Curing the Unphysical Behaviour of NLO Quarkonium Production at the LHC and Its Relevance to Constrain Gluon PDF at Low Scales

Home , CLEO (particle detector), Quarkonium

Eur. Phys. J. C manuscript No. (will be inserted by the editor)

Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales

Jean-Philippe Lansberg1 , Melih A. Ozcelik1

1 Universite´ Paris-Saclay, CNRS, IJCLab, 91405 Orsay, France

E-mail: [email protected] & [email protected]

Version of December 2, 2020

Abstract We address the unphysical energy dependence at the interplay between its perturbative and nonpertuba- of quarkonium-hadroproduction cross sections at Next-to- tive regimes, quarkonium production –once theoretically Leading Order (NLO) in αs which we attribute to an over- understood– should in principle allow us to probe the proton subtraction in the factorisation of the collinear singularities gluon content in terms of PDFs (see e.g. [7–10]) or TMDs inside the PDFs in the MS scheme. Such over- or under- (see e.g. [11–21]). subtractions have a limited phenomenological relevance in most of the scattering processes in particle physics. On the contrary, it is particularly harmful for PT -integrated charmo- What we have learnt in the recent years with the ad- nium hadroproduction which renders a wide class of NLO vent of NLO computations of PT -differential cross sections, NLO results essentially unusable. Indeed, in such processes, αs dσ /dPT , of J/ψ and Υ [22–28] is that the inclusion is not so small, the PDFs are not evolved much and can be of NLO corrections in any data-theory comparison is ab- rather flat for the corresponding momentum fractions and, solutely mandatory to extract qualitatively reliable state- finally, some process-dependent NLO pieces are either too ments. This is particularly true in two of the three most used small or too large. We propose a scale-fixing criterion which approaches, the Colour-Singlet Model (CSM) [29–31] and avoids such an over-subtraction. We demonstrate its effi- Non-Relativistic QCD (NRQCD) [32], where PT -enhanced ciency for ηc,b but also for a fictitious light elementary scalar NLO contributions notably affect observables like dσ/dPT boson. Having provided stable NLO predictions for ηc,b PT - and the yield polarisation as a function of PT . In fact, for integrated cross sections, σNLO, and discussed the options to ηQ S -wave quarkonia, the CSM is the leading NRQCD con- study ηb hadroproduction, we argue that their measurement tribution in the heavy-quark velocity, v. For the Colour- at the LHC can help better determine the gluon PDF at low Evaporation Model [33, 34], the impact of NLO correc- scales and tell whether the local minimum in conventional tions [35, 36] to dσ/dPT is limited. In the latter model, all NLO gluon PDFs around x = 0.001 at scales below 2 GeV the spin and colour contributions of the heavy-quark pair are is physical or not. summed over and the possible additional gluon radiations at NLO do not open new production channels at variance with arXiv:2012.00702v1 [hep-ph] 1 Dec 2020 the CSM and NRQCD. 1 Introduction

The production of quarkonia (Q) in inclusive proton-proton When one integrates over PT , these NLO channels, and electron-proton collisions when the protons break apart which are PT -enhanced in the CSM and NRQCD and which is one of most often studied process at high-energy col- are precisely responsible for the large impact of the NLO liders. Yet one still does not agree on how these heavy corrections at mid and large PT , are just suppressed by one quark-antiquark bound states are produced. The interested power of αs without any PT -enhancement factor. In this con- reader will ﬁnd it useful to consult the following reviews text, in 2015, we studied [37] the energy dependence of the NLO [1–4] addressing HERA and Tevatron results and more re- J/ψ and Υ PT -integrated cross section at NLO, σ , in cent ones [5,6] as what regards the recent advances in NRQCD to verify the coherence with the NRQCD predic- NLO the ﬁeld with the RHIC and LHC. Besides probing QCD tions for dσ /dPT . 2

Beside the confirmation of a possible breakdown of PDF fits to analyse how degraded global fits would be if 1 NRQCD universality –as first claimed [38] by F. Maltoni et xg(x; µF ) at NLO is required to be monotonous for x < 0.01 al. based on a partial NLO study–, we found out that, for all at µF ∼ mc. the NRQCD contributions2, the energy dependence became The structure of the article is as follow. In section 2, unphysical once the αs corrections were added. The same we outline the structure of the NLO ηQ production cross observation was made for the ηc which is at the centre of this sections and explain how to reproduce the existing results. study. More precisely, the charmonium cross sections would On the way, we provide analytical expressions in terms of become negative at increasing energies for a wide class of the partonic cross section and of the partonic luminosities factorisation and renormalisation scales. In the J/ψ case, the needed to compute the rapidity differential cross section, NLO corrections significantly reduce the predicted yields dσNLO/dy, which are not available in the literature. In sec- NLO close to RHIC energies [39] and σJ/ψ already becomes neg- tion 3, we make a brief historical survey of the past phe- ative at a couple of hundred GeV at central rapidities, y, for nomenology of NLO ηQ hadroproduction and of the at- µF ≥ Mψ [37]. Such observations were already made in the tempts to identify the origin of these negative NLO cross 1990’s regarding the ηc independently by Schuler [40] and sections and we explain that they come from the subtraction then by Mangano & Petrelli [41] but were then essentially procedure in the factorisation of the collinear singularities forgotten, see e.g. [42]. in the MS scheme. Section 4 is devoted to our factorisation- For bottomonia and for some –small– µF scale choices scale choice. Section 5 gathers our resulting cross sections NLO for charmonia (see [37] for details), σ would not become for ηc and ηb. We first demonstrate that our proposal works NLO NLO LO NLO η negative but the K factor, defined as σ√ /σ , would by discussing the behaviour of the K factors for Q and steadily deviate from unity for increasing s. As discussed elementary scalar bosons. Then, we discuss the interplay be- above, large KNLO factors have already been observed in tween the gluon luminosity and our obtained cross sections quarkonium production at finite PT but they can then be ex- and finally we present what we believe to be the best possi- plained by kinematical factors scaling like PT /mQ. These ble NLO predictions. Section 6 gathers our conclusions and are absent when PT is integrated over. As we noted√ such an outlook at other quarkonium-production processes. an intriguing behaviour can already be observed for s on 200 ∼ 300 GeV [37], so not necessarily at very high energies where large logarithms of the colliding gluon momen- 2 ηQ production up to NLO in the collinear and tum fraction x should be accounted for. Indeed, such ener- NRQCD factorisations gies typically corresponds to x = 0.01 and even above. In this article, we propose a solution to this issue which 2.1 CSM, NRQCD and collinear factorisation we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDF in the MS scheme. The present study essentially bears on collinear factorisa- As such, it may appear in any NLO computations once a tion [43] whereby the hadronic cross-section to produce a Q couple of unfavourable factors combine. In general, such quarkonium is factorised into a convolution of PDFs and Q over-subtractions indeed have a limited phenomenological a partonic cross section,σ ˆ ( ). Through NRQCD factorisa- relevance. It is clearly not the case for charmonium production [32], the latter is further factorised into short-distance tion which therefore offers a neat study case. As we will perturbative parts, computable with Feynman graphs, and discuss, we propose a simple solution which consists in a long distance non-perturbative parts. As a result, one starts Q factorisation-scale choice based on the high-energy limit of for the production of a quarkonium in a collision of two the partonic cross section and we demonstrate how well it hadrons A and B from: Z works for η and η production and for the production of a X c b dσ = dx dx f (x ; µ ) f (x ; µ ) × fictitious elementary light boson, whose production mecha- AB 1 2 a/A 1 F b/B 2 F ab nism is at odds with the production of a non-relativistic pair X σˆ ¯ [ ] + { } (µ , µ , µ )hOn i , (1) of heavy quarks then forming a quarkonium. d ab QQ n k R F Λ Q µΛ n Having proposed a way to get sound NLO perturbative | {z } results, we discuss the interplay between the behaviour of dσˆ ab(Q) the gluon PDFs at low scales and, in particular, the ηc pro- where e.g. fa/A is the PDF of the parton a inside the hadron duction cross sections. This motivates us to encourage a vig- A, dσˆ ab(QQ¯ [n] + {k}) are proportional to the partonic dif- orous experimental effort to measure it and, before data are ferential cross-section to produce a QQ¯ pair in the (spin available to fit them, we are tempted to suggest experts in and colour) quantum number n, possibly with other parti- n 1 cles {k} from the scattering of the partons ab and hO )i is The LDME values obtained by fitting dσ/dPT are ten times larger Q than those fit from σ. an NRQCD Long-Distance Matrix Element (LDME) for the 2Both the Colour-Singlet (CS) and Colour-Octet (CO) contributions. non-perturbative hadronisation of the pair in the state n into 3 the quarkonium Q. NRQCD factorisation stems from an ex- pansion in the relative velocity v between the QQ¯ pair in the η quarkonium rest frame. In this work, we focus on the terms ηQ ηQ Q leading in v and sub-leading in αs. As such, we only need 1 [1] to consider the colour singlet S 0 state for pseudo-scalar quarkonia, which is thus equivalent to the CSM. In such a case, the sum over n in dσˆ (Q) reduces to a single term. ab (a) (b) (c) The purpose of the next sections is to explain how dσˆ ab(Q) can computed up to NLO accuracy in order to explain the appearance of negative cross sections in past computations. ηQ ηQ

ηQ 2.2 ηQ hadroproduction at LO

2 At LO (α ), ηQ hadroproduction proceeds through gluon s (d) (e) (f) fusion, g(k1) + g(k2) → ηQ(P), which can be computed via Feynman diagrams like Fig. 1a. In the CSM [29–31], Fig. 1 Representative diagrams contributing to ηQ hadroproduction via 1 CS channels at orders α2 (a), α3 (b,c,d,e,f). The quark and antiquark the matrix element to create a S 0 pseudoscalar quarko- s s attached to the ellipsis are taken as on-shell and their relative velocity nium η with a momentum , possibly accompanied by Q P v is set to zero. other partons, noted {k}, is obtained from the product of the amplitude to create the corresponding heavy-quark pair, squared [45], M(ab → QQ¯ + {k}), a spin projector, N(Ps.|s1, s2) and R(0), the ηQ radial wave function at the origin in the configura- 2 2 2 N − 1 16α π|R0| tion space. The CSM being the leading v contribution to |M|2 = c s µ4 (1 − )(1 − 2) , (4) N M R NRQCD, R(0) can naturally be related to a NRQCD LDME c Q as follows: and for the partonic cross section (where we have set MQ = 2mQ as expected within NRQCD and Nc = 3), 1 [1] | |2 S 0 2(2J + 1)Nc R(0) hOη i = . (2) Q 4π π 1 1 M2 LO |M|2 − Q σˆ gg = 2 2 2 δ 1 By virtue of heavy-quark spin symmetry, R(0) is identical sˆ 8 (2 − 2) sˆ 2 π 1 1 for the ηc and J/ψ for instance up to v corrections. It can |M|2 − (5) = 2 2 2 δ 1 z then be obtained from the well measured leptonic width of MQ 8 (2 − 2) the /ψ computed in the CSM/NRQCD or from potential | {z } J LO 2 3 σˆ 0 models. In what follows, we will use |Rηc (0)| = 1 GeV and 2 3 |Rηb (0)| = 7.5 GeV [39]. 2 where we have defined z = MQ/sˆ. Overall, one has 2 The hadronic section then reads with τ0 = MQ/s = X N(Ps.|s , s ) 4m2 /s and τ = τ /z, M(ab → Q(P) + {k}) = √ 1 2 × Q 0 0 mQ s1,s2,i,i LO √ Z (3) dσ ∂Lgg LO ii0 − δ R(0) ( s, y; µF ) = dτ σˆ 0 τ δ τ τ0 , M → s1 ¯ s2 { } dy ∂y∂τ √ √ (ab Qi Qi0 (p = 0) + k ), (6) N 4π √ Z c LO ∂Lgg LO σ ( s; µF ) = dτ σˆ 0 τ δ τ − τ0 , − ∂τ where P = pQ + pQ¯ , p = (√pQ pQ¯ )/2, s1 and s2 are the ii0 heavy-quark spins, and δ / Nc is the projector onto a CS in terms of the following differential gluon luminosities: state. For v → 0, the spin projector on a pseudoscalar state, √1 P P ∂L √ √ N(Ps.|si, s j) = v¯( 2 , s j)γ5u( 2 , si). After one sums over gg y −y 2 2mQ (τ, y; µF ) = fg( τe , µF ) fg( τe ; µF ), the quark spin, one obtains traces which can be evaluated in ∂y∂τ L Z −1/2 log τ √ √ a standard way. ∂ gg y −y (τ; µF ) = dy fg( τe ; µF ) fg( τe ; µF ). However we note here the explicit appearance of γ5 in ∂τ 1/2 log τ N(Ps.|si, s j) which can cause issues within the framework (7) of dimensional regularisation. Here we employ the standard ’t Hooft-Veltman scheme to deal with γ5 in D = 4 − 2- These fully encapsulate the energy and rapidity dependences dimensions [44]. We obtain for the LO matrix element of the ηQ yields at LO. 4

2.3 ηQ hadroproduction at NLO For bothσ ˆ and dσ/ˆ dy, after combining the virtual with the real corrections, the soft singularities vanish5 and Let us now outline how to compute the ηQ cross section up to we are left, as usual, with the initial-state collinear diver- NLO accuracy [40, 45, 46] which we will then use through- gences which originate from diagrams such as in Fig. 1d out our study. and Fig. 1e. The occurrence of these divergences is a con- NLO contributions involve both virtual(-emission) and sequence of the fact that the initial states are fixed by the real(-emission) corrections via gg fusion that can be repre- kinematics and therefore not integrated over. sented by diagrams in Fig. 1b, Fig. 1c and Fig. 1d. In addi- Under the collinear factorisation, the divergences arising tion to the gg fusion, qg and qq¯ channels contribute at order from such collinear emissions from specific initial partons 3 αs as shown in Fig. 1e and Fig. 1f. Both real and virtual con- are subtracted in the factorised PDFs via the corresponding tributions individually exhibit singularities. In order to deal Altarelli-Parisi (AP) Counter-Terms (CTs) which introduce with these singularities, we employ dimensional regularisa- the factorisation scale µF in the partonic cross section [48]. tion where we define D = 4 − 2. In the MS scheme, the AP CT forσ ¯ ag reads, As usual, the virtual contributions exhibit both Ultra- bare 4πµ2  Violet (UV) and Infra-Red (IR) divergences. The former (AP-CT) 1 αs  R  LO σ¯ =   Γ [1 + ] σˆ zPag (z) , (9) ag 2π  2  0 are removed via the renormalisation procedure. To do so, IR µF we apply the on-shell (OS) renormalisation scheme for where Γ [1 + ] is the Gamma function and ( ) are the the gluon/quark wave functions and the heavy-quark mass Pag z splitting functions between parton and a gluon. We have counter-term while, for the coupling, we perform the renor- a given their expressions in Appendix A. malisation δZMS within the MS-scheme and we take [47] g Using these standard procedures, we have reproduced     3 α  1 2  µ2   the expressions of the partonic cross sections up to αs [40, OS − s   R   δZ2 = CF  + + 3 log  2  + 4 , 45, 46]. It does not generate any specific complications to 4π UV IR m   Q fold these with PDFs. We have collected in Appendix B !! α 1 1 NLO OS s the final expressions for the integrated cross section ση δZ3 = β0 − 2CA − , Q 4π UV IR in terms of the partonic luminosities. On the contrary, the (8)   2   analytical expressions needed to obtain σNLO/ are ab- α  1  µ  4 d ηQ dy δZOS = −3C s  + log  R  +  , m F 4π   2  3 sent in the literature. We have gathered them in terms of the UV mQ ! partonic luminosities for the three channels gg, qg and qq¯ in MS β0 αs 1 Appendix C. The codes which we have derived from these δZg = − , 2 4π UV expressions and which we have used to generate the results presented later have been successfully cross-checked versus where β = 11 C − 4 T n with n being the number of 0 3 A 3 F f f the semi-automatic code FDC [49]. active light flavours. We have above made a distinction between UV and IR to label the poles coming from UV and IR divergences respectively. In the following we will only 3 On the origin of unphysical ηQ cross section at high label the poles to show their UV/IR character but not the energies appearing in exponents. The with and without labels ultimately originate from the regulator in D = 4 − 2. We have 3.1 The NLO partonic cross section and its HE behaviour also absorbed a global factor of e−γE (4π) inside the MS- renormalised αs coupling. In this section, we focus on the partonic high-energy (HE) As what regards the virtual corrections, we are thus only limit (ˆs → ∞ or equally z → 0) and show how this limit can left with soft IR divergences. In contrast to the virtual con- help us understand the origin of the unphysical cross-section tributions where the singularities are already manifest in the results which we referred to in the introduction. 3 ηc–gluon form factor , the divergences in the real-emission The first NLO computation for pseudo-scalar quarko- part only reveal themselves after taking the phase-space in- nium production was done [45] by Kuhn¨ & Mirkes in 1992 tegration4. for toponium. At the time, it was not known that a toponium For dσ/ˆ dy, the phase-space integration is slightly less state could not bind. Their NLO results were confirmed by straightforward to be performed analytically than forσ ˆ G. Schuler [40] two years later who performed the first phe- where one can just integrate over the full phase-space with- nomenological application for charmonia. He was the√ first out separating out the rapidity y and the transverse momen- to report negative cross sections for ηc production at s just tum PT . above 1 TeV for the central scale choice. He explained this

3 5 That is the contribution proportional to δ(1 − z). But for a soft singularity proportional to β0 that arises through renor- 4z → 1 for soft and tˆ, uˆ → 0 for collinear divergences. malisation. This factor will be absorbed inside the PDFs, see later. 5 unphysical behaviour by the fact that the partonic gg cross CT (see Eq. (9)) to subtract the initial-state collinear diver- section was approaching a negative constant for µF = MQ at gences. On the other side, Aa is clearly process-dependent highs ˆ. When folded with PDFs, such negative contributions as Table1 illustrates it. coming from real emissions would become larger than the Born contributions for too flat low-x gluon PDFs. However, as what regarded the reason why the gg-partonic cross section was approaching a negative constant at highs ˆ, he did Aa µˆ F not provide any explanation, only a suggestion of a possible η[1,8] −1 M√Q = 0.607 side effect of the restriction in the heavy-quark kinematics Q e MQ for them to be at threshold to form a non-relativistic bound χ[1,8] −43/27 0.451M state like a quarkonium. Q, J=0 Q [1,8] − In 1996, while presenting preliminary NLO cross- χQ, J=2 53/36 0.479MQ section results within NRQCD, Mangano & Petrelli dis- Fictitious H˜ 0 cussed in a proceedings contribution [41] similar issues; −0.147 0.93MH˜ 0 (2m /m ˜ = 1) they then attributed these negative cross sections to a pos- Q H Fictitious H˜ 0 sible over-subtraction of the collinear divergences inside the 1.61 2.43MH˜ 0 (mQ/mH˜ = 1) PDFs, thus rendering the partonic cross section negative in Real H0 2.28 3.12MH0 the HE limit. Quoting them, “there is nothing wrong in prin- (2mt/mH = 2.76) ciple with these [partonic] cross sections turning negative in Table 1 The process-dependent constants Aa [For quarkonia, Aa is the small-[z] region, as what is subtracted is partly returned identical for CS and CO states] along with the µF value cancelling the to the gluon density via the evolution equations”. They how- HE limit. For the quarkonia, Aa is identical for CS and CO states. For ever also noted that, for processes like charmonium produc- the scalar particles, these value haves been derived for the HE expressions of [50]. [M represents here, and in what follows, the mass of the tion occurring at scales near where the PDF evolution is produced particle, be it a quarkonium or a, elementary scalar boson.] initiated, it is insufficient in practice – hence the negative hadronic cross sections then observed by Schuler. An important observation we would like to make here for our following reasoning is that the magnitude of these negative partonic cross sections for z → 0 is process depen- As a consequence, if Aa < 0, the HE limit of the partonic dent. As such, the universal PDF evolution, for a given scale, cross section thus gets negative for the natural scale choice cannot thus possibly fix the issue in a global manner. µF = MQ and above. Whether this can make the hadronic To further assess this, let us indeed focus on the small-z cross section turn negative is then a matter of a complex limit ofσ ˆ ab which we obtained in the previous section, as interplay between the hadronic energy, the PDFs, µR and the done by Schuler, Mangano and Petrelli. Taking this limit in size of the (process-dependent) virtual corrections. Eq. (B.6)&(B.8)6, one gets Before discussing this interplay, let us however go back  2  to the notion of over-subtraction to trace back the origin of α  MQ  NLO s LO   2 lim σˆ ag (z) = Ca σˆ 0 log + Aa , (10) these negative limits. Away froms ˆ = M , only the real emis- z→0 π  µ2  F sions contribute. In fact, at larges ˆ, the sole tˆ-channel gluon- where Cg = 2CA, Cq = CF and MQ is the mass of the pro- exchange topologies depicted by Fig. 1d& 1e contribute; the duced quarkonium (or 2mQ). Equivalent expressions were other real-emission graphs depicted by Fig. 1c& 1f–which obtained for P-wave quarkonium [40, 41]. are not divergent in the collinear region– are suppressed at 2 One can also consider such a limit [50] for the pro- least one power of MQ/sˆ. At this stage, the amplitude square duction of the Brout-Englert-Higgs (BEH) scalar boson H0 can only be positive-definite by construction as it is a full using NLO expressions [51–53] or a fictitious elementary Hermitian square fors ˆ , M2.7 scalar boson, dubbed H˜ 0, whose coupling to gluons also oc- When integrating over ˆ, one will encounter the afore- curs through a loop of heavy quarks and get a similar limit. t mentioned collinear divergences which are to be absorbed In all these cases, we stress, since it will be essential for in the PDF via the AP CT. Anticipating this subtraction, we our forthcoming discussion, that this limit [50] in fact yields can exhibit the corresponding divergence and recast the un- Ag = Aq. We further note the presence of the factorisation scale M2 µF inside log 2 in these limits. This term is in fact uni- µF versal and process-independent as it originates from the AP 7Fors ˆ = M2, we note that the virtual contributions are not squared as 6 dσˆ 4 The following discussion applies to bothσ ˆ and dy . their square contributes at αs . 6 renormalised cross sectionσ ¯ as reduce the weight of these regions in z where the partonic cross sections are negative, and eventually avoid negative Z dσ¯ NLO,z,1 NLO,z,1 ag hadronic cross sections. Yet, it is hard to believe that they σ¯ ag = dtˆ dtˆ would do so for all possible processes where this can oc- bare  2  1 α 4πµ  cur as the coefficients Aa are process-dependent while the = − s  R  Γ [1 + ] σˆ LOzP (z) D 2π  2  0 ag a DGLAP evolution is process-independent. IR MQ αbare + s σˆ LOC A¯ (z) , π 0 a a (11) 3.2 From negative partonic cross sections to negative (or positive) hadronic cross sections 8 where above we have split the collinear part from A¯a (z) which is free of divergences for any 0 ≤ < 1. We have mul- z Having now identified the origin of the negative cross sec- tiplied the first term by a factor Da = 1 + δag to account for tions, we can discuss their relevance to the past phenomenol- the fact that one has collinear singularities for each gluon in ogy which we recalled in the previous subsection. (AP-CT) the gg channel. Therefore one would need to take 2σ ¯ gg , First, we note that the ηb phenomenology, for which i.e. for each parton, to eliminate the poles. From the equa- σNLO remains positive in the LHC range, is less patholog- ηb tion above, it follows thatσ ¯ NLO,z,1 is positive-definite due ical. We have indeed found out [37] that σNLO only slightly ag ηb 9 LO to the fact that the first term evaluates to positive infinite as deviates from ση in the LHC range. It thus seems that it − b IR → 0 irrespective of A¯a (z). is less sensitive to the limit of Eq. (10). Both charmonia Clearly, other schemes to absorb these collinear diver- and bottomonia have the same partonic cross section but for 10 gences inside the PDFs would yield different A¯a (z) . In the three changes: the mass shift and a trivial rescaling of the ¯ DIS scheme for instance, Aa (z) [45] exhibits a log z depen- LDME and n f which plays a minor role here. This mass shift dence, which does not create any issue once integrated over however has three immediate effects : (i) a given z = M2 /sˆ √ Q z and this different z dependence should in principle be com- value for bottomonia corresponds to 3 times larger sˆ. Con- ¯ pensated by a different evolution of the PDFs. Yet, Aa (z , 0) sidering the rescaling on the integration bounds, [M2 /s, 1], should remain finite. Q when convoluted√ with PDFs, this effectively corresponds to What we wish to argue here is that, since this subtraction a 3 times larger s, which is thus easily outside the range of is the only possible source of negative numbers at z , 1, if past studies, (ii) however, even at fixed z, the results would A¯ (z) happens to be negative in a given scheme where PDFs a differ since αs(µR ' MQ) is smaller and this reduces the are supposedly positive (see [54] for MS), this signals that 3 impact of αs contributions compared to the (positive) Born the AP CT have likely over-subtracted some collinear con- 2 ones at αs , (iii) the evolved gluon PDFs up to a larger µF be- tributions from the real-emission contributions, and this can come steeper which reduces the relative importance of the yield the observed negative hadronic cross sections. This is small-z domain compared to the threshold contribution at indeed what happens for quarkonia since the NLO threshold z = 1 which remains positive. Taken together, these 3 points 2 contributions (ˆs = M ) are found to be positive-definite for explain very well why the charmonium case, at low scales, 11 ηQ and several other states at least√ for µF = µR . Note that is the most pathological one and that the issue of a possi- σNLO also goes negative at large for µ = µ . ηQ s F R ble over-subtraction of the collinear divergences is usually Let us re-iterate at this stage that contributions of type considered to be rather academical with a limited impact on 2 full square |M| like the real emissions are always positive- other hadronic cross sections. definite by construction at any kinematical point z. The only It is however legitimate to wonder if further aspects spe- way to render them negative is the over-subtraction via the cific to the modelling of quarkonium production renders its AP-CT inside the PDFs. We agree that evolved PDFs can phenomenology particular. Our answer tends to be negative. 8We remark at this stage that if the form factor of the Born cross section Indeed, as Table1 shows, CO and CS states are equally is resolved, i.e. considering the top-quark loop with a finite mass in the affected, in agreement with the past phenomenology [37]. 0 ¯ case of H production via gluon fusion, Aa = Aa(z = 0) is a constant. This confirms that neglecting such higher-order v correc- On the contrary, the coupling is tree-level type-like as in the Higgs i.e. tions is not the source the issue. Since CS and CO are EFT with mt → ∞, then we have an additional log z dependence and a different off set for the gg and qg channel. It is not very surprising as both computed in the non-relativistic limit, one may wonder for z → 0, mt ands ˆ are both large and HEFT cannot be applied. whether that this limit is also a source of issues, as suggested 9 For IR poles, one has that IR < 0, while for UV poles UV > 0. by Schuler. Yet, we anticipate that the phenomenology of a 10In principle, one could thus look for a scheme where the partonic 0 H˜ with m = M ˜ 0 /2 should also be affected since A is also cross sections simply do not become negative. This is left for future Q H a investigations as it would entail refitting the PDFs with different evo- negative. As our numerical results will show, it is indeed the lution equations. case and this will thus confirm that this is not a quarkonium 11This is also the case for H0 and H˜0. issue per se. 7

4 A scale choice as solution crucial to damp down a contribution which should not be the leading one in any case. We now come to our proposal to solve this unnatural be- Physics wise, our scale choice essentially amounts, in haviour of the cross section. In fact, it simply amounts to the partonic HE limit, to reshuffle the entirety of the real 12 set the factorisation scale µF such that partonic cross-section emissions inside the PDFs . From a HE viewpoint, such vanish at larges ˆ, instead of risking it to become negative. Of contributions are expected to be important, in particular at course, such a scale choice is only possible provided that it small tˆsince they are supposed to be enhanced by logarithms is the same for all the partonic channels. Dubbing our scale ofs ˆ, which should eventually be resummed. In fact, as our choiceµ ˆ F , we just define it as discussion has illustrated, the prominent effect of such contributions is a source of issues in a fixed-order computation Aa/2 µˆ F = Me , (12) as it jeopardises the convergence of the perturbative series with NLO contributions being more important than the Born ones. In this sense, our scale setting amounts to include all having in mind that that Aq = Ag. It is clear, from our def- these possible HE effects in the PDFs. This makes sense as inition that, since Aa is a process-dependent quantity,µ ˆ F will be process-dependent. We have listed some values of the PDFs are ultimately determined by fitting data – contain- ing all type of higher order corrections. In fact, recent PDF µˆ F in Table1 for the di fferent particles we considered. It analyses have been made taking into account HE effects in is important to note that theµ ˆ F values we have found are within or close to the usual ranges of values anyway taken their evolution [56, 57]. in phenomenological studies. Let us now turn to the physical picture of our reasoning. Our motivation is clear as it amounts to avoid negative cross 5 Results and discussion sections which we attribute to an over-subtraction in the MS factorisation scheme. Our scale choice avoids thatσ ˆ ab be 5.1 A word on our PDF choice negative at small z. It makes sense to base its construction from this limit as it becomes more and more relevant at large Given the importance of the PDF shape at low scales in the , precisely where σNLO can become negative. previous discussions, we have employed on purpose, thanks s ηQ Even if A¯a(z) becomes more negative than its limiting to LHAPDF6 [58], 3 NLO sets which show rather different value, Aa, our results will show that cancellingσ ˆ ab(z → 0) features: : with µ = µˆ will be sufficient to get much more sound F F 1. a representative13 set of the conventional NLO PDFs, results, in particular to avoid σNLO < 0. Going further, we ηQ PDF4LHC15 nlo 30 [59], stress that A¯ (z) also contains real emissions from the heavy- g 2. a dynamical PDF set, JR14NLO08VF [60], where glu- quark line (see Fig. 1c) and thus differs from A¯ (z). Working q ons are radiatively generated from a valence-like posi- at finite z where A¯ (z) , A¯ (z) does not allow us to derive g q tive input distributions at a low scale which is optimally an equally simple gauge-invariant solution based on a scale chosen, and choice. In the quarkonium case, the latter contributions to 3. a set taking into account HE effects in the evolution, σˆ are relatively suppressed by M2 /sˆ = z and thus their gg Q NNPDF31sx nlonllx as 0118 [56], effect disappear at small z. Another reason to focus on the small-z limit is that when in order to perform our NLO cross-section evaluations. foldingσ ˆ ab(z) with the PDFs (see Eq. (1) or Eq. (B.6)), the These are plotted on Fig.2 along with CT14nlo [61], Jacobian to transform the integration measure from dx1dx2 MMHT14nlo [62], NNPDF31 nlo as 0118 [63] for com- to a measure involving dz will comprise a multiplicative parison, for two scale choices, 1.55 GeV and 3 GeV. We factor 1/z2. As a result, the impact of the small-z region note that x g(x) from PDF4LHC15 nlo 30, MMHT14nlo, certainly depends much on whetherσ ˆ ab(z → 0) is zero CT14nlo and NNPDF31 nlo as 0118 all show a maximum or not, even though the z range has a lower bound set by around 0.02 and then a local mininum below 0.001. 2 τ0 = M /s. Indeed, for nonzeroσ ˆ ab(z → 0), the PDFs are 12 the key element regulating the integral. By virtue of evo- Understanding the possible connection with a recent study of the pos- itivity of the PDFs in the MS scheme [54] and the collinear factorisa- lution, they should become steep enough as to essentially tion scheme [55] is left for a future study. damp down the contribution of the small-z region. However, 13Our choice has mainly been driven by technical reasons. For instance, at low µF , the PDFs can be rather flat. This can give a large at low scales, as can be seen on Fig.2 (a) NNPDF31 nlo as 0118 − weight to this small-z region where the real-emission con- seems to suddenly saturate at x = 5 × 10 5 whereas CT14nlo (like tributions are negative for large µ / , hence the possibility CT18nlo) comprises two outstanding eigensets (one low and one high) F M which are extremely different from the others. We stress that our forth- that σNLO < 0. Now, ifσ ˆ ( → 0) = 0 as our µ choice ηQ ab z F coming physical conclusions would not be affected if we made other entails, the PDF shape at low scales is suddenly much less choices. 8

x g(x,µF) NNPDF30 nlo. Under such assumptions, the presence of a 20 local minimum below 0.001 is unlikely as it would yield 18 ______NNPDF30 constrained from J/ψ exclusive photoproduction to a decrease in the J/ψ exclusive production cross section, 16 PDF4LHC15_nlo_30 MMHT14nlo which is absent in the data. At this stage, since it is not an 14 JR14NLO08VF NNPDF31sx_nlollx_as_0118 CT14nlo actual PDF ﬁt and since the exclusive cross sections are not 12 NNPDF31_nlo_as_0118

10 µF = 1.55 GeV directly related to the PDFs, we consider this ﬁnding as a

8 enerated with APFEL 2.7.1 Web guidance, yet a very interesting one anticipating our results. 6 4 2 Adapted from a plot g 5.2 Assessing the perturbative convergence withµ ˆ F using 0 − − − − − NLO 10 5 10 4 10 3 10 2 10 1 1 the K factors x (a) µ = 1.55 GeV F We have found so far that signiﬁcant NLO contributions to η production are expected to appear if the hadronic cross x g(x,µF) Q 2 20 section becomes sensitive to z = M /sˆ values far away from Q √ 18 threshold. This would result in a signiﬁcant s dependence 16 NLO PDF4LHC15_nlo_30 of the NLO/LO hadronic cross-section ratio ( ). As we MMHT14nlo K 14 JR14NLO08VF NNPDF31sx_nlollx_as_0118 explained, it is due to an over subtraction, in the MS scheme, CT14nlo 12 NNPDF31_nlo_as_0118 µF = 3 GeV of collinear contributions from the real-emission NLO con- 10

enerated with APFEL 2.7.1 Web tributions inside the PDFs. A relative constant oﬀset is how- 8 ever expected from the virtual corrections at z = 1, like for 6 the decay widths, in particular for reactions where α is not 4 s very small.

2 Adapted from a plot g To mitigate this ﬁxed-order treatment shortcoming, we 0 − − − − − 10 5 10 4 10 3 10 2 10 1 1 x have thus proposed a speciﬁc scale choice which corre-

(b) µF = 3.0 GeV sponds to the inclusion, in the z → 0 limit, of the entirety of such NLO contributions in the PDFs. The logic behind Fig. 2 Gluon PDFs as encoded in PDF4LHC15 nlo 30 [59], JR14NLO08VF [60], NNPDF31sx nlonllx as 0118 [56], is that PDFs are fit to data which incorporate all such emis- CT14nlo [61], MMHT14nlo [62], NNPDF31 nlo as 0118 [63] sions. This is probably not a perfect solution but, beside of for two scale values : (a) 1.55 GeV and (b) 3 GeV. In addition, we corresponding to perfectly acceptable µF values, it indeed have added on (a) (solid black lines) the resulting constraints on avoids erratically varying KNLO factors and negative and un- NNPDF3.0 obtained by Flett et al. under some asumptions [64] from J/ψ exclusive photoproduction. [These plots have been adapted from physical hadronic cross section, as the results of this section plots generated by APFEL web [65, 66]]. show. Before discussing our results for KNLO, let us describe Such features are absent in both JR14NLO08VF and our set-up. We have evaluated them at y = 0 using Eq. (6) NNPDF31sx nlonllx as 0118 whereas they have a signifi- for LO and Eq. (C.10), (C.11)&(C.12) for NLO. The same cant impact on the phenomenology as we will show later PDF have been used for both. We have set mc = 1.5 GeV on. However, we stress that the local minimum has al- for the ηc and mb = 4.75 GeV for ηb. We have used the αs ready disappeared once the gluon PDFs are evolved up corresponding to our PDF choice thanks to LHAPDF6. to 3 GeV, where the 3 sets we have used display simi- As for a fictitious H˜0, we have set its mass at 3 GeV, lar features but for the size of the uncertainties. At µ = F close to that of ηc. Having at our disposal, the small-z limit 1.55 GeV, we note that both for PDF4LHC15 nlo 30 for different MH˜0 /mQ ratio, we have chosen three values and NNPDF31sx nlonllx as 0118, the shape can be very for the mass of the heavy-quark active in the loop, namely different within the uncertainty spanned by their PDF 0.5×MH˜0 , MH˜0 and (mt/mH0 )×MH˜0 . As can be seen from Ta- eigensets. At µF = 3 GeV, this only remains the case for ble1, mQ = 0.5 × MH˜0 renders Aa slightly negative, −0.147, PDF4LHC15 nlo 30. These different behaviours will in fact whereas it is large and positive, 2.28, for the SM H0 with be very useful to study the interplay between the scale and NLO mH = 125 GeV and mt = 173 GeV. The plotted K fac- the PDF choices. tors have been computed with the publicly available code We further note that a recent study by Flett et al. ggHiggs by Bonvini [67–69] based on [50,70] for the NLO has shown that one could extract, under specific assump- result with a finite heavy-quark mass in the loop. Other than tions [64], actual constraints on the gluon PDF at low scales this, we have run with its default setup. from J/ψ exclusive production data. These are represented Let us first discuss the ηQ results. Fig.3 gathers our re- NLO by the solid black lines in Fig.2 (a) when applied to sult for the K factor computed at y = 0 for ηc (top) and ηb 9

3 3 3 y = 0 y = 0 y = 0 2 2 2 / dy / dy / dy LO LO LO

ηc PDF4LHC15_30_nlo ηc JR14NLO08VF ηc NNPDFNLL / d σ / d σ / d σ μ =μ =M/2 1 μR=μF=M/2 1 μR=μF=M/2 1 R F

y = 0 μR=M/2,μ F=M y = 0 μR=M/2,μ F=M y = 0 μR=M/2,μ F=M

μR=M,μ F=M/2 μR=M,μ F=M/2 μR=M,μ F=M/2

μR=μF=M μR=μF=M μR=μF=M / dy / dy / dy μ =M,μ =2M 0 μR=M,μ F=2M 0 μR=M,μ F=2M 0 R F

NLO μR=2M,μ F=M NLO μR=2M,μ F=M NLO μR=2M,μ F=M

μR=μF=2M μR=μF=2M μR=μF=2M d σ d σ d σ

μR=μF=M/ e μR=μF=M/ e μR=μF=M/ e -1 -1 -1 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c)

ηb PDF4LHC15_30_nlo ηb JR14NLO08VF ηb NNPDFNLL μ =μ =M/2 μ =M/2,μ =M 3 μR=μF=M/2 μR=M/2,μ F=M 3 μR=μF=M/2 μR=M/2,μ F=M 3 R F R F y = 0 y = 0 y = 0 μR=M,μ F=M/2 μR=μF=M μR=M,μ F=M/2 μR=μF=M μR=M,μ F=M/2 μR=μF=M

μR=M,μ F=2M μR=2M,μ F=M μR=M,μ F=2M μR=2M,μ F=M μR=M,μ F=2M μR=2M,μ F=M / dy / dy / dy

μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e LO LO LO 2 2 2 / d σ / d σ / d σ y = 0 y = 0 y = 0

/ dy 1 / dy 1 / dy 1 NLO NLO NLO d σ d σ d σ 0 0 0 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (d) (e) (f)

NLO Fig. 3 K |y=0 √for ηc (top) and ηb (bottom) (for PDF4LHC15 nlo 30 (left), JR14NLO08VF (middle) and NNPDF31sx nlonllx as 0118 (right)) as a function of s for the usual 7-point scale choices and ourµ ˆ F scale with µR = µF . (down) and for the central eigenset of our 3 NLO PDF sets, the low scale gluon distribution at low scales encoded in namely PDF4LHC15 nlo 30 (left), JR14NLO08VF (mid- PDF4LHC15 nlo 30. So far, these results confirm the afore- dle) and NNPDF31sx nlonllx as 0118 (right). We have used mentioned past phenomenology. the conventional 7-point scale-choice values obtained by in- On the other hand, when adopting our scale choice, dependently varying µR and µF by a factor of 2 about a de- µF = µˆ F (= µR), the behaviour is smooth and seems to fault value which we simply chose here to be the mass of the slowly converge towards a constant value slightly above the quarkonium, MQ. We stress that LO cross sections used to unity. In fact, we have checked that varying µR at fixed NLO compute K were obtained with the same PDF and scale µF = µˆ F would simply shift the curve without changing its as those used for the NLO cross sections. In addition, we shape. This is connected to the fact that the limiting val- NLO have plotted K for µF = µˆ F which we expect to provide ues is also driven by the threshold contribution at z = 1 the best behaviour. We have only plotted it for µR = µF . where the virtual corrections which are sensitive on µR sit. We now discuss the qualitative features of the results. Now, summarising our result for a conventional PDF like First, we note that, for PDF4LHC15 nlo 30, negative cross PDF4LHC15 nlo 30, we can claim that the scale choice sections (KNLO < 0) appear as expected as early as 1 TeV. which we advocate provides a very simple solution to avoid This happens first for µF = MQ and µR = 0.5MQ, then pathological behaviour of the PT -integrated ηc cross sec- for µF = 2MQ and µR = MQ, and then for µF = 2MQ tions at NLO. NLO and µR = 2MQ, while K essentially converges to 0 for All the above observations can be made again for ηb (see NLO µF = 2MQ and µR = MQ, which is also not acceptable. Fig. 3d) but for the fact that the K factor does not get In short, all the results with µF equal or larger than the de- negative. Nonetheless, it gets so small for the large scale fault choice are pathological and the situation is worsened √choices that the results remain meaningless. Presumably at by a lower value of µR which comes along with a larger size s above those of a FCC, the cross section for µF =√2MQ NLO of αs. On the other hand, for µF = 0.5MQ, K does not and µR = MQ would turn negative. However, at such s, we get negative, neither particularly small, but shows a peak admit that it is rather an academic example. Yet, we stress at the top LHC energies which is related to the peak in that KNLO varying by a factor of 10 from fixed-target en- 10

20 20 10 0 0 Fictitious-H production with mH=3 GeV, mQ=mH Fictitious H production with mH=3 GeV, mQ=mH x 173/125 at NLO computed with ggHiggs and PDF4LHC30 at NLO computed with ggHiggs and PDF4LHC30 µR/mH=0.93, µF/mH=0.93 µR/mH=0.5, µF/mH=0.5 8 µR/mH=1, µF/mH=1 µR/mH=2.0, µF/mH=1.0 µ /m =1, µ /m =2 µ /m =2.0, µ /m =2.0 µR/mH=2.23, µF/mH=2.23 µR/mH=2.0, µF/mH=1.0 R H F H R H F H 15 15 µ /m =3.13, µ /m =3.13 µ /m =0.5, µ /m =0.5 µ /m =1, µ /m =0.5 µ /m =0.5, µ /m =1.0 µR/mH=1, µF/mH=1 µR/mH=2.0, µF/mH=2.0 R H F H R H F H R H F H R H F H µ /m =1, µ /m =1 µ /m =2.0, µ /m =1.0 µR/mH=1, µF/mH=2 µR/mH=0.5, µF/mH=1.0 R H F H R H F H µ /m =1, µ /m =2 µ /m =2.0, µ /m =2.0 6 µR/mH=1, µF/mH=0.5 µR/mH=3.0, µF/mH=3.0 R H F H R H F H µ /m =1, µ /m =0.5 µ /m =0.5, µ /m =1.0 µR/mH=0.5, µF/mH=0.5 R H F H R H F H

NLO 10 NLO 10 NLO K K K 4

5 5 2 Fictitious-H0 production for mH=3 GeV and mQ=1.5 GeV in the loop computed with ggHiggs and PDF4LHC15NLO 0 0 0 10 100 1000 10000 100000 100 1000 10000 100000 1e+06 100 1000 10000 100000 1e+06

sqrt(s)/mH sqrt(s)/mH sqrt(s)/mH (a) (b) (c)

3 3 3

2.5 2.5 2.5

2 2 2

NLO 1.5 0 NLO 1.5 0 NLO 1.5

K K K 0 H production with a fictitious mQ=mH/2 H production with mt=mH at NLO H production with finite top quark mass at NLO at NLO computed with ggHiggs and PDF4LHC30 computed with ggHiggs and PDF4LHC30 computed with ggHiggs and PDF4LHC30 1 1 1 µR/mH=0.93, µF/mH=0.93 µR/mH=0.5, µF/mH=0.5 µR/mH=2.23, µF/mH=2.23 µR/mH=2.0, µF/mH=1.0 µR/mH=3.13, µF/mH=3.13 µR/mH=0.5, µF/mH=0.5 µR/mH=1, µF/mH=1 µR/mH=2.0, µF/mH=1.0 µR/mH=1, µF/mH=1 µR/mH=2.0, µF/mH=2.0 µR/mH=1, µF/mH=1 µR/mH=2.0, µF/mH=1.0 0.5 µR/mH=1, µF/mH=2 µR/mH=2.0, µF/mH=2.0 0.5 µR/mH=1, µF/mH=2 µR/mH=0.5, µF/mH=1.0 0.5 µR/mH=1, µF/mH=2 µR/mH=2.0, µF/mH=2.0 µR/mH=1, µF/mH=0.5 µR/mH=0.5, µF/mH=1.0 µR/mH=1, µF/mH=0.5 µR/mH=3.0, µF/mH=3.0 µR/mH=1, µF/mH=0.5 µR/mH=0.5, µF/mH=1.0 µR/mH=0.5, µF/mH=0.5 0 0 0 100 1000 10000 100000 100 1000 10000 100000 100 1000 10000 100000

sqrt(s)/mH sqrt(s)/mH sqrt(s)/mH (d) (e) (f) √ NLO 0 0 Fig. 4 K for (top) fictitious H˜ and (bottom) H with different (fictitious) heavy-quark masses for PDF4LHC15 nlo 30 as a function of s/MH for the usual 7-point scale choices and ourµ ˆ F scale with µR = µF . [Only the case (f) is realistic, all the other are academical examples.] ergies to FCC energies is the sign of a bad convergence of In conclusion, even with a priori the steepest possible the NLO computation for such scales. Besides, we do not gluon PDFs compatible with a global NLO PDF analysis, observe any more a peak µF = 0.5MQ, for which the en- one gets negative or strongly suppressed NLO ηc cross sec- ergy dependence of KNLO starts to be acceptable. Choosing tions for a majority of the conventional scale choices (5 out µF = µˆ F gives the best trend with a quasi constant value, of 7), whereas that obtained with our scale choice µF = µˆ F close to 1, for 1 TeV and above. Such a choice completely is remarkably stable. For ηb, the 3 PDFs essentially yield stabilises the KNLO energy dependence as it results that go- the same KNLO factors which also shows the most stable being to higher energies does not give an artificial importance haviour for µF = µˆ F . 3 to the αs corrections. Having demonstrated the efficiency of our scale choice From the early studies of Schuler, Mangano and Petrelli to avoid anomalously large NLO corrections to pseudoscalar one expects a strong sensitivity of the PDF shape on the quarkonium production attributed to an over-subtraction of impact of the NLO corrections (see also [42]). We have the collinear divergence inside the PDFs, let us now investi- checked that the PDF uncertainty on KNLO derived from gate whether it works for elementary scalar bosons coupling the 30 PDF4LHC15 nlo 30 eigensets is indeed smaller for to gluons via heavy quarks. If our argumentation√ is correct, NLO µF = µˆ F than for larger scales, despite the fact that the K should, first, be rather µF - and s-dependent for a PDF uncertainty themselves usually decrease for growing scalar boson of similar mass than the ηc and, second, be- scales. Actually to assess the PDF-shape sensitivity, it can come stable for µF = µˆ F . more insightful to compare the trend with the central set Our results, shown on Fig. 4a, exactly confirms our ex- of JR14NLO08VF and NNPDF31sx nlonllx which show ˜ 0 pectation for H with MH˜ 0 = 3 GeV√ and mQ = 1.5 GeV. a clear different shape in particular close to 1.5 GeV (see If we were to work at even larger s, the larger µF choices Fig.2). These are respectively shown on Fig. 3b& 3e and would eventually yield very small KNLO. They would proba- Fig. 3c& 3f. For ηc, the trend is very similar compared to bly not become negative but we recall that |Aa| is smaller for 0 what we obtained with PDF4LHC15 nlo 30 except for the H˜ than for ηc rendering the HE limit slightly less harmful. absence of the peak for µF = 0.5MQ. As we wrote above, On the other hand, too small scales yield strongly growing such a peak resulted from the local maximum and minimum in the central PDF4LHC15 nlo 30 eigenset14. with flatter PDFs, σ is in principle more sensitive to the larges ˆ behaviour ofσ ˆ gg. For the considered scale, µF = 0.5MQ, this limit is 14Two effects can come into play here. First, the average momentum positive, thus KNLO is expected to get larger, precisely right after the fraction of the gluons in the NLO contributions is slightly larger than bump in the PDF luminosity. It is likely that the latter effect actually for the LO one. As such, if the gluon PDF oscillates, it could happen dominates. Indeed, for µF = µˆ F , the limiting value ofσ ˆ is set to 0 while that the PDFs product multiplying the gg NLO partonic cross section the bump in KNLO has nearly disappeared although there is still a slight could be larger than the LO one. Second, as we previously discussed, bump in the PDF. 11 √ NLO K at large s. Finally, setting µF = µˆ F , or close to it we have plotted on Fig.5 and Fig.6 the corresponding dif- with µF = MH˜ 0 sinceµ ˆ F = 0.93MH˜ 0 , gives remarkably sta- ferential gluon luminosity which would normally multiply ble KNLO. We consider this to be a confirmation that the in- a simple gg fusion process at LO. We have done so for our stabilities in NLO computations of quarkonium production 3 chosen PDF sets, for 2 masses M (3 and 9.5 GeV) and 3 are not connected to the modelling of quarkonium produc- √corresponding scales µF (0.5M, M and 2M) as a function of tion. The situation is equally good if we set consider, as an s and y in the ranges which will correspond to the NLO academic example, MH0 = 125 GeV and mQ = MH˜ 0 /2. In- cross section plots which we will show in the next sections. deed, µF = µˆ F yields the most stable results on Fig. 4d. It clearly appears that for PDF4LHC15 nlo 30, which is 0 On the other side, for the real H case with mt = representative of usual PDF sets, and to√ a lesser extent for 173 GeV –for which the situation is of course not NNPDF31sx nlonllx as 0118,√ both the s dependence at NLO = 0 and the dependence at = 14 TeV will strongly be problematic–, we observe√ on Fig. 4f that K tends to y y s clearly increase with s for the smaller scale choices, like distorted for µF = 1.5 GeV. Not only one observes a strong 16 µF = 0.5MH0 . Choosingµ ˆ F = 2MH0 or µF = µˆ F = 3.3MH0 scale sensitivity in the luminosity magnitude , but the dis- yield to more stable trends. In principle, we would expect tributions are very different. More importantly, it is very im- µF = µˆ F to yield the most stable curve. More investigations probable that any measured differential cross sections, even are needed to understand why µF = 2MH0 shows the best at low scales, would follow the trend of Fig. 5f (and Fig. 5d) trend. One observes the same for MH˜ 0 = 3 GeV and the with a yield showing a global maximum around y = 5 at same mQ/MH˜ 0 ratio on Fig. 4c. Fig. 4b and Fig. 4e, in be- the LHC or a differential cross section at y = 0 essentially tween both cases, hint at an effect which would scale like constant between the Tevatron and the top LHC energy like mQ. A possible explanation could come from contributions on Fig. 5a. As we will see, the corresponding NLO cross of the box diagrams which would yield A¯g(z) , A¯g(z) down sections will be driven by this behaviour of the gluon lumi- to very low z. nosity. Yet, the phenomenology of H0 being now made at N3LO accuracy [71, 72] in the infinite top-quark-mass limit, it is rather an academical question. For the validation of our scale 5.4 A digression on the ηb detectability ˜ 0 proposal, the success of the light H case with MH˜ 0 = 3 GeV NLO and mQ = 1.5 GeV, with strongly decreasing K for large Having now at our disposal reliable NLO predictions for ηb µF and a very stable one for µF = µˆ F is much more telling cross sections at hadron colliders, let us address the question since it perfectly confirms what we observed with the ηc. of the feasibility of such studies and in particular of the extraction of cross sections. Whereas prompt ηc production at the LHC has now been the object of two experimental studies [73, 74] by the LHCb collaboration, the prospects for η 5.3 A word on gluon luminosities at NLO and low scales b production studies are however less clear. Before moving to our NLO predictions for the cross sec- Compared to the ηc, we do not know much on the tions, we find it useful to make a short digression on the ηb properties which was only discovered in 2008 by a ar gluon luminosities. Indeed, we would not want the reader to B B [75] and whose mass is 9.4 GeV, while its 2S excita- be confused by some unnatural cross-section behaviours as tion, the ηb(2S ), was discovered in 2012 by Belle [76] with √ a mass of 10.0 GeV. The η in fact has so far been observed a function of s or y which we will show and to attribute b + − a ar these behaviours to QCD corrections to the hard scattering. only in e e annihilations, by B B [75, 77], CLEO [78] and Belle [76,79]. Most likely, future measurements will be Contrary to KNLO where the PDF impact is indirect15 because of a large cancellation of their effects in the ratio, the carried out by Belle II [80]. Its width has been measured to be on the order of 10 MeV. We note the good agreement with gg contribution to the hadronic cross section will essentially 2 2 2 the LO estimate for Γ(η → gg) = 8α |R0| /3M assuming be proportional to the square of gluon PDFs at low scales. b s ηb ∼ → As we have seen when discussing Fig.2, the conventional Γ(ηb) Γ(ηb gg). The agreement is confirmed up to PDF sets typically exhibit a local minimum below 0.001 at NNLO [81]. So far no measurement of any branching frac- scales below 2 GeV. In this region, however, the gluon PDFs tions are reported in the PDG [82] and it is thus not clear are only poorly constrained by scarce data sensitive to glu- in which decay channel it could be measured in hadroproduction. In addition, one needs to know the branching value ons at smaller x and larger scales. with an acceptable uncertainty to derive a cross section to To illustrate the√ typical effects that such a shape can in- duce on both the s or y distribution on a low-scale process, 16 In a sense, this is acceptable since µF /µF,0 (with µF,0 being where the evolution starts) significantly varies between the µF values we took. 15Yet, we have observed a bump in KNLO because of the changing Hence, the evolution can generate significantly more gluons at our shape of PDF4LHC15 nlo 30 for low scales. larger scale choice. 12

104 104 104

ηc PDF4LHC15_30_nlo ηc JR14NLO08VF ηc NNPDF31NLL

μF=1.5GeV μF=1.5GeV μF=1.5GeV 3 3 3

( y = 0 ) 10 μF=3.0GeV ( y = 0 ) 10 μF=3.0GeV ( y = 0 ) 10 μF=3.0GeV

μF=6.0GeV μF=6.0GeV μF=6.0GeV

GeV M = 3 102 GeV M = 3 102 GeV M = 3 102

101 101 101 dL / d τ dy / s dL / d τ dy / s dL / d τ dy / s 2 2 2 M M M 100 100 100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c) 104 104 104

ηc, s=14TeV, PDF4LHC15 ηc, s=14TeV, JR14NLO08VF ηc, s=14TeV, NNPDF31NLL ======103 μF 1.5GeV μF 3.0GeV μF 6.0GeV 103 μF 1.5GeV μF 3.0GeV μF 6.0GeV 103 μF 1.5GeV μF 3.0GeV μF 6.0GeV

102 102 102

1 1 1

dL / d τ dy / s 10 dL / d τ dy / s 10 dL / d τ dy / s 10 2 2 2 M M M

100 100 100

- - - 10 1 10 1 10 1 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 y y y (d) (e) (f) √ √ dL dL Fig. 5 τ0 dτdy as function of energy s and at y = 0 (top) and dτdy as function of y at s = 14 TeV (bottom) for M = 3 GeV (for PDF4LHC15 nlo 30 (left), JR14NLO08VF (middle) and NNPDF31sx nlonllx as 0118 (right)) for 3 µF values (0.5M, M and 2M).

105 105 105

ηb ηb PDF4LHC15_30_nlo ηb JR14NLO08VF NNPDF31NLL 4 4 4 μF 10 μF=4.75GeV 10 μF=4.75GeV 10 =4.75GeV

( y = 0 ) ( y = 0 ) ( y = 0 ) μF μF=9.5GeV μF=9.5GeV =9.5GeV 103 103 103 μF μF=19.0GeV μF=19.0GeV =19.0GeV 2 2 2

. GeV M = 9.5 10 GeV M = 9.5 10 GeV M = 9.5 10

101 101 101

100 100 100 dL / d τ dy / s dL / d τ dy / s dL / d τ dy / s 2 10-1 2 10-1 2 10-1 M M M

0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c) 104 104 104

103 103 103

102 102 102

101 101 101 dL / d τ dy / s dL / d τ dy / s dL / d τ dy / s

2 0 2 0 2 0 ηc, s=14TeV, PDF4LHC15 ηc, s=14TeV, JR14NLO08VF ηc, s=14TeV, NNPDF31NLL M 10 M 10 M 10

μF=4.75GeV μF=9.5GeV μF=4.75GeV μF=9.5GeV μF=4.75GeV μF=9.5GeV -1 -1 -1 10 μF=19.0GeV 10 μF=19.0GeV 10 μF=19.0GeV

- - - 10 2 10 2 10 2 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 y y y (d) (e) (f) √ √ dL dL Fig. 6 τ0 dτdy as function of energy s and at y = 0 (top) and dτdy as function of y at s = 14 TeV (bottom) for M = 9.5 GeV (for PDF4LHC15 nlo 30 (left), JR14NLO08VF (middle) and NNPDF31sx nlonllx as 0118 (right)) for 3 µF values (0.5M, M and 2M).). test state-of-the-art computations which do not address the Diﬀerent theoretical ideas have been pushed forward decay but only the production. about the usable decay channels at the Tevatron and the 13

LHC. For a long time, the decay into a pair of J/ψ po- easier to deal with as what regards the combinatoric back- tentially clearly signalled by 4-muon events was consid- ground. Tools like EtabFDC [94] should also definitely be ered to be a discovery channel in the busy environment of helpful for experimental prospective studies. pp collisions. Even though, from the beginning, physicists As what regards the experimental setups where ηb were aware that the branching fraction into this channel hadroproduction cross sections could be measured in pp could be very small, we should stress here that the produc- collisions, let us cite the LHC in the collider and fixed- tion cross sections for ηb at colliders are not small at all, target modes, in particular with the LHCb detector. The lat- as they are comparable to those for Υ( ) which are rou- nS ter mode has been studied in details in [95√–98] as what tinely studied at the LHC. As such, small branching frac- regards quarkonium production. Its nominal s for 7 TeV tions could still yield observable rates. First estimations [83] proton beams reaches 114.6 GeV. Another possibility is the 17 −4±1 √ reported B(ηb → J/ψ + J/ψ) = 7 × 10 , using mass- SPD detector at the NICA facility up to s = 27 GeV [99]. rescaling arguments applied to ηc → φφ. However, this es- We will provide predictions for these 3 setups. timate was then questioned and searches via the detection of 2 charmed mesons were suggested [84]. Actual computations based on NRQCD [85,86] later yielded a much smaller 5.5 Cross section predictions −8 B(ηb → J/ψ + J/ψ), as low as 5 × 10 . It was however suggested that final-state interactions, beyond the effects in- We are now finally in position to show our results for the cluded in NRQCD computations, could enhance the di-J/ψ √ cross sections√ as a function of s and of y for selected val- decay width by up 2 orders of magnitude [87]. It is thus clear ues of s which correspond to experimental setups where that until B(ηb → J/ψ+J/ψ) is actually measured elsewhere, we believe the challenging measurement of PT -integrated it could not provide a way to derive cross-section measure- ηQ yield could be performed in the future. These are the ments. Yet, given the current intense activity in J/ψ + J/ψ LHC at 14 TeV in the collider mode and in the fixed-target studies with the observation [88] of a di-J/ψ resonance, a mode at 114.6 GeV, having in mind in particular the LHCb search for this decay channel at the LHC could still be an detector, and to the SPD experiment at NICA which could option as it may also be looked for at Belle II. Another chan- run up to 27 GeV. nel subject of debates is that into 2 charmed mesons as the As our study mainly addresses the interplay between the first branching-fraction estimate [84], on the order of a few size of the NLO corrections and the choice of scales, mainly per cent, was then also questioned [89]. Finally, let us men- NLO that of µ , we start by showing, dσ /dy| =0 as a func- √F ηc y tion the inclusive channel ηb → J/ψX which however suf- tion of s, for the 7-point scale choice (in black) andµ ˆ F fers from large uncertainties owing to possible CO contri- (in green) for the central eigensets of our 3 PDF sets. These butions [90]. Other ideas could be found by looking at the NLO plots are the exact counterpart of the K |y=0 plots of Fig.3 many χb decay channels which have been analyses so far, which allowed us to assess the much better convergence of see [82]. the NLO results at high energies when taking µF = µˆ F . The An alternative to these hadronic decays is the exclusive essential difference here is that the PDF effects do not can- → radiative decay, ηb J/ψγ, whose branching is computed cel. to be on the order of 2×10−7 [85], which remains admittedly As what regards the η results shown on the first row of small. In principle, it suffers from smaller theory uncertain- c Fig.7, we first note that 3 out of the 7 scale choices leads ties compared to those above. An even simpler branching to to negative cross section (see the incomplete curves) irre- predict is that of η → γγ, whose partial width is in fact b spective of the PDF choice. No matter what the PDF shape known up to two loops [91, 92] in NRQCD18. In addition, is, too “large” a value of µ inevitably leads to unaccept- some theoretical uncertainties cancel with those of the to- F able results. This observation is obviously in line with pre- tal width computed in NRQCD by assuming the dominance vious studies [37, 40, 41]. We now know that it is related of the decay into two gluons, which is also known up to to large negative contributions away from threshold due to NNLO [81] (see above), yielding a process-dependent oversubtraction of the collinear diver- −5 gences which cannot be compensated by the PDF evolution, B(ηb → γγ) = (4.8 ± 0.7) × 10 . (13) in particular at low scales where they are not much evolved. NLO We thus believe that this channel should seriously be con- Now, as anticipated during the discussion of K |y=0 plots, sidered for a first extraction of the ηb hadroproduction cross choosing µF = µˆ F provides much more sound results. The section even though channels involving J/ψ are probably results are particularly good up to the FCC energies with the JR14NLO08VF PDF as can be seen on Fig. 7b. 17Note that this should then be multiplied by the square of B(J/ψ → `+`−), i.e. 6 %. Yet, as we discussed in section 5.3, most of the conven- 18Other approaches may sometimes give different results, but not by tional PDFs exhibit, at low scales, a local minimum for x factors differing by more than 2 [93]. around or below 0.001. This results in gluon luminosities 14

5×105 5×105 5×105

ηc PDF4LHC15_30_nlo ηc JR14NLO08VF ηc NNPDFNLL μR=μF=M/2 μR=M/2,μ F=M μR=μF=M/2 μR=M/2,μ F=M μR=μF=M/2 μR=M/2,μ F=M μ =M,μ =M/2 μ =μ =M μ =M,μ =M/2 μ =μ =M μ =M,μ =M/2 μ =μ =M 105 R F R F 105 R F R F 105 R F R F μR=M,μ F=2M μR=2M,μ F=M μR=M,μ F=2M μR=2M,μ F=M μR=M,μ F=2M μR=2M,μ F=M [ nb ] [ nb ] [ nb ]

μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e y = 0 y = 0 y = 0 / dy / dy / dy 104 104 104 NLO NLO NLO d σ d σ d σ

103 103 103 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c)

104 104 104

103 103 103 [ nb ] [ nb ] [ nb ] y = 0 y = 0 y = 0 102 102 102 / dy / dy / dy

1 ηb PDF4LHC15_30_nlo 1 ηb JR14NLO08VF 1 ηb NNPDFNLL NLO 10 NLO 10 NLO 10 μR=μF=M/2 μR=M/2,μ F=M μR=μF=M/2 μR=M/2,μ F=M μR=μF=M/2 μR=M/2,μ F=M d σ d σ d σ μR=M,μ F=M/2 μR=μF=M μR=M,μ F=M/2 μR=μF=M μR=M,μ F=M/2 μR=μF=M 0 0 0 10 μR=M,μ F=2M μR=2M,μ F=M 10 μR=M,μ F=2M μR=2M,μ F=M 10 μR=M,μ F=2M μR=2M,μ F=M

μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e μR=μF=2M μR=μF=M/ e

0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (d) (e) (f) √ dσNLO | Fig. 7 dy y=0 for ηc (top) and ηb (bottom) (for PDF4LHC (left), JR14NLO08VF (middle), NNPDFsxNLONLL (right)) as a function of s for the usual 7-point scale choices and ourµ ˆ f scale with µR = µF .

5×105 5×105

ηc NLO:μ R=μF=M/ e ηc LO:μ R=μF=M/ e PDF4LHC15 JR14NLO08VF PDF4LHC15 JR14NLO08VF NNPDFNLL ABM11_3n_nlo NNPDFNLL ABM11_3n_nlo 5 5 10 CT14nlo NNPDF31_nlo_as_0118 10 CT14nlo NNPDF31_nlo_as_0118

[ nb ] MMHT2014nlo68cl [ nb ] MMHT2014nlo68cl y = 0 y = 0 / dy / dy 104 104 NLO NLO d σ d σ

103 103 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] (a) (b) √ dσNLO | Fig. 8 dy y=0 for ηc (a) at NLO and (b) LO for various NLO PDF choice as a function of s forµ ˆ f scale with µR = µF . –which correspond the expected kinematical dependencies in order to yield monotonous gluon PDFs at scale below for a simple LO gg fusion– essentially constant in the TeV 2 GeV. It is very important to realise that such a distorted range. Without any surprise, this what we observe for theµ ˆ F shape is not at all due to the NLO corrections, but entirely curve using PDF4LHC15 nlo 30 (Fig. 7a) and to a lesser due to the PDFs, as the JR14NLO08VF with monotonous extent for NNPDF31sx nlonllx as 0118 (Fig. 7c). As ex- gluon PDF case shows (Fig. 7b and Fig. 8a). pected, we observe the same with MMHT14nlo, CT14nlo and NNPDF31 nlo as 0118 on Fig.8 both at LO and NLO. Let us now turn to the ηb case for which we know that the issue of over-subtraction is less problematic. Indeed, The ηc NLO energy dependence admittedly does√ not one only sees, on the second row of Fig.7, a slight devi- make sense when it remains a constant between s = ation at the FCC energies for the curves for µF = 2M and 10 GeV and 10 TeV ! We urge the global ﬁtters to examine µF = 0.5M which admittedly is the most critical one accord- NLO whether global NLO ﬁts cannot in fact be slightly amended ing to our K |y=0 analysis (see Fig.3). Even though the 15

5×105 5×105 5×105

ηc PDF4LHC15_30_nlo,μ R=μF=M/ e ηc JR14NLO08VF,μ R=μF=M/ e ηc NNPDFsxNLL,μ R=μF=M/ e 105 NLO: PDF Uncertainty 105 NLO: PDF Uncertainty 105 NLO: PDF Uncertainty

[ nb ] NLO:μ R Uncertainty [ nb ] NLO:μ R Uncertainty [ nb ] NLO:μ R Uncertainty y = 0 y = 0 y = 0 / dy / dy / dy 104 104 104 NLO NLO NLO d σ d σ d σ

103 103 103 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c)

η PDF4LHC15_30_nlo,μ =μ =M/ e η JR14NLO08VF,μ =μ =M/ e η NNPDFsxNLL,μ =μ =M/ e 4 b R F 4 b R F 4 b R F 10 NLO: PDF Uncertainty 10 NLO: PDF Uncertainty 10 NLO: PDF Uncertainty

NLO:μ R Uncertainty NLO:μ R Uncertainty NLO:μ R Uncertainty 103 103 103 [ nb ] [ nb ] [ nb ] y = 0 y = 0 y = 0 102 102 102 / dy / dy / dy 1 1 1 NLO 10 NLO 10 NLO 10 d σ d σ d σ 100 100 100

0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (d) (e) (f) √ dσNLO | Fig. 9 dy y=0 for ηc (top) and ηb (bottom) (for PDF4LHC (left), JR14NLO08VF (middle), NNPDFsxNLONLL (right)) as a function of s for ourµ ˆ f scale. The green bands indicate the PDF uncertainty (for µR = µF ) and the red band, the µR uncertainty (for µR ∈ [M/2 : 2M]).

ηc PDF4LHC15_30_nlo,μ F=M/ e ηc JR14NLO08VF,μ F=M/ e ηc NNPDFsxNLL,μ F=M/ e 200 200 200 LOμ R Uncertainty LOμ R Uncertainty LOμ R Uncertainty

NLOμ R Uncertainty NLOμ R Uncertainty NLOμ R Uncertainty NLO PDF Uncertainty NLO PDF Uncertainty NLO PDF Uncertainty 100 100 100 [%] Δσ / σ [%] Δσ / σ [%] Δσ / σ 0 0 0

-100 -100 -100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (a) (b) (c)

ηb PDF4LHC15_30_nlo,μ F=M/ e ηb JR14NLO08VF,μ F=M/ e ηb NNPDFsxNLL,μ F=M/ e 200 200 200 LOμ R Uncertainty LOμ R Uncertainty LOμ R Uncertainty

NLOμ R Uncertainty NLOμ R Uncertainty NLOμ R Uncertainty NLO PDF Uncertainty NLO PDF Uncertainty NLO PDF Uncertainty 100 100 100 [%] Δσ / σ [%] Δσ / σ [%] Δσ / σ 0 0 0

-100 -100 -100 0.1 1 10 100 0.1 1 10 100 0.1 1 10 100 s[TeV] s[TeV] s[TeV] (d) (e) (f)

Fig. 10 Relative uncertainties√ ∆σ/σ from µR (for µR ∈ [M/2 : 2M]; red band) and PDF (for µR = µF ; green band) at NLO for ηc (top) and ηb (bottom) as a function of s compared to the µR uncertainty at LO for PDF4LHC (left), JR14NLO08VF (middle) and NNPDFsxNLONLL (right). NLO| are signiﬁcantly µ -dependent, the σNLO/ | are much less µ -dependent as the diﬀerence in the – Kηb y=0 F d ηb dy y=0 F 16

4 6 10 104 10

ηc s=24GeV, JR14NLO08VF, , 5 μR=μF=M/ e 10 103 LO: PDF Uncertainty 103 NLO: PDF Uncertainty 4 LO:μ R Uncertainty 10

NLO:μ R Uncertainty 2 2 3 [ nb ] / dy [ nb ] / dy 10 [ nb ] / dy 10 10 s ηc, s=114.6GeV, JR14NLO08VF, ηc, =14TeV, JR14NLO08VF, e μR=μF=M/ e NLO μR μF NLO NLO = =M/ LO: PDF Uncertainty 2 LO: PDF Uncertainty d σ d σ

d σ 10 NLO: PDF Uncertainty NLO: PDF Uncertainty 1 1 10 10 LO:μ R Uncertainty LO:μ R Uncertainty NLO:μ R Uncertainty 101 NLO:μ R Uncertainty

100 100 100 0 1 2 0 1 2 3 4 0 1 2 3 4 5 6 7 8 y y y (a) (b) (c) 2 4 100 10 10

s ηb, s=24GeV, JR14NLO08VF, ηb, =114.6GeV, JR14NLO08VF, e e -1 μR=μF=M/ μR=μF=M/ 10 LO: PDF Uncertainty LO: PDF Uncertainty 103 NLO: PDF Uncertainty NLO: PDF Uncertainty 1 LO:μ R Uncertainty 10 LO:μ R Uncertainty -2 10 NLO:μ R Uncertainty NLO:μ R Uncertainty 2 [ nb ] / dy

[ nb ] / dy 10 [ nb ] / dy ηb, s=14TeV, JR14NLO08VF,

μR=μF=M/ e

-3 NLO NLO NLO 10 0 LO: PDF Uncertainty d σ

d σ 10 d σ NLO: PDF Uncertainty 1 10 LO:μ R Uncertainty -4 10 NLO:μ R Uncertainty

10-5 10-1 100 0 1 0 1 2 0 1 2 3 4 5 6 7 y y y (d) (e) (f) √ dσNLO Fig. 11 dy for ηc (top) and ηb (bottom) as a function of y at s = 24 GeV(left), 114.6 GeV (middle), 14 TeV (right), for µF = µˆ F = µR for JR14NLO08VF at LO and NLO. The green (gray) bands indicate the PDF uncertainty (for µR = µF ) and the red (beige) band, the µR uncertainty (for µR ∈ [M/2 : 2M]) at NLO (LO). process-independent– gluon evolution induced by the dif- of this set. It is admittedly much smaller than the conven- ferent chosen µF values efficiently compensates the explicit tional ones. Possible experimental data should be able to test µF dependence of the –process-dependent– hard scattering such PDFs rather straightforwardly. for ηb. In such a case, the conventional scale choices would To go further in the analysis of our improved NLO re- range between 5 and 20 GeV. Clearly, for ηc, such a com- sults with µF = µˆ F , we have plotted on Fig. 10 the relative pensation does not work at NLO and the only solution to the uncertainties, dubbed as ∆σ/σ, from the PDF uncertainties issue remains our scale choice µF = µˆ F . and from µR variations, obtained by normalising the upper This is why for the following plots we stick to this and lower values of dσηQ /dy|y=0 by the respective central scale choice which we consider to give, at the moment, ones, i.e. from the central eigenset for the PDF uncertainty the best possible NLO predictions for η production in c and from µR = MQ for the µR uncertainty. The resulting NLO the TeV range. On Fig.9, we show again dσ /dy| =0 ηQ y (green and red) bands are compared to the LO µR uncertainty but with the PDF uncertainty associated with our PDF set (horizontal dotted blue line)–which is obviously a constant. choices for µR = µˆ F (green band) and the µR uncertainty Our first observation is that the renormalisation scale uncer- for µR ∈ [M/2 : 2M] (red band). The same observa- tainty is clearly reduced at NLO, which is a good sign of the tion as above can be made. Sinceµ ˆ = 1.82 GeV for η , F c αs convergence even at these low scales. Second we note PDF4LHC15 nlo 30 results show a large distortion due the √ that the scale uncertainty for ηc is smaller, for s > 2 TeV, PDF eigenset shapes and, clearly, nobody would expect to than that from PDF4LHC15 nlo 30, on the order of 30 % 19 see it in any experimental data in the future . The situation and then steadily growing; thich is representative of what is better for NNPDF31sx nlonllx as 0118 and remains very NLO global fits would give. This means that forthcoming ηc good for JR14NLO08VF. For ηb, the 3 PDF sets yield simi- data at the LHC with a precision of 10 %, or lower, should lar results. The smaller uncertainty band for JR14NLO08VF already be enough to improve PDF fits, even taking into ac- (Fig. 9e) simply comes from the very small PDF uncertainty count the µR uncertainty. For ηb, both are of similar√ sizes NLO and one should probably look at dση /dy at fixed s or at 19 NLO √ Q We stress that dσ /dy| =0 for µ = µˆ only get negative on Fig. 9a ηc y F F various if available to get more discriminating power on and Fig. 9c because some gluon NLO PDF eigensets get negative. Neg- s ative cross sections would disappear at LO since the gluon PDF are the PDF along the lines of [100–102]. The same holds for ηc squared at y = 0. at lower energies, which should then be differential in y. At 17 this point, we wish to stress that we have decided not to vary scales. This over-subtraction cannot be returned in a global the charm and beauty quark masses20 which are also usually manner by the PDFs and this mismatch badly affects the considered to yield additional theoretical uncertainties. We charmonium phenomenology as αs is not very small and the however note that√ the induced variations are highly corre- PDF evolution is limited which results in flat PDF shapes in lated in y (and s) and such correlations can expediently be the mid and low-x region. used to alleviate their effect in a possible PDF fit. Our solution to this issue is to propose a new scale set- In view of the above observations,√ we have decided to ting,µ ˆ F , which is based on the simple criterion that the par- show σNLO/ at fixed on Fig. 11 only for our µ d ηQ dy s F tonic cross-section vanish at larges ˆ (or small z). This is to scale choice, only using JR14NLO08VF and its uncertainty be understood as that the real-emission contributions com- (green band), along with the µR uncertainty (red band) coming from the initial partons are entirely absorbed in the PDF. pared to the LO results (gray and beige bands). Our objective Although less ambitious, this is somewhat equivalent to a with these plots is to provide NLO predictions to motivate resummation picture, yet much simpler to implement. prospects for measurements and then the extraction of con- We have demonstrated the efficiency of this new scale straints on PDFs, rather than to test quarkonium-production setting for a wide class of rapidity and energy shapes. We models. From a kinematical viewpoint, σNLO/ measure- d ηc dy have applied this scale setting to ηc,b production, a fictitious ment should offer reliable constraints on the gluon PDF x light elementary scalar boson H˜ 0 and also for the real BEH −2 dependence in the approximate range [5×10 : 1] for SPD, boson H0 with different mass values of a fictitious heavy- − − − [10 2 : 1] for AFTER@LHCb and [10 6 : 5 × 10 2] for quark active in the loop. The success of this scale demon- LHCb (at 14 TeV). At SPD, the ηc production cross section strates that the issue we tackle is not in principle limited to is expected to be on the order of 1 pb. At AFTER@LHC, it quarkonium but rather to processes occurring at low scales, would grow to 10 pb to reach 200 pb at the LHC. Of course, in particular when some unfavourable effects add up. to these, one should apply rapidity-acceptance cuts, besides Having cured the NLO ηc,b phenomenology from these the appropriate branching fractions. As for the ηb, the y- negative cross sections, we have then provided predictions integrated cross section, σNLO( ) is respectively expected to ηb s for SPD, AFTER@LHC, LHC up to FCC energies. Natu- be 0.5 nb, 60 nb and 10 pb. rally, our NLO ηc predictions bore on PDFs at low√ scales. Indeed, our scale choice for ηc amounts to 2mc/ e, thus 1.82 GeV which is well within the usual scale range for a 6 Conclusion and outlook produced system whose mass is 3 GeV. However, at such a scale, conventional PDFs such as PDF4LHC15 nlo 30 In this work, we have addressed the unphysical predic- which we used exhibit a local minimum for x close or be- tions of the collinear and NRQCD factorisations for the PT - low 0.001. This translates into a distortion of the energy and integrated quarkonium production, whereby negative cross rapidity dependence of the cross-section in the TeV range sections are obtained for most of the conventional factori- that nobody would expect to see in experimental data. sation and renormalisation scale choices at LHC energies We thus encourage the PDF community to see what down to RHIC energies in some cases. In particular, we have would be the outcome of a global NLO PDF fit prevent- focused on the pseudoscalar case, which is by far the sim- ing a local minimum in xg(x), which is by the way ab- plest to tackle with analytical results available for the total sent in NNPDF31sx nlonllx as 0118, where resummation cross section available since the mid nineties. On the way, effects have been taken into account in the PDF fit, and in we have provided analytical results for the rapidity differen- JR14NLO08, where gluons are evolved from a lower scale. tial cross sections, which were not available elsewhere. This local minimum also seems to contradict the recent anal- We have shown that this unphysical behaviour can be ysis of Flett et al. [64]. explained through the high-energy limit of the partonic Similarly, we encourage the LHC experimental commu- cross section, which is negative unless µ is chosen to be F nity to study such P -integrated quarkonium cross section, relatively small compared to the quarkonium mass. This T despite the likely challenging decay channels which should negative limit can be ultimately traced back to an over- be used. η has already been studied by LHCb at finite P subtraction of the Altarelli-Parisi counterterm in the MS c T (P > 6 GeV), they can definitely push down to 0 with scheme to absorb the collinear divergences inside the PDFs. T a targeted effort, hopefully motivated by our study. As for This over-subtraction should usually be returned via the η , which remains unobserved in hadroproduction, we have DGLAP evolution with steeper PDFs. However, the high- b gathered some suggestions on how to extract its production energy-limit values A are process-dependent while the a cross section at the LHC. Indeed, these cross sections are DGLAP evolution is clearly process-independent at fixed definitely very large. Once they are available, we believe 20 them to be ideal to better constrain the gluon PDF at low Typical variations for quarkonium production studies are mc ∈ [1.4 : 1.6] GeV and mb ∈ [4.5 : 5.0] GeV. scales, and thus the gluon content of the proton in general, 18 despite the remaining theoretical uncertainties inherent to 8. A. D. Martin, C. K. Ng, and W. J. Stirling, “Inelastic quarkonium production. Extension to nuclei could then be Leptoproduction of J/ψ as a Probe of the Small X Behavior of considered along the lines of [103, 104]. the Gluon Structure Function,” Phys. Lett. B191 (1987) 200. 9. A. D. Martin, R. G. Roberts, and W. J. Stirling, “Structure In the near future, it remains to be investigated the effect Function Analysis and psi, Jet, W, Z Production: Pinning Down of such a scale setting for χc and J/ψ production. The sit- the Gluon,” Phys. Rev. D37 (1988) 1161. uation for the χ and J/ψ is much worse [37, 39] than that 10. H. Jung, G. A. Schuler, and J. Terron, “J / psi production c mechanisms and determination of the gluon density at HERA,” for η as one already encounters negative yields in hadropro- c √ Int. J. Mod. Phys. A7 (1992) 7955–7988. duction at s as low as a couple of hundreds of GeV. With 11. D. Boer and C. Pisano, “Polarized gluon studies with our scale setting, we expect that both χc and J/ψ NLO cross charmonium and bottomonium at LHCb and AFTER,” Phys. section will stabilise and give physical cross-section results Rev. D86 (2012) 094007, arXiv:1208.3642 [hep-ph]. 12. W. J. den Dunnen, J. P. Lansberg, C. Pisano, and M. Schlegel, which can then be used in NLO PDF fits. “Accessing the Transverse Dynamics and Polarization of Gluons inside the Proton at the LHC,” Phys. Rev. Lett. 112 Acknowledgements (2014) 212001, arXiv:1401.7611 [hep-ph]. 13. D. Boer, “Gluon TMDs in quarkonium production,” Few Body We are indebted to Y. Feng for cross checks of our codes Syst. 58 no. 2, (2017) 32, arXiv:1611.06089 [hep-ph]. with FDC. We thank S. Abreu, N. Armesto, S. Barsuk, 14. J.-P. Lansberg, C. Pisano, F. Scarpa, and M. Schlegel, “Pinning M. Becchetti, V. Bertone, D. Boer, M. Bonvini, C. Duhr, down the linearly-polarised gluons inside unpolarised protons using quarkonium-pair production at the LHC,” Phys. Lett. M.G. Echevarria, C. Flett, C. Flore, M. Garzelli, B. Gong, B784 (2018) 217–222, arXiv:1710.01684 [hep-ph]. R. Harlander, L. Harland-Lang, T. Kasemets, A. Kusina, [Erratum: Phys. Lett.B791,420(2019)]. M. Mangano, S. Marzani, M. Nefedov, C. Pisano, J.W. Qiu, 15. J.-P. Lansberg, C. Pisano, and M. Schlegel, “Associated H. Sazdjian, I. Schienbein, H.S. Shao, A. Signori, M. Spira, production of a dilepton and a Υ(J/ψ) at the LHC as a probe of gluon transverse momentum dependent distributions,” Nucl. L. Szymanowski, A. Usachov, S. Wallon, J.X. Wang, for Phys. B920 (2017) 192–210, arXiv:1702.00305 [hep-ph]. their help, insightful comments and useful suggestions on 16. J.-P. Lansberg, C. Pisano, F. Scarpa, and M. Schlegel, “Probing our study. the gluon TMDs with quarkonia,” PoS DIS2018 (2018) 159, arXiv:1808.09866 [hep-ph]. This project has received funding from the European 17. A. Bacchetta, D. Boer, C. Pisano, and P. Taels, “Gluon TMDs Union’s Horizon 2020 research and innovation programme and NRQCD matrix elements in J/ψ production at an EIC,” Eur. under the grant agreement No.824093 in order to con- Phys. J. C80 no. 1, (2020) 72, arXiv:1809.02056 [hep-ph]. tribute to the EU Virtual AccessNLOAccess. This work 18. U. D’Alesio, F. Murgia, C. Pisano, and P. Taels, “Azimuthal asymmetries in semi-inclusive /ψ + jet production at an EIC,” was also partly supported by the French CNRS via the J Phys. Rev. D100 no. 9, (2019) 094016, arXiv:1908.00446 IN2P3 project GLUE@NLO and via the Franco-Chinese [hep-ph]. LIA FCPPL (Quarkonium4AFTER), by the Paris-Saclay U. 19. R. Kishore, A. Mukherjee, and S. Rajesh, “Sivers asymmetry in via the P2I Department and by the P2IO Labex via the Glu- the photoproduction of a J/ψ and a jet at the EIC,” Phys. Rev. D101 no. 5, (2020) 054003, arXiv:1908.03698 [hep-ph]. odynamics project. M.A.O.’s work was partly supported by 20. F. Scarpa, D. Boer, M. G. Echevarria, J.-P. Lansberg, C. Pisano, the ERC grant 637019 “MathAm”. and M. Schlegel, “Studies of gluon TMDs and their evolution using quarkonium-pair production at the LHC,” Eur. Phys. J. References C80 no. 2, (2020) 87, arXiv:1909.05769 [hep-ph]. 21. D. Boer, U. D’Alesio, F. Murgia, C. Pisano, and P. Taels, “J/ψ meson production in SIDIS: matching high and low transverse 1. M. Kraemer, “Quarkonium production at high-energy colliders,” momentum,” JHEP 09 (2020) 040, arXiv:2004.06740 47 Prog. Part. Nucl. Phys. (2001) 141–201, [hep-ph]. arXiv:hep-ph/0106120 [hep-ph] . 22. M. Kraemer, “QCD corrections to inelastic J / psi 2. Quarkonium Working Group Collaboration, N. Brambilla photoproduction,” Nucl. Phys. B459 (1996) 3–50, et al., “Heavy quarkonium physics,” arXiv:hep-ph/0412158 arXiv:hep-ph/9508409 [hep-ph]. [hep-ph]. 23. P. Artoisenet, J. M. Campbell, J. Lansberg, F. Maltoni, and 0 3. J. P. Lansberg, “J/ψ, ψ and Υ production at hadron colliders: A F. Tramontano, “Υ Production at Fermilab Tevatron and LHC Review,” Int. J. Mod. Phys. A21 (2006) 3857–3916, Energies,” Phys. Rev. Lett. 101 (2008) 152001, arXiv:hep-ph/0602091 [hep-ph]. arXiv:0806.3282 [hep-ph]. 4. N. Brambilla et al., “Heavy Quarkonium: Progress, Puzzles, and 24. J. Lansberg, “On the mechanisms of heavy-quarkonium Opportunities,” Eur. Phys. J. C71 (2011) 1534, hadroproduction,” Eur. Phys. J. C 61 (2009) 693–703, arXiv:1010.5827 [hep-ph]. arXiv:0811.4005 [hep-ph]. 5. A. Andronic et al., “Heavy-flavour and quarkonium production 25. J. Lansberg, “Real next-to-next-to-leading-order QCD in the LHC era: from proton–proton to heavy-ion collisions,” corrections to J/ψ and Upsilon hadroproduction in association Eur. Phys. J. C76 no. 3, (2016) 107, arXiv:1506.03981 with a photon,” Phys. Lett. B 679 (2009) 340–346, [nucl-ex]. arXiv:0901.4777 [hep-ph]. 6. J.-P. Lansberg, “New Observables in Inclusive Production of 26. B. Gong, J.-P. Lansberg, C. Lorce, and J. Wang, Quarkonia,” Phys. Rept. 889 (2020) 1–106, “Next-to-leading-order QCD corrections to the yields and arXiv:1903.09185 [hep-ph]. polarisations of J/Psi and Upsilon directly produced in 7. F. Halzen, F. Herzog, E. W. N. Glover, and A. D. Martin, “The association with a Z boson at the LHC,” JHEP 03 (2013) 115, J/ψ as a Trigger inpp ¯ Collisions,” Phys. Rev. D30 (1984) 700. arXiv:1210.2430 [hep-ph]. 19

27. J.-P. Lansberg and H.-S. Shao, “Production of J/ψ + ηc versus 48. G. Altarelli and G. Parisi, “Asymptotic Freedom in Parton 5 J/ψ + J/ψ at the LHC: Importance of Real αs Corrections,” Language,” Nucl. Phys. B 126 (1977) 298–318. Phys. Rev. Lett. 111 (2013) 122001, arXiv:1308.0474 49. J.-X. Wang, “Progress in FDC project,” Nucl. Instrum. Meth. A [hep-ph]. 534 (2004) 241–245, arXiv:hep-ph/0407058. 28. J.-P. Lansberg and H.-S. Shao, “J/psi -pair production at large 50. R. V. Harlander, H. Mantler, S. Marzani, and K. J. Ozeren, 5 momenta: Indications for double parton scatterings and large αs “Higgs production in gluon fusion at next-to-next-to-leading contributions,” Phys. Lett. B751 (2015) 479–486, order QCD for finite top mass,” Eur. Phys. J. C 66 (2010) arXiv:1410.8822 [hep-ph]. 359–372, arXiv:0912.2104 [hep-ph]. 29. C.-H. Chang, “Hadronic Production of J/ψ Associated With a 51. S. Dawson, “Radiative corrections to Higgs boson production,” Gluon,” Nucl. Phys. B172 (1980) 425–434. Nucl. Phys. B 359 (1991) 283–300. 30. E. L. Berger and D. L. Jones, “Inelastic Photoproduction of J/psi 52. D. Graudenz, M. Spira, and P. Zerwas, “QCD corrections to and Upsilon by Gluons,” Phys. Rev. D23 (1981) 1521–1530. Higgs boson production at proton proton colliders,” Phys. Rev. 31. R. Baier and R. Ruckl, “Hadronic Collisions: A Quarkonium Lett. 70 (1993) 1372–1375. Factory,” Z. Phys. C19 (1983) 251. 53. M. Spira, A. Djouadi, D. Graudenz, and P. Zerwas, “Higgs 32. G. T. Bodwin, E. Braaten, and G. P. Lepage, “Rigorous QCD boson production at the LHC,” Nucl. Phys. B 453 (1995) 17–82, analysis of inclusive annihilation and production of heavy arXiv:hep-ph/9504378. quarkonium,” Phys. Rev. D51 (1995) 1125–1171, 54. A. Candido, S. Forte, and F. Hekhorn, “Can MS parton arXiv:hep-ph/9407339 [hep-ph]. [Erratum: Phys. distributions be negative?,” arXiv:2006.07377 [hep-ph]. Rev.D55,5853(1997)]. 55. F. Maltoni, T. McElmurry, R. Putman, and S. Willenbrock, 33. H. Fritzsch, “Producing Heavy Quark Flavors in Hadronic “Choosing the Factorization Scale in Perturbative QCD,” Collisions: A Test of Quantum Chromodynamics,” Phys. Lett. arXiv:hep-ph/0703156 [HEP-PH]. 67B (1977) 217–221. 56. R. D. Ball, V. Bertone, M. Bonvini, S. Marzani, J. Rojo, and 34. F. Halzen, “Cvc for Gluons and Hadroproduction of Quark L. Rottoli, “Parton distributions with small-x resummation: 78 Flavors,” Phys. Lett. 69B (1977) 105–108. evidence for BFKL dynamics in HERA data,” Eur. Phys. J. C arXiv:1710.05935 [hep-ph] 35. J.-P. Lansberg and H.-S. Shao, “Associated production of a no. 4, (2018) 321, . xFitter Developers’ Team quarkonium and a Z boson at one loop in a quark-hadron-duality 57. Collaboration, H. Abdolmaleki , “Impact of low- resummation on QCD analysis of approach,” JHEP 10 (2016) 153, arXiv:1608.03198 et al. x C78 [hep-ph]. HERA data,” Eur. Phys. J. no. 8, (2018) 621, arXiv:1802.00064 [hep-ph]. 36. J.-P. Lansberg, H.-S. Shao, N. Yamanaka, Y.-J. Zhang, and 58. A. Buckley, J. Ferrando, S. Lloyd, K. Nordstrom,¨ B. Page, C. Nous,ˆ “Complete NLO QCD study of single- and M. Rufenacht,¨ M. Schonherr,¨ and G. Watt, “LHAPDF6: parton double-quarkonium hadroproduction in the colour-evaporation density access in the LHC precision era,” Eur. Phys. J. C 75 model at the Tevatron and the LHC,” Phys. Lett. B807 (2020) (2015) 132, arXiv:1412.7420 [hep-ph]. 135559, arXiv:2004.14345 [hep-ph]. 59. J. Butterworth et al., “PDF4LHC recommendations for LHC 37. Y. Feng, J.-P. Lansberg, and J.-X. Wang, “Energy dependence of Run II,” J. Phys. G43 (2016) 023001, arXiv:1510.03865 direct-quarkonium production in pp collisions from fixed-target [hep-ph]. to LHC energies: complete one-loop analysis,” Eur. Phys. J. 60. P. Jimenez-Delgado and E. Reya, “Delineating parton C75 no. 7, (2015) 313, arXiv:1504.00317 [hep-ph]. distributions and the strong coupling,” Phys. Rev. D89 no. 7, 38. F. Maltoni , “Analysis of charmonium production at et al. (2014) 074049, arXiv:1403.1852 [hep-ph]. fixed-target experiments in the NRQCD approach,” Phys. Lett. 61. S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, arXiv:hep-ph/0601203 [hep-ph] B638 (2006) 202–208, . J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, “New parton 39. S. J. Brodsky and J.-P. Lansberg, “Heavy-Quarkonium distribution functions from a global analysis of quantum Production in High Energy Proton-Proton Collisions at RHIC,” chromodynamics,” Phys. Rev. D93 no. 3, (2016) 033006, Phys. Rev. D 81 (2010) 051502, arXiv:0908.0754 [hep-ph]. arXiv:1506.07443 [hep-ph]. 40. G. A. Schuler, “Quarkonium production and decays,” 1994. 62. L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. 41. M. L. Mangano and A. Petrelli, “NLO quarkonium production Thorne, “Parton distributions in the LHC era: MMHT 2014 in hadronic collisions,” Int. J. Mod. Phys. A12 (1997) PDFs,” Eur. Phys. J. C75 no. 5, (2015) 204, arXiv:1412.3989 3887–3897, arXiv:hep-ph/9610364 [hep-ph]. [hep-ph]. 42. M. A. Ozcelik, “Constraining gluon PDFs with quarkonium 63. NNPDF Collaboration, R. D. Ball et al., “Parton distributions production,” PoS DIS2019 (2019) 159, arXiv:1907.01400 from high-precision collider data,” Eur. Phys. J. C77 no. 10, [hep-ph]. (2017) 663, arXiv:1706.00428 [hep-ph]. 43. CTEQ Collaboration, R. Brock et al., “Handbook of 64. C. A. Flett, A. D. Martin, M. G. Ryskin, and T. Teubner, “Very perturbative QCD: Version 1.0,” Rev. Mod. Phys. 67 (1995) low x gluon density determined by LHCb exclusive J/ψ data,” 157–248. arXiv:2006.13857 [hep-ph]. 44. G. ’t Hooft and M. Veltman, “Regularization and 65. V. Bertone, S. Carrazza, and J. Rojo, “APFEL: A PDF Evolution Renormalization of Gauge Fields,” Nucl. Phys. B 44 (1972) Library with QED corrections,” Comput. Phys. Commun. 185 189–213. (2014) 1647–1668, arXiv:1310.1394 [hep-ph]. 45. J. H. Kuhn and E. Mirkes, “QCD corrections to toponium 66. S. Carrazza, A. Ferrara, D. Palazzo, and J. Rojo, “APFEL Web: production at hadron colliders,” Phys. Rev. D 48 (1993) a web-based application for the graphical visualization of parton 179–189, arXiv:hep-ph/9301204. distribution functions,” J. Phys. G 42 no. 5, (2015) 057001, 46. A. Petrelli, M. Cacciari, M. Greco, F. Maltoni, and M. L. arXiv:1410.5456 [hep-ph]. Mangano, “NLO production and decay of quarkonium,” Nucl. 67. R. D. Ball, M. Bonvini, S. Forte, S. Marzani, and G. Ridolfi, Phys. B 514 (1998) 245–309, arXiv:hep-ph/9707223. “Higgs production in gluon fusion beyond NNLO,” Nucl. Phys. 47. M. Klasen, B. Kniehl, L. Mihaila, and M. Steinhauser, “J/ψ B 874 (2013) 746–772, arXiv:1303.3590 [hep-ph]. plus jet associated production in two-photon collisions at 68. M. Bonvini, R. D. Ball, S. Forte, S. Marzani, and G. Ridolfi, next-to-leading order,” Nucl. Phys. B 713 (2005) 487–521, “Updated Higgs cross section at approximate N3LO,” J. Phys. G arXiv:hep-ph/0407014. 41 (2014) 095002, arXiv:1404.3204 [hep-ph]. 20

69. M. Bonvini, S. Marzani, C. Muselli, and L. Rottoli, “On the 88. LHCb Collaboration, R. Aaij et al., “Observation of structure in Higgs cross section at N3LO+N3LL and its uncertainty,” JHEP the J/ψ-pair mass spectrum,” Sci. Bull. 2020 (2020) 65, 08 (2016) 105, arXiv:1603.08000 [hep-ph]. arXiv:2006.16957 [hep-ex]. 70. R. Bonciani, G. Degrassi, and A. Vicini, “Scalar particle 89. Y. Jia, “Which hadronic decay modes are good for eta(b) contribution to Higgs production via gluon fusion at NLO,” searching: double J/psi or something else?,” Phys. Rev. D78 JHEP 11 (2007) 095, arXiv:0709.4227 [hep-ph]. (2008) 054003, arXiv:hep-ph/0611130 [hep-ph]. 71. C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger, 90. G. Hao, C.-F. Qiao, and P. Sun, “Investigate the Bottomonium “Higgs Boson Gluon-Fusion Production in QCD at Three Ground State eta(b) via Its Inclusive Charm Decays,” Phys. Rev. Loops,” Phys. Rev. Lett. 114 (2015) 212001, D76 (2007) 125013, arXiv:0710.3339 [hep-ph]. arXiv:1503.06056 [hep-ph]. 91. A. Czarnecki and K. Melnikov, “Charmonium decays: J / psi 72. C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, —¿ e+ e- and eta(c) —¿ gamma gamma,” Phys. Lett. B519 F. Herzog, A. Lazopoulos, and B. Mistlberger, “High precision (2001) 212–218, arXiv:hep-ph/0109054 [hep-ph]. determination of the gluon fusion Higgs boson cross-section at 92. F. Feng, Y. Jia, and W.-L. Sang, “Can Nonrelativistic QCD ∗ the LHC,” JHEP 05 (2016) 058, arXiv:1602.00695 Explain the γγ → ηc Transition Form Factor Data?,” Phys. Rev. [hep-ph]. Lett. 115 no. 22, (2015) 222001, arXiv:1505.02665 73. LHCb Collaboration, R. Aaij et al., “Measurement of the [hep-ph]. ηc(1S ) production cross-section in proton-proton collisions via 93. J. P. Lansberg and T. N. Pham, “Two-photon width of eta(b), the decay ηc(1S ) → pp¯,” Eur. Phys. J. C75 no. 7, (2015) 311, eta-prime(b) and eta-prime-prime(b) from Heavy-Quark Spin arXiv:1409.3612 [hep-ex]. Symmetry,” Phys. Rev. D75 (2007) 017501, arXiv:hep-ph/0609268 [hep-ph]. 74. LHCb Collaboration, R. Aaij et al., “Measurement√ of the ηc(1S ) production cross-section in pp collisions at s = 13 94. C.-F. Qiao, J. Wang, J. X. Wang, and Y. Zheng, “EtabFDC: An TeV,” Eur. Phys. J. C80 no. 3, (2020) 191, arXiv:1911.03326 eta(b) event generator in hadroproduction at LHC,” Comput. [hep-ex]. Phys. Commun. 180 (2009) 61–68, arXiv:0804.1288 75. BaBar Collaboration, B. Aubert et al., “Observation of the [hep-ph]. 95. S. J. Brodsky, F. Fleuret, C. Hadjidakis, and J. P. Lansberg, bottomonium ground state in the decay υ3S → γetab,” Phys. Rev. Lett. 101 (2008) 071801, arXiv:0807.1086 [hep-ex]. “Physics Opportunities of a Fixed-Target Experiment using the [Erratum: Phys. Rev. Lett.102,029901(2009)]. LHC Beams,” Phys. Rept. 522 (2013) 239–255, arXiv:1202.6585 [hep-ph] 76. Belle Collaboration, R. Mizuk et al., “Evidence for the ηb(2S ) . 96. J. P. Lansberg, S. J. Brodsky, F. Fleuret, and C. Hadjidakis, and observation of hb(1P) → ηb(1S )γ and hb(2P) → ηb(1S )γ,” Phys. Rev. Lett. 109 (2012) 232002, arXiv:1205.6351 “Quarkonium Physics at a Fixed-Target Experiment using the [hep-ex]. LHC Beams,” Few Body Syst. 53 (2012) 11–25, 77. BaBar Collaboration, B. Aubert et al., “Evidence for the arXiv:1204.5793 [hep-ph]. eta(b)(1S) Meson in Radiative Upsilon(2S) Decay,” Phys. Rev. 97. L. Massacrier, B. Trzeciak, F. Fleuret, C. Hadjidakis, D. Kikola, Lett. 103 (2009) 161801, arXiv:0903.1124 [hep-ex]. J. P. Lansberg, and H. S. Shao, “Feasibility studies for 78. CLEO Collaboration, G. Bonvicini et al., “Measurement of the quarkonium production at a fixed-target experiment using the eta(b)(1S) mass and the branching fraction for Upsilon(3S) —¿ LHC proton and lead beams (AFTER@LHC),” Adv. High gamma eta(b)(1S),” Phys. Rev. D81 (2010) 031104, Energy Phys. 2015 (2015) 986348, arXiv:1504.05145 arXiv:0909.5474 [hep-ex]. [hep-ex]. 79. Belle Collaboration, U. Tamponi et al., “First observation of the 98. C. Hadjidakis et al., “A Fixed-Target Programme at the LHC: Physics Case and Projected Performances for Heavy-Ion, hadronic transition Υ(4S ) → ηhb(1P) and new measurement of Hadron, Spin and Astroparticle Studies,” arXiv:1807.00603 the hb(1P) and ηb(1S ) parameters,” Phys. Rev. Lett. 115 no. 14, (2015) 142001, arXiv:1506.08914 [hep-ex]. [hep-ex]. 80. Belle-II Collaboration, W. Altmannshofer et al., “The Belle II 99. A. Arbuzov et al., “On the physics potential to study the gluon Physics Book,” PTEP 2019 no. 12, (2019) 123C01, content of proton and deuteron at NICA SPD,” arXiv:1808.10567 [hep-ex]. [Erratum: arXiv:2011.15005 [hep-ex]. PTEP2020,no.2,029201(2020)]. 100. PROSA Collaboration, O. Zenaiev et al., “Impact of 81. F. Feng, Y. Jia, and W.-L. Sang, heavy-flavour production cross sections measured by the LHCb “Next-to-Next-to-Leading-Order QCD Corrections to the experiment on parton distribution functions at low x,” Eur. Phys. Hadronic width of Pseudoscalar Quarkonium,” Phys. Rev. Lett. J. C75 no. 8, (2015) 396, arXiv:1503.04581 [hep-ph]. 119 no. 25, (2017) 252001, arXiv:1707.05758 [hep-ph]. 101. PROSA Collaboration, O. Zenaiev, M. V. Garzelli, K. Lipka, 82. Particle Data Group Collaboration, P. A. Zyla et al., “Review S. O. Moch, A. Cooper-Sarkar, F. Olness, A. Geiser, and of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01. G. Sigl, “Improved constraints on parton distributions using 83. E. Braaten, S. Fleming, and A. K. Leibovich, “NRQCD analysis LHCb, ALICE and HERA heavy-flavour measurements and of bottomonium production at the Tevatron,” Phys. Rev. D63 implications for the predictions for prompt atmospheric-neutrino (2001) 094006, arXiv:hep-ph/0008091 [hep-ph]. fluxes,” JHEP 04 (2020) 118, arXiv:1911.13164 [hep-ph]. 84. F. Maltoni and A. D. Polosa, “Observation potential for eta(b) at 102. R. Gauld and J. Rojo, “Precision determination of the small-x the Tevatron,” Phys. Rev. D70 (2004) 054014, gluon from charm production at LHCb,” Phys. Rev. Lett. 118 arXiv:hep-ph/0405082 [hep-ph]. no. 7, (2017) 072001, arXiv:1610.09373 [hep-ph]. 85. G. Hao, Y. Jia, C.-F. Qiao, and P. Sun, “Hunting eta(b) through 103. A. Kusina, J.-P. Lansberg, I. Schienbein, and H.-S. Shao, radiative decay into J/psi,” JHEP 02 (2007) 057, “Gluon Shadowing in Heavy-Flavor Production at the LHC,” arXiv:hep-ph/0612173 [hep-ph]. Phys. Rev. Lett. 121 no. 5, (2018) 052004, arXiv:1712.07024 86. B. Gong, Y. Jia, and J.-X. Wang, “Exclusive eta(b) decay to [hep-ph]. double J / psi at next-to-leading order in alpha(s),” Phys. Lett. 104. J.-P. Lansberg and H.-S. Shao, “Towards an automated tool to B670 (2009) 350–355, arXiv:0808.1034 [hep-ph]. evaluate the impact of the nuclear modification of the gluon 87. P. Santorelli, “Long-distance contributions to the density on quarkonium, D and B meson production in etab—¿J/psiJ/psi decay,” Phys. Rev. D77 (2008) 074012, proton-nucleus collisions,” Eur. Phys. J. C77 no. 1, (2017) 1, arXiv:hep-ph/0703232 [HEP-PH]. arXiv:1610.05382 [hep-ph]. 21

Appendix A: Splitting Functions

In this Appendix we deﬁne the splitting functions used in the text, ! 1 − z z Pgg (z) = 2CA + + z (1 − z) + b0δ (1 − z) , (A.1) z (1 − z) +

1 + (1 − z)2 P (z) = C , (A.2) gq F z

2−1 where C = N and C = Nc are the Casimirs of the adjoint and the fundamental representation. In order to apply the A c F 2Nc splitting functions we need to deﬁne the plus distribution 1 that regulates the pole at z = 1. This distribution can be (1−z) + applied to any arbitrary function f (z) that is ﬁnite at z = 1,

Z 1 Z 1 1 f (z) − f (1) dz f (z) = dz . (A.3) 0 (1 − z) + 0 1 − z In cross-section computations, one usually integrates z from a non-zero value that is bounded by for example the center of 2 MQ mass energy τ0 = s . The plus distributions are however defined for the integral over the entire region from [0, 1]. In order to deal with a modified bound [τ0, 1] however, we need to make a modification. Defining the following modified plus distribution as,

Z 1 Z 1 1 f (z) − f (1) dz f (z) = dz , (A.4) − − τ0 (1 z) τ0 τ0 1 z we can thus make the replacement, 1 1 = + log (1 − τ ) δ (1 − z) . (A.5) − − 0 (1 z) + (1 z) τ0

Appendix B: Analytical Results for σ

For completeness and for the sake of the discussion we present here the result for σ as well. To obtain these, one can for instance take the expressions for the partonic cross sections from Refs [40, 45, 46] and fold these with the PDFs. Recalling 2 that z = τ0/τ and τ0 = 4mQ/s, one gets

2 2| |2 ∂L    µ2   µ2  αs π R0  gg  αs τ0  2  R  2  F  σgg =  (τ0) τ0 + −44 + 7π + 12b0 log   + 72 log (1 − τ0) − 72 log (1 − τ0) log   5  ∂τ  π 12   2  4 2  96mc µF mc "Z 1 ! α 1 ∂Lgg 1 12 log z + s dτ (τ) 24 log (1 − z) (1 − z) z2 − 2z + 1 − z2 5 + z 2 + z + 3z3 + 2z4 − 3 π 2 τ0 ∂τ 1 z (1 − z)(1 + z) !! Z 1 " ! !# 1 2 log (1 − z) ∂Lgg ∂Lgg − 12 + z2 23 + z 24 + 2z + 11z3 + 12 1 + z3 log z + 24 dτ (τ) − z2 (τ ) 2 − 0 (1 + z) τ0 1 z ∂τ ∂τ Z 1   2  ! !   µ  1 2 ∂Lgg ∂Lgg  −12 dτ log  F  1 − z + z2 (τ) − z2 (τ )  ,   2  − 0  τ0 4mc 1 z ∂τ ∂τ (B.6)

Z 1 3 2 ∂L ¯ α π|R | σ = τ qq (τ) s 0 2 (1 − ) , (B.7) qq d 5 z z τ0 ∂τ 81mc

Z 1  3 2  ! !  2   ∂Lqg α π|R0| 1 1 1  µ  1  σ = τ (τ)  s  2 + − 1 + 2 2 − + 1 log (1 − ) − 2 − + 1 log  F  − 2 log  , qg d  5  z z z z z z z  2  z z τ0 ∂τ 72mc 2 2 2 4mc 2 22

(B.8) where have deﬁned L Z −1/2 log τ √ √ ∂ gg y −y (τ) = dy fg τe , µF fg τe , µF , ∂τ 1/2 log τ Z −1/2 log τ ∂Lqq¯ X √ √ √ √ (τ) = dy f τey, µ f τe−y, µ + f τey, µ f τe−y, µ , ∂τ q F q F q F q F (B.9) q=u,d,s 1/2 log τ L X Z −1/2 log τ √ √ √ √ ∂ qg y −y y −y (τ) = dy fq τe , µF fg τe , µF + fg τe , µF fq τe , µF . ∂τ 1/2 log τ q=u,d,s,u,d,s

Appendix C: Analytical Results for dσ/dy

In this appendix, we provide the analytical expressions in terms of convolution of PDFs for the rapidity-diﬀerential cros section for ηc hadron-production. These are not available in the literature. As discussed above, 3 channels should be considered. The formulae below hold for y ≥ 0 in order to perform the integration-boundary decomposition. For symmetric collisions, dσ/dy is just symmetric. For asymmetric hadron A - hadron B collisions, one can obtain dσ/dy for y < 0, by assigning the PDF depending on x1 to hadron B and conversely. We start with the gg-channel:

2 2 2 " dσ , α π R α n o NLO gg = s 0 L˜ (τ , ) τ 1 + s −44 + 7π2 + 12 + 36 log(1 − η ) + log (1 − η ) + η ↔ η 5 gg 0 y 0 b0LRF 1 LMF 1 1 2 dy 96mc 12π Z 1 Z t2 Z η1 Z 1 ˜ ˜ ˜ 3α 2a hLgg(τ, y3) a1 − Lgg(τ, y4) a2 Lgg(τ, y3) a1 i + s dτ dw L˜ (τ, y ) 1 + dτ dw + gg 3 2 − π η1 t1 1 − w η2 t1 1 w 1 + w ( Z η1 ˜ 2 ˜ ) hLgg(τ, y1) c1 LMxF − z Lgg(τ0, y)LMF log (1 − z) i + dτ + 2 L˜ (τ, y ) c − z2L˜ (τ , y) + (η , y ) ↔ (η , y ) − − gg 1 1 gg 0 1 1 2 2 τ0 1 z 1 z Z η1 Z η2 Z 1 ˜ ˜ ˜ ˜ !# 1 − t hLgg(τ, y3) a1 − Lgg(τ, y4) a2 Lgg(τ, y3) a1 − Lgg(τ, y5) a2 i + dτ a L˜ (τ, y ) log 1 + dτ dw + 2 gg 4 − η2 2 τ0 −1 1 w 1 + w (C.10) √ √ 2 L˜ ∂Lgg y˜ −y˜ where z = τ0/τ and τ0 = 4mc/s. The deﬁnition for gg (τ, y˜) = ∂τ∂y (τ, y˜) = fg( τe , µF ) fg( τe , µF ), where fg(x1, µF ) is the gluon PDF with the factorisation scale µF and x1-value. We now turn to the qg-channel:

3 2 "Z η Z η dσ , + α πR 1 2 NLO gq qg = s 0 τ L˜ (τ, ) + 2 log (1 − ) + 2 + τ L˜ (τ, ) + 2 log (1 − ) + 2 5 d gq y1 c2 LMxF x z d qg y2 c2 LMxF z z dy 144mc τ0 τ0 Z 1 Z ˜ ˜ Z η ! t2 2 Lgq (τ, y3) a4 + Lqg (τ, y3) a5 1 1 − t L˜ ( ) 1 + dτ dw 2 + dτ c2 gq τ, y4 log η1 t1 1 − w η2 2   Z η1 Z 1 ˜ ˜ ˜ L˜ L˜ Lgq (τ, y3) a4 − Lgq (τ, y4) c2 + Lqg (τ, y3) a5 gq (τ, y3) a4 + qg (τ, y3) a5  + dτ dw  +   −  η2 t1 1 w 1 + w

Z η2 Z 1  ˜ ˜ ˜ ˜ ˜ ˜  Lgq (τ, y3) a4 − Lgq (τ, y4) c2 + Lqg (τ, y3) a5 Lgq (τ, y3) a4 + Lqg (τ, y3) a5 − Lqg (τ, y5) c2  + dτ dw  +   −  τ0 −1 1 w 1 + w (C.11) √ √ ∂Lgq P − ∂Lqg where L˜ (τ, y˜) = (τ, y˜) = f ( τey˜, µ ) f ( τe y˜, µ ), L˜ (τ, y˜) = (τ, y˜) = gq √ √∂τ∂y q=u,d,s,u,d,s g F q F qg ∂τ∂y P y˜ −y˜ q=u,d,s,u,d,s fq( τe , µF ) fg( τe , µF ), with fq(x1, µF ) being quark PDF and fg(x1, µF ) the gluon PDF with the factorisation scale µF and x1-value. Finally, we have for qq-channel:

3 2 Z 1 Z t Z η Z 1 Z η Z 1 ! dσ , α πR 2 1 2 NLO qq = s 0 τ L˜ (τ, ) + τ L˜ (τ, ) + τ L˜ (τ, ) 5 d dw qq y3 a3 d dw qq y3 a3 d dw qq y3 a3 dy 216mc η1 t1 η2 t1 τ0 −1 (C.12) 23 √ √ L˜ ∂Lqq P y˜ −y˜ where qq (τ, y˜) = ∂τ∂y (τ, y˜) = q=u,d,s,u,d,s fq( τe , µF ) fq( τe , µF ), with fq(x1, µF ) being the quark PDF at the factorisation scale µF and x1 value. In the above expressions, we have adopted the following deﬁnitions:  µ2   2   2  √ 11 2  R  4mc  4mc  ±y b0 = CA − nlTF ; LRF = log  ; LMF = log  ; LMxF = log  ; η1,2 = τ0e ; 6 3  2   2   2  µF µF µF z ! ! ! ! ! 1 1 + z 1 1 − z 1 − z y1,2 = y ± log z; t1,2 = tanh y ± log τ ; y3 = y − arctanh w ; y4,5 = y ∓ arctanh ; 2 1 − z 2 1 + z 1 + z  2  2 2 2 4 3 2 4 4 4 2 (C.13) z −zw + z + w + 3 9z − 4z + 6z + (z − 1) w + 6(z − 1) w − 4z + 9  =   a1  2   16(1 − z) (z + 1)2 − (z − 1)2w2  2 ((z − 1)z + 1) 2 2 2 2 a2 = ; a3 = (1 − z) z 1 + w ; c1 = z − z + 1 ; c2 = ((z − 2) z + 2) ; 1 − z

z2(w + 1) z2(w + 1)2 − 2z(w + 1)2 + w(w + 2) + 5 z2(1 − w) z2(w − 1)2 − 2z(w − 1)2 + (w − 2)w + 5 a4 = ; a5 = 2((z − 1)w + z + 1)2 2(z(−w) + z + w + 1)2 (C.14)