press, Oxford, 1996). (1251.3) is closer to the analytic result of 1424.3than [18] L. Durand and G. Jaczko, Phys. Rev. D 58, 113002 the predictions from the [2/1] and [1/2] Padé approxi- (1998). mants which yield values of 1133.5 and 1187.5, respec- [19] H. Kleinert, J. Neu, V. Schulte-Fröhlich, K. G. tively. For the unknown sixth coefficient, a prediction of Chetyrkin, and S. A. Larin, Phys. Lett. B 272, 39 (1991).

3 TABLE I. Resummation of the perturbation series (6) for gB = 10. Results are given in terms of the dimen- 2 2 2 2 sionless function S¯B =10 (8π )/( e B gB) SB. Apparent convergence is indicated by underlining. − (0) (0) nsn [[ ( n+1)/2]]/[[ n/2]] dn 1,s0(gB) δn 1,s0(gB) 110.476 10.476 190 476 1.617 535 903 1.617 535 903 2 243.492 1.617 535 903 − 0.820 833 551 −0.820 833 551 3− 530.918− 4.627 654 271 −0.588 575 814 −0.659 817 926 4 774.888 1.401 288 801 −0.864 617 071 −0.733 843 307 58.674 647− 107 − 2.773 159 300 −0.839 447 096 −0.780 249 512 ...× ...... − ...− ... 60 3.652 544 10201 0.920 487 125 1.385 114 1013 0.805 633 980 61− 5.553 434 × 10205 −0.400 319 939 4.131 495 × 1013 −0.805 633 979 62 8.721 566 × 10209 −0.918 054 104 −8.500 694 × 1013 −0.805 633 978 63− 1.414 066 × 10214 −0.411 140 364− 2.890 004 × 1014 −0.805 633 977 64 2.365 759 × 10218 −0.915 746 814 5.272 267 × 1014 −0.805 633 976 65− 4.082 125 × 10222 −0.421 331 007 2.050 491 × 1015 −0.805 633 975 66 7.261 275 × 10226 −0.913 555 178 −3.296 170 × 1015 −0.805 633 975 67− 1.330 921 × 10231 −0.430 946 630− 1.475 230 × 1016 −0.805 633 975 ...× ...− ...× ...− ... exact 0.805 633 975 0.805 633 975 0.805 633 975 0.805 633 975 − − − −

TABLE II. Prediction of perturbative coefficients for the power series (6). Results are given for the scaled 2 2 2 dimensionless power series SB0 = (8π )/( e B gB) SB. First column: order of perturbation theory. Second column: exact coefficients. Third and fourth− column: predictions obtained by re-expanding Padé approximants and Weniger transforms, respectively.

(0) n exact [[ n/2]]/[[ ( n 1)/2]] δn 2 1,s0(gB) − − ...... 14 2.181 588 772 1015 2.170 458 614 1015 2.181 574 607 1015 15 +2− .055 682 756 × 1017 +2− .049 236 087 × 1017 +2− .055 678 921 × 1017 16 2.199 481 257 × 1019 2.194 962 521 × 1019 2.199 480 091 × 1019 ...− × ...− × ...− × ... 24 1.711 360 421 1037 1.711 272 235 1037 1.711 360 421 1037 25− +4.421 625 118 × 1039 +4− .421 484 513 × 1039 +4− .421 625 118 × 1039 26 1.234 699 825 × 1042 1.234 674 716 × 1042 1.234 699 825 × 1042 ...− × ...− × ...− × ...

TABLE III. Prediction of perturbative coeﬃcients cn0 for the electric background ﬁeld (10). Results are given for the 2 2 2 scaled dimensionless power series SE0 = (8π )/(e E gE) SE.

(0) n exact δn 2 1,s0(gE) ...... − ... 14 2.181 588 1015 2.181 574 1015 15 2.055 682 × 1017 2.055 678 × 1017 16 2.199 481 × 1019 2.199 480 × 1019 ...× ...× ... 24 1.711 360 1037 1.711 360 1037 25 4.421 625 × 1039 4.421 625 × 1039 26 1.234 699 × 1042 1.234 699 × 1042 ...× ...× ...