The top-quark pair production cross section at next-to-next-to-leading logarithmic order
aa r X i v : . [ h e p - ph ] D ec The top-quark pair production cross section atnext-to-next-to-leading logarithmic order
Martin Beneke
Institute für Theoretische Teilchenphysik und Kosmologie,RWTH Aachen University, D–52056 Aachen, Germany
Pietro Falgari ∗ † Institute for Theoretical Physics and Spinoza Institute,Utrecht University, 3508 TD Utrecht, The NetherlandsE-mail: [email protected]
Sebastian Klein
Institute für Theoretische Teilchenphysik und Kosmologie,RWTH Aachen University, D–52056 Aachen, Germany
Christian Schwinn
Albert-Ludwigs Universität Freiburg, Physikalisches Institut,D-79104 Freiburg, Germany
We present predictions for the total t ¯ t production cross section s t ¯ t at the Tevatron and LHC,which include the resummation of soft logarithms and Coulomb singularities through next-to-next-to-leading logarithmic order, and t ¯ t bound-state contributions. Resummation effects amountto about 8% of the next-to-leading order result at Tevatron and about 3% at LHC with 7 TeVcentre-of-mass energy. They lead to a significant reduction of the theoretical uncertainty. With m t = . s Tevatron t ¯ t = . + . + . − . − . pb s LHC t ¯ t = . + . + . − . − . pb , in good agreement with the latest experimental measurements. ∗ Speaker. † Preprint numbers: TTK-11-60, ITP-UU-11/45, SPIN-11/35, FR-PHENO-2011-024, SFB/CPP-11-79 c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ he top-pair cross section at NNLL order
Pietro Falgari
1. Introduction
The total top-pair production cross section s t ¯ t has been measured at Tevatron with an accuracy D s t ¯ t / s t ¯ t of about ±
7% [1, 2] and the two LHC experiments have already reached similar sensitivity[3, 4], with the accuracy of some analyses already at the ± .
5% level. With more statistics beingcollected, the experimental error is bound to reduce even further, opening a new era of precisiontop phenomenology. The total cross section, in particular, can be used to measure the top-quarkpole mass m t in a theoretically clean way, to constrain new physics and to extract informationon the gluon distribution function (PDF) of the proton. This clearly requires a precise theoreticalunderstanding of the t ¯ t -production dynamics; more specifically, predictions beyond next-to-leadingorder (NLO) in standard fixed-order perturbative QCD are necessary.Near the partonic production threshold, b ∼
0, where b ≡ p − m t / ˆ s is the velocity of thetwo top quarks, the partonic cross sections ˆ s pp ′ → t ¯ tX are enhanced due to suppression of soft-gluonemission and exchange of potential (Coulomb) gluons between the non-relativistic top and anti-top. These two effects give rise to singular terms, at all orders in perturbation theory, of the form a s ln , b and a s / b , respectively. While the hadronic cross section, s t ¯ t ( s ) = (cid:229) p , p ′ = q , ¯ q , g Z m t / s d t L pp ′ ( t , m f ) ˆ s pp ′ → t ¯ tX ( t s , m f ) , (1.1)where L pp ′ are parton luminosity functions, receives contributions from regions where b is notnecessarily small, especially for LHC centre-of-mass energies, the region with b . . s t ¯ t , due to the rapid fall off of the parton luminosity functions at large t .One therefore argues that an all-order resummation of soft and Coulomb corrections provides moreaccurate predictions than the fixed-order calculation.Leading logarithmic (LL) and next-to-leading logarithmic (NLL) resummation in the so-calledMellin-space formalism has been known for a while [5]. More recently, thanks also to the calcu-lation of the relevant soft anomalous dimensions [6], next-to-next-to-leading logarithmic (NNLL)resummations [7, 8, 9], and approximated NNLO cross sections constructed from the re-expansionof the resummed result [10, 11, 12, 13], have become available. Of the aforementioned works, onlyRef. [8] provides a simultaneous resummation of soft and Coulomb corrections, obtained through ageneral formalism [14] based on the factorization of soft and Coulomb effects in the context of soft-collinear effective theory (SCET) and potential non-relativistic QCD (PNRQCD). The formalism,and the results of [8], will be reviewed in the following.
2. Factorization and resummation of the t ¯ t total cross section The basis for resummation is the factorization of the partonic cross section into short-distancecontributions, related to physics at the hard scale ∼ m t , and effects associated with soft-gluonemission, which naturally "live" at a much smaller scale ∼ m t b . In [14] it was shown that, atthreshold, the partonic cross section in fact factorizes into three different contributions,ˆ s pp ′ → t ¯ tX ( ˆ s , m ) = (cid:229) R = , H Rpp ′ ( m t , m ) Z d w J R ( E − w ) W Ri ( w , m ) . (2.1)2 he top-pair cross section at NNLL order Pietro Falgari
The hard function H Rpp ′ encodes the model-specific short-distance effects, the Coulomb function J R describes the internal evolution of the t ¯ t pair, driven by Coulomb exchange, and the soft function W Ri contains soft-gluon contributions. Here R denotes the irreducible colour representation, eithersinglet ( R = ) or octet ( R = ), of the t ¯ t pair.In the approach adopted here [14, 15], the hard and soft functions, H Rpp ′ and W Ri , are resummedthrough renormalization-group (RG) evolution equations directly in momentum space, contraryto the conventional formalism, where resummation is performed in Mellin-moment space. Therelevant RG equations and their solutions were derived, for the general case of massive particlepairs HH ′ in arbitrary colour representations R , R ′ , in [14]. For the soft function W Ri the solutionto the evolution equation reads W R , res i ( w , m ) = exp [ − S ( m s , m ) + a RW , i ( m s , m )] ˜ s Ri ( ¶ h , m s ) w (cid:18) wm s (cid:19) h q ( w ) e − g E h G ( h ) . (2.2)In (2.2), ˜ s Ri represents the Laplace transform of the fixed-order soft function W Ri . The function S controls the resummation of double logarithms, while a RW , i and h resum single logarithms. Toobtain NNLL accuracy, these functions must be included at the three-loop ( S ) or two-loop ( a RW , i and h ) order, while the fixed-order soft function ˜ s Ri is required at one loop. An expression analogous to(2.2) can be derived for the hard function H Rpp ′ .Resummation of Coulomb effects has been extensively studied in the context of PNRQCD andquarkonia physics. The potential function J R is related to the Green’s function of the Schrödingeroperator − ~ (cid:209) / m t − ( − D R ) a s / r [ + O ( a s )] , J R ( E ) = h G ( ) C , R ( , E ) D nC ( E ) + G ( ) C , R ( , E ) + . . . i . (2.3)In Eq. (2.3), G ( ) C , R denotes the LO Coulomb Green’s function, G ( ) C , R = − m t p (r − Em t + ( − D R ) a s (cid:20)
12 ln (cid:18) − m t E m C (cid:19) − + g E + y (cid:18) − ( − D R ) a s p − E / m t (cid:19)(cid:21)) , (2.4)with E = √ ˆ s − m t ∼ m t b . G ( ) C , R is the correction to the Green function from the a s terms inthe Coulomb potential, and D nC = + O ( a s ln b ) represents non-Coulomb contributions whichenter the cross section first at NNLL order [10]. Notice that for E <
0, i.e. below the productionthreshold, Eq. (2.3) contains a series of t ¯ t bound-state resonances when D R <
0, which are includedin the NNLL results presented in Section 4.It is often convenient to re-expand the resummed results to construct higher-order approxima-tions at fixed order in a s . In particular, at NNLL all singular terms in the limit b → O ( a s ) canbe correctly predicted. Hence, one can define an approximated NNLO prediction asˆ s NNLO app pp ′ = (cid:229) R ( ˆ s NLO pp ′ , R + s ( ) pp ′ , R (cid:16) a s p (cid:17) (cid:229) n = , f ( , n ) pp ′ , R ln n m f m t ) , (2.5)with ˆ s NLO pp ′ , R the exact colour-separated NLO cross sections [16], and s ( ) pp ′ , R the Born contributions.The NNLO functions f ( , n ) pp ′ , R incorporate exactly all terms (and only those) of the form ln , , , b , ln , bb , b , , and were first correctly obtained in [10].3 he top-pair cross section at NNLL order Pietro Falgari
3. Scale choices and theoretical uncertainties
The starting point of the evolution of the soft function W Ri , represented by the soft scale m s appearing in (2.2), has to be chosen such that logarithms in the expansion in a s of ˜ s Ri are small,giving a stable perturbative behaviour at the low scale m s . For soft interactions, the natural scale isset by the kinetic energy of the t ¯ t pair, m t b . Accordingly, in [8] the soft scale in the resummedcross section was set to m s = max [ k s m t b , k s m t b ] , (3.1)with the constant k s chosen by default as k s =
2. In the upper interval, b > b cut , the soft logarithmsln b in the partonic cross section are correctly resummed by the running soft scale k s m t b . If b cut is not too big, in the lower interval, b < b cut , the frozen soft scale k s m t b still correctly resumsthe dominant contributions ( ∼ ln b cut ) to the hadronic cross section, at the same time avoidingambiguities related to the Landau pole in a s . A prescription for the choice of b cut is detailed in [8].Contrary to the soft function, the hard function H Rpp ′ naturally lives at the parametrically largerscale given by the invariant mass of the t ¯ t pair. The default value for the hard scale m h appearingin the resummed hard function is therefore chosen as m h = m t . On the other hand, the scaleof Coulomb interactions is set by the typical virtuality of potential gluons, q ∼ m t b . For theCoulomb scale m C in (2.3) we thus choose m C = max [ C F a s m t , m t b ] , (3.2)where the frozen scale at low values of b , equal to the inverse Bohr radius C F a s m t , signals theonset of t ¯ t bound-state effects.There is clearly some degree of ambiguity in the choice of the scales appearing in the re-summed result. Besides these, NNLL and approximated NNLO predictions are affected by uncer-tainties related to constant and power suppressed terms which are not controlled by resummation.Therefore, to reliably ascertain the residual theoretical error of the predictions presented below, weconsider the following sources of ambiguity: Scale uncertainty: we vary all scales m i in the interval [ ˜ m i / , m i ] around their central values ˜ m i . m C is varied while keeping the other scales fixed. m h and m f are allowed to vary simultane-ously, imposing the additional constraint 1 ≤ m h / m f ≤
4. For the fixed-order results (NLOand NNLO app ) the factorization scale m f and renormalization scale m r are varied simultane-ously, with the constraint 1 / ≤ m r / m f ≤
2. The errors from varying { m f , m h } and m C areadded in quadrature. Resummation ambiguities: we consider three different sources of ambiguities: i) the differencebetween the default setting E = m t b compared to E = √ ˆ s − m t in (2.3), ii) the differencebetween the NNLL implementation for the soft scale choices k s = , k s =
2, iii) the effect of varying b cut by 20% around the default value for k s = NNLO-constant: by default, the O ( a s ) constant in (2.5), C ( ) pp ′ , R , is set to zero. We estimate theeffect of a non-vanishing constant by considering variations C ( ) pp ′ , R = ±| C ( ) pp ′ , R | , with C ( ) pp ′ , R the constants in the threshold expansion of the NLO cross sections.4 he top-pair cross section at NNLL order Pietro Falgari s t ¯ t [pb] Tevatron LHC ( √ s = √ s =
14 TeV)NLO 6 . + . + . − . − . . + . + . − . − . + + − − NNLO app . + . + . − . − . . + . + . − . − . + + − − NNLL . + . + . − . − . . + . + . − . − . + + − − Table 1: t ¯ t cross section at Tevatron and LHC from NLO, NNLO app and NNLL approximations, for m t = . a s error. In Section 4 the total theory error is obtained summing in quadrature the three aforementioneduncertainties. Additionally, we estimate the error due to uncertainties in the PDFs and the strongcoupling, using the 90% confidence level set of the MSTW08NNLO PDFs and the five sets forvariations of a s provided in [17]. This will be quoted separately from the theoretical error.
4. Results
We define our default prediction, NNLL , by matching the NNLL resummed result to theapproximated NNLO cross section, i.e.NNLL = NNLL − NNLL ( a s ) + NNLO app , (4.1)where NNLL ( a s ) is the expansion, up to order a s , of the resummed result. The top-quark mass isset to m t = . m f = m r = m t .The other scales are treated as explained in Section 3. For the convolution of the partonic crosssection with the parton luminosity functions, Eq. (1.1), we use the MSTW08 PDF set [17].Numerical results for the cross section at the Tevatron and LHC for NLO, NNLO app and theresummed NNLL implementation are given in Table 1, with the first error corresponding to thetotal theoretical uncertainty, computed as described in Section 3, and the second error the PDF+ a s uncertainty. The genuine NNLL corrections are sizeable and positive both at Tevatron ( ∼ + ∼ + app and NNLL results, which results in a negative shift of the cross section.The bulk of the corrections beyond NLO is accounted for by the O ( a s ) terms, as evident from acomparison of NNLO app and NNLL . One can also notice a significant reduction of the theoreticaluncertainty from NLO to NNLO app /NNLL , both at Tevatron and the LHC, to the extent that forNNLO app and NNLL the total error is dominated by the PDF+ a s error.The theory uncertainty of different approximations is shown in Figure 1, where the totaltheoretical error bands for NLO (dashed black), NNLO app (dot-dashed blue) and NNLL (solidred) are plotted as functions of m t . At the LHC, one observes a nice convergence of the seriesNLO → NNLO app → NNLL , with the resummed result having the smallest theoretical uncertainty,corresponding to about ± . app result exhibits the smallesterror. We interpret this as an indication that the NNLO app error is accidentally underestimated bythe scale variation procedure. For NNLL , the residual theoretical uncertainty is + . − . he top-pair cross section at NNLL order Pietro Falgari
NLONNLO app
NNLL
166 168 170 172 174 176 178 180 m t @ GeV D Σ tt @ pb D Tevatron
NLONNLO app
NNLL
166 168 170 172 174 176 178 180 m t @ GeV D Σ tt @ pb D LHC H L Figure 1:
NLO (dashed black), NNLO app (dot-dashed blue) and NNLL (solid red) as function of m t . Thebands correspond to the total theoretical uncertainty (that is, excluding the PDF+ a s error) of the prediction. and in whether the total cross section is resummed, or differential distributions in different kine-matics limits, which are then integrated to obtain predictions for the inclusive cross section. Theyalso differ in the treatment of Coulomb effects beyond NLO and of the constant terms at O ( a s ) ,and some include sets of power-suppressed contributions in b . A comparison of the various resultscan thus give an estimate of the ambiguities inherent to resummation. This is shown in Figure 2,where the predictions for NNLO and NNLL given here (black circles, [8]) are compared to thenumbers by Kidonakis (blue up-pointing triangles, [13]), Ahrens et al. (red diamonds, [12] forNNLO and [7] for NNLL in both 1PI and PIM kinematics) and Cacciari et al. (green squares,[9]). The experimental measurements (purple down-pointing triangles, [1, 2, 3, 4]) are also givenfor comparison. At the LHC the predictions of different groups show a good agreement, with asomewhat better agreement for the NNLO results. At NNLL, the total spread of the four resultsis still smaller than the theoretical uncertainty of the NLO result. At Tevatron there appears to bea stronger tension between different results (both for central values and error estimates), with theenvelope of all predictions at NNLO and NNLL being almost as large as the uncertainty of theNLO result. In [8] it was argued that this might be a consequence of the dominance, at Tevatron,of the q ¯ q production channel, which is less well approximated by its threshold expansion than the gg channel dominant at the LHC, thus leading to larger ambiguities in resummation.
5. Mass determination
Precise theoretical predictions of s t ¯ t can be translated into measures of the top-quark pole mass m t . In first approximation, this can be done by comparing the mass dependence of the theoreticaland experimental cross sections, and extracting the top mass at the intersection point. In a moresophisticated approach, one defines a likelihood function f ( m t ) (cid:181) Z d s f th ( s | m t ) · f exp ( s | m t ) , (5.1)where f th and f exp represent normalized gaussian distributions centred around the theoretical pre-diction and measured experimental cross section, respectively, and extracts m t from the maximumof the probability distribution. A suitable parameterization of the experimental cross section in6 he top-pair cross section at NNLL order Pietro Falgari æ æ æààì ììòò ôô
BFKS ( m t = L Kidonakis H m t =
173 GeV L Ahrens et al. H m t = L Cacciari et al. H m t = L NLO NNLO NNLL CDF/D056789 Σ @ pb D Tevatron æ æ æààì ììòò ôô
BFKS ( m t = L Kidonakis ( m t =
173 GeV L Ahrens et al. ( m t = L Cacciari et al. ( m t = L NLO NNLO NNLL Atlas/CMS100120140160180200220 Σ @ pb D LHC H L Figure 2:
Comparison of different NNLO and NNLL predictions, see the text for explanation and refer-ences. The error bands include theoretical uncertainties, but no PDF+ a s errors. The rightmost set of pointsrepresents the most recent experimental measurements, which assume a top mass of 172.5 GeV. Σ ttth H m t L Σ ttexp H m t L
160 165 170 175 180 m t @ GeV D Σ tt H m t L@ pb D Figure 3: m t dependence of the experimental t ¯ t cross section [3] (solid black) and of the NNLL approxi-mation presented here (solid red). The dashed lines represent the total uncertainties of the two curves. terms of m t has been provided by the ATLAS collaboration [3], and is plotted in Figure 3. Usingthe NNLL prediction presented here as theoretical input in (5.1) and the data from [3] gives thefollowing value for the top pole mass, m t = . + . − . GeV , (5.2)which agrees with the direct-reconstruction measurement of Tevatron, m t = . ± . m t . Allowing for a ± ± .
65 GeV on the extracted mass.
6. Conclusions
We presented new predictions [8] for the t ¯ t total production cross section which include NNLLresummation of soft and Coulomb effects. Resummation increases the cross section relative to thefixed-order NLO result, and leads to a significant reduction of the theoretical uncertainty. Our7 he top-pair cross section at NNLL order Pietro Falgari numbers are in good agreement with experimental measurements at the Tevatron and LHC. At theLHC they also show a reasonably good agreement with other NNLL predictions, though biggerdifferences are found at the Tevatron.
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