Wing Sheung Chan

Event selection and classification 77 left for the ` – τ had - vis – E miss T syste m † : ˆ p z ( ` ) , ˆ E ( ` ) , ˆ p x ( τ had - vis ) , ˆ p z ( τ had - vis ) , ˆ E ( τ had - vis ) and ˆ E miss T (the hats indicate that these are variables measured in the transformed frame of reference). They are direct inputs to the NNs. The high-level variables include the visible mass m vis ( `, τ ) , the collinear mass m coll ( `, τ ) , the track-based invariant mass m ( `, τ track ) , and a kinematic discriminant known as ∆ α [61] . The track-based invariant mass is similar to the visible mass, except that the momentum of the τ had - vis candidate is replaced by that of the leading- p T track associated to the τ had - vis candidate. This variable carries especially useful information for identifying Z → `` events with ` misidentified as τ had - vis . For Z → ee events with e → τ had - vis fakes, it is common for the electron to emit bremsstrahlung before entering or in the inner detector. In such cases, the momentum of the inner-detector track would be smaller than the original momentum of the electron before radiation. However, the momentum of the τ had - vis candidate, which is reconstructed with energy deposits in the calorimeters taken into account, would likely recover the original momentum of the electron as the calorimeters are able to capture the energy carried away by the bremsstrahlung. Therefore, for these events, one can expect m vis ( `, τ had - vis ) ≈ m Z but m ( `, τ track ) 6 ≈ m Z , where m Z is the rest mass of the Z boson. On the other hand, for Z → µµ events with µ → τ had - vis fakes, the situation could be different. Since muons do not usually deposit much energy in the calorimeters, they are usually misidentified as τ had - vis candidate only when the inner-detector track is wrongly matched with some random energy deposits in the calorimeters from other sources. For these events, one can instead expect m vis ( `, τ had - vis ) 6 ≈ m Z but m ( `, τ track ) ≈ m Z . In any case, the difference between m vis ( `, τ had - vis ) and m ( `, τ track ) is a piece of valuable information for the NN Z `` classifier. The kinematic discriminant ∆ α ( `, τ ) is defined by the equation ∆ α ( `, τ ) = m 2 Z − m 2 τ 2 p ( ` ) · p ( τ had - vis ) − p T ( ` ) p T ( τ had - vis ) , (4.4) where m Z and m τ are the nominal rest masses of the Z boson and τ lepton respectively, and p ( ` ) and p ( τ had - vis ) are the measured four-momenta of the ` and τ had - vis candidates respectively. In the event of a signal Z → `τ decay, one can reasonably assume that the momentum of the τ lepton before it decays ( p ( τ ) ) is collinear with the τ had - vis momentum ( p ( τ had - vis ) ), i.e. p ( τ ) = α p ( τ had - vis ) , (4.5) where α is some constant. The value of α can be estimated with two different approaches. First, since the ` and τ lepton are the products of a Z boson decay, m 2 Z − m 2 τ = 2 p ( ` ) · p ( τ ) = 2 α p ( ` ) · p ( τ had - vis ) . (4.6) † The τ had - vis is assumed to have a zero rest mass