Studies of $X(3872)$ and $\psi(2S)$ production in $p\bar{p}$ collisions at 1.96 TeV

We present various properties of the production of the $X(3872)$ and $\psi(2S)$ states based on 10.4 fb$^{-1}$ collected by D0 experiment in Tevatron $p \bar p$ collisions at $\sqrt{s}$ = 1.96 TeV. For both states, we measure the nonprompt fraction $f_{NP}$ of the inclusive production rate due to decays of $b$-flavored hadrons. We find the $f_{NP}$ values systematically below those obtained at the LHC. The $f_{NP}$ fraction for $\psi(2S)$ increases with transverse momentum whereas for the X(3872) it is constant within large uncertainties, in agreement with the LHC results. The ratio of prompt to nonprompt $\psi(2S)$ production, $(1 - f_{NP}) / f_{NP}$, decreases only slightly going from the Tevatron to the LHC, but for the $X(3872)$ this ratio decreases by a factor of about 3. We test the soft-pion signature of the $X(3872)$ modeled as a weakly-bound charm-meson pair by studying the production of the $X(3872)$ as a function of the kinetic energy of the $X(3872)$ and the pion in the $X(3872)\pi$ center-of-mass frame. For a subsample consistent with prompt production, the results are incompatible with a strong enhancement in the production of the $X(3872)$ at small kinetic energy of the $X(3872)$ and the $\pi$ in the $X(3872)\pi$ center-of-mass frame expected for the $X$+soft-pion production mechanism. For events consistent with being due to decays of $b$ hadrons, there is no significant evidence for the soft-pion effect but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale $\Lambda=M(\pi)$ is not ruled out.

We present various properties of the production of the X(3872) and ψ(2S) states based on 10.4 fb −1 collected by D0 experiment in Tevatron pp collisions at √ s = 1.96 TeV. For both states, we measure the nonprompt fraction fNP of the inclusive production rate due to decays of b-flavored hadrons. We find the fNP values systematically below those obtained at the LHC. The fNP fraction for ψ(2S) increases with transverse momentum whereas for the X(3872) it is constant within large uncertainties, in agreement with the LHC results. The ratio of prompt to nonprompt ψ(2S) production, (1 − fNP )/fNP , decreases only slightly going from the Tevatron to the LHC, but for the X(3872) this ratio decreases by a factor of about 3. We test the soft-pion signature of the X(3872) modeled as a weakly-bound charm-meson pair by studying the production of the X(3872) as a function of the kinetic energy of the X(3872) and the pion in the X(3872)π center-of-mass frame. For a subsample consistent with prompt production, the results are incompatible with a strong enhancement in the production of the X(3872) at small kinetic energy of the X(3872) and the π in the X(3872)π center-of-mass frame expected for the X+soft-pion production mechanism. For events consistent with being due to decays of b hadrons, there is no significant evidence for the soft-pion effect but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale Λ = M (π) is not ruled out.

I. INTRODUCTION
Fifteen years after the discovery of the state X(3872) [1] (also named χ c1 (3872) [2]) its nature is still * with visitors from a Augustana College, Sioux debated. Its proximity to the D 0D * 0 threshold suggests a charm-meson molecule loosely bound by the pion exchange potential, first suggested by Tornqvist [3]. The molecular model also explains the isospin breaking decay to J/ψρ that is not allowed for a pure charmonium state. However, the copious prompt production of the X(3872) at hadron colliders has been used as an argument against a pure molecule interpretation [4]. With the binding energy less than 1 MeV, the average distance between the two components is a few fm. It has been argued that production of such an extended object in the hadron collision environment is strongly disfavored and is better described by a compact charm-anticharm or diquarkantidiquark structure. Meng, Gao and Chao [5] proposed that the X(3872) is a mixture of the conventional charmonium state χ c1 (2P ) and a D 0D * 0 molecule. In this picture, the short-distance production proceeds through the χ c1 (2P ) component while the D 0D * 0 component is responsible for hadronic decays. An evaluation of the production cross section of the X(3872) [6] through its χ c1 (2P ) component gives a good description of the dif-ferential cross section for prompt production of X(3872) measured by CMS [7] and ATLAS [8].
Recently Braaten et al. [9,10] have revised the calculation of the production of the X(3872) under the purely molecular hypothesis by taking into account formation of D * D * at short distances followed by rescattering of the charm mesons onto Xπ. According to the authors, such process should be easily observable at hadron colliders as an increased event rate at small values of the kinetic energy T (Xπ) of the X(3872) and the "soft" pion in the X(3872)π center-of-mass frame and should provide a clean test of the molecular structure of the X(3872).
In this article we present production properties of the X(3872) in Tevatron pp collisions at the energy √ s = 1.96 TeV and compare them with those of the conventional charmonium state ψ(2S). Section II describes relevant experimental details and the event selections. In Section III we present the the transverse momentum p T and pseudorapidity η dependence of the fraction f N P of the inclusive production rate due to nonprompt decays of b-flavored hadrons. In Section IV we study the hadronic activity around the X(3872) and ψ(2S). We also test the soft-pion signature of the X(3872) as a weakly-bound charm-meson pair by studying the production of X(3872) plus a co-moving pion at small T (Xπ). As a control process, we use the production of the charmonium state ψ(2S) for which this production mechanism does not apply. We summarize the findings in Section V.

II. THE D0 DETECTOR, EVENT RECONSTRUCTION AND SELECTION
The D0 detector has a central tracking system consisting of a silicon microstrip tracker and the central fiber tracker, both located within a 1.9 T superconducting solenoidal magnet [11,12]. A muon system, covering the pseudorapidity interval |η| < 2 [13], consists of a layer of tracking detectors and scintillation trigger counters in front of 1.8 T iron toroidal magnets, followed by two similar layers after the toroids [14]. Events used in this analysis are collected with both single-muon and dimuon triggers. Single-muon triggers require a coincidence of signals in trigger elements inside and outside the toroidal magnets. All dimuon triggers require at least one muon to have track segments after the toroid; muons in the forward region are always required to penetrate the toroid. The minimum muon transverse momentum is 1.5 GeV. No minimum p T requirement is applied to the muon pair, but the effective threshold is approximately 4 GeV due to the requirement for muons to penetrate the toroids, and the average value for accepted events is 10 GeV.
We select two samples, referred to as 4-track and 5track selections. To select 4-track candidates, we reconstruct J/ψ → µ + µ − decay candidates accompanied by two particles of opposite charge assumed to be pions, with transverse momentum p T with respect to the beam axis greater than 0.5 GeV. We perform a kinematic fit under the hypothesis that the muons come from the J/ψ and the J/ψ and the two particles originate from the same space point. In the fit, the dimuon invariant mass is constrained to the world-average value of the J/ψ meson mass [2]. The track parameters (p T and position and direction in three dimensions) readjusted according to the fit are used in the calculation of the invariant mass M (J/ψπ + π − ) and the decay length vector L xy , which is the transverse projection of the vector directed from the primary vertex to the J/ψπ + π − production vertex. The two-pion mass for each accepted J/ψπ + π − candidate is required to be greater than 0.35 GeV (0.5 GeV) for ψ(2S) (X(3872)) candidates. These conditions have a signal acceptance of more than 99% while reducing combinatorial background. The transverse momentum of the J/ψπ + π − system is required to be greater than 7 GeV. All tracks in a given event are considered and all combinations of tracks satisfying the conditions stated are kept. The mass windows 3.62 < M (J/ψπ + π − ) < 3.78 GeV and 3.75 < M (J/ψπ + π − ) < 4.0 GeV are used for ψ(2S) and X(3872) selections, respectively. The rates of multiple entries within these ranges are less than 10%.
Fits to the M (J/ψπ + π − ) distribution for the 4-track selection are shown in Fig. 1. In the fits, the signal is modeled by a Gaussian function with a free mass and width. Background is described by a fourth-order Chebyshev polynomial. The fits yield 126, 891 ± 770 and 16, 423 ± 1031 events of ψ(2S) and X(3872), with mass parameters of 3684.88±0.07 MeV and 3871.0±0.2 MeV, and mass resolutions of 9.7±0.1 MeV and 16.7±0.9 MeV, respectively. These mass resolutions are used in all subsequent fits.
For the 5-track sample, we require the presence of an additional charged particle with p T > 0.5 GeV, consistent with coming from the same vertex. We assume it to be a pion and set a mass limit M (J/ψπ + π + π − ) < 4.8 GeV. Charge-conjugate processes are implied throughout this article. To further reduce background, we allow up to two sets of three hadronic tracks per event, with an additional requirement that M (J/ψπ + π − ) be less than 4 GeV. With up to two accepted J/ψπ + π − combinations per set, there are up to four accepted combinations per event. Because tracks are ordered by descending p T , this procedure selects the highest-p T tracks of each charge. Fits to the M (J/ψπ + π − ) distribution for the 5-track selection are shown in Fig. 2. The fits yield 75406±1435 and 8192±671 signal events of ψ(2S) and X(3872). The 5-track sample is used in the studies presented in Section IV. In this Section we study the pseudo-proper time distributions for the charmonium states ψ(2S) and X(3872) using the 4-track sample. These states can originate from the primary pp interaction vertex (prompt production), or they can originate from a displaced secondary vertex corresponding to a beauty hadron decay (nonprompt production). The pseudo-proper time t pp is calculated using the formula t pp = L xy · p T m/(p 2 T c), where p T and m are the transverse momentum and mass of the charmonium state ψ(2S) or X(3872) expressed in natural units and c is the speed of light. We note that the true lifetimes of b hadrons decaying to ψ(2S) or X(3872) mesons are slightly different from the pseudo-proper time values obtained from the formula because the boost factor of the charmonium is not exactly equal to the boost factor of the parent. Therefore, the nonprompt pseudo-proper charmonium time distributions will have effective exponential lifetime values, which are close to but not equal to the lifetime for an admixture of B 0 , B − , B 0 s , B − c mesons, and b baryons.
To obtain the t pp distributions, the numbers of events are extracted from fits for the ψ(2S) and X(3872) signals in mass distributions. This method removes combinatorial backgrounds and yields background-subtracted numbers of ψ(2S) or X(3872) signal events produced in each time interval. The bin width of the pseudo-proper time distributions is chosen to increase exponentially to reflect the exponential shape of the lifetime distributions.
The fit function used to describe the ψ(2S) mass distribution includes two terms: a single Gaussian used to model the signal and a third-order Chebyshev polynomial used to describe background. In the specific p T and η intervals the statistics in some t pp bins may be insufficient for the fit to converge. In the case of a low number of background events, a second-order or a first-order Chebyshev polynomial is used. If the number of signal events is small, the signal Gaussian mass and width are fixed to the cental values obtained in the fit to the distribution including all accepted events. Possible variations in the parameters appearing in this approach are estimated and are included in the systematic uncertainty.
The t pp distribution for the ψ(2S) sample is shown in Fig. 3. The numbers of events/0.0207 ps shown in Fig. 3 are obtained from fits to the mass distribution and corrected to the bin center to account for the steeply falling distribution. The bin width 0.0207 ps resulted from 0.05 ps bin divided in 2 bins by the exponential factor 1/(1+ √ 2). To correct to the bin center, the values in the bin center and the bin integral are calculated for the fitting function. The obtained t pp distributions include prompt and nonprompt contributions. The prompt production is assumed to have a strictly zero lifetime, whereas the nonprompt component is assumed to be distributed exponentially starting from zero. These ideal signal distributions are smeared by the detector vertex resolution. The shape of the smearing function is expected to be the same for prompt and nonprompt production. Negative time values are possible due to the detector resolution of primary and secondary vertices. The pseudo-proper time distribution parameterization method is similar to that used in the ATLAS analysis [8]. For the ψ(2S) sample the t pp distributions are fitted using the χ 2 method with a model that includes prompt and nonprompt components: Here N is a free normalization factor, f N P is a free parameter corresponding to the nonprompt contribution fraction, and F P (t) and F N P (t) are the shapes of the prompt and nonprompt components. The shape of the prompt component is modeled by a sum of three Gaussian functions with zero means and free normalizations and widths: where g 1 , g 2 and g 3 are normalization parameters and G 1 , G 2 and G 3 are Gaussian functions. The ψ(2S) time distribution fit yields the three Gaussian widths σ 1 = 0.0476 ± 0.0016 ps, σ 2 = 0.1059 ± 0.0047 ps, and σ 3 = 0.264 ± 0.021 ps, and the relative normalization factors g 1 = 0.491 ± 0.035, g 2 = 0.447 ± 0.039, and g 3 = 0.062 ± 0.013. The shape of the nonprompt function F N P (t) includes two terms, a short-lived (SL) component and a long-lived (LL) component: ( 3) The f SL is a free parameter in the fit. The long-lived and short-lived shape functions F LL (t) and F SL (t) are described by single exponential functions with slopes τ LL and τ SL , convolved with the resolution shape function that is the same as for the prompt component: The long-lived component corresponds to charmonium production from B 0 , B + , B 0 s , and other b hadron decays, whereas the short-lived component is due to the B + c decays. The production rate of the B + c mesons in the pp collisions at 1.96 TeV is not well known. Theoretically the ratio of B + c meson production over all bhadrons is expected to be about 0.1-0.2 % [2]. However the production ratio of B + c to B + mesons has been measured by CDF [15] and an unexpectedly large value for this ratio between 0.9 % and 1.9 % was obtained; this ratio was calculated using theoretical predictions for the branching fraction B(B + c → J/ψµ + ν) to be in the range 1.15-2.37 % [15]. Assuming that the ψ(2S) production rate in B + c decays is enhanced by a factor of ∼ 20 compared with B + , B 0 and B 0 s decays, we expect a value of f SL in the range of about 0.08-0.15. Such factor can be estimated taking into account that the B + c meson decays to charmonium states via the dominant "tree" diagram, whereas other B hadrons produce charmonium via the "color-suppressed" diagram. On the other hand, the short-lived component f SL was measured by ATLAS [8] in pp collisions at the center-of-mass (CM) energy 8 TeV and a value of a few percent was obtained for ψ(2S) and of 0.25 ± 0.13 ± 0.05 for X(3872). Because of the range of possible values we include the short-lived term with a free normalization in the lifetime fit for the ψ(2S) sample.
The t pp distribution of the ψ(2S) sample shown in Fig. 3 is well described by the function discussed above, where the exponential dependence is clearly seen in the large-time region. The fit quality is reasonably good, χ 2 /NDF = 24.5/14, corresponding to a p-value of 4 %. This fit quality is adequate in view of the large range of numbers of events per bin and the simplicity of the pseudo-proper time fitting function. The fitted value of the short-lived component is f SL = 0.218 ± 0.025. If the short-lived component is neglected, a significantly larger value of χ 2 = 112 is obtained. The parameters obtained from the fit shown in Fig. 3 are listed in Table I. A similar method is used to obtain the pseudo-proper time distribution for the X(3872) sample. The numbers  Fig. 4. Because the number of X(3872) events is an order of magnitude smaller and combinatorial background under the signal is slightly larger than for the ψ(2S) sample, the number of t pp bins for the mass fits is reduced from 24 to 12. The following assumptions are applied in the fit procedure: the vertex reconstruction resolution is the same for the X(3872) and ψ(2S) states and the short-lived and long-lived component lifetimes and relative rates are fixed for the X(3872) to the values obtained from the ψ(2S) fit. These assumptions are based on similarity in production kinematics and an only 5 % difference in the mass of these states. The relative short-lived and long-lived rate is expected to be similar, if the ratio of inclusive branching fractions from the B + c and other B hadrons is similar for the X(3872) and ψ(2S) states. The uncertainties of these assumptions are estimated and included in systematics. These systematic uncertainties are significantly smaller than the statistical uncertainties, because the f N P values for X(3872) are small and the statistical uncertainties are large. Therefore, in the X(3872) t pp fit procedure all parameters are fixed to the values obtained in the ψ(2S) pseudo-proper time fit, except the f N P parameter. The prompt signal Gaussian widths are scaled by the mass ratio M (X(3872))/M (ψ(2S)) to correct for the difference in the boost factors of the X(3872) sample relative to the ψ(2S) sample which results in a different time resolution for the same spatial resolution. We obtain f N P = 0.139 ± 0.025 from the fit with χ 2 / NDF = 8.1 / 10.
The systematic uncertainties on f N P estimated for the full p T region are listed in Table II. They include the uncertainty due to (1) the muon reconstruction and identification efficiencies; (2) variation of the pion reconstruction efficiency in the low and high t pp regions; (3) uncertainties due to different p T distribution shapes for the prompt and nonprompt events; (4) variation of the mass fit model parameters; (5) variation of the time resolution function; (6) variation of the short-lived function shape; (7) variation of the long-lived function shape; and (8) production ratio of the short-lived and long-lived components.
For the full p T range studied, we obtained f N P = 0.328 ± 0.006 +0.010 −0.013 for the ψ(2S) meson sample and f N P = 0.139 ± 0.025 ±0.009 for the X(3872) meson sample, where the first uncertainty is statistical and the second is systematic.
The large sample sizes allow us to study the t pp distributions in several p T intervals. We choose six p T intervals  for the ψ(2S) and three for the X(3872). In addition, the fit procedure is performed by dividing the full data samples into two ψ(2S) and X(3872) pseudorapidity intervals: |η| < 1 and 1 < |η| < 2. The method used to obtain parameters is the same as for the full data sample. For a given p T or η interval, we first fit the ψ(2S) t pp distribution and obtain the free parameters. Then, these parameters are fixed in the fit of the X(3872) t pp distribution. For both mesons, the fraction f N P of the nonprompt component is allowed to vary in each p T or η interval. Figure 5 shows the p T dependence of f SL for the ψ(2S); the values of this parameter are larger than the values of a few percent obtained by the ATLAS col-laboration [8].
[GeV] For all measured f N P values the systematic uncertainties are calculated applying the same procedure and the same variation intervals as for the whole data sample. The values of nonprompt fractions for the ψ(2S) and X(3872) states in different p T or η intervals with the statistical and systematic uncertainties are given in Table III. Figure 6 shows f N P as a function of p T for the ψ(2S), compared with the ATLAS [8] Figure 7 shows similar distributions for the X(3872) obtained in this analysis, together with the ATLAS [8] and CMS [7] measurements. The D0 measurements of f N P are systematically below the ATLAS [8] and CMS [7] points obtained at higher CM energies although the LHC measurements are restricted to more central pseudorapidity regions. The small differences between the CDF and D0 ψ(2S) measurements can be ascribed to differences in pseudorapidity acceptance. However, the general tendencies are very similar: the f N P values increase with p T in the case of ψ(2S) state production, whereas the f N P values for X(3872) are independent of p T within large uncertainties.
We summarize the measurements of this Section as: • The nonprompt fractions for ψ(2S) increase as a function of p T whereas those for X(3872) are consistent with being independent of p T . These trends are similar to those seen at the LHC. The Tevatron values tend to be somewhat smaller than those measured by ATLAS and CMS.
• The ratio of prompt to nonprompt ψ(2S) production, R p/np = (1 − f N P )/f N P , decreases only slightly going from the Tevatron to the LHC. As can be seen in Fig. 6 the f N P value in the 9 -10 GeV range is about 0.3 for Tevatron data and 0.35 for LHC data, resulting in  [17]. The uncertainties shown are total uncertainties, except for the CDF points, for which only the statistical uncertainties are displayed. The D0 and ATLAS analyses are performed using ψ(2S) → J/ψπ + π − decay channel, whereas the CMS and CDF data are obtained throught the ψ(2S) → µ + µ − decay.
[GeV] ∼25% increase in R p/np . For the X(3872) the D0 measurement gives f N P ∼0.14 in comparison to f N P ∼ 0.33 for ATLAS (Fig. 7), resulting in increase of the R p/np ratio by about three times. It has to be noted that this difference may be partially compensated by the larger rapidity interval covered by D0. This increase of the R p/np value indicates that the prompt production of the ex- otic state X(3872) relative to the b-hadron production is strongly suppressed at the LHC in comparison with the Tevatron conditions. This suppression is possibly due to more particles produced in the primary collision at LHC that increase the probability to disassociate the nearly unbound and possibly spatially extended X(3872).

IV. HADRONIC ACTIVITY AROUND THE ψ(2S) AND X(3872) STATES
In this section we study the association of the ψ(2S) or X(3872) states with another particle assumed to be a pion using the 5-track sample. We study the dependence of the production of these two states on the surrounding hadronic activity. We also test the soft-pion signature of the X(3872) as a weakly-bound charm-meson pair by studying the production of X(3872) at small kinetic energy of the X(3872) and the π in the X(3872)π center of mass frame.
The data are separated into a "prompt" sample, defined by the conditions L xy < 0.025 cm and L xy /σ(L xy ) < 3 and a "nonprompt" sample defined by L xy > 0.025 cm and L xy /σ(L xy ) > 3 where L xy is the decay-length of the J/ψπ + π − system in the transverse plane.
In these studies the uncertainties in the results are dominated by the statistical uncertainties in the fitted X(3872) yields. In the limited mass range around the ψ(2S) or X(3872), the background is smooth and monotonic, and is well described by low-order Chebyshev polynomials. Depending on the size of a given subsample, the polynomial order is set to two or three. In all cases, the difference between the yields for the two background choices is less than 30% of the statistical uncertainty. The small systematic uncertainties are ignored.

A. ψ(2S) and X(3872) isolation
The LHCb Collaboration has studied [18] the dependence of production cross sections of the X(3872) and ψ(2S) on the hadronic activity in an event, which is approximated using a measure of the charged particle multiplicity. The authors found the ratio of the cross sections for promptly produced particles, σ(X(3872))/σ(ψ(2S)), to decrease with increasing multiplicity and observed that this behaviour is consistent with the interpretation of the X(3872) as a weakly bound state, such as a D 0D * 0 hadronic molecule. In this scenario, interactions with comoving hadrons produced in the collision dissociate the large, weakly bound X(3872) state more than the relatively compact conventional charmonium state ψ(2S).
In this study of the production of charmonium-like states, we introduce Isolation as an observable quantifying the hadronic activity in a restricted cone in the φ − η space around the candidate, ∆R = ∆φ 2 + ∆η 2 . We define the Isolation as a ratio of the candidate's momentum to the scalar sum of momenta of all charged particles pointing to the primary vertex produced in a cone of ∆R = 1 around the candidate and the candidate itself. Distributions of Isolation for prompt ψ(2S)π and X(3872)π normalized to unity are shown in Fig. 8 and the ratio of the unnormalized distributions is shown in Fig. 9. The shapes of the two Isolation distributions are similar. The difference between the χ 2 values obtained for fits to the ratio as a function of Isolation assuming a free slope and zero slope corresponds to 1.2σ. This gives modest support for the hypothesis that increased hadronic activity near X(3872) depresses its production.

B. Search for the soft-pion effect
Recent theoretical work [9], [10] predicts a sizable contribution to the production of the X(3872), both directly in the hadronic beam collisions and in b-hadron decays, from the formation of the X(3872) in association with a co-moving pion. According to the authors, the X(3872), assumed to be a DD * molecule, is produced by creation of DD * at short distances. But it can also be produced by creation of D * D * at short distances, followed by a rescattering of the charm-meson pair into a X(3872)π pair by exchanging a D meson. The cross section from this mechanism would have a narrow peak in the X(3872)π invariant mass distribution near the D * D * threshold from a triangle singularity that occurs when the three particles participating in a rescattering are all near the mass shell.
A convenient variable to quantify this effect is the kinetic energy T (Xπ) of the X(3872) and the π in the X(3872)π center-of-mass frame. The authors define the peak region to be 0 ≤ T (Xπ) ≤ 2δ 1 where δ 1 = M (D * + ) − M (D 0 ) − M (π + ) = 5.9 MeV. The effect is sensitive to the DD * binding energy whose current estimated value is (−0.01 ± 0.18) MeV. The peak height is expected to decrease with increasing binding energy. It also depends on the value of the momentum scale Λ expected to be of the order of M (π + ). For the conservative choice of a binding energy of 0.17 MeV, the yield in the peak region is predicted to be smaller than the yield without a soft-pion by a factor ∼ 0.14(M (π + )/Λ) 2 . For Λ = M (π + ), this ratio is equal to 0.14. We search for this effect separately in the "prompt" and "nonprompt" samples.

Prompt production
As a benchmark, we use the ψ(2S) for which no soft-pion effect is expected. We select combinations J/ψπ + π + π − that have a J/ψπ + π − combination in the mass range 3.62 < M (J/ψπ + π − ) < 3.74 GeV. The to-tal number of entries is 310,636 and the ψ(2S) signal has 48, 711 ± 511 events. The mass distributions and fits are shown in Fig. 10. After the T (ψ(2S)π) < 11.8 MeV cut, the number of entries is 368 and the signal yield is 44 ± 14 events. The cut T (ψ(2S)π) < 11.8 MeV keeps a fraction 0.0009 ± 0.0003 of the signal, in agreement with the measured reduction of the combinatorial background by a factor of 0.0012. As expected, there is no evidence for a soft pion effect for ψ(2S). Then, we select J/ψπ + π + π − combinations that have a J/ψπ + π − combination in the mass range 3.75 < M (J/ψπ + π − ) < 4.0 GeV that includes the X(3872). The total number of selected entries is 749,179 and the X(3872) signal yield is 6157 ± 599 events. The mass distributions and fits are shown in Fig. 11. The signal consists of a X(3872) meson produced together with a charged particle. It includes possible pairs of a X(3872) meson and an associated soft-pion from the triangle singularity. Background is due to random combinations of a J/ψ meson and three charged particles. The cut T (Xπ) < 11.8 MeV should remove the bulk of random X(3872)-particle combinations while keeping the events due to the triangle singularity. For this subsample of 730 events, the fitted signal yield is 18 ± 16 events. Thus, the cut T (Xπ) < 11.8 MeV keeps a fraction 0.003 ± 0.003 of the signal, consistent with the background reduction by a factor of 0.00097±0.00004. In the absence of the soft-pion process, the expected yield at small T (Xπ) is N = 6157×0.00097 = 6 events. With the measured yield of 18 ± 16 events, the net excess is 12±16 events. The 90% C.L. upper limit is 43 events which is less than 0.007 of the total number of accepted events.
To compare this result with the expected number of accepted soft-pion events, we make a rough estimate of the kinematic acceptance for events above and below the 11.8 MeV cutoff. The main factor is the loss of pions produced with p T < 0.5 GeV that strongly depends on T (Xπ), given the p T distribution of the X(3872).
The transverse momentum distributions of pions in the two subsamples are shown in Fig. 12. Above 0.5 GeV, the distributions fall exponentially. Below the 0.5 GeV threshold, the spectrum must rise from the minimum kinematically allowed value to a peak followed by the exponential fall-off. For events with T (Xπ) >11.8 MeV, we fit the distribution to the function N · p T · exp(−p T /p T 0 ) and define the acceptance A as the ratio of the integral from 0.5 GeV to infinity to the integral from zero to infinity. The result is 0.6. With alternate functions, the acceptance values vary from 0.3 to 0.9. Figure 12 shows two fits with similar behavior above threshold but different below threshold, the default function and the function N · (1 − exp(−p T /p T 1 )) · exp(−p T /p T 2 ). For events with T (Xπ) < 11.8 MeV, the p T distribution of the accompanying pion is closely related to the p T of the X(3872). To determine the pion acceptance, we employ a simplified MC model, starting with the differential cross section as a function of T (Xπ) <11.8 MeV given in Ref. [9]. For a X(3872) with a given p T (X), the X and pion are distributed isotropically in the Xπ rest frame. The transverse momentum of the pion p T (π) in the laboratory frame is determined by transforming to the X(3872) rest frame, using the chosen p T (X) and a rapidity y(X) chosen from a uniform distribution |y| < 2, and then transforming to the laboratory frame. The pion acceptance as a function of p T (X), A(p T (X)), is then convolved with the fitted X(3872) yield dN/dp T (X) as a function of p T (X) to determine the overall acceptance for pions.
Our observed dN/dp T distribution for the X(3872) is found by dividing the mass distribution for J/ψπ + π − in Fig. 11(a) for the 5-track sample into seven p T bins each 2 GeV wide, between 7 and 21 GeV, and fitting for the yield of the X(3872) for each bin. This produces a background-subtracted sample, however with relatively large statistical uncertainties. These seven dN/dp T yield points for the X(3872) are plotted in Fig. 13. The higher statistics and finer binned yield for inclusive J/ψπ + π − events over the mass range 3850-3900 MeV of Fig. 11(a) as a function of p T is used to check the shape of the p T distribution of the X(3872). After scaling to equal areas, dN/dp T (J/ψπ + π − ) shows a good agreement within statistical uncertainties with the X(3872) spectrum, thus indicating a comparable behavior of the X(3872) signal and background. The transverse momentum distribution for the background-subtracted mass fitted X(3872) (filled circles) and two fits representing the high and low range of the acceptance for the accompanying pion. The dashed curves represent A(pT ) · dN/dpT (X3872)). The overall acceptance for the accompanying pion is the ratio of the areas below A · dN/pT (X) curves and the corresponding dN/dpT (X) fits. For comparison, the scaled pT distribution of the inclusive J/ψππ for 3.85 < M (J/ψπ + π − ) < 3.9 GeV (open blue circles) are overlaid illustrating their similarity in shape.
Fits of the background-subracted yields using functions p b T · exp(a + c · p T ) and (p T − b) · exp(a + c · p T ) are shown in Fig. 13 , along with the products A(p T )·dN/dp T which allow the calculation of the acceptance for p T (π) > 0.5 GeV for events with p T (X) > 7 GeV. We find the acceptances A = 0.278±0.031 and 0.296±0.036 for the two functions respectively, where the uncertainties are due to the statistical uncertainty in the determination of the dN/dp T (X) distribution. Additional functions were used to fit dN/dp T (X). The aformentioned functions yield the lowest and higest pion acceptances obtained from the different forms. Their difference is considered as the systematic uncertainty associated with the choice of parameterization. We average the two results to obtain A = 0.29 ± 0.03 (stat) ±0.02 (syst).
For the prompt case, this leads to the expected number of produced X(3872) events at N = 18/0.29 + 6139/0.6 ≈ 10, 000 with an uncertainty of about ± 50%. With N = N 1 + N 0 , where N 1 is the number of events with a soft-pion, and the relation N 1 = 0.14 · N 0 , N ≈ 10, 000 · 0.14/1.14 ≈ 1, 300 events would be produced through the soft-pion process with an uncertainty of about 650 events and between 245 and 730 would be accepted. That is much larger than the observed 12±16 events. We conclude that there is no evidence for the soft-pion effect in the prompt sample.

Nonprompt production
The kinematics of the prompt and nonprompt samples are sufficiently similar to use the acceptance derived for the prompt case for both samples. Calculations analogous to those for the prompt case give the following results for the nonprompt sample. For the ψ(2S), the kinetic energy cut keeps a fraction of 0.004 ± 0.001 of the signal, in agreement with the reduction by a factor of 0.003 of the total number of entries.
For the X(3872), the signal yields before and after the cut are 703 ± 25 and 27 ± 12, respectively. The cut accepts a fraction 0.04 ± 0.02 of the signal. The corresponding reduction in the total number of events in the distribution is by a factor of 0.0029 ± 0.0001. For a random pairing of the X(3872) with a pion, the expected yield at small T (Xπ) is N = 703 × 0.0029 = 2 events, leading to a net excess of 25±12 events. The statistical significance of the excess, based on the χ 2 difference between the fit with a free signal yield and the fixed value of N = 2 expected for the "random-pairing only" case, is 2σ. The expected number of produced soft-pion events is ≈150. With the acceptance of 0.29±0.04, the expected number of accepted soft-pion events is between 31 and 87. The measured excess yield of 25 ± 12 events is in agreement with this expectation however the 2σ excess prevents drawing a definite conclusion.
For further details on the distribution of the nonprompt signal versus T (Xπ), we fit the X(3872) mass distributions in 2-MeV bins of T (Xπ) from 0 to 10 MeV and in 40-MeV bins from 10 to 490 MeV. The resulting distribution of events/2 MeV is shown in Fig. 14. Above ∼10 MeV, the observed spectrum is consistent with the pairing of a X(3872) with a random particle. It is similar to the T (Xπ) distribution of all nonprompt X(3872) candidates. At lower T (Xπ), there is a small excess, with a significance of 2σ, above the random pairing, at the level consistent with the predictions of Ref. [9]. We again conclude that there is no significant evidence for the soft-pion effect but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale Λ = M (π)) is not ruled out.

V. SUMMARY AND CONCLUSIONS
We have presented various properties of the production of the ψ(2S) and X(3872) in Tevatron pp collisions. For both states, we have measured the fraction f N P of the inclusive production rate due to decays of b-flavored hadrons as a function of the transverse momentum p T . Our nonprompt fractions for ψ(2S) increase as a function of p T whereas those for X(3872) are consistent with being independent of p T . These trends are similar to those seen at the LHC. The Tevatron values tend to be somewhat smaller than those measured by ATLAS and CMS but this difference can at least partially be accounted for by the larger rapidity interval covered by D0. The ratio of prompt to nonprompt ψ(2S) production, (1−f N P )/f N P , decreases only slightly going from the Tevatron to the LHC, but comparing the 8 TeV ATLAS data to the 1.96 TeV D0 data for the X(3872) production this ratio decreases by a factor of approximately 3. This indicates that the prompt production of the exotic state X(3872) is suppressed at the LHC, possibly due to the production of more particles in the primary collision that increase the probability to disassociate the nearly unbound and possibly more spatially extended X(3872) state.
We have tested the soft-pion signature of the X(3872) modeled as a weakly-bound charm-meson pair by studying the production of the X(3872) as a function of the kinetic energy of the X(3872) and the pion in the Xπ center-of-mass frame. For a subsample consistent with prompt production, the results are incompatible with a strong enhancement in the production of the X(3872) at small T (Xπ) expected for the X+soft-pion production mechanism. For events consistent with being due to decays of b hadrons, there is no significant evidence for the soft-pion effect but its presence at the level expected for the binding energy of 0.17 MeV and the momentum scale Λ = M (π) is not ruled out.
This document was prepared by the D0 collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.