Phenomenological analysis of associated production of $Z^0+b$ in the $b \rightarrow J/\psi X$ decay channel at the LHC

The ATLAS collaboration recently reported on the first observation of associated-production of a $Z^0$ boson with a $J/\psi$. We recently claimed that the corresponding yield of the {\it prompt} $J/\psi$ was dominated by double parton scatterings in the ATLAS acceptance with a somewhat small value of $\sigma_{\rm eff}$. We also found out that single parton scatterings were only dominant at large transverse momenta. We present here the first phenomenological analysis of another part of the ATLAS data sample, namely of a $Z^0$ boson plus a {\it non-prompt} $J/\psi$. Our study is performed at next-to-leading order in $\alpha_s$ and includes parton-shower effects via the {\sc\small MadGraph5_aMC@NLO} framework. We find out that the data, unlike the case of prompt $J/\psi+Z^0$, do not hint at significant DPS contributions. Owing to the current experimental and theoretical uncertainties, there is still a room for these but with a lower limit of $\sigma_{\rm eff}$ close to 5 mb. We stress the importance of QCD corrections to account for the ATLAS data.


Introduction
Thanks to the large luminosities of the LHC, the study of differential distributions of associated production of vector bosons with open-and hidden-heavy flavour became accessible. These are particularly interesting because they can give access to information on double parton scatterings (DPS). These, as opposed to the conventional single parton scatterings (SPS), consist in two simultaneous partonic scatterings during a single proton-proton collision.
The relevance of DPS is known to increase for increasing energies at hadron-hadron colliders which explains why they have only started to be systematically studied with the advent of the Tevatron and the LHC. This is also why they remain poorly understood. Yet, the measurement of fundamental SM parameters like the bottom-quark Yukawa coupling via H 0 and vector-boson associated production requires the reanalysis of these processes by taking into account DPS. New physics searches via same-sign W boson-pair production also requires a good control on the DPS.
Recent DPS studies based on quarkonium-pair production [1,2,3,4,5] indicate a smaller σ eff -a parameter characterising the importance of the DPS (σ DPS ∝ 1/σ eff )-than the jet-related final states. We should however note that all these extractions were carried out under a simplified -but The structure of the article is as follows. Next section contains a short description of the computation set-up including the definition of the fiducial cuts as well as our results. We will compare our theoretical results with the ATLAS data and extract the information of σ eff . Besides, a theoretical prediction will be given for the ongoing CMS measurement. Finally, we draw our conclusions in section 3.

Framework
Let us now describe how we have computed the (differential) yields to be compared to the measurement of the ATLAS collaboration recently reported in [34]. In order to generate the (N)LO event sample for pp → Z 0 + bb + X in the 3-initial-quark-flavour scheme, we have used MadGraph5 aMC@NLO [52]. For the record, this single framework includes MadLoop [54] and MadFKS [55] to handle the virtual and real pieces respectively; the former module uses the OPP method [56,57] whereas the latter uses the FKS subtraction method [58]. It also automatically uses the MC@NLO approach [59] to match NLO matrix elements to parton showers. The spinentangled Z 0 → e + e − decays were then performed by the MadSpin module [60] and we have used Pythia 8.1 [53] to account for the parton showers, the hadronisation and the other decays. All this allowed us to compute the yield in the ATLAS and CMS fiducial regions (see Table 1).
As what regards the choice of the renormalisation scale µ R and factorisation scale µ F , we have chosen as a central value µ 0 = H T 2 , where H T is the transverse mass sum of the final states. For the PDFs, we have used CTEQ6L1 (CTEQ6M) [62] for the LO (NLO) computation. The integrated fragmentation fraction of b-hadrons to J/ψ was taken to be 1.15% from Ref. [63], which is also close to other estimations in the literature (see e.g. Refs. [64,65]). The other relevant Standard Model parameters are reported in Table 2.

Parameter Value
Parameter Value

Our results for the SPS contributions
In the ATLAS fiducial and inclusive regions (defined in Table 1), we have obtained the following (N)LO SPS cross sections for non-prompt J/ψ + Z production at the LHC for √ s = 8 TeV: where the theoretical uncertainty includes the renormalisation scale µ R and factorisation scale µ F uncertainty µ 0 2 ≤ µ R , µ F ≤ 2µ 0 , varied independently. In absence of such predictions in the literature, no comparison was made by ATLAS in Ref. [34]. We proceed now to the comparison to their results presented in the form of yield ratio in order to reduce some systematical uncertainties attached to the detection of the Z boson, namely : We therefore also need to use the Z production cross section with the ATLAS cuts and have decided to simply take σ ATLAS (Z) × Br(Z → e + e − ) = 533.4 fb used by ATLAS for the comparison with the prompt J/ψ + Z theory predictions. The latter was estimated at the next-to-next-to leading order by FEWZ [66]. We have also checked this estimation with MadGraph5 aMC@NLO by including the parton shower effects via the MC@NLO [59] approach and have obtained a similar value within the theory uncertainty.
Our results and those of ATLAS are shown in Fig.1 and in Table 3. The NLO corrections in α s increase the Z+ non-prompt J/ψ SPS yield by a factor of 1.46. Overall the SPS yield ends up to be close to the ATLAS measurement and hence leaves a small room for the DPS yield. We also note that the relative scale uncertainty is also reduced from 30% at LO to 13% at NLO.   Table 3: Comparison of the cross section ratio np R(J/ψ + Z) between the theoretical calculations and the experimental data [34] at 8 TeV LHC.

Discussion about the DPS contributions
Let us now turn to the discussion of the DPS contributions. ATLAS [34] has made an estimation of the DPS yield using the data for single Z and non-prompt J/ψ production and using the simple "pocket formula" 3 .
In fact, this also means that we can extract a lower limit on σ eff , corresponding to a maximum allowed DPS yields, now that we have at disposal a SPS computation. Contrary to other cases which we previously analysed [2,46], we cannot extract an upper limit since the SPS yield alone is compatible with the data. We evaluate the {68%; 95%} confidence level upper limit on the SPS yield simply as follows: where σ generically denotes the central value of the J/ψ + Z cross section and δσ is the standard deviation of this cross section. The lower value of σ eff at 68% (95%) confidence level is then 5.0 mb (2.3 mb), which is compatible with the σ eff extraction from the other quarkonium-related measurements [1,31,2,3,4,46] and it is close to the range σ eff = 4.7 +2.4 −1.5 mb we obtained for prompt J/ψ + Z production [46].

Comparison with differential distributions
Let us now turn to the comparison of the differential distributions between the theoretical results and the ATLAS data, which in fact allows us to draw similar conclusions. Still by lack of SPS predictions, ATLAS could only compare its measurement of the transverse-momentum P T spectrum of non-prompt J/ψ to their estimation of the DPS yield. As expected from the near dominance of SPS for this process (see above), they found out a very large discrepancy (gap between the data and the blue histogram).
Adding the SPS contribution which we have computed largely fills the gap between theory and experiment as can be noted in Fig. 2a. Only remains a small discrepancy in the last P J/ψ T bin which should however be confirmed by forthcoming measurements as well as more accurate theoretical calculations, with e.g. an improved description of the b quark fragmentation, an account of even higher order QCD corrections, and a matching between different initial-quark flavour number schemes. Similar to the prompt J/ψ + Z production, the DPS contributions exhibit a softer P J/ψ T spectrum than the SPS ones.
Unlike the P J/ψ T spectrum, ATLAS did not provide the efficiency-corrected azimuthal angle correlation between J/ψ and Z, ∆φ Z−J/ψ . Although such a distribution may significantly be smeared by non-perturbative intrinsic initial parton k T [2,38] in the low P T region, we do not think that such a smearing effect will be large here because of the ATLAS P J/ψ T cut which is as large as 8 GeV. Therefore, the investigation of this correlation may indeed reveal the importance of DPS directly. However, in order to make a fair comparison, one has to unfold the efficiency since it is largely dependent on P J/ψ T and hence impacts the collected number of events reported in this distribution. Along the same lines as Ref. [46] and thus by assuming the background events, B, and the signal events, S , to scale like B/S = 17/P J/ψ T , we can evaluate the number of non-prompt J/ψ + Z events 6  Table 4: The estimation of the number of the signal events S (before the efficiency corrections) for non-prompt J/ψ + Z in each P J/ψ T bin with the assumption B/S = 17/P J/ψ T .
in each P J/ψ T , see Table (4). Overall, we evaluate the total number of non-prompt signal events to be 112 vs the ATLAS found 95 ± 12 ± 8, which are roughly consistent. Of course, a more precise comparison will be possible if ATLAS releases a cross section differential in ∆φ Z−J/ψ . Once the number of signal events is known in each bin, one can fill the ∆φ Z−J/ψ distribution according to these numbers using the ratio DPS/SPS in each bin as well as the expected azimuthal DPS and SPS distributions. The former is assumed to be flat (uncorrelated production) see the blue line of Fig. 2b. The latter is obtained from our NLO computation just as the SPS P J/ψ T spectrum was, see the red histogram of Fig. 2b. The resulting comparison ends up to be satisfactory and confirms that the non-prompt J/ψ + Z yield in the ATLAS acceptance is dominated by SPS contributions contrary to the prompt J/ψ + Z yield [46].
As aforementioned, the ratio DPS/SPS increases in the ATLAS acceptance for decreasing P J/ψ T . For the lowest P J/ψ T bin, it even reaches 20 % with σ eff = 15 mb (thus 3 times larger for 5 mb). With higher statistics, it will be possible to measure the yield differential in the rapidity difference, ∆y, between the Z and the J/ψ. For increasing, ∆y, the DPS yield, i.e. from two independent scatterings, is favoured with respect to the SPS yield and could eventually be dominant like for quarkonium-pair production [38,2].

Conclusions
By providing the first phenomenological analysis of (SPS contributions to) Z + b production in the b → J/ψX decay channel, we have filled a gap in the literature and could make the first comparison between theory and the corresponding measurement by the ATLAS collaboration [34]. Unlike the case of prompt J/ψ + Z production [46], we have found out that the SPS contributions to non-prompt J/ψ + Z production happen to be dominant and very close to experimental data. This therefore sets up an upper (lower) limit of the DPS yield (σ eff ).
Our conclusion is based on a computation including NLO QCD corrections and parton-shower effects using MadGraph5 aMC@NLO and Pythia 8.1. Our comparison between the theory and the experiment also shows the importance of the QCD corrections, which not only results in a smaller scale uncertainty but also improves the agreement with data. An improved determination of σ eff requires a better control on both the theoretical and the experimental uncertainties. This of course holds only if one sticks to the simple "pocket formula" where factorisation between both parton scatterings is implied as done for all the existing σ eff experimental extractions.
Based on the ATLAS measurement [34] at √ s = 8 TeV, we thus set the lower limit of σ eff to be 5.0 mb at 68% confidence level and 2.3 mb at 95% confidence level. A comparison with other extractions from both quarkonium associated production [46,2,1,32,3] and jet production [67,68,69,70,71] is displayed in Fig. 3. The values of σ eff from quarkonium production are in general lower than those from jet production, although no strong conclusion can be drawn at the moment due to the remaining large uncertainties, some of them inherent to our incomplete knowledge of the quarkonium-production mechanisms [72,73,74]. If such an observation is confirmed in the future, it may reveal a nontrivial transverse correlations between sets of two partons from a proton or a violation of DPS factorisation, even at high energies. In general, processes as the one discussed here are very important because both scatterings probe different initial states -despite the expected small impact of DPS on the current data set. We emphasise that such a test is very important and, as illustrated here, is feasible at the LHC with higher statistics which would allow one to reach parts of the phase space where DPS contributions are expected to be larger.  Figure 3: Extractions of σ eff from quarkonium associated production [46,2,1,32,3] and jet production [67,68,69,70,71] processes at the Tevatron and the LHC. The symbol "np" in the legend refers to non-prompt J/ψ, otherwise it refers to prompt J/ψ. The lower limit (green arrow) σ eff ≥ 5.0 mb is determined from the present analysis of the ATLAS non-prompt J/ψ + Z sample [34]. The upper limit of σ eff ≤ 8.2 mb in D0 J/ψ + Υ [3] is refined in Ref. [4].