Threshold Improved QCD Corrections for Stop-Antistop production at Hadron colliders
aa r X i v : . [ h e p - ph ] J u l Threshold Improved QCD Corrections for Stop-Antistop production atHadron colliders
U. Langenfeld
Humboldt-Universität zu BerlinNewtonstraße 15, D–12489 Berlin, GermanyandJulius-Maximilians-Universität WürzburgAm Hubland, D–97074 Würzburg, Germany [email protected]
Abstract
I present improved predictions for the total hadronic cross section of stop-antistop production athadron colliders including next-to-next-to-leading-order threshold corrections and approximatedCoulomb corrections. The results are based on soft corrections, which are logarithmically en-hanced near threshold. I present analytic formulas for the NNLO scaling functions at thresholdand explicit numbers for the total hadronic cross sections for the Tevatron and the LHC. Finally Idiscuss the systematic error, the scale uncertainty and the PDF error of the hadronic cross section. g ˜ t i ˜ t ∗ i a b c d q ¯ q ˜ t i ˜ t ∗ i e Figure 1: LO production of a ˜ t i ˜ t ∗ i pair via gg annihilation (diagrams a-d) and q ¯ q annihilation (diagram e). The Minimal Supersymmetric Standard model (MSSM) is an attractive extension [1, 2] of thevery successful Standard Model. One property of this theory is its rich spectrum of new heavyparticles which might be discovered at the LHC if they are lighter than ≈ pp / p ¯ p → ˜ t i ˜ t ∗ i X , i =
1, 2, (1)with its partonic subprocesses gg → ˜ t i ˜ t ∗ i and q ¯ q → ˜ t i ˜ t ∗ i , q = u , d , c , s , b , (2)including NNLO threshold contributions. The relevant leading order (LO) Feynman diagramsare shown in Fig. 1. The production of mixed stop pairs ˜ t ˜ t ∗ or ˜ t ˜ t ∗ starts at next to leadingorder (NLO) [5] and is therefore suppressed. This case will not be considered in this paper.The top parton density distribution in a proton is assumed to be zero, in contrast to the otherquark parton density distributions. As a consequence, there is no gluino exchange diagram asfor squark antisquark hadroproduction. For that reason, the q ¯ q channel is suppressed by a largerpower of β = q − m t / s due to P -wave annihilation. The final state must be in a state withangular momentum l = P ) to balance the spin of the gluon. Therefore, the caseof stop-antistop hadroproduction needs a special treatment. At NLO, there is the gq channel asan additional production mechanism. At the LHC with a center of mass energy of 7 TeV, onecan expect for a luminosity of 1 fb − , 100 to 10 events; even 10 events are possible, if the stopis sufficiently light. At the LHC with a final center of mass energy of 14 TeV, even more eventsare expected to be collected. Hence it is necessary to predict the hadronic cross section with highaccuracy.So far, stop pairs have been searched for at the CDF [6–9] and D0 experiments [10, 11] at theTevatron using different strategies, for details see Tab. 1.Squarks carry colour charge, so it is not surprising that processes involving Quantum Chro-modynamics (QCD) obtain large higher-order corrections. For the production of colour-charged1uperymmetric particles, the NLO corrections have been calculated in Ref. [12], NLL and ap-proximated NNLO corrections can be found in the Refs [13–16]. It has been found that thesecorrections are quite sizeable. Electroweak NLO corrections to stop-antistop production are dis-cussed in Ref. [17, 18].The theoretical aspects of ˜ t i ˜ t ∗ i - production up to NLO have been discussed in Ref. [5] andof its NLL contributions in Ref. [19]. The hadronic LO and NLO cross section can be evaluatednumerically using the programme Prospino [20].In this paper, I calculate and study soft gluon effects to hadronic stop-antistop production inthe framework of the R -parity- conserving MSSM. I use Sudakov resummation to generate theapproximated NNLO corrections and include approximated two-loop Coulomb corrections andthe exact scale dependence. I follow the approach for top-antitop production at the LHC and theTevatron [21, 22].This paper is organised as follows. I review the LO and NLO contributions to the cross section.Then I describe the necessary steps to construct the approximated NNLO corrections. Using theseresults I calculate the approximated NNLO cross section and discuss the theoretical uncertaintydue to scale variation and the error due to the parton density functions (PDFs). I give an examplehow these NNLO contributions reduce the scale uncertainty and improve exclusion limits.Ref. Process Exclusion limit Assumptions or comments[6] ˜ t → c ˜ χ m ˜ t <
100 GeV m χ >
50 GeV[10] p ¯ p → ˜ t ˜ t ∗
130 GeV < m ˜ t <
190 GeV Comparison of theor. predictionswith experimental and observed limits[7, 8] ˜ t → b ˜ χ ± → b ˜ χ ℓ ± ν ℓ
128 GeV < m ˜ t <
135 GeV[9, 11] ˜ t → b ℓ + ˜ ν ℓ m ˜ t >
180 GeV m ˜ ν ≥
45 GeV m ˜ t =
100 GeV 75 GeV ≤ m ˜ ν ≤
95 GeV
Table 1: Exclusion limits for stop searches at the Tevatron.
I focus on the inclusive hadronic cross section of hadroproduction of stop pairs, σ pp → ˜ t i ˜ t ∗ i X , whichis a function of the hadronic center-of-mass energy √ s , the stop mass m ˜ t , the gluino mass m ˜ g , therenormalisation scale µ r and the factorisation scale µ f . In the standard factorisation approach ofperturbative QCD, it reads σ pp / p ¯ p → ˜ t ˜ t ∗ X ( s , m t , m g , µ f , µ r ) = X i , j = q , ¯ q , g s Z m t d ˆ s L ij ( ˆ s , s , µ f ) ˆ σ ij → ˜ t ˜ t ∗ ( ˆ s , m t , m g , µ f , µ r ) (3)where the parton luminosities L ij are given as convolutions of the PDFs f i / p defined through L ij ( ˆ s , s , µ f ) = s s Z ˆ s dzz f i / p (cid:16) µ f , zs (cid:17) f j / p (cid:18) µ f , ˆ sz (cid:19) . (4)2ere, ˆ s denotes the partonic center of mass energy and µ f , µ r are the factorisation and the renor-malisation scale. The partonic cross section is expressed by dimensionless scaling function f ( kl ) ij ˆ σ ij = α s m t (cid:20) f ( ) ij + πα s (cid:16) f ( ) ij + f ( ) ij L N (cid:17) + ( πα s ) (cid:16) f ( ) ij + f ( ) ij L N + f ( ) ij L N (cid:17)(cid:21) (5)with L N = ln (cid:16) µ m t (cid:17) . The LO scaling functions are given by [5] f ( ) q ¯ q = π β ρ = π β + O ( β ) , (6) f ( ) gg = π ρ (cid:20) β − β + (cid:16) − β + β (cid:17) log (cid:18) − β + β (cid:19)(cid:21) = π β + O ( β ) . (7)Formulas for the higher orders of the gg -channel and its threshold expansions can be found inRefs [5, 14, 23], if one takes into account that in the case of stop-antistop production no sum overflavours and helicities is needed. f ( ) gg has been calculated numerically using Prospino [20]. Afit to this function for an easier numerical handling can be found in [14]. At NLO, f ( ) q ¯ q is givenat threshold by [5, 19] f ( ) q ¯ q = f ( ) q ¯ q π (cid:18)
83 log (cid:0) β (cid:1) − (cid:0) β (cid:1) − π β + π a q ¯ q (cid:19) . (8)The constant a q ¯ q can be determined from a fit and is approximately given as a q ¯ q ≈ ± gq -channel is absent at tree level. Its NLO contribution has been extracted from Prospino .This channel is strongly suppressed at threshold.The ln β terms which appear in the threshold expansions of the NLO scaling functions canbe resummed systematically to all orders in perturbation theory using the techniques describedin [24–28]. Logarithmically enhanced terms for the hadronic production of heavy quarks ad-mitting an S -wave are also studied in Ref. [29] for arbitrary SU ( ) colour representations . Theresummation is performed in Mellin space after introducing moments N with respect to the vari-able ρ = m t / ˆ s of the physical space:ˆ σ ( N , m t ) = Z d ρ ρ N − ˆ σ ( ˆ s , m t ) . (9)The resummed cross section is obtained for the individual color structures denoted as I fromthe exponential ˆ σ ij , I ( N , m t ) ˆ σ Bij , I ( N , m t ) = g ij , I ( m t ) · exp h G ij , I ( N + ) i + O ( N − log n N ) , (10)where all dependence on the renormalisation and factorisation scale µ r and µ f is suppressed andthe respective Born term is denoted ˆ σ Bij , I . The exponent G ij , I contains all large Sudakov logarithmslog k N and the resummed cross section (10) is accurate up to terms which vanish as a power forlarge Mellin- N . To NNLL accuracy, G ij , I is commonly written as G ij , I ( N ) = log N · g ij ( λ ) + g ij , I ( λ ) + α s π g ij , I ( λ ) + . . . , (11)3here λ = β log N α s / ( π ) . The exponential exp (cid:2) G ij , I ( N + ) (cid:3) in Eq. (11) is independent fromthe Born cross section [21, 28]. The functions g kij , k =
1, 2, 3, for the octet color structure areexplicitly given in Ref. [21] and can be taken over from the case of top-quark hadroproduction,the function g q ¯ q is given by Eq. (36) in the App. A.2. All g kij , 0 ≤ k ≤
3, depend on a numberof anomalous dimensions, i.e. the well-known cusp anomalous dimension A q , the functions D Q ¯ Q and D q controlling soft gluon emission, and the coefficients of the QCD β -function. The strengthof soft gluon emission is proportional to the Casimir operator of the SU ( ) colour representation ofthe produced state. This is identical for t ¯ t and ˜ t i ˜ t ∗ i - production. Expressions for A q and D q aregiven in the Refs. [30, 31], and for D Q ¯ Q in Ref. [32]. At higher orders, they also depend on thechosen renormalisation scheme, thus on the dynamical degrees of freedom.For my fixed order NNLO calculation, I extracted the α s -terms from the right hand side ofEq. (10). At the end, I used Eqs (30) - (35) given in App. A.1 to convert the Mellin space resultback to the physical ρ space. I kept all those terms which are of the order β ln k β , 0 ≤ k ≤ f ( ) q ¯ q : f ( ) q ¯ q = f ( ) q ¯ q ( π ) " β + (cid:18) − + + n f (cid:19) ln β + (cid:18) − − π + C ( ) q ¯ q + + n f − n f − π β (cid:19) ln β + (cid:18) − + − C ( ) q ¯ q − + ζ + π − π ln 2 + C ( ) q ¯ q ln 2 − D ( ) Q ¯ Q + (cid:16) − π +
256 ln + − (cid:17) n f + (cid:16) − − n f (cid:17) π β (cid:19) ln β + (cid:18) − +
223 ln 2 + (cid:16) −
49 ln 2 (cid:17) n f (cid:19) π β + π β + C ( ) q ¯ q . (12) C ( ) q ¯ q is given as C ( ) q ¯ q = π a q ¯ q − , C ( ) q ¯ q is the unknown 2-loop matching constant, which isset to zero in the numerical evaluation and D ( ) Q ¯ Q = − π + ζ − n f , see Ref. [32]. The β -behaviour of the threshold expansion of the LO cross section comes only from the P -wave ofthe final state ˜ t i ˜ t ∗ i as mentioned in the introduction and does not spoil the factorisation propertiesin the threshold region of the phase space. Note that the formulas given in Ref. [28] can easilyextended to Mellin transformed cross sections ω , which vanish as a power β k with k ≥
1. Thesetwo more powers of β in the q ¯ q -channel of ˜ t i ˜ t ∗ i - production lead to an additional 1/ N factorin Mellin space. Eq. (10) reproduces the known NLO threshold expansion given in Ref. [5, 19]for the q ¯ q channel. This is a check that the approach works. Logarithmically enhanced termswhich are suppressed by an additional 1/ N factor appear in the resummation of the (sub)leading4og k ( N ) / N -terms of the corrections for the structure function F L and are studied in detail in theRefs [33, 34]. For these reasons I apply the formulas derived for heavy quark hadroproduction.The coefficients of the ln β , ln β , and ln β terms depend only on first order anomalousdimensions and on the constant C ( ) q ¯ q which is related to the NLO constant a q ¯ q , see the equationabove. The linear log β term depends on C ( ) q ¯ q as well and on other first order (NLO) contributions,but also on second-order anomalous dimensions and non-Coulomb potential contributions [29].In Tab. 2, I show for four examples how these parts contribute to the hadronic NNLO thresholdcorrections. The numbers show that terms which have an NLO origin contribute most and thatNNLO contributions have a small but sizeable effect.I also included the Coulomb corrections up to NNLO. For the singlet case, the Coulomb contri-butions are studied in Ref. [35]. Generalisation to other colour structures requires the substitutionof the corresponding group factors and decomposition of the colour structures of the consideredprocess in irreducible colour representations. The last step will not be necessary for stop pairproduction in the q ¯ q -annihilation channel. The NLO Coulomb corrections agree with the NLOCoulomb corrections for top antitop production [5, 14]. In both cases, only the colour octet con-tributes to the scaling function at the corresponding leading order in β . Therefore, I have usedas an approximation for the NNLO Coulomb contributions for ˜ t ˜ t ∗ -production the same NNLOCoulomb contributions as for t ¯ t -production [14, 21, 35]. Gauge invariance together with Super-symmetry support this approximation. Note that the log β / β -term comes from interference ofthe NLO Coulomb-contribution with the NLO threshold logarithms. Tab. 2 shows also the NLOand the pure NNLO Coulomb contributions to the NNLO threshold corrections at the hadroniclevel.The scale-dependent scaling functions are derived by renormalisation group techniques fol-lowing Refs [36, 37]: f ( ) ij = π (cid:16) β f ( ) ij − f ( ) kj ⊗ P ( ) ki − f ( ) ik ⊗ P ( ) kj (cid:17) , (13) f ( ) ij = ( π ) (cid:16) β f ( ) ij − f ( ) kj ⊗ P ( ) ki − f ( ) ik ⊗ P ( ) kj (cid:17) + π (cid:16) β f ( ) ij − f ( ) kj ⊗ P ( ) ki − f ( ) ik ⊗ P ( ) kj (cid:17) , (14) f ( ) ij = ( π ) (cid:18) f ( ) kl ⊗ P ( ) ki ⊗ P ( ) lj + f ( ) in ⊗ P ( ) nl ⊗ P ( ) lj + f ( ) nj ⊗ P ( ) nk ⊗ P ( ) ki + β f ( ) ij − β f ( ) ik ⊗ P ( ) kj − β f ( ) kj ⊗ P ( ) ki (cid:19) , (15)where ⊗ denotes the standard Mellin convolution; these are ordinary products in Mellin spaceusing Eq. (9). Repeated indices imply summation over admissible partons. However, I restrictmyself for phenomenological applications to the numerically dominant diagonal parton channelsat two-loop. Note that the scale dependence is exact at all energies, even away from threshold,because the Eqs (13)-(15) depend on functions which are at least one order lower than they them-selves have. The functions P ij ( x ) are called splitting functions and govern the PDF evolution.They have the expansion P ij ( x ) = α s π P ( ) ij ( x ) + (cid:16) α s π (cid:17) P ( ) ij ( x ) + . . . . (16)Explicit expressions for the P ( k ) ij can be found in Refs [30, 38].5nalytical results for f ( ) gg and f ( ) q ¯ q are given in Ref. [23]. For the gq -channel, Eq. (13) simpli-fies to f ( ) gq = − π (cid:18) P ( ) gq ⊗ f ( ) gg + n f P ( ) qg ⊗ f ( ) q ¯ q (cid:19) . (17)The integration can be done explicitly yielding f ( ) gq = π (cid:20) β (cid:16) − − ρ + ρ (cid:17) + ρ (cid:16) − ( −
24 ln 2 ) ρ − ρ (cid:17) L + ρ (cid:16) L − L (cid:17)(cid:21) , (18)where the functions L , L , and L [23] are defined as L = log (cid:16) + β − β (cid:17) , L = Li (cid:16) − β (cid:17) − Li (cid:16) + β (cid:17) , L = log ( − β ) − log ( + β ) . (19)The high energy limit of this scaling function islim β → f ( ) gq = − π , (20)which agrees with the result given in Ref. [5].The threshold expansions of the NNLO-scale-dependent scaling functions of the q ¯ q channelread f ( ) q ¯ q = − f ( ) q ¯ q ( π ) (cid:20) β + (cid:18) n f + − (cid:19) ln β + (cid:18) − + + − n f − π − π β + C ( ) + n f ln 2 (cid:19) ln β + n f − C ( ) + + − + C ( ) n f − + ζ + π − π ln 2 +
192 ln n f + π β + C ( ) ln 2 − n f ln 2 − π n f − n f π β (cid:21) , (21) f ( ) q ¯ q = f ( ) q ¯ q ( π ) (cid:20) β + (cid:18) − + n f + (cid:19) ln β − n f + − + + n f ln 2 − π + n f (cid:21) (22)with C ( ) = π a q ¯ q . 6ollider m ˜ t P ln ( β ) ln ( β ) ln ( β ) log ( β ) C NLO C NNLO nC D ( ) Q ¯ Q restLHC 14 TeV 300 GeV 74.54 5.33 18.80 31.87 27.71 -9.45 0.29-2.23 17.90 12.04LHC 14 TeV 600 GeV 2.93 0.24 0.81 1.28 0.97 -0.37 -0.01-0.08 0.63 0.42LHC 7 TeV 300 GeV 17.5 1.42 4.83 7.66 5.85 -2.21 -0.06-0.47 3,78 2.54Tevatron 300 GeV 1.41 0.16 0.48 0.62 0.34 -0.17 -0.02-0.03 0.22 0.15 Table 2: Individual hadronic contributions of the log-powers and Coulomb-corrections to the NNLOthreshold contributions of the q ¯ q -channel in fb. ln ( β ) has to be understood as f ( ) q ¯ q ( π ) · ln ( β ) , anal-ogously for the other terms. The linear log-term is decomposed into contributions coming from non-Coulomb potential terms, from the two loop anomalous dimension D ( ) Q ¯ Q , and, finally, the rest. TheCoulomb contribution are decomposed into contributions coming from the interference of NLO thresh-old logarithms with NLO Coulomb corrections C NLO and pure NNLO Coulomb corrections C
NNLO . P denotes the sum over all NNLO threshold contributions. The PDF set used is MSTW 2008 NNLO [39]. In Fig. 2, I show the LO, NLO, and NNLO scaling functions. The scaling functions f ( ) q ¯ q , f ( ) q ¯ q f ( ) q ¯ q , and f ( ) q ¯ q depend only on the dimensionless variable η = ˆ s m t −
1, but f ( ) q ¯ q and f ( ) q ¯ q dependalso mildly on the masses of the squarks and the gluino and the stop mixing angle [5]. At thehadronic level, the effect for the NLO + NLL cross section is smaller than 2% [19], so I neglectthem.As example point, I have chosen the following masses: m ˜ t =
300 GeV, m ˜ q =
400 GeV = m ˜ t , m ˜ t =
480 GeV = m ˜ t , m ˜ g =
500 GeV = m ˜ t , and θ = π /2, i.e. m ˜ t = m ˜ t R and m ˜ t = m ˜ t L . When varying the stop mass I conserve these mass relations. I restrict myselfto the lighter stop, but the results also apply to the heavier stop, because the gluon-stop-stopinteractions entering my process do not distinguish between the left-handed and the right-handedstop squarks. I start with the discussion of the total hadronic cross section, which is obtained by convolutingthe partonic cross section with the PDFs, see Eq. (3). I keep the gg and the q ¯ q channel at all ordersup to NNLO, and for the scale-dependent terms, only contributions coming from diagonal partonchannels are considered. Only the NLO contributions of the gq channel are considered, whichare the leading contributions of this channel. The scale dependence of this channel is given byEq. (17).I define the NLO and NNLO K factors as K NLO = σ NLO σ LO , K NNLO = σ NNLO σ NLO . (23)7 .0001 0.001 0.01 0.1 1 10 100 1000 10000 η = s/(4 m ) -100.0020.0040.0060.0080.010.012 f (00) qq f (10) qq f (11) qq η = s/(4 m ) -1-0.000200.00020.00040.0006 f (20) qq f (21) qq f (22) qq η = s/(4 m ) -100.0020.0040.0060.0080.010.012 f (00) qq f (10)qq f (20)qq Figure 2: Scaling functions f ( ij ) q ¯ q with i =
0, 1, 2 and j ≤ i . The masses are m ˜ t =
300 GeV, m ˜ q =
400 GeV, m ˜ t =
480 GeV m ˜ g =
500 GeV. K factorsaccount only for the pure higher order corrections of the partonic cross section (convoluted withthe PDFs) and not for higher order corrections of the PDFs and the strong coupling constant α s .In the left column of Fig. 3, I show the total hadronic cross section for the LHC (7 TeV firstrow, 14 TeV second row) and the Tevatron (third row) as a function of the stop mass. Similar totop-antitop [21, 22] and squark-antisquark production [14], the total cross section shows a strongmass dependence. At the LHC (14 TeV), the cross section decreases within the shown stop massrange from about 1000 pb to 10 − pb. In the right column of Fig. 3, I show the corresponding K factors.For example, for a stop mass of 300 GeV produced at the LHC with 14 TeV, I have a total crosssection of 6.57 pb, 9.96 pb, 10.92 pb at LO, NLO, NNLO approx , respectively. For a stop mass of600 GeV, I find 0.146 pb, 0.216 pb, 0.244 pb. The K -factors are K NLO ≈ K NNLO ≈ ≤ m ˜ t ≤
600 GeV. At the Tevatron, I have K NLO = K NNLO ≈ ≤ m ˜ t ≤
300 GeV.In Tabs 3 - 5, see App. B, values for the total hadronic cross section for different masses,PDF sets, scales and colliders are shown. The values for the PDF sets Cteq6.6 [40], MSTW 2008NNLO [39],and CT10 [41] show only small differences, whereas the ABKM09 NNLO (5 flavours)PDFs differ in the treatment of the gluon PDF from the other PDF sets. This leads to sizeabledifferences in the total cross sections.
In this section, I address the following sources for errors: the systematic theoretical error, the scaleuncertainty, and the PDF error.In Tab. 2, I listed the individual ln k β contributions to the total NNLO contributions. Note thatthe NNLO matching constants C ( ) ij are unknown and set to zero. Compared to the total NNLOcontributions the ln β term is quite sizeable, this translates to a roughly 3 −
5% contribution to theNNLO cross section. To estimate the systematic error coming from the NNLO matching constants C ( ) ij , I proceed as described in Ref. [22]. I find for the ratio σ NLL+Coul / σ exact = − µ with m ˜ t /2 ≤ µ ≤ m ˜ t , where I haveidentified the factorisation scale with the renormalisation scale. The width of the band indicatesthe scale uncertainty, which becomes smaller when going from LO to NLO and NNLO. On theright-hand side of Fig. 4, the scale dependence for the example point is shown in more detail. Iquote as theoretical uncertaintymin σ ( µ ) ≤ σ ( m ˜ t ) ≤ max σ ( µ ) , (24)where the min and max are to be taken over the interval [ m ˜ t /2, 2 m ˜ t ] . At LO and NLO, theminimal value is taken at µ = m ˜ t and the maximal value at µ = m ˜ t /2. However, this is notlonger true at NNLO. For the theoretical error, one finds σ LO = + − pb, σ NLO = + − pb, σ NNLO = + − pb . (25)As one can see, there is a strong scale dependence at LO, becoming weaker at NLO, and is9
00 200 300 400 500 600 m t~ [GeV] -2 -1 σ [ pb ] LONLONNLO approx
LHC @ 7 TeV, MSTW2008 NNLO
100 200 300 400 500 600 m t~ [GeV] K f ac t o r K NNLO K NLO
200 400 600 800 1000 m t~ [GeV] -2 -1 σ [ pb ] LONLONNLO approx
LHC @ 14 TeV, MSTW2008 NNLO
200 400 600 800 1000 m t~ [GeV] K f ac t o r K NNLO K NLO
100 150 200 250 300 m t~ [GeV] -2 -1 σ [ pb ] LONLONNLO approx
Tevatron, MSTW2008 NNLO
100 150 200 250 300 m t~ [GeV] K f ac t o r K NNLO K NLO
Figure 3: Total hadronic cross section at LO, NLO, and NNLO approx at the LHC 7 TeV(first row) and14 TeV(second row) and the Tevatron (1.96 TeV, third row). The right column shows the corresponding K factors. The PDF set used is MSTW2008 NNLO [39]. flattend out at NNLO within the considered range. This flattening gives a hint that the approachis reliable.Using renormalisation group techniques, one recovers the full dependence on the renormal-isation scale µ r and factorisation scale µ f . I have done this for the example point for the NLOand the NNLO cross section, see Fig. 5. I define the theoretical uncertainty coming from an10
00 400 600 800 100010 -2 -1 σ [ pb ] LO
200 400 600 800 100010 -2 -1 σ [ pb ] NLO
200 400 600 800 1000 m t~ [GeV]10 -2 -1 σ [ pb ] NNLO approx
LHC @ 14 TeV, MSTW2008 NNLOLHC @ 14 TeV, MSTW2008 NNLOLHC @ 14 TeV, MSTW2008 NNLO µ /m t~ σ [ pb ] LONLONNLO approx
LHC @ 14 TeV, MSTW 2008 NNLOm t~ = 300 GeVm q~ = 400 GeVm g~ = 500 GeV Figure 4: Left hand side: Theoretical uncertainty of the total hadronic cross section at the LHC (14 TeV) atLO (upper figure, blue band), NLO (central figure, green band), NNLO approx (lower figure, purple line).At NNLO approx , the theoretical uncertainty has shrunk to a small band. Right hand side: Scale dependenceof the total hadronic cross section for the example point m ˜ t =
300 GeV, m ˜ q =
400 GeV, m ˜ t =
480 GeV, m ˜ g =
500 GeV. The vertical bars indicate the total scale variation in the range [ m ˜ t /2, 2 m ˜ t ] . independent variation of µ r and µ f in the standard range µ r , µ f ∈ [ m ˜ t /2, 2 m ˜ t ] asmin σ ( µ r , µ f ) ≤ σ ( m ˜ t ) ≤ max σ ( µ r , µ f ) . (26)The contour lines of the total cross section for the example point with an independent variationof µ r and µ f are shown in Fig. 5. Note that the range of the axes is from log ( µ r , f / m ˜ t ) = − ( µ r , f / m ˜ t ) =
1. The scale variation with fixed scales µ r = µ f proceeds along the diagonalfrom the lower left to the upper right corner of the figure. The gradient of the NLO contourlines lies approximately in the µ r = µ f direction, meaning that the theoretical error from thedefinition in Eq. (26) is the same as if one sets µ r = µ f . For the NNLO case one observes theopposite situation: the contour lines are nearly parallel to the diagonal µ r = µ f . I obtain a largeruncertainty in that case: σ NLO = + − pb, σ NNLO = + − pb. (27)Another source of error to discuss is the PDF error. I calculated the PDF uncertainty accordingto Ref. [40] for the two PDF sets CT10 and MSTW2008 NNLO (90% C.L.). In both cases, theuncertainty increases with higher stop masses due to large uncertainties of the gluon PDF inhigh x -ranges. For CT10, I find as relative errors ≈
3% for m ˜ t =
100 GeV and ≈
18% for m ˜ t = ≈
3% for m ˜ t =
100 GeV and ≈ m ˜ t = - - - H Μ _r (cid:144) m_stop L l og2 H Μ _ f (cid:144) m _ s t op L p b . p b . p b . p b p b . p b . p b . p b p b - - - - H Μ _r (cid:144) m_stop L l og2 H Μ _ f (cid:144) m _ s t op L . p b . p b . p b . p b p b . p b . p b . p b . p b . p b Figure 5: Contour lines of the total hadronic NLO (left) and NNLO (right) cross section from the inde-pendent variation of the renormalisation and factorisation scale µ r and µ f for LHC, 14 TeV, with PDF setMSTW2008 NNLO [39] for the example point with m ˜ t =
300 GeV. The dot in the middle of the figureindicates the cross section for µ r = µ f = m ˜ t , and the range corresponds to µ f , µ r ∈ [ m ˜ t /2, 2 m ˜ t ] .
200 400 600 800 1000 m t~ [GeV] -2 -1 σ [ pb ] PDF uncertainty CT10LHC @ 14 TeVNNLO approx 200 400 600 800 1000 m t~ [GeV] -2 -1 σ [ pb ] PDF uncertainty MSTW2008 NNLOLHC @ 14 TeVNNLO approx
Figure 6: PDF uncertainty of the total NNLO cross section for the two PDF sets CT10 [41] (left figure) andMSTW2008 NNLO [39] (right figure) at the LHC (14 TeV). σ NNLO = + − pb ( scale ) + − pb ( MSTW2008 NNLO ) (28) σ NNLO = + − pb ( scale ) + − pb ( CT10 ) . (29) The approximated NNLO contributions enlarge the stop-antistop production cross section by ≈ − m ˜ t =
120 GeV has a NLO production cross section of 5.05 pb, the same cross section correspondsat NNLO to a stop with a mass of 123.5 GeV. At higher stop masses, the exclusion limit is evenfurther enhanced: 0.164 pb corresponds at NLO to a stop with a mass of 210 GeV, but at NNLOto a mass of 215 GeV. At the LHC (14 TeV) one would have a similar situation. An NLO crosssection of 750 pb corresponds to m ˜ t =
120 GeV, but to m ˜ t = m ˜ t =
300 GeV at NLO, but to m ˜ t = In this paper, I computed the NNLO threshold contributions including Coulomb corrections forstop-antistop production at hadron colliders. • I presented analytical formulas for the threshold expansion of the NNLO scaling functionusing resummation techniques for the scale independent scaling function and RGE tech-niques for the scale-dependent scaling functions. • After convolution with suitable PDF sets, the NNLO corrections are found to be about 20%for the Tevatron and 10 −
20% for the LHC compared to the hadronic NLO cross section. ThePDF sets Cteq6.6 [40], MSTW 2008 NNLO [39], and CT10 [41] show only small differencesin the total cross section, whereas the values obtained with the PDF set ABKM09 NNLO (5flavours) [42] differ by 10 −
35% due to differences in the gluon PDF. • I calculated the exact scale dependence and I found a remarkable stabilisation of the crosssection under scale variation. For my example point, the theoretical error is reduced from12% at NLO to better than 2% at NNLO. • I discussed three types of errors: systematic theoretical errors, uncertainties due to scalevariation and PDF errors. The systematic error was estimated to be about 3 − −
18% depending on the stop massand the PDF set used. • Finally, I demonstrated how NNLO cross sections could enlarge exclusion limits. The im-provement of the lower exclusion limit were about a few GeV.
Acknowledgments
I would like to thank P. Uwer and A. Kulesza, W. Porod, and S. Uccirati for useful discussionsand S. Moch and M. Krämer for reading the manuscript and giving helpful comments. This work13s supported in part by the Helmholtz Alliance “Physics at the Terascale” (HA-101) and by theresearch training group GRK 1147 of the Deutsche Forschungsgemeinschaft.
A Useful Formulas
A.1 Mellin Transformations Z d ρρ N − β ln β = − √ π h − + + e N i N (cid:0) + O ( N ) (cid:1) (30) Z d ρρ N − β ln β = √ π h π + −
64 ln 2 +
24 ln + (cid:0) − + (cid:1) ln e N + e N i × N (cid:0) + O ( N ) (cid:1) (31) Z d ρρ N − β ln β = − √ π h
32 ln 2 −
64 ln +
16 ln − π + π ln 2 + ζ + (cid:0) + π −
64 ln 2 +
24 ln (cid:1) ln e N + (cid:0) − + (cid:1) ln e N + e N i N (cid:0) + O ( N ) (cid:1) (32) Z d ρρ N − β ln β = √ π h π + π (cid:0) −
32 ln 2 +
12 ln (cid:1) + (cid:0)
192 ln −
256 ln +
48 ln − ζ + ζ ln 2 (cid:1) + (cid:0) − π + π ln 2 +
32 ln 2 −
64 ln +
16 ln + ζ (cid:1) ln e N + (cid:0) + π −
64 ln 2 +
24 ln (cid:1) ln e N + (cid:0) − + (cid:1) ln e N +
12 ln e N i N (cid:0) + O ( N ) (cid:1) (33) Z d ρρ N − β ln β = −
12 ln e N N (cid:0) + O ( N ) (cid:1) (34) Z d ρρ N − β ln β = h π −
12 ln e N + e N i N (cid:0) + O ( N ) (cid:1) (35)with ln e N = ln N + γ e . A detailed overview of Mellin transformations is given in Ref. [43].14 .2 Resummation For the sake of clarity, I specify the function g q ¯ q ,8 I have used in my resummation. The func-tion g q ¯ q ,8 differs a little bit from the function given in Ref. [21] due to Mellin transformation offunctions of the type β log k ( β ) instead of β log k ( β ) , 0 ≤ k ≤
4, as it is in the case of t ¯ t -production. g q ¯ q ,8 = + α s h C F (cid:0) − + γ e + π −
32 ln 2 + (cid:1) + C A ( − + γ e − ) + C ( ) q ¯ q + C F ( − γ e ) ln 2 i + α s (cid:20) C F C ( ) q ¯ q (cid:18) − + γ e + π +
512 ln 2 − γ e ln 2 −
256 ln (cid:19) + C F n f (cid:18) − ζ − γ e − γ e − γ e − π − + γ e ln 2 + γ e ln 2 + π ln 2 + − γ e ln − (cid:19) + C F (cid:18) − + ζ − γ e + γ e + π + π γ e + π + − ζ ln 2 + γ e ln 2 − γ e ln 2 − π ln 2 − π γ e ln 2 − + γ e ln + π ln + − γ e ln − (cid:19) + C A C ( ) q ¯ q (cid:18) − + γ e +
32 ln 2 (cid:19) + C A n f (cid:18) − γ e − γ e − π − + γ e ln 2 +
643 ln (cid:19) + C A C F (cid:18) − + ζ − γ e − γ e ζ − γ e + γ e + π + π γ e − π γ e − π + − ζ ln 2 + γ e ln 2 − γ e ln 2 − π ln 2 + π γ e ln 2 − + γ e ln + π ln + (cid:19) + C A (cid:18) − + γ e + γ e + π − π γ e + − γ e ln 2 − π ln 2 − − ζ + ζ ln 2 + γ e ζ (cid:19) + C ( ) q ¯ q (cid:21) . (36) γ e denotes the Euler constant and ζ is the Riemannian Zeta function ζ ( x ) evaluated at x = Numerical Results m ˜ t σ ( LO )[ pb ] σ ( NLO )[ pb ] σ ( NNLO )[ pb ][ GeV ] x = x = x = x = x = x = x = x = x = Table 3: Numerical values for the stop pair-production cross section at LHC with √ s =
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