Indication for Double Parton Scatterings in W + Prompt J/Psi Production at the LHC

We re-analyse the associated production of a prompt J/psi and a W boson in pp collisions at the LHC following the results of the ATLAS Collaboration. We perform the first study of the Single-Parton-Scattering (SPS) contributions at the Next-to-Leading Order (NLO) in alpha_s in the Colour-Evaporation Model (CEM), an approach based on the quark-hadron-duality. Our study provides clear indications for Double-Parton-Scattering (DPS) contributions, in particular at low transverse momenta, since our SPS CEM evaluation, which can be viewed as a conservative upper limit of the SPS yields, falls short compared to the ATLAS experimental data by 3.1 standard deviations. We also determine a finite allowed region for sigma_eff, inversely proportional to the size of the DPS yields, corresponding to the otherwise opposed hypotheses, namely our NLO CEM evaluation and the LO direct Colour-Singlet (CS) Model contribution. In both cases, the resulting DPS yields are significantly larger than that initially assumed by ATLAS based on jet-related analyses but is consistent with their observed raw-yield azimuthal distribution and with their prompt J/psi+J/psi and Z + prompt J/psi data.


Introduction.
The simultaneous production of vector bosons and quarkonia at high-energy colliders is a very useful observable to study perturbative and nonperturbative aspects of Quantum Chromodynamics (QCD). It also provides original means to search for new physics beyond the standard model via the Higgs sector, as illustrated by the pioneering study of CDF Collaboration [1,2]. Recently, the final states J/ψ + W [3] and J/ψ + Z [4] were observed by the ATLAS Collaboration. Similar processes with bottomonia have however not been observed yet 1 .
In the case of J/ψ + Z production, the yield observed by ATLAS happens to be up to nearly one order of magnitude larger than the theory evaluations from NRQCD (with CS and/or Colour-Octet (CO) -higher order in v 2 -contributions) [53,54]. A natural explanation for such a gap would be -like for quarkonium pairs at large ∆y ψψ -associated production through DPS but for the fact that the azimuthal distribution of the observed events shows a significant back-to-back peak pointing at a large SPS. In a recent paper [55], we have solved this apparent conflict by invoking a misinterpretation of this raw-yield azimuthal distribution when it results from two different sources with very different transverse-momentum distributions in a detector like ATLAS with a strongly transverse-momentum-dependent acceptance. In a further study [56], we have shown that the production of a Z with a non-prompt J/ψ (i.e. from a b-hadron decay), as reported by ATLAS in [4], is well accounted by the SPS predictions with a limited impact of DPSs.
In the case of J/ψ + W production, the observed yield of ATLAS [3] also ends up to be nearly one order of magnitude larger than the recent theory SPS evaluations from NRQCD (with CS and/or CO contributions) [57,58] (see [59,60] for earlier studies). Just like the J/ψ + Z case, the ATLAS rawyield azimuthal distribution seems to exhibit a non-trivial structure hinting at the presence of SPS events.
In this context, we have decided to have another look at prompt-J/ψ + W production at the LHC, in particular relying on an analysis of the SPS yield at NLO under the assumption of quark-hadron duality [the Colour-Evaporation Model (CEM)] [61,62] as in our earlier study [55]. As we demonstrated then, the CEM provides a conservative upper limit to the SPS yield which we use to draw definite conclusions about the importance of the DPSs. This paper is organised as follows. In Section 2, we briefly explain our NLO CEM SPS computation which, as an upper theory limit for the SPS, allows us to claim for a clear indication of the contributions of DPSs at a 3 σ level. In Section 3, we extract the DPS yield and determine σ eff with its uncertainty range. We then show that the azimuthal distribution of the prompt J/ψ + W events collected by AT-LAS remains compatible with a yield highly dominated by DPSs. Section 4 is devoted to our conclusions and outlook.

Our NLO evaluation of the SPS yield
In this work, we thus focus on the production of a W with a prompt J/ψ (which does not result from b-hadron decays) at the LHC in the kinematical region accessible by the ATLAS detector (see Table 1).
As said above, the SPS contributions have been studied in the past within NRQCD, considering either CS [58] or CO [57] channels. The latter CO evaluation at NLO accuracy in α s is however perfectible as it relies on artificially low scales ignoring the emission of a W boson and with NRQCD Long Distance Matrix Elements (LDMEs) which are significantly larger than the NLO LDMEs compatible with the LHC yield and polarisation data (see e.g. [63]). That being said, since the direct CS yield does not depend on free tunable parameters, it provides a strict lower limit on the SPS J/ψ + W production cross section.
On the contrary, the CEM most certainly sets an upper limit on the SPS yield. Indeed, the CEM tends to overshoot the experimental data for single quarkonium production at large P J/ψ T [64][65][66] 3 . This follows from the precocious dominance of the gluon-fragmentation topologies whose strength is therefore too high at large P J/ψ T . For J/ψ + W, the dominant CEM channel is quark-antiquark annihilation with the emission of a W and an off-shell gluon fragmenting into a quarkonium. As such, the CEM likely results in an overestimation of the prompt J/ψ + W cross section. It can thus be considered as a conservative upper limit of the theoretical prediction of the SPS yield. In other words, any set of NRQCD LDMEs which would be compatible with the yield predicted here would severely overshoot the very precise single-J/ψ data. Let us also add that, due to the simplicity of the model, the CEM avoids the complexities encountered in NLO analyses of NRQCD, where a large theoretical uncertainty in any case remains. Overall, we will consider that the difference between both (direct CSM & CEM) represents our best evaluation of the theoretical uncertainty on the SPS cross section.
In the CEM, one considers the integral of the cross section for QQ pair production with an invariant-mass between the quark mass threshold 2m Q and that of open-heavyflavour hadrons 2m H , where the hadronisation of heavy quarks into a quarkonium is likely. To obtain the cross section to the hadron state (Q), this integral is multiplied by a phenomenological factor for the probability of the quark states to hadronise into a given quarkonium state. In short, we would consider P Q can be paralleled to the LDMEs in NRQCD. With the approach just described, the direct or prompt yields result from the same computation but with a different overall factor. In practice, we use P LO,prompt J/ψ = 0.014 ± 0.001 and P NLO,prompt = 0.009 ± 0.0004 which we have fit in [55] on the differential cross section for the production of single J/ψ as measured by ATLAS [71] using m c = 1.27 GeV (see [72] for the mass choice). We refer to [73] for the first NLO CEM fits using the P T -integrated cross sections.
In view of Eq. (1), computing J/ψ+W production at NLO can be performed with modern tools of automated NLO frameworks, with some slight tunings. In practice, we have used MADGRAPH5 AMC@NLO [74] 4 .
The hard scattering process to be considered is 5 i j → cc+W ± +k with i, j and k standing for g, q orq. As what regards the parton distribution function (PDF), we have used 3 To cure this issue, different mechanisms [67][68][69][70] were proposed but are not the object of a consensus. In our recent study [55], we have shown that the NLO corrections to the P J/ψ T spectrum reduce the scale sensitivity but confirm the issue. 4 We stress that, for the CEM, there is no need to use a specific automated tool like MADONIA [75] and HELAC-ONIA [76,77] which are by the way currently not able to treat loop corrections. 5 Unlike the CSM case, channels such as sg → W − J/ψc negligibly contribute to the CEM yield and can safely be neglected.
σ(pp→W ± ) with our results of calculations in the CEM. We also show the normalised LO CS direct yield [58] and the DPS yield ratios extracted from [3] as a function of σ eff . The unit is in 10 −7 . The theoretical uncertainty for the (N)LO SPS is from the renormalisation and factorisation scales.
As already stated above, we have taken m c = 1.27 GeV, while choosing m c = 1.5 GeV would generate negligible changes to our results provided that the non-perturbative CEM parameter is chosen coherently. We note that the heavy-quark-mass dependence is de facto absorbed in the CEM parameter, hence the main theoretical uncertainties result from the renormalisation µ R and factorisation µ F scale variations which are believed to account for the unknown higher-order α s corrections. In practice, we have varied them independently within 1 2 For the current analysis, ATLAS preferred to consider the ratio of the cross section for W ± +J/ψ to that for W ± in order to discard some efficiency and acceptance corrections related to the W ± detection. For our theory evaluation, we use the NNLO W boson production cross section in the ATLAS acceptance corrected by the branching ratio for the decay to muons σ(pp → W ± )BR(W → µν) = 5.08 nb [80][81][82]. In such a case, the W ± + J/ψ cross section in the numerator should also be multiplied by the branching BR(W → µν) which cancels with that in the numerator. Overall one only needs to consider: where y J/ψ is the J/ψ rapidity, BR(J/ψ → µ + µ − ) = 0.05961 ± 0.00033 [83]. We have also assumed BR(W → µν) = 0.1063 and the J/ψ-rapidity range of 4.2 units to evaluate dσ/dy J/ψ . Our results (LO, NLO CEM) for the (inclusive) total cross section ratio are gathered in Table 2 along with the experimental results by ATLAS and the LO direct CS yield ratio computed in [58]. One notes that the NLO CEM ratio 6 is -as expected-nearly three times larger than the direct CSM ratio. In spite of the large experimental uncertainties, the SPS predictions strongly underestimate the measurements, by more than three standard deviations 7 . The same features can be observed in the distribution in P J/ψ T , shown in Fig. 1 with the black (green) hatched histograms, resulting in a 3.1 standard-deviation discrepancy. As such we claim that these are clear indications of the existence of a DPS yield in this process which is supported by the analysis of the transverse momentum and azimuthal dependences as we will show now.

DPSs and the P J/ψ T -integrated cross section
The DPS yield results from two uncorrelated scatterings within one proton-proton collision. As such, it is usually parametrised by the rudimentary pocket formula which, for the process under study, reads where σ eff is a supposedly universal parameter with the dimension of a surface. Recent experimental analyses are pointing at a visible impact of the DPSs in many reactions [15,17,[25][26][27][28][29][30][31]40]. As a reference value, ATLAS used in their data-theory comparison [3] σ eff = 15 ± 3(stat.) +5 −3 (sys.) mb (from W+ 2jet data [30]) which results, with σ(J/ψ) and σ(W) obtained from their data, in a normalised DPS cross section about 3 times smaller than their measurements (compare the first and last column of Table 2).
In the present analysis, we will instead extract the DPS yield by assuming that the difference between the ATLAS value and our SPS evaluation is entirely due to DPSs. As such, the σ eff associated to the present observable can also be constrained in a relatively precise manner 8 and confronted to other extractions. Let us recall that the SPS CEM evaluation should be regarded as an upper limit of the SPS yield. Let us now describe how we evaluate σ eff and its uncertainty. Since the DPS yield is simply obtained by subtracting the SPS yield from the (inclusive) total cross section of ATLAS, the uncertainty on the DPS yield, and consequently 9 that of σ eff , will then depend on the (statistical and systematic) uncertainties of the data [3] and on the range spanned by the SPS evaluations -the NLO CEM and the direct LO CSM [58] 10 . Since the SPS values are in both cases much smaller than the data, the theoretical uncertainties are in fact nearly irrelevant for the determination of σ eff . Our results are reported on Table 3 and on Fig. 2.
In particular, our combined result for σ eff is then which is nearly three times smaller than the ATLAS assumption (15 mb) with somewhat smaller uncertainties than found by other works. It is also consistent with our latest extraction from prompt J/ψ + Z production [55] in Fig. 3, where we also compare our extraction with other measurements [15, 17, 25-31, 40, 48, 49]. Such a low value may hint at the non-universality of σ eff in different processes, with a dependence on the flavour of the initial state, on the kinematics of the final state or on the energy of the protonproton collision. For example, the 3 values of σ eff from 8 Assuming the same single particle cross sections, σ(J/ψ) and σ(W), as ATLAS. 9 The uncertainties on the measured single-particle cross sections, σ(J/ψ) and σ(W), are negligible compared to that on σ(J/ψ + W). 10 We note that the NLO NRQCD evaluation [57] -despite the aforementioned drawbacks-lies within this range. the ATLAS J/ψ-associated-production measurements are in general smaller than those from the LHCb measurements at forward rapidities. Such an observation seems to follow -at least qualitatively-the lines of the mean-field approximation [85].

Consistency with the transverse momentum and azimuthal distributions
Our analysis shows that the DPSs are by far dominant and the SPSs could even be omitted in the analysis of the P J/ψ T -integrated cross section. Since our procedure simply amounted to assume that any gap between the predicted SPS yield and the data was from DPS, it is important to check its consistency with differential cross sections.
Looking back at Fig. 1, we see that the introduction of a somewhat larger DPS yield perfectly fills the gap where needed and does not create any surplus at large P J/ψ T where the SPS was closest to the data. We note that the plotted DPS P J/ψ T spectrum follows from that of ATLAS [3] except for a different normalisation owing to the smaller value of σ eff which we have just discussed. Like for the P J/ψ T -integrated DPS cross section, ATLAS used the singleparticle cross sections, here dσ(J/ψ)/dP J/ψ T , which they obtained from their own data.
Indeed, the sum of the SPS and DPS yields, in red, gives a reasonable account of the ATLAS differential yield. This agreement should however not be over-interpreted in view of the large experimental uncertainties. This observation is rather a consistency check than a test. The good agreement at low P J/ψ T simply follows from the fit of σ eff to the total yield but resolves the 3-σ discrepancy between "theory" and "experiment".
This also helps illustrate that the low-P J/ψ T yield is only from DPS contributions (as is the total yield) and that, at high P J/ψ T , DPS and SPS contributions could be of the same size. This is an important point to discuss the azimuthal dependence.
If initial-state-radiation effects are not too important, the SPS yield (dominated by 2 → 2 scatterings) tends to peak at ∆φ Wψ ∼ π -back-to-back scatterings-whereas the DPS one is believed to be evenly spread in ∆φ Wψ . Fig. 4 shows the event azimuthal distribution for prompt J/ψ + W ± production. At first sight, it seems improbable that a yield largely dominated by DPS could be produced in agreement with such a distribution.
Just like for J/ψ + Z [55], we note that the ATLAS acceptance for high-P J/ψ T events is significantly higher than at low P J/ψ T . This, along with the very different SPS/DPS ratios as a function of P J/ψ T , can lead to a misinterpretation of such an azimuthal dependence which is not corrected in acceptance.
In particular, the peak near ∆φ Wψ ∼ π, visible in the raw event distribution, could artificially be accentuated due to a large acceptance of high-P J/ψ T events. Since we are not in the position of correcting the data, the simplest way to proceed is to mimic the folding of the theory event distribution with an approximated ATLAS acceptance. To do so, we use the J/ψ-acceptance dependence  Table 3: Different extractions of σ eff (in units of mb). The experimental uncertainties on our combination conservatively account for the "spin" uncertainty (see the remark above). inferred in [55] where we assumed the background-oversignal ratio to be like B/S ∝ 1/P J/ψ T . Reading out the statistical uncertainties for 4 bins of [3], we can derive the yield in each P J/ψ T bin. In these, we can then compute the DPSover-SPS ratio, derive the corresponding azimuthal dependence and finally combine them for the 4 bins using the yield in each. The resulting "theory" distribution is shown on Fig. 4 and agrees within uncertainties with the uncorrected ATLAS distribution. On the way, we note that the number of SPS events following our computation at NLO CEM is 2 ± 1 to be compared to 29 +8 −7 events observed by ATLAS [3]. We expected these 2 ± 1 SPS events to lie at ∆φ Wψ ∼ π. Conversely, we expect an updated ATLAS ∆φ Wψ analysis in 2 P J/ψ T bins to show a flat behaviour in the "low" P J/ψ T bins and a slightly peaked one for the "high" P J/ψ T bins. It however seems that more statistics is needed to draw final conclusions.

Conclusions
We have re-analysed the associated production of a prompt J/ψ with a W boson at the LHC in view of the AT-LAS data [3]. To do so, we have performed the first NLO calculation of the SPS cross sections in the CEM which we consider to be a conservative upper limit of the SPS yield. This has allowed us to claim a 3-σ deviation between SPS theory and the ATLAS data, which we interpret as a strong indication for DPSs. Our conclusion does not depend on the quarkonium-production model used to compute the SPS yields.
We have then determined a finite range for σ eff around 6 mb, consistent with extractions from other experiments and other quarkonium-related processes. In particular, we emphasise that our result is consistent with σ eff extracted from our analysis of prompt J/ψ+Z production as measured by ATLAS. Whereas the azimuthal dependence could be very useful in separating the SPSs from the DPSs, one has to be very cautious with uncorrected raw event distribution. A further analysis with higher statistics allowing one to study the rapidity-separation spectrum and a fully corrected azimuthal distribution will further constrain the range for σ eff since the theoretical uncertainties on the SPS happen to be quasi irrelevant at low P J/ψ T .