Parton Recombination Primer

The development of the parton recombination model was driven by a series of observations, made by the RHIC experiments, which lacked a consistent explanation in the prevalent physics picture at that time. This primer will provide a brief overview of the recombination approach and guide the reader to references needed for a more extensive study.

The RHIC observations which triggered the development of the recombination approach were:

  • the two component (thermal + power-law) shape of the hadronic transverse momentum spectra;
  • the strikingly different behavior of the nuclear suppression factors for baryons and mesons at intermediate transverse momentum;
  • the unusually large baryon-to-meson ratios at intermediate/high transverse momenta;
  • the different transverse momentum dependence of the elliptic flow for baryons and mesons.

All of these phenomena could be explained by postulating that the formation of hadrons at intermediate transverse momenta (approx. 2 – 5 GeV/c) occurs by recombination of valence (constituent) quarks. Mathematically, the recombination probability depends on the overlap of a quark-antiquark or three quark state (which is usually assumed to be a product state, neglecting interactions among the quarks) with the meson or baryon state, respectively.

If one assumes a parton momentum distribution which is thermal for low and intermediate transverse momenta and which exhibits a pQCD power-law shape at large transverse momenta, the recombination mechanism will push thermal features of the parton distribution out to higher transverse momenta for baryons than for mesons (due to the parton momenta being additive). Such a spectral shape for partons would certainly explain the likewise shape observed for hadrons at RHIC. The shift in the onset of pQCD features with transverse momentum for baryons compared to mesons can directly be used to explain the difference in the nuclear suppression factors and the large baryon to meson ratios at intermediate transverse momentum.

These key features of the recombination (or rather the recombination plus fragmentation) approach were first realized by the groups at Duke, Texas A&M and Oregon and published in the following papers:

The fourth observation, namely the different transverse momentum dependence of baryon and meson elliptic flow, was proposed to be a consequence of the recombination picture somewhat earlier by Sergey Voloshin, who mentioned an approximate version of a recombination scaling law in his talk at QM2002. The recombination scaling law for elliptic flow is currently one of the most compelling direct observations of partonic degrees of freedom at RHIC – it has been discussed in detail in a previous post.

One should note, however, that the idea of parton recombination as hadronization mechanism is not new to RHIC or even heavy-ion physics: the concept has been used as far back as 1977 to describe the leading particle effect in proton-proton collisions by K.P. Das and R.C. Hwa. In heavy-ion physics, the first incarnation of the recombination approach was the ALCOR model, which was developed to describe the bulk hadro-chemistry of the heavy-ion reaction, but suffered from pertinent criticism regarding the entropy balance during the hadronization process.

With the increasing amount of RHIC data available, the recombination plus fragmentation model was put to test (or challenged) in a number of ways, leading to a substantial body of new work within its framework in order to address the new data:

  • Jet-like two particle correlations:
    One of the biggest challenge for the recombination models has been the measurement of dynamic two-particle correlations. The picture of quarks recombining from a collectively flowing, deconfined thermal quark plasma appears to be at odds with the observation of jet-like correlations of hadrons observed in the same transverse momentum range of 2 to 5 GeV/c. Obviously, the observation is incompatible with any model which assumes that no correlations exist among the quarks before recombination. Such correlations require deviations from a global thermal equilibrium in the quark phase. However, it was shown by both by the Duke group (for arbitrary correlations among partons) as well as by the Oregon group (in the context of the recombination of shower partons with thermal partons) that if jet-like correlations exist in the deconfined phase, they survive hadronization via recombination and manifest themselves in jet-like correlations among hadrons.
  • Entropy balance at hadronization:
    to zero’th order, people commonly equate entropy conservation with particle number conservation, which would be problematic for the recombination approach since 2 (3) partons are required to form one meson (baryon). The original formulation of parton recombination sidestepped this issue by restricting its range of applicability to the intermediate transverse momentum range, leaving the bulk as heat-bath which could be used to satisfy the 2nd law of thermodynamics. However, the compelling success of the recombination model at intermediate transverse momenta has lead to ongoing speculation whether the model could also be used to describe bulk hadronization. A more sophisticated analysis of the entropy balance between the deconfined and confined phase by the Duke group as well as an analysis on the effect of resonances on the entropy balance by the TAMU group show that the 2nd law of thermodynamics need not be violated in bulk recombination, thus opening the door to extending the formalism into the low momentum domain.
  • Reco scaling beyond the simple valence quark picture:
    the inclusion of higher Fock states that add sea quarks and gluons to the hadron structure may significantly complicate the simple valence quark recombination picture and its original counting-rules. Fortunately it was shown that, when recombination occurs from a thermal medium, hadron spectra remain unaffected by the inclusion of higher Fock states. However, the quark number scaling for elliptic flow is affected, with reco scaling violated on the order of 5-10%.
  • Charged particle fluctuations:
    Fluctuations of the net electric charge of all particles emitted into a specified rapidity window have been proposed as a possible signal for the formation of deconfined quark matter in relativistic heavy ion collisions (see e.g. here and here). The most widely used measure for the entropy normalized net-charge fluctuations is the D measure, which assumes a value of approx. 1 for a free gas of quarks and gluons and approx. 4 for a free pion gas. Measurements at SPS and RHIC have generally yielded values of D somewhat smaller than 4, but much larger than the value predicted for a QGP, thus posing somewhat of an inconsistency will all the other observations at RHIC which are in line with the deconfinement hypothesis. However, applying the recombination formalism to charged particle fluctuations shows that (within systematic errors) parton recombination yields values for D compatible with the measurements. Thus fluctuations are affected by hadronization (at least in the recombination approach) such that the D measure does not survive as meaningful deconfinement indicator.
  • Resonance decays and reco scaling:
    the concept of recombination is not unique to the formation of hadrons out of valence quarks. In principle every pion-nucleon scattering into a Delta resonance is recombination process on the hadronic level. Therefore one has to wonder how hadron resonance formation affects the reco scaling law predicted by the parton recombination model. Two different investigations have been carried out in that domain. The TAMU group found that resonance decays can explain the observed deviation of the pions from the general reco scaling law, wheras the Duke group focused on constraining the amount of hadronic rescattering via the degree of scaling law violation observed in the data.

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