The conversion algorithm

 The algorithm for reconstructing conversions is relatively simple. It is assumed that all the primary photons have zero impact parameter, thus ignoring the beam spread of 15 $\mu$ m in the transverse plane. No interesting areas with high energy photons from secondary vertices have yet been found. A conversion is, with this constraint, fully defined with the parameter set

 

$$\varphi_{c}^{}$$ (90)

where the first three parameters define the conversion point in polar coordinates, T is the slope cot $\theta$ of the photon and C_i the curvature of the two electrons signed with the charge. The parameters, P _i^C, of the electrons making up the conversion are uniquely given as
  $$\begin{align} d_i &= \frac{\sqrt{1+r_{c}^2 C_i^2}-1}{C_i} \\ \varphi_i &= \var... ...eld and} a_i &= \frac{C_i}{2(1+C_i d_i)} \\ b_i &= d_i (1-a_i d_i).\end{align}$$

The reconstructed conversion is obtained as a $\chi^{2}_{}$ fit to the track parameters, P _i^PR, from the pattern recognition. The $\chi^{2}_{}$ is defined as

 

$$\chi^{2}_{} \sum_{i=1}^{2} \top$$ (91)

which is minimised by varying the conversion parameters C as defined in (6.18). W _i is the inverse error matrix of the track with parameters P _i^PR. It is assumed that the individual track fits from the pattern recognition are uncorrelated. The minimisation of the 10 parameters with 6 degrees of freedom is performed with the MINUIT package [65].

Since a MINUIT fit is quite time consuming a preselection is made on the basis of the distance between the 2 tracks in the transverse plane. The distance is calculated at the radius of the hit closest to the primary vertex on the two tracks. The time consumed for conversion finding is at high luminosity in any case much lower than the time spent in the pattern recognition.

With the detector design as presented in the Inner Detector TDR [5] 10.6% of all photons with 50 GeV in transverse momentum convert below a radius of 40 cm. This fraction varies only slowly with the photon momentum. For H $\rightarrow$ $\gamma$$\gamma$ events this leads to at least one conversion in 20% of the events. Since the conversion electrons curve in the magnetic field, the cluster width in the r$\varphi$ direction is larger for converted photons leading to a worse energy resolution. To some extent this loss can be regained by an improved resolution in the position of the primary vertex from converted photons (section 7.3).

The identification of conversions has been tested mainly on a sample of photons and $\pi^{0}_{}$ 's simulated with transverse momentum of 50 GeV over all pseudorapidities. This is close to the average p_T of photons expected from triggered H $\rightarrow$ $\gamma$$\gamma$ decays.

The efficiency and fake rates for conversions are normalised to conversions with r_c < 80 cm and |z_c| < 280 cm. Outside this region, the efficiency for finding conversions decreases quickly to zero as the amount of the Inner Detector crossed decreases.

The efficiency is almost flat across the Inner Detector volume, with an exception of conversions taking place close to the transition region between the barrel and end-cap TRT. The track search is performed down to a transverse momentum of 0.5 GeV, below which tracks begin to loop; tracks are found with high efficiency down to p_T $\sim$ 1 GeV. For photons with 50 GeV p_T this leads to a loss of 2% in efficiency. The distributions of the efficiency for recovering converted photons are shown in fig. 6.10 and fig. 6.11. The fall at large radii is mainly caused by conversions lost in the transition region between the barrel and end-cap TRT.

  

Figure: The efficiency for reconstructing converted photons with 50 GeV p_T as a function of the conversion radius.

  

Figure: The efficiency for reconstructing converted photons with 50 GeV p_T as a function of the pseudorapidity of the photon.

The efficiency as a function of the transverse momentum of the lowest energy conversion electron p_{T min} is independent of this for p_{T min} greater than 1 GeV. Early conversions have a reconstruction efficiency integrated over $\eta$ of 85%. The efficiency to reconstruct the high- p_T electron in this study is 95%.

How well the photon is reconstructed depends strongly on the radius of conversion r_c . Various reconstructed parameters for converted photons with 50 GeV transverse momentum (integrated over all pseudorapidities) are summarised in table 6.1. The resolution in transverse momentum is shown both before the conversion fit and after, where the photon p_T before the fit is simply defined as the summed p_T of the two reconstructed electrons. It can be seen how the $\chi^{2}_{}$ fit improves the p_T resolution by approximately 20%. The tails in the p_T resolution are measured as the fraction of the reconstructed photons with the p_T in the conversion fit above 2$\sigma$ from the true value.

 

$$\sigma \sigma \sigma \sigma \varphi_{0}^{} \sigma$$ r_c Tails

$$\pm \sigma$$ (cm) before fit after fit (cm) (mrad) (cm)

0-20 0.051 0.040 0.46 0.88 0.17 0.03

20-40 0.17 0.14 0.20 1.03 0.10 0.54

40-60 0.31 0.23 0.15 4.09 0.96 -

60-80 0.28 0.23 0.09 4.29 1.16 -

 

Reconstructed momentum distributions after the conversion fit are in fig. 6.12 shown separately for r_c below 40 cm and r_c above 40 cm which more or less corresponds to the tracks reconstructed with XKALMAN (section 6.2) or the pattern recognition working in the TRT separately (section 6.3.1). It can be seen how the inclusion of hits from the silicon tracker and the longer track in the magnetic field greatly improves the p_T resolution. The long tails towards low reconstructed p_T is caused by the electrons emitting bremsstrahlung in the material of the Inner Detector. For isolated electrons it is possible, to some extent, to correct for this using the position of the electromagnetic cluster but this will not be possible for the electrons from converted photons.

  

Figure 6.12: The reconstructed transverse momentum of the reconstructed conversions relative to the transverse momentum of the original photons with a transverse momentum of 50 GeV.

  

Figure 6.13: The distribution in transverse momentum of the fake conversions found at high luminosity.

Total efficiencies for finding reconstructed conversions are meaningful only when a comparison is made with the number of fake conversions identified in the presence of pile-up. Fake conversions arise mainly from pairs of uncorrelated charged pions, and it is desirable that this rate should be well below the rate of real conversions from $\pi^{0}_{}$ decays in the pile-up.

The strongest handle to reject fake conversions is the number of transition radiation hits on the tracks. This is especially true for tracks crossing the end-cap TRT, where the transition radiation yield is higher than in the barrel TRT. The rate of fake conversions is defined as the rate of conversion candidates in pile-up events at full luminosity where true conversions in the pile-up have been subtracted. Rates are normalised to a road size of ($\eta$,$\varphi$) = (0.20,0.20) and are shown in table 6.2. No cut was applied on the transverse momentum of the reconstructed conversions except the implicit cut of 0.5 GeV p_T on the individual tracks in the pattern recognition. The transverse momentum of the fake conversions are peaked towards low momentum as shown in fig. 6.13.

 

$$\eta$$ Efficiency Rate in pile-up Fake rate

$$\cdot \cdot$$ 0.0-0.6 0.87

$$\cdot \cdot$$ 0.6-1.2 0.70

$$\cdot \cdot$$ 1.2-1.8 0.85

$$\cdot \cdot$$ 1.8-2.4 0.85

 

In the region 1.2 < |$\eta$| < 1.8 , where the TRT has its best performance, the conversion fake rate is at a satisfactory level far below the conversion rate in the pile-up. Opposed to this the situation in the other pseudorapidity regions is not at all as good. The effect of this relatively high fake rate on the physics performance has not yet been evaluated. As most fake conversions have low transverse momentum the most critical area will be b-tagging where the identification of low energy converted photons is required.