Extracting A_LL and DeltaG

Determining state of beam polarization for Monte Carlo events

While Pythia does a pretty good job of simulating prompt photon production in p+p collisions, it does not include polarization for the colliding protons nor partons. A statistical method for assigning polarization states for each event based on A_LL [1] is demonstrated in this section. For an average number of interactions for each unpolarized bunch crossing, N_eff, the occurence of an event with a particular polarization state obeys a Poisson distribution with average yield of events per bunch crossing:

For simplicity, the polarizations of the blue and yellow beams are assumed to be P

_B

=P

_Y

=0.7 and N

_eff

=0.01. The "+" spin state defines the case where both beams have the same helicities and the "-" spin state for the case of opposite helicities. The asymmetry A

_LL

is calculated from the initial states polarized and unpolarized parton distribution functions and parton-level asymmetry:

The algorithm then consists in alternatively drawing a random value N

_int

from the Poisson distributions with mean μ

₊

and μ

_-

until N

_int

>0 at which point an interaction has occured and the event is assigned the current spin state. The functioning of the algorithm is illustrated in Figure 1a where an input A

_LL

=0.2 was fixed and N

_trials

=500 different asymmetries were calculated. Each trial integrated N

_total

=300 events. It is then expected that the mean A

_LL

~0.2 and the statistical precision~0.1:

Indeed, both the A

_LL

and its error are reproduced. In addition, variations on the number of events per trial were investigated (N

_total

) in Figure 1b. The extracted width of the Gaussian distribution for A

_LL

is consistent with the prediction for the error (red curve).

• ROOT macro used to generate Figure 1a SimALL.C
• ROOT macro used to generate Figure 1b SimALL2.C

Figure 1a	Figure 1b

Event reconstruction

For this study, the gamma-jets Monte Carlo sample for all partonic p_T were used. As an example, the prompt photon processes for the partonic p_T bin 9-11 GeV and their total cross sections are listed in the table below. Each partonic p_T bin was divided into 15 files each of 2000 events.

 ==============================================================================
 I                                  I                            I            I
 I            Subprocess            I      Number of points      I    Sigma   I
 I                                  I                            I            I
 I----------------------------------I----------------------------I    (mb)    I
 I                                  I                            I            I
 I N:o Type                         I    Generated         Tried I            I
 I                                  I                            I            I
 ==============================================================================
 I                                  I                            I            I
 I   0 All included subprocesses    I         2000          9365 I  3.074E-06 I
 I  14 f + fbar -> g + gamma        I          331          1337 I  4.930E-07 I
 I  18 f + fbar -> gamma + gamma    I            2             8 I  1.941E-09 I
 I  29 f + g -> f + gamma           I         1667          8019 I  2.579E-06 I
 I 114 g + g -> gamma + gamma       I            0             1 I  1.191E-10 I
 I 115 g + g -> g + gamma           I            0             0 I  0.000E+00 I
 I                                  I                            I            I
 ==============================================================================

The cross sections for the different partonic p_T bins has been tabulated by Michael Betancourt and is reproduced here for convenience.

These events were processed through the 2006 pp200 analysis chain, albeit without any cuts on the SMD. The simulated quantities were taken from the Pythia record and the reconstructed ones from the analysis.

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Partonic kinematics reconstruction

PartonicKinematics.C

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15 (a)

Figure 15 (b): quark from proton 1 (blue beam, +z direction)

Figure 15 (c): quark from proton 1 (blue beam, +z direction) and q + g -> q + gamma subprocess (29)

Figure 15 (d): quark from proton 1 (blue beam, +z direction) and q + qbar -> gamma + g subprocess (14)

Predictions for A_LL and direct determination of DeltaG(x)

DeltaG.C

Figure 16 (a)

Figure 16 (b): quark from proton 1 (blue beam, +z direction)

Figure 16 (c): quark from proton 1 (blue beam, +z direction) and q + g -> q + gamma subprocess (29)

Figure 16 (d): quark from proton 1 (blue beam, +z direction) and q + qbar -> gamma + g subprocess (14)

References

1. Appendix Simulation Studies of Direct Photon Production at STAR
2. DeltaG(x,mu^2) from jet and prompt photon production at RHIC arXiv:hep-ph/0005320