Radial scaling in inclusive jet production at hadron colliders

Inclusive jet production in p-p and ¯ p - p collisions shows many of the same kinematic systematics as observed in single-particle inclusive production at much lower energies. In an earlier study (1974) a phenomenology, called radial scaling, was developed for the single-particle inclusive cross sections that attempted to capture the essential underlying physics of pointlike parton scattering and the fragmentation of partons into hadrons suppressed by the kinematic boundary. The phenomenology was successful in emphasizing the underlying systematics of the inclusive particle productions. Here we demonstrate that inclusive jet production at the Large Hadron Collider (LHC) in high-energy p-p collisions and at the Tevatron in ¯ p - p inelastic scattering shows similar behavior. The ATLAS inclusive jet production plotted as a function of this scaling variable is studied for p s of 2.76, 7 and 13 TeVand is compared to ¯ p - p inclusive jet production at 1.96 TeV measured at the CDF and D0 at the Tevatron and p-Pb inclusive jet production at the LHC ATLAS at p s NN ¼ 5 . 02 TeV. Inclusive single-particle production at Fermi National Accelerator Laboratory fixed target and Intersecting Storage Rings energies are compared to inclusive J / ψ production at the LHC measured in ATLAS, CMS and LHCb. Striking common features of the data are discussed.


I. INTRODUCTION
Single-particle inclusive productions were studied extensively in the early 1970s as a hadronic analogue to deep inelastic electron-nucleon scattering studies conducted at SLAC. The theoretical underpinnings of single-particle inclusive production were developed by Field and Feynman [1], Field et al. [2] and others [3], who described the production of the detected particle to originate from the hard-elastic scattering of a pair of incoming partons which subsequently fragment and hadronize into the inclusively detected particles. The same general quest has been followed in inclusive jet production at hadron colliders (LHC and Tevatron) to test quantum chromodynamics (QCD) and to provide the standard model foundation for searches for a phenomenon beyond the standard model. In the case of inclusive jet production, incoming partons hard scatter, fragment, then hadronize into cones of particles that form jets where the jet itself is analyzed as the inclusively detected particle. Some 40 years ago, in the early time of the operation of the Fermilab synchrotron and at the SPS synchrotron and the Intersecting Storage Ring at CERN, singleparticle inclusive productions, such as p þ p → π 0 þ X, p þ p → π AE þ X, p þ p → K AE þ X, were studied [4][5][6][7]. When the data were analyzed in terms of the transverse momentum p T and the radial scaling variable x R , the kinematic form of the Lorentz invariant cross section was greatly simplified. The radial scaling variable is defined by x R ¼ E/E Max , where E is the detected singleparticle total energy in the center-of-momentum (COM) frame; E Max is the corresponding maximum energy and is roughly ¼ p s/2 in the p-p COM frame. The radial scaling variable describes the phase space suppression as the single-particle production approaches the kinematic boundary where E ¼ E Max . Note that this suppression is independent of the angle of the emitted particle in the COM frame and depends only on the radial distance in energy-momentum space to the kinematic boundary.
The earlier analyses of data indicated that the singleparticle inclusive cross section Ed 3 σ/dp 3 had power-law dependences on the transverse momentum p T and on the variable (1-x R ) that roughly factorizes in the form, where Λ is a transverse mass term, potentially important at low p T , n pT , n xR are the power-law indices and α is the parameter that controls the magnitude of the invariant cross section and fixes the dimensions to ½momentum −4 (e.g. GeV −4 ). In principle, all the parameters of Eq. (1) can be functions of p s, as well as dependent on the inclusively detected particle. However, in the limited energy range of data analyzed in this earlier work, the s dependence of the inclusive cross section was found to be mostly in the x R variable itself; namely, for fixed p T and x R , the inclusive cross sections were roughly constant as p s was varied. The transverse momentum dependence was found to be approximately independent of the inclusively detected particle, but the (1-x R ) dependence varied for different inclusive particle productions. However, more extensive data taken at the ISR showed that there is an overall s dependence beyond that embodied in the x R variable [8] and this narrowly defined radial scaling was violated. Nevertheless, even with this additional s dependence, the radial scaling formulation was helpful in revealing systematics of the single-particle inclusive cross sections.
The question naturally arises whether or not the radial scaling phenomenology has any utility in uncovering systematics in inclusive jet and charm production in p-p and heavy ion collisions at the LHC. After all, the theoretical underpinning of single-particle inclusive production and jet inclusive production is the same-namely, both are described by hard scattering of incoming partons, followed by fragmentation and hadronization, only in the case of jet production an ensemble of particles carrying the scattered parton momentum is collimated and forms a jet. The following questions are therefore quite natural: (i) Is the p T dependence of inclusive jet production at the LHC a power law? (ii) How does the p T dependence of inclusive jet production compare with single-particle inclusive production? (iii) Is there a power-law dependence in (1-x R ) as was observed in single-particle inclusive processes such as the one given in Eq. (1)? On general grounds, one may expect that all the parameters of this simple formulation (α and the power-law indices n pT and n xR ) depend on p s and that there would be no simplification in the much more complex process of inclusive jet production at TeV energies. Nevertheless, it is interesting to seek answers to these questions.

II. JET COLLIDER DATA
There is now agreement between perturbative quantum chromodynamics (pQCD) calculations to next-to-leading order and inclusive jet production at the LHC to better than ∼20%, except at high rapidity and high p T , in effect explaining the jet production in terms of scattered partons and subsequent scattered parton hadronizations [9]. It is a success of the underlying theory that the simulations based on pQCD calculations show such good agreement. Further improvements in the data-theory agreement are expected with the future consideration of higher order effects [10] and better methods of calculating various subprocesses such as by amplitude methods [11].
Inclusive jet production at hadron colliders is conventionally described by p T and the rapidity y ¼ 1 2 lnð Eþp z E−p z Þ, which is roughly equal to the pseudorapidity defined by η ¼ − ln½tanðθ/2Þ, where θ is the angle of the emitted jet in the p-p COM frame with respect to the incoming beams. The invariant inclusive jet (single-particle) cross section can by written in terms of the jet transverse momentum, p T , and jet rapidity, y, after integrating over the azimuthal angle ϕ as In this formulation, the invariant cross section is a function of three variables, the COM energy p s, the transverse momentum p T 1 and the rapidity y. The jet mass has been integrated into the rapidity variable y through the value of the jet total energy E.
The invariant cross section for inclusive jet or singleparticle production could just as well be written in terms of other groupings of three variables such as p s, p T and a combination of y, p T and p s assembled together to express the radial scaling variable x R . In the limit of high energy and small particle or jet mass with respect to p s, the radial scaling variable x R ≈ 2p T coshðηÞ/ p s, where η is the pseudorapidity of the jet in the COM frame. Note that coshðyÞ ∼ coshðηÞ ¼ 1/ sinðθÞ; hence x R ∼ 2p/ p s∼ E Ã /EÃ max . Figure 1 shows the relations of the radial FIG. 1. The lines of constant η are plotted on the p T -x R plane for p s ¼ 13 TeV starting at top jηj ¼ 0, 1, 2, 3, 4. The region above and to the left of the η ¼ 0 line (solid black) is kinematically forbidden. Holding η (y) constant mixes p T and x R and therefore does not control the radial distance from the kinematic limit x R ¼ 1. 1 We have chosen the p T differential to be p T dp T rather than 2p T dp T ¼ dp T 2 .
scaling variable x R to p T to η for p s ¼ 13 TeV. Note that lines of constant η (η ∼ y) mix p T and the scaling variable x R . Thus, the kinematic boundary suppression, controlled by x R , is convoluted with the p T and η (y) dependence.

A. Inclusive jets at the LHC
It is interesting to analyze inclusive jet production at the LHC in the simplest terms by seeing if there are kinematic generalities like those observed in the single-particle production in p-p collisions. As a typical example, Fig. 2 shows the inclusive jet production at 13 TeV measured by the ATLAS Collaboration at the LHC [12][13][14].
It is evident that the inclusive jet cross sections agree with the NLOJET þ þðCT14nloÞ [15,16] and corrections. However, we note that plotting the cross section for constant y as a function of p T , as in Fig. 2, involves changing the value of x R as was demonstrated in Fig. 1. Therefore, the presentation of the data for constant y obscures a putative power-law behavior in p T and (1-x R ) that we would expect if inclusive jet production in p-p scattering has a similar behavior to that of single-particle inclusive cross sections.
Examining Fig. 2, it is evident that the cross section decreases with increasing p T and y, but it is not obvious that there are any power laws in p T and (1-x R ) as were observed in single-particle inclusive production in p-p collisions. However, we can roughly test the hypothesis that the invariant cross section has the factorized form of Eq. (1) by plotting the resultant invariant cross section d 2 σ/p T dp T dy multiplied by a function of p T , where we find ∼p T 6 works reasonably well. The resultant behavior is shown in Fig. 3. Notice that p T 6 ðd 2 σ/p T dp T dyÞ is mostly a function of x R in the sense that the data for different values of jyj fall on top of each other and therefore tend to radially scale for a fixed p s. For a deeper view of the p T and x R dependencies of the 13 TeV inclusive jet data, we analyze the cross section using the form suggested by single-particle inclusive data, by plotting the cross section in slices of constant p T as a function of 1-x R . In Eq. (3) we have not assumed a specific form for the p T function, Aðp T ; sÞ, except to posit that it is not a function of x R . Since the 13 TeV ATLAS inclusive jet data are binned in rapidity (y), thereby requiring the jet mass to be known in order to determine the angle of the jet in the p-p COM, we approximate the radial scaling variable as FIG. 2. The ATLAS inclusive jet cross section [12] is plotted as a function of p T for various rapidity regions. The data were taken at p s ¼ 13 TeV and the jets defined by the anti-k T algorithm. Note that the suppression when close to the kinematic boundary is evident on the rhs of the plot, e.g. for the highest rapidity bin 2.5 < jyj < 3.0 the maximum p T is less than 1 TeV/c. The data are in good agreement with simulations. 6 is plotted as a function x R for various jyj values. Note that the data tend to roughly scale as a function of x R . The error bars represent the statistical and systematic errors added in quadrature. The point with x R > 1 is due to the finite bin corrections in p T and y not performed. The data so plotted roughly follow ð1-x R Þ 4.5 .
where the jet mass m J has been adsorbed in the variable y but is bounded using the prescription of Ref. [17] by m J /p T < R/ p 2 ¼ 0.28 for the jet cone size R ¼ 0.4 given by where Δϕ and Δη are the jet cone widths in ϕ and η, respectively, defined with respect to the colliding beams axis. The finite y bin size was treated by assuming that the published data value for a bin corresponds to the midpoint of the lower and upper limits-a valid assumption for low y, where the rapidity distribution is approximately flat. However, the bin center so calculated is slightly larger by <1.4% from a more valid data-weighted value for the highest rapidity bin (2.5 < jyj < 3.0) resulting in the computed value of x R larger by <3.8%. This putative finite bin correction was ignored. The 13 TeV inclusive jet data so analyzed are shown in Fig. 4. We would expect that the x R behavior would be complicated and, even if a power law were operative, the indices n pT and n xR would be functions of p s, p T and y. Note that x R ¼ 0, where Aðp T ; sÞ evaluated in the (1-x R ) power-law fits corresponds to the limit when p s → ∞ for constant p T and thus is beyond the minimum value (η ¼ 0) of x R min ¼ 2p T / p s for finite p s. This small extrapolation assumes that the functional form of Eq. (3) is valid in the small region from x R min to x R ¼ 0.
The power-law fits of ð1-x R Þ nxR were performed by a least-squares linear method on the natural logarithms of the cross section as a function of the lnð1-x R Þ using the statistical and systematic errors added in quadrature. The slopes of these linear fits are the exponents n xR and the constant terms are the logs of Aðp T ; sÞ for the fixed p T values. In general, we would expect that n xR would be a function of p T and p s. Figure 5 shows the values of n xR plotted as a function of p T where it is evident that n xR → ∼4 for high p T but has a higher value for low p T . Figure 6 is a plot of Aðp T ; sÞ as a function of p T where it is clear that the data follow a power law as suggested by early radial scaling studies of singleparticle inclusive scattering denoted by Eq. (1) above. The fits, represented by the dotted red lines in the figures below, have the following forms: Aðp T ; sÞ ¼ αðsÞ where n xR0 and D are the fit parameters for the power of (1-x R ); and α and n pT are the fit parameters for the powerlaw fit to Aðp T ; sÞ. Note that at the high p T values of these data the Λ term was not necessary. Encouraged by the simplicity of the 13 TeV inclusive jet cross section when analyzed with the radial scaling variable, we now examine the ATLAS jet data taken at p s ¼ 2.76 [18] and 7 TeV [19]. Both analyses at these lower energies used the same anti-kT jet defining algorithm TeV, respectively. The dotted red lines are power-law fits of the form Aðp T ; sÞ ð1-x R Þ nxR described in the text. Note that the data are consistent with a power law in (1-x R ) as in the case of single-particle inclusive cross sections measured at much lower energies but that the power-law indices nxR are a function of p T . The (1-x R ) power index is larger for lower p T . The displayed error bars are statistical and systematic, added in quadrature. The overall error in the luminosity normalization has been neglected. Note that the 1/p T term of the n xR dependence grows with increasing s, but the p T power-law exponent, n pT , is constant. The overall magnitude of the cross section, governed by the α term, increases with s.

B. CDF and D0 inclusive jet¯p p data
The CDF [20] and D0 [21] inclusive jet data taken at 1.96 TeV collisionsp-p were analyzed in the same way as the ATLAS inclusive jet data. The results are shown in Figs. 9 and 10. The power indices of (1-x R ) tend to be flatter in rough agreement with the trend seen in Fig. 7, that is, the 1/p T slope of n xR assumes a smaller value for lower COM energies. In Fig. 10 we notice the same p T power-law behavior as seen in p-p inclusive jets. Hence, we conclude that the p-p jets have a behavior consistent with trends shown in Figs. 7 and 8 for the LHC p-p jets.
The fit parameters of the data using Eqs. (6) and (7) shown in Figs 7-10 are given in Tables I and II below where we have added a measurement by the CMS Collaboration of inclusive jets at 8 [22] and 13 TeV [23] to the ATLAS measurements discussed above. Notice that the quality of the n xR fit is reasonable (χ 2 < 1.7/d:f:),  7). We observe that Aðp T ; sÞ functions for the three energies have the same power index, but the overall magnitude of Aðp T ; sÞ grows with increasing s.
whereas that of the power-law fit to Aðp T ; sÞ has a large χ 2 /d:f: This is discussed later.
Examining Tables I and II we conclude that most of the variation of the parameters of these fits to inclusive jet production is in the overall normalization term controlled by the parameter α, which increases with increasing p s, and the p T dependence of the power of (1-x R ) given by the term, D, which also increases with increasing p s. The parameters np T and nx R 0 do not show such large systematic p s dependences. The parameters shown in Tables I and II are plotted in Fig. 11 below. It is interesting to note that the D and α terms grow linearly with s, whereas the n xR0 and n pT terms are roughly constant. The units of α in Table II are [pb/GeV 2 TeV npT ], which for n pT ∼ 6 become [energy 2 ], the same units as the Mandelstam variable, s. Hence it is not surprising that α grows linearly with increasing s in order to preserve the dimensions of the invariant cross section d 2 σ/p T dp T dy to be [pb/GeV 2 ].

C. Inclusive jet production in p-Pb collisions
For another view of inclusive jet production at the LHC we analyze the jet data taken in p-Pb collisions at a nucleon-nucleon COM energy of p s NN ¼ 5.02 TeV [24]. Here we examine the two sides of the collision separately-namely the side where the incoming proton FIG. 9. The exponents of the (1-x R ) power-law fits are plotted as a function of 1/p T for the CDF (circles) and D0 (triangles) inclusive jet production at 1.96 TeV inp-p collisions. The dotted red line is the straight-line fits in 1/p T of the form given by Eq. (6) to the CDF and D0 data considered as one data set. The data show considerable scatter, especially at high p T (low 1/p T ). Only data for x R < 0.9 were included in the fits.
FIG. 10. The p T power law of CDF (circles) and D0 (triangles) for inclusive jet production at the FNAL at 1.96 TeVp-p collisions. The systematic and statistical errors have been added in quadrature. The dotted red line is a power-law fit to both data sets taken together. Only data with x R < 0.9 were considered. The parameters of the fit are in Table II. fragments (y > 0) and the side where the Pb nucleus fragments (y < 0). Rather than arbitrarily assigning the central rapidity bin −0.3 < y < 0.3 to either the proton forward or the Pb-forward data, this central bin was included in both sides of the data. If there were an underlying hard partonparton scattering that initiates the formation of the detected jet, we would naively expect the same power law in the transverse momentum p T as observed in jet production in p-p collisions. On the other hand, the fragmentation part of the cross section expressed by the x R dependence may be different since the jet formation on the Pb fragmentation side would have to contend with many nucleus fragments, whereas the jet formation on the proton side would be similar to p-p scattering. A difference would be an expression of the well-established jet quenching [25] observed in heavy ion collisions. The comparison of inclusive jets in p-Pb scattering of the power of (1-x R ) for the two fragmentation cases is shown in Fig. 12. The corresponding p T dependences are shown in Fig. 13. The fit parameters are listed in Tables III and IV.  From Tables III and IV we conclude that the p T dependences are approximately the same in the two fragmentation cases (the values of n pT are within 5% of each other), whereas the D terms of the (1-x R ) exponent parametrizations by Eq. (6) depend strongly on the type of fragmentation side. In fact, such a D value for the Pb fragmentation side would correspond to inclusive jets at ≈10 TeV in p-p collisions, implying that the quenching of jets observed in heavy ion collisions is also operative in p-p collisions, but at higher energies. This interpretation suggests an equivalency between the formation and quenching of jets at lower energies in A-A collisions with jet production at higher energies in p-p collisions and could be systematically studied by performing this analysis for jets produced in A-A collisions as a function of centrality.

D. Comparison with inclusive jet simulations
It is not the object of this paper to appraise the quality of the pQCD simulations of inclusive jet production, but it is of interest to check that the simulations show the same power-law behaviors. Of the data examined in this work, from the CDF inclusive jets at 1.96 TeV published in 2009 to ATLAS inclusive jets at 13 TeV published in 2016 there is good agreement with simulations. The CDF analysis used the midpoint jet clustering algorithm with a cone size R ¼ 0.7 and proton and antiproton parton distribution functions (PDFs) from [26] in conjunction with PYTHIA 6.2 [27]. The ATLAS Collaboration used an anti-k T clustering algorithm with R ¼ 0.4 and a more refined PDF set in PYTHIA 8.186 [28,29].
As a demonstration of the agreement, the Monte Carlo (MC) simulation SHERPA [30] has been compared with the 7 TeV ATLAS data, where the MC data were analyzed in the same way as the ATLAS 7 TeV inclusive jet data using the radial scaling formulation. The comparison of the ratios (MC/data) of the respective fit parameters is given in Table V. All parameters of the data-SHERPA comparison are consistent with each other, with the exception of D, which is smaller by about 30% in the SHERPA simulation (3σ). However, we note that this comparison of data vs MC  7). Note that the p T dependence is consistent within errors. As in Fig. 12, the error bars represent the statistical and systematic errors added in quadrature. The systematic errors dominate. Aðp T ; sÞ units are arbitrary since the cross section is self-normalized.

III. SINGLE-PARTICLE INCLUSIVE DATA
Since we find that the inclusive jet production at the LHC in p-p, p-Pb collisions and inpp collisions at the Fermi National Accelerator Laboratory (FNAL) collider has power law dependences in both p T and (1-x R ), it is interesting to analyze single hadron and prompt photon production.

a. Prompt photon production
In prompt photon production, the photon is believed to come directly from the primordial hard parton scattering such as q g → q γ and higher order processes, such as the fragmentation process q g → q g γ. Unlike jet production, prompt photon production has no final state interaction other than the radiative fragmentation process above. Hence, the E T dependence as well as the (1-x R ) dependence are important measures of the production mechanism without the influence of the final state processes.
A number of authors (for example see [31]) have extensively analyzed direct photon production in p-p collisions as a means to determine the nucleon gluon distribution, but not with our variables (p T , x R ). For this study, we consider the prompt photon data determined by CMS at 7 TeV [32] and that of ATLAS at 8 [33] and 13 TeV [34]. The photon data are analyzed in the same manner as the inclusive jet data-namely we compute the invariant cross sections d 2 σ/E T dE T dη and plot the results as a function of (1-x R ) for fixed E T in order to determine AðE T ; sÞ and the (1-x R ) power-law indices. The outcomes of the analysis for all three data sets are shown in Figs. 14 and 15.
As in the analysis of the inclusive jet data, we have determined the power indices n xR of Fig. 14 by fits of the function of Eq. (6), where E T replaced p T and the function AðE T ; sÞ in Fig. 15 with Eq. (7). The results are given in Tables VI and VII. It is interesting to note in Table VI that the parameter D for prompt photons is negative and appears to grow more negative with increasing p s. This is in contrast with the behavior for inclusive jets at the LHC where D is positive and increases with increasing p s. The parameter n xR 0 has an average value hn xR 0i ¼ 4.2 AE 0.4 which does not show a systematic energy dependence, although the dispersion of the data is large.
In Table VII, where values of α and n ET are given, we see that the overall direct photon cross section grows with increasing p s as indicated by the fitted values of the α parameter. This is the same general behavior observed in our analysis of inclusive jet production cross sections.
It is notable that the power-law index n ET is less than the corresponding value for inclusive jets. Averaging over the three measurements in Table VII (7 to 13 TeV) we find hn ET i ¼ 5.6 AE 0.2 unweighted average, whereas the average of the corresponding parameter for inclusive jets in the energy range 7 to 13 TeV is hn pT i ¼ 6.4 AE 0.2. This suggests that the prompt photon leaves the scene of the primordial collision unencumbered; whereas jets must  IV. The parameters of the power-law fits to Aðp T ; sÞ according to Eq. (7) are tabulated. The power indices np T are the same within 5% for the p-forward and Pb-forward cases and are consistent with the index for p-p scattering given in Table II tear themselves free of the QCD color fields. If we assume that E T dependence of AðE T ; sÞ for prompt photons is a measure of the primordial hard parton scattering, then the fragmentation and hadronization operative in the production of jets in p-p and p-p collisions contribute to the jet and hadron p T dependence by roughly one more term ∼1/p T (a 3σ difference with these data).

b. Hadron production
A host of other inclusive production data were analyzed in the same way. Since the data extend to lower transverse momenta, we expect the parton intrinsic transverse momentum (k T ) to be an influence as well as transverse mass effects for heavy particle production. Moreover, for the case of charm production at the LHC we might expect a similar term that could arise from a production mechanism from the decay of a heavier "parent" particle. Hence, we fit the p T dependence with the form given in Eq. (8) below, where the values of Λ, α and n pT are determined by a minimum χ 2 fit [35]. A typical result is shown in Fig. 16 for π þ using the compilation of Refs. [5].
The results of the p T power-law fits are listed in Table VIII. We note that all the processes considered, including the five direct photon measurements, have an average p T power-law dependence Aðp T ; sÞ ∼ 1/p T npT with an index np T ≈ 6.1 AE 0.6. The inclusive production of light hadrons up to K þ has a Λ value consistent with the parton intrinsic k T ∼ 0.6 GeV [36], whereas the J/ψ and ψð2SÞ production are consistent with a larger Λ value (Λ ≥ 3.6 GeV) which must provide an important clue about their production mechanism [37]. FIG. 16. The inclusive π þ data [5] are plotted with respect to the measured p T (black triangles) and with respect to ðΛ 2 þ p T 2 Þ 1/2 closed black circles. The red dotted line is the power-law fit corresponding to np T ¼ 6.94 AE 0.04 as given in Table VIII. A minimum χ 2 fit of Λ from Eq. (8) yields minimum at Λ ¼ 0.602 AE 0.012 GeV consistent with intrinsic parton k T of the nucleon [36]. The power-law indices average ¼ 6.1 AE 0.6. The values of Λ for the single light (π AE;0 , K þ ) particle inclusive production are consistent with the intrinsic k T . Note that hΛi tends to grow with increasing mass of the inclusively detected particle, and tends to be larger for J/ψ nonprompt production in comparison to J/ψ prompt production. VIII. Tabulated are the values of the power-law fits to various processes through Eq. (8). All processes are for inclusive production in p-p collisions except for Ag-Ag collisions of BRAHAMS RHIC data [38]. For those entries of the table where p s ¼ 0.063 TeV the tabulated p s value is the maximum of the data set, which also includes lower p s values down to 10 GeV in some entries [5]. The values of Λ and associated errors are determined by the curvature χ 2 function about its minimum. The 7 TeV CMS prompt J/ψ data [39,40] are consistent with Λ ¼ 0, unlike the other measurements, but with a large error and for this reason Λ and σðΛÞ for this entry are left blank. Entries 1 through 5 for direct γ have a minimum E T ≫ k T ∼ 0.6 GeV and thus no sensitivity to the k T (Λ) value. The last column indicates the average value of Λ for each category of inclusive processes.

IV. LINE COUNTING, HIGHER TWISTS, AND DIQUARKS
Using the radial scaling formulation discussed above and examining Tables II, IV, and VIII, it is remarkable that the p T factorized part of the invariant cross sections is a power law with the behavior Aðp T ; sÞ≈αðsÞ/p T 6 with essentially all the s dependence confined in the term αðsÞ. This is true for inclusive jet production in high-energy p-p,p-p collisions and inclusive single-particle production in p-p and inclusive π þ production in Ag-Ag collisions. (Direct photon production favors ∼1/p T 5.6AE0.2 .) Further, the exponent n xR of (1-x R ) for LHC inclusive jet production is found to be a linear function of 1/p T with the slope parameter D increasing with increasing p s. Inclusive jet production in p-Pb collisions shows the same behavior but has a significantly different D value depending on the fragmentation side (proton forward or the Pb forward).
It is well known that the dimensions of the invariant cross section for partons are dependent on the number of active fields that hard scatter to produce the detected jet or the particle in nucleon-nucleon scattering. By this argument the matrix element for the hard-scattering M ∼ ðMassÞ 4-nA , where nA is the number of active fields that scatter [50][51][52][53][54] andŝ is the total parton-parton COM energy squared. Since the invariant cross section has the form given by Eq. (9) we would expect the p T dependence of the invariant cross section by this argument to be given by Eq. (10).
Referring to Fig. 17 below, we note, for example, that u-d elastic scattering, or g-u elastic scattering, involves four active fields and therefore would have a p T dependence given by Eq. (11a). Whereas, for example the diagram at the bottom-left part of the figure involves quark-quark scattering with two radiated gluons, one in the final state and one that forms a diquark. This diagram involves five fundamental fields and consequently would have a p T dependence given by Eq. (11b).
It is noteworthy that all the processes tabulated above seem to favor the p T dependence given in Eq. (11b) over a wide range of energies, rather than the lowest order scattering which has a p T dependence given by Eq. (11a). Other diagrams, such as ones with a radiated gluon from a scattered quark, would correspond to a 2 → 3 scattering resulting in a higher p T power. The p T power-law index ∼6 is a surprise since one would expect an index of ∼4 for parton-parton (2 → 2) hard elastic scattering at lowest order.
A number of authors have observed [55][56][57][58][59][60]that the effective p T power is larger than the expected dimensional limit of 2 → 2 scattering but some researchers find that the p T power seems to depend on the process. Some of these analyses explore the limit to scaling as a function of x T → 0, which we have shown does not respect the kinematic boundary and therefore mixes the kinematic boundary suppression with the underlying p T dependence. An appraisal of one of these studies [51] is given in the Appendix. On the contrary, we find that the average p T power is hn pT i ¼ 6.2 AE 0.6 for all 48 inclusive single photon/hadron/jet data sets considered in p-p,p-p, p-Pb collisions (for inclusive jets in p-p, p-p and p-Pb hn pT i ¼ 6.5 AE 0.3 (Tables II and IV) and single-particle inclusive cross sections hn pT i ¼ 6.1 AE 0.6) ( Table VIII). The invariant cross section dimensional limit n pT ¼ 4 is therefore disfavored by 3.8σ. Even direct γ production disfavors n pT ¼ 4 by 8σ (hn ET i ¼ 5.6 AE 0.2 vs 4) (Table VII). This behavior is consistent with a dominant 2 → 3 hard scattering that is saturated at a relatively low p s, thereby becoming independent of process and COM energy (see Feynman et al. [2]).
The dominant 2 → 3 scattering is consistent with an intrinsic diquark inside the nucleon, although a 2 → 2 scattering with a radiated gluon from one of the final quark legs would also have a cross section of the same p T dimension. Evidence of diquark correlations in the proton has been discussed for some time [61]. Recently, data from JLab [62] supported the notion of diquarks affecting the proton elastic form factors. Lattice QCD calculation also indicates that there is a strong association of the u-d quarks in the proton that forms a singlet (diquark) state [63,64].

V. Aðp T ;sÞ FOR JETS AS A QUADRATIC IN lnðp T Þ
We have noted in Tables II and IV that the p T powerlaw fits for inclusive jets had rather unlikely χ 2 values. A close examination of the fit-data relation reveals a systematic deviation from a pure power law-namely, there is a small curvature making the p T dependence less steep at low p T than at high p T . The effect is illustrated in Fig. 18 where we plot the residuals of the power-law fit of the 13 TeV ATLAS inclusive jet data as a function of lnðp T Þ. In order to make the discrepancy clear, the baseline power-law fit was determined by treating all errors the same. The error bars in the figure represent the statistical and systematic errors added in quadrature as in Fig. 6.
We notice that the residuals of the single power-law fit plotted in Fig. 18 can be quite well fitted to a quadratic in lnðp T Þ. This suggests that the p T dependence of the invariant cross section for inclusive jets at 13 TeV is a function of the type lnðAðp T ; sÞÞ ¼ βðsÞ ln ðp T Þ 2 − n p T lnðp T Þ þ ρðsÞ: An equivalent form of Eq. (12) makes evident the underlying p T power law with a moderating term controlled by the parameter β and is given by where αðsÞ ¼ expðρðsÞÞ.
Fitting Aðp T ; sÞ of ATLAS and CMS inclusive jets and the inclusive jets of CDF and D0 1.96 TeV to Eq. (12), we find the parameter values given in Table IX. The residuals of the ATLAS 13 TeV inclusive jets of this quadratic lnðp T Þ fit are shown in Fig. 19.
Adding the β term of Eqs. (12) and (13) improves the χ 2 of the Aðp T ; sÞ fits for inclusive jet production quite significantly as noted in Table IX compared with  Table II and seen in Fig. 19. However, the data are good enough to draw only rough conclusions about the systematics of the s dependence. The β and n pT terms are roughly independent of p s, whereas the α term grows roughly linearly with increasing s.
While we have interpreted the deviations from a pure p T -power law as real, an uncorrected nonlinearity in the jet energy calibration could also be contributing. The  The s dependence of our jet fit parameters, shown in Fig. 11 and in Tables I and II, is an indication that jets at low pT and high p s are strongly quenched. This may be an important factor in planning experiments at a 100 TeV p-p collider.
Without a detailed analysis of the various experimental systematic errors, which is beyond the scope of this work, it is not clear in some cases whether the relatively small differences in parameter values seen are evidence of real differences, such as the different of p T powers of prompt photon production from inclusive jets, or uncorrected systematic effects. Better data and more sophisticated analyses, which for example involve corrections for finite p T and x R bins, would help resolve these issues. In fact, an examination of the fit values, by comparing the parameters for ATLAS and CMS inclusive jets of n pT for inclusive jets in Table I, indicates that Δðn pT Þ ∼ 0.4 is within the systematic errors of this analysis.
One aspect of this analysis not explored in detail is the x R dependences. Unlike the p T behavior, the x R side is process, as well as s dependent through the D term, and is therefore rich phenomenologically. In quark-line counting schemes the exponent of (1-x R ) is dependent on the number of spectator fields and is given by 2n spectator − 1 [51]. Examining this feature of the inclusive charm cross section should offer important information about the production mechanism.
The inclusive jet and prompt photon invariant cross sections are well replicated by simulation. In fact, pQCD and various MC programs, such as HERWIG and PHYTHIA [27,28] throughout their historical development, show power laws in p T as well as in the variable (1-x R ). Hence, the elementary behavior revealed in this analysis is already deeply embedded in the simulations and therefore understood. However, the factorized form of the invariant inclusive cross sections, as worked out by this analysis using the x R variable to control phase space, shows a simple structure that may be useful in uncovering nontrivial signatures independent of kinematic effects.
In the original formulation of radial scaling [5], it was posited that all the s dependence of the inclusive invariant cross sections was in the scaling variable, x R ≈ 2p T coshðηÞ/ p s, and that the p T and x R dependences of the invariant cross sections completely factorized. This turned out to be not generally true. Data taken at higher collision energies showed that there is an additional s dependence in the αðsÞ term, beyond the x R function, that arises from the QCD evolution of the parton, fragmentation and hadronization functions. Moreover, we found in our analysis that the (1-x R ) power index, n xR , has a p T dependence that is controlled in our formulation by the D term. Thus, the rudimentary factorization of the invariant inclusive cross sections into a p T part and an x R part is broken.
The x R variable, unlike x T or x k , has utility in that it quantifies the fraction of the energy of the jet or particle with respect to the kinematic limit in inclusive cross sections that is independent of angle in the COM frame. Controlling this faction breaks the conflation of a purely kinematic effect from a deeper dynamical behavior that seems to have confused several authors. The approximate scaling variable, developed over 40 years ago in the analysis of inclusive particle production in p-p collisions, still finds utility in uncovering simple power laws in inclusive jets, photons and charm in both p-p and p-Pb collisions at the LHC. Now that the data from the LHC are reaching maturity in broad kinematic ranges, it will be interesting to analyze their broad trends using our formulation.

APPENDIX: CRITIQUE OF ARLEO et al.
Arleo et al. [51] have analyzed a number of inclusive measurements, such as inclusive single-particle production in p-p scattering and inclusive jet production at the SPS and FNAL collider. They find that the p T power depends on the process as given in Fig. 20 and is strikingly different from our analysis, which finds all processes examined to be clustered around n exp ∼ 6.5. Of particular note is the FIG. 20. The effective p T power n exp from the analysis of Arleo et al. [51] is shown for various processes. This result is strikingly different from this analysis, which finds n pT for all processes examined to be clustered around n exp ∼ 6.4 AE 0.5. Note that the highest exponents determined by Arleo et al. are from comparisons of the lowest p s data.
analysis of inclusive jets at CDF and D0 (triangles in the figure below) where the exponent n exp ∼ 4.5 is found. Arleo et al. posit that the invariant cross sections depend on p T and x T defined by x T ¼ 2p T / p s. By computing the ratios of cross sections at different values of p s they are able to extract the effective p T power denoted by n exp . In their analysis, the invariant cross section is given in Eq. (A1), The analysis rests on the assumption that the function Fðx T ; θ 1 Þ ≈ Fðx T Þ and that the s dependence of the cross section is entirely through the x T ¼ 2p T / p s term so that the invariant cross section can be written as The n exp value is determined by the ratio of the cross sections and the respective ratio of the COM energies. From Eqs. (A1) and (A2) using p T ¼ x T p s/2, But by the radial scaling hypothesis, the form in Eq. (A2) is generally not true since the function Fðx T Þ is really a function of p T , θ (x R ) and importantly of p s through the αðsÞ term.
In order to show the flaw in this analysis at least for LHC inclusive jets, we take our parametrization of the LHC inclusive jets at p s ¼ 13 TeV and 2.76 TeV given in Tables I and II to determine n exp in the same way. We compute the ratio of 13 to 2.76 TeVATLAS inclusive jets to examine the cross section ratio as x T → 0. The result is shown in Fig. 21 where we plot n exp given by Eq. (A3) as a function of x T for various fixed jet COM angles θ. We find that our evaluation of Eq. (A1) yields an effective p T power of ∼4 in the limit x T → 0 consistent with the analysis of Arleo et al. [51] for CDF and D0 inclusive jets at 1.8/0.63 TeV. Similar results are obtained when we compare 13 TeV jets to 7 TeV jets.
It is interesting to note that this result for the ATLAS data depends strongly on the s dependence of the cross sections through the α term of Eq. (7). Setting the term α ¼ 1, but leaving the other parameters (np T , D and nx R0 ) at their fit values one finds n exp ≈ 6.3 as x T → 0. Setting all parameters to the same value (α ¼ 1, n pT ¼ 6.3, D ¼ 0 and nx R0 ¼ 3.5) we find n exp ≈ 6.3. But putting in the measured s dependence of α and leaving the other parameters of the cross section the same (np T ¼ 6.3, D ¼ 0 and nx R 0 ¼ 3.5) we find n exp ≈ 4.4. These results indicate that n exp ≈ 4 of Arleo et al. is a result of the s dependence of the cross section, which in our parametrization is mostly through the α term for small x R , and not a true measure of the intrinsic p T dependence.
In order to compute the true p T power exponent, n, of the invariant cross section given in Eq. (A1), we must include not only the θ dependence, or equivalently the x R dependence of the cross section, but also the α-term s dependence. Thus, Eq. (A1) becomes σ inv ≡ E d 3 σ dp 3 ðAB → CXÞ ¼  Tables I and II for p s ¼ 2.76 and 13 TeV are used to evaluate Eq. (A1) above. The various lines are for fixed COM angles starting at θ ¼ π/2 down to 0.196 radians. All lines converge to n exp ∼ 4.3 even though the underlying p T dependence is ∼1/p T 6.3 .