Single-top transverse-momentum distributions at approximate NNLO
aa r X i v : . [ h e p - ph ] M a r Single-top transverse-momentum distributions atapproximate NNLO
Nikolaos Kidonakis
Department of Physics, Kennesaw State University,Kennesaw, GA 30144, USA
Abstract
I present approximate next-to-next-to-leading-order (aNNLO) transverse-momentum( p T ) distributions in single-top production processes. The aNNLO results are derivedfrom next-to-next-to-leading-logarithm (NNLL) resummation of soft-gluon corrections inthe differential cross section. Single-top as well as single-antitop p T distributions areshown in t -channel, s -channel, and tW production for LHC energies. The study of top quarks is a central part of the current collider programs. While the main pro-duction mode at the LHC is top-antitop pair production, single-top production is an importantset of processes which have been observed at the Tevatron and the LHC.The production of single tops or single antitops can proceed via three different partonic-channel processes. The numerically dominant one is the t -channel, which involves the exchangeof a space-like W boson, i.e. processes of the form qb → q ′ t and ¯ qb → ¯ q ′ t for single-topproduction, as well as q ¯ b → q ′ ¯ t and ¯ q ¯ b → ¯ q ′ ¯ t for single-antitop production. The numericallysmallest single-top process at the LHC is the s -channel, which involves the exchange of a time-like W boson, i.e. processes of the form q ¯ q ′ → ¯ bt for single-top production and q ¯ q ′ → b ¯ t forsingle-antitop production. The third channel is associated tW production, via the processes bg → tW − and ¯ bg → ¯ tW + , which is quite significant at the LHC.The complete next-to-leading order (NLO) corrections to the differential cross section for t -channel and s -channel production appeared in Ref. [1] and for tW production in Ref. [2].Further results for the NLO top transverse-momentum, p T , distributions at LHC energies in t -channel production have appeared in [3, 4, 5, 6, 7]; for s -channel production in [8]; and for tW production in [9].Higher-order corrections (beyond NLO) can be calculated from soft-gluon resummation.Such calculations for single-top production appeared first at next-to-leading-logarithm (NLL)accuracy in Ref. [10]. More recently, calculations were performed at next-to-next-to-leading-logarithm (NNLL) accuracy in the resummation in Ref. [11]. Approximate next-to-next-to-leading-order (aNNLO) corrections were calculated from the NNLL result for t -channel, s -channel, and tW production [11]. As shown and discussed in detail in [10, 11], the soft-gluoncorrections numerically dominate the cross section, and thus the soft-gluon approximationworks very well. The approximate and exact NLO cross sections for single-top production inall channels are within a few percent of each other for all LHC and Tevatron energies, and forthe t -channel this is also known to hold at NNLO since the recent results in [12].1he resummation in [10, 11] is performed for the double-differential cross section, and thusit enables the calculation of differential distributions in addition to total cross sections. Thetransverse-momentum distribution of the top or antitop is very interesting because effects dueto new physics may appear at large p T . The calculation of these p T distributions in all threesingle-top channels at LHC energies is the topic of this paper. Results for the t -channel p T distributions have been published before in [13], so here we update them and give new resultsfor the new 13 TeV LHC energy. For the s -channel and tW production we provide new results.Our work follows the standard moment-space perturbative QCD resummation formalism.Results for t -channel p T distributions based on another approach, soft-collinear effective theory(SCET), have appeared in [14], and the differences between the moment-space and SCETapproaches have been described in [11].In the next section I describe the kinematics and give some details for the calculation ofthe aNNLO corrections. We present numerical results for the single-top and single-antitop p T distributions in the t -channel in Section 3, in the s -channel in Section 4, and in the tW -channelin Section 5. We conclude in Section 6. We study single-top production in collisions of protons A and B with momenta p A + p B → p + p .The hadronic kinematical variables are S = ( p A + p B ) , T = ( p A − p ) , and U = ( p B − p ) .The partonic reactions have momenta p + p → p + p . The partonic kinematical variablesare s = ( p + p ) , t = ( p − p ) , and u = ( p − p ) , with p = x p A and p = x p B . Wealso define the threshold variable s = s + t + u − m − m . If we denote the top-quark massby m t and the W -boson mass by m W , then for t -channel and s -channel production m = 0and m = m t , while for tW production m = m t and m = m W . We note that s vanishes atpartonic threshold for each process.The resummation of soft-gluon corrections follows from the factorization of the differentialcross section into hard, soft, and jet functions in the partonic processes [10, 11]. The resummedresult is then used to generate approximate higher-order corrections. The soft-gluon correctionshave the form of logarithmic plus distributions, [ln k ( s /m t ) /s ] + , where 0 ≤ k ≤ n − n th order perturbative QCD corrections. The approximate NNLO soft-gluon correctionsto the double-differential partonic cross section, d ˆ σ/ ( dt du ), are of the form d ˆ σ (2) dt du = F LO α s π X k =0 C (2) k " ln k ( s /m t ) s + (2.1)where α s is the strong coupling, and F LO denotes the leading-order (LO) contributions, i.e. d ˆ σ (0) / ( dt du ) = F LO δ ( s ). The aNNLO coefficients C (2) k are in general different for eachpartonic process. The leading coefficient, C (2)3 , depends only on color factors and it equals3 C F for t -channel and s -channel production, and 2( C F + C A ) for tW production, where C F =( N c − / (2 N c ) and C A = N c , with N c = 3 the number of colors. The subleading coefficients C (2)2 , C (2)1 , and C (2)0 are in general functions of s , t , u , m t , and the factorization scale µ F , and (for C (2)1 and C (2)0 ) also the renormalization scale µ R . These coefficients have been determined from2wo-loop calculations for all partonic processes contributing to these channels [10, 11]. NLLresummation [10] is sufficient to calculate all aNNLO coefficients except C (2)0 , which is fullydetermined only by NNLL resummation [11]. The one-loop soft-anomalous dimension [10, 11]for each process contributes to all subleading coefficients, while the two-loop soft-anomalousdimension [11] for each process contributes to C (2)0 .For the t -channel processes qb → q ′ t the LO terms are F qb → q ′ t LO = πα V tb V qq ′ sin θ W ( s − m t )4 s ( t − m W ) . (2.2)Here α = e / (4 π ), V ij are elements of the CKM matrix, and θ W is the weak mixing angle.For the t -channel processes ¯ qb → ¯ q ′ t the LO terms are F ¯ qb → ¯ q ′ t LO = πα V tb V qq ′ sin θ W [( s + t ) − ( s + t ) m t ]4 s ( t − m W ) . (2.3)For the s -channel processes q ¯ q ′ → ¯ bt the LO terms are F q ¯ q ′ → ¯ bt LO = πα V tb V qq ′ sin θ W t ( t − m t )4 s ( s − m W ) . (2.4)For the associated production process bg → tW − the LO terms are F bg → tW − LO = πV tb α s α m W sin θ W s A ( u − m t ) − A ( u − m t ) s + 2 A s ! , (2.5)where A = − ( u − m W )( s − m t − m W )(2 m W + m t ) / − ( t − m t )( − m W + m W m t + m t ) / − u − m W ) m t (2 m W + m t ); A = − ( t − m t )( − m W + m t ) m W − ( u − m W )( t − m t ) m t / − ( u − m t )( u − m W ) m t / − sm W m t − sm t /
2; and A = − s ( u − m t )(2 m W + m t ) / dσdp T = 2 p T Z Y + Y − dY Z x − dx Z s max ds x x Sx S + T φ ( x ) φ ( x ) d ˆ σdt du (2.6)where φ denotes the pdf; Y is the top-quark rapidity, Y ± = ± (1 /
2) ln[(1 + β T ) / (1 − β T )],with β T = [1 − m + p T ) S/ ( S + m − m ) ] / ; x = ( s − m + m − x U ) / ( x S + T ),with T = T − m = −√ S ( m + p T ) / e − Y and U = U − m = −√ S ( m + p T ) / e Y ; x − = ( m − T ) / ( S + U ); and s max = x ( S + U ) + T − m . In particular, using Eq. (2.1)3nd the properties of plus distributions, the aNNLO corrections to the p T distribution can bewritten as dσ (2) dp T = α s π p T Z Y + Y − dY Z x − dx φ ( x ) × (Z s max ds X k =0 s ln k s m t ! " F LO C (2) k x x Sx S + T φ ( x ) − F elLO C (2)el k x el1 x Sx S + T φ (cid:16) x el1 (cid:17) + X k =0 k + 1 ln k +1 s max m t ! F elLO C (2)el k x el1 x Sx S + T φ (cid:16) x el1 (cid:17)) . (2.7)Here the elastic versions of x , F LO , and C (2) k , denoted by the superscript “el”, refer to thesevariables calculated with the constraint s = 0. We note that the total cross section can be ob-tained by integrating the p T distribution from 0 to p T max = [( S − m − m ) − m m ] / / (2 √ S ),and we have checked for consistency that we find the total cross section results of [11], whichare also in excellent agreement with LHC and Tevatron data in all three channels (see Ref. [17]for comparisons with recent data). t -channel p T distributions We begin with t -channel single-top production. The total cross section at 13 TeV energy atthe LHC for a top-quark mass m t = 173 . +3 − ± +2 − ± t -channel using MSTW 2008 NNLO pdf [15].The central results are with µ F = µ R = m t . The theoretical uncertainty of the cross sectionconsists of two parts: the first one is from scale variation by a factor of two (i.e. from m t / m t ); and the second one is from the MSTW pdf [15] uncertainties at 90% C.L. It is seen thatthe pdf uncertainties are somewhat larger than the scale uncertainties.We note that the difference is very small if instead we use MMHT 2014 NNLO pdf [16];in that case we find 138 +3 − ± +2 − ± t -channel for the top-quark p T distribution in the left plot as well as for the antitop p T distribution in the right plot. The p T range displayed is up to 500 GeV and the verticallogarithmic scales in the two plots are chosen the same for ease of comparison of the relativemagnitude of the distributions.In Fig. 2 we present linear plots for the aNNLO p T distribution for the top (left) and theantitop (right) in t -channel production at 13 TeV LHC energy. We also show the theoreticaluncertainty by providing upper and lower values (dashed lines). As we noted for the total crosssection, the majority of the uncertainty is due to the pdf. The top p T distributions peak ata p T of around 36 GeV, and the aNNLO corrections provide a small enhancement of 1% over4
100 200 300 400 500 p T top (GeV) -3 -2 -1 d σ / dp T ( pb / G e V )
14 TeV13 TeV 8 TeV 7 TeV t-channel top p T distribution at LHC aNNLO m t =173.3 GeV p T antitop (GeV) -3 -2 -1 d σ / dp T ( pb / G e V )
14 TeV13 TeV 8 TeV 7 TeV t-channel antitop p T distribution at LHC aNNLO m t =173.3 GeV Figure 1: Approximate NNLO top (left) and antitop (right) t -channel p T distributions at 7, 8,13, and 14 TeV LHC energy. p T top (GeV) d σ / dp T ( pb / G e V )
13 TeVaNNLOt-channel top p T distribution at LHC m t =173.3 GeV
150 200 250 300 350 40011.005 aNNLO/NLO p T antitop (GeV) d σ / dp T ( pb / G e V )
13 TeVaNNLOt-channel antitop p T distribution at LHC m t =173.3 GeV
150 200 250 300 350 40011.005 aNNLO/NLO
Figure 2: Approximate NNLO top (left) and antitop (right) t -channel p T distributions at 13TeV LHC energy with theoretical uncertainty displayed by the dashed lines.5
100 200 300 400 500 p T top (GeV) -5 -4 -3 -2 ( / σ ) d σ / dp T ( G e V - )
14 TeV13 TeV 8 TeV 7 TeV t-channel normalized top p T distribution at LHC aNNLO m t =173.3 GeV p T antitop (GeV) -5 -4 -3 -2 ( / σ ) d σ / dp T ( G e V - )
14 TeV13 TeV 8 TeV 7 TeV t-channel normalized antitop p T distribution at LHC aNNLO m t =173.3 GeV Figure 3: Approximate NNLO top (left) and antitop (right) t -channel normalized p T distribu-tions at 7, 8, 13, and 14 TeV LHC energy.the NLO result calculated with the same pdf. The inset plot shows the ratio of the aNNLOand NLO distributions at high p T values. The enhancement diminishes at large p T ; we note,however, that we do not perform a targeted large- p T resummation.We note that the shape of the distributions is unaffected (to the per mille level) if MMHT2014 pdf are instead used. There is only a very small overall normalization change as for thecross section. If one plots the normalized distribution (1 /σ ) dσ/dp T using the two different pdf,then the two curves are indistinguishable. In Fig. 3 we plot the t -channel top (left) and antitop(right) normalized p T distributions at 7, 8, 13, and 14 TeV LHC energies. s -channel p T distributions We continue with s -channel single-top production. The total cross section at 13 TeV energy atthe LHC for a top-quark mass m t = 173 . . ± . +0 . − . pb for single-top productionand 4 . ± . +0 . − . pb for single-antitop production in the s -channel. As before, the theoreticaluncertainty consists of two parts: the first one is from scale variation by a factor of two, andthe second and larger one is from the MSTW pdf [15] 90% C.L. uncertainties. Again, we notethat the difference is very small if instead we use MMHT 2014 NNLO pdf [16]; in that case wefind 7 . ± . +0 . − . pb for single-top production and 4 . ± . ± .
10 pb for single-antitopproduction, where the pdf uncertainty is at 68% C.L.In the s -channel, the enhancement from the NNLO soft-gluon corrections is significant, incontrast to the t -channel. We find an enhancement of over 8% for the total aNNLO s -channelcross section relative to NLO.In Fig. 4 we present the s -channel central aNNLO results for the top-quark p T distributionin the left plot as well as for the antitop p T distribution in the right plot at 7, 8, 13, and 14TeV LHC energy. The p T range displayed is up to 320 GeV and the vertical logarithmic scalesin the two plots are again chosen to be identical.6
50 100 150 200 250 300 p T top (GeV) -3 -2 -1 d σ / dp T ( pb / G e V )
14 TeV13 TeV 8 TeV 7 TeV s-channel top p T distribution at LHC aNNLO m t =173.3 GeV p T antitop (GeV) -3 -2 -1 d σ / dp T ( pb / G e V )
14 TeV13 TeV 8 TeV 7 TeV s-channel antitop p T distribution at LHC aNNLO m t =173.3 GeV Figure 4: Approximate NNLO top (left) and antitop (right) s -channel p T distributions at 7, 8,13, and 14 TeV LHC energy. p T top (GeV) d σ / dp T ( pb / G e V )
13 TeVaNNLOs-channel top p T distribution at LHC m t =173.3 GeV
150 200 250 30011.021.04 aNNLO/NLO p T antitop (GeV) d σ / dp T ( pb / G e V )
13 TeVaNNLOs-channel antitop p T distribution at LHC m t =173.3 GeV
150 200 250 30011.021.04 aNNLO/NLO
Figure 5: Approximate NNLO top (left) and antitop (right) s -channel p T distributions at 13TeV LHC energy with theoretical uncertainty displayed by the dashed lines.7
50 100 150 200 250 300 p T top (GeV) -4 -3 -2 ( / σ ) d σ / dp T ( G e V - )
14 TeV13 TeV 8 TeV 7 TeV s-channel normalized top p T distribution at LHC aNNLO m t =173.3 GeV p T antitop (GeV) -4 -3 -2 ( / σ ) d σ / dp T ( G e V - )
14 TeV13 TeV 8 TeV 7 TeV s-channel normalized antitop p T distribution at LHC aNNLO m t =173.3 GeV Figure 6: Approximate NNLO top (left) and antitop (right) normalized s -channel p T distribu-tions at 7, 8, 13, and 14 TeV LHC energy.In Fig. 5 we present linear plots for the aNNLO p T distribution for the top (left) andthe antitop (right) in s -channel production at 13 TeV LHC energy. As before, we show thetheoretical uncertainty by providing upper and lower values. The top p T distributions peakat a p T of around 28 GeV, and the aNNLO corrections provide a large enhancement over theNLO result. The inset plot shows the ratio aNNLO/NLO at high p T where the enhancementis smaller.Again, we note that the shape of the distributions is unaffected if MMHT 2014 pdf areinstead used. In Fig. 6 we plot the s -channel top (left) and antitop (right) normalized p T distributions, (1 /σ ) dσ/dp T , at 7, 8, 13, and 14 TeV LHC energies. tW -channel p T distributions Finally, we discuss tW production. The total cross section at 13 TeV energy at the LHC fora top quark mass m t = 173 . . ± . +1 . − . pb for tW − production, and it is the same for ¯ tW + production. Again, the theoretical uncertainty comesfrom scale variation by a factor of two, and from the 90% C.L. pdf uncertainty; the latter isalmost a factor of two larger. Again, we note that the difference is very small if instead weuse MMHT 2014 NNLO pdf [16]; in that case we find 36 . ± . ± . tW channel.We find an 8% increase for the total aNNLO tW cross section relative to NLO.In the left plot of Fig. 7 we present the central aNNLO results for the top-quark p T distribution in tW − production at 7, 8, 13, and 14 TeV LHC energy. In the right plot ofFig. 7 we present a linear plot for the aNNLO top p T distribution in tW − production at 13TeV LHC energy. We also show the theoretical uncertainty by providing upper and lowervalues. The top p T distributions peak at a p T of around 56 GeV, and the aNNLO corrections8
100 200 300 400 p T top (GeV) -3 -2 -1 d σ / dp T ( pb / G e V )
14 TeV13 TeV 8 TeV 7 TeV
Top p T distribution in tW - production at LHC aNNLO m t =173.3 GeV p T top (GeV) d σ / dp T ( pb / G e V )
13 TeV aNNLOTop p T distribution in tW - production at LHC m t =173.3 GeV
150 200 250 300 350 40011.051.1 aNNLO/NLO
Figure 7: Approximate NNLO top-quark p T distributions in the tW channel at (left) 7, 8, 13,and 14 TeV LHC energy, and (right) at 13 TeV with theoretical uncertainty displayed.provide a substantial enhancement of 8.5% over the NLO result. The inset plot shows the ratioaNNLO/NLO at high p T . The p T distributions for the antitop in this channel are the same asfor the top.Once again, we note that the shape of the distributions is unaffected if MMHT 2014 pdf areinstead used. In Fig. 8 we plot the tW -channel normalized top p T distributions, (1 /σ ) dσ/dp T ,at 7, 8, 13, and 14 TeV LHC energies. I have presented the single-top and single-antitop transverse-momentum distributions at ap-proximate NNLO by including soft-gluon corrections derived from NNLL resummation. Resultswere presented at 7, 8, 13, and 14 LHC energies for t -channel, s -channel and tW production. Wehave paid particular attention to the current 13 TeV LHC energy and also provided theoreticaluncertainties. The corrections are large and very significant in s -channel and tW productionbut they are rather small in t -channel production. Acknowledgements
This material is based upon work supported by the National Science Foundation under GrantNo. PHY 1519606.
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