Uncertainties in the underlying physics

 The main uncertainty in the primary vertex determination method described above is the model describing the Higgs production. The physics responsible for the Higgs vertex to be any different from the minimum bias vertices is the transverse energy flows in the initial state radiation and the effect of multiple interactions.

The two gluons in the gluon fusion process creating the Higgs will be a part of the initial state radiation. In PYTHIA as the two gluons are a part of a virtual parton cascade inside each proton. The hard scattering prevents the virtual cascade from being re-absorbed by the partons again and instead time-like partons are created from the remaining part of the virtual cascade. With a mass of the Higgs particle of 100 GeV the Higgs particle obtains in this process a transverse momentum of the order of 50 GeV which cancels the scalar sum of the remaining partons in the cascade. It should be noted that it is the scalar sum and hence a low transverse momentum of the Higgs does not imply that the event had no initial state radiation.

In the PYTHIA simulation the evolution of the initial stage radiation is controlled by a parameter $\kappa$ which defines the maximum virtuality of the space-like cascades in the initial state radiation as

 

$$\kappa^{2}_{}$$ (102)

where Q ² = m_r² for a resonance of mass m_r and Q ² = (m_t1² + m_t2²)/2 for 2 $\rightarrow$ 2 processes.

When two partons interact, there is a finite probability that another pair of partons will interact as well in the same proton-proton collision. This is called multiple interactions and should not be confused with pile-up which is multiple interactions in a bunch crossing but in different proton-proton collisions. The existence of multiple interactions have been controversial but has now been identified clearly in analysis from H1 at HERA in $\gamma$ p collisions [66] and in the analysis of $\gamma$/$\pi^{0}_{}$ + 3jet events from the CDF detector at the Tevatron [67]. In the CDF analysis is was shown that 14% of all inelastic proton-proton collisions have multiple interactions at $\sqrt{s}$ = 1.8 TeV.

To look into the uncertainties in the simulation of the Higgs vertex selection two different models for the multiple interactions has been tested [68]. In the first model all proton-proton collisions are treated in the same way and the number of parton interactions follows a Poisson distribution. In the more advanced model the protons are simulated with a parton distribution as a double Gaussian i.e. a narrow and dense core which can be understood as a resolved valence quark and a wider less dense distribution around it. The distribution in the number of multiple interactions now depends on how central the collision is; for a central collision the multiple interaction rate is higher than for a peripheral collision. An event with a low cross section compared to the inelastic cross section will be most likely in a central collision which gives a bias where seldom events have a larger probability for multiple interactions.

In [68] it was shown that jet events from UA1 at $\sqrt{s}$ = 630 GeV favours the model with a double Gaussian distribution. With the recent articles published from CDF and D0 on the confirmation of multiple interactions, results on the structure of these events could be expected quite soon. At the Tevatron it is also possible to look at the underlying event structure of W events but they are mainly produced in quark fusion events while the minimum bias events they should be compared to are dominantly from gluon fusion processes. For this reason the comparison will have problems with systematic errors.

Particle level simulations of minimum bias events and Higgs events were performed to evaluate the uncertainty in the identification of the Higgs vertex from the uncertainties in the theoretical description. The sensitivity to the level of initial state radiation was evaluated by varying the $\kappa$ parameter from (7.13) in the interval from half to the double of the default value. The value best describing the level of initial state radiation is strongly believed to be within those limits.

In another simulation the two models describing multiple interactions were compared. The first model has a uniform density of the protons while the second model as described above have hard partons inside a wider Gaussian distribution. The effect of changing between the two models is for minimum bias and H $\rightarrow$ $\gamma$$\gamma$ events shown in fig. 7.10 for the number of tracks with transverse momentum above 0.5 GeV and in fig. 7.11 for the maximum p_T from the vertex. As expected it is mostly the spectrum of low p_T particles that is affected, such that the track counting with the p_T limit of 0.5 GeV is strongly affected while the maximum p_T of a charged track from the vertex is almost unaffected by the choice of model.

  

Figure: The probability for a minimum bias event (a) or H $\rightarrow$ $\gamma$$\gamma$ event (b) to have above a given number of charged tracks from the primary vertex with p_T > 0.5 GeV as a function of the threshold. The distributions are shown both with the assumption of a homogeneous proton (white) and a proton with a hard core (shaded).

  

Figure: The probability for a minimum bias event (a) or H $\rightarrow$ $\gamma$$\gamma$ event (b) to have a charged track with p_T above a given threshold as a function of the same threshold. The distributions are shown both with the assumption of a homogeneous proton (white) and a proton with a hard core (shaded).

In the comparisons between the different models, the thresholds on the number of tracks and maximum p_T were varied in a systematic way to keep the probability for the minimum bias events to reach above both thresholds constant. In this way, it is expected that the fraction of minimum bias vertices misidentified as H $\rightarrow$ $\gamma$$\gamma$ vertices are kept constant in a full simulation, while the fraction of correctly selected H $\rightarrow$ $\gamma$$\gamma$ vertices varies.

Setting the maximum virtuality of the initial state radiation to half the default value results for a fixed probability to accept the minimum bias events, in a 25% reduction of the H $\rightarrow$ $\gamma$$\gamma$ efficiency while a doubling of the maximum virtuality causes a 10% increase in the H $\rightarrow$ $\gamma$$\gamma$ efficiency. As the probability for the minimum bias events to get accepted was kept constant it can be expected that the weight falling outside the central peak in fig. 7.7 will remain constant for the different models of the underlying event. For the H $\rightarrow$ $\gamma$$\gamma$ events used for the full detector simulation 30% of the weight entered into the correct primary vertex and based on the arguments above the theoretical uncertainty from initial state radiation is estimated to be ^{+ 3}_{- 8}% on this value.

The events entering into the full simulation of H $\rightarrow$ $\gamma$$\gamma$ events with pile-up were simulated with the uniform proton model for the H $\rightarrow$ $\gamma$$\gamma$ events and with the hard core model for the superimposed minimum bias events. As seen from fig. 7.10 and fig. 7.11 this results in a pessimistic estimate in the power of global tracking to improve the H $\rightarrow$ $\gamma$$\gamma$ significance. Using the uniform proton model for both the H $\rightarrow$ $\gamma$$\gamma$ and minimum bias events results in an insignificant change of the results from the full simulation. On the other hand the use of the hard core model changes the H $\rightarrow$ $\gamma$$\gamma$ distribution in the number of tracks from the vertex radically and the probability to select the correct H $\rightarrow$ $\gamma$$\gamma$ rise by 65% when the probability to select a wrong vertex is kept constant. For the full simulation this means that the total weight entering into the correct H $\rightarrow$ $\gamma$$\gamma$ vertex raise from 30% to 50%.

To summarise, the theoretical uncertainties in the kinematics of minimum bias and H $\rightarrow$ $\gamma$$\gamma$ events are large for the particles with the lowest transverse momentum. The theoretical uncertainty in the model for initial state radiation is of minor importance compared to the uncertainties in the description of multiple interactions. The full simulations used for the primary vertex selection was performed with a pessimistic set of parameters and can thus be taken essentially as a lower limit in the power of finding the correct primary vertex with global tracking. Simulations performed at particle level indicate that the weight entering into the correct H $\rightarrow$ $\gamma$$\gamma$ vertex will be in the interval from 23% to 55%, with 14% of the weight entering into wrong vertex positions as determined from the full detector simulation of H $\rightarrow$ $\gamma$$\gamma$ events with pile-up corresponding to high luminosity.