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.     

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