Non-identical particle correlation functions - fitting

There is no analytic form of the correlation function for non-identical particles. The theoretical correlation function is defined as:

C = \int S(r,k) \Psi(r,k) d^4 r

Where S(x,p) is the emission function or the probability to emit a pair of particles with relative momentum k at an initial separation r. Psi is the pair wave function, which describes how the two particles interact with each other. For non-identical particle pairs of interest (pion-kaon, pion-proton, kaon-proton) Psi contains parts arising from Coulomb and strong interactions. Its mathematical form is quite complex, so although they are perfectly calculable numerically, the integral above cannot be carried out analytically.

General fitting procedure

We are therefore left with carrying out the fitting procedure numerically. It is as follows:

(1) We assume some functional form of S(x,p). These form should be parametric, preferably with few parameters P that are directly related to our physics quantities of interest.

(2) We postulate some set of values for parameters P and calculate the theoretical function CP from the integration above

(3) We compare the theoretical function CP with the measured one CM, e.g. by the chi2 test

(4) We repeat the points (2) and (3) for many sets of parameters P, the one that fits our measured correlation function best is taken as the best fit.

The procedure is more involved numerically that the standard HBT Bowler-Sinyukov method, but is also more general as it allows much more freedom in choosing S(r,k).

Fitting prcedure for STAR

We follow the procedure above, making the following selections.

(1) Following almost all other femtoscopic analysis we assume the source is a 3D sphere with gaussian density profile. The source is defined in the Longitudinally Co-Moving System (LCMS), again just like for pion HBT. The source has 3 gaussian radii in the out, side and long direction. However the quality of non-id data does not allow for a fit with three free parameters. Therefore ratios of Rside/Rout and Rlong/Rout are fixed to the values obtained from pion HBT and only Rout remains as a free parameter of the fit. There is also one more free parameter - the emission is allowed to be asymmetric - that is the mean emission point difference between pions and kaons is allowed to be non-zero. This shift is another free parameter of the fit. So in summary the assumed source function is:

S(r,k) = exp( - (rout-Mout)^2/Rout^2 - rside^2/(Rout ss)^2 - rlong^2/(Rlong sl)^2)

where Rout and Mout are free parameters (source size and emission asymmetry respectively), while ss and sl are fixed multipliers, in our case fixed to 1.0 and 1.3. Please note that Rout should be compared to the pion HBT radii, multiplied by a factor of sqrt(2.0), since it described the with of "two-particle separation distribution", while the HBT radius is a width of "single particle distribution"

Please note that the source function above is static - it does not depend at all on the momentum variables. However pair momentum needs to be provided if the relative wave function is to be calculated. We have used the momenta of true pairs, taken directly from the experiment in our calculations.

(2),(3),(4) We peform the calculation on a 2D mesh of parameters Rout vs Mout. Once the point is found where the chi2 is minimum, it is checked whether the point is at the edge of the mesh. If yes, the mesh is extended in this direction and the procedure is repeated. Once the minimum point is within the mesh, we repeat the procedure around that point, with smaller mesh size. We have performed all the fits up to a mesh size of 0.1 fm for the size Mout and 0.2 fm for the shift Rout, which is the expected minimum statistical error.

The procedure described in points (2), (3) and (4) is quite complex numericall and progrmatically. A separate software program, CorrFit has been created for this purpose. It is able to read STAR data in many forms (traditional double ratio, spherical harmonics representation etc.). It also enables to use many different functional forms of S(r,k), but the standard one mentioned above is used thruough this analysis. It is integrated with the package from Richard Lednicky, that is responsible for calucalting the pair wave function values. It is able to calculate pion-kaon, pion-proton and kaon-proton pair weights, therefore the same CorrFit program can be used for all pair types.

Example fitting procedure

Below I show an example fitting procedure input and results.