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Summation of divergent series: Order-dependent mapping JeanZinn-Justin CEA,IRFUandInstitutdePhysiqueTheórique,CentredeSaclay,F91191Gif-sur-YvetteCedex,France Abstract Summationmethodsplayaveryimportantroleinquantumfieldtheorybecauseallperturbationseriesaredivergent 0 1 andtheexpansionparameterisnotalwayssmall. Anumberofmethodshavebeentriedinthiscontext,mostnotably 0 Pade´ approximants,Borel–Pade´ summation,Boreltransformationwithmapping,whichwebrieflydescribeandone 2 onwhichweconcentratehere, Order-DependentMapping(ODM).Werecallthebasisofthemethod, foraclassof n serieswegiveintuitiveargumentstoexplainitsconvergenceandillustrateitspropertiesbyseveralsimpleexamples. a Sincethemethodwasproposed,somerigorousconvergenceproofsweregiven. Themethodhasalsofoundanumber J ofapplicationsandweshalllistafew. 5 Keywords: Divergentseries;Summationmethods;Boreltransformation;Quantummechanics. ] h p - h 1. Theinitialmotivation: Perturbativequantumfieldtheory t a m In quantum field theory, the main analytic calculation tool is the perturbative expansion. As an illustration, we [ considertheimportantexampleoftheφ4 fieldtheory[24]. Inthestatisticalformulation,oneconsiderstheEuclidean (orimaginarytime)action ,localfunctionalofthefieldφ(x),x Rd, 1 S ∈ v d 5 (φ)= ddx 1 ∂ φ(x) 2+ 1rφ2(x)+ gφ4(x) , (1) 7 S Z 2 µ 2 4!  06  Xµ=1h i  1. whererandgaretwoparameters. Tothisactionisassociatedafunctionalmeasuree−S(φ)/Z,whereZisthepartition functiongivenbythefieldintegral 0 0 = [dφ]e (φ). (2) −S 1 Z Z v: Thelimitd =0correspondstoasimpleintegral. i Thecased =1correspondstothequantumquarticanharmonicoscillator. X Dimensionsd >1correspondtoquantumfieldtheoryandtheexpression(1)isthensomewhatsymbolicsincethe r a theoryhastobemodifiedatshortdistancetoregularizeUVdivergencesandrenormalizedtocancelthem. In particular, the dimensions d = 2,3 are especially relevant to classical statistical physics and the theory of phase transitions. Finally, d = 4 isrelevant tothe theory of fundamental interactions atthe microscopic scale. The correspondingrelativisticquantumfieldtheoryispartoftheso-calledHiggsmechanism. Forthefieldtheory(1),theperturbativeexpansionamountstoanexpansioninpowersofthepositiveparameterg. Ford >1,thedifficultyofevaluatingthesuccessiveperturbativetermsincreasesveryrapidly.Moreover,questions like regularization and renormalization arise. Therefore, the calculation of renormalization group functions in the d =3(φ2)2fieldtheoryuporderg7[1]isaremarkableachievement. Emailaddress:[email protected](JeanZinn-Justin) PreprintsubmittedtoAppliedNumericalMathematics January5,2010 1.1. Largeorderbehaviourofperturbativeseries In the φ4 field theory (1), g = 0 corresponds to a singularity since the integral (2) is not defined for g < 0. The perturbative series is divergent. For d < 4, the large order behaviour can be inferred from a steepest descent calculationofthefieldintegral(2)[19][7].Forthequarticanharmonicoscillator(d =1)theresultwasderivedearlier fromtheSchrdingerequation[3]. Foranyphysicalobservable f,theresultshavethegeneralstructure f ( 1)kkbakk!, (3) k k∝ − →∞ where a depends only on d and b is a half-integer that depends on the observable. The coefficient A = 1/a has the value d =0: A =3/2, (4) d =1: A =8 , (5) d =2: A =35.10268957367896(1) (Zinn-Justin), (6) d =3: A =113.38350781527714(1) (Zinn-Justin). (7) Ford =4,tothecontributioncomingfromthesteepestdescentcalculation,acontributionduetothelargemomentum singularitiesofFeynmandiagramshasingeneraltobeadded. Finally,noticethatford <4,Borelsummabilityhasbeenproved. Similar results can be obtained for a number of quantum field theories. When the formal expansion parameter is Planck’s constant, a divergence of the form (3) is in general found (except for some fermion theories), but the parameteramaybecomplex. Foranearlyreview,see[23]. Itfollowsfromthelargeorderbehaviouranalysisthat,whentheexpansionparameterisnotsmall,asummation oftheperturbativeexpansionisindispensable. 1.2. Seriessummation Inthestudyofthefundamentalinteractionsatthemicroscopicscale,itwasrealizedthatinthecaseofthestrong nuclear force, unlike QED, the expansion parameter was large and, therefore, perturbation theory useless, leading manyphysicistseventorejectquantumfieldtheoryasaframeworktodescribesuchphenomena. Beforethelargeorderbehaviourwasevenknown,in[5]itwasproposed,instead,tosumtheperturbativeexpan- sion, using Pade´ approximants and the idea was applied to a phenomenological model, the φ4 field theory in d = 4 dimensions. Sinceonlytwoorthreetermscouldbecalculated,thepossibleconvergenceofthePade´summationcould notbecheckedverywell. However, theresultsobtainedinthiswaymademuchbetterphysicalsensethanthoseof plainperturbationtheory. Forareviewsee[22]. In the seventies, one outstanding problem for which summation methods was required, is the determination of criticalexponentsandothercriticalquantitiesinthetheoryofsecondorderphasetransitions. FollowingWilson,for awholeclassofphysicalsystems,thesequantitiescanbeobtainedfromthe(φ2)2 fieldtheoryind = 3dimensions. One verifies immediately that the expansion parameter, the renormalized interaction g, is of order 1 and a series r summationisrequired(wedonotdiscussheretheε=4 dexpansion,buttheproblemisanalogous). − To deal with the practical problem of series summation, a method was proposed based on Borel–Pade´ approximants[1]. Withtheknowledgeofthelargeorderbehaviour,amoreefficientmethodcouldbedeveloped,combining aBoreltransformation(actuallyBorel–Leroy)andaconformalmapping[17],[12],whichwebrieflypresentinnext section. However, anothermethodbasedonlyontheanalyticpropertiesoftheseries, theorder-dependentmapping wasalsoinvestigated,whichwedescribeinmoredetailinsection3(ageneralreferenceis[24]). 2. Boreltransformationandconformalmapping Thevaluesofcriticalexponentsinalargeclassofcontinuous(orsecondorder)phasetransitionscanbeinferred from so-called renormalization group (RG) functions of the (φ2)2 quantum field theory. One important function is 2 Table1: SeriessummedbythemethodbasedonBoreltransformationandmappingforthezerog˜ oftheRGβ(g)functionand ∗ theexponentsγandνintheφ4fieldtheory. 3 k 2 3 4 5 6 7 g˜ 1.8774 1.5135 1.4149 1.4107 1.4103 1.4105 ∗ ν 0.6338 0.6328 0.62966 0.6302 0.6302 0.6302 γ 1.2257 1.2370 1.2386 1.2398 1.2398 1.2398 the RG β-function whose zeros determine the RG fixed points. For example, in the case of the φ4 theory in d = 3 dimensions,Nickel[1]hascalculated 308 β˜(g˜) = g˜+g˜2 g˜3+0.3510695977g˜4 0.3765268283g˜5+0.49554751g˜6 − − 729 − 0.749689g˜7+O g˜8 , (8) − (cid:16) (cid:17) where g˜ = 3g/(16π) and g is the so-called renormalized interaction, related to the parameter that appears in the r r action(1)byg =g+O(g2)and r 16π 1 β(g)= β˜(g˜)= . (9) r 3 −dlng/dg r Theperturbativeexpansionisdivergent(equation(3)). Fortheβ-functioninthreedimensions, β˜(g˜)= β˜ g˜k, k Xk thelargeorderbehaviour,impliedbytheestimate(7),isgivenby β˜ ( a)kk7/2k! k k∝ − →∞ witha=0.147774232.... Tocharacterizethelargedistancepropertiesofstatisticalsystemsatthephasetransition,onemustfirstdetermine thenon-trivialzerog˜ oftheβ-functionandthencalculatevariousphysicalquantitieslikecriticalexponentsforg˜ =g˜ . ∗ ∗ Onediscoversthatg˜ isanumberoforder1and,thus,anumericaldeterminationfromtheseries8clearlyrequiresa ∗ summationoftheseries. Inthreedimensions,theperturbativeexpansionisprovedtobeBorelsummable. Itisthusnaturaltointroducethe Borel–Laplacetransformation(here,Borel–Leroy): β B (g)= k gk, σ Γ(k+σ+1) Xk whereσisafreeparameter. Then,formallyinthesenseofpowerseries + β(g)= ∞tσe tB (gt)dt. − σ Z 0 Thefunction B (g)isanalyticinacircleofradius1/a. TheseriesissaidBorelsummableif, inaddition, B (g)is σ σ analyticinaneighbourhoodoftherealpositivesemi-axisandtheintegralconverges. The series defines the function in a circle. It is thus necessary to perform an analytic continuation. In practice, with a small number of terms, the continuation requires a domain of analyticity larger than rigorously established. 3 LeGuillouandZinn-Justin[17]haveassumedmaximalanalyticity,i.e.,analyticityinacut-plane. Thecontinuation hasthenbeobtainedbyaconformalmappingofthecut-planeontoacircle. Finally,variousmodificationshasbeen introducedtooptimizethesummationmethod(fordetailssee[17]). Further optimization of the summation technique and the additional seven-loop contributions have led to new estimatesofcriticalexponents[12]. Someresultsaredisplayedintable1. 3. Order-dependentmapping Theorder-dependentmapping(ODM)summationmethod[20]isbasedonsomeknowledgeoftheanalyticprop- ertiesofthefunctionthatisexpanded. Itappliesbothtoconvergentanddivergentseries,althoughitismainlyuseful inthelattercase. 3.1. Thegeneralmethod Let f(z)beananalyticfunctionthathastheTaylorseriesexpansion f(z)= f zℓ. ℓ Xℓ=0 (the=signhastounderstoodinthesenseofseriesexpansion.) WhentheTaylorserieshasafiniteradiusofconvergence,tocontinuethefunctioninthewholedomainofanalyt- icity,onecanmapthedomainontoacircle,whilepreservingtheorigin. Divergentseries: theintuitiveidea. Inacaseofadivergentseries,oneaddstothedomainofanalyticityadisk z < r ofvariableradiusr andappliesasimilarmapping. Ofcourse,thetransformedseriesisstilldivergent. Then, | | onerecallstheempiricalrulethat,foradivergentseries,oneisinstructedtotruncatetheseriesatthetermofminimal modulus, the last term giving an order of magnitude of the error. By adjusting the radius r order by order, one can managetosettheminimumalwaysjustatthelastcalculatedterm. r 7−→ 1 − − Figure1:Mappingz λ:exampleofafunctionanalyticinacut-plane. 7→ In what follows, we consider only functions analytic in a sector (as in the example of figure 1) and mappings z λoftheform 7→ z=ρζ(λ), ζ(λ)=λ+O λ2 , (cid:16) (cid:17) whereζ(λ)isanexplicitanalyticfunctionandρanadjustableparameter. Althoughthetransformedseriesisstilldivergentatρfixed,weshallverifyonafewexamplesthat,byadjustingρ orderbyorder(here,welimitourselvestoBorelsummableexamples)onecanconstructaconvergentalgorithm. Afterthetransformation, f isgivenbyaTaylorseriesinλoftheform f z(λ) = P (ρ)λk, k (cid:0) (cid:1) Xk=0 4 wherethecoefficientsP (ρ)arepolynomialsofdegreekinρ.Sincetheresultisformallyindependentoftheparameter k ρ,theparametercanbechosenfreely. The k-th approximant f(k)(z) is constructed in the following way: one truncates the expansion at order k and choosesρastocancelthelastterm. SinceP (ρ)haskroots(realorcomplex),onechoosesforρthelargestpossible k root(inmodulus)ρ forwhichP (ρ)issmall. Thisleadstoasequenceofapproximants k ′k k f(k)(z)= P (ρ )λℓ(ρ ,z) with P (ρ )=0. ℓ k k k k Xℓ=0 Inthecaseofconvergentseries,itisexpectedthatρ hasanon-vanishinglimitfork . Bycontrast,fordivergent k →∞ seriesitisexpectedthatρ goestozeroforlargekas k ρk =O fk−1/k . (cid:16) (cid:17) Theintuitiveideahereisthatρ correspondstoa‘local’radiusofconvergence. k Sinceρ goestozero,thefunctionζ(λ)mustdivergeforafinitevalueofλ.Below,wechooseλ=1byconvention. k Remark. Inthecaseofrealfunctions,whentherelevantzerosarecomplexitisoftenconvenienttochooseminima ofthepolynomialsP ,whichsatisfy k P (ρ )=0, ′k k choosing,ingeneral,thelargestzeroforwhich P issmall. Othermixedcriteriainvolvingacombinationof P and k k P canalsobeused. Indeed, theapproximantisnotverysensitivetotheprecisevalueofρ , withinerrors. Finally, ′k k Pk+1(ρk)givesanorderofmagnitudeoftheerror. 3.2. Functionsanalyticinacut-plane: Heuristicconvergenceanalysis Althoughsomerigorousconvergenceresultshavebeenobtained[10], therearenotoptimal. Therefore, wegive here heuristic but quantitative arguments that show the nature of the convergence of the ODM method. Following [20],tosimplifyweconsiderarealfunctionanalyticinacut-planewithacutalongtherealnegativeaxis(figure1) andaCauchyrepresentationoftheform 1 0 ∆(g) E(g)= −dg ′ , ′ πZ g g ′− butthegeneralizationissimple. Moreover,weassumethat ∆(g) gbeA/g, A>0. (10) g∝0 → − ThefunctionE(g)canbeexpandedinpowersofg: 1 0 dg E(g)= E gk withE = − ∆(g). k k πZ gk+1 Xk Theassumption(10)thenimpliesalargeorderbehaviour E ( A) kΓ(k b) ( A) kk b 1k!, k − − − − k∝ − − ∼ − →∞ exactlyoftheformdisplayedinsection1.1. Weintroducethemapping λ g=ρ , α>1. (11) (1 λ)α − TheCauchyrepresentationthencanbewrittenas 1 0 ∆ g(λ) E g(λ) = −dλ ′ +R(λ), ′ πZ λ(cid:0) λ (cid:1) (cid:0) (cid:1) − ′ 5 whereR(λ)isasumofcontributionsfromcutsatfinitedistancefromtheorigin. Weexpand E g(λ) = P (ρ)[λ(g)]k (12) k (cid:0) (cid:1) Xk with 1 0 P (ρ)= −dλ∆ g(λ) λ k 1+ finitedistancecontributions. (13) k πZ − − (cid:0) (cid:1) Fork ,thefactorλ kfavourssmallvaluesofλbutfortoosmallvaluesofλtheexponentialdecayof∆ g(λ)takes − →∞ over. Thus,Pk(ρk)canbeevaluatedbythesteepestdescentmethod. WiththeAnsatzthatatthesaddlepoi(cid:0)ntλ<0is independentofkand ρ R/k, R>0, k ∼ whichimpliesg(λ) 0,∆(g)canbereplacedbyitsasymptoticform(10)forg 0 . Atleadingorder,thesaddle → → − pointequationreducesto d A (1 λ)α ln λ =0. (14) dλ(cid:18)Rλ − − | |(cid:19) Inwhatfollows,wesetR/A=µ,sincethisistheonlyparameter. Theequationcanberewrittenas 1 µ+ (1 λ)α 1 (α 1)λ+1 =0. − λ − − (cid:0) (cid:1) If the mapping (11) does not cancel all singularities (and this excludes the case of the integral of section 4), then P (ρ )cannotdecreaseexponentiallywithk. Thisimpliesanotherequation k k 1 (1 λ)α µln λ =0. (15) λ − − | | Thisisindeedtheregionwherethecontributioncomingfromthecutattheoriginandfromtheotherfinitedistance singularitiesarecomparableandwherethezerosofP (ρ)canlie. k Returningtotheexpansion(12),atgfixed,fromthebehaviourofρ weinfer k 1 λ (R/kg)1/α λk e−k1−1/α(R/g)1/α. − ∼ ⇒ ∼ Inagenericsituation,wethenexpectP (ρ )tobehavelike k k Pk(ρk)=O(eCk1−1/α) andthedomainofconvergencedependsonthesignoftheconstantC. ForC >0,thedomainofconvergenceis g <RC α[cos(Argg/α)]α. − | | For α > 2, this domain extends beyond the first Riemann sheet and requires analyticity of the function E(g) in the correspondingdomain. ForC <0,thedomainofconvergenceistheunionofthesector Argg <πα/2andthedomain | | g >RC α[ cos(Argg/α)]α. − | | | | − Againforα>2,thisdomainextendsbeyondthefirstRiemannsheet. 3.3. Examples Forα=3/2,combiningequations(14)and(15),onefinds µ=4.031233504, λ= 0.2429640300. − 6 Table2: α 3/2 2 5/2 3 4 µ 4.031233504 4.466846120 4.895690188 5.3168634291 6.1359656420 λ 0.2429640300 0.2136524524 0.1896450439 0.1699396648 0.14003129119 − Forα=2,equation(14)becomes λ2 µλ 1=0 λ= 1(µ µ2+4). − − ⇒ 2 − q Forµ=3.017759126...onerecoverstheexponentialrateofconvergence(16). Inthecaseofadditionalsingularities,withtheadditionalequation(15),oneobtains µ=4.466846120... , λ= 0.2136524524.... − Togiveafewotherexamples,againcombiningequations(14)and(15)onefindstheresultsdisplayedintable2. 4. Application: Thesimpleintegrald = 0 Forr =1,theintegral(2)inthecased =0reducestothesimpleintegral 1 Z(g)= dx e x2/2 gx4/4!, − − √2πZ andtheconvergenceoftheODMmethodcanbestudiedanalytically. 4.1. Theoptimalmapping Analyticpropertiessuggestthattheoptimalmappingisgivenbysetting λ g=ρ andZ(g)=(1 λ)1/2f(λ). (1 λ)2 − − Then, f hasanexpansionoftheform f(λ)= P (ρ)[λ(g)]k. k Xk Convergencecanbestudiedanalytically. First, f hastherepresentation 1 f(λ)= ds e s2/2+λ(s2/2 ρs4/24) = P (ρ)[λ(g)]k − − k √2πZ with 1 1 P (ρ)= ds e s2/2(s2/2 ρs4/24)k. k k! √2πZ − − Setting s2/2=kt, ρ=R/k, onecanrewritetheexpressionas kk+1/2 1 dt P (ρ)= e kt(t Rt2/6)k. k k! rπZ √t − − 7 Table3: ODM for the integral d = 0 for g : Z(g) g 1/4(1/2) 241/4 √π/Γ(3/4). We define [Z(g)g1/4] − exact [Z(g)g1/4] =δ → ∞ ∼ ∗ ∗ − ODM k 5 10 15 20 25 30 1/ρ 1.131726 2.35036 3.34050 4.5594 5.5495 6.8614 δ 5.7 10 3 2.5 10 5 3.7 10 6 2.2 10 8 3.4 10 9 5.1 10 11 − − − − − − − ∗ ∗ ∗ ∗ ∗ ∗ ln δ 5.1578 10.5921 12.5008 17.5923 19.4855 23.6818 | | − − − − − − k 35 40 45 50 55 60 1/ρ 7.7586 8.9778 9.9678 11.1869 12.1769 13.3958 δ 3.5 10 12 2.5 10 14 3.9 10 15 2.9 10 17 4.5 10 18 3.4 10 20 − − − − − − − ∗ ∗ ∗ ∗ ∗ ∗ ln δ 26.3535 31.2859 33.1625 38.0643 39.9364 44.8208 | | − − − − − − Fork ,theintegralcanbeevaluatedbythesteepestdescentmethod. Thesaddlepointequationis →∞ Rt2/6 (1+R/3)t+1=0 − and,thus, 3 t= 1+R/3 1+R2/9 . R ± (cid:16) p (cid:17) Forkodd,thezerocorrespondstoacancellationbetweenthetwosaddlepoints. Thisyieldstheequation √R2+9+R 1 e2√R2+9/R = e√R2+9/R = R2+9+R . √R2+9 R ⇔ 3(cid:18)p (cid:19) − Asexpected,onefinds R ρ withR=4.526638689... (R/A=3.017759126...). k ∼ k Theminimumisgivenbyoneofthesaddlepoints P (ρ) e 3k/R =(0.5154353381...)k. (16) k − ∝ Atgfixed,λconvergesto1. Moreprecisely, λ=1 R/kg+O(1/k) λk e √Rk/g. − − ⇒ ∼ p TheapproximantsconvergegeometricallyontheentireRiemannsurface,asituationpossibleonlybecausethefunc- tionZ(g)hasnoothersingularityatfinitedistance. Numerical verifications. With about 60 terms, the slope is found to be 1/kρ 0.2209 in agreement with the k prediction1/R = 0.2209(onceeven–oddorderoscillationsaretakenintoaccount).≈Thelogarithmoftheerrorhasa slope0.696/0.685tobecomparedwiththeprediction3/R=0.66(seetable3). 4.2. Analternativemapping Anothermappingthatalsoregularizesthepointatinfinityis λ g=ρ . (1 λ)4 − 8 Table4: ODMfortheintegrald =0forg=5: wedefine[Z(g)] [Z(g)] =δ exact ODM − k 5 10 15 20 25 30 1/ρ 0.5918 1.0297 1.5627 2.0779 2.5865 3.1376 δ 1.1 10 3 3.7 10 5 1.7 10 6 9.2 10 7 1.1 10 7 3.5 10 9 − − − − − − | | ∗ ∗ ∗ ∗ ∗ ∗ ln δ 6.7454 10.2069 13.2837 13.8898 16.0103 19.4614 | | − − − − − − k 35 40 45 50 55 60 1/ρ 3.6167 4.1877 4.6557 5.3021 5.6959 6.2458 δ 3.4 10 9 8.0 10 10 1.0 10 10 2.9 10 11 1.2 10 11 2.4 10 12 − − − − − − − ∗ ∗ ∗ ∗ ∗ ∗ ln δ 19.4706 20.9453 22.9796 24.2450 25.0907 26.7433 | | − − − − − − Numericalresultsforg=5aredisplayedintable4. Finally,fork=65,70, 1/ρ =6.7479, 7.5724, ln δ = 30.3657, 28.9505 k | | − − Ontheaveragebetweenk=5andk=70, ρ R/kwithR=9.75. k ∼ Theresultsdisplayedintable2leadtotheprediction R=9.2039. Theerrorisabout k3/4 δ= 1.59 − g1/4 and1.59hastobecomparedwiththeexpectedasymptoticvalue1.74ifoneassumesconvergenceforallg>0. 5. Thequarticanharmonicoscillator: d = 1 Forr =1,thepathintegralcorrespondstothequantumHamiltonian g H = 1p2+ 1x2+ x4. 2 2 4! TheeigenvaluesEofHaregivenbythesolutionofthetime-independentSchrdingerequation g 1ψ (x)+ 1x2+ x4 ψ(x)= Eψ(x), −2 ′′ (cid:18)2 4! (cid:19) whereψ(x)isasquare-integrablefunction. Asanexample,weconsidertheperturbativeexpansionofthelowesteigenvalue,thegroundstateenergy. Varia- tionalargumentsandscalingsuggestthemapping λ g=ρ , E = E . (17) (1 λ)3/2 (1 λ)1/2 − − Then, = P (ρ)[λ(g)]k. k E Xk 9 Largeorderbehaviour(section1.1)andasteepestdescentevaluation(table2)leadtotheprediction ρ R/kwithR=µA=32.25.... k ∼ Then,λconvergesto1as R 2/3 λ=1 +O(k 4/3) λk e R2/3k1/3/g2/3 withR2/3 =10.131.... − − − kg! ⇒ ∼ Anunbiasedfitofthenumericaldatafork 60yieldsresultswithin10%ofthepredictedvalues. Finally,afitofthe ≤ relativeerrorforg yields[20] →∞ Pk+1(ρk) e−9.6k1/3. ∝ Therelativeerroratorderkisthusofordere−k1/3(9.6+R2/3g−2/3). Onefindsconvergencefor 0.95+ g 2/3cos 2Argg >0. | |− 3 (cid:16) (cid:17) ThecorrespondingdomaincontainsasectionofthefirstRiemannsheetandextendstothesecondRiemannsheetfor g largeenough. | | 6. φ4fieldtheoryind = 3dimensions In[20],theODMmethodhasbeenappliedonfunctionsoftheinitialparametergoftheaction(1)rathertherenor- malizedparameterg introducedinsection2. Thenthepointofphysicalinterestisg ,whichcorrespondstothe r →∞ zerog˜ oftheβ-function(8). DuetoUVdivergences,aneededregularizationandrenormalization,scalingarguments ∗ are no longer applicable to determine an appropriate mapping. The relation (9) between initial and renormalized parameter shows that, for g , physical observables have an expansion in powers of g ω, where the exponent − ω=β˜ (g˜ ). Thisthensuggest→sth∞emapping ′ ∗ λ g=ρ , (18) (1 λ)1/ω − but the difficulty is that ω has to be inferred from the series (8) itself. The results obtained in this way [20] are consistentwiththoseobtainedin[12](ω=0.80(1)fromBoreltransformationandmapping),butempiricalerrorsare moredifficulttoassess. Alsotheexpectedrateofconvergenceisofordere−const.k1−ω =e−const.k0.2,whichisratherslow (seetable5). Seereference[20]fordetails. Finally,theinformationaboutthelargeorderbehaviourcannoteasilybe incorporated. Table5: SeriesfortheexponentωsummedbyODMintheφ4fieldtheorywithω =0.79anddω /dω = 0.6. 3 in cal. in − k 2 3 4 5 6 ω 0.552 0.754 0.711 0.767 0.759 k Here, toillustratetheflexibilityofthemethod, weworkdirectlywithfunctionsofg˜. Wealsotakeintoaccount thecovarianceoftheβ-functionsunderachangeofparametrization: dg β (g )= 1β (g ). (19) 1 1 2 2 dg 2 This transformation law is such that the derivative of the β-function at a zero (a fixed point), which is a physical observable,remainsunchanged. 10