Exploring the Potential Role of Diquarks in Hadronization Using Semi-inclusive Deep Inelastic Scattering on Nuclear Targets

Hadronization, which is the process by which an energetic colored quark evolves into a color-singlet hadron in Quantum Chromodynamics, can be studied both in small nuclei and large nuclei, and the comparison of the differences between those two systems provides information about the hadronization mechanisms on femtometer distance scales. It can be argued that this process is well understood for light meson production on nuclei, but data from HERMES and CLAS for baryon production present puzzling patterns that are not well described by models. Herein we suggest that this might be due to the presence of diquarks in the protons and neutrons making up these nuclei.


Introduction
The process of hadronization, where an energetic colored quark evolves into a color singlet hadron, can be studied by implanting the process into atomic nuclei of different sizes.This is particularly cleanly studied in deep inelastic scattering by leptons, where initial state interactions between the probe and the medium are minimal.Measurements of this type have been performed by the HERMES collaboration Airapetian et al. [1] and more recently by the CLAS collaboration Morán et al. [8], Paul et al. [9], both for electro-nuclear interactions.
The picture that emerges from these data for meson production is particularly clear.The virtual photon emitted by the scattered electron is fully absorbed by one of the valence quarks, and ejected with high energy out of the nucleon.It then can pass through the nuclear medium and emerge out of the nucleus, or it can fluctuate into a color-single hadron that subsequently interacts hadronically with the nuclear medium.In the HERMES data fitted to one space-time model, the timescale of the propagating quark stage ranges from 2 fm/c to 8 fm/c in the sense that those are the characteristic times of a decreasing exponential decay probability function Brooks and López [3].The same approach suggests that for mesons the dominant mechanism for hadron attenuation is hadronic interaction with the medium, not partonic energy loss.
Other evidence that meson production from nuclei is well understood is the agreement of the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) Buss et al. [4] predictions with the recent CLAS results for positive pion production.The GiBUU code successfully describes a wide variety of interactions, from neutrino beams to proton and electron beams to heavy ion beams.In the recent CLAS results, GiBUU can even describe threefold differential measurements with comparatively good agreement.A third model Guiot and Kopeliovich [6] also enjoys success over a more limited range of kinematics.
However, the HERMES data for proton and antiproton production, and new CLAS data for lambda hyperon production, show behavior that is qualitatively very different from the meson production.This strongly suggests that the hadronization mechanisms must be at least partially different.In the following, the reasoning behind this statement will be described in more detail.

Description of the Observables and Data
Both the HERMES data and the CLAS data feature the same two experimental observables.The first is the hadronic multiplicity ratio R h M , which is defined as follows: where N h is the number of hadrons produced as a function of the usual semi-inclusive deep inelastic scattering (SIDIS) variables in larger nucleus A and in smaller nucleus D, and N e is the number of scattered electrons as a function of four-momentum transfer and energy transfer.
The second observable is the transverse momentum broadening: where "transverse" is with respect to the direction of the virtual photon (not to the beamline).This second observable is conventionally thought to be sensitive to partonic-level processes, at least for meson production.In Fig. 1 are shown the HERMES data for two-fold dimensional multiplicity ratios for xenon for four different mesons and two different baryons.(Figure taken from Barabanov et al. [2].)As discussed in more detail below, the qualitative features of the four mesons are similar, but the qualitative features for the two baryons are different from each other and also different from the mesons.
In Fig. 2 is shown the multiplicity ratio for the lambda hyperon in the new data from CLAS taken for Jefferson Lab experiment E-02-104 William K. Brooks [11].It is shown in two figures with different z h ranges because the magnitude of the multiplicity ratio is very different above and below z h = 0.5.In the region below z h = 0.5, the lambda multiplicity ratio grows to a value of nearly 8, in very strong contrast to the case for pions.In the published HERMES data, charged pion multiplicity ratios that were integrated over p T never exceeded 1.2.In the published CLAS data, charged pion multiplicity ratios that were integrated over p T never exceeded 1.3 Morán et al. [8].This is the first and most dramatic observation of the extremely strong difference between meson multiplicity ratios and lambda hyperon multiplicity ratios.

Direct Diquark Scattering
In this paper we wish to make an argument for the direct interaction with a diquark in the proton or neutron by the virtual photon, as a new scattering mechanism.We will call it Direct Diquark Scattering (DDS).If this mechanism exists, it will occur both in the small nucleus D and in the large nucleus A, but in the large nucleus  z-binned multiplicity ratios for carbon, iron, and lead (the results are horizontally shifted for clarity).The outer error bars are the p2p systematic uncertainties added in quadrature with the statistical uncertainties.The inset contains the total normalization uncertainties for each nucleus.The plots illustrate the results of the low (left) and high (right) z ranges corresponding, respectively, to the target and current fragmentation regions.The curves correspond to GiBUU model calculations Buss et al. [4] A this mechanism will be detectable by a modified final state, as in the case when a single quark absorbs the momentum and energy of the virtual photon and passes through a large nucleus.
There are at least two pieces of experimental evidence that DDS may be occurring.First, it is well known that in SIDIS, the kinematic variable z h = E h /ν, sometimes referred to as the relative energy of the hadron, takes on values from zero to unity.It does not exceed 1.0 except for very small resolution effects, at the level of a few percent.The interpretation of this variable is that when a single quark absorbs the energy and momentum of a virtual photon with energy ν, the maximum possible energy of the quark, as well as of the produced hadron, is ν.Therefore measured values of z h can never be greater than 1.0.
However, in preliminary data from CLAS on proton production in SIDIS kinematics, which is analogous to the lambda hyperon production, there is a very clear extension of the data to z h = 1.2.It is not a decreasing tail, rather, it is a constant and substantial value from 1.0 to 1.2.What can be the cause of this?If DDS occurs, and the virtual photon is absorbed by a diquark pair, the total energy of the diquark that is knocked out of the nucleon can be greater than ν because the mass of the struck diquark pair can be substantial, unlike the typical case of single-quark scattering.In fact, in many models of diquarks, the effective mass can be quite large, even exceeding the mass of the proton for the spin 1 diquark case Barabanov et al. [2].If the diquark is not broken up by the scattering, then the two-quark pair will have a strong interaction field binding them together, and will thus carry much more mass than an isolated quark.In this way, the total energy of the emerging hadron has both a momentum term and a mass term, and it can exceed ν, and thus z h can exceed 1.0.
A second piece of evidence that DDS may exist can be seen in the transverse momentum broadening of the produced lambda.This was measured by the CLAS Collaboration and the results can be seen in Fig. 3.The maximum broadening in the left panel of that figure is 0.3 GeV 2 .That is a solid order of magnitude greater than the broadening seen in CLAS data for charged pions Molina [7].The conventional view is that partonic multiple scattering in the nuclear medium is the primary cause of p T broadening, although one can argue that it could also be elastic scattering of the hadron at highest values of z h .In either case, whatever is moving through the medium is so disruptive that it causes an order of magnitude more broadening.One may speculate that the propagating object that becomes a lambda has a more extended color field than the propagating object in the case of pion production.Perhaps it is a color dipole that is propagating?Whatever it is, the manifestation it produces in this observable is a very strong signal.
A third, and less direct, piece of evidence of DDS is shown in Fig. 2 in that the multiplicity ratio is mildly suppressed for z h ≥ 0.5 but enormously enhanced for z h ≤ 0.5.For the higher z h the amount of suppression is similar to that seen in charged pion production Morán et al. [8].This suggests that an object with QCD color is propagating through the nucleus in that kinematic region for both pions and lambdas.However, in the lower z h region, the enhancement in the multiplicity ratio for the lambda, of nearly 8, is much larger than the maximum value seen for pions, which is 1.3.If there were single-quark scattering that left behind a remnant interacting strongly with the medium, it would happen in pion production as well.
In this same figure, another piece of information can be found.If the mechanism involved a high-z h colored system moving through the medium, losing energy via gluon emission, then whatever part of the cross section at high z h would push lambdas to lower z h .To put it differently, the integral of the number of missing lambdas for z h ≥0.5 would be an upper bound for the number of excess lambdas for ≥ 0.5.This statement can only be proven by looking at the cross section, not the ratio in this figure, and we have looked at it and there are far more excess lambdas at lower z h than higher z.What can this mean?We believe it means there are at least two production mechanisms for the lambda, in distinction to the case for pion production, which is fully consistent with the solitary mechanism of single-quark scattering Brooks and Lopéz [3].A plausible supposition is that for the lower z h region, single-quark scattering dominates, and many more lambdas are produced in this region than for higher z h , because the probability of single-quark scattering is much higher than that of DDS.
A few more comments can be made to illustrate that single-quark scattering is not dominating the high-z h production of protons at HERMES, nor for the high-z h production of lambdas in CLAS.First, as can be seen in both lambda figures, the statistical uncertainties are still quite small at z h , reflecting a high rate of lambda production at high z h .In single-quark scattering, the quark is violently torn away from the residual system.In that case it would require a very exotic mechanism to impart the large momentum to the lambda needed to push it to highest z h .By contrast, in DDS it is very natural to transfer the majority of the available momentum to the diquark, so the region at high z h will naturally be well-populated.
Another argument for why single-quark scattering is not dominating at high z h in HERMES proton production can be seen in the HERMES figure, for the plots that show the multiplicity ratio behavior with p 2 T and with ν.The p 2 T behavior at high p 2 T as a function of z h is radically different for the proton compared to the mesons.For the mesons, low-z h is suppressed at high p 2  T .In a space-time model Brooks and Lopéz [3] this is obvious, because the color lifetime goes to zero for high z h and so the hadron forms immediately and interacts with the nuclear medium hadronically.In distinction, for the proton at high p 2  T , the multiplicity ratio is independent of z h .This does not directly imply diquarks, but it suggests that single quark scattering is not exclusively happening at high z h on the proton, otherwise the z h behavior at high p 2  T would be the same as for the mesons.
Another piece of evidence that single-quark scattering is not occurring for the HERMES protons at high z h can be seen by inspecting the multiplicity ratio behavior with ν.For pion production with single-quark scattering, ν is interpreted as the energy transferred to the quark, and therefore it is the initial energy of the quark immediately following the scattering.It therefore is a measure of the relativistic boost of the quark, which for larger values of ν the color lifetime will be time dilated.At infinite energy, the quark will pass through the system without interacting at all (QCD factorization theorem) and the multiplicity ratio will go exactly to unity.In this picture, the multiplicity ratio for single-quark scattering can never be greater than unity.In the HERMES figure for the multiplicity ratios vs. ν it can be seen that for the mesons, the ratios are bounded from above by unity, but for the protons the ratio rises to 1.3.Without a detailed model one cannot say much more than that something is extremely different for the protons than for the pions.

Future
In the near term, progress in the experimental area will consist of finishing the analysis of the CLAS SIDIS data for production of protons.It is very clear from the preliminary data for the proton that the multiplicity ratio looks essentially the same in every respect as that for the lambda baryon, which would be the case if the scattering takes place on the spin zero "ud" diquark, because that diquark is thought to be dominant Yin et al. [12] in the target protons and neutrons, and in the produced lambda and produced protons.This interpretation can be tested by analyzing other baryons, such as sigma and cascade baryons, for which the dominant diquark according to models is spin zero "us", so producing those requires a different and more complicated mechanism, and thus their production rate, especially at high z h , will probably be suppressed compared to that of the proton and the lambda.Such studies will be feasible within the decade, because there is an approved and scheduled experiment The CLAS Hadronization Collaboration [10] to measure production of nine baryons (and ten mesons) from nuclear targets from deuterium to carbon, aluminum, copper, tin, and lead.
It is clear that improved modeling is needed to begin to understand more details of the hadronization mechanisms.For example, if one were to be able to simultaneously reproduce the HERMES proton data and the CLAS proton and lambda data (as well as future CLAS12 data), it would give great confidence that we understand the main ingredients of the hadronization mechanisms.
In the longer term, if there is a substantial increase in the CEBAF beam energy, these studies can be extended to heavier baryons and mesons, getting into the charm sector.They can also be performed at the future Electron-Ion Collider, with much greater kinematic reach and higher mass hadrons.

Conclusions
Baryon SIDIS data on nuclei from HERMES and CLAS behave qualitatively differently from mesons, in multiplicity ratios and in transverse momentum broadening.Our hypothesis is that Direct Diquark Scattering may be one mechanism for formation of protons and lambdas, for z h ≥0.5.Protons, neutrons and lambdas should behave the same if this is actually a valid mechanism, based on advanced models of diquarks, which predict that the spin zero [ud] diquark dominates in their structure.Two mechanisms are needed to explain the observed lambda hyperons produced nuclei.More theoretical work is needed to determine feasibility and plausibility of this interpretation, and distinguish it from any other possible mechanisms.The planned and approved CLAS12 Color Propagation program is ideal for testing these ideas: access to production of nine long-lived baryons.

Fig. 1
Fig. 1 Figure taken from Barabanov et al. [2].Data from the HERMES Collaboration showing two-dimensional multiplicity ratios for positively charged hadrons as a function of z h , p 2 T , and ν, for three bins in a second variable Airapetian et al. [1].Column 1a and 1b: R h M (z h ; p 2 T ).Column 2: R h M ( p 2 T ; z h ).Column 3: R h M (ν; z h ).Each of the top panels corresponds to positive or negative pions, the middle panels correspond to positive or negative kaons, and the bottom panels correspond to protons or antiprotons, as labelled.The multiplicity ratios shown are for the xenon target data compared to the deuteron target data, which show the most pronounced nuclear effects of the various targets discussed in Ref. Airapetia et al.[1]

Fig. 2
Fig. 2 Figure taken from Chetry et al.[5] z-binned multiplicity ratios for carbon, iron, and lead (the results are horizontally shifted for clarity).The outer error bars are the p2p systematic uncertainties added in quadrature with the statistical uncertainties.The inset contains the total normalization uncertainties for each nucleus.The plots illustrate the results of the low (left) and high (right) z ranges corresponding, respectively, to the target and current fragmentation regions.The curves correspond to GiBUU model calculations Buss et al.[4]

Fig. 3
Fig. 3 Figure taken Chetry et al. [5] Left (right): The z (nuclear radius)-dependent p 2 T results for the three nuclei (results are horizontally shifted for clarity).The outer error bars are the p2p systematic uncertainties added in quadrature with the statistical uncertainties, while the normalization uncertainties are presented in the inset for the z-dependence and found to be less than 1% for the A-dependence.The GiBUU model calculations are represented by the colored (left) and shaded (right) bands obtained by interpolating the model points and their statistical uncertainties