MADPH–96–945 UCD–96–17 May 1996
arXiv:hep-ph/9605430v1 29 May 1996
Signals for Double Parton Scattering at the Fermilab Tevatron
Manuel Dreesa and Tao Hanb
Physics Department, University of Wisconsin, 1150 University Ave., Madison, WI 53706, USA b Department of Physics, University of California, Davis, CA 95616, USA
Abstract Four double-parton scattering processes are examined at the Fermilab Tevatron energy. With optimized kinematical cuts and realistic parton level simulation for both signals and backgrounds, we ?nd large samples of four-jet and three-jet+one-photon events with signal to background ratio being 20%-30%, and much cleaner signals from two-jet+two-photon and two-jet+e+ e? ?nal states. The last channel may provide the ?rst unambiguous observation of multiple parton interactions, even with the existing data sample accumulated by the Tevatron collider experiments.
There are good reasons to believe that multiple partonic interactions, where two or more pairs of partons scatter o? each other, occur in many, or even most, p? collisions at the p √ Tevatron ( s = 1.8 TeV). On the theoretical side, multiple partonic interactions are an integral part of the eikonalized minijet model , which attempts to describe the observed increase of the total p? cross section with energy in terms of the rapidly growing cross section p for the production of (mini)jets with transverse momentum pT ≥ pT,min ? 2 GeV. Sj¨strand o and van Zijl  also pointed out that including multiple interactions in the PYTHIA event generator greatly improves the description of the “underlying event” in p? collisions. A p similar result was found recently by the H1 collaboration  in a study of γp collisions. However, hadronic event generators have many ingredients. This makes it di?cult to draw unambiguous conclusions from such studies. It is therefore desirable to search for more direct evidence for multiple partonic interactions, using ?nal states that are amenable to a perturbative treatment. Clearly the cross section will be largest if only strong interactions are involved. The simplest signal of this kind is the production of four high?pT jets in independent partonic scatters within the same p? collision  (4 → 4 reactions). Since energy p and momentum are assumed to be conserved independently in each partonic collision, the signal for a 4 → 4 reaction is two pairs of jets with the members of each pair having equal and opposite transverse momentum. Various hadron collider experiments have searched for this signature. The AFS collaboration at the CERN ISR reported  a strong signal. However, the exact matrix elements for the QCD background 2 → 4 processes were not used and the size of the signal claimed was considerably larger than expected. The UA2 collaboration at the CERN SppS collider saw a hint of a signal, but preferred to only quote an upper bound . More recently, the CDF collaboration at the Fermilab Tevatron found evidence at the 2.5σ level that 4 → 4 processes contribute about 5% to the production of four jets with pT ≥ 25 GeV . While ?nal states consisting only of jets o?er large cross sections, they su?er from severe backgrounds. There are three possible ways to group four jets into two pairs. Further, the experimental error on the energy of jets with pT ? 20 GeV is quite large. Hence even fourjet events that result from 2 → 4 background processes often contain two pairs of jets with transverse momenta that are equal and opposite within the experimental errors. The study of “cleaner” ?nal states has therefore been advocated: The production of two pairs of leptons (double Drell–Yan production) has been studied in refs., the production of two J/ψ mesons in refs., and the production of a W boson and a pair of jets in refs.. However, in our opinion none of these processes is ideally suited for studying multiple partonic interactions. Double Drell–Yan production o?ers a very clean ?nal state, but the cross section at Tevatron energies is very small once simple acceptance cuts have been applied. The cross section for double J/ψ production is quite uncertain, since it depends on several poorly known hadronic matrix elements . Finally, W +jets events can only be identi?ed if the W boson decays leptonically, which makes it impossible to fully reconstruct the ?nal state. Here we study mixed strong and electroweak ?nal states: ? three jets and an isolated photon (jjjγ); ? two jets and two isolated photons (jjγγ); ? two jets and an e+ e? pair (jjee). 1
For comparison, we also include ? four-jet ?nal states (denoted by 4-jet). We try to be as close to experiment as possible within a parton level calculation. To this end we not only apply acceptance cuts, but also allow for ?nite energy resolution, and try to model transverse momentum “kicks” due to initial and ?nal state radiation. We ?nd that the jjjγ ?nal state o?ers an only slightly better signal to background ratio than the 4-jet ?nal state does; note that the combinatorial background is the same in these two cases. This combinatorical background does not exist for the jjγγ and jjee ?nal states, which o?er much better signal to noise ratios, at the price of small cross sections. The calculation of our signal cross sections is based on the standard assumption [1, 2, 8, 9, 10] that the two partonic interactions occur independently of each other. The cross section for a 4 → 4 process is then simply proportional to the square of the 2 → 2 cross section: σ(4 → 4) = [σ(2 → 2)]2 /σ0 . (1)
This assumption cannot be entirely correct, since energy–momentum conservation restricts the available range of Bjorken?x values of the second interaction, depending on the x values of the ?rst one. We include this (small) e?ect using the prescription of ref.. In the eikonalized minijet model  σ0 is related to the transverse distribution of partons in the proton. Unfortunately total cross section data do not allow to determine this quantity very precisely. We ?nd values between about 20 and 60 mb, depending on the choice of the numerous free parameters of the model. The recent CDF study  found σ0 = 24.2+21.4 mb, ?10.8 within the range that can be accommodated in minijet models. We will take σ0 = 30 mb in our numerical analysis; the results can be scaled trivially to other values of σ0 . The for us relevant 2 → 2 cross section can be written as a sum of di?erent terms: σ(2 → 2) = σ(p? → jjX) + σ(p? → jγX) + σ(p? → γγX) + σ(p? → e+ e? X), p p p p (2)
where j stands for a high?pT jet. Inserting eq.(2) into eq.(1) gives a 4 → 4 cross section that sums over many di?erent states; it should be obvious which terms in the sum are of relevance to us. Note that this procedure gives an extra factor of 2 in the cross section for the production of ?nal states made up from two di?erent 2 → 2 reactions (e.g, jjjγ) compared to those produced from two identical reactions. Partly for this reason we only consider jjγγ con?gurations where the two jets are produced in one partonic scatter and the two photons in another. The other possible con?guration (jγjγ), where each jet pairs up with one photon, also su?ers from larger backgrounds, since there are two ways to form such pairs. We use leading order matrix elements in eq.(2), but we include the contribution √ from gg → γγ, which enhances the total p? → γγX cross section by about 50% at s = 1.8 p TeV. We take MRSA’ structure functions ; other modern parametrizations give very similar results. We use the leading order expression for αs , with ΛQCD = 0.2 GeV, and take the (average) partonic pT as factorization and renormalization scale. We use exact leading order matrix elements to compute the backgrounds from 2 → 4 processes. These have been computed in ref. for the 4-jet ?nal state, in ref. for the jjjγ ?nal state, in ref. for jjγγ production, and in ref. for jjee production. 2
also require the isolation cut ?Rij ≡ (yi ? yj )2 + (φi ? φj )2 ≥ 0.7 for all combinations ij of ?nal state particles. We generally ?nd that the 4 → 4 signal decreases more quickly than the 2 → 4 background when the (transverse) momentum of the outgoing particles is increased. The reason is that the signal cross section contains four factors of parton densities, while the background only has two. We therefore try to keep the minimal acceptable pT as small as possible, subject to the constraint that the event can still be triggered on. Speci?cally, we chose i) for 4-jet: pT (j1 , j2 ) ≥ 20 GeV, pT (j3 , j4 ) ≥ 10 GeV ii) for jjjγ: pT (γ, j1 ) ≥ 15 GeV, pT (j2 , j3 ) ≥ 10 GeV; iii) for jjγγ: pT (γ1 , γ2 , j1 , j2 ) ≥ 10 GeV; iv) for jjee: pT (e1 , e2 ) ≥ 15 GeV, pT (j1 , j2 ) ≥ 10 GeV. The signal and background cross sections with only these basic acceptance cuts included are listed in column 2 of Table 1 for the 4-jet and jjjγ ?nal states, and Table 2 for the jjγγ and jjee ?nal states. We see that without further cuts, 4 → 4 processes only contribute between 7% (4-jet) and 18% (jjee), so additional cuts are clearly needed to extract the signal. As expected from our previous discussion, the signal to background ratio is worst for the 4-jet ?nal state. As mentioned earlier, in 4 → 4 processes two pairs of particles are produced with equal and opposite transverse momenta, pT (1) = ?pT (2) and pT (3) = ?pT (4). However, additional radiation can change the kinematics signi?cantly, and the ?nite resolution of real detectors means that we can require momenta to be equal only within the experimental uncertainty. In the presence of initial or ?nal state radiation the transverse momenta within a pair no longer balance exactly even if the resolution was perfect. We include this e?ect only for the signal, since in the background the ?nal state particles in any case only pair up “accidentally”; we do therefore not expect large e?ects on the backgrounds. We randomly generate transverse “kicks” for each of the 2 → 2 processes in the signal. We assume that the direction of the kick is not correlated with the plane of the hard scattering. The absolute values qT of these additional transverse momenta are generated according to the distribution
2 f (qT ) ∝ exp ? (q0 /qT )0.7 /qT ,
In order to approximately mimic the acceptance of the CDF and D0 detectors, we require all jets to have rapidity |yjet | ≤ 3.5, while we require |ye,γ | ≤ 2.5 for electrons and photons. We
with 0 < qT ≤ qT,max . This function describes the transverse momentum distribution  √ of W bosons produced at s = 1.8 TeV quite well, with q0 = 9 GeV. We adopt this choice of q0 for the jjee ?nal state, which is dominated by the production of real Z bosons, but use the smaller value q0 = 4.5 GeV for the other ?nal states, which are characterized by a smaller momentum scale. Finally, we take qT,max = 8 GeV as our default value; this assumes that one can reliably veto against jets with transverse momentum exceeding this value. We simulate ?nite energy resolutions by ?uctuating the energies of all outgoing particles (keeping the 4–vectors light–like), using Gaussian smearing functions. The width of the Gaussian is given by √ δ(E) = a · E ⊕ b · E, (4) 3
where ⊕ stands for addition in quadrature and E is in GeV. We take ajet = 0.80, bjet = 0.05, ae,γ = 0.20, be,γ = 0.01, (5) which roughly corresponds to the performance of the CDF detector. We do not ?uctuate the directions of the outgoing particles in this step. These are, however, a?ected by the transverse “kicks” mentioned earlier. For this reason, and in order to allow for an error in the determination of jet axes, we apply a relatively mild cut on the azimuthal opening angle of each pair: cos(φi ? φj ) ≤ ?0.9. (6)
This allows an opening angle as small as 154? . As emphasized earlier, in 4 → 4 processes, the members of a pair should also have equal absolute values of pT . As our ?nal cut, we therefore require ||pT (i)| ? |pT (j)|| ≤ cij δ 2 [|pT (i)] + δ 2 [|pT (j)|], (7) with δ(|pT |) = a · |pT | ⊕ b · |pT | as in eqs.(4) and (5). Our results for signal and background with these additional cuts included are summarized in the Tables. For the 4-jet and jjjγ ?nal states (Table 1) we always take c12 = c34 ≡ c, but we occasionally allow cee,γγ > cjj in the jjee and jjγγ ?nal states. The reason is that the cut (7) is much more severe for e+ e? and γγ pairs than for jet pairs, due to the better resolution of electromagnetic calorimeters, see eq.(5). Inclusion of the transverse “kick” therefore leads to a signi?cant loss of signal if we take cee,γγ = 1. Although the stronger cut still gives a slightly better signal to noise ratio, given the limited available event sample employing a looser cut might give a statistically more signi?cant signal. We do not attempt to quantify this statement here, since we have not included any reconstruction e?ciencies in our calculation. Finally, in the last three columns of Table 1 we increase the cut on ?Rij from 0.7 to 1.2. This enhances the signal to background ratio by about 20 to 25%. Switching on energy smearing and transverse momentum kicks, and imposing the cuts (6) and (7) with c = 5, reduces the signal by typically a factor of 2. This reduction is almost entirely due to the energy smearing. Ignoring the transverse kicks for the moment, in the signal both members of a pair have equal |pT |. If it falls below the cut–o? value, both energies have to ?uctuate upwards for the event to be accepted. In contrast, the downwards ?uctuation of one energy can be su?cient to remove an event from the sample. The reduction is smaller for jjee production since most electrons have typically pT ? MZ /2, well above the lower limit. Fortunately the background is reduced even more in this step, by a factor of 4 for 4-jet and jjjγ and 9 for jjee and jjγγ ?nal states, mainly due to the cut (6). Making the cut (7) stricter, i.e. decreasing c, only slightly enhances the signal to background ratio in Table 1. This is partly due to the transverse kicks. Without them, the jjjγ signal for c = 1.0 would be about 50% larger. This indicates that restricting additional jet activity as much as possible is quite important. Although in Table 1 the optimized S/B ratios are only about 0.23 for 4-jet and 0.31 for jjjγ, the signals are statistically quite signi?cant; recall that the CDF and D0 experiments together have accumulated about 200 pb?1 of data. We do not attempt to further optimize the S/B ratio for these two processes because we do not trust our parton level analysis, with a simpli?ed treatment of ?nite detector resolutions and the e?ect of parton showering, suf?ciently to extrapolate into the tails of distributions. Nevertheless, given the normalization 4
uncertainties of leading order QCD predictions, one will have to study the shapes of various distributions, such as the opening angle cos φij , ?Rij and pT balancing etc. in order to convince oneself that a signal is indeed present. Clearly the S/B ratio is much more favorable for the jjee and jjγγ ?nal states (Table 2). For these ?nal states reducing c from its starting point c = 5 does increase this ratio signi?cantly. Recall that for a ?xed value of c the cut (7) is much more restrictive for e+ e? and γγ pairs than for jj pairs; this reduces the background more than the signal. On the other hand, this also has the e?ect that after imposing the cut (7) with cjj = cγγ = 1, the size of the jjγγ signal depends quite sensitively on the treatment of the transverse kick. Had we used q0 = 9 GeV in eq.(3), as appropriate for W production, the signal would have been reduced by a factor of about 0.7, while without any transverse kick it would have been larger by a factor 1.6. Clearly this uncertainty can be reduced by using the actual measured pT distribution of γγ pairs produced at the Tevatron. Fortunately the jjee signal is less sensitive to the “kick”, since the electrons are usually so hard that adding or subtracting a few GeV does not matter very much. This ?nal state therefore o?ers our most promising and robust signal. In summary, we have studied four di?erent ?nal states with a view of establishing an unambiguous signal for multiple partonic interactions in p? collisions at the Tevatron. The p 4-jet and jjjγ ?nal states o?er very large event samples, but with a S/B ratio about 0.2 0.3. One must study the shapes of various kinematical distributions for con?rmation of the existence of the signal, as was indeed done by the CDF collaboration in their study of the 4-jet ?nal state . The situation is much more favorable for the jjγγ and, especially, jjee ?nal states; in the latter case one can increase the event sample by including muon pairs as well. Although even in these channels the signal to noise ratio is less favorable than what we found for four-jet production in γγ collisions , a clear signal should be visible already in the present data sample. Once a signal is found, it would be important to establish if the normalization σ0 in eq.(1) is indeed the same for di?erent processes, and independent of the Bjorken?x range probed, as assumed in minijet models. Further, it would be very interesting to reduce the pT cut for at least some of the jets as much as possible, so that one can get closer to the actual minijet region. This could greatly enhance our understanding of “minimum bias” physics, and give us some con?dence that we can trust extrapolations to LHC energies, where the understanding of overlapping minimum bias events becomes a crucial issue in the assessment of the viability of various “new physics” signals. Finally, such studies might shed new light on the thirty-year old problem of the rising total hadronic cross sections.
We thank Walter Giele for sending us a computer code based on the results of ref.. We also thank J. Huston for information on experimental capabilities, and H. Baer for discussions of the transverse kick. The work of M.D. was supported in part by the U.S. Department of Energy under grant No. DE-FG02-95ER40896, by the Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, as well as by a grant from the Deutsche Forschungsgemeinschaft under the Heisenberg program. T.H. was supported in part by DOE under grant DE–FG03–91ER40674.
 D. Cline, F. Halzen and J. Luthe, Phys. Rev. Lett. 31, 491 (1973); L. Durand and H. Pi, Phys. Rev. Lett. 58, 303 (1987); M.M. Block, F. Halzen and B. Margolis, Phys. Rev. D45, 839 (1992).  T. Sj¨strand and M. van Zijl, Phys. Rev. D36, 2019 (1987). o  H1 collab., S. Aid et al., Z. Phys. C70, 17 (1996).  P.V. Landsho? and J.C. Polkinghorne, Phys. Rev. D18, 3344 (1978); C. Goebel, F. Halzen and D.M. Scott, Phys. Rev. D22, 2789 (1980); N. Paver and D. Treleani, Z. Phys. C28, 187 (1985); B. Humpert and R. Odorico, Phys. Lett. 154B, 211 (1985).  AFS collab., T. Akesson et al., Z. Phys. C34, 163 (1987).  UA2 collab., J. Alitti et al., Phys. Lett. B268, 145 (1991).  CDF collab., F. Abe et al., Phys. Rev. D47, 4857 (1993).  M. Mekh?, Phys. Rev. D32, 2371 (1985); F. Halzen, P. Hoyer and W.J. Stirling, Phys. Lett. 188B, 375 (1988).  R. Ecclestone and D.M. Scott, Z. Phys. C19, 29 (1983); R.W. Robinett, Phys. Lett. B230, 153 (1989).  B. Humpert, Phys. Lett. 135B, 179 (1984); R.M. Godbole, S. Gupta and J. Lindfors, Z. Phys. C47, 69 (1990).  G.T. Bodwin, E. Braaten and G.P. Lepage, Phys. Rev. D51, 1125 (1995).  A. Martin, R.G. Roberts and W.J. Stirling, Phys. Rev. D50, 6734 (1994).  F.A. Berends, W.T. Giele and H. Kuijf, Phys. Lett. B232, 266 (1989); F.A. Berends and H. Kuijf, Nucl. Phys. B353, 59 (1991).  V. Barger, T. Han, J. Ohnemus and D. Zeppenfeld, Phys. Lett. B232, 371 (1989).  V. Barger, T. Han, J. Ohnemus and D. Zeppenfeld, Phys. Rev. D41, 2782 (1990).  V. Barger, T. Han, J. Ohnemus and D. Zeppenfeld, Phys. Rev. Lett. 62, 1971 (1989), and Phys. Rev. D40, 2888 (1989).  H. Baer and M.H. Reno, Phys. Rev. D45, 1503 (1992).  M. Drees and T. Han, Phys. Rev. Lett. 76, 3076 (1996).
Table 1: Signal and background cross sections, as well as their ratios (S/B), for 4-jet production (in nb) and jjjγ production (in pb) at the Tevatron. In the ?rst column only the basic acceptance cuts on the transverse momenta, rapidities and on ?Rij have been applied. In the second column we in addition apply the cuts (6) and (7), with c = 5. In the last three columns we sharpen the ?R cut to ?Rij ≥ 1.2, and gradually reduce c as indicated. Note that the “basic” cross sections have been computed ignoring ?nite energy resolution and transverse “kicks”; these e?ects have been included in the other columns, as described in the text. basic 266 3,990 0.067 515 5,370 0.096 ?Rij ≥ 0.7, ?Rij ≥ 1.2 c=5 c=5 c=2 c=1 131 91 87 57 878 485 442 246 0.15 0.19 0.20 0.23 265 169 158 97 1,310 611 571 311 0.20 0.28 0.28 0.31
σ(4j)(S) σ(4j)(B) S/B σ(jjjγ)(S) σ(jjjγ)(B) S/B
Table 2: Signal and background cross sections in pb, as well as their ratios, for jjγγ production and jje+ e? production at the Tevatron. The notation is as in Table 1, except that we use the basic isolation cut ?Rij ≥ 0.7 everywhere, and allow di?erent values for c1 ≡ cjj and c2 ≡ cee or cγγ . σ(jjγγ)(S) σ(jjγγ)(B) S/B σ(jjee)(S) σ(jjee)(B) S/B basic c1 1.86 20.8 0.089 3.45 19.0 0.18 = c2 = 5 c1 0.96 2.34 0.41 2.01 1.94 1.04 = c2 = 2 0.71 1.16 0.61 1.42 1.00 1.42 c1 = 1, c2 = 2 c1 0.59 0.94 0.63 1.07 0.70 1.53 = c2 = 1 0.37 0.52 0.71 0.62 0.37 1.68