NLO Monte Carlo predictions for heavy-quark production at the LHC: pp collisions in ALICE

Next-to-leading order (NLO) QCD predictions for the production of heavy quarks in proton-proton collisions are presented within three different approaches to quark mass, resummation and fragmentation effects. In particular, new NLO and parton shower simulations with POWHEG are performed in the ALICE kinematic regime at three different centre-of-mass energies, including scale and parton density variations, in order to establish a reliable baseline for future detailed studies of heavy-quark suppression in heavy-ion collisions. Very good agreement of POWHEG is found with FONLL, in particular for centrally produced D^0, D^+ and D^*+ mesons and electrons from charm and bottom quark decays, but also with the generally somewhat higher GM-VFNS predictions within the theoretical uncertainties. The latter are dominated by scale rather than quark mass variations. Parton density uncertainties for charm and bottom quark production are computed here with POWHEG for the first time and shown to be dominant in the forward regime, e.g. for muons coming from heavy-flavour decays. The fragmentation into D_s^+ mesons seems to require further tuning within the NLO Monte Carlo approach.


GM-VFNS
Fragmentation approach in FONLL perturbative FF satisfying DGLAP evolution in the scale non-perturbative part describes hadronisation of heavy quark into heavy hadron (fitted from LEP data) assessed clearly and unambiguously. In Ref.
[10] the CDF Collaboration compares its data to a theoretical prediction obtained by convoluting the NLO cross section for bottom quarks with a Peterson fragmentation function. They use = 0.006 ± 0.002, which is the traditional value proposed in Ref. [20]. They claim that their data is a factor of 2.9 higher than the QCD calculation.
The purpose of this Letter is precisely to implement correctly the effect of heavy quark fragmentation in the QCD calculation. Several ingredients are necessary in order to do this: • A calculation with resummation of large transverse momentum logarithms at the next-to-leading level (NLL) should be used for heavy quark production [21], in order to correctly account for scaling violation in the fragmentation function.
• A formalism for merging the NLL resummed results with the NLO fixed order calculation (FO) should be used, in order to account properly for mass effects [22]. This calculation will be called FONLL in the following.
• A NLL formalism should be used to extract the non-perturbative fragmentation effects from e + e − data [23-29].
We begin by pointing out that, as shown in Refs. [27,28], the value = 0.006 is appropriate only when a leading-log the region of interest ranges from 3 to 5. It is therefore clear that, when fitting e + e − data, getting a good determination of the moments of the non-perturbative fragmentation function between 3 and 5 is more important than attempting to describe the whole z spectrum.
FIG. 1. Moments of the measured B meson fragmentation function, compared with the perturbative NLL calculation supplemented with different D(z) non-perturbative fragmentation forms. The solid line is obtained using a one-parameter form fitted to the second moment. Fig. 1 shows the moments calculated from the x E (the B meson energy fraction with respect to the beam energy) distribution data for weakly decaying B mesons Non-perturbative fragmentation fitted using moments

NLO cross-sections
The first term in the square bracket represents the production of an event w kinematics, and phase space Φ B . In the Higgs example, it represents a Higgs bo transverse momentum. The second term represents the full real process, with a Higgs and a parton, balanced in transverse momentum. The above formula r probability that either event is produced. The shower unitarity Equation 11 is then written in the general form where it is intended that the dΦ rad integration is limited to the region where p In Figure 2 we give a pictorial representation of the distribution of the tr mentum of the Higgs boson at fixed rapidity at NLO order (i.e. O(α 3 S )), from algorithm, and from a MEC shower algorithm. For the NLO result, one should the NLO curve diverges at small p T up to a tiny cutoff, and that a tiny bin large, negative value is located at p T = 0. The resummation of collinear and s ties performed by the shower algorithm using the exact real emission cross sec differ from the LO one at p T around 40 GeV, and for smaller p T it tames the the NLO cross section. The shower approximation has the same behaviour for small p T , but it drops rapidly as p T approaches the maximum scale of radiatio the shower algorithm (an exact implementation of Equation 10 would imply t section vanishes exactly for p T ≥ Q. Subsequent emissions in the shower proc to smear the region of p T ≈ Q). The area under the two shower curves equ cross section.
The main objective of a NLO+PS implementation is to improve the showe tion, in such away that it achieves NLO accuracy for inclusive quantities. Thu 7 NLO cross-sections complicated objects -combining 2 types of processes real emission corrections -containing IR divergence -phase-space with n+1 particles R Cancellation of UV divergence 'simple' through renormalization of couplings constants etc.

Cancellation of IR divergence only in sufficiently inclusive quantities (!)
To cancel IR singularities in each part separately, one introduces auxiliary subtraction terms & one has to factorize the phase-space R ( B , rad ) Imperfect cancellation of singularities for exclusive quantities e.g. in a Monte Carlo

NLO cross-sections & parton shower
How to use NLO cross-sections in parton showers ?
In parton shower language an equivalent of a NLO cross-section is a cross-section with one emission

virtual corrections real corrections
Shower cross-section contains approximate virtual & real corrections in the collinear limit NOTE: Sudakov form-factor resums universal part of the virtual(!) correction Goal of NLO Monte Carlos is to recover exact NLO cross-sections when we expand the parton shower cross-section in ↵ s

POWHEG method
Main idea -replace the parton shower approximation for no radiation and the first (hardest) emission by the full NLO calculation Modified Sudakov form-factor & modified shower generating emission only with lower p T than the first emission In GM-VFNS the decay of a B-hadron into lepton parametrized as a "lepton fragmentation"

Comparison with ALICE data
Heavy flavor decay into electrons @ TeV and central rapidity |y| < 0.8 p s = 5.023 Possible baseline for future heavy quark production measurements in pPb collisions

Conclusions
All three methods describe the data within experimental and theoretical errors Different treatment of fragmentation functions might explain small discrepancies