We used Pythia 8 to produce b-bbar events. First we used the default Pythia 8. Macro for running with default parameters is here. We then used the STAR Heavy Flavor tune v1.1 for Pythia 8.  The macro for running with the STAR HF tune is here.

The cross-sections reported by Pythia (numbers after 5M events) using the default parameters:

  *-------  PYTHIA Event and Cross Section Statistics  -------------------------------------------------------------*
 |                                                                                                                 |
 | Subprocess                                    Code |            Number of events       |      sigma +- delta    |
 |                                                    |       Tried   Selected   Accepted |     (estimated) (mb)   |
 |                                                    |                                   |                        |
 |-----------------------------------------------------------------------------------------------------------------|
 |                                                    |                                   |                        |
 | g g -> b bbar                                  123 |    19262606    4198826    4198275 |   6.971e-04  1.854e-07 |
 | q qbar -> b bbar                               124 |     3126270     801174     800981 |   1.331e-04  8.216e-08 |
 |                                                    |                                   |                        |
 | sum                                                |    22388876    5000000    4999256 |   8.303e-04  2.028e-07 |
 |                                                                                                                 |
 *-------  End PYTHIA Event and Cross Section Statistics ----------------------------------------------------------*

So gg initiated subprocess has a 0.697 ub cross section and the q-qbar initiated subprocess has a 0.133 ub cross section. The sum for both subprocesses pp -> b bbar is 0.830 ub. 

Using the STAR HF Tune, the cross section statistics reported by Pythia change to the following:

 *-------  PYTHIA Event and Cross Section Statistics  -------------------------------------------------------------*
 |                                                                                                                 |
 | Subprocess                                    Code |            Number of events       |      sigma +- delta    |
 |                                                    |       Tried   Selected   Accepted |     (estimated) (mb)   |
 |                                                    |                                   |                        |
 |-----------------------------------------------------------------------------------------------------------------|
 |                                                    |                                   |                        |
 | g g -> b bbar                                  123 |    31956918    4520459    4520459 |   9.247e-04  2.542e-07 |
 | q qbar -> b bbar                               124 |     2259563     479541     479541 |   9.817e-05  8.544e-08 |
 |                                                    |                                   |                        |
 | sum                                                |    34216481    5000000    5000000 |   1.023e-03  2.682e-07 |
 |                                                                                                                 |
 *-------  End PYTHIA Event and Cross Section Statistics ----------------------------------------------------------*
 

The cross section increases to 1.023 ub with the STAR HF Tune v1.1.  The main changes to the default parameters are the reduction of the bottom quark mass from 4.8 (default) to 4.3 GeV/c², the change of PDF from CTEQ5L (default) to the LHAPDF set MRSTMCal.LHgrid, and the choice of renormalization and factorization scales.

The selection of e+e- in the final state is done by following the fragmentation of the b or bbar quark into a B meson or baryon, and then looking at its decay products to find an electron or positron.  The pT distribution of the genrated b quarks is shown below.

Fig. 1: Generated b quarks.

The <pT> of the b quarks is 3.3 GeV.  These then fragment into B mesons and baryons.  As an example, we plot here the B0 and B0-bar pT distribution, below.

Fig. 2:Pt distribution of B0 and B0-bar mesons.

The <pT> of the B mesons is 3.055 GeV/c, one can estimate the peak of the Z distribution (most of the momentum of the b quark is carried by the meson, so it should be close to 1) as 3.055/3.331=0.92.

After the beauty hadrons are produced, they can decay producing electrons and positrons.  We search for the e+e- daughters of the beauty hadrons, their pT distribution is shown below.

Fig. 3: pT distribution of the e+ e- daughters of the b quarks.

When an event has both an electron and positron from the b-bbar pair this can generate a trigger.  However, these are generated in all of phase space, and we mainly have acceptance at mid-rapidity.  The full rapidity distribution of the e+e- pairs is shown below:

Fig. 4: Rapidity distribution of the e+e- pairs from b decay.

The distribution is well approximated by a Gaussian with mean ~ 0 and width close to 1 (off by 4.3%).

We calculate the invariant mass. This is shown below:

Fig 6. Invariant mass spectrum of e-e+ pairs originating from b-bbar pairs.

The red histogram is for all e+e- pairs generated by Pythia.  The blue histogram is for pairs with |y_pair|<0.5, which is the region of interest. The distributions are fit to a function to parameterize the shape, shown in the black lines.  This is inspired by using a QCD tree-level power-law distribution multiplied with a phase-space factor in the numerator. The fit parameters for the blue line are:

• b = 1.59 +/- 0.06
• c = 27.6 +/- 5.8
• m0 = 29.7 +/- 7.8

Using the STAR HF Tune, the parameters are:

• b = 1.45 +/- 0.05
• c = 64.2 +/- 26.1
• m0 = 49.7 +/- 18.0

With the default parameters, in mass region 8 < m < 11 GeV/c² and for |y|<0.5 the Pythia prediction is for a cross section of 29.5 pb.

With the STAR HF Tune, in the same phase space region the Pythia prediciton is for a cross section of 46.9 pb.

One can calculate from the Pythia cross section, the STAR efficiency*acceptance and the integrated luminosity the expected yield in the region 8< m < 11 GeV/c². This gives 12 expected counts for trigger 137603, assuming the trigger doesn't affect the invariant mass shape.

However, tince the trigger has a turn-on region, we need to take this into account.  The turn on can be obtained by looking at the background counts in the real data.  By modeling the background with an error function of the form (erf((m-m0)/sigma)+1)/2 and multiplying by an exponential, we obtain the parameters m0=8.07 +/- 0.74 GeV/c² and sigma = 1.75 +/- 0.45 GeV/c². The fit to obtain the error function is shown below (it is one of the figures in the paper):

Fig. 7 Unlike-sign and like-sign invariant mass distributions from data. The like-sign is fit with and exponential multiplied by an erf.

We then need to apply this function to parameterize the turn-on region of the trigger to the b-bbar e+e- invariant mass spectrum.  We have one additional piece of information from the efficiency estimation: the overall acceptance * trigger efficiency * tracking efficiency and including additional PID cuts for the Upsilon(12) is 5.4%, we can use this to normalize the function after including the trigger turn-on so that at M=10 GeV/c2 it gives 5.4% of the yield before applying the trigger turn-on.  This way we take care of the trigger turn-on shape and the overall normalization given by the acceptance, efficiency, etc. obtained from the upsilon embedding.  This assumes that an e+e- pair with invariant mass identical to the upsilon will have identical efficiency and acceptance.  Using this, we estimate the yield in the region 8<m<11 including the trigger turn-on and acceptance and efficiency to be 19 counts from b-bbar in the Upsilon mass region in the entire dataset.

For the STAR HF Tune, the cross section is larger and the expected counts are larger: