The top-pair total cross section at NNLL order

The top-quark pair production cross section at next-to-next-to-leading logarithmic order 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 Netherlands E-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 σt t¯ at the Tevatron and LHC, which include the resummation of soft logarithms and Coulomb singularities through next-tonext-to-leading logarithmic order, and t t¯ bound-state contributions. Resummation effects amount to about 8% of the next-to-leading order result at Tevatron and about 3% at LHC with 7 TeV centre-of-mass energy. They lead to a significant reduction of the theoretical uncertainty. With mt = 173.3 GeV, we find +0.31+0.71 σtTevatron = 7.22−0.47−0.55 pb t¯ +7.4+15.4 σtLHC = 162.6−7.5−14.7 pb , t¯ in good agreement with the latest experimental measurements. 10th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) - Radcor2011 September 26-30, 2011 Mamallapuram, India ∗ Speaker. † Preprint numbers: TTK-11-60, ITP-UU-11/45, SPIN-11/35, FR-PHENO-2011-024, SFB/CPP-11-79 c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ PoS(RADCOR2011)024 Martin Beneke Pietro Falgari The top-pair cross section at NNLL order 1. Introduction σt t¯(s) = ∑ Z 1 2 p,p0 =q,q̄,g 4mt /s d τ L pp0 (τ , µ f )σ̂ pp0 →t t¯X (τ s, µ f ) , (1.1) where L pp0 are parton luminosity functions, receives contributions from regions where β is not necessarily small, especially for LHC centre-of-mass energies, the region with β . 0.3 still gives a sizeable contribution to σt t¯, due to the rapid fall off of the parton luminosity functions at large τ . One therefore argues that an all-order resummation of soft and Coulomb corrections provides more accurate predictions than the fixed-order calculation. Leading logarithmic (LL) and next-to-leading logarithmic (NLL) resummation in the so-called Mellin-space formalism has been known for a while [5]. More recently, thanks also to the calculation 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-expansion of the resummed result [10, 11, 12, 13], have become available. Of the aforementioned works, only Ref. [8] provides a simultaneous resummation of soft and Coulomb corrections, obtained through a general formalism [14] based on the factorization of soft and Coulomb effects in the context of softcollinear 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-distance contributions, related to physics at the hard scale ∼ mt , and effects associated with soft-gluon emission, which naturally "live" at a much smaller scale ∼ mt β 2 . In [14] it was shown that, at threshold, the partonic cross section in fact factorizes into three different contributions, R σ̂ pp0 →t t¯X (ŝ, µ ) = ∑ Hpp 0 (mt , µ ) R=1,8 2 Z d ω JR (E − ω )W R (ω , µ ) . 2 i (2.1) PoS(RADCOR2011)024 The total top-pair production cross section σt t¯ has been measured at Tevatron with an accuracy ∆σt t¯/σt 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 ±6.5% level. With more statistics being collected, the experimental error is bound to reduce even further, opening a new era of precision top phenomenology. The total cross section, in particular, can be used to measure the top-quark pole mass mt in a theoretically clean way, to constrain new physics and to extract information on the gluon distribution function (PDF) of the proton. This clearly requires a precise theoretical understanding of the t t¯-production dynamics; more specifically, predictions beyond next-to-leading order (NLO) in standard fixed-order perturbative QCD are necessary. p Near the partonic production threshold, β ∼ 0, where β ≡ 1 − 4mt2 /ŝ is the velocity of the two top quarks, the partonic cross sections σ̂ pp0 →t t¯X are enhanced due to suppression of soft-gluon emission and exchange of potential (Coulomb) gluons between the non-relativistic top and antitop. These two effects give rise to singular terms, at all orders in perturbation theory, of the form αs ln2,1 β and αs /β , respectively. While the hadronic cross section, Pietro Falgari The top-pair cross section at NNLL order In (2.2), s̃Ri represents the Laplace transform of the fixed-order soft function WiR . The function R and η resum single logarithms. To S controls the resummation of double logarithms, while aW,i R and obtain NNLL accuracy, these functions must be included at the three-loop (S) or two-loop (aW,i η ) order, while the fixed-order soft function s̃Ri is required at one loop. An expression analogous to R . (2.2) can be derived for the hard function Hpp 0 Resummation of Coulomb effects has been extensively studied in the context of PNRQCD and quarkonia physics. The potential function JR is related to the Green’s function of the Schrödinger operator −~∇ 2 /mt − (−DR ) αs /r [1 + O(αs )], h i (0) (1) JR (E) = 2 Im GC,R (0, 0; E) ∆nC (E) + GC,R (0, 0; E) + . . . . (2.3) (0) In Eq. (2.3), GC,R denotes the LO Coulomb Green’s function, (r  )    2 α m 1 1 E 4 m E (−D ) t R s (0) GC,R = − t , − + (−DR )αs ln − 2 − + γE + ψ 1 − p 4π mt 2 2 µC 2 −E/mt (2.4) √ (1) with E = ŝ − 2mt ∼ mt β 2 . GC,R is the correction to the Green function from the αs2 terms in the Coulomb potential, and ∆nC = 1 + O(αs2 ln β ) represents non-Coulomb contributions which enter the cross section first at NNLL order [10]. Notice that for E < 0, i.e. below the production threshold, Eq. (2.3) contains a series of t t¯ bound-state resonances when DR < 0, which are included in the NNLL results presented in Section 4. It is often convenient to re-expand the resummed results to construct higher-order approximations at fixed order in αs . In particular, at NNLL all singular terms in the limit β → 0 at O(αs4 ) can be correctly predicted. Hence, one can define an approximated NNLO prediction as ) (  α 2 NNLOapp s (0) (2,n) n µ f NLO σ̂ pp0 (2.5) = ∑ σ̂ pp0 ,R + σ pp0 ,R ∑ f pp0 ,R ln mt , 4π n=0,2 R (0) NLO the exact colour-separated NLO cross sections [16], and σ with σ̂ pp 0 ,R pp0 ,R the Born contributions. (2,n) The NNLO functions f pp0 ,R incorporate exactly all terms (and only those) of the form ln4,3,2,1 β , ln2,1 β 1 β , β 2,1 , and were first correctly obtained in [10]. 3 PoS(RADCOR2011)024 R encodes the model-specific short-distance effects, the Coulomb function J The hard function Hpp 0 R describes the internal evolution of the t t¯ pair, driven by Coulomb exchange, and the soft function WiR contains soft-gluon contributions. Here R denotes the irreducible colour representation, either singlet (R = 1) or octet (R = 8), of the t t¯ pair. R and W R , are resummed In the approach adopted here [14, 15], the hard and soft functions, Hpp 0 i through renormalization-group (RG) evolution equations directly in momentum space, contrary to the conventional formalism, where resummation is performed in Mellin-moment space. The relevant RG equations and their solutions were derived, for the general case of massive particle pairs HH 0 in arbitrary colour representations R, R0 , in [14]. For the soft function WiR the solution to the evolution equation reads   e−2γE η 1 ω 2η R (µs , µ )] s̃Ri (∂η , µs ) WiR,res (ω , µ ) = exp[−4S(µs , µ ) + 2aW,i . (2.2) θ (ω ) ω µs Γ(2η ) Pietro Falgari The top-pair cross section at NNLL order 3. Scale choices and theoretical uncertainties The starting point of the evolution of the soft function WiR , represented by the soft scale µs appearing in (2.2), has to be chosen such that logarithms in the expansion in αs of s̃Ri are small, giving a stable perturbative behaviour at the low scale µs . For soft interactions, the natural scale is set by the kinetic energy of the t t¯ pair, mt β 2 . Accordingly, in [8] the soft scale in the resummed cross section was set to 2 (3.1) µs = max[ks mt βcut , ks mt β 2 ] , µC = max[CF αs mt , 2mt β ] , (3.2) where the frozen scale at low values of β , equal to the inverse Bohr radius CF αs mt , signals the onset of t t¯ bound-state effects. There is clearly some degree of ambiguity in the choice of the scales appearing in the resummed result. Besides these, NNLL and approximated NNLO predictions are affected by uncertainties 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, we consider the following sources of ambiguity: Scale uncertainty: we vary all scales µi in the interval [µ̃i /2, 2µ̃i ] around their central values µ̃i . µC is varied while keeping the other scales fixed. µh and µ f are allowed to vary simultaneously, imposing the additional constraint 1 ≤ µh /µ f ≤ 4. For the fixed-order results (NLO and NNLOapp ) the factorization scale µ f and renormalization scale µr are varied simultaneously, with the constraint 1/2 ≤ µr /µ f ≤ 2. The errors from varying {µ f , µh } and µC are added in quadrature. Resummation ambiguities: we consider three different sources of ambiguities: i) the difference √ between the default setting E = mt β 2 compared to E = ŝ − 2mt in (2.3), ii) the difference between the NNLL implementation for the soft scale choices ks = 1, 4 to the default choice ks = 2, iii) the effect of varying βcut by 20% around the default value for ks = 2 (see [8] for details). The resulting errors are added in quadrature. (2) NNLO-constant: by default, the O(αs4 ) constant in (2.5), Cpp0 ,R , is set to zero. We estimate the (2) (1) (1) effect of a non-vanishing constant by considering variations Cpp0 ,R = ±|Cpp0 ,R |2 , with Cpp0 ,R the constants in the threshold expansion of the NLO cross sections. 4 PoS(RADCOR2011)024 with the constant ks chosen by default as ks = 2. In the upper interval, β > βcut , the soft logarithms ln β in the partonic cross section are correctly resummed by the running soft scale ks mt β 2 . If βcut 2 still correctly resums is not too big, in the lower interval, β < βcut , the frozen soft scale ks mt βcut the dominant contributions (∼ ln βcut ) to the hadronic cross section, at the same time avoiding ambiguities related to the Landau pole in αs . A prescription for the choice of βcut is detailed in [8]. R naturally lives at the parametrically larger Contrary to the soft function, the hard function Hpp 0 scale given by the invariant mass of the t t¯ pair. The default value for the hard scale µh appearing in the resummed hard function is therefore chosen as µh = 2mt . On the other hand, the scale of Coulomb interactions is set by the typical virtuality of potential gluons, q2 ∼ mt2 β 2 . For the Coulomb scale µC in (2.3) we thus choose Pietro Falgari The top-pair cross section at NNLL order σt t¯[pb] Tevatron √ LHC ( s =7 TeV) √ LHC ( s =14 TeV) NLO NNLOapp NNLL2 6.68+0.36+0.51 −0.75−0.45 7.06+0.27+0.69 −0.34−0.53 7.22+0.31+0.71 −0.47−0.55 +18.5+13.9 158.1−21.2−13.1 +12.3+15.2 161.1−11.9−14.5 162.6+7.4+15.4 −7.5−14.7 884+107+65 −106−58 891+76+64 −69−63 896+40+65 −37−64 Table 1: t t¯ cross section at Tevatron and LHC from NLO, NNLOapp and NNLL2 approximations, for mt = 173.3 GeV. The first error set denotes the total theoretical uncertainty, the second the PDF+αs error. 4. Results We define our default prediction, NNLL2 , by matching the NNLL resummed result to the approximated NNLO cross section, i.e. NNLL2 = NNLL − NNLL(αs4 ) + NNLOapp , (4.1) where NNLL(αs4 ) is the expansion, up to order αs4 , of the resummed result. The top-quark mass is set to mt = 173.3GeV, and the renormalization and factorization scale are chosen as µ f = µr = mt . The other scales are treated as explained in Section 3. For the convolution of the partonic cross section 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, NNLOapp and the resummed NNLL2 implementation are given in Table 1, with the first error corresponding to the total theoretical uncertainty, computed as described in Section 3, and the second error the PDF+αs uncertainty. The genuine NNLL corrections are sizeable and positive both at Tevatron (∼ +13%) and LHC (∼ +9%), though their effect is partially compensated by switching from NLO to NNLO PDF sets for the NNLOapp and NNLL2 results, which results in a negative shift of the cross section. The bulk of the corrections beyond NLO is accounted for by the O(αs4 ) terms, as evident from a comparison of NNLOapp and NNLL2 . One can also notice a significant reduction of the theoretical uncertainty from NLO to NNLOapp /NNLL2 , both at Tevatron and the LHC, to the extent that for NNLOapp and NNLL2 the total error is dominated by the PDF+αs error. The theory uncertainty of different approximations is shown in Figure 1, where the total theoretical error bands for NLO (dashed black), NNLOapp (dot-dashed blue) and NNLL2 (solid red) are plotted as functions of mt . At the LHC, one observes a nice convergence of the series NLO→NNLOapp →NNLL2 , with the resummed result having the smallest theoretical uncertainty, corresponding to about ±4.7%. At the Tevatron, however, the NNLOapp result exhibits the smallest error. We interpret this as an indication that the NNLOapp error is accidentally underestimated by the scale variation procedure. For NNLL2 , the residual theoretical uncertainty is +4.3 − 6.5%. Predictions for the total cross section at the NNLL or approximated NNLO level have been published recently by several groups. These results differ in the resummation formalism adopted 5 PoS(RADCOR2011)024 In Section 4 the total theory error is obtained summing in quadrature the three aforementioned uncertainties. Additionally, we estimate the error due to uncertainties in the PDFs and the strong coupling, using the 90% confidence level set of the MSTW08NNLO PDFs and the five sets for variations of αs provided in [17]. This will be quoted separately from the theoretical error. Pietro Falgari The top-pair cross section at NNLL order Tevatron LHC H7 TeVL Σtt @pbD Σtt @pbD NLO NLO NNLOapp NNLL2 9 220 NNLOapp NNLL2 200 8 180 7 160 6 140 5 120 168 170 172 174 176 178 180 mt @GeVD 166 168 170 172 174 176 178 180 mt @GeVD Figure 1: NLO (dashed black), NNLOapp (dot-dashed blue) and NNLL2 (solid red) as function of mt . The bands correspond to the total theoretical uncertainty (that is, excluding the PDF+αs error) of the prediction. and in whether the total cross section is resummed, or differential distributions in different kinematics limits, which are then integrated to obtain predictions for the inclusive cross section. They also differ in the treatment of Coulomb effects beyond NLO and of the constant terms at O(αs4 ), and some include sets of power-suppressed contributions in β . A comparison of the various results can 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 the numbers by Kidonakis (blue up-pointing triangles, [13]), Ahrens et al. (red diamonds, [12] for NNLO 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 given for comparison. At the LHC the predictions of different groups show a good agreement, with a somewhat better agreement for the NNLO results. At NNLL, the total spread of the four results is still smaller than the theoretical uncertainty of the NLO result. At Tevatron there appears to be a stronger tension between different results (both for central values and error estimates), with the envelope of all predictions at NNLO and NNLL being almost as large as the uncertainty of the NLO result. In [8] it was argued that this might be a consequence of the dominance, at Tevatron, of the qq̄ 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 σt t¯ can be translated into measures of the top-quark pole mass mt . In first approximation, this can be done by comparing the mass dependence of the theoretical and experimental cross sections, and extracting the top mass at the intersection point. In a more sophisticated approach, one defines a likelihood function f (mt ) ∝ Z d σ fth (σ |mt ) · fexp (σ |mt ) , (5.1) where fth and fexp represent normalized gaussian distributions centred around the theoretical prediction and measured experimental cross section, respectively, and extracts mt from the maximum of the probability distribution. A suitable parameterization of the experimental cross section in 6 PoS(RADCOR2011)024 166 Pietro Falgari The top-pair cross section at NNLL order Tevatron Σ@pbD 9 8 LHC H7 TeVL Σ@pbD 220 BFKS (mt = 173.3 GeVL Kidonakis Hmt = 173 GeVL Ahrens et al. Hmt = 173.1 GeVL Cacciari et al. Hmt = 173.3 GeVL 200 BFKS (mt = 173.3 GeVL Kidonakis (mt = 173 GeVL Ahrens et al. (mt = 173.1 GeVL Cacciari et al. (mt = 173.3 GeVL 180 ôô ô æ æ ò 7 æ 160 æ ò ì æ à ì æ ô à ìì ìì 140 6 120 NNLO NNLL NLO CDF/D0 NNLO NNLL Atlas/CMS 100 Figure 2: Comparison of different NNLO and NNLL predictions, see the text for explanation and references. The error bands include theoretical uncertainties, but no PDF+αs errors. The rightmost set of points represents the most recent experimental measurements, which assume a top mass of 172.5 GeV. Σtt Hmt L@pbD th 260 Σtt Hmt L 240 exp Σtt Hmt L 220 200 180 160 140 mt @GeVD 160 165 170 175 180 Figure 3: mt dependence of the experimental t t¯ cross section [3] (solid black) and of the NNLL2 approximation presented here (solid red). The dashed lines represent the total uncertainties of the two curves. terms of mt has been provided by the ATLAS collaboration [3], and is plotted in Figure 3. Using the NNLL2 prediction presented here as theoretical input in (5.1) and the data from [3] gives the following value for the top pole mass, +4.9 mt = 169.8−4.7 GeV , (5.2) which agrees with the direct-reconstruction measurement of Tevatron, mt = 173.3 ± 1.1 GeV. Note that the error in (5.2) does not include the uncertainty deriving from identifying the Monte Carlo mass in the experimental input with the pole mass mt . Allowing for a ±2 GeV difference would correspond to an additional uncertainty of ±0.65 GeV on the extracted mass. 6. Conclusions We presented new predictions [8] for the t t¯ total production cross section which include NNLL resummation of soft and Coulomb effects. Resummation increases the cross section relative to the fixed-order NLO result, and leads to a significant reduction of the theoretical uncertainty. 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