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MIT–CTP 4123 May 21, 2010 The Beam Thrust Cross Section for Drell-Yan at NNLL Order Iain W. Stewart, Frank J. Tackmann, and Wouter J. Waalewijn Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA At the LHC and Tevatron strong initial-state radiation (ISR) plays an important role. It can significantlyaffect thepartonicluminosityavailable tothehardinteraction orcontaminateasignal withadditional jetsandsoft radiation. Anideal processtostudyISRisisolated Drell-Yanproduc- tion, pp→Xℓ+ℓ− without central jets, where thejet veto is provided by thehadronic event shape beam thrust τB. Most hadron collider event shapes are designed to study central jets. In contrast, requiring τB ≪1 provides an inclusive veto of central jets and measures the spectrum of ISR. For τB ≪1wecarryout aresummation of αns lnmτB corrections at next-to-next-to-leading-logarithmic order. This is the first resummation at this order for a hadron-hadron collider event shape. Mea- 1 surementsofτB attheTevatronandLHCcanprovidecrucialtestsofourunderstandingofISRand 1 of τB’s utility as a central jet veto. 0 2 Introduction. Event shapes play a vital role in the jets occurs at measurable rapidities. In this limit τB n success of QCD measurements at e+e− colliders. This provides similar information as thrust in e+e− jets a → J includes the measurements of αs(mZ), the QCD β func- with the thrust axis fixed to the proton beam axis. For tion and color factors [1], and the tuning and testing of τ 1, the final state has energetic jets at central ra- 5 B ∼ 1 Monte Carlo event generators (see e.g. Refs. [2]). Event pidities. Thus, requiring small τB provides an inclusive shapes for the more complicated environment at hadron veto on central jets, while allowing ISR in the forward ] colliders have been designed and studied in Refs. [3–6]. direction, as depicted in Fig. 1. h p There is much anticipation that they can play a signif- Experimentally, beam thrust is one of the simplest - icant role at the Tevatron and LHC by improving our hadronic observables at a hadron collider. It requires p understanding of basic aspects of QCD in high-energy nojetalgorithms,isboostinvariantalongthebeamaxis, e h collisions such as the underlying event and initial- and and can be directly compared to theory predictions that [ final-state radiation, as well as nonperturbative effects. require no additional parton showering or hadronization Herewefocusoninitial-stateradiation(ISR).StrongISR from Monte Carlo programs. Beam thrust is defined 2 v can significantly alter the partonic luminosity available as [6] 0 for the hard interaction. Additional jets from ISR can 6 also contaminate the jet signature for a specific signal. τ = 1 p~ e−|ηk−Y|, (1) 0 An ideal process to study ISR is isolated Drell-Yan pro- B Q | kT| 4 Xk duction, pp Xℓ+ℓ− with a veto on central jets. By 5. vetoing hard→central jets, the measurement becomes di- where Q2 and Y are the dilepton invariant mass 0 rectlysensitivetohowenergeticandsoftISRcontributes and rapidity, respectively. The sum runs over all 0 to the hadronic final state X. (pseudo)particles in the final state except the two sig- 1 : Recently an inclusive hadron collider event shape τB nal leptons, where p~kT and ηk are the measured trans- v | | was introduced, called “beam thrust” [6]. For τ 1, verse momenta and rapidities with respect to the beam i B ≪ X the hadronic final state consists of two back-to-backjets axis, and all particles are considered massless. The ab- r centered around the beam axis. The ISR causing these solutevalueinthe exponentinEq.(1)effectivelydivides a all particles into two hemispheres η > Y and η < Y, k k wheretheformergives p~ e−ηk =E pz andthelatter b Soft Y ℓ+ a |p~kT|eηk =Ek+pzk: | kT| k− k 1 τ = eY (E pz)+e−Y (E +pz) . (2) Jetb Jeta B Q(cid:20) ηXk>Y k− k ηXk<Y k k (cid:21) p p The dependence on Y explicitly takes into account the boost of the partonic center-of-mass frame, i.e. the fact that the collinear ISR in the direction of the boost is Soft narrower, as depicted in Fig. 1. From Eqs. (1) and (2) we see that soft particles with energies E Q as k ℓ− well as energetic particles in the forward directio≪ns with E pz Q contribute only small amounts to τ . k − | k| ≪ B FIG.1: Isolated Drell-Yanproduction with avetoon central In particular, unmeasured particles beyond the rapid- jets. ity reach of the detector are exponentially suppressed, 2 p~ e−|ηk| 2E e−2|ηk|, and give negligible contribu- and analogously for B . Here, the sum runs over parton kT k j t|ions| to τ .≈On the other hand, energetic particles in species k = g,u,u¯,d,d¯,... and f (ξ ,µ ) denotes the B k a B { } the central region with E pz E Q give an (1) standard parton distribution function (PDF) for parton k ± k ∼ k ∼ O contribution to τ . Hence, a cut τ τcut 1 vetoes k with momentum fraction ξ . The Wilson coefficients B B ≤ B ≪ a central energetic jets without requiring a jet algorithm. (t ,z ,µ ) describe the collinearvirtualand realISR ik a a B I Beam thrust is also theoretically clean. It is infrared emitted by this parton at the beam scale µ2 t B ≃ a ≃ safe, and an all-orders factorization theorem exists for τ Q2. TherealISRcausestheformationofajetpriorto B thecrosssectionatsmallτ [6]. Thisallowsforahigher- thehardcollisionwhichisobservedasradiationcentered B order summation of large logarithms, αnlnmτ , and the aroundthebeamaxis. ThePDFsinEq.(5)areevaluated s B calculation of perturbative and estimation of nonpertur- at the beam scale µ , because the measurement of τ B B bativecontributionsfromsoftradiation. Thestateofthe introduces sensitivity to the virtualities t τ Q2 of the B ≃ artforresummationinhadroncollidereventshapesisthe colliding hard partons, giving large logarithms ln(µ2/t). B next-to-leadinglogarithm(NLL)plusnext-to-leadingor- For small τ , we have B der (NLO) analysis in Ref. [4]. In this Letter we present t t k results for the beam thrust cross section for τ 1 at τ = a + b + B + (τ2), (6) B ≪ B Q2 Q2 Q O B next-to-next-to-leading-logarithmic (NNLL) order. This representsthe firstcomplete calculationto this order for where the last term is the contribution from soft radia- a hadron collider event shape. Letting v i0 be the tionatthe scaleµ k τ Q andis describedby the B S B B − ≃ ≃ Fourierconjugatevariabletoτ ,theFourier-transformed soft function S (k ,µ ) in Eq. (4). The collinear and B B B S cross section exponentiates and has the form soft contributions are not separately measurable, which leads to the convolution of S , B , and B in Eq. (4). B i j dσ ln L(α L)k+(α L)k+α (α L)k+ , (3) SB(kB,µS)includes the effects ofhadronizationandsoft s s s s dvB ∼ ··· radiation in the underlying event. For ΛQCD µS, ≪ it is perturbatively calculable with power corrections of where L=lnv andwe sumoverk 1. Here, the three sets ofterms arBethe leadinglogarith≥mic(LL), NLL, and O(TΛhQeCDla/rµgSe).logarithms αnlnmτ , with m 2n, are NNLL corrections. s B ≤ summed in Eq. (4) as follows. The hard, beam, and Beam Thrust Factorization Theorem. The Drell-Yan soft functions are each evaluated at their natural scale beam thrust cross section for small τ obeys the factor- B µH = Q, µB = √τBQ, and µS = τBQ, respectively, ization theorem [6] | | where they contain no large logarithms and can be com- puted in fixed-order perturbation theory. They are then dσ 8πα2 = em H (Q2,µ )U (Q2,µ ,µ) evolvedto anarbitrarycommonscaleµ by the evolution dQdYdτB 9Ec2mQXij ij H H H kernels UH, UBi,j, and US, and this sums logarithms of the three scale ratios µ/µ , µ/µ , and µ/µ , respec- dt dt′ B (t t′,x ,µ )Ui(t′,µ ,µ) H B S ×Z a a i a− a a B B a B tively. The combination of the different evolution ker- nelsinEq.(4)isµ-independentandsumsthe logarithms ×Z dtbdt′bBj(tb−t′b,xb,µB)UBj(t′b,µB,µ) of τB. The hard function for Drell-Yan production is a timelike form factor and for µ Q contains large dkQS τ Q ta+tb k,µ π2 terms from ln2( iQ/µ ). WHe s≃um these π2 terms B B S H ×Z (cid:16) − Q − (cid:17) by taking µ = iQ−[9]. We estimate perturbative un- H ×US(k,µS,µ), (4) certainties by var−ying µH, µB, and µS about the above values. The complete summation at NNLL requires the where xa = (Q/Ecm)eY and xb = (Q/Ecm)e−Y, Ecm is NLO expressions for Hij, qq, qg, and SB, as well as the total center-of-mass energy, and the sum runs over the NNLL expressionsfor UI , UIi,j, and U . See Ref. [8] quark flavors ij = uu¯,u¯u,dd¯,... . The hard function H B S foradiscussionandreferencesoftherequiredanomalous { } H (Q2,µ ) contains virtual radiation at the hard scale ij H dimensions and fixed-order computations. Q (and also includes the leptonic process). Resultsat NNLL. Inournumericalresultsweusethe The beam functions B (t ,x ,µ ) and B (t ,x ,µ ) i a a B j b b B MSTW2008NLOPDFs[10]withtheirα (m )=0.1208. s Z in Eq. (4) depend on the momentum fractions x and a,b WealsointegrateoverY inEq.(4). InFig.2,weshowthe virtualities t of the partons i and j annihilated in the a,b Drell-Yan cross section dσ/dQ with no cut at NLO and hardinteraction. They canbe calculatedinanoperator- with cuts τ 0.1,0.02 at NNLL, for the LHC with B product expansion [7, 8] ≤ { } E = 7TeV and for the Tevatron. The Z resonance is cm visible at Q=m . 1 dξ x Z Bi(ta,xa,µB)=Xk Zxa ξaa Iik(cid:16)ta, ξaa,µB(cid:17)fk(ξa,µB), a fTahcteorcuotfτaBrou≤nd0.11.3readbuocvese tthhee cZropsseaskec(toiorn5–o1n.l5y fboyr (5) τ 0.02),showingthatmostofthe crosssectioncomes B ≤ 3 In Fig. 4 we show the cross section integrated up to 100 no cut (NLO) τ τcut as a function of Qτcut for Q = m and V] 10 τBcut=0.1 (NNLL) QB=≤30B0GeV. We see again thBat the logarithmZs are e 1 τBcut=0.02 (NNLL) important at small τcut and need to be resummed. G B /b 0.1 σ (E =7TeV) In Figs. 3 and 4, the perturbative scale uncertainties p pp cm are given by bands from varying µ , µ , and µ . The [ 0.01 H B S Q independent variationofthese three scaleswouldoveres- d 10−3 timatetheuncertainty,sinceitdoesnottakeintoaccount / σ 10−4 the parametric relation µ2 µ µ and the hierarchy d B ≃ S H 10−5 0.1σpp¯(Ecm=1.96TeV) µS ≪µB ≪µH. On the other hand, their simultaneous 10−6 variation [case (a) in Eq. (7)] can produce unnaturally 50 100 200 300 500 1000 small scale uncertainties. Hence, the perturbative un- Q [GeV] certainties in all figures are the envelope of the separate scale variations FIG. 2: Drell-Yan cross section dσ/dQ with cuts τB ≤ 0.1 (dashed lines) and τB ≤ 0.02 (dotted lines) at NNLL. The (a) µH =−riQ, µB =r√τBQ, µS =rτBQ, (7) solid lines show the total NLO cross section without a cut. (b) µ = iQ, µ =r−(lnτB)/4√τ Q, µ =τ Q, H B B S B Forbetterdistinction,theTevatroncrosssectionismultiplied − by 0.1. (c) µH = iQ, µB =√τBQ, µS =r−(lnτB)/4τBQ, − with r = 1/2,2 , and r = 1 corresponding to the 40 { } ] central-value curves. The exponent of r for cases (b) 2V35 Ecm=7TeV and (c) is chosen such that for τ = e−4 the scales µ Ge Q=mZ or µ vary by factors of 2, withBsmaller variations foBr /30 S b NNLL increasing τ and no variation for τ 1. In this limit, p B B → [25 NLL there should only be a single scale µ =µ =µ , and H B S B | | τd20 LL thus the only scale variation should be case (a). For the Q sing.NLO integrated cross section we replace τB in Eq. (7) with d15 τcut. In both Figs. 3 and 4, we see good convergence of / B σ d10 theperturbativeseriesandasubstantialreductioninthe ) perturbative uncertainties at NNLL. The convergence is Q / 5 improvedappreciablybythesummationoftheπ2 terms. 1 ( 0 In Fig. 5, we plot percent differences for several cross 0 5 10 15 20 sections relative to the NNLL result. All results are in- QτB [GeV] tegrated up to τcut =0.1 and are plotted versus Q. The B dark orange bands show the NNLL perturbative uncer- FIG. 3: Cross section differential in τB at Q = mZ for the taintiesandthelightyellowbandsthe90%C.L.PDF+α s LHCwithEcm =7TeV. ShownaretheLL,NLL,andNNLL uncertainties using the procedure from Ref. [10]. The results, wherethebandsindicatetheperturbativeuncertain- dashed line shows the NNLL result without the gluon tiesas explainedin thetext. Forcomparison, thedottedline shows thesingular NLOresult with no resummation. contribution to the quark beam function, qg in Eq. (5). I ThegluoncontributionissignificantattheLHCandless prominent at the Tevatron, because the gluon PDF is from small τ . The cross section differential in τ at more important for pp than pp¯collisions. In the dotted B B fixed Q = m is shown in Fig. 3, where we can see line we further neglect all terms in the quark contribu- Z explicitly that the cross section is dominated by small tion thataresubleadinginthethresholdlimitx 1. qq I → τ . To see the effect of the higher-order resummation Except for the Tevatron at large Q the threshold result B we plot the LL, NLL, and NNLL results. The impor- isapoorapproximationtothefullresult,beingwellout- tance of resummation is illustrated by comparing them side the perturbative uncertainties. The dark band and to the singular part of the fixed NLO result (dashed solidline showthe NLO resultwiththe perturbativeun- line), which is obtained from our NNLL result by set- certainties from varying the common scale between Q/2 ting µ = µ = µ = Q. (The full NLO result con- and2Q. ItsdifferencefromtheresummedNNLLresultis H B S tains additional nonsingular terms that are not numeri- genericallylargeandnotcapturedbythefixed-orderper- cally relevant at small τ .) Results are not plotted be- turbativeuncertainties,showingthattheresummationis B low Qτ = µ 1GeV, where the soft function be- important not only to get an improvedcentralvalue but B S ≤ comesnonperturbativeandweexpectlargecorrectionsof also to obtain reliable perturbative uncertainties. (Λ /µ ) to our purely perturbative results. Corre- Beam thrust in Drell-Yan production provides an ex- QCD S O spondingly,theperturbativeuncertaintiesgetlargehere. perimentally and theoretically clean measure of ISR in 4 280 3.5 E =7TeV E =7TeV V]240 cQm=m V] 3 cQm=300GeV e Z e G200 G 2.5 / / b b [p160 [f 2 ) ) cutB120 cutB1.5 τ τ Q( NNLL Q( NNLL d 80 NLL d 1 NLL / / σ LL σ LL d 40 d 0.5 sing.NLO sing.NLO 0 0 0 5 10 15 20 0 5 10 15 20 Qτcut [GeV] Qτcut [GeV] B B FIG. 4: Integrated cross section with a cut τB ≤ τBcut as a function of τBcut at Q = mZ (left panel) and Q = 300GeV (right panel) for theLHC. The curves havethesame meaning as in Fig. 3. 20 20 no g scale unc. scale unc. x→1 PDF+α PDF+α 10 sing.NLO s 10 s ] ] % % [ [ ) ) Q Q 0 0 d d / / σ σ d d ( E =1.96TeV ( δ−10 cm δ−10 no g τcut=0.1 B E =7TeV x→1 cm τcut=0.1 sing.NLO −20 −20 B 50 100 200 300 500 1000 50 100 200 300 500 1000 Q [GeV] Q [GeV] FIG. 5: Percent difference relative tothecentral NNLL result, δ(dσ/dQ)=(dσ/dQ)/(dσNNLL/dQ)−1, at theTevatron (left panel) and theLHC (right panel). Here we always take τBcut =0.1 in both the numeratorand denominator. qq¯ ℓ+ℓ−, similar to how thrust measures final-state [2] DELPHI Collaboration, P. Abreu et al., Z. Phys. C 73, rad→iationine+e− qq¯. The experimentalmeasurement 11 (1996); A. Buckley et al., Eur. Phys. J. C 65, 331 of beam thrust wil→l contribute very valuable information (2010). [3] Z. Nagy, Phys. 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