Precision predictions for the t+t(bar) production cross section at hadron colliders
Valentin Ahrens, Andrea Ferroglia, Matthias Neubert, Ben D. Pecjak, Li Lin Yang
aa r X i v : . [ h e p - ph ] M a y MZ-TH/11-11ZU-TH 10/11
Precision predictions for the t ¯ t production cross section at hadron colliders Valentin Ahrens, Andrea Ferroglia, Matthias Neubert, Ben D. Pecjak, and Li Lin Yang Institut f¨ur Physik (THEP), Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany New York City College of Technology, 300 Jay Street, Brooklyn, NY 11201, USA Institute for Theoretical Physics, University of Z¨urich, CH-8057 Z¨urich, Switzerland (Dated: August 7, 2018)We make use of recent results in effective theory and higher-order perturbative calculations toimprove the theoretical predictions of the top-quark pair production cross section at hadron colliders.In particular, we supplement the fixed-order NLO calculation with higher-order corrections from softgluon resummation at NNLL accuracy. Uncertainties due to power corrections to the soft limit areestimated by combining results from single-particle inclusive and pair invariant-mass kinematics.We present our predictions as functions of the top-quark mass in both the pole scheme and the MSscheme. We also discuss the merits of using threshold masses as an alternative, and calculate thecross section with the top-quark mass defined in the 1S scheme as an illustrative example.
PACS numbers: 14.65.Ha, 12.38.Cy
I. INTRODUCTION
The total t ¯ t production cross section is an important observable at hadron colliders such as the Tevatron and LHC.For instance, it provides information about the top-quark mass, which is an input for electroweak fits [1] used toconstrain the mass of the Higgs boson. Extractions of the top-quark mass from the production cross section havethe advantage that the perturbative calculations used in the analysis are carried out in a well-defined renormalizationscheme for the top-quark mass. The pole mass as well as the MS mass have already been extracted from the productioncross section at the Tevatron [2]. The use of a short-distance mass such as the MS mass is theoretically favored overthe pole mass, which can be defined only up to a renormalon ambiguity of order Λ QCD . Moreover, it has been proposedin [3] that the apparent convergence and scale uncertainties of the perturbative series for the total cross section isimproved in the MS scheme compared to the pole scheme, already at low orders in the perturbative expansion.According to the QCD factorization theorem [4], the hadronic cross section can be obtained from the partonic oneafter convolution with parton distribution functions (PDFs). For t ¯ t production, it is conventional to express thisfactorization theorem in the form σ ( s, m t ) = α s ( µ f ) m t X i,j Z s m t d ˆ ss ff ij (cid:18) ˆ ss , µ f (cid:19) f ij (cid:18) m t ˆ s , µ f (cid:19) , (1)where µ f is the factorization scale, i, j ∈ { q, ¯ q, g } , and the parton luminosities are defined as ff ij ( y, µ f ) = Z y dxx f i/N ( x, µ f ) f j/N ( y/x, µ f ) . (2)Here and throughout this letter the running coupling is defined in the MS scheme with five active (massless) flavors.The scaling functions f ij are proportional to the partonic cross sections and can be expanded in powers of α s /π .Numerical results for the next-to-leading order (NLO) term have been known for over two decades [5–7], and morerecently fully analytic results were obtained in [8]. Predictions based on NLO calculations typically exhibit scaleuncertainties larger than 10%. In order to further reduce the theoretical uncertainties to match the experimentalprecision, it is necessary to go beyond NLO. Consequently, the calculation of the next-to-next-to-leading order (NNLO)corrections is an active area of research. Many efforts have been made in the calculation of two-loop virtual corrections[9–14], one-loop interference terms [15–17], and double real emission [18–21]. Due to very recent progress in developinga new subtraction scheme for double real emission in the presence of massive particles [22–26], a calculation of theNNLO corrections now seems feasible, and its completion will be a major accomplishment.An important way to improve on the fixed-order calculations is to supplement them with threshold resummation[27, 28]. In such an approach, which in general works at the level of differential cross sections, one identifies a partonicthreshold parameter which vanishes in the limit where extra real radiation is soft, and sums a certain tower oflogarithmic corrections in this parameter to all orders in the strong coupling. Such resummed formulas neglect powercorrections which vanish when the threshold parameter goes to zero, but these can be taken into account, up to a givenorder in the strong coupling, by matching with the fixed-order results. In this way, one obtains predictions which notonly make full use of the fixed-order calculations, but also resum a class of logarithmic corrections to all orders. Inthe limit where the higher-order corrections are dominated by these logarithmic terms, such a resummation is clearlyan improvement. For this reason, many different implementations of threshold resummation have been considered inthe literature, where the current frontier is next-to-next-to-leading logarithmic (NNLL) accuracy [3, 29–35].The purpose of this letter is to consolidate results for the total cross section based on threshold resummation ineffective field theory. These are obtained from two different threshold limits for the differential cross section. The first,referred to as pair invariant-mass (PIM) kinematics, uses the top-pair invariant-mass distribution as the fundamentalobservable [32, 33]. The second, referred to as single-particle inclusive (1PI) kinematics, works at the level of thetransverse-momentum or rapidity distributions of the top quark [35]. The total partonic cross section is obtained byintegrating the resummed distributions over the appropriate phase space. In both kinematics, the predictions in theeffective field-theory framework include resummation effects to NNLL order, and are matched with the fixed-orderresults at NLO in order to achieve NLO+NNLL accuracy for the total cross section. Such formulas can be evaluatednumerically using specific values of the matching scales appearing in the effective-theory calculations, or otherwisere-expanded in a fixed-order expansion, in the form of approximate NNLO predictions. We have considered bothscenarios in [33, 35], finding that when the results in PIM and 1PI kinematics are combined as in the present work,the numerical differences between the NLO+NNLL and approximate NNLO results are rather small. In this letterwe focus on the approximate NNLO results for concreteness.In what follows, we briefly review the formalism, and then give our best numerical predictions for the total crosssection in the pole scheme for the top-quark mass, in the form of numerical fits as functions of m t . We then explainhow to convert the results to the MS scheme and present numerical results as a function of the MS mass. The resultsin both schemes are calculated with a Fortran implementation of our approximate NNLO formulas, which we includewith the electronic submission of this letter. Contrary to [3], we do not find a strong improvement of the perturbativeseries in the MS scheme compared to the pole scheme. We suggest that the group of short-distance masses known asthreshold masses (see e.g. [36]) may actually be the more appropriate choice, and we explicitly consider the case ofthe mass defined in the 1S scheme [37]. However, we believe that the poor large-order behavior of the perturbativeseries in the pole scheme is unlikely to be of practical importance in the foreseeable future. II. INGREDIENTS OF THE CALCULATION
Our calculation is based on [33, 35], where threshold resummation using soft-collinear effective theory (SCET)[38–40] is applied to t ¯ t production at hadron colliders. The partonic scattering process is i ( p ) + j ( p ) → t ( p ) + ¯ t ( p ) + ˆ X ( k ) , (3)where i , j indicate the incoming partons and ˆ X is a partonic final state. In [33], the invariant-mass distribution dσ/dM of the t ¯ t pair was considered, where M ≡ ( p + p ) , and the threshold region is defined by z = M / ˆ s → s = ( p + p ) (so-called PIM kinematics). In [35], on the other hand, the transverse-momentum and rapiditydistributions of the top-quark were considered. In this case, the threshold region is defined by s = ( p + k ) − m t → α ns (cid:20) ln m ξξ (cid:21) + ; m = 0 , . . . , n − , (4)where ξ = (1 − z ) / √ z for PIM and ξ = s / ( m t p m t + s ) for 1PI kinematics.In the threshold limit ξ →
0, the partonic differential cross section can be factorized into a product of matrix-valuedhard and soft functions, d ˆ σ ( ξ, µ f ) ∝ Tr (cid:20) H ( µ f ) S (cid:18) E s ( ξ ) µ f , α s ( µ f ) (cid:19)(cid:21) , (5)where we have suppressed the dependence on the other kinematic variables, and E s ( ξ ) is the energy of the soft gluonradiation, which is given by E s = M ξ in the partonic center-of-mass frame for PIM kinematics, and E s = m t ξ in the¯ t rest frame for 1PI kinematics. The hard function H is related to virtual corrections and is thus evaluated at ξ = 0.The ξ dependence of the cross section is encoded in the soft function S via the ratio E s ( ξ ) /µ f . One can therefore usethe renormalization-group (RG) equation of the soft function to resum the singular distributions in ξ to all orders in α s . A technical complication is that the RG equation of the soft function is non-local. We solve this equation usingthe technique of Laplace transforms [41], and then carry out the inverse Laplace transform analytically to obtainexpressions in momentum space. The final result for the resummed cross section reads d ˆ σ ( ξ, µ f ) ∝ exp (cid:2) a γ φ ( µ s , µ f ) (cid:3) Tr h U ( µ h , µ s ) H ( µ h ) U † ( µ h , µ s ) ˜ s ( ∂ η , α s ( µ s )) i e − γ E η Γ(2 η ) ξ − η , (6)where ˜ s is the Laplace transform of the soft function, U is an evolution matrix which resums large logarithmsbetween the hard and soft scales µ h and µ s , and η = 2 a Γ ( µ s , µ f ) and a γ φ ( µ s , µ f ) are related to certain integrals overanomalous-dimension functions. Explicit expressions for these objects in RG-improved perturbation theory can befound in [33]. For NNLL accuracy, one needs the anomalous-dimension matrices for the hard and soft functions totwo-loop order (computed in [42]), as well as the hard and soft functions to one-loop order (computed in [33, 35]).One must also specify a procedure for choosing the hard and soft scales. In the SCET approach, these are chosensuch that the contributions of the NLO matching functions to the hadronic cross section are minimized; details canbe found in [33, 35].The effective theory is a powerful tool for separating physics at the formally very different scales µ h and µ s and forsumming logarithms of their ratio. In practice, however, in areas of phase space where the differential cross sectionis largest, the numerical hierarchy between the two scales is only moderate. Therefore, an alternative to using theNLO+NNLL formulas directly is to re-expand them in a fixed-order series in α s ( µ f ), constructing what are referredto as approximate NNLO formulas. The full NNLO correction to the cross section takes the form d ˆ σ (2) ( ξ, µ f ) ∝ α s ( µ f ) X m =0 D m (cid:20) ln m ξξ (cid:21) + + C δ ( ξ ) + R ( ξ ) ! . (7)The approximate NNLO formulas derived from the NLO+NNLL results contain an unambiguous answer for the D m coefficients in the limit ξ →
0. Only parts of the coefficient C are determined, and the regular piece R is subleading inthe threshold limit and not determined. There are thus ambiguities in what to include in the C term, and a freedomto shuffle terms between the D i and R away from the exact limit ξ →
0. For our method of dealing with theseambiguities, we refer the reader to [35], which specifies all details relevant for our implementation of the approximateNNLO formulas used in the numerical analysis.Equations (1) and (5)–(7) are written for the case where the factorization and renormalization scales are setequal, µ f = µ r . Later on we will consider the case where the running coupling is instead evaluated at an arbitraryrenormalization scale µ r . To derive such expressions we use α s ( µ f ) = α s ( µ r ) " α s ( µ r ) π L rf + (cid:18) α s ( µ r ) π (cid:19) (cid:18) L rf + 2912 L rf (cid:19) , (8)with L rf = ln( µ r /µ f ), and re-expand the formulas to NNLO in powers of α s ( µ r ).Although PIM and 1PI kinematics are used to describe different differential distributions, one can always integrateover the distributions to obtain the total hadronic cross section or the partonic scaling functions f ij using the twoapproaches. Since the two kinematics both account for soft gluon emission, they give rise to the same results for f ij in the limit ˆ s → m t . When ˆ s deviates from 4 m t , the difference between the two kinematics is formally subleadingin ξ . However, the numerical differences may be visible or even significant if ˆ s is much larger than 4 m t . In [35], wehave shown that the effective-theory predictions for the total cross section from the two kinematics actually agreequite well, as long as the exact dependence on the energy of soft gluon radiation is kept in the factorization formula(5). The differences between these two kinematics can be regarded as another source of theoretical uncertainty,namely that due to power corrections to the soft limit, and used along with scale uncertainties to estimate the totaluncertainty associated with the calculation of the total cross section at NLO+NNLL or approximate NNLO. In thefollowing sections we define our procedure for combining the two types of uncertainties and give our final results forthe total cross section as a function of the top-quark mass in the pole, MS, and 1S schemes.
III. CROSS SECTIONS IN THE POLE SCHEME
We now specify our method for estimating the theoretical uncertainties from scale variations, PDF variations andvariations of α s in the calculation of the top-pair production cross section, using either PIM or 1PI kinematics. We Previous calculations expanded E s ( ξ ) in the limit z → s → then define a procedure for combining the results obtained in these two schemes. By default, the pole mass m t is usedin the calculation of the partonic cross sections. We comment later on alternative schemes for defining the top-quarkmass.To estimate the uncertainties associated with scale variations, we view the cross section as a function of therenormalization and factorization scales, which by default are chosen as µ f = µ r = m t . We then consider twomethods of scale variations: correlated variations with µ f = µ r varied up and down by a factor of two from thedefault value, and independent variations of µ f and µ r by factors of two, with the uncertainties added in quadrature.We use as our final answer the larger uncertainty from these two methods.To combine the results from PIM and 1PI kinematics, we first compute the cross sections and scale uncertaintiesin the PIM and 1PI schemes separately, and obtain six quantities σ PIM , ∆ σ +PIM , ∆ σ − PIM , σ , ∆ σ +1PI , ∆ σ − . Thecentral value and perturbative uncertainties for the combined results are then determined by σ = 12 ( σ PIM + σ ) , ∆ σ + = max (cid:0) σ PIM + ∆ σ +PIM , σ + ∆ σ +1PI (cid:1) − σ , (9)∆ σ − = min (cid:0) σ PIM + ∆ σ − PIM , σ + ∆ σ − (cid:1) − σ . In this way, the central value is the average of the two, and the perturbative uncertainties reflect both the variationof the scales and the difference between the two types of kinematics. The PDF uncertainties are estimated as usualby evaluating the average of the 1PI and PIM results using the PDF error sets at a particular confidence level.We quote in Table I the approximate NNLO predictions obtained with the above procedure at m t = 173 . µ r = µ f . The one exceptionis the upper error at the Tevatron, which is instead determined by the independent variations of µ r and µ f added inquadrature. Even though the perturbative uncertainty in the approximate NNLO result includes both scale variationsand an estimate of power corrections to the soft limit through the difference of 1PI and PIM kinematics, it is stillreduced compared to that in the NLO calculation, which by definition is due only to scale variations. We note thatthe central value and uncertainties of the approximate NNLO results are well contained within the uncertainty rangepredicted by the NLO results, so that the perturbative series to this order is well behaved in the pole scheme. TheNNLO results are also within the uncertainties of the LO calculation, although the NLO results are slightly higherthan the LO ones in the case of the LHC. Tevatron LHC7 LHC14MSTW CTEQ MSTW CTEQ MSTW CTEQLO 6 . +2 . . − . − (0 . . +2 . . . − . − . . +49+(6) − − (7) +35+9(7) − − +228+(26) − − (34) +157+25(18) − − NLO 6 . +0 . . . − . − . . . +0 . . . − . − . . +20+14(8) − − +18+13(11) − − +107+66(31) − − +97+41(27) − − NNLO approx. 6 . +0 . . . − . − . . . +0 . . . − . − . . +8+14(8) − − +8+13(11) − − +52+60(30) − − +51+40(26) − − TABLE I: Total cross sections in pb for m t = 173 . α s uncertainty. The numbers in parenthesis show the PDF uncertainty only. For comparison, we also include the results using CTEQ6.6 PDFs [44] in Table I. Since the CTEQ PDFs are basedon a NLO fit, the same set is used at LO, NLO and approximate NNLO. The statements based on the analysis withMSTW PDFs above, including those concerning the moderate size of the NNLO corrections, are also true for theanalysis with CTEQ PDFs. In this case, however, the LO results at the LHC are significantly lower than the NLOand NNLO results. To a certain extent, this shows the potential benefit of switching PDFs as appropriate to the orderof perturbation theory. On the other hand, LO calculations are usually considered unreliable, so the more importantobservation for the perturbative convergence is the modest size of the NNLO correction.The perturbative uncertainties in the approximate NNLO predictions are about the same size at both the Tevatronand the LHC. An additional source of uncertainty is related to the experimental value of α s ( M Z ) (where M Z denotesthe Z -boson mass), which is an input parameter for the running of the strong coupling constant. We estimate thisuncertainty in combination with the PDF one by employing the method proposed in [45, 46]. Table I shows that theuncertainty on α s ( M Z ) adds an error of ± (3 – 4)% to the pair-production cross section when the calculation is carriedout with MSTW2008 PDFs. The error is somewhat smaller, ± (1 – 2)%, when CTEQ6.6 PDFs are used. The reasonis that CTEQ6.6 assigns a 90% CL error of ± .
002 to α s ( M Z ), while for MSTW2008 it is ± . α s ( M Z ) induced uncertainty is of the same order of magnitude as the perturbative and PDF uncertainties,and should not be neglected.For an extraction of the top-quark mass through a comparison with the experimental cross section, we also provideour results as a function of m t . We parametrize the mass dependence of the approximate NNLO cross section usingthe simple polynomial fit σ ( m t ) = c + c x + c x + c x + c x , (10)where x = m t / GeV − c i are fit coefficients which depend on the collider and the PDF set. The results for thefit coefficients including upper and lower errors due to perturbative uncertainties are shown in Table II, again usingMSTW2008 NNLO PDFs. A Mathematica implementation of the fit coefficients can be found with the electronicversion of this letter, where the combined PDF and α s uncertainties as well as the fit coefficients using CTEQ6.6PDFs are also included. These fits reproduce the approximate NNLO calculations to 1 permille or better in the range m t ∈ [150 , m t in the range m t ∈ [150 , m t = 173 . c [pb] c [pb] c [pb] c [pb] c [pb]Tevatron σ . × − . × − . × − − . × − . × − σ + ∆ σ + . × − . × − . × − − . × − . × − σ + ∆ σ − . × − . × − . × − − . × − . × − LHC7 σ . × − . × . × − − . × − . × − σ + ∆ σ + . × − . × . × − − . × − . × − σ + ∆ σ − . × − . × . × − − . × − . × − LHC14 σ . × − . × . × − − . × − . × − σ + ∆ σ + . × − . × . × − − . × − . × − σ + ∆ σ − . × − . × . × − − . × − . × − TABLE II: Fit coefficients in (10) for the total cross sections with perturbative uncertainties at approximate NNLO, usingMSTW2008 NNLO PDFs.
IV. CROSS SECTIONS IN THE MS AND 1S SCHEMES
It is well-known that the pole mass of a quark cannot be defined unambiguously in QCD due to confinement;the perturbatively defined pole mass is sensitive to long-distance physics and suffers from renormalon ambiguities oforder Λ
QCD [47, 48]. In perturbative calculations, the renormalon ambiguity is associated with large higher-ordercorrections to the pole mass, and thus to any observable calculated in this scheme. Therefore, it is worth investigatingshort-distance mass definitions which are free from these shortcomings. In this section, we analyze the cross sectionas a function of the running top-quark mass defined in the MS scheme, and of the threshold top-quark mass definedin the 1S scheme [37].It is possible to calculate the cross section using the MS mass from the beginning, by performing mass renormal-ization in that scheme. However, since we already have the cross section in the pole scheme, it is simpler to convertfrom one scheme to another using the perturbative relation between the pole mass and MS mass. This relation iscurrently known to three-loop order [49]. To perform the conversion to the MS scheme, we take that result for QCDwith five active flavors and write it in the form m t = m (¯ µ ) (cid:20) α s ( µ r ) π d (1) + α s ( µ r ) π d (2) + O ( α s ) (cid:21) , (11)where d (1) = 43 + L m , d (2) = 8 . L m + 3724 L m + 2312 d (1) L r , (12)with L m = ln(¯ µ /m (¯ µ )) and L r = ln( µ r / ¯ µ ). We then decompose the NNLO cross section in the pole scheme as σ NNLO ( m t ) = (cid:20) α s ( µ r ) π (cid:21) σ (0) ( m t , µ r ) + (cid:20) α s ( µ r ) π (cid:21) σ (1) ( m t , µ r ) + (cid:20) α s ( µ r ) π (cid:21) σ (2) ( m t , µ r ) , (13)eliminate m t through the relation (11), and re-expand the result in powers of α s ( µ r ). The resulting cross section inthe MS scheme can be written as¯ σ NNLO ( m ) = (cid:20) α s ( µ r ) π (cid:21) ¯ σ (0) ( m (¯ µ ) , ¯ µ, µ r ) + (cid:20) α s ( µ r ) π (cid:21) ¯ σ (1) ( m (¯ µ ) , ¯ µ, µ r ) + (cid:20) α s ( µ r ) π (cid:21) ¯ σ (2) ( m (¯ µ ) , ¯ µ, µ r ) , (14)where¯ σ (0) ( m (¯ µ ) , ¯ µ, µ r ) = σ (0) ( m (¯ µ ) , µ r ) , ¯ σ (1) ( m (¯ µ ) , ¯ µ, µ r ) = σ (1) ( m (¯ µ ) , µ r ) + m (¯ µ ) d (1) (cid:20) dσ (0) ( m t , µ r ) dm t (cid:21) m t = m (¯ µ ) , (15)¯ σ (2) ( m (¯ µ ) , ¯ µ, µ r ) = σ (2) ( m (¯ µ ) , µ r )+ m (¯ µ ) " d (1) dσ (1) ( m t , µ r ) dm t + d (2) dσ (0) ( m t , µ r ) dm t + (cid:0) d (1) (cid:1) m (¯ µ )2 d σ (0) ( m t , µ r ) dm t m t = m (¯ µ ) . The derivatives can be taken either at the level of the hadronic cross section, using fits such as the one in (10), orat the level of the differential cross section before carrying out the phase-space integrations. We have checked ourcalculations by verifying the agreement between the two methods. We note that our method of converting resultsfrom the pole scheme to the MS scheme is similar to that used in [3, 50]. Indeed, our approximate NNLO results inthe MS scheme for the choice ¯ µ = m agree with those in the HATHOR program [50], apart from the piece related tothe NNLO correction σ (2) , which is of course different since we are not working in the ˆ s → m t limit of the partoniccross section.Our procedure for combining the results from 1PI and PIM kinematics in the MS scheme is analagous to that forthe pole scheme described above. In the present case, we use by default µ f = µ r = m ( m ). We must also specify thescale in the running top-quark mass, for which we use ¯ µ = m . We provide results for the cross sections as a functionof m ( m ) using the fit σ ( m ) = ¯ c + ¯ c ¯ x + ¯ c ¯ x + ¯ c ¯ x + ¯ c ¯ x , (16)where ¯ x = m/ GeV − α s uncertainties also with CTEQ6.6 PDFs are included in the Mathematica notebook mentioned above. ¯ c [pb] ¯ c [pb] ¯ c [pb] ¯ c [pb] ¯ c [pb]Tevatron σ . × − . × − . × − − . × − . × − σ + ∆ σ + . × − . × − . × − − . × − . × − σ + ∆ σ − . × − . × − . × − − . × − . × − LHC7 σ . × − . × . × − − . × − . × − σ + ∆ σ + . × − . × . × − − . × − . × − σ + ∆ σ − . × − . × . × − − . × − . × − LHC14 σ . × − . × . × − − . × − . × − σ + ∆ σ + . × − . × . × − − . × − . × − σ + ∆ σ − . × − . × . × − − . × − . × − TABLE III: Fit coefficients (16) for the cross section with perturbative uncertainties at approximate NNLO in the MS scheme,using MSTW2008 NNLO PDFs. Variations of ¯ µ around values close to m , which would correspond to sampling over different mass definitions, could potentially be usedas an additional means of estimating systematic uncertainties. However, a numerical analysis shows that our approximate NNLO resultsare very stable for variations of ¯ µ around the default value. The results for m ( m ) = 164 . m t = 173 . µ f and µ r are varied independently, rather than the scheme with correlated µ r = µ f variations,as was the case in the pole scheme. Tevatron LHC7 LHC14MSTW CTEQ MSTW CTEQ MSTW CTEQLO 8 . +3 . . − . − (0 . . +2 . . . − . − . . +64+(8) − − (9) +45+11(9) − − +291+(32) − − (43) +199+30(21) − − NLO 7 . +0 . . . − . − . . . +0 . . . − . − . . +11+15(10) − − +10+15(12) − − +79+71(35) − − +71+44(29) − − NNLO approx. 6 . +0 . . . − . − . . . +0 . . . − . − . . +9+13(8) − − +9+13(11) − − +56+54(30) − − +56+37(26) − − TABLE IV: Total cross sections in pb in the MS scheme, for m ( m ) = 164 . α s uncertainty. The numbers in parenthesis show the PDF uncertainty only. We observe that the results obtained from the approximate NNLO formulas are quite close to those in the polescheme shown in Table I, both in the central values and in the errors. Given this good agreement, which is roughlyindependent of the exact value of the top-quark mass as shown by the fits, it makes little practical difference whetherone extracts the pole mass using the approximate NNLO results, and then determines the MS mass using the per-turbative conversion (11), or whether one determines the MS mass directly, using the experimental results along withthe fits at approximate NNLO. This statement would not be true at very high orders in perturbation theory, since therenormalon ambiguity inherent to the pole mass would lead to large corrections not present in a short-distance schemesuch as the MS scheme. But given the present accuracy of perturbative calculations and experimental measurements,this does not yet appear to be an issue.It is of course still interesting to study whether even at low orders the perturbative expansion is better behavedin the MS scheme than in the pole scheme. We observe that the perturbative uncertainties at NLO are generallysmaller in the MS scheme than in the pole scheme, and that the central values are relatively higher compared to theapproximate NNLO calculation. For this reason, the overlap between the NLO and approximate NNLO results isactually better in the pole scheme than in the MS scheme. These results differ from those obtained in the ˆ s → m t limit, where the approximated NNLO corrections and the perturbative uncertainties at that order are significantlysmaller in the MS scheme than in the pole scheme [3].To elaborate further on these results, we note that the re-organization of the perturbative expansion in the MSscheme compared to the pole scheme is accomplished by the terms in square brackets in (15). To understand whetherthese terms are expected to cancel againt unphysically large corrections in the pole scheme, we note that the mainsource of mass dependence in the Born level cross section is due to phase-space factors: the lower limit of integration in(1), and an overall factor of p − m t / ˆ s in the partonic cross section related to two-body phase space and multiplyingthe Born-level matrix element. The derivatives contained in the terms in square brackets are mainly sensitive to thosesources of m t dependence. However, the phase space of the pair production is more indicative of the pole massthan of an MS mass. Indeed, we are calculating the cross section for on-shell quarks according to the narrow widthapproximation. If the cross section is instead calculated in the MS scheme, the terms in the square brackets of theNLO and NNLO pieces of (15) give sizeable negative corrections, which are accounted in the pole scheme by using anumerically higher value of the mass in the LO and NLO cross sections. Since the most appropriate mass scheme fora given process is the one where the higher-order corrections are expected to be smallest on physical grounds, it doesnot seem to us that the MS scheme is the optimal choice for this case.As an alternative to the MS mass, we consider the group of short-distance masses known as threshold masses[36]. At lower orders in perturbation theory, these are closer numerically to the pole mass, but they do not sufferfrom renormalon ambiguities at higher orders. The cross section in these schemes can be easily calculated from thepole-scheme results, using an analogous procedure to the MS scheme calculation. It is evident that at approximateNNLO the numerical difference between these results and the MS and pole-scheme results will be quite small oncethe numerical value of the mass is adjusted appropriately, but we nonetheless illustrate this with a specific example. The overlap between LO and NLO is worse at the Tevatron and improved at the LHC compared to the pole scheme, but as mentionedearlier we consider the more important issue the overlap between the NLO and approximate NNLO results.
In particular, we consider the cross section as a function of the 1S mass introduced in [37]. The 1S mass is definedthrough the perturbative contribution to the mass of a hypothetical n = 1, S toponium bound state. To performthe conversion to this scheme, we write its relation with the pole mass in the form [37] m t = m t ( α s ( µ r ) π πα s ( µ r ) + (cid:18) α s ( µ r ) π (cid:19) (cid:20) πα s ( µ r ) (cid:18)
233 ln 3 µ r α s ( µ r ) m t + 18118 + 29 πα s ( µ r ) (cid:19)(cid:21) + O (cid:18) α s π (cid:19)) , (17)and follow the same procedure as for the MS scheme calculation with the appropriate replacements, cf. (11). Notethat in the above relation πα s is counted as O (1) and is not expanded. The results are listed in Table V for the value m t = 172 . m t = 173 . Tevatron LHC7 LHC14MSTW CTEQ MSTW CTEQ MSTW CTEQLO 6 . +3 . . − . − (0 . . +2 . . . − . − . . +50+(6) − − (7) +35+9(7) − − +223+(26) − − (34) +160+25(18) − − NLO 6 . +0 . . . − . − . . . +0 . . . − . − . . +19+14(9) − − +17+14(11) − − +106+66(32) − − +96+41(27) − − NNLO approx. 6 . +0 . . . − . − . . . +0 . . . − . − . . +7+14(9) − − +7+13(11) − − +47+59(30) − − +46+39(25) − − TABLE V: Total cross sections in pb in the 1S scheme, for m t = 172 . α s uncertainty. The numbers in parenthesis show the PDF uncertainty only. V. CONCLUSIONS
We have presented predictions for the total inclusive cross section for top-quark pairs at hadron colliders at ap-proximate NNLO in QCD. Our calculations are based on soft gluon resummation to NNLL order in PIM and 1PIkinematics, carried out within the context of effective field theory. They represent the state-of-the-art, combining allknowledge presently available about higher-order QCD corrections to the production cross section. The perturbativeuncertainties associated with our results are estimated in two ways: through the standard method of variations offactorization and renormalization scales, and also through the difference between the two types of kinematics. Thelatter gives a means of estimating the size of perturbative power corrections to the soft limits in which the approximateNNLO formulas are derived. The results presented here consolidate those previously presented in [33, 35]. We havealso provided a computer program which calculates the total cross section within our approach.The total production cross section can be used along with experimental measurements to extract the top-quarkmass. An advantage of such extractions is that the theory calculations are carried out in a well-defined renormalizationscheme for m t . For very precise extractions of the top-quark mass the pole mass is disfavored, because it is only definedup to a renormalon ambiguity of order Λ QCD . In practice, however, we have not observed a poor convergence of theperturbative series up to NNLO in the pole scheme compared to the MS scheme, and pointed out that the group ofshort-distance masses known as threshold masses may be equally appropriate. We have provided numerical fits of ourresults as a function of the mass in both the pole and MS schemes, including perturbative and PDF uncertainties, inaddition to those from the strong coupling constant, which are non-negligible at this level of accuracy.The results presented in this letter can be used directly by the experimental collaborations at the Tevatron andLHC in top-quark mass measurements from the corresponding production cross sections.
Acknowledgements:
This research was supported in part by the State of Rhineland-Palatinate via the Research Cen-ter
Elementary Forces and Mathematical Foundations , by the German Federal Ministry for Education and Researchunder grant 05H09UME, by the German Research Foundation under grant NE398/3-1, 577401, by the EuropeanCommission through the
LHCPhenoNet
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