Compelling evidence of renormalons in QCD from high order perturbative expansions
aa r X i v : . [ h e p - ph ] M a y Compelling evidence of renormalons in QCD from high order perturbative expansions
Clemens Bauer, Gunnar S. Bali, and Antonio Pineda Institut f¨ur Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany Grup de F´ısica Te`orica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain (Dated: September 13, 2018)We compute the static self-energy of SU(3) gauge theory in four spacetime dimensions to order α in the strong coupling constant α . We employ lattice regularization to enable a numericalsimulation within the framework of stochastic perturbation theory. We find perfect agreement withthe factorial growth of high order coefficients predicted by the conjectured renormalon picture basedon the operator product expansion. PACS numbers: 11.15.Bt,12.38.Cy,12.38.Bx,11.10.Jj,12.39.Hg
Little is known about properties of quantum field the-ories from first principles. This is particularly so forasymptotically free gauge theories such as quantum gluo-dynamics. One of the most salient features of this theoryis the confinement of charged objects. Yet this prop-erty has not been proven, and the best evidence comesfrom the linearly rising static potential at large distancesobtained in lattice simulations. Another expected prop-erty is the asymptotic nature of perturbative weak cou-pling expansions. In four dimensional non-Abelian gaugetheories one particular pattern of asymptotic divergenceshould be determined by the structure of the operatorproduct expansion (OPE). It is usually named renor-malon [1] or, more specifically, infrared renormalon. Itsexistence has also not been proven but only tested as-suming the dominance of β -terms, which amounts toan effective Abelianization of the theory, or in the twodimensional O( N ) model [2], where it is suppressed bypowers of 1 /N . Moreover, the possible non-existence orirrelevance of renormalons in Quantum Chromodynamicshas been suggested in several papers, see, e.g. [3, 4] andreferences therein. This has motivated dedicated highorder perturbative expansions of the plaquette, e.g. [5–8], in lattice regularization, with conflicting conclusions.Powers as high as α were achieved in the most recentsimulation [9]. However, the expected asymptotic be-haviour was not seen. If confirmed, this non-observationwould cast doubt on the well-accepted lore of the OPEand renormalon physics (see [10] for a comprehensive re-view), and would significantly affect the phenomenologi-cal analysis of data from high energy physics experimentson the decay of heavy hadrons, heavy quark masses, therunning coupling parameter, parton distributions, etc..Therefore, this issue should be clarified unambiguously.In this letter we present compelling numerical evidencethat the expected renormalons indeed exist not only inmodels but in real gluodynamics. We also argue whyprevious analyses based on the plaquette have failed todetect them. The vital and new ingredients of our studyare as follows.(a) We consider a perturbative series whose leadingrenormalon is dictated by a dimension d = 1 operator, rather than by the d = 4 plaquette.(b) Using a higher order integrator and employingtwisted boundary conditions, among other improve-ments, we are able to obtain results of unprecedentedprecision on an extensive set of spacetime volumes.(c) We carefully extrapolate to the infinite volume limit,thoroughly investigating finite size effects.Perturbative expansions in powers of α , K = X n k n α n , (1)are believed to be asymptotic and not Borel summablein QCD, due to the existence of singularities in the Borelplane (renormalons). Typically k n will diverge like a nd n !,with a constant a d . This divergence pattern of k n shouldnot be arbitrary but consistent with the OPE associ-ated to a physical observable. Even though this factorialgrowth was originally discovered analyzing the Feynmandiagrams that contribute to the large β approximation,the correct divergent structure can only be inferred byassuming that the perturbative series is asymptotic andcomplies with the OPE. The OPE fixes the positions andthe structure of the renormalon singularities in the com-plex Borel plane, resulting in a more intricate patternthat cannot be obtained from the large β approxima-tion alone. Successive contributions k n α n decrease forsmall orders n down to a minimum at n ∼ / ( | a d | α ).Higher order contributions should be neglected and in-troduce an ambiguity of the order of this minimum term, k n α n ∼ exp[ − / ( | a d | α )].Within the OPE an observable R ( q, Λ) can be factor-ized into short distance Wilson coefficients C i ( q, µ ) andnon-perturbative matrix elements h O i ( µ, Λ) i of dimen-sion i : R = C ( q, µ ) h O ( µ, Λ) i + C d ( q, µ ) h O d ( µ, Λ) i (cid:18) Λ q (cid:19) d + · · · . (2) q , Λ and µ denote a perturbative, low momentum andfactorization scale, respectively, so that q ≫ µ ≫ Λ.For the plaquette, h O i = 1 and the next higher non-vanishing operator is the dimension d = 4 gluon con-densate. In this case, the perturbative expansion of C cannot be more accurate than O (Λ /q ) which is exactlyof the size of the k n α n term since (cid:18) Λ q (cid:19) d ≃ exp (cid:18) − | a d | α (cid:19) , where a d = β πd , (3)with β = 11. The so-called leading infrared renormalonof this expansion cancels the ultraviolet ambiguity of thenext order non-perturbative matrix element so that thephysical observable R is well-defined.From this discussion it is evident that we should studyseries expansions with the smallest possible n or, equiv-alently, d . For d = 1 the perturbative expansion shouldstart to diverge at an order n that amounts to aboutone fourth of that for the plaquette. This applies to thepole mass (see [11, 12]) and to the associated self-energyof a static source, which we consider here. The latterdoes not have a continuum limit, as it linearly dependson the ultraviolet regulator. Here we consider lattice reg-ularization with the Wilson gauge action [13] and writethe self-energy in the following way: δm = 1 a X n ≥ c n α n +1 (1 /a ) . (4) a − , the inverse lattice spacing, provides the ultravio-let cut-off. The large n behaviour of the coefficients c n isregulator independent, universal and equal to the asymp-totic behaviour of the pole mass up to O ( e − /n ) terms(due to subleading renormalons): c n n →∞ = N m (cid:18) β π (cid:19) n Γ( n + 1 + b )Γ(1 + b ) (5) × (cid:18) b ( n + b ) s + · · · (cid:19) . The coefficients b and s were computed in [14]. Theyread (see [15] for details) b = β β , s = 14 β b (cid:18) β β − β (cid:19) . (6)For a static source in the fundamental (triplet) repre-sentation the normalization constant N m is exactly thesame as for the leading renormalon of a heavy quark polemass. This renormalon is also related to a renormalonof the singlet static potential since these contributionscancel from the energy E ( r ) = 2 m + V ( r ) [16–18]. Foradjoint sources it corresponds to a specific combination ofpole mass and adjoint static potential renormalons [19].The factor N m is cancelled in the ratios c n c n − n = β π (cid:20) bn − (1 − b s ) b s n + O (cid:18) n (cid:19)(cid:21) . (7)We obtain the expansion coefficients c n of the staticenergy from the temporal Polyakov line on hypercubic TABLE I. Lattice geometries. Volumes with boldface timeextents are expanded up to O ( α ), the others up to O ( α ). N S N T N S N T , , , 12, 16 12
10 8, , 12, 16, 20 16 12, 16, 20 lattices. We investigate volumes of N T lattice points inthe time direction and spatial extents of N S points. For-mally we may introduce an anisotropy a t = a s . In thiscase the lattice action, that is invariant under time orparity reversal, agrees with the continuum action up to O ( a t , a s ) terms. The temporal and spatial lattice ex-tents in physical units are given by a t N T and a s N S , re-spectively, so that the only dimensionless combinationsconsistent with the leading order lattice artefacts are a t / ( a t N T ) = 1 /N T and 1 /N S . Therefore, within per-turbation theory, where we cannot dynamically generateadditional scales, the leading order lattice artefacts areindistinguishable from O (1 /N T , /N S ) finite size effects.We choose periodic boundary conditions in time and,to eliminate zero modes and to improve the numericalstability, twisted boundary conditions [20–23] in all spa-tial directions. The Polyakov line is defined by L ( R ) ( N S , N T ) = 1 N S X n d R tr " N T − Y n =0 U R ( n ) , (8)where U Rµ ( n ) ≈ e igA Rµ [( n +1 / a ] ∈ SU(3) denotes a gaugelink in representation R , connecting the sites n and n + ˆ µ , n i ∈ { , . . . , N S − } , n ∈ { , . . . , N T − } and g = √ πα .We implement triplet and octet representations R of di-mensions d R = 3 and 8. The link U ( n ) appears withinthe covariant derivative of the static action ¯ ψD ψ , thediscretization of which is not unique. We use singlystout-smeared [24] (smearing parameter ρ = 1 /
6) covari-ant transporters instead of U ( n ) as a second, alternativechoice, to demonstrate the universality of our findings.We remark that neither the lattice spacing nor thestrong coupling parameter α enter our simulationsexplicitly. Numerical stochastic perturbation theory(NSPT) [25–27] enables us to directly calculate coeffi-cients of perturbative expansions. We employ the vari-ant of the Langevin algorithm introduced in [28] thatonly quadratically depends on a time step ∆ τ . Extrap-olations to ∆ τ = 0 were performed on a subset of latticevolumes where we found agreement within statistical er-rors between all our extrapolated expansion coefficientsand those obtained at ∆ τ = 0 .
05. For the geometrieslisted in Table I we restrict ourselves to this fixed value,which, within errors, effectively corresponds to ∆ τ = 0.We expand the logarithm of the smeared and un-smeared Polyakov lines in different representations to ob-tain the corresponding static energies: P ( N S , N T ) = − ln h L ( N S , N T ) i aN T N S ,N T →∞ −→ δm . (9)Fortunately, the dependence of this logarithm on N T and N S can be deduced and only a few parameters need tobe fitted at each order: aP = X n ≥ (cid:20) c n α n +1 (cid:0) a − (cid:1) − f n N S α n +1 (cid:16) ( aN S ) − (cid:17) + O (cid:18) N T , N S (cid:19)(cid:21) ≈ X n ≥ h c n + ∆ (1) n ( N S ) + ∆ (2) n ( N S , N T ) i α n +1 (cid:0) a − (cid:1) , ∆ (1) n = − N S h f n + logs ( c ) n ( N S ) i , (10)∆ (2) n = 1 N T (cid:26) v n − N S h f ( v ) n + logs ( v ) n ( N S ) i(cid:27) + 1 N S (cid:26) w n − N S h f ( w ) n + logs ( w ) n ( N S ) i(cid:27) . The logs ( c ) n ( N S ) are polynomials of ln( N S ) of order n − f j and the β -function coefficients β j where j ≤ n −
1. These terms areentirely determined by the renormalization group run-ning of α . The logs ( v/w ) n ( N S ) are obtained in the sameway. In the N T → ∞ limit ∆ (1) n is the dominant cor-rection while ∆ (2) n includes the leading O (1 /N T , /N S )lattice artifacts discussed above.The term ∆ (1) n originates from interactions with mirrorimages, see also [29]. This effectively produces a staticpotential between charges separated at distances aN S ,but without self-energies. Therefore, we expect the highorder behaviour of f n and c n to be dominated by oneand the same renormalon. This can also be illustratedconsidering the leading dressed gluon propagator D ( k ) ∝ /k , where k = 0. With the (formal) ultraviolet cut-off1 /a and an infrared cut-off 1 / ( aN S ) this can be writtenas (ignoring lattice corrections), P ∝ Z /a / ( aN S ) dk k D ( k ) (11) ∼ a X n c n α n +1 (cid:0) a − (cid:1) − aN S X n c n α n +1 (cid:0) ( aN S ) − (cid:1) , after perturbatively expanding D ( k ). When re-expressing α (( aN S ) − ) in terms of α ( a − ) we may con-sider two situations:(a) N S > e n . In this limit the last term of Eq. (11)is exponentially suppressed in n and the renormalon candirectly be obtained from a large order expansion of aP . (b) N S < e n . The last term of Eq. (11) is important andthe renormalon cancels order-by-order in n .In present-day numerical simulations N S < e n , and theterm ∆ (1) n needs to be taken into account, in combinationwith c n . A similar phenomenon was numerically observedfor the static singlet energy E ( r ) = 2 m + V ( r ) [19, 30].This teaches us that to correctly identify the renormalonstructure of δm , it is compulsory to incorporate the 1 /N S corrections. So far, in studies of high order perturbativeexpansions of the plaquette the corresponding finite sizeterms have been neglected. Our fits indeed yield f n ≃ c n for large n , in clear support of the renormalon dominancepicture.In the lattice scheme β , β and β are known [31].The effects of higher β j start at O ( α ), but this uncer-tainty in our parametrization quickly becomes negligibleat high orders where the coefficients f j , governed by the d = 1 renormalon, will dominate. This can be quanti-fied systematically in a large n analysis [32], where anypossible renormalon of the lattice β -function is sublead-ing ( d > β we have performed fits including β j for j ≤ , O ( α ), one may expect additional finite sizeterms ∝ ln( N T /N S ) /N S from a possible mixing of the an-titriplet interaction between mirror charges with sextetand higher representations, mediated by ultrasoft gluons,in analogy to the mixing of singlet and octet static poten-tials in potential nonrelativistic QCD (pNRQCD) [33].These terms are subleading from the renormalon pointof view ( d = 3). Moreover, aN S provides an infraredcut-off to gluon momenta so that one would only expectsuch contributions in the limit N S ≫ N T that we do notinvestigate and, indeed, we see no numerical evidence ofthem.Our data are sensitive to the 1 /N T correction termswithin ∆ (2) n . However, including w n or f ( w ) n as additionalfit parameters did not significantly improve the χ -valueand so we decided to omit the 1 /N S and 1 /N S terms.Note that these contributions, if present, can numericallyeasily be distinguished from 1 /N S and become irrelevantat relatively small N S , unlike logs( N S ) /N S terms.As a cross-check we calculate diagrammatically, c = 2 . . . . , (12) c = 11 . , f = 0 . , (13) f ( w )0 = 0 . , w = v = f ( v )0 = 0 , (14)for the unsmeared Polyakov line. In this case f ( w )0 , the1 /N S coefficient, does not vanish but it is small. Forfundamental sources, c and c were known diagrammat-ically before and c numerically [29, 34]. Our fit repro-duces these values. For adjoint sources the above coeffi-cients need to be multiplied by the factor C A /C F = 9 /
6 7 8 9 10 8 10 12 14 16 18 20 c / N T N S =16N S =14N S =12N S =11N S =10N S = 9N S = 8N S = 7 FIG. 1. Comparison between the global fit and data for n = 9. c n / ( n c n - ) nsmearedunsmearedNNLONLOLO FIG. 2. The ratio c n / ( nc n − ) for the smeared and unsmearedfundamental static self-energies, compared to the predictionEq. (7) at different orders of the 1 /n expansion. Table I) and orders of perturbation theory (with four pa-rameters per order) in Fig. 1, where a comparison to the n = 9 data is shown. We find smeared and unsmeareddata to be well described by the fits, with reasonable χ /N DF ≈ .
29 and 1.46, respectively. Note that the fac-torial growth found (and expected) for the coefficients f n produces very sizable 1 /N S terms at high orders.In Fig. 2 we compare the infinite volume extrapolatedratios c n / ( nc n − ) to the theoretical prediction, Eq. (7).LO, NLO and NNLO refer to this prediction, truncatedat O (1), O (1 /n ) and O (1 /n ), respectively. The data arerobust to subtracting lattice artefacts (the ∆ (2) termsof Eq. (10)) or to truncating at different orders in β j ,see Fig. 3. Particularly reassuring is the universalityof the result; fits to smeared and unsmeared Polyakovloop expansions give the same large n behaviour, fullyconsistent with the dominance and universality of theinfrared renormalon; smearing only affects the ultravi-olet behaviour. Fits to the octet representation data c n / ( n c n - ) n β β , β β , β , β NNLOLO
FIG. 3. The same as Fig. 2 for the unsmeared data, truncatingat different orders in β j . also show exactly the same behaviour, again in agree-ment with the renormalon dominance picture. Also notethat NSPT data for different orders are statistically cor-related. These correlations work in our favour. We post-pone the details of this to [32].Finally we determine the normalization of the polemass renormalon, see Eq. (5), and obtain N lat m =18 . N lat m = 19 . N MS m =Λ lat N lat m / Λ MS = 0 . N MS m ≃ .
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