A Conservative Approach

A wide variety of thermodynamic and/or statistical techniques have been applied to describe particle spectra and yields in relativistic heavy ion collisions. Not having done  a detailed, ‘statistical’ analysis of the hundreds of such papers, it is still safe to say the vast majority of such calculations do not start from the microcanonical ensemble, that is, they do not incorporate the effects of global energy-momentum conservation.  In Conservation Laws and the Multiplicity Evolution of Spectra at the Relativistic Heavy Ion Collider Chajecki and Lisa study the role of Energy and Momentum Conservation-Induced Constraints (EMCIC’s) on single particle spectra at RHIC.  (In a nice piece of acronym-overloading, this work builds on their previous studies of Energy and Momentum Conservation-Induced Correlations in femtoscopic measurements.) Their studies suggest that the effects of EMCIC’s can lead to surprisingly large shifts in the momentum distributions between low and high multiplicity states. 

The key plot in this well-written paper is Figure 3, which shows the ratio of the yields in  p+p collisions to central Au+Au collisions as a function of pT, compared to curves calculated on the basis of energy-momentum conservation alone. Even light particles such as pions show 50% effects in the low pT region 0.2 to 0.7 GeV/c; the effect is much larger (a factor of ~5) for protons in the same transverse momentum regime.   The structure is consistent with one’s naive expectations- the presence of a larger ‘reservoir’ in the Au+Au case makes it easier to access high transverse momentum than in p+p collisions. 

There is additional information in this figure that the authors may wish to extract. This low momentum regime is where one expects participant scaling to hold, and this seems to be (roughly) the case for the lowest momentum pions, but is badly violated for the protons. Protons with  pT < 0.7 GeV/c are suppressed in Au+Au collisions relative to participant scaling; protons with momenta greater than this value are enhanced. The conventional explanation for this is radial flow, which has a larger effect on higher mass particles, and which will lead to a depletion of the low  pT region. This paper suggests that identical trends can result from EMCIC’s, leading the authors to state “Extracting physics messages from the changing spectra, while ignoring kinematic effects of the same order as the observed changes themselves, seems unjustified.”


In reading this paper, I was reminded of a very clever analysis by  T.T. Chou, C.N. Yang and E. Yen: Single Particle Momentum Distribution At High-Energies And Concept Of Partition Temperature . These authors noted that energy-momentum conserving delta-function in the microcanonical ensemble ‘inevitably’ leads to an exponential distribution of single-particle energies, with an exponential slope they labeled the ‘partition temperature’ (and took pains to note was not necessarily a real temperature). They applied this idea to the analysis of the rapidity distributions measured by UA5 in proton-antiproton collisions at center-of-mass energy 540 GeV. After introducing another, independent, ansatz, i.e., that confinement leads to  an exponential distribution in transverse momentum, they obtained a striking good description of how “phase space” considerations (aka EMCIC’s) described the systematic variation of the pseudo-rapidity distributions with collision multiplicity.     

2008 Nobel Prize in Physics

The 2008 Nobel Prize in Physics was awarded, in half, to Yoichiro Nambu for his groundbreaking work on the mechanism responsible for the spontaneous breaking of symmetries in elementary particle physics. Nambu’s work, published in 1961 in two seminal articles co-authored by Giovanni Jona-Lasinio:

showed that a strong interaction among (nearly) massless fermions will lead the formation of a condensate of fermion-antifermion pairs in the vacuum, which breaks the symmetry associated with massless fermions, i.e. chiral symmetry, endowes the fermions with a dynamical mass, and leads to the emergence of massless bosons called Goldstone bosons. When applied to the isospin doublet of up- and down-quarks, the NJL mechanism explains the relative lightness and other properties of the pions, π+, π0, and π, as a consequence of their Goldstone boson character.

Nambu’s mechanism today is understood to be realized in the fundamental interaction of the strong nuclear interaction, quantum chromodynamics (QCD). Because of the analytical intractability of QCD, the NJL model is still used extensively an an effective model of chiral symmetry breaking in QCD. The central goal of relativistic heavy ion collisions is to heat the QCD vacuum to such a high temperature that the interaction among quarks is weakened and the broken symmetry is restored.

It has been recognized in recent years that chiral symmetry breaking is intimately linked to the other fundamental property of the QCD vacuum, quark confinement. Naïvely, the restoration of chiral symmetry and quark-deconfinement could occur at different temperatures. Numerical simulations of lattice QCD have shown that they occur together. An extension of the NJL model to include symmetry aspects of the color force, the PNJL model

provides an explanation of the mechanism responsible for the linkage of the two phenomena and thus for the existence of a single critical temperature threshold for the formation of a quark-gluon plasma exhibiting the full symmetries of QCD.