Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation
EEPJ manuscript No. (will be inserted by the editor)
Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation
R.A. Davison a and B.R. Webber Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK19 September 2008
Abstract.
We re-evaluate the non-perturbative contribution to the thrust distribution in e + e − → hadrons,in the light of the latest experimental data and the recent NNLO perturbative calculation of this quantity.By extending the calculation to NNLO+NLL accuracy, we perform the most detailed study to date of theeffects of non-perturbative physics on this observable. In particular, we investigate how well a model basedon a low-scale QCD effective coupling can account for such effects. We find that the difference betweenthe improved perturbative distribution and the experimental data is consistent with a 1 /Q -dependentnon-perturbative shift in the distribution, as predicted by the effective coupling model. Best fit values of α s (91 . . +0 . − . and α (2 GeV) = 0 . ± .
03 are obtained with χ / d.o.f. = 1 .
09. This isconsistent with NLO+NLL results but the quality of fit is improved. The agreement in α is non-trivialbecause a part of the 1 /Q -dependent contribution (the infrared renormalon) is included in the NNLOperturbative correction. PACS. e − e + interactions – 12.38.Cy Summation of QCD perturbationtheory – 12.38.Lg Other nonperturbative QCD calculations One of the most common and successful ways of test-ing QCD has been by investigating the distribution ofevent shapes in e + e − → hadrons, which have been mea-sured accurately over a range of centre-of-mass energies(14 GeV ≤ Q ≤
207 GeV), and provide a useful way ofevaluating the strong coupling constant α s .The main obstruction to obtaining an accurate valueof α s from these distributions is not due to a lack of pre-cise data but to dominant errors in the theoretical calcu-lation of the distributions. In particular, there are non-perturbative effects that cannot yet be calculated fromfirst principles but cause power-suppressed corrections thatcan be significant at experimentally accessible energy scales.In the case of the thrust distribution dσ/dT , previous workhas shown that matching α s with a low-scale effective cou-pling α eff which extrapolates below some infra-red match-ing scale µ I results in a 1 /Q -dependent shift in the distri-bution that accounts well for the discrepancy between theexperimental and perturbative results [1].The presence of 1 /Q corrections in event shapes is ageneric expectation based on the renormalon analysis ofperturbation theory, which implies an ambiguity of thatorder in the perturbative predictions for these observables(see [2,3] for reviews). The low-scale effective coupling hy- a Address after 1 October 2008:
Rudolf Peierls Centre forTheoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK pothesis [4] leads to universality relations between the cor-rections to different observables, valid to lowest order inthe effective coupling, and to a well-defined prescriptionfor matching the perturbative and non-perturbative con-tributions.The calculation of Ref. [1] was performed to NLO+NLLaccuracy, i.e. terms up to O (cid:0) α s (cid:1) were retained exactlywhile exponentiating logarithmically-enhanced terms ofthe form α ns ln n +1 (1 − T ) and α ns ln n (1 − T ) were summedto all orders. In the present paper, the recent evaluation ofthe NNLO term (i.e O (cid:0) α s (cid:1) ) in the fixed-order perturba-tion series expansion of the thrust distribution [5,6] is usedto refine the perturbative calculation of the distribution toNNLO+NLL accuracy and thus to reduce the uncertaintypresent in the theoretical prediction. A low-scale effectivecoupling is then introduced and matched to NNLO. This isagain found to be a good method for dealing with the non-perturbative shift. By comparing the NNLO+NLL+shiftresults with the latest experimental distributions, valuesof α s and α = 1 µ I (cid:90) µ I dµ α eff ( µ ) (1)are obtained. These are consistent with those determinedto NLO+NLL accuracy. The agreement is non-trivial be-cause a part of the 1 /Q -dependent contribution – the in-frared renormalon – is included in the NNLO perturbativecorrection.The organisation of the paper is as follows. In Sect. 2we briefly recall the relevant properties of the thrust distri- a r X i v : . [ h e p - ph ] N ov R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation bution, the fixed-order calculation and the resummation oflarge logarithms. Sect. 3 presents the predictions of pertur-bative NNLO+NLL matching and the power dependenceof the discrepancy with experimental data. The matchingto the low-scale effective coupling and comparisons withdata are performed in Sect. 4, and our conclusions arepresented in Sect. 5.
We recall that the thrust T is a measure of the distributionof momenta of the final state hadrons: T = max −→ n (cid:32) (cid:80) Ni =1 |−→ p i . −→ n | (cid:80) Ni =1 |−→ p i | (cid:33) , (2)where −→ n is a unit vector and we sum over the 3-momentumof each final-state hadron in the centre-of-mass frame.Theoretical calculations of thrust are performed by sum-ming over the individual final state partons, as the hadro-nisation process is still not well understood. T can varybetween the limits T = 1 for back-to-back jets and T = for a uniform angular distribution of hadrons.For comparison with experiments, it is the thrust dis-tribution 1 σ dσdT , (3)which is relevant, where σ is the total cross-section fore + e − → hadrons. In calculations it is more convenient touse the event shape variable t ≡ − T, (4)which has the two-jet limit t = 0. The distribution awayfrom this limit therefore depends directly upon the pro-duction of extra final-state partons at QCD vertices, andhence is ideal for testing QCD and evaluating α s . Thenormalised thrust cross section is then defined as R ( t ) = (cid:90) t dt σ dσdt = (cid:90) − t dT σ dσdT . (5) The perturbative expansion of the normalised thrust crosssection has the general form R ( t ) = 1 + ¯ α s R ( t ) + ¯ α s R ( t ) + ¯ α s R ( t ) + . . . , (6)where R ( t ) is the leading order (LO) coefficient, R ( t ) isthe next-to-leading order (NLO) coefficient, R ( t ) is thenext-to-next-to-leading order (NNLO) coefficient etc. and¯ α s ≡ α s / π . Solving the renormalisation group equationfor the running coupling to NNLO gives α s ( µ R ) = 2 πβ L (cid:32) − β ln Lβ L + 1 β L (cid:20) β β (cid:0) ln L − ln L − (cid:1) + β β (cid:21)(cid:33) , (7) where µ R is some chosen renormalisation scale (we take µ R = Q except where stated otherwise), β = 11 N − N F β = 17 N − N N F − C F N F ,β = 1432 (2857 N + 54 C F N F − N C F N F − N N F + 66 C F N F + 79 N N F ) , (8)with C F = ( N − / N for an SU( N ) gauge theory with N F active flavours ( N = 3 for QCD and N F = 5 at allenergies considered here) and L = ln( µ R /Λ (5) 2 MS ), Λ (5) MS being the 5-flavour QCD scale in the modified minimalsubtraction renormalisation scheme.A numerical Monte Carlo program, EERAD3 [7], has re-cently been developed which computes the process e + e − → jets to NNLO in α s via the decay of a virtual neutral gaugeboson ( γ or Z ) to between three and five partons [5,6]. The
EERAD3 predictions for the thrust distribution at avariety of centre-of-mass energies Q spanning the range14 GeV to 206 GeV are shown by the green/lighter curvesin Figs. 1-3. The values of α s ( Q ) were calculated using Λ (5) MS = 0 .
204 GeV, corresponding to the world average α s (91 . . The enhancement of the distribution at low t due to softor collinear gluon emission (as seen in Figs. 1- 3) is presentat all orders in perturbation theory: the dominant term at n th order is typically of the form1 σ dσdt ∼ α ns t ln n − (cid:18) t (cid:19) . (9)Thus we see that at low t the condition α s (cid:28) α s L (cid:28)
1, where L ≡ ln(1 /t ). To obtain accurate predictions in the two-jet limit t →
0, we must therefore take account of theseenhanced terms at all orders in perturbation theory byresumming them.Resummation of large logarithms is possible for eventshape variables y that exponentiate [10], i.e. their corre-sponding normalised cross section can be written in theform R ( y ) = C ( α s ) Σ ( y, α s ) + D ( y, α s ) , (10) A recent calculation [8] finds some discrepancies withRefs. [5,6], but these are not significant in the kinematic re-gions that we consider..A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation 3 / ! d ! / d t t Q=14 GeV " s=0.1644 NNLONNLO+NLLTASSO (14 GeV) / ! d ! / d t t Q=35 GeV " s=0.1375 NNLONNLO+NLLTASSO (35 GeV)JADE (35 GeV) / ! d ! / d t t Q=44 GeV " s=0.1322 NNLONNLO+NLLTASSO (44 GeV)JADE (44 GeV)DELPHI (45 GeV) / ! d ! / d t t Q=55 GeV " s=0.1273 NNLONNLO+NLLAMY (54.5 GeV)L3 (55.3 GeV) / ! d ! / d t t Q=66 GeV " s=0.1236 NNLONNLO+NLLL3 (65.4 GeV)DELPHI (66.0 GeV)
Fig. 1.
Fixed-order (NNLO), resummed (NNLO+NLL) andexperimental thrust distributions: Q = 14 −
66 GeV. / ! d ! / d t t Q=91.2 GeV " s=0.1176 NNLONNLO+NLLOPAL (91.0 GeV)DELPHI (91.2 GeV)ALEPH (91.2 GeV)SLD (91.2 GeV) / ! d ! / d t t Q=133 GeV " s=0.1113 NNLONNLO+NLLOPAL (133 GeV)ALEPH (133 GeV) / ! d ! / d t t Q=161 GeV " s=0.1084 NNLONNLO+NLLALEPH (161.0 GeV)L3 (161.3 GeV) / ! d ! / d t t Q=172 GeV " s=0.1074 NNLONNLO+NLLALEPH (172.0 GeV)L3 (172.3 GeV) / ! d ! / d t t Q=183 GeV " s=0.1065 NNLONNLO+NLLL3 (182.8 GeV)ALEPH (183.0 GeV)DELPHI (183.0 GeV)
Fig. 2.
Fixed-order (NNLO), resummed (NNLO+NLL) andexperimental thrust distributions: Q = 91 −
183 GeV. R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation / ! d ! / d t t Q=189 GeV " s=0.1060 NNLONNLO+NLLL3 (188.6 GeV)ALEPH (189.0 GeV)DELPHI (189.0 GeV) / ! d ! / d t t Q=200 GeV " s=0.1052 NNLONNLO+NLLL3 (200 GeV)ALEPH (200 GeV)DELPHI (200 GeV) / ! d ! / d t t Q=206 GeV " s=0.1048 NNLONNLO+NLLALEPH (206.0 GeV)L3 (206.2 GeV)DELPHI (207.0 GeV)
Fig. 3.
Fixed-order (NNLO), resummed (NNLO+NLL) andexperimental thrust distributions: Q = 189 −
207 GeV. where C ( α s ) = 1 + ∞ (cid:88) n =1 C n ¯ α ns , ln Σ ( y, α s ) = ∞ (cid:88) n =1 n +1 (cid:88) m =1 G nm ¯ α ns L m = Lg ( α s L ) + g ( α s L ) + α s g ( α s L ) + . . . , (11) L = ln(1 /y ) and D ( y, α s ) is a remainder function thatvanishes order-by-order in perturbation theory in the two-jet limit y →
0. The functions g i ( α s L ) are power se-ries in α s L (with no leading constant term) and hence Lg ( α s L ) sums all leading logarithms α ns L n +1 , g ( α s L )sums all next-to-leading logarithms (NLL) α ns L n and thesubdominant logarithmic terms α ns L m with 0 < m < n are contained in the g , g , . . . terms. The functions g i thusresum the logarithmic contributions at all orders in per-turbation theory, and knowledge of their form allows usto make accurate perturbative predictions in the range α s L (cid:46) α s L (cid:28) coherent branching formalism [11,12], which uses consecutive branchings from an initial quark-antiquark state to produce multi-parton final states toNLL accuracy. The results of this calculation depend uponthe jet mass distribution J (cid:0) Q , k (cid:1) – the probability ofproducing a final state jet with invariant mass k from aparent parton produced in a hard process at scale Q –and its Laplace transform ˜ J ν (cid:0) Q (cid:1) . To the required accu-racy, the thrust distribution is1 σ dσdt = Q πi (cid:90) C dνe tνQ (cid:104) ˜ J µν (cid:0) Q (cid:1)(cid:105) , (12)where the contour C runs parallel to the imaginary axison the right of all singularities of the integrand,ln ˜ J µν (cid:0) Q (cid:1) = (cid:90) duu (cid:16) e − uνQ − (cid:17) (cid:34)(cid:90) uQ u Q dµ µ C F α s ( µ ) π (cid:18) − K α s ( µ )2 π (cid:19) − + . . . (cid:35) , (13)and K = N (cid:18) − π (cid:19) − N F . (14)This expression demonstrates explicitly that the diver-gence of α s ( µ ) at low µ will affect the perturbative thrustdistribution – such effects are related to the renormalonmentioned earlier. To NLL accuracy, however, we can ne-glect the low µ region (although we will return to it inSect. 4) to give the thrust resummation functions [10] g ( α s L ) = 2 f ( β ¯ α s L ) ,g ( α s L ) = 2 f ( β ¯ α s L ) − ln Γ [1 − f ( β ¯ α s L ) − β ¯ α s Lf (cid:48) ( β ¯ α s L )] , (15)where f ( x ) = − C F β x [(1 − x ) ln (1 − x ) − − x ) ln (1 − x )] ,f ( x ) = − C F Kβ [2 ln (1 − x ) − ln (1 − x )] − C F β ln (1 − x ) − C F γ E β [ln (1 − x ) − ln (1 − x )] − C F β β (cid:2) ln (1 − x ) − − x ) + 12 ln (1 − x ) − ln (1 − x ) (cid:3) , (16) By writing the K dependence in the form shown in (13), wechange from the MS renormalisation scheme to the so-calledbremsstrahlung scheme [13]..A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation 5 with Γ the Euler Γ -function, γ E the Euler constant, and C F , K and β n the constants previously defined.By combining these with the fixed-order calculation,we can obtain a new estimate of the normalised cross sec-tion to NLL accuracy. This should particularly improvethe fixed-order estimate in the two-jet region, where L becomes large. Naively we would simply calculate R ( t ) asdefined in Eq. (10), but it turns out to be considerablysimpler to consider ln R ( t ), as we recall next. In the log-R matching scheme , we rewrite the exponenti-ation formula asln R ( t ) = F ( α s ) + ln Σ ( t, α s ) + H ( t, α s ) , (17)where F ( α s ) is a power series in α s and H ( t, α s ) denotesthe remainder function which vanishes as t → R ( t ) toorder M , we can write Eq. (6) asln R ( t ) = ln (cid:32) M (cid:88) n =1 ¯ α ns R n ( t ) (cid:33) = M (cid:88) n =1 ¯ α ns R n ( t ) − (cid:32) M (cid:88) n =1 ¯ α ns R n ( t ) (cid:33) + 13 (cid:32) M (cid:88) n =1 ¯ α ns R n ( t ) (cid:33) − . . . . (18)The matched estimate is obtained by combining the M thorder perturbative result with the resummed contribu-tions and subtracting the terms of order ≤ M in ln Σ (asthese are already accounted for in the fixed-order terms).Thus for a fixed-order calculation to order α s , the matchedestimate after resumming large logarithms to NLL accu-racy isln R ( t ) = Lg ( α s L ) + g ( α s L )+ ¯ α s (cid:0) R ( t ) − G L − G L (cid:1) + ¯ α s (cid:18) R ( t ) −
12 [ R ( t )] − G L − G L (cid:19) + ¯ α s (cid:18) R ( t ) − R ( t ) R ( t ) + 13 [ R ( t )] − G L − G L (cid:19) . (19)The coefficients G nm can be extracted by expandingthe functions g ( α s L ) and g ( α s L ) as power series in α s L and comparing them with the definition (11) of G nm : G = 3 C F ,G = − C F ,G = − C F (cid:2) π C F + (cid:0) − π (cid:1) N − N F (cid:3) ,G = − C F N − N F ) ,G = C F (cid:2) ζ (3) C F − π N C F − (cid:0) − π (cid:1) N + (cid:0)
108 + 144 π (cid:1) C F N F + (cid:0) − π (cid:1) N N F − N F (cid:3) ,G = − C F (11 N − N F ) , (20)where ζ (3) = 1 . . . . .There are two reasons why it is simpler to use thislog- R matching scheme rather than R matching (i.e. eval-uating Eq. (10) explicitly to NLL precision). Firstly, wedo not have to be concerned with the C ( α s ) and D ( t, α s )terms in (10), for which we do not have analytic expres-sions but which contribute to the fixed-order calculation– these are contained in R ( t ), R ( t ), etc. Secondly, itis easier to impose physical boundary conditions on thenormalised cross section, namely R ( t = t max ) = 1 , (21)by definition of the normalised cross section, and dRdt ( t = t max ) = 0 , (22)as there is an upper kinematic limit t max on the thrustfor a given number of final-state partons. Although theresummed logarithmic terms are small at high t , dR/dt is also small and so these terms can cause relatively largeunphysical effects if we do not impose these conditions.The above constraints are automatically obeyed bythe fixed-order terms R n ( t ) but not by the resummedterms, as we have neglected the subdominant logarithms g ( α s L ), g ( α s L ) etc. To satisfy these constraints, wetherefore require Q ( t ) = Lg ( α s L ) + g ( α s L ) − ¯ α s (cid:0) G L + G L (cid:1) − ¯ α s (cid:0) G L + G L (cid:1) − ¯ α s (cid:0) G L + G L (cid:1) (23)and its first derivative to vanish at t = t max . Q ( t ) cor-responds to the resummed logarithmic terms of order L and higher and hence at small L , t dQdt = aL + bL + cL + . . . . (24)By making the replacement L → ˜ L = ln (cid:18) t − t max (cid:19) , (25) R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation the boundary conditions are satisfied as ˜ L ( t max ) = 0. Thisdoes introduce corrections to the expression for ln R ( t )but these are power-suppressed at small t :˜ L ( t ) = ln (cid:18) t (cid:19) + ln (cid:18) − tt max + t (cid:19) = L ( t ) + (cid:18) t − tt max (cid:19) − (cid:18) t − tt max (cid:19) + . . . , (26)and so ˜ L ( t ) → L ( t ) in the important limit t → To perform the matching, the integrated perturbation se-ries coefficients are required as in Eq. (19). For R ( t ), theanalytic result is R ( t ) = −
83 ln (cid:18) t − t (cid:19) − − t ) ln (cid:18) t − t (cid:19) + 4 π −
103 + 8 t + 6 t −
163 Li (cid:18) t − t (cid:19) , (27)where Li ( z ) ≡ (cid:90) z dx ln (1 − x ) x (28)is the dilogarithm function. R ( t ) and R ( t ) were ob-tained by interpolating the differential results from EERAD3 and then numerically integrating them. For R ( t ), the EERAD3 results were first smoothed by taking dR dt ( t i ) → (cid:20) dR dt ( t i +1 ) + dR dt ( t i ) + dR dt ( t i − ) (cid:21) , (29)repeatedly until a smooth curve was obtained. The peaknear t = 0 had to be reintroduced by hand, as this smooth-ing technique always results in the peak value being re-duced. R ( t ) was computed to NNLO+NLL precision usingEqs. (19) and (26) with t max = 0 .
42 in ˜ L , as this is themaximum value of t kinematically allowed in the five par-ton limit. The differential cross section was then obtainedby numerically differentiating R ( t ). The results at a rangeof energies are shown by the red/darker curves in Figs 1-3.The values of α s ( Q ) were calculated as described earlierfor the unresummed NNLO (green/lighter) curves. Theshaded area around each line shows the renormalisationscale uncertainty found by taking µ R ∈ (cid:2) Q / , Q (cid:3) . The matched, resummed differential thrust distributionwas compared with data from a wide range of experiments,as listed in Table 1. The points in Figs. 1-3 show the dataat an illustrative selection of energies. The error bars rep-resent the experimental statistical and systematic errors,added in quadrature.
Experiment Q /GeV Ref. No. Pts. χ TASSO 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1.
Data sets used and best-fit χ contributions. There are a few features common to the graphs atall energies. Firstly, the resummed distribution and theNNLO distribution are almost identical away from thetwo-jet region. However, in this low- t limit the resummeddistribution peaks, in line with the experimental data,whereas the NNLO distribution carries on increasing. Thusresummation has significantly improved the theoreticalprediction in the two-jet limit, as we had expected. .A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation 7
It should be noted that the kink around t = 0 .
33 inall of the curves is due to the LO term vanishing herefor kinematic reasons. One would expect that with manyhigher-order perturbation theory terms taken into account(i.e. more partons present in the final state), this wouldgradually smoothen, in line with the experimental data.At all energies, the overall shape of the theoretical dis-tribution is similar to that of the data, but is shifted toa lower value of t . This apparent shift δt has a clear en-ergy dependence – at the upper end of the energy rangeconsidered here, the NNLO+NLL and experimental dis-tributions are fairly close and the shift δt is a very smallcorrection. On decreasing the energy, the shift becomesmore pronounced and at low energies the theoretical dis-tribution is clearly not consistent with the data. There isno obvious way that this could be remedied by the inclu-sion of sub-leading logarithms or higher fixed-order terms,and so we now turn to considering non-perturbative effectsfor an explanation. The increasing discrepancy at low en-ergies is also consistent with this interpretation, as we ex-pect such effects to have a 1 /Q dependence, as mentionedin Sect. 1. To verify that these discrepancies are due tonon-perturbative effects, the exact form of their energydependence was investigated. As both the experimental data and
EERAD3 results aregiven as histograms, and not in terms of individual val-ues of t , the integrated thrust distribution R ( t ) should beslightly more accurate than dσ/dt as it does not involvethe assumption of a uniform distribution over the widthof each histogram bin ∆t .Graphs of ln ( R theory − R expt ) against ln ( Q/ GeV) wereplotted for 0 . ≤ t ≤ .
24 and, anticipating correctionsproportional to an inverse power of Q , a linear fit wasmade to each plot such that the gradient n of the straightline gives the power dependence of the required correction( ∝ Q n ). The results are shown in Figs. 4-6. This t rangewas chosen since at lower values of t there is no obviousstraight line (due to the distributions peaking), and athigher values of t the percentage errors on the gradientbecome large due to R theory − R expt quickly decreasing tozero (as both normalised cross-sections converge to 1).Although not totally conclusive, these results are con-sistent with power corrections of the form 1 /Q , and weturn now to considering the quantitative form of thesenon-perturbative corrections to the thrust distribution. Although there are various ways to phenomenologicallytreat non-perturbative effects in QCD, one of the most in-tuitive is by means of a low-scale effective coupling [1]. Inthis approach, the running coupling (7) is replaced by aneffective coupling α eff ( µ ), which differs from the standard -6-5-4-3-2-1 0 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.025n = -0.85 ± L3DELPHIBest Fit -5-4-3-2-1 0 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.05n = -1.45 ± DELPHIALEPHOPALL3AMYBest Fit -6-5-4-3-2-1 0 2.5 3 3.5 4 4.5 5 5.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.06n = -0.88 ± DELPHIALEPHJADETASSOSLDBest Fit -6-5-4-3-2-1 0 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.07n = -1.49 ± DELPHIALEPHOPALBest Fit -6-5-4-3-2-1 0 2.5 3 3.5 4 4.5 5 5.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.08n = -1.16 ± DELPHIALEPHJADETASSOSLDBest Fit
Fig. 4.
Power dependence of corrections required to resolvetheory/data discrepancy: t = 0 . − .
08. R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation -6-5-4-3-2-1 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.09n = -1.15 ± DELPHIALEPHOPALBest Fit -6-5-4-3-2-1 2.5 3 3.5 4 4.5 5 5.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.10n = -1.02 ± DELPHIALEPHJADETASSOAMYL3Best Fit -6-5-4-3-2-1 2.5 3 3.5 4 4.5 5 5.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.12n = -1.23 ± DELPHIALEPHJADETASSOOPALSLDBest Fit -6-5-4-3-2-1 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.14n = -0.80 ± DELPHIALEPHJADEBest Fit -6-5-4-3-2-1 2.5 3 3.5 4 4.5 5 5.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.16n = -1.14 ± DELPHIALEPHJADETASSOSLDBest Fit
Fig. 5.
Power dependence of corrections required to resolvetheory/data discrepancy: t = 0 . − . -6-5-4-3-2-1 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.18n = -0.66 ± DELPHIALEPHJADEBest Fit -7-6-5-4-3-2-1 2.5 3 3.5 4 4.5 5 5.5 6 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.20n = -0.99 ± DELPHIALEPHJADETASSOSLDAMYL3Best Fit -6-5-4-3-2-1 2.5 3 3.5 4 4.5 l n ( R t heo r y - R e x p t ) ln(Q/GeV) t = 0.24n = -1.11 ± DELPHIALEPHTASSOBest Fit
Fig. 6.
Power dependence of corrections required to resolvetheory/data discrepancy: t = 0 . − . perturbative α s ( µ ) in the infra-red region where the latterdiverges. Using this finite effective coupling allows us touse the formalism of perturbation theory to describe non-perturbative effects which cannot be probed using stan-dard perturbative QCD.Various forms for α eff ( µ ) have been proposed [22,23]that have high-energy behaviour consistent with α s , butwe will not be concerned with their details here. The onlyparameter we require is the ‘average’ value of the effectivecoupling below the infra-red matching scale µ I where α s and α eff begin to differ: α ( µ I ) ≡ µ I (cid:90) µ I dµ α eff ( µ ) . (30)We make the additional assumption that α eff is smallenough in the infra-red region that we can neglect termsof order α and higher. .A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation 9
In deriving the form of the NNLO+NLL prediction usedearlier, the low µ region in Eq. (13) was neglected asit produced a subleading contribution. We now includethis region by subtracting the fixed-order NNLO contri-bution from µ ≤ µ I and replacing it with a contributiondue to the effective coupling. We are thus removing therenormalon contributions to the perturbation series (up toNNLO) and incorporating all 1 /Q -dependent behaviourinto α eff .Firstly, we note that the order of integration in (13)can be changed, to giveln ˜ J µν (cid:0) Q (cid:1) = 2 C F π (cid:90) Q dµµ α s ( µ ) (cid:18) − K α s ( µ )2 π (cid:19) − (cid:90) µQµ Q duu (cid:16) e − uνQ − (cid:17) . (31)Inserting the NNLO perturbative running coupling α s ( µ ) = α s ( µ R ) + α s ( µ R ) β π ln µ R µ + α s ( µ R ) (cid:34)(cid:18) β π (cid:19) ln µ R µ + β π ln µ R µ (cid:35) , (32)expanding the exponential to first order and integratingover the range 0 ≤ µ ≤ µ I gives an NNLO contribution of − C F π µ I Q (cid:40) α s ( µ R ) + α s ( µ R ) β π (cid:18) ln µ R µ I + K β + 1 (cid:19) + α s ( µ R ) (cid:18) β π (cid:19) (cid:20) ln µ R µ I + (cid:18) ln µ R µ I + 1 (cid:19)(cid:18) β β + Kβ (cid:19) + K β (cid:21)(cid:41) νQ . (33)It should be noted that t is the conjugate variable to νQ in the Laplace transform (12) and thus the first-orderexpansion of the exponential will only be a valid approxi-mation in the limit t (cid:29) µ I /Q . Below this, we would needto retain higher order terms in the expansion, which wouldrequire us to have a specific form for α eff ( µ ).Following a similar procedure with α eff ( µ ) in the placeof α s ( µ ) gives a non-perturbative contribution of − C F π (cid:90) µ I dµ α eff ( µ ) νQ ≡ − C F π µ I Q α ( µ I ) νQ , (34)where we have neglected terms of order α as previouslynoted.By adding this, after subtracting the perturbative con-tribution (33), we obtain the change in the quark jet massdistribution caused by changing from a perturbative to aneffective coupling in the low-scale region below µ I . Higher-order terms in the expansion would give correctionsof order 1 /Q , which we neglect. Substituting the result into Eq. (12), we see that theeffect of this non-perturbative contribution is to shift thethrust distribution by an amount δt , such that1 σ dσdt (cid:12)(cid:12)(cid:12)(cid:12) t = (cid:18) σ dσdt (cid:19) pert. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t + δt , (35)where δt = − C F π µ I Q (cid:40) α ( µ I ) − α s ( µ R ) − α s ( µ R ) β π (cid:18) ln µ R µ I + K β + 1 (cid:19) − α s ( µ R ) (cid:18) β π (cid:19) (cid:20) ln µ R µ I + (cid:18) ln µ R µ I + 1 (cid:19)(cid:18) β β + Kβ (cid:19) + K β (cid:21)(cid:41) , (36)to NNLO. This 1 /Q -dependent shift is precisely what isrequired to account for the differences between the per-turbative and experimental distributions seen in Sect. 3. α s and α By applying the shift (36) to the perturbative results, weexpect to reduce significantly the differences between thetheoretical and experimental distributions. Comparison ofthese differences to the predicted form of δt allows us toestimate the values of α and α s .Maximum accuracy was obtained by comparing theexperimental distribution with a discretely-defined theo-retical distribution R ( t + ∆t ) − R ( t ) ∆t , (37)where ∆t is the bin width of the experimental data.The NNLO+NLL+shift distribution was calculated asa function of α and Λ (5) MS . This calculation was performedfor 0 . ≤ t ≤ .
33, at the centre-of-mass energies listedpreviously in Table 1 (i.e. in the range 14 GeV ≤ Q ≤ χ was calculated for each pair of input parameters,with its minimum corresponding to the best-fit values.The upper limit for the fits was chosen as t = 0 . t (cid:29) µ I /Q ; in fact we found satisfactory fits usingan energy-dependent lower cut-off t ≥ max { µ I /Q, . } .For infra-red matching scale µ I = 2 GeV, best fit val-ues of α (2 GeV) = 0 . ± . ,Λ (5) MS = 0 . +0 . − . GeV (38) e + e − Annihilation ! " (5) =5 =10 =20 =30 =50 =100Best Fit Fig. 7. χ contour plot in “ Λ (5) MS , α ” space. were obtained, with χ / d.o.f. = 466 . / ≈ .
09. Thequoted errors correspond to one standard deviation, com-puted as recommended by the Particle Data Group [9]: thevalue of χ corresponding to the 1 σ (68.3% C.L.) contourwas rescaled by the value of χ / d.o.f., giving χ = 480 . ∆χ = 14 . χ from each data set is shown inTable 1. It should be noted that the few data sets with χ / no. pts. (cid:29) α and Λ (5) MS which give fits within ∆χ of the best-fit value of χ ,and also demonstrates the correlation between these twoparameters.Varying the renormalisation scale µ R ∈ (cid:2) Q / , Q (cid:3) gave best fit values in the range α (2 GeV) = 0 . Λ (5) MS = 0 .
173 GeV to α (2 GeV) = 0 . Λ (5) MS = 0 . Λ (5) MS = 0 . +0 . . − . − . GeV (39)where the first error is the combined experimental statis-tical and systematic error and the second is due to thetheoretical renormalisation scale uncertainty. The corre-sponding strong coupling constant is α s (91 . . +0 . . − . − . , (40)or, combining all the errors in quadrature, α s (91 . . +0 . − . , (41)in good agreement with the world average value of 0.1176 [9].To assess the importance of the NNLO terms, the anal-ysis was repeated with all those terms omitted, i.e. com-bining NLO+NLL in perturbation theory with Eq. (36)without the O ( α s ) contribution. The resulting best fit val-ues were α (2 GeV) = 0 . ± . ,Λ (5) MS = 0 . +0 . . − . − . GeV ,α s (91 . . +0 . . − . − . (42) with χ / d.o.f. = 515 . / ≈ .
20. Thus the NLO andNNLO results are consistent but the inclusion of NNLOterms consistently in both the perturbative prediction andthe power correction improves the quality of the fit andreduces the errors.The most complete previous NLO study along similarlines [24], combining NLO+NLL in perturbation theorywith the NLO equivalent of Eq. (36) and covering a varietyof event shapes but a slightly narrower range of energiesthan that used here, obtained the overall best fit at α s (91 . . +0 . − . ,α (2 GeV) = 0 . +0 . − . (43)in good agreement with our results. Their fit to the thrustdistribution alone gave α s (91 . . +0 . − . ,α (2 GeV) = 0 . +0 . − . (44)also in good agreement.In the recent NNLO analysis [25], a range of eventshapes at energies at and above 91.2 GeV were fittedwithout resummation; non-perturbative effects were es-timated using Monte Carlo event generators. The valueobtained for the strong coupling was α s (91 . . ± . µ I = 3 GeV was made,yielding α (3 GeV) = 0 . ± .
025 and Λ (5) MS = 0 . +0 . − . ,with χ / d.o.f. ≈ .
09. Thus the fit remains good and thevalue obtained for Λ (5) MS is stable under variation of µ I ,while the value of α decreases as expected for a runningeffective coupling. Indeed, the implied mean value of α eff in the range 2-3 GeV, α eff = 3 α (3 GeV) − α (2 GeV) = 0 . ± .
10 (45)is consistent with the perturbative value α s (2 . . Figures 8-10 show the final (NNLO+NLL+shift) theoret-ical distributions in comparison to the experimental ones,with the best-fit values of α and α s assumed. The shadedarea around the unshifted distribution is the renormalisa-tion scale uncertainty found by varying µ R ∈ (cid:2) Q / , Q (cid:3) ,and the shaded area around the shifted distribution isthe corresponding error found by varying between thebest fit limits obtained previously ( α (2 GeV) = 0 . Λ (5) MS = 0 .
173 GeV and α (2 GeV) = 0 . Λ (5) MS = 0 . α s is very close to the world average, the unshifted distri-butions here are essentially the same as those in Figs. 1-3. .A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation 11 / ! d ! / d t t Q=14 GeV " s=0.1619 UnshiftedShiftedTASSO (14 GeV) / ! d ! / d t t Q=35 GeV " s=0.1358 UnshiftedShiftedTASSO (35 GeV)JADE (35 GeV) / ! d ! / d t t Q=44 GeV " s=0.1306 UnshiftedShiftedTASSO (44 GeV)JADE (44 GeV)DELPHI (45 GeV) / ! d ! / d t t Q=55 GeV " s=0.1259 UnshiftedShiftedAMY (54.5 GeV)L3 (55.3 GeV) / ! d ! / d t t Q=66 GeV " s=0.1223 UnshiftedShiftedL3 (65.4 GeV)DELPHI (66.0 GeV)
Fig. 8.
Comparison of shifted, unshifted and experimentalthrust distributions: Q = 14 −
66 GeV. / ! d ! / d t t Q=91.2 GeV " s=0.1164 UnshiftedShiftedOPAL (91.0 GeV)DELPHI (91.2 GeV)ALEPH (91.2 GeV)SLD (91.2 GeV) / ! d ! / d t t Q=133 GeV " s=0.1102 UnshiftedShiftedOPAL (133 GeV)ALEPH (133 GeV) / ! d ! / d t t Q=161 GeV " s=0.1073 UnshiftedShiftedALEPH (161.0 GeV)L3 (161.3 GeV) / ! d ! / d t t Q=172 GeV " s=0.1064 UnshiftedShiftedALEPH (172.0 GeV)L3 (172.3 GeV) / ! d ! / d t t Q=183 GeV " s=0.1055 UnshiftedShiftedL3 (182.8 GeV)ALEPH (183.0 GeV)DELPHI (183.0 GeV)
Fig. 9.
Comparison of shifted, unshifted and experimentalthrust distributions: Q = 91 −
183 GeV.2 R.A. Davison, B.R. Webber: Non-Perturbative Contribution to the Thrust Distribution in e + e − Annihilation / ! d ! / d t t Q=189 GeV " s=0.1050 UnshiftedShiftedL3 (188.6 GeV)ALEPH (189.0 GeV)DELPHI (189.0 GeV) / ! d ! / d t t Q=200 GeV " s=0.1042 UnshiftedShiftedL3 (200 GeV)ALEPH (200 GeV)DELPHI (200 GeV) / ! d ! / d t t Q=206 GeV " s=0.1038 UnshiftedShiftedALEPH (206.0 GeV)L3 (206.2 GeV)DELPHI (207.0 GeV)
Fig. 10.
Comparison of shifted, unshifted and experimentalthrust distributions: Q = 189 −
207 GeV.
We have seen that the extension of the NNLO perturba-tive distribution to NNLO+NLL accuracy results in animproved matching with experiment, particularly in thelow t region.Analysis of the difference between the perturbativeand experimental distributions over a range of energiesshowed that 1 /Q power corrections were required to ac-count for this difference. Replacement of the perturba-tive strong coupling with an effective coupling below aninfra-red matching scale was used to include such non-perturbative corrections in our theoretical calculation andresulted in a 1 /Q -dependent shift in the distribution. Withbest-fit values α (2 GeV) = 0 . ± .
03 and α s (91 . . +0 . − . , this gave a significantly improved matchingwith the experimental distributions in the range 14 GeV ≤ Q ≤
207 GeV. These values are consistent with thoseachieved in similar analyses to NLO, as well as with theworld-average value of α s . The agreement of the α and α s values from the analy-sis at NNLO+NLL with those obtained at NLO+NLL is anon-trivial test of the low-scale effective coupling hypoth-esis. The presence of the O ( α s ) term in Eq. (36), whichamounts to about 80% of the O ( α s ) term, means thatwe are not simply adding a 1 /Q correction to the pertur-bative result, but rather that we are regularizing the di-vergent renormalon contribution by modifying the strongcoupling at low scales. This implies that the explicit non-perturbative 1 /Q shift applied to the perturbative predic-tion becomes smaller as higher orders are computed, andwould eventually change sign at sufficiently high orders,as the renormalon contribution grows indefinitely.A similar analysis to that in this work could be re-peated for other event shape variables whose distributionshave been determined perturbatively to NNLO and forwhich resummation of large logarithms is possible. Pertur-bative resummed calculations of such distributions havebeen performed [26] but non-perturbative effects have notbeen included in the way advocated here – they are notnecessarily simple shifts as in the case of thrust. It wouldalso be of interest to combine the present approach tonon-perturbative effects with soft-collinear effective the-ory, which permits the resummation of next-to-next-to-leading logarithms [27]. Acknowledgements
We are grateful to the authors of Refs. [5,6] for provid-ing results of their calculations and for helpful comments.BRW thanks the CERN Theory Group for hospitalitywhile part of this work was performed. This research wassupported in part by the UK Science and Technology Fa-cilities Council.
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