J.G. Cramer

In CERN Experiment NA49 the 33 TeV Pb beam strikes a Pb target, producing thousands of pions and other particles that are tracked and momentum-analyzed in the four time-projection chambers of the experiment. In this environment the detected particle multiplicity of a single event is large enough to permit single-event physics. 'Single event physics' here means pre-selecting an event containing 1/100 to 1/1000 of the total events using a narrow range of values of a particular global variable (e.g., distribution temperature) and using this reduced ensemble in further physics analysis (e.g., strange particle ratios or HBT interferometry).

The distribution's 'temperature' (or inverse slope) is usually extracted by binning the particle transverse mass distribution function into a histogram, taking the logarithm of the vertical scale of this histogram, and then fitting a straight line to the resulting distribution. The inverse slope of the fitted line is the temperature. Here we present an algorithm which is a faster alternative to this procedure.

The transverse mass of a particle with rest mass m₀ and momentum components p_x and p_y in the directions transverse to the beam (z) direction is defined as: m_t=. In ultra-relativistic heavy ion experiments at CERN it has been found that the distribution of transverse masses of pions produced in a collision is well described by the probability density function (PDF):

(1)

where m_tm₀. We note the values of the following integrals over this PDF:

(2)

(3)

(4)

Therefore, R₁₂=I₁/I₂ = 2T² + 2T_m0 + m₀², and this quadratic equation can be solved for the distribution temperature, i.e:

(5)

This suggests that, given a set of tracked particles from a given event, the temperature can be extracted by computing sums over all tracks of the extracted values of m_t and 1/m_t, taking the ratio R₁₂ of these sums, and substituting this ratio into Eq. (5) above.

We have tested this procedure using a Monte Carlo simulation of NA49 events with particles having a Gaussian distribution of width of 3.2 in rapidity y, a uniform distribution in the azimuthal angle , and a transverse-mass PDF that is an order 2 gamma distribution as given by Eq. (1) above. We find that using this procedure the temperature of the PDF of a set of particles can be reliably extracted to a relative precision of about 1/ where N is the number of particles in the event.