SUSY QCD corrections to electroweak gauge boson production with an associated jet at the LHC
PPreprint typeset in JHEP style - HYPER VERSION
MADPH-11-1575
SUSY QCD corrections to electroweak gauge bosonproduction with an associated jet at the LHC
Ryan Gavin
Phenomenology Institute, Department of Physics, University of Wisconsin-Madison,1150 University Avenue, Madison, Wisconsin 53706, USAandPaul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandEmail: [email protected]
Maike K. Trenkel
Phenomenology Institute, Department of Physics, University of Wisconsin-Madison,1150 University Avenue, Madison, Wisconsin 53706, USAEmail: [email protected]
Abstract:
We study the stability of the neutral- and charged-current Drell–Yan processin association with a jet as a standard candle at the LHC under the inclusion of O ( α s )supersymmetric QCD (SQCD) corrections within the MSSM. We include the decay ofthe electroweak gauge boson into dileptons, i. e. we consider the production of chargedlepton–anti-lepton or lepton-neutrino final states with one hard jet. We find that theSQCD corrections are negligible for the integrated cross section. Only at high leptontransverse momentum can they induce effects of the percent level. Keywords:
Supersymmetry Phenomenology, NLO Computations, Hadronic Colliders. a r X i v : . [ h e p - ph ] S e p ontents
1. Introduction 12. Details of the calculation 3
3. Numerical results 6
Z/γ ∗ +jet integrated cross section results 83.3 Z/γ ∗ +jet kinematic distributions 93.4 W +jet integrated cross section results 103.5 W +jet kinematic distributions 12
4. Conclusions 14A. Counterterms and renormalization constants 15
1. Introduction
Electroweak gauge boson production with subsequent leptonic decays plays a crucial rolein the physics studies performed at hadron colliders. With large cross sections and a clearcollider signature, the Drell–Yan processes have become a standard candle at the LHC andmany precise measurements are currently being performed [1–4]. Their study can be usedto help calibrate detectors and place constraints on parton distribution functions (PDFs),see e. g. [5–7]. It has also been proposed that measured electroweak boson production crosssections can be used as a luminosity monitor [8]. Drell–Yan processes also constitute animportant source of background for searches for new physics, such as Z (cid:48) and W (cid:48) bosonproduction and other high mass dilepton resonances [9–12].Given the large amount of data that the LHC will deliver, measurement errors willeventually be dominated by systematics rather than statistics, see e. g. [13]. Consequently,electroweak boson production needs to be known to high precision. Much effort has beenmade in this area. Indeed, it was the first hadronic scattering process to be computed atnext-to-next-to-leading (NNLO) in QCD [14, 15]. Now also the differential cross sectionsfor Drell–Yan processes are known through NNLO in QCD [16–21], and the NLO QCD cor-rections have been matched to parton showers [22–25]. Electroweak corrections have beenknown for sometime [26–29], and continue to be refined [24, 30–34]. Attempts to approxi-mate QCD × EW corrections to Drell–Yan have also appeared in the literature [34,35] andvery recently the mixed QCD × EW two-loop virtual corrections have been calculated [36].– 1 –t hadron colliders, the high center-of-mass energy (c. m.) typically leads to produc-tion of electroweak gauge bosons in association with QCD radiation. Here, we focus onDrell–Yan processes with one additional, hard jet. The intermediate gauge boson thenrecoils against the jet and can be strongly boosted. If the jet has large transverse momen-tum, the leptons originating from the gauge boson decay are produced at high transversemomentum as well and provide a source for either high-energy dilepton pairs of oppositecharge (
Z/γ ∗ +jet production) or high missing transverse energy in combination with acharged lepton ( W +jet production). Theoretical predictions for the Drell–Yan+jet crosssection include NLO QCD corrections [37–39] and parton shower matching [22], as well asin the electroweak sector the NLO corrections in the on-shell approximation [40–44] andthe full NLO calculation, including photonic corrections and including the leptonic decayof the gauge boson [45, 46]. Also for Drell–Yan cross sections with higher jet multiplicities,the NLO QCD corrections are now available [39, 47–50].Often, precision measurements of Standard Model (SM) processes are used to constrainnew physics, not by direct detection, but by their influence through quantum fluctuations.One may ask the question how radiative corrections from physics beyond the StandardModel (BSM) would effect such well-known processes. To allow for precision tests of theSM and its perturbative expansions, it is crucial to know if there were large BSM correctionssince otherwise information about the underlying physics cannot reliably be extracted fromthe data. An interesting paradigm for physics beyond the SM is supersymmetry (SUSY).Many collider studies have been performed to examine the phenomenology of SUSY pro-duction at hadron colliders, and the possibility of their detection. The investigation ofnew physics contributions to SM processes via higher-order corrections is a logical nextstep. Only in recent years, Drell–Yan processes have been studied within the frameworkof the minimal supersymmetric Standard Model (MSSM). The NLO supersymmetric QCD(SQCD) and electroweak corrections within the MSSM to the charged- and neutral-currentDrell–Yan processes with no final state parton were calculated in [30,31] and the NLO elec-troweak corrections for the on-shell W +jet production process within MSSM have beencomputed in [51]. The impact of SUSY corrections has been found to be small.In this work, we further complete the one-loop picture and calculate the NLO SQCDcorrections within the MSSM to Drell–Yan processes in association with a hard jet in thefinal state. We consider both Z/γ ∗ +jet and W +jet production at the LHC, including theleptonic decay of the gauge bosons, and take all off-shell effects due to the finite widths ofthe Z and W boson and all contributions of an intermediate photon into account. Our aimis to show the stability of Drell–Yan+jet processes under the inclusion of SQCD correctionswithin the MSSM.This paper is organized as follows. The setup of the calculation is detailed in section 2.The computation of the SQCD corrections to the neutral- and charged-current Drell–Yanprocesses proceed in close analogy and are discussed in parallel. In section 3 we present thenumerical results for both Z/γ ∗ +jet and W +jet production at the LHC. We summarize ourfindings in section 4. Details on the renormalization procedure are given in appendix A.– 2 – q ℓℓgqZ/γ qq ℓ ℓgq Z/γ Figure 1:
Feynman diagrams for the quark–anti-quark initial state process, Eq. (2.1).
2. Details of the calculation
In this section we describe the technical setup of our calculation of SQCD corrections forthe neutral- and charged-current Drell–Yan process with an associated hard jet. We alwaysinclude the decay of the gauge bosons into leptons and take all off-shell effects due to thefinite width of the Z or W boson into account. We first address the leading order (LO)processes in section 2.1 and then discuss the NLO calculation in section 2.2. In view ofSQCD corrections, the computations for Z/γ ∗ +jet and W +jet production are very similarand proceed in close analogy. At the LHC, the Drell–Yan process in association with a hard jet is initiated by three differ-ent partonic processes at LO. Including the leptonic decay of the intermediate electroweakgauge boson, they are in the case of the neutral-current Drell–Yan process, q ¯ q → Z/γ ∗ g → (cid:96) − (cid:96) + g , (2.1) g q → Z/γ ∗ q → (cid:96) − (cid:96) + q , (2.2) g ¯ q → Z/γ ∗ ¯ q → (cid:96) − (cid:96) + ¯ q , (2.3)and for the charged-current Drell–Yan process, q ¯ q (cid:48) → W g → (cid:96) ν (cid:96) g , (2.4) g q → W q (cid:48) → (cid:96) ν (cid:96) q (cid:48) , (2.5) g ¯ q → W ¯ q (cid:48) → (cid:96) ν (cid:96) ¯ q (cid:48) . (2.6)For W + production, a positively charged lepton is produced together with an anti-neutrino( (cid:96) + ¯ ν (cid:96) ), for W − production it is a negatively charged lepton and a neutrino ( (cid:96) − ν (cid:96) ).The tree-level Feynman diagrams for the process Eq. (2.1) can be found in Figure 1,those for W +jet production follow analogously by replacing the Zqq ( Z(cid:96)(cid:96) ) vertex witha
W qq (cid:48) ( W (cid:96)ν (cid:96) ) vertex. Diagrams for the (anti-)quark–gluon processes are obtained bycrossing the gluon with an initial-state (anti-)quark.The quarks considered are q, q (cid:48) = u, c, d, s with q and q (cid:48) being quarks of oppositeisospin. The quarks are treated as massless throughout the calculation. The Z/γ ∗ +jet LOcross section is quark generation independent, aside from the appropriate PDF inclusionand convolution. The only up- and down-type quark dependence comes from the vector– 3 –nd axial couplings in the Zq ¯ q vertex. The W +jet cross sections do depend on the quarkflavor due to the quark mixing parameterized by the CKM matrix. However, the CKMquark mixing factorizes from the tree-level matrix elements and, for the inclusive crosssection, there needs only one squared amplitude to be calculated for each of three partonicchannels, weighted by the sum of squares of the respective absolute value elements of theCKM matrix, as well as the corresponding PDFs.Throughout the calculation, we exclude bottom-quark initiated processes due to theirPDF-suppressed small contribution to the cross section and we neglect the mixing betweenfirst and third generation quarks. A fixed width is included in the propagator of theintermediate Z or W boson, giving the expected Breit-Wigner resonance.Already at LO, there exists the possibility that the parton in the final state may becomesoft or collinear to the gluon or quark that it couples to. If this occurs, the cross sectionbecomes infinite. Divergences of this type are regulated by requiring the jet to be hard.This is achieved by applying a minimum transverse-momentum ( p T ) cut on the jet. Alsowe treat the leptons as massless throughout the calculation and require a minimum lepton p T as well as a minimum lepton-pair invariant mass, M (cid:96)(cid:96) , in case of Z/γ ∗ +jet production. We calculate the one-loop SQCD corrections to the partonic processes in Eqs. 2.1-2.3 andEqs. 2.4-2.6, which are purely virtual. The SUSY particles charged under SU (3) C arethe squarks, ˜ q i , and gluinos, ˜ g , the superpartners of the SM chiral quarks and gluons,respectively. In the SM, left- and right-handed quarks transform differently under SU (2) L .Consequently, under supersymmetric transformations, there are separate superpartners,˜ q L , ˜ q R , for quarks of different handedness, q L , q R . After electroweak symmetry breaking(EWSB), left- and right-handed squark eigenstates mix to form squark mass eigenstates,˜ q , ˜ q . Squark mixing, however, is proportional to the mass of their SM quark partner.In this calculation, only squarks of the same flavor as the initial state quarks are present.Since the masses of the quarks considered ( u , d , c , s ) are small, we can neglect any left-rightmixing between squark eigenstates, and the two gauge eigenstates of a given flavor q arealso mass eigenstates that we denote by ˜ q i , i = L, R .In the calculation of
Z/γ ∗ +jet production, the relevant interactions between SM andSUSY particles are the γ ˜ q i ˜ q j , Z ˜ q i ˜ q j , g ˜ q i ˜ q i , g ˜ g ˜ g , and q ˜ q i ˜ g vertices. Although there arediagrams with the γg ˜ q i ˜ q j or Zg ˜ q i ˜ q j vertex, their contribution to the virtual corrections iszero, and they have not been explicitly included in this manuscript. There are 44 diagramsper partonic channel that give non-zero contributions to the differential cross section (22for Z boson and photon mediation each). They divide into self-energy insertions, gluon-and Z/γ ∗ -vertex corrections, and box contributions and are organized in Figure 2. Ineach diagram, the two squark eigenstates ˜ q i of the same quark flavor q as the initial statecan run in the loop. The Feynman diagrams for W +jet production can easily be inferredfrom Figure 2 by again replacing the Zqq ( Z(cid:96)(cid:96) ) vertex with a
W qq (cid:48) ( W (cid:96)ν (cid:96) ) vertex andby replacing the Z ˜ q i ˜ q i by W ˜ q i ˜ q (cid:48) i vertices. Of course the W boson couples to left-handedparticles only, and only the left-handed components of the squark eigenstates contribute(i. e. ˜ q i = ˜ q L in case of no left-right mixing).– 4 –a) Self-energy insertions: qq ℓℓgq Z/γq ˜ g ˜ q i qq ℓ ℓgZ/γqq ˜ g ˜ q i (b) Gluon-vertex corrections: qq ℓℓgZ/γq ˜ g ˜ q i ˜ g qq ℓℓgZ/γq ˜ q i ˜ g ˜ q i q q ℓℓgZ/γq ˜ g ˜ q i ˜ g q q ℓℓgZ/γq ˜ q i ˜ g ˜ q i (c) Z/γ ∗ -vertex corrections: qq ℓℓgq Z/γ ˜ g ˜ q i ˜ q i q q ℓℓgq Z/γ ˜ g ˜ q i ˜ q i (d) Box contributions: q q ℓ ℓgZ/γ ˜ g ˜ q i ˜ q i ˜ q i qq ℓ ℓgZ/γ ˜ g ˜ q i ˜ q i ˜ q i qq ℓ ℓgZ/γ ˜ g ˜ q i ˜ g ˜ q i Figure 2:
Feynman diagrams for the self-energy, gluon- and
Z/γ ∗ -vertex, and box contributionsto the partonic process q ¯ q → (cid:96) + (cid:96) − g , mediated by Z/γ ∗ exchange, at SQCD NLO. Due to only massive particles propagating in the loop, the virtual corrections are com-pletely infrared (IR) safe. Therefore, there are no real emission processes to be consideredto cancel singularities. However, as noted above, we require a hard jet in the final state.This statement implies a minimum cut has been placed on the transverse momentum ofthe jet to avoid IR divergences that would otherwise appear at LO.Ultraviolet (UV) divergences arise from the self-energy insertions and the gluon and
Z/γ ∗ or W vertex corrections, while the box diagrams are IR and UV finite. Dimensional– 5 –egularization is used to regulate the UV divergences, where the number of dimensions is d = 4 − (cid:15) and the singularities then take the form of 1 /(cid:15) poles. In order to remove the UVdivergences one has to include the proper counterterms for SQCD one-loop renormalization.Our renormalization procedure closely follows ref. [52]. We impose on-shell conditions tofix the renormalization constants in the quark sector and use the MS scheme, modified todecouple the heavy SUSY particles [53, 54], for the renormalization of the strong couplingconstant and the gluon field. The explicit expressions in terms of the renormalizationconstants can be found in appendix A.We set up the calculation in the conventional diagrammatic approach. As an im-portant cross-check on the numerical results, two completely independent computationswere performed using different techniques. The first calculation is based on the frameworkof FeynArts 3.6 [55], FormCalc 7.0, and LoopTools 2.6 [56]. In the second calculation,Feynman diagrams are generated with QGRAF [57]. Algebraic manipulation is performedwith an in-house software comprised of Maple and Form [58]. Then AIR [59] is used forthe reduction of tensor loop integrals to scalar ones. The numerical integration is per-formed using Vegas of the Cuba 1.7 [60] library, as well as the scalar loop integral libraryQCDLoop [61].The Zqq , W qq (cid:48) , Z ˜ q i ˜ q j , W ˜ q i ˜ q (cid:48) j , and q ˜ q ˜ g vertices contain axial pieces. Special attentionshould be paid to the the treatment of γ when working in d dimensions. In the firstcalculation the naive anticommuting scheme [56, 62] was used. Here, it is assumed that (cid:8) γ , γ µ (cid:9) = 0 as in four space-time dimensions and that the four-dimensional, non-zerotrace, Tr[ γ µ γ ν γ ρ γ σ γ ] = − i(cid:15) µνρσ , (2.7)remains. This FormCalc-based approach has been justified in [63]. In the second calcula-tion, the treatment of γ as described in [64] is used, where γ = i (cid:15) µνρσ γ µ γ ν γ ρ γ σ . (2.8)Both approaches have been shown to give consistent results in our NLO calculation.
3. Numerical results
In this section we investigate the numerical influence of the SQCD one-loop corrections toboth the neutral- and charged current Drell–Yan process with an associated hard jet. Weexamine these processes at the LHC with a c. m. energy of 7 TeV, unless noted otherwise.We list the relevant input parameters in section 3.1. The integrated cross section resultsand differential distributions for the
Z/γ ∗ +jet process are given in sections 3.2 and 3.3,respectively. The results for W +jet production are discussed in sections 3.4 and 3.5.– 6 – .1 Input parameters The SM input parameters are chosen in accordance with [65], M Z = 91 . , M W = 80 .
339 GeV , Γ Z = 2 . , Γ W = 2 .
085 GeV , cos θ W = M W /M Z , α − = α ( M Z ) − = 128 . , | V ud | = 0 . , | V us | = 0 . , | V cd | = 0 . , | V cs | = 0 . . (3.1)Quarks and leptons are considered massless and we do not further specify the flavor ofthe leptons in the final state as the SQCD corrections do not depend on the lepton flavor.Factorization and renormalization scales are fixed and identified with the vector bosonmass, µ F = µ R = M Z, W . We use the central MSTW 2008 NLO (68% CL) PDF set [66] inits LHAPDF implementation [67], with the strong coupling α s ( µ R ) they provide, yielding α s ( M Z ) = 0 . , α s ( M W ) = 0 . . (3.2)We require a set of basic kinematic cuts to be satisfied. As previously mentioned, aminimum p T of the final state parton is required to render the cross section finite. We alsodemand the two final-state leptons to have a minimum transverse momentum. Furthermorewe require the leptons and the jet to be produced in the central region of the detector andapply a cut on their rapidities. For the Z/γ ∗ +jet process we also require a minimumlepton-pair invariant mass, M (cid:96)(cid:96) . We choose the following numerical values, Z/γ ∗ +jet : p T, jet >
25 GeV , | y jet | < . ,p T,(cid:96) ± >
25 GeV , | y (cid:96) ± | < . , M (cid:96)(cid:96) >
50 GeV .W +jet : p T, jet >
25 GeV , | y jet | < . ,p T,(cid:96) , /p T >
25 GeV , | y (cid:96) | < . . (3.3)However, only the integrated cross sections depend strongly on the specific cuts chosen,while we have found that the relative importance of the SQCD corrections does not varymuch when the cuts are tightened or loosened.The only SUSY particles that appear in the loops are squarks and the gluino. TheSQCD corrections are flavor- and chirality-blind and no other SUSY parameters than thesquark and gluino masses enter the calculation. For simplicity we neglect the squark left-right mixing, set all squark masses equal and use a common squark mass, m ˜ q , and thegluino mass, m ˜ g , as direct input. There is no need to define a complete set of SUSY inputparameters and we do not consider commonly used benchmark scenarios (as e. g. SPS1a’)here. As we will see below, this approach is sufficient for the purpose of our study toshow the stability of Drell–Yan+jet under the inclusion of SQCD corrections. We use thefollowing values for our numerical studies, if not otherwise noted, m ˜ q = 600 GeV , m ˜ g = 500 GeV . (3.4)These sparticle masses are already at the lower limit of the mass region that is currentlyinvestigated by LHC experiments, see e. g. [68, 69], and allow for a conservative estimate ofthe typical size of SQCD corrections to Drell–Yan+jet processes.– 7 – /γ ∗ +jet partonic LO SQCDproduction channel cross section contributions δ √ s = 7 TeV q ¯ q gq + g ¯ q √ s = 14 TeV q ¯ q gq + g ¯ q Table 1:
Numerical results for integrated cross sections for the neutral-current Drell–Yan processmediated by a Z boson or virtual photon γ ∗ in association with a hard jet at the LHC, with √ s = 7 TeV and √ s = 14 TeV. Shown are the leading order results in picobarn (pb), the SQCDcontributions in femtobarn (fb) and the relative corrections δ for the partonic subchannels and theinclusive result (incl.). Light quarks are implicitly summed over in the initial state, q = u, d, c, s .We consider m ˜ q = 600 GeV, m ˜ g = 500 GeV and the cuts listed in Eq. (3.3) have been applied.Factorization and renormalization scale are set to µ = M Z (with MSTW 2008 NLO) . Z/γ ∗ +jet integrated cross section results Here we present the integrated cross section for charged dilepton production with a hardjet at the LHC. Table 1 shows the LO cross sections and SQCD contributions for the q ¯ q and gluon-initiated partonic processes, at a proton-proton c. m. energy of √ s = 7 TeV and √ s = 14 TeV.The prominent production modes at the LHC are the gluon-induced initial states,enhanced by the large gluon densities at small parton momentum fractions. With thekinematic constraints in Eq. (3.3), the total integrated cross section is 54.27 pb at 7 TeVand 123.1 pb at 14 TeV ( q = u, d, c, s ). The SQCD corrections, for squark and gluinomasses of 600 and 500 GeV, respectively, account for an increase of 0 . − . Z/γ ∗ +jet production, we show in Figure 3 the relative corrections as a function of thecommon squark mass m ˜ q , for different values of the gluino mass. As one would expect, theimpact of the SQCD corrections increases as the SUSY particle masses decrease. However,even for very light squarks and gluinos of only 100 GeV the relative corrections are stillsignificantly below the 1% level. Figure 3 shows that the Z/γ ∗ +jet integrated cross sectionis stable under SQCD corrections, over a broad range of low-mass squarks and gluinos.For TeV-range SUSY particles the SQCD corrections are completely negligible. This is– 8 – [GeV]q~m(
100 200 300 400 500 600 1000 [ % ] d *+jet g Z/ Figure 3:
Relative SQCD corrections of the integrated cross section, δ = ( σ NLO − σ LO ) /σ LO , forcharged lepton-pair production mediated by a Z boson or virtual photon in association with a hardjet at the LHC, with √ s = 7 TeV, as a function of a common squark mass, m ˜ q , for different gluinomasses, m ˜ g . The cuts listed in Eq. (3.3) have been applied. comforting in the sense that if Z/γ ∗ +jet is used as a normalization process, no SUSY BSMphysics should be masked in this normalization. In terms of our calculation, this stabilityalso justifies our neglecting of squark left-right mixing, and use of a degenerate squark massscheme. Z/γ ∗ +jet kinematic distributions We have seen above that the SQCD contributions to the integrated cross section are sub-tle. However, the corrections can be more pronounced in the differential distributions ofkinematic variables. We consider the LO and NLO differential cross sections, and definethe relative corrections δ , δ = O NLO −O LO O LO , for a given observable O , where O NLO is thesum of the LO contributions and the SQCD contributions. We present distributions in thelepton-pair invariant mass, M (cid:96)(cid:96) , the lepton transverse momentum and rapidity, p T,(cid:96) and y (cid:96) , and the jet transverse momentum and rapidity, p T, jet and y jet . All distributions shownhave been subject to the kinematic constraints given in Eq. (3.3).In Figure 4, distributions in the lepton-pair invariant mass M (cid:96)(cid:96) are displayed. We findthat the SQCD corrections hardly affect the shape of the LO distribution and are below the1% level for M (cid:96)(cid:96) < Z boson resonance, shown inthe upper right panel, they are completely negligible and do not distort the SM result. Forlarger values of M (cid:96)(cid:96) (lower right panel) the relative corrections can reach several percent,until a SUSY mass threshold is reached and then the relative corrections begin to fall.The lepton differential distributions in transverse momentum, p T,(cid:96) , and rapidity, y (cid:96) ,are given in Figure 5. In the low- and intermediate- p T range, the SQCD contributions tothe p T,(cid:96) distribution are positive but small, increasing only to a maximum of about 1%at around the threshold of the average sparticle mass. In the high- p T region, the relativecorrections become non-negligible, on the order of a few negative percent. Concerning the– 9 – [GeV] ll M [ pb / G e V ] ll / d M s d -5 -4 -3 -2 -1
101 LO distributionNLO distribution *+jet g Z/ [GeV] ll M [ % ] d -1012 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( [GeV] ll M
50 60 70 80 90 100 110 120 130 140 [ % ] d *+jet g Z/ [GeV] ll M [ % ] d -101234 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( *+jet g Z/ Figure 4:
Differential distributions for the neutral-current Drell–Yan process in assocation with ahard jet at the LHC. NLO SQCD corrections are calculated for m ˜ q = 600 GeV and m ˜ g = 500 GeVand the cuts given in Eq. (3.3) have been placed. Shown are the absolute differential distributions forthe LO and NLO processes (top left) and the relative difference between NLO and LO distributions(bottom left) with respect to lepton-pair invariant mass M (cid:96)(cid:96) . In the right panels, the relativecorrections around the Z boson resonance and in the high- M (cid:96)(cid:96) region are shown, respectively. lepton rapidity, Figure 5 (right), we find that the relative corrections are nonzero but farbelow 1% and thus do not have any impact on the shape of the SM-only distribution. Itis important to observe the discrepancies between the lepton and anti-lepton distributionsin Figure 5. These differences occur already at the LO process and are a result of the Z boson’s axial coupling to fermions. Setting the axial coupling to zero would render thelepton and antilepton curves identical.The jet distributions, Figure 6, exhibit the same behavior as the lepton distributions.We see the threshold effect around ( m ˜ q + m ˜ g ) / p T, jet region ( p T > . − m ˜ q = 400 GeV and m ˜ g = 350 GeV, the SQCDcorrections become only slightly more important, and reach the 1 −
2% level at the sparticlethresholds and the −
5% level for p T ≈ W +jet integrated cross section results In Table 2 we show the results for the integrated cross section for lepton-neutrino productionwith a hard jet, mediated by a W boson, at the LHC. The LO cross sections and SQCDcontributions for the q ¯ q (cid:48) and gluon-initiated partonic processes are given, at a proton-proton– 10 – lep) [GeV] T p [ pb / G e V ] T / dp s d -7 -6 -5 -4 -3 -2 -1
101 LO distribution (l-)NLO distribution (l-)LO distribution (l+)NLO distribution (l+) *+jet g Z/ (lep) [GeV] T p [ % ] d -3-2-101 relative NLO corrections (l-) relative NLO corrections (l+))=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( y (lep) -2 0 2 / d y [ pb ] s d
10 LO distribution (l-)NLO distribution (l-)LO distribution (l+)NLO distribution (l+) *+jet g Z/ y (lep) -3 -2 -1 0 1 2 3 [ % ] d -0.0100.010.02 relative NLO corrections (l-) relative NLO corrections (l+))=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( Figure 5:
Lepton transverse momentum, p T,(cid:96) , (a) and lepton rapidity, y (cid:96) , (b) distributions for the Z/γ ∗ +jet process at the LHC with √ s = 7 TeV. The black line represents the lepton while thered line represents the antilepton. NLO SQCD corrections are calculated for m ˜ q = 600 GeV and m ˜ g = 500 GeV and the cuts given in Eq. (3.3) have been placed. (jet) [GeV] T p [ pb / G e V ] T / dp s d -6 -5 -4 -3 -2 -1
101 LO distributionNLO distribution *+jet g Z/ (jet) [GeV] T p [ % ] d -3-2-101 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( y (jet) -2 0 2 / d y [ pb ] s d *+jet g Z/ y (jet) -3 -2 -1 0 1 2 3 [ % ] d -0.0100.010.02 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( Figure 6:
Jet transverse momentum, p T, jet , (a) and jet rapidity, y jet , (b) distributions for the Z/γ ∗ +jet process at the LHC with √ s = 7 TeV. NLO SQCD corrections are calculated for m ˜ q =600 GeV and m ˜ g = 500 GeV and the cuts given in Eq. (3.3) have been placed. – 11 – +jet partonic LO SQCDproduction channel cross section contributions δ √ s = 7 TeV q ¯ q (cid:48) gq + g ¯ q √ s = 14 TeV q ¯ q (cid:48) gq + g ¯ q Table 2:
Numerical results for integrated cross sections for the charged-current Drell–Yan processmediated by a W boson in association with a hard jet at the LHC, with √ s = 7 TeV and √ s =14 TeV. Shown are the leading order results in picobarn (pb), the SQCD contributions in femtobarn(fb) and the relative corrections δ for the partonic subchannels and the inclusive result (incl.). Lightquarks are implicitly summed over in the initial state, q = u, d, c, s . We consider m ˜ q = 600 GeV, m ˜ g = 500 GeV and the cuts listed in Eq. (3.3) have been applied. Factorization and renormalizationscale are set to µ = M W (with MSTW 2008 NLO) . c. m. energy of √ s = 7 TeV and √ s = 14 TeV. Similar to the Z/γ ∗ +jet case, the SQCDcorrections contribute between 0 . − . W +jet kinematic distributions The differential distributions for W +jet production at the LHC with √ s = 7 TeV are shownin Figs. 7, 8, and 9. They correspond to distributions in transverse mass, lepton transversemomentum and rapidity, and jet transverse momentum and rapidity.The transverse mass, M T ( (cid:96)ν ), and the p T (cid:96) distributions are particularly relevant forthe measurement of the W boson mass at hadron colliders. Here, the transverse mass isdefined as M T ( (cid:96)ν ) = [( | p T,(cid:96) | + | /p T | ) − ( p T,(cid:96) + / p T ) ] / .Again, the results are very similar to those for Z/γ ∗ +jet production. The M T ( (cid:96)ν )distributions, displayed in Figure 7, receive percent-level corrections due to SQCD effects,which are minimal and almost vanishing when the W boson is on-shell and maximal aroundsquark and gluino thresholds in the high- M T ( (cid:96)ν ) region.The relative corrections to the p T,(cid:96) and p T, jet distributions, Figure 8 (left) and Figure 9(left), peak around the average sparticle mass where they amount to about 1%, and grownegative in the high- p T region, reaching a couple percent in the TeV region. The SQCDcorrections in the lepton and jet rapidity distributions, Figure 8 (right) and Figure 9 (right),are flat and can safely be neglected in an experimental analysis.– 12 – [GeV] n (l, T M [ pb / G e V ] T / d M s d -5 -4 -3 -2 -1 W+jet ) [GeV] n (l, T M [ % ] d -1012 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( ) [GeV] n (l, T M [ % ] d W+jet) [GeV] n (l, T M [ % ] d -101234 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( W+jet
Figure 7:
Differential distributions for the charged-current Drell–Yan process in assocation with ahard jet at the LHC. NLO SQCD corrections are calculated for m ˜ q = 600 GeV and m ˜ g = 500 GeVand the cuts given in Eq. (3.3) have been placed. Shown are the absolute differential distributions forthe LO and NLO processes (top left) and the relative difference between NLO and LO distributions(bottom left) with respect to transverse mass M T ( (cid:96)ν ). In the right panels, the relative correctionsaround the W boson resonance and in the high- M T ( (cid:96)ν ) region are shown, respectively. (lep) [GeV] T p [ pb / G e V ] T / dp s d -6 -5 -4 -3 -2 -1 n LO distribution ( ) n NLO distribution (
W+jet (lep) [GeV] T p [ % ] d -3-2-101 relative NLO corrections (l) ) n relative NLO corrections ()=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( y (lep) -2 0 2 / d y [ pb ] s d n LO distribution ( ) n NLO distribution (
W+jet y (lep) -3 -2 -1 0 1 2 3 [ % ] d -0.0100.010.02 relative NLO corrections (l) ) n relative NLO corrections ()=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( Figure 8:
Lepton transverse momentum, p T,(cid:96) , (a) and lepton rapidity, y (cid:96) , (b) distributions for the W +jet process at the LHC with √ s = 7 TeV. The red line represents the charged lepton while theblack line represents the neutrino (missing transverse momentum, /E T ). – 13 – jet) [GeV] T p [ pb / G e V ] T / dp s d -5 -4 -3 -2 -1 W+jet (jet) [GeV] T p [ % ] d -3-2-101 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( y (jet) -2 0 2 / d y [ pb ] s d W+jet y (jet) -3 -2 -1 0 1 2 3 [ % ] d -0.0100.010.02 relative NLO corrections)=500 GeV, LHC 7 TeVg~)=600 GeV, m(q~m( Figure 9:
Jet transverse momentum, p T, jet , (a) and jet rapidity, y jet , (b) distributions for the W +jet process at the LHC with √ s = 7 TeV.
4. Conclusions
In this manuscript, we present a full treatment of the SQCD corrections to the neutral- andcharged-current Drell–Yan process in association with a hard jet. We include the decay ofthe electroweak gauge bosons into a lepton pair or a lepton-neutrino pair, respectively, andtake all off-shell effects due to the finite Z and W boson widths into account, as well asthe contributions and interference effects of the photon-mediated diagrams in case of theneutral Drell–Yan process.Even though the Drell–Yan process, with an associated jet, can not be used in anypractical manner to set indirect limits on SUSY particle masses or SUSY parameters, itis important to study the impact of SUSY (and other BSM physics) contributions to SMprocesses. As standard candles, and important backgrounds to new physics at the LHC, itis necessary to understand the stability of EW gauge bosons production processes againstBSM contributions.We find that the relative corrections to the integrated cross section due to SQCDcorrections are small, below the 0.5% level even for very light SUSY particles. Examiningthe differential distributions in conventional kinematic variables such as p T , M (cid:96)(cid:96) , or M T ( (cid:96)ν ),we find vanishing effects in the vicinity of the Z or W boson resonance peak, while therelative SQCD corrections can increase to 2 −
5% when the electroweak gauge bosons arefar off-shell. – 14 – cknowledgments
We are grateful to T. Hahn, S. Kallweit, and F. Petriello for useful discussions and help-ful comments. This work was supported by the US DOE under contract No. DE-FG02-95ER40896.
A. Counterterms and renormalization constants
Here we list the counterterms for the renormalization of vertices and propagators in theSQCD one-loop amplitudes for Drell–Yan+jet production. All quarks are considered mass-less.By using multiplicative renormalization, we replace in the QCD Lagrangian the left-and right-handed quark fields, Ψ
L,R , the gluon field, G a , and the strong coupling constant, g s , with Ψ L,R → (cid:112) Z L,R Ψ L,R , G aµ → (cid:112) Z G G aµ , g s → Z g s g s . (A.1)Expanding Z i = 1 + δZ i , and introducing δZ V,A = ( δZ L ± δZ R ), we find the follow-ing Feynman rules for the self-energy and vertex counterterms that are relevant in ourcalculation. q q −→ p : i (cid:54) p ( δZ V − δZ A γ ) , (A.2) γ, µ qq : − iee q γ µ ( δZ V − δZ A γ ) , (A.3) Z, µ qq : − iec W s W γ µ ( g qV − g qA γ ) ( δZ V − δZ A γ ) , g qV = I q − e q s W ,g qA = I q , (A.4) W, µ q ′ q : − ie √ s W V CKM qq (cid:48) γ µ (1 − γ ) ( δZ V + δZ A ) , (A.5) g, a, µ qq : − ig s T a γ µ ( δZ V − δZ A γ + δZ g s + 12 δZ G ) , (A.6)where e q is the electric charge of quark q and I q denotes the eigenvalue of the third com-ponent of its weak isospin. T a , a = 1 ..
8, are the color matrices of SU (3) C , while we haveomitted the color indices of the quarks. We use the abbreviation s W and c W for the sineand cosine of the electroweak mixing angle θ W . As usual, q and q (cid:48) are quarks of oppositeisospin and V CKM qq (cid:48) is the corresponding entry of the CKM quark mixing matrix.For the renormalization of the gluon field and the strong coupling we use the M S scheme, modified to decouple the squarks and the gluino [53, 54]. The renormalization– 15 –onstant of the strong coupling is then given by, see also [52], δZ g s = − α s π β (cid:88) ˜ q log (cid:32) m q µ (cid:33) + log (cid:32) m g µ (cid:33) , (A.7)where the sum runs over the twelve squark eigenstates, µ is the renormalization scale,∆ = 1 /(cid:15) − γ E + ln 4 π ; and β = − N − n ˜ q , (A.8)with N = 3 and n ˜ q = 6 for the six squark flavors, includes only the gluino and squarkcontributions. In this case, there is a simple relation between the gluon field and strongcoupling renormalization constants, δZ G = − δZ g s , (A.9)and they actually cancel out in Eq. (A.6) and do not enter our calculation.In the quark sector, we use the on-shell scheme to fix the renormalization constants δZ V,A of the (massless) quarks, i. e. the renormalization constants are obtained by evalu-ating the vector and axial components of the quark self-energy at the on-shell quark mass, δZ V,A = − Σ V,A ( p = 0) . (A.10)The SQCD corrections to the quark self-energy consist of a squark-gluino bubble insertionin the quark line and can be expressed in the following general, compact form [52],Σ V ( p ) = − α s π (cid:88) i (cid:16)(cid:0) g iS (cid:1) + (cid:0) g iP (cid:1) (cid:17) B ( p , m ˜ g , m ˜ q i ) , Σ A ( p ) = α s π (cid:88) i g iS g iP B ( p , m ˜ g , m ˜ q i ) . (A.11)Here, B ( p , m , m ) is the two-point function as defined in [56]. i sums over the twosquark mass eigenstates and g iS and g iP are scalar and pseudoscalar couplings. In the limitof no left-right mixing in the squark sector, g L,RS = ± g L,RP = 1. When consideringthe mixing, then g , S = cos θ ˜ q g L,RS ± sin θ ˜ q g R,LS ,g , P = cos θ ˜ q g L,RP ± sin θ ˜ q g R,LP , (A.12)where θ ˜ q is the squark mixing angle (notation as in [52]).In our numerical study, we neglect the left-right squark mixing, and consider degen-erate squark masses, m ˜ q = m ˜ q = m ˜ q . In this limit, we find Σ A ( p ) = 0 and the onlyrenormalization constant that remains is δZ V , with δZ V = − α s π (cid:40) ∆ + 3 m g − m q m g − m q ) − m g ( m g − m q ) ln (cid:32) m g µ R (cid:33) + m q (2 m g − m q )( m g − m q ) ln (cid:32) m q µ R (cid:33)(cid:41) . (A.13)– 16 – eferences [1] CMS, Measurement of the Drell-Yan Cross Section in pp Collisions at sqrt(s) = 7 TeV , arXiv:1108.0566 .[2] CMS
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