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Effects of azimuth-symmetric acceptance cutoffs on the measured asymmetry in unpolarized Drell-Yan fixed target experiments

Fixed-target unpolarized Drell-Yan experiments often feature an acceptance depending on the polar angle of the lepton tracks in the laboratory frame. Typically leptons are detected in a defined angular range, with a dead zone in the forward region. If the cutoffs imposed by the angular acceptance are independent of the azimuth, at first sight they do not appear dangerous for a measurement of the cos(2\phi)-asymmetry, relevant because of its association with the violation of the Lam-Tung rule and with the Boer-Mulders function. On the contrary, direct simulations show that up to 10 percent asymmetries are produced by these cutoffs. These artificial asymmetries present qualitative features that allow them to mimic the physical ones. They introduce some model-dependence in the measurements of the cos(2\phi)-asymmetry, since a precise reconstruction of the acceptance in the Collins-Soper frame requires a Monte Carlo simulation, that in turn requires some detailed physical input to generate event distributions. Although experiments in the eighties seem to have been aware of this problem, the possibility of using the Boer-Mulders function as an input parameter in the extraction of Transversity has much increased the requirements of precision on this measurement. Our simulations show that the safest approach to these measurements is a strong cutoff on the Collins-Soper polar angle. This reduces statistics, but does not necessarily decrease the precision in a measurement of the Boer-Mulders function.


Introduction
Drell-Yan experiments have a long history (the idea is introduced in [1], a brief introduction to later developments may be found in chapter 5 of [2] and in the reviews [3] and [4], while an extensive data collection has been organized in the Hepdata database [5]. A more modern general scheme may be found in [6]). One of the most interesting and controversial observables measured in unpolarized Drell-Yan is the so-called ν−coefficient [7] associated with the cos(2φ)−azimuthal asymmetry in the rest frame of the dilepton pair. This quantity is relevant because of its association with the violation of the PQCD Lam-Tung relation [7], and with the T-odd Boer-Mulders distribution function [8,9].
The cross section for Drell-Yan dilepton production in scattering of unpolarized hadrons (charged pion, proton or antiproton vs a target nucleus) may be approximately written in the parton model form where W (x 1 , x 2 , Q T ) depends on variables associated with the virtual photon (equivalently: the dilepton pair as an overall). Even if it is not written explicitly, A(θ, φ) depends on x 1 , x 2 , and Q T as well. The angles θ and φ characterize one of the two leptons (since now on: the positive one) in a suitable frame where the virtual photon is at rest. Although different choices exist for such a frame, here we will use the Collins-Soper frame [10].
We have to do with three relevant frames here: (1) The Lab frame, where the target is at rest, (2) the "Collider" frame, i.e. the center of mass frame of the projectile and of the hit nucleon, (3) the Collins-Soper frame.
The angular cutoff that is relevant in this work is imposed on the polar angle of both the leptons in the Laboratory frame. We will indicate such angles as θ Lab . Without suffix, θ and φ will be the angles of the positive lepton in the Collins-Soper frame. The Collider frame is necessary for calculating the longitudinal fractions and some auxiliary variables needed to define the Collins-Soper frame (for any event, we have a different Collins-Soper frame). The difference p 1 − p 2 of the 3-momenta of the colliding hadrons in the Collider frame identifies a direction, that is chosen as the z axis of the Collins-Soper frame. The momentum q of the virtual photon (of the dilepton pair) in the collider frame identifies the xz plane in the Collins-Soper frame. The angles θ and φ are the polar and the azimuthal angles of the positive lepton in this frame.
In unpolarized Drell-Yan, A(θ, φ) may be written as a linear sum, where we focus 1 on the νcos(2φ) term [7]: The cos(2φ)−asymmetry is a "quadrupolar" azimuthal asymmetry. In the laboratory frame, let us name "lepton plane" the plane containing the two lepton momenta q + and q − , and collision plane the one containing the beam axisẑ and the dilepton (or virtual photon) momentum q = q + + q − . In events with positive cos(2φ) the two planes are parallel, in events with negative cos(2φ) the two planes are perpendicular. There is no sensitivity to dipolar (lepton exchange) effects 2 In a fixed target Drell-Yan experiment it is common to have a dead forward cone in the laboratory, and often a dead backward cone, where lepton tracks are invisible or submerged by noise. In other words, if θ Lab is the angle of a lepton in the laboratory frame, it is quite normal to have the acceptance limits θ f orward < θ Lab < θ backward . In the forward direction, the cone occupied by the beam and by the diffraction products of the hadron collisions is a zero acceptance region. For a beam energy of magnitude 100 GeV this normally means a few degrees, decreasing at increasing beam energy.
Although both a forward and a backward cutoff produce effects like the ones discussed in the following, in this work we will limit the discussion to the effects of a forward cutoff. We will also restrict to the beam energy region 10-300 GeV. This region includes the available measurements [11,12,13] of a nonzero value for ν, 3 and some proposals for measuring it in the near future [15,16,17,18].
The key qualitative statement of this work is that a forward dead cone removes preferentially events where the lepton plane and the collision plane are parallel. This may be seen in the very peculiar example shown in fig. 1.
The relevant elements appearing in this figure are: 1) a forward dead cone (with enlarged opening with respect to reality, for 1 Other terms are present contributing to a non-flat φ−distribution, but not directly concerning the following considerations. Also, the parameter λ has been taken at its value 1, that is equivalent to assuming full dominance of Born-1 vector boson exchange. So the present analysis does not extend to e.g. Drell-Yan in the Charmonium mass region. 2 Exchanging the two leptons, nothing changes since cos(2φ) → cos(2φ + 2π). 3 More recently ν has been found compatible with zero by E866 [14], but in a highenergy/small-x regime that makes this measurement peculiar. We will not care such situation here. illustration purposes); 2) two possible lepton pairs (A + A − and B + B − ) with the same q = q + + q − , but different space orientation for the lepton-antilepton plane; 3) q is chosen so to be tangent to the dead cone surface.
In the A + A − case one of the two leptons falls inside the dead cone. Since the pair is accepted only if two opposite-charge lepton tracks are detected, this pair is removed from the event collection of the experiment. In the B + B − case neither B + nor B − is inside the dead cone, so this pair is detected and enters the event collection of the experiment. The A + A − pair has negative cos(2φ), the B + B − pair positive cos(2φ).
Generalizing this example, a forward dead cone introduces a systematic anysotropy with respect to cos(2φ) into the distribution of the detected dilepton pairs. How large is this effect? This will be studied in a systematic way by a suitable MonteCarlo simulation.
In the following, we simulate the effects of a forward dead cone in some typical conditions for a fixed target Drell-Yan experiment withp and π − projectiles, and a target nucleus with Z/A = 0.4. In these simulations, the ν term is constrained to be zero, so any produced cos(2φ) asymmetry originates in statistical fluctuations or in systematic errors.
We will show several simulation results. The first one among the presented sim-ulations ( fig.2) considers a case where there is no forward acceptance cutoff, so that we can have an idea of the magnitude of a false asymmetry due to statistical effects only. The following figures 3 to 6 present examples of simulated measurements where an artificial asymmetry is produced that sometimes is larger than statistical fluctuations. Then we devote the remaining figures and sections 3, 4, and 5 to study the dependence of the false asymmetry on some experimental parameters and some further cutoffs that are often imposed. In particular we show that a (strong) cutoff on polar angle in the Collins-Soper frame reduces the effects of a laboratory frame angular cutoff. This should have "protected" the results of at least one of the past quoted experiments (NA10) from the problem discussed here. The MonteCarlo simulation of Drell-Yan pairs is performed with the same generator code [19,20] and general technique used in [21] and other works. All details about the event generation technique may be found in the related references. A critical discussion of the underlying formalism may be found in [22].
First, 50,000 Drell-Yan µ + µ − pairs produced in antiproton-nucleus collisions have been generated with the kinematic cutoffs described below. To estimate statistical fluctuations this 50,000−event generation has been repeated six times, with different sequencies of random numbers. We name "experiment" a single set of 50,000 events. So e.g., fig. 2 summarizes the information extracted from 6 "experiment repetitions". Each point in the plots is the value of an observable, averaged over the six repetitions, whereas each error bar is the standard deviation that quantifies the corresponding fluctuation. It was tested [21] that the average value and the size of the fluctuations are stable if the number of repetitions is increased over six.
The general simulation technique is the same for figures 2, 3, 4, 5, and 6. What makes each figure peculiar is a given set of kinematic and acceptance details.
In the case of figs. 2, 3, 4 the 50,000 events forming a single experiment satisfy the following conditions: -) The squared c.m. energy is s = 300 GeV 2 .
In these figures, no cutoffs are imposed on the Collins-Soper frame angle θ. The cuts on s, Q and Q T previously described are imposed on the virtual photon before, to allow for the µ + , µ − decay. In each "experiment" the MonteCarlo simulation is stopped after 50,000 events have been sorted with the described cutoffs.
The resulting x distribution and the corresponding asymmetry for the 50,000 events are presented in fig. 2. Figs. 3 and 4 show the same distributions obtained after a further acceptance cutoff on the laboratory angle of any of the individual lepton tracks: This way of calculating the ν−related asymmetry is especially simple, and in absence of any forward dead cone, it would lead to measured asymmetries that are roughly ∼ ν/2. As it is evident from eq.2, ν coincides with (f when this quantity is estimated from a subset of events with θ ≈ 90 o . To include the full phase space (or at least the region |cos(θ)| < 0.8-0.9, as it is usual) means to reduce the resulting asymmetry by about a factor 2. since sin 2 (θ) reduces f + − f − while the factor 1 + cos 2 (θ) increases f + + f − in eq.2.
As previously stressed, all the simulations of this work were performed with ν of events and makes the simulation work much slower. 5 Binning the events with respect to x target is a procedure inspired by those planned single-spin Drell-Yan experiments where the target is polarized and one will measure at the same time unpolarized and single-spin asymmetries with the aim of extracting transversity [23] from the combination of the two measurements [9] = 0, in such way to extract a pure error. So, to compare the estimated errors in the following figures, with the expected "true" asymmetries in a real experiment, one should assume "true asymmetry" ∼ ν/2, meaning asymmetries like 2-10 %. As a reference simulation, fig. 2 has been produced without cutoff on the forward acceptance. The obtained asymmetry is of purely statistical origin, and indeed its values are zero within 1σ (represented by the error bars).
In fig. 2 the overall event number is 50,000. The low number of events at small x is due to the cut Q > 4 GeV, while at large x the decrease of the bin population follows the fact that all parton distributions are small at large x 6 In fig. 3 the cut applied to the forward acceptance is θ Lab± > 0.02 π radians (3.6 o ), i.e. the single leptons are not detected when scattered at angles smaller than 3.6 o in the laboratory frame. At the considered kinematics, a false 5-10 % asymmetry is obtained in the three most populated event bins. This asymmetry is by far larger than the statistical error, and its magnitude makes it competitive with possible asymmetry expectations, and comparable with the measured asymmetries [11,12,13].
For the same s, Q and Q T ranges, different forward angular cutoffs produce 6 For details on the chosen K−factors, on the x and Q T distributions adopted, and on how they fit Drell-Yan data, see [19]). similar false asymmetries for cutoff angles ranging from 0.01π to 0.03π radians (1.8 to 5.4 degrees). For larger angular cutoffs the fake asymmetry disappears, but this coincides with the fact that the event numbers become dramatically small. This is evident in fig. 4, calculated with forward angle cutoff 0.05π = 9 o . The cutoff on the forward angles affects mostly the most populated bins as can be deduced by comparing fig. 3 and fig. 4.
To begin exploring the dependence on Q T , fig. 5 has been calculated with the same parameters as fig. 3 but with a smaller Q T : 0 < Q T < 1 GeV/c 2 . The effect of the cut on Q T appears to be practically negligible. So, at small Q T the fake asymmetry is small.
As a different example, fig. 6 has been calculated for the experimental conditions of a fixed target experiment at FAIR/HESR like the ones of the Panda experiment [15], where the beam energy, the dilepton mass and the transverse momentum are comparatively small. The values considered in the simulations are s = 30 GeV 2 , 1.8 GeV/c 2 < Q < 2.7 GeV/c 2 , 0.5 GeV/c < Q T < 2 GeV/c. The opening angle of the dead forward cone has been set to 0.06 π, i.e. about 11 o . In Panda this angular forward region will be covered by a forward spectrometer [15].
Because of the relatively weak boost effect, the event distributions are shifted towards larger average x. 7 The simulation suggests that the forward cutoff effect could depend critically on the experiment details. In the following we shall try to establish some general trends. All the figures show x−integrated populations and asymmetries. For sake of comparison the integrated asymmetry in the case of fig. 3 is about 0.05. In general, for the simulations performed here the integrated asymmetry is 50-70 % of the peak value reached in the x−distribution.
3 Effects of the forward angular cutoffs -examples:pN at s = 30 GeV 2 and π − N at s = 500 GeV 2 .
In figs. 7 and 8 we show the behavior the integrated asymmetry when the opening angle of the forward dead cone is progressively increased. Fig. 7 refers to the PANDA kinematic conditions (the same used to produce fig. 6) while GeV 2 ) and NA10 [12] running at the maximum beam energy (s = 537 GeV 2 ). The same cut 1 GeV/c < Q T < 2.5 GeV/c was used in both figures.
The effect of the cut is large when the number of events removed by the cutoff is relevant but not very large, i.e. when ∼ 10 − 30 % of the total events are rejected by the cutoff.
For cutoff at larger angles events with positive or negative cos(2φ) signature are going equally removed but the acceptance decreases correspondingly.
Let us consider first the events where the lepton-antilepton invariant mass Q is comparatively small. For these events the the opening angle between the two particles in the laboratory frame is small with respect to the dead cone opening angle. From fig. 1 we guess that in this case the strongest rejection is observed when q is near the surface of the dead cone. When q lies far away from the cone the event is accepted. When q is close to the cone axis most events are lost, regardless to the orientation of the dilepton plane in the laboratory frame. We deduce that the effect of the cutoff must depend strongly on Q T .
Increasing the dilepton invariant mass Q, the dilepton opening angle increases. When it overcomes the dead cone angle, the effect of the dead cone on the acceptance should be reduced. For this reason we also expect a relevant dependence of the cutoff effect on Q/ √ s.

Interaction with other cuts affecting the forward region
Typically in Drell-Yan experiments additional cutoffs in |x F | and in |cos(θ)| are applied to the data, x F being the difference between the longitudinal fractions of the colliding hadrons, and θ is the lepton angle in the Collins-Soper frame. In most cases these variables do not reach their maximum value 1. Since cuts on both parameters reduce the number of forward tracks collected, one may imagine that a cutoff like e.g. cos(θ) < 0.8 could affect the fake asymmetry due the forward dead cone. So we have repeated the previous simulations imposing additional cutoffs on in |x F | and in |cos(θ)|.
The usual cut |x F | < 0.9 does not seem to introduce qualitative differences on the fake asymmetry. Stricter cuts on |x F | are not customary.
More interesting is the effect of a cut on |cos(θ)|, since limits |cos(θ)| < 0.8−0.9 are usually applied in Drell-Yan experiments to remove effects due to the rescattering in the nuclear target (see e.g. the discussion on the NA3 data analysis in [11]). This cutoff is azimuth-symmetric in the Collins-Soper frame, and it may remove events that would be unsymmetrically affected by the dead cone in the laboratory frame. This is evident in fig. 11, where the previous case of a negative pion beam on tungsten at s = 500 GeV 2 with a 1.8 o (0.01 radians) dead cone is simulated eight times, each simulation leading to a different point in fig.11. In each simulation the effect of two overlapping cutoffs is reproduced.
For each point a different cutoff on |cos(θ)| is applied. This set of simulations is organized so to get each time 50,000 events af ter applying the cutoff on |cos(θ)|. After generating 50,000 events with a given cut on |cos(θ)|, in each simulation the effect of a dead cone in the laboratory (0.01 radians = 1.8 o for all points) is added, and this leads to removing some extra events, so that each point in fig.11 summarizes a group of about 45,000−50,000 events. 8 The percentage of the events removed by the latter cut (the forward dead cone) is reported in the lower panel of fig.11. Let us consider the case of negative pion interaction on tungsten at s = 500 GeV 2 , with |cos(θ)| constrained to be lower than 0.8. From fig. 11 we observe that this cutoff reduces the integrated fake asymmetry to 60 % of the maximum value, that is reached when the cut on |cos(θ)| is not applied. In this example, the cut |cos(θ)| < 0.8 rejects 30% events. The additional reduction on the number of accepted events due the forward dead cone comes out to be 4 % only, and to this 4 % we must attribute the residual 2 % false asymmetry effect. It must be remarked that this still means something like ν f alse ∼ 0.5.
From fig. 11 it can be also deduced that a hard |cos(θ)| cutoff assures a clean asymmetry measurement even without model-dependent MonteCarlo corrections, but at the cost of a relevant reduction in the experiment statistics.

Discussion
The fake asymmetries induced by cuts on the acceptance in the forward region feature a behavior very similar to the one of the asymmetries measured in the experiments previously cited: both decrease at increasing Q/ √ s, both increase at increasing Q T . Moreover, the size of the integrated asymmetries is similar. So, the discussion about the measured ν values deserves some attention to the details of the analysis of the experimental data.
With more than 10,000 useful events, and in the same kinematical regime considered in this work (beam energies far below 1000 GeV), ν has been measured by the collaborations NA3 [11], NA10 [12], E615 [13] in the decade 1980-90, leading to results that are similar in magnitude (ν ∼ 0.1 at Q T > 1 GeV/c) and present similar qualitative behaviors, within error bars. Although the related papers quoted here report a quite careful description of the data analysis procedure followed, it is impossible to get from them some detailed information like that nowadays accessible via on-line PhD Theses. However, in at least one paper for each collaboration a figure is presented showing a non-flat acceptance as a function of φ, so it is likely that all the acceptance problems have been implicitly matched.
Because of the limited phase space covered, in the NA10 experiment the |cos(θ)| minimum was set to 0.5 or 0.6, depending on the run (see [12]). The corresponding fake asymmetry is negligible, as it can be deduced from fig. 11. The other experiments previously considered [11,13] do not put such severe constraints on θ, but their results are similar to those from NA10.
Summarizing, we have shown that a forward dead cone in the laboratory frame, in a fixed target unpolarized Drell-Yan experiment in an intermediate  Fig. 11. Effect of an additional cutoff on the Collins-Soper polar angle θ for negative pion on tungsten at s = 500 GeV 2 , 4.5 < Q < 5.5 GeV, 1 < Q T < 2.5 GeV/c. The angular cutoff in the laboratory is 0.01π radians (1.8 o ) as in the previous two figures. Each point corresponds to an additional cutoff on the CS polar angle: |cos(θ)| < M AX(|cos(θ)|). Upper panel: the false asymmetry. error bars from statistics. Lower panel: the percentual of the events removed by the forward cutoff in the laboratory from a sample of 50,000 simulated events selected by the |cos(θ)| cutoff. energy range, seriously distortes a measurement of the Lam-Tung-style lepton asymmetry. This does not seem to be a problem when considering the results of past experiments. It could be a more serious problem for future experiments aiming at much smaller error bars than in the past. The most obvious and model-independent technique to overcome this problem is to artificially reduce the acceptance in terms of Collins-Soper frame variables, but this has a relevant cost in terms of final available statistics.