Confronting current NLO parton fragmentation functions with inclusive charged-particle spectra at hadron colliders
David d'Enterria, Kari J. Eskola, Ilkka Helenius, Hannu Paukkunen
CConfronting current NLO parton fragmentation functions with inclusivecharged-particle spectra at hadron colliders
David d’Enterria
CERN, PH Department, CH-1211 Geneva 23, Switzerland
Kari J. Eskola, Ilkka Helenius, Hannu Paukkunen
Department of Physics, University of Jyv¨askyl¨a, P.O. Box 35, FI-40014, FinlandHelsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland
Abstract
The inclusive spectra of charged particles measured at high transverse momenta ( p T (cid:38) √ s = 200–7000 GeV are compared with next-to-leading order perturbative QCD calculations using sevenrecent sets of parton-to-hadron fragmentation functions (FFs). Accounting for the uncertaintiesin the scale choices and in the parton distribution functions, we find that most of the theoreticalpredictions tend to overpredict the measured LHC and Tevatron cross sections by up to a factor oftwo. We identify the currently too-hard gluon-to-hadron FFs as the probable source of the problem,and justify the need to refit the FFs using the available LHC and Tevatron data in a region oftransverse momenta, p T (cid:38)
10 GeV/c, which is supposedly free from additional non-perturbativecontributions and where the scale uncertainty is only modest.
Keywords:
Charged-hadron fragmentation functions, NLO computations, Hadronic collisions
1. Introduction
The inclusive production of large-transverse-momentum ( p T (cid:38) √ s ) and hadron p T [6], are particularlyuseful for studying the FFs and their universality. From an experimental perspective, the bestprecision and widest kinematic reach in the p T -differential cross sections are achieved when no Email addresses: [email protected] (David d’Enterria), [email protected] (Kari J. Eskola), [email protected] (Ilkka Helenius), [email protected] (Hannu Paukkunen)
Preprint submitted to Nuclear Physics B October 29, 2018 a r X i v : . [ h e p - ph ] A p r article identification is performed. The pre-LHC measurements for unidentified charged-hadronspectra in p-p and p-p collisions range from fixed-target experiments at √ s (cid:46)
60 GeV [7, 8]to collider experiments covering a wide range of center-of-mass energies √ s = 200 − p T (cid:46)
10 GeV / c, and the accuracy for p T ≥
10 GeV / c is rather poor. The importance of the newLHC data to overcome such limitations is underscored by the confusion [17, 18, 19] triggered bythe original CDF data [16] which seemed to display deviations from next-to-leading order (NLO)perturbative QCD (pQCD) calculations up to three orders of magnitude at p T (cid:39)
150 GeV / c, butwhich was later on identified as an experimental issue (see the erratum of Ref. [16]).Interestingly, the NLO pQCD predictions presented along with the recently published CMS [20,21] and ALICE [22] inclusive charged hadron spectra, appear to overshoot the data by up to afactor of two in the kinematical region where effects such as e.g. intrinsic transverse momentum ofthe colliding partons (intrinsic k T ) [23, 24, 25], soft-gluon resummation [26, 27], or small- z instabil-ities of the FFs [28] should not play a major role. Especially since the recent LHC measurementsof the p T -differential cross sections for inclusive jets [29, 30] and prompt photons [31, 32] are inperfect agreement with the NLO pQCD expectations [33, 34, 35], the data-vs-theory discrepanciesfor inclusive charged-hadrons come totally unexpected. Resolving such inconsistencies is also ofrelevance for other QCD analyses such as those related to the suppression of high- p T hadrons in ul-trarelativistic heavy-ion collisions where the p-p spectra are required as baseline measurements [36].In this paper, we present a systematic comparison of the theoretical predictions for unidenti-fied charged-hadron production to experimental data with a special emphasis on the latest LHCmeasurements. Our aim is to demonstrate that such a process in hadron colliders is predominantlysensitive to the gluon-to-hadron FFs which are presently not well determined and, consequently,large discrepancies among the modern sets of FFs exist. These differences not only translate intoa significant scatter in the corresponding predictions for the cross sections, but none of the currentFF sets can consistently reproduce the current LHC and Tevatron data at p T (cid:38)
10 GeV/c. As thedata measured by different experiments at the same collision energies are in mutual agreement, itseems excluded that such discrepancies are due to an experimental issue. Instead, this hints to asevere problem in the gluon-to-hadron FFs in most of the existing sets. We conclude that the gluonFF, which is currently mildly constrained by charged-hadron spectra from hadronic collisions atRHIC and SppS energies, should be refitted by using the LHC and Tevatron hadron spectra in theregion p T (cid:38)
10 GeV/c, where the theoretical scale uncertainties appear tolerable and which shouldbe free from additional, non-perturbative hadron production mechanisms.
2. The pQCD framework for inclusive hadron production
The cross section for the inclusive production of a single hadron h with a momentum p inthe collision of two hadrons h and h carrying momenta p and p respectively, can be expressed,2ifferentially in transverse momentum p T and (pseudo)rapidity η , as [24, 37] dσ ( h + h → h + X ) dp T dη = (cid:88) ijl (cid:90) dx (cid:90) dx (cid:90) dzz f h i ( x , µ ) f h j ( x , µ ) D l → h ( z, µ ) d ˆ σ (ˆ p i + ˆ p j → ˆ p l , µ , µ , µ ) d ˆ p T dη (cid:12)(cid:12)(cid:12)(cid:12) ˆ p = x p ˆ p = x p ˆ p = p /z . (1)In this expression, f h k i ( x k , µ ) denote the PDFs of the colliding hadrons evaluated at partonfractional momenta x k and scale µ . We will use the ct10nlo [38] parametrization throughoutthis work. The parton-to-hadron FFs are denoted by D l → h ( z, µ ) where z is the fraction of theparton momentum carried out by the outgoing charged hadron. The PDFs and FFs are convolutedwith the partonic coefficient functions d ˆ σ for which we use their fixed-order NLO O ( α s ) expres-sions [37, 39, 40], treating the partons and hadrons as massless particles. In practice, we evaluatethese cross sections employing the INCNLO [37, 41] program which we have modified to improve theconvergence at small values of p T . The fixed-order calculations are supposed to be adequate for p T (cid:29) / c but still sufficiently away from the phase-space boundary p max T ∼ √ s/ η ≈ O ( α s ) leads to the well-known scale dependenceof the pQCD calculations. For inclusive hadron production, there are three independent scales:the renormalization scale µ ren , factorization scale µ fact , and the fragmentation scale µ frag . Thesensitivity of the computed cross sections to the variations of these scales is typically taken as anindication of the size of the missing higher-order corrections. Our default choice is µ ren = µ fact = µ frag = p T , and we take the scale uncertainty as the envelope enclosed by the following 16 scalevariations [42] (cid:18) µ fact p T , µ ren p T , µ frag p T (cid:19) = ( , , ) , ( , , , ( , , ) , ( , , , ( , , , (1 , , ) , (1 , , , (1 , , ) , (1 , , , (1 , , , (1 , , , (2 , , ) , (2 , , , (2 , , , (2 , , , (2 , , . (2)We omit the combinations in which µ ren and µ fact or µ ren and µ frag are pairwise scaled by a factorof two in opposite directions due to the appearance of potentially large contributions of the formlog( µ /µ ) and log( µ /µ ) in the calculation. The next-to-NLO (NNLO) calculations areexpected to definitely reduce the scale dependence. However, although the PDF analyses can benowadays carried out partly at NNLO level [43, 44, 45, 46], the time-like splitting functions neededin the NNLO evolution of the FFs are not yet fully known [47, 48], nor are the NNLO coefficientfunctions needed in Eq. (1), although the latter could finally emerge through the work currentlycarried out for jets [49, 50, 51].
3. Comparison of the parton fragmentation functions
Table 1 lists the seven commonly used sets of NLO parton-to-charged-hadron FF parametriza-tions: Kretzer ( kre ) [52], kkp [53], bfgw [54], hkns [55], akk05 [56] dss [57, 58], and akk08 [59],3hat we employ in our calculations. In a few of these analyses the charged hadron FFs are con-structed as a sum of the individual FFs for pions ( π ± ), kaons ( K ± ), plus (anti)protons; but e.g. in dss there is still a small “residual” contribution on top of these. There are other sets of FFs withrestricted particle species [60, 57, 61], but we do not consider them here as we focus only on theinclusive sum which should be better constrained than the FFs for individual hadron species. Forreviews, see e.g. [4, 5]. Table 1: Characteristics of the existing sets of parton-to-charged-hadron FFs. The hadron species included, use ofdifferent collision systems, attempts to estimate the FF errors, minimum value of z considered, and the available Q -range are indicated.FF set Species Fitted data Error estimates z min Q (GeV )Kretzer ( kre ) [52] π ± , K ± , h + + h − e + e − no 0.01 0.8–10 kkp [53] π + + π − , K + + K − e + e − no 0.1 1 –10 p + ¯ p , h + + h − bfgw [54] h ± e + e − yes 10 − · akk05 [56] π ± , K ± , p , ¯ p e + e − no 0.1 2–4 · hkns [55] π ± , K ± , p + ¯ p e + e − yes 0 .
01 1–10 akk08 [59] π ± , K ± , p , ¯ p e + e − , p-p no 0.05 2–4 · dss [57, 58] π ± , K ± , p , ¯ p , h ± e + e − , p-p, e-p yes 0.05 1–10 z D h + + h − u ( z , Q = G e V ) z z D h + + h − g ( z , Q = G e V ) z Figure 1: Top: Charged-hadron fragmentation functions as a function of z for u -quarks (left) and gluons (right) at Q = 20 GeV. Bottom: Ratio between different FFs over the Kretzer FFs. The main constraints in global fits of FFs come from the inclusive hadron production in e + e − collider experiments, from which the data are abundant [62]. These experiments are, however,mainly sensitive to the quark FFs leaving the gluon-to-hadron FFs largely unconstrained. Similarly,the data for semi-inclusive hadron production in deeply inelastic scattering [63], used in the dss dss and akk08 include various datasets (in different combinations) from hadronic collisions at RHIC [10, 11, 64],SppS [12, 13, 14] and Tevatron [15, 65]. The bulk of these measurements is, however, concentratedat rather small values of transverse momentum p T (cid:46) / c (dictating the hard scales of theprocess) where the sensitivity to the gluon FFs is certainly present, but where the NLO pQCDcalculations are not well under control due to the large scale uncertainty (see later). Therefore,the extraction of the gluons FFs based on these data cannot be considered completely safe, and,therefore, we believe that the “older” sets of FFs with only e + e − data should not be blindlydiscarded. Although the uncertainties in FFs due to the experimental errors (or lack of data)have been addressed in some FF extractions (see also Ref. [66]), only hkns provides the necessaryinformation to fully propagate these uncertainties to further observables. The bfgw packageprovides three alternative sets with somewhat different gluon FFs, but the mutual variation betweenthese sets translates only to a few-percent difference in the observables we discuss here. The erroranalysis of dss was performed via the method of Lagrange multipliers [67] which does not allowthe end user to estimate the propagation of their FF errors. For these reasons, in what follows, wewill present the central results from all the parametrizations including also the hkns error bands. d σ h ++ h − d p T d y d z / d σ h ++ h − d p T d y z p T = 5 . p T = 20 GeV p T = 5 . p T = 20 GeV √ s = 900 GeV | η | < . d σ h ++ h − d p T d y d z / d σ h ++ h − d p T d y z p T = 5 . p T = 20 GeV p T = 5 . p T = 20 GeV √ s = 7000 GeV | η | < . Figure 2: Normalized cross section for charged-hadron production as a function of z for √ s = 900 GeV (left) and √ s = 7000 GeV (right) for p T = 5 GeV / c (solid) and p T = 20 GeV / c (dashed) at midrapidity, obtained with Kretzer(dark blue) and dss (orange) FFs. In Fig. 1, we present a comparison of the up-quark and gluon FFs into charged hadrons for allthe available FFs at a common scale Q = 20 GeV. The spread among the gluon FFs is significantlylarger than in the case of quarks — a clear indication of lack of definitive constraints — in particularfor moderate and large z > hkns error band is not broad enough tocover all different sets above z ∼ .
5. We believe this is mainly due to the small amount of fitparameters that could be left free in the absence of strict gluon constraints from the e + e − dataalone, as discussed above. To understand at which z values the hadron production at LHC energiesprobes the FFs, and to what extent this depends on the hardness of the gluon FFs, we examine theshape of the differential z distributions. A couple of typical examples corresponding to the LHCkinematics are shown in Fig. 2. The old Kretzer and the modern dss FFs are considered here as thehardness of their gluon FFs is quite different. The z distributions appear to be rather broad andto depend significantly on the specific set of FFs used. The spectra in the range p T = 5–20 GeV/c5robe average hadron fractional momenta (cid:104) z (cid:105) ≈ √ s = 900 GeV, decreasing to (cid:104) z (cid:105) ≈ √ s = 7 TeV. The behaviour of these z distributions can be understood approximating theˆ p T dependence of the convolution between the PDFs and the partonic cross sections d ˆ σ by a powerlaw, as follows [68]: (cid:88) ij (cid:90) dx dx f h i ( x , µ ) f h j ( x , µ ) d ˆ σ (ˆ p i + ˆ p j → ˆ p (cid:96) , µ , µ , µ ) d ˆ p T dη ≈ C (cid:96) ˆ p − n T , (3)where C (cid:96) is a p T -independent constant, so that one can write dσ ( h + h → h + X ) dp T dηdz ≈ (cid:88) (cid:96) C (cid:96) z ˆ p − n T D (cid:96) → h ( z, µ ) = p − n T (cid:88) (cid:96) C (cid:96) z n − D (cid:96) → h ( z, µ ) . (4)Due to the factor z n − , the contributions from small values of z are efficiently suppressed, andthe average value of z becomes much larger than the kinematic lower limit z min ≈ p T / √ s atmidrapidity. However, towards larger √ s , the p T distributions are flatter (the power n is smaller),and the z distributions become on average shifted towards smaller values of z . In any case, thecontributions from the small- z region, z (cid:46) . − .
1, which is more difficult to treat in DGLAP-based approaches [28, 6], remain very small and the use of the standard FF framework is welljustified. d σ g / q d p T d η / d σ g + q d p T d η p T → h + + h − → h + + h − √ s = 900 GeV | η | < . d σ g / q d p T d η / d σ g + q d p T d η p T → h + + h − → h + + h − √ s = 7000 GeV | η | < . Figure 3: Relative contributions of quark (dashed) and gluon (solid) fragmentation to the inclusive charged-hadroncross section at √ s = 900 GeV (left) and √ s = 7000 GeV (right) at midrapidity, obtained with Kretzer (dark blue)and dss (orange) FFs. The relative contributions from the quark and gluon fragmentations are plotted in Fig. 3 for √ s = 900 GeV (left) and √ s = 7000 GeV (right). At small p T , the gluon fragmentation clearlydominates but towards large values of p T the quark fragmentation becomes eventually predominant.In any case, the gluon contribution is always significant and therefore the LHC promises to be a good“laboratory” to determine the gluon FFs in the region of p T where the NLO pQCD calculationscan be considered to be well under control.
4. Comparison of NLO pQCD to high- p T charged-hadron collider data In this section, we compare the data from various experiments with the NLO calculations usingthe seven FF sets listed in Table 1. Our main attention will be on the latest data from CMS [20, 21]6nd ALICE [22] for p-p collisions at the LHC, as well as the CDF measurements [16] in p-p collisionsat Tevatron. We do not include the similar ATLAS measurements [69] here as their results havenot been given as invariant cross sections, but only in terms of the absolute yields. However, wehave checked that the shapes of the ATLAS p T -spectra are in agreement with those measured bythe other LHC experiments. In order to compare with some earlier hadron-collider data, includedin the akk08 and dss global fits, we consider also the p-p results from UA1 [12, 13, 14] as well asthe p-p spectra measured by STAR [10]. E d σ c h / d p [ m b G e V − c ] √ s = 7000 GeV √ s = 2760 GeV √ s = 900 GeV µ = p T | η | < . p T -13 -11 -9 -7 -5 -3 -1
10 0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0
CMSCMSCMSKretzer, D a t a / N L O [GeV/c] E d σ c h / d p [ m b G e V − c ] √ s = 7000 GeV √ s = 2760 GeV √ s = 900 GeV µ = p T | η | < . p T Figure 4: Top: Charged-hadron invariant cross sections measured as a function of p T by CMS [20, 21] at √ s = 0 . √ s = 2 .
76 TeV (red squares), and √ s = 7 TeV (blue circles), compared to NLO calculations with dss [58] (left) and Kretzer (right) [52] FFs. The point-to-point systematic and statistical errors are indicated bycolored rectangles and error bars. Bottom: Ratio between the data and the respective calculations. The boxes atthe beginning of the p T -axis mark the luminosity uncertainties of each measurement. As an example of the p T -differential cross sections, Fig. 4 presents the CMS measurements forinclusive charged hadrons at √ s = 0 . , . , p T , are compared to the NLO calculations us-ing two sets of FFs, dss and Kretzer. While dss clearly overshoots the data (by up to a factorof 2), the Kretzer FFs do a much better job in describing the spectra both in shape and absolutenormalization. The apparent difference between the two parametrizations derives from the factthat the large- z Kretzer gluon FFs are much softer than those of dss (Fig. 1).The central result of our paper is shown in Fig. 5 which presents a comprehensive compari-son of the charged-hadron world-data in the TeV-range to the NLO predictions using the sevenmost recent sets of FFs. The panels show the ratio of the data to the predictions obtained withthe Kretzer FF (data points), as well as the ratios of cross sections obtained with various FFsover those from Kretzer (curves). The light-blue band denotes the scale uncertainty envelope (cf.Eq. (2)), and the dark-blue band the uncertainty derived from the ct10nlo
PDF (90% confidencelevel) error sets. The hkns error band is shown in light-brown color. The scale uncertainty below7 T ≈
10 GeV / c is prohibitively large making it difficult to draw any strict conclusion regardingthe level of agreement between the data and the calculations there (although the central NLOprediction obtained with the Kretzer FFs agrees very well with the data). The scale dependence,however, stabilizes below ±
20% beyond p T ≈
10 GeV / c and it is rather this region where the NLOcalculations are to be fully trusted. In all cases, the PDF errors are almost negligibly small incomparison to the scale uncertainty. p T | η | < . | η | < . √ s = 7 . p T | η | < . √ s = 1960 . p T | η | < . | η | < . √ s = 2760 GeV p T | η | < . | η | < . | η | < . √ s = 900 GeV Figure 5: Ratio of the inclusive charged-hadron spectra measured by CMS (circles) [20, 21], ALICE [22] (diamonds),CDF (squares) [16], and UA1 (triangles) [13] at √ s = 900–7000 GeV, over the corresponding NLO calculations usingthe Kretzer FFs. The curves show the NLO predictions obtained with other FF sets: kkp (pink scarcely dashed), dss (green dashed), bfgw (brown long-dashed), hkns (purple dashed-dotted), akk08 (yellow dotted-dashed), and akk05 (red long-dashed short-dashed) relative to Kretzer FFs. The point-to-point systematic and statistical errorsare indicated by colored rectangles (gray for CMS and CDF, brown for ALICE, green for UA1) and error bars. Theboxes at the beginning of the p T -axis mark the overall normalization uncertainty. The light-blue bands correspondto the scale uncertainty envelopes while the dark-blue ones indicate the variations derived from the ct10nlo PDFerror sets. The hkns uncertainties are shown by the light-brown bands.
All the LHC data are in mutual agreement within their systematic and statistical uncertainties,although especially the CMS data at 2.76 TeV seem to show larger fluctuations than those inferredfrom the typical size of the quoted point-to-point experimental uncertainty. The UA1 spectrumat √ s = 0 . p T distribution is not incompatible with the rest, this disagree-ment could well be an issue of the experimental determination of the overall normalization in theoldest measurement. 8he results of Fig. 5 exhibit clear systematic trends as a function of √ s and p T : As √ s increasesfrom 0 . p T ≈ p T ≈
10 GeV / c. Indeed, theflatness of the data/theory ratio is worth noticing, although the absolute spectra span many ordersof magnitude. This suggests that the underlying pQCD dynamics of the hadron production isindeed correctly understood and that the data-theory disagreement lies rather in the current setsof FFs. On average, the Kretzer FFs used as reference for the data/NLO ratios shown in Fig. 5,seem to do the best job in describing the data, the results from all other FFs being practicallyenclosed by the hkns error bands. We quantify the data-to-theory correspondence by computingthe χ values for each FF set (in the case of hkns only the central predictions are considered)defined by χ ≡ (cid:88) i (cid:18) D i − T i δ tot i (cid:19) , (5)where D i corresponds to the data point with total error δ tot i (correlated and uncorrelated point-to-point uncertainties added in quadrature), and the theory values T i are specific for each set of FFs.The sum runs over all the data shown in Fig. 5 using three different cuts for the hadron transversemomenta: p min T = 1 . , ,
10 GeV / c. Such thresholds are chosen so as to reduce the weight ofthe lower- p T data which would otherwise dominate the χ due to their larger cross sections andassociated smaller statistical uncertainties. The calculations are run for three scale choices, µ ≡ ( µ ren /p T , µ fact /p T , µ frag /p T ) = (cid:18) , , (cid:19) , (1 , , , (2 , , , which, above p T ≈ / c, practically cover the larger selection of scale variations used earlier,Eq. (2). The results from this exercise are shown in Table 2, and numerically confirm what is seenin Fig. 5: The lowest χ value is almost always obtained with the Kretzer FFs and the highest onewith akk05 . The preferred choice of scale is specific for each set of FFs and depends on the p min T cut imposed. However, on average, the choice µ = (2 , ,
2) is preferred as this set of values tendsto reduce the computed cross sections and thereby moderate the data overshooting.The same conclusion is reached if the χ is computed accounting separately for the correlatedsystematic errors. In this particular case, the only known systematic parameter is the overallnormalization (given by all but UA1) and the χ can be expressed [67] as χ = (cid:88) k ∈ data sets χ k (6) χ k = (cid:88) i (cid:18) f k D i − T i δ uncor i (cid:19) + (cid:18) − f k δ norm i (cid:19) , (7)where δ uncor i is the uncorrelated error, and δ norm i the quoted normalization error. The parameter f k is found by minimizing the χ . The corresponding results are listed in Table 3. In comparisonto the uncorrelated-uncertainties case, the χ values become generally somewhat lower, since, thecalculated values are usually quite above the data, and the improvement attained in the first termin Eq. (7) exceeds the growth of the latter. This procedure, however, often also leads to unnaturally9 able 2: Values of χ /N characterizing how the NLO calculations agree with the data at √ s = 0 . − p T values have been considered. N is the totalnumber of the LHC, Tevatron and UA1 data points above the p min T cut. In obtaining the χ , all the data uncertaintieshave been added in quadrature. µ (1,1,1) (1/2,1/2,1/2) (2,2,2) p min T [GeV / c] 1.3 5.0 10.0 1.3 5.0 10.0 1.3 5.0 10.0 N
368 169 103 368 169 103 368 169 103 kre kkp dss hkns akk05 akk08 bfgw | − f k | as the disagreement between the calculations and the data tends to be muchbeyond the normalization uncertainty quoted by the experiments. In any case, the values in Table 3lead to the same conclusions as those extracted from Table 2 and the final outcome is the sameregardless of the way the χ -function is defined, namely that NLO predictions generally overpredictthe experimental charged-hadron spectra by a factor of two. Similar discrepancies have been foundfor high- p T neutral pion and η meson production at LHC energies [70] implying that the problemis not limited to the total ( g → h + + h − ) fragmentation function but affects the identified ( π ± , K ± , p , p ) gluon FFs individually. Table 3: As Table 2, but accounting for the normalization uncertainty of the data as in Eq. (7). µ (1,1,1) (1/2,1/2,1/2) (2,2,2) p min T [GeV / c] 1.3 5.0 10.0 1.3 5.0 10.0 1.3 5.0 10.0 N
368 169 103 368 169 103 368 169 103 kre kkp dss hkns akk05 akk08 bfgw √ s = 200–630 GeV, where thesituation is different than that found at the LHC energies. Fig. 6 presents a comparison of NLOcalculations to the UA1 [12, 13, 14] and STAR [10] charged-hadron spectra. For these datasets, thecalculation obtained with the Kretzer FFs — the preferred set at the Tevatron and LHC energies— gives a bad description of the spectra. On the other hand, the dss set describes now these10easurements reasonably well. This is, however, not too surprising as these datasets were includedin their actual fit. In this case, also the hkns error band is wide enough to enclose these data. p T √ s = 630 GeV , | η | < . √ s = 540GeV , | η | < . p T √ s = 200 GeV , | η | < . Figure 6: Ratios data/NLO[ kre ] (data points) and NLO[FFs]/NLO[ kre ] (curves) as in Fig. 5 but for lower-energyUA1 ( √ s = 540,630 GeV, left panel) [12, 13, 14] and STAR ( √ s = 200 GeV, right panel) [10] p-p and p-p collisions. Looking back to Fig. 1, one can see that the lower-energy collisions prefer much harder gluons atlarge z than the LHC data. That is, any set of FFs that can, more or less, reproduce the lower- √ s data (preferring hard gluon FFs), will disastrously overshoot the LHC measurements (preferringsofter gluon FFs). As the variation in the probed range in z is only mild as a function of √ s and p T (see Fig. 2) such a result hints that it may be difficult to tensionlessly include all these existingdata into a global FF fit with a p T cut as low as e.g. p T ≥ / c. Indeed, as the very largescale-uncertainty indicates, the fixed-order NLO calculations are not stable below p T ≈
10 GeV / c,and it is questionable whether these lower- p T data points should be even considered in such a fitin the absence of full NNLO corrections. It should be also recalled that within the RHIC kinemat-ics reach, the threshold logarithms can still play a role by increasing the NLO cross sections forincreasingly-high p T values, although such effects should die out towards larger √ s (at fixed p T ).In any case, the fact that such effects cannot be seen in Fig. 6, signals that threshold resummationsare likely not the main cause for the different √ s -dependence of the NLO calculations and the data.In addition, the good agreement between fixed-order NLO calculations [33, 34] and the jet dataat LHC [29, 30], Tevatron [71, 72] and RHIC [73], and the fact that the single high- p T chargedparticle spectrum is dominated by leading hadrons carrying out a large fraction, (cid:104) z (cid:105) ≈ p T the whole picture of independent parton-to-hadronfragmentation may not be adequate, especially in the case of production of heavier baryons. Higher-twist effects, where the hadron is produced directly (i.e. more exclusively) in the hard subprocessrather than by gluon or quark jet fragmentation, may contribute to the cross sections at RHICenergies in the range of transverse momenta experimentally studied [74] and hence “contaminate”the extraction of FFs in global fits that use such data. Even at LHC energies, the proton-to-pionratio below p T ≈ / c (see e.g. [75]) appears to behave qualitatively very differently than the11aon-to-pion ratio or the pQCD expectations. While the kaon-to-pion ratio increases smoothlytowards larger p T , the proton-to-pion ratio contains a clear “bump“ around p T ≈ / c. Toreproduce such a behaviour, additional effects outside the pQCD toolbox are called for. Assumingthat the behaviour of the perturbative proton-to-pion ratio is qualitatively similar to the thekaon-to-pion ratio, one could crudely estimate that there is a roughly 5% “non-fragmentation”enhancement around p T ≈ / c which then diminishes towards higher p T . Subtracting such anon-perturbative contribution from the LHC data would make the disagreement between the dataand e.g. dss even worse. On the contrary, in comparison to the description with the Kretzer FFs,the data-to-theory ratio would be flatter and thereby improve the compatibility. For lower √ s the surplus of (anti)protons could be even more pronounced and remain important up to highervalues of p T . This could partly explain the strong √ s -dependence of the data-to-theory ratio.In Ref. [22], it was observed that the ratios of the ALICE cross sections between different butnearby √ s become rather well reproduced — also at low p T — by the dss FFs. However, it isimportant to note that as the √ s dependence of the z distributions (see Fig. 2) is only mild, partof the FF dependence is bound to cancel in such ratios. Despite such cancellations, it appears thatsome √ s dependence still remains at low p T , supporting our conjecture regarding the presence ofa √ s -dependent non-perturbative component.
5. Summary and outlook
We have examined the LHC, Tevatron, SppS and RHIC data for inclusive unidentified charged-hadron production at √ s = 0 . − p T region, p T ≥
10 GeV / c, where the fixed-order NLO pQCD calculations should betrustworthy as their scale dependence is modest. The best overall agreement with the data isobtained using the relatively old Kretzer FFs in which the gluon-to-hadron FFs are the softest ofall. Below p T ≈
10 GeV / c, the √ s -dependence of the data appears too strong to be reproduced bycalculations based solely on collinear factorization (especially if also lower values of √ s are includedin the comparison), although the NLO scale uncertainties become there very large preventing onefrom drawing definitive conclusions. However, this may not be a problem of the pQCD calculationitself as below p T ≈
10 GeV / c there are increasing indications of additional non-perturbativeeffects in the case of (anti)proton production even at the LHC energies, which may have a non-negligible impact on the total hadron yield. These observations indicate that only the region above p T ≈
10 GeV / c of these charged-hadron data, with theoretical scale uncertainties below ± Acknowledgments
We acknowledge the financial support from the Magnus Ehrnrooth Foundation (I.H.) and from theAcademy of Finland, Project No. 133005.
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