Phenomenological NLO analysis of eta(c) production at the LHC in the collider and fixed-target modes

In view of the good agreement between the LHCb prompt-eta(c) data at sqrt(s)=7 and 8 TeV and the NLO colour-singlet model predictions --i.e. the leading v^2 NRQCD contribution--, we provide predictions in the LHCb acceptance for the forthcoming 13 TeV analysis bearing on data taken during the LHC Run2. We also provide predictions for sqrt(s)=115 GeV for proton-hydrogen collisions in the fixed-target mode which could be studied during the LHC Run3. Our predictions are complemented by a full theoretical uncertainty analysis. In addition to cross section predictions, we elaborate on the uncertainties on the p bar-p branching ratio --necessary for data-theory comparison-- and discuss other usable branching fractions for future studies.


Introduction
In 2014, LHCb released the first experimental study of prompt-η c hadroproduction at the LHC [1] at √ s = 7 and 8 TeV. It was found that the cross section measured by LHCb was compatible with a negligible contribution of Colour-Octet (CO) transitions. More quantitatively, this observation combined with Heavy-Quark-Spin Symmetry (HQSS) yielded severe constraints on the corresponding CO transitions at work on J/ψ production [2][3][4][5]. These are so stringent that only one fit [2] currently survives these constraints at the expense of a slight tension with the CDF polarisation data [6] 1 . For reviews on quarkonium production, the reader is referred to Refs. [8][9][10][11][12][13] In this paper, we provide predictions for prompt-η c hadroproduction at √ s = 13 TeV to further test the compatibility between the Colour-Singlet (CS) contributions and the data and then in turn to refine the constraints on the Long-Distance Matrix Elements (LDMEs) associated with the dominant CO contributions. See [14] for a recent similar study for the η c case for which forthcoming data will also be invaluable. Since such constraints need to be extracted taking a proper account of both theoretical and experimental uncertainties, we also elaborate on our knowledge of the branching fractions of the decay channels which can be used by LHCb as well as on the scale and Parton-Distribution Functions (PDFs) uncertainties of the CS cross-section predictions.
The structure of the article is as follows. Section 2 is devoted to the discussion on the decay channels. Section 3 explains the theory framework we have used to provide CS NLO predictions and gathers our predictions both for the collider kinematics and for the fixed-target kinematics. Section 4 gathers our conclusion.

Discussion on the decay channels
The decays of non-1 −− charmonium states to the experimentally clean di-muon channel are strongly suppressed and hence these states can only be reconstructed using decays to hadrons or their radiative transitions to underlying charmonium states. In this section we discuss possible decay channels to study η c , h c and η c , which cannot be accessed using their decays to µ + µ − or J/ψγ. The known branching fractions [15] of the decays discussed below are summarised in Tab. 1. Many of these branching fractions can be measured more precisely at Belle, Belle II, BES III, or the super tau-charm experiments. The pp decays of charmonia have been investigated as a possible channel to measure charmonium production at the LHC [16]. The measurement of the η c production at the LHCb experiment has been performed using the η c → pp decay [1], which demonstrated that the pp final state is powerful to reconstruct the η c state and measure the η c production rate relative to that of the J/ψ, even though the η c hadroproduction rate is measured only for η c with transverse momenta (P T ) larger than 6.5 GeV due to the available trigger bandwidth. Also, this decay is used to study exotic candidates decaying to η c π − [17]. The branching fraction of the η c → pp is known to about 10% precision [15]. The studies of the η c would benefit from a more precise measurement of B(η c → pp) or B(η c → pp)/B(J/ψ → pp). Branching fractions of χ cJ → pp decays and ψ → pp have been measured to about 3-5% precision. Recently, LHCb has observed the decay η c → pp using a data sample of exclusive B + → ppK + decays [18]. Together with the measurement of B(B + → η c K + ) by Belle [19], the branching fraction of η c → pp is indirectly determined to be about 0.7 × 10 −4 . Therefore, the decay η c → pp is promising for the η c hadroproduction measurement.
The other promising final state to study prompt production of charmonium is φφ. The 1 − charmonium states are forbidden to decay to φφ. LHCb measured the χ c0,1,2 and η c production in inclusive b-hadron decays using the φφ final state with the first evidence of the η c → φφ decay [20]. In the latter analysis, a possible problem was highlighted, namely the PDG fit value of B(η c → φφ) differs from the PDG average value [15] by a factor close to 2. In addition, the ratio of the branching fractions B(η c → φφ)/B(η c → pp) was measured. More measurements are needed to establish a robust value of the B(η c → φφ). Also, due to the evidence of the η c → φφ, this channel is promising to measure the hadroproduction of the η c . Similarly, the φK + K − and the φπ + π − final states could be used.
The branching fractions of charmonium decays to long-lived baryons such as ΛΛ and Ξ + Ξ − are measured for most charmonium states. The reconstruction of these decay channels is challenging for LHCb due to the large lifetimes of these baryons such that they escape the Vertex Locator (VELO), which cause a small reconstruction and trigger efficiency. Decays involving short-lived baryons are reconstructed by LHCb with better efficiency.
The least studied charmonium state is the h c meson, and not many of h c decays have been observed so far. The h c meson is expected to decay to pp, however, the upper limit on the B(h c → pp) reported by the BES III collaboration [23] is more than one order of magnitude smaller than the theoretical prediction [16]. Also, the h c can be measured using its radiative transition h c → η c γ with branching fraction about 50%, which requires reconstruction of the η c state. Recently, LHCb observed the very clean decays χ c1,2 → J/ψµ + µ − , and precisely measured the χ c2 mass and its natural width [24]. The h c → η c µ + µ − decay can be searched similarly. Also, the BES III has observed the h c → ppπ + π − decay and measured its branching fraction [25] to be (2.89 ± 0.32 ± 0.55) × 10 −3 , which is promising for searches by LHCb.

Framework
The present NLO analysis was performed thanks to the FDC framework [26,27] 3 which generates the Born, real-emission and virtual contributions, ensures the finiteness of their sum, performs the partonic-phase-space integration and that over the PDFs. As announced, we performed a full study of the scale uncertainty by varying both µ R and µ F about the default value µ 0 = m 2 η c + P 2 T as (µ R , µ F ) = µ 0 × (1, 1; 0.5, 0.5; 2, 2; 0.5, 1; 1, 0.5; 1, 2; 2, 1). As for the CS LDME, we have taken O η c ( 1 S [1] 0 ) = 0.39 GeV 3 which corresponds to |R(0)| 2 = 0.81 GeV 3 for the radial wave function at the origin. In order to study the impact of the PDF uncertainties at NLO, we have used the CT14 set [28] which is included in LHAPDF5 [29]. The corresponding uncertainties follow from the 57 eigensets of CT14.

Results for the collider mode at
√ s = 13 TeV Our predictions at √ s = 13 TeV follow from the expected kinematical range of the forthcoming LHCb study performed on data taken during the Run2 in 2015-2016. They correspond to 2 fb −1 of data at √ s = 13 TeV. We have therefore considered the same rapidity acceptance as that used for the first LHCb analysis [1], namely 2 < y cms < 4.5 without any additional fiducial cuts on the decay product of the η c . Fig. 1a displays our predictions for the P T -differential prompt-η c cross section at NLO accuracy along with their associated scale and PDF uncertainties. It is clear that the latter are negligible in this energy range as compared to those from the scales. Fig. 1b shows the ratio of the NLO/LO cross sections with the scale uncertainties only and points at a K factor slightly increasing with P T . This is the expected behaviour with leading P T channels opening up at α 4 s . It also shows that the scale uncertainty is as large as 50%.
Assuming a recorded luminosity of 2 fb −1 , B(η c → pp) = 1.52 × 10 −3 and an efficiency on the order of 2 %, the 100-count limit per GeV correspond to 2 pb and located around P T 20 GeV. Without any surprise, the increase in the energy should allow LHCb to push their measurements at 13 TeV to slightly larger P T compared to 7 and 8 TeV. Limitation may come from the range where the J/ψ → pp yield is measured as well as from systematical uncertainties. In view of the other branching fractions on Table (1), let us add that other decays are also within the reach of LHCb measurements. 3 The FDC (standing for Feynman Diagram Calculation) package have been developed to automated HEP computations. It is based on the LISP symbolic programming language in order to produce FORTRAN codes. The Lagrangian are formed by the code, following the user requirement, from which are derived the corresponding Feynman rules. The package generates all possible Feynman diagrams contributing to a given process up to one loop in a given model. It can in particular deal quarkonium production within NRQCD. The amplitude of the process are analytically manipulated to generate FORTRAN codes of the squared amplitudes up to one loop. Numerical results for the (differential) cross sections are then computed by performing the phase-space integrals using the phase-space slicing method. We refer to [27] for explanations relevant to quarkonium-production applications.

Results for the fixed-target mode at
The use of the proton LHC beam in the fixed-target mode has lately be the object of intense investigation both in terms of feasibility and in terms of physics reach, see e.g. . In particular, a wide variety of measurements in different possible implementations were discussed in [30]. We will limit ourselves here to a few statements on the kinematics. First, 7 TeV protons impinging fixed targets release a center-of-mass system (cms) energy close to 115 GeV ( √ s = 2E p m N ). Second, the boost between the cms and the laboratory is γ lab cms = √ s/(2m p ) 60 yielding a rapidity shift as large as tanh −1 β lab cms 4.8. As such, the nominal acceptance of the LHCb detector in the cms approximates to −2.8 < y cms < −0.3. Physics wise, the LHCb detector probes backward physics in the fixed-target mode.
Nowadays, the first analysed fixed-target data based on the SMOG LHCb system -initially designed to improve the luminosity determination in LHCb, now used as a low-density-unpolarisedgas target-are coming in [53,54] and confirm that the particle multiplicity in the LHCb detector is such that its performance in the fixed-target mode remains intact. One can thus consider that similar decay channels of the η c as those discussed for the collider mode could be studied if sufficient luminosities can be achieved.
Until now, the LHCb-SMOG statistical samples for J/ψ taken with different noble gases (He, Ar, Ne) remain too small -on the order of hundreds-to expect any η c counts. The situation could significantly get better in the future with proposed SMOG2 system [55,56] with achievable yearly luminosities on the order of 10 pb −1 during the LHC Run3. It is however crucial to further constrain NRQCD LDMEs -as we propose here-to have a H target available as opposed as to nuclear -noble gas-targets. For the LHC Run4, yearly luminosities as high as few fb −1 will be within experimental reach as discussed in [30]. Fig. 1a displays the P T -differential η c cross section at √ s = 115 GeV in the expected acceptance of LHCb in the fixed-target mode. As above, we separated out the uncertainties from the scale variations (µ F and µ R ) and from the PDFs which are a little larger here since one probes slightly larger x values. As a matter of fact, dedicated rapidity-differential measurements at very negative y cms could provide specific constraints on the gluon PDFs [30,51,52]. Fig. 2b shows the ratio of the NLO/LO cross sections with the scale uncertainties only and points at a K factor slightly increasing with P T . This is the expected behaviour with leading P T channels opening up at α 4 s . It also shows that the scale uncertainty is as large as 5 at low energies. Assuming an integrated luminosity of 10 pb −1 , B(η c → pp) = 1.52 × 10 −3 and an efficiency on 50 %, the one-count limit per 2.5 GeV for dσ/dP T is on the order 0.08 pb, which corresponds according to our results to a P T upper limit of 8.5 ÷ 10 GeV. It precisely happens to be the range accessed at 7 and 8 TeV. With 10 fb −1 , the reach would simply be equivalent to that of the collider mode. We further note that, thanks to the reduced multiplicities in fixed-target mode, lower P T 's should be accessible. This would allow one to measure the gluon Transverse-Momentum-Dependent functions (TMDs) along the lines of [57][58][59].

Conclusions and outlook
We have computed the prompt η c -production cross section at one loop accuracy in QCD and in the CSM (LO in v 2 of NRQCD) for the LHCb kinematics in the collider mode at √ s = 13 TeV and in the fixed-target mode at √ s = 115 GeV. In addition, we have provided an up-to-date discussion of the possible decay channels to be used for such studies and performed an original analysis of the theoretical analysis including that from the factorisation and renormalisation scales and from the PDFs.
In addition, let us stress that the understanding and the measurements of η c production go well beyond the determination of NRQCD LDMEs. Its production in proton-nucleus collisions (see [60] for predictions of the corresponding nuclear modification factors at LHC energies) can provide complementary means to probe the distribution of gluons inside nuclei along the lines of [61,62]. In proton-deuteron collisions at extreme x F , it can also give us some handle on the gluon distribution in the deuteron at very large x [63].