Precision determination of r_0Lambda_MS from the QCD static energy
Nora Brambilla, Xavier Garcia i Tormo, Joan Soto, Antonio Vairo
aa r X i v : . [ h e p - ph ] J un Alberta Thy 04-10, TUM-EFT 8/10, UB-ECM-PF 10/15, ICCUB-10-029
Precision determination of r Λ MS from the QCD static energy Nora Brambilla, Xavier Garcia i Tormo, Joan Soto, and Antonio Vairo Physik Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 ∗ Departament d’Estructura i Constituents de la Mat`eria and Institut de Ci`encies del Cosmos,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain (Dated: June 11, 2018)We use the recently obtained theoretical expression for the complete QCD static energy at next-to-next-to-next-to leading-logarithmic accuracy to determine r Λ MS by comparison with availablelattice data, where r is the lattice scale and Λ MS is the QCD scale. We obtain r Λ MS = 0 . +0 . − . for the zero-flavor case. The procedure we describe can be directly used to obtain r Λ MS in theunquenched case, when unquenched lattice data for the static energy at short distances becomesavailable. Using the value of the strong coupling α s as an input, the unquenched result wouldprovide a determination of the lattice scale r . PACS numbers: 12.38.Aw, 12.38.Bx, 12.38.Cy, 12.38.Gc
The energy between a static quark and a static an-tiquark is a fundamental object to understand the be-havior of quantum chromodynamics (QCD) [1]. Its long-distance part encodes the confining dynamics of the the-ory while the short-distance part can be calculated tohigh accuracy using perturbative techniques. Perturba-tive computations of the short-distance part have beenperformed for many years [2, 3] and the two-loop cor-rections have been known for quite some time now [4–6]. When using perturbation theory to calculate theshort-distance part, the virtual emission of gluons thatcan change the color state of the quark-antiquark pair(so-called ultrasoft gluons) produce infrared divergences,which induce logarithmic terms, ln α s (1 /r ), in the staticenergy. Those effects, which first appear at the three-loop order, were identified in Ref. [7] and calculatedin Ref. [8, 9] using an effective field theory framework[10, 11]. That framework also allows for resummation ofthe ultrasoft logarithms [12], which may be large at smalldistances r . Very recently, the complete three-loop cor-rections to the static energy have become available [13–15]. Combining the results of those calculations with theresummation of the ultrasoft logarithms at sub-leadingorder [16, 17], the static energy at next-to-next-to-next-to leading-logarithmic (N LL) accuracy, i.e. includingterms up to order α n s ln n α s with n ≥
0, is now com-pletely known.In the first part of the letter, we compare the static en-ergy at N LL accuracy with lattice data. The comparisonshows that, after subtracting the leading renormalon sin-gularity, perturbation theory reproduces very accuratelythe lattice data at short distances, thus confirming atan unprecedented precision level the conclusions reachedin previous analyses [16, 18, 19]. In the second part ofthe letter, the excellent agreement of perturbation the-ory with lattice data allows us to obtain a precise deter-mination of the quantity r Λ MS , where r is the latticescale and Λ MS is the QCD scale (in the MS scheme), a key ingredient to relate low energy hadronic physics withhigh energy collider phenomenology. This constitutes themain result of our work.The static energy E ( r ) at short distances can be writ-ten as E ( r ) = V s + Λ s + δ US , (1)where V s and Λ s are matching coefficients in potentialNon-Relativistic QCD (pNRQCD) [11] and δ US containsthe contributions from ultrasoft gluons. V s correspondsto the static potential and Λ s inherits the residual massterm from the Heavy Quark Effective Theory Lagrangian.In order to obtain a rapidly converging perturbative se-ries for the static potential in the short-distance regime,it has been argued that it is necessary to implement ascheme that cancels the leading renormalon singularity[20]. The use of any such scheme introduces an addi-tional dimensional scale (which we call ρ ), upon which allthe quantities in Eq. (1) depend. We will employ the so-called RS scheme [21], in the same way as it was done inRef. [16]. The explicit expressions for E at N LL accu-racy were presented in Ref. [16] and will not be repeatedhere. We refer to that paper for details. The only new in-gredient is that the three-loop gluonic contribution to thestatic potential is now known. At three-loop order thestatic potential presents infrared divergences, which can-cel in the physical observable E after the inclusion of theultrasoft effects. Therefore, it is necessary to consistentlyuse the same scheme to factorize the ultrasoft contribu-tions for all the terms in Eq. (1). That way one obtainsthe correct three-loop coefficient for the static energy E ,which is independent of the scheme used to factorize theultrasoft contributions. Refs. [14, 15] present the resultfor the purely-gluonic three-loop coefficient of the staticpotential in momentum space, which we call a (0)3 (fol-lowing the notation of Ref. [15]). We emphasize againthat a (0)3 is scheme dependent. The corresponding coeffi-cient in the static energy can be obtained by taking the d -dimensional Fourier transform of the momentum-spacepotential (as calculated in Refs. [14, 15]) and adding toit the ultrasoft contribution. In the factorization schemeused in Refs. [14, 15], the ultrasoft contribution is givenby Eq. (8) of Ref. [14] (which we confirm). Note thatthe scheme used in Refs. [14, 15] is different from the oneused in Ref. [16]. If we then subtract Eq. (34) of Ref. [16](the ultrasoft contribution in the scheme of that paper)from the static energy we get the three-loop gluonic con-tribution to the static potential in the scheme of Ref. [16](which we will denote as a (0)3 , Ref.[16] ). By doing that we ob-tain [15] c := a (0)3 , Ref.[16] / = 222 . LL accuracyat the time that paper was written. Note that the valueof c above is within the range (215,350) predicted inRef. [16], and considerably lower than the Pad´e estimate c = 313 commonly used in the literature [22].We now compare the perturbative results for the staticenergy with the n f = 0 lattice data of Ref. [23]. All re-sults are presented in units of r . As it is explained inRef. [16], an appropriate quantity to plot for this com-parison is E ( r ) − E ( r min ) + E latt . ( r min ) , (2)where r min is the shortest distance at which lattice datais available and r E latt . ( r min ) = − .
676 [23]. Now thatthe three-loop static potential is known the normalizationof the u = 1 / R s , which is nec-essary to implement the RS scheme, can be determinedusing one order more in the perturbative expansion ofthe potential. We obtain R s = − .
333 + 0 . − . − .
033 = − . , (3)which is the value we will use. The rest of the scalesand parameters are set as in Ref. [16]. We note that,in particular, this means that we have ρ = 3 . r − and r Λ MS = 0 .
602 [24]. The comparison of the static energywith lattice data is presented in Fig. 1(a). We recall thatall the curves coincide with the lattice point at r = r min by construction [as one can see from Eq. (2)]. We notethat, due to the singlet-octet mixing in the renormaliza-tion group equation for Λ s (at order r in the multipoleexpansion of pNRQCD), the N LL curve (solid black) de-pends on a constant, which we call K . Power countingtells us that | K | ∼ Λ MS , but the constant is otherwisearbitrary. We fix it by a fit to the lattice data, whichdelivers r K = − .
06. The bands in Fig. 1(a) illustratethe effect of variations in r Λ MS = 0 . ± .
048 [24](yellow lighter band) and the effect of adding the term ± C F α /r , which is representative of the neglected higherorder corrections (green darker band); the remaining pa-rameters are kept at their original values to obtain thebands. The figure confirms that perturbation theory (inthe RS scheme) reproduces very well the lattice data for - - - r (cid:144) r r H E H r L - E H r m i n L + E l a tt . H r m i n LL (a) - - - r (cid:144) r r H E H r L - E H r m i n L + E l a tt . H r m i n LL (b)FIG. 1. (a) Comparison of the singlet static energy with lat-tice data. We plot r o (cid:0) E ( r ) − E ( r min ) + E latt . ( r min ) (cid:1) as afunction of r/r and the lattice data of Ref. [23] (red points).The dotted blue curve is at tree level, the dot-dashed ma-genta curve is at one loop, the dashed brown curve is at twoloop plus leading ultrasoft logarithmic resummation and thesolid black curve is at three loop plus next-to-leading ultrasoftlogarithmic resummation. The yellow (lighter) band is ob-tained by varying r Λ MS according to r Λ MS = 0 . ± . ± C F α /r , for the solid black curve. (b) Same but using r Λ MS = 0 . +0 . − . (see text). the static energy at short distances. We also note that theband due to the neglected higher order terms is smallerthan the one due to the uncertainty in r Λ MS . Thosefacts indicate that we should be able to use the latticedata to obtain a more precise determination of r Λ MS , aswe describe later. It is worth emphasizing that the aimof Fig. 1 is to compare with lattice data for the specific r range displayed in the figure and to see if the theoreti-cal curves follow the lattice data points starting from thepoint at the shortest distance. The scale ρ was set to thefixed value 3 . r − , which is at the center of the range,and it was kept the same for all curves. Moreover, thestatic energy was expressed as a series in α s ( ρ ), ratherthan α s (1 /r ), to reduce the uncertainty associated with R s and the specific implementation of the renormalonsubtraction.We will now assume that perturbation theory by itself(after canceling the leading renormalon) is indeed enoughto accurately describe the lattice data for the range of r we are considering, i.e. 0 . r ≤ r < . r . That is, weare assuming that non-perturbative effects are small andcan be totally neglected . With this assumption, we canuse the lattice data for the static energy to determine r Λ MS . We initially consider the set of r Λ MS for which:(i) the perturbative series for the static energy appears toconverge, and (ii) the agreement with lattice is improvedwhen increasing the perturbative order of the calculation.If we implement the condition (ii) by demanding thatthe reduced χ of the curves decreases when we increasethe perturbative order of the calculation, we obtain therange (0.58,0.8) for r Λ MS . We now proceed to improvethe precision of that determination. First we recall thatthe use of the RS scheme introduced the scale ρ in theexpressions for the static energy. We have used the value ρ = 3 . r − because it corresponds to the inverse of thecentral value of the r range we are comparing with latticedata. That way we keep ln rρ terms from becoming large[16], but in principle any ρ around that value is valid. Wewill exploit this freedom to find a set of ρ values which are“optimal” for the determination of the parameter r Λ MS by following the procedure we describe next:1. We vary ρ by ±
25% around ρ = 3 . r − , i.e. from ρ = 2 .
44 to ρ = 4 .
2. For each value of ρ and at each order in the pertur-bative expansion of the static energy, we performa fit to the lattice data. The parameters of the fitare r Λ MS for the curves from tree level to next-to-next-to leading logarithmic (N LL) accuracy (1-parameter fits), and r Λ MS and r K for the curveat N LL accuracy (2-parameter fit).3. We select those ρ values for which the reduced χ of the fits decreases when increasing the order ofthe perturbative expansion.4. Finally, from the set of ρ values obtained above,we select the ones for which the fitted value of K is compatible with the power counting (we require | r K | ≤ ρ . Wethen consider the set of fitted values of r Λ MS at N LL The leading genuine non-perturbative contribution, which is pro-portional to the gluon condensate, is of order Λ r /α s (1 /r ),and hence parametrically suppressed according to the countingof Ref. [16], which assumes Λ MS ∼ α /r . We have used steps of 1 . × − r − to do that. accuracy (denoted as x i below) for that range. In orderto give more significance to the better fits, we assign aweight to each of the x i . We choose those weights to begiven by the inverse of the reduced χ of the fit. We takethe weighted average of the x i as our central value forthe determination of r Λ MS and obtain¯ x := X i w i x i = 0 . , (4)where w i := ˜ w i / ( P j ˜ w j ) and ˜ w i is the weight of thepoint x i . To estimate the error that we should associateto that number, we first consider the weighted standarddeviation of that set of values and assign it as an errorto the weighted average , we obtain σ := s − P j w j X i w i ( x i − ¯ x ) = 0 . . (5)Fig. 2 shows the obtained fit values of r Λ MS at N LLaccuracy and at N LL accuracy, for the different valuesof ρ . The size of each point in the plot is proportionalto its weight. Additionally, we also consider the dif-ference between the weighted averages computed usingthe N LL result and the N LL result and assign it as asecond error to the weighted average at N LL accuracy(we compute the weighted average at N LL accuracy inthe ρ range obtained after step 3 of the procedure de-scribed above, since step 4 applies only to the N LL re-sult). The value of that difference is 0.011. We sumthose two errors linearly and obtain 0 . ± . r Λ MS follow-ing this procedure at different orders of accuracy. Weemphasize that the error assigned to the result must ac-count for the uncertainties associated to the neglectedhigher order terms in the perturbative expansion of thestatic energy; in that sense, we note that Table I showsthat our errors at a certain perturbative order always in-clude the central value at the next order, which gives usconfidence in the reliability of the procedure. In order tofurther assess the systematic errors steming from our pro-cedure, we have redone the analysis using two additionalweight assignements: (1) p -value weights, this analysisgives r Λ MS = 0 . ± . r Λ MS = 0 . ± . x i coming from the variance of the lattice data points . A similarly motivated procedure to estimate theoretical errorshas been used, for example, in Ref. [25]. To obtain this error we consider 1-parameter fits, with K fixed. N LL N LL3.5 3.6 3.7 3.8 3.90.600.610.620.630.640.65 r Ρ r L M S FIG. 2. Fit values of r Λ MS for different values of ρ . Therange of ρ displayed in the plot corresponds to that obtainedafter step 3 (N LL points) or step 4 (N LL points) of theprocedure described in the text. The size of each point isproportional to its assigned weight.Accuracy r Λ MS next-to-leading 0 . ± . ± . LL 0 . ± . ± . LL (no p.c.c.) 0 . ± . ± . LL 0 . ± . ± . r Λ MS obtained at different levels of ac-curacy. N LL (no p.c.c.) stands for N LL accuracy withoutimposing the power counting constraint on K , i.e. omittingstep 4 in the procedure described in the text. The first er-ror corresponds to the weighted standard deviation and thesecond one to the difference with the previous order. The error induced in this way in ¯ x is much smaller thanthe other ones we are considering (due to the fact thatthe lattice points have very small error bars) and can beneglected.According to the discussion above, our final result for r Λ MS reads r Λ MS = 0 . +0 . − . . (6)Our result is compatible with the value r Λ MS = 0 . ± .
048 given in Ref. [24], which we had been using previ-ously, but has a smaller error. We also mention that anew (preliminary) lattice calculation, which determines r Λ MS from the ghost and gluon propagators, obtains aresult, r Λ MS = 0 . ± .
01, which is similar to ours [26].We present in Fig. 1(b) a comparison of the static en-ergy with lattice data using the value obtained in Eq. (6)for r Λ MS , and the best fit value for r K when thatvalue of r Λ MS is used, namely r K = − .
39. We notethat the error bands due to uncertainties in r Λ MS andhigher-order terms in Fig. 1(b) are of comparable size.We would like to emphasize that exactly the same pro-cedure we have described here could be used in the un-quenched case. If unquenched lattice data for the static energy at short distances were available we could obtainan unquenched value for r Λ MS from it. Combining thatresult with the value of the strong coupling α s , deter-mined from other sources, would provide a determina-tion of the lattice scale r . This determination would bemodel independent and alternative to other determina-tions like, for instance, the one using the 1S-2S bottomo-nium radial excitation energy [27].In summary, we have updated the comparison of thestatic energy at N LL accuracy with lattice data by in-cluding the recently calculated purely-gluonic three-loopcontribution to the static potential, which was the lastmissing ingredient, (see Fig. 1). We confirmed that, af-ter canceling the leading renormalon singularity, pertur-bation theory can accurately reproduce the lattice dataat short distances. By taking advantage of this fact, wehave obtained a new determination of r Λ MS . Our resultfor the zero-flavor case is r Λ MS = 0 . +0 . − . , whichimproves the precision of the N ∗ Current address: Institut f¨ur Theoretische Physik, Uni-versit¨at Bern, Sidlerstrasse 5, CH-3012 Bern, Switzer-land.[1] K. G. Wilson, Phys. Rev. D , 2445 (1974).[2] W. Fischler, Nucl. Phys. B , 157 (1977).[3] A. Billoire, Phys. Lett. B , 343 (1980).[4] M. Peter, Phys. Rev. Lett. , 602 (1997).[5] M. Peter, Nucl. Phys. B , 471 (1997).[6] Y. Schr¨oder, Phys. Lett. B , 321 (1999).[7] T. Appelquist, M. Dine and I. J. Muzinich, Phys. Rev.D , 2074 (1978).[8] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Phys.Rev. D , 091502 (1999).[9] B. A. Kniehl and A. A. Penin, Nucl. Phys. B (1999)200.[10] A. Pineda and J. Soto, Nucl. Phys. Proc. Suppl. , 428(1998).[11] N. Brambilla, A. Pineda, J. Soto and A. Vairo, Nucl.Phys. B , 275 (2000).[12] A. Pineda and J. Soto, Phys. Lett. B , 323 (2000).[13] A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys.Lett. B , 293 (2008). [14] C. Anzai, Y. Kiyo and Y. Sumino, Phys. Rev. Lett. ,112003 (2010).[15] A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys.Rev. Lett. , 112002 (2010).[16] N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo,Phys. Rev. D , 034016 (2009).[17] N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo,Phys. Lett. B , 185 (2007).[18] A. Pineda, J. Phys. G , 371 (2003).[19] Y. Sumino, Phys. Rev. D , 114009 (2007).[20] M. Beneke, Phys. Lett. B , 115 (1998).[21] A. Pineda, JHEP , 022 (2001). [22] F. A. Chishtie and V. Elias, Phys. Lett. B , 434(2001)[23] S. Necco and R. Sommer, Nucl. Phys. B , 328 (2002).[24] S. Capitani, M. L¨uscher, R. Sommer and H. Wittig [AL-PHA Collaboration], Nucl. Phys. B , 669 (1999).[25] S. Durr et al. , Phys. Rev. D , 054507 (2010); A. Ramos,private communication.[26] A. Sternbeck, E. M. Ilgenfritz, K. Maltman, M. Muller-Preussker, L. von Smekal and A. G. Williams, PoS LAT2009 , 210 (2009).[27] C. T. H. Davies, E. Follana, I. D. Kendall, G. P. Lepageand C. McNeile [HPQCD Collaboration], Phys. Rev. D81