Double-parton scattering effects in associated production of charm mesons and dijets at the LHC
Rafal Maciula Antoni Szczurek
hep-ph hep-ex






rafal.maciula@ifj.edu.pl Institute of Nuclear
Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31-342 Kraków, Poland


antoni.szczurek@ifj.edu.pl Institute of Nuclear
Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31-342 Kraków, Poland




We calculate several differential distributions for the production of charm and dijets.
Both single-parton scattering (SPS) and double-parton scattering (DPS) contributions are calculated
in the k_T-factorization approach. The Kimber-Martin-Ryskin unintegrated parton distributions
are used in our calculations. Relatively low cuts on jet transverse momenta are imposed to enhance 
the double-parton scattering mechanism contribution. We find dominance of the DPS contribution 
over the SPS one. We have found regions of 
the phase space where the SPS contribution 
is negligible compared to the DPS contribution.
The distribution in transverse momentum of charm quark/antiquark or
charmed mesons can be used to observe transition from the dominance of 
DPS at low transvsverse momenta to the dominance of SPS at large
transverse momenta. Very distinct azimuthal correlation patterns (for c c̅, c-jet,
jet-jet, D^0-jet, D^0 D^0) are predicted as a result of
the competition of the SPS and DPS mechanisms.



13.87.Ce, 14.65.Dw

Double-parton scattering effects in associated production
 of charm mesons and dijets
at the LHC

    Antoni Szczurek[also at University of Rzeszów, PL-35-959 Rzeszów, Poland]
    December 30, 2023
=================================================================================================






§ INTRODUCTION



The cross section for production of charm quarks or mesons is known to 
be very large especially at high energies which is caused by relatively 
small mass of the charm quark. On the other hand the mass of the charm
quark is large enough to use perturbative methods of quantum chromodynamics. 
Charm quarks are also produced abundantly in double <cit.>
or multiple <cit.> parton scattering.  The cross section for double 
c c̅  production was shown to grow considerably with the collision
energy <cit.>. We have explained that total rates as well as several
differential distributions cannot be explained without inclusion of
double parton scattering. 
Many processes in association with charm quarks or mesons are possible
and can be studied at the LHC. Recently we discussed inclusive production
of single jet associated with c c̅ or charmed mesons <cit.>.
Quite large cross sections were found there. Here we discuss inclusive
production of dijets in association with c c̅ production. 

We wish to include both single parton scattering (SPS) and 
double parton scattering (DPS) processes.
We wish to discuss whether the process can be used to extract the
so-called σ_eff parameter which governs the strength of double
parton scattering. The genral theoretical picture allows that this
quantity may depend on kinematical quantities as well as type of the process.
However, surprisingly similar values of σ_eff were
obtained from different reactions. There are exceptions, much smaller
values were obtained for double J/ψ charmonium production at 
large transverse momenta
<cit.> but there the mechanism of the reaction is not yet fully 
understood <cit.>.
We focus on how to disantangle single and double parton scattering
contributions for a simultaneous production of c c̅ (or charmed
mesons) and dijets.
In the current paper we present first predictions for the associated
production of charm and dijets as well as many differential distributions, 
many of them of the correlation character.




§ A SKETCH OF THE THEORETICAL FORMALISM





 §.§ Single-parton scattering






Within the k_T-factorization approach the SPS cross section for 
pp → cc̅ + 2jets  X reaction, sketched in Fig. <ref>, can be written as


    d σ_p p → cc̅ + 2jets = ∑_ij∫ d x_1 d^2 k_1t/π d x_2 d^2 k_2t/π F_i(x_1,k_1t^2,μ^2)  F_j(x_2,k_2t^2,μ^2)
    d σ̂_ij → c c̅ + 2part.  .


In the formula above F_i(x,k_t^2,μ^2) is a unintegrated
parton distribution function (uPDF) for a given type of parton i = g, u, d, s, u̅, d̅, s̅. The uPDFs depend on longitudinal momentum fraction x, transverse momentum squared k_t^2 of the partons entering the hard process,
and in general also on a (factorization) scale of the hard process μ^2.
The elementary cross section in Eq. (<ref>) can be written
somewhat formally as:


    d σ̂_ij → c c̅ + 2part. =
    ∏_l=1^4d^3 p_l/(2 π)^3 2 E_l 
    (2 π)^4 δ^4(∑_l=1^4 p_l - k_1 - k_2) ×1/flux| M_i^* j^* → c c̅ + 2part.(k_1,k_2)|^2  ,


where E_l and p_l are energies and momenta of final state particles. 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 partons entering the hard
process are off-shell with virtualities k_1^2 = -k_1t^2 and k_2^2 = -k_2t^2.
We take into account all 9 channels of the 2 → 4 type contributing to the cross section at the parton-level:



#1 = g   g → g   g   c  c̅  #2 = g   g → q  q̅  c  c̅
 #3 = g   q → g   q   c  c̅

#4 = q   g → q   g   c  c̅  #5 = q  q̅→ q'  q̅'   c  c̅
 #6 = q  q̅→ g   g   c  c̅

#7 = q   q → q   q   c  c̅  #8 = q   q' → q   q'   c  c̅
 #9 = q  q̅→ q  q̅  c  c̅.



The off-shell matrix elements are well known only in the leading-order (LO) and only for limited types of QCD 2 → 2 processes (see e.g. heavy quarks <cit.>, dijet <cit.>, Drell-Yan <cit.>). Some first steps to calculate NLO corrections in the k_T-factorization framework have been done only very recently for diphoton production <cit.>.
For higher final state parton multiplicities, relevant amplitudes can be calculated analytically applying suitably defined Feynman rules <cit.> or recursive methods, like  generalised  BCFW recursion <cit.>, or numerically with the help of methods of numerical BCFW recursion <cit.>. The latter method was already successfully applied for 2 → 3 production mechanisms in the case of cc̅ + jet <cit.> and even for 2 → 4 processes in the case of cc̅ cc̅ <cit.> and four-jet <cit.> final states.

In this paper we follow the same numerical techniques. The calculation has been performed with the help of KaTie <cit.>, 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 up to four final-state particles and beyond, and for any initial-state partons on-shell or off-shell. The scattering amplitudes are calculated numerically as a function of the external four-momenta via Dyson-Schwinger recursion <cit.> generalized also to tree-level off-shell amplitudes. The phase space integration in KaTie is done with the help of a full Monte Carlo program with an adaptive phase space generator, previously incorporated as a part of the AVHLIB library  <cit.>, that deals with the integration variables related to both the initial-state momenta and the final-state momenta. KaTie can be used for single-parton scattering as well as
for multi-parton scattering processes. 

In the present calculation, we use μ^2  = m_1t^2+m_2t^2+p_3t^2+p_4t^2/4 as the renormalization/factorization scale, where m_1t, m_2t are the transverse mass of the outgoing c-quark and c̅-antiquark and p_3t, p_4t are the transverse momenta of outgonig jets. Furthermore, we take running α_s at next-to-leading order (NLO) and
charm quark mass m_c = 1.5 GeV. The parameters are the same for both k_t-factorization and in the reference
collinear-factorization calculations. Uncertainties related to the choice of the parameters were discussed
e.g. in Ref. <cit.> and will be not considered here. We use the Kimber-Martin-Ryskin (KMR) <cit.> unintegrated distributions for quarks and gluon calculated from the MMHT2014nlo PDFs <cit.>. The above choices are kept the same also in the case of double-parton scattering calculation except of the scales.  





 §.§ Double-parton scattering


According to the general form of the multiple-parton scattering theory (see e.g. Refs. <cit.>)
the DPS cross sections can be expressed in terms of the double parton distribution functions (dPDFs).
These objects should fulfill sum rules and take into account all the correlations between the two partons, including transverse and longitudinal momenta correlations as well as color, flavour and spin correlations. The theory of dPDFs is well established but still not fully applicable for phenomenological studies. The currently available models of the dPDFs are still rather 
at a preliminary stage. So far they are formulated exlusively for gluon or for valence quarks and only in a leading order framework
which may be not sufficient for many processes, especially when charm production is considered.    

Instead of the general form, one usually follows the assumption of the factorization of the DPS cross section.
Within this framework, the dPDFs are taken in the following factorized form:  


    D_1, 2(x_1,x_2,μ) = f_1(x_1,μ)  f_2(x_2,μ)  θ(1-x_1-x_2)   ,


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. 





According to the above, the differential cross section for pp → cc̅ + 2jets  X reaction within the DPS mechanism, sketched in Fig. <ref>, can be expressed as follows: 


    dσ^DPS(c c̅ + 2jets)/dξ_1dξ_2 = ∑_i,j 1/σ_eff·dσ^SPS(g g → c c̅)/dξ_1·σ^SPS(i j →2jets)/dξ_2,


where ξ_1 and ξ_2 stand for generic phase space kinematical variables for the first and second scattering, respectively.
When integrating over kinematical variables one recovers the commonly used pocket-formula:


    σ^DPS(c c̅ + 2jets) = ∑_i,j σ^SPS(g g → c c̅) ·σ^SPS(i j →2jets)/σ_eff  .



The effective cross section σ_eff provides a proper normalization of the DPS cross section and can be roughly interpreted 
as a measure of the transverse correlation of the two partons inside 
the hadrons. The longitudinal parton-parton correlations should become far less
important as the energy of the collision is increased, due to the increase in the parton multiplicity. It is belived that for small-x partons and for low and intermediate scales the possible longitudinal correlations can be safely
neglected (see e.g. Ref. <cit.>). 
In this paper we use world-average value of σ_eff = 15 mb provided by 
several experiments at Tevatron <cit.> and LHC <cit.>. Future experiments may verify this value.

There are several effects that may lead to a violation of the factorized anstaz, which is a severe approximation.
The flavour, spin and color correlations lead, in principle, to interference effects
that result in breaking of the pocket-formula (see e.g. Refs. <cit.>. 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 <cit.> but in this case the corresponding effects were found to be very small.
Moreover, including perturbative parton splitting mechanism <cit.> and/or imposing sum rules <cit.> also leads to a breaking of the pocket-formula.  
However, taken the above and looking forward to further improvements in this field, we choose to
limit ourselves to a more pragmatic approach in this paper.

In our present analysis cross sections for each step of the DPS mechanism are calculated in the
k_T-factorization approach, that is:


    d σ^SPS(p p → c c̅  X_1)/d y_1 d y_2 d^2 p_1,t d^2 p_2,t       = 1/16 π^2 ŝ^2∫d^2 k_1t/πd^2 k_2t/π| M_g^* g^*→ c c̅|^2
          ×  δ^2 ( k⃗_1t + k⃗_2t - p⃗_1t - p⃗_2t)
     F_g(x_1,k_1t^2,μ^2)  F_g(x_2,k_2t^2,μ^2),




    d σ^SPS(p p →2jets  X_2)/d y_3 d y_4 d^2 p_3,t d^2 p_4,t       = 1/16 π^2 ŝ^2∑_ij∫d^2 k_3t/πd^2 k_4t/π| M_i^* j^*→2part.|^2
          ×  δ^2 ( k⃗_3t + k⃗_4t - p⃗_3t - p⃗_4t)
     F_i(x_3,k_3t^2,μ^2)  F_j(x_4,k_4t^2,μ^2).


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 elements M_g^* g^*→ c c̅ and M_i^* j^*→2part. are obtained numerically. Their useful analytical form can be found in Ref. <cit.> for c c̅ and in Ref. <cit.> for dijet production.  
Here, the strong coupling constant α_S and uPDFs are taken the same as in the case of the calculation of the SPS mechanism for cc̅ + 2jets production. The factorization and renormalization scales for the two single scatterings are μ^2  = m_1t^2+m_2t^2/2 for the first, and μ^2  = p_3t^2+p_4t^2/4 for the second subprocess.
The framework of the k_T-factorization approach together with the KMR uPDFs was used with success in describing inclusive spectra of
D, DD̅ correlations <cit.> as well as in the case of dijet production <cit.> and therefore can be expected to be a good starting point for the DPS predictions for the cc̅ + 2jets final state.




§ NUMERICAL RESULTS





 §.§ c c̅ + 2jets


We start presentation of the numerical predictions with the results for production of charm quark-antiquark pair associated with two jets at √(s) = 13 TeV.
Here, the phase space for charm quarks is almost unlimited, with broad range of rapidities |y_c| < 8 and without any cuts on their transverse momenta. For jets we keep the kinematical regime relevant for the ATLAS/CMS experiments, with |y_jet| < 4.9 and with transverse momentum cut
p_T^jet > p_T,cut^jet = 20 GeV for leading and subleading jet.    





In Fig. <ref> we show transverse momentum distributions of charm quark/antiquark for single-parton scattering (left panel) and for double-parton scattering (right panel) mechanism at √(s) = 13 TeV. The three different histograms correspond to different approaches used in the calculations: LO collinear approximation (dotted), k_T-factorization (dashed) and k_T-factorization with extra cut on incident parton transverse momenta k_T < 20 GeV (solid). The KMR model for uPDFs, due to its special construction, allows for additional emission of hard gluon or quark from uPDF when the initial parton that enters the hard scattering has transverse momentum k_T > μ. To make predictions for the final state with c c̅-pair and with exactly two jets one needs to introduce the special limitation: k_T < p_T,cut^jet. We see that the effects related with this cut become important only when going to larger transverse momenta of charm quark and are much stronger in the case of the DPS results, where the cut is applied for all four incident partons. In both cases, for the SPS and the DPS calculation, the LO collinear approximation leads to a significantly smaller cross sections than those obtained within the k_T-factorization in the whole considered range of charm quark transverse momenta. 





In the left panel of Fig. <ref> we show again the transverse momentum distribution of charm quark/antiquark. Here, the DPS (dashed histogram) and the SPS (dotted histogram) contributions calculated within the k_T-factorization approach are shown together on the same plot.
The DPS contribution clearly dominates over the SPS one in the region of c-quark p_T < 15 GeV. In the right panel of Fig. <ref>
we present rapidity distribution of charm quark/antiquark. The two upper histograms correspond to the DPS mechanism and the two lower histograms   
correspond to the SPS contribution. Here, results for the k_T-factorization approach (solid histograms) are shown together with results obtained with the LO collinear approach (dotted histograms). The DPS component significantly dominates in the whole considered range of rapidities and the relative contribution of the SPS mechanism becomes even smaller when moving to forward/backward region.  


Figure <ref> shows transverse momentum distributions of the leading (left panel) and the subleading (right panel) jet calculated in the k_T-factorization approach for the SPS (lower histograms) and for the DPS mechanism (upper histograms). The DPS component dominates in the whole range of the considered jet transverse momenta.





In Fig. <ref> we present some cc̅ correlation observables. The left panel shows distribution in azimuthal angle φ_cc̅ between the c-quark and the c̅-antiquark. Again, the DPS mechanism dominates over the SPS one in the whole range of relative azimuthal angle. The shape of the distribution for the DPS is determined directly by the inclusive single cc̅-pair production mechanism.
In the case of the SPS component we observe an evident enhancement of the cross section in the region of rather small angles, which is not a typical behaviour e.g. in the case of inclusive charm production. In the right panel of Fig. <ref> we present differential cross section as a function of invariant mass of the cc̅-system M_cc̅. Both mechanisms have different slope of the distribution. At small invariant masses the DPS component clearly dominates over the SPS one. In the region of M_cc̅ < 20 GeV the difference is bigger than one order of magnitude. Both of the mechanisms become comparable starting from M_cc̅≈ 70 GeV.     
   




Figure <ref> presents differential distributions as a function of the azimuthal angle φ_jj between the two jets (left panel)
and as a function of the dijet invariant mass M_jj (right panel). The SPS component is much more decorrelated in azimuthal angle than the DPS one.
Both of them have a similar shape of the dijet invariant mass and differ only as far as only the normalization is considered. The DPS mechanism dominates in the whole range of the dijet invariant mass. 





Considering the cc̅ + 2jets final state one can also look at the correlations between c-quark and associated jet.
For example, in Fig. <ref> we show the correlation distributions in the azimuthal angle φ_c-jet (left panel)
and the rapidity difference Δ Y_c-jet between the c-quark (c̅-antiquark) and the leading jet (right panel). 
Both correlation observables are predicted to be dominated by the DPS mechanism in the whole range of φ_c-jet and Δ Y_c-jet , respectively.







 §.§ D^0 + 2jets


Now we move to the predictions for single D^0 meson production in association with exactly two jets.
The effects of the c → D^0 hadronization are taken into account via standard fragmentation function technique.
For this purpose, we employ the scale-independent Peterson model of fragmentation function <cit.> with ε_c = 0.05 which is commonly-used in the literature in the context of heavy quark fragmentation.  
Details of the fragmentation procedure applied here and useful discussion of the uncertainties related to the choice of the fragmentation function
can be found e.g. in Ref. <cit.>. In the last step, the cross section for meson is normalized by the relevant branching fraction BR(c → D^0) = 0.565.    

In this analysis, the D^0 meson is required to have |y^D^0| < 2.5 and p_T^D^0 > 3.5 GeV and the rapidities of both associated jets are |y^jet| < 4.9, which corresponds to the ATLAS detector acceptance. In Table <ref> we collect the corresponding integrated cross sections for inclusive D^0+2 jets production in pp-scattering at √(s) = 13 TeV for different cuts on transverse momenta of the associated jets, specified in the left column. The predictions are obtained within the k_T-factorization approach for the KMR uPDFs with the k_T < p_T,cut^jet constrain. We found large cross sections, of the order of a few, and up to even tens of microbarns, depending on the cuts on transverse momenta of the associated jets. The cross sections are dominated by the DPS mechanism with the relative DPS contribution
at the level of 70 - 80 %.  





In Fig. <ref> we show the differential cross section as a function of transverse momenta of the D^0 meson for two different sets of cuts on transverse momenta of the associated jets (left and right panel). The DPS (dashed line) and the SPS (dotted line) components are shown separately together with their sum (solid line). We observe that in the region of D^0 meson transverse momenta p_T < 10 GeV the DPS mechanism significantly dominates over the SPS one.





Figure <ref> shows a very interesting distributions in the azimuthal angle φ_D^0-jet
between the D^0 meson (D^0 antimeson) and the leading jet, again for two different sets of cuts on transverse momenta of the associated jets (left and right panel). We see that the presence and the dominant role of the DPS component leads to a significant enhancement of the cross section and to a visible decorrelation of the distribution in contrast to the pure SPS-based predictions.   







 §.§ D^0D̅^̅0̅ + 2jets


In the present analysis, we also consider the case of production of the D^0D^0-pair in association with two jets.
So now both, D^0-meson and D^0-antimeson are required to enter the ATLAS detector acceptance.
The corresponding theoretical cross sections are collected in Table <ref>. Here, the predicted cross sections for  D^0D^0+2 jets are slightly
smaller than in the case of D^0+2 jets production (see Table <ref>) but still large (in the best scenario, of the order of a few microbarns).
Also the relative DPS contribution is somewhat reduced and varies at the level of 50 - 70 %.   





In the case of the D^0D^0+2 jets final state we also find a very interesting correlation observable that may be useful to distinguish between the DPS and SPS mechanisms. 
Figure <ref> presents the distributions in azimuthal angle φ_D^0D^0
between the D^0 meson and D^0 antimeson in the case of D^0D^0+2 jets production.
One can observe an evident enhancement of the cross section in the region of φ_D^0D^0 > π/2 caused by the presence of the DPS mechanism.








§ CONCLUSIONS



In the present paper we have calculated for a first time cross sections 
for simultanous production of c c̅ (or D mesons) and dijets. 
Rather low transverse momentum cuts on jets have been used in order to 
enhance the DPS contribution. Both single and double parton scatttering
mechanisms have been included.

Several differential distributions have been shown and discussed.
The calculation have been performed for current LHC collision energy
√(s) = 13 TeV within the k_t-factorization approach using the
Kimber-Martin-Ryskin unintegrated gluon distributions. The same
formalism turned out previously to be very successfull for description 
of production of one and two pairs of c c̅ and production of dijets.

It was shown that the DPS contribution considerably dominates over the
SPS contribution for small transverse momenta of c and/or c̅.  
At larger transverse momenta of c and/or c̅ the SPS contribution
takes over.
The distribution in transverse momentum of D mesons could be used to
pin down the competition of both mechanisms.
Very interesting are also azimuthal correlations between jets, between
jet and charm quark/antiquark or jet and D mesons or even between charmed
mesons. The corresponding experimental distributions can be used
to further pin down underlying production mechanism.

We have identified regions of the phase space where the DPS
contributions significantly dominate. A future experimental cross sections in
these regions could be used to determine the σ_eff parameter. 
In general, this quantity can dependend on several kinematical
quantities. Surprisingly similar values were obtained from the analysis of different
processes. We have discussed conditions how to extract σ_eff 
from the discused in the present paper associated production of charm and dijets. 
Our analysis performed here shows that the extraction should be almost
free of systematic errors as the DPS contribution clearly dominantes
over the SPS one for the discussed reaction.





Acknowledgments

We are particularly indebted to Andreas van Hameren for a help in using KaTie code.
This study was partially
supported by the Polish National Science Center grant
DEC-2014/15/B/ST2/02528 and by the Center for Innovation and
Transfer of Natural Sciences and Engineering Knowledge in
Rzeszów.



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