# Draft: Faraway (viz. T-Track) vertex reconstruction & Lambda selection line

# Introduction

This MR introduces a faraway tracks (viz. t-tracks) vertexing algorithm and dedicated HLT1 line for long-lived particles. The algorithm uses Faraway tracks (viz. t-tracks) reconstructed by HybridSeeding as an input.

# 1. Faraway Tracks (viz. T-tracks) Extrapolation Model

The new vertexing algorithm for faraway tracks (viz. T-Tracks) considers the non-linear extrapolation of tracks from SciFi in the upstream direction. This algorithm will substantially improve the mass resolution of composite particles and provide a fit chi2, which is highly discriminant for background rejection in selection lines.

The non-linear extrapolation is described as follows:
`x(z) = x_0 + tx (z-z_0) + \frac{q}{p} (z-z_0) f(z,\frac{q}{p})`

`y(z) = y0 + ty (z-z0)`

where
`f(z,\frac{q}{p}) = a0 \frac{e^{a1(\frac{q}{p}-a2)} + e^{-a1(\frac{q}{p}+a2)}}{1+e^{a3(z-a4)}}`

`a0`

, `a1`

, `a2`

, `a3`

, `a4`

- constants, determined from MC;

# 2. Faraway (viz. T-track) vertex reconstruction algorithm

Vertex reconstruction is done in three steps:

- POCA
`z`

estimation using neural network. - Estimating vertex position with a given POCA
`z`

as a mean point between track coordinates. - Clarification of Faraway track (viz. T-Track) and vertex parameters using vertex position hypothesis.

##
`z`

estimation using neural network

2.1. POCA A two-layer neural network is used for POCA `z`

estimation.

The size of the layers is 14 and 5. Both layers utilize the `tanh`

activation function.

The six inputs are calculated using SciFi track variables: `y^A - y^B`

, `ty^A - ty^B`

, `x^A - x^B`

, `tx^A - tx^B`

, `qop^A`

, `qop^B`

.

## 2.2. Estimation of vertex position

The track extrapolation model, described in this section is used for initial vertex position estimation. Given two tracks are extrapolated to a found POCA `z`

. The `x`

and `y`

vertex positions are calculated as a mean of the corresponding track coordinates.

## 2.3. Clarification of Faraway track/vertex (viz. T-Track) parameters

The unsatisfactory resolution of `qop`

estimation by HybridSeeding leads to the need for clarification to improve the mass resolution of the mother particle. In addition, track slopes `tx`

are another parameters, that are improved. The procedure is performed with a given vertex position hypothesis, which acts as a constraint on tracks.

The clarified values are found as follows:

`(\frac{q}{p})^{A,B} = -\frac{tx_{sf}^{A,B} (z_{ov} - z_{sf}) + x_{sf}^{A,B} - x_{ov} }{f^{A,B}(z_{ov} - z_{sf})}`

`tx^{A,B} = - \frac{x_{sf}^{A,B} - x_{ov}}{z_{ov}-z_{sf}} + (\frac{q}{p})^{A,B} (z_{ov} - z_{sf}) f'^{A,B}`

where `A,B`

- denotes either one of tracks, `f`

- function, described in extrapolation model section, `f' = df/dz`