View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server DTP/95/36 April 1995 All-Orders Renormalon Resummations for some QCD Observables C.N.Lovett-Turner and C.J.Maxwell Centre for Particle Theory, University of Durham South Road, Durham, DH1 3LE, England Exact large-N results for the QCD Adler D -function and Deep Inelastic Scattering f sum rules are used to resum to all orders the p ortion of QCD p erturbative co ecients 1 containing the highest p ower of b= (11N {2N ), for SU(N ) QCD with N quark avours. f f 6 These terms corresp ond to renormalon singularities in the Borel plane and are exp ected asymptotically to dominate the co ecients to all orders in the 1=N expansion. Remark- f ably,we note that this is already apparent in comparisons with the exact next-to-leading order (NLO) and next-to-NLO (NNLO) p erturbative co ecients. The ultra-violet (UV ) and infra-red (IR) renormalon singularities in the Borel transform are isolated and the Borel sum (principal value regulated for IR) p erformed. Resummed results are also ob- + tained for the Minkowski quantities related to the D -function, the e e R-ratio and the analogous -lepton decay ratio, R . The renormalization scheme dep endence of these par- tial resummations is discussed and they are compared with the results from other groups [1{3] and with exact xed order p erturbation theory at NNLO. Prosp ects for improving the resummation by including more exact details of the Borel transform are considered. HEP-PH-9505224 1 1 Intro duction In several recent pap ers [1{3] the p ossibility of resummation to all orders of the part of p erturbative corrections contributed by QCD renormalons has b een explored. The QCD p erturbative corrections to some generic QCD Green's function or current correlator (to the Adler D -function of QCD vacuum p olarization for instance) can b e written: 2 3 k+1 D = a + d a + d a + + d a + ; (1) 1 2 k where a = is the renormalization group (RG) improved coupling; and the p erturba- s tive co ecients d can themselves b e written as p olynomials of degree k in the number k of quark avours, N ;we shall assume massless quarks. f [k ] [k 1] [0] k k 1 d = d N + d N + + d : (2) k f k k f k [k r ] The \N -expansion" co ecients, d , will consist of sums of multinomials in the ad- f k 2 joint and fundamental Casimirs, C =N , C =(N {1)=2N , of SU(N ) QCD; and will have A F k r s s the structure C C . The terms in this \N -expansion" will corresp ond to Feynman f A F diagrams with di ering numb ers of vacuum p olarization lo ops. By explicit evaluation of [k ] diagrams with chains of such lo ops inserted, it has b een p ossible to obtain the leading d k co ecient exactly to all orders for the Adler D -function [4{6] (and hence its Minkowski + continuations, the e e QCD R-ratio and the -decay ratio, R ); the Gross Llewellyn- Smith (GLS) sum rule corrections [6]; and heavy quark decay widths and p ole masses [k ] [7]. A general pro cedure enabling d to b e obtained from knowledge of the one-lo op k correction with a ctitious gluon mass has b een develop ed [1, 2]. In a recent pap er [8]we p ointed out that the large order b ehaviour of p erturbative co ecients is most transparently discussed in terms of an expansion of the p erturbative co ecients in p owers of b=(11C {2N )=6, the rst QCD b eta-function co ecient: A f (k ) (k 1) (0) k k 1 d = d b + d b + + d : (3) k k k k 11 This \b-expansion" is uniquely obtained by substituting N =( C 3b) in equation (2). f A 2 [k ] (k ) [k ] k d =(1=3) d and so exact knowledge of the leading-N d to all orders implies exact f k k k (k ) knowledge of the d . k In QCD one exp ects the large-order growth of p erturbative co ecients to b e driven by Borel plane singularities at z =z =2`=b, with `=1; 2; 3;:::. The singularities on the ` negative real axis are the so-called ultra-violet renormalons, UV , and those on the p ositive ` real axis are the infra-red renormalons, IR . These singularities result in the large-order ` k b ehaviour of the co ecients d b k !. Indeed we showed in reference [8] that, given a set k of renormalon singularities at the exp ected p ositions in the Borel plane, the leading terms (k ) k in the b-expansion, d b , should, if expanded in p owers of N , asymptotically repro duce f k [k r ] the d co ecients of equation (2) up to O (1=k ) accuracy.We conversely checked that k (k ) k the exact d b results corresp onded to a set of renormalon singularities at the exp ected k p ositions. We shall demonstrate in section 2 of the present pap er that, for the Adler D -function and the GLS sum rule, the N -expansion co ecients obtained by expanding f (1) (2) 2 3 d b and d b are in go o d (10{20% level) agreement with those of the exact O (a ) next- 1 2 to-next-to-leading order (NNLO) p erturbative calculations for these quantities, so that 2 the anticipated asymptotic dominance of the leading-b term is already apparentinlow orders. Given the dominance of the leading-b terms, an obvious prop osal is to sum them to all orders. That is to split D into two comp onents: (L) (NL) D = D + D (4) where `L' and `NL' sup erscripts refer to leading and non-leading terms in the b-expansion. 1 X (k ) (L) k k +1 D d b a (5) k k =1 and 1 k 1 X X (`) (NL) k +1 ` D a d b : (6) k k=1 `=0 The summation of terms can b e achieved by using the Borel sum. The Borel integral can itself b e split into two comp onents and is well de ned for the UV singularities on the ` (L) negative axis, which contribute p oles to the Borel transform of D . The integral can b e p erformed explicitly in terms of exp onential integral functions and other elementary (L) functions. The piece of the Borel integral for D involving the IR singularities on ` the p ositive real axis is formally divergent; but a principal value or other prescription can b e used to go around the p oles. The sp eci cation of this prescription is intimately linked to the pro cedure needed to combine the non-p erturbativevacuum condensates in the op erator pro duct expansion (OPE) with the p erturbation theory in order to arriveat awell-de ned result for D [9]. In recent pap ers by Neub ert [3] and by Ball, Beneke and Braun [1, 2] a summation of the leading-b terms has also b een considered. In these pap ers it has b een motivated as a generalisation of the BLM scale xing prescription [10 ] and termed \nave non- ab elianization" [11]. The Neub ert pro cedure uses weighted integrals over a running cou- pling. For the Euclidean Adler D -function this representation is equivalent to splitting the Borel integral into ultra-violet renormalon and infra-red renormalon singularities and principal value regulating the latter. When one continues to Minkowski space to obtain + the e e R-ratio and the -decay ratio, R , there are several inequivalentways to p erform the continuation of the running coupling representation; and hence apparent additional non-p erturbativeambiguities are claimed. In our view only a consideration of the singu- larities in the Borel integral provides a satisfactory way of combining p erturbative e ects with non-p erturbative condensates, along the lines discussed in reference [9]; and these extra uncertainties are spurious. + Our intention in this pap er is to fo cus on the Adler D -function, the e e R-ratio, R and the GLS sum rule (the latter was not considered in references [1{3]). For all of these quantities there exist exact NNLO xed order p erturbative calculations and our interest is in comparing the leading-b resummation with these exact xed order results. The large-b results provide partial information ab out the Borel transform and the question is how this can b est b e utilised. We discuss the renormalization scheme (RS) dep endence of (L) (NL) (L) the split b etween D and D in equation (4), the relative contribution of D b eing RS-dep endent. This RS uncertainty needs to b e kept in mind and is carefully discussed. 3 The organisation of the pap er is as follows. In section 2 we shall discuss the b-expansion for the Adler D -function and the GLS sum rule and will consider the extent to which the dominance of the leading-b term is RS-dep endent. In section 3 the exact large-b results are used to determine partially the Borel transforms for these quantities; and, having split them into UV and IR renormalon pieces, a resummation along the lines discussed ab ove is p erformed.