Further exploration of top pair hadroproduction at NNLO
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Vol. ?, N. ? ? Further exploration of top pair hadroproduction at NNLO
M. Czakon ( ) , P. Fiedler ( ) , A. Mitov ( )( ∗ ) and J. Rojo ( ) ( ) Institut f¨ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen UniversityD-52056 Aachen, Germany ( ) Theory Division, CERN, CH-1211 Geneva 23, Switzerland
Summary. — Top quark pair production is one of the cornerstones of the physicsprogram at hadron colliders. In this contribution, we further explore the phe-nomenological implications of the recent NNLO calculation of the total inclusivecross-section. We provide a comparison of the scale dependence of the top pairhadroproduction cross section at different perturbative orders and study its pertur-bative convergence (with and without soft-gluon resummation). We also sketch howthe NNLO top quark cross section could be used to improve searches of physicsbeyond the Standard Model.PACS: 14.65.Ha, 13.85.Lg, 12.38.BxPreprint: CERN-PH-TH/2013-100, TTK-13-15
1. – Introduction
Top quark pair production is one of the cornerstones of the Standard Model (SM)program at hadron colliders, and a number of precision calculations of this process haveappeared in the recent past. In this writeup, we focus our attention on the total inclusivecross-section which, during the last year, became known in full NNLO [1-4], and presentanalyses based on the NNLO calculation that are not available in the literature. ( )This writeup is organized as follows: in section we introduce our notation. Insection we give the explicit results for the collinear factorization contribution and forthe scale dependent terms in the gg reaction, both of which were not explicitly presentedin Ref. [4]. In section we present a number of results that illustrate the convergenceproperties of perturbation theory with and without soft-gluon resummation. Finallyin section we present some preliminary results that illustrate how precision top pairproduction can be relevant for searches of BSM physics. ( ∗ ) Speaker( ) For a broader recent overview of theoretical developments in top quark physics see, forexample, Ref. [5]. c (cid:13) Societ`a Italiana di Fisica M. CZAKON, P. FIEDLER, A. MITOV AND J. ROJO
2. – The t ¯ t total cross-section: notations We follow the notation established in Refs. [1-4]. The total inclusive top pair produc-tion cross-section is defined as σ tot = X i,j Z β max dβ Φ ij ( β, µ F ) ˆ σ ij ( α S ( µ R ) , β, m , µ F , µ R ) . (1)The indices i, j run over all possible initial state partons; β max ≡ p − m /S ; √ S isthe c.m. energy of the hadron collider and β = √ − ρ , with ρ ≡ m /s , is the relativevelocity of the final state top quarks with pole mass m and partonic c.m. energy √ s .The function Φ in Eq. (1) is the partonic fluxΦ ij ( β, µ F ) = 2 β − β L ij (cid:18) − β − β , µ F (cid:19) , (2)expressed through the partonic luminosity L ij ( x, µ F ) = x ( f i ⊗ f j ) ( x, µ F ) = x Z dy Z dz δ ( x − yz ) f i ( y, µ F ) f j ( z, µ F ) . (3) As usual, µ R,F are the renormalization and factorization scales. Setting µ F = µ R = µ ,the partonic cross-section can be expanded through NNLO asˆ σ ij = α S m ( σ (0) ij + α S h σ (1) ij + L σ (1 , ij i + α S h σ (2) ij + L σ (2 , ij + L σ (2 , ij i ) . (4)In the above equation L = ln (cid:0) µ /m (cid:1) , α S is the MS coupling renormalized with N L = 5active flavors at scale µ and σ ( n ( ,m )) ij are functions only of β .All partonic cross-sections are known through NNLO [1-4]. The scaling functions σ (2 , ij and σ (2 , ij can be computed from σ (1) ij , see section . The dependence on µ R = µ F can be trivially restored in Eq. (4) by re-expressing α S ( µ F ) in powers of α S ( µ R ); see forexample Ref. [6].
3. – Collinear factorization and scale dependence of the partonic cross-section
We follow the setup and notation described in Ref. [2] and denote the collinearlyunrenormalized partonic cross-sections as ˜ σ ( n ) ij ( ε, ρ ). Then, introducing the functions ˜ s ( n ) ij and s ( n ) ij defined as ˜ s ( n ) ij ( ε, ρ ) ≡ ˜ σ ( n ) ij ( ε, ρ ) /ρ and s ( n ) ij ( ρ ) ≡ σ ( n ) ij ( ρ ) /ρ , the MS–subtracted gg -initiated cross-section s ( n ) gg reads through NNLO: s (1) gg = ˜ s (1) gg + 2 ǫ (cid:18) π (cid:19) ˜ s (0) gg ⊗ P (0) gg , (5) s (2) gg = ˜ s (2) gg + (cid:18) π (cid:19) ( ε h − β ˜ s (0) gg ⊗ P (0) gg + 2˜ s (0) gg ⊗ P (0) gg ⊗ P (0) gg (6) URTHER EXPLORATION OF TOP PAIR HADROPRODUCTION AT NNLO +2 N L (cid:16) ˜ s (0) gg ⊗ P (0) gq ⊗ P (0) qg + ˜ s (0) q ¯ q ⊗ P (0) qg ⊗ P (0) qg (cid:17)i + 1 ǫ ˜ s (0) gg ⊗ P (1) gg ) + 1 ǫ (cid:18) π (cid:19) ( N L ˜ s (1) qg ⊗ P (0) qg + 2˜ s (1) gg ⊗ P (0) gg ) , with β = 11 C A / − N L / β ∈ (0 , s (1) gg through order O ( ǫ ) has been detailed in Ref. [3].The evaluation of the scale dependent functions σ (2 , gg and σ (2 , gg in Eq. (4) is ratherstraightforward, see [2] for details. In terms of the functions s ( n ( ,m )) ij ( ρ ) ≡ σ ( n ( ,m )) ij ( ρ ) /ρ we get: s (2 , gg = 1(2 π ) ( β s (0) gg − β s (0) gg ⊗ P (0) gg + 2 s (0) gg ⊗ P (0) gg ⊗ P (0) gg +2 N L (cid:16) s (0) gg ⊗ P (0) qg ⊗ P (0) gq + s (0) q ¯ q ⊗ P (0) qg ⊗ P (0) qg (cid:17) ) ,s (2 , gg = 2(2 π ) ( β s (0) gg − s (0) gg ⊗ P (1) gg ) + 12 π ( β s (1) gg − s (1) gg ⊗ P (0) gg − N L s (1) qg ⊗ P (0) qg ) , (7)with β = 17 C A / − C A N L / − C F N L / σ (2 , gg and σ (2 , gg have been implementedin version 2.0 of the program Top++ [7] ( ) and can be read off from there.
4. – Perturbative convergence of the hadronic cross-section
The size of the scale dependence of the t ¯ t cross-section at NNLO and NNLO+NNLLhas been studied in Ref. [4], while a detailed breakdown of the various sources of theo-retical uncertainty (PDFs, scale, α s and m top ) was provided in Ref. [9]. In the followingwe will study the changes of the scale dependence of the total cross-section as a functionof the perturbative order. As a representative case, we focus our discussion on LHC 8TeV. We also update the corresponding plot for the Tevatron from Ref. [1].We begin by first comparing the pure fixed order predictions i.e. not including softgluon resummation. We compare the LO, NLO and NNLO results, and each one iscomputed with a PDF set of matching accuracy. For consistency with our earlier pre-sentations we use everywhere the MSTW2008 (68cl) family of PDF sets [10]. Similarresults are obtained if other PDF sets such as CT10 [11] and NNPDF2.3 [12] are used,see Ref. [9] for a detailed comparison of the predictions from the various sets. ( ) Fits implemented in the program Hathor [8] have also been utilized. M. CZAKON, P. FIEDLER, A. MITOV AND J. ROJO σ t o t [ pb ] m top [GeV]PPbar → tt+XIndependent µ F,R variationMSTW2008(68c.l.) LO; NLO; NNLONNLO (scales)NLO (scales)LO (scales)CDF+D0 (8.8fb -1 ) 50 100 150 200 250 300 350 164 166 168 170 172 174 176 178 180 182 σ t o t [ pb ] m top [GeV]PP(8 TeV) → tt+XIndependent µ F,R variationMSTW2008(68c.l.) LO; NLO; NNLONNLO (scales)NLO (scales)LO (scales)CMS, 8TeV
Fig. 1. – Scale dependence of the total cross-section at LO (blue), NLO (red) and NNLO(black) as a function of m top at the Tevatron (left) and the LHC 8 TeV (right). No softgluon resummation is included. For reference the most precise experimental measurements arealso shown. In fig. 1 (left) we show the scale dependence of the predicted cross-section at theTevatron, as a function of the top quark mass. We note the significant and consistentimprovement in the theoretical precision due to inclusion of corrections at higher per-turbative orders. We also note the agreement between the theoretical prediction ( ) andthe latest Tevatron measurement [13].Next we turn to the LHC. In fig. 1 (right) we show the scale dependence of thepredicted cross-section at the LHC 8 TeV as a function of m top . Similarly to the caseof the Tevatron, we observe a very good perturbative convergence of the theoreticalprediction and good agreement with the available measurement [14].In fig. 2 (left) we show the scale dependence of the predicted cross-section at the LHCas a function of the collider energy. We note that the perturbative convergence observedat 8 TeV is consistently present in the whole range of relevant LHC energies. Moreover,the good agreement of the NNLO theoretical prediction with the available data persistsat all energies where data is currently available [15-17].Next we study the impact of soft-gluon resummation on the size of the scale depen-dence and the central value of the theoretical prediction. In fig. 2 (right) we show thescale dependence of the predicted cross-section at the LHC 8 TeV for a number of caseswith different fixed order and logarithmic accuracy: LO, NLO, NLO+LL, NLO+NLL,NLO+NNLL, NNLO, NNLO+LL, NNLO+NLL and NNLO+NNLL. In all cases we fol-low the resummation procedure of Ref. [18]. We set the constant A = 0 (introduced inRef. [19]), m top = 173 . ( ) Recall that only the scale dependence is shown. The full theoretical uncertainty is, roughly,about twice as large as the scale dependence. URTHER EXPLORATION OF TOP PAIR HADROPRODUCTION AT NNLO
50 100 150 200 250 300 350 6.5 7 7.5 8 8.5 σ t o t [ pb ] √ s [TeV]Indep. µ F,R variationPP → tt+X; m top =173.3 GeVMSTW2008(68c.l.) LO; NLO; NNLONNLO (scales)NLO (scales)LO (scales)CMS, 7TeVATLAS+CMS, 7TeVATLAS, 7TeVCMS, 8TeV 120 140 160 180 200 220 240 260 280 σ t o t [ pb ] Scale variationLO NLO NNLOLL NLLNNLL LL NLLNNLLLHC 8 TeV; m top =173.3 GeV; A=0MSTW2008 LO; NLO; NNLOFixed OrderNLO+resNNLO+res
Fig. 2. – Scale dependence of the predicted cross-section at LO, NLO and NNLO at the LHCas a function of √ s (left). On the right plot: detailed breakdown of scale uncertainty for LHC8 TeV at LO, NLO and NNLO including also soft-gluon resummation at LL, NLL and NNLL. resummation. Inclusion of resummation with logarithmic accuracy at NLL or NNLLalso noticeably decreases the scale dependence of the theoretical prediction, as expected.The absolute size of the resulting reduction in scale dependence is also at the 2% level.An alternative way of assessing the impact of soft-gluon resummation is shown infig. 3 (which updates fig. 1 of Ref. [18] by including the exact NNLO result). Plottedis the relative error of the cross-section at the LHC as a function of the collider energy.We consider a broad range of energies, starting from slightly above the t ¯ t productionthreshold and going up to 45 TeV which is far above threshold. In all cases we observethat the inclusion of soft gluon resummation extends the validity of the perturbativeprediction closer to threshold. For large collider energies the enhanced t ¯ t thresholdcontribution gets reduced and, indeed, we observe that the resummed and unresummedpredictions converge to each other in this case. We also notice that the difference betweenNLL and NNLL is small and is more pronounced when added on top of the NLO result(as anticipated). Finally we note that the inclusion of soft-gluon resummation on topof the NNLO result makes the relative scale uncertainty practically independent of thecollider energy, except of course for the immediate threshold region which, a posteriori ,is another justification for the use of soft-gluon resummation.
5. – Application to searches for physics beyond the Standard Model
In addition to being a powerful tool for testing the Standard Model, the high precisionof the total inclusive t ¯ t production cross-section presents an opportunity for devising newstrategies for searches of physics beyond the Standard Model. A first exploration of theimprovements in BSM searches arising from NNLO top data was presented in Ref. [9],where it was shown that the use of top quark data in a NNLO global PDF fit leads toan improved determination of the poorly known large- x gluon PDF. This improvementthen translates into more accurate predictions for BSM heavy particle production andfor the large mass tail of the M tt distribution, the latter used in searches of new heavyresonances which decay into top quarks.While the above examples illustrate the indirect improvement in BSM searches dueto top quark data, high-precision top production can also impact BSM studies directly, M. CZAKON, P. FIEDLER, A. MITOV AND J. ROJO ( σ m a x - σ m i n ) / σ c en t r √ s [TeV] NLONLO+NLLNLO+NNLLNNLONNLO+NLLNNLO+NNLL 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 30 35 40 45 ( σ m a x - σ m i n ) / σ c en t r √ s [TeV] NLONLO+NLLNLO+NNLLNNLONNLO+NLLNNLO+NNLL Fig. 3. – The relative scale uncertainty of the t ¯ t cross-section, computed as a function of theLHC collider energy at fixed order (NLO and NNLO) and including with soft-gluon resummation(NLL and NNLL). for example, in the search for supersymmetric top partners - the stops. The basic ideais rather simple [20]: in searches for stops with mass that is only slightly above the topmass, the stops decay to either a pair of top quarks or to the decay products of the topquark. Either way, the conventional stop searches require separation of the stop signalfrom the very similar and much larger top background. The ratio of the stop over topcross-sections is shown in fig. 4 (left) for LHC 8 and 14 TeV. The computation of thetop cross-section is done at NNLO+NNLL with the program Top++ (2.0) [7], whilethe stop cross-section is computed at NLO with the program
Prospino(2.1) [21], usingconsistently MSTW2008 in both programs. For a stop mass equal to the top mass theratio of cross sections is about 15%, decreasing quickly as the stop mass increases.In fig. 4 (right) we show the “double” ratio R / (top + stop) /R / (top), where Stop Mass ( GeV ) 170 180 190 200 210 220 230 240 250
Ratio of Stop over Top cross sections
Top: NNLO+NNLL (top++ v2.0)Stop: NLO (prospino v2.1)
Ratio of Stop over Top cross sections
Stop Mass ( GeV ) 180 200 220 240 260 280 300
Enhacement of 14/8 ratio from Stop production ( X) ] σ ( X ) / σ (X) = [ R ( Top ) ( Top + Stop) / R R Enhacement of 14/8 ratio from Stop production
Fig. 4. – Stop production at LHC 8 and 14 TeV. Left plot: the ratio of the stop and topproduction cross-sections. Right plot: the double ratio of the sum of top and stop cross-sectionsat 8 and 14 TeV normalized to pure top pair cross-section at 8 and 14 TeV. The top paircross-section is evaluated at NNLO+NNLL with
Top++(2.0) while the stop pair cross-sectionis evaluated at NLO with the help of the program
Prospino(2.1) . URTHER EXPLORATION OF TOP PAIR HADROPRODUCTION AT NNLO R / ( X ) is the ratio of the cross-section for producing final state X at the LHC 14and 8 TeV. Such cross-section ratios have been introduced [22] due to their very hightheoretical precision (since most of the theoretical uncertainties cancel), and becausethey can be accurately measured. Unfortunately, as can be seen from fig. 4 (right),this particular double ratio has size that is at most few permil, which likely makes itexperimentally inaccessible.The reason for this double ratio’s smallness is that top and stop production are bothdominated by gg scattering and scale in a similar way with the center of mass energy,which is the result of two competing factors. First, as discussed in Ref. [22], the BSMcontribution can be accessed in such a ratio when the BSM signal and the SM backgroundare dominated by different parton luminosities (which is not the case here). Second, thedifferent masses of tops and stops lead to different scalings with the c.m. energy. Thislatter factor, alone, ensures that in the general case the cross section ratios have somesensitivity to BSM dynamics even if it is initiated by the same parton luminosity as theSM background. ∗ ∗ ∗ We thank Michelangelo Mangano for many insightful discussions and helpful sugges-tions. The work of M.C. and P.F. was supported by the DFG Sonderforschungsbere-ich/Transregio 9 Computergest¨utzte Theoretische Teilchenphysik. M.C. was also sup-ported by the Heisenberg programme of the Deutsche Forschungsgemeinschaft. Thework of A.M. is supported by ERC grant 291377 “LHCtheory: Theoretical predictionsand analyses of LHC physics: advancing the precision frontier”. J.R. is supported by aMarie Curie Intra–European Fellowship of the European Community’s 7th FrameworkProgramme under contract number PIEF-GA-2010-272515.
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