Capitulo II. La educación básica en México: Programa Nacional de Inglés,
2.9 Programa Nacional de Inglés en Educación Básica (PNIEB) y
The efficiency of the conversion finder is dependent on th e m om entum sym m etry of th e electron and positron tracks produced. For sym m etric tracks th e efficiency
Extraction of Semi-leptonic Decays to Electrons C hapter 7
is high, b u t for asym m etric tracks, one of th e tracks may lie outside the reconstruction region of the CTD . Therefore only one track is reconstructed and is indistinguishable from our quark decays and so an estim ation of th is efficiency is essential. To estim ate this we look a t th e d istribution of the energy fraction of the electron pair-produced from a photon by considering the calculation of Tsai [102]. T his sta rts from the exact calculation for pair-production, accounting for screening and applying it to heavy atom s.
For a photon w ith m om entum k producing an electron w ith m om entum p = X ’ k
(and positron w ith m om entum p' = k — p), the cross-section for atom ic num ber, Z > 5 is given by:
da art
(1 — x) {<1)1 — ( 1 ) 2 ) Z{ipi — 1P2) 1 5 (7.10)
2
where a is the fine stru ctu re constant, Tq th e classical electron radius and;
01 {7 ) = 20.863 - 21og [1 + (0.558467)^] - 4 [l - - 0.4e“ '-^ ] ,
2 —1
4 > 2 (7) = (7) “ 3 (1 + 6.57 -f 6 7^) ,
(e) = 28.340 - 21og [l 4- (3.621e)^] - 4 [l - O.Te"^" - 0.3e-^^'^'] ,
”02 (^) = "01 (^) — - ( l + 40s -j- 4006^j , oo 1 3 f ( z ) = z ^ ~
1
- 1.0369z^ + 1.008 n^ T t ( T ^2 + z) (1 + z ) ' . lOOmgA: lOOrUek / Z ^ Ë I Ë Ï F ^ ’ ^ ^ E Æ '. Z V ^ ’ ^ i m jwhere is th e energy of the electron and E'^ th e energy of th e positron in the lab o rato ry system.
The cross-section, d a /d x , is shown in Figure 7.19 as a function of E ^ - / E ^ for % = 13 (alum inium ) for different photon energies.
Chapter 7 7.5 Background from Conversion Electrons •1.75 1.25 0.75 0.5 0.25
Figure 7.19; Differential cross-section as a function of E e -/E ^ for aluminium using the equations from Tsai with photon energies, E^ of oo (top curve), 5, 2.5, 1.2, 0.5, 0.1 (bottom curve) GeV.
Using this theoretical prediction, the hope is then to determine the efficiency of the conversion finder for asymm etrically produced pairs. To do this it has to be dem onstrated th a t the ZEUS conversion d a ta is in agreement with the theoretical prediction. In this way a model independent efficiency can be determ ined by not relying on Monte Carlo expectations.
W ithin the range under study, —1.1 < ri{e~) < 1.1, the CTD reconstructs tracks w ith a momentum independent efficiency above 200 M eV/c. Below this momentum value, the efficiency is greatly reduced and so using the Monte Carlo introduces model-dependence. To estim ate the number missed we therefore apply a cut to the positive track from the pair to be above this m omentum value (the negative track is already required to be > 1 .6 GeV/c) and compare to the situation when no momentum cut is applied.
To dem onstrate the consistency of the ZEUS d ata and the theory, we take a sample of clean conversions, as defined previously, and a conversion energy,
E . y > I GeV with the additional criteria th a t the momentum of each of the two tracks is greater than 200 M eV/c. We then have a sample of conversions with different photon energies and we compare this to the theory by taking the photon distribution for the d ata and using it as the input for T sai’s equations.
Extraction of Semi-leptonic Decays to Electrons C hapter 7
The results can be seen in Figure 7.20 where the points are the d a ta and the solid line th e theoretical prediction.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
X = Ee/E,
Figure 7.20:
Comparison of the number of events as found in the data (points) with the prediction from theory (solid line) for clean conversions with energy, Ery > 1 GeV and the momentum of both tracks greater than 200 M eV/c. The prediction is also shown for no cut on the momentum of the tracks (dashed line).Also shown is th e theoretical prediction if no track m om entum cuts were imposed (dashed line). The d a ta and theory describe each other well in b o th shape and norm alisation. The dip a t central values w ith a slow rise and then a sharp drop (due to th e track m om entum cuts) are all well-produced. W ith confidence in the com patibility of the d a ta and th e theory, we can assign an efficiency, which is the ratio of th e two curves.
The ratio of these two curves is an overall efficiency which is not directly applicable to th e m easurem ent being made. Due to th e cut on our candidate tracks of
p^^ > 1 . 6 G eV /c, the efficiency is dependent on the energy of th e converting photon. T he ratio of the two curves was then found as a function of photon energy (Figure 7.21) for different m inim um track cut-offs. As can be seen for th e curve w ith th e cut-off a t 200 M eV /c (thick line), the minimum photon energy is (1.6 -F 0.2 = ) 1.8 GeV, w ith the efficiency rising rapidly before reaching a reasonably constant value around 3 GeV. Therefore, for a given photon energy, a weight taken from th is curve can be assigned to ascertain the num ber of conversions corrected for asym m etric pairs which were not tagged.
C hapter 7 7.5 Background from Conversion Electrons
0.8
0.6
0.4
0.2
Figure
7.21: The fraction of findable coversions (efficiency) as a function of photon energy for different track momentum cuts of 100 (top curve), 200 and 300 MeV/c (bottom curve).The procedure yielded an efficiency of 80%, thereby increasing th e estim ated num ber of conversions from 599 ± 26 to 748 ± 33.