Forward charm-production models and prompt neutrinos at IceCube

We investigate the prompt neutrino background at IceCube, as determined from forward charm. We consider the role of intrinsic charm and of a recombination model and show that the contribution of these mechanisms is at most a factor two.


Introduction
This century is witnessing the birth of a new astronomy making use of all the fundamental interactions to explore hidden processes in the Universe: the multi-messenger approach which enjoyed its first successes with the observation of GW170817/GRB170817A [1], and with the recent IceCube breakthrough [2] which identified the blazar TXS 0506+056 as the source of IceCube-170922A. This method will facilitate easier characterisation of sources, as well as provide a new tool to discover unknown objects [3]. It may also open a window to particle physics discoveries [4,5]. Important players in this multi-messenger program are the neutrino telescopes IceCube and ANTARES/KM3NET, which use large volumes of ice/water in the Earth. Their main drawback is that the Earth comes with an atmosphere, which produces an insurmountable background from the collisions of cosmic rays on the air. These collisions, via the weak decay of light mesons, produce a large neutrino background. However, at high energy, these light mesons interact with air molecules before having the chance to decay. Hence, the neutrino background is increasingly suppressed with higher energies and higher time dilation. However, there is one exception to this-mesons containing heavy quarks will quickly decay and produce neutrinos. Heavier quarks are produced through more energetic gluons, which are scarce compared to low-energy ones. This means that bb and tt production is suppressed compared to cc. Consequently, the main background at high energies comes from decays of charmed mesons.
The highest-energy neutrino background comes from forward production, and this implies that the charm mass is the only available scale for perturbative QCD. Inclusive QCD calculations of quark production start to make sense at the charm mass scale but require at least a next-to-leading order evaluation [6][7][8][9]. However, we are interested in the semi-inclusive calculation as the charm quark is singled out at the high momentum, and this implies we are sensitive to non-perturbative effects which cannot be factored out into fragmentation or structure functions. This makes the highest-energy charm background model dependent. Recent perturbative calculations of charmed-meson production are already consistent with observations from the latest collider experiments, including ALICE, ATLAS, and CMS at low rapidities 1 This only leaves a small window of rapidities where the diffractive cross-section can significantly contribute, viz. at very high rapidities.
There are currently no data corresponding to D ±,0 production at rapidities beyond those accessible to LHCb at √ s = 13 TeV; as a consequence the constraints on theoretical parameters governing the forward production of these mesons are rather weak. At these very high rapidities, two classes of models are usually considered to describe the production of heavy mesons: a) models where the charm can intrinsically carry a large momentum that can then be inherited by the D mesons [11][12][13], and b) those that assume that the D is boosted because its light quark is a spectator valence quark from one of the initial nucleons [14,15]. The former is straightforwardly implemented via modified structure functions, whereas the latter require dedicated computations involving new fragmentation mechanisms. Our goal in this work is to estimate the maximum prompt atmospheric neutrino flux that may be seen at IceCube by allowing the parameters in these models to go as high as possible whilst still maintaining consistency with data at central rapidities.
This paper is organised as follows: in Section I, we consider the predictions of the leading-twist NLLO charm production calculation for very high rapidities, and carefully evaluate the uncertainties. In Section II, we consider and constrain the possible modification to the charm momentum from intrinsic charm [11]. In Section III, we consider the recombination process of Braaten, Jia and Mehen [14]. Finally, in Section IV, we estimate the corresponding prompt neutrino fluxes, and show that this component of the background remains small in all cases.

High-rapidity D-meson from the high-x F NLO contribution
As previously noted, the perturbative treatment of Bhattacharya-Enberg-Reno-Sarcevic-Stasto (BERSS) [9] has been tuned to the experimental results for D-meson production cross-sections across a wide range of energies. At the low-energy end of the spectrum, this includes data from fixed-target, proton and pion beam experiments at the CERN SPS at a few hundred GeV, and the 920 GeV fixed-target experiment HERA-B; further up the energy ladder are the somewhat discordant results from STAR and PHENIX fixed-target experiments, both involving a beam of energy 200 TeV; and finally, constraints at the highest energies come from D-meson production data at the LHC at √ s = 7 and 13 TeV. The total number of experimental points is thirteen.
The tunable parameters in the theory are the charm mass and the factorisation and renormalisation scales. By fitting the theoretical cross-sections, computed perturbatively to the next-to-leading order, to observed data at these widely disparate energies, one is able to obtain the best-fit and limiting values of these scales consistent with observations. The overall fit has a χ 2 /d.o.f. of 1.31. Allowing the factorisation and renormalisation scales to run proportionally to the transverse mass of the charm quark m T = m 2 c + p 2 T , where p T is its transverse momentum, the BERSS analysis finds best fit values at (M F /m T , µ R /m T ) = (2.1, 1.6) for the choice of fixed m c = 1.27 GeV. The upper and lower limits are obtained with reference to 1σ errors on experimental data, as a convolution of the 1σ uncertainty from PDF's at high x (as an example, Fig. 1 shows leading-order PDFs) and cross sections computed by letting both scales vary freely.

Intrinsic Charm
The simplest way to boost charm is to recognise that sometimes the proton has a fluctuation that produces a cc pair, as shown in Fig. 2. This was recognised a long time ago [11] and lead to the introduction of intrinsic charm. It is the part of the structure functions that comes from the non-perturbative large-distance gluon field and it cannot be described via DGLAP evolution. Modern structure functions include this contribution as an initial parametrisation in some of their sets. Here we will consider the latest parametrisation due to the CT collaboration [16]. They have looked at the intrinsic charm content in the proton at the leading (LO), next-to-leading (NLO) and next-to-next-to-leading orders parametrisations of the intrinsic charm distribution function (as shown in Fig. 1), and we choose the one with the highest charm content.
As the contribution of intrinsic charm will turn out to be small, it is sufficient to look at the leading-order calculation, for which we use the expressions of [13]. There is a large  uncertainty attached to the value of the quark mass, and we choose here a small charm mass, m c = 1.27 GeV, that corresponds to a high neutrino flux, as in [4,5]. We also use leading-order structure functions, and keep the Kramer-Kniehl fragmentation functions [17]. The results are shown in Fig. 3, for a representative value of √ s = 60 GeV, where we add the intrinsic charm contribution to the NLO gluon-fusion cross section. We see that intrinsic charm makes a difference only at very high x F > 0.9 where it can change the cross section by a factor of two of larger. However, this is the region of x F in which the cross section is suppressed, and we shall see that the effect on the neutrino flux is small. Our prediction for the prompt neutrino flux when including intrinsic charm contribution is consistent with results from other recent works [18][19][20].

The BJM recombination formalism
In the Braaten-Jia-Mehen (BJM) formalism for heavy-quark recombination with an active light quark, the D-production cross section is expressed to leading order as product of two factors: The first factor is the usual perturbative term. Whereas the standard calculation (BERSS) calculates this factor to the next-to-leading order and then multiplies it by fragmentation functions to make D-mesons, an extra contribution is considered here, in which the light quarks do not come from the vacuum but rather from the proton. This contribution is parametrised by the non-perturbative numerical factor ρ [(qc) n → D], which is calculated from an effective Lagrangian. This formalism allows for coloured bound states or spin-flips. For example, the production of a colour singlet, spin-conserving D + meson is given by the product of and ρ sm 1 = ρ cd 1 S (1) 0 → D + , where, S, T , and U are modified Mandelstam variables that can be expressed in terms of the final state transverse momentum p ⊥ and rapidity ∆y as Expressions for other cross-sections are given in [14]. Note that if one includes fragmentations cd 1 S (1) 0 → D + X then ρ becomes a free positive parameter, that can be adjusted to the data. BJM found that one only needs the colour-singlet spin-conserving term to reproduce the high-x F charge asymetries in D + /D − production observed in π − p by the E791 experiment, with a value ρ 1 = 0.06. They also fitted photoproduction data, which leads to ρ 1 ≈ 0.15. More recently, these expressions have been used [21] to reproduce the D asymmetry observed at LHCb. There it was found, at somewhat lower values of x F , that coulour-singlet spin-non-flip terms were not enough to reproduce the asymmetry. As we concentrate on the high-x F data, we shall adopt the BJM   We see in Fig. 4 that the inclusion of the recombination enhances the forward D ±,0 production by up to a factor of 2 for the higher value of ρ = 0.15 at x F ≈ 0.9. Note that the BJM model is valid at high x F only and that it may get substantial corrections at low and medium values of x F , hence the beginning of the curve of Fig. 4 is only indicative.

Prompt neutrino fluxes from the best-fit cross-section
Using the BJM cross-sections with the best-fit parameters above, we can determine the corresponding neutrino flux expected at Earth as a consequence of interactions of cosmic rays with atmospheric nuclei assumed to contain A nucleons. As an estimate of the incident cosmic-ray proton flux, we use the models of Gaisser [22], specifically his protonrich estimates designated as H3p. The computation of the corresponding neutrino flux follows the semi-analytical procedure outlined in standard literature (see e.g., [23], [24]). Briefly, working in the exponential atmosphere approximation, where the column density as a function of height is given by ρ(h) = ρ 0 exp(h/h 0 ) with ρ 0 = 2.03 × 10 −3 g/cm 3 and h 0 = 6.4 km, the low-and high-energy lepton fluxes may be expressed in simple semianalytic forms in terms of spectrum-weighted Z-moments. These Z-moments relate to the conversion of the incoming proton content in the cosmic-ray flux to the heavy-meson flux produced therefrom (Z ph ), and then from the latter to the final leptonic flux reaching the detector (Z h ). Additional moments Z pp and Z hh describe respectively the energy losses of the protons in collisions with air nuclei not leading to meson production, and the energy losses of the meson before their decay resulting in leptons . The full procedure is described in [8], and, for brevity, we refer the reader to the discussion in Sec. 3 thereof rather than repeat it here. Using this machinery, but with additional contributions to D ±,0 production from non-perturbative diffractive processes in the forward x F region, we have computed the total prompt neutrino flux in the intrinsic-charm and scenarios. These are shown in Fig. 5.

Event rates
IceCube has been steadily accumulating events over the last seven years. Its latest results [25] see 56 high-energy starting events within the energy range of 10 TeV-2.1 PeV. By looking at the angular distribution of the incident background neutrinos, IceCube is capable of distinguishing between those from the prompt flux, which shows a largely flat distribution, and those from the conventional flux which is dominated by the vertical flux and suppressed toward the horizons. With present data, IceCube sees no evidence of the prompt flux yet, and accordingly sets a 90% confidence level upper bound at about 0.52 times the best-fit from [26]. The total prompt flux, even when including that from the BJM recombination and intrinsic charm, is consistent with this limit.
The overall modification to the prompt atmospheric neutrino background in terms of IceCube 6-year event rates plotted against the energy (E dep ) deposited in the detector, courtesy interactions of the lepton with detector nuclei, is shown in Fig. 6. While we have not re-evaluated the modification to the statistical significance of the non-atmospheric signal in light of the modified background, it is clear from looking at the figure that the change will be negligible.

Conclusions
We have evaluated the upper limit to the contribution to prompt neutrino background from diffractive forward-rapidity cross sections by surveying existing models in the literature. As the rapidities where such contributions can be significant are limited in range, viz. at very high x F , their contribution to the overall prompt-neutrino flux is limited to the very high energies at IceCube E 200 TeV. As such, the background, even when accounting for novel diffractive production mechanisms, is a rather minor player in comparison to the flux of nonterrestrial neutrinos. We have evaluated the upper limit to this component, maintaining consistency with constraints at low and middle x F from accelerator experiments, and have estimated the expected event rates.
It is evident, from analyses at IceCube and ultra-high-energy cosmic-ray observatories, that an understanding of the origin of the extragalactic particles seen at energies of hundreds of TeV and higher requires quantitatively enhanced data, and more precision estimates of the various ingredients involved in their theoretical modeling. The background is an important part of this understanding.
We have shown here that there are large uncertainties in the QCD modeling of the prompt signal. However, in all considered cases, the contribution of novel mechanisms does not contribute significantly to the prompt neutrino signal.