It is argued that power contributions of short distance origin naturally
arise in the infrared finite coupling approach. A phenomenology of $1/Q^2$
power corrections is sketched.Comment: 5 pages, latex, no figures. Expanded version of a talk presented at
  the EPS HEP97 Conference (Jerusalem, August 1997

Grunberg, G.

English

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It is argued that power contributions of short distance origin naturally arise in the infrared finite coupling approach. A phenomenology of $1/Q^2$ power corrections is sketched.It is argued that power contributions of short distance origin naturally arise in the infrared finite coupling approach. A phenomenology of $1/Q^2$ power corrections is sketched

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POWER CORRECTIONS FROM SHORT DISTANCESG. GRUNBERGyTheory Division, CERN, CH-1211 Geneva 23, Switzerland(grunberg@pth.polytechnique.fr)ABSTRACTIt is argued that power contributions of short distance origin natu-rally arise in the infrared nite coupling approach. A phenomenologyof 1=Q2power corrections is sketched.CERN-TH/97-340 November 1997Expanded version of a talk presented at the EPS HEP97 Conference (Jerusalem, August 1997).yOn leave of absence from Centre de Physique Theorique de l'Ecole Polytechnique, 91128 PalaiseauCedex, France.1. IntroductionPower-behaved contributions to hard processes not amenable to operator prod-uct expansion (OPE) have been derived1in recent years through various techniques(renormalons, nite gluon mass, dispersive approach), which all share the assump-tion that these contributions are of essentially infrared (IR) origin. In this talk, Ipoint out that the IR nite coupling approach2;3naturally suggests the existence ofadditionnal non-standard contributions of ultraviolet (UV) origin, hence not relatedto renormalons (but which may be connected4;5to the removal of the Landau polefrom the perturbative coupling).2. Power corrections and IR regular couplingConsider the contribution to an Euclidean (quark dominated) observable aris-ing from dressed virtual single gluon exchange, which takes the generic form (aftersubtraction of the Born term):D(Q2) =Z10dk2k2s(k2) ' k2Q2!(1)The \physical" coupling s(k2) is assumed to be IR regular, and thus must dier fromthe perturbative coupling PTs(k2) ( which is assumed to contain a Landau pole) bya non-perturbative piece s(k2):s= PTs+ s(2)To determine the various types of power contributions, it is appropriate2;6todisentangle long from short distances \a la SVZ" with an IR cuto I:D(Q2) =Z2I0dk2k2s(k2) ' k2Q2!+Z12Idk2k2PTs(k2) ' k2Q2!+Z12Idk2k2s(k2) ' k2Q2!(3)The rst integral yields, for large Q2, \long distance " power contributions whichcorrespond to the standard OPE \condensates". If the Feynman diagram kernel'k2Q2is O ((k2=Q2)n) at small k2, this piece contributes an O ((2=Q2)n) term froma dimension n condensate, with the normalization given by a low energy average ofthe IR regular coupling s. The integral over the perturbative coupling in the shortdistance part represents a form of \regularized perturbation theory " (choosing the IRcut-o Iabove the Landau pole). The last integral in eq.(3) usually yields (unlesss(k2) is exponentially supressed) new power contributions at large Q2of shortdistance origin , unrelated to the OPE. Assume for instance a power law decrease:s(k2) ' c 2k2!p(4)2The short distance integral will then contribute a piece:Z1Q2dk2k2s(k2) ' k2Q2!' A c 2Q2!p(5)where A R1Q2dk2k2Q2k2p'k2Q2is a number. If one assumes moreover that p < n,the lower integration limit in eq.(5) and in A can actually be set to zero7, and onegets a parametrically leading O ((2=Q2)p) power contribution of UV origin, unrelatedto the OPE.3. Power contributions to the running couplingA power law decrease of s(k2) is a natural expectation for a coupling which isassumed to be dened at the non-perturbative level, and could eventually be derivedfrom the OPE itself as the following QED analogy shows. In QED, the couplings(k2) should be identied, in the present dressed single gluon exchange context, tothe Gell-Mann-Low eective charge , related to the photon vacuum polarisation(k2) by:(k2) =1 +  (k2=2; )(6)One expects (k2), hence (k2), to receive power contributions from the OPE. Ofcourse, this cannot happen in QED itself, which is an IR trivial theory, but mightoccur in the \large 0", Nf=  1 limit of QCD . Instead of (k2), it is convenient tointroduce the related (properly normalized) renormalisation group invariant \Adlerfunction" (with the Born term removed):R(k2) =10 dd log k2 dd log k2j = 0!(7)which contributes the higher order terms in the renormalisation group equation:dsd lnk2=  0(s)2(1 +R) (8)where 0is (minus) the one loop beta function coecient. Consider now theNf=  1limit in QCD. Then R(k2) is expected to be purely non-perturbative, since in this limitthe perturbative part of sis just the one-loop coupling PTs(k2) = 1=0ln(k2=2).Indeed, OPE-renormalons type arguments suggest the general structure8at large k2:R(k2) =1Xp=1 aplogk22+ bp! 2k2!p(9)where the log enhanced power corrections reect the presence of double IR renor-malons poles9. Eq.(8) with R as in eq.(9) can be easily integrated to give:s(k2) = PTs(k2) +2k2a1PTs(k2) + 0(a1+ b1)PTs(k2)2+O(4=k4)(10)3One actually expects : a1= b1= 0 (corresponding to the absence of d = 2 gaugeinvariant operator ), and a2= 0 (corresponding to the gluon condensate which yieldsonly a single renormalon pole). It is amusing to note that keeping only the p = 2(gluon condensate) contribution in eq.(9) with a2= 0, eq.(8) yields :s(k2) =10lnk22+b224k4(11)which coincides with a previously suggested ansatz10based on dierent arguments.4. 1=Q2power correctionsThe previous QED - inspired model (with a1= b1= 0 ) remains of academicinterest, since it is clear that the short distance power corrections induced in QCDobservables by the OPE-generated corrections in sare then parametrically consis-tent with those expected from the OPE, and are actually probably numerically smallcompared to those originating directly from the long distance piece in eq.(3) (al-though it is still an interesting question whether such short distance contributionswill not mismatch the expected OPE result for the coecient functions). The situ-ation changes if one assumes4;5the existence of 1=k2power corrections of non-OPEorigin in s. Evidence for such corrections has been found in a lattice calculation11of the gluon condensate, and physical arguments have been given5;12for their actualoccurence. For instance, consider the case where n = 2 in the low energy behaviorof the kernel 'k2Q2, i.e. where the leading OPE contribution has dimension 4 (thegluon condensate). Then, setting p = 1 in eq.(4), the parametrically leading powercontribution will be a 1=Q2correction of short distance origin given by the right-handside eq.(5) with:A Z10dk2k2Q2k2' k2Q2!(12)Assuming further the physical coupling sis universal2;3, so is the non-perturbativeparameter c in eq.(4). Then the process dependance of the strength of the 1=Q2correction is entirely contained in the computable parameter A. In particular, it isinteresting to check13whether A in the pseudoscalar channel is substantially largerthen in the vector channel, which could help resolve14a long standing QCD sum rulepuzzle15, in addition to provide further evidence for 1=Q2corrections. Note that theproposed mechanism is dierent from the (in essence purely perturbative) one basedon UV renormalons14;5. The latter yields16an enhancement factor of 18 already inthe single renormalon chain approximation (consistent with the present dressed singlegluon exchange picture), but is subject to unknown arbitrarily large corrections frommultiple renormalons chains, at the dierence of the present argument.4Acknowledgments I am grateful to M. Beneke, Yu.L. Dokshitzer, G. Marchesiniand V.I. Zakharov for useful discussions.References1. For recent reviews, see R. Akhoury and V.I. Zakharov, Nuclear Physics Proc.Suppl. 54A (1997) 217; M. Beneke, hep-ph/9706457.2. Yu.L. Dokshitzer and B.R. Webber, Physics Letters B352 (1995) 451.3. Yu.L. Dokshitzer, G. Marchesini and B.R. Webber, Nuclear Physics B469(1996) 93.4. G. Grunberg, hep-ph/9705290; hep-ph/9705460.5. R. Akhoury and V.I. Zakharov, hep-ph/9705318; hep-ph/9710257.6. M. Neubert, Physical Review D51 (1995) 5924.7. I am indepted to M. Beneke for a related observation.8. G. Grunberg, Physics Letters B349 (1995) 469.9. M. Beneke, Nuclear Physics B405 (1993) 424.10. V.N. Gribov (private communication from Yu.L. Dokshitzer).11. G. Burgio, F. di Renzo, G. Marchesini and E. Onofri, hep-ph/9706209.12. R. Akhoury and V.I. Zakharov, hep-ph/9710487.13. G. Grunberg, in preparation.14. K. Yamawaki and V.I. Zakharov, hep-ph/9406373.15. V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, NuclearPhysics B191 (1981) 301.16. S. Peris and E. de Rafael, Nuclear Physics B500 (1997) 325.5

Power corrections from short distances

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