Double-parton scattering effects in $D^{0}B^{+}$ and $B^{+}B^{+}$ meson-meson pair production in proton-proton collisions at the LHC

We extend our previous studies of double-parton scattering (DPS) to simultaneous production of $c \bar c$ and $b \bar b$ and production of two pairs of $b \bar b$. The calculation is performed within factorized ansatz. Each parton scattering is calculated within $k_T$-factorization approach. The hadronization is done with the help of fragmentation functions. Production of $D$ mesons in our framework was tested in our previous works. Here we present our predictions for $B$ mesons. A good agreement is achieved with the LHCb data. We present our results for $c \bar c b \bar b$ and $b \bar b b \bar b$ final states. For completeness we compare results for double- and single-parton scattering (SPS). As for $c \bar c c \bar c$ final state also here the DPS dominates over the SPS, especially for small transverse momenta. We present several distributions and integrated cross sections with realistic cuts for simultaneous production of $D^0 B^+$ and $B^+ B^+$, suggesting future experimental studies at the LHC.


I. INTRODUCTION
Phenomena of multiple-parton interaction (MPI) have become very important for precise description of high-energy proton-proton collisions in the ongoing LHC era. There are several experimental and theoretical studies of soft and hard MPI effects in progress (see e.g. Refs. [1,2]), so far mostly concentrated on double-parton scattering (DPS). In many cases exploration of DPS mechanisms for different processes needs dedicated experimental analysis and is strongly limited because of large background coming from standard single-parton scattering (SPS).
Some time ago we proposed and discussed double open charm meson production pp → DD X as a potentially one of the best reaction to study hard double-parton scattering effects at the LHC [3]. This conclusion was further confirmed by the LHCb collaboration that has reported surprisingly large cross sections for DD meson-meson pair production in pp-scattering at 7 TeV [4]. As we have shown in our subsequent studies the LHCb double charm data cannot be explained without the DPS mechanism [5]. In this case the standard SPS contribution is much smaller and the data sample is clearly dominated by the DPS component [6,7]. Subsequently, we have done similar phenomenological studies for other final states.
We identified optimal conditions for exploring DPS effects in pp → 4jets X [8,9] as well as in pp → D 0 + 2jets X and pp → D 0 D 0 + 2jets X [10] reactions for the ATLAS experiment. Very recently, we have also discussed for the first time possible observation of triple-parton scattering (TPS) mechanism in triple open charm meson production with the LHCb detector [11]. Some rather general features of double-parton scattering were discussed previously both for bbbb [12] and ccbb [13] final states. Here we extend the discussion by including also single-parton scattering mechanism for a first time.
In this paper, we wish to present results of phenomenological studies of DPS effects bottom pp → B + B + X production. In particular, we will show theoretical predictions of integrated and differential cross sections for different energies that could help to conclude whether and how the DPS effects for these two cases can be observed experimentally by the LHCb/CMS collaborations.

A. Single-parton scattering
In Fig. 1 we show a diagrammatic representation of the dominant SPS mechanism for double heavy quark pair production. In particular, in the following we consider mixed ccbb (left panel) and double bottom bbbb (righ panel) final states, however, here the production mechanism is the same as was discussed by us in the case of double charm production (see e.g. Ref. [7]). In the k T -factorization approach [14][15][16][17] the SPS cross section for pp → QQQQ X reaction can be written as In the formula above F g (x, k 2 t , µ 2 ) is the unintegrated gluon distribution function (uGDF). The uGDF depends on longitudinal momentum fraction x, transverse momentum squared k 2 t of the gluons entering the hard process, and in general also on a (factorization) scale of the hard process µ 2 . The elementary cross section in Eq. (2.1) can be written somewhat formally as: where E l and p l are energies and momenta of final state heavy quarks. Above only dependence of the matrix element on four-vectors of incident partons k 1 and k 2 is made explicit. In general all four-momenta associated with partonic legs enter. The matrix element takes into account that both gluons entering the hard process are off-shell with virtualities k 2 1 = −k 2 1t and k 2 2 = −k 2 2t . In numerical calculations we limit ourselves to the dominant gluon-gluon fusion channel of the 2 → 4 type parton-level mechanism.
We checked numerically that the channel induced by the qq-annihilation can be safely neglected in the kinematical region under consideration here.
The off-shell matrix elements for higher final state parton multiplicities, at the treelevel are calculated analytically applying well defined Feynman rules [18] or recursive methods, like generalised BCFW recursion [19], or numerically with the help of methods of numerical BCFW recursion [20]. The latter method was already applied for 2 → 3 production mechanisms in the case of cc + jet [21] and even for 2 → 4 processes in the case of cccc [7], four-jet [22] and cc + 2jets [10] final states.
In this paper we use the same numerical methods. The calculation is performed with the help of KaTie [23], which is a complete Monte Carlo parton-level event generator for hadron scattering processes. It can can be applied to any arbitrary processes within the Standard Model, for several final-state particles, and for any initial partonic state with on-shell or off-shell partons. The scattering amplitudes are calculated numerically as a function of the external four-momenta via Dyson-Schwinger recursion [24] generalized also to tree-level off-shell amplitudes. The phase space integration is done with the help of a Monte Carlo program with an adaptive phase space generator, previously incorporated as a part of the AVHLIB library [25,26].
In the present calculation, we use µ 2 = ∑ 4 i=1 m 2 it /4 as the renormalization/factorization scale, where m it 's are the transverse masses of the outgoing heavy quarks. We take running α s at next-to-leading order (NLO), charm quark mass m c = 1.5 GeV and bottom quark mass m b = 4.75 GeV. Uncertainties related to the choice of the parameters were discussed very recently in Ref. [10] and will be not considered here. We use the Kimber-Martin-Ryskin (KMR) [27,28] unintegrated distributions for gluon calculated from the MMHT2014nlo PDFs [29]. The above choices are kept the same also in the case of doubleparton scattering calculation except of the scales.
The effects of the c → D 0 and b → B + hadronization are taken into account via standard fragmentation function (FF) technique. We use the scale-independent Peterson model of FF [30] with ε c = 0.05 and ε b = 0.004 which is commonly used in the literature in the context of heavy quark fragmentation. Details of the fragmentation procedure together with discussion of the uncertainties related to the choice of the FF model can be found e.g. in Ref. [31]. In the last step, the cross section for meson is normalized by the relevant branching fractions BR(c → D 0 ) = 0.565 and BR(b → B + ) = 0.4.

B. Double-parton scattering
A formal theory of multiple-parton scattering (see e.g. Refs. [32,33]) is rather well established but still not fully applicable for phenomenological studies. In general, the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs). However, the currently available models of the dPDFs are still rather at a preliminary stage. So far they are formulated only for gluon or for valence quarks and only in a leading-order framework which is for sure not sufficient for many processes, especially when heavy quark production is considered.
Instead of the general form, one usually follows the assumption of the factorization of the DPS cross section. Within the factorized ansatz, the dPDFs are taken in the following form: where D 1,2 (x 1 , x 2 , µ) is the dPDF and f i (x i , µ) are the standard single PDFs for the two generic partons in the same proton. The factor θ(1 − x 1 − x 2 ) ensures that the sum of the two parton momenta does not exceed 1.
and for the pp → bbbb X (right panel) reactions.
The differential cross section for pp → QQQQ X reaction within the DPS mechanism, sketched in Fig. 2, can be then expressed as follows: where ξ 1 and ξ 2 stand for generic phase space kinematical variables for the first and second scattering, respectively. The combinatorial factor m is equal 1 for ccbb and 0.5 for bbbb case. When integrating over kinematical variables one recovers the commonly used pocket-formula: The can be safely neglected (see e.g. Ref. [34]). In this paper we use world-average value of σ eff = 15 mb provided by several experiments at Tevatron [35][36][37] and LHC [4,[38][39][40][41].
Future experiments may verify this value and establish a systematics.
There are several effects that may lead to a violation of the factorized ansatz (2.4), which seems a priori a severe approximation. The flavour, spin and color correlations lead, in principle, to interference effects that result in breaking the pocket-formula (see e.g. Refs. [32,33]). In any case, the spin polarization of the two partons from one hadron can be mutually correlated, especially when the partons are relatively close in phase space (having comparable x's). The two-parton distributions have a nontrivial color structure which also may lead to a non-negligible correlations effects. Such effects are usually not included in phenomenological analyses. They were exceptionally discussed in the context of double charm production [42] but in this case the corresponding effects were found to be very small. Moreover, including perturbative parton splitting mechanism [43][44][45] and/or imposing sum rules [46] also leads to a breaking of the pocket-formula.
However, taken the above and looking forward to further improvements in this field, here we limit ourselves to a more pragmatic approach.
In our present analysis cross sections for each step of the DPS mechanism are calcu-lated in the k T -factorization approach, that is: The numerical calculations for both SPS mechanisms are also done within the KaTie code, where the relevant fully gauge-invariant off-shell 2 → 2 matrix element M g * g * →QQ is obtained numerically. Its useful analytical form can be found e.g. in Ref. [15]. We get a very good agreement with the experimental points for both, the transverse momentum (left panel) and rapidity (right panel) B 0 meson distributions. Only the cross section in the lowest rapidity bin y ∈ (2.0, 2.5) seems to be slightly overestimated, however the experimental uncertainties in this case are noticeably larger than in other rapidity intervals. Similar high-level agreement between the k T -factorization predictions and experimental data has been also reported by us in the case of inclusive open charm meson production (see e.g. Ref. [48]). This approach was found to be very efficient also for more exclusive correlation observables [31,49]. Having those conclusions in mind we expect that the chosen theoretical framework should provide a reliable predictions also for simultaneous production of charm and bottom as well as for double bottom production. Now we go to the case of simultaneous production of charm and bottom particles.
We start with the parton-level predictions for inclusive production of ccbb final state at √ s = 13 TeV. In Fig. 4   To summarize the situation for the LHCb experiment, in Table I, we collect the integrated cross sections for D 0 B + and B + B + meson-meson pair production in nanobarns within the relevant acceptance: 2 < y D 0 ,B + < 4 and 3 < p D 0 ,B + T < 12 GeV. We predict quite large cross sections, in particular, at √ s = 7 TeV the calculated cross section for D 0 B + pair production is only 5 times smaller than the cross section already measured by the LHCb for D 0 D 0 final state [4]. The cross sections for B + B + are order of magnitude smaller than in the mixed charm-bottom mode, however, still seems measurable. In both  bottom production in the region of |y B ± | < 2.2 and 10 < p B ± T < 100 GeV. Here, crucial is the lower cut on meson transverse momenta which is quite large (much larger than in the case of the LHCb). This may lead to damping of the relative DPS contribution to the cross section under consideration.  however, the effect is not so strong as in the case of the LHCb experiment. Finally, in Table II  In our previous studies we discussed in detail production of cccc and cc + 2jets final states in order to test and explore double-parton scattering effects. In general the processes with charm production and/or jets with small transverse momenta have large contribution of double-parton scatterings. Here we have tried to complete the first stage of exploration of DPS effects in the heavy flavour sector.
In the present paper we have extended our previous studies to simultaneous produc- Next we have considered distributions for simultaneous production of charmed and bottom mesons. The DPS mechanism have been shown to dominate for small invariant masses of the DB systems. We have predicted only a small decorrelation in relative azimuthal angle, typical for DPS dominance.
The situation for bbbb and two B + B + meson production is rather similar as for the mixed heavy flavour production, but here the dominance of the DPS over SPS is limited to smaller corners of the phase space. A good description of future data will therefore require to include both DPS and SPS mechanisms simultaneously. All the considered reactions should be easily measured as the corresponding cross sections are rahter large.
A comment on possible in principle measurements is in order. Usually experimental subgroups specialize exclusively either in the production of D mesons or B mesons, simultaneous production of D and B mesons will require some coordination of the action of such different subgroups. In our opinion it would be a valueble effort. An experimental extraction of the σ eff parameter for different reactions and a comparison for different processes studied here and in our previous papers would be a simple but necessary step to better understand double scattering in a more precise way. Also a compilation of the σ eff would be important phenomenological knowledge. The factorized ansatz is an approximation and a possible deviations from it were discussed in the literature. Once such studies as discussed here are completed one can try to explore deviations from the simple approach. No clear deviations were found so far. The only exception is production of quarkonia pairs were very small values of σ eff were extracted from experimental data.
The situation in quarkonia pair production is however more complex. As disussed recently in Ref. [50] there are several single-parton mechanisms with DPS characteristics.
Such processes were not considered so far in theoretical calculations so the extraction of σ eff for these reactions is not reliable. Therefore in DPS studies one should concentrate first rather on processes with heavy quark/meson production.